Møller scattering in 2+1 of generalized quantum electrodynamics in the heisenberg picture
MMøller scattering in 2 + David Montenegro Instituto de F´ısica Te´orica (IFT), Universidade Estadual PaulistaRua Dr. Bento Teobaldo Ferraz 271, Bloco II Barra Funda, CEP 01140-070 S˜ao Paulo, SP, Brazil
September 16, 2020
Abstract
In this paper, we investigate from the framework of generalized electrodynamics the differen-tial cross section of the electron-electron scattering process e − e − → e − e − , i.e., Møller scattering,in ( + ) dimensions in the Heisenberg picture. To this goal, one starts within the stable andunitary framework of planar generalized electrodynamics, instead of Maxwell one. We argue theHaag’s theorem strongly suggests the study of the differential cross section in the Heisenbergrepresentation. Afterward, we explore the influence of Podolsky mass cutoff and calculate thedifferential cross section considering data based on condensed matter systems. a r X i v : . [ qu a n t - ph ] S e p Introduction
Quantum electrodynamics ( QED ) is a gauge theory of remarkable success with highly attractivefrom the theoretical and experimental viewpoint. The satisfactory conciliation between them is pos-sible due to the fact, among others, the perturbative evaluation of cross section in different levelsof energy. The great accomplishment in almost everything we know concerning particles has beenavailable because of scattering methods. Such a picture has been supported by phenomena in highenergy as well as condensed matter that rely on different information through experiments. One of themost fundamental observations is the electron-electron interaction which in the perspective of Møller ( e − e − ) scattering characterizes somewhat special since its large cross section contributes for a goodstatistic to examine the standard model and test new emergent physics [1].As it is well known, the QED structure and symmetries are inherent restricted by the dimen-sionality of the system. Particularly, planar
QED has attracted attention as a potential framework todiscuss a large class of systems in the perturbative regime. The recent advances suggest as a qualita-tive explanation for the mechanism generating the formation of cooper pair in high- T c superconduc-tors [2]. Other notable examples are the prominent features of the fractional quantum Hall effect [3],the Heisenberg model of quantum spin [4], the quantum simulator of cold atoms [5], and the bandstructure of graphene [6]. Furthermore, we also consider as a toy model to explore dynamical chiralsymmetry breaking in quantum chromodynamics (QCD) [7].Despite the remarkable application in many areas, planar physics belongs to the group of effectivefield theory. Thus, at this point, we are in the position to generalize the quantum electrodynamicsin ( + ) dimensions ( QED ) and include higher order terms in such a way to leave the originalsymmetries unchanged.Higher order field equations (third derivative or higher at the level of dynamics) have been presentin many physical situations, for example, the extension of gravity models [8], the coupling of superstring theory [9], and supersymmetry (SUSY) [10]. It seems an attractive way to investigate questionswhere low energy theories are insufficient to archive, here low energy corresponds to a defined scale inthe system. This scenario may then lead us to assume that these models are incomplete or inaccurate.Higher derivative field theories draw useful insights in the clarification of underlying fundamentalaspects of low-energy models and there is a wide class of new higher order theories that we can extractinformation and obtain interesting consequences. One remarkable misleading should be highlightedin this procedure before mentioning the aim of our paper. A first approach could induce that theoriginal theory would appear small perturbed with higher derivative corrections even if assuminga small coupling. However, in contrast to lower derivative corrections, it is possible to recognizealong with this paper that the higher theories are entirely different from the original ones and morefundamental due to the increase of new degrees of freedom and families of solutions.At early in 1940, Podolsky and Schwed proposed [11] an extension of Maxwell-Lorentz electrody-namics. Initially, they intended to lead with classical problems as the 4 / r − singularity) [11]. The2odolsky’s non-singular Lagrangian exhibits a natural regulator (cutoff) encoded in the massive pa-rameter, which becomes relevant from classical and quantum view and admits, as a natural extensionof Maxwell electrodynamics ( ME ) , a strength field quadratic divergent term, which conserves thelinearity (superposition principle), Lorentz, and Gauge invariance. We can ask if assuming theseproperties, the GQED would be unique. One rigorous way to answer this question is by Utiyamassystematic method that proved the uniqueness apart from a surface term [13]. The Podolsky theory isoften called generalized quantum electrodynamics ( GQED ) .The novel feature inherent from any Higher derivative Lagrangian is the unboundedness of thecanonical Noether energy that yields dynamic instability and negative norm states at the classical andquantum level, respectively. Ostrogradski has already evidenced this undesirable behavior when de-veloped a canonical program to deal with non-degenerate higher derivative Hamiltonian [14]. How-ever, as pointed in [15], it should be stressed that the positivity of energy is not a necessary but asufficient requirement for the stability of equations of motion. Following this line of thought, weshould update the criteria of energy positivity from Noether’s theorem to validate the stability. It hasbeen demonstrated in [15] the formalism of the Lagrangian anchor established the proper connectionbetween the positive integral of motion and time translation invariance, so the boundness motion en-sures classical and quantum stability. Also, this enables us to identify a safe room of stable higherorder derivative field theories even if their unboundedness Noether’s energy restricts them, as thePodolsky one. Moreover, the BRST symmetry revealed Podolsky as a unitary model [16]. The fact GQED also assures unitarity and stability turns out a suitable candidate to deal with fermion andphoton interaction.Scattering theories provide the bridge for comparison between experimental data and quantumfield theory, however, a mathematical problem may concern the characterization of these techniques.It is not