Path integral quantization of a spinning particle

Following the idea of Alekseev and Shatashvili we derive the path integral quantization of a modified relativistic particle action that results in the Feynman propagator of a free field with arbitrary spin. This propagator can be associated with the Duffin, Kemmer, and Petiau (DKP) form of a free field theory. We show explicitly that the obtained DKP propagator is equivalent to the standard one, for spins 0 and 1. We argue that this equivalence holds also for higher spins.


II. THE SCALAR PATH INTEGRAL
Consider the standard, free, scalar relativistic particle moving in four-dimensional spacetime, between the spacetime point with coordinates x 1 and the one with coordinates x 2 . The transition amplitude from the initial state in| = x 1 | to the final one |out = |x 2 is given by the path integral 1 for the trajectories beginning at x 1 and ending in x 2 : where the action S is and D(N (τ )/Diff) denotes the measure on the Lagrange multiplier N up to worldline reparametrization, under which it transforms as a one-dimensional metric determinant. Now we integrate over x(τ ). In order to do that we must first rewrite it in the form that conveniently takes into account the boundary conditions: x(τ ) = x 1 + (x 2 − x 1 )τ + y(τ ) , y(0) = y(1) = 0. ( Clearly D(x(τ )) = D(y(τ )). Now we can integrate by parts the action (2), and then integrate over y(τ ) obtaining Noticing now that δ(ṗ) enforces the momenta to be τ -independent, so that D(p(τ ))δ(ṗ) = d 4 p, and that L ≡ where we added the −i term to regularize the integral, as usual. The Fourier transform of the transition amplitude (5) is the (Feynman) propagator of the quantum scalar field.
The approach outlined above cannot be directly applied to the case of fields with higher spin. For example, in the case of spin 1/2 field the propagator is the inverse of an expression linear in momentum, ( / p − m) −1 , instead of the inverse of quadratic expression, (p 2 − m 2 ) −1 , as in (5). It was not long after Dirac's formulation of a theory of spin 1/2-fields, when a similar (unified) formulation for fields of spin-0 and spin-1 has been put forward by Duffin, Kemmer, and Petiau (DKP) [10][11][12]. While the details of the DKP theory needed for our analysis will be discussed in Sec. IV, let us present here a brief introduction to this approach for the scalar (spin-0) fields.
The very reason behind the p 2 − m 2 term in the scalar field propagator is the form of scalar field equations that follow from the lagrangian In order to get the DKP propagator, inverse proportional to momentum (instead of its square), we must rewrite the lagrangian (6) in the form linear in space-time derivatives, similar to the form of Dirac lagrangian. This can be achieved by turning from the second order formulation (with second order derivatives) to the first order one, in which the field φ and its derivatives ∂φ are treated as independent field components of a multi-component field ψ = (π 0 , π 1 , π 2 , π 3 , ϕ) T .
1 In this manuscript we denote the four dimensional indices by µ = 0, 1, 2, 3,, raised and lowered by the 4D Minkowski metric ηµν = diag (1, −1, −1, −1); u · v and v 2 are shorthand for u µ vµ and vµv µ respectively, while u · v = 3 i=1 u i v i . 2 It is essential at this point that N , being the one dimensional Euclidean metric is positive. See [9] for the recent detailed discussion of this issue.
For a real scalar field ϕ † = ϕ the DKP lagrangian takes the form 3 where β µ are the so-called DKP β-matrices, playing, for the spin 0 and spin 1 theories, a role analogous to that of Dirac γ matrices for spin 1/2, and the adjoint fieldψ are defined as with η 0 ≡ 2β 2 0 − η 00 1.
Varying the DKP Lagrangian with respect to π µ and π † µ , we obtain the expression for the conjugate momenta which, substituted into the DKP lagrangian, gives back, after the identification ϕ = √ mφ, the quadratic lagrangian (6). This shows that the two lagrangians are equivalent (both classically and quantum mechanically), and one concludes that, for free fields, the DKP formalism is nothing but using the first order lagrangian.
