BRST quantization of Yang-Mills theory: A purely Hamiltonian approach on Fock space
BBRST quantization of Yang-Mills theory: A purely Hamiltonian approach on Fockspace
Hans Christian ¨Ottinger ∗ ETH Z¨urich, Department of Materials, Polymer Physics, HCP F 47.2, CH-8093 Z¨urich, Switzerland (Dated: March 2, 2018)We develop the basic ideas and equations for the BRST quantization of Yang-Mills theories inan explicit Hamiltonian approach, without any reference to the Lagrangian approach at any stageof the development. We present a new representation of ghost fields that combines desirable self-adjointness properties with canonical anticommutation relations for ghost creation and annihilationoperators, thus enabling us to characterize the physical states on a well-defined Fock space. TheHamiltonian is constructed by piecing together simple BRST invariant operators to obtain a minimalinvariant extension of the free theory. It is verified that the evolution equations implied by theresulting minimal Hamiltonian provide a quantum version of the classical Yang-Mills equations.The modifications and requirements for the inclusion of matter are discussed in detail.
PACS numbers: 03.70.+k
I. INTRODUCTION
BRST quantization is a pivotal tool in developing the-ories of the fundamental interactions, where the acronymBRST refers to Becchi, Rouet, Stora [1] and Tyutin [2].This method for handling constraints in the quantiza-tion of field theories usually requires a broad viewpointbecause it covers a number of important aspects. Theconstraints are related to gauge symmetry, which sug-gests that a Lagrangian approach is preferable, in partic-ular, as also Lorentz symmetry needs to be incorporated.By Noether’s theorem, symmetries come with conservedquantities, which suggest to focus on time evolution andhence to favor the Hamiltonian approach. Practical cal-culations, for example in perturbation theory, are mostconveniently done in terms of the path-integral formula-tion and hence on the Lagrangian side. The identificationof physical states, which requires gauge fixing as an addi-tional aspect of introducing constraints, is most naturallydone on Hilbert space (as the space of all states). The is-sue of signed inner products, to be considered simultane-ously with canonical inner products, requires particularattention in constructing the physical states and shouldclearly benefit from a simple and intuitive approach toBRST quantization.The purpose of this paper is to show in the context ofYang-Mills theories how all the above facets can be han-dled entirely within the Hamiltonian approach, where ex-plicit constructions on a suitable Fock space allow for amaximum of intuition. The focus on Fock space impliesa (quantum) particle interpretation rather than a fieldidealization. The signed and canonical inner productsare particularly transparent on Fock space. By includ-ing temporal and longitudinal in addition to transversegauge bosons (we typically think of photons or gluons),Lorentz symmetry is enabled at the early stage of con- ∗ [email protected]; structing the underlying Fock space. All symmetry argu-ments are based on the BRST charge, the construction ofwhich relies on its role as the generator of BRST transfor-mations, the quantum version of gauge transformations.Of course, the BRST charge has to respect also Lorentzsymmetry.In this paper, we present a new way of introducingghost particles. Whereas one usually has to make thechoice between natural anticommutation relations for theghost creation and annihilation operators on the onehand (see, e.g., [3]) and self-adjointness of the BRSTcharge on the other hand (see, e.g., [4]), we here pro-pose a representation of ghost particles that combinesboth properties. This is a crucial advantage because “Inthe non-Abelian case, the removal of unphysical gaugeboson polarizations is more subtle [than in the Abeliancase], and we have seen that it involves the ghosts inan essential way” (see p. 520 of [5]). We simultaneouslyhave a well-defined Fock space and the powerful tools re-quired to select the physical states of a gauge theory inthe BRST approach.After constructing a number of simple BRST invari-ant operators, these operators can be used to build upthe BRST invariant Hamiltonian. By piecing togetherinvariant operators to reproduce the proper Hamiltonianof the free theory for vanishing interaction strength, oneobtains the Hamiltonian of Yang-Mills theory as a min-imal BRST invariant extension of the free theory. Thevalidity of the Hamiltonian can be verified by comparingthe time evolution implied by this Hamiltonian to theclassical evolution equations. Matter can be includedinto the Hamiltonian approach with the help of the cur-rent algebra. a r X i v : . [ h e p - t h ] M a r II. CLASSICAL YANG-MILLS THEORY
Yang-Mills theory introduces antisymmetric fields F aµν that are defined in terms of four-vector potentials A aµ , F aµν = ∂ µ A aν − ∂ ν A aµ − gf abc A bµ A cν , (1)where the superscripts a, b, c label the generators of theunderlying symmetry group and the indices µ, ν label thespace-time components; the parameter g is the strengthof the interaction and the set of numbers f abc are thestructure constants of the underlying Lie group. Thefield equations are given by ∂ µ F aµν − gf abc A bµ F cµν = − J aν , (2)where the four-vector J aν is an external current. Theparameter g usually occurs with the opposite sign be-cause we here choose the signature ( − , + , + , +) for theMinkowski metric, contrary to the more common con-vention (+ , − , − , − ); the four-vectors ∂ µ and A aµ areindependent of the signature of the Minkowski metric.The term ˜ J aν = − gf abc A bµ F cµν may be interpreted as thecurrent carried by the gauge bosons. The structure con-stants can be assumed to be completely antisymmetricin the indices a, b, c (see Sec. 15.4 of [5]). They moreoversatisfy the Jacobi identity f ads f bcs + f bds f cas + f cds f abs = 0 . (3)This identity is repeatedly needed in analyzing the gaugetransformation behavior of the classical and quantumYang-Mills equations.Let us consider gauge transformations, which are givenby (see, e.g., pp. 490f of [5] or Section 15.1 of [6]) A aµ → A aµ + ∂ µ Λ a − gf abc A bµ Λ c . (4)Unlike for the Abelian case, the resulting transformation F aµν → F aµν − gf abc F bµν Λ c + O (Λ ) , (5)implies that the fields are not gauge invariant for non-Abelian Yang-Mills theories. However, by consideringthe combined transformation law ∂ µ F aµν − gf abc A bµ F cµν → ∂ µ F aµν − gf abc A bµ F cµν − gf abc (cid:0) ∂ µ F bµν − gf bde A dµ F eµν (cid:1) Λ c + O (Λ ) , (6)we realize that the field