2. Preliminaries

This section considers the formal description of a charged spin zero (scalar) field in a static external electric field. (1) The following sections use this formalism to discuss the detailed behaviour of the ``vacuum'' in the presence of an electric field.

The Klein-Gordon Lagrangian for a complex scalar field with mass and charge minimally coupled to an electromagnetic field , is

where is an external electromagnetic current. This yields the equations of motion

where

is the conserved electric current for the field. The charge density is defined as . Choosing Coulomb gauge . Eq. (2.1) yields the Hamiltonian density for the field

where is the canonical momentum

With Coulomb gauge, the longitudinal (Coulomb) electric field in the penultimate term of Eq. (2.3) is constrained by

In the last term of Eq. (2.3), and are the transverse electric field and the magnetic field, and represent dynamical degrees of freedom in the electromagnetic field. (A total divergence term . () does not contribute in the charge zero sector, and is dropped in Eq. (2.3) and below.) Introducing the electric displacement and polarization according to

the Hamiltonian density (2.3a) becomes

Sections 3 and 4 and the remainder of this section consider the external field approximation. This approximation consists in taking a quantized field with a fixed classical external electric field, and ignoring the back reaction of the charged field on the electric field and transverse photon contributions. The effective Hamiltonian in this case is obtained from Eq. (2.6) by setting and to zero, and using the relation between and in Coulomb gauge

The total charge of the field (in space dimensions)

for solutions to Eq. (2.2a) is conserved if satisfies appropriate gauge invariant boundary conditions. In the systems considered below, the electric charge is usually confined to a finite volume with boundary , so that

where is the outward normal to . The field must thus satisfy a boundary condition of the form

where is a real function.

The boundary condition (2.10) may be implemented in the Lagrangian (2.1) by the introduction of a source term

We take a classical static external electric field . A gauge may be chosen in this case so that is time independent, and determined up to a constant by

The linearity of (2.2a) then allows its solutions to decomposed into normal modes of definite frequency:

The addition of a constant to corresponds to a gauge transformation with parameter , leading to a phase change in and hence a displacement of :

The hermiticity of the operator with the boundary condition (2.10) (and ) implies the relation

For real , this yields

For complex ,

In classifying solutions, it is convenient to rewrite Eq. (2.2) as an ordinary first order (Schrodinger-like) eigenvalue equation:

With the boundary conditions (2.10), is an hermitean operator with respect to the indefinite scalar product

Comparison with Eq. (2.8) reveals that the norm gives the total charge (in units of ) for the state. Similarly, Eq. (2.3) shows that .

Equation (2.16) expresses the orthogonality of the eigenfunctions with respect to the scalar product (2.19) for real eigenvalues .

For some , eigenfunctions with complex eigenvalues exist [3,4,6]. The fields associated with these eigenfunctions exhibit a real exponential time dependence. Equation (2.17) implies that all eigenfunctions with complex eigenvalues have zero norm. If is an eigenfunction with eigenvalue is an eigenfunction with conjugate eigenvalue . The scalar product is almost always non-zero. It vanishes only for exceptional , at which an additional class of solutions to Eq. (2.18) with time dependence exist. These solutions correspond to the ``associated eigenfunctions'' satisfying the equation [7]

The first step in the second quantization of the field is to determine the normalization of the eigenfunctions . The eigenfunctions with real eigenvalues may be normalized to have total charge . (Note that while the magnitude of the charge associated with can be modified by normalization, its sign cannot.) Normal modes with positive charge are interpreted as ``particle'' solutions, and those with negative charge as ``antiparticle'' solutions. The energy of a ``particle'' solution is given by its corresponding eigenvalue , while for an ``antiparticle'' solution, the energy is .

At the exceptional points for which the charge deduced from the scalar product (2.19) vanishes, the modes may be normalized so that . For complex eigenvalues, a possible normalization is .

We shall assume that the eigenfunctions (including, when necessary, ``associated'' and complex eigenfunctions) form a complete set of functions with the boundary conditions (2.10). A proof of this result is outlined in Ref. [17].

The second quantization of the field is simplest when , is such that all eigenvalues are real, and positive and negative norm eigenfunctions have eigenvalues and . respectively, such that for all . This case is often realized in weak electric fields, as discussed in Section 3. Decomposing in terms of positive and negative norm eigenfunctions

the equal-time canonical commutation relations

imply

for the operator-valued expansion coefficients .

The vacuum state in the presence of the external electric field is defined such that

The ``induced charge density'' in this state is given by

Hence

where are the charge densities associated with the modes As expected, each possible mode of the field contributes with weight in the vacuum state.

The ``vacuum polarization'' is given in terms of the induced charge density (2.25) in analogy with the classical equation (2.5) by

where is the ``dielectric constant of the vacuum.'' The electric field is taken here to be classical, as assumed below. Since the net charge of the field is conserved, the electric displacement satisfies

The presence of the external electric field causes a shift in the energy eigenvalues, and hence in the vacuum energy, given by

where the are the energy eigenvalues for zero external electric field. Note that while and both change under a gauge transformation, their difference is gauge invariant. An induced charge density modifies the classical electric field, as in Eq. (2.27). The energy (2.28) may be considered as the effective potential for the electric field calculated to the one loop approximation in the field.

Even when positive and negative charge eigenvalues do not form separated sequences, the expansion (2.21) remains formally correct so long as all eigenvalues are real and non-zero. However, as discussed below, the energy in such a case is not bounded below, and no ground state may therefore be identified [3].

When complex eigenvalues exist, straightforward second quantization of the scalar field is no longer possible. An expansion of the form (2.21) may still be used if terms corresponding to the complex modes are added [7]:

The charge of the field remains quantized in this case, but the energy is no longer discrete. The energy operator for the complex modes is , and has a continuous spectrum covering all real eigenvalues. Again, no ground state may be defined.

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