It follows that the DKP Lagrangian leads to the quantum propagator of the form It is expected that an analogous construction can be made for higher spins. In that case the propagators for higher spins will have the same form, but with appropriately chosen matrices replacing the β matrices of spin-0/spin-1 theory.
A natural question arises as if it is possible to obtain this propagator from the path integral with some form of the particle action, as it was in the case of the scalar field above, (5). The answer is positive, and in the next section we present the explicit construction.

III. THE SPINNING PARTICLE PATH INTEGRAL
In this section we will discuss how the path integral for spinning particle can be written in the form proposed by Alekseev and Shatashvili [7], whose construction is, in turn, a generalization of the one of Polyakov [8]. We will omit the more technical aspects of the argument, presenting them in details in the Appendix, stressing here the motivations and the meaning of the final result.
Our starting point is the path integral (4) in momentum representation Our goal is to generalize the form of (12) so as to make it describe a particle of an arbitrary spin. We start with the observation that the action in (12) can be rewritten as where we introduced a new variable υ that replaces p/m. The variable υ is, of course, nothing but the four velocity, satisfying υ 2 = 1 and therefore belonging to the 3-dimensional pseudo-sphere P S 3 . Let us stop for a moment to contemplate on the meaning of (13). We replaced the second order constraint of the particle action p 2 − m 2 = 0 with the first order one pυ − m = 0. The former leads to the standard scalar propagator (5), and it is natural to expect that the latter will lead to the DKP one (11) if we force the path integral to replace υ with the DKP matrices β. Now, the β matrices, similarly to the Dirac matrices, are defined to satisfy (among others) the requirement that their commutator has the form where S µν generate Lorentz transformations [13] U 1 + 1 2 ω µν S µν . Since the commutator (14) must come as a result of quantization of a classical theory, the kinetic term (symplectic form) of the latter should be such that the associated Poisson bracket has the form where, again, j are Lorentz generators, satisfying so(3, 1) algebra. After quantization (as we will show below), the associated quantum operators satisfy the commutators and one gets (14) after identifying It turns out that in order to get the correct properties for the β (as well as for the Dirac matrices γ, see for instance [14] Ch. 5.4), the operatorsv µ must be generators of Clifford algebra Cl 3,1 . Their operatorsĵ µν generate the so(3, 1) Lorentz algebra, and one can show that together,v µ andĵ µν generate the so(3, 2) Lie algebra, the anti de Sitter algebra. In turn, the matrices β µ (or γ µ ) and S µν , obtained by the substitution (17) generate the so(4, 1) de Sitter algebra. Now, since in the Poisson-Lie theory there is a one-to-one correspondence between commutators of the algebra and the Poisson structure on the dual algebra, it is natural to to identify υ with elements of the Lie algebra so(3, 2) * "dual" to the one spanned by the generators of so (3,2). Let us discuss in details how this comes about.