equations (2) in the absence of ex-ternal currents are gauge invariant. Moreover, the gaugetransformation behavior of external currents required toobtain gauge invariant field equations becomes evident, J aµ → J aµ − gf abc J bµ Λ c . (7)For the gauge bosons, we here impose the covariantLorenz gauge condition ∂ µ A aµ = 0 . (8) We finally rewrite the Yang-Mills equations in a partic-ular inertial system in the Maxwellian form which, in con-trast to all the above equations, is no longer manifestlyLorentz covariant. Such a reformulation is a straightfor-ward exercise (see, e.g., [7, 8] with some deviations in thechoice of signs).The counterparts of electric and magnetic fields areobtained by the convention( F aµν ) = − E a − E a − E a E a B a − B a E a − B a B a E a B a − B a . (9)As pointed out before, for the non-Abelian case, thesefields are not gauge invariant, B a → B a − gf abc B b Λ c + O (Λ ) , (10)and E a → E a − gf abc E b Λ c + O (Λ ) . (11)For convenience, we define the additional component E a = ∂ µ A aµ , (12)so that the Lorenz gauge condition (8) can be expressedas E a = 0.The gauge-dependent field definitions (1) become B a = ∇ × A a − gf abc A b × A c , (13) E a = ∇ A a − ∂ A a ∂t − gf abc A b A c , (14)and the field equations (2) can be written as ∇ · E a − gf abc A b · E c = − J a = J a , (15) ∂ E a ∂t − ∇ × B a − gf abc ( A b E c − A b × B c ) = − J a . (16)Equations (13)–(16) correspond to Eqs. (3.15), (4.2c),(4.2a) and (4.2b) of [7]. In Eqs. (15) and (16), we recog-nize the temporal and spatial components of the current − gf abc A bµ F cµν associated with the fields,˜ J a = gf abc A b · E c , (17)˜ J a = − gf abc ( A b E c − A b × B c ) . (18)Equation (13) can be used to eliminate B a and B c from Eq. (16). After eliminating B from the picture, wehave the evolution equations ∂ A a ∂t = − E a + ∇ A a − gf abc A b A c , (19)and ∂ E a ∂t = − J a − ∇ A a + ∇∇ · A a + gf abc A b E c + gf abc [2 A b · ∇ A c − A b ∇ · A c + ( ∇ A b ) · A c ] − g f abs f cds A b · A c A d , (20)together with the relation (15). These equations defineclassical Yang-Mills theories in a formulation that doesnot manifestly exhibit Lorentz or gauge symmetry. Theseequations imply a conservation law with source term, ∂ ˜ J a ∂t + ∇ · ˜ J a = − gf abc A bµ J cµ . (21)A wave equation for A a is obtained by subtractingEq. (16) from the time derivative of Eq. (14). Similarly,a the wave equation for A a is obtained by subtractingEq. (15) from the divergence of Eq. (14). We thus findthe wave equations (cid:18) ∂ ∂t − ∇ (cid:19) A a = J a + gf abc A bµ ( ∂ A cµ − ∂ µ A c )+ g f abs f cds A b · A c A d − ( δ ac ∂ − gf abc A b ) ∂ µ A cµ , (22)and (cid:18) ∂ ∂t − ∇ (cid:19) A a = J a + gf abc A bµ ( ∇ A cµ − ∂ µ A c )+ g f abs f cds A bµ A cµ A d − ( δ ac ∇ − gf abc A b ) ∂ µ A cµ , (23)in each of which the last term disappears for the Lorenzgauge (8) (cf. appendix of [8]). A closer look at Eqs. (22)and (23) reveals that one recovers Lorentz covariance inthese wave equations. III. FOCK SPACE
We introduce a Fock space together with a collection ofadjoint operators a aα † q and a aα q creating and annihilatingfield quanta, such as photons or gluons, with momentum q ∈ ¯ K × and polarization state α ∈ { , , , } . The spaceof momentum vectors is given by the simple cubic lattice¯ K × = { q = K L ( z , z , z ) | z , z , z ∈ Z } , (24)where K L is a lattice constant in momentum space, whichis assumed to be small because it is given by the inversesystem size. The additional label a is associated withthe infinitesimal generators of an underlying Lie group[assuming 3 values for SU (2) corresponding to the W + ,W − , and Z bosons mediating weak interactions, 8 val-ues for SU (3) corresponding to the gluon “color octet”mediating strong interactions, and N − SU ( N )]. For simplicity, we occasionally refer to the gauge or vector bosons associated with the Yang-Millsfield as gluons, for which a labels eight different “color”states. The gluon creation and annihilation operatorssatisfy canonical commutation relations.Details on the construction of Fock spaces can befound, e.g., in Secs. 1 and 2 of [9], in Secs. 12.1 and12.2 of [10], or in Sec. 1.2.1 of [11]. We assume that thecollection of the states created by multiple application ofall the above creation operators on a ground state, whichis annihilated by all the corresponding annihilation op-erators, is complete.In the following, we repeatedly need the properties ofthe polarization states. We hence give our explicit rep-resentations. The temporal unit four-vector,(¯ n µ q ) = , (25)is actually independent of q . The three orthonormal spa-tial polarization vectors are chosen as(¯ n µ q ) = 1 (cid:112) q + q q − q , (26)(¯ n µ q ) = 1 q (cid:112) q + q q q q q − q − q , (27)and (¯ n µ q ) = 1 q q q q , (28)where q = | q | . The polarization vectors ¯ n q and ¯ n q cor-respond to transverse gluons, ¯ n q corresponds to longitu-dinal gluons. Note the symmetry property¯ n α − q = ( − α ¯ n α q . (29)In the BRST approach, one introduces the additionalpairs B a † q , B a q and D a † q , D a q of creation and annihila-tion operators associated with ghost particles and theirantiparticles. They are assumed to satisfy canonical an-ticommutation relations.Fock spaces come with canonical inner products, s can .The superscript † on the creation operators can actuallybe interpreted as the adjoint with respect to the canoni-cal inner product. In gauge theories, the canonical innerproduct is not the physical one. We need an additionalsigned inner product, s sign , for which states with negativenorm exist (so that it is not an inner product in a strictsense). In both inner products, the natural base vectors TABLE I: Properties associated with the various creation op-erators used for constructing the Fock spacecommuting anticommuting- metric + metric - metric + metric a a † q a aj † q B a † q D a † q of a Fock space, which are characterized by the occupa-tion numbers of the various particles in the system, areorthogonal. If the canonical norm of a Fock base vec-tor is one, the signed norm of that vector is obtained byintroducing a factor − B a † q . The adjointof an operator with respect to the signed inner productis indicated by the superscript ‡ . The commutation andsigned product properties of all particles are compiled inTable I. IV. BASICS OF BRST APPROACH
A brief discussion of BRST quantization in terms ofcreation and annihilation operators can be found onpp. 239–240 of [3]. For quantum electrodynamics, all thedetails have been elaborated in Section 3.2.5 of [11]. Wehere generalize these ideas to non-Abelian gauge theories.
