Finite temperature Casimir effect of massive fermionic fields in the presence of compact dimensions

We consider the finite temperature Casimir effect of a massive fermionic field confined between two parallel plates, with MIT bag boundary conditions on the plates. The background spacetime is $M^{p+1}\times T^q$ which has $q$ dimensions compactified to a torus. On the compact dimensions, the field is assumed to satisfy periodicity boundary conditions with arbitrary phases. Both the high temperature and the low temperature expansions of the Casimir free energy and the force are derived explicitly. It is found that the Casimir force acting on the plates is always attractive at any temperature regardless of the boundary conditions assumed on the compact torus. The asymptotic limits of the Casimir force in the small plate separation limit are also obtained.Comment: 10 pages, accepted by Phys. Lett.

The Casimir effect for parallel plates at finite temperature in the presence of one fractal extra compactified dimension

We discuss the Casimir effect for massless scalar fields subject to the Dirichlet boundary conditions on the parallel plates at finite temperature in the presence of one fractal extra compactified dimension. We obtain the Casimir energy density with the help of the regularization of multiple zeta function with one arbitrary exponent and further the renormalized Casimir energy density involving the thermal corrections. It is found that when the temperature is sufficiently high, the sign of the Casimir energy remains negative no matter how great the scale dimension $\delta$ is within its allowed region. We derive and calculate the Casimir force between the parallel plates affected by the fractal additional compactified dimension and surrounding temperature. The stronger thermal influence leads the force to be stronger. The nature of the Casimir force keeps attractive.Comment: 14 pages, 2 figure

Conformal Mappings and Dispersionless Toda hierarchy

Let $\mathfrak{D}$ be the space consists of pairs $(f,g)$, where $f$ is a univalent function on the unit disc with $f(0)=0$, $g$ is a univalent function on the exterior of the unit disc with $g(\infty)=\infty$ and $f'(0)g'(\infty)=1$. In this article, we define the time variables $t_n, n\in \Z$, on $\mathfrak{D}$ which are holomorphic with respect to the natural complex structure on $\mathfrak{D}$ and can serve as local complex coordinates for $\mathfrak{D}$. We show that the evolutions of the pair $(f,g)$ with respect to these time coordinates are governed by the dispersionless Toda hierarchy flows. An explicit tau function is constructed for the dispersionless Toda hierarchy. By restricting $\mathfrak{D}$ to the subspace $\Sigma$ consists of pairs where $f(w)=1/\bar{g(1/\bar{w})}$, we obtain the integrable hierarchy of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since every $C^1$ homeomorphism $\gamma$ of the unit circle corresponds uniquely to an element $(f,g)$ of $\mathfrak{D}$ under the conformal welding $\gamma=g^{-1}\circ f$, the space $\text{Homeo}_{C}(S^1)$ can be naturally identified as a subspace of $\mathfrak{D}$ characterized by $f(S^1)=g(S^1)$. We show that we can naturally define complexified vector fields \pa_n, n\in \Z on $\text{Homeo}_{C}(S^1)$ so that the evolutions of $(f,g)$ on $\text{Homeo}_{C}(S^1)$ with respect to \pa_n satisfy the dispersionless Toda hierarchy. Finally, we show that there is a similar integrable structure for the Riemann mappings $(f^{-1}, g^{-1})$. Moreover, in the latter case, the time variables are Fourier coefficients of $\gamma$ and $1/\gamma^{-1}$.Comment: 23 pages. This is to replace the previous preprint arXiv:0808.072

Casimir Effect in Spacetime with Extra Dimensions -- From Kaluza-Klein to Randall-Sundrum Models

In this article, we derive the finite temperature Casimir force acting on a pair of parallel plates due to a massless scalar field propagating in the bulk of a higher dimensional brane model. In contrast to previous works which used approximations for the effective masses in deriving the Casimir force, the formulas of the Casimir force we derive are exact formulas. Our results disprove the speculations that existence of the warped extra dimension can change the sign of the Casimir force, be it at zero or any finite temperature.Comment: 9 pages, 3 figure. Final version accepted by Phys. Lett.