We start from the so(3, 2) Lie algebra, generated by the antisymmetric matrices (A, B = 0, 1, 2, 3, 4) and defined by the Lie brackets With the redefinition Υ µ = M µ4 , J µν = M µν (µ, ν = 0, 1, 2, 3), the so(3, 2) algebra takes the form An arbitrary element X of the dual algebra so(3, 2) * is spanned by the generators {Υ µ ,J µν } (J νµ = −j µν ), dual to {Υ µ , J µν } in the sense that (see Appendix A) Υ µ , Υ ν = δ µ ν , J µν , J ρσ = δ µ ρ δ ν σ − δ ν ρ δ µ σ , Υ µ , J ρσ = J µν , Υ ν = 0 , and it has the form (j νµ = −j µν ) Using the definitions presented in Appendix A one can check that the coadjoint orbit of υ ≡ υ µΥ µ under the action of the Lorentz subgroup is exactly the pseudosphere P S 3 . The orbits are characterized by the values (c 1 , c 2 ) of the two polynomials of v µ , j µν invariant under the coadjoint action of SO (3,2), corresponding to the two Casimirs of so(3, 2): with The action in the path integral should therefore consist of two pieces. The first is given by (13), and the second is an action S to be defined so as to impose the condition (15), and which leads to its quantization. It is given by [15] where ω = X(g), dgg −1 , g being an element of SO (3,2), is the Liouville form associated with Kirillov symplectic two-form, discussed in details in Appendix A. This action leads to the following expressions for the Poisson brackets of the dynamical variables This is exactly what we want, because after quantization the first equation above will become the defining equation for the β matrices of the DKP formalism. The final form of the momentum space propagator is therefore It is shown in the Appendix (B) that the term exp (iS (v, j)) 'quantizes' the values of the invariant polynomials in {v µ , j µν } defining the orbits, so that the corresponding operators {v µ ,ĵ µν } belong to an irreducible representation {π(υ µ ), π(ĵ µν )} of so(3, 2) (or, through the substitutionŜ µν = iĵ µν , to an irreducible representation {π(v µ ), π(Ŝ µν )} of so(4, 1)). In other words the path integral in (26) computes, for given boundary conditions, the correlation function between states |i belonging to a particular representation of the so(4, 1) algebra, corresponding to the particular choice of integral orbit. In the formula (28)v µ is the quantum operator corresponding to v µ , and, depending on the spin representation |i , it is given by a Dirac γ matrix (for spin 1/2) or a DKP β matrix for spin 0 or 1, and, presumably, to matrix representations for higher spins. In particular, these matrices must satisfy (14), and we will show now that this is indeed the case. Let us denote by l a ≡ {v µ , j µν } the coordinates on the dual Lie algebra so(3, 2) * , so that the Poisson brackets (25) can be written concisely in terms of the so(3, 2) structure constant f c ab defined by (20) as {l a , l b } = f c ab . The transformation is a symmetry of the classical action, and the resulting Ward identity reads (see Appendix C for details) wherel a k (t k ) indicates that the particular term is missing. In order to derive the equal time commutators (ETC) for the corresponding field operators we can apply the BJL (Bjorken-Johnson-Low) procedure to the correlation function, stating that the 1/p 0 term in the matrix element of the two point function, at large p 0 , determines the commutator: where [l a ,l a1 ] is the ETC between field operators corresponding to l a , l a1 . Integrating the left hand side in the last expression by parts we rewrite it as where we neglected boundary terms. From (30) the last expression is equal to so that from (31) one gets or, expanding in terms of the operatorsυ,ĵ, Finally, using the substitution (17), we can rewrite the commutators as This is nothing but the Lie algebra (with real structure constants) so(4, 1) of SO(4, 1), which proves that, after computing the path integral in the formula (28), the operatorsυ µ can be taken to be a matrix of a particular representation of the so(4, 1) algebra. We will show in the next session how, depending on the specific so(4, 1) representation, one gets in this way the Dirac (spin 1/2) or the DKP (spin 0 or 1) propagator (and presumably the propagator for higher spins as well).
To complete the derivation we need yet another property of the correlation function (28) derived in the Appendix C, Now we can integrate (37) over L to find the momentum space propagator

IV. THE PROPAGATOR FOR DIFFERENT SPINS
Depending on the specific choice of representation for the so(4,1) generators, expression (38) gives the propagator for different spin values in the first order formalism. As shown in Appendix B, the spinning term exp i ω , upon appropriate choice of coadjoint orbits, decomposes the path-integral into matrix elements between states belonging to the finite dimensional representations of SO(4,1) labeled by the highest weights of the irreducible representations of the maximally compact subgroup SO(4) SU(2)⊗SU(2), parametrized by a set of ordered integer or half-integer numbers 4 Following the argument worked out in [20] for the Euclidean case we can define the algebras B (k) arising from the so(4, 1) matrix representations π p,q (υ µ ) defined in App. B, satisfying the commutation relations, following from (36).
