aa r X i v : . [ qu a n t - ph ] N ov Version 2
Spacetime Path Formalism for Massive Particles of Any Spin
Ed Seidewitz ∗ (Dated: November 14, 2018) Abstract
Earlier work presented a spacetime path formalism for relativistic quantum mechanics arisingnaturally from the fundamental principles of the Born probability rule, superposition, and space-time translation invariance. The resulting formalism can be seen as a foundation for a number ofprevious parameterized approaches to relativistic quantum mechanics in the literature. Becausetime is treated similarly to the three space coordinates, rather than as an evolution parameter,such approaches have proved particularly useful in the study of quantum gravity and cosmology.The present paper extends the foundational spacetime path formalism to include massive, non-scalar particles of any (integer or half-integer) spin. This is done by generalizing the principle oftranslational invariance used in the scalar case to the principle of full Poincar´e invariance, leadingto a formulation for the nonscalar propagator in terms of a path integral over the Poincar´e group.Once the difficulty of the non-compactness of the component Lorentz group is dealt with, the sub-sequent development is remarkably parallel to the scalar case. This allows the formalism to retain aclear probabilistic interpretation throughout, with a natural reduction to non-relativistic quantummechanics closely related to the well known generalized Foldy-Wouthuysen transformation.
PACS numbers: 03.65.Pm, 03.65.Fd, 03.30.+p, 11.10.Ef ∗ Electronic address: [email protected] . INTRODUCTION Reference 1 presented a foundational formalism for relativistic quantum mechanics basedon path integrals over parametrized paths in spacetime. As discussed there, such an approachis particularly suited for further study of quantum gravity and cosmology, and it can be givena natural interpretation in terms of decoherent histories [2]. However, the formalism as givenin [1] is limited to scalar particles. The present paper extends this spacetime path formalismto non-scalar particles, although the present work is still limited to massive particles.There have been several approaches proposed in the literature for extending the pathintegral formulation of the relativistic scalar propagator [3, 4, 5, 6] to the case of non-scalar particles, particularly spin-1/2 (see, for example, [7, 8, 9, 10, 11]). These approachesgenerally proceed by including in the path integral additional variables to represent higherspin degrees of freedom. However, there is still a lack of a comprehensive path integralformalism that treats all spin values in a consistent way, in the spirit of the classic work ofWeinburg [12, 13, 14] for traditional quantum field theory. Further, most earlier referencesassume that the path integral approach is basically a reformulation of an a priori traditionalHamiltonian formulation of quantum mechanics, rather than being foundational in its ownright.The approach to be considered here extends the approach from [1] to non-scalar particlesby expanding the configuration space of a particle to be the Poincar´e group (also known asthe inhomogeneous Lorentz group). That is, rather than just considering the position of aparticle, the configuration of a particle will be taken to be both a position and a Lorentztransformation. Choosing various representations of the group of Lorentz transformationsthen allows all spins to be handled in a consistent way.The idea of using a Lorentz group variable to represent spin degrees of freedom is notnew. For example, Hanson and Regge [15] describe the physical configuration of a relativisticspherical top as a Poincar´e element whose degrees of freedom are then restricted. Similarly,Hannibal [16] proposes a full canonical formalism for classical spinning particles using theLorentz group for the spin configuration space, which is then quantized to describe bothspin and isospin. Rivas [17, 18, 19] has made a comprehensive study in which an elementaryparticle is defined as “a mechanical system whose kinematical space is a homogeneous spaceof the Poincar´e group”. 2ivas actually proposes quantization using path integrals, but he does not provide anexplicit derivation of the non-scalar propagator by evaluating such an integral. A primarygoal of this paper to provide such a derivation.Following a similar approach to [1], the form of the path integral for non-scalar particleswill be deduced from the fundamental principles of the Born probability rule, superposition,and Poincar´e invariance. After a brief overview in Sec. II of some background for thisapproach, Sec. III generalizes the postulates from [1] to the non-scalar case, leading to apath integral over an appropriate Lagrangian function on the Poincar´e group variables.The major difficulty with evaluating this path integral is the non-compactness of theLorentz group. Previous work on evaluating Lorentz group path integrals (going back to[20]) is based on the irreducible unitary representations of the group. This is awkward, since,for a non-compact group, these representations are continuous [21] and the results do notgeneralize easily to the covering group SL (2 , C ) that includes half-integral spins.Instead, we will proceed by considering a Wick rotation to Euclidean space, which replacesthe non-compact Lorentz group SO (3 ,
1) by the compact group SO (4) of rotations in fourdimensions, in which it is straightforward to evaluate the path integral. It will then beargued that, even though the SO (4) propagator cannot be assumed the same as the trueLorentz propagator, the propagators should be the same when restricted to the commonsubgroup SO (3) of rotations in three dimensions. This leads directly to considerations ofthe spin representations of SO (3).Accordingly, Sec. IV develops the Euclidean SO (4) propagator and Sec. V then considersthe reduction to the three-dimensional rotation group and its spin representations. However,rather than using the usual Wigner approach of reduction along the momentum vector[22], we will reduce along an independent time-like four-vector [23, 24]. This allows fora very parallel development to [1] for antiparticles in Sec. VI and for a clear probabilityinterpretation in Sec. VII.Interactions of non-scalar particles can be included in the formalism by a straightforwardgeneralization of the approach given in [1]. Section VIII gives an overview of this, thoughfull details are not included where they are substantially the same as the scalar case.Natural units with ~ = 1 = c are used throughout the following and the metric has asignature of ( − + ++). 3 I. BACKGROUND
Path integrals were originally introduced by Feynman [25, 26] to represent the non-relativistic propagation kernel ∆( x − x ; t − t ). This kernel gives the transition amplitudefor a particle to propagate from the position x at time t to the position x at time t .That is, if ψ ( x ; t ) is the probability amplitude for the particle to be at position x at time t , then the amplitude for it to propagate to another position at a later time is ψ ( x ; t ) = Z d x ∆( x − x ; t − t ) ψ ( x ; t ) . A specific path of a particle in space is given by a position function q ( t ) parametrizedby time (or, in coordinate form, the three functions q i ( t ) for i = 1 , , q ( t ) = x and ending at q ( t ) = x . The path integral form forthe propagation kernel is then given by integrating over all these paths as follows:∆( x − x ; t − t ) = ζ Z D q δ ( q ( t ) − x ) δ ( q ( t ) − x )e i S [ q ] , (1)where the phase function S [ q ] is given by the classical action S [ q ] ≡ Z t t d t L ( ˙ q ( t )) , with L ( ˙ q ) being the non-relativistic Lagrangian in terms of the three-velocity ˙ q ≡ d q / d t .In Eq. (1), the notation D q indicates a path integral over the three functions q i ( t ). TheDirac delta functions constrain the paths integrated over to start and end at the appropriatepositions. Finally, ζ is a normalization factor, including any limiting factors required to keepthe path integral finite (which are sometimes incorporated into the integration measure D q instead).As later noted by Feynman himself [5], it is possible to generalize the path integralapproach to the relativistic case. To do this, it is necessary to consider paths in spacetime ,rather than just space. Such a path is given by a four dimensional position function q ( λ ),parametrized by an invariant path parameter λ (or, in coordinate form, the four functions q µ ( λ ), for µ = 0 , , , x − x ) = − i(2 π ) − Z d p e i p · ( x − x ) p + m − i ǫ . (2)4t can be shown (in addition to [5], see also, e.g., [1, 3]) that this propagator can be expressedin path integral form as∆( x − x ) = Z ∞ λ d λ ζ Z D q δ ( q ( λ ) − x ) δ ( q ( λ ) − x )e i S [ q ] , (3)where S [ q ] ≡ Z λλ d λ ′ L ( ˙ q )( λ ′ )) , and L ( ˙ q ) is now the relativistic Lagrangian in terms of the the four-velocity ˙ q ≡ d q/ d λ .Notice that the form of the relativistic expression differs from the non-relativistic one byhaving an additional integration over λ . This is necessary, since the propagator must, inthe end, depend only on the change in position, independent of λ . However, as noted in [1],Eq. (3) can be written as ∆( x − x ) = Z ∞ λ d λ ∆( x − x ; λ − λ ) , (4)where the relativistic kernel ∆( x − x ; λ − λ ) = ζ Z D q δ ( q ( λ ) − x ) δ ( q ( λ ) − x )e i S [ q ] (5)now has a form entirely parallel with the non-relativistic case. The relativistic kernel can beconsidered to represent propagation over paths of the specific length λ − λ , while Eq. (4)then integrates over all possible path lengths.Given the parallel with the non-relativistic case, define the parametrized probability am-plitudes ψ ( x ; λ ) such that ψ ( x ; λ ) = Z d x ∆( x − x ; λ − λ ) ψ ( x ; λ ) . Parametrized amplitudes were introduced by Stueckelberg [27, 28], and parametrized ap-proaches to relativistic quantum