Unitary quantization of a scalar charged field and Schwinger effect

Abstract

Quantum field theory in curved spacetimes suffers in general from an infinite ambiguity in the choice of Fock representation and associated vacuum. In cosmological backgrounds, the requirement of a unitary implementation of the field dynamics in the physical Hilbert space of the theory is a good criterion to ameliorate such ambiguity. Indeed, this criterion, together with a unitary implementation of the symmetries of the equations of motion, leads to a unique equivalence class of Fock representations. In this work, we apply the procedure developed for fields in cosmological settings to analyze the quantization of a scalar field in the presence of an external electromagnetic classical field in a flat background. We find a natural Fock representation that admits a unitary implementation of the quantum field dynamics. It automatically allows to define a particle number density at all times in the evolution with the correct asymptotic behavior, when the electric field vanishes. Moreover we show the unitary equivalence of all the quantizations that fulfill our criteria. Although we perform the field quantization in a specific gauge, we also show the equivalence between the procedures taken in different gauges.

Adiabatic regularization for Dirac fields in time-varying electric backgrounds

“…Plugging the ansatz (30) in the mode equations (22) and (23) and also in the normalization condition (26) we get a system of equations for the functions FðtÞ, GðtÞ and ΩðtÞ…”

“…In this context, the most important physical local expectation value is the electric current hj μ i, which also possesses ultraviolet divergences and has to be renormalized in a proper way. Recent discussions on foundational issues related to the particle number density of the created particles, adiabatic invariance, and unitary evolution can be seen in [29][30][31].…”

Quantum unitary dynamics of a charged fermionic field and Schwinger effect

“…S with a complex Hilbert space structure. We find here the fundamental difference with respect to the case of a charged scalar field in an electromagnetic background: the space of solutions of the Klein-Gordon field coupled to an external electromagnetic potential has no natural inner product defined on it, but an antisymmetric symplectic form which fails to be positive definite [12,13]. In that case, the introduction of a complex structure is necessary in order to construct a Hilbert space of solutions.…”

“…Thus, we need to impose physical criteria on the complex structure in order to reduce the ambiguity in the quantization. Motivated by previous studies in cosmology [7,8] and in the Schwinger effect for a charged scalar field [13], our central work in section 5 will be to characterize the complex structures which preserve the symmetries of the system and unitarily implement the dynamical evolution of a charged fermionic field in presence of a homogeneous time-dependent electromagnetic background. Therefore, let us describe how can time evolution be treated as a Bogoliubov transformation.…”

“…The aim of this work is to study the canonical quantization of a massive charged Dirac field in a flat spacetime coupled to a homogeneous time-dependent electric field, extending to the fermionic case the results achieved in [13] for the scalar field. In additon, we will consider the electric field to be intense enough so that the test matter field does not modify its dynamics; i.e., we will neglect the backreaction.…”