Towards a nonperturbative path integral in gauge theories |
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| Institution: | 1. Fakultät für Physik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria;2. Department of Mathematics, Faculty of Science Division II, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan;3. Mathematisches Institut der Westfälischen Wilhelms-Universität, Einsteinstraße 62, D-48149 Münster, Germany;1. Department of Theoretical Physics, University of Szeged, Tisza Lajos krt 84-86, H-6720 Szeged, Hungary;2. Department of Theoretical Physics, WIGNER RCP, RMKI, H-1525 Budapest, P.O.B. 49, Hungary;1. LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France;2. Department of Physics, City College of New York, NY 10038, USA;1. National Research University Higher School of Economics, Russian Federation;2. Institute of Biochemical Physics of Russian Academy of Sciences, Kosygina str. 4, 119334, Moscow, Russian Federation;3. Skolkovo Institute of Science and Technology, 143026 Moscow, Russian Federation;4. Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russian Federation;5. ITEP, B. Cheremushkinskaya 25, Moscow 117218, Russian Federation;6. Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny, Moscow region, 141700, Russian Federation;1. Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata-700064, India;2. Department of Physics, Banaras Hindu University, Varanasi-221005, India;3. Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India |
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| Abstract: | We propose a modification of the Faddeev–Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato–Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few explicit examples are given to illustrate new features of the formalism developed. The method is applied to the Kogut–Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang–Mills theory. Feynman's conjecture about a relation between the mass gap and the orbit space geometry in gluodynamics is discussed in the framework of the modified path integral. |
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