ar X iv : h ep - t h / 96 08 01 6 v 1 2 A ug 1 99 6 On Quantum Cohomology

We discuss a general quantum theoretical example of quantum cohomology and show that various mathematical aspects of quantum cohomology have quantum mechanical and also observable significance. 1 The quantum cohomology is one of the most fundamental and intressting mathematical-physical fields and although it is introduced according to certain physical models [1], however it should be considered as a general invariant geometrical tool for all quantum theories. Nevertheless, in view of various mathematical dificulties [2] its physical foundations are not well discussed yet. The main reason for this situations lies on the non-well understood toplogical or differential geometric structure of quantization as a general sheme. It is importent to mention that if one take the fact serious that classical mechanics is a classical limit of quantum mechanics, then a fundamental part of topology which is based on the globalization of classical mechanical results, e. g. Morse theory and symplectic topology, should be considered as a classical limit of some quantum topological originals. In view of the fact that the main difference between quantum and classical mechanics is the global (topological) character of states and accordingly the observables of quantum mechanics despite of local character of classical observables [3]; It is natural that the main difference in the classical and quantum geometries also arise in the topological scope. In other words, in view of the genuin topological character of quantization it is quite natural that quantization has such an influence like a quantum deformation of cohomology on the topology of the quantized system. Moreover, in view of the necessary symplectic background of quantization it is also not surprising that the quantum cohomology becomes equivalent to some generalization of certain results on invariant structures of symplectic mechanics (in quantum theoretical sense), i. e. to the so called Floer cohomology [4]. Briefly speaking the quantum cohomology should be considered as a result of existence of flat connections, which are related with quantization, together with the multiply connectedness of the quantum phase space which is related with multivalued functions. Equivalently, a closed " path " (circle) surrounding the minimum cell of the quantized phase space with an area h > 0 can not be shrunk to a point. It can be considered also as a result of finiteness of some relevant measures like position, i. e. position uncertainty δq which are prevented to become zero in quantum mechanics (δq > 0). In this sence, for …