p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously. We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allow us to evaluate classical observables at any point of phase space simultaneously to evaluating quantum probability amplitudes. The example of the forced harmonic oscilator is used to demonstrate these concepts.
p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously. We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allow us to evaluate classical observables at any point of phase space simultaneously to evaluating quantum probability amplitudes. The example of the forced harmonic oscilator is used to demonstrate these concepts.
This is an up-to-date survey of the p-mechanical construction (see funct-an/9405002, quant-ph/9610016, math-ph/0007030, quant-ph/0212101, quant-ph/0303142), which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. Observables in p-mechanics are defined to be convolution operators on the Heisenberg group H^n. Under irreducible representations of H^n the p-observables generate corresponding observables in classical and quantum mechanics. p-States are defined as positive linear functionals on p-observables. It is shown that both states and observables can be realised as certain sets of functions/distributions on the Heisenberg group. The dynamical equations for both p-observables and p-states are provided. The construction is illustrated by the forced and unforced harmonic oscillators. Connections with the contextual interpretation of quantum mechanics are discussed. Keywords: Classical mechanics, quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, deformation quantisation, symplectic group, representation theory, metaplectic representation, Berezin quantisation, Weyl quantisation, Segal--Bargmann--Fock space, coherent states, wavelet transform, Liouville equation, contextual interpretation, interaction picture, forced harmonic oscillator.
This thesis is concerned with the representation theory of the Heisenberg group and its applications to both classical and quantum mechanics. We continue the development of p-mechanics which is a consistent physical theory capable of describing both classical and quantum mechanics simultaneously. p-Mechanics starts from the observation that the one dimensional representations of the Heisenberg group play the same role in classical mechanics which the infinite dimensional representations play in quantum mechanics. In this thesis we introduce the idea of states to p-mechanics. p-Mechanical states come in two forms: elements of a Hilbert space and integration kernels. In developing p-mechanical states we show that quantum probability amplitudes can be obtained using solely functions/distributions on the Heisenberg group. This theory is applied to the examples of the forced, harmonic and coupled oscillators. In doing so we show that both the quantum and classical dynamics of these systems can be derived from the same source. Also using p-mechanics we simplify some of the current quantum mechanical calculations. We also analyse the role of both linear and non-linear canonical transformations in p-mechanics. We enhance a method derived by Moshinsky for studying the passage of canonical transformations from classical to quantum mechanics. The Kepler/Coulomb problem is also examined in the p-mechanical context. In analysing this problem we show some limitations of the current p-mechanical approach. We then use Klauder's coherent states to generate a Hilbert space which is particularly useful for the Kepler/Coulomb problem.
p-mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how nonlinear canonical transformations affect p-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of p-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.
Wigner representation In classical optics (Born and Wolf, 1999), the state of the electromagnetic oscillator is perfectly described by the statistics of the classical amplitude α . The amplitude may be completely fixed (then the field is coherent), or α may fluctuate (then the field is partially coherent or incoherent). In classical optics as well as in classical mechanics, we can characterize the statistics of the complex amplitude?or, equivalently, the statistics of the components position q and momentum p introducing a phase space distribution W ( q , p ). (As explained in Section 3.1, the real and the imaginary part of the complex amplitude α can be regarded as the position and the momentum of the electromagnetic oscillator.) The distribution W ( q , p ) quantifies the probability of finding a particular pair of q and p values in their simultaneous measurement. Knowing the phase space probability distribution, all statistical quantities of the electromagnetic oscillator can be predicted by calculation. In this sense the phase space distribution describes the state in classical physics. All this is much more subtle in quantum mechanics. First of all, Heisenberg's uncertainty principle prevents us from observing position momentum simultaneously and precisely. So it seems there is no point in thinking about quantum phase space. But wait! In quantum mechanics we cannot directly observe quantum states either. Nevertheless, we are legitimately entitled to use the concept of states as if they were existing entities (whatever they are). We use their properties to predict the statistics of observations.
The quantum harmonic oscillator is a quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Coherent state is physically interesting because it is a minimum uncertainty state at which the Heisenberg uncertainty relation reaches its minimum. If a charged particle of charge moves in an electromagnetic field, there is a Lorentz force acting on the particle. The electric and magnetic are invariant under the gauge transformation. In the presence of an external vector potential, the Hamiltonian is changed, so is the probability current operator.