1. Worldline Numerics for Casimir Energies
Jef Wagner
Aug
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)

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

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

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

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

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