Gauge Field Theories and Propagators in Curved Space-Time
aa r X i v : . [ h e p - t h ] J a n GAUGE FIELD THEORIES ANDPROPAGATORS IN CURVED SPACE-TIME
Roberto Niardi ORCID: 0000-0001-9216-3322 ∗ Dipartimento di Fisica “Ettore Pancini”,Universit`a degli Studi di Napoli “Federico II ”, ItalyJanuary 20, 2021
Abstract
In this paper DeWitt’s formalism for field theories is presented; itprovides a framework in which the quantization of fields possessing in-finite dimensional invariance groups may be carried out in a manifestlycovariant (non-Hamiltonian) fashion, even in curved space-time. An-other important virtue of DeWitt’s approach is that it emphasizes thecommon features of apparently very different theories such as Yang-Mills theories and General Relativity; moreover, it makes it possible toclassify all gauge theories in three categories characterized in a purelygeometrical way, i.e., by the algebra which the generators of the gaugegroup obey; the geometry of such theories is the fundamental reasonunderlying the emergence of ghost fields in the corresponding quantumtheories, too. These “tricky extra particles”, as Feynman called themin 1964, contribute to a physical observable such as the stress-energytensor, which can be expressed in terms of Feynman’s Green functionitself. Therefore, an entire section is devoted to the study of the Greenfunctions of the neutron scalar meson: in flat space-time, the choice ofa particular Green’s function is the choice of an integration contour inthe “momentum” space; in curved space-time the momentum space isno longer available, and the definition of the different Green functionsrequires a careful discussion itself. After the necessary introductionof bitensors, world function and parallel displacement tensor, an ex-pansion for the Feynman propagator in curved space-time is obtained.Most calculations are explicitly shown. ∗ E-mail: [email protected] Introduction
The study of quantum gauge field theories and gravitation is both an intell-ectual pursuit and a necessity connected to black hole theory and quantumcosmological models. On one hand, electroweak and strong interactions aredescribed by Yang-Mills theories, and are satisfactorily set in a quantumframework; on the other hand, although General Relativity is in many waysa gauge theory too, it stands apart from the other three forces of nature, and aquantum theory of gravity does not yet exist , at least as a coherent discipline.Nevertheless, it is possible to discuss the influence of the gravitational fieldon quantum phenomena: one can study the regime for quantum aspects ofgravity in which the gravitational field is described as a classical background through Einstein’s theory while matter fields are quantized ; this is reasonableas long as length and time scales of quantum processes of interest are greaterthan the Planck values ( l P lanck ≡ ( G ~ /c ) / ∼ . × − cm, t P lanck ≡ ( G ~ /c ) / ∼ . × − s). Since Planck length is so small (twenty ordersof magnitude below the size of an atomic nucleus), one can hope that sucha “semiclassical” approach has some predictive power. Therefore, one isnaturally led to the subject of quantum field theory in a curved backgroundspacetime. Its basic physical prediction is that strong gravitational fieldscan “polarize” the vacuum and, when time-dependent, lead to pair creation ;moreover, in a curved space-time, notions of “vacuum” and “particles” needa deeper discussion than in the flat case. These two fundamental resultsare strongly linked to the most important predictions of the theory, i.e.,Hawking and Unruh effects (see [1], [2] and [3]): according to the Hawkingeffect, a classical, spherically symmetric black hole of mass M has the samespectrum of emission of a black body having the temperature T Hawking ≡ πM ; according to the Unruh effect, from the point of view of an acceleratingobserver , empty space contains a gas of particles at a temperature proportionalto the acceleration . For a detailed treatise on these subjects, see also [4], [5]and [6].This paper is structured in six sections, including the introduction;in Sec. 2, gauge field theories are introduced in DeWitt’s formalism [7], [8],[9], [10], and the set of all fields is presented as an infinite-dimensional man-ifold , on which an action functional is defined; then gauge transformationsare viewed as flows which leave the action functional unchanged; wheneverthe gauge group realization is linear (this is the case for Yang-Mills theoriesand General Relativity; see [11] and [12] for precious results in the finite-dimensional case), manifest covariance is ensured, i.e., the group transfor-mation laws for the various symbols that appear in the theory may be in-ferred simply from the position and nature of their indices, and both sides of2ny equation transform similarly. Last, the theory of small disturbances isdiscussed, Green’s functions are introduced in a general fashion and Peierlsbrackets are defined, in light of their importance in the quantization proce-dure.In Sec. 3, quantization of non-gauge field theories is discussed in theframework of DeWitt’s formalism: problems with the heuristic quantizationrules are stressed, and Schwinger’s variational principle is introduced as a wayto get around them; then the operator dynamical equations are presented,with the necessary introduction of the measure functional, which, at the sim-plest level, may be thought of as correcting the lack of self-adjointness of the“time-ordered” version of the classical dynamical equations; nevertheless, itplays a far deeper role, closely linked to the Wick rotation for the evalua-tion of divergent integrals. Last, the path integral for non-gauge theories isderived.In Sec. 4, the treatment is extended to gauge theories: some insight isgiven about the geometrical structure of the infinite-dimensional manifold offield theories belonging to the same class as Yang-Mills and General Rela-tivity; under the group action, this space is separated into orbits ; one couldsay that it is in the space of orbits that the real physics of the system takesplace; then one can choose on the manifold a new set of coordinates madeof two parts: the first one labelling the orbit, and the second one labellinga particular field configuration belonging to the specified orbit; when onewrites down the functional integral and reverts to the original coordinates,one finds out that a new term appears: it is the ghost contribution , whichinvolves two ghost fields; they are fermionic for a bosonic theory and bosonicfor a fermionic theory. Therefore, an “extended” space may be introduced,where ghost fields appear too and a new action functional, containing alsoghost terms, may be defined: then one is fascinated to find out that thegauge transformations for the original theory correspond to a set of rigid in-variance transformations for the theory on the extended space, i.e., the BRSTtransformations . It is important to stress that, altough many manipulationscontained in this paper are highly formal, many rigorous results concerningfunctional integration and path integral can be found in [13], [14], [15].Section 5 is dedicated to the study of the
Green functions of free, massive,scalar field, first in flat, then in curved space-time; it is shown that all Greenfunctions can be derived from Feynman’s Green function: in flat space-time,one can pass to the “momentum” space and easily verify that the choiceof the Green function is the choice of an integration contour which passesaround two poles; in curved space-time , where the momentum space is nolonger available , one can nevertheless define the Feynman’s Green function:it is the one that obeys the same variational law as finite square matrices and3s symmetric. In sight of the curved space-time treatment, it is of fundamen-tal importance to derive, even in the flat case, a formula for the Feynmanpropagator involving space-time coordinates only ; therefore, after the intro-duction of some necessary mathematical tools such as bitensors and somegeometric quantities such as the world function (see [16], [17], [18]), an ex-pansion for the Feynman Green function in curved space-time is obtained, valid for small values of the geodetic distance . It is important to observe thatthe expansion obtained ceases to exist when the massless limit is taken: themassless case has to be treated with alternative methods.In the conclusions some applications are outlined, with their related lit-erature.
Basic to the whole of quantum field theory is the assumption that space-time,which we shall denote by M (for manifold), has the topological structure M = R ×
Σ (1)where R is the real line and Σ is some connected three-dimensional manifold,compact or non-compact. In particular, space-time will be assumed to beendowed with a hyperbolic metric g which admits a foliation of space-timeinto spacelike sections, each being a complete Cauchy hypersurface (i.e., aspacelike surface which intersects every non spacelike curve exactly once ) anda topological copy of Σ. Being characterized by a manifold and a tensor fielddefined on the manifold, it is more accurate to say that a “space-time” is anequivalence class of pairs ( M, g ): the equivalence relation is the following:(
M, g ) ∼ ( N, h ) ←→ ∃ ψ ∈ Dif f ( M ) | N = ψ ( M ) , g = ψ ∗ h, (2)where ψ ∗ is the pullback map associated to ψ . Throughout this work, thefollowing sign convention will be assumed for the signature of the metrictensor: ( − . + , + , ... ). In this section, DeWitt’s formalism for field theories will be introduced, bothbosonic and fermionic ones.Denote by Φ the set, or space, of all possible field histories; it will be usefulto view Φ as an infinite-dimensional manifold; in this work the coordinates4 i will be assumed to be real-valued, whether c -type or a -type (see [19] and[20]). The concept of differentiation on Φ, on which the idea of tangent spaceat φ ∈ Φ is based, can be introduced through functional derivative :Let F : Φ ∋ φ F [ φ ] ∈ Λ ∞ , where Λ ∞ is the supernumber algebra ; F iscalled a supernumber-valued scalar field or functional on Φ, and its value ata point φ of Φ is denoted by F [ φ ]. Let δφ be an infinitesimal variation in φ ;it can be represented by a set of functions δφ i on the manifold M , where, ateach point x ∈ M , the δφ i ( x ) are components, in the appropriate chart of Φ,of an infinitesimal vector in its tangent space at the point having coordinates φ i ( x ). Let the δφ i ( x ) be C ∞ and have compact support in M , and let δF [ φ ]denote the change in value that F [ φ ] undergoes in shifting from φ to φ + δφ .If, for all φ ∈ Φ and for all C ∞ variations δφ of compact support, δF [ φ ] canbe written in the form δF [ φ ] = Z M δφ i ( x ) i ( x ) , F [ φ ] d n x = Z M F ,i ( x ) [ φ ] δφ i ( x ) d n x, (3)where i ( x ) , F [ φ ], F ,i ( x ) [ φ ] in the integrands are independent of δφ i and dependat most on φ , then F is called a differentiable functional on Φ, and i ( x ) , F [ φ ], F ,i ( x ) [ φ ] are called left and right functional derivatives of F , respectively: i ( x ) , F [ φ ] ≡ −→ δδφ ( x ) i F [ φ ] , (4) F ,i ( x ) [ φ ] ≡ F [ φ ] ←− δδφ ( x ) i . (5)Differentiation will be indicated by a comma followed by one or more indices:Greek indices will denote differentiation with respect to the chart coordinates x µ in M , while Latin indices will denote differentiation with respect to thefield coordinates.In a repeated functional derivative it does not matter whether the leftdifferentiations or the right differentiations are performed first, but the orderof the induced indices on either side is important. That is, although leftdifferentiations commute with right differentiations, left differentiations donot generally commute with each other, nor do right differentiations. Thelaws for interchanging are ij ′ , F = ( − ij ′ j ′ i, F F ,ij ′ = ( − ij ′ F ,j ′ i , (6)where the convention is here adopted that an index or symbol appearingin an exponent of ( −
1) is to be understood as assuming the value 0 or 1according as the associated quantity is c -type or a -type. The summation convention over repeated indices is assumed throughout this work.
5f the functional F is pure , i.e., either c -number-valued or a -number-valued, then (3) implies that its left and right functional derivatives arerelated by i, F = ( − i ( F +1) F ,i . (7)In the previous equation too, the symbol F in the exponent of ( −
1) assumesthe value 0 if F is a c -type quantity, the value 1 if F is a a -type quantity.When indices appear in exponents of − − In developing the general formalism of field theory we shall find it oftenconvenient to lump the symbol x with the generic index i and to make thelatter do double duty as a discrete label for the field components and as acontinuous label for the points of space-time. With this notation, the symbol δ ij ′ should be understood as a combined δ -distribution Kronecker delta, whileKronecker deltas shall be δ ij for the sake of clarity: φ i ←− δδφ j ′ ≡ φ i ( x ) ←− δδφ j ( x ′ ) = δ ij ′ ≡ δ ij δ ( x, x ′ ) (8)Therefore, it seems natural to establish a new convention: the summationover repeated field indices includes (by virtue of their role as continuouslabels) integration over M . Hence (3) takes the form: δF [ φ ] = δφ i i, F [ φ ] = F ,i [ φ ] δφ i . (9)Sometimes, even this condensed notation is cumbersome, and a supercon-densed notation is used: the indices themselves are suppressed and the fol-lowing replacement is made: i r ...i , F j ...j s → r F s . (10) Remark . The condensed and supercondensed notations must be used withcare because the associative law of multiplication does not always hold. Forexample, the value of an expression such as χ i i, F ,j ψ j may depend on whichsummation-integration is performed first. They give the same results onlyin certain cases . When the law does not hold, ambiguities in condensedexpressions will be removed by the use of parentheses or arrows. For example, if F is given by the expression F [ φ ] ≡ R M K µij ( x ) φ i ( x ) φ j,µ ( x ) d n x then .4 A few words on space-time (II) Use of the condensed notation underscores the following point:
The manifold M of space-time, independently of any physical fields that may be imposed onit, is an index set . Its points are labels that may be lumped together withthe indices for field components.When M is viewed in this way the notion that alternative topologies forspace-time may be alternative dynamical possibilities for a given universemakes no sense. Changing the topology of M corresponds to changing theindex set, and one cannot change the index set of a theory in midstream. Adifferent index set means a different theory.Transitions from one topology to another could be followed if space-timewere embedded in a higher dimensional manifold endowed with physical prop-erties. But then space-time and its contents would not be all there is; the“universe” would be something bigger. Since nobody has yet developed asuccessful embedding theory of space-time we shall assume that space-timeis the universe and leave its topology fixed. Throughout this work, the following (fundamental) principle will be postu-lated: the nature and dynamical properties of a classical dynamical systemare completely determined by specifying an action functional S for it.The action functional is a differentiable real- c -number-valued scalar fieldon the space of histories Φ, i.e., a functionally differentiable mapping S : Φ ∋ φ S [ φ ] ∈ R c .The choice of action functional for a given system is not unique but de-pends on the choice of dynamical variables φ i used to describe the systemand on the boundary conditions that one imposes on the φ i at the time limitsand at spatial infinity. However, all the possible action functionals for a givensystem must yield equivalent families of dynamical histories . A dynamicalhistory is any stationary point of S , i.e., any point φ of Φ that satisfies i, S [ φ ] = 0 or, equivalently S ,i [ φ ] = 0 . (11)The set of all stationary points is called the dynamical subspace of Φ, or the dynamical shell , and all histories satisfying (11) are said to be on shell . the reader may easily verify that if the i summation-integration is performed first one getsa result that differs by an amount − R M ( K µij φ i φ j ) ,µ d n x from that obtained when the j summation-integration is performed first. In order to get one result from the other onehas to carry out an integration by parts, and this is legitimate only in certain cases, forexample if the intersection of the supports of χ i and ψ j is compact in M . dynamical equations of the system. Theywill be assumed to be local in time, i.e., involving no time integrals and notmore than a finite number of time derivatives. In a relativistic theory thisimplies that they must also be local in space. This greatly limits the possiblechoices for S . Throughout this work S will have the general form S [ φ ] = Z M L ( φ i , φ i,µ , x ) d n x + boundary terms . (12)The integrand of this expression, known as the Lagrange function or La-grangian , is a scalar density of unit weight. If the gravitational field is notnumbered among the φ i , then L usually has an explicit dependence on a fixedbackground metric. Expression (12) immediately yields0 = i, S [ φ ] ≡ −→ δδφ i L [ φ ] − −→ δδφ i,µ L [ φ ] ! ,µ , (13)in which the boundary terms do not appear.It is worth remarking already at this point that the condition of locality,which is imposed on the dynamical equations largely in order to have easycontrol over causality, is by no means the only condition that is imposed inpractice. Even when the action functional has the structure (12) there areadditional criteria, of a physical nature, that greatly restrict the Lagrangefunction itself. For example L must satisfy the constraints imposed by rela-tivistic invariance, either special or general; it should lead to an energy thatis bounded from below and the stationary points of the action should benon-trivial. For many of the most interesting dynamical systems there exists, on the spaceof histories Φ, a set of flows that leave the value of the action invariant. Thatis, there exists a set of nowhere vanishing vector fields Q α on Φ such that SQ α ≡ . (14)Here the vector fields are written as operators acting from the right. They canbe expressed in terms of the basis vectors ←− δδφ i associated to the coordinatepatch on Φ defined by the φ i : Q α = ←− δδφ i i Q α . (15)8n terms of the components i Q α , (14) becomes S ,i i Q α ≡ . (16)Alternative forms are α QS ≡ , or, equivalently α Q ∼ i i, S ≡ , (17)where α Q acts from the left, and “ ∼ ” denotes the supertranspose: α Q = ( − α Q α , α Q ∼ i = ( − α ( i +1) i Q α . (18)It will be noted that allowance has been made in the previous equation forthe possibility that some of the Q α may be a -type. The index α is said to be c -type or a -type according as the vector field that it designates is c -type or a -type. It should also be noted that eq. (14) will generally lead to difficultiesat the boundary of the action integral (12) unless, for each α , the Q α havecompact support in M . If the Q α must be independent of both boundaryconditions and any special coordinate frame in space-time, then they canonly be δ -distributions or derivatives of δ -distributions times local functionsof the fields and their derivatives. This means that the index α , like theindex i , must include a space-time point and hence range over a continuousinfinity of values.Because of the invariance equation, the value of the action remains in-variant under infinitesimal changes in the dynamical variables of the form δφ i = i Q α δξ α . (19)The infinitesimal parameters δξ α of these transformations are C ∞ functionsover space-time, c -number-valued or a -number-valued according as the index α is c -type or a -type. The δξ α will be assumed to be real (i.e., taking theirvalues in either R c or R a ), and since the dynamical variables φ i are real-valued this implies that the vector fields Q α are real or imaginary accordingas α is c -type or a -type. The δξ α are additionally required to have compactsupport in space-time or else to satisfy such conditions at the time boundariesand at spatial infinity as are needed in order that the integrations in0 ≡ δS = S ,i δφ i = S ,i i Q α δξ α (20)be performable in any order.In the following sections, two important examples of such theories will bepresented: Yang-Mills theories and
