Worldline Numerics for Casimir Energies

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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.

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Worldline Numerics for Casimir Energies

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Worldline Numericsfor 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.
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