surprising that we can construct a unitary and invertible mapping of the canonical com-mutation relation between the Interaction picture (IP) and Heisenberg picture (HP) if we work in thenon-relativistic quantum mechanics (finite degrees of freedom). In this sense, each hermitian operatormatches the same empirical result from cross section scattering. Nonetheless, the situation is not thecase if switching the framework to quantum field theory (infinite degrees of freedom), furthermore,we cannot assure the existence of such an equivalent representation. In addition, the famous Haag’stheorem [17] undermines the perturbative evaluation of operators in the IP approach since there isno unitary representation from the Hilbert space of free and interacting vacuum state. To avoid thismathematical inconsistency, we choose to capture the perturbative scattering calculation under theHeisenberg representation.The preceding declarations denote that a coherent alternative to solve quantum electrodynamicsstill misses. In a series of papers, Klln preferred to tackle
QED problems in four spacetime dimen-sions in the HP [18, 19]. Instead of working directly with Hamiltonian, the Klln method applies anexpansion for every operator in the HP as a power series in the gauge coupling and substitutes intothe equations of motion to deduce the rules of S -matrix. Such method has been applied in Thirringmodel [20], Scalar Quantum Electrodynamics (SQED) [21], quantum correction at one loop level in3 ED [22].Motivated by these considerations, the main goal of this paper is to evaluate the differential crosssection of Møller scattering in the GQED within the framework of Heisenberg representation. Thispaper is outlined as follows. In section II, we begin a brief revision of GQED and exhibit the freesolution and gauge field propagator. In section III, we review Dirac quantization. We devote the mainaspects of S -matrix and the calculation at second-order approximation in section IV. The main partof this work we evaluate the Møller scattering in section V. Finally, we dedicate the section VI toconcluding remarks. In this section, we intend to review the basic results of
GQED and obtain the analytic gaugepropagators. The starting point is the Podolsky lagrangian density function in a generalized d space-time dimensions with a matter-current density j µ ( x ) coupled to a gauge field A µ ( x ) L GQED d = − e F µν F µν − a e ∂ µ F µβ ∂ α F αβ + j µ A µ (2.1)Here, we assume A µ ( x ) as continuous partial derivative up to fourth order in a d -dimensionalvolume element in Minkowski space-time . The field-strength tensor defined as F µν = ∂ µ A ν − ∂ ν A µ and its two-form leads to Bianchi identity dF =
0. The Podolsky mass is m P = a − and we canrecover the ordinary QED d if taking the limit m P → ∞ .To help visualize the behavior of GQED , one adopts the Lagrangian above to reveal the di-mensional peculiarity by a simple redefinition of couplings where the effective nature of couplingmay show the dimension signature. We should then examine the effective coupling of Podolskymass. A systematic look in ( + ) and ( + ) dimensions conduct to ( e (cid:48) m (cid:48) P ) = ( em P ) / E and ( e (cid:48) m (cid:48) P ) = ( em P ) / E , respectively, where the energy scale is E . To understand these physical con-sequences, we observe gauge field has [ A ] = M , in any dimension, with M mass unity and the di-mensionless of coupling e in GQED shows the effective coupling [ m (cid:48) P ] = M and e (cid:48) are independentcontributions, while in GQED they are connected because of [ e ] = M .Furthermore, according to already known result in QED , which photon are free in higher energyand strongly coupled in low energy, we prove the m P can archive faster than QED the asymptoticultraviolet and infrared photon states. Therefore, GQED serves as a ground framework to IR systemas mechanism of confinement as pair cooper and quarks into hadrons.Returning now to the Lagrangian field in eq. (2.1) where d =
3, the principle of least action states ( − a (cid:3) ) ∂ µ F µν = , (2.2)where (cid:3) ≡ ∂ α ∂ α . As it is clear from the perspective of any gauge theory, we must demand We should adopt the Greek indices run from 1 to 3 and natural units ¯ h = c = Using x = ix o = ict and − ds = dx α . et all. based on covariant assumption of ME imposed the well known Lorentzgauge Ω L [ A ] = ∂ µ A µ [11]. Indeed, GQED presents new constrains on the gauge field due to factthere are more physical variables than ME, furthermore, we can argue that eq. (2.2) and its initialcondition destroy the Ω L [ A ] constrain [23]. Following the form of (2.1), we would guess Ω G [ A ] = ( − a (cid:3) ) ∂ µ A µ as a natural choice, nevertheless, the main evidence against shows the order of equationsof motion are no longer preserved. Continuing in this direction, we accomplish the problem of gaugefixing by the notorious nonmixing gauge Ω P [ A ] = ( √ − a (cid:3) ) ∂ µ A µ [24]. Therefore, incorporatingthe gauge framework Ω P [ A ] into the (2.1) with d =
3, we have L GQED = − F µν F µν − a ∂ µ F µβ ∂ α F αβ + ( ∂ µ A µ )( − a (cid:3) )( ∂ µ A µ ) . (2.3)The equations of motion now are ( − a (cid:3) ) (cid:3) A µ ( x ) = . (2.4)Here, the main aspect of free theory is that we can split only in the free case the gauge fieldinto two families of solutions A µ ( x ) = A µ Max ( x ) + A µ Pod ( x ) , where A µ Max ( x ) and A µ Pod ( x ) are Maxwelland Podolsky gauge field, respectively. Within this factorization, each dynamical field is furtherunderstood upon equation (2.4) as ( − a (cid:3) ) A µ ( x ) = A µ Max ( x ) , a (cid:3) A µ ( x ) = A µ Pod ( x )( − a (cid:3) ) A µ Pod ( x ) = , (cid:3) A µ Max ( x ) = A µ ( x ) = (cid:90) d p ( π ) ∑ λ = (cid:26) ε λµ ( p )( a ( p ) e ipx + a ∗ ( p ) e − ipx ) + η µ ( p ) ¯ a ( p ) e i ¯ px + η ∗ µ ( p ) ¯ a ∗ ( p ) e − i ¯ px ) (cid:27) (2.6)where p α = ( p , ip o ) and ¯ p α = ( p , i ¯ p o ) with p o = p and ¯ p o = ( + a p o ) / / a . The only non-vanishing commutation relations are [ a λ ( p ) , a ∗ λ (cid:48) ( p (cid:48) )] = δ p , p (cid:48) δ λλ (cid:48) = [ ¯ a λ ( p ) , ¯ a ∗ λ (cid:48) ( p (cid:48) )] . The masslessand massive polarization vector are ε λµ ( p ) and η µ ( p ) , respectively, with ε µλ ( p ) ε λ (cid:48) µ ( p ) = δ λ (cid:48) λ and η µ ( p ) η ∗ µ ( p ) = −