General Relativity .9 .7 Yang-Mills theories (I) The dynamical object in the N -dimensional Yang-Mills theory is a Lie-algebra-valued 1-form, which can be expressed, in a suitable chart, as follows: A ≡ A αµ T α ⊗ dx µ (21) ≡ A µ dx µ , (22)where the T α are a basis for the Lie algebra of SU ( N ), which will be denotedby su ( N ); for N ≥
2, it is the algebra of square anti-hermitian tracelessmatrices with Lie bracket the commutator; this algebra can be endowed withthe Euclidean metric γ αβ ≡ − tr ( T α T β ) (23)which will be used to lower/raise Lie algebra indices.Therefore the following Lie-algebra-valued 2-form is defined: F ≡ F αµν T α ⊗ ( dx µ ∧ dx ν ) (24) ≡ F αµν T α ⊗ dx µ ⊗ dx ν (25) ≡ (cid:0) A αν ; µ − A αµ ; ν + f αβγ A βµ A γν (cid:1) T α ⊗ dx µ ⊗ dx ν , (26)where the semicolon denotes covariant differentiation associated with theLevi-Civita connection ∇ , and the f αβγ are the structure constants of su ( N )associated to the basis of the T α , i.e., they satisfy the equation[ T α , T β ] = T γ f γαβ . (27)The dynamical equations follow from the action functional S Y M [ A αµ ] ≡ − Z M p | g | F αµν F µνα d n x, (28)where g is defined to be det( g µν ).It can be readily seen that such an action is invariant under the transfor-mation A µ ( x ) A U µ ( x ) ≡ U † ( x ) A µ ( x ) U ( x ) + U † ( x ) U ,µ ( x ) (29)for every U : M ∋ x U ( x ) ∈ SU ( N ). Consider an element T ≡ ω α T α ∈ su ( N ); as is well known, exponentiation yields an element in SU ( N ); let now ω α ( x ) be a set of real funcions on M ; then, for every x ∈ M , U ( x ) ≡ e ω α ( x ) T α this metric is nonsingular if and only if the group is semisimple; additionally, wheneverthe group is compact, as in this case, then it is positive definite. SU ( N ); if the ω α ( x ) are infinitesimal, then U ( x ) is close to the identicaltransformation, and (29) reads A γµ A U γµ ≡ A γµ + A αµ ω β f γαβ + ω γ,µ , (30) δA γµ ≡ A U γµ − A γµ = A αµ ω β f γαβ + ω γ,µ . (31)With DeWitt’s notation, eq. (31) can be written δA γµ = Q γµ α δξ α , (32) Q γµ α ( x, x ′ ) ≡ δ ( x, x ′ ) (cid:0) A βµ ( x ) f γβα + δ γα ∂ µ ′ (cid:1) (33)= δ ( x, x ′ ) A βµ ( x ) f γβα − δ γα δ ( x, x ′ ) ,µ ′ . (34) The dynamical object is the Lorentzian metric tensor defined on a manifold M ; it can be expressed, in a suitable chart, as follows: g = g µν dx µ ⊗ dx ν . (35)It is important to observe that, if a theory describes nature in terms ofa manifold M and tensor fields T ( i ) defined on the manifold, then if φ : M → N is a diffeomorphism, the solutions ( M, T ( i ) ) and ( N, φ ∗ T ( i ) ) havephysically identical properties. Any physically meaningful statement about( M, T ( i ) ) will hold with equal validity for ( N, φ ∗ T ( i ) ). On the other hand, if( N, φ ∗ T ( i ) ) is not related to ( M, T ( i ) ) by a diffeomorphism, then ( N, φ ∗ T ( i ) )will be physically distinguishable from ( M, T ( i ) ).Thus, the diffeomorphisms comprise the gauge freedom of any theoryformulated in terms of tensor fields on a manifold. In particular, diffeomor-phisms comprise the gauge freedom of general relativity.Now consider the case where N = M ; let φ : M ∋ x φ ( x ) ∈ M be adiffeomorphism; its pullback acts on tensors of type (0 ,
2) as follows: φ ∗ : T ∗ φ ( x ) M ⊗ T ∗ φ ( x ) M ∋ g (cid:12)(cid:12) φ ( x ) φ ∗ g (cid:12)(cid:12) φ ( x ) ∈ T ∗ x M ⊗ T ∗ x M, (36) φ ∗ g (cid:12)(cid:12) φ ( x ) = g µν (cid:12)(cid:12) φ ( x ) ∂φ µ ∂x ρ (cid:12)(cid:12)(cid:12)(cid:12) x ∂φ ν ∂x σ (cid:12)(cid:12)(cid:12)(cid:12) x dx ρ (cid:12)(cid:12) x ⊗ dx σ (cid:12)(cid:12) x . (37)If φ is close to the identical diffeomorphism, then it can be always seen as anelement of the flow σ ξ ( t, · ) ≡ σ t associated to a vector field ξ defined on M ;11ence, for an “infinitesimal” diffeomorphism, eq. (37) reads: φ ∗ g (cid:12)(cid:12) φ ( x ) = σ ξ ∗ ǫ g (cid:12)(cid:12) σ ξǫ ( x ) = g µν (cid:12)(cid:12) σ ξǫ ( x ) ∂σ ξ µǫ ∂x ρ (cid:12)(cid:12)(cid:12)(cid:12) x ∂σ ξ νǫ ∂x σ (cid:12)(cid:12)(cid:12)(cid:12) x dx ρ (cid:12)(cid:12) x ⊗ dx σ (cid:12)(cid:12) x = g (cid:12)(cid:12) x + ǫ ( ξ α ∂ α g ρσ + g ρµ ∂ σ ξ µ + g µσ ∂ ρ ξ µ ) (cid:12)(cid:12) x dx ρ (cid:12)(cid:12) x ⊗ dx σ (cid:12)(cid:12) x = g (cid:12)(cid:12) x + ǫ ( ξ ρ ; σ + ξ σ ; ρ ) (cid:12)(cid:12) x dx ρ (cid:12)(cid:12) x ⊗ dx σ (cid:12)(cid:12) x = g (cid:12)(cid:12) x + ǫ L ξ g (cid:12)(cid:12) x . (38)Equation (38) shows a remarkable result: the Lie algebra of the diffeomor-phism group of M , Dif f ( M ), is the space of vector fields on M with Liebracket Lie derivative.Therefore, General Relativity is invariant under the trasformation g g + ǫ L ξ g, (39) δg = ǫ L ξ g. (40)for every vector field ξ on M .With DeWitt’s notation, eq. (40) can be written δg µν = Q µν ρ δξ ρ ,Q µν ρ ( x, x ′ ) = δ ( x, x ′ ) ( g µρ ∇ ν + g νρ ∇ µ ) (cid:12)(cid:12) x ′ (41)= ∂ ρ g µν (cid:12)(cid:12) x δ ( x, x ′ ) − g µρ (cid:12)(cid:12) x δ ( x, x ′ ) ,ν − g νρ (cid:12)(cid:12) x δ ( x, x ′ ) ,µ . (42) Let B be a functional on Φ; by applying two invariance transformations Q α , Q β to G , one arrives at B ( Q α Q β ) ≡ ( BQ α ) Q β = (cid:0) B ,i Q i α (cid:1) Q β = B ,i Q i α,j Q j β + ( − jα + ij B ,ij Q i α Q j β (43)while ( BQ β ) Q α = B ,i Q i β,j Q j α + ( − jβ + ij B ,ij Q i β Q j α = B ,i Q i β,j Q j α + ( − iβ + ji B ,ji Q j β Q i α = B ,i Q i β,j Q j α + ( − iβ B ,ij Q j β Q i α = B ,i Q i β,j Q j α + ( − iβ + iβ + ij + αβ + αj B ,ij Q i α Q j β = B ,i Q i β,j Q j α + ( − ij + αβ + αj B ,ij Q i α Q j β . (44)12ence one obtains B (cid:0) Q α Q β − ( − αβ Q β Q α (cid:1) = B ,i Q i α,j Q j β − ( − αβ B ,i Q i β,j Q j α = B ,i (cid:0) Q i α,j Q j β − ( − αβ Q i β,j Q j α (cid:1) . (45)Thus, given two fields Q α , Q β , their supercommutator or super Lie bracket [ Q α , Q β ] is itself a vector field:[ Q α , Q β ] ≡ Q α Q β − ( − αβ Q β Q α , (46)[ Q α , Q β ] i = Q i α,j Q j β − ( − αβ Q i β,j Q j α . (47)In the particular case where B is the action functional, it is immediatelyobvious that S [ Q α , Q β ] = 0 . (48)Hence, the commutator of two invariance transformations is an invariancetransformation itself.It must be pointed out at once that, for every dynamical system thereexist, on the space of histories Φ, vector fields that, like the Q α , yield zerowhen acting on the action, i.e., vector fields V of the form V i = S ,j T j i , (49)where T is any antisupersymmetric tensor field: T j i = − ( − ij T i j . (50)Such vector fields, however, vanish on the dynamical shell and are not trueflows. They will be called skew fields .It will be assumed that all true flows can be expressed, at each point of Φ,as linear combinations of the Q α ’s and skew fields at that point, i.e., that the Q α ’s form a pointwise complete set of flows modulo skew fields. Pointwisecompleteness of the Q α ’s and eq. (48) imply that the supercommutator in(46) must have the general structure[ Q α , Q β ] = Q γ c γαβ + T αβ S , (51)or, in component form:[ Q α , Q β ] i = Q i γ c γαβ + T i jαβ S j, , (52) The components of T should also have the necessary support or rate-of-fall-off prop-erties in space-time for the implicit summation integration in (49) to converge. c γαβ are scalar fields on Φ and the T αβ are tensor fields, havingthe symmetries: c γαβ = − ( − αβ c γβα (53) T i jαβ = − ( − αβ T i jβα = − ( − ij +( α + β )( i + j ) T j iαβ . (54)In addition to these symmetries the c γαβ and T αβ must satisfy functionaldifferential conditions imposed by the Jacobi identity[ Q α , [ Q β , Q γ ]] ǫ γβα = 0 , (55)the ǫ αβγ being any coefficients completely antisupersymmetric in their in-dices: ǫ αβγ = − ( − αβ ǫ βαγ = − ( − βγ ǫ αγβ . (56) One may easily verify that the super Lie bracket of any two skew fields is askew field. By functionally differentiating (16), one obtains:0 = ( S ,j Q j α ) ,i = S ,j Q j α,i + ( − ij + iα S ,ji Q j α = S ,j Q j α,i + ( − ij + iα + ij S ,ij Q j α = S ,j Q j α,i + ( − iα S ,ij Q j α ,S ,j Q j α,i = − ( − iα S ,ij Q j α . (57)By using this identity, one can easily verify that the super Lie Bracket of a Q α with a skew field is again a skew field: S ,i i [ Q α , S ,j T j • ] = S ,i (cid:16) Q i α,k S ,j T j k − ( − αT (cid:0) S ,j T j i (cid:1) ,k Q k α (cid:17) = S ,i Q i α,k S ,j T j k − ( − αT S ,i S ,j T j i,k Q k α − ( − αT + ki + kT + kj S ,i S ,jk T j i Q k α = S ,i Q i α,k S ,j T j k − ( − αT + ki + kT + kj S ,i S ,jk T j i Q k α = S ,i Q i α,k S ,j T j k − ( − αT + αi + αT + αj S ,i S ,jk Q k α T j i = S ,i Q i α,k S ,j T j k + ( − αi S ,i S ,k Q k α,j T j i = ( − jk + jα + ji S ,i S ,j Q i α,k T j k + ( − αi S ,i S ,k Q k α,j T j i = ( − ik + iα + ji S ,j S ,i Q j α,k T i k + ( − αi S ,i S ,j Q j α,k T k i = ( − ik + iα S ,i S ,j Q j α,k T i k + ( − αi S ,i S ,j Q j α,k T k i = ( − iα (cid:2) S ,i S ,j Q j α,k (cid:0) ( − ik T i k + T k i (cid:1)(cid:3) = 0 . (58)14rom these facts, together with eq. (51), it follows that the set of all vectorfields on Φ of the form Q i α ξ α + T i j S j, , (59)the ξ α being arbitrary ( φ -dependent) coefficients, T being an arbitrary antisu-persymmetric tensor field, form a closed algebra under the super Lie bracketoperation. When true flows exist this algebra is called a gauge algebra .The vector fields Q α characterizing the flows on Φ are evidently notunique. They are defined only up to transformations of the form¯ Q i α = Q i β X βα + T i jα S j, , (60)where the X βα are functionally differentiable scalar fields on Φ which, ateach point of Φ, form the elements of an invertible matrix, whose inverse isformed by functionally differentiable scalar fields, too, while T i jα obey T i jα = − ( − ij +( i + j ) α T j iα . (61)It is easy to see that such transformations leave eq. (51) unchanged. It isalso easy to see that even when the Q α are fixed, the T αβ in eq. (51) are notunique but are determined only up to transformations of the form¯ T i jαβ = T i jαβ + Q i γ U γ δαβ Q ∼ jδ , (62)where the coefficients U γ δαβ satisfy U γ δαβ = − ( − αβ U γ δβα = − ( − γδ +( γ + δ )( α + β ) U δ γαβ . (63)By carrying out these transformations, one may often simplify the relationssatisfied by the Q α . Three cases may be distinguished: The Q α and the T αβ may be chosen in such a way that the latter vanish andthe c γαβ are φ -independent; then eq. (51) becomes[ Q α , Q β ] = Q γ c γαβ , (64) c γαβ,i = 0 , (65)and the Jacobi identity implies: c ηαδ c δβγ ǫ γβα = 0 . (66)15n this case the c γαβ are the structure constants of an infinite dimensional Liegroup known as the gauge group of the system. The gauge group, or morecorrectly, the proper gauge group is defined as the set of transformationsof Φ into itself obtained by exponentiating the transformation (19) with φ -independent ξ α and taking products of the resulting exponential maps.The proper gauge group is viewed as acting on Φ, and its actions leave S invariant. The full gauge group is obtained by appending to the propergauge group all other φ -independent transformations that leave S invariantand do not arise from global symmetries. Elements of the proper group aresometimes called little gauge transformations , while elements of the full groupoutside the proper group are called big gauge transformations . When biggauge transformations exist the gauge group has disconnected components.It should be remarked that for the systems encountered in practice a choiceof flow vectors Q α satisfying (64) (65) is usually given a priori, and it is notnecessary to carry out transformations of the forms (60) (62) to find them.The closure property expressed by (64), which is stronger than eq. (51),implies that the gauge group decomposes Φ into subspaces to which the Q α are tangent. These subspaces are known as orbits, and the point-wise linearindependence of the Q α implies that each orbit is a copy of the gauge groupsupermanifold. If the gauge group has disconnected components, then so doesΦ itself. Φ may be viewed as a principal fibre bundle of which the orbits arethe fibres. The base space of this bundle is a supermanifold of which theorbits may be regarded as the points. It is called the space of orbits . Theaction functional is a scalar field on the space of orbits, and one might betempted to say that it is in this space that the real physics of the system takesplace. However, there may exist physical observables that remain invariantunder little gauge transformations but not big ones, so a separate “physical”base space should in principle be assigned to each component of Φ. But inpractice the amounts by which physical observables change under a big gaugetransformation are always dynamically inert. Therefore we shall from nowon focus solely on the proper gauge group. The T αβ can be made to vanish but the c γαβ cannot be made φ -independentglobally on Φ. Equation (64) continues to hold, and the space of historiesis again decomposed into orbits to which the Q α are tangent, but the orbitsare not group supermanifolds. If the Q α are pointwise linearly independent,then the components of the orbits are all topologically identical, and each isa parallelizable supermanifold. The space of histories may again be viewedas a fibre bundle, and the real physics of the system takes place in the space16f orbit components. The Jacobi identity, in this case, implies( c ηαδ c δβγ − c ηαβ,i Q i γ ) ǫ γβα = 0 . (67) The T αβ cannot be made to vanish globally on Φ. Flow vectors of the form Q α ξ α , where the ξ α are φ -dependent, do not by themselves form a closedsystem under the super Lie bracket operation, except on the dynamical shell.Only the dynamical shell, not the full space of histories Φ, is decomposed intoorbits. The space of histories cannot be viewed as a fibre bundle; only thedynamical shell can. This means that although the real physics takes placein the space of orbit components as usual, the dynamics cannot be derivedfrom an action functional on this space. The full space Φ is needed. In this section it will be shown that Yang-Mills theories are Type-I theories,and their structure functions will be calculated explicitly. By recalling eq.(34), one obtains Q γ ν ′′ µ α ′ , δ ′′ ≡ Q γµ α ′ ←− δδA ( x ′′ ) δν = δ ( x, x ′ ) δ ν ′′ µ δ ρδ f γρα ′ = δ ( x, x ′ ) δ ν ′′ µ f γδα ′ . (68)Hence Q γ ν ′′ µ α ′ , δ ′′ Q δ ′′ ν ′′ β ′′′ == Z M dx ′′ (cid:16) δ ( x, x ′ ) δ ν ′′ µ f γδα ′ (cid:17) (cid:16) δ ( x ′′ , x ′′′ ) A ρ ′′ ν ′′ f δρ ′′ β ′′′ − δ δ ′′ β ′′′ ,ν ′′ (cid:17) = δ ( x, x ′ ) δ ( x, x ′′′ ) A ρµ f γδα ′ f δρβ ′′′ − δ ( x, x ′ ) f γδα ′ δ δβ ′′′ ,µ ; (69)by swapping ( α ′ , x ′ ) and ( β ′′′ , x ′′′ ), one obtains: Q γ ν ′′ µ β ′′′ , δ ′′ Q δ ′′ ν ′′ α ′ == δ ( x, x ′ ) δ ( x, x ′′′ ) A ρµ f γδβ ′′′ f δρα ′ − δ ( x, x ′′′ ) f γδβ ′′′ δ δα ′ ,µ , (70)17herefore [ Q α , Q β ] γµ == Q γ ν ′′ µ α ′ , δ ′′ Q δ ′′ ν ′′ β ′′′ − Q γ ν ′′ µ β ′′′ , δ ′′ Q δ ′′ ν ′′ α ′ = δ ( x, x ′ ) δ ( x, x ′′′ ) A ρµ (cid:0) f γδα ′ f δρβ ′′′ − f γδβ ′′′ f δρα ′ (cid:1) − (cid:0) δ ( x, x ′ ) f γδα ′ δ δβ ′′′ ,µ − δ ( x, x ′′′ ) f γδβ ′′′ δ δα ′ ,µ (cid:1) . (71)The first term contains f γδα ′ f δρβ ′′′ − f γδβ ′′′ f δρα ′ ; by using the Jacobi identityfor the structure constants and their antisimmetry in the lower indices, oneobtains: f γδα ′ f δρβ ′′′ − f γδβ ′′′ f δρα ′ = f γρδ f δα ′ β ′′′ , (72)therefore the first term is δ ( x, x ′ ) δ ( x, x ′′′ ) A ρµ f γρδ f δα ′ β ′′′ == Z M dx ′′ f δ ′′ α ′ β ′′′ δ ( x ′′ , x ′ ) δ ( x ′′ , x ′′′ ) (cid:0) δ ( x, x ′′ ) A ρµ f γρδ ′′ (cid:1) (73)the second term is, instead: − (cid:0) δ ( x, x ′ ) f γδα ′ δ δβ ′′′ ,µ − δ ( x, x ′′′ ) f γδβ ′′′ δ δα ′ ,µ (cid:1) == − (cid:0) δ ( x, x ′ ) f γβ ′′′ α ′ δ ( x, x ′′′ ) ,µ − δ ( x, x ′′′ ) f γα ′ β ′′′ δ ( x, x ′ ) ,µ (cid:1) = − f γβ ′′′ α ′ (cid:0) δ ( x, x ′ ) δ ( x, x ′′′ ) ,µ + δ ( x, x ′ ) ,µ δ ( x, x ′′′ ) (cid:1) = f γα ′ β ′′′ ( δ ( x, x ′ ) δ ( x, x ′′′ ) + δ ( x, x ′ ) δ ( x, x ′′′ )) ,µ = Z M dx ′′ f γα ′ β ′′′ δ ( x, x ′′ ) ( δ ( x ′′ , x ′ ) δ ( x ′′ , x ′′′ )) ,µ = − Z M dx ′′ f γα ′ β ′′′ δ ( x, x ′′ ) ,µ ( δ ( x ′′ , x ′ ) δ ( x ′′ , x ′′′ ))= − Z M dx ′′ f δ ′′ α ′ β ′′′ δ γδ ′′ ,µ δ ( x ′′ , x ′ ) δ ( x ′′ , x ′′′ )= Z M dx ′′ f δ ′′ α ′ β ′′′ δ ( x ′′ , x ′ ) δ ( x ′′ , x ′′′ ) (cid:0) − δ γδ ′′ ,µ (cid:1) (74)Putting it all together, one obtains, eventually:[ Q α , Q β ] γµ == Z M dx ′′ f δ ′′ α ′ β ′′′ δ ( x ′′ , x ′ ) δ ( x ′′ , x ′′′ ) (cid:0) δ ( x, x ′′ ) A ρµ f γρδ ′′ − δ γδ ′′ ,µ (cid:1) = Z M dx ′′ Q γµ δ ′′ f δ ′′ α ′ β ′′′ δ ( x ′′ , x ′ ) δ ( x ′′ , x ′′′ ) (75)18quation (75) shows that Yang-Mills theories are Type-I theories, with struc-ture constants c δ ′′ α ′ β ′′′ ≡ f δαβ δ ( x ′′ , x ′ ) δ ( x ′′ , x ′′′ ) . (76) By recalling the commutation law for the Lie derivative, one obtains[ L X , L Y ] = L [ X,Y ] ; (77)But [ X, Y ] (cid:12)(cid:12) x = ( X ν ∂ ν Y µ − Y ν ∂ ν X µ ) (cid:12)(cid:12) x ∂ µ (cid:12)(cid:12) x can be expressed as[ X, Y ] (cid:12)(cid:12) x = Z M dx ′ Z M dx ′′ c σµ ′ ν ′′ X µ ′ Y ν ′′ , (78) c σµ ′ ν ′′ = δ σµ ′ ; τ δ τν ′′ − δ σν ′′ ; τ δ τµ ′ , (79)as is straightforward to verify. These equations show that General Relativityis a Type-I theory too, and its structure functions are given by (79). A change in the dynamical variables of the form (19) leaves the action func-tional invariant. Such changes therefore play no role in determining thedynamical shell. Moreover, they map the dynamical shell into itself, as maybe seen by varying the dynamical equations and making use of eq. (57): δ S j, = S j, ,i δφ i = S j, ,i Q i α δξ α = − ( − jα + j S ,i Q i α,j δξ α , (80)therefore S j, = 0 = ⇒ δ S j, = 0 . (81)Hence transformations generated by the Q α are unphysical. No functionalof the dynamical variables that is affected by them can be a physical quan-tity. Conversely, any functional that is invariant under (19) will be called a physical observable . In the classical theory this nomenclature constitutes anabuse of language because both c -type and a -type quantities can be invariantunder (19), and of course nobody can observe an a -number. However, thequantum counterpart of a real-valued classical observable, whether c -type or a -type, will, for any valid physical theory, be a self-adjoint linear operatorin the super Hilbert space of the full quantum theory, having ordinary realnumbers as eigenvalues. 19t is useful to distinguish two types of invariants under (19): absoluteinvariants and conditional invariants . An absolute invariant A is a functionalof the φ i that is invariant under (19) at all points of Φ. It satisies AQ α = A ,i Q i α = 0 ∀ φ ∈ Φ . (82)The action functional is always an absolute invariant. A conditional invari-ant B is a funcional of the φ i that is invariant under (19) on shell but noteverywhere on Φ. It typically satisfies BQ α = B ,i Q i α = S ,i b i α ∀ φ ∈ Φ , (83)where the b i α are certain φ i -dependent coefficients. A simple example of aconditional invariant is ¯ A = A + S ,i a i , (84)where A is an absolute invariant and a i are arbitrary φ i -dependent coeffi-cients.A physical observable may be either an absolute invariant or a condi-tional invariant. In a physical situation (i.e., when φ is on shell) there isin fact no distinction between the two. As a functional of the φ i a physicalobservable is really defined only modulo the dynamical equations, i.e., up totransformations of the form (84). Manifest covariance refers to the following facts: the group transformationlaws for the various symbols that appear in the theory may be inferred simplyfrom the position and nature of their indices, and both sides of any equa-tion transform similarly. This applies for Type-I theories when the grouprealization is linear, i.e. Q i α,jk = 0 . (85)In fact, by deriving (64), one obtains: Q i α,j Q j β,k − ( − αβ Q i β,j Q j α,k = ( − k ( α + β + γ ) Q i γ,k c γαβ , (86)which implies that the matrices ( Q i α,j ) (which, in view of eq. (85), are φ -independent) generate a representation of the Lie algebra of the gauge groupand, by exponentiation, of the gauge group itself. Call this representationthe defining representation and call the contragradient representation (gener-ated by the negative (super)transposes of the above matrices) the co-definingrepresentation . Similarly, eq. (66) implies that ( c γαβ ) generate a representa-tion of the Lie algebra of the gauge group too: call it adjoint representation ;20all co-adjoint representation the representation generated by the negative(super)transpose of the structure constants.Given an absolute invariant A , one can take subsequent derivatives of eq.