[Paper Review] Green's function for a n-dimensional closed, static universe and with a spherical boundary

Mustafa Özcan|arXiv (Cornell University)|Jun 26, 2001

Quantum Electrodynamics and Casimir Effect3 references4 citations

TL;DR

This paper constructs the Hadamard Green's function for a massless conformal scalar field in an n-dimensional closed, static universe and in a half-space with a spherical boundary, using eigenfunction expansion and the method of images. The key result is explicit analytical expressions for the Green's functions under Dirichlet and Neumann boundary conditions, confirming consistency with image method predictions and providing a foundation for Casimir energy calculations in curved, compactified spacetimes.

ABSTRACT

We construct the Hadamard Green's function by using the eigenfunction, which are obtained by solving the wave equation for the massless conformal scalar field on the S^n-1 of a n-dimensional closed, static universe. We also consider the half space case with both the Dirichlet and the Neumann boundary conditions. Solving of eigenfunction and eigenvalues of the corresponding field equation is interesting since the Casimir energy could be calculated analytically by various methods.

Motivation & Objective

• To derive the Hadamard Green's function for a massless conformal scalar field in an n-dimensional closed, static universe with global $S^{n-1}$ topology.
• To extend the Green's function construction to a half-space geometry with a spherical boundary at $\chi_0 = \pi/2$, modeling a physical region bounded by a reflective surface.
• To compute the Green's function under both Dirichlet and Neumann boundary conditions using eigenfunction solutions of the wave equation on $S^{n-1}$.
• To establish a correspondence between the Green's function and the image method on the double manifold $M \cup \partial M \cup M^*$, ensuring consistency with known regularization techniques.
• To provide a foundation for analytical computation of the Casimir energy in curved, compact spacetimes with non-trivial topology and boundary conditions.

Proposed method

• Solving the wave equation for a massless conformal scalar field on the $S^{n-1}$ sphere in n-dimensional spacetime using hyperspherical coordinates.
• Expanding the Green's function in terms of eigenfunctions derived from the scalar wave equation, with eigenvalues determined by the quantum numbers of the $S^{n-1}$ Laplacian.
• Applying the mode-sum method to construct the positive Wightmann function and then deriving the Hadamard Green's function via $G^{(1)}(x,x') = 2\operatorname{Re} G^{(+)}(x,x')$.
• Using the method of images to model boundary conditions: placing an image source in the dual region $M^*$ to satisfy Dirichlet ($G = 0$) or Neumann ($\partial G/\partial n = 0$) conditions on $\partial M$.
• Deriving the full Green's function as a superposition: $G(x,x') = D(x,x') \pm D(x,\tilde{x}')$, where $D$ is the Green's function on the full $S^{n-1}$, and $\tilde{x}'$ is the image point with $\chi' \to \pi - \chi'$.
• Expressing the Green's functions in terms of the chordal distance $\Delta s_{n-2}$ and its image $\Delta \tilde{s}_{n-2}$, with the cosine of the angular separation given recursively via $\cos\gamma_{n-\mu-2}$.

Experimental results

Research questions

• RQ1 How can the Hadamard Green's function be constructed for a massless conformal scalar field in an n-dimensional closed, static universe with $S^{n-1}$ spatial topology?
• RQ2 What is the form of the Green's function in a half-space geometry bounded by a spherical surface, under Dirichlet and Neumann boundary conditions?
• RQ3 How do the eigenfunctions of the wave equation on $S^{n-1}$ contribute to the construction of the Green's function in higher dimensions?
• RQ4 To what extent do the results from the eigenfunction method agree with the image method in the double manifold construction?
• RQ5 Can the Green's function be expressed in a closed analytical form that enables the calculation of the Casimir energy in curved, compact spacetimes?

Key findings

• The Hadamard Green's function for the n-dimensional closed, static universe is derived as a sum over eigenfunctions of the scalar wave equation on $S^{n-1}$, with explicit dependence on the spacetime interval and angular separation.
• For the half-space case with a spherical boundary, the Green's function is expressed as a superposition of the full-space Green's function and its image, with the sign determined by the boundary condition: $D_N$ for Neumann and $D_D$ for Dirichlet.
• The explicit form of the Green's function is given by $D_N(x,x') = \frac{\alpha_n}{4\pi^{n-2}R_0^{n-2}} \left[ \frac{1}{(\cos\frac{\Delta t}{R_0} - \cos\frac{\Delta s_{n-2}}{R_0})^{(n-2)/2}} + \frac{1}{(\cos\frac{\Delta t}{R_0} - \cos\frac{\Delta \tilde{s}_{n-2}}{R_0})^{(n-2)/2}} \right]$, and similarly for $D_D$ with a minus sign.
• The angular dependence is encoded in the recursive cosine expressions $\cos\gamma_{n-\mu-2}$, which generalize the spherical harmonic structure to $n$ dimensions, culminating in $\cos\gamma_1 = \cos\theta_{n-3}\cos\theta'_{n-3} + \sin\theta_{n-3}\sin\theta'_{n-3}\cos(\phi - \phi')$.
• The image method is confirmed as valid: the Green's function on the physical region $M$ is equivalent to $D(x,x') \pm D(x,\tilde{x}')$, where $\tilde{x}'$ is the image point with $\chi' \to \pi - \chi'$, and $\cos\frac{\Delta \tilde{s}_{n-2}}{R_0} = -\cos\chi\cos\chi' + \sin\chi\sin\chi'\cos\gamma_{n-3}$.
• The results are locally valid and independent of the global exterior geometry, making them suitable for applications such as the bag model in quantum chromodynamics, where only the interior region and boundary conditions matter.