Quantum unitary dynamics of a charged fermionic field and Schwinger effect

Abstract

In quantum field theory, particle creation occurs, in general, when an intense external field, such as an electromagnetic field, breaks time translational invariance. This leads to an ambiguity in the definition of the vacuum state. In cosmological backgrounds this ambiguity has been reduced by imposing that the quantization preserves the symmetries of the system and that the dynamics is unitarily implemented. In this work, we apply these requirements to the quantization of a massive charged fermionic field coupled to a classical time-dependent homogeneous electric field, extending previous studies done for a scalar field. We characterize the quantizations fulfilling the criteria above and we show that they form a unique equivalence class of unitarily related quantizations, which provide a well-defined number of created particles at all finite times.

Generalized quantum Vlasov equation for particle creation and unitary dynamics

“…In this case, the Fourier transform of the Dirac equation yields fermionic modes formed by four real variables satisfying (2.3) with time-dependent frequencies (see e.g. [17,28])…”

“…However, these variables are not completely decoupled and minor manipulations inspired in references [17,28] should be applied to the procedure followed here in order to extend these results to the fermionic case.…”

“…For deeper analyses covering a wide range of systems of the type described in section 2, see e.g. [13,25,27,28,37].…”

“…When it no longer possesses Poincaré symmetry, as it is the case, this requirement is not restrictive enough to select particular functions (ζ k (t), ρ k (t)) in general and ambiguities emerge in the canonical quantization (see section 3). In order to reduce these ambiguities, previous studies for both scalar and fermionic fields in homogeneous cosmological settings [13,[24][25][26] as well as in the context of the Schwinger effect [27,28] impose that the canonical time evolution of the fields be unitarily implemented in the quantum theory. Physically, this translates into a well-defined total number of created particles throughout the evolution of fields at all finite times.…”

“…In order to reduce the ambiguities in the quantization of classical theories with no time-translational invariance, a criterion has been proposed in various settings, from homogeneous cosmologies [13,[24][25][26] to the Schwinger effect [27,28]: the unitary implementation in the quantum theory of matter fields dynamics. Weaker conditions are also found in the literature, imposing that only the in and out states (at asymptotic past and future times) are related by a unitary S-matrix [29,30].…”