Abstract
This paper reports certain ambiguities in the calculation of the ensemble average $\left<T_\mu{}_\nu\right>$ of the stress-energy-momentum tensor of an arbitrarily coupled massless scalar field in one-dimensional boxes in flat spacetime. The study addresses a box with periodic boundary condition (a circle) and boxes with reflecting edges (with Dirichlet's or Neumann's boundary conditions at the endpoints). The expressions for $\left<T^\mu{}^\nu\right>$ are obtained from finite-temperature Green functions. In an appendix, in order to control divergences typical of two dimensions, these Green functions are calculated for related backgrounds with arbitrary number of dimensions and for scalar fields of arbitrary mass, and specialized in the text to two dimensions and for massless fields. The ambiguities arise due to the presence in $\left<T^\mu{}^\nu\right>$ of double series that are not absolutely convergent. The order in which the two associated summations are evaluated matters, leading to two different thermodynamics for each type of box. In the case of a circle, it is shown that the ambiguity corresponds to the classic controversy in the literature whether or not zero mode contributions should be taken into account in computations of partition functions. In the case of boxes with reflecting edges, it results that one of the thermodynamics corresponds to a total energy (obtained by integrating the non homogeneous energy density over space) that does not depend on the curvature coupling parameter $\xi$ as expected; whereas the other thermodynamics curiously corresponds to a total energy that does depend on $\xi$. Thermodynamic requirements (such as local and global stability) and their restrictions to the values of $\xi$ are considered.
Highlights
Over the last decades, since the discovery that a black hole behaves very much like a blackbody—with entropy and temperature [1], and even radiation [2]—the study of quantum fields at finite temperature near boundaries and in spacetimes with nontrivial topologies and geometries has received increasing attention in the literature
The message seems to be that one may learn a great deal about the nature of gravity itself by looking at boundary quantum field theory at finite temperature, especially in lower dimensions as holography suggests [3]
The appearance of ξ in Eq (7) is a bit surprising since, for a massless scalar field φ in flat spacetime with one spatial dimension x, the classical expression for the energy density Ttt depends on the curvature coupling parameter ξ only through the term −2ξ∂xðφ∂xφÞ, which does not contribute when integrating Ttt for Dirichlet or Neumann boundary conditions [21]
Summary
Since the discovery that a black hole behaves very much like a blackbody—with entropy and temperature [1], and even radiation [2]—the study of quantum fields at finite temperature near boundaries and in spacetimes with nontrivial topologies and geometries has received increasing attention in the literature. We have not found any study of hTμνi for hot scalar radiation with Dirichlet or Neumann boundary conditions at the end points of a one-dimensional box.2 The appearance of ξ in Eq (7) is a bit surprising since, for a massless scalar field φ in flat spacetime with one spatial dimension x, the classical expression for the energy density Ttt depends on the curvature coupling parameter ξ only through the term −2ξ∂xðφ∂xφÞ, which does not contribute when integrating Ttt for Dirichlet or Neumann boundary conditions [21].
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