Schwinger-Dyson equations and currents in lattice gauge theory

After introducing appropriate derivatives, the structure of Schwinger-Dyson equations, currents and Ward-Takahashi identities (including the anomalous ones) on a finite lattice is completely clarified. A general relation between correlation functions without and with gauge fixing is given.     

Normalized Schwinger-Dyson equations in Quantum Electrodynamics

We obtain Schwinger-Dyson equations for the normalized two-point propagators in Quantum Electrodynamics. Exploiting the spectral representations of the Green functions, we deal with normalized quantities everywhere. The explicit form we obtain for the propagator equations allows for non-perturbative approaches.1 On leave from Instituto de Física, UNAM, Mexico.     

Quantum gauge equivalence in QED

We discuss gauge transformations in QED coupled to a charged spinor field, and examine whether we can gauge-transform the entire formulation of the theory from one gauge to another, so that not only the gauge and spinor fields, but also the forms of the operator-valued Hamiltonians are transformed. The discussion includes the covariant gauge, in which the gauge condition and Gauss's law are not primary constraints on operator-valued quantities; it also includes the Coulomb gauge, and the spatial axial gauge, in which the constraints are imposed on operator-valued fields by applying the Dirac-Bergmann procedure. We show how to transform the covariant, Coulomb, and spatial axial gauges to what we call “common form,” in which all particle excitation modes have identical properties. We also show that, once that common form has been reached, QED in different gauges has a common time-evolution operator that defines time-translation for states that represent systems of electrons and photons. By combining gauge transformations with changes of representation from standard to common form, the entire apparatus of a gauge theory can be transformed from one gauge to another.     

Curing fermion mass gauge variance in QED

The problem of the gauge dependence of the fermion mass in the Maxwell-Chern-Simons QED is revisited. Using Proca mass term as an intermediate infrared regulator we are demonstrating gauge-invariance of the fermion mass shell in QED in all orders of the perturbation theory.     

Manifest gauge and Poincaré covariance

Manifest gauge invariance is known to be incompatible with manifest Poincaré covariance (Strocchi's theorem). By extending the notion of gauge invariance to that of gauge covariance, we circumvent that incompatibility, at least for free electromagnetic potentials. In the new formulation the potentials, A_G, for all permissible gauges G. act on a common Hilbert space. This formulation is shown to be inequivalent to the more conventional ones. (In particular, the Coulomb gauge is now inaccessible.) The abstract gauges G are represented by c-number potentials V_G, which play a central role in the theory. Even without interaction, they obey a field equation with a source, and thus they anticipate the existence of electric charges.     

Non-linear equations and covariance

A discussion about the connection between non-linear wave equations covariant under the action of a Lie group on one hand, and the theory of non-linear representations of the covariance group developed elsewhere [1] on the other hand, is presented here.Applications to a class of relativistic non-linear wave-equations are then suggested.     

Hamiltonian formulation of QED in the superaxial gauge

We present a hamiltonian formulation of QED in a fully fixed axial gauge. The equal-time commutators for all field variables are computed and are shown to lead to the correct equations of motion. The constraints and gauge conditions hold as strong operator relations.     

Perturbative QED in terms of gauge invariant fields

A formulation of QED using only gauge invariant fields acting on a physical state space is discussed. The fields are the electromagnetic tensor F_μν and a non-local electron field ψ_f depending on a quadruple {f^μ} of auxiliary functions. The f-ambiguity is physically meaningful: the f^μ contain information on the asymptotic configuration of the electromagnetic field accompanying charged particles. Equations of motion are introduced and solved perturbatively, in the sense that expressions for the Wightman functions of the theory are derived. No information on the commutation relations between the basic fields is needed.     

Schwinger-Dyson equations and dynamic symmetry violation in two-dimensional quantum quadratic gravitation

The dynamic violation of symmetry in two-dimensional quantum gravitation with the action quadratic in the curvature over a flat background is studied. The Schwinger-Dyson equation method is used in the ladder approximation. A numerical analysis of the equations for the structural functions defining the precise fermion propagator is given. The existence of a critical value of the coupling constant, which corresponds to dynamic violation of symmetry and the appearance of the bifermion condensate, is shown. The dependence of the dynamic mass on the coupling constant is found. Kharkov State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 40–44, March, 1997.     

Computing the couplings of Ising systems from Schwinger-Dyson equations

We study the problem of determining the couplings of Ising systems in terms of expectation values. We find that all previously known methods can be based on Schwinger-Dyson equations. Indeed, new methods can be derived with relative advantages with respect to all of them. Application for the renormalization group flow of the Ising model on a square lattice is presented.     

Admissibility of the axial gauge in QED2

The question recently raised by Tyburski as to whether the axial gauge can be used in two-dimensional quantum electrodynamics is examined. It is shown that even if the arguments leading to A₁≠0 are accepted, the consequences is no more than the existence of an uncoupled massless field in the theory. Thus QED₂ calculations based on the axial gauge do not require reconsideration.     

Ward identities and radiative corrections in QED with conformal gauge

A model of QED with conformally invariant gauge is considered. This gauge, being essentially nonlocal, is written in a local form by means of two nonphysical scalar fields. Using the BRST symmetry and the additional residual symmetry, a system of Ward identities is derived. These Ward identities are applied to prove the renormalizability of the model as well as to investigate the radiative corrections. A new class of conformal anomalies arises, connected with the absence of radiative corrections to the propagators including auxiliary fields.     

Hopf-algebraic renormalization of QED in the linear covariant gauge

In the context of massless quantum electrodynamics (QED) with a linear covariant gauge fixing, the connection between the counterterm and the Hopf-algebraic approach to renormalization is examined. The coproduct formula of Green’s functions contains two invariant charges, which give rise to different renormalization group functions. All formulas are tested by explicit computations to third loop order. The possibility of a finite electron self-energy by fixing a generalized linear covariant gauge is discussed. An analysis of subdivergences leads to the conclusion that such a gauge only exists in quenched QED.