(82) and use (85); the first two derivatives yield:( A ,i Q i α ) ,j = 0 , ( − ij + jα A ,ij Q i α = − A ,i Q i α,j ; (87)( A ,i Q i α ) ,jk = 0 , ( A ,i Q i α,j + ( − ij + jα A ,ij Q i α ) ,k = 0 , ( − ij + jα + αk + ik A ,ijk Q i α = − ( − jk + jα + ij A ,ik Q i α,j − ( − ij + jα A ,ij Q i α,k . (88)These identities relate functional derivatives of any absolute invariant of ad-jacent order; when the functional under observation is the action functional,these derivatives are called vertex functions , and these identities are called bare Ward identities . They imply the transformation laws δA ,j ≡ A ,ji δφ i = A ,ji Q i α δξ α = ( − ij A ,ij Q i α δξ α = ( − jα ( − ij + jα A ji Q j α δξ α = ( − jα (cid:2) ( A ,i Q i α ) ,j − A ,i Q i α,j (cid:3) δξ α = − ( − jα A ,i Q i α,j δξ α ; (89) δA ,jk ≡ A ,jki δφ i = A ,jki Q i α δξ α = ( − ij + ik A ,ijk Q i α δξ α = ( − jα + kα ( − ij + ik + jα + kα A ,ijk Q i α δξ α = ( − jα + kα [( A ,i Q i α ) ,jk − ( − jk + jα + ij A ,ik Q i α,j − ( − ij + jα A ,ij Q i α,k ] δξ α = − ( − jα + kα [( − jk + jα + ij A ,ik Q i α,j +( − ij + jα A ,ij Q i α,k ] δξ α = − ( − kα + kj + ij A ,ik Q i α,j δξ α − ( − kα A ,ji Q i α,k δξ α ; (90)analogous equations hold for higher order derivatives.21he above equations show that the functional derivatives of absolute in-variants transform according to direct products of the codefining representa-tion. Equation (64) may itself be regarded as a transformation law: δ Q i α ≡ Q i α,j Q j β δξ β = ( [ Q α , Q β ] i + ( − αβ Q i α,j Q j β ) δξ β = ( Q i γ c γαβ + ( − αβ Q i β,j Q j α ) δξ β = ( − Q i γ c γβα + ( − αβ Q i β,j Q j α ) δξ β , (91)which says that Q i α transforms according to the direct product of the definingrepresentation and the coadjoint representation.Hence, when the group realization is linear, quite generally, field indices(Latin) and group indices (Greek) signal respectively the defining represen-tation and the adjoint representation when they are in the upper positionand the contragradient representations when they are in the lower position.One may then wonder whether the realization can be always made linearfor Type-I theories: for Yang-Mills theories and General Relativity, this ispossible, as has been shown in the previous sections, but the answer is notknown in general. However, certain results in the theory of finite-dimensionalcompact Lie groups are suggestive in this connection. Palais [11] and Mostow[12] showed that if a manifold is acted on by a compact Lie group with finitelymany orbit types, then it can be embedded into some finite-dimensionallinear, homogeneous, orthogonal representation. Moreover, results of thiskind can usually be extended to the case of finite-dimensional semisimple Liegroups whether compact or not.If similar results could be extended to field realizations of gauge groups (which are infinite-dimensional), then one could simply add enough extrafields to Type-I systems to make the realization linear. The extra fieldscould be made dynamically innocuous by inclusion of appropriate Lagrange-multiplier fields in the action. There is one difference between the finite-dimensional and field theoretical cases that apparently cannot be eliminated:in the case of fields the variables φ i cannot always be chosen in such a wayas to yield a realization that is simultaneously linear and homogeneous. Inthe case of the Yang-Mills field the infinitesimal gauge transformation law(31) includes an inhomogeneous term that cannot be removed by any choiceof variables. In the case of the Maxwell field the inhomogeneous term is allthere is. 22 .18 Equation of small disturbances Let φ i and φ i + δφ i be two neighboring solutions of the dynamical equations(11): 0 = S i, [ φ ] , (92)0 = S i, [ φ + δφ ] = S i, [ φ ] + S i, ,j [ φ ] δφ j + ..., (93)where the dots stand for terms which are at least quadratic in δφ i .Evidently, to first order in δφ i , we have: S i, ,j [ φ ] δφ j = 0 . (94)This is called homogeneous equation of small disturbances . Its solutions areknown as Jacobi fields relative to the on-shell field φ . In the following equa-tions the argument φ will often be suppressed. In practice a small disturbanceis produced by a weak external agent, which may be described by a smallchange in the functional form of the action. Let A be a pure real-valuedscalar field on Φ and ǫ be an infinitesimal real c -number or imaginary a -number according as A is c -type or a -type; therefore Aǫ is a real valued c -type scalar field, and the following change in the functional form of theaction is admissible: S [ φ ] S [ φ ] + A [ φ ] ǫ (95)Let δφ i be a solution of S i, ,j δφ j = − Aǫ i, . (96)It is easy to see that, neglecting higher order terms, φ i + δφ i satisfies thedynamical equations of the system S + Aǫ if and only if φ i satisfies those ofthe system S :( S + Aǫ ) i, [ φ + δφ ] = S i, [ φ + δφ ] + A i, [ φ + δφ ] ǫ = S i, [ φ ] + S i, ,j [ φ ] δφ j + A i, [ φ ] ǫ + A i, ,j [ φ ] δφ j ǫ + ... = S i, + ( S i, ,j δφ j + A i, ǫ ) + .... (97)Equation (96) is called inhomogeneous equation of small disturbances . Itsgeneral solution is obtained by adding to a particular solution an arbitraryJacobi field.When the dynamical equation (11) is satisfied, eq. (57) reads: Q ∼ iα S i, ,j = 0 ( S ,j = 0) (98)When applied to (96), by virtue of the arbitrariness of ǫ , this equation implies Q ∼ iα A i, = 0 ( S ,j = 0) . (99)23herefore eq. (96) is seen to be inconsistent unless A is a conditional in-variant, i.e., a physical observable. If this is not the case, then the solutionof ( i, S + Aǫ ) = 0 cannot differ from φ by infinitesimal amounts. Evidentlysmall changes in the action functional will produce small changes in the on-shell dynamical variables only if they leave intact the flow invariances of thetheory. Equation (98) implies that, on the dynamical shell S i, ,j Q j α δξ α = 0 ( S ,j = 0) (100)for every δξ α of compact support in space-time: this implies that S is notan invertible operator; hence, it has no Green’s functions.When Q i α δξ α is added to a solution of eq. (96), the result is anothersolution: however, they are physically identical, since they differ merely byan invariance transformation (19). It is convenient to remove this redundancyby imposing a differential supplementary condition on the small disturbances δφ i , of the form P α i δφ i = 0 . (101)The supplementary condition is effective as long as the operator F α β ≡ P α j Q j β (102)is nonsingular and has Green’s functions; in fact, given a Jacobi field δφ i , allthe physically identical solutions can be written as: δφ i + Q i α δξ α ; (103)by imposing the supplementary condition (101), one obtains P α i ( δφ i + Q i β δξ β ) = 0 P α i δφ i + P α i Q i β δξ β = 0 P α i δφ i + F α β δξ β = 0 . (104)Being F α β invertible, this equation determines δξ α and, therefore, the solu-tion δφ i + Q i α δξ α . Position and nature of the indices of the auxiliary distributions introduced throughoutthis work is not accidental: when dealing with Type-I theories with linear gauge group re-alization, they show how these distributions must transform under gauge transformations. η be a local, continuous, nonsingular, supersymmetric matrix whoseelements are: η αβ = ( − α + β η βα ; (105)introduce the following differental operator: F i j ≡ S i, ,j + P ∼ i α η αβ P β j , (106)where P ∼ i α is the supertranspose of P α i : P ∼ i α = ( − i + α + iα P α i . (107)It is easy to see that F i j has the same supersymmetry properties as S i, ,j , i.e.it is supersymmetric: F i j = ( − i + j + ij F j i . (108)From now on, we will assume that the kernel of the linear differential operator S i, ,j consists of the fields of the form Q i α δξ α , with δξ α of compact support inspace-time: this is true in all the practical cases. Hence, the folloing holds: Theorem 2. If F α β is nonsingular, then F i j is nonsingular too.Proof. Suppose that there is a set of functions X j of compact support suchthat F i j X j = 0 . (109)By suppressing all indices and recalling the definition (106), one obtains:( S + P ∼ ηP ) X = 0 . (110)By taking the supertranspose and recalling the supersymmetry propertiesstated above, one can write: X ∼ ( S + P ∼ ηP ) = 0 . (111)By applying Q α from the right and using (98), which holds on shell, onearrives at: X ∼ ( S + P ∼ ηP ) Q α = 0 , (112) X ∼ P ∼ ηP Q α = 0 . (113)By restoring indices and recalling the definition (102), one can write: X ∼ j P ∼ j α η αβ F β γ = 0 . (114)25eing η αβ , F β γ nonsingular, the previous equation implies X ∼ j P ∼ j α = 0 , (115)or equivalently P α i X i = 0 , (116) P X = 0 . (117)Hence, from the first equation (109), it follows F X = ( S + P ∼ ηP ) X = S X = 0 . (118)But the kernel of S consists of the fields of the form Q i α δξ α , with δξ α ofcompact support in space-time, then X must be of the form Qξ ; thereforeone can write again eq. (117) and recall the definition (102): P X = 0 ,P Qξ = 0 , F ξ = 0 . (119)Since F is non singular, this equation implies ξ = 0 and, as a consequence, X = 0. Therefore the kernel of the operator F i j consists of the null field X i = 0 only: then F i j is nonsingular.This ends the proof. F When the supplementary condition (101) is satisfied, eq. (96) may be re-placed by F i j δφ j = − Aǫ i, . (120)In fact, if both (101) and (96) are satisfied, it is obvious that (120) is satisfiedtoo; on the other hand, if (120) is satisfied, by applying Q ∼ iα from the leftone obtains: Q ∼ iα F i j δφ j = Q ∼ iα ( − Aǫ i, ) Q ∼ iα ( S i, ,j + P ∼ i α η αβ P β j ) δφ j = Q ∼ iα ( − Aǫ i, ) Q ∼ iα P ∼ i α η αβ P β j δφ j = 0 F ∼ α β η αβ P β j δφ j = 0 , (121)26here (98) have been used, plus the fact that A is a conditional invariant;Being η αβ , F β γ nonsingular, the previous equation implies P β j δφ j = 0 , (122)i.e., the supplementary condition is satisfied. Hence, the following holds: F i j δφ j = − Aǫ i, S i, ,j δφ j + P ∼ i α η αβ P β j δφ j = − Aǫ i, S i, ,j δφ j = − Aǫ i, (123)i.e., the inhomogeneous equation of small disturbances is satisfied too.Since F is a nonsingular operator this equation has unique solutions forgiven boundary conditions. These solutions can be expressed in terms ofGreen’s functions.We shall consider in this section only retarded and advanced boundaryconditions. Denote by δ − φ i and δ + φ i respectively the corresponding solu-tions. Then δ ± φ i = G ± ij Aǫ j, (124)where G − ij and G + ij are the retarded and advanced Green’s functions of F i j ,respectively : F i k G ± kj = − δ ji , (125) G − ij = 0 if i < j, G + ij = 0 if i > j, (126)where “ i < j ” means “the time associated with the index i lies in the past ofthe time associated with the index j ” and “ i > j ” means “the time associatedwith the index i lies in the future of the time associated with the index j ”. Consequently the kinematical conditions (126) imply that G − ij ( G + ij )is nonvanishing only when the space-time point associated with i lies onor inside the future (past) light cone emanating from the space-time pointassociated with j .A minus sign appears on the right of eq. (125) for historical reasons, andthe symbol δ ji represents a combined Kronecker delta δ -distribution. In thesupercondensed notation (125) is written F G ± = − . (127)It should be remarked that the summation-integration involved on the rightside of eq. (124) will generally not converge unless the functional form of A is such that the functions A j, do not increase in magnitude too rapidly27oward the past or future. A sufficient condition for convergence, of course,is that supp A j, be limited in time, where “supp A j, ” denotes the union ofthe supports of all the A j, . If (124) does not converge, then the solutions of S i, ,j δφ j + Aǫ i, = 0 do not lie close (in Φ) to those of S i, = 0 no matter howsmall ǫ may be chosen. In eqs. (125), (127) the G ± appear as right Green’s functions. They arealso left
Green’s functions. To prove this let us temporarily distinguish leftGreen’s functions from right Green’s functions by employing subscripts L and R . Let X i be arbitrary functions of compact support in space-time andlet Y i ≡ ( G − ijR − G − ijL ) F j k X k . (128)Since the X k have compact support it does not matter whether the j summation-integration or the k summation-integration is performed first in this expres-sion. That is, F j k may be regarded as acting either to the right or to the left.By performing the j summation-integration first and using G − ijL F j k = − δ i k ,one obtains Y i = G − ijR F j k X k + δ i k X k = G − ijR F j k X k + X i . (129)By applying F m i from the the left, performing the i summation-integrationfirst and using F m i G − ijR = − δ jm , one obtains: F m i Y i = F m i G − ijR F j k X k + F m i X i F m i Y i = − δ jm F j k X k + F m i X i F m i Y i = − F m k X k + F m i X i F m i Y i = 0 . (130)But Y i vanishes if i < supp X ≡ ∪ j supp X j . This means that the boundarydata for the above equation vanish to the past of supp X , and hence Y i must vanish everywhere . Since the X i are arbitrary it follows that0 = ( G − ijR − G − ijL ) F j k = G − ijR F j k + δ i k ,G − ijR F j k = − δ i k . (131)28ut this is just the condition that G − ijR be a left Green’s function. Therefore G − ijR = G − ijL . In a similar manner one may show that G + ijR = G + ijL . Thuseqs. (125), (127) imply G ± ik F k j = − δ i j (132)or, in supercondensed notation G ± ←− F = − . (133)It is important to stress that this proof holds regardless of the symmetry of F . The actual supersymmetry of F gives rise to simple relations between theretarded and advanced Green’s functions. Consider the expression( − ki G − ki F k l G + lj . (134)Because of the kinematical conditions (126), the intersection of the sup-ports of G − ki and G + lj , with i and j held fixed, is compact, since it is theintersection of a forward light cone with a backward light cone. There-fore it makes no difference whether the k summation-integration or the l summation-integration is performed first. Using this fact, together with thesupersymmetry law (108), one obtains:0 = ( − ki G − ki ( F k l − ( − k + l + kl F l k ) G + lj = − ( − ki G − ki δ jk − ( − ki + k + l + kl + il + ik + kl + k F l k G − ki G + lj = − ( − ji G − ji + ( − l + il δ il G + lj = − ( − ji G − ji + G + ij . (135)Therefore ( − ji G − ji = G + ij , (136)or, equivalently G ± ij = ( − ij G ∓ ji . (137)Equations (137) are called reciprocity relations for the Green’s functions . Inthe supercondensed notations, they take the form: G ± = G ∓∼ . (138)29 .23 Relation between Green’s functions of F and F In this section an important relation between the Green’s Functions of F and F will be derived; attention will be confined, for now, to the retardedand advanced Green’s functions, those of F being denoted by G − and G + ,respectively: F α γ G ± γβ = − δ βα . (139)Also in this case, G ± are both right and left Green’s functions, and obeysimilar reciprocity relations to the ones shown for G .By writing down the definition of F (106) and using (98), that is validon shell, one obtains: F j k Q k β = S j k Q k β + P ∼ j α η αγ P γ k Q k β = 0 + P ∼ j α η αγ F γ β . (140)Multiplying this equation on the left by G ± and on the right by G ± , andnoting that the intersection of the supports of these extra factors (with theouter suppressed indices held fixed) is compact so that F and F may act ineither direction, one gets G ± ij F j k Q k β G ± βθ = G ± ij P ∼ j α η αγ F γ β G ± βθ ,δ i k Q k β G ± βθ = G ± ij P ∼ j α η αγ δ θγ ,Q i β G ± βθ = G ± ij P ∼ j α η αθ , (141)or, equivalently: Q G ± = G ± P ∼ η. (142)Now, if η is chosen to be ultralocal , i.e., in η no undifferentiated δ distributionsappear, then its negative inverse λ is unique and supersymmetric; on theother hand, if η is not ultralocal, then its negative inverse is not unique: theyare Green’s functions, and they will be assumed to obey the same kinematicalrelations as G ± , G ± ; its elements will be indicated as λ α β = ( − α + β + αβ λ β α . (143)Hence eq. (142) may be written − Q G ± λ = G ± P ∼ , (144)or, taking the supertranspose and using (137) and the symmetry properties: − λ G ∓∼ Q ∼ = P G ± . (145)30iven this equation, one can prove in another way that (124) is the solu-tion for the inhomogeneous equation of small disturbances which obeys thesupplementary conditions: P α i δ ± φ i = P α i G ± ij Aǫ j, = − λ α β G ± βγ Q ∼ jγ Aǫ j, = 0 , (146)where the last line follows from the fact that A is a physical observable. Given φ ∈ Φ , the entire tangent space at φ , T φ Φ, is spanned by vectors ofthe form G ± ij D j, , with D of compact support. Equation (146) shows that,in order to get the subspace of T φ Φ obeying the supplementary conditions,only D which are physical observables have to be considered.Another possible choice to obtain the same result without imposing con-ditions on D is to “modify” the Green’s function: the task is complete if onefinds an object B ± ij such that ( P ( G ± + B ± ) = 0 .S B ± = 0 (147)But, by using (145), the first equation reads P B ± = λ G ∓∼ Q ∼ . (148)Using P Q G = F G = − B ± = − Q G ± λ G ∓∼ Q ∼ (149)is a solution which satisfies the second equation of the system, too. Thereforeone is led to define Landau Green’s functions G ±∞ G ±∞ ≡ G ± + B ± = G ± − Q G ± λ G ∓∼ Q ∼ , (150)or, with restored indices: G ± ij ∞ ≡ G ± ij + B ± ij = G ± ij − Q i α G ± αβ λ β γ G ∓∼ γδ Q ∼ jδ . (151)31he Landau Green’s functions are defined only on shell and, when appliedto a physical observable, G ± and G ±∞ yield trivially the same results.It is important to observe that G ±∞ is no longer a negative inverse for F ;in fact G ±∞ F = ( G ± − Q G ± λ G ∓∼ Q ∼ ) F = − − Q G ± λ G ∓∼ Q ∼ F = − − Q G ± λ G ∓∼ Q ∼ ( S + P ∼ ηP )= − − Q G ± λ G ∓∼ Q ∼ P ∼ ηP = − − Q G ± λ G ∓∼ F ∼ ηP = − Q G ± ληP = − − Q G ± P. (152)Call this operator Π ± ≡ − G ±∞ F ; it can be easily seen that it is a projectionoperator whose kernel is the subspace of T φ Φ that is tangent to the invarianceflows Q α : Π ± = Π ± Π ± = (1 + Q G ± P )(1 + Q G ± P )= 1 + 2 Q G ± P + Q G ± P Q G ± P = 1 + 2 Q G ± P + Q G ± F G ± P = 1 + 2 Q G ± P − Q G ± P = 1 + Q G ± P = Π ± , (153)Π ± Q = (1 + Q G ± P ) Q = Q + Q G ± P Q = Q + Q G ± F = Q − Q = 0 , (154)and, obviously P Π ± = P (1 + Q G ± P )= P + P Q G ± P = P + F G ± P = P − P = 0 . (155)32inally, noticing that Π ± ≡ − G ±∞ F = − G ±∞ S = − G ± S , and recallingthat the restriction of a projection operator on its image is the identity op-erator, it can be said that the Landau Green’s functions are the negativeinverses of the restriction on Ran (Π ± ) of the singular operator S . Let B be a physical observable; call δ ± B the changes in value of B under thedisturbance (124) caused by the change in the action functional (95). Then δ ± B = B ,i δ ± φ i = B ,i ( G ± ij A j, ) ǫ = ( − AB ( A ,j G ∓ ji ) B i, ǫ, (156)in which eq. (7) and the reciprocity relations (137) have been used in ob-taining the final expression. Parentheses have been inserted because it isnot guaranteed that if the summation-integration over i is performed be-fore the summation-integration over j the same result will be obtained. Weshall assume that the functions B ,i do not increase in magnitude too rapidlyfor convergence either in the past or in the future. In fact we shall assumethat these functions are well enough behaved that the