Lattice gauge fields and noncommutative geometry

Journal of Geometry and Physics

Volumn 24, Issue 4, 1998, Pages 353-385

  Balachandran, A P ^a,b   Bimonte, G ^a,c   Landi, G ^a,d,e   Lizzi, F ^a,e,f   Teotonio Sobrinho, P ^a,g,h  

^a Erwin Schrodinger International Institute for Mathematical Physics   (Austria)

^b SYRACUSE UNIVERSITY   (United States)

^c UNIVERSITY OF ZARAGOZA   (Spain)

^d UNIVERSITY OF TRIESTE   (Italy)

^e INFN SEZIONE DI NAPOLI   (Italy)

^f UNIVERSITY OF NAPLES FEDERICO II   (Italy)

^g UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^h UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

Chern simons terms; Geometry and topology of complexes; Lattice gauge theory; Noncommutative geometry; Topological actions

Indexed keywords

EID: 0032018187     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(97)00017-X     Document Type: Article

References (48)

1. [1] J. Donoghue, E. Golowich and B. Holstein, Dynamics of the Standard Model (Cambridge University Press, Cambridge, 1992).
2. [2] I. Montvay and G. Münster, Field Theories on the Lattice (Cambridge University Press, Cambridge, 1994).
3. [3] A.P. Balachandran, G. Bimonte, E. Ercolessi and P. Teotonio-Sobrinho, Finite approximations to quantum physics: Quantum points and their bundles, Nucl. Phys. B418 (1994) 477.
4. [4] A.P. Balachandran, G. Bimonte, E. Ercolessi, G. Landi. F, Lizzi, G. Sparano and P. Teotonio-Sobrinho, Finite quantum physics and noncommutative geometry. Nucl. Phys. B (Proc. Suppl.) 37C (1995) 20 and hep-th/9403067.
5. [5] A.P. Balachandran, G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, Noncommutative lattices as finite approximations, J. Geom. Phys. 18 (1996) 163 and hep-th/9510217.
6. [6] R.D. Sorkin, Finitary substitute for continuous topology, Int. J. Theor. Phys. 30 (1991) 323.
7. [7] J. Kijowski, Sympleczic geometry and second quantization, Reports Math. Phys. 11 (1977) 97; A.V. Evako, Dimension on Discrete Spaces, Int. J. Theor. Phys. 33 (1994) 1553 and gr-qc/9402035;
8. [7] J. Kijowski, Sympleczic geometry and second quantization, Reports Math. Phys. 11 (1977) 97; A.V. Evako, Dimension on Discrete Spaces, Int. J. Theor. Phys. 33 (1994) 1553 and gr-qc/9402035;
9. A.A. Grib and R.R. Zapatrin, Topologimeter and the problem of physical interpretation of topology lattice, Int. J. Theor. Phys. 35 (1996) 593 and gr-qc/9507033.
10. [8] P. Teotonio-Sobrinho, Self-Intersection numbers and random surfaces on the lattice, Nucl. Phys. B 452 (3995) 526 and hep-th/9505028.
11. [9] J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley, Reading, MA, 1984).
12. [10] E. Seiler, I.O. Stamatescu and D. Zwazinger, Monte Carlo simulation of noncompact QCD with stochastic gauge fixing, Nucl. Phys. B 239 (1984) 177; Numerical evidence for a barrier at the Gribov horizon, Nucl. Phys. B 239 (1984) 201.
13. [10] E. Seiler, I.O. Stamatescu and D. Zwazinger, Monte Carlo simulation of noncompact QCD with stochastic gauge fixing, Nucl. Phys. B 239 (1984) 177; Numerical evidence for a barrier at the Gribov horizon, Nucl. Phys. B 239 (1984) 201.
14. [11] D. Birmingham and P. Rakowski, A star product in lattice gauge theory, Phys. Lett. B 299 (1993) 299.
15. [12] J.M.G. Fell and R.S. Doran, Representations of *-Algebras, Locally Compact Groups and Banach *-Algebraic Bundles (Academic Press, New York, 1988).
16. [13] A. Connes and J. Lott, Particle models and noncommutative geometry, Nucl. Phys. B 18B (Proc. Suppl.) (1990) 29.
17. [14] A. Connes, Noncommutative Geometry (Academic Press, New York, 1994).
18. [15] J. Varilly and J.M. Gracia-Bondìa, Connes' noncommutative differential geometry and the standard model, J. Geom. Phys. 12 (1993) 223.
19. [16] P. di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Preliminary evidence for a U (1)-breaking in QCD from lattice, Nucl. Phys. B 192 (1981) 392; M. Luscher, Topology of lattice gauge fields, Comm. Math. Phys. 85 (1982) 39.
