Quantization by coherent states

In addition to direct applications such as quantum optics, spin waves, superfluidity, and solitons, coherent states have been used to address many fundamental issues in quantum mechanics. Coherent states offer several advantages over standard quantization techniques: they give a well regularized Wiener measure for the path integral, provide a natural relation between classical functions and quantum operators, and place quantization on a geometrical foundation. I will start by reviewing the definition of coherent states, the construction of the coherent path integral, and the use of the Wiener measure to regularize the path integral.; Recent work has been done to include constraints into coherent state quantization. For constrained systems, coherent states again offer several advantages over conventional quantization techniques: no Gribov problems, no need to eliminate second class constraints, no ambiguous determinants. They might also help determine the difference between Dirac and reduced phase space quantization methods. I will review my work and work done by others in this area, including Klauder's projection operator approach. I will then give some examples of constrained coherent states. All of these examples will be time reparameterization invariant theories for which the path integral is simple to solve. The first example is the free particle in one dimension. The second is the one dimensional harmonic oscillator. The third is a system of two harmonic oscillators.; The system of two harmonic oscillators is an important example in the study of quantum gravity. Because the theory of gravity must be time reparameterization invariant due to general covariance, there is no externally defined time coordinate for this system. If one has a pair of harmonic oscillators, the motion of one of the oscillators can give a natural clock to measure motion of the other oscillator. This model is a good toy model in which to study quantization of such systems. After discussing these models, I will give a natural projection method for operators using coherent states and Klauder's projection operator. Finally, I will discuss how this projection operator may be used to construct a natural time coordinate in terms of a clock determined from the motion of one of the oscillators.