parentheses may be re-moved. The following are some sufficient (although not necessary) conditionsfor the parentheses to be absent:1. With the retarded solution δ − φ , if ( ∪ j supp A j, ) ∩ ( ∪ i supp B ,i ) iscompact and there exist spacelike hypersurfaces Σ + ,Σ − such that( ∪ i supp B ,i ) < Σ + and ( ∪ j supp A j, ) > Σ − . In this case δ − B vanishes unless Σ + > Σ − .2. With the advanced solution δ + φ , if ( ∪ j supp A j, ) ∩ ( ∪ i supp B ,i ) iscompact and there exist spacelike hypersurfaces Σ + ,Σ − such that( ∪ j supp A j, ) < Σ + and ( ∪ i supp B ,i ) > Σ − . In this case δ + B vanishes unless Σ + > Σ − .3. With either solution, if ( ∪ j supp A j, ) and ( ∪ i supp B ,i ) are both com-pact. Remark . When, as now, we are working with φ on shell, a question arisesregarding the meaning of the expressions supp A j, and supp B ,i . When φ is33n shell the functional form of a physical observable is defined only modulo the dynamical equations, and hence the expressions supp A j, and supp B ,i would seem to be ambiguous. The following clarification is necessary: everyphysical observable has an expression in terms of the fields φ i the functionalform of which is independent of that of S . This is the form that is tobe understood in the expressions supp A j, and supp B ,i . This form remainsinvariant under the change (95) in the action. Only its value changes, becausethe values of the dynamical variables themselves change. It will be useful to introduce the notation D ± A B ≡ A ,i G ∓ ij B j, . (157)Equation (156) may then be written: δ ± B = D ± Aǫ B = ( Aǫ ) ,i G ∓ ij B j, , (158)as may be seen by noting that A and ǫ have the same type.Colloquially, D − A B may be called the “retarded effect of A on B ” and D + A B the “advanced effect of A on B ”. It is a consequence of the reciprocityrelations (137) that D ± A B ≡ A ,i G ∓ ij B j, = ( − AB B ,i G ± ij A j, = ( − AB D ∓ B A. (159)In words: The retarded effect of A on B equals ( − AB times the advancedeffect of B on A (and vice versa) . This is known as the reciprocity relationfor physical observables. So far we have used the distributions P α i , η αβ only on shell. However, they,like the Q i α , S i j , etc. have specific functional forms (as functionals of φ )and are defined also off shell. Thus the operators F i j and F α β and Green’sfunctions G ± ij and G ± αβ are defined off shell as well. Off shell, eq. (98)no longer necessarily holds and the relations (142) (144) fail to be satisfiedgenerically. However, the operators F i j and F α β continue to be nonsingular34nd the Green’s functions G ± ij and G ± αβ continue to exist, at least in anopen neighborhood of the dynamical shell.We need G ± ij and G ± αβ off shell because we need to be able to calculatetheir functional derivatives, in order to discuss the invariance properties of D ± A B . We begin by considering an arbitrary infinitesimal variation δF in theoperator F . This variation may arise either by shifting the point φ in Φ,or by varying the functional forms of P , η , or even the action S (the vectorfields Q α will be left untouched). It leads to corresponding variations δG ± in the advanced and retarded Green’s functions.The δG ± satisfy a differential equation that is obtained by varying eq.(127): δ ( F G ± ) = δ ( − ,δF G ± + F δG ± = 0 ,F δG ± = − δF G ± . (160)This equation has the following unique solution: δG ± = G ± δF G ± (161)which is determined by the kinematical conditions (126) that the Green’sfunctions satisfy. It will be noted that the intersection of the supports ofthe two factors G ± (with outer suppressed indices fixed) in the previousequation is compact, so that the operator δF in this equation may act ineither direction.Equation (161) is just the equation one would get if F were a finite squarematrix and G ± were its negative inverse. There are important differences,however, between the present case and the case of finite matrices. First, F has many “inverses”, or Green’s functions, not just one. Second, most of itsGreen’s functions do not satisfy variational equations having the structure(161). For example, the average¯ G ≡ ( G + + G − ) (162)is a Green’s function of F : F ¯ G = F ( G + + G − )= F G + + F G − = − − = − , (163)but it satisfies δ ¯ G = ( G + δF G + + G − δF G − ) , (164)35hich is not equal to ¯ GδF ¯ G .If a Green’s function does satisfy δG = GδF G (165)it will be called a coherent
Green’s function. In many of the equations ofthis work the operators F , F , S , δF , etc. will appear sandwiched betweenfactors, the intersections of the supports of which are not compact. Theseoperators will nevertheless be able to act in either direction because theGreen’s functions in these factors are all in the same coherence class andestablish an over-all set of boundary conditions that are preserved regardlessof the direction of action.Suppose φ is on shell and suppose the variation δF arises from variations δP and δη in P and η : then δF = δP ∼ ηP + P ∼ δηP + P ∼ ηδP. (166)Inserting this expression in (161) and making use of eqs. (144) and (145), aswell as δη = η δλ η, (167)one obtains δG ± = G ± ( δP ∼ ηP + P ∼ δηP + P ∼ ηδP ) G ± = G ± δP ∼ ηP G ± + G ± P ∼ ( η δλ η ) P G ± + G ± P ∼ ηδP G ± = − G ± δP ∼ ηλ G ∓∼ Q ∼ − Q G ± λ ( η δλ η )( − λ G ∓∼ Q ∼ ) − Q G ± ληδP G ± = G ± δP ∼ G ∓∼ Q ∼ + Q G ± δλ G ∓∼ Q ∼ + Q G ± δP G ± . (168)If the variation δF arises instead from a variation in φ , then one is led to theformula δG ± ij = G ± ij,k δφ k = G ± il δF l m G ± mj = G ± il F l m,k δφ k G ± mj = ( − km + kj G ± il F l m,k G ± mj δφ k , (169)and then G ± ij,k = ( − km + kj G ± il F l m,k G ± mj (170)Of course this formula is valid off shell. Going on shell one can proceed just36s in the derivation of eq. (168) and obtain: G ± ij,k = ( − km + kj G ± il ( S l, ,m + P ∼ l α η αβ P β m ) ,k G ± mj = ( − km + kj G ± il ( S l, ,mk + P ∼ l α η αβ P β m,k +( − km + kβ P ∼ l α η αβ,k P β m +( − km + kα P ∼ l α,k η αβ P β m ) G ± mj = ( − km + kj [ G ± il S l, ,mk G ± mj + Q i α G ± αβ P β m,k G ± mj − ( − km + kβ Q i α G ± αβ η β γ η γθ,k λ θ ι G ∓∼ ικ Q ∼ jκ +( − km + kα G ± il P ∼ l α,k G ∓∼ αβ Q ∼ jβ ]= ( − km + kj [ G ± il S l, ,mk G ± mj + Q i α G ± αβ P β m,k G ± mj +( − km + kβ Q i α G ± αβ λ β γ,k G ∓∼ γθ Q ∼ jθ +( − km + kα G ± il P ∼ l α,k G ∓∼ αβ Q ∼ jβ ] . (171)It will be noted that summation-integrations can be performed in any orderin the previous equations because all the Green’s functions are in the samecoherence class. D ± A B Expression (157) for D ± A B involves the Green’s functions G ± , which dependon specific choice for the operators P and η . Since D ± A B represent the phys-ical effects of physical changes in the action they must be P - and η - inde-pendent. To verify this write the variation of (157), under changes in P and η , in the supercondensed notation: δD ± A B = A ,i δG ∓ ij B j, . (172)If φ is on shell, one can insert (168) into the right-hand side. The result isimmediately seen to vanish, δD ± A B = 0 , (173)because of the on shell invariance conditions A Q = 0 Q ∼ B = 0 (174)satisfied by A and B as physical observables.37or each on shell φ the values of the D ± A B are physical, i.e., P - and η - independent. However, as functionals of the φ (off shell as well as on)the D ± A B turn out not to be physical observables unless both A and B areabsolute invariants. To see this introduce parameters ξ α of compact supportin M , and make use of the supercondensed notation. Then if one evaluatesthe following quantity on shell, one obtains:( D ± A B ) Qξ = ( A G ∓ B ) Qξ = A G ∓ B Qξ + A G ∓ Qξ B + ξ ∼ Q ∼ A G ∓ B = A G ∓ B Qξ + A G ∓ S Qξ G ∓ B + ξ ∼ Q ∼ A G ∓ B , (175)where eqs. (171) and (174) have been used.In order to evaluate B Qξ and ξ ∼ Q ∼ A , multiply A Q = S a B Q = S b (176)by ξ and functionally differentiate; the result is (on shell): A Qξ = − ( ξ ∼ Q ∼ ) A + S aξ B Qξ = − ( ξ ∼ Q ∼ ) B + S bξ. (177)Functionally differentiating twice S Qξ = 0 and then going on shell, oneobtains the third Ward identity:First functional derivative: S Qξ = − ( ξ ∼ Q ∼ ) S (178)Second functional derivative: S Qξ + S ( Qξ ) = − ( ξ ∼ Q ∼ ) S − ( ξ ∼ Q ∼ ) S (179)Going on shell: S Qξ = − S ( Qξ ) − ( ξ ∼ Q ∼ ) S . (180)Hence, inserting eqs. (177) and (180) in (175), one obtains:( D ± A B ) Qξ = A G ∓ ( − ( ξ ∼ Q ∼ ) B + S bξ )+ A G ∓ ( − S ( Qξ ) − ( ξ ∼ Q ∼ ) S ) G ∓ B +( − A ( Qξ ) + ( ξ ∼ a ∼ ) S ) G ∓ B = − A G ∓ ( ξ ∼ Q ∼ ) B + A ( − Π ∓ ) bξ − A ( − Π ∓ ) ( Qξ ) G ∓ B − A G ∓ ( ξ ∼ Q ∼ ) ( − Π ±∼ ) B − A ( Qξ ) G ∓ B + ξ ∼ a ∼ ( − Π ±∼ ) B = − A bξ − ξ ∼ a ∼ B , (181)38n which the properties of the projection operator Π ± and eq. (174) havebeen exploited in obtaining the final expression.This remarkably simple expression has two important properties. First, itvanishes if A and B are absolute invariants, i.e., if a and b vanish. Second, itis independent of the ± signs, being the same for both retarded and advanceddisturbances.Another result that is independent of the ± signs is the following, whichshows explicitly that D ± A B is not invariant under changes in A and B of theform (84); on shell: D ± ¯ A ¯ B − D ± A B = ( A + a l S l, ) ,i G ∓ ij ( B j, + S ,k b ) k − A ,i G ∓ ij B j, = ( A + a ∼ S ) G ∓ ( B + S b ) − A G ∓ B = A G ∓ ( S b )+( a ∼ S ) G ∓ ( B )+( a ∼ S ) G ∓ ( S b )= A G ∓ S b + A G ∓ S b + a S G ∓ B + a ∼ S G ∓ B + a ∼ S G ∓ S b + a ∼ S G ∓ S b + a ∼ S G ∓ S b + a ∼ S G ∓ S b = A ( − Π ∓ ) b + a ∼ ( − Π ±∼ ) B + a ∼ S ( − Π ∓ ) b = − A b − a ∼ B − a ∼ S b, (182)where ¯ A = A + a i S i, ¯ B = B + S ,k b k . (183)Here a i and b k are assumed to have the properties (e.g., rapid fall-off in thepast and future) that are necessary for the associative law of multiplicationto hold. Let A and B be two physical observables. Their Peierls Bracket is definedto be (
A, B ) ≡ D − A B − ( − AB D − B A. (184)39sing the definition (157) and the reciprocity relation for physical observables(159) one may re-express this bracket in the form( A, B ) = D − A B − D + A B = A ,i G + ij B j, − A ,i G − ij B j, = A ,i ˜ G ij B j, , (185)where ˜ G ij ≡ G + ij − G − ij . (186)In anticipation of its role in quantum theory ˜ G wll be called the supercom-mutator function ; it has the symmetry properties˜ G ji = G + ji − G − ji = ( − ij ( G − ij − G + ji )= − ( − ij ˜ G ij , (187)or, in supercondensed notation ˜ G ∼ = − ˜ G. (188)Unlike D ± A B the Peierls bracket , as a functional of φ , is always a physicalobservable no matter whether A and B are absolute invariants or conditionalinvariants. This follows from (175), which yields( A, B ) Qξ = ( D − A B ) Qξ − ( D + A B ) Qξ = 0 . (189)Moreover, we also have, using (182)( ¯ A, ¯ B ) = D − ¯ A ¯ B − D +¯ A ¯ B = ( D − A B − A b − a ∼ B − a ∼ S b ) − ( D + A B − A b − a ∼ B − a ∼ S b )= D − A B − D + A B = ( A, B ) , (190)where ¯ A , ¯ B are given by (183). Equation (190) shows that it is immate-rial whether the dynamical equations are used before or after computing thePeierls bracket. That is, use of Peierls bracket commutes with use of any onshell conditions or restrictions .If the dynamical system possesses no invariant flows, then the φ are them-selves physical observables, and eq. (185) implies( φ i , φ j ) = ˜ G ij (191)40hen invariant flows are present the Peierls bracket of the φ is not defined.However, in computing the brackets of observables one may proceed as if itwere given by eq. (191). Let A α and B α be any two families of physical observables, and let U ( A )and V ( B ) be two any functions of these families. Equation (185) has theimmediate corollary( U ( A ) , V ( B )) = U ( A ) ←− ∂∂A α ( A α , B β ) −→ ∂∂B β V ( B ) . (192)The following properties are called simple identities satisfied by the PeierlsBracket; their proof is straightforward:( A, B + C ) = ( A, B ) + (
A, C ) , (193)( A, BC ) = (
A, B ) C + ( − AB B ( A, C ) , (194)( A, B ) = − ( − AB ( B, A ) . (195) Given a classical field theory, how does one pass from the classical theory tothe quantum theory? Traditionally, one attempts to answer the first questionby starting from a classical (or super classical) dynamical system and obtain-ing from it a corresponding quantum system. Each real dynamical variable φ is replaced by a self-adjoint linear operator ˆ φ of the same type ( c -typeor a -type), and these operators are assumed to satisfy differential equationssimilar, if not identical, in form to the dynamical equations of the classicaltheory. The only differences are: (i) a particular choice of factor orderingmay have to be made, (ii) renormalization constants may have to be inserted,and (iii) in order to maintain consistency one may have to add extra termsthat do not appear in the (super)classical theory.In the same way, a functional R [ φ ] on Φ is replaced by an operator ˆ R ≡ R [ ˆ φ ]. The super Hilbert or Fock space on which the operators act is notgiven a priori but is constructed in such a way as to yield a representation41f the operator superalgebra satisfied by the ˆ φ . This superalgebra is alwaysdetermined in some way by the heuristic quantization rule ( φ j , φ k ) ( ˆ φ j , ˆ φ k ) ≡ − i [ ˆ φ j , ˆ φ k ] ( ~ = 1) , (196)which tries to identify, up to a factor i , each Peierls bracket with a supercom-mutator. When the (super)classical action S possesses invariant flows, thevariables ˆ φ i , and hence the supercommutator (196), are defined only mod-ulo invariance transformations. If there are no invariant flows one may inprinciple write [ ˆ φ j , ˆ φ k ] = i ˆ˜ G jk , (197)but there is a difficulty. When the dynamical equations are nonlinear thequantum supercommutator function ˆ˜ G ij is not just the identity operatortimes the classical ˜ G ij but depends on the ˆ φ i . It is usually difficult if notimpossible to give a simple factor-ordering prescription for passing from ˜ G ij to ˆ˜ G ij . The difficulty is even greater with the more general quantization rule[ ˆ A, ˆ B ] = i ( ˆ A, ˆ B ) ? = i ˆ A ,j ˆ˜ G jk ˆ B k, , (198)which is applicable in principle to all physical observables (i.e., flow invari-ants). Even though simple factor-ordering prescriptions may exist for defin-ing ˆ A and ˆ B there will often be no simple prescription for passing from theclassical A ,i ˜ G ij B j, to its quantum analog.A possible way out of these difficulties will be discussed in the following. In classical physics the dynamical equations are of central importance be-cause their solutions correspond directly to reality. In quantum physics thesituation is different. Solutions of the dynamical equations represent the sys-tem only in a generic sense. Correspondence to reality can be set up onlywhen the state vector has been specified.Instead of making direct use of the operator dynamical equations one canexpress the dynamical content of the quantum theory in another form, whichbrings the state vector into the picture and which is often more useful inapplications. Let ˆ A , ˆ B be any two physical observables of a given systemwhich satisfy supp ˆ A ,i > supp ˆ B ,i . (199)That is, ˆ A is constructed out of ˆ φ taken from a region of space-time thatlies to the future of the region from which the ˆ φ making up ˆ B are taken.42et | a i and | b i be normalized physical eigenvectors of ˆ A and ˆ B respectively,corresponding to physical eigenvalues a and b . The inner product h a | b i isoften called a transition amplitude . If the state vector of the system is | b i ,then h a | b i is the probability amplitude for the system to be found in the staterepresented by | a i , i.e., for the value a to be obtained when ˆ A is measured.The probability itself is |h a | b i| . Suppose the action functional of the classical theory suffers an infinitesimalchange δS ; the quantum theory will change accordingly, and we shall postu-late that the associated change δ ˆ S in the quantum action ˆ S is self-adjoint.This produces a change in the quantum dynamical equations and hence achange in their solutions ˆ φ i . Suppose the forms of ˆ A and ˆ B as functionalof the ˆ φ i remain unchanged. As operators, ˆ A and ˆ B will nevertheless bechanged because the ˆ φ i have changed. Denote these changes by δ ˆ A and δ ˆ B respectively. The eigenvectors | a i and | b i too will suffer changes δ | a i and δ | b i .The precise nature of these changes will depend on boundary conditions.Suppose δ ˆ S satisfies the conditionsupp ˆ A ,i > δ ˆ S ,i > supp ˆ B ,i . (200)That is, suppose δ ˆ S is constructed out of ˆ φ ’s taken from a region of space-time that lies to the past of the region associated with ˆ A and to the future ofthe region associated with ˆ B . Suppose furthermore that retarded boundaryconditions are adopted. Then at times to the past of the region associatedwith δ ˆ S the dynamical variables ˆ φ i will remain unchanged. This means that δ ˆ B = 0 . (201)The observable ˆ A , on the other hand, suffers a change which, with DeWitt’snotation, can be written as δ ˆ A = D − δ ˆ S ˆ A. (202)In view of the kinematical relation (200) one has also: D − ˆ A δ ˆ S = 0 , (203)and hence δ ˆ A = D − δ ˆ S ˆ A − D − ˆ A δ ˆ S ≡ ( δ ˆ S, ˆ A ) . (204)Therefore, imposing the heuristic quantization rule, one finds δ ˆ A = − i [ δ ˆ S, ˆ A ] . (205)43ince the relation between Peierls brackets and supercommutators is onlyheuristic, the derivation of eq. (205) is hardly rigorous. Indeed, if an ar-bitrary operator ordering is chosen for the dynamical equations, eq. (205)need not hold. However, there is an inevitability and elegance about thisequation which suggests that one turn the problem around and demand thatthe dynamics be such that it does hold, whatever operator ordering may bechosen for δ ˆ S as a functional of the ˆ φ ’s. Note that if it holds for ˆ A = ˆ φ j ,with j > δ ˆ S ,i , (205) readsˆ φ j + δ ˆ φ j = ˆ φ j − i [ δ ˆ S, ˆ φ j ] ≡ − iδ ˆ S ˆ φ j + i ˆ φ j δ ˆ S = ˆ u − ˆ φ j ˆ u, (206)with ˆ u ≡ iδ ˆ S, (207)hence (206) is a unitary transformation, and (205) holds for all ˆ A satisfyingthe kinematical inequality (200):ˆ A + δ ˆ A = A [ ˆ φ + δ ˆ φ ]= A [ˆ u − ˆ φ ˆ u ]= ˆ u − A [ ˆ φ ]ˆ u = ˆ A − i [ δ ˆ S, ˆ A ] . (208)In this work the previous equations will be postulated as rigorous statementsof quantum dynamics; moreover, we shall try to costrain the structure δ ˆ S so that corresponding to each classical theory (i.e., to each action functional S ) there is a virtually unique quantum theory or at most a unique family ofquantum theories. Modulo an ignorable phase change iδθ | a i , with δθ real c -number, thechange (205) induces a change in the eigenvector | a i given by | a i + δ | a i = ˆ u − | a i , (209)as is straightforward to verify:( ˆ A + δ ˆ A )(ˆ u − | a i ) = (ˆ u − A [ ˆ φ ]ˆ u )(ˆ u − | a i )= ˆ u − A [ ˆ φ ] | a i = a (ˆ u − | a i ) . (210)Equation (209) is equivalent to δ | a i = − iδ ˆ S | a i . (211)44quation (201), on the other hand, implies: δ | b i = 0 (212) modulo a similar ignorable phase change. Hence δ h a | b i = ( δ h a | ) | b i + h a | ( δ | b i )= i h a | δ ˆ S | b i . (213)Equation (213) is known as the Schwinger Variational Principle . Althoughthe Schwinger variational principle was “derived” through imposition of re-tarded boundary conditions, it is in fact independent of boundary conditions.For example, if advanced boundary conditions are imposed and use is madeof the reciprocity relation (159), then eqs. (205) and (201) get replaced by δ ˆ A = 0 , (214) δ ˆ B = D + δ ˆ S ˆ B = D − ˆ B δ ˆ S − D − δ ˆ S ˆ B = ( ˆ B, δ ˆ S ) = − i [ ˆ B, δ ˆ S ] , (215)which imply δ | a i = 0 , δ | b i = iδ ˆ S | b i (216)again leading to (213).Whether one imposes retarded or advanced boundary conditions, or some-thing in between, the following statements are always true:1. The unperturbed dynamical equations continue to hold in the regionsto the past and to the future of supp δ ˆ S ,i .2. The ˆ φ + δ ˆ φ in these regions are related to the unperturbed ˆ φ by unitarytransformations. Remark . Being the variation (213) a unitary transformation the Schwingerprinciple is guaranteed to preserve both the probability interpretation of thequantum theory and the unit normalization of total probability.