20. [16] P. di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Preliminary evidence for a U (1)-breaking in QCD from lattice, Nucl. Phys. B 192 (1981) 392; M. Luscher, Topology of lattice gauge fields, Comm. Math. Phys. 85 (1982) 39.
21. [17] M. Göckeler, A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.-J. Wiese, Can the topological susceptibility be calculated from SU (N) lattice gauge theories, Phys. Lett. B 233 (1989) 192.
22. [18] M. Teper, Pure gauge theories, Nucl. Phys. B (Proc. Sup.) 20 (1991) 159.
23. [19] R. Jackiw, Introduction to the Yang-Mills quantum theory, Rev. Mod. Phys. 52 (1980) 661; M. Bos and V.P. Nair, Coherent state quantization of Chern-Simons theory. Int. J. Mod. Phys. A 5 (1990) 959; R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990) 393.
24. [19] R. Jackiw, Introduction to the Yang-Mills quantum theory, Rev. Mod. Phys. 52 (1980) 661; M. Bos and V.P. Nair, Coherent state quantization of Chern-Simons theory. Int. J. Mod. Phys. A 5 (1990) 959; R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990) 393.
25. [19] R. Jackiw, Introduction to the Yang-Mills quantum theory, Rev. Mod. Phys. 52 (1980) 661; M. Bos and V.P. Nair, Coherent state quantization of Chern-Simons theory. Int. J. Mod. Phys. A 5 (1990) 959; R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990) 393.
26. [20] A.V. Phillips and D.A Stone, Chern-Simons an a lattice, Nucl. Phys. B (Proc. Supp.) 20 (1991) 28.
27. [21] D. Eliezer and G. Semenoff, Canonically quantized lattice Chern-Simons theory, in: Quantum Topology, eds., L.H. Kauffman and R.A. Baadhio (World Scientific, Singapore, 1993) p. 139.
28. [22] S.G. Rajeev, Universal gauge theory, Phys. Rev. D 42 (1990) 2779.
29. [23] G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, Lattices and their continuum limits, J. Geom. Phys. 20 (1996) 318 and hep-th/9507147; Noncommutative lattices and their continuum limits, 20 (1996) 329 and hep-th/9507148.
30. [23] G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, Lattices and their continuum limits, J. Geom. Phys. 20 (1996) 318 and hep-th/9507147; Noncommutative lattices and their continuum limits, 20 (1996) 329 and hep-th/9507148.
31. [24] P.S. Aleksandrov, Combinatorial Topology, Vols. 1-3 (Greylock, 1960); R.P. Stanley, Combinatorics, Vol. 1 (Woodsworth and Brooks, Advanced Books and Software, 1986); P.J. Hilton and S. Wyke, Homology Theory (Cambridge University Press, Cambridge, 1966).
32. [24] P.S. Aleksandrov, Combinatorial Topology, Vois. 1-3 (Greylock, 1960); R.P. Stanley, Combinatorics, Vol. 1 (Woodsworth and Brooks, Advanced Books and Software, 1986); P.J. Hilton and S. Wyke, Homology Theory (Cambridge University Press, Cambridge, 1966).
33. [24] P.S. Aleksandrov, Combinatorial Topology, Vois. 1-3 (Greylock, 1960); R.P. Stanley, Combinatorics, Vol. 1 (Woodsworth and Brooks, Advanced Books and Software, 1986); P.J. Hilton and S. Wyke, Homology Theory (Cambridge University Press, Cambridge, 1966).
34. [25] L. Bombelli, J. Lee, D. Meyer and R. Sorkin, Space-Time as a causal set, Phys. Rev. Lett. 59 (1987) 521.
35. [26] P.J. Hilton and S. Wyke, in [24]; W.S. Massey, Homology and Cohomology Theory (Marcel Dekker, New York. 1978).
36. [26] P.J. Hilton and S. Wyke, in [24]; W.S. Massey, Homology and Cohomology Theory (Marcel Dekker, New York. 1978).
37. [27] J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, LMS 206, 1995: A. Dimakis and F. Müller-Hoissen, Differential calculus and gauge theory on finite sets, J. Phys. A 27 (1994) 3159; Discrete differential calculus, graphs, topologies and gauge theory, J. Math. Phys. 35 (1994) 6703; A. Dimakis, F. Müller-Hoissen and T. Striker, Noncommutative differential calculus and lattice gauge theory, J. Phys. A 26 (1993) 1927; Umbral calculus, discretization and quantum mechanics on a lattice, quant-ph/9509014; A. Sitarz, Noncommutative differential geometry and connections on simplicial manifolds, hep-th/9603121.