1. Now, we demand the construction of free propagator in terms of the commuta-tion relation of the gauge field A µ ( x ) at equal time [ A µ ( x ) , A ν ( x (cid:48) )] = − i δ µν D P ( x (cid:48) − x ) (2.7)where the expression in momentum space is D P ( x (cid:48) − x ) = − i ( π ) (cid:90) d pe ip ( x (cid:48) − x ) ( δ ( p ) − δ ( p + a − )) ε ( p ) . (2.8)5he propagator is a c -number. Subsequently, we are interesting in the Feynman propagator. Beforedoing so, it is important to separate (2.8) into the retarded D RP ( p ) = − Θ ( x o ) D P ( p ) and advanced D AP ( p ) = Θ ( − x o ) D P ( p ) propagator D RP ( x (cid:48) − x ) = ( π ) (cid:90) d pe ip ( x (cid:48) − x ) (cid:18) P p − P p + a − + i π ( δ ( p ) − δ ( p + a − )) ε ( p ) (cid:19) , (2.9a) D AP ( x (cid:48) − x ) = ( π ) (cid:90) d pe ip ( x (cid:48) − x ) (cid:18) P p − P p + a − − i π ( δ ( p ) − δ ( p + a − )) ε ( p ) (cid:19) , (2.9b)where P means the principal value, and ε ( p ) and Θ ( x o ) are defined as ε ( p ) ≡ p o | p o | , Θ ( p ) ≡ + ε ( p ) . (2.10)For completeness, we should determine the vacuum expectation value of anticommutation relationat equal time (cid:104) |{ A µ ( x ) , A ν ( x (cid:48) ) }| (cid:105) = δ µν D ( ) ( x (cid:48) − x ) (2.11)where D ( ) P ( x (cid:48) − x ) = ( π ) (cid:90) d pe ip ( x (cid:48) − x ) ( δ ( p ) − δ ( p + a − )) (2.12)The equations (2.7) and (2.11) can be further combined into the Feynman propagator D FP ( x )( x (cid:48) − x ) = − i ε ( x (cid:48) − x ) D P ( x (cid:48) − x ) + D ( ) P ( x (cid:48) − x ) = i ( π ) (cid:90) d pe ip ( x (cid:48) − x ) (cid:26) P p − P p + a − + i π ( δ ( p ) − δ ( p + a − )) (cid:27) . (2.13)In fact, this propagator regards as a superposition of positive and negative frequency propagatingin the future and past light cone, respectively. As we should see later, this causal correlation functionwill be a fundamental ingredient, in section 5, to explore the Møller scattering. Moreover, we are ina position to solve an inhomogeneous differential equation by elementary methods [25]. From theLagrangian (2.1), we find the Euler-Lagrange equation ( − a − (cid:3) ) (cid:3) A µ ( x ) = j µ ( x ) (2.14)where the general solution is A µ ( x ) = A ( ) µ ( x ) + (cid:90) d x (cid:48) D R ( x − x (cid:48) ) j µ ( x (cid:48) ) (2.15)where A ( ) µ ( x ) obeys the free equation (2.4). In the next section, we should address the basicstructures of Fermion propagator in HP. 6 The Dirac Field
To calculate the transition amplitude of Møller scattering, we should consider the Fermionic sectorfor a single spinor governed by the charge-symmetric Dirac Lagrangian L ψ = − [ ¯ ψ , ( γ · ∂ + m ) ψ ] − [ ¯ ψ ( γ · ←− ∂ + m ) , ψ ] (3.1)where ¯ ψ ≡ ¯ ψ b ( x ) and ψ ≡ ψ b ( x ) are Grassmannian operators . By the Hamilton’s principle, weobtain the dynamic of the system ( γ ab µ ∂ µ + δ ab m ) ψ b ( x ) = , (3.2a)¯ ψ a ( x ) ( γ ab µ ←− ∂ µ + δ ab m ) = ψ a ( x ) = ψ ∗ a ( x ) γ where γ µ = ( i σ , σ , σ ) with Pauli matrix σ i [25]. The solutions forequations of motion above are ψ a ( x ) = ∑ r = (cid:90) d p ( π ) ( ˆ a r ( p ) u ra ( p ) e − ipx + ˆ b † r ( p ) v ra ( p ) e ipx ) , ψ a ( x ) = ∑ r = (cid:90) d p ( π ) ( ˆ a † r ( p ) ¯ u ra ( p ) e ipx + ˆ b r ( p ) ¯ v ra ( p ) e − ipx ) (3.3)where ¯ u ( q ) = u ∗ ( q ) γ . In order to maintain the stability and unitarity of the free system, the cre-ation and annihilation operators of particle ( a r , a † r ) and antiparticle ( b r , b † r ) , respectively, must obeythe Fermi-Dirac distribution from the spin-statistic theorem [25], and thus fulfill the commutationrules at equal times given by { ˆ a ( p ) , ˆ a † ( p (cid:48) ) } = δ ( p − p (cid:48) ) , { ˆ b ( p ) , ˆ b † ( p (cid:48) ) } = δ ( p − p (cid:48) ) . (3.4)Moreover, we should formulate the necessary tools to interpret and calculate the perturbative ele-ment of S -matrix. We proceed in complete analogy with electromagnetism developed in the previoussection. Hence, one can see explicitly the anticommutation of fermion operators at equal time through(3.4), then (cid:8) ¯ ψ a ( x ) , ψ b ( x (cid:48) ) (cid:9) = − iS ba ( x (cid:48) − x ) . (3.5)Note that S ba is a invariant c -number and only a matrix in spinor space, where the expression inmomentum space is S ab ( x − x (cid:48) ) = − i ( π ) (cid:90) d pe ip ( x − x (cid:48) ) ( i γ p − m ) ab δ ( p + m ) ε ( p ) (3.6) The spinor indices b runs from 1 to 2. S ab is a free solution of Dirac equation ( γ · ∂ + m ) S ab ( x ) =
0. In asimilar fashion, we take the vacuum expectation value of spinor field commutation since one verifiesthe [ ¯ ψ a ( x ) , ψ b ( x (cid:48) )] is no longer a c -number. (cid:104) | [ ¯ ψ a ( x ) , ψ b ( x (cid:48) )] | (cid:105) = S ( ) ba ( x (cid:48) − x ) (3.7)where the matrix S ( ) ba is S ( ) ba ( x − x (cid:48) ) = ( π ) (cid:90) d pe ip ( x − x (cid:48) ) ( i γ p − m ) ba δ ( p + m ) . (3.8)Alternatively, we can identify the equation (3.5) as superposition of retarded S R ( x ) = − Θ ( x o ) S ( x ) and advanced S A ( x ) = Θ ( − x o ) S ( x ) propagators, for sake of simplicity the spinor indices are implicit.We then obtain the integral representation S R ( x ) = ( π ) (cid:90) d pe ixp ( ip γ − m )[ P p + m + i πε ( p ) δ ( p + m )] , (3.9a) S A ( x ) = ( π ) (cid:90) d pe ixp ( ip γ − m )[ P p + m − i πε ( p ) δ ( p + m )] . (3.9b)It will be clear that these representations together with eqs. (2.9a) and (2.9b) are the base to solvescattering amplitude in the framework of the Heisenberg picture. We are now in the position to brieflydrawn the general structure from the interacting system. We begin with ( γ · ∂ + m ) ψ ( x ) = g ( x ) (3.10)where g ( x ) is a product of field operators satisfying the requirement of locality and relativisticinvariance. It is possible to check with the aid of the standard techniques of the Green function andadequate boundary condition that the solution can be written as ψ ( x ) = ψ ( ) ( x ) − (cid:90) d x (cid:48) S R ( x − x (cid:48) ) g ( x (cid:48) ) (3.11)where ψ ( ) ( x ) is the solution of (3.2a). It is natural to explore the conjugated equation from (3.10)¯ ψ ( x )( γ · ←− ∂ + m ) = g ( x ) (3.12)where the general solution reads¯ ψ ( x ) = ¯ ψ ( ) ( x ) − (cid:90) d x (cid:48) g ( x (cid:48) ) S A ( x (cid:48) − x ) (3.13)where ¯ ψ a ( x ) solves the eq. (3.2b). According to Noether’s theorem in QED , the conservationof charge is connected with the abelian symmetry group U ( ) . The properties of the field in GQED left the abelian symmetry unchanged and requires by Lorentz invariant reason that the lagrangian ofparticles and interaction are equal to ME. As it is clear from U ( ) , the status of symmetry naturally8pens the way for the charge current density becomes j µ = e ¯ ψγ µ ψ [25] but we can also reduce