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.
References:
G. Sardanashvily, “Lectures on Differential Geometry of Modules
and Rings. Application to Quantum Theory" (Lambert Academic
Publishing, Saarbrucken
G. Sardanashvily, Lectures on differential geometry of modules and
rings, arXiv: 0910.1515
WikipediA: Connection (algebraic framework)
No comments:
Post a Comment