A. Ghost particle operators
In the usual interpretation, the operators B a † q and D a † q in Table I create massless ghost particles and their an-tiparticles. The Fourier modes of the corresponding fieldsare hence given by c a q = 1 √ q ( D a † q − B a − q ) , ¯ c a q = 1 √ q ( B a † q + D a − q ) . (30)If we further define the fields˙ c a q = i (cid:114) q D a † q + B a − q ) , ˙¯ c a q = i (cid:114) q B a † q − D a − q ) , (31)the only non-vanishing anticommutation relations amongthe operators introduced in Eqs. (30) and (31) are givenby { c a q , ˙¯ c b q (cid:48) } = − i δ ab δ q + q (cid:48) , , { ¯ c a q , ˙ c b q (cid:48) } = i δ ab δ q + q (cid:48) , . (32)Further note the adjointness properties with respect tothe signed inner product, c a ‡ q = ¯ c a − q , ˙ c a ‡ q = ˙¯ c a − q , (33)and the simple transformation rule for the energy of non-interacting massless ghosts (neglecting an irrelevant con- stant to achieve normal ordering), (cid:88) q ∈ ¯ K × q (cid:0) B a † q B a q + D a † q D a q (cid:1) = (cid:88) q ∈ ¯ K × (cid:0) ˙ c a q ˙¯ c a − q + q c a q ¯ c a − q (cid:1) . (34)At this point a comment on notation should be use-ful. One might be tempted to interpret the dots in thesymbols ˙ c a q , ˙¯ c a q introduced in Eq. (31) as time derivatives.However, this interpretation works only for massless freeghost particles. For quantum electrodynamics, this in-terpretation would actually be justified. For non-Abeliangauge theories, however, the ghost particles are no longerfree and ˙ c a q , ˙¯ c a q should be recognized as nothing but theoperators defined in Eq. (31); they do not coincide withtime derivatives that could be defined in terms of the fullHamiltonian. B. Alternative representation of ghosts
It has been pointed out by Kugo and Ojima [4] that thelack of self-adjointness of the ghost operators expressedin Eq. (33) is a serious disadvantage in constructing thephysical states. In non-Abelian Yang-Mills theories, itmoreover keeps the Hamiltonian from being self-adjoint.Those authors hence recommend an alternative represen-tation of ghosts that, however, requires non-canonical an-ticommutation relations for the creation and annihilationoperators B a † q , B a q and D a † q , D a q . In order to base ourdevelopment on a well-defined Fock space, we stronglyprefer to keep canonical anticommutation relations. Wetherefore propose a new representation of the ghost fieldsin terms of two creation and two annihilation operators, c a q = 12 √ q ( B a † q + D a † q − B a − q + D a − q ) , (35)¯ c a q = 12 √ q ( B a † q − D a † q + B a − q + D a − q ) , (36)˙ c a q = i √ q B a † q + D a † q + B a − q − D a − q ) , (37)and ˙¯ c a q = i √ q B a † q − D a † q − B a − q − D a − q ) . (38)A straightforward calculation shows that the resultinganticommutation relations for the operators defined inEqs. (35)–(38) are identical to the previously found ones,where the only nontrivial anticommutators are given inEq. (32). Therefore, also most of the subsequent calcu-lations for the two ways of introducing ghost fields areidentical. Moreover, Eq. (34) remains valid.The big advantage of the new definitions are the self-adjointness properties c a ‡ q = c a − q , ˙ c a ‡ q = ˙ c a − q , ¯ c a ‡ q = − ¯ c a − q , ˙¯ c a ‡ q = − ˙¯ c a − q . (39)These properties imply the self-adjointness of productssuch as (cid:0) c a q ¯ c b q (cid:48) (cid:1) ‡ = c a − q ¯ c b − q (cid:48) . (40)The systematic occurrence of c ¯ c pairs is attractive alsofrom another viewpoint: We obtain a symmetry underrescaling c and ¯ c by inverse factors and hence an addi-tional conserved quantity in the ghost domain (see Sec-tion VI C for details). In view of Eqs. (33) and (39), werefer to the two options for introducing ghost operatorsas “cross-adjointness” (option 1) and “self-adjointness”(option 2), respectively. We once more emphasize thatboth options are realized on exactly the same Fock space. C. BRST charge and transformations
To keep track of sums and differences of photon cre-ation and annihilation operators, to evaluate commuta-tors and anticommutators in an efficient manner, and tofacilitate the comparison with the classical theory, we in-troduce the operators α a q = 1 √ q (cid:104) ¯ n q ( a a † q − a a − q ) + ¯ n q ( a a † q − a a − q )+ ¯ n q ( a a † q + a a − q ) + ¯ n q ( a a † q − a a − q ) (cid:105) , (41)and ε a q = − i (cid:114) q (cid:104) ¯ n q ( a a † q + a a − q + a a † q − a a − q )+ ¯ n q ( a a † q + a a − q ) + ¯ n q ( a a † q − a a − q )+ ¯ n q ( a a † q + a a − q + a a † q − a a − q ) (cid:105) . (42)According to Eqs. (3.73) and (3.81) of [11], α a q corre-sponds to the four-vector potential and ε a q to the electricfield [augmented by the time component introduced inEq. (12)]. These four-vectors satisfy canonical commuta-tion relations [ ε aµ p , α bν q ] = i η µν δ ab δ p + q , , (43)[ α aµ p , α bν q ] = [ ε aµ p , ε bν q ] = 0 , (44)where η µν represents the Minkowski metric, and possessthe following self-adjointness properties with respect tothe signed inner product, α a ‡ q = α a − q , ε a ‡ q = ε a − q . (45)The energy of temporal and longitudinal noninteract-ing vector bosons can be written as (cid:88) q ∈ ¯ K × q (cid:0) a a † q a a q + a a † q a a q (cid:1) = (cid:88) q ∈ ¯ K × i q j (cid:0) ε aj q α a − q + α aj q ε a − q (cid:1) + 12 (cid:88) q ∈ ¯ K × (cid:18) q j q k q ε aj q ε ak − q − ε a q ε a − q (cid:19) . (46) In the following, we use the notation( AB ) q = 1 √ V (cid:88) q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q (cid:48) + q (cid:48)(cid:48) , q A q (cid:48) B q (cid:48)(cid:48) , (47)for any two q dependent operators A and B . This nota-tion for convolutions allows us to obtain more compactequations, where working on the infinite lattice ¯ K × (thatis, in the thermodynamic limit) is crucial for obtainingthe usual properties of convolutions (the relevance of highmomenta for a symmetry involving derivatives has beenpointed out on p. 169 of [11]; on a finite lattice, rigorousBRST symmetry cannot be expected).The following expression for the BRST charge is in-spired, for example, by Eq. (5.10) of [3] or Eq. (2.22) of[12], but the proper formulation in the present setting isnot entirely straightforward, Q = (cid:88) q ∈ ¯ K × (cid:0) ε a q ˙ c a − q − i q j ε aj q c a − q (cid:1) + gf abc (cid:88) q ∈ ¯ K × (cid:16) ε aµ α bµ − ˙ c a ¯ c b + 12 c a ˙¯ c b (cid:17) q c c − q . (48)Rearranging the factors in the three-particle collisionterms proportional to g in Eq. (48) is unproblematic be-cause, whenever a nonzero contribution might arise froma nontrivial anticommutation relation, the correspondingstructure constant with two equal labels vanishes. In par-ticular, normal ordering is not an issue in three-particlecollision terms, which is an appealing feature of such in-teractions.The BRST charge (48) implies the following character-istic BRST transformations for non-Abelian gauge theo-ries, δα a q = ˙ c a q − gf abc ( α b c c ) q ,δα aj q = − i q j c a q − gf abc ( α bj c c ) q ,δε aµ q = − gf abc ( ε bµ c c ) q .