×

Simplicial gauge theory and quantum gauge theory simulation. (English) Zbl 1229.81182

Summary: We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several \(SU(2)\) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of \(\beta \).

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81T25 Quantum field theory on lattices
81V05 Strong interaction, including quantum chromodynamics
35Q40 PDEs in connection with quantum mechanics
65C05 Monte Carlo methods
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Yang, C.-N.; Mills, R. L., Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev., 96, 191-195 (1954) · Zbl 1378.81075
[2] Weinberg, S., The Quantum theory of Fields, vol. 1: Foundations (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK
[3] Weinberg, S., The Quantum Theory of Fields, vol. 2: Modern Applications (1996), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK
[4] Peskin, M. E.; Schroeder, D. V., An Introduction to Quantum Field Theory (1995), Addison-Wesley: Addison-Wesley Reading, USA
[5] Wilson, K. G., Confinement of quarks, Phys. Rev. D, 10, 8, 2445-2459 (1974)
[6] Creutz, M., Quarks, Gluons and Lattices, Cambridge Monographs on Mathematical Physics (1986), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK
[7] Christ, N. H.; Friedberg, R.; Lee, T. D., Weights of links and plaquettes in a random lattice, Nucl. Phys. B, 210, 337 (1982)
[8] Christ, N. H.; Friedberg, R.; Lee, T. D., Gauge theory on a random lattice, Nucl. Phys. B, 210, 310 (1982)
[9] Christ, N. H.; Friedberg, R.; Lee, T. D., Random lattice field theory: General formulation, Nucl. Phys. B, 202, 89 (1982)
[10] Drouffe, J. M.; Moriarty, K. J.M.; Mouhas, C. N., Monte Carlo simulation of pure \(U(N)\) and \(SU(N)\) gauge theories on a simplicial lattice, Comput. Phys. Commun., 30, 249 (1983)
[11] Drouffe, J. M.; Moriarty, K. J.M.; Mouhas, C. N., \(U(1)\) four-dimensional gauge theory on a simplicial lattice, J. Phys. G, 10, 115 (1984)
[12] Drouffe, J. M.; Moriarty, K. J.M., Gauge theories on a simplicial lattice, Nucl. Phys. B, 220, 253-268 (1983)
[13] Drouffe, J. M.; Moriarty, K. J.M., \(U(2)\) four-dimensional simplicial lattice gauge theory, Z. Phys. C, 24, 395 (1984)
[14] Cahill, K. E.; Reeder, R., Comparison of the simplicial method with Wilsonʼs Lattice Gauge Theory for \(U(1)\) in three-dimensions, Phys. Lett. B, 168, 381 (1986)
[15] URL http://stacks.iop.org/0305-4616/10/i=10/a=001; URL http://stacks.iop.org/0305-4616/10/i=10/a=001
[16] Ardill, R. W.B.; Clarke, J. P.; Drouffe, J. M.; Moriarty, K. J.M., Quantum chromodynamics on a simplicial lattice, Phys. Lett. B, 128, 203 (1983)
[17] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications, vol. 4 (1978), North-Holland Publishing Company · Zbl 0445.73043
[18] Monk, P., Finite Element Methods for Maxwellʼs Equations (2006), Oxford Science Publications
[19] Nédélec, J.-C., Mixed finite elements in \(R^3\), Num. Math., 35, 315-341 (1980) · Zbl 0419.65069
[20] Whitney, H., Geometric Integration Theory (1957), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0083.28204
[21] Hiptmair, R., Finite elements in computational electromagnetism, Acta Numerica, 11, 237-339 (2002) · Zbl 1123.78320
[22] Bender, C. M.; Milton, K. A., Approximate determination of the mass gap in quantum field theory using the method of finite elements, Phys. Rev. D, 34, 10, 3149-3155 (1986)
[23] Bender, C. M.; Milton, K. A.; Sharp, D. H., Gauge invariance and the finite-element solution of the Schwinger model, Phys. Rev. D, 31, 2, 383-388 (1985)
[24] Halvorsen, T. G.; Sorensen, T. M., Simplicial gauge theory on spacetime · Zbl 1283.81106
[25] Flyvbjerg, H.; Petersen, H. G., Error estimates on averages of correlated data, Journal of Chemical Physics, 91, 1, 461-466 (1989)
[26] Creutz, M., Monte Carlo study of quantized \(SU(2)\) gauge theory, Phys. Rev. D, 21, 2308-2315 (1980)
[27] Christiansen, S. H.; Munthe-Kaas, H. Z.; Owren, B., Topics in structure-preserving discretization, Acta Numerica, 20, 1-119 (2011) · Zbl 1233.65087
[28] MPICH2, URL http://www.mcs.anl.gov/mpi/mpich2; MPICH2, URL http://www.mcs.anl.gov/mpi/mpich2
[29] H. Bauke, TINA pseudo-RNG library, URL http://trng.berlios.de; H. Bauke, TINA pseudo-RNG library, URL http://trng.berlios.de
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.