and, for k ≥ p, the equation (following from (B11)) Different values of k then correspond to different spin sectors. For k = 1/2 one has a four dimensional π( 1 2 , 1 2 ) representation (see App. B) of B ( 1 2 ) corresponding to the Dirac algebra. Indeed Eq. (41) becomesυ 2 µ = − 1 4 η µµ , and defining γ µ = 2υ µ , we find from (40) that Plugging this to (38) we get the spin 1 2 propagator It appears that the spin-1 2 propagator has its usual form expressed in terms of p momenta. For k = 1 the matrices β µ (no summation) satisfy the relations that define the DKP (Duffin-Kemmer-Petiau) algebra [10][11][12] The derivation of Eq. (44) is carried out in App. D. In this case one has three irreducible representations (see App. B), the trivial one-dimensional π (0, 0), the five dimensional π (1, 0), and the ten dimensional π (1, 1). Several results (see for instance [13] and [21]) have been obtained showing that for these two latter irreducible representations the DKP field equations reduce respectively to the equations of motion for a spin-0 scalar field (the Klein-Gordon equation) and for a spin-1 vector field (the Proca equations). However to our knowledge the reduction of the propagator to the standard expressions for the spin-0 and spin-1 fields have not been treated thoroughly, and we devote next section to this task.
IV.1. The propagator for spin-0 and spin-1 Let us start noticing that it follows from (38) that the propagator in momentum space is (apart from the term i ) Using the properties (44) of the DKP matrices, one can prove that (see for instance [22]) Indeed, from (44), Then and it follows We consider first the 5 dimensional representation π (1, 0) describing the spin-0 sector. The field equations for spin-0 are obtained with the help of a projection operator [13] so that the field decomposes into the vector field V µ = Pβ µ ψ and the scalar one Φ = Pψ. Indeed one can show that Φ and V µ transform, respectively, as a (pseudo)-scalar and a (pseudo)-vector under Lorentz transformations, where infinitesimal transformations are generators by S µν = [β µ , β ν ] as (ω µν = −ω µν ) Moreover, one can show that upon imposing the DKP equation for the free field ψ the components of ψ are not independent, and that one can define (see App. D) a specific representation of the β µ such that ψ 4 = φ and ψ µ = ∂ µ φ, making explicit the fact that ψ describes in this case the scalar φ and its derivatives ∂ µ φ.
We can obtain the propagator for the scalar field S (p) by projecting the propagator (46) on the scalar field sector with P, so that S (p) is defined by the matrix element in (the mass factor is for dimensional reasons) As discussed in App. B since we are in Lorentzian metric the β 0 and β j matrices must have opposite hermiticity, and in particular in our notations we have that β 0 is Hermitian and β j anti-Hermitian: β † 0 = β 0 , β † j = −β j . In DKP theory the adjoint field is given byψ = ψ † η 0 , where η µ is the operator such that β † µ = η 0 β µ η 0 , andΦ = ψ † η 0 P † . Noticing also that from the defining properties of the β matrices (44), setting µ = ν in (44), follow the relations one finds that Using again (44) (setting ν = ρ = µ in (44) ) we find the relations and multiplying last relation by β µ from the left and from the right we find From last relation, the Hermiticity of β's and (56) we find also that while using (59) and (56) it follows From last relations we find Plugging (57) and (62) together with (46) in (54) we finally obtain so that The DKP propagator, projected on the scalar field sector, has the standard form. We can repeat a similar procedure to derive the propagator for the spin-1 representation. In this case the projection operators are where now the β matrices are to be taken in the 10 dimensional irreducible representation (we give an explicit realization in App. D). The beta matrices maintain the same hermiticity of the scalar case, and one can show that R µ ψ transforms, under the infinitesimal Lorentz transformation (52), like a (pseudo)vector while R µν ψ = R µ β ν ψ like a (pseudo)tensor. Upon imposing the DKP equation for ψ one can then show that R µν is proportional to the strength tensor of the vector field R µ ψ (see for instance [13] and [21]). We define then the vector field A µ and its adjoint as with η 0 given by (55). It is possible to show that with this definition the fields A µ andĀ µ transform respectively with covariant and contravariant indexes. Thus we may identify ( µĀ µ A µ = A µ A µ transforms as a (pseudo)scalar). The spin-1 propagator S µν (p) for the vector field is then obtained by projection as the matrix element in We can use the properties of the β matrices (56), (58) and (59) to find the following relations Using these relations it follows that and finally where we used the above relations together with R 0 A µ = R 0 R µ ψ = R µ ψ = A µ . Thus the spin-1 propagator reduces to the standard propagator in unitary gauge

V. CONCLUSIONS
In this paper, following the idea of Alekseev and Shatashvili of adding to the first order action the Kirillov presymplectic form, which forces the path integral to select a particular representation of the de Sitter group, we derive the DKP propagator for fields of spin 0 and 1 (as well as the Dirac propagator for spin 1/2) in the path integral formalism. We then show that the obtained DKP propagators are equivalent to the standard ones.