δc a q = 12 gf abc ( c b c c ) q ,δ ˙ c a q = gf abc ( ˙ c b c c ) q ,δ ¯ c a q = ε a q + gf abc (¯ c b c c ) q ,δ ˙¯ c a q = − i q j ε aj q + gf abc ( ε bµ α cµ − ˙ c b ¯ c c + c b ˙¯ c c ) q , (49)where δ · = i[ Q, · ] for boson and δ · = i { Q, ·} for fermionoperators. The first two lines of Eq. (49) correspond tothe gauge transformation (4), the third line of Eq. (49)corresponds to the transformation (11).With the BRST charge (48) we have the tool to dis-cuss all aspects of BRST symmetry. In particular, we canconstruct a BRST invariant Hamiltonian and the physi-cal states in the large Fock space involving ghosts. Thecompatibility of BRST invariance with Lorentz symme-try is visible in the first two lines of Eq. (49), whichrelates the BRST transformations of the four-vector po-tential to the time and space derivatives of the ghost field c . The nilpotency of the BRST charge ( Q = 0), whichis crucial for the handling of BRST symmetry, is verifiedin Appendix A. As a somewhat simpler alternative, onecan verify δQ = 0 by means of the BRST transforma-tions (49). The first two lines of Eq. (49), which expressthe essence of the classical gauge transformations (4),and the nilpotency of Q provide the deeper reasons forwriting the BRST charge in the form given in Eq. (48). V. CONSTRUCTION OF BRST INVARIANTOPERATORS
A simple way of constructing BRST invariant opera-tors is based on the identity[ Q, i { Q, X } ] = 0 , (50)which, for any operator X , follows trivially from thenilpotency of Q . In other words, any operator i { Q, X } is BRST invariant as it commutes with Q . In practice,one chooses X to produce a desirable term and one auto-matically gets all the additional terms required for BRSTinvariance. We illustrate the idea for some simple choicesof X , which consist of an α or ε paired with a c or ¯ c .To produce terms of the type ˙ c ˙¯ c , we choose X = (cid:88) q ∈ ¯ K × α a q ˙¯ c a − q . (51)Straightforward calculations based on the product ruleand the results in Eq. (49) givei { Q, X } = (cid:88) q ∈ ¯ K × (cid:0) ˙ c a q ˙¯ c a − q + i q j ε aj q α a − q (cid:1) + gf abc (cid:88) q ∈ ¯ K × (cid:0) ε aj α bj − ˙ c a ¯ c b (cid:1) − q α c q . (52)Similarly, to produce terms of the type c ¯ c , we choose X = (cid:88) q ∈ ¯ K × i q j α aj q ¯ c a − q , (53)and we obtain the BRST invariant operatori { Q, X } = (cid:88) q ∈ ¯ K × (cid:0) q c a q ¯ c a − q + i q j α aj q ε a − q (cid:1) − gf abc (cid:88) q ∈ ¯ K × i q j c a q (¯ c b α c ) − q . (54)Another interesting choice is given by X = (cid:88) q ∈ ¯ K × ε a q ¯ c a − q . (55)It leads to the simple BRST invariant operatori { Q, X } = − (cid:88) q ∈ ¯ K × ε a q ε a − q . (56) In summary, we have used bilinear operators X to pro-duce a number of BRST invariant operators i { Q, X } .A comparison with Eqs. (34) and (46) shows that, for g = 0, all our examples contain contributions from thefree Hamiltonian. This observation is very useful for thesubsequent construction of a BRST invariant Hamilto-nian. VI. YANG-MILLS HAMILTONIAN
We now construct a BRST invariant Hamiltonian anddiscuss some implications. The construction is based en-tirely on symmetry considerations.
A. Construction of Hamiltonian
We start from the energy of the noninteracting trans-verse polarizations of the vector bosons, (cid:88) q ∈ ¯ K × q (cid:0) a a † q a a q + a a † q a a q (cid:1) = 12 (cid:88) q ∈ ¯ K × (cid:0) ¯ n j q ¯ n k q + ¯ n j q ¯ n k q (cid:1)(cid:0) q α aj q α ak − q + ε aj q ε ak − q (cid:1) . (57)A more convenient starting point is actually given byΦ = (cid:88) q ∈ ¯ K × q (cid:0) a a † q a a q + a a † q a a q (cid:1) + 12 (cid:88) q ∈ ¯ K × q j q k q ε aj q ε ak − q = 12 (cid:88) q ∈ ¯ K × (cid:104)(cid:0) q δ jk − q j q k (cid:1) α aj q α ak − q + ε aj q ε aj − q (cid:105) . (58)A straightforward calculation yieldsi[ Q, Φ] = − gf abc (cid:88) q ∈ ¯ K × ( q δ jk − q j q k ) α aj q ( α bk c c ) − q , (59)so that Φ is not yet BRST invariant for g (cid:54) = 0. As acompensating term we considerΦ (cid:48) = gf abc √ V (cid:88) q , q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q + q (cid:48) + q (cid:48)(cid:48) , ( − i q j ) α ak q α bk q (cid:48) α cj q (cid:48)(cid:48) , (60)for which we findi[ Q, Φ (cid:48) ] = gf abc (cid:88) q ∈ ¯ K × ( q δ jk − q j q k ) α aj q ( α bk c c ) − q + g f abs f cds V (cid:88) q , q (cid:48) , p , p (cid:48) ∈ ¯ K × δ q + q (cid:48) + p + p (cid:48) , × i q j c a q α bk q (cid:48) α cj p α dk p (cid:48) . (61)The term proportional to g indeed cancels the contri-bution from Φ in Eq. (59), however, a new second-orderterm in g arises. This second-order term has been writtenin a compact form by arranging the three contributionsresulting from the product rule in a standard form where,for rearranging the last contribution, the Jacobi identityis required. In this compact form we easily find the finalcompensating termΦ (cid:48)(cid:48) = g f abs f cds V (cid:88) q , q (cid:48) , p , p (cid:48) ∈ ¯ K × δ q + q (cid:48) + p + p (cid:48) , α aj q α bk q (cid:48) α cj p α dk p (cid:48) , (62)in which all for spatial gauge boson operators are on anequal footing. In order to show that no leftover higher-order terms occur one needs to use the Jacobi identity.The operator Φ + Φ (cid:48) + Φ (cid:48)(cid:48) hence is BRST invariant.The operator Φ contains the energy of the transversegauge bosons and, according to Eq. (46), also part ofenergy of temporal and longitudinal gauge bosons. Themissing parts are found in the BRST invariant operatorsestablished in Section V. Moreover, these BRST invariantoperators contain the energy (34) of noninteracting ghostparticles. By collecting terms, we get the BRST invarianttotal Hamiltonian H = H freegb + H collgb , (63)consisting of the energy of free massless gauge bosons andghost particles H freegb = (cid:88) q ∈ ¯ K × q (cid:16) a aα † q a aα q + B a † q B a q + D a † q D a q (cid:17) , (64)and the interaction (confer to Eq. (3) of [13]), H collgb = gf abc √ V (cid:88) q , q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q + q (cid:48) + q (cid:48)(cid:48) , (cid:16) ε aj q α bj q (cid:48) α c q (cid:48)(cid:48) − ˙ c a q ¯ c b q (cid:48) α c q (cid:48)(cid:48) − i q j α ak q α bk q (cid:48) α cj q (cid:48)(cid:48) − i q j c a q ¯ c b q (cid:48) α cj q (cid:48)(cid:48) (cid:17) + g f abs f cds V (cid:88) q , q (cid:48) , p , p (cid:48) ∈ ¯ K × δ q + q (cid:48) + p + p (cid:48) , α aj q α bk q (cid:48) α cj p α dk p (cid:48) . (65)By making sure that all four gauge boson polarizationsoccur on an equal footing we have taken into account theLorentz invariance of the free theory. The collisional con-tribution consists mostly of three-particle interactions,but the second-order term in g represents a four-particlecollision. Note that, for the construction of self-adjointghost particle operators, the Hamiltonian is self-adjointwith respect to the signed inner product. B. Evolution equations
