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
• 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.
A loop only contributes if it
touches both potentials.
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.
• Sphere and a plane.
QuickTime™ and a
TIFF (U ncompressed) decompressor
are needed to see this picture.
Worldlines as a test for the
Validity of the PFA.
• Cylinder and a plane.
QuickTi me™ and a
TIFF (U ncompressed) decompressor
are needed to see this pi cture.
Casimir Density and Edge
Effects
• Two semi-infinite plates.
QuickTi me™ and a
TIFF (U ncompressed) decompressor
are needed to see thi s picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405
Casimir Density and Edge
Effects
• Semi-infinite plate over infinite plate.
QuickTime™ and a
TIFF (U ncompressed) decompressor
are needed to see this picture.
QuickTi me™ and a
TIFF (U ncompressed) decompressor
are needed to see this pi cture.
Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405
Casimir Density and Edge
Effects
• Semi-infinite plate on edge.
QuickTime™ and a
TIFF (U ncompressed) decompressor
are needed to see this picture.
QuickTi me™ and a
TIFF (U ncompressed) decompressor
are needed to see this picture.
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