Re: Harmonic Osillator and Least Action

On Feb 26, 1:49 am, Arnold Neumaier <Arnold.Neuma...@xxxxxxxxxxxx>
wrote:

I don't have much time at the moment to consider this deeply, but
superficially, the construction seems to be alright.

However, I deny that Igor's construction is a standard Hamiltonian
description. Nowhere in the literature I know is the standard
Hamiltonian approach used with observables at different times.
Thus it is a true generalization of the Hamiltonian approach.
Whether it is useful needs to be seen.

On the Lagrangian side, there have been discussions of Lagrangians
that are nonlocal in time, so the new Lagrangian approach deserves
this name. But nobody seems to have made much progress in using
these nonlocal ideas. In particular, no one was successful in
quantizing these theories.

I haven't seen such a Hamiltonian before either, with a convolution
term,
but it does seem to work fine. Some further remarks, on what's good,
bad and indifferent:

The action
S = int_0^t dt [ m (x')^2 - k x^2 + x (K*x)) + xF ] ,
without further constraints, certainly works fine in the classical
case.
Moreover, it appears to generalise, using Igor's construction, to the
interaction between any two systems, providing only that the coupling
between the systems is linear. It is also interesting to note that
the
equation of motion,
m x'' + k x = (K + K#)*x + F = G*x + F
for the above action, with K#(t):=K(-t), is time-symmetric whenever
the
external force F(t) is time-symmetric. In particular, the
Hamiltonian
formalism is forced to model the memory kernel in a time-symmetric
manner. This is consistent with the notion that if one throws away
information at time t=0, the effect of doing propagates both ways in
time, similar to standard coarse-graining.

In the quantum case, while again the approach appears to work fine
(for the oscillator at least), dropping Igor's constraints does have
one
important consequence: the above equation of motion, rewritten for
for the position operator X, can only model CLASSICAL noise added
to a quantum system. One way of seeing this is to note that in
Igor's construction, the specification of the functions F and G
requires
the initial values to be given for the position and momentum of the
the external system. This is fine when adding classical noise to a
classical system. However, to add quantum noise to a quantum
system, one must quantise the external system at some level -
implying that sharp intial values for the latter cannot be imposed,
and
hence that F and G must be replaced by something more general.
So, "dissipation without decoherence" is only possible if the external
system is treated classically. However, this point does give some
insight into why the density operator is needed to model fully
quantum dissipation.

Igor's model with the further constraints on the action does not
suffer the above classical-quantum restriction, at least at first
sight.
Presumably, since F and G under these constraints satisfy certain
Poisson bracket relations, then one must, when quantising, replace
F and G by operators (instead of c-numbers). It would be of some
interest to demonstrate that this can be done so as to lead to the
usual master equation formulation.

Speculating further, Igor's construction could, possibly, lead to a
simplified master equation formulation. My reasons for thinking this
are:
(i) for finite quantum systems, one can always model any master
equation on the Hilbert space H_N as a Schrodinger equation on
the Hilbert space H_N x H_N, no matter what the physical
description of the external system is that leads to the master
equation.
(ii) Igor's formalism, with the further constraints, still appears to
throw away a good deal of information about the external system,
which may map to throwing away 'irrelevant' dimensions of the
external system when effectively quantised. In particular,
replacing F and G by operators may lead to quantisation on a
smaller Hilbert space than the tensor product
H_oscillator x H_external.
.

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