By recognizing i[
H, A ] as the time derivative of an op-erator A , we can now formulate the evolution equations for various operators and, in particular, we can comparethem to the classical equations compiled in Section II.We begin with the evolution of the four-vector potential,i[ H, α aj q ] = − ε aj q − i q j α a q − gf abc ( α bj α c ) q , (66)which reassuringly is the quantum version of the Fouriertransformed classical evolution equation (19). For thetemporal component of the four-vector potential, we findi[ H, α a q ] = i (cid:114) q a † q + a − q ) = − ε a q − i q j α aj q , (67)which is the quantum version of the definition (12) of ε a q ,relating ε a q to the time derivative of α a q .We next turn to the electric field type operators andfindi[ H, ε aj q ] = q α aj q − q j q k α ak q + i q j ε a q + gf abc ( α b ε cj ) q + i gf abc √ V (cid:88) q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q (cid:48) + q (cid:48)(cid:48) , q [(2 q (cid:48) k + q (cid:48)(cid:48) k ) α bj q (cid:48) − q (cid:48) j α bk q (cid:48) ] α ck q (cid:48)(cid:48) − g f abs f cds V (cid:88) p , p (cid:48) , p (cid:48)(cid:48) ∈ ¯ K × δ p + p (cid:48) + p (cid:48)(cid:48) , q α bk p α ck p (cid:48) α dj p (cid:48)(cid:48) − gf abc √ V (cid:88) q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q (cid:48) + q (cid:48)(cid:48) , q i q (cid:48)(cid:48) j ¯ c b q (cid:48) c c q (cid:48)(cid:48) , (68)where the same simplification as in Eq. (67) has beenused. Except for the terms involving ε a and ¯ c b c c , werecognize that exactly the same terms as in the classi-cal evolution equation (20) in the absence of an externalcurrent. In view of Eq. (12), ε a vanishes in the Lorenzgauge. Also the ghost term ¯ c b c c is clearly related to theproper handling of gauge conditions. The same remarkshold for the temporal componenti[ H, ε a q ] = i q j ε aj q + gf abc ( α bj ε cj + ¯ c b ˙ c c ) q , (69)which contains Eq. (15) in the absence of an external cur-rent. The consistency of Eqs. (66)–(69) with the classicalfield equations implies that our minimal BRST invari-ant extension of the Lorentz covariant free theory indeedleads to the quantized Yang-Mills field theory.For completeness, we also look at the evolution equa-tions for the ghost operators,i[ H, c a q ] = ˙ c a q , (70)and i[ H, ¯ c a q ] = ˙¯ c a q + gf abc ( α b ¯ c c ) q . (71)This last equation clearly shows that the operator ˙¯ c a q in-troduced in Eq. (31) is not the full time derivative of ¯ c a q ,as pointed out before. For non-Abelian gauge theories,interactions between ghost particles and vector bosonsoccur. More of these interactions are implied byi[ H, ˙ c a q ] = − q c a q + gf abc ( α b ˙ c c ) q + gf abc √ V (cid:88) q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q (cid:48) + q (cid:48)(cid:48) , q i q (cid:48)(cid:48) j α bj q (cid:48) c c q (cid:48)(cid:48) , (72)andi[ H, ˙¯ c a q ] = − q ¯ c a q + gf abc √ V (cid:88) q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q (cid:48) + q (cid:48)(cid:48) , q i q j α bj q (cid:48) ¯ c c q (cid:48)(cid:48) . (73) C. Additional conserved charge
It can easily be verified that an operator R possessingthe propertiesi[ R, c a q ] = c a q , i[ R, ¯ c a q ] = − ¯ c a q , (74)and i[ R, ˙ c a q ] = ˙ c a q , i[ R, ˙¯ c a q ] = − ˙¯ c a q , (75)as well as vanishing commutators with all gauge bosonoperators, commutes with the full Hamiltonian H . Suchan operator R can be interpreted as the generator of thesymmetry associated with the rescaling of ghost opera-tors with and without bars by reciprocal factors.As suggested in Eq. (2.17) of [12], we have the followingnatural candidate for R , R = (cid:88) q ∈ ¯ K × (cid:0) c a q ˙¯ c a − q − ˙ c a q ¯ c a − q (cid:1) . (76)One can easily verify by means of the anticommutationrelations (32) that this operator R indeed leads to theproperties in Eqs. (74), (75). The simple form of R (compared to Eq. (2.17) of [12]) is a consequence of thefact that the dots don’t indicate full time derivatives butrather the operators defined in Eq. (31) or Eqs. (37) and(38).We further find the commutatori[ R, Q ] = Q. (77)On the kernel of Q , the operator R commutes also withthe BRST charge. We can hence restrict the physicalstates to an eigenspace of R (most conveniently witheigenvalue zero). VII. CONSTRUCTION OF PHYSICAL STATES
In Section III, we had introduced two inner products,the canonical and the signed ones, where the latter im-plies negative-norm states and hence is not truly an inner
Ker Q Ker Q † Im Q Im Q † Ker Q † Ker Q FIG. 1: Decomposition of Fock space into three mutuallyorthogonal subspaces. product. As the signed inner product serves for definingthe physical bra- in terms of ket-vectors, we need to iden-tify a subspace of the full Fock space on which the signedproduct becomes a true inner product.If the ghost operators are introduced according toEqs. (35)–(38), the BRST charge (48) is self-adjoint withrespect to the signed inner product, Q ‡ = Q . However,the BRST charge is not self-adjoint with respect to thecanonical inner product, Q † (cid:54) = Q . We here decomposethe Fock space into three mutually orthogonal spaces (interms of the canonical inner product) and discuss thephysical implications (associated with the signed innerproduct). A. Decomposition of states
A decomposition of Fock space into three mutually or-thogonal subspaces is achieved in terms of the images andkernels of the BRST operator Q and of the co-BRST op-erator Q † . Our discussion is based on Section 2.5 of [14].A toy illustration of the essential features of the subse-quent development is offered in Appendix B.Relying on the canonical inner product, the full Fockspace F can be expressed as the direct sum of three mu-tually orthogonal spaces (the characteristic features ofthis decomposition are illustrated in Figure 1), F = (Ker Q ∩ Ker Q † ) ⊕ Im Q ⊕ Im Q † . (78)In view of the nilpotency properties Q = ( Q † ) = 0, wehave Im Q ⊂ Ker Q, Im Q † ⊂ Ker Q † . (79)As a next step, we prove the representationsKer Q = (Im Q † ) ⊥ , Ker Q † = (Im Q ) ⊥ . (80)We only prove the first identity in Eq. (80) because thesecond one can be shown in exactly the same way. If | ϕ (cid:105) ∈ Ker Q , that is Q | ϕ (cid:105) = 0, we have 0 = s can ( Q | ϕ (cid:105) , | ψ (cid:105) ) = s can ( | ϕ (cid:105) , Q † | ψ (cid:105) ) for all | ψ (cid:105) ∈ F . Conversely, if | ϕ (cid:105) ∈F is such that s can ( | ϕ (cid:105) , Q † | ψ (cid:105) ) = 0 for all | ψ (cid:105) ∈ F ,then s can ( Q | ϕ (cid:105) , | ψ (cid:105) ) = 0 for all | ψ (cid:105) ∈ F , which implies Q | ϕ (cid:105) = 0.Equations (79) and (80) imply that Im Q and Im Q † areorthogonal spaces and that Ker Q ∩ Ker Q † is orthogonalto both images and exhausts the rest of F . We havethus established Eq. (78) and the situation depicted inFigure 1. This equation implies that every state | ϕ (cid:105) ∈ F can be written as | ϕ (cid:105) = | χ (cid:105) + Q | ψ (cid:105) + Q † | ψ (cid:105) with Q | χ (cid:105) = Q † | χ (cid:105) = 0 , (81)where the three contributions to | ϕ (cid:105) are mutually orthog-onal in the canonical inner product. B. BRST Laplacian
