Worldline Numerics for Casimir Energies

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.

X0 changes the location of the loop

T changes the size of the loop

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.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

• 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.




Casimir Density and Edge Effects

• Two semi-infinite plates.





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


• Semi-infinite plate over infinite plate.


are needed to see this picture. QuickTime™ and aTIFF (Uncompressed) decompressor




• Semi-infinite plate on edge.






Works Cited

• Holger Gies, Klaus Klingmuller. Phys.Rev.Lett. 97 (2006) 220405arXiv:quant-ph/0606235v1

• Holger Gies, Klaus Klingmuller. Phys.Rev.Lett. 96 (2006) 220401 arXiv:quant-ph/0601094v1


Worldline Numerics for Casimir Energies

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