The dynamics of systems composed of a classical sector plus a quantum sector is studied. We show that, even in the simplest cases, (i) the existence of a consistent canonical description for such mixed systems is incompatible with very basic requirements related to the time evolution of the two sectors when they are decoupled, (ii) the classical sector cannot inherit quantum fluctuations from the quantum sector, and (iii) a coupling among the two sectors is incompatible with the requirement of physical positivity of the theory, i.e., there would be positive observables with a nonpositive expectation value.
The dynamics of systems composed of a classical sector plus a quantum sector is studied. We show that, even in the simplest cases, (i) the existence of a consistent canonical description for such mixed systems is incompatible with very basic requirements related to the time evolution of the two sectors when they are decoupled. (ii) The classical sector cannot inherit quantum fluctuations from the quantum sector. And, (iii) a coupling among the two sectors is incompatible with the requirement of physical positivity of the theory, i.e., there would be positive observables with a non positive expectation value. PACS numbers: 03.65.Bz, 03.65.Sq, 03.65.Fd Typeset using REVTEXI. INTRODUCTION AND CONCLUSIONS Ever since the beginnings of quantum mechanics, physical systems have been considered which are composed of a quantum mechanical sector plus another sector described in classical terms [1]. For instance, this issue is central in the quantum theory of the measurement when the apparatus is treated classically. The same situation appears also at a less fundamental
Quantum mechanics is not logically closed with respect to the classical world.Its formalism unfolds as the quantization of a sub-set of classical Hamiltonians.The interpretation of quantum theory in terms of the measurement process inevitably requires to deal with systems composed by a mixture of both classical and quantum degrees of freedom.Moreover, when energy can flow between the quantum and classical degrees of freedom (i.e., in the case of nonadiabatic dynamics), there are more theoretical difficulties in order to obtain a fully consistent quantum-classical formalism.In order to perform calculations, one can renounce to the usual Lie algebraic structure of well-established physical theories, adopt non-Hamiltonian brackets, and obtain a formalism for the dynamics and statistics of quantum-classical systems that has an affordable computational complexity.Recent progress in the algorithms for the sampling of nonadiabatic dynamics of quantum-classical systems at long time is reviewed here.
We propose an extended quantum theory, in which the number of degrees of freedom K behaves as fourth power the number N of distinguishable states. As the simplex of classical N–point probability distributions of mixed quantum states, one can further increase the dimensionality constructing the set of extended quantum states. The embedding proposed corresponds to an assumption that the physical system described in N dimensional Hilbert space is coupled with an auxiliary subsystem of the same dimensionality. The extended theory works for simple quantum systems and is shown to be a non-trivial generalisation of the standard quantum theory for which K = N 2. Imposing certain restrictions on initial conditions and dynamics allowed in the quartic theory one obtains quadratic theory as a special case. By imposing even stronger constraints one arrives at the classical theory, for which K = N. can be embedded inside a higher dimensional convex body M Q N
In Newtonian spacetime, the canonical description of a classical
N-particle system requires
3N degrees of freedom. Not all of these are physical, however, since the conservation of the net momentum implies that only
3(N−1) accelerations are independent. Hence, three constraints can be used to eliminate the unphysical centre-of-mass variables, at the level of the Lagrangian, leaving only the subset of observable displacements and momenta, which are relational. Imposing the constraints does not change the dynamics of these variables, at the classical level, and is analogous to a gauge-fixing procedure, which removes redundancy in the description of the system. In classical physics, therefore, the number of physical degrees of freedom equals the number of independent relational degrees of freedom. Here, we show that this is not the case in quantum mechanics. While an operator-analogue of the classical net momentum exists, it cannot be used to impose constraints that restrict the degrees of freedom in the theory, without a loss of physical information. This means that all
3N canonical degrees of freedom are physical, even though only
3(N−1) of them are relational. We explore the physical consequences of the non-relational variables and show that they give rise to generalised uncertainty relations (GURs), for the relational quantities that define the quantum reference frame (QRF). Hence, it is shown that the non-relational degrees of freedom refer to the frame itself and that the non-Heisenberg terms in the GURs define its Galilean-invariant spreads, in both real space and momentum space. The implications of this result for recent work on relational models, including the ``perspective neutral'' framework for QRFs, are discussed. Its implications for the wider relational program, and, in particular, the relevance of the latter to quantum gravity research, are also critically assessed