Finite temperature Casimir effect in piston geometry and its classical limit

We consider the Casimir force acting on a $d$-dimensional rectangular piston due to massless scalar field with periodic, Dirichlet and Neumann boundary conditions and electromagnetic field with perfect electric conductor and perfect magnetic conductor boundary conditions. It is verified analytically that at any temperature, the Casimir force acting on the piston is always an attractive force pulling the piston towards the interior region, and the magnitude of the force gets larger as the separation $a$ gets smaller. Explicit exact expressions for the Casimir force for small and large plate separations and for low and high temperatures are computed. The limits of the Casimir force acting on the piston when some pairs of transversal plates are large are also derived. An interesting result regarding the influence of temperature is that in contrast to the conventional result that the leading term of the Casimir force acting on a wall of a rectangular cavity at high temperature is the Stefan--Boltzmann (or black body radiation) term which is of order $T^{d+1}$, it is found that the contributions of this term from the interior and exterior regions cancel with each other in the case of piston. The high temperature leading order term of the Casimir force acting on the piston is of order $T$, which shows that the Casimir force has a nontrivial classical $hbarto 0$ limit.

Repulsive Casimir force at zero and finite temperature

We study the zero and finite temperature Casimir force acting on a perfectly conducting piston with arbitrary cross section moving inside a closed cylinder with infinitely permeable walls. We show that at any temperature, the Casimir force always tends to move the piston away from the walls and towards its equilibrium position. In the case of rectangular piston, exact expressions for the Casimir force are derived. In the high temperature regime, we show that the leading term of the Casimir force is linear in temperature and therefore the Casimir force has a classical limit. Due to duality, all these result also hold for an infinitely permeable piston moving inside a closed cylinder with perfectly conducting walls.

Three dimensional Casimir piston for massive scalar fields

We consider Casimir force acting on a three dimensional rectangular piston due to a massive scalar field subject to periodic, Dirichlet and Neumann boundary conditions. Exponential cut-off method is used to derive the Casimir energy in the interior region and the exterior region separated by the piston. It is shown that the divergent term of the Casimir force acting on the piston due to the interior region cancels with that due to the exterior region, thus render a finite well-defined Casimir force acting on the piston. Explicit expressions for the total Casimir force acting on the piston is derived, which show that the Casimir force is always attractive for all the different boundary conditions considered. As a function of a -- the distance from the piston to the opposite wall, it is found that the magnitude of the Casimir force behaves like $1/a^4$ when $ato 0^+$ and decays exponentially when $ato infty$. Moreover, the magnitude of the Casimir force is always a decreasing function of a. On the other hand, passing from massless to massive, we find that the effect of the mass is insignificant when a is small, but the magnitude of the force is decreased for large a in the massive case.

Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Two types of Gaussian processes, namely the Gaussian field with generalized Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy covariance (GSGCC) are considered. Some of the basic properties and the asymptotic properties of the spectral densities of these random fields are studied. The associated self-similar random fields obtained by applying the Lamperti transformation to GFGCC and GSGCC are studied.

On the minima and convexity of Epstein Zeta function

Let $Z_n(s; a_1,..., a_n)$ be the Epstein zeta function defined as the meromorphic continuation of the function sum_{kinZ^nsetminus{0}}(sum_{i=1}^n [a_i k_i]^2)^{-s}, text{Re} s>frac{n}{2} to the complex plane. We show that for fixed $s eq n/2$, the function $Z_n(s; a_1,..., a_n)$, as a function of $(a_1,..., a_n)in (R^+)^n$ with fixed $prod_{i=1}^n a_i$, has a unique minimum at the point $a_1=...=a_n$. When $sum_{i=1}^n c_i$ is fixed, the function $$(c_1,..., c_n)mapsto Z_n(s; e^{c_1},..., e^{c_n})$$ can be shown to be a convex function of any $(n-1)$ of the variables ${c_1,...,c_n}$. These results are then applied to the study of the sign of $Z_n(s; a_1,..., a_n)$ when $s$ is in the critical range $(0, n/2)$. It is shown that when $1leq nleq 9$, $Z_n(s; a_1,..., a_n)$ as a function of $(a_1,..., a_n)in (R^+)^n$, can be both positive and negative for every $sin (0,n/2)$. When $ngeq 10$, there are some open subsets $I_{n,+}$ of $sin(0,n/2)$, where $Z_{n}(s; a_1,..., a_n)$ is positive for all $(a_1,..., a_n)in(R^+)^n$. By regarding $Z_n(s; a_1,..., a_n)$ as a function of $s$, we find that when $ngeq 10$, the generalized Riemann hypothesis is false for all $(a_1,...,a_n)$.

Finite temperature Casimir energy in closed rectangular cavities: a rigorous derivation based on zeta function technique

We derive rigorously explicit formulas of the Casimir free energy at finite temperature for massless scalar field and electromagnetic field confined in a closed rectangular cavity with different boundary conditions by zeta regularization method. We study both the low and high temperature expansions of the free energy. In each case, we write the free energy as a sum of a polynomial in temperature plus exponentially decay terms. We show that the free energy is always a decreasing function of temperature. In the cases of massless scalar field with Dirichlet boundary condition and electromagnetic field, the zero temperature Casimir free energy might be positive. In each of these cases, there is a unique transition temperature (as a function of the side lengths of the cavity) where the Casimir energy change from positive to negative. When the space dimension is equal to two and three, we show graphically the dependence of this transition temperature on the side lengths of the cavity. Finally we also show that we can obtain the results for a non-closed rectangular cavity by letting the size of some directions of a closed cavity going to infinity, and we find that these results agree with the usual integration prescription adopted by other authors.

Finite Temperature Casimir Effect for a Massless Fractional Klein-Gordon field with Fractional Neumann Conditions

This paper studies the Casimir effect due to fractional massless Klein-Gordon field confined to parallel plates. A new kind of boundary condition called fractional Neumann condition which involves vanishing fractional derivatives of the field is introduced. The fractional Neumann condition allows the interpolation of Dirichlet and Neumann conditions imposed on the two plates. There exists a transition value in the difference between the orders of the fractional Neumann conditions for which the Casimir force changes from attractive to repulsive. Low and high temperature limits of Casimir energy and pressure are obtained. For sufficiently high temperature, these quantities are dominated by terms independent of the boundary conditions. Finally, validity of the temperature inversion symmetry for various boundary conditions is discussed.

Topological symmetry breaking of self--interacting fractional Klein--Gordon field on toroidal spacetime

Quartic self--interacting fractional Klein--Gordon scalar massive and massless field theories on toroidal spacetime are studied. The effective potential and topologically generated mass are determined using zeta function regularization technique. Renormalization of these quantities are derived. Conditions for symmetry breaking are obtained analytically. Simulations are carried out to illustrate regions or values of compactified dimensions where symmetry breaking mechanisms appear.

Fractional oscillator process with two indices

We introduce a new fractional oscillator process which can be obtained as solution of a stochastic differential equation with two fractional orders. Basic properties such as fractal dimension and short range dependence of the process are studied by considering the asymptotic properties of its covariance function. The fluctuation--dissipation relation of the process is investigated. The fractional oscillator process can be regarded as one-dimensional fractional Euclidean Klein-Gordon field, which can be obtained by applying the Parisi-Wu stochastic quantization method to a nonlocal Euclidean action. The Casimir energy associated with the fractional field at positive temperature is calculated by using the zeta function regularization technique.