mechanics have been developed by a number of subsequentauthors [23, 29, 30, 31, 32, 33, 34, 35, 36]. The approach is developed further in the contextof spacetime paths of scalar particles in [1].In the traditional presentation, however, it is not at all clear why the path integrals ofEqs. (1) and (2) should reproduce the expected results for non-relativistic and relativisticpropagation. The phase functional S is simply chosen to have the form of the classicalaction, such that this works. In contrast, [1] makes a more fundamental argument thatthe exponential form of Eq. (5) is a consequence of translation invariance in Minkowski5pacetime. This allows for development of the spacetime path formalism as a foundationalapproach, rather than just a re-expression of already known results.The full invariant group of Minkowski spacetime is not the translation group, though,but the Poincar´e group consisting of both translations and Lorentz transformations. Thisleads one to consider the implications of applying the argument of [1] to the full Poincar´egroup.Now, while a translation applies to the position of a particle, a Lorentz transformationapplies to its frame of reference . Just as we can consider the position x of a particle to be atranslation by x from some fixed origin O , we can consider the frame of reference of a particleto be given by a Lorentz transformation Λ from a fixed initial frame I . The full configurationof a particle is then given by ( x, Λ), for a position x and a Lorentz transformation Λ—thatis, the configuration space of the particle is also just the Poincar´e group. The applicationof an arbitrary Poincar´e transformation (∆ x, Λ ′ ) to a particle configuration ( x, Λ) results inthe transformed configuration (Λ ′ x + ∆ x, Λ ′ Λ).A particle path will now be a path through the Poincar´e group, not just through space-time. Such a path is given by both a position function q ( λ ) and a Lorentz transformationfunction M ( λ ) (in coordinate form, a Lorentz transformation is represented by a matrix,so there are sixteen functions M µν ( λ ), for µ, ν = 0 , , , , ). The remainder of this paperwill re-develop the spacetime path formalism introduced in [1] in terms of this expandedconception of particle paths. As we will see, this naturally leads to a model for non-scalarparticles. III. THE NON-SCALAR PROPAGATOR
This section develops the path-integral form of the non-scalar propagator from the con-ception of Poincar´e group particle paths introduced in the previous section. The argumentparallels that of [1] for the scalar case, motivating a set of postulates that lead to the ap-propriate path integral form.To begin, let ∆( x − x , ΛΛ − ; λ − λ ) be the transition amplitude for a particle to go fromthe configuration ( x , Λ ) at λ to the configuration ( x, Λ) at λ . By Poincar´e invariance,this amplitude only depends on the relative quantities x − x and ΛΛ − . By parametershift invariance, it only depends on λ − λ . Similarly to the scalar case (Eq. (4)), the full6ropagator is given by integrating over the kernel path length parameter:∆( x − x , ΛΛ − ) = Z ∞ d λ ∆( x − x , ΛΛ − ; λ ) . (6)The fundamental postulate of the spacetime path approach is that a particle’s transitionamplitude between two points is a superposition of the transition amplitudes for all possiblepaths between those points. Let the functional ∆[ q, M ] give the transition amplitude for apath q ( λ ) , M ( λ ). Then the transition amplitude ∆( x − x , ΛΛ − ; λ − λ ) must be given bya path integral over ∆[ q, M ] for all paths starting at ( x , Λ ) and ending at ( x, Λ) with theparameter interval [ λ , λ ]. Postulate 1.
For a free, non-scalar particle, the transition amplitude ∆( x − x , ΛΛ − ; λ − λ ) is given by the superposition of path transition amplitudes ∆[ q, M ] , for all possible Poincar´epath functions q ( λ ) , M ( λ ) beginning at ( x , Λ ) and ending at ( x, Λ) , parametrized over theinterval [ λ , λ ] . That is, ∆( x − x , ΛΛ − ; λ − λ ) = ζ Z D q Z D Mδ ( q ( λ ) − x ) δ ( M ( λ )Λ − − I ) δ ( q ( λ ) − x ) δ ( M ( λ )Λ − − I )∆[ q, M ] , (7) where ζ is a normalization factor as required to keep the path integral finite. As previously noted, the notation D q in Eq. (7) indicates a path integral over the fourpath functions q µ ( λ ). Similarly, D M indicates a path integral over the Lorentz groupfunctions M µν ( λ ). While a Lorentz transformation matrix [Λ µν ] has sixteen elements, anysuch matrix is constrained by the conditionΛ αµ η αβ Λ βν = η µν , (8)where [ η µν ] = diag( − , , ,
1) is the flat Minkowski space metric tensor. This equation issymmetric, so it introduces ten constraints, leaving only six actual degrees of freedom fora Lorentz transformation. The Lorentz group is thus six dimensional, as indicated by thenotation D in the path integral.To further deduce the form of ∆[ q, M ], consider a family of particle paths q x , Λ , M x , Λ ,indexed by the starting configuration ( x , Λ ), such that q x , Λ ( λ ) = x + Λ ˜ q ( λ ) and M x , Λ ( λ ) = Λ ˜ M ( λ ) , q ( λ ) = 0 and ˜ M ( λ ) = I . These paths are constructed by, in effect, applying thePoincar´e transformation given by ( x , Λ ) to the specific functions ˜ q and ˜ M defining thefamily. (Note how the ability to do this depends on the particle configuration space beingthe same as the Poincar´e transformation group.)Consider, though, that the particle propagation embodied in ∆[ q, M ] must be Poincar´einvariant. That is, ∆[ q ′ , M ′ ] = ∆[ q, M ] for any q ′ , M ′ related to q, M by a fixed Poincar´etransformation. Thus, all members of the family q x , Λ , M x , Λ , which are all related to ˜ q. ˜ M by Poincar´e transformations, must have the same amplitude ∆[ q x , Λ , M x , Λ ] = ∆[˜ q, ˜ M ],depending only on the functions ˜ q and ˜ M .Suppose that a probability amplitude ψ ( x , Λ ) is given for a particle to be at in an initialconfiguration ( x , Λ ) and that the transition amplitude is known to be ∆[˜ q, ˜ M ] for specificrelative configuration functions ˜ q, ˜ M . Then, the probability amplitude for the particle totraverse a specific path ( q x , Λ ( λ ) , M x , Λ ( λ )) from the family defined by the functions ˜ q, ˜ M should be just ∆[ q x , Λ , M x , Λ ] ψ ( x , Λ ) = ∆[˜ q, ˜ M ] ψ ( x , Λ ).However, the very meaning of being on a specific path is that the particle must propa-gate from the given starting configuration to the specific ending configuration of the path.Further, since the paths in the family are parallel in configuration space, the ending config-uration is uniquely determined by the starting configuration. Therefore, the probability forreaching the ending configuration must be the same as the probability for having startedout at the given initial configuration ( x , Λ ). That is, | ∆[˜ q, ˜ M ] ψ ( x , Λ ) | = | ψ ( x , Λ ) | . But, since ∆[˜ q, ˜ M ] is independent of x and Λ , we must have | ∆[ q, M ] | = 1 in general.This argument therefore suggests the following postulate. Postulate 2.
For any path ( q ( λ ) , M ( λ )) , the transition amplitude ∆[ q, M ] preserves theprobability density for the particle along the path. That is, it satisfies | ∆[ q, M ] | = 1 . (9)The requirements of Eq. (9) and Poincar´e invariance mean that ∆[ q, M ] must have theexponential form ∆[ q, M ] = e i S [˜ q, ˜ M ] , (10)8or some phase functional S of the relative path functions˜ q ( λ ) ≡ M ( λ ) − ( q ( λ ) − q ( λ )) and ˜ M ( λ ) ≡ M ( λ ) − M ( λ ) . As discussed in [1], we are actually justified in replacing these relative functions withpath derivatives under the path integral, even though the path functions q ( λ ) and M ( λ )may not themselves be differentiable in general. This is because a path integral is defined asthe limit of discretized approximations in which path derivatives are approximated as meanvalues, and the limit is then taken over the path integral as a whole, not each derivativeindividually. Thus, even though the individual path derivative limits may not be defined,the path integral has a well-defined value so long as the overall path integral limit is defined.However, the quantities ˜ q and ˜ M are expressed in a frame that varies with the M ( λ ) ofthe specific path under consideration. We wish instead to construct differentials in the fixed“laboratory” frame of the q ( λ ). Transforming ˜ q and ˜ M to this frame gives M ( λ )˜ q ( λ ) = q ( λ ) − q ( λ ) and M ( λ ) ˜ M ( λ ) M ( λ ) − = M ( λ ) M ( λ ) − . Clearly, the corresponding derivative for q is simply ˙ q ( λ ) ≡ d q/ d λ , which is the tangentvector to the path q ( λ ). The derivative for M needs to be treated a little more carefully. Sincethe Lorentz group is a Lie group (that is, a continuous, differentiable group), the tangentto a path M ( λ ) in the Lorentz group space is given by an element of the corresponding Lie algebra [37, 38]. For the Lorentz group, the proper such tangent is given by the matrixΩ( λ ) = ˙ M ( λ ) M ( λ ) − , where ˙ M ( λ ) ≡ d M/ d λ .Together, the quantities ( ˙ q, Ω) form a tangent along the path in the full Poincar´e groupspace. We can then take the arguments of the phase functional in Eq. (10) to be ( ˙ q, Ω).Substituting this into Eq. (7) gives∆( x − x , ΛΛ − ; λ − λ ) = ζ Z D q Z D Mδ ( q ( λ ) − x ) δ ( M ( λ )Λ − − I ) δ ( q ( λ ) − x ) δ ( M ( λ )Λ − − I )e i S [ ˙ q, Ω] , (11)which reflects the typical form of a Feynman sum over paths.Now, by dividing a path ( q ( λ ) , M ( λ )) into two paths at some arbitrary parameter value λ and propagating over each segment, one can see that S [ ˙ q, Ω; λ , λ ] = S [ ˙ q, Ω; λ , λ ] + S [ ˙ q, Ω; λ, λ ] , (12)9here S [ ˙ q, Ω; λ ′ , λ ] denotes the value of S [ ˙ q, Ω] for the path parameter range restricted to[ λ, λ ′ ]. Using this property to build the total value of S [ ˙ q, Ω] from infinitesimal incrementsleads to the following result (whose full proof is a straightforward generalization of the proofgiven in [1] for the scalar case).
