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.

众所周知，由非相互作用的无质量量子产生的量子真空 (Casimir) 能量会引起长程力，而对于大于所涉及的量子的反质量的大量场和分离，则以指数方式衰减。在这里，我们展示了场论中维数和非线性之间的相互作用以一种非平凡的方式改变了这种行为。我们认为这些变化与 Mermin-Wagner-Hohenberg-Coleman 定理密切相关，并使用非线性 sigma 模型作为工作示例来说明这种情况。我们计算了量子真空能量，它由通常的 Casimir 贡献加上半经典贡献组成，并发现真空诱导力在大距离上是长距离的，而在小间距时表现出复杂的行为。最后，即使对于这个相对简单的设置，