There are several interesting problems that could be addressed in follow-up investigations. First, although it seems pretty obvious that an analogous construction should work for spins higher than 1, it would be illuminating to do it explicitly.
Second, the construction presented here can, presumably, be extended to the case of κ-deformation (in the sense of κ-Poincaré Hopf symmetries) [23], [24], [25], [26], a scenario that has attracted much interest especially in relation with quantum gravity phenomenology. In this case momentum space is not the ordinary flat (Minkowskian) momentum space, but it is described as a curved manifold (specifically the group AN 3 , corresponding to half of de Sitter space, see [26]), whose scale of curvature 1/κ is taken to be proportional to the (inverse) Planck energy (1/E pl ∼ 10 −19 GeV ). As (four-dimensional) κ-momentum space can be also described in terms of flat embedding 'momentum coordinates' in five dimensions, with some additional constraint enforcing physical momenta to live on the de Sitter hyperboloid, one can think of extending the formalism described in this manuscript, which is not restricted to four-dimensional momentum space, exploiting the use of embedding coordinates. The construction of a Dirac (spin 1/2) action with κ-Poincaré symmetries has been already addressed in some previous works (see for instance [24,[27][28][29]). It would be interesting to compare the spin 1/2 propagator for κ-momentum space resulting from our approach with previous results. Moreover, if working, our construction would allow in principle to study higher spin propagators for κmomentum space, setting the stage for constructing a higher-spin field theory action based on κ-deformed symmetries.
ACKNOWLEDGMENT For JKG this work is supported by Polish National Science Centre, projects number 2017/27/B/ST2/01902.

Appendix A: The action functional on the orbits
We here discuss the construction of the action (24) needed to implement the spin degrees of freedom in the pathintegral. We refer the reader to the characterization outlined for instance in [30], based on the Kirillov symplectic form [31]. Consider a (matrix) Lie group G. Let g be Lie algebra of G and g * its dual Lie algebra: for a basis {e a } of g and {ẽ a } of g * , the duality relations are canonically given by ẽ a , e b = δ a b . The coadjoint representation of G is defined by where g ∈ G, X ∈ g * , u, v ∈ g. Let us parametrize the orbits by group variables fixing the point X 0 so that a generic point on the orbit is In the basis {e a } and {ẽ a } we will write a generic point on the orbit as X = l aẽ a . Define the action Here Y = dgg −1 is the Maurer-Cartan form on the group. It is possible to show that the following equivalent equations are satisfied where X = l aẽ a , and Y = Y a e a , and f c ab are the structure constant of the Lie algebra 5 g [e a , e b ] = f c ab e c . Let us first show the equivalence of Eq.(A6) and Eq.(A5): Let's now prove Eq. (A5): dX, e a =d Ad * (g) X 0 , e a = X 0 , g −1 e a dg + dg −1 e a g where we used definition (A2) and d1 = d g −1 g = dg −1 g + g −1 dg = 0, from which follows dg −1 = −g −1 dgg −1 .