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 $a\to 0^+$ and decays exponentially when $a\to \infty$. 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.Comment: 22 pages, 8 figure

Finite temperature Casimir effect for massive scalar field in spacetime with extra dimensions

We compute the finite temperature Casimir energy for massive scalar field with general curvature coupling subject to Dirichlet or Neumann boundary conditions on the walls of a closed cylinder with arbitrary cross section, located in a background spacetime of the form $M^{d_1+1}\times \mathcal{N}^n$, where $M^{d_1+1}$ is the $(d_1+1)$-dimensional Minkowski spacetime and $\mathcal{N}^n$ is an $n$-dimensional internal manifold. The Casimir energy is regularized using the criteria that it should vanish in the infinite mass limit. The Casimir force acting on a piston moving freely inside the closed cylinder is derived and it is shown that it is independent of the regularization procedure. By letting one of the chambers of the cylinder divided by the piston to be infinitely long, we obtain the Casimir force acting on two parallel plates embedded in the cylinder. It is shown that if both the plates assume Dirichlet or Neumann boundary conditions, the strength of the Casimir force is reduced by the increase in mass. Under certain conditions, the passage from massless to massive will change the nature of the force from long range to short range. Other properties of the Casimir force such as its sign, its behavior at low and high temperature, and its behavior at small and large plate separations, are found to be similar to the massless case. Explicit exact formulas and asymptotic behaviors of the Casimir force at different limits are derived. The Casimir force when one plate assumes Dirichlet boundary condition and one plate assumes Neumann boundary condition is also derived and shown to be repulsive.Comment: 28 pages, 4 figure

Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations

In this article, we classify the solutions of the dispersionless Toda hierarchy into degenerate and non-degenerate cases. We show that every non-degenerate solution is determined by a function $\mathcal{H}(z_1,z_2)$ of two variables. We interpret these non-degenerate solutions as defining evolutions on the space $\mathfrak{D}$ of pairs of conformal mappings $(g,f)$, where $g$ is a univalent function on the exterior of the unit disc, $f$ is a univalent function on the unit disc, normalized such that $g(\infty)=\infty$, $f(0)=0$ and $f'(0)g'(\infty)=1$. For each solution, we show how to define the natural time variables $t_n, n\in\Z$, as complex coordinates on the space $\mathfrak{D}$. We also find explicit formulas for the tau function of the dispersionless Toda hierarchy in terms of $\mathcal{H}(z_1, z_2)$. Imposing some conditions on the function $\mathcal{H}(z_1, z_2)$, we show that the dispersionless Toda flows can be naturally restricted to the subspace $\Sigma$ of $\mathfrak{D}$ defined by $f(w)=1/\overline{g(1/\bar{w})}$. This recovers the result of Zabrodin.Comment: 25 page

Effective Electromagnetic Lagrangian at Finite Temperature and Density in the Electroweak Model

Using the exact propagators in a constant magnetic field, the effective electromagnetic Lagrangian at finite temperature and density is calculated to all orders in the field strength B within the framework of the complete electroweak model, in the weak coupling limit. The partition function and free energy are obtained explicitly and the finite temperature effective coupling is derived in closed form. Some implications of this result, potentially interesting to astrophysics and cosmology, are discussed.Comment: 14 pages, Revtex

Repulsive Casimir Force from Fractional Neumann Boundary Conditions

This paper studies the finite temperature Casimir force acting on a rectangular piston associated with a massless fractional Klein-Gordon field at finite temperature. Dirichlet boundary conditions are imposed on the walls of a $d$-dimensional rectangular cavity, and a fractional Neumann condition is imposed on the piston that moves freely inside the cavity. The fractional Neumann condition gives an interpolation between the Dirichlet and Neumann conditions, where the Casimir force is known to be always attractive and always repulsive respectively. For the fractional Neumann boundary condition, the attractive or repulsive nature of the Casimir force is governed by the fractional order which takes values from zero (Dirichlet) to one (Neumann). When the fractional order is larger than 1/2, the Casimir force is always repulsive. For some fractional orders that are less than but close to 1/2, it is shown that the Casimir force can be either attractive or repulsive depending on the aspect ratio of the cavity and the temperature.Comment: 9 pages, 3 figure

The Casimir effect for parallel plates in the spacetime with a fractal extra compactified dimension

The Casimir effect for massless scalar fields satisfying Dirichlet boundary conditions on the parallel plates in the presence of one fractal extra compactified dimension is analyzed. We obtain the Casimir energy density by means of the regularization of multiple zeta function with one arbitrary exponent. We find a limit on the scale dimension like $\delta>1/2$ to keep the negative sign of the renormalized Casimir energy which is the difference between the regularized energy for two parallel plates and the one with no plates. We derive and calculate the Casimir force relating to the influence from the fractal additional compactified dimension between the parallel plates. The larger scale dimension leads to the greater revision on the original Casimir force. The two kinds of curves of Casimir force in the case of integer-numbered extra compactified dimension or fractal one are not superposition, which means that the Casimir force show whether the dimensionality of additional compactified space is integer or fraction.Comment: 9 pages, 3 figure