Realizing \(3\)-cocycles as obstructions. (English) Zbl 0847.17019
Authors’ abstract: The occurrence of a 3-cocycle in quantum mechanics or quantum field theory has been interpreted somewhat paradoxically as a breakdown of the Jacobi identity. The main result of this paper is that the 3-cocycle in chiral QCD arises as an obstruction which prevents the existence of a certain extension of one Lie algebra by another. The obstruction may be avoided by constructing a modified Lie algebra extension consisting of derivations on the algebra generated by the fields. However the 3-cocycle then appears when an attempt is made to implement these derivations by commutation with unbounded operators in the canonical equal-time formalism. Assuming the existence of these unbounded operators is what leads to the violation of the Jacobi identity.
Reviewer: Ma Zhong-Qi (Beijing)
MSC:
| 17B56 | Cohomology of Lie (super)algebras |
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
| 81V05 | Strong interaction, including quantum chromodynamics |
References:
| [1] | DOI: 10.1016/0370-2693(87)90540-5 · doi:10.1016/0370-2693(87)90540-5 |
| [2] | DOI: 10.1016/0370-2693(84)90952-3 · doi:10.1016/0370-2693(84)90952-3 |
| [3] | DOI: 10.1016/0370-2693(85)91146-3 · doi:10.1016/0370-2693(85)91146-3 |
| [4] | DOI: 10.1016/0370-2693(85)91472-8 · Zbl 1177.81097 · doi:10.1016/0370-2693(85)91472-8 |
| [5] | DOI: 10.1103/PhysRevLett.54.159 · doi:10.1103/PhysRevLett.54.159 |
| [6] | DOI: 10.1007/BF01213402 · Zbl 1223.81143 · doi:10.1007/BF01213402 |
| [7] | DOI: 10.1007/BF01218587 · Zbl 0688.22003 · doi:10.1007/BF01218587 |
| [8] | Ryzhov A. B., JETP Lett. 47 pp 153– (1988) |
| [9] | DOI: 10.1103/PhysRevLett.60.680 · doi:10.1103/PhysRevLett.60.680 |
| [10] | DOI: 10.1016/0370-2693(85)91147-5 · doi:10.1016/0370-2693(85)91147-5 |
| [11] | DOI: 10.1016/0370-2693(89)91091-5 · doi:10.1016/0370-2693(89)91091-5 |
| [12] | DOI: 10.1016/0370-2693(89)91091-5 · doi:10.1016/0370-2693(89)91091-5 |
| [13] | DOI: 10.1016/0370-2693(89)91091-5 · doi:10.1016/0370-2693(89)91091-5 |
| [14] | Wightman A. S., Ann. Inst. Henri Poincare 1 pp 403– (1964) |
| [15] | DOI: 10.1016/0370-2693(85)90295-3 · doi:10.1016/0370-2693(85)90295-3 |
| [16] | Grümm H. H.-R., Acta Phys. Austriaca 53 pp 113– (1981) |
| [17] | DOI: 10.1016/0003-4916(68)90316-3 · Zbl 0162.58202 · doi:10.1016/0003-4916(68)90316-3 |
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