Remark . The particular choice of physical observables ˆ A and ˆ B in thestatement of the Schwinger principle is irrelevant. Only the condition (200)is important. Since the eigenvalues of more than one observable usually haveto be specified in order to determine a quantum state uniquely, it will beconvenient from now on to replace (213) by the more general statement δ h out | in i = i h out | δ ˆ S | in i , (217)where | in i and | out i are state supervectors determined by some unspecifiedconditions on the dynamics in regions respectively to the past and to thefuture of the region in which one may wish to vary the action.45 .4 External sources and chronological products When the action possesses no invariant flows, a particularly convenient wayto vary ˆ S is to append to it a term of the form J i ˆ φ i , where the J i are puresupernumber-valued functions over space-time, c -type and real when ˆ φ i is c -type, a -type and imaginary when ˆ φ i is a-type. The J i are called externalsources .Let the external sources suffer variations δJ i , whose supports are confinedto the space-time region lying, in time, between the regions associated withthe state supervectors | in i and | out i . Then the transition amplitude h out | in i suffers the change δ h out | in i = i h out | δJ j ˆ φ j | in i , (218)which implies −→ δiδJ j h out | in i = ( − jF h out | ˆ φ j | in i , (219)where F is the fermionic number of | out i , i.e., F is 0 or 1 according as | out i is c -type or a -type. Let | φ i be a complete set of normalized physical eigen-vectors of ˆ φ j , corresponding to the eigenvalues φ j . Such eigenvectors existsince, when no invariant flows are present, the ˆ φ j are physical observables.The previous equation may then be rewritten in the form −→ δiδJ j h out | in i = ( − jF X h out | φ i φ j h φ | in i , (220)where the summation is over all the | φ i . Now let δJ i be a second variationin the sources, and suppose supp δJ i > j . Then the factor h φ | in i in (220)remains unchanged, and δ −→ δiδJ j h out | in i = ( − jF X ( δ h out | φ i ) φ j h φ | in i = ( − jF X ( h out | δJ k ˆ φ k | φ i ) φ j h φ | in i = ( − jF h out | δJ k ˆ φ k ˆ φ j in i . (221)Therefore −→ δiδJ i −→ δiδJ j h out | in i = ( − ( i + j ) F h out | ˆ φ i ˆ φ j | in i . (222)46f, on the other hand, supp δJ i < j , then δ −→ δiδJ j h out | in i = ( − jF X h out | φ i φ j ( δ h φ | in i )= i ( − jF X h out | φ i φ j ( h φ | δJ k ˆ φ k | in i )= i ( − jF h out | ˆ φ j δJ k ˆ φ k | in i )= i ( − jF h out | ˆ φ j δJ k ˆ φ k | in i ) , (223)and −→ δiδJ i −→ δiδJ j h out | in i = ( − ( i + j ) F + ij h out | ˆ φ j ˆ φ i | in i . (224)Continuing in this manner one obtains, quite generally, −→ δiδJ i ... −→ δiδJ i n h out | in i = ( − ( i + ...i n ) F h out |T ( ˆ φ i ... ˆ φ i n ) | in i (225)where T is the chronological ordering operator , which rearranges the factorsˆ φ i ... ˆ φ i n so that the times associated with the indices appear in chronologicalsequence, increasing from right to left, and which inserts an additional factor − a -type indices that occurs in the carryingout of this rearrangement.In the above derivation of (225) the times associated with the indiceswere assumed to be in a well defined chronological order. However, we shallultimately need to give meaning to the chronological product T ( ˆ φ i ... ˆ φ i n ) forarbitrary relative orientations of the space-time points associated with theindices. When the points associated with an index pair i, j are separated by aspacelike interval there is no ambiguity in the chronological product becausethe supercommutator function ˆ˜ G ij then vanishes. Problems arise in the limitwhen two points coincide: it will be seen later that these problems are allresolved by requiring the T -operation to commute with both differentiationand integration with respect to space-time coordinates: this requirement isequivalent to first imposing the linearity condition T (( α ˆ φ i + β ˆ φ j ) ˆ φ k ˆ φ l ... ) = α T ( α ˆ φ i ˆ φ k ˆ φ l ... ) + β T ( ˆ φ j ˆ φ k ˆ φ l ... ) (226)for all α, β ∈ Λ ∞ and then requiring the T -operation to commute with certainoperations involving passage to a limit. Remark . The above requirements have the consequence that an expressionlike h out |T ( ˆ S ,i ˆ φ i ˆ φ j ) | in i does not generally vanish despite the fact that ˆ S ,i iszero when the operators that compose it are ordered appropriately for theoperator dynamical equations. 47ow let A [ φ ] be any functional of the classical φ for which supp A ,i liesbetween the “in” and “out” regions, and which possesses a functional Taylorexpansion about φ = 0 with a nonzero radius of convergence: A [ φ ] = A [0] + A [0] φ + 12 A [0] φφ + ... . (227)It then follows from eq. (225) and the requirements illustrated by eq. (226)that h out |T ( A [ ˆ φ ]) | in i = ( − AF A " −→ δiδJ h out | in i , (228) A " −→ δiδJ ≡ A [0] + A [0] −→ δiδJ + 12 A [0] −→ δiδJ −→ δiδJ + ... . (229)The previous equations provide a way of associating an operator, i.e., T ( A [ ˆ φ ]),with each classical functional A [ φ ] having appropriate properties. If A [ φ ] hasonly a finite radius of convergence, then, strictly speaking, T ( A [ ˆ φ ]) is not de-fined by eqs. (228) and (229), but it can often be given a meaning, for given“in” and “out” states, by analytic continuation. One can also frequently give T ( A [ ˆ φ ]) a meaning, even when A [ φ ] is singular at φ = 0, by expanding abouta different point and continuing analytically. Equation (228) finds a widevariety of applications in quantum field theory. The association established by eq. (228), between a classical functional anda quantum operator, suggests that when A [ φ ] is a classical observable theoperator T ( A [ ˆ φ ]) might be taken as its quantum counterpart. This sugges-tion is often valid when T ( A [ ˆ φ ]) is self-adjoint. However, the chronologicalproduct is not self-adjoint in general, even when A [ φ ] is real because the op-eration of taking the adjoint reverses the order of all factors, placing them inantichronological order .In order for T ( A [ ˆ φ ]) to be self-adjoint A [ φ ] must usually be local , i.e.,built out of φ ’s and their derivatives all taken at the same space-time point.But this is not sufficient to guarantee the self-adjointness of T ( A [ ˆ φ ]). For ex-ample, T ( S ,i [ ˆ φ ]) is not generally self-adjoint (or anti-self-adjoint if the index i is a -type rather than c -type). Hence the operator dynamical equations ofthe quantum theory should not be taken in the form T ( S ,i [ ˆ φ ]) = 0 or, when48xternal sources are present, in the form T ( S ,i [ ˆ φ ]) = − J i . What we shallassume instead is the validity of the following postulate: there exists a functional µ [ φ ] , determined by the classical action S [ φ ] ,such that the operator dynamical equations take the form T { S [ ˆ φ ] − i log µ [ ˆ φ ] } ←− δδφ k ! = − J k . (230)The functional µ [ φ ] is known as the measure functional . At the simplestlevel it may be thought of as correcting for the lack of self-adjointness oranti-self-adjointness of T ( S ,i [ ˆ φ ]). But it plays a far deeper role than this:once it is chosen the quantum theory is completely determined up to a finite-parameter family. Establishment of a correspondence between each classi-cal theory and a unique quantum theory (or family of quantum theories) istherefore achieved by making µ [ φ ] depend in a definite way on S [ φ ]. Howthis dependence is itself to be chosen is a major question, which has to beapproached in steps, but there is an easy argument that leads quickly to atleast an approximate answer.In the quantum theory, as in the classical theory, it is often convenient toseparate the dynamical variables ˆ φ i into a background φ i and a remainder ˆ φ i .If the background φ i is classical, i.e., it is a pure supernumber-valued functiontimes the identity operator, then the ˆ φ i satisfy the same supercommutationrelations as the ˆ φ i : [ ˆ φ j , ˆ φ k ] = i ˆ˜ G jk . (231)In terms of the ˆ φ i the operator dynamical equations take the form S ,i [ φ ] + S ,ik [ φ ] ˆ φ k + 12 S ,ikl [ φ ] ˆ φ l ˆ φ k + O ( ~ ) + O ( ˆ φ ) = − J i (232)The terms of order 0 and 1 in ˆ φ on the left-hand side are unambiguous.Terms of higher order are not unambiguous since we do not yet know thecomplete factor-oredering rules. The coefficients of the higher order termswill generally not be just the classical coefficients S ,ijk... . However, they willdiffer from the classical coefficients by terms of order ~ , which arise whenthe ˆ φ ’s are ordered as in (232) and which will be dropped in the presentapproximate analysis.Taking the supercommutator of (232) with ˆ φ j , remembering that the s-ources J i are supernumber-valued, and moving the index i to the left, one49nds S ,ik [ φ ][ ˆ φ k , ˆ φ j ] + 12 S ,ikl [ φ ][ ˆ φ l ˆ φ k , ˆ φ j ] + ... = 0 ,iS ,ik [ φ ] ˆ˜ G kj − ( − j ( k + l ) S ,ikl [ φ ][ ˆ φ j , ˆ φ l ˆ φ k ] + .... = 0 ,iS ,ik [ φ ] ˆ˜ G kj − ( − j ( k + l ) S ,ikl [ φ ]([ ˆ φ j , ˆ φ l ] ˆ φ k +( − jl ˆ φ l [ ˆ φ j , ˆ φ k ]) + ... = 0 ,iS ,ik [ φ ] ˆ˜ G kj − ( − j ( k + l ) S ,ikl [ φ ] (cid:16) i ˆ˜ G jl ˆ φ k + i ( − jl ˆ φ l ˆ˜ G jk (cid:17) + ... = 0 ,S ,ik [ φ ] ˆ˜ G kj − ( − j ( k + l ) S ,ikl [ φ ] (cid:16) ˆ˜ G jl ˆ φ k + ( − jl ˆ φ l ˆ˜ G jk (cid:17) + ... = 0 ,S i, ,k [ φ ] ˆ˜ G kj − ( − j ( k + l ) S i, ,kl [ φ ] ˆ˜ G jl ˆ φ k − ( − j ( k + l )+ jl S i, ,kl [ φ ] ˆ φ l ˆ˜ G jk + ... = 0 ,S i, ,k [ φ ] ˆ˜ G kj + ( − jk S i, ,kl [ φ ] ˆ˜ G lj ˆ φ k + 12 S i, ,kl [ φ ] ˆ φ l ˆ˜ G kj + ... = 0 . (233)The quantum supercommutator function, like the classical supercommutatorfunction, can be expressed as the difference between an advanced Green’sfunction and a retarded Green’s function, both now operator-valued:ˆ˜ G ij = ˆ G + ij − ˆ G − ij , (234)where S i, ,k [ ˆ φ ] ˆ G ± kj = − δ ji ,S i, ,k [ φ ] ˆ G ± kj + ( − jk S i, ,kl [ φ ] ˆ G ± lj ˆ φ k ++ 12 S i, ,kl [ φ ] ˆ φ l ˆ G ± kj + ... = − δ ji . (235)The previous equation, as is straightforward to verify, can be solved by iter-ation, yielding:ˆ G ± ij = G ± ij [ φ ] + 12 ( − jk G ± im [ φ ] S m, ,kl G ± lj [ φ ] ˆ φ k + 12 G ± im [ φ ] S m, ,kl ˆ φ l G ± kj [ φ ] + ... = G ± ij [ φ ] + G ± ij,k [ φ ] ˆ φ k + ... . (236)50gain it is not possible to specify the forms of the higher terms in the ex-pansion. The coefficients of these terms will generally not be just functionalderivatives G ± ij,kl... [ φ ] of the classical Green’s functions.Introduce now the space-time generalization of the step function: θ ( i, j ′ ) ≡ x > x ′ if x = x ′ x < x ′ (237)where x is a global timelike coordinate and the submanifolds x = constantare complete spacelike Cauchy hypersurfaces. If the points x and x ′ are notin the immediate vicinity of one another, thenˆ φ i ˆ φ j ′ − T ( ˆ φ i ˆ φ j ′ ) = [ θ ( i, j ′ ) + θ ( j ′ , i )] ˆ φ i ˆ φ j ′ − θ ( i, j ′ ) ˆ φ i ˆ φ j ′ − ( − ij ′ θ ( j ′ , i ) ˆ φ j ′ ˆ φ i = θ ( j ′ , i ) h ˆ φ i , ˆ φ j ′ i = iθ ( j ′ , i ) ˆ˜ G ij ′ = i ˆ G + ij ′ . (238)We shall assume that this equation in fact holds for all x , x ′ , at least upto the order needed in our analysis of the expanded dynamical equations.Equation (232) may then be written in the form − J l = S ,l [ φ ] + S ,lk [ φ ] ˆ φ k + 12 S ,ljk [ φ ] h T ( ˆ φ k ˆ φ j ) + i ˆ G + kj i + ... = T ( S ,l [ ˆ φ ] + 12 iS ,ljk [ ˆ φ ] G + kj [ ˆ φ ] + ... ) , (239)where the dots in the second line stands not only for the unwritten terms inthe first line but also for the error in replacing S ijk [ φ ] ˆ G + kj by T ( S ,ijk [ ˆ φ ] G + kj [ ˆ φ ).The previous expression can be simplified by recalling that S is a negativeinverse of G + ; therefore: S ,ijk G + kj = − S ,jk G + kj,i + ( S ,jk G + kj ) ,i = − S ,jk G + kj,i + ( − δ kk ) ,i = − S ,jk G + kj,i = − ( − j S j, ,k G + kj,i = log | sdet G + | ) ,i . (240)Thus − J k = T ( S ,k [ ˆ φ ] + 12 i (log | sdet G + [ ˆ φ ] | ) ,k + ... ) . (241)51omparison of eqs. (230) and (241) yields µ [ φ ] ≈ const · | sdet G + [ φ ] | − . (242) The transition amplitude h out | in i is a functional of the external sources. Letus try to express it as a functional Fourier integral: h out | in i = Z X [ φ ] e iJφ [ dφ ] = Z e iJφ X [ φ ][ dφ ] , (243)[ dφ ] ≡ Y i dφ i . (244)Here the φ i are supernumber-valued variables of integration, and the productin eq. (244) is a continuous infinite one. Both it and the integral itself arethus formal expressions. Integration over the c -number variables is to beunderstood as patterned on ordinary integration. Integration over the a -number variables is to be understood as an infinite limit of a multiple Berezinintegral (see [19] and [20]). The integral is to be understood as taken overa certain subspace of the space of field histories Φ, whose properties will beindicated below. Assuming the validity of integrating by parts, and making52se of eqs. (228) and (230), one may write Z X [ φ ] ←− δiδφ k e iJφ [ dφ ] == − Z X ( e iJφ ←− δiδφ k )[ dφ ]= − Z X ( e iJφ ←− δiδφ k )[ dφ ]= − Z Xe iJφ J k [ dφ ]= − Z XJ k e iJφ [ dφ ]= − ( − kX J k Z Xe iJφ [ dφ ]= − ( − kX J k Z Xe iJφ [ dφ ]= − ( − kX J k h out | in i = ( − k ( X + F ) h out | − J k | in i = ( − k ( X + F ) h out |T { S [ ˆ φ ] − i log µ [ ˆ φ ] } ←− δδφ k ! | in i = ( − kX { S ,k h −→ δ /iδJ i − iµ − h −→ δ /iδJ i µ ,k h −→ δ /iδJ i }h out | in i = ( − kX { S ,k h −→ δ /iδJ i − iµ − h −→ δ /iδJ i µ ,k h −→ δ /iδJ i } Z e iJφ X [ φ ][ dφ ]= ( − kX Z { S [ φ ] − i (log µ [ φ ]) } ←− δδφ k e iJφ X [ φ ][ dφ ] . (245)Because of the uniqueness of Fourier integral representations the integrandsin the first and last lines must be equal: X [ φ ] ←− δiδφ k = ( − kX { S [ φ ] − i (log µ [ φ ]) } ←− δδφ k X [ φ ] . (246)One possible solution of this equation is X [ φ ] = N e iS [ φ ] µ [ φ ] , (247)where N is a constant of integration. This leads to h out | in i = N Z e i ( S [ φ ]+ Jφ ) µ [ φ ][ dφ ] . (248)53 .7 The Schwinger variational principle revisited Expression (248), when combined with eq. (228), yields immediately a func-tional integral expression for the “in-out” matrix elements of chronologicalproducts: h out |T ( A [ ˆ φ ]) | in i = ( − A ( F + N ) N Z A [ φ ] e i ( S [ φ ]+ Jφ ) µ [ φ ][ dφ ] . (249)The previous equation can be used to obtain a partial check on the con-sistency of the Feynman integral with the Schwinger variational principlewhich was used to derive it in the first place. Under a variation δS in thefunctional form of the action, such that supp δS ,i lies in the “in between”region, eqs. (248) and (249) yield δ h out | in i = N Z (cid:0) iδS [ φ ] e i ( S [ φ ]+ Jφ ) µ [ φ ] + e i ( S [ φ ]+ Jφ ) δµ [ φ ] (cid:1) [ dφ ]= N Z (cid:0) iδS [ φ ] e i ( S [ φ ]+ Jφ ) µ [ φ ] + e i ( S [ φ ]+ Jφ ) µ [ φ ] δ log µ [ φ ] (cid:1) [ dφ ]= iN Z ( δS [ φ ] − iδ log µ [ φ ]) e i ( S [ φ ]+ Jφ ) µ [ φ ]= i h out |T ( δS [ ˆ φ ] − iδ log µ [ ˆ φ ]) | in i . (250)But from (242) one has δ log µ [ φ ] = µ − [ φ ] δµ [ φ ] ≈ | sdet G + [ φ ] | δ | sdet G + [ φ ] | − = | sdet G + [ φ ] | (cid:18) − (cid:19) | sdet G + [ φ ] | − δ | sdet G + [ φ ] | = | sdet G + [ φ ] | (cid:18) − (cid:19) | sdet G + [ φ ] | − | sdet G + [ φ ] | str( G + [ φ ] δ S [ φ ])= −
12 str( G + [ φ ] δ S [ φ ]) . (251)Hence δ h out | in i = i h out |T ( δS [ ˆ φ ] + i G + [ ˆ φ ] δ S [ ˆ φ ]) + ... ) | in i . (252)By the same kind of rearrangement as was used in obtaining eq. (241) oneeasily sees that the chronological product is, at least approximately, just δS [ ˆ φ ] in its self-adjoint operator form.54 .8 Expressions involving S ,i Let A [ φ ] be an arbitrary functional of φ such that the support of A ,j liesbetween the “in” and “out” regions. Since the functional integral respectsthe procedure of integration by parts, the following identity holds:0 = ( − ( A + i )( F + N ) N Z [ dφ ] (cid:0) A [ φ ] µ [ φ ] e i ( S [ φ ]+ Jφ ) (cid:1) ←− δδφ k = h out |T ( A ,k [ ˆ φ ] + A [ ˆ φ ](log µ ) ,k [ ˆ φ ] + iA [ ˆ φ ] S ,k [ ˆ φ ]) | in i . (253)Since the “in” and “out” state supervectors may be chosen arbitrarily thisis in fact a statement about chronological products of operators involving ˆ S ,i when the sources vanish: T ( S [ ˆ φ ] − i log µ [ ˆ φ ]) ←− δδφ k A [ ˆ φ ] ! = i ( − iA T (cid:16) A ,k [ ˆ φ ] (cid:17) . (254)This expression does not generally vanish despite eq. (230). The task of extending the previous treatment to gauge theories, i.e., fieldtheories whose action functional features invariance flows, requires a basicunderstanding of the properties of the space Φ of field histories, its structureand its geometry. We shall only focus on Type-I theories in which the co-efficients c αβγ are structure constants of an infinite dimensional Lie group,the gauge group, which we shall denote by G . As already stated, Φ may beviewed as a principal bundle having G as its typical fibre. Real physics takesplace in the base space of this bundle, i.e., the space of orbits (or fibres),denoted by Φ / G .Since G is a group manifold it admits an invariant Riemannian or pseudo-Riemannian metric. This metric can be extended in an infinity of ways to agroup (or flow) invariant metric on Φ. But it turns out that if one requiresthe extended metric to be ultralocal (a requirement that greatly simplifiesthe analysis of any formalism in which it is used) then, up to a scale factor,it is unique in the case of the Yang-Mills field and belongs to a one-parameterfamily in the case of gravity, these being the two primary gauge systems ofinterest. 