Journal of Physics: Conference Series PAPER • OPEN ACCESS Generalized Scalar Duffin-Kemmer-Petiau Electrodynamics (GSDKP) To cite this article: R Bufalo et al 2016 J. Phys.: Conf. Ser. 706 052002   View the article online for updates and enhancements. Recent citations Transition amplitude, partition function and the role of physical degrees of freedom in gauge theories A.A. Nogueira et al - This content was downloaded from IP address 186.217.236.64 on 18/04/2019 at 18:28 https://doi.org/10.1088/1742-6596/706/5/052002 http://dx.doi.org/10.1016/j.nuclphysb.2018.07.024 http://dx.doi.org/10.1016/j.nuclphysb.2018.07.024 http://dx.doi.org/10.1016/j.nuclphysb.2018.07.024 https://oasc-eu1.247realmedia.com/5c/iopscience.iop.org/198725545/Middle/IOPP/IOPs-Mid-JPCS-pdf/IOPs-Mid-JPCS-pdf.jpg/1? Generalized Scalar Duffin-Kemmer-Petiau Electrodynamics (GSDKP) R Bufalo, T R Cardoso, A A Nogueira and B M Pimentel Instituto de Fı́sica Teórica (IFT), Universidade Estadual Paulista; Rua Dr. Bento Teobaldo Ferraz 271, Bloco II Barra Funda, CEP 01140-070 São Paulo, SP, Brazil. E-mail: bufalo@ift.unesp.br, cardoso@ift.unesp.br, nogueira@ift.unesp.br, pimentel@ift.unesp.br Abstract. The main goal of this work is to investigate the quantum interaction between scalar field and gauge field in the context of Generalized Scalar Duffin-Kemmer-Petiau Electrodynamics (GSDKP) by a quantum theory in the functional approach. The Hamiltonian structure is obtained with the Dirac method and the Faddeev-Senjanovic procedure is established in order to write the transition amplitude in an alternative gauge fixing, known as the non-mixing gauge. As a consequence, the Schwinger-Dyson- Fradkin equations and the Ward-Takahashi-Fradkin identities are obtained. 1. Introduction The purpose of this work is to study the covariant dynamics of systems with constraints and the quantization procedure in a particular case (GSDKP), so let us make a brief introduction to the subject. The quantization problem of a classical dynamics was first put in a formal language by Dirac [1] because he observed that the classical dynamics described in the phase space by the observable and Poisson brackets was associated to a quantum dynamics described in the Hilbert space by the operators and commutators\anti-commutators via correspondence principle {,}P→−i[, ]±. (1) Later the existence of constraints in the hamiltonian dynamics led Dirac to extend its mechanical analysis of the phase space by the parenthesis of Dirac and the classification (first class\second class) always being able to make connection to quantum dynamics with the correspondence principle [2], {,}D→−i[, ]±. (2) This is the first look at the connection between classical\quantum dynamics. The second look begins in a Dirac study [3] about the connection between a dynamics described in the configuration space and its resulting quantum dynamics. In this study we see the emergence of an object very important called transition amplitude Z = ∫ Dµ exp[iS], (3) where Dµ is the integration measure and S is the action. XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052002 doi:10.1088/1742-6596/706/5/052002 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 Feynman uses the idea of Dirac to formulate a way to describe the quantum mechanics via path integral [4]. However elegantly Schwinger infers that because the quantum equations in the Heisenberg description preserve its classic form it should be a principle of quantum action [5]. We have now that