R. E. Gamboa Saraví

Chiral symmetry and functional integral

Abstract The change in the fermionic functional integral measure under chiral rotations is analyzed. Using the ζ-function method, the evaluation of chiral Jacobians to theories including nonhermitian Dirac operators D , can be extended in a natural way. (This being of interest, for example, in connection with the Weinberg-Salam model or with the relativistic string theory.) Results are compared with those obtained following other approaches, the possible discrepancies are analyzed and the equivalence of the different methods under certain conditions on D is proved. Also shown is how to compute the Jacobian for the case of a finite chiral transformation and this result is used to develop a sort of path-integral version of bosonization in d = 2 space-time dimensions. This result is used to solve in a very simple and economical way relevant d = 2 fermionic models. Furthermore, some interesting features in connection with the θ-vacuum in d = 2,4 gauge theories are discussed.

ζ‐function method and the evaluation of fermion currents

Using the ζ‐function method, a prescription for the evaluation of fermion currents in the presence of background fields is given. The method preserves gauge invariance at each step of the computation and yields to a finite answer showing the relevant physical properties for arbitrary background configurations. Examples for n=2,3 dimensions are worked out, emphasizing the connection between preservation of gauge invariance and violation of other symmetries (chiral symmetry for n=2, parity for n=3).

THE ζ FUNCTION ANSWER TO PARITY VIOLATION IN THREE-DIMENSIONAL GAUGE THEORIES

We study parity violation in (2+1)-dimensional gauge theories coupled to massive fermions. Using the ζ function regularization approach we evaluate the ground state fermion current in an arbitrary gauge field background, showing that it gets two different contributions which violate parity invariance and induce a Chern–Simons term in the gauge field effective action. One is related to the well-known classical parity breaking produced by a fermion mass term in three dimensions; the other, already present for massless fermions, is related to peculiarities of gauge-invariant regularization in odd-dimensional spaces.

Dirac operator on a disk with global boundary conditions

We compute the functional determinant for a Dirac operator in the presence of an Abelian gauge field on a bidimensional disk, under global boundary conditions of the type introduced by Atiyah–Patodi–Singer. We also discuss the connection between our result and the index theorem.

Determinants of Dirac operators with local boundary conditions

We study functional determinants for Dirac operators on manifolds with boundary. We give, for local boundary conditions, an explicit formula relating these determinants to the corresponding Green’s functions. We finally apply this result to the case of a bidimensional disk under baglike conditions.

Bosonization of fermion currents in two-dimensional quantum chromodynamics

Abstract Bosonization rules for fermion currents in QCD 2 are constructed using a very simple approach within the path-integral framework, based on the existence of a gauge condition in which fermions can be decoupled.

On Chiral Rotations and the Fermionic Path Integral

Abstract We show the equivalence between the ζ-function regularization and other techniques used to compute jacobians arising from chiral rotations in the path-integral. The proof is valid for theories with hermitian Dirac operators including γ5 couplings and hence justifies the path integral derivation of anomalous Ward identities in this context. Some implications on chiral rotations and the θ-vacuum are discussed.

P determinants and boundary values

It is shown that a regularized determinant based on Hilbert’s approach (which we call the ‘‘p determinant’’) of a quotient of elliptic operators defined on a manifold with boundary is equal to the ‘‘p determinant’’ of a quotient of pseudodifferential operators. The last ones are entirely expressible in terms of boundary values of solutions of the original differential operators. It is argued that, in the context of quantum field theory, these boundary values also determine the subtractions (i.e., the counterterms) to which this regularization scheme gives rise.

On the global version of Euler–Lagrange equations

The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler–Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version.

On the quotient of the regularized determinant of two elliptic operators

We study the quotient of the regularized determinants of two elliptic operators having the same principal symbol. We prove that, under general conditions, a method recently proposed by Tamura coincides with the ζ-function approach.