Geometry in quantum theory I: Why familiar differential geometry contributes to quantum theory

Geometry of classical mechanics and field theory is mainly differential geometry of finite-dimensional smooth manifolds, fibre bundles and Lie groups.

The key point why geometry plays a prominent role in classical field theory lies in the fact that it enables one to deal with invariantly defined objects. Gauge theory has shown clearly that this is a basic physical principle. At first, a pseudo-Riemannian metric has been identified to a gravitational field in the framework of Einstein's General Relativity.  In 60-70^th of XX century, one has observed that connections on a principal bundle provide the mathematical model of classical gauge potentials. Furthermore, since the characteristic classes of principal bundles are expressed in terms of the gauge strengths, one can also describe the topological phenomena in classical gauge models. Spontaneous symmetry breaking and Higgs fields have been explained in terms of reduced G-structures. A gravitational field seen as a pseudo-Riemannian metric exemplifies such a Higgs field. In a general setting, differential geometry of smooth fibre bundles gives the adequate mathematical formulation of classical field theory, where fields are represented by sections of fibre bundles and their dynamics is phrased in terms of jet manifolds.

Autonomous classical mechanics speaks the geometric language of symplectic and Poisson manifolds. Non-relativistic time-dependent mechanics can be formulated as a particular field theory on fibre bundles over R.

At the same time, the standard mathematical language of quantum mechanics and quantum field theory has been long far from geometry. In the last twenty years the incremental development of new physical ideas in quantum theory (including super- and BRST symmetries, geometric and deformation quantization, topological field theory, anomalies, non-commutativity, strings and branes) has called into play advanced geometric techniques, based on the deep interplay between algebra, geometry and topology.

Let us start with familiar differential geometry. There are the following reasons why this geometry contributes to quantum theory.

(i) Most of the quantum models comes from quantization of the original classical systems and, therefore, inherits their differential geometric properties. First of all, this is the case of canonical quantization  which replaces the Poisson bracket {f,f'} of smooth functions with the bracket [f, f'] of Hermitian operators in a Hilbert space. Let us mention Berezin--Toeplitz quantization and geometric quantization of symplectic, Poisson and Kahler manifolds.

(ii) Many quantum systems are considered on a smooth manifold equipped with some background geometry. As a consequence, quantum operators are often represented by differential operators which act in a pre-Hilbert space of smooth functions. A familiar example is the Schrodinger equation. The Kontsevich deformation quantization is based on the quasi-isomorphism of the differential graded Lie algebra of multivector fields (endowed with the Schouten--Nijenhuis bracket and the zero differential) to that of polydifferential operators (provided with the Gerstenhaber bracket and the modified Hochschild differential).

(iii) In some quantum models, differential geometry is called into play as a technical tool. For instance, a suitable U(1)-principal connection is used in order to construct the operators f in the framework of geometric quantization. Another example is Fedosov's deformation quantization where a symplectic connection plays a similar role. Let us note that this application has stimulated the  study of symplectic connections.

(iv) Geometric constructions in quantum models often generalize the classical ones, and they are build in a similar way. For example, connections on principal superbundles, graded principal bundles, and quantum principal bundles are defined by means of the corresponding one-forms in the same manner as connections on smooth principal bundles with structure finite-dimensional Lie groups.

References:

G. Giachetta, L. Mangiarotty, G. Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (WS, 2005)
G. Sardanashvily, G. Giachetta, What is geometry in quantum theory arXiv: hep-th/0401080