Before we discuss the signed norm of states in the var-ious subspaces, we elaborate some details of the decom-position (78). This can be done in terms of the BRSTLaplacian ∆ = ( Q + Q † ) = QQ † + Q † Q. (82)As ∆ is the square of a self-adjoint operator, its eigen-values λ must be real and nonnegative. All states inKer Q ∩ Ker Q † are eigenstates of ∆ with eigenvalue zero.A nonzero eigenvalue can only be obtained for vectorsfrom one of the images, say | ψ (cid:105) ∈ Im Q . To obtain∆ | ψ (cid:105) (cid:54) = 0, | ϕ (cid:105) = Q † | ψ (cid:105) has to lie in (Ker Q ) ⊥ = Im Q † .Therefore, eigenstates of ∆ with nonzero eigenvalues canonly be obtained by flipping back and forth between thetwo images when applying the two factors in the defini-tion of ∆. By rescaling | ϕ (cid:105) , we find the following doubletof states, Q | ϕ (cid:105) = λ | ψ (cid:105) Q † | ψ (cid:105) = λ | ϕ (cid:105) . (83)For λ = 0 we would end up in the kernel of ∆, so thatKer ∆ = Ker Q ∩ Ker Q † . (84)According to Eq. (83), both | ϕ (cid:105) ∈ Im Q † and | ψ (cid:105) ∈ Im Q are eigenstates of ∆ with the same eigenvalue λ .We further note the property( Q + Q † )( | ϕ (cid:105) ± | ψ (cid:105) ) = λ ( | ψ (cid:105) ± | ϕ (cid:105) ) , (85)so that | ϕ (cid:105)±| ψ (cid:105) are eigenvectors of Q + Q † with eigenval-ues ± λ . In summary, we have shown that the eigenvec-tors of ∆ with nonzero eigenvalues are nicely organizedin doublets. C. Physical subspace
As a next step, we wish to identify Ker ∆ = Ker Q ∩ Ker Q † as the physical subspace of F in which s sign be-comes a valid inner product, to be taken as the physical inner product. In other words, physical states are char-acterized by Q | ϕ (cid:105) = Q † | ϕ (cid:105) = 0.In the natural base of our Fock space, the canonicalinner product is represented by the unit matrix, whereasthe signed inner product is represented by a diagonalmatrix σ with diagonal elements ±
1. If A † and A ‡ arethe matrices representing the two adjoints of an operator A in the natural basis, the definition of these adjointsimplies σA † = A ‡ σ and, in view of σ = 1, also σA ‡ = A † σ . In view of the self-adjointness property Q ‡ = Q , weconclude that we can choose a canonically orthonormalbasis of eigenvectors of Q + Q † which all possess signednorm +1 or − Q or Q † vanishes. For example, s sign ( Q | ϕ (cid:105) , Q | ϕ (cid:105) ) = s sign ( | ϕ (cid:105) , Q ‡ Q | ϕ (cid:105) ) = 0 as Q ‡ = Q and Q = 0. By superposition of states from the twoimages one can produce states of negative norm. For theeigenvectors of Q + Q † found in Eq. (85), we have s sign ( | ϕ (cid:105) ± | ψ (cid:105) , | ϕ (cid:105) ± | ψ (cid:105) ) = ± ( (cid:104) ϕ | ψ (cid:105) + (cid:104) ψ | ϕ (cid:105) ) . (86)This result nicely shows that the (properly scaled) eigen-vectors of Q + Q † come in pairs with signed norms +1and − Q ⊕ Im Q † ) ⊥ ? The toy exam-ple of Appendix B shows that the answer is ‘no.’ In theBRST construction we have to make sure that the num-ber of negative-norm states matches exactly the pairs inIm Q ⊕ Im Q † . We need to verify that every negative-norm state can be written as a linear combination ofzero-norm states from Im Q and Im Q † .The above construction is usually presented in twosteps. For the physical states, one first imposes thecondition Q | ϕ (cid:105) = 0. By excluding Im Q † one avoidsthe above construction of negative-norm states from thedoublets (83), but zero-norm states clearly still exist inIm Q . In a second step, one considers equivalence classesof states that differ only by zero-norm states. Selectingrepresentatives of the equivalence classes is often referredto as gauge fixing. Our gauge fixing condition thus is Q † | ϕ (cid:105) = 0. According to Eq. (81), the states satisfyingthe first condition Q | ϕ (cid:105) = 0 possess the representation | ϕ (cid:105) = | χ (cid:105) + Q | ψ (cid:105) with Q | χ (cid:105) = Q † | χ (cid:105) = 0 . (87)We can take | χ (cid:105) as the unique representative of an equiv-alence class of states that differ by the zero-norm states Q | ψ (cid:105) for some | ψ (cid:105) ∈ F , so that the physical subspaceindeed is Ker ∆. The physical norms are independent ofthe choice of the representative. For Hamiltonians com-muting with Q and Q † , the physical subspace Ker ∆ isinvariant under Hamiltonian dynamics. VIII. INCLUSION OF MATTER
In the terms proportional to g in the BRST charge (48)we recognize a term that contains the temporal compo-0nent (17) of the current four-vector resulting from thegauge bosons. The simplest way to incorporate matterinto the BRST charge is to add the corresponding terminvolving the temporal component of the current four-vector associated with matter, Q J = − (cid:88) q ∈ ¯ K × J a q c a − q . (88)This idea is consistent with the expression for the BRSTcharge in quantum electrodynamics (see, for example,Eq. (3.125) of [11]). Nilpotency of the extended BRSTcharge Q + Q J requires Q J + { Q, Q J } = 0 . (89)Both terms can be calculated under the assumption that J a q commutes with all gauge boson and ghost operators: { Q, Q J } = i2 gf abc (cid:88) q ∈ ¯ K × J c q ( c a c b ) − q , (90)and Q J = 12 (cid:88) q , q (cid:48) ∈ ¯ K × [ J a q , J b q (cid:48) ] c a − q c b − q (cid:48) . (91)After using Eq. (47), the nilpotency condition can bewritten as [ J a q , J b q (cid:48) ] = − i gf abc √ V J c q + q (cid:48) . (92)With this commutator we obtain the BRST transfor-mation δJ a q = − gf abc ( J b c c ) q , (93)which is exactly what we expect in view of the classicalgauge transformation (7). To recover also the propertransformation behavior of the spatial components of thecurrent four-vector, we need to generalize Eq. (92) to[ J a q , J bµ q (cid:48) ] = − i gf abc √ V J cµ q + q (cid:48) . (94)Equation (94) is a simple case of a current algebra,in which neither an axial current nor a Schwinger termis considered. Additional terms would require a changeof the BRST charge and/or Hamiltonian to make surethat the BRST approach can still be used to handle theconstraints associated with gauge theories. For example,the Schwinger term [15] has been discussed in the contextof Yang-Mills theory in