School of Mathematics,
University of Leeds,
Leeds LS2 9JT,
UK
and
Odesa National University,
Odesa,
UKRAINE
E-mail: kisilv@maths.leeds.ac.uk

Mixing Quantum and Classical States

Abstract:
There is a strong persistent interest over decades in the following problem: provide a self-consistent model for an aggregate system, which combines components with both quantum and classical behaviour.

This typically reduces to a search of "quantum-classical brackets" which should combine properties of quantum commutator $[\cdot, \cdot]$ and Poisson brackets $\{\cdot, \cdot\}$ in the corresponding sectors. Some simple algebraic combinations like $[A,B] +\frac{1}{2}(\{A,B\}-\{B,A\}\)$ were guessed during the last thirty years but neither of them turned to be completely satisfactory. Moreover some "no-go" theorems in that direction were proved over the last ten years. Thus the prevailing opinion now is that no mathematically consistent "quantum-classical" brackets is possible. This seems to close the whole problem.

However this question can be attacked from the direction which uses common symmetries of quantum and classical dynamics. In a sequence of paper started in [1] (see [2] for a recent treatment) we are developing a so-called $p$-mechanics, which naturally embed both quantum and classical descriptions. Observables are given by convolutions on the Heisenberg group. Then the Schr\"odinger (or  the Segal-Bargmann) representations map such observables into operators on Hilbert spaces. On the other hand one-dimensional (commutative) representations produce functions on the phase space.

The important step [2] is the definition of the universal brackets between convolutions on the Heisenberg group which are transformed by the above mentioned representations into the quantum commutator and the Poisson brackets correspondingly. Consequently it is sufficient to solve the dynamic equation in $p$-mechanics in order to obtain both quantum and classical dynamics.

States in $p$-mechanics are naturally defined [3] as linear functionals on $p$-observables. The choice of a pure state produces an irreducible representation which is either quantum or classic. But we may take a mixed state as well. A properly designed mixed state may evaluate some observables as quantum and other --- as classic, i.e. produce a mixing of quantum and classical mechanics. There is no need to search new "quantum-classical" brackets --- the universal brackets from $p$-mechanics will do the job, consequently there are no doubts in their consistency as well.

The proposed approach looks even more attractive in the light of generalisations of $p$-mechanics to field theory [4].

1. V.V. Kisil, Plain Mechanics: Classical and Quantum, J. of Natural Geometry, 9 (1996), no. 1, pp. 1-14; MR # 96m:81112; Zbl # 836.22010; funct-an/9405002.
2. V.V. Kisil, p-Mechanics as a Physical Theory: an Introduction, J. Phys. A, 37 (2004), no. 1, pp. 183-204; MR # 2005c:81078; quant-ph/0212101.
3. A. Brodlie and V.V. Kisil, Observables and states in p-mechanics, Advances in Mathematics Research, V, Nova Sci., 2003, pp. 101-136; quant-ph/0304023.
4. V.V. Kisil, p-Mechanics and De Donder-Weyl theory, The Fifth International Conference "Symmetry in Nonlinear Mathematical Physics", Proc. of Institute of Mathematics of NAS of Ukraine, v. 50 (part 3), 2004, pp. 1108-1115; quant-ph/0306101.