quantum analogue of classical mechanics

David Macmanus

My question involves a comparison between classical and quantum
mechanics. In classical mechanics of solid bodies there are two schemes;
call them D(iscrete) and C(ontinuous). In D one studies systems of
points - point masses and their movement under the influence of gravity
and more general Newtonian attraction. We write down the equations of
motion for such discrete systems.

Now in scheme C we don't repeat the analysis in D for infinitely many
points. Briefly, one does not repeat for a continuum of points the
analysis used for D: the sums in D are replaced by integrals in C.

In quantum mechanics we know the analogue of D: essentially
Schrodinger's equation for one particle or for a finite number of
particles.

My question is, what is a the QM analogue of (C) ?
Also, how can one apply quantum mechanics without having some idea of an
answer to this question ?
Hope this makes sense,
Thanks,
David.


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Arnold Neumaier

David Macmanus wrote:
> My question involves a comparison between classical and quantum
> mechanics. In classical mechanics of solid bodies there are two schemes;
> call them D(iscrete) and C(ontinuous). In D one studies systems of
> points - point masses and their movement under the influence of gravity
> and more general Newtonian attraction. We write down the equations of
> motion for such discrete systems.
>
> Now in scheme C we don't repeat the analysis in D for infinitely many
> points. Briefly, one does not repeat for a continuum of points the
> analysis used for D: the sums in D are replaced by integrals in C.
>
> In quantum mechanics we know the analogue of D: essentially
> Schrodinger's equation for one particle or for a finite number of
> particles.
>
> My question is, what is a the QM analogue of (C) ?

Quantum field theory.


Arnold Neumaier

Souvik

David Macmanus wrote:
> My question is, what is a the QM analogue of (C) ?

If your continuous medium can be represented as a field, and an
effective Lagrangian that simulates its classical behaviour be written
for it, you could order a quantum theory for the field.

The classical trajectory of the field being one that minimizes its
Lagrangian, its quantum 'trajectory' may be written down by summing
over all possible field trajectories with (complex) weights directly
proportional to the amplitude of each trajectory. The amplitude of each
trajectory is ~ exp(i*S) where S = integral over relevant time and
space of the Lagrangian.

If you are thinking of quantum versions of incompressible fluid flow
etc, I bet they could be modelled by some sophisticated
self-interacting quantum field theory or the like.

Hope this helps, or opens up further discussion.

-Souvik

markwh04@yahoo.com

David Macmanus wrote:
> My question involves a comparison between classical and quantum
> mechanics. In classical mechanics of solid bodies there are two schemes;
> call them D(iscrete) and C(ontinuous). [...] My question is, what is a the QM
> analogue of (C) ?

Hilbert spaces, the same as in Quantum Mechanics.

Let P be the classical phase space, P_0 any countable dense subset, and
let H_P the Hilbert space generated by P_0 (i.e., as the closure of the
complex vector space that is freely generated from P_0 as an
orthonormal basis). For each p in P_0, the corresponding state is |p>.
A classical observable is then defined as a bounded linear operator, A,
over H_P such that A commutes with all the projections |p><p|:
A |p><p| = |p><p| A, for all p in
P_0.

This is a quantum theory with supersection. The subspaces C |p> (for
each p in P_0) give you the superselection sectors. Since each sector
is 1-dimensional, there is no (meaningful) coherent superpositions.

Let B_P be the corresponding algebra of operators. Then the state space
associated with B_P is just P, itself. The algebra B_P is just the
algebra of classical observables over P.

(A state is defined formally as a linear functional W over the operator
algebra with W[o] giving you the "expectation" of operator o, such that
W[a] >= 0 if a >= 0; and W[1] = 1).

Each quantum degree of freedom has a dynamics given by the Heisenberg
equation. As Hojman and Shepley (1990) showed, such dynamics must
always have a classical Hamiltonian dynamics in the limit as h-bar -> 0
... meaning it's a quantization of a Hamiltonian dynamics. However, the
degrees of freedom corresponding to superselection parameters need not
be so constrained and may be governed by any type of law of motion,
Lagrangian/Hamiltonian/or otherwise. Since the system above has only
superselection degrees of freedom, there is no necessity of any
Hamiltonians appearing anywhere.

Going the other way, starting out with a classical phase space P, a
quantum theory (which has non-trivial coherent sectors) comes about by
smearing the states |p> of P_0 so that <p|q> != 0 is allowed for p !=
q. This is an instance of Berezin quantization and yields an equivalent
foundation for quantum theory.

The general states in the Berezin phase space are positive-definite and
may be interpreted as probability densities. The space is effectively
related to the Wigner "phase space" (which does not have
positive-definite distributions for its states) by a Gaussian smear.

Since the basis P_0 is not orthonormal, then it's no longer possible to
exactly produce the identity operator I = sum (|p><p|: p in P_0), as
you could in the classical case. The closest approximation to the
identity operator is directly related to the "reproducing kernel" and
underlies the smearing mentioned twice above.