the amplitude transition, or functional generator is a solution of the functional equation arising from a dynamic [ϕ̇−{ϕ,H0−ϕJ}D]|ϕ= δ iδJ Z[J] = 0 (4) where H0 is the initial hamiltonian. Continuing, Faddeev explores the integration measure properties Dµ to study connections between classical\quantum dynamics of the physical systems with first-class constraints [6]. Later Senjanovic extends the ideas of Faddeev [7] for second-class constraints. Finally Fradkin and Vilkovisky make an overview of the problem in view of covariant formalisms and its connection with BRST symmetry [8]. See, for more details on quantization of systems with constraints [9]. 2. Constraint analysis and transition amplitude To construct the transition amplitude we will use the Faddeev-Senjanovic [6, 7] analysis where we must first do a short study of constraint. The Lagrangian density describing the GSDKP is defined by L = i 2 ψ̄β µ ( ∂µψ ) − i 2 ( ∂µ ψ̄β µ ) ψ−mψ̄ψ + eAµ ψ̄β µ ψ− 1 4 FµνFµν + a2 2 ∂ µFµβ ∂αFαβ (5) where Fµν = ∂µAν−∂νAµ is the usual electromagnetic field-strength tensor and β µ are the DKP matrices that obey the algebra [10] β µ β ν β θ +β θ β ν β µ = β µ η νθ +β θ η νµ . (6) This theory is invariant under local gauge transformations ψ → eiα(x) ψ, Aµ → Aµ + 1 e ∂µα (x) . (7) The matter (scalar field, mesons) are described by the Duffin-Kemmer-Petiau (DKP) theory [11], a first- order relativistic theory for the description of spin 0 and spin 1 bosons with a similar form as the Dirac equation, and the radiation (vector field, photons) are described by the Podolsky theory [12]. We can see this gauge theory with U(1) symmetry as an extension of the work on DKP Quantum Electrodynamics [13]. The translational space-time invariance of the Lagrangian density leads us to write the canonical Hamiltonian Hc = ∫ d3x [ (∂0ψ̄) ∂L ∂ (∂0ψ̄) + ∂L ∂ (∂0ψ) (∂0ψ)+ ∂L ∂ (∂0Aν) (∂0Aν) −∂θ ( ∂L ∂ (∂0∂θ Aν) ) (∂0Aν)+ ∂L ∂ (∂0∂θ Aν) (∂θ ∂0Aν)−L ] , (8) where we define the momenta expressions p = ∂L ∂ (∂0ψ̄) =− i 2 β 0 ψ, (9) p̄ = ∂L ∂ (∂0ψ) = i 2 ψ̄β 0, (10) Π ν = Fν0 +a2[η iν ∂i∂αFα0−∂0∂αFαν ], (11) Φ ν = a2[∂αFαν −η ν0 ∂αFα0]. (12) XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052002 doi:10.1088/1742-6596/706/5/052002 2 From the above momentum expressions, we shall study the constraint structure of the theory following Dirac’s approach to singular systems [2]. In this approach it is possible to obtain the set of first-class constraints ϕ1 =Φ0 ≈ 0, (13) ϕ2 =Π0−∂kΦ k ≈ 0, (14) ϕ3 =eψ̄β 0 ψ−∂ k Πk ≈ 0 (15) and the set of second-class constraints χ (1) = p+ i 2 β 0 ψ ≈ 0, (16) χ̄ (1) = p̄− i 2 ψ̄β 0 ≈ 0, (17) χ (2) = [1− (β 0)2][iβ i ∂iψ(x)−mψ(x)+ eβ iAi(x)ψ(x)]≈ 0, (18) χ̄ (2) = [−i∂iψ̄(x)β i +mψ̄(x)− eψ̄(x)β iAi(x)][1− (β 0)2]≈ 0. (19) The weak equality ≈ is understood in according to Dirac’s sense. With the full set of first-class and second-class constraints determined, we are now in a position to obtain the functional generator. The transition amplitude in the Hamiltonian form is written in the following way: Z = N ∫ Dµ exp { i ∫ d4x [(∂0ψ̄) p+ p̄(∂0ψ)+Π ν (∂0Aν)+Φ ν (∂0Γν)−H c] } (20) where the canonical Hamiltonian is given by Hc = Π0Γ 0 +ΠkΓ k +Φk(∂ k Γ0−∂lF lk + Φk 2a2 )− i 2 ψ̄β i←→ ∂ iψ +mψ̄ψ − eψ̄Âψ + 1 4 Fk jFk j + 1 4 (Γ j−∂ jA0) 2− a2 2 (∂ j Γ j−∂ j ∂ jA0) 2, (21) and the integration measure Dµ = DΦ νDΓνDΠ µDAµDψ̄DψDp̄Dpδ (Θl)det‖{Θl,Θm}‖ 1 2 . (22) We have that the complete set of constraints for the GSDKP is Θl = { χ (1), χ̄(1),χ(2), χ̄(2),ϕ1,ϕ2,ϕ3,Σ1,Σ2,Σ3 } , (23) in which suitable gauge conditions for the first-class constraints are chosen as the generalized radiation conditions [14] Σ1 = Γ0(x)≈ 0, Σ2 = A0 ≈ 0, Σ3 = (1+a2�)(~∇.