thecurrent to guarantee a symmetrized definition [18, 19] j µ ( x ) ≡ ie [ ¯ ψ ( x ) , γ µ ψ ( x )] (3.14)this approach turns out to be convenient and should be instructive writing down explicitly thequantized operator current into normal products [ ψ ( x ) , γ µ ψ ( x )] = : ψ ( x ) γ µ ψ ( x ) : where ”::” corre-sponding to chronological ordering [25]. It is straightforward to notice the vacuum expectation valueof this quantized current vanishes (cid:104) | j µ ( x ) | (cid:105) = . (3.15)Before attempting to construct the Møller amplitude in the Podolsky frame, we should discuss theperturbative methods to find out the S -matrix. The objective of this section is to describe the S -matrix, which relates the structure of the physicalprocess, in the Heisenberg representation in order to circumvent the inconsistency already pointedout by Haag’s theorem [17]. Here, we follow the Klln methodology [18, 19] to shed some light onthe cross section of GQED experiments. One of the approaches towards a description of S -matrix isthrough the perturbative framework, in other words, expressing the matrix as a series in the power ofsmall coupling. We therefore focus our attention on how to build up the perturbative terms of S -matrixin the HP for GQED , which has a straight extension from
QED .Before beginning, the idea behind the IP is that we may separate the total Hamiltonian into afree and an interacting part. On the other hand, the HP admits one analogous process where wecan construct each operator in the Heisenberg representation as a homogeneous and inhomogeneoussuperposition of the solution equation [26].We now move on to present how S -matrix originates in the HP. To do this, we are interested inthe differential equations of motion without any reference to the spacelike surface as in IP [26, 27].Summing over eqs. (2.3), (3.1), and the minimal coupling j µ ( x ) A µ ( x ) , where j µ ( x ) is (3.14), wearrive at dynamics ( − a (cid:3) ) (cid:3) A µ ( x ) = − ie [ ¯ ψ ( x ) , γ µ ψ ( x )] = − j µ ( x ) , ( γ · ∂ + m ) ψ ( x ) = ieA µ ( x ) γ µ ψ ( x ) . (4.1)From systematic methods used in the previous sections, we can write down the set of solution forequations of motion above 9 ( x ) = ψ ( in ) ( x ) − ie (cid:90) d x (cid:48) S R ( x − x (cid:48) ) γ ν A ν ( x (cid:48) ) ψ ( x (cid:48) ) A µ ( x ) = A ( in ) µ ( x ) + ie (cid:90) d x (cid:48) D RP ( x − x (cid:48) )[ ¯ ψ ( x (cid:48) ) , γ µ ψ ( x (cid:48) ) ] (4.2)Since the dynamical variable above contain retarded functions, we can associate the incomingfield operators ( A ( in ) µ , ψ ( in ) ) as the initial values of ( A µ ( x ) , ψ ( x )) at x o → − ∞ . Quite naturally, wepossible consider solving the differential equations (4.1) by advanced singular function ψ ( x ) = ψ ( out ) ( x ) − ie (cid:90) d x (cid:48) S A ( x − x (cid:48) ) γ ν A ν ( x (cid:48) ) ψ ( x (cid:48) ) A µ ( x ) = A ( out ) µ ( x ) + ie (cid:90) d x (cid:48) D AP ( x − x (cid:48) )[ ¯ ψ ( x (cid:48) ) , γ µ ψ ( x (cid:48) )] (4.3)where ( A ( out ) µ , ψ ( out ) ) are defined as the limit x o → + ∞ for free field operator ( A ( ) µ , ψ ( ) ) andoften called ”outgoing” fields. Physically, they corresponds to the final value of ( A µ , ψ ) whenwe switching off adiabatically the interaction. In other words, the interaction vanishes in the limit x o → + ∞ and so the operator are governed by free field equations.In addition, it is also essential that incoming and outgoing fields obey the same commutationrelation of free-fields { ¯ ψ ( in ) a ( x ) , ψ ( in ) b ( x (cid:48) ) } = − iS ba ( x (cid:48) − x ) , [ A ( in ) µ ( x ) , A ( in ) ν ( x (cid:48) )] = − i δ µν D P ( x (cid:48) − x ) , [ A ( in ) µ ( x ) , ψ ( in ) b ( x (cid:48) )] = , { ¯ ψ ( out ) a ( x ) , ψ ( out ) b ( x (cid:48) ) } = − iS ba ( x (cid:48) − x ) , [ A ( out ) µ ( x ) , A ( out ) ν ( x (cid:48) )] = − i δ µν D P ( x (cid:48) − x ) , [ A ( out ) µ ( x ) , ψ ( out ) b ( x (cid:48) )] = . The justification of these commutation rules regards the fact that the free, incoming, and outgoingoperators share the differential equations of motion (4.1) without interactions. This scenario alsoillustrate the free field operators converge at ( x o → − ∞ ) and ( x o → + ∞ ) to incoming and outgoingfields, respectively, [26].We thus reach, without difficult, an important conclusion where a canonical transformation mustconnect the asymptotic fields ( A ( in ) µ , ψ ( in ) ) and ( A ( out ) µ , ψ ( out ) ) since both of them settle down thesame canonical commutation relation. Then it follows ψ ( out ) ( x ) = S − ψ ( in ) ( x ) SA ( out ) µ ( x ) = S − A ( in ) µ ( x ) S (4.4)the operator S must be unitary since it relates two sets of orthogonal operators SS ∗ = S ∗ S = (cid:49) (4.5)In analogous fashion with (4.4), we emphasize the Hamiltonian at x o → + ∞ can be decomposedas function of the Hamiltonian at x o → − ∞ in the form10 ( ) ( ψ ( out ) , A ( out ) µ ) = S − H ( ) ( ψ ( in ) , A ( in ) µ ) S , (4.6)then one should be aware that the incoming and outgoing fields live in the same Hilbert space[18, 19]. At this point, we establish a general formulation as a first step towards the perturbativeformulation of the scattering matrix by replacing eqs. (4.2) into (4.3) to get ψ ( out ) ( x ) = S ( − ) ψ ( ) ( x ) S = ψ ( ) ( x ) − ie (cid:90) d x (cid:48) S ( x − x (cid:48) ) γ ν A ν ( x (cid:48) ) ψ ( x (cid:48) ) A ( out ) µ ( x ) = S ( − ) A ( ) µ ( x ) S = A ( ) µ ( x ) + ie (cid:90) d x (cid:48) D P ( x − x (cid:48) )[ ¯ ψ ( x (cid:48) ) , γ µ ψ ( x (cid:48) )] (4.7)or rewriting to have a systematic view of S -matrix as [ S , ψ ( ) ] = − S (cid:90) d x (cid:48) S ( x − x (cid:48) ) ie γ µ A µ ( x (cid:48) ) ψ ( x (cid:48) ) , [ S , A ( ) µ ] = − S (cid:90) d x (cid:48) D ( x − x (cid:48) ) ie [ ¯ ψ ( x (cid:48) ) , γ µ ψ ( x (cid:48) )] (4.8)For practical computation and without losing the physical meaning, we adopted the notation ( A ( ) , ψ ( ) ) to incoming fields. Using the assumption of the small gauge coupling ( e / π ∼ / ) and expanding the operators ( S , ψ , A µ ) in power series of e . We get S = (cid:49) + eS ( ) + e S ( ) + . . . ψ ( x ) = ψ ( ) ( x ) + e ψ ( ) ( x ) + e ψ ( ) ( x ) + . . . , A µ ( x ) = A ( ) µ ( x ) + eA ( ) µ ( x ) + e A ( ) µ ( x ) + . . . . (4.9)Substituting (4.9) into (4.8), the first approximation reads [ S ( ) , ψ ( ) ( x )] = − S (cid:90) d x (cid:48) S ( x − x (cid:48) ) ie γ µ A ( ) µ ( x (cid:48) ) ψ ( ) ( x (cid:48) ) , (4.10a) [ S ( ) , A ( ) µ ( x )] = − S (cid:90) d x (cid:48) D ( x − x (cid:48) ) ie [ ¯ ψ ( ) ( x (cid:48) ) , γ µ ψ ( ) ( x (cid:48) )] . (4.10b)After some manipulation, the form of S -matrix is S ( ) = − ie (cid:90) d x : ¯ ψ ( ) ( x ) γ µ ψ ( ) ( x ) : A ( ) µ ( x ) . (4.11)To help visualize this result, see eq (2.7). The calculation for the second order of S -matrix isstraightforward S ( ) = e (cid:90) d x (cid:48) d x (cid:48)(cid:48) T ( : ¯ ψ ( ) ( x (cid:48) ) γ ν ψ ( ) ( x (cid:48) ) :: ¯ ψ ( ) ( x (cid:48)(cid:48) ) γ ν ψ ( ) ( x (cid:48)(cid:48) ) : ) × T ( A ( ) ν ( x (cid:48) ) A ( ) ν ( x (cid:48)(cid:48) )) . (4.12)Despite the considerations above appear complicated, the S ( n ) terms are very direct. We shouldunderline that the link between HP and IP has only the same mathematical structural form of pertur-bative expansion but the underlying physical concepts involved turn these framework into different11ddress scattering matrix. The advantage of the HP is clear, there is no need to recover the space-likesurfaces [27].In the remainder of this paper, the core idea from Haag’s theorem is that the free and interactingHamiltonian from IP act on orthogonal Hilbert spaces, in other words, the mathematical inconsistencyresumes in the lack of a global unitary transformation relating both of them [17], as the argumentshown in eq. (4.4). Thus, the S -matrix in the HP is not ruined by Haag’s theorem and thus the HPframework is appropriated to carry out scattering process.In the next section, we are ready to calculate the analytical solution of Møller scattering at tree-level. Now that we are more familiar with the Podolsky theory and display essential ideas to deal withthe scattering matrix. We should move on to a better understanding of the scattering process in
GQED . We focus on showing how Klln methodology [18, 19] in the HP is an attractive viewpoint toaddress the perturbative apparatus of the scattering process without worrying about Haag’s theorem.The main objective is to determine the Podolsky corrections for Møller scattering ( e − e − → e − e − ) attree-level. Starting with the incoming | p , q (cid:105) and outgoing | p (cid:48) , q (cid:48) (cid:105) states of electrons | p , q (cid:105) = a ∗ ( r i ) ( p ) a ∗ ( s i ) ( q ) | (cid:105)| p (cid:48) , q (cid:48) (cid:105) = a ∗ ( r f ) ( p (cid:48) ) a ∗ ( s f ) ( q (cid:48) ) | (cid:105) (5.1)where ( p , r i ) and ( q , s i ) are the momentum and spin of ingoing particle, and ( p (cid:48) , r f ) and ( q (cid:48) , s f ) are the momentum and spin of outgoing particle. Being | (cid:105) the vacuum state. By the spin-statisticstheorem, the interchanges of identical particles must follows the rule | p , q (cid:105) = −| q , p (cid:105) . (5.2)After that we are able evaluate the cross section for the process e − e − → e − e − . The first non-vanishing approximation involves the second order of S -matrix element (4.12), mapping the asymp-totic initial states onto final ones (cid:104) q (cid:48) , p (cid:48) | S | p , q (cid:105) = − e (cid:90) d x (cid:48) d x (cid:48)(cid:48) [ (cid:104) q (cid:48) | ¯ ψ ( x (cid:48) ) | (cid:105) γ ν (cid:104) | ψ ( x (cid:48) ) | q (cid:105)(cid:104) p (cid:48) | ¯ ψ ( x (cid:48)(cid:48) ) | (cid:105) γ ν (cid:104) | ψ ( x (cid:48)(cid:48) ) | p (cid:105)− (cid:104) q (cid:48) | ¯ ψ ( x (cid:48) ) | (cid:105) γ ν (cid:104) | ψ ( x (cid:48) ) | p (cid:105)(cid:104) p (cid:48) | ¯ ψ ( x (cid:48)(cid:48) ) | (cid:105) γ ν (cid:104) | ψ ( x (cid:48)(cid:48) ) | q (cid:105) ] δ ν ν D FP ( x (cid:48) − x (cid:48)(cid:48) ) . (5.3)Substituting the Feynman propagator (2.13) and matrix elements (3.3), we obtain (cid:104) q (cid:48) , p (cid:48) | S | p , q (cid:105) = ie A (cid:20) ¯ u ( q (cid:48) ) γ λ u ( q ) ¯ u ( p (cid:48) ) γ λ u ( p )( p − p (cid:48) ) ( + ( p − p (cid:48) ) m P ) − ¯ u ( q (cid:48) ) γ λ u ( q ) ¯ u ( p (cid:48) ) γ λ u ( p )( p − q (cid:48) ) ( + ( p − q (cid:48) ) m P ) (cid:21) × ( π ) δ ( p + q − p (cid:48) − q (cid:48) ) (5.4)12here A is the area. In what follows, the omission of factor 2 occurs because of two possibilitiesfor the combination of operators ¯ ψ ( x (cid:48) ) and ¯ ψ ( x (cid:48)(cid:48) ) and the negative signal proceeds from eq. (5.2). Inthe present context, the Podolsky propagator above shows the separation in two different families ofthe solution is no longer possible for interacting system, as we discussed in the eq. (2.5) .We can infer the leading order of S -matrix elements in HP may be identified with the Feynmanrules and thus the interpretation suggests a possible one-to-one interplay between eq. (5.3) and Feyn-man diagrams. Nevertheless, as we well know, the standard quantum field theory in IP relies onthe direct association between Feynman techniques and expansion ”a l” Dyson. This statement hasnot been proved mathematically and remains more ignored when infrared divergences appear in theFeynman amplitude. We therefore continue to work with the position specified in the Heisenbergrepresentation to obtain the differential cross section.Now, we should infer the construction of the cross section with sufficient care to conduct thesituation from a reduced dimension standpoint. The operation to determine this physical variable ofinterest could be applied in the same way as ( + ) dimension. After taking the probability per unitof time from (5.4) and dividing by the flux of incident particle v rel / A (flux rate across a curve), where v rel is the relative velocity. The incoming and outgoing states are then normalized to one particle perunit of area. We rather get σ = α v rel (cid:90) (cid:90) d p (cid:48) d q (cid:48) E p E q E p (cid:48) E q (cid:48) (cid:20) A ( p − p (cid:48) ) ( + ( p − p (cid:48) ) m P ) + B ( p − q (cid:48) ) ( + ( p − q (cid:48) ) m P ) + C ( p − p (cid:48) ) ( p − q (cid:48) ) ( + ( p − p (cid:48) ) m P )( + ( p − q (cid:48) ) m P ) (cid:21) δ ( p + q − p (cid:48) − q (cid:48) ) . (5.5)For sake of clarity, we averaged over ingoing spin states since they are unpolarized and sum ofall possible spin state of outgoing particles. The α = e / π is the fine constant structure. With theaid of the equations (3.3), the completeness relation of quantized wave equation are ∑ r = ¯ u α u β =( i γ · q + − m ) β α / E and ∑ r = ¯ v α v β = − ( i γ · q − − m ) β α / E , where q + = ( q , iE ) and q − = ( q , − iE ) .The capital letters above indicate the summing over spin indices A = Tr (cid:2) γ λ ( i γ · p − m ) γ ν (cid:0) i γ · p (cid:48) − m (cid:1)(cid:3) · Tr (cid:104) i γ λ ( i γ · q − m ) γ ν (cid:0) i γ · q (cid:48) − m (cid:1)(cid:105) , (5.6a) B = Tr (cid:2) γ λ ( i γ · p − m ) γ ν (cid:0) i