A Brief Introduction to Quantum Geometry

Quantum Tori and Finite Quantum Spaces

By definition, a quantum 2-torus is based on a C*-algebra A generated by the elements U and V and relations

where z is a fixed complex unit. The algebra A will be commutative if and only if z=1, and in this case it describes the classical 2-torus. If z is different from 1 then A describes a purely quantum object. This space is called a quantum torus. It has no points at all. The quantum torus has the same U(1)xU(1) symmetry of the classical torus, because the group U(1)xU(1) acts by automorphisms on A, as in the classical case--by phase shifts of generators U and V. It is worth noticing that if z is not a root of unity, then (and only then) the algebra A is simple.

The quantum torus can be naturally viewed as a quantum principal U(1)-bundle over the base M=U(1). This is a nice example of a quantum principal bundle, with the classical base and the classical structure group. The existence of such bundles have a deep impact to the classification problematics of bundles in non-commutative geometry. For example, the classifying space for U(1) is still a quantum object.

Let us now consider algebras of the form

Quantum spaces based on such algebras are finite, because the algebras are finite-dimensional (and a classical space is finite iff its function algebra is finite-dimensional, and in this case the dimension of the algebra is the same as the number of points of the space). The space X has no points at all (because matrix algebras are always simple, and so do not admit characters). A purely quantum phenomena is that finite quantum spaces possess nontrivial continuous symmetries. In the case of X, all automorphisms of A are inner, given by the similarity transformations by unitary matrices. So the group of symmetries of X is Aut(A)=U(n)/U(1). Another purely quantum phenomena consists in a topological non-triviality of such spaces, and the nontriviality of differential structures over them. For example, the calculus based on derivations as explained above. The matrix spaces can be viewed as quantum counterparts of a 2-sphere, where the classical sphere is replaced by a quantum object consisting of n elementary quantum `cells' in such a way that the classical SO(3) symmetry is preserved. A detailed analysis of such quantum spaces can be found in [Mad].

A very interesting possible application of these examples in physics is in formulating a Kaluza-Klein type theory [DKM2,Mad], where the internal space-time manifold is one of the quantum spaces based on matrix algebras. One of new purely quantum phenomenas appearing in such Kaluza-Klein theories is a possibility to interpret Higgs fields as parts of Yang-Mills multiplets.