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Wednesday, 15 January 2014

Foundations of Modern Physics: 1. The differential calculus on modules and rings

Differential geometry of smooth fiber bundles provides the comprehensive formulation  of classical field theory and mechanics.

At the same time, geometry in quantum systems speaks mainly the algebraic language of rings, modules and sheaves due to the fact that basic ingredients in the differential calculus and differential geometry of smooth manifolds can be restarted in a pure algebraic way.

Let X be a smooth manifold and C(X) a ring of smooth real functions on X. A key point is that, by virtue of the well-known Serre--Swan theorem, a C(X)-module is finitely generated and projective iff it is isomorphic to a module of sections of some smooth vector bundle over X. Moreover, this isomorphism is a categorial equivalence. Therefore, differential geometry of smooth vector bundles can be adequately formulated in algebraic terms of a ring C(X), its derivations and the Koszul connections.

In a general setting, let K be a commutative ring, A an arbitrary commutative K-ring, and P,Q some A-modules. The K-linear Q-valued differential operators on P can be defined. The representative objects of functors Q-> Dif(P,Q) are jet modules JP of P. Using the first order jet module J^1P, one also restarts the notion of a connection on an A-module P.

As was mentioned above, if P is a C(X)-module of sections of a smooth vector bundle Y-> X, we come to the familiar notions of a linear differential operator on Y, the jets of sections of Y-> X and a linear connection on Y-> X.

Let K be a commutative ring, A a (commutative or non-commutative) K-ring, and Z(A) the center of A. Derivations of A constitute a Lie K-algebra DA. Let us consider the Chevalley-Eilenberg complex of K-multilinear morphisms of DA to A, seen as a DA-module. Its subcomplex O*(A) of Z(A)-multilinear morphisms is a differential graded algebra, called the Chevalley – Eilenberg differential calculus over A. If A is an R-ring C(X) of smooth real functions on a smooth manifold X, the module DC(X) of its derivations is a Lie algebra of vector fields on X, and the Chevalley-Eilenberg differential calculus over C(X) is exactly an algebra of exterior forms on a manifold $X$ so that the Chevalley-Eilenberg coboundary operator d coincides with an exterior differential, i.e., O*(A) is the familiar de Rham complex. In a general setting, one therefore can think of elements of the Chevalley Eilenberg differential calculus over an algebra A as being differential forms over A.

Similarly, the differential calculus over a Grassmann-graded commutative ring is constructed. This is the case of supergeometry. In supergeometry, connections on graded manifolds and supervector bundles are defined as those on graded modules over a graded commutative ring and graded local-ringed spaces.

Note that a Grassmann-graded commutative ring is a particular non-commutative ring. However, the definition of its derivations differs from the non-commutative Leibniz rule. Therefore, supergeometry is not particular non-commutative geometry.

Non-commutative geometry also is developed as a generalization of the calculus in commutative rings of smooth functions. In a general setting, any non-nommutative K-ring Aover a commutative ring K can be called into play. One can consider the above mentioned Chevalley – Eilenberg differential calculus over A. However, the definition of differential operators on modules over commutative rings fails to be straightforwardly extended to the non-commutative ones. A key point is that A-module morphisms fail to be zero order differential operators if A is non-commutative. In this case, several nonequivalent definitions of differential operators have been suggested. Accordingly, there are also different definitions of a connection on modules over a non-commutative ring.


G. Sardanashvily, “Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory" (Lambert Academic Publishing, Saarbrucken, 2012) (#)

G. Sardanashvily, Lectures on differential geometry of modules and rings, arXiv: 0910.1515

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