55et us denote this metric tensor by γ and its components in the chartspecified by the dynamical variables φ i by γ i j . Let x and x ′ be the space-timepoints specified by the indices i and j . Ultralocality of γ is the conditionthat γ i j be equal to the undifferentiated delta distributon δ ( x, x ′ ) times acoefficient that involves no space-time derivatives of fields. Group invarianceof γ is the statement L Q α γ = 0 . (255)The Q α are Killing vectors for the metric γ and vertical vector fieldsfor the principal bundle Φ. In the following, we shall rely on the followingconvention: in gauge theories the indices from the first part of the Greekalphabet are always c -type. Therefore they need never appear in exponentsof ( − a from the middle of the Latin alphabet refer tofermion fields that may be coupled to the basic gauge fields.Choice of an invariant metric on Φ immediately singles out a naturalfamily of connection 1-forms ω α on Φ: ω αi ≡ Q i β N βα , (256)where Q i α = γ i j Q j α and N βα is any coherent Green’s function of the realself-adjoint (and hence symmetric) operator M αβ ≡ − Q i α γ i j Q j β , (257)which, in all cases of interest, turns out to be globally nonsingular, i.e., overthe whole of Φ. In view of the relation M αβ N βγ = − δ γα one easily sees that Q i α ω βi = Q i α Q i γ N γβ = Q i α γ i j Q j γ N γβ = −M αγ N γβ = δ βα , (258)and that horizontal vectors on Φ are those that are perpendicular (under themetric γ ) to the fibres. A horizontal vector may be obtained from any vectorby application of the horizontal projection operator :Π ij ≡ δ ij − Q i α ω αj , ω αi ≡ ( − i ω αi . (259)56he name is justified by the following properties:Π ij Π jk = ( δ ij − Q i α ω αj )( δ jk − Q j α ω αk )= δ ik − Q i α ω αk − Q i α ω αk + Q i α ω αj Q j β ω βk = δ ik − Q i α ω αk + ( − j Q i α Q j γ N γα Q j β ω βk = δ ik − Q i α ω αk + ( − j Q i α γ j l Q l γ N γα Q j β ω βk = δ ik − Q i α ω αk + ( − j + j Q i α Q j β γ j l Q l γ N γα ω βk = δ ik − Q i α ω αk − Q i α M βγ N γα ω βk = δ ik − Q i α ω αk + Q i α δ αβ ω βk = δ ik − Q i α ω αk + Q i α ω αk = δ ik − Q i α ω αk = Π ik , (260) ω αi Π ij = ω αi ( δ ij − Q i β ω βj )= ω αj − ω αi Q i β ω βj = ω αj − ( − i + j ω αi Q i β ω βj = ω αj − ( − j Q i β ω αi ω βj = ω αj − ( − j δ αβ ω βj = ω αj − ( − j ω αj = ω αj − ω αj = 0 , (261)Π ij Q j α = ( δ ij − Q i β ω βj ) Q j α = Q i α − Q i β ω βj Q j α = Q i α − ( − j Q i β ω βj Q j α = Q i α − ( − j + j Q i β Q j β ω αj = Q i α − Q i β δ βα = Q i α − Q i α = 0 . (262)57 .2 Fibre-adapted coordinate patches When dealing with Φ it is convenient to consider a transformation φ i → I A , K α (263)to a set of fibre-adapted coordinates I A , K α . Here the I ’s label the fibres(i.e., the points in Φ / G ) and are gauge invariant (i.e., flow invariant): I A Q α = I A,i Q i α = 0 . (264)The K ’s label the points within each fibre. Because there is no canonicalway of associating points on one fibre with those on another, transformationsbetween fibre-adapted coordinate patches have the general structure I ′ A = I ′ A [ I ] , K ′ α = K ′ α [ I, K ] , (265)which is still special enough so that the Jacobian of the transformation splitsinto factors: δ ( I ′ , K ′ ) δ ( I, K ) = δ ( I ′ ) δ ( I ) δ ( K ′ ) δ ( K ) . (266)One often makes specific choices for the K ’s. One usually singles out a basepoint φ ∗ in Φ and chooses the K ’s to be local functionals of the φ ’s of sucha form that the matrix M αβ ≡ K α Q β = K α,i Q i β (267)is a nonsingular differential operator at and in a neighborhood of φ ∗ . Ty-pical convenient choices for φ ∗ are A αµ ∗ ( x ) = 0 in pure Yang-Mills theory and g µν ∗ ( x ) = some well studied background metric (Minkowski, Friedmann-Le-maˆıtre-Robertson-Walker, black hole, etc.) in pure gravity theory.One can also make specific choices for the I ’s, but in general such invari-ants depend nonlocally on the φ ’s and are clumsy to work with; no specificchoices will be made here, so in what follows the I ’s will remain purely con-ceptual.In the region of Φ where the operator M is nonsingular it is easy to showthat ←− δδK α = − Q β N βα , (268)where N is a Green’s function of M . In fact, by multiplying (267) first on58he right by N and then on the left by ←− δδK , and using (264), one obtains M αβ N βγ = K α,i Q i β N βγ , − δ αγ = K α,i Q i β N βγ , − ←− δδK α δ αγ = ←− δδK α K α,i Q i β N βγ , − ←− δδK γ = ←− δδK α K α ←− δδφ i Q i β N βγ , − ←− δδK γ = ←− δδK α K α ←− δδφ i + ←− δδI A I A ←− δδφ i ! Q i β N βγ , − ←− δδK γ = ←− δδφ i Q i β N βγ , − ←− δδK γ = Q β N βγ . (269)It is important to stress that when G is non-Abelian it is impossible for the K α to be valid coordinates globally: in fact if they were, then the ←− δ /δK α ,which are vertical vector fields that commute with each other, would generateAbelian orbits (fibres). This means that M , unlike M , cannot be nonsingularglobally on Φ. Since the physics of a gauge theory takes place in the base space Φ / G it isnatural to try to write a functional integral for “in-out” amplitudes in theform h out | in i = Z µ I [ I ][ dI ] e iS [ I ] . (270)All functional integrals encountered in previous sections were purely formal.Equation (270) is even more formal, for the following reasons:1. The labels I A are not chosen explicitly but used only conceptually.2. Since all known usable explicit choices depend nonlocally on the φ ’s itis hard to know what one can mean by an advanced Green’s functionof the Jacobi field operator S A, ,B (or its superdeterminant) and hencehow to determine the measure µ I [ I ] even approximately.3. It is also hard to know how to set boundary conditions.59o bring the local variables φ i into the theory one must first introduce theremaining variables K α of a fibre-adapted coordinate system and then trans-form to the φ ’s. Let Ω[ I, K ] be a real scalar function(al) on Φ such that theintegral ∆[ I ] ≡ Z e i Ω[ I,K ] µ K [ I, K ][ dK ] (271)exists and is nonvanishing for all I , the measure µ K [ I, K ] being assumed totransform under changes (generally I -dependent) of the fibre-adapted coor-dinates K α according to µ K ′ [ I, K ′ ] = µ K [ I, K ] δKδK ′ . (272)Then ∆[ I ] is invariant under such coordinate changes and one may write: h out | in i = Z [ dI ] Z [ dK ] e i ( S [ I ]+Ω[ I,K ]) ∆[ I ] − µ I,K [ I, K ] , (273)where µ I,K [ I, K ] ≡ µ I [ I ] µ K [ I, K ] . (274)In a similar way one may write the analog of eq. (228) in the forms h out |T ( A [ I ]) | in i = Z µ I [ I ][ dI ] A [ I ] e iS [ I ] (275)= Z [ dI ] Z [ dK ] A [ I ] e i ( S [ I ]+Ω[ I,K ]) ∆[ I ] − µ I,K [ I, K ] . (276)Under changes of fibre-adapted coordinates the measure µ I [ I ] must obvi-ously transform according to µ I ′ [ I ′ ] = µ I [ I ] δIδI ′ , (277)and hence the total measure µ I,K [ I, K ] transforms as it should: µ I ′ ,K ′ [ I ′ , K ′ ] = µ I,K [ I, K ] δIδI ′ δKδK ′ = µ I,K [ I, K ] δ ( I, K ) δ ( I ′ , K ′ ) . (278)To make the transformation from the I A , K α to the local coordinates φ i one must include also the formal Jacobian J [ φ ] ≡ δ ( I, K ) δφ = sdet (cid:18) I A,i K α,i (cid:19) . (279)60hen the functional integrals (273) and (276) take the forms: h out | in i = Z [ dφ ] e i ( S [ φ ]+Ω[ φ ]) ∆[ φ ] − J [ φ ] µ I,K [ φ ] , (280) h out |T ( A [ φ ]) | in i = Z [ dφ ] A [ φ ] e i ( S [ φ ]+Ω[ φ ]) •• ∆[ φ ] − J [ φ ] µ I,K [ φ ] , (281)in which we have abused notation somewhat by simply writing ∆[ I ] = ∆[ φ ], S [ I ] = S [ φ ], µ I,K [ I ] = µ I,K [ φ ], Ω[ I, K ] = Ω[ φ ] and A [ I ] = A [ φ ]. The lastabuse in fact allows a certain generalization of the formalism. In eqs. (228),(229) the functional A was an invariant, i.e., a physical observable. The inte-gral (281) may be regarded as a generalized average which can give meaningto h out |T ( A [ φ ]) | in i even when A is not gauge invariant. True physical am-plitudes, of course, only involve A ’s that are gauge invariant. Note thatwhen (and only when) A is gauge invariant the average (281) is completelyindependent of the choice of the functional Ω[ φ ]. J [ φ ] Suppose we carry out an infinitesimal transformation of the fibre-adaptedcoordinates K α : K ′ α = K α + δK α [ I, K ] . (282)Formally this will produce the following change in the Jacobian J [ φ ]: δJ [ φ ] = δ sdet (cid:18) I A,i K α,i (cid:19) = J str "(cid:18) I A,j K α,j (cid:19) − (cid:18) δI A,i δK α,i (cid:19) = J str (cid:20)(cid:18) φ j,A φ j,α (cid:19) (cid:18) δK α,i (cid:19)(cid:21) = ( − i J φ i,α δK α,i , (283)Here we encounter an immediate problem: the meaning to be given to thefactor φ i,α . If we apply the operator (268) to the fields φ i we get φ i,α = − Q i β N βα , (284)and we have to decide which Green’s function of M to use. Different choicescorrespond to different possible interpretations of the Jacobian itself. Ten-tatively we choose N and J to be coherent with the boundary conditionsappropriate to the functional integral.61ow note that δ log J = J − δJ = ( − i φ i,α δK α,i = − ( − i Q i β N βα δK α,i = −N βα δK α,i Q i β = −N βα δ M αβ = − δ log det N ; (285)therefore δ ( J det N ) = 0 , (286)i.e.,the product J det N is independent of how the coordinates K α are cho-sen. The same is also true of the products J ± det N ± , where the J ± arethe Jacobians interpreted according to advanced or retarded boundary con-ditions. This does not automatically mean that these products depend onlyon the I ’s and are hence gauge invariant: in fact, these products are notscalar functionals, but scalar densities of unit weight; therefore:( J det N ) ←−L Q α = J det N str Q i α,k + ( J det N ) Q α = ( − i J det N Q i α,i + ( J det N ) ,i Q i α = ( − i J det N Q i α,i +( J det N ) ,A I A,i Q i α +( J det N ) ,β K β,i Q i α = ( − i J det N Q i α,i +( J det N ) ,A ( I A Q α )+( J det N ) ,β K β,i Q i α = ( − i J det N Q i α,i , (287)where the following facts have been used:1. the I ’s are flow invariant, i.e., I A Q α = 0,2. J det N is independent of the K ’s, i.e., ( J det N ) ,β = 0,3. ←− δ /δφ i = ←− δ /δI A I A,i + ←− δ /δK β K β,i .Taking the Lie derivative of J det N with respect to Q α is seen to be thesame as multiplying it by ( − i Q i α,i , which is a constant. This constant,62hich essentially describes a rescaling of J det N as one moves up and downthe fibres, depends in no way on the choice made for the K ’s. For practical purposes it may be taken as zero , for two reasons:1. In Yang-Mills theory the zero value may follow formally from the com-pactness of the finite dimensional Lie group with which it is associated,which implies f γγα = 0: Q γ µµ α ′ , γ = Z dx δ ( x, x ′ ) δ ( x, x ) δ µµ f γγα = N δ ( x ′ , x ′ ) f γγα = 0 . (288)2. In both Yang-Mills and gravity theories it is also formally either a δ -distribution, or the derivative of a δ -distribution, with coincident ar-guments. In dimensional regularization such formal expressions vanish.From now on therefore we set ( − i Q i α,i = 0 , (289)and similarly c βαβ = 0 . (290) Ω[ φ ] and a new measure func-tional Although we know that the K α cannot be global coordinates when the gaugegroup is non-Abelian, in the loop expansion we can pretend that they are.That is, we pretend that they are coordinates in a tangent space. A favoritechoice for the functional Ω[ I, K ] is thenΩ = 12 κ αβ K α K β (291)where ( κ αβ ) is a symmetric ultralocal invertible continuous real matrix whichcan be chosen either to be constant or to depend on the base point φ ∗ in theneighborhood of which the operator M αβ of (267) is nonsingular. The K ’sthemselves may be chosen to vanish at this base point.Since we are staying in a single chart it is simplest to choose µ K [ I, K ] = 1 (292)63o that (271) reduces to ∆ = const · (det κ ) − / . (293)Equations (280) and (281) then take the forms h out | in i = Z µ [ φ ][ dφ ] e i ( S [ φ ]+ κ αβ K α K β ) (det N ) − , (294) h out |T ( A [ φ ]) | in i = Z µ [ φ ][ dφ ] A [ φ ] e i ( S [ φ ]+ κ αβ K α K β ) (det N ) − , (295)where µ [ φ ] = const · µ I [ φ ](det κ ) / J [ φ ]det N (296)The previous expression may be regarded as a new measure functional, whichis to be used when the integration is carried out over the whole space ofhistories Φ rather than just the base space Φ / G . By virtue of eq. (287), theconstancy of κ , and the fact that µ I [ φ ] depends only on the I ’s, it followsthat this measure satisfies µ ←−L Q α = 0 . (297) The functional integrals (294) and (295) do not differ greatly in form fromexpressions (248) and (249). The chief difference is the presence of the K ’sand κ ’s, and the curious factor (det N ) − which comes ultimately from theJacobian J . By introducing the a -type ghost fields χ α , ψ β , one obtains(det N ) − = Z [ dχ ] Z [ dψ ] e iχ α M αβ ψ β . (298)Therefore eq. (294) may be written h out | in i = Z µ [ φ ][ dφ ] Z [ dχ ] Z [ dψ ] e i ( S [ φ ]+ κ αβ K α K β + χ α M αβ ψ β ) . (299)It is important to emphasize that the ghost fields arise entirely from thefiber-bundle structure of Φ, from the Jacobian of the transformation fromthe fiber-adapted coordinates to the conventional local fields φ i . Hence, the theory may be seen as a field theory on an “extended” space offield histories ¯Φ, where the ghost field χ α , ψ β appears in addition to φ ’s; how-ever, the new fields have to be considered non-physical, since their “fermionic These “tricky extra particles” were first introduced by R.P. Feynman (see [21]) as away of compensating for the propagation of nonphysical modes in one-loop order. α, β are bosonic indices. It is clear that the fullargument of the exponential in (299) is no longer invariant under gaugetransformations, because of the “gauge-averaging” term κ αβ K α K β and theghost term χ α M αβ ψ β .Nevertheless, both the full action functional¯ S [ φ, χ, ψ ] ≡ S [ φ ] + κ αβ K α K β + χ α M αβ ψ β (300)and the measure ¯ µ [ φ, χ, ψ ][ dφ ][ dχ ][ dψ ] ≡ µ [ φ ][ dφ ][ dχ ][ dψ ] (301)are invariant under a group of global transformations whose infinitesimal formis δφ i = Q i α ψ α δλ, (302) δχ α = κ αβ K β δλ, (303) δψ α = − c αβγ ψ β ψ γ δλ, (304)where δλ is an infinitesimal a -number. They are called Becchi-Rouet-Stora-Tyutin (BRST) transformations.As is clear, for the fields φ i , they are gauge transformations with an a -type parameter; therefore, from the invariance of µ [ φ ] under gauge trans-formations, ¯ µ [ φ, χ, ψ ] is invariant under BRST transformations; one can showthat ¯ S [ φ, χ, ψ ] is invariant too: δS [ φ ] = 0 , (305) δ ( κ αβ K α K β ) = · κ αβ δK α K β = κ αβ ( K α,i δφ i ) K β = κ αβ ( K α,i Q i γ ψ γ δλ ) K β = κ αβ ( K α,i Q i γ ψ γ ) K β δλ = κ αβ M αγ ψ γ K β δλ, (306)65 ( χ α M αβ ψ β ) = ( δχ α ) M αβ ψ β + χ α M αβ ( δψ β ) + χ α ( δ M αβ ) ψ β = κ αγ K γ δλ M αβ ψ β + χ α M αβ ( − c βαγ ψ α ψ γ δλ )+ χ α ( δK α,i Q i β ) ψ β + χ α ( K α,i δ Q i β ) ψ β = − κ αβ M αγ ψ γ K β δλ − χ α M αβ c βζγ ψ ζ ψ γ δλ + χ α ( K α,ij δφ j Q i β ) ψ β + χ α ( K α,i Q i β,j ) δφ j ψ β = − κ αβ M αγ ψ γ K β δλ − χ α M αβ c βζγ ψ ζ ψ γ δλ + χ α ( K α,ij Q j η ψ η δλ Q i β ) ψ β + χ α ( K α,i Q i β,j ) Q j ζ ψ ζ δλψ β = − κ αβ M αγ ψ γ K β δλ − χ α M αβ c βζγ ψ ζ ψ γ δλ + χ α K α,ij Q j η Q i β ψ β ψ η δλ + χ α K α,i Q i β,j Q j ζ ψ β ψ ζ δλ. (307)In the last equation, consider the third term: χ α K α,ij Q j η Q i β ψ β ψ η δλ = χ α K α,ji Q i η Q j β ψ β ψ η δλ = ( − ij χ α K α,ij Q i η Q j β ψ β ψ η δλ = χ α K α,ij Q j β Q i η ψ β ψ η δλ = − χ α K α,ij Q j β Q i η ψ η ψ β δλ = − χ α K α,ij Q j η Q i β ψ β ψ η δλ, (308)therefore χ α K α,ij Q j η Q i β ψ β ψ η δλ = 0 . (309)Consider again (307); adding the second and the fourth terms, one obtains − χ α M αβ c βζγ ψ ζ ψ γ δλ + χ α K α,i Q i β,j Q j ζ ψ β ψ ζ δλ = − χ α K α,i Q i β c βζγ ψ ζ ψ γ δλ + χ α K α,i Q i ζ,j Q j γ ψ ζ ψ γ δλ = − χ α K α,i Q i β c βζγ ψ ζ ψ γ δλ + χ α K α,i Q i ζ,j Q j γ ψ ζ ψ γ δλ − χ α K α,i Q i γ,j Q j ζ ψ ζ ψ γ δλ = − χ α K α,i Q i β c βζγ ψ ζ ψ γ δλ + χ α K α,i ( Q i ζ,j Q j γ − Q i γ,j Q j ζ ) ψ ζ ψ γ δλ = − χ α K α,i Q i β c βζγ ψ ζ ψ γ δλ + χ α K α,i Q i β c βζγ ψ ζ ψ γ δλ = 0 . (310)Noting that the first term in (307) is the opposite of the rhs of (306), it isproven that δ ¯ S = δS + δ ( κ αβ K α K β ) + δ ( χ α M αβ ψ β ) = 0 . (311)66oreover, BRST invariance is often a good substitute for the original gaugeinvariance. For example, if A is a functional of φ , but not of the ghost field,it is easy to see that A is gauge invariant if and only if it is BRST invariant: δ BRST A = A ,i δ BRST φ i = A ,i Q i α ψ α δλ ; (312)In this case, since the ψ α are arbitrary functions on space-time (althoughnot necessarily of compact support) BRST invariance of A implies gaugeinvariance in the original sense, and vice versa.Therefore, any Type-I gauge theory with action functional S [ φ ] may beviewed as a non-gauge, BRST-symmetric theory on an extendend space offield histories where ghost fields appear, with action functional ¯ S [ φ, χ, ψ ]given by (300). Remark . In the general case when no assumption on the bosonic nature ofthe indices from the first part of the Greek alphabet, sdet( N αβ ) − appears inplace of det( N αβ ) − ; therefore (see [19] and [20]) if the indices from the firstpart of the Greek alphabet are bosonic, one obtainssdet( N αβ ) − = det( N αβ ) − = Z [ dχ ] Z [ dψ ] e iχ α M αβ ψ β , (313)where χ α , ψ β are fermionic fields, as shown in the previous section. Onthe other hand, if the indices from the first part of the Greek alphabet arefermionic, thensdet( N αβ ) − = det( N αβ ) = Z [ dχ ] Z [ dψ ] e iχ α M αβ ψ β , (314)where, in this case, χ α , ψ β are bosonic fields.Hence, we infer that the fermionic nature of the ghost fields is always opposite to the one suggested by their indices, and this is another clue thatghost fields do not represent physical particles. The measure functional was introduced as a device for correcting the possi-ble failure of chronological ordering to yield Hermitian (or skew-Hermitian)operator field equations. It arose from the noncommutativity (or nonanti-commutativity) of field operators and hence is a purely quantum construct.But the measure functional plays a far deeper role, and we shall briefly outlinethe reason in this section. 67s is well known, the main tool to evaluate transition amplitudes inan interacting field theory is renormalized perturbation theory; in order toobtain renormalized observables, one has to choose a renormalization schemeand has to deal with divergent Feynman diagrams, up to a chosen order.Consider now one-loop perturbation theory for some field in Minkowski space-time; in Minkowski space-time one can use the Fourier transform and passto momentum space; therefore the task is to evaluate a graph consisting of asingle closed loop with r external prongs. Let the momenta assigned to theinternal lines all have the same orientation around the loop. Then, makinguse of the so-called “Feynman’s trick” to combine the factors contributedby the internal lines, i.e., by the propagators, and appropriately shifting theintegration zero point, one finds for the Feynman-propagator contribution tothe value of the graph an expression having the general form I ( C ) = constant · Z d r − y Z d n k P m ( y, k, p )[ k − iǫ + Q m ( y, k, p )] r (315)= constant · Z d r − y Z C d n k P m ( y, k, p )[ k + Q m ( y, k, p )] r (316)in which external space-time and/or spinor indices have been suppressed.Here “ y ” denotes the parameters y , y , ..., y r − needed to implement “Feyn-man’s trick” and R d r − y is a schematic symbol for the integrations in whichthese parameters are involved. The incoming momenta at the external prongsare ( − p − p − ... − p r ) , p , ..., p r . Q m is a quadratic function of these mo-menta, which also depends on the y ’s and on the masses m associated withthe internal lines. P m is a polynomial in the k ’s and p ’s, which depends onthe y ’s and m ’s. C in (316) denotes the contour in the complex plane of thetime component k of the k -variable which is appropriate to the Feynmanpropagator: this contour runs from −∞ to 0 below the negative real axis (inthe complex k -plane) and from 0 to + ∞ above the positive real axis: it isthe same as integrating on the real line with the iǫ prescription used in (315). If the integral were convergent , the contour could be rotated so that it wouldrun along the imaginary axis. One would set k = ik n , and (316) would be-come an integral over Euclidean momentum- n -space. Generically, however,this rotation, which is known as Wick rotation , is not legitimate. Contri-butions from arcs at infinity, which themselves diverge or are nonvanishing,have to be included.
