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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

Worldline Numericsfor Casimir Energies

Jef Wagner

Aug 6 2007

Quantum Vacuum Meeting 2007

Texas A & M

casimir energy
Casimir Energy
  • Assume we have a massless scalar field with the following Lagrangian density.
  • The Casimir Energy is given by the following formula.
casimir energy1
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
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
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
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
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
Dirichlet Potentials
  • If the potentials are delta function potentials, and we take the Dirichlet limit, the expression for the energy simplifies greatly.
ideal evaluation
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
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 plates1
Parallel Plates
  • We need to evaluate the following:
  • The integral of this over x0 and T gives a final energy as follows.
error
Error
  • There are two sources of error:
    • Representing the ratio of path integrals as a sum.
error1
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
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 pfa1
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
Casimir Density and Edge Effects
  • Two semi-infinite plates.

Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405

casimir density and edge effects1
Casimir Density and Edge Effects
  • Semi-infinite plate over infinite plate.

Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405

casimir density and edge effects2
Casimir Density and Edge Effects
  • Semi-infinite plate on edge.

Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405

works cited
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