Proposition A (Form of the Phase Functional) . The phase functional S must have theform S [ ˙ q, Ω] = Z λ λ d λ ′ L [ ˙ q, Ω; λ ′ ] , where the parametrization domain is [ λ , λ ] and L [ ˙ q, Ω; λ ] depends only on ˙ q , Ω and theirhigher derivatives evaluated at λ . Clearly, the functional L [ ˙ q, Ω; λ ] plays the traditional role of the Lagrangian. The simplestnon-trivial form for this functional would be for it to depend only on ˙ q and Ω and no higherderivatives. Further, suppose that it separates into uncoupled parts dependent on ˙ q and Ω: L [ ˙ q, Ω; λ ] = L q [ ˙ q ; λ ] + L M [Ω; λ ] . The path integral of Eq. (11) then factors into independent parts in q and M , such that∆( x − x , ΛΛ − ; λ − λ ) = ∆( x − x ; λ − λ )∆(ΛΛ − ; λ − λ ) . (13)If we take L q to have the classical Lagrangian form L q [ ˙ q ; λ ] = L q ( ˙ q ( λ )) = 14 ˙ q ( λ ) − m , for a particle of mass m , then the path integral in q can be evaluated to give [1, 3]∆( x − x ; λ − λ ) = (2 π ) − Z d p e ip · ( x − x ) e − i( λ − λ )( p + m ) . (14)Similarly, take L M to be a Lorentz-invariant scalar function of Ω( λ ). Ω is an antisymmetricmatrix (this can be shown by differentiating the constraint Eq. (8)), so the scalar tr(Ω) =Ω µµ = 0. The next simplest choice is L M [Ω; λ ] = L M (Ω( λ )) = 12 tr(Ω( λ )Ω( λ ) T ) = 12 Ω µν ( λ )Ω µν ( λ ) . Postulate 3.
For a free non-scalar particle of mass m , the Lagrangian function is given by L ( ˙ q, Ω) = L q ( ˙ q ) + L M (Ω) , here L q ( ˙ q ) = 14 ˙ q − m and L M (Ω) = 12 tr(ΩΩ T ) . Evaluating the path integral in M is complicated by the fact that the Lorentz groupis not compact , and integration over the group is not, in general, bounded. The Lorentzgroup is denoted SO (3 ,
1) for the three plus and one minus sign of the Minkowski metric η in the defining pseudo-orthogonality condition Eq. (8). It is the minus sign on the timecomponent of η that leads to the characteristic Lorentz boosts of special relativity. But sincesuch boosts are parametrized by the boost velocity, integration of this sector of the Lorentzgroup is unbounded. This is in contrast to the three dimensional rotation subgroup SO (3)for the Lorentz, which is parameterized by rotation angles that are bounded.To avoid this problem, we will Wick rotate [39] the time axis in complex space. Thisreplaces the physical t coordinate with i t , turning the minus sign in the metric to a plus sign,resulting in the normal Euclidean metric diag(1 , , , SO (4) ofrotations in four-dimensional Euclidean space. The group SO (4) is compact, and the pathintegration over SO (4) can be done [20].Rather than dividing into boost and rotational parts, like the Lorentz group, SO (4)instead divides into two SO (3) subgroups of rotations in three dimensions. Actually, ratherthan SO (3) itself, it is more useful to consider its universal covering group SU (2), the groupof two-dimensional unitary matrices, because SU (2) allows for representations with half-integral spin [38, 40, 41]. (The covering group SU (2) × SU (2) for SO (4) in Euclidean spacecorresponds to the covering group SL (2 , C ) of two-dimensional complex matrices for theLorentz group SO (3 ,
1) in Minkowski space.)Typically, Wick rotations have been used to simplify the evaluation of path integralsparametrized in time, like the non-relativistic integral of Eq. (1). In this case, replacing t byi t results in the exponent in the integrand of the path integral to become real. Unlike thiscase, the exponent in the integrand of a spacetime path integral remains imaginary, sincethe Wick rotation does not affect the path parameter λ . Nevertheless, the path integral canbe evaluated, giving the following result (proved in the Appendix).11 roposition B (Evaluation of the SO (4) Path Integral) . Consider the path integral ∆(Λ E Λ E − ; λ − λ ) = η E Z D M E δ ( M E ( λ )Λ − E − I ) δ ( M E ( λ )Λ E − − I )exp (cid:20) i Z λλ d λ ′
12 tr(Ω E ( λ ′ )Ω E ( λ ′ ) T ) (cid:21) (15) over the six dimensional group SO (4) ∼ SU (2) × SU (2) , where Ω E ( λ ′ ) is the element of theLie algebra so (4) tangent to the path M E ( λ ) at λ ′ . This path integral may be evaluated toget ∆(Λ E Λ E − ; λ − λ )= X ℓ A ,ℓ B exp − i(∆ m ℓA +∆ m ℓB )( λ − λ ) (2 ℓ A + 1)(2 ℓ B + 1) χ ( ℓ A ℓ B ) (Λ E Λ E − ) , (16) where the summation over ℓ A and ℓ B is from to ∞ in steps of / , ∆ m ℓ = ℓ ( ℓ + 1) and χ ( ℓ A ,ℓ B ) is the group character for the ( ℓ A , ℓ B ) SU (2) × SU (2) group representation. The result of Eq. (16) is in terms of the representations of the covering group SU (2) × SU (2). A (matrix) representation L of a group assigns to each group element g a matrix D ( L ) ( g ) that respects the group operation, that is, such that D ( L ) ( g g ) = D ( L ) ( g ) D ( L ) ( g ).The character function χ ( L ) for the representation L of a group is a function from the groupto the reals such that χ ( L ) ( g ) ≡ tr( D ( L ) ( g )) . The group SU (2) has the well known spin representations , labeled by spins ℓ =0 , / , , / , . . . [40, 41] (for example, spin 0 is the trivial scalar representation, spin 1/2 isthe spinor representation and spin 1 is the vector representation). A ( ℓ A , ℓ B ) representationof SU (2) × SU (2) then corresponds to a spin- ℓ A representation for the first SU (2) componentand a spin- ℓ B representation for the second SU (2) component.Of course, it is not immediately clear that this result for SO (4) applies directly to SO (3 , SO (4) and SO (3 ,
1) (in which an odd number, three, of the sixgenerators of SO (4) are multiplied by i to get the boost generators for SO (3 , SO (4) and SO (3 ,
1) both have compact SO (3) subgroups, which are iso-morphic. Therefore, the restriction of the SO (4) propagator to its SO (3) subgroup shouldcorrespond to the restriction of the SO (3 ,
1) propagator to its SO (3) subgroup. This willprove sufficient for our purposes. In the next section, we will continue to freely work withthe Wick rotated Euclidean space and the SO (4) propagator as necessary. To show clearlywhen this is being done, quantities effected by Wick rotation will be given a subscript E , asin Eq. (16). IV. THE EUCLIDEAN PROPAGATOR
For a scalar particle, one can define the probability amplitude ψ ( x ; λ ) for the particleto be at position x at the point λ in its path [1, 27, 28]. For a non-scalar particle, thiscan be extended to a probability amplitude ψ ( x, Λ; λ ) for the particle to be in the Poincar´econfiguration ( x, Λ), at the point λ in its path. The transition amplitude given in Eq. (7)acts as a propagation kernel for ψ ( x, Λ; λ ): ψ ( x, Λ; λ ) = Z d x Z d Λ ∆( x − x , ΛΛ − ; λ − λ ) ψ ( x , Λ ; λ ) . The Euclidean version of this equation has an identical form, but in terms of Euclideanconfiguration space quantities: ψ ( x E , Λ E ; λ ) = Z d x E Z d Λ E ∆( x E − x E , Λ E Λ E − ; λ − λ ) ψ ( x E , Λ E ; λ ) . (17)Using Eq. (13), substitute into Eq. (17) the Euclidean scalar kernel (as in Eq. (14), butwith a leading factor of i) and the SO (4) kernel (Eq. (16)), giving ψ ( x E , Λ E ; λ ) = X ℓ A ,ℓ B Z d x E Z d Λ E ∆ ( ℓ A ,ℓ B ) ( x E − x E ; λ − λ ) χ ( ℓ A ,ℓ B ) (Λ E Λ E − ) ψ ( x E , Λ E ; λ ) , (18)where∆ ( ℓ A ,ℓ B ) ( x E − x E ; λ − λ ) ≡ i(2 π ) − Z d p E e ip E · ( x E − x E ) e − i( λ − λ )( p E + m +∆ m A +∆ m B ) . Since the group characters provide a complete set of orthogonal functions [40], the function ψ ( x E , Λ E ; λ ) can be expanded as ψ ( x E , Λ E ; λ ) = X ℓ A ,ℓ B χ ( ℓ A ,ℓ B ) (Λ E ) ψ ( ℓ A ,ℓ B ) ( x E ; λ ) . χ ( ℓ A ,ℓ B ) (Λ E ) = Z d Λ E χ ( ℓ A ,ℓ B ) (Λ E Λ E − ) χ ( ℓ A ,ℓ B ) (Λ E )(see [40]) gives ψ ( x E , Λ E ; λ ) = X ℓ A ,ℓ B χ ( ℓ A ,ℓ B ) (Λ E ) ψ ( ℓ A ,ℓ B ) ( x E ; λ ) , where ψ ( ℓ A ,ℓ B ) ( x E ; λ ) = Z d x E ∆ ( ℓ A ,ℓ B ) ( x E − x E ; λ − λ ) ψ ( ℓ A ,ℓ B ) ( x E ; λ ) . (19)The general amplitude ψ ( x E , Λ E ; λ ) can thus be expanded into a sum of terms in thevarious SU (2) × SU (2) representations, the coefficients ψ ( ℓ A ,ℓ B ) ( x E ; λ ) of which each evolveseparately according to Eq. (19). As is well known, reflection symmetry requires that a realparticle amplitude must transform according to a ( ℓ, ℓ ) or ( ℓ A , ℓ B ) ⊕ ( ℓ B , ℓ A ) representation.That is, the amplitude function ψ ( x E , Λ E ; λ ) must either have the form ψ ( x E , Λ E ; λ ) = χ ( ℓ,ℓ ) (Λ E ) ψ ( ℓ,ℓ ) ( x E ; λ )or ψ ( x E , Λ E ; λ ) = χ ( ℓ A ,ℓ B ) (Λ E ) ψ ( ℓ A ,ℓ B ) ( x E ; λ ) + χ ( ℓ B ,ℓ A ) (Λ E ) ψ ( ℓ B ,ℓ A ) ( x E ; λ ) . Assuming one of the above two forms, shift the particle mass to m ′ = m + 2∆ m ℓ or m ′ = m + 2∆ m ℓ A + 2∆ m ℓ B , so that ψ ( x E , Λ E ; λ ) = Z d x , Z d Λ χ ( L ) (Λ E Λ E − )∆( x E − x E ; λ − λ ) ψ ( x E , Λ E ; λ ) , where ∆ here is (the Euclidean version of) the scalar propagator of Eq. (14), but now forthe shifted mass m ′ , and ( L ) is either ( ℓ, ℓ ) or ( ℓ A , ℓ B ). That is, the full kernel must havethe form ∆ ( L ) ( x E − x E , Λ E Λ E − ; λ − λ ) = χ ( L ) (Λ E Λ E − )∆( x E − x E ; λ − λ ) . (20)As is conventional, from now on we will use four-dimensional spinor indices for the(1 / , ⊕ (0 , /
2) representation and vector indices (also four dimensional) for the (1 , SU (2) × SU (2) indices ( ℓ A , ℓ B ) (see, for example, [41]). Let D l ′ l (Λ E ) be a matrix representation of the SO (4) group using such indices. Define corre-spondingly indexed amplitude functions by ψ l ′ l ( x E ; λ ) ≡ Z d Λ E D l ′ l (Λ E ) ψ ( x E , Λ E ; λ ) (21)14note the double indexing of ψ here).These ψ l ′ l are the elements of an algebra over the SO (4) group for which, given x E and λ , the ψ ( x E , Λ E ; λ ) are the components , “indexed” by the group elements Λ E (see SectionIII.13 of [40]). The product of two such algebra elements is (with summation implied overrepeated up and down indices) ψ l ′ ¯ l ( x E ; λ ) ψ ll ( x E ; λ ) = Z d Λ E Z d Λ E D l ′ ¯ l (Λ E ) D ¯ ll (Λ E ) ψ ( x E , Λ E ; λ ) ψ ( x E , Λ E ; λ )= Z d Λ E D l ′ l (Λ E ) Z d Λ E ψ ( x E , Λ E ; λ ) ψ ( x E , Λ E − Λ E ; λ )= ( ψ ψ ) l ′ l ( x E ; λ ) , where the second equality follows after setting Λ E = Λ E − Λ E from the invariance of theintegration measure of a Lie group (see, for example, [37], Section 4.11, and [40], SectionIII.12—this property will be used regularly in the following), and the product components( ψ ψ )( x E , Λ E ; λ ) are defined to be( ψ ψ )( x E , Λ E ; λ ) ≡ Z d Λ ′ E ψ ( x E , Λ ′ E ; λ ) ψ ( x E , Λ ′− E Λ E ; λ ) . Now substitute Eq. (17) into Eq. (21) to get ψ l ′ l ( x E ; λ ) = Z d Λ E Z d x E Z d Λ E D l ′ l (Λ E )∆( x E − x E , Λ E Λ E − ; λ − λ ) ψ ( x E , Λ E ; λ ) . Changing variables Λ E → Λ ′ E Λ E then gives ψ l ′ l ( x E ; λ ) = Z d x (cid:20)Z d Λ ′ E D l ′ ¯ l (Λ ′ E )∆( x E − x , Λ ′ E ; λ − λ ) (cid:21)Z d Λ E D ¯ ll (Λ E ) ψ ( x E , Λ E ; λ )= Z d x ∆ l ′ ¯ l ( x E − x ; λ − λ ) ψ ¯ ll ( x ; λ ) , where the kernel for the algebra elements ψ l ′ l ( x E ; λ ) is thus∆ l ′ l ( x E − x E ; λ − λ ) = Z d Λ E D l ′ l (Λ E )∆( x E − x E , Λ E ; λ − λ ) . Substituting Eq. (20) into this, and using the definition of the character for a specific rep-resentation, χ (Λ E ) ≡ tr( D (Λ E )), gives∆ l ′ l ( x E − x E ; λ − λ ) = (cid:20)Z d Λ E D l ′ l (Λ E ) D ¯ l ¯ l (Λ E ) (cid:21) ∆( x E − x E ; λ − λ ) . Z d Λ E D l ′ l (Λ E ) D ¯ l ′ ¯ l (Λ E ) = δ l ′ ¯ l ′ δ l ¯ l , where the SO (4) integration measure has been normalized so that R d Λ E = 1 (see [40],Section 11), to get ∆ l ′ l ( x E − x E ; λ − λ ) = δ l ′ l ∆( x E − x E ; λ − λ ) . (22)The SO (4) group propagator is thus simply δ l ′ l . As expected, this does not have thesame form as would be expected for the SO (3 ,
1) Lorentz group propagator. However, asargued at the end of Sec. III, the propagator restricted to the compact SO (3) subgroup of SO (3 , is expected to have the same form as for the SO (3) subgroup of SO (4). So weturn now to the reduction of SO (3 ,
1) to SO (3). V. SPIN
In traditional relativistic quantum mechanics, the Lorentz-group dependence of non-scalar states is reduced to a rotation representation that is amenable to interpretation asthe intrinsic particle spin. Since, in the usual approach, physical states are considered tohave on-shell momentum, it is natural to use the 3-momentum as the vector around whichthe spin representation is induced, using Wigner’s classic “little group” argument [22].However, in the spacetime path approach used here, the fundamental states are notnaturally on-shell, rather the on-shell states are given as the time limits of off-shell states[1]. Further, there are well-known issues with the localization of on-shell momentum states[43, 44]. Therefore, instead of assuming on-shell states to start, we will adopt the approachof [23, 24], in which the spin representation is induced about an arbitrary timelike vector.This will allow for a straightforward generalization of the interpretation obtained in thespacetime path formalism for the scalar case [1].First, define the probability amplitudes ψ l ′ l ( x ; λ ) for a given Lorentz group representationsimilarly to the correspondingly indexed amplitudes for SO (4) representations from Sec. IV.Corresponding to such amplitudes, define a set of ket vectors | ψ i l , with a single Lorentz-group representation index. The | ψ i l define a vector bundle (see, for example, [38]), of thesame dimension as the Lorentz-group representation, over the scalar-state Hilbert space.16he basis position states for this vector bundle then have the form | x ; λ i l , such that ψ l ′ l ( x ; λ ) = G l ′ ¯ l ¯ l h x ; λ | ψ i l , with summation assumed over repeated upper and lower indices and G being the invariantmatrix of a given Lorentz group representation such that D † GD = DGD † = G , for any member D of the representation, where D † is the Hermitian transpose of the matrix D . For the scalar representation, G is 1, for the (Weyl) spinor representation it is the Diracmatrix β and for the vector representation it is the Minkowski metric η .In the following, G will be used (usually implicitly) to “raise” and “lower” group repre-sentation indices. For instance, l ′ h x ; λ | ≡ G l ′ l l h x ; λ | , so that ψ l ′ l ( x ; λ ) = l ′ h x ; λ | ψ i l . (23)The states | x ; λ i l are then normalized so that l ′ h x ′ ; λ | x ; λ i l = δ l ′ l δ ( x ′ − x ) , (24)that is, they are orthogonal at equal λ .Consider an arbitrary Lorentz transformation M . Since ψ ( x, Λ; λ ) is a scalar, it shouldtransform as ψ ′ ( x ′ , Λ ′ ; λ ) = ψ ( M − x ′ , M − Λ ′ ; λ ). In terms of algebra elements, ψ ′ l ′ l ( x ′ ; λ ) = Z d Λ ′ D l ′ l (Λ ′ ) ψ ( M − x ′ , M − Λ ′ ; λ )= Z d Λ D l ′ ¯ l ′ ( M ) D ¯ l ′ l (Λ) ψ ( M − x ′ , Λ; λ )= D l ′ ¯ l ′ ( M ) ψ ¯ l ′ l ( M − x ; λ ) . (25)Let ˆ U (Λ) denote the unitary operator on Hilbert space corresponding to the Lorentztransformation Λ. Then, from Eq. (23), ψ ′ l ′ l ( x ′ ; λ ) = l ′ h x ′ ; λ | ψ ′ i l = l ′ h x ′ ; λ | ˆ U (Λ) | ψ i l . U (Λ) − | x ′ ; λ i l = | Λ − x ′ ; λ i l ′ [ D (Λ) − ] l ′ l , or ˆ U (Λ) | x ; λ i l = | Λ x ; λ i l ′ D l ′ l (Λ) . (26)Thus, the | x ; λ i l are localized position states that transform according to a representationof the Lorentz group.Now, for any future-pointing, timelike, unit vector n ( n = − n >
