ETD

• Bogoliubov transformation
• classical background
• heat kernel
• pair production
• Schwinger effect

This thesis explores the vacuum instability in quantum field theory, which leads to the spontaneous creation of particle–antiparticle pairs from the vacuum under the influence of external classical background fields, such as the electromagnetic field or the gravitational field.

The first chapter is devoted to the paradigmatic case of this phenomenon, known as the Schwinger effect, which describes a quantum field coupled to a constant, homogeneous electric field. Presented in two complementary approaches: the ``effective Lagrangian method'' introduced by Schwinger, and a canonical quantization approach first used by Nikishov.


The effect is first derived through the proper-time formalism. Starting from the effective Lagrangian of quantum electrodynamics in an external electromagnetic background, the one-loop determinant is evaluated via the heat-kernel method. The vacuum decay probability and pair-production rate are obtained fom the imaginary part of the effective action.
Special attention is given to the assumptions underlying this approach, in particular the use of asymptotic free states, which may limit its validity.


The same effect is then re-derived through canonical quantization of the full interacting theory, thereby avoiding the conceptual issues of the effective Lagrangian method.
However, in this approach, in the view of the author, the choice of asymptotic vacua in the existing literature—though intuitively justified—still lacks a clear physical foundation.

As an original contribution, a diagonalization procedure of the Hamiltonian density is introduced to define an instantaneous ground state, providing a natural and physically meaningful candidate for the vacuum.
The equivalence between this approach and Schwinger’s method confirms the robustness of the result.

The analysis employs the method of Bogoliubov coefficients to connect different mode decompositions and to quantify particle production. Since the Bogoliubov transformations for a complex scalar field were required, in the thesis they are recovered from a straightforward extension of the real scalar case.


The second part of the thesis addresses the pair creation in curved spacetime, where the geometry itself plays the role of the classical background. To do so, a covariant approach, known as the heat-kernel method, is applied to compute the imaginary part of the effective action. In The Schwinger–DeWitt expansion, the local (low-energy) and nonlocal (high-energy) regimes are examined, highlighting the limits and application range of each one, using formal arguments.

The integration of the nonlocal form factor of the heat-kernel is then explicitly re-derived, confirming the effective action in the form presented by Boasso and Viñas,

Finally, some physical consequences and applications of the obtained expressions are discussed in the conclusion of the thesis.