Eq. (3.16) of [16] or Eq. (1.6) of[17]. It is typically associated with sums that are notabsolutely convergent so that regularization is required.For electromagnetic fields and massless fermions in onespace dimension (known as the Schwinger model [18]),the Schwinger term is finite and well-defined. Possible modifications of the Hamiltonian and BRST charge arediscussed in Section 3.3 of [11]. As at least temporalphotons have to acquire mass, the Schwinger term leadsto chiral symmetry breaking.If we choose the Hamiltonian for the interaction ofgauge bosons and matter as H J = − (cid:88) q ∈ ¯ K × J aµ − q α aµ q , (95)we find i[ H J , ε aµ q ] = − J aµ q , (96)and hence the proper occurrence of the current inEqs. (15) and (16). In order to check whether H + H J is BRST invariant with respect to the new BRST charge Q + Q J , we calculate the commutatorsi[ Q, H J ] = (cid:88) q ∈ ¯ K × (cid:0) J a q ˙ c a − q − i q j J aj q c a − q (cid:1) − gf abc (cid:88) q ∈ ¯ K × ( J bµ c c ) − q α aµ q , (97)i[ Q J , H ] = − (cid:88) q ∈ ¯ K × J a q ˙ c a − q , (98)and i[ Q J , H J ] = gf abc (cid:88) q ∈ ¯ K × ( J bµ c c ) − q α aµ q . (99)The incomplete compensation of terms in Eqs. (97)–(99)implies that H + H J is not BRST invariant with respectto the new BRST charge Q + Q J . This is not surprising asthe Hamiltonian for non-interacting matter is still miss-ing. BRST invariance of the full Hamiltonian requiresi[ Q J , H freemat ] = (cid:88) q ∈ ¯ K × i q j J aj q c a − q , (100)or, by means of Eq. (88), (cid:88) q ∈ ¯ K × (cid:16) i[ H freemat , J a q ] − i q j J aj q (cid:17) c a − q = 0 . (101)The current four-vector must be defined such that theoperator identityi[ H freemat , J a q ] − i q j J aj q = 0 (102)is satisfied (see, e.g., Eq. (3.95) of [11] for quantum elec-trodynamics). In other words, the current four-vectormust be constructed such that a continuity equationholds. However, the condition (102) is not a completebalance equation, which would require occurrence of the1full Hamiltonian instead of H freemat . An additional contri-bution i[ H J , J a q ] = gf abc ( α bµ J cµ ) q (103)appears in the complete balance equation. In view of theclassical continuity equation (21) one might assume thatthis source term is compensated by including the currentof the field. However, if we define the quantum versionof the current four-vector,˜ J a q = gf abc ( α bj ε cj ) q , (104)and˜ J aj q = − gf abc (cid:104) ( α b ε cj ) q + gf cde ( α bk α dj α ek ) q (cid:105) + i gf abc √ V (cid:88) q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q (cid:48) + q (cid:48)(cid:48) , q ( q (cid:48) j α bk q (cid:48) − q (cid:48) k α bj q (cid:48) ) α ck q (cid:48)(cid:48) , (105)a lengthy calculation shows that additional gauge termsappear,i[ H + H J , ˜ J a q ] − i q j ˜ J aj q = − gf abc ( α bµ J cµ ) q + gf abc √ V (cid:88) q (cid:48) , q (cid:48)(cid:48) ∈ ¯ K × δ q (cid:48) + q (cid:48)(cid:48) , q (cid:110) i[ H, ε b q (cid:48) ] α c q (cid:48)(cid:48) − i q (cid:48) j ε b q (cid:48) α cj q (cid:48)(cid:48) − gf bde (cid:2) ˙ c d q (cid:48) (¯ c e α c ) q (cid:48)(cid:48) − i q (cid:48) j c d q (cid:48) (¯ c e α cj ) q (cid:48)(cid:48) (cid:3)(cid:111) . (106)Therefore, a more careful analysis of say color conserva-tion on the physical space is required. IX. SUMMARY AND DISCUSSION
We have elaborated all the details of the BRST quan-tization of Yang-Mills theory in a strictly Hamiltonianapproach. A new representation of ghost-field operatorsin terms of canonical creation and annihilation operatorshas been introduced in Eqs. (35)–(38). This represen-tation combines two pivotal advantages: (i) there existsa well-defined Fock space that serves as the underlyingHilbert space for carrying out the Hamiltonian approachand (ii) the BRST charge (48) and the Hamiltonian (63)–(65) turn out to be self-adjoint operators with respect tothe physical inner product. The Fock space actually car-ries two inner products, a canonical and a signed one,where the restriction of the latter to a suitable subspaceserves as the physical inner product. The occurrence ofnegative-norm states in the unrestricted space explainswhy ghost particles can combine the anticommutationrelations for fermions with zero spin, in an apparent vio-lation of the spin-statistics theorem.The construction of the BRST charge is based on thefollowing two properties: (i) its role as the generator ofthe operator version of classical gauge transformationsand (ii) its nilpotency. Lorentz symmetry is taken into account in the construction of the Fock space (with fourgauge boson polarizations), in the formulation of theBRST charge (inherited from classical gauge transforma-tions), and in the construction of the Hamiltonian as theminimal BRST invariant extension of the free theory (inwhich all four gauge boson polarizations are on an equalfooting). The final Hamiltonian found in Eqs. (63)–(65)reproduces the field equations of classical Yang-Mills the-ory.The interaction with matter has been included in termsof the current four-vector by adding the contributions(88) and (95) to the BRST charge and Hamiltonian, re-spectively. In order to make the BRST approach work,a current algebra has to be postulated, where a particu-larly simple one is given in Eq. (94). Any change in thecurrent algebra, say by a Schwinger term, requires mod-ifications of the BRST charge and the BRST invariantHamiltonian. We have also discussed the formulation ofthe conservation law associated with BRST symmetry.Whereas the Hamiltonian approach to BRST quantiza-tion on Fock space has the educational advantage of beingnicely explicit and transparent, it has serious disadvan-tages in practical calculations. In particular, perturba-tion theory becomes very cumbersome (see Appendix Aof [11] for some simplifications). Our motivation for elab-orating the Hamiltonian approach stems from the formu-lation of dissipative quantum field theory [11, 19], whichis based on quantum master equations for evolving den-sity matrices in time as an irreversible generalization ofHamiltonian dynamics. Dissipative quantum field the-ory, which is based on the idea that the elimination ofdegrees of freedom in renormalization procedures leadsto the emergence of irreversibility, adds rigor, robustnessand intuition to the field and hence has the potential toclarify the foundations of quantum field theory. Dissi-pation can easily be introduced in the Hamiltonian ap-proach (see Section 1.2.3.2 of [11]).Stochastic simulation techniques developed for quan-tum master equations [20] can then be used to simulatequantum field theories (see Sections 1.2.8.6 and 3.4.3.3 of[11] for details). With the present paper, the simulationideas so far applied only in a rudimentary way to quan-tum electrodynamics [21], become applicable to quantumchromodynamics.
