worldline numerics for casimir energies
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/1.jpg)
Worldline Numericsfor Casimir Energies
Jef Wagner
Aug 6 2007
Quantum Vacuum Meeting 2007
Texas A & M
![Page 2: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/2.jpg)
Casimir Energy
• Assume we have a massless scalar field with the following Lagrangian density.
• The Casimir Energy is given by the following formula.
![Page 3: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/3.jpg)
Casimir Energy
• We write the trace log of G in the worldline representation.
• The Casimir energy is then given by.
![Page 4: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/4.jpg)
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).
![Page 5: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/5.jpg)
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.
![Page 6: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/6.jpg)
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.
![Page 7: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/7.jpg)
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.
![Page 8: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/8.jpg)
Dirichlet Potentials
• If the potentials are delta function potentials, and we take the Dirichlet limit, the expression for the energy simplifies greatly.
![Page 9: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/9.jpg)
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.
![Page 10: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/10.jpg)
X0 changes the location of the loop
![Page 11: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/11.jpg)
T changes the size of the loop
![Page 12: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/12.jpg)
A loop only contributes if it touches both potentials.
![Page 13: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/13.jpg)
A loop only contributes if it touches both potentials.
![Page 14: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/14.jpg)
A loop only contributes if it touches both potentials.
![Page 15: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/15.jpg)
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.
![Page 16: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/16.jpg)
Parallel Plates
• We need to evaluate the following:
• The integral of this over x0 and T gives a final energy as follows.
![Page 17: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/17.jpg)
Error
• There are two sources of error:– Representing the ratio of path integrals as
a sum.
![Page 18: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/18.jpg)
Error
• There are two sources of error:– Discretizing the loop y(u) into a piecewise
linear function.
![Page 19: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/19.jpg)
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
![Page 20: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/20.jpg)
Worldlines as a test for the Validity of the PFA.
• Cylinder and a plane.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Gies Klingmuller Phys.Rev.Lett. 96 (2006) 220401
![Page 21: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/21.jpg)
Casimir Density and Edge Effects
• Two semi-infinite plates.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405
![Page 22: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/22.jpg)
Casimir Density and Edge Effects
• Semi-infinite plate over infinite plate.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture. QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405
![Page 23: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/23.jpg)
Casimir Density and Edge Effects
• Semi-infinite plate on edge.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Gies KlingMuller Phys.Rev.Lett. 97 (2006) 220405
![Page 24: Worldline Numerics for Casimir Energies](https://reader036.vdocuments.mx/reader036/viewer/2022062422/56813b3c550346895da41090/html5/thumbnails/24.jpg)
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
Gies Klingmuller Phys.Rev.Lett. 96 (2006) 220401