1. Introduction

This is the first in a series of papers on the ``bulk'' properties of quantized fields. These properties are determined from the ``response'' of the ``vacuum'' state to classical external fields or constraints. Such investigations complement perturbative treatments of small disturbances in quantized fields.

This paper discusses some ``mechanical'' and ``thermodynamic'' properties of quantized fields. Sections 2 through 6 consider the mechanical forces associated with containment of quantized fields in finite regions. Sections 7 and 8 discuss thermodynamic properties of fields in finite regions at non-zero temperatures. In later papers, we shall consider ``electrodynamics [1] and ``gravitational'' properties of quantized fields. We shall for the most part use the ``free'' approximation in which interactions of the fields with themselves are ignored.

The mechanical properties of the vacuum are unlike those of ordinary matter. As mentioned in Section 5, a region of vacuum may under certain circumstances have a negative energy, so that ``empty'' cavities may apparently attain negative masses.

The vacuum (ground) states of several highly non-linear field theories may have a foam like structure with ``bubbles'' of low field strength separated by walls of high field strength. It is plausible that fluctuations of the field within a bubble may exert forces which determine the shape of its walls. We discuss this possibility in Section 6 and show that the equilibrium shapes of such bubbles may be isotropic or tubular.

In Section 8 we consider the relation between the entropy and energy density of quantized fields at finite temperatures, and show that a recently suggested entropy bound [2] is incorrect.

We shall usually consider fields contained in regions of space some of whose directions are finite in extent. In many cases, the fields may be viewed as confined by hyper-cuboidal (hyper-rectangular-parallelipedal) boundaries. The necessary constraints on the fields are implemented either by modifying the spacetime on which field exists (so that, for example, certain coordinates have finite ranges, or are periodically identified) or through physical boundary conditions as would result from interaction with extended external sources of infinite strength. Such constraints modify the zero-point modes of the fields, and change the energy of the vacuum (ground state). This change is manifest as an observable Casimir energy [3]. The Casimir energy may be either positive or negative. The corresponding mechanical pressure exerted on the walls of a ``container'' may be either positive or negative, and is often anisotropic. The result depends on the exact form of the constraints and on the nature of the field.

Section 2 describes in detail the calculation of Casimir energy for the simple case of a scalar (spin 0) field between two plates. It is convenient and illuminating to take the dimensionality of spacetime as an arbitrary continuous parameter. In this way, divergences associated with high frequency fluctuations in the field are regularized.

Sections 3 and 4 generalize the results to confining regions of other shapes, and to vector (spin 1) fields.

Attractive van der Waals ``dispersion'' forces between electrically neutral macroscopic objects may be attributed to Casimir energies of the electromagnetic field (e.g. [4]). However, as discussed in Section 5, not all Casimir energies may be associated with sums of van der Waals forces. For example, the electromagnetic Casimir energy of a three-dimensional cubic cavity with perfectly-conducting walls leads to an outward pressure [5,6,7,8], while a sum of van der Waals forces would suggest inward pressure. Sections 3 and 4 consider a variety of systems exhibiting repulsive Casimir forces. The existence and nature of these examples appears to have caused confusion in previous investigations.

previous  l  next