Abstract
The quantum vacuum (Casimir) energy arising from noninteracting massless quanta is known to induce a long-range force, while decays exponentially for massive fields and separations larger than the inverse mass of the quanta involved. Here, we show that the interplay between dimensionality and nonlinearities in the field theory alters this behaviour in a nontrivial way. We argue that the changes are intimately related to the Mermin–Wagner–Hohenberg–Coleman theorem, and illustrate this situation using a nonlinear sigma model as a working example. We compute the quantum vacuum energy, which consists of the usual Casimir contribution plus a semiclassical contribution, and find that the vacuum-induced force is long-ranged at large distance, while displays a complex behaviour at small separations. Finally, even for this relatively simple set-up, we show that nonlinearities are generally responsible for modulations in the force as a function of the coupling constant and the temperature.
Highlights
In quantum field theory, a continuous symmetry cannot be spontaneously broken in D = 1 spatial dimension
If we fix the dimensionality to be D = 1, the verdict of the MWHC theorem is final: no transition to a massless phase can occur and M = 0 for any value of the separation. This implies that quantum fluctuations are effectively massive and, according to the above discussion, the Casimir force arising from such fluctuations should decay exponentially for separations larger than the inverse mass-gap, that is in the regime M 1
Fields), the mass suppression appearing in the Casimir force is dictated by the gap equation, inducing in the Casimir force an additional nonlinear dependence on the separation
Summary
If we fix the dimensionality to be D = 1, the verdict of the MWHC theorem is final: no transition to a massless phase can occur and M = 0 for any value of the separation This implies that quantum fluctuations are effectively massive and, according to the above discussion, the Casimir force arising from such fluctuations should decay exponentially for separations larger than the inverse mass-gap, that is in the regime M 1. Fields), the mass suppression appearing in the Casimir force is dictated by the gap equation (that determines how M 2 depends on the size of the system, , or any other external forcing eventually present), inducing in the Casimir force an additional nonlinear dependence on the separation It is this implication of the MWHC theorem that causes a dependence of the effective mass M on the separation and modifies the exponential behaviour in the Casimir force. In the following, we numerically calculate the total quantum vacuum force and show that it consists of the usual Casimir term, analogous in 1 spatial dimension to that of Ref. [18] plus a contribution proportional to M 2 (this second contribution is of semiclassical nature, since M 2 is determined by the one-loop effective equations) and show that, despite the relative simplicity of the set-up, the resulting quantum vacuum force displays a nontrivial behaviour
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