~A)≈ 0. (24) After integrating over the gauge and fermionic momenta, the transition amplitude Z is explicitly written Z = N ∫ DAµDψ̄Dψ det ∥∥∥(1+a2~∇2)~∇2 ∥∥∥δ ((1+a2�)(~∇.~A)) exp[i ∫ d4x{ψ̄(iβ µ ∇µ −m)ψ− 1 4 FµνFµν + a2 2 ∂ µFµβ ∂αFαβ}]. (25) XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052002 doi:10.1088/1742-6596/706/5/052002 3 Hence, with the Faddeev-Popov-DeWitt ansatz in the non-mixing gauge condition [15] Ω(A) = ( 1+a2� ) 1 2 ∂ µAµ (26) the transition amplitude can be written in a covariant form Z = N ∫ DAµDψ̄Dψ exp{i ∫ d4x[ψ̄ ( iβ µ ∇µ −m ) ψ− 1 4 FµνFµν + a2 2 ∂ µFµβ ∂αFαβ − 1 2ξ ( ∂ µAµ )( 1+a2� )( ∂ µAµ ) ]}. (27) Although initially Podolsky used the Lorenz gauge fixing condition Ω(A) = (∂µAµ) to fix the physical degrees of freedom in generalized electrodynamics, lately with a rigorous study of constraint analysis we see that this is not really true [14]. As a consequence, the natural way to fix the physical degrees of freedom was the generalized Lorenz gauge fixing condition Ω(A) = (1+ a2�)(∂µAµ), however it complicates the theory’s quantization once it increases the order of the field equation. Today we think that the non-mixing gauge fixing condition [15] Ω(A) = ( 1+a2� ) 1 2 ∂ µAµ combines with the theory because it preserves the order of the field equation, but it has a pseudo-differential structure. 3. The Schwinger-Dyson-Fradkin equations The most elegant way of studying the set of field equations in the Heisenberg description is in a functional formulation, consisting in an infinite chain of differential equations that relate different Green’s function in a exact manner. This infinite tower of equations is referred to as the Schwinger-Dyson-Fradkin (SDF) equations [17]. 3.1. The Schwinger-Dyson-Fradkin equations for the gauge-field propagator The complete expression of the gauge-field propagator can be determined by means of the functional generator leading to the Schwinger variational equation for the gauge field, δS δAγ (x) ∣∣∣∣ δ δ iη , δ δ iη̄ , δ δ iJµ + Jγ (x) Z [ η , η̄ ,Jµ ] = 0. (28) In terms of the functional Z [J] = exp{iW [J]} for the connected Green’s functions we obtain the equation −Jγ (x) =− ie δ δη (x) β γ ( δW δ η̄ (x) ) + e δW δη (x) β γ δW δ η̄ (x) + [ T γµ + 1 ξ Lγµ ]( 1+a2� ) � δW δJµ (x) . (29) The last equation can be interpreted as the complete Podolsky field equation subjected to an external source Jγ . On this equation T γµ and Lγµ are differential projectors T γµ +Lγµ = gγµ , Lγµ = ∂ γ∂ µ � . (30) In order to obtain the complete gauge-field propagator it proves convenient to introduce also the generating functional for the one particle irreducible (1PI) Green’s functions, which is related to W by a functional Legendre transformation Γ [ ψ, ψ̄,Aµ ] =W [ η , η̄ ,Jµ ] − ∫ d4x ( ψ̄η + η̄ψ +AµJµ ) . (31) XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052002 doi:10.1088/1742-6596/706/5/052002 4 Hence, rewriting (29) in terms of the 1PI functional δ 2Γ δAν (y)δAγ (x) = [ T γν + 1 ξ Lγν ]( 1+a2� ) �δ (4) (x,y) + ie2 ∫ d4ud4wTr [S (x,u;A)β γS (w,x;A)Γ ν (u,w;y)] . (32) The second term of (32) can be identified with the polarization operator, Pγν , Pγν (x,y) = ie2 ∫ d4ud4wTr [S (x,u;A)β γS (w,x;A)Γ ν (u,w;y)] . (33) The expression for the inverse of the gauge-field complete propagator in momentum representation is (Dγν)−1 (p) = (Dγν)−1 (p)+Pγν (p) . (34) This equation can be viewed as = + -1 -1 Figure 1. The SDF equation for the gauge-field propagator. The expression (34) can be solved in order to find iDγν (p) =− ( ηγν − pγ pν p2 ) [P(p)− (1−a2 p2) p2] + ξ p2 (1−a2 p2) pγ pν p2 , (35) in which the scalar polarization P(p) it is related to the polarization tensor Pγν (p), through the structure Pγν (p) = ( −p2 η γν + pγ pν ) P(p) . (36) For the free propagator, namely taking P(p) = 0 on (35) and a = m−1 p , one has the expression iDγν (p) = [ η γν − (1−ξ ) pγ pν m2 p ][ 1 p2 − 1 p2−m2 p ] − (1−ξ ) pγ pν (p2)2 (37) 3.2. The Schwinger-Dyson-Fradkin equations for the DKP propagator The expression to the full DKP propagator can be derived starting from Schwinger variational equation δS δψ̄ (x) ∣∣∣∣ δ δ iη , δ δ iη̄ , δ δ iJµ +η (x) Z [ η , η̄ ,Jµ ] = 0. (38) Now, writing the equation (38) in terms of the generating functional W and then differentiating the resulting expression with respect to the source η (y), we obtain iδ (4) (x− y) =− [ iβ µ ∂µ −m+ eβ µ 〈Aµ〉− ieβ µ δ δJµ (x) ] S (x,y;A) . (39) XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052002 doi:10.1088/1742-6596/706/5/052002 5 By solving the derivative of the last term we can identify Σ(x,z) = ie2 β µ ∫ d4ud4vDα µ (u,x)S (x,v;A)Γα (v,z;u) , (40) as the DKP self-energy function Σ. Hence, by taking the limit of null sources, we find that iδ (4) (x− y) =− [ iβ µ ∂µ −m ] S (x,y;A)+ ∫ d4zΣ(x,z)S (z,y;A) . (41) In momentum representation the above equation becomes S −1 (p) = S−1(p)+Σ(p) . The later equation can be viewed as = + -1 -1 Figure 2. The SDF equation for the scalar propagator. The equation (39) can formally be written as S (p) = i β µ pµ −M(p) (42) where we defined the mass operator M, M(p) = m+Σ(p). (43) Besides, the expression for the DKP free propagator can be obtained with help of the DKP algebra (6), S(p) = i 1 m [ p̂(p̂+m) (p2−m2) −1 ] , p̂ = β µ pµ . (44) 3.3. The Schwinger-Dyson-Fradkin equations for the vertex part The starting point to get the vertex function is the equation (39). In a similar way, we take the derivative of the resulting expression with respect to the field Aσ (z), and after some manipulations, we find the following expression for the vertex function iΓσ (q, p;k) =−β σ (2π)4 δ (q− p− k)+ iΛσ (k, p;q) (45) where we have introduced a new quantity, the vertex part iΛσ (q, p;k) = ie2 β µ 1 (2π)4 ∫ d4tDµρ (t)S (t + k)Φ σρ (t + k, p;q, t)+ +e2 β µ 1 (2π)4 ∫ d4 p1d4 p2Dµρ (p1)S (p1 + k)Γ σ ((p1 + k) , p2;q)S (p2)Γ ρ (p2, p; p1) , (46) in this expression we have defined the four-point vertex function e2 Φ σρ (a,w;z,s) = δ 4Γ δAρ (s)δAσ (z)δψ (w)δψ̄ (a) . (47) From Eq.(46) we clearly see that the three-point vertex function depends on the four-point one. Diagrammatically, the irreducible vertex part can be visualized XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052002 doi:10.1088/1742-6596/706/5/052002 6 = + + Figure 3. The SDF equation for the vertex function. 4. Ward-Takahashi-Fradkin identities It is of our understanding that when we formulate the equations of electrodynamics in a covariant way we can describe them by a Lagrangian that has at its cornerstone a gauge symmetry. Classically the matter field (scalar) has a global U(1) gauge symmetry. When we impose a local gauge symmetry we need intermediaries fields of interaction (vector). Therefore when we write the dynamics of this electrodynamics (GSDKP) in a covariant way we have a local gauge symmetry U(1). Similarly we can