38. [27] J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, LMS 206, 1995: A. Dimakis and F. Müller-Hoissen, Differential calculus and gauge theory on finite sets, J. Phys. A 27 (1994) 3159; Discrete differential calculus, graphs, topologies and gauge theory, J. Math. Phys. 35 (1994) 6703; A. Dimakis, F. Müller-Hoissen and T. Striker, Noncommutative differential calculus and lattice gauge theory, J. Phys. A 26 (1993) 1927; Umbral calculus, discretization and quantum mechanics on a lattice, quant-ph/9509014; A. Sitarz, Noncommutative differential geometry and connections on simplicial manifolds, hep-th/9603121.
39. [27] J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, LMS 206, 1995: A. Dimakis and F. Müller-Hoissen, Differential calculus and gauge theory on finite sets, J. Phys. A 27 (1994) 3159; Discrete differential calculus, graphs, topologies and gauge theory, J. Math. Phys. 35 (1994) 6703; A. Dimakis, F. Müller-Hoissen and T. Striker, Noncommutative differential calculus and lattice gauge theory, J. Phys. A 26 (1993) 1927; Umbral calculus, discretization and quantum mechanics on a lattice, quant-ph/9509014; A. Sitarz, Noncommutative differential geometry and connections on simplicial manifolds, hep-th/9603121.
40. [27] J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, LMS 206, 1995: A. Dimakis and F. Müller-Hoissen, Differential calculus and gauge theory on finite sets, J. Phys. A 27 (1994) 3159; Discrete differential calculus, graphs, topologies and gauge theory, J. Math. Phys. 35 (1994) 6703; A. Dimakis, F. Müller-Hoissen and T. Striker, Noncommutative differential calculus and lattice gauge theory, J. Phys. A 26 (1993) 1927; Umbral calculus, discretization and quantum mechanics on a lattice, quant-ph/9509014; A. Sitarz, Noncommutative differential geometry and connections on simplicial manifolds, hep-th/9603121.
41. [27] J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, LMS 206, 1995: A. Dimakis and F. Müller-Hoissen, Differential calculus and gauge theory on finite sets, J. Phys. A 27 (1994) 3159; Discrete differential calculus, graphs, topologies and gauge theory, J. Math. Phys. 35 (1994) 6703; A. Dimakis, F. Müller-Hoissen and T. Striker, Noncommutative differential calculus and lattice gauge theory, J. Phys. A 26 (1993) 1927; Umbral calculus, discretization and quantum mechanics on a lattice, quant-ph/9509014; A. Sitarz, Noncommutative differential geometry and connections on simplicial manifolds, hep-th/9603121.
42. [27] J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, LMS 206, 1995: A. Dimakis and F. Müller-Hoissen, Differential calculus and gauge theory on finite sets, J. Phys. A 27 (1994) 3159; Discrete differential calculus, graphs, topologies and gauge theory, J. Math. Phys. 35 (1994) 6703; A. Dimakis, F. Müller-Hoissen and T. Striker, Noncommutative differential calculus and lattice gauge theory, J. Phys. A 26 (1993) 1927; Umbral calculus, discretization and quantum mechanics on a lattice, quant-ph/9509014; A. Sitarz, Noncommutative differential geometry and connections on simplicial manifolds, hep-th/9603121.
43. [28] A.P. Balachandran, G Bimonte, G. Marmo and A. Simoni, Topology change and quantum physics, Nucl. Phys. B 446 (1995) 299.
44. [29] A. Dimakis, F. Müller-Hoissen and T. Striker, in [27].
45. [30] H.B. Lawson and M.-L. Michelsohn, Spin Geometry (Princeton University Press. Princeton. NJ, 1989).
46. [31] Cf. S. Coleman. 1/N, in Aspects of Symmetry, Selected Erice Lectures (Cambridge University Press, Cambridge, 1985).
47. [32] G. 't Hooft, How instantons solve the U (1) problem, Phys. Rep. 142 (1986) 257.
48. [33] A. Stern et al., in preparation.