0) define thestandard Lorentz transformation L ( n ) ≡ R ( n ) B ( | n | ) R − ( n ) , where R ( n ) is a rotation that takes the z -axis into the direction of n and B ( | n | ) is a boostof velocity | n | in the z direction. Then n = L ( n ) e , where e ≡ (1 , , , Wigner rotation for n and an arbitrary Lorentz transformation Λ to be W (Λ , n ) ≡ L (Λ n ) − Λ L ( n ) , (27)such that W (Λ , n ) e = e . That is, W (Λ , n ) is a member of the little group of transformationsthat leave e invariant. Since e is along the time axis, its little group is simply the rotationgroup SO (3) of the three space axes.Substituting the transformationΛ = L (Λ n ) W (Λ , n ) L ( n ) − , into Eq. (26) gives ˆ U (Λ) | x ; λ i l = | Λ x ; λ i l ′ (cid:2) D (cid:0) L (Λ n ) W (Λ , n ) L ( n ) − (cid:1)(cid:3) l ′ l . Defining | x, n ; λ i ( W ) l ≡ | x ; λ i l ′ [ L ( n )] l ′ l , (28)where L ( n ) ≡ D ( L ( n )), we see that | x, n ; λ i ( W ) l transforms under ˆ U (Λ) asˆ U (Λ) | x, n ; λ i ( W ) l = | Λ x, Λ n ; λ i ( W ) l ′ [ D ( W (Λ , n ))] l ′ l , (29)that is, according to the Lorentz representation subgroup given by D ( W (Λ , n )), which isisomorphic to some representation of the rotation group.18he irreducible representations of the rotation group (or, more exactly, its covering group SU (2)) are just the spin representations, with members given by matrices D σ ′ σ , where the σ are spin indices. Let | ψ i σ be a member of a Hilbert space vector bundle indexed by spinindices. Then there is a linear, surjective mapping from | ψ i l to | ψ i σ given by | ψ i σ = | ψ i l u lσ , where ( u lσ ′ ) ∗ u lσ = δ σ ′ σ . (30)The isomorphism between the rotation subgroup of the Lorentz group and the rotation groupthen implies that, for any rotation W , for all | ψ i l , | ψ i l ′ u l ′ σ ′ [ D ( W )] σ ′ σ = | ψ i l ′ [ D ( W )] l ′ l u lσ (with summation implied over repeated σ indices, as well as l indices) or u l ′ σ ′ [ D ( W )] σ ′ σ = [ D ( W )] l ′ l u lσ , (31)where D ( W ) is the spin representation matrix corresponding to W .Define | x, n ; λ i σ ≡ | x, n ; λ i ( W ) l u lσ . (32)Substituting from Eq. (28) gives | x, n ; λ i σ = | x ; λ i l u lσ ( n ) . (33)where u lσ ( n ) ≡ [ L ( n )] ll ′ u l ′ σ . (34)Then, under a Lorentz transformation Λ, using Eqs. (29) and (31),ˆ U (Λ) | x, n ; λ i σ = | Λ x, Λ n ; λ i ( W ) l ′ [ D ( W (Λ , n ))] l ′ l u lσ = | Λ x, Λ n ; λ i ( W ) l ′ u l ′ σ ′ [ D ( W (Λ , n ))] σ ′ σ = | Λ x, Λ n ; λ i σ ′ [ D ( W (Λ , n ))] σ ′ σ , that is, | x, n ; λ i σ transforms according to the appropriate spin representation.Now consider a past-pointing n ( n = − n < − n is future pointingso that − n = L ( − n ) e , or n = L ( − n )( − e ). Taking L ( − n ) to be the standard Lorentz19ransformation for past-pointing n , it is thus possible to construct spin states in terms ofthe future-pointing − n . However, since the spacial part of n is also reversed in − n , it isconventional to consider the spin sense reversed, too. Therefore, define v lσ ( n ) ≡ ( − j + σ u l − σ ( − n ) , (35)for a spin- j representation, and, for past-pointing n , take | x, n ; λ i σ = | x ; λ i l v lσ ( n ) . The matrices u lσ and v lσ are the same as the spin coefficient functions in Weinberg’sformalism in the context of traditional field theory [12] (see also Chapter 5 of [41]). Notethat, from Eq. (31), using Eq. (27), u l ′ σ ′ [ D ( W (Λ , n ))] σ ′ σ = [ D ( W (Λ , n ))] l ′ l u lσ = [ L (Λ n ) − D (Λ) L ( n )] l ′ l u lσ , so, using Eq. (34), u lσ ′ (Λ n )[ D ( W (Λ , n ))] σ ′ σ = [ D (Λ)] l ′ l u lσ ( n ) . (36)Using this with Eq. (35) gives[ D (Λ)] l ′ l v lσ ( n ) = ( − σ − σ ′ v l ′ σ ′ (Λ n )[ D ( W (Λ , n ))] − σ ′ − σ . Since ( − σ − σ ′ D ( W ) − σ ′ − σ = [ D ( W ) σ ′ σ ] ∗ (which can be derived by integrating the infinitesimal case), this gives, v l ′ σ ′ (Λ n )[ D ( W (Λ , n )) σ ′ σ ] ∗ = [ D (Λ)] l ′ l v lσ ( n ) . (37)As shown by Weinberg [12, 41], Eqs. (36) and (37) can be used to completely determine the u and v matrices, along with the usual relationship of the Lorentz group scalar, spinor andvector representations to the rotation group spin-0, spin-1/2 and spin-1 representations.Since, from Eqs. (30) and (34), u lσ ′ ( n ) ∗ u lσ ( n ) = [ L ( n ) l ¯ l ′ ] ∗ ( u ¯ l ′ σ ′ ) ∗ [ L ( n )] l ¯ l u ¯ lσ = ( u ¯ l ′ σ ′ ) ∗ [ L ( n ) − ] ¯ l ′ l [ L ( n )] l ¯ l u ¯ lσ = ( u ¯ lσ ′ ) ∗ u ¯ lσ = δ σ ′ σ , σ ′ h x ′ , n ; λ | x, n ; λ i σ = δ σ ′ σ δ ( x ′ − x ) (38)(and similarly for past-pointing n with v lσ ), so that, for given n and λ , the | x, n ; λ i σ forman orthogonal basis. However, for different λ , the inner product is σ ′ h x, n ; λ | x , n ; λ i σ = ∆ σ ′ σ ( x − x ; λ − λ ) , (39)where ∆ σ ′ σ ( x − x ; λ − λ ) is the kernel for the rotation group. As previously argued, thisshould have the same form as the Euclidean kernel of Eq. (22), restricted to the rotationsubgroup of SO (4). That is∆ σ ′ σ ( x − x ; λ − λ ) = δ σ ′ σ ∆( x − x ; λ − λ ) . (40)As in Eq. (6), the propagator is given by integrating the kernel over λ :∆ σ ′ σ ( x − x ) = δ σ ′ σ ∆( x − x ) , where (using Eq. (14))∆( x − x ) = Z ∞ λ d λ ∆( x − x ; λ − λ ) = − i(2 π ) − Z d p e i p · ( x − x ) p + m − i ǫ , the usual Feynman propagator [1]. Defining | x, n i σ ≡ Z ∞ λ d λ | x, n ; λ i σ then gives σ ′ h x, n | x , n ; λ i σ = ∆ σ ′ σ ( x − x ) . (41)Finally, we can inject the spin-representation basis states | x, n ; λ i σ back into the Lorentzgroup representation by | x, n ; λ i l ≡ | x, n ; λ i σ u lσ ( n ) ∗ , (and similarly for past-pointing n with v lσ ). Substituting Eq. (33) into this gives | x, n ; λ i l = | x ; λ i l ′ P l ′ l ( n ) , (42)where P l ′ l ( n ) ≡ u l ′ σ ( n ) u lσ ( n ) ∗ = v lσ ( n ) v lσ ( n ) ∗ (43)21the last equality following from Eq. (35)). Using Eqs. (38) and (39), the kernel for thesestates is l ′ h x, n ; λ | x , n ; λ i l = P l ′ l ( n )∆( x − x ; λ − λ ) . However, using Eqs. (36) and (37), it can be shown that the | x, n ; λ i l transform like the | x ; λ i l : ˆ U (Λ) | x, n ; λ i l = | Λ x, Λ n ; λ i l ′ D l ′ l ( λ ) . Taking | x, n i l ≡ Z ∞ λ d λ | x, n ; λ i l and using Eq. (41) gives the propagator l ′ h x, n | x , n ; λ i l = P l ′ l ( n )∆( x − x ) . (44)Now, the | x, n ; λ i l do not span the full Lorentz group Hilbert space vector bundle of the | x ; λ i l , but they do span the subspace corresponding to the rotation subgroup. Therefore,using Eq. (42) and the idempotency of P l ′ l ( n ) as a projection matrix, | x, n i l = Z d x l ′ h x , n ; λ | x, n i l | x , n ; λ i l ′ = Z d x P l ′ l ( n )∆( x − x ) ∗ P ¯ l ′ l ′ ( n ) | x ; λ i ¯ l ′ = Z d x P l ′ l ( n )∆( x − x ) ∗ | x ; λ i l ′ . (45) VI. PARTICLES AND ANTIPARTICLES
Because of Eq. (41), the states | x, n i σ allow for a straightforward generalization of thetreatment of particles and antiparticles from [1] to the non-scalar case. As in that treatment,consider particles to propagate from the past to the future while antiparticles propagate fromthe future into the past [27, 28, 45]. Therefore, postulate non-scalar particle states | x + , n i σ and antiparticle states | x − , n i σ as follows. Postulate 4.