γ · q (cid:48) − m (cid:1)(cid:3) · Tr (cid:104) i γ λ ( i γ · q − m ) γ ν (cid:0) i γ · p (cid:48) − m (cid:1)(cid:105) , (5.6b) C = − Tr (cid:104) γ λ ( i γ · p − m ) γ ν (cid:0) i γ · q (cid:48) − m (cid:1) γ λ ( i γ · q − m ) γ ν (cid:0) i γ · p (cid:48) − m (cid:1)(cid:105) . (5.6c)The new features of GQED manifest in the properties of γ -matrices, in the sense, they obeythe so ( , ) algebra [ γ µ , γ ν ] = − i ε µνλ γ λ and { γ µ , γ ν } = g µν [25]. We now define the Mandelstam13ariables to describe future Lorentz invariant quantities s = ( p + q ) = ( p (cid:48) + q (cid:48) ) = − m + ( p . q ) , (5.7a) t = ( p − p (cid:48) ) = ( q − q (cid:48) ) = − m − ( p . p (cid:48) ) , (5.7b) u = ( p (cid:48) − q ) = ( p − q (cid:48) ) = − m − ( p (cid:48) . q ) . (5.7c)Finally, after applying eqs. (5.7a) to (5.7c) into eqs. (5.6a) to (5.6c) together with the algebra ofDirac matrices, we get A = ( s + u − t / ) + m ( s + u − t / ) + m , (5.8a) B = ( s + t − u / ) + m ( s + t − u / ) + m , (5.8b) C = ( s − u − t ) + m ( s − u − t ) + m . (5.8c)Next, one has to calculate the relative velocity to compute the scattering. The ordinary physicalmeaning of this concept infers as the velocity of a particle in relation to an observer in the rest frameof the other one. If the observer is at rest frame of q particle. We have v rel E p E q = mE p | p | E p = (cid:113) m E p − m . (5.9)However, this argument is not physically well defined to relativistic phenomena. Adopting v rel E p E q in a relativistic invariant way. It is easy to see that v rel = E q E p (cid:113) ( p · q ) − m . (5.10)We now turn our attention to the analysis of the invariant differential cross section from eq. (5.5)by using the ”normalized” Lorentz invariant quantity λ = t / m to get d σ dt = r o (cid:112) ( p · q ) − m (cid:20) At ( + tm P ) + Bu ( + um P ) + Cut ( + tm P )( + um P ) (cid:21) · I (5.11)where the classical electron radius r o = α m is dimensionless. Let us first start with the Lorentzinvariant phase space integral I that is given by I = (cid:90) (cid:90) d p (cid:48) d q (cid:48) E p (cid:48) E q (cid:48) δ ( p + q − p (cid:48) − q (cid:48) ) δ ( λ − t m )= (cid:90) (cid:90) d p (cid:48) d q (cid:48) δ ( p (cid:48) + m ) δ ( q (cid:48) + m ) Θ ( p (cid:48) ) Θ ( q (cid:48) ) δ ( p + q − p (cid:48) − q (cid:48) ) δ ( λ − t m )= m (cid:112) tm | q | + st Θ ( t m ) Θ ( E q m − − t m ) (5.12)14n the rest frame of the particle p . Moving to an arbitrary system of coordinate I = m √− s √ tu Θ ( t m ) Θ ( u m ) . (5.13)This result enables us to write the differential cross section (5.11) in a relativistic invariant fashion.The influence of factor ( √ tu ) concerns the transverse asymmetry even if the radiative correctionfrom soft and hard photons are absent. This characteristic is reminiscent of ( + ) dimensions,whereas in ( + ) dimensions the effect is only supposed to be presented when infrared radiations areincluded [28]. Moreover, we recover the usual QED cross section from (5.11) in the limit m P → ∞ because the Podolsky Feynman propagator turns into the Maxwell one. The effects in the higherenergy regime m (cid:28) | s | ∼ t ∼ u . (5.14)Note that in this present situation, the form of Lorentz-invariant cross section (5.11) is d σ dt = α ( √− s ) √ tu (cid:20) s + u − t / t ( + tm P ) + s + t − u / u ( + um P ) + s − u − t ut ( + tm P )( + um P ) (cid:21) . (5.15)The reader may wonder about this energy level where we find the direct influence of the Podolskyparameter. This regime is not interesting since the particles may archive high values of energy andthe Podolsky mass cannot play the UV cutoff role. On the order hand, it leads immediately to a subtleinvestigation of the bound value of m P . The next step will be favorable for our purposes since theparticles have the energy value restrict to the following bounds m ≤ | s | ∼ t ∼ u < m P . (5.16)This regime points m P as a natural higher-frequency cutoff. Although this condition of energycould appear inadequate to conduct analysis in physical system of interest, we should recall the re-search in condensed matter with relativistic behavior is recent, for instance, considering the investi-gation on electron-electron interaction in graphene structure [32].As a result of the leading contribution √ tm P in the eq. (5.15), we have d σ dt = α ( √− s ) √ tu (cid:20) s + u − t / t + s + t − u / u + s − u − t ut − m P (cid:18) s + u − t t + s + t − u u (cid:19)(cid:21) . (5.17)It is quite interesting to evaluate in the center mass system p = ( E i , p , ) , q = ( E i , − q , ) , p (cid:48) = ( E f , p cos θ , p sin θ ) , q (cid:48) = ( E f , − q cos θ , − q sin θ ) (5.18)15here θ is the scattering angle in the center of mass frame and energy-momentum conservationresults E i = E f and p = q , i.e. ( p · q ) = p cos θ . Hence, we can rewrite the Mandelstam variable(5.7a)-(5.7c) as s = − E , u = p cos θ , t = p sin θ , (5.19)we thus obtain the cross section in the center of mass d σ d θ = α E ( + cos 2 θ ) sin θ − α m P E ( − θ + cos θ ) sin θ . (5.20)At this point, the second term plays the central role of the correction in the relativistic limit from GQED , while the first one concerns the usual QED contribution. Moreover, the small deviation forMøller scattering at tree-level cross section may be calculated with the formula δ = (cid:18) d σ d θ (cid:19) GQED (cid:30)(cid:18) d σ d θ (cid:19) QED − . (5.21)The most experimental advances in particle physics require the study of Møller scattering withhigher precision even in higher energy reactions [30]. The quantum ideas developed so far fit nicelywith quasiplanar structures of condensed matter and renew the view of charge confinement and screen-ing coulomb potential since a notorious quality of GQED comes from the contribution of ”Yukawa”positive electrostatic potential in ( − e / r )( − e − r m P ) [31]. Then under certain circumstances we canachieve an attractiveness global interaction and expect the theoretical features of GQED be signifi-cant to tackle with an effective description of condensed matter.The leading order correction for small angles θ (cid:28) δ = − (cid:18) sm P (cid:19) θ + O ( θ ) + . . . . (5.22)The small angles report the kinematical region of low energy where the dominant process is t -channel. The GQED lower order effect for the cross section in terms of small θ angle is d σ d θ = α E θ (cid:20) − (cid:18) + E m a (cid:19) θ + O ( θ ) + . . . (cid:21) (5.23)In the above result, the GQED contribution turns out at second order and decreases the differen-tial cross section. We start by taking the electron mass m = , MeV and considering the energylimit where Møller scattering is relevant < MeV [33]. Thus, this constraint could inform the realaccuracy of correction suggested by the generalized electrodynamics for θ = ◦ , i.e., | δ | ≤ . − %.An recent measurement of e − e − in cold atoms lead us to E (cid:39) . MeV , within the estimation ofMøller regime, and the order of correction for θ = ◦ , | δ | ≤ − % [5]. One could argue the experi-ments to Møller scattering are not appropriate to detect the Podolsky mass effects, otherwise it wouldalready be detected. 