Eq. (A5) (or (A6)) ensures that the action (A3) generates on the orbit the canonical 2-form where dΩ is closed on the orbit. The last equation can be rewritten explicitly as where the Maurer-Cartan equation d(gdg −1 ) = −gdg −1 ∧ gdg −1 has been used. One can show that the Poisson brackets of the restriction of the linear functions on the orbit reproduce the algebraic commutation relation. Defining the linear functions u (X) = X, u , where on the r.h.s. u = u a e a (so that u (X) = u a l a ), we get where we used that the Maurer-Cartan form evaluated on an element of the basis of the Lie algebra gives Y (e a ) = e a . It follows in particular, for u a = δ a b , that on the orbit, Appendix B: Coadjoint orbits and irreps for SO(4,1) 1. Integral orbits for SO (3,2) In the spirit of geometric quantization (see for instance [32,33] the orbit method [31] can be used to "quantize" the values of some parameters labeling the orbits of the action of the group on its dual Lie algebra. This mechanism can be realized [7,15] by the requirement for the action exponential exp (iS ( )) to be single valued, so that the pathintegral is well defined. Indeed the 1-form ω is singular, and the action S = ω is multivalued, and the requirement of uniqueness of the expression exp (iS ( )) over closed path leads to integral orbits.
In our specific case, starting from the so(3, 2) algebra (20), so that a generic element can be parametrized as u =υ µ Υ µ + 1 2j µν J µν (j νµ = −j µν ), we can fix the orbits considering the action of the Lorentz subgroup SO(3,1) generated by J µν . Rewriting the generators as R i = − 1 2 ijk J jk and P i = Υ i , we can rewrite the so(3, 1) subalgebra as so that an element of the so(3, 1) subalgebra is u so (3,1) ing (see [14] Ch. 5.6) an element of so(3, 1) as u so (3,1) the algebra splits into a direct sum of 2 mutually commuting complex (conjugate) su(2): so(3, 1) ≈ su(2) C ⊕ su(2) C ≈ sl(2, C) ⊕ sl(2, C). Thus we have reduced the problem to fixing the orbits of the two Sl(2, C) subgroups of SO(3, 1). Finally, we notice that each of the two Sl(2, C) admits SU (2) as (maximal) compact subgroup, and we can use it to fix the orbits for each of the two copies.
Representing the SU(2) generators in terms of Pauli matrices, A i , B i ≡ − i 2 σ i , for each SU(2) copy we can parametrize an element of the group by Euler angles as with On each copy the Maurer-Cartan connection Y SU(2) = g −1 SU(2) dg SU (2) can be evaluated to The orbits can be chosen fixing the value of the coordinates in g * su (2) , Re(ã i ) = (0, 0, m), Re(b i ) = (0, 0, n), and we thus find respectively the action (s = m or n) where we renamed the azimuthal angle χ = γ φ for some constant γ and γ = γ s. The action is multivalued as it counts the windings around the axis passing through the poles θ = 0, π of the sphere, where the 1-form cos θdφ is singular. For infinitesimal closed contours around the poles θ = 0, π, the action gives the value 2π (γ ± s), so that, if γ ± s is an integer, the action exp i ω does not contribute to the path-integral, which is then well-defined. We can choose γ = 0 for s integer and γ = 1 2 for s semi-integer. Thus the condition for single-valuedness of exp i ω translates into the condition of "quantization" of the values of s = (m, n), which take only integer or semi-integer values ( 1 , 2 ).

(B8)
Thus, the orbit quantization method exhibited in the previous section, singling out integer or semi-integer values ( 1 , 2 ), selects the values of the invariant polynomials C 1 and C 2 of l a ≡ (υ µ , j µν ), defining the orbits, which occur in the irreducible representations of SO (4,1), and in particular in the discrete series.