These contributions cannot be handled by dimensionalregularization .When the measure is included it contributes to the generic one-loop graph68n amount equal to the negative of the integral I ( C + ) = constant · Z d r − y Z d n k P m ( y, k, p )[ − ( k − iǫ ) + ( ~k ) + Q m ( y, k, p )] r = constant · Z d r − y Z C + d n k P m ( y, k, p )[ k + Q m ( y, k, p )] r (317)where C + is the contour (in the complex k -plane) appropriate to the ad-vanced Green function: it runs from −∞ to + ∞ below the real axis; it isthe same as integrating on the real line with the iǫ prescription used on thefirst line of the previous equation. These two contributions, taken together,yield I ( C ) − I ( C + ) as the correct value of the graph. This corresponds totaking a contour that runs from + ∞ to 0 below the positive real axis andthen back to + ∞ again above the positive real axis, and yields an integralthat can be handled by dimensional regularization.The remarkable fact is that I ( C ) − I ( C + ) is equal precisely to the valuethat is obtained by Wick rotation. This means that the measure justifies theWick-rotation procedure . Although it has never been proved, one may spec-ulate that the exact measure functional, whatever it is, will justify the Wickrotation to all orders and will establish a rigorous connection between quan-tum field theory in Minkowski space-time and its corresponding euclideanizedversion. Green’s functions have been introduced in section 2 : in gauge theories,they are the negative inverses of the differential operator F i j , while in fieldtheories with no gauge transformations, they are the negative inverses of thenon-singular operator S . The prototypes of the Green’s functions of interestin quantum field theory are those of the neutral scalar meson in Minkowskispace-time; by spin-statistics theorem, it has to be a bosonic particle, andits Lagrangian is L = − (cid:0) φ µ, φ ,µ + m φ (cid:1) . (318)Therefore S = S = δδφ ( x ) S = (cid:0) φ µ, µ − m φ (cid:1) , (319)69nd S = S = δδφ ( x ′ ) δδφ ( x ) S = (cid:0) δ ( x, x ′ ) µ, µ − m δ ( x, x ′ ) (cid:1) = (cid:0) ∂ µ ∂ µ − m (cid:1) δ ( x, x ′ ) . (320)Hence (125) takes the form Z dy (cid:2)(cid:0) ∂ µ ∂ µ − m (cid:1) δ ( x, y ) (cid:3) G ( y, x ′ )= (cid:0) ∂ µ ∂ µ − m (cid:1) G ( x, x ′ ) = − δ ( x, x ′ ) . (321)This equation is most easily solved in “momentum” space, i.e., using theFourier transform; as is well known, it is a linear, invertible operator whichturns derivative operators into multiplication operators, i.e., the Fouriertransform of a linear, differential equation for a function is a linear, alge-braic equation which is easily solved; therefore the desired function can beobtained by inverse Fourier transform; in our case, the result is G ( x, x ′ ) = 1(2 π ) Z dp e ip ( x − x ′ ) m + p . (322)In the previous equation the contours in the p , p , p planes are confined tothe real axis and the choice of Green’s function is determined by selecting acontour in the p plane which passes in an appropriate fashion around thepoles at ± E where E ≡ p m + ~p ≡ ω, (323) ~p ≡ ( p , p , p ) , (324)( ~p ) ≡ ( p ) + ( p ) + ( p ) . (325)The most important contours are shown in Fig. 1. From these contours thefollowing relations between the various Green’s functions are easily estab-lished: ¯ G = (cid:0) G + + G − (cid:1) = ˜ G + G − = − ˜ G + G + , (326)˜ G = G + − G − = G (+) + G ( − ) , (327) G (1) = i (cid:0) G (+) − G ( − ) (cid:1) , (328) G = ¯ G + iG (1) = G − + G ( − ) = G + − G (+) , (329) G ∗ = ¯ G − iG (1) = G − + G (+) = G + − G ( − ) . (330)70 ✞ ✞(cid:0)✒☛✁ (cid:0)✒✂✁(cid:0)☎ (cid:0)(cid:0)☛(cid:0)✂✄(cid:0)✒✆✁ (cid:0)✝ (cid:0)✟ Figure 1: Contours in the complex p -plane for the integral representation ofthe Green’s function of the neutron scalar meson.71y closing the contours for G + and G − at infinity it is easy to see thatthese functions satisfy the kinematical conditions (126) and hence are theadvanced and retarded Green’s functions. Their uniqueness is also evident.We may therefore write the further relations G + ( x, x ′ ) = 2 θ ( x ′ , x ) ¯ G ( x, x ′ ) = θ ( x ′ , x ) ˜ G, (331) G − ( x, x ′ ) = 2 θ ( x, x ′ ) ¯ G ( x, x ′ ) = − θ ( x, x ′ ) ˜ G, (332)˜ G ( x, x ′ ) = − ǫ ( x, x ′ ) ¯ G ( x, x ′ ) , (333)¯ G ( x, x ′ ) = − ǫ ( x, x ′ ) ˜ G ( x, x ′ ) , (334)where θ ( x, x ′ ) and ǫ ( x, x ′ ) are the step functions, defined by θ ( x, x ′ ) ≡ ( x > x ′ , x < x ′ , (335)= 1 − θ ( x ′ , x ) = [1 + ǫ ( x, x ′ )] , (336) ǫ ( x, x ′ ) = θ ( x, x ′ ) − θ ( x ′ , x ) = ( x > x ′ , − x < x ′ , = − ǫ ( x ′ , x ) . (337)We also have G ( x, x ′ ) = − θ ( x, x ′ ) G (+) ( x, x ′ ) + θ ( x ′ , x ) G ( − ) ( x, x ′ ) , (338) G ∗ ( x, x ′ ) = θ ( x ′ , x ) G (+) ( x, x ′ ) − θ ( x, x ′ ) G ( − ) ( x, x ′ ) , (339) G (+) ( x, x ′ ) = − θ ( x, x ′ ) G ( x, x ′ ) + θ ( x ′ , x ) G ∗ ( x, x ′ ) , (340) G ( − ) ( x, x ′ ) = θ ( x ′ , x ) G ( x, x ′ ) − θ ( x, x ′ ) G ∗ ( x, x ′ ) , (341)which follow from (327), (333), (334), (336), and the identities θ ( x, x ′ ) θ ( x ′ , x ) = 0 (342)[ θ ( x, x ′ )] = θ ( x, x ′ ) (343)[ ǫ ( x, x ′ )] = 1 (344)Care should be exercised in the use of the step functions. Strictly speaking,all the equations where θ ( x, x ′ ), ǫ ( x, x ′ ) appear and their corollaries can beinferred to hold only when one of the two points x, x ′ is clearly to the futureor the past of the other. When the two points are separated by a space-like interval further investigation is needed. We shall see presently that thefunctions G ± ( x, x ′ ) vanish for finite space-like separations, and hence theinvestigation reduces to a study of the behavior of the Green’s functionswhen x ′ is in the immediate neighborhood of x . The study is complicated72y the fact that the Green’s functions are actually distributions rather thanordinary functions. It turns out, in the present case, that the above relationsare in fact valid everywhere. Analogous relations, for the Green’s functionsof systems more complicated than the neutral scalar meson, however, do notalways similarly hold when x = x ′ . In this work we shall avoid this difficultyby using the step functions only when x = x ′ . We may also remark thatthere will never be any ambiguity about the Green’s functions themselves.In the present case they are well defined by the integral representation (322),once the contour is chosen. For the neutral scalar meson, the reciprocity relations (137) and (187) read G ± ( x, x ′ ) = G ∓ ( x ′ , x ) , (345)˜ G ( x, x ′ ) = − ˜ G ( x ′ , x ) , (346)¯ G ( x, x ′ ) = ¯ G ( x ′ , x ) . (347)The contour for the function ¯ G corresponds to performing a principal value integration along the real axis. In light of the reality of all the integrationvariables in this case, and because of the symmetry (in p ) of the denominatorof the integrand of (322), we may infer the reality of ¯ G :¯ G ∗ = ¯ G. (348)Similarly, by performing the transformation p
7→ − p , paying attention to thecontour and using the previous relations, we may infer G ( x, x ′ ) = G ( x ′ , x ) , (349) G (1) ( x, x ′ ) = G (1) ( x ′ , x ) , (350) G ( ± ) ( x, x ′ ) = − G ( ∓ ) ( x ′ , x ) , (351)and G ±∗ = G ± , (352)˜ G ∗ = ˜ G, (353) G (1) ∗ = G (1) , (354) G ( ± ) ∗ = − G ( ± ) , (355)i.e., G ± , ˜ G, G (1) are all real, while G ( ± ) is imaginary; it follows that G ∗ , asdefined above, is the complex conjugate of G .73e note, finally, the differential equations satisfied by the various func-tions: (cid:0) ∂ µ ∂ µ − m (cid:1) G ( x, x ′ ) = (cid:0) ∂ µ ∂ µ − m (cid:1) ¯ G ( x, x ′ )= (cid:0) ∂ µ ∂ µ − m (cid:1) G ± ( x, x ′ )= − δ ( x, x ′ ) , (356) (cid:0) ∂ µ ∂ µ − m (cid:1) ˜ G ( x, x ′ ) = (cid:0) ∂ µ ∂ µ − m (cid:1) G (1) ( x, x ′ )= (cid:0) ∂ µ ∂ µ − m (cid:1) G ( ± ) ( x, x ′ )= 0 . (357) In harmony with section 2, ˜ G is known as the commutator function , and G (+) , G ( − ) are called its positive and negative frequency parts , respectively. G (1) is known as Hadamard’s elementary function , and G is called the Feynmanpropagator. From the relations given above it may be seen that all of thefunctions which we have introduced may be obtained from the Feynmanpropagator by splitting it into its real, imaginary, advanced, and retardedparts, and recombining these parts in various ways. It suffices therefore toevaluate the Feynman propagator in order to obtain all the rest. From Fig.1 it is not hard to see that the contour for the Feynman propagator may bedisplaced to the real axis provided we give to the mass m in eq. (322) aninfinitesimal negative imaginary part, i.e., we move up the negative pole andmove down the positive one in the complex p -plane. We therefore write G ( x, x ′ ) = 1(2 π ) Z dp e ip ( x − x ′ ) m − iǫ + p , ǫ > . (358)with the understanding that the limit ǫ → unique Green’s functionwhich the operator ( ∂ µ ∂ µ − m ) possesses when the x -manifold has a positivedefinite metric. This fact is responsible for many of the remarkable propertieswhich characterize the Feynman propagator, and the analytic continuationmethod is often employed to obtain it.Making use of the integral identities Z + ∞ ds e − is ( ξ − iǫ ) = 1 i ( ξ − iǫ ) , ǫ > Z + ∞−∞ dx e iax = p ( π/ | a | ) e i sgn( a )( π/ , a real , (360)74where sgn is the signum function: sgn( a ) ≡ a/ | a | ), one obtains G ( x, x ′ ) = i (2 π ) Z + ∞ ds Z dp e ip ( x − x ′ ) e − is ( m + p − iǫ ) = i (2 π ) Z + ∞ ds Z dp e − i [ ( m + p − iǫ ) s − p ( x − x ′ )) ]= i (2 π ) Z + ∞ ds e − is ( m − iǫ ) Z dp e − isp + ip ( x − x ′ ) = i (2 π ) Z + ∞ ds e − is ( m − iǫ ) Z dp e − isp + ip ( x − x ′ ) − i ( x − x ′ √ s ) e i ( x − x ′ √ s ) = i (2 π ) Z + ∞ ds e − is ( m − iǫ ) e i (cid:18) x − x ′ √ s (cid:19) Z dp •• e − i sp − p ( x − x ′ )+ (cid:18) x − x ′ √ s (cid:19) ! = i (2 π ) Z + ∞ ds e − is ( m − iǫ )+ i (cid:18) x − x ′ √ s (cid:19) Z dp e − i (cid:18) √ sp − x − x ′ √ s (cid:19) = i (2 π ) Z + ∞ ds e − is ( m − iǫ )+ i (cid:18) x − x ′ √ s (cid:19) Z dp e − is (cid:18) p − x − x ′ s (cid:19) = i (2 π ) Z + ∞ ds e − is ( m − iǫ )+ i (cid:18) x − x ′ √ s (cid:19) •• (cid:16)p ( π/s ) (cid:17) (cid:0) e i sgn( − s )( π/ (cid:1) = i (2 π ) Z + ∞ ds e − is ( m − iǫ )+ i (cid:18) x − x ′ √ s (cid:19) π s e − i ( π/ = i (2 π ) Z + ∞ ds e − is ( m − iǫ )+ i (cid:18) x − x ′ √ s (cid:19) π s e − iπ/ = − i (4 π ) Z + ∞ ds s e − is ( m − iǫ )+ i (cid:18) x − x ′ √ s (cid:19) = 1(4 π ) Z + ∞ ds s e − is ( m − iǫ )+ i (cid:18) x − x ′ √ s (cid:19) = 1(4 π ) Z + ∞ ds s e − i (cid:18) m s − ( x − x ′ ) s (cid:19) . (361)In the final form the negative imaginary part − iǫ attached to m has beendropped, with the understanding that G ( x, x ′ ) has to be regarded as the boundary value (on the real axis) of a function of m and ( x − x ′ ) which75s analytic in the lower half m plane and in the upper half ( x − x ′ ) plane.The fact that G ( x, x ′ ) depends on x and x ′ only through the combination( x − x ′ ) is a consequence of Lorentz invariance and the homogeneity of flatspace-time. We shall see later that in a curved space-time the dependence of G ( x, x ′ ) on x and x ’ will not be so simple.When ( x − x ′ ) < z = − m ( x − x ′ ) > , z > u = − im sz , (363)which convert (361) to G ( x, x ′ ) = 116 π Z − i ∞ du (cid:16) − z im (cid:17) (cid:18) − m u z (cid:19) e − i (cid:18) − m uz im + im ( x − x ′ ) uz (cid:19) = 116 π Z − i ∞ du (cid:18) iz m (cid:19) (cid:18) m u z (cid:19) e (cid:18) uz m ( x − x ′ ) uz (cid:19) = 116 π im z Z − i ∞ du u e z (cid:16) u − u (cid:17) = im π z Z − i ∞ du u e z (cid:16) u − u (cid:17) (364)The contour of integration may be deformed in the manner shown in Fig. 2,and in virtue of the well known integral representation H (2)1 ( z ) = 1 iπ Z C du u e z (cid:16) u − u (cid:17) (365)of the Hankel function of the second kind, of order 1, we finally have G ( x, x ′ ) = − m π H (2)1 ( z ) z . (366) For small values of z (i.e., near the light cone ) it is convenient to use thepower series expansions H (2)1 ( z ) = J ( z ) − iY ( z ) , (367)76igure 2: Contour for the Hankel function H (2)1 ( z ). J ( z ) = z − z z ..., (368) Y ( z ) = 2 π (cid:20) − z J ( z ) + ( γ + log z ) J ( z ) − z z ) − z + ) + ... (cid:21) , (369) J ( z ) = 1 − z + z − ..., (370) γ = 0 , ... . (371)Remembering that analytic continuation should be performed in the lowerhalf z plane, and making use of the identities, which hold ∀ z ∈ R :lim ǫ → + z − iǫ = 1 z + iπδ ( z ) , (372)lim ǫ → + log( z − iǫ ) = log | z | − iπθ ( − z ) , (373)77e find, on splitting the Feynman propagator into its real and imaginaryparts ¯ G ( x, x ′ ) = Re( G )( x, x ′ ) = 14 π δ (( x − x ′ ) ) − m π θ ( − ( x − x ′ ) ) (cid:20) + m ( x − x ′ ) m ( x − x ′ ) ... (cid:21) , (374) G (1) ( x, x ′ ) = 2Im( G )( x, x ′ ) = m π (cid:26) m ( x − x ′ ) + (cid:2) γ − log2 + log m + log | ( x − x ′ ) | (cid:3) (cid:20) + m ( x − x ′ ) ... (cid:21) − − m ( x − x ′ ) ) + m ( x − x ′ ) + ) − ... (cid:27) . (375)The Green’s function G (and hence also G + and G − ) is seen to have a δ -distribution type singularity on the light cone [( x − x ′ ) = 0] and to vanishoutside the light cone [( x − x ′ ) > m = 0 . In this case we have¯ G ( x, x ′ ) = 14 π δ (( x − x ′ ) ) , (376) G (1) ( x, x ′ ) = 12 π ( x − x ′ ) , (377)whence, in virtue of eqs. (331) and (332) and the identity δ ( ξ − a ) = 12 a [ δ ( ξ − a ) + δ ( ξ + a )] , a > G ± ( x, x ′ ) = 14 π | ~x − ~x ′ | δ ( x − x ′ ± | ~x − ~x ′ | ) . (379) Consider now the scalar field theory obtained by applying the minimal cou-pling to the gravitational field to the field theory examined in the previoussections:1. Replace the Minkowski metric η µν by g µν . This property no longer holds when space-time is curved.
78. Replace ordinary space-time derivatives by the covariant derivativesassociated to the (unique) Levi-Civita connection determined by themetric.3. Multiply the Lagrange function by | g | / , where g = det( g µν ).Then the action functional for the theory is S [ φ ] = − Z dx | g | / (cid:0) φ µ ; φ ; µ + m φ (cid:1) . (380)Hence S [ φ ] = | g | / (cid:0) φ µ ; µ − m φ (cid:1) , (381)and S = | g( x ) | / (cid:0) δ ( x, x ′ ) µ ; µ − m δ ( x, x ′ ) (cid:1) = | g( x ) | / (cid:0) ∇ µ ∇ µ − m (cid:1) δ ( x, x ′ ) . (382)Therefore the equation for the Green’s functions is | g( x ) | / (cid:0) ∇ µ ∇ µ − m (cid:1) G ( x, x ′ ) = − δ ( x, x ′ ) . (383)A very elegant method for solving this equation exists, which is due to Sch-winger. One regards the Green’s function as the matrix element of an oper-ator G in an abstract (nonphysical) Hilbert space: G ( x, x ′ ) = h x | G | x ′ i , (384)the basis vectors | x ′ i being eigenvectors of a commuting set of Hermitianoperators x µ x µ | x ′ i = x ′ µ | x ′ i , h x ′′ | x ′ i = δ ( x ′′ , x ′ ) . (385)The differential eq. (383) may then be recast in the operator form( p µ | g | / g µν p ν + m | g | / ) G = 1 , (386)where the p µ are Hermitian operators which satisfy the commutation relations[ x µ , p ν ] = iδ µν , [ p µ , p ν ] = 0 . (387)79 .6 General definition of the Feynman propagator In order to solve the operator equation (383) we must first decide whichGreen’s function we want. As in the previous sections, we shall choose theFeynman propagator as the basic Green’s function of interest. However, thisimmediately begs the question of what we mean by the Feynman propagatorwhen space-time is curved and non-empty. In a flat empty space-time theFeynman propagator can be defined as that Green’s function which propa-gates positive frequencies into the future and negative frequencies into thepast (see eq. (338)). The same definition can be used when space-time iscurved provided it becomes asymptotically flat at large space-like and time-like distances and the words “future” and “past” are replaced by “remotefuture” and “remote past” respectively. Under these circumstances the samevariational law holds for the Feynman propagator as well as the retarded andadvanced Green’s functions: δG ij = G ik δF k l G lj , (388)for this law immediately permits the expansion about the flat-empty-space-time values, F and G , of the operators F and G , G − G = δG = ( G + δG )( F − F )( G + δG )= G U G + δG F G + G F δG + δG F δG = G U G + G U G U G + ... (389)where U ≡ F − F ; hence one obtains G = G + G U G + G U G U G + ... = G (1 − U G ) − (390)= (1 − G U ) − G . (391)We see that the first G standing on the left and the last G standing on theright, in each term of the expansion, do indeed ensure that ultimately onlypure positive frequencies are found in the remote future and pure negativefrequencies in the remote past, owing to the effectively limited domain overwhich U is non-vanishing.A word is perhaps in order at this point regarding the very special proper-ties the Feynman propagator possesses. When F is symmetric (and we havealways assumed it is) the Feynman propagator is symmetric. Since it alsosatisfies the variational law (388) it is the only Green’s function which, whenregarded as a continuous matrix, obeys all the rules of finite matrix theory .In a certain sense it may therefore be regarded as the inverse of the matrix80 − F i j ). In flat space-time its special properties stem from the fact (alreadynoted) that it may be obtained by analytic continuation from the unique inverse which ( − F i j ) possesses in a Euclidean space. When space-time iscurved these properties may themselves be used to define the Feynman prop-agator even when space-time is not asymptotically flat . From the results of the previous sections, we shall obtain the Feynman prop-agator, in curved space-times as well as flat, simply by giving the mass pa-rameter m an infinitesimal negative imaginary part. This has the effect ofrendering the operator in (386) nonsingular so that inverses may be takenin a simple and direct fashion. It also emphasizes once again that Green’sfunctions are boundary values of analytic functions. Multiplying equation(386) on the left by | g | − / and on the right by | g | / , we obtain | g | − / ( p µ | g | / g µν p ν + m | g | / ) G | g | / = | g | − / | g | / , | g | − / p µ | g | / g µν p ν G | g | / + m | g | / G | g | / = 1 , | g | − / p µ | g | / g µν p ν | g | − / | g | / G | g | / + m | g | / G | g | / = 1 , ( | g | − / p µ | g | / g µν p ν | g | − / + m ) | g | / G | g | / = 1 , ( H + m ) | g | / G | g | / = 1 , (392)where H ≡ | g | − / p µ | g | / g µν p ν | g | / . (393)Therefore, with the correct prescription for the Feynman propagator: | g | / G | g | / = 1 H + m − iǫ = i Z + ∞ ds e − is ( H + m ) . (394)Taking matrix elements of the previous equation we obtain | g(x ′ ) | / G | g( x ′′ ) | / = i Z + ∞ ds (cid:10) x ′ | e − isH | x ′′ (cid:11) e − ism , | g ′ | / G | g ′′ | / = i Z + ∞ ds h x ′ , s | x ′′ , i e − ism , (395)with h x ′ , s | x ′′ , i ≡ (cid:10) x ′ | e − isH | x ′′ (cid:11) . (396)Thus we are led to an associated dynamical problem governed by the “Hamil-tonian” H . 81he “transition amplitude” h x ′ , s | x ′′ , i satisfies the Schr¨odinger equation i ∂∂s h x ′ , s | x ′′ , i = h x ′ , s | H | x ′′ , i = − h x ′ , s | x ′′ , i µ ′ ; µ ′ (397)and the boundary condition h x ′ , | x ′′ , i = δ ( x ′ , x ′′ ) . (398)In flat empty space-time, this equation is solved by h x ′ , s | x ′′ , i Minkowski = − i π s e i ( x ′− x ′′ )24 s , (399)which agrees with (361).In order to discuss the generalization to curved space-time, some insighton auxiliary geometric quantities is necessary. k -point tensors As is well known, a ( r, s )-tensor field T on a manifold M is a map whichassigns to every point p ∈ M an element from the direct product of thetangent space T p M , taken r times, and the cotangent space T ∗ p M , taken s times: T : M → ( T M ) r ⊗ ( T ∗ M ) s ,p T ( p ) = T µ ...µ r ν ...ν s ( p ) ∂∂x µ (cid:12)(cid:12)(cid:12)(cid:12) p ⊗ ... ⊗ ∂∂x µ r (cid:12)(cid:12)(cid:12)(cid:12) p ⊗ dx ν (cid:12)(cid:12)(cid:12)(cid:12) p ⊗ ... ⊗ dx ν s (cid:12)(cid:12)(cid:12)(cid:12) p . (400)The tensor field concept can be generalized in this way: we shall call k -pointtensor on a manifold M a map which assigns to every k -tuple of points in M an element from the direct product of the tensor spaces built upon those82oints: T : M k → [( T M ) r ⊗ ( T ∗ M ) s ] ⊗ ... ⊗ [( T M ) r k ⊗ ( T ∗ M ) s k ] , ( p (1) , p (2) , ..., p ( k ) ) T µ (1)1 ...µ (1) r ... µ ( k )1 ...µ ( k ) rk ν (1)1 ...ν (1) s ... ν ( k )1 ...