Abstract
In this work we consider the generalized zeta function method to obtain temperature corrections to the vacuum (Casimir) energy density, at zero temperature, associated with quantum vacuum fluctuations of a scalar field subjected to a helix boundary condition and whose modes propagate in ($3+1$)-dimensional Euclidean spacetime. We find closed and analytical expressions for both the two-point heat kernel function and free energy density in the massive and massless scalar field cases. In particular, for the massless scalar field case, we also calculate the thermodynamics quantities internal energy density and entropy density, with their corresponding high- and low-temperature limits. We show that the temperature correction term in the free energy density must suffer a finite renormalization, by subtracting the scalar thermal blackbody radiation contribution, in order to provide the correct classical limit at high temperatures. We check that, at low temperature, the entropy density vanishes as the temperature goes to zero, in accordance with the third law of thermodynamics. We also point out that, at low temperatures, the dominant term in the free energy and internal energy densities is the vacuum energy density at zero temperature. Finally, we also show that the pressure obeys an equation of state.
Highlights
The Casimir Effect was predicted for the first time back in 1948 by Hendrik Casimir [1]
In this work we have overviewed how the partition function can be obtained from the path integral for the scalar field and how it is connected with the generalized zeta function by means of Eq (13)
By linking the generalized zeta function with the partition function we have been able to obtain the closed expressions (26) and (27) for the free energy and free energy density, respectively. These expressions explicitly show that they are composed by a term for the zero temperature obtained from the usual calculations for the vacuum energy, and a term for temperature corrections—these are important expressions since experiments are normally performed at a finite temperature
Summary
The Casimir Effect was predicted for the first time back in 1948 by Hendrik Casimir [1]. The Casimir effect arises due to perturbations in the vacuum state fluctuations of a given quantum field, generating a force that could be attractive or repulsive depending. Critchley [21] and reshaped by Hawking [17], a connection can be made between a generalized ζ-function and the determinant of differential operators such as the D’Alambertian and the Laplace-Beltrami operators In this context, the path integral approach shows to be of great importance since it allows us to obtain a regularized and, renormalized expression for the energy density from the partition function of the field, in terms of the corresponding operator, providing a way of exploiting temperature contributions. In this work we seek to calculate temperature contributions for the Casimir energy density, in a flat spacetime, arising from perturbations on the scalar field vacuum as a consequence of the imposition of a helix boundary condition.
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