Apart from the irreducible unitary (infinite dimensional) representations, one can obtain finite dimensional, nonunitary, representations that can be understood as a "Wick-rotation" of the irreducible unitary representations of SO(5) discussed for instance in [19]. In this case the generators can be represented through finite dimensional (mixed Hermitian and anti-Hermitian) matrices, and we will use this representations to construct the projection operators for the different spin sectors of the propagator. Denoting such finite dimensional representations with π p,q , they are labeled now by p = max( 1 + 2 ) and q = max( 1 − 2 ), whose ranges are such that p and q are all integers or all semi-integers and p ≥ q ≥ 0, while, for given (p, q), 1 + 2 and 1 − 2 range respectively from q to p and from −q to q by steps of 1 (thus 1 , 2 range from 0 to 1 2 (p + q) by steps of 1 2 ). The π p,q are are characterized by the values of the Casimir operators π p,q (Ĉ 1 ) = p (p + 3) + q (q + 1) , π p,q (Ĉ 2 ) = (p + 1) (p + 2) q (q + 1) , and their dimension is given by the formula The range of the highest weights ( 1 , 2 ) for the lowest order representations π p,q is depicted in Fig. 1 in terms of the allowed ( 1 , 2 ) values. From the formula (B10) we find that they have respectively the dimensions d The matrices that form the π p,q representation can be considered to be obtained by "Wick rotation" from the SO(5) matrices derived in [19]. To better characterize this definition consider first the defining 5 dimensional representation of SO(4,1) matrices. They are the matrices preserving the bilinear form given by the five dimensional Lorentzian metric η (5) ≡ (+, −, −, −, −), so that the generators are matrices M AB such that M T AB = −η (5) M AB η (5) , and are related with the so(5) (skew-symmetric) matricesM AB byM AB = −η 5 M AB . We can set then (M AB ) KL = η  (5) to so(4, 1) matrices one can easily see that the matrices M 0A become symmetric (and Hermitian) while, the remaining ones stay skew-symmetric (and anti-Hermitian). In particular, ifλ are the (imaginary) eigenvalues of thẽ M AB , from the spectral theorem, the effect of the map on the eigenvalues isλ → λ = iλ for the matrices M 0A , while for the remaining M iA (i = 1, 2, 3; A = 0) they stay the same. With this in mind we can generalize this definition of Wick-rotation to representations of any dimensions: one can always choose the matrix generators M 0A

n-point correlation function
One can show that the n-point correlation function is given by the time ordered product of the field operators as where on the r.h.s.l a are matrix operators for a specific representation of so (4, 1), and summation is over all permutations {i 1 · · · i n } of the numbers (1 · · · n) (θ (x) is the Heaviside step function). Indeed the Ward identities (30) are satisfied, as can be shown explicitly by deriving Eq. (C7). For instance for n = 3, denoting δ ij = δ (t i − t j ) and θ ij = θ (t i − t j ), and using that ∂/∂x θ (±x) = ±δ (±x), ∂ t1 i|l a1 (t 1 )l a2 (t 2 )l a3 (t 3 ) |j =δ 12 θ 23 i| l a1 ,l a2 l a3 |j + θ 32 i|l a3 l a1 ,l a2 |j + δ 13 θ 32 i| l a1 ,l a3 l a2 |j + θ 23 i|l a2 l a1 ,l a3 |j where we used the ETC (34), and in general (30) are verified. One can also check that the BJL limit is satisfied by (C7). Indeed (see [35]) using the integral representation of the step function and (C7) for n = 2, Eq. (31) can be rewritten as Taking the limit for p 0 → ∞ it becomes i dtδ (t − t 1 ) i| l a ,l a1 |j = i i| l a ,l a1 (t 1 ) |j .
2. 5-dimensional representation of DKP matrices for spin-0 The DKP matrices (44) admit a five dimensional irreducible representation that carries the degrees of freedom of a spin-0 field theory. We here report an explicit expression for the matrices in this representation [13] . These are given as Notice that with this definition the spin-0 β matrices coincide with the defining representation for the generators corresponding to the "momentum sector" of SO(4, 1), i.e. β µ ≡ M µ4 , where (M AB ) KL := η AK δ KL − η BK δ AL (where here η AB = diag(1, −1, −1, −1, −1)). In this representation the projection of the field is The ten dimensional representations for the β matrices can be given as [21] with these representations the field ψ is projected as whileĀ µ = −ψ † η 0 R † µ = 0 0 0 0 0 0 0 0 0 η µµ A µ (D11) so that the scalar product is given by