Abelian gerbes as a gauge theory of quantum mechanics on phase space (original) (raw)
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A Gauge Theory of Quantum Mechanics
Modern Physics Letters A, 2007
An Abelian gerbe is constructed over classical phase space. The 2-cocycles defining the gerbe are given by Feynman path integrals whose integrands contain the exponential of the Poincaré-Cartan form. The U(1) gauge group on the gerbe has a natural interpretation as the invariance group of the Schroedinger equation on phase space.
Classical and quantum mechanics of non-abelian gauge fields
Nuclear Physics B, 1984
We i n v estigate the classical and quantum properties of a system of SU(N) non-Abelian Chern-Simons (NACS) particles. After a brief introduction to the subject of NACS particles, we rst discuss about the most general phase space of SU(N) i n ternal degrees of freedom or isospins which can be identied as one of the coadjoint orbits of SU(N) group by the method of symplectic reduction. A complete Dirac's constraint analysis is carried out on each orbit and the Dirac bracket relations among the isospin variables are calculated. Then, the spatial degrees of freedom and interaction with background gauge eld are introduced by considering the phase space of associated bundle which has one of the coadjoint orbit as the ber. Finally, the theory is quantized by using the coherent state method and various quantum mechanical properties are discussed in this approach. In particular, a coherent state representation of the Knizhnik-Zamolodchikov equation is given and possible solutions in this representation are discussed.
Bundle Gerbes Applied to Quantum Field Theory
Reviews in Mathematical Physics, 2000
This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and a proof that this leads in a simple manner to the known Schwinger terms (Mickelsson-Faddeev cocycle) for the gauge group action. This gives an explicit computation of the Dixmier-Douady class of the associated bundle gerbe. The method works also in other cases of fermions in external fields (external gravitational field, for example) provided that the APS theorem can be applied; however, we have worked out the details only in the case of vector potentials. Another example, in which the bundle gerbe curvature plays a role, arises from the WZW model on Riemann surfaces. A further example is the 'existence of string structures' question. We conclude by showing how global Hamiltonian anomalies fit within this framework.
Topological Quantum Field Theory on non-Abelian gerbes
Journal of Geometry and Physics, 2007
The infinitesimal symmetries of a fully decomposed non-Abelian gerbe can be generated in terms of a nilpotent BRST operator, which is here constructed. The appearing fields find a natural interpretation in terms of the universal gerbe, a generalisation of the universal bundle. We comment on the construction of observables in the arising Topological Quantum Field Theory. It is also shown how the BRST operator and the trace part of a suitably truncated set of fields on the non-Abelian gerbe reduce directly to the coboundary operator and the pertinent cochains of the underlyingČech-de Rham complex.
On the Relation Between Gauge and Phase Symmetries
We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state (e.g. from q and p to q or p in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau's moment map, the Mardsen-Weinstein symplectic reduction, the symplectic "category" introduced by Weinstein, and the conjecture (proposed by Guillemin and Sternberg) according to which "quantization commutes with reduction". By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit (the extreme cases of) this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points.
Gauge theory on a quantum phase space
Physics Letters B, 2001
In this note we present a operator formulation of gauge theories in a quantum phase space which is specified by a operator algebra. For simplicity we work with the Heisenberg algebra. We introduce the notion of the derivative (transport) and Wilson line (parallel transport) which enables us to construct a gauge theory in a simple way. We illustrate the formulation by a discussion of the Higgs mechanism and comment on the large N masterfield.
The Universal Gerbe, Dixmier–Douady Class, and Gauge Theory
Letters in Mathematical Physics - LETT MATH PHYS, 2002
We clarify the relation between the Dixmier–Douady class on the space of self-adjoint Fredholm operators (“universal B-field”) and the curvature of determinant bundles over infinite-dimensional Grassmannians. In particular, in the case of Dirac type operators on a three dimensional compact manifold we obtain a simple and explicit expression for both forms.
International Journal of Theoretical Physics, 2019
The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.