16o discuss this challenge, we should map the phenomenological objects in different physical con-texts by suitable choices. Since adopting GQED as the analogous model for planar structure in thecondensed matter system, we should link up the free parameters of theory with important observable.To carry this idea, we should therefore be able to identify the theoretical similarity between GQED cutoff with experimental data. These results must be compatible with the standard interpretation ofthe planar quantum field theory.Before attempting further progress, we remark the boundness limit of Podolsky parameter in GQED was found from the anomalous magnetic momentum m P ≥ .
59 GeV [29] and the groundstate of Hydrogen m P ≥ . MeV [31]. To propose a correspondent model for condensed matter,we should replace the cutoff m P by the uncertainty in determining the interatomic distance since theknown ME results are in agreement with experiments that depend on interatomic distance in crystallattice [34]. So considering the measuring from [36] we roughly argue a ≤ f m or m P ≥ . MeV .It is therefore possible to estimate the
GQED corrections, | δ | ≤ . θ = ◦ [5].Finally, we explore the nonrelativistic limit where the long-distance physics is independent of m P .In this approximation, the energy of particles are E q = E p = E q (cid:48) = E p (cid:48) = m + p m and the cross sectionin the center of mass system reads d σ d θ = r o m p (cid:20)(cid:18) θ + θ − θ sin θ (cid:19) + p m P (cid:18) θ + θ (cid:19)(cid:35) (5.24) θ ° σ Figure 1: Scattering electrons compared with scattering angle. The blue and orange lines are thecross section using α = . α = .
5, respectively.The first term represents the nonrelativistic ordinary
QED , whereas the second one shows theleading ”Yukawa” correction for the differential cross section from the massive contribution of Podol-sky’s potential [11]. Judging the form of differential cross section, unlike what happens in the fermion17ector where the algebra of gamma matrices set up the planar dynamic, the gauge propagator merelyplays the same similar characteristic in GQED , namely, living in a spatial bulk space [35].We show a simulation for (5.24) in the graph of Fig. 1 based on the data [32], where the effectivefine structure is α (cid:48) = e π v F ∼ . − . v F . The typical form relates a coulombdifferential cross section in which diverges at forwarding angles.It is essential to realize which aspects from a Chern-Simons (CS) term would imply in our model.We would like to emphasize which in QED even though the CS is absent in the lagrangian (2.3),this term can be induced by photon self-energy at one loop order [37]. Beyond that, we can find,without any loss of generality, the solution from the spontaneous breaking of parity, which is thesame physical predictions if the CS is present in the eq. (2.3) [38]. Given the last remarks, if wetake into account radiative correction for evaluating the differential cross section, we should expect afavorable scenario involving results from the induced CS term. In this paper, we have analyzed the Møller scattering in the framework of generalized quantumelectrodynamics in ( + ) dimensions. One immediate feature of the GQED framework was theunitarity and stability. We also derived the gauge propagator, which preserves the U ( ) symmetry,with a massless and massive photon .We have presented the covariant formalism of Klln’s method where the S -matrix can be con-structed directly in the Heisenberg picture. The main advantage of this representation was to removethe mathematical inconsistency barrier, so far demonstrated by Haag’s theorem, associated with thecanonical approach of the Interaction picture. We analyzed the structure of S -matrix, which was fa-cilitated through the integral equation of motion, and collected the S ( ) matrix amplitude to evaluatethe tree-level Møller scattering.The richness outcomes offered by QED allowed an understanding of the elusive effects of con-densed matter. However, due to a lack of concrete picture in some of these crucial phenomena, weproposed the GQED framework where ”short-ranged force” encoded by massive photon arisen natu-rally. Moreover, even though GQED easily copes with higher energy mechanism, Podolsky’s theorywas first investigated in the classical regime and the parameter m P played an important role in reg-ularizing classical problems, as mentioned in the introduction. In this spirit, we hoped the cutoffparameter m P translated into condensed matter language could facilitate the investigation into the roleof planar matter interactions.The remarkable contribution in this paper was to show the Møller differential cross section in QED and ensuing GQED . As we known, a topological mass is generated in quantum radiation atone loop order and if this mass exceeds the electron one, we have a favorable scenario to attractivepotential (cooper pair) in condensed matter. Hence, we showed GQED instead of QED createsthe proper conditions to work in the perturbative approach due to the energy regime: m ≤ p < m P ,where the cutoff is m P ≥ . MeV , and offer a better physical estimation of the future deviations18rom ME.By fundamental arguments of quantum field theory, we paved the road to establish a systematicinvestigation of radiative effects in the differential cross section. We can extend this formalism toincorporate quantum fluctuations since we already known the principal ideas of generalized electro-dynamics in the Heisenberg representation at tree-level process. We believe the higher orders of crosssection from
GQED can also be interesting to understand the attractive mechanism of Cooper pair,charge confinement, dynamical mass generation and the connection of the Chern-Simons term withphysical observable in the condensed matter. A further study in this formalism will be elaborated in aforthcoming work. Acknowledgments
David Montenegro thanks to CAPES for full support.
