Quantum vacuum, rotation, and nonlinear fields

In this paper, we extend previous results on the quantum vacuum or Casimir energy, for a noninteracting rotating system and for an interacting nonrotating system, to the case where both rotation and interactions are present. Concretely, we first reconsider the noninteracting rotating case of a scalar field theory and propose an alternative and simpler method to compute the Casimir energy based on a replica trick and the Coleman-Weinberg effective potential. We then consider the simultaneous effect of rotation and interactions, including an explicit breaking of rotational symmetry. To study this problem, we develop a numerical implementation of zeta-function regularization. Our work recovers previous results as limiting cases and shows that the simultaneous inclusion of rotation and interactions produces nontrivial changes in the quantum vacuum energy. Besides expected changes (where, as the size of the ring increases for fixed interaction strength, the angular momentum grows with the angular velocity), we notice that the way rotation combines with the coupling constant amplifies the intensity of the interaction strength. Interestingly, we also observe a departure from the typical massless behavior where the Casimir energy is proportional to the inverse size of the ring.

Figure 1

Both panels represent a ring rotating with an angular velocity $\mathrm{\Omega }$. On the left, periodic boundary conditions are imposed at any one point, and the two observers (represented by the two reference frames $O$ and $O^{\prime }$, the latter rotating with the ring) cannot detect that the ring is rotating. The ring on the right has an impurity (represented by a black dot) that breaks the rotational symmetry. In this second case, for the observer $O^{\prime }$ at rest with the ring (i.e., rotating with the ring) the position of the impurity would not change (i.e., the boundary conditions are time independent), while for the laboratory observer $O$, rotation is evident and the boundary conditions imposed at the impurity are time dependent.

Figure 2

Numerical eigenvalue analysis. (a) Comparison of the polynomial fit [Eq. (62)] and the numerically computed eigenvalues [Eqs. (49) and (50)] eigenvalues up to cubic order. The individual fitting coefficients are defined in the legend as ${\tilde{\omega }}_j=({\tilde{\omega }}_0,{\tilde{\omega }}_1,{\tilde{\omega }}_2,{\tilde{\omega }}_3)$. (b) Correction $\delta E_r$ [Eq. (59)] computed for each of the cases presented in (a), i.e., for fixed $\lambda =1$, $L=2$ with $\mathrm{\Omega }R=0.2$, 0.4, 0.6, 0.8. Inset: example of the eigenvalue difference ${\omega }_n-{\tilde{\omega }}_n$ for $\mathrm{\Omega }R=0.2$.

Figure 3

Rotation-interaction heat map. (a) Casimir energy $E_r$ [Eq. (68)] in the $(\mathrm{\Omega }R,\lambda )$ parameter space. (b) $E_r(\lambda )$ for $\mathrm{\Omega }R=0$, 0.2, 0.4, 0.6, 0.8, 0.9. The size of the ring is $L=20$.

Figure 4

Casimir energies. (a) $E_r$ [Eq. (68)] for varying $L$ for various fixed interaction strengths $\lambda =1$, 10, 25, 50, 75, 100. The rotation parameter is $\mathrm{\Omega }R=0.75$. (b) $E_r$ for fixed $\lambda =10$ for $\mathrm{\Omega }R=0$, 0.5, 0.6, 0.7, 0.8, 0.9. Inset: enlarged portion between $L={10}^{-3}:{10}^{-2}$. (c), (d) $E_r$ as a function of the rotation parameter $\mathrm{\Omega }R$ for a fixed ring length $L=20$ in (c) and a fixed interaction strength $\lambda =0.5$ in (d). Here we have fixed $\mu =1$ throughout.

Figure 5

Angular momentum. (a), (b) Angular momentum calculated per Eq. (20). (a) $L_{\mathrm{am}}$ computed from the data in Fig. 4 as a function of $\mathrm{\Omega }R$ for fixed ring length $L=20$ in (a) and from the data in Fig. 4 for fixed interaction strength $\lambda =0.5$ in (b).