Normal particle states | x + , n i σ are such that σ ′ h x + , n | x , n ; λ i σ = θ ( x − x )∆ σ ′ σ ( x − x ) = θ ( x − x )∆ + σ ′ σ ( x − x ) , and antiparticle states | x − , n i σ are such that σ ′ h x − , n | x , n ; λ i σ = θ ( x − x )∆ σ ′ σ ( x − x ) = θ ( x − x )∆ − σ ′ σ ( x − x ) , here θ is the Heaviside step function, θ ( x ) = 0 , for x < , and θ ( x ) = 1 , for x > , and ∆ ± σ ′ σ ( x − x ) = δ σ ′ σ (2 π ) − Z d p (2 ω p ) − e i[ ∓ ω p ( x − x )+ p · ( x − x )] , with ω p ≡ p p + m . Note that the vector n used here is timelike but otherwise arbitrary, with no commitmentthat it be, e.g., future-pointing for particles and past-pointing for antiparticles.This division into particle and antiparticle paths depends, of course, on the choice of aspecific coordinate system in which to define the time coordinate. However, if we take thetime limit of the end point of the path to infinity for particles and negative infinity for an-tiparticles, then the particle/antiparticle distinction will be coordinate system independent.In taking this time limit, one cannot expect to hold the 3-position of the path end pointconstant. However, for a free particle, it is reasonable to take the particle asbeing fixed. Therefore, consider the state of a particle or antiparticle with a 3-momentum p at a certain time t . Postulate 5.
The state of a particle ( + ) or antiparticle ( − ) with 3-momentum p is givenby | t, p ± , n i σ ≡ (2 π ) − / Z d x e i( ∓ ω p t + p · x ) | t, x ± , n i σ . Now, following the derivation in [1], but carrying along the spin indices, gives | t, p + , n i σ = (2 ω p ) − Z t −∞ d t | t , p + , n ; λ i σ and | t, p − , n i σ = (2 ω p ) − Z + ∞ t d t | t , p − , n ; λ i σ , (46)where | t, p ± , n ; λ i σ ≡ (2 π ) − / Z d x e i( ∓ ω p t + p · x ) | t, x , n ; λ i σ . (47)Since σ ′ h t ′ , p ′± , n ; λ | t, p ± , n ; λ i σ = δ σ ′ σ δ ( t ′ − t ) δ ( p ′ − p ) , we have, from Eq. (46), σ ′ h t, p ± , n | t , p ± , n ; λ i σ = (2 ω p ) − δ σ ′ σ θ ( ± ( t − t )) δ ( p − p ) . Defining the time limit particle and antiparticle states | p ± , n i σ ≡ lim t →±∞ | t, p ± , n i σ , (48)23hen gives σ ′ h p ± , n | t , p , n ± ; λ i σ = (2 ω p ) − δ σ ′ σ δ ( p − p ) , (49)for any value of t .Further, writing | t , p ± , n ; λ i σ = (2 π ) − / e ∓ i ω p t Z d p e i p t | p, n ; λ i σ , where | p, n ; λ i σ ≡ (2 π ) − Z d x e i p · x | x, n ; λ i σ (50)is the corresponding 4-momentum state, it is straightforward to see from Eq. (46) that thetime limit of Eq. (48) is | p ± , n i σ ≡ lim t →±∞ | t, p ± , n i σ = (2 π ) / (2 ω p ) − | ± ω p , p ± , n ; λ i σ . (51)Thus, a normal particle (+) or antiparticle ( − ) that has 3-momentum p as t → ±∞ is on-shell , with energy ± ω p . Such on-shell particles are unambiguously normal particles orantiparticles.For the on-shell states | p ± , n i σ , it now becomes reasonable to introduce the usualconvention of taking the on-shell momentum vector as the spin vector. That is, set n p ± ≡ ( ± ω p , p ) /m and define | p ± ) σ ≡ | p ± , n p ± i σ and | t, p ± ) σ ≡ | t, p ± , n p ± i σ , so that | p ± ) σ = lim t →±∞ | t, p ± ) σ . Further, define the position states | x + ) l ≡ (2 π ) − / Z d p e i( ω p x − p · x ) | x , p + ) σ u lσ ( n p + ) ∗ and | x − ) l ≡ (2 π ) − / Z d p e i( − ω p x − p · x ) | x , p − ) σ v lσ ( n p − ) ∗ . (52)Then, working the previous derivation backwards gives l ′ ( x ± | x ; λ i l = θ ( ± ( x − x ))∆ ± l ′ l ( x − x ) , ± l ′ l ( x − x ) ≡ (2 π ) − Z d p P l ′ l ( n p ± )(2 ω p ) − e i[ ± ω p ( x − x ) − p · ( x − x )] . Now, it is shown in [12, 41] that the covariant non-scalar propagator∆ l ′ l ( x − x ) = − i(2 π ) − Z d p P l ′ l ( p/m ) e i p · ( x − x ) p + m − i ε , in which P l ′ l ( p/m ) has the polynomial form of P l ′ l ( n ), but p is not constrained to be on-shell,can be decomposed into∆ l ′ l ( x − x ) = θ ( x − x )∆ + l ′ l ( x − x ) + θ ( x − x )∆ − l ′ l ( x − x ) + Q l ′ l (cid:18) − i ∂∂x (cid:19) i δ ( x − x ) , where the form of Q l ′ l depends on any non-linearity of P l ′ l ( p/m ) in p . Then, defining | x ) l ≡ Z d x ∆ l ′ l ( x − x ) ∗ | x ; λ i l ′ , | x + ) l and | x − ) l can be considered as a particle/antiparticle partitioning of | x ) l , in a similarway as the partitioning of | x, n i σ into | x, n + i σ and | x, n − i σ : θ ( ± ( x − x )) l ′ ( x | x ; λ i l = θ ( ± ( x − x ))∆ l ′ l ( x − x )= θ ( ± ( x − x ))∆ ± l ′ l ( x − x )= l ′ ( x ± | x ; λ i l . Because of the delta function, the term in Q l ′ l does not contribute for x = x .The states | x, n i l and | x ) l both transform according to a representation D l ′ l of the Lorentzgroup, but it is important to distinguish between them. The | x, n i l are projections back intothe Lorentz group of the states | x, n i σ defined on the rotation subgroup, in which thatsubgroup is obtained by uniformly reducing the Lorentz group about the axis given by n .The | x ) l , on the other hand, are constructed by inverse-transforming from the momentumstates | t, p ± ) σ , with each superposed state defined over a rotation subgroup reduced alonga different on-shell momentum vector.One can further highlight the relationship of the | x ) l to the momentum in the positionrepresentation by the formal equation (using Eq. (45)) | x ) l = Z d x P l ′ l (cid:18) i m − ∂∂x (cid:19) ∆( x − x ) ∗ | x ; λ i l ′ = | x, i m − ∂/∂x i l = P l ′ l (cid:18) i m − ∂∂x (cid:19) | x i l ′ . The | x ) l correspond to the position states used in traditional relativistic quantum mechanics,with associated on-shell momentum states | p ± ). However, we will see in the next section thatthe states | x, n i l provide a better basis for generalizing the scalar probability interpretationdiscussed in [1]. 25 II. ON-SHELL PROBABILITY INTERPRETATION
Similarly to the scalar case [1], let H ( j,n ) be the Hilbert space of the | x, n ; λ i σ for thespin- j representation of the rotation group and a specific timelike vector n , and let H ( j,n ) t be the subspaces spanned by the | t, x , n ; λ i σ , for each t , forming a foliation of H ( j,n ) . Now,from Eq. (47), it is clear that the particle and antiparticle 3-momentum states | t, p ± , n ; λ i σ also span H ( j,n ) t . Using these momentum bases, states in H ( j,n ) t have the form | t, ψ ± , n ; λ i σ = Z d p ψ σ ′ σ ( p ) | t, p ± , n ; λ i σ ′ , for matrix functions ψ such that tr( ψ † ψ ) is integrable. Conversely, it follows from Eq. (49)that the probability amplitude ψ σ ′ σ ( p ) is given by ψ σ ′ σ ( p ) = (2 ω p ) σ ′ h p ± , n | t, ψ ± , n ; λ i σ . (53)Let H ′ ( j,n ) t be the space of linear functions dual to H ( j,n ) t . Via Eq. (53), the bra states σ h p + | can be considered as spanning subspaces H ′ ( j,n ) ± of the H ′ ( j,n ) t , with states of the form σ h ψ ± , n | = Z d p ψ σ ′ σ ( p ) ∗ σ ′ h p ± , n | . The inner product( ψ , ψ ) ≡ σ h ψ ± , n | t, ψ ± , n ; λ i σ = Z d p ω p ψ σ ′ σ ( p ) ∗ ψ σ ′ σ ( p )gives ( ψ, ψ ) = Z d p ω p X σ ′ σ | ψ σ ′ σ ( p ) | ≥ , so that, with this inner product, the H ( j,n ) t actually are Hilbert spaces in their own right.Further, Eq. (49) is a bi-orthonormality relation with the corresponding resolution of theidentity (see [46] and App. A.8.1 of [47]) Z d p (2 ω p ) | t, p ± , n ; λ i σ σ h p ± , n | = 1 . The operator (2 ω p ) | t, p ± , n ; λ i σ σ h p , n ± | represents the quantum proposition that an on-shell, non-scalar particle or antiparticle has 3-momentum p .Like the ψ l ′ l discussed in Sec. IV for the Lorentz group, the ψ σ ′ σ form an algebra overthe rotation group with components ψ ( p , B ), where B σ ′ σ is a member of the appropriaterepresentation of the rotation group, such that ψ σ ′ σ ( p ) = Z d B B σ ′ σ ψ ( p , B ) , (54)26ith the integration taken over the 3-dimensional rotation group. Unlike the Lorentz group,however, components can also be reconstructed from the ψ σ ′ σ ( p ) by ψ ( p , B ) = β − ( B − ) σσ ′ ψ σ ′ σ ( p ) (55)where β ≡ j + 1 Z d B , for a spin- j representation, is finite because the rotation group is closed. Plugging Eq. (55)into the right side of Eq. (54) and evaluating the integral does, indeed, give ψ σ ′ σ ( p ), asrequired, because of the orthogonality property Z d B B σ ′ σ ( B − ) ¯ σ ¯ σ ′ = βδ σ ′ ¯ σ ′ δ σ ¯ σ (see [40], Section 11). We can now adjust the group volume measure d B so that β = 1.The set of all ψ ( p , B ) constructed as in Eq. (55) forms a subalgebra such that each ψ ( p , B ) is uniquely determined by the corresponding ψ σ ′ σ ( p ) (see [40], pages 167ff). Wecan then take | ψ ( p , B ) | = | ( B − ) σσ ′ ψ σ ′ σ ( p ) | to be the probability density for the particleor antiparticle to have 3-momentum p and to be rotated as given by B about the axis givenby the spacial part of the unit timelike 4-vector n . The probability density for the particleor antiparticle in 3-momentum space is Z d B | ψ ( p , B ) | = ψ σ ′ σ ( p ) ∗ ψ σ ′ σ ( p )with the normalization ( ψ, ψ ) = Z d p ω p ψ σ ′ σ ( p ) ∗ ψ σ ′ σ ( p ) = 1 . Next, consider that | t, x , n ; λ i σ is an eigenstate of the three-position operator ˆ X , repre-senting a particle localized at the three-position x at time t . From Eq. (53), and using theinverse Fourier transform of Eq. (50) with Eq. (51), its three momentum wave function is(2 ω p ) σ ′ h p ± , n | t, x ; λ i σ = (2 π ) − / δ σ ′ σ e i( ± ω p t − p · x ) . (56)This is just a plane wave, and it is an eigenfunction of the operatore ± i ω p t i ∂∂ p e ∓ i ω p t , ∂/∂ p of the three-position operator ˆ X , translated to time t .This result exactly parallels that of the scalar case [1]. Note that this is only so becauseof the use of the independent vector n for reduction to the rotation group, rather than thetraditional approach of using the three-momentum vector p . Indeed, it is not even possibleto define a spin-indexed position eigenstate in the traditional approach, because, of course,the momentum is not sharply defined for such a state [23, 24].On the other hand, consider the three-position states | x ± ) l introduced at the end ofSec. VI. Even though these are Lorentz-indexed, they only span the rotation subgroup.Therefore, we can form their three-momentum wave functions in the σ ( p ± | bases. UsingEqs. (52) and (49), (2 ω p ) σ ( p ± | x ± ) l = (2 π ) − / u lσ ( n p ) ∗ e i( ± ω p t − p · x ) . (57)At t = 0, up to normalization factors of powers of (2 ω p ), this is just the Newton-Wignerwave function for a localized particle of non-zero spin [43]. It is an eigenfunction of theposition operator represented as u l ′ σ ′ ( n p ) ∗ e i ω p t i ∂∂ p e − i ω p t u lσ ′ ( n p ) (58)for the particle case, with a similar expression using v lσ in the antiparticle case. Other thanthe time translation, this is essentially the Newton-Wigner position operator for non-zerospin [43].Note that Eq. (56) is effectively related to Eq. (57) by a generalized Foldy-Wouthuysentransformation [48, 49]. However, in the present approach it is Eq. (56) that is seen to be theprimary result, with a natural separation of particle and antiparticle states and a reasonablenon-relativistic limit, just as in the scalar case [1]. VIII. INTERACTIONS
It is now straightforward to extend the formalism to multiparticle states and introduceinteractions, quite analogously to the scalar case [1]. In order to allow for multiparticlestates with different types of particles, extend the position state of each individual particlewith a particle type index ν , such that l ′ h x ′ , ν ′ ; λ | x, ν ; λ i l = δ l ′ l δ ν ′ ν δ ( x ′ − x ) . N single particle states: | x , ν , λ ; . . . ; x N , ν N , λ N i l ··· l N ≡ ( N !) − / X perms P δ P | x P , ν P ; λ P i l P · · ·| x P N , ν P N ; λ P N i l P N , where the sum is over permutations P of 1 , . . . , N , and δ P is +1 for permutations with aneven number of interchanges of fermions and − | x , ν ; . . . ; x N , ν N i l ··· l N as similarly symmetrized/antisymme-trized products of | x i l states. Then, l ′ ··· l ′ N h x ′ , ν ′ ; . . . ; x ′ N , ν ′ N | x , ν , λ ; . . . ; x N , ν N , λ i l ··· l N = X perms P δ P N Y i =1 δ ν ′P i ν i ∆ l ′P i l i ( x ′P i − x i ) , (59)where each propagator is also implicitly a function of the mass of the appropriate type ofparticle. Note that the use of the same parameter value λ for the starting point of eachparticle path is simply a matter of convenience. The intrinsic length of each particle path isstill integrated over separately in | x , ν ; . . . ; x N , ν N i l ··· l N , which is important for obtainingthe proper particle propagator factors in Eq. (59). Nevertheless, by using λ as a commonstarting parameter, we can adopt a similar notation simplification as in [1], defining | x , ν ; . . . ; x N , ν N ; λ i l ··· l N ≡ | x , ν , λ ; . . . ; x N , ν N , λ i l ··· l N . It is also convenient to introduce the formalism of creation and annihilation fields forthese multiparticle states. Specifically, define the creation field ˆ ψ † l ( x, ν ; λ ) byˆ ψ † l ( x, ν ; λ ) | x , ν , λ ; . . . ; x N , ν N , λ N i l ··· l N = | x, ν, λ ; x , ν , λ ; . . . ; x N , ν N , λ N i l,l ··· l N , with the corresponding annihilation field ˆ ψ l ( x, ν ; λ ) having the commutation relation[ ˆ ψ l ′ ( x ′ , ν ′ ; λ ) , ˆ ψ † l ( x, ν ; λ )] ∓ = δ ν ′ ν ∆ l ′ l ( x ′ − x ; λ − λ ) , where the upper − is for bosons and the lower + is for fermions. Further defineˆ ψ l ( x, ν ) ≡ Z ∞ λ d λ ˆ ψ l ( x, ν ; λ ) , so that [ ˆ ψ l ′ ( x ′ , ν ′ ) , ˆ ψ † l ( x, ν ; λ )] ∓ = δ ν ′ ν ∆ l ′ l ( x ′ − x ) , special adjoint ˆ ψ ‡ byˆ ψ ‡ l ( x, ν ) = ˆ ψ † l ( x, ν ; λ ) and ˆ ψ ‡ l ( x, ν ; λ ) = ˆ ψ † l ( x, ν ) , (60)which allows the commutation relation to be expressed in the more symmetric form[ ˆ ψ l ′ ( x ′ , ν ′ ) , ˆ ψ ‡ l ( x, ν )] ∓ = δ ν ′ ν ∆ l ′ l ( x ′ − x ) . We can now readily generalize the postulated interaction vertex operator of [1] to thenon-scalar case.