References [1] P.L. Anthonyet al.(The SLAC E158 Collaboration), Phys. Rev. Lett.95, 081601 (2005).[2] M. M. Ferreira, Jr., Phys. Rev. D , 045013 (2004) doi:10.1103/PhysRevD.70.045013; H. Be-lich, O. M. Del Cima, M. M. Ferreira, Jr. and J. A. Helayel-Neto, J. Phys. G , 1431 (2003)doi:10.1088/0954-3899/29/7/309.[3] ”Quantum Hall Effect in 2+1 Quantum Electrodynamics,” with R. Acharya, Nuovo CimentoB 107B, 351 (1992); Relativistic QED in 2+1 space-time dimensions: Integer and FractionalQuantized Hall Effects,” with R. Acharya, Int. Journal Mod. Phys. A9 861, 1994.[4] R. Dillenschneider and J. Richert, Phys. Rev. B , 144404 (2006)doi:10.1103/PhysRevB.74.144404 [cond-mat/0606721].[5] Klar, Leonhard, Nikodem Szpak, and Ralf Schtzhold. ”Quantum simulation of spontaneous paircreation in 2D optical lattices.”[6] T. Uehlinger, G. Jotzu, M. Messer, D. Greif, W. Hofstetter, U.Bissbort, and T. Esslinger,Phys.Rev. Lett., 111 185307 (2013).[7] P. Narayana Swamy, Nuovo Cim. A , 45 (1996). doi:10.1007/BF02734428[8] Myers, Robert C. ”Higher-derivative gravity, surface terms, and string theory.” Physical ReviewD 36.2 (1987): 392.[9] Hatefi, E. (2012). Higher derivative corrections to Wess-Zumino and tachyonic actions in typeII super string theory. Physical Review D, 86(4), 046003.1910] Dine, Michael, and Nathan Seiberg. ”Comments on higher derivative operators in some SUSYfield theories.” arXiv preprint hep-th/9705057 (1997).[11] B. Podolsky, Phys. Rev. , 68 (1942); B. Podolsky and C. Kikuchy, Phys. Rev. , 228 (1944);B. Podolsky and P. Schwed, Rev. Mod. Phys. , 40 (1948).[12] J. Frenkel, Phys. Rev. E , 5859 (1996). doi:10.1103/PhysRevE.54.5859[13] R. R. Cuzinatto, C. A. M. de Melo and P. J. Pompeia, Annals Phys. , 1211 (2007)doi:10.1016/j.aop.2006.07.006 [hep-th/0502052].[14] M. Ostrogradski, Mem. Ac. St. Petersburg VI , 385 (1850); R. Weiss, Proc. R. Soc. A , 102(1938); J. S. Chang, Proc. Cambridge Philos. Soc. , 76 (1948).[15] D. S. Kaparulin, S. L. Lyakhovich and A. A. Sharapov, Eur. Phys. J. C , no. 10, 3072 (2014)doi:10.1140/epjc/s10052-014-3072-3 [arXiv:1407.8481 [hep-th]].[16] A. A. Nogueira and B. M. Pimentel, Phys. Rev. D , no. 6, 065034 (2017)doi:10.1103/PhysRevD.95.065034 [arXiv:1607.01361 [hep-th]].[17] R. Haag, Dan. Mat. Fys. Medd. (1955) 12; A. S. Wightman and Schweber, Phys. Rev. (1955) 812; R. F. Streater, A. S. Wightman, PCT, Spin and Statistics, and All That, Ben-jamin/Cummings, Advanced Book Program, Reading, Massachusetts (1978).[18] G. K¨all´en. Arkiv f¨or Fysik, , 187 (1950); G. Klln. Arkiv fr Fysik, , (1950) 371.[19] G. Klln, Quantum Electrodynamics (Springer-Verlag, Berlin, 1972).[20] J. T. Lunardi, L. A. Manzoni, B. M. Pimentel, Int. J. Mod. Phys. A, , 3263 (2000).[21] J. Beltran, N. T. Maia and B. M. Pimentel, Int. J. Mod. Phys. A , no. 10, 1850059 (2018)oi:10.1142/S0217751X18500598 [arXiv:1710.01571 [hep-th]].[22] B. M. Pimentel, A. T. Suzuki, J. L. Tomazelli, Int. J. Theor. Phys. , 2199 (1994).[23] M. C. Bertin, B. M. Pimentel and G. E. R. Zambrano, J. Math. Phys. , 102902 (2011)doi:10.1063/1.3653510 [arXiv:0907.1078 [hep-th]].[24] C. Lmmerzahl, J. Math. Phys. , 9 (1993); R. S. Chivukula, A. Farzinnia, R. Foadi andE.H.Simmons, Phys. Rev. D , 035015 (2010); R. Bufalo, B. M. Pimentel and D. E. Soto,Phys.Rev. D , 085012 (2014)[25] G. Scharf, Finite Quantum Electrodynamics: The Causal Approach , Third Edition. (Dover Pub-lications, 2014).[26] Yang, Chen-Ning, and David Feldman. ”The S-matrix in the Heisenberg representation.” Physi-cal Review 79.6 (1950): 972. 2027] Dyson, F. J. ”Heisenberg operators in quantum electrodynamics. I.” Physical Review 82.3(1951): 428.[28] N. M. Shumeiko and J. G. Suarez, hep-ph/9712407.[29] R. Bufalo, B.M. Pimentel and G.E.R. Zambrano, Phys. Rev. D , 125023 (2012).[30] A. Aleksejevs, S. Barkanova, A. Ilyichev and V. Zykunov, Phys. Rev. D , 093013 (2010)doi:10.1103/PhysRevD.82.093013 [arXiv:1008.3355 [hep-ph]].[31] R. R. Cuzinatto, C. A. M. de Melo, L. G. Medeiros and P. J. Pompeia, Int. J. Mod. Phys. A ,3641 (2011) doi:10.1142/S0217751X11053961 [arXiv:0810.4106 [quant-ph]].[32] Effect of Coulomb interactions on the physical observables of graphene M.A.H. Vozmediano, F.Guinea Phys.Scripta T 146 (2011) 014015.[33] C. Epstein, R. Johnston, S. Lee, J. Bernauer, R. Corliss, K. Dow, P. Fisher, I. Friscic, D. Hasell,R. Milner, P. Moran, S. Steadman, Y. Wang, J. Dodge, E. Ihloff, J. Kelsey, C. Vidal and C. Cooke,[arXiv:1903.09265 [nucl-ex]].[34] Slezak, J. A., et al. ”Imaging the impact on cuprate superconductivity of varying the interatomicdistances within individual crystal unit cells.” Proceedings of the National Academy of Sciences105.9 (2008): 3203-3208.[35] D. Dudal, A. Mizher and P. Pais, Phys. Rev. D , no.6, 065008 (2018)doi:10.1103/PhysRevD.98.065008.[36] Razik, N.A. Precise lattice constants determination of cubic crystals from x-ray powder diffrac-tometric measurements. Appl. Phys. A 37, 187189 (1985).[37] A. N. Redlich, ”Gauge Noninvariance and Parity Violation of Three-DimensionalFermions”,Phys. Rev. Lett.52