Worldline Numerics for Casimir Energies Jef Wagner Aug 6 2007 Quantum Vacuum Meeting 2007 Texas A & M

Casimir Energy Assume we have a massless scalar field with the following Lagrangian density. The Casimir Energy is given by the following formula.

Casimir Energy We write the trace log of G in the worldline representation. The Casimir energy is then given by.

Interpretation or the Path Integrals We can interpret the path integral as the expectation value, and take the average value over a finite number of closed paths, or loops, x(u).

Interpretation of the Path integrals To make the calculation easier we can scale the loop so they all have unit length. Now expectation value can be evaluated by generating unit loops that have Gaussian velocity distribution.

Expectation value for the Energy We can now pull the sum past the integrals. Now we have something like the average value of the energy of each loop y(u). Let I be the integral of potential V.

Regularizing the energy To regularize the energy we subtract of the self energy terms A loop y(u) only contributes if it touches both loops, which gives a lower bound for T.

Dirichlet Potentials If the potentials are delta function potentials, and we take the Dirichlet limit, the expression for the energy simplifies greatly.

Ideal evaluation Generate y(u) as a piecewise linear function Evaluate I or the exponential of I as an explicit function of T and x0. Integrate over x0 and T analytically to get Casimir Energy.

X0 changes the location of the loop

T changes the size of the loop

A loop only contributes if it touches both potentials.

A loop only contributes if it touches both potentials.

A loop only contributes if it touches both potentials.

Parallel Plates Let the potentials be a delta function in the 1 coordinate a distance a apart. The integrals in the exponentials can be evaluated to give.

Parallel Plates We need to evaluate the following: The integral of this over x0 and T gives a final energy as follows.

Error There are two sources of error: Representing the ratio of path integrals as a sum.

Error There are two sources of error: Discretizing the loop y(u) into a piecewise linear function.

Worldlines as a test for the Validity of the PFA. Sphere and a plane. Gies Klingmuller Phys.Rev.Lett. 96 (2006) 220401

Worldlines as a test for the Validity of the PFA. Cylinder and a plane. Gies Klingmuller Phys.Rev.Lett. 96 (2006) 220401

Casimir Density and Edge Effects Two semi-infinite plates. Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405

Casimir Density and Edge Effects Semi-infinite plate over infinite plate. Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405

Casimir Density and Edge Effects Semi-infinite plate on edge. Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405

Works Cited Holger Gies, Klaus Klingmuller. Phys.Rev.Lett. 97 (2006) 220405 arXiv:quant-ph/0606235v1 Holger Gies, Klaus Klingmuller. Phys.Rev.Lett. 96 (2006) 220401 arXiv:quant-ph/0601094v1 Gies Klingmuller Phys.Rev.Lett. 96 (2006) 220401