Postulate 6.
An interaction vertex, possibly occurring at any position in spacetime, withsome number a of incoming particles and some number b of outgoing particles, is representedby the operator ˆ V ≡ g l ′ ··· l ′ a l ··· l b Z d x a Y i =1 ˆ ψ ‡ l ′ i ( x, ν ′ i ) b Y j =1 ˆ ψ l j ( x, ν j ) , (61) where the coefficients g l ′ ··· l ′ a l ··· l b represent the relative probability amplitudes of various com-binations of indices in the interaction and ˆ ψ ‡ is the special adjoint defined in Eq. (60) . Given a vertex operator defined as in Eq. (61), the interacting transition amplitude, withany number of intermediate interactions, is then G ( x ′ , ν ′ ; . . . ; x ′ N ′ , ν ′ N ′ | x , ν ; . . . ; x N , ν N ) l ′ ··· l ′ N ′ l ··· l N = l ′ ··· l ′ N ′ h x ′ , ν ′ ; . . . ; x ′ N , ν ′ N | ˆ G | x , ν ; . . . ; x N , ν N ; λ i l ··· l N , (62)where ˆ G ≡ ∞ X m =0 ( − i) m m ! ˆ V m = e − i ˆ V . Each term in this sum gives the amplitude for m interactions, represented by m applicationsof ˆ V . The ( m !) − factor accounts for all possible permutations of the m identical factors ofˆ V . Clearly, we can also construct on-shell multiparticle states | p ′ ± , ν ′ ; . . . ; p ′ N ′ ± , ν ′ N ′ i σ ′ ··· σ ′ N ′ and | t , p ± , ν ; . . . ; t N , p N ± , ν N ; λ i σ ··· σ N from the on-shell particle and antiparticle states | p ± i σ and | t, p ± ; λ i σ . Using these with the operator ˆ G : G ( p ′ ± , ν ′ ; . . . ; p ′ N ′ ± , ν ′ N ′ | p ± , ν ; . . . ; p N ± , ν N ) σ ′ ··· σ ′ N ′ σ ··· σ N ≡ " N ′ Y i =1 ω p ′ i σ ′ ··· σ ′ N ′ h p ′ ± , ν ′ ; . . . ; p ′ N ′ ± , ν ′ N ′ | ˆ G | t , p ± , ν ; . . . ; t N , p N ± , ν N ; λ i σ ··· σ N , (63)30esults in a sum of Feynman diagrams with the given momenta on external legs. Notethat use of the on-shell states requires specifically identifying external lines as particles andantiparticles. For each incoming and outgoing particle, + is chosen if it is a normal particleand − if it is an antiparticle. (Note that “incoming” and “outgoing” here are in terms ofthe path evolution parameter λ , not time.)The inner products of the on-shell states for individual incoming and outgoing particleswith the off-shell states for interaction vertices give the proper factors for the external linesof a Feynman diagram. For example, the on-shell state | p ′ + i σ is obtained in the + ∞ timelimit and thus represents a final (i.e., outgoing in time ) particle. If the external line for thisparticle starts at an interaction vertex x , then the line contributes a factor(2 ω p ′ ) σ ′ h p ′ + | x ; λ i l = (2 π ) − / e i(+ ω p ′ x − p ′ · x ) u lσ ′ ( p ′ ) ∗ . For an incoming particle on an external line ending at an interaction vertex x ′ , the factorfor this line is (assuming x ′ > t )(2 ω p ) l ′ h x ′ | t, p + ; λ i σ = (2 π ) − / e i( − ω p x ′ + p · x ′ ) u l ′ σ ( p ) . Note that this expression is independent of t , so we can take t → −∞ and treat the particleas initial (i.e., incoming in time). The factors for antiparticles are similar, but with thetime sense reversed. Thus, the effect is to remove the propagator factors from external lines,exactly in the sense of the usual LSZ reduction [50].Now, the formulation of Eq. (63) is still not that of the usual scattering matrix, sincethe incoming state involves initial particles but final antiparticles, and vice versa for theoutgoing state. To construct the usual scattering matrix, it is necessary to have multi-particle states that involve either all initial particles and antiparticles (that is, they arecomposed of individual asymptotic particle states that are all consistently for t → −∞ )or all final particles and antiparticles (with individual asymptotic states all for t → + ∞ ).The result is a formulation in terms of the more familiar scattering operator ˆ S , which canbe expanded in a Dyson series in terms of a time-dependent version ˆ V ( t ) of the interactionoperator. The procedure for doing this is exactly analogous to the scalar case. For detailssee [1]. 31 X. CONCLUSION
The extension made here of the scalar spacetime path approach [1] begins with theargument in Sec. II on the form of the path propagator based on Poincar´e invariance. Thismotivates the use of a path integral over the Poincar´e group, with both position and Lorentzgroup variables, for computation of the non-scalar propagator. Once the difficulty with thenon-compactness of the Lorentz group is overcome, the development for the non-scalar caseis remarkably parallel to the scalar case.A natural further generalization of the approach, particularly given its potential applica-tion to quantum gravity and cosmology, would be to consider paths in curved spacetime. Ofcourse, in this case it is not in general possible to construct a family of parallel paths overthe entire spacetime, as was done in Sec. III. Nevertheless, it is still possible to considerinfinitesimal variations along a path corresponding to arbitrary coordinate transformations.And one can certainly construct a family of “parallel” paths at least over any one coordinatepatch on the spacetime manifold. The implications of this for piecing together a completepath integral will be explored in future work.Another direction for generalization is to consider massless particles, leading to a com-plete spacetime path formulation for Quantum Electrodynamics. However, as has beenshown in previous work on relativistically parametrized approaches to QED (e.g., [51]), theresulting gauge symmetries need to be handled carefully. This will likely be even more so ifconsideration is further extended to non-Abelian interactions. Nevertheless, the spacetimepath approach may provide some interesting opportunities for addressing renormalizationissues in these cases [1].In any case, the present paper shows that the formalism proposed in [1] can naturallyinclude non-scalar particles. This is, of course, critical if the approach is to be given thefoundational status considered in [1] and the cosmological interpretation discussed in [2].32
PPENDIX: EVALUATION OF THE SO (4) PATH INTEGRAL
Proposition.
Consider the path integral ∆(Λ E Λ E − ; λ − λ ) = ζ E Z D M E δ ( M E ( λ )Λ − E − I ) δ ( M E ( λ )Λ E − − I )exp (cid:20) i Z λλ d λ ′
12 tr(Ω E ( λ ′ )Ω E ( λ ′ ) T ) (cid:21) over the six dimensional group SO (4) ∼ SU (2) × SU (2) , where Ω E ( λ ′ ) is the element of theLie algebra so (4) tangent to the path M E ( λ ) at λ ′ . This path integral may be evaluated toget ∆(Λ E Λ E − ; λ − λ )= X ℓ A ,ℓ B exp − i(∆ m ℓA +∆ m ℓB )( λ − λ ) (2 ℓ A + 1)(2 ℓ B + 1) χ ( ℓ A ℓ B ) (Λ E Λ E − ) , (A.1) where the summation over ℓ A and ℓ B is from to ∞ in steps of / , ∆ m ℓ = ℓ ( ℓ + 1) and χ ( ℓ A ,ℓ B ) is the group character for the ( ℓ A , ℓ B ) SU (2) × SU (2) group representation.Proof. Parametrize a group element M E by a six-vector θ such that M E = exp( X i =1 θ i J i ) , where the J i are so (4) generators for SO (4). Then tr(Ω E Ω T E ) = ˙ θ , where the dot denotesdifferentiation with respect to λ . Dividing the six generators J i into two sets of three SU (2)generators, the six-vector θ may be divided into two three-vectors θ A and θ B , parametrizingthe two SU (2) subgroups. The path integral then factors into two path integrals over SU (2):∆(Λ E Λ E − ; λ − λ )= ζ / E Z D W A δ ( W A ( λ ) B − A − I ) δ ( W A ( λ ) B − A − I ) exp (cid:20) i Z λλ d λ ′
12 ˙ θ A ) (cid:21) × ζ / E Z D W B δ ( W B ( λ ) B − B − I ) δ ( W B ( λ ) B − B − I ) exp (cid:20) i Z λλ d λ ′
12 ˙ θ B ) (cid:21) , where Λ E = B A ⊗ B B and Λ E = B A ⊗ B B .The SU (2) path integrals may be computed by expanding the exponential in group33haracters [20, 52]. The result is ζ / E Z D W δ ( W ( λ ) B − − I ) δ ( W ( λ ) B − − I ) exp (cid:20) i Z λλ d λ ′
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