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Worldline Numerics for Casimir Energies

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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  1. Worldline Numericsfor Casimir Energies Jef Wagner Aug 6 2007 Quantum Vacuum Meeting 2007 Texas A & M

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

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

  4. 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).

  5. 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.

  6. 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.

  7. 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.

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

  9. 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.

  10. X0 changes the location of the loop

  11. T changes the size of the loop

  12. A loop only contributes if it touches both potentials.

  13. A loop only contributes if it touches both potentials.

  14. A loop only contributes if it touches both potentials.

  15. 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.

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

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

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

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

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

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

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

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

  24. 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

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