ν ( k ) s ( k ) ( p (1) , p (2) , ..., p ( k ) ) ·· ∂∂x µ (1)1 (cid:12)(cid:12)(cid:12)(cid:12) p (1) ⊗ ... ⊗ ∂∂x µ (1) r (1) (cid:12)(cid:12)(cid:12)(cid:12) p (1) ⊗ dx ν (1)1 (cid:12)(cid:12)(cid:12)(cid:12) p (1) ⊗ ... ⊗ dx ν (1) s (1) (cid:12)(cid:12)(cid:12)(cid:12) p (1) ⊗ ... ⊗ ∂∂x µ ( k )1 (cid:12)(cid:12)(cid:12)(cid:12) p ( k ) ⊗ ... ⊗ ∂∂x µ ( k ) r ( k ) (cid:12)(cid:12)(cid:12)(cid:12) p ( k ) ⊗ dx ν ( k )1 (cid:12)(cid:12)(cid:12)(cid:12) p ( k ) ⊗ ... ⊗ dx ν ( k ) s ( k ) (cid:12)(cid:12)(cid:12)(cid:12) p ( k ) . (401)Roughly speaking, whenever k − k -point tensor becomes a tensor field. For a detailed discussion on geodesics on a Riemannian manifold, see [17]; fora Lorentzian manifold, see [16]. Here we will only introduce tools necessaryfor later treatise. As is well known, given a connection ∇ on a manifold M , there is exactly one parallel transport on M , i.e., exactly one way toparallel transport a given vector along any curve; one shall define geodesic (associated to that connection) any curve whose tangent vector is paralleltransported along the curve. If M is a (pseudo-) Riemannian manifold withmetric tensor g , and ∇ is the unique Levi-Civita connection associated tothat metric, then, given two close enough points, the curve for which thelength functional (associated to the metric g ) is stationary (on the curveswhich connect those points) is a geodesic.In fact the equations for a geodesic x ( τ ) with affine parameter,¨ x µ ( τ ) + Γ µρσ ( x ( τ )) ˙ x ρ ( τ ) ˙ x σ ( τ ) = 0 (402)are precisely the Euler-Lagrange equations associated to the functional S [ x ( τ )] = Length[ x ( τ )] = Z dτ L ( x ( τ ) , ˙ x ( τ ) = Z dτ [ ± ˙ x ρ g ρσ ˙ x σ ] / δSδx µ = ∂L∂x µ − ddτ ∂L∂ ˙ x µ == [ ± ˙ x ρ g ρσ ˙ x σ ] − / (cid:20) ± g ρσ,µ ˙ x ρ ˙ x σ ∓ ddτ (2 g µσ ˙ x σ ) (cid:21) = ∓ L (cid:20) ddτ ( g µσ ˙ x σ ) − g ρσ,µ ˙ x ρ ˙ x σ (cid:21) = ∓ L (cid:18) g µσ ¨ x σ + ddτ ( g µσ ) ˙ x σ − g ρσ,µ ˙ x ρ ˙ x σ (cid:19) = ∓ L (cid:0) g µσ ¨ x σ + g µσ,ρ ˙ x ρ ˙ x σ − g ρσ,µ ˙ x ρ ˙ x σ (cid:1) = ∓ L (cid:0) g µσ ¨ x σ + g µσ,ρ ˙ x ρ ˙ x σ + g µρ,σ ˙ x ρ ˙ x σ − g ρσ,µ ˙ x ρ ˙ x σ (cid:1) = 0 ,g µσ ¨ x σ + g µσ,ρ ˙ x ρ ˙ x σ + g µρ,σ ˙ x ρ ˙ x σ − g ρσ,µ ˙ x ρ ˙ x σ = 0 . (403)Raising the index µ , we obtain¨ x µ + g µν ( g νσ,ρ + g νρ,σ − g ρσ,ν ) ˙ x ρ ˙ x σ = 0 , (404)which are exactly the geodesic equations, beingΓ µσρ = g µν ( g νσ,ρ + g νρ,σ − g ρσ,ν ) . (405)As is straightforward to verify, the same equations are obtained consideringthe functional S [ x ( τ )] = R dτ L ( x ( τ ) , ˙ x ( τ )) = R dτ ˙ x ρ g ρσ ˙ x σ .Since a variational principle has been introduced, the theory of geodesicsmay be viewed formally as a dynamical theory, and all the results of Hamilton-Jacobi theory can be immediately applied to it. The “conjugate momenta”and “Hamiltonian” are given by p µ ≡ ∂L∂ ˙ x µ = g µν ˙ x ν ≡ ˙ x µ , (406) H ≡ p µ ˙ x µ − L = ˙ x µ ˙ x µ − ˙ x µ g µν ˙ x ν = L. (407)Hence we are led to the “energy integral” for the geodesics: H = L = ˙ x ρ g ρσ ˙ x σ = (cid:18) dsdτ (cid:19) = const , (408)where s is the arc length defined on the curve; the action functional on asolution, i.e., a geodesic, whose endpoints are x ( τ ) ≡ x , x ( τ ′ ) ≡ x ′ , reduces84o S ( x, τ | x ′ , τ ′ ) = Z ττ ′ dτ ′′ ˙ x ρ g ρσ ˙ x σ = Z ττ ′ dτ ′′ (cid:18) dsdτ ′′ (cid:19) = Z xx ′ ds dτ ′′ ds (cid:18) dsdτ ′′ (cid:19) = Z xx ′ ds (cid:18) dsdτ ′′ (cid:19) = (cid:18) dsdτ (cid:19) Z xx ′ ds = (cid:18) dsdτ (cid:19) ( s ( x ) − s ( x ′ ))= (cid:18) s ( x ) − s ( x ′ ) τ − τ ′ (cid:19) ( s ( x ) − s ( x ′ ))= σ ( x, x ′ ) τ − τ ′ , (409)where the bi-scalar σ ( x, x ′ ), which we shall call geodetic interval or worldfunction , is equal to one half the square of the distance along the geodesicbetween x and x ′ .The bi-scalar of geodetic interval satisfies an important differential equa-tion which follows immediately from the Hamilton-Jacobi equation for theaction S ; we have p µ = ∂S∂x µ = σ ; µ τ − τ ′ , (410)0 = ∂S∂τ + H = − σ ( x, x ′ )( τ − τ ′ ) + p µ p µ , (411)where p µ is now the “momentum” at x corresponding to the geodesic definedby the endpoints x , x ′ ; therefore the world function is the solution of theCauchy problem ( σ ; µ σ ; µ = σ,σ ( x ′ , x ′ ) = 0 . (412)Obviously, the Hamilton-Jacobi equation holds on the other endpoint too;then σ ; µ σ ; µ = σ ; µ ′ σ ; µ ′ = σ. (413)85n other words, σ ; µ is a vector of length equal to the distance along thegeodesic between x and x ′ , tangent to the geodesic at x , and oriented inthe direction x ′ → x , while σ ; µ ′ is a vector of equal length, tangent to thegeodesic at x ′ , and oriented in the opposite direction. The geodetic intervalitself is obviously a symmetric function of x and x ′ : σ ( x, x ′ ) = σ ( x ′ , x ) . (414) In a general Riemannian manifold the geodetic interval is not single-valued,except when x and x ′ are sufficiently close to one another. The geodesicsemanating from a given point will often, beyond a certain distance, begin tocross over one another. The locus of points at which the onset of overlapoccurs forms an envelope of the family of geodesics, known as a caustic sur-face . The equation for the caustic surface relative to a given point can beexpressed in terms of the quantity det( σ ; µν ′ ).In fact, a geodesic can be specified by means of its endpoints or by meansof one of its endpoints together with a tangent vector at that point having alength equal to the length of the geodesic. Therefore we can vary σ ; µ ′ helding x ′ fixed, and evaluate the resulting variation in x ; it is straightforward toobtain δσ ; µ ′ = σ ; µ ′ ν δx ν ; (415)therefore δx µ = − D − µν ′ δσ ; ν ′ , (416)where D − µν ′ is the inverse transpose of the finite matrix having the elements D ρν ′ = − σ ; ν ′ ρ , i.e., D − µν ′ D ρν ′ = − D − µν ′ σ ; ν ′ ρ = δ µρ . (417)When D − µν ′ is a singular matrix, it is possible to choose a variation in σ ; µ ′ which produces no variation in the x . The point x then lies on the causticsurface relative to x ′ , and the condition for this is evidently D − = 0, where D = − det( D µν ′ ) , (418)the minus sign expressing a convention appropriate to the metric of space-time. In 4-dimensional space-time the caustic surface will usually be a 3-dimensional hypersurface, but degenerate forms having fewer dimensions,including zero (focal points) can occur. It will be noted that variations of σ ; µ ′ which leave x unchanged must be orthogonal to σ ; µ ′ ; that is, the length86f the geodesic itself must remain unchanged. This may be inferred by takingthe derivative of the Hamilton-Jacobi equation (412): σ ; µ σ ; µν ′ = σ ; ν ′ , (419)and recasting it in the form − D − µν ′ σ ; ν ′ = σ ; µ = 0 , (420)which shows, together with (416) that changing the length of σ ; µ ′ withoutchanging its direction necessarily shifts x a proportional distance: in fact, bytaking δσ ; µ ′ = ǫσ ; µ ′ , one obtains δx µ = − D − µν ′ δσ ; ν ′ = − ǫD − µν ′ σ ; ν ′ = − ǫσ ; µ . (421) The determinant D is a bi-density, of unit weight at both x and x ′ . Notsurprisingly it plays a fundamental role in the description of the rate at whichgeodesics emanating from fixed points diverge from or converge toward oneanother. If we differentiate eq. (419) with respect to x ρ and we note thatthe indices µ and ρ commute, we get σ µ ; ρ σ ; µν ′ + σ ; µ σ ; µν ′ ρ = σ ; ν ′ ρ ,σ µ ; ρ σ ; µν ′ + σ ; µ σ ; ρν ′ µ = σ ; ν ′ ρ , − σ µ ; ρ D µν ′ − σ ; µ D ν ′ ρ ; µ = − D ρν ′ ,D ρν ′ = σ µ ; ρ D µν ′ + σ ; µ D ν ′ ρ ; µ , (422)which, on multiplication by D − ρν ′ , gives D ρν ′ D − ρν ′ = σ µ ; ρ D µν ′ D − ρν ′ + σ ; µ D ν ′ ρ ; µ D − ρν ′ − δ ρρ = − σ µ ; ρ δ ρµ − σ ; µ DD ; µ σ µ ; µ + σ ; µ D − D ; µ (423) D − ( Dσ ; µ ) ; µ = 4 . (424)The significance of this equation may be made transparent by first replacing D with the bi-scalar ∆ ≡ | g | − / D | g ′ | − / (425)87nd observing that the operator σ ; µ ∂ µ gives the derivative of any functionalong the geodesic from x ′ . Thus σ ; µ ∂ µ f = ( τ − τ ′ ) ˙ f (426)where f is any scalar. Arbitrarily setting τ ′ = 0, we may recast eq. (424) inthe form σ µ ; µ = 4 − d (log ∆) d (log τ ) . (427)In fact σ ; µ D − D ; µ = σ ; µ (cid:0) | g | / ∆ | g ′ | / (cid:1) − (cid:0) | g | / ∆ | g ′ | / (cid:1) ; µ = σ ; µ | g | − / ∆ − | g ′ | − / | g | / ∆ ; µ | g ′ | / = σ ; µ ∆ − ∆ ; µ = σ ; µ (log∆) ; µ = τ ddτ (log∆)= d (log ∆) d (log τ ) . (428)From (427) it follows immediately that ∆ increases or decreases along eachgeodesic from x ′ according as the rate of divergence of the neighboringgeodesics from x ′ , which is measured by σ µ ; µ , is less than or greater than4, the rate in flat space-time. If the divergence rate becomes negativelyinfinite a caustic surface develops and ∆ blows up. Another geometrical quantity of fundamental importance is the geodetic par-allel displacement bi-vector , g µν ′ , which is defined by the differential equations σ η ; g µν ′ ; η = 0 (429)together with the boundary conditionlim x ′ → x g µν ′ = g µν . (430)The bi-vector g µν ′ gets its name from the fact that the result of applyingit, for example, to a local contravariant vector A µ ′ at x ′ , is to obtain thecovariant form or the vector which results from displacing A µ ′ in a parallelfashion along the geodesic from x ′ to x . This follows from the defining eq.88429), which requires the covariant derivative of g µν ′ to vanish in directionstangent to the geodesic: in fact ddτ (cid:16) g µν ′ A ν ′ (cid:17) = σ ρ ; τ (cid:16) g µν ′ A ν ′ (cid:17) ; ρ = τ σ ρ ; g µν ′ ; ρ A ν ′ = 0 , (431)and lim x → x ′ g µρ g ρν ′ A ν ′ = g µ ′ ρ ′ g ρ ′ ν ′ A ν ′ = δ µ ′ ν ′ A ν ′ = A µ ′ . (432)From its geometrical significance and the fact that tangents to a geodesic areself-parallel the following properties of g µν ′ are obvious: g ν ′ µ σ ; ν ′ = − σ ; µ , g νµ ′ σ ; ν = − σ ; µ ′ , (433) σ η ′ ; g µν ′ ; η ′ = 0 , (434) g µν ′ = g ν ′ µ , (435) g µρ ′ g ρ ′ ν ′ = g µν , g ρµ ′ g ρν ′ = g µ ′ ν ′ , (436)det( − g µν ′ ) = | g | / | g ′ | / . (437)In a similar manner one may define a geodetic parallel displacement bi-spinor I ( x, x ′ ) which satisfies σ µ ; I ; µ = 0 , (438)lim x ′ → x I = unity matrix , (439)and which transforms like ψ ≡ ψ ( x ) at x and like ψ ′ ≡ ψ ( x ′ ) at x ′ . Now we are ready to propose an ansatz to solve (397), (398): in (399), replace1 s D ( x, x ′ ) / s (440)( x − x ′ ) σ ( x, x ′ ) (441)and multiplicate by a power series in ( is ): + ∞ X n =0 a n ( x, x ′ )( is ) n , (442)lim x ′ → x a ( x, x ′ ) = 1 , (443)89hen the ansatz is h x ′ , s | x ′′ , i = − i π D ( x, x ′ ) / s e i σ ( x,x ′ )2 s + ∞ X n =0 a n ( x, x ′ )( is ) n , (444)lim x ′ → x a ( x, x ′ ) = 1 . (445)A requirement for the ansatz to be meaningful is the existence of a recurrencerelation for the unknown a n ’s. Inserting (444) in (397), the lhs is: i ∂∂s h x ′ , s | x ′′ , i = i ∂∂s − i π D / s e i σ s + ∞ X n =0 a n ( is ) n ! = D / π ∂∂s s e i σ s + ∞ X n =0 a n ( is ) n ! = D / π e i σ s (cid:18) − s ∞ X n =0 a n ( is ) n + 1 s − iσ s ∞ X n =0 a n ( is ) n + 1 s ∞ X n =1 ina n ( is ) n − (cid:19) = D / π e i σ s (cid:18) + ∞ X n =0 ( − i a n )( is ) n − + + ∞ X n =0 ( − ii σa n / is ) n − + + ∞ X n =1 ( ii na n )( is ) n − (cid:19) = iD / π e i σ s (cid:18) + ∞ X n =0 (2 a n )( is ) n − + + ∞ X n =0 ( − σa n / is ) n − + + ∞ X n =1 ( − na n )( is ) n − (cid:19) , (446)90hile the rhs is − h x ′ , s | x ′′ , i µ ; µ = − − i π D / s e i σ s + ∞ X n =0 a n ( is ) n ! µ ; µ = i π s D / e i σ s + ∞ X n =0 a n ( is ) n ! µ ; µ = i π s e i σ s (cid:18) D / µ ; µ + ∞ X n =0 a n ( is ) n + D / i s σ µ ; µ + ∞ X n =0 a n ( is ) n + D / (cid:18) i s (cid:19) σ ; µ σ ; µ + ∞ X n =0 a n ( is ) n + D / ∞ X n =0 a µn ; µ ( is ) n + 2 D / µ (cid:18) i s (cid:19) σ ; µ + ∞ X n =0 a n ( is ) n +2 D / µ + ∞ X n =0 a ; µn ( is ) n + 2 D / (cid:18) i s (cid:19) σ ; µ + ∞ X n =0 a ; µn ( is ) n (cid:19) = − iD / π e i σ s (cid:18) + ∞ X n =0 D / µ ; µ D / a n ( is ) n − + + ∞ X n =0 − σ µ ; µ a n ( is ) n − + + ∞ X n =0 σ ; µ σ ; µ a n ( is ) n − + + ∞ X n =0 a µn ; µ ( is ) n + + ∞ X n =0 − D / µ D / σ ; µ a n ( is ) n − + + ∞ X n =0 D / µ D / a ; µn ( is ) n − + + ∞ X n =0 ( − σ ; µ a ; µn )( is ) n − (cid:19) ; (447)91xploiting (412) and (423), one obtains − h x ′ , s | x ′′ , i µ ; µ = − iD / π e i σ s (cid:18) + ∞ X n =0 D / µ ; µ D / a n ( is ) n − + + ∞ X n =0 ( − a n ( is ) n − + + ∞ X n =0 σ a n ( is ) n − + + ∞ X n =0 a µn ; µ ( is ) n − + + ∞ X n =0 D / µ D / a ; µn ( is ) n − + + ∞ X n =0 ( − σ ; µ a ; µn )( is ) n − (cid:19) . (448)Finally, equating lhs and rhs one obtains0 = iD / π e i σ s (cid:18) + ∞ X n =0 ( − na n )( is ) n − + + ∞ X n =0 ∆ − / (∆ / a n ) µ ; µ ( is ) n − + + ∞ X n =0 ( − σ ; µ a ; µn )( is ) n − (cid:19) . (449)The necessary and sufficient condition for this equation to be satisfied for ev-ery s is the multiplicative coefficient of every monomial ( is ) k be zero; there-fore ( is ) − : σ ; µ a ; µ = 0 , (450)( is ) k , k > − σ ; µ a ; µn +1 + ( n + 1) a n +1 = ∆ − / (∆ / a n ) µ ; µ . (451)In view of (438) and (439) the equations for a are solved by a ( x, x ′ ) = I ( x, x ′ ) , (452)while the recurrence relation (451) may be solved by integrating along eachgeodesic emanating from x ′ ; in fact, multiplying the lhs by τ ′′ n , where τ ′′ isthe parameter labeling a point x ′′ on the geodesic between x ′ ( τ ′′ = 0) and92 ( τ ′′ = τ ), one obtains τ ′′ n σ ; µ a ; µn +1 ( x ( τ ′′ ) , x ′ ) + ( n + 1) τ ′′ n a n +1 ( x ( τ ′′ ) , x ′ ) == τ ′′ n +1 ddτ ′′ a ( x ( τ ′′ ) , x ′ ) + (cid:18) ddτ ′′ τ ′′ n +1 (cid:19) a n +1 ( x ( τ ′′ ) , x ′ )= ddτ ′′ (cid:16) τ ′′ n +1 a n +1 ( x ( τ ′′ ) , x ′ ) (cid:17) , (453)therefore a n +1 ( x, x ′ ) = τ − n − Z τ dτ ′′ τ ′′ n ∆ ′′ − / (∆ ′′ / a ′′ n ) µ ′′ ; µ ′′ (454) Inserting (444) in (395), we now get G ( x, x ′ ) = ∆ / (4 π ) Z ∞ ds s e − i ( m s − σ s ) + ∞ X n =0 a n ( is ) n = ∆ / (4 π ) ∞ X n =0 a n (cid:18) − ∂∂m (cid:19) n Z ∞ ds e − i ( m s − σ s ) . (455)The latter integral has already been evaluated (eqs. (361) and (366)). Break-ing the Feynman propagator into its real and imaginary parts and makinguse of the expansions (374) and (375), we obtain, upon carrying out the93ifferentiations with respect to m , G ( x, x ′ ) = ¯ G ( x, x ′ ) + 12 iG (1) ( x, x ′ ) , ¯ G ( x, x ′ ) = ∆ / I π δ ( σ ) − ∆ / π θ ( − σ ) (cid:20) ( m I − a )+ 2 σ m I − m a + 2 a )+ (2 σ ) m I − m a + 6 m a − a ) + ... (cid:21) , (456) G (1) ( x, x ′ ) = ∆ / I π σ + ∆ / π ( γ − log2 + log | m σ | ) •• (cid:20) ( m I − a ) + 2 σ m I − m a + 2 a ) + ... (cid:21) − ∆ / π (cid:20) m I + 2 σ m I − m a + a )+ (2 σ ) (cid:18) m I − m a + 152 m a − a (cid:19) + ... (cid:21) + ∆ / π (cid:20) (cid:16) a m + a m + a m + ... (cid:17) − σ (cid:16) a m + a m + ... (cid:17) + ... (cid:21) . (457)Several comments must be made about these expansions. First, there isthe obvious remark that they are useful only for small values of σ . However,this is precisely the domain in which we are often interested, particularly inrenormalization theory. We note that the Feynman propagator has, at σ = 0,the same types of singularity in the presence of a gravitational field as it hasin a flat empty space-time. The Green’s function ¯ G , which can be split intothe advanced and retarded Green’s functions, has a δ -distribution singularityon the light cone and vanishes outside. We note, however, that when m = 0,it no longer vanishes inside the light cone as it does when space-time is flatand empty. Instead, we have¯ G = ∆ / I π δ ( σ ) + ∆ / π θ ( − σ ) (cid:18) a − σ a + σ · a − ... (cid:19) . (458)lt is important to observe in this connection that although the expansion interms of the a ’s can be used for ¯ G when m = 0, it cannot be used for G (1) .94his may be seen from the last line of (457) which shows that an expansion ininverse powers of m is involved. When m is vanishing, alternative methods,based either on special properties of the fields or on perturbation theory,must be found for evaluating the Feynman propagator. This work has pedagogical purposes, hence the particular care in calculations,most of which are explicitly shown. A powerful formalism for gauge field the-ories has been described and has been used to obtain a manifesly covariantquantization of such theories, even in curved space-time; an important ap-plication is the evaluation of physical observables, such as the stress-energytensor through point-split regularization: in any theory of interacting fieldsthe set of currents that describe the interaction is of fundamental importance;in General Relativity, these currents are the components of the stress-energytensor, therefore the main problem in developing a quantum field theory incurved space-time is precisely to understand the stress-energy tensor (see [9]).One fundamental result is that it can always be expressed through Feyn-man’s Green function and its derivatives (see [22], [23], [24], [9]) but theactual task is to give it meaning by some subtraction process. A regulariza-tion, or subtraction, process conventionally makes use of the vacuum state,but in a curved space-time, this notion is not trivial, as already stressed.Compared to the flat space-time case, in a curved background the resulting renormalized stress-energy tensor is covariantly conserved, of course, but it possesses a state-independent anomalous trace (see [25]).Currently the author is working on the study of the stress-energy tensorfor Maxwell’s theory, along the lines of Christensen’s work, which is a corner-stone in this area (see [23] and [24]). Last, it is important to emphasize thatsimilar calculations appear interestingly also in the context of effective actionin curved space-time, whose divergent part is essential to discuss renormaliza-tion group equations for the Newton constant and the cosmological constant(see [26]).
Acknowledgments
The author is grateful to G. Esposito for his guidance and support; his loveof physics is deeply inspiring. 95 eferences [1] S. W. Hawking, Black hole explosions?,
Nature (1974), 30–31.[2] S. W. Hawking, Particle creation by black holes,
Commun. Math. Phys. (1975), 199–200.[3] W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D (1976),870–892.[4] N. D. Birrell and P. C. W. Davies, Quantum fields in curved space (Cam-bridge University Press, Cambridge, 1982).[5] S. A. Fulling et al. , Aspects of quantum field theory in curved spacetime ,(Cambridge University Press, Cambridge, 1989).[6] L. Parker and D. Toms,
Quantum field theory in curved spacetime: quan-tized fields and gravity (Cambridge University Press, Cambridge, 2009).[7] B. S. DeWitt, The spacetime approach to quantum field theory, in
Rela-tivity, Groups and Topology II , eds. B. S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984) pp. 381–738.[8] B. S. DeWitt,
Dynamical theory of groups and fields (Gordon & Breach,New York, 1965).[9] B. S. DeWitt, Quantum field theory in curved spacetime,
Phys. Rep. (1975), 295–357.[10] B. S. DeWitt, The global approach to quantum field theory (Oxford Uni-versity Press, Oxford, 2003).[11] R. S. Palais, Imbedding of compact, differentiable transformation groupsin orthogonal representations,
J. Math. Mech. (1957), 673–678.[12] G. D. Mostow, Equivariant embeddings in Euclidean space, Ann. Math. (1957), 432–446.[13] B. Simon, Functional integration and quantum physics (Academic Press,New York, 1979).[14] J. Glimm and A. Jaffe,
Quantum physics: a functional integral point ofview (Springer, Berlin, 2012).[15] P. Cartier and C. DeWitt-Morette,
Functional integration: action andsymmetries (Cambridge University Press, Cambridge, 2006).9616] S. W. Hawking and G. F. R. Ellis,
Functional integration: action andsymmetries (Cambridge University Press, Cambridge, 2006).[17] J. W. Milnor, M. Spivak and R. Wells,
Morse theory (Princeton Univer-sity Press, Princeton, 1969).[18] J. L. Synge,
Relativity: The general theory (North-Holland, Amsterdam,1960).[19] F. A. Berezin,
The method of second quantization (Elsevier, Amsterdam,2012).[20] B. S. DeWitt,
Supermanifolds (Cambridge University Press, Cambridge,1992).[21] R. P. Feynman, Quantum theory of gravitation,
Acta Phys. Pol. (1963), 697–722.[22] G. Bimonte, E. Calloni, L. Di Fiore, G. Esposito, L. Milano and L. Rosa,On the photon Green functions in curved spacetime, Class. QuantumGravity (2004), 647–659.[23] S. M. Christensen, Vacuum expectation value of the stress tensor inan arbitrary curved background: the covariant point-separation method, Phys. Rev. D (1976), 2490-2501.[24] S. M. Christensen, Regularization, renormalization, and covariantgeodesic point separation, Phys. Rev. D17