Appendix A: Proof of Q = 0 In order to prove the nilpotency of the BRST chargedefined in Eq. (48), we write Q = Q + Q + Q + Q with Q = (cid:88) q ∈ ¯ K × (cid:0) ε a q ˙ c a − q − i q j ε aj q c a − q (cid:1) , (A1) Q = gf abc (cid:88) q ∈ ¯ K × ( ε aµ α bµ ) q c c − q , (A2)2 Q = − gf abc (cid:88) q ∈ ¯ K × ( ˙ c a ¯ c b ) q c c − q , (A3)and Q = 12 gf abc (cid:88) q ∈ ¯ K × ˙¯ c a − q ( c b c c ) q . (A4)Based on trivial canonical commutation and anticommu-tation relations, we find Q = 0 , (A5)exactly as for Abelian gauge theory. The evaluation of Q is based on the commutation relations (43) and (44);the result is Q = − i g f ads f bcs (cid:88) q ∈ ¯ K × ( ε dµ α cµ ) − q ( c b c a ) q . (A6)By means of Eq. (32) we find Q = i g f cds f abs (cid:88) q ∈ ¯ K × (¯ c a ˙ c d ) q ( c b c c ) − q . (A7)To obtain Q = 0 , (A8)we need the Jacobi identity (3).As a consequence of { Q , Q } = 0 , (A9)the nonzero contribution to Q in Eq. (A6) can only becompensated by { Q , Q } = − i2 g f cds f abs (cid:88) q ∈ ¯ K × ( ε dµ α cµ ) − q ( c b c a ) q . (A10)Indeed, the Jacobi identity (3) implies Q + { Q , Q } = 0 . (A11)The nonzero contribution to Q in Eq. (A7) can only becompensated by { Q , Q } = i2 g f ads f bcs (cid:88) q ∈ ¯ K × (¯ c a ˙ c d ) q ( c b c c ) − q . (A12)By once more using the Jacobi identity (3), we find Q + { Q , Q } = 0 . (A13)We still need to evaluate the anticommutators of Q with all the other contributions to Q . We first evaluate { Q , Q } = i gf abc (cid:88) q ∈ ¯ K × ε a q ( ˙ c b c c ) − q + 12 gf abc (cid:88) q ∈ ¯ K × q j ε aj q ( c b c c ) − q . (A14) The next anticommutator is given by { Q , Q } = − i gf abc (cid:88) q ∈ ¯ K × ε a q ( ˙ c b c c ) − q , (A15)which cancels the first term on the right-hand-side ofEq. (A14). A final straightforward calculation yields { Q , Q } = − gf abc (cid:88) q ∈ ¯ K × q j ε aj q ( c b c c ) − q . (A16)As { Q , Q } cancels the second term on the right-hand-side of Eq. (A14), we obtain { Q , Q } + { Q , Q } + { Q , Q } = 0 . (A17)By summing up Eqs. (A11), (A13), (A17) and usingthe results (A5), (A8), (A9), we indeed obtain the desirednilpotency of the BRST charge defined in Eq. (48), Q =0. Appendix B: Construction of physical states – a toyversion
We here sketch a toy version of the construction ofphysical states in the BRST approach. The generalidea is nicely illustrated in a three-dimensional cartoonversion of temporal, longitudinal and transverse gaugebosons (where physical states don’t contain any rightbosons, equivalent physical states differ by left bosons,and unique representatives of equivalence classes do notcontain any left photons).For the linear operators Q = − − , Q † = − − , (B1)one easily verifies Q = ( Q † ) = 0. The image and kernelof the operator Q are given byIm Q = λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ∈ R , (B2)andKer Q = λ + λ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ, λ (cid:48) ∈ R , (B3)illustrating the general relation Im Q ⊂ Ker Q . We simi-larly have Im Q † = λ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ∈ R , (B4)3Ker Q † = λ − + λ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ, λ (cid:48) ∈ R , (B5)and Im Q † ⊂ Ker Q † . We also find Im Q † = (Ker Q ) ⊥ andIm Q = (Ker Q † ) ⊥ , implying the orthogonality of Im Q and Im Q † . The kernel of ∆ = ( Q + Q † ) = QQ † + Q † Q ,∆ = , (B6)coincides with Ker Q ∩ Ker Q † , so that Ker ∆, Im Q andIm Q † are three mutually orthogonal spaces. The totalspace is the direct sum of these three vector spaces. Anyvector can uniquely be written as the sum of three vec-tors, one from each of these spaces.Introducing a signed inner product by σ = − , (B7) we realize the self-adjointness property Q ‡ = σQ † σ = Q .For this signed inner product, states from Im Q or Im Q † have zero norm, states from Ker ∆ have positive norm.Had we chosen a signed inner product with σ (cid:48) = − − , (B8)we would still have the self-adjointness property Q ‡ = σ (cid:48) Q † σ (cid:48) = Q . However, we would have introduced toomany states with a negative norm leading to an unphys-ical inner product on Ker ∆. [1] C. Becchi, A. Rouet, and R. Stora, Ann. Phys. (N.Y.) , 287 (1976).[2] I. V. Tyutin (1975), preprint of P. N. Lebedev PhysicalInstitute, No. 39, 1975, arXiv:0812.0580.[3] D. Nemeschansky, C. Preitschopf, and M. Weinstein,Ann. Phys. (N.Y.) , 226 (1988).[4] T. Kugo and I. Ojima, Phys. Lett. B , 459 (1978).[5] M. E. Peskin and D. V. Schroeder, An Introduction toQuantum Field Theory (Perseus Books, Reading, MA,1995).[6] S. Weinberg,
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