Path integral quantization of generalized quantum electrodynamics

Abstract

In this paper, a complete covariant quantization of generalized electrodynamics is shown through the path integral approach. To this goal, we first studied the hamiltonian structure of system following Dirac's methodology and, then, we followed the Faddeev-Senjanovic procedure to obtain the transition amplitude. The complete propagators (Schwinger-Dyson-Fradkin equations) of the correct gauge fixation and the generalized Ward-Fradkin-Takahashi identities are also obtained. Afterwards, an explicit calculation of one-loop approximation of all Green's functions and a discussion about the obtained results are presented. * rbufalo@ift.unesp.br † pimentel@ift.unesp.br ‡ gramos@udenar.edu.co [3]. Also, the use of higher derivative terms becomes interesting regulatorr, by the fact that it improves the convergence of the Feynman diagrams [4].More examples of systems treated with high-order Lagrangians that we can mention are: the study of the problem of color confinement on the infrared sector of QCD 4 [5], the attempts to solve the problem of renormalization of the gravitational field [6], and a generalization of Utiyma's theory to second-order theories [7]. Although all these works improve the use of higher-order terms, the ones that most contributed to show the effectiveness of such terms in field theory was the contributions of Bopp [8], and Podolsky and Schwed [9], where they proposed a generalization of the Maxwell electromagnetic field. They wanted to get rid of the infinities of the theory, such as the electron self-energy (r −1 singularity) and the vacuum polarization current present on the Maxwell theory. The modification suggested by Podolsky and Schwed handle these unsolved problems and, also, gives a positive definite energy in the electrostatic case; also, as showed by Frenkel [10], it gives the correct expression for the self-force of charged particles. In [7], it was shown that the Podolsky Lagrangian is the only possible generalization of Maxwell electrodynamics that preserves invariance under U (1).On theoretical and experimental framework, efforts have been made to determine an upper-bound value for the mass of the photon [11], the existence of a massive sector being a prediction of generalized electrodynamics. Along this line of thought, we believe that a way to set limits over Podolsky parameter will be to study the Podolsky's photons interacting with standard model particles, and compare the obtained results with high-energy experiments. This idea and other purposes led Podolsky and some of his students to study the interaction of electrons with the Podolsky photons, which they called generalized quantum electrodynamics (GQED 4 ) [12]. Among the points dealt with in their thesis, the most interesting was the calculation of electron self-energy at a one-loop approximation. They expected that the contribution of massive photons lead to a finite result; nevertheless, in the end, they found, as in the usual QED 4 , a divergent expression. Analyzing, now, the thesis results, we found a mistake in their tr...

Renormalizability of generalized quantum electrodynamics

“…This is a major issue and certainly cannot be neglected, and requires a detailed study; nevertheless, once we have identified the primitive divergences on the 1P I functions [18] and the theory rendered UV finite (at the light of effective theories) we are motivated to retain some attention regarding the underlying aspects of the renormalized behavior of the present theory. We expect to obtain from comparison of our results to the regular expressions for the effective coupling and one-loop contribution for the anomalous magnetic moment, an energy regime of validity of the theory and a bound value for the free parameter, respectively.…”

“…According to the Lagrangian (2) we obtain here new Schwinger-Dyson-Fradkin equations for the theory, which we shall denote by the suffix (R) . More precisely, the original self-energy functions [18] are changed by adding the counter-terms δ Zi . Now, by a formal derivation it is not complicated to show that the photon sector has the following renormalized self-energy function:…”

“…We begin this section by presenting the explicit expressions for the one-loop radiative corrections functions at a general ξ [18]. The expression for the polarization scalar reads as [23]:…”

“…[18] through functional methods. In contrast to the results obtained in the usual Lorenz condition [19] it was found that, in the one-loop approximation, the electron self-energy and the vertex function are both ultraviolet finite.…”

“…Thus, at one-loop calculation, the electron self-energy expression can be casted into the following form [18]:…”