Abstract
Confinement in non-abelian gauge theory involves the idea that the vacuum state is disordered at large scales; our best evidence that this is true comes from Monte Carlo simulations of lattice gauge theories. So to begin with, I need to explain what is meant by
-
a disordered state,
-
a lattice gauge theory,
-
a Monte Carlo simulation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The construction was first introduced by Wegner [2].
- 2.
This leading-order (in 1/m 2) result is obtained by neglecting the one-link term in S matter everywhere except along the time-like links from t = 0 to t = T, at x = 0 and x = R. On these links, expand \(\exp[\phi^\dagger U_0 {\phi}+\hbox{h.c.}] \approx 1 + \phi^{\dagger}U_0 {\phi}+\hbox{h.c.}\). Integration over the scalar field then yields the result (2.32).
- 3.
One approach, based on the non-abelian Stokes Law, derives an area law for a large Wilson loop from an assumed finite range behavior of field strength correlators, which means that field strengths are uncorrelated, i.e. disordered, at sufficiently large separations. This “field correlator” approach to magnetic disorder has been pursued by Simonov and co-workers [10].
- 4.
The Wilson loop calculation is a little easier in two dimensions with free boundary conditions. Periodic boundary conditions introduce a correction which is irrelevant for N p large.
References
Elitzur, S.: Impossibility of spontaneously breaking local symmetries. Phys. Rev. D 12, 3978–3982 (1975)
Wegner, F.J.: Duality in generalized Ising models and phase transitions without local order parameter. J. Math. Phys. 12, 2259–2272 (1971)
DeGrand, T., DeTar, C.E.: Lattice Methods for Quantum Chromodynamics. World Scientific, Singapore (2006)
Seiberg, N.: Electric–magnetic duality in supersymmetric non-Abelian gauge theories. Nucl. Phys. B 435, 129–146 (1995) (arXiv:hep-th/9411149)
Intriligator, K.A., Seiberg, N.: Phases of N = 1 supersymmetric gauge theories and electric–magnetic triality. In: Strings 95: Future Perspectives in String Theory (arXiv:hep-th/9506084)
Diakonov, D., Petrov, V.Y.: A formula for the Wilson loop. Phys. Lett. B 224, 131–135 (1989)
Karp, R.L., Mansouri, F., Rno, J.S.: Product integral formalism and non-Abelian stokes theorem. J. Math. Phys. 40, 6033–6043 (1999) (arXiv:hep-th/9910173)
Arefeva, I.: Non-Abelian Stokes formula. Theor. Math. Phys. 43, 353–356 (1980) (Teor. Mat. Fiz. 43, 111 (1980))
Fishbane, P.M., Gasiorowicz, S., Kaus, P.: Stokes's theorems for non-Abelian fields. Phys. Rev. D 24, 2324–2329 (1981)
DiGiacomo, A., Dosch, H.G., Shevchenko, V.I., Simonov, Yu. A.: Field correlators in QCD: Theory and applications. Phys. Rep. 372, 319–368 (2002) (arXiv:hep-ph/0007223)
Halpern, M.B.: Field strength and dual variable formulations of gauge theory. Phys. Rev. D 19, 517–530 (1979)
Batrouni, G.G., Halpern, M.B.: String, corner and plaquette formulation of finite lattice gauge theory. Phys. Rev. D 30, 1782–1790 (1984)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Greensite, J. (2010). Global Symmetry, Local Symmetry, and the Lattice. In: An Introduction to the Confinement Problem. Lecture Notes in Physics, vol 821. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14382-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-14382-3_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14381-6
Online ISBN: 978-3-642-14382-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)