describe the equations of quantum electrodynamics in a covariant way in the Heisenberg representation. On the other hand when we quantize the theory covariantly using the functional formalism we lost the gauge symmetry, to fix the physical degrees of freedom and define adequately the measure of integration∫ Dµ . To maintain gauge symmetry we impose it in the functional generator and therefore this generates certain constraints on Green’s functions, relationships between them known as Ward-Takahashi-Fradkin identities (WTF) [18]. The gauge symmetry is a consequence of a covariant quantum dynamics, the symmetry is implicit in the quantum equations of motion discussed previously. The derivation of the WTF identities is formally given in terms of the following identity upon the functional generator δZ [ η , η̄ ,Jµ ] δα(x) ∣∣∣∣∣ α=0 = 0. (48) This leads to the equation of motion satisfied by Z [ η , η̄ ,Jµ ] [ −i � eξ ( 1+a2� ) ∂ µ δ δJµ − δ δη η + η̄ δ δ η̄ − 1 e ∂ µJµ ] Z = 0. (49) Finally, one can get the desired quantum equation of motion for the theory by writing (49) first in terms of W , and then as an expression for the 1PI-generating functional Γ [ ψ, ψ̄,Aµ ] through the relation (31). Then −i � eξ ( 1+a2� ) ∂ µ x Aµ − ψ̄ δΓ δψ̄ + δΓ δψ ψ + 1 e ∂ µ x δΓ δAµ = 0. (50) This is the equation that will supply all the WFT identities. The first identity comes by applying the derivatives of (50) with respect to ψ (y) and ψ̄ (z), yielding ∂ µ Γµ(z,y;x) =−δ (x− z)Γ(x,y)+Γ(x,z)δ (x− y). (51) In momentum representation kµΓ µ ( p, p′,k = p− p′ ) = S −1 (p′ ) −S −1 (p) . (52) Furthermore, we may consider the limit of this equation as k→ 0, in which we find that the vertex part is related to the DKP self-energy function as Λ µ (p, p,k = 0) =− ∂ ∂ pµ Σ −1 (p) . (53) XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052002 doi:10.1088/1742-6596/706/5/052002 7 On the other hand, upon the differentiation of (50) with respect to Aν(y), it follows the identity ∂µΓ µν (x,y) = � ξ ( 1+a2� ) ∂ ν δ (4) (x− y) (54) which together with equation (33) implies that kµP µν (k) = 0. (55) 5. Conclusions In this work we have analyzed the covariant dynamics of interaction between scalar particles and generalized photons. The first point to note is that we have implemented the non-mixing gauge in the constraint analysis. The second point to note is that we have used the DKP field to describe the scalar particles. The complete quantum structure of the scalar field, seen by means of SDF diagrams, is exactly as those from GQED4. In this case the same diagram phenomenology for the electromagnetic interaction between scalar or fermionic fields happens. But the DKP fields are described by the DKP algebra (mesonic algebraic structure), while the fermionic field obeys a Clifford algebra (fermionic algebraic structure). This work complements previous studies on covariant quantum dynamics [19] and opens the door to a more complete approach, with the study of renormalization, first radiative corrections and quantum covariant dynamics in equilibrium (Matsubara-Fradkin formalism). Acknowledgements R.B. thanks FAPESP for full support, T.R.C. and A.A.N. thank CAPES for full support, B.M.P. thanks CAPES and CNPq for partial suport. References [1] Van Der Waerden B L 1967 Sources of Quantum Mechanics (North-Holland Publishing Company); Dirac P A M 1958 The Principle of Quantum Mechanics (Oxford University Press) [2] Dirac P A M 1950 Can. J. 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