Re: Interpretation of canonical/contact transformations

From: Igor Khavkine (igor.kh_at_gmail.com)
Date: 01/18/05 

Date: Tue, 18 Jan 2005 19:21:17 +0000 (UTC)

On Fri, 14 Jan 2005 17:01:47 +0000, Eugene Stefanovich wrote:

> Exactly. No physical predictions would change if all elements in the
> Hilbert (or Fock) space (state vectors and operators) are subjected by the
> same unitary transformation (that's the point of the Wigner theorem). I
> was talking about the situation in which only Hamiltonian is transformed,
> while other operators of observables and state vectors are not touched.

If you count the Hamiltonian + other observables as a complete theory,
then yes in general you are creating a different theory.

> With two Hamiltonians unitarily connected to each other we have two
> different theories. That was my point.

It is not important that the Hamiltonian is transformed unitarily, only
that it is transformed independently of all other observables. However, in
the case of this kind of transformation, you have to be aware of the
possibility that UHU^* may differ from H, while UXU^* = X, where X
represents other observables. This kind of transformation is really no
more significant than a change of basis.

> However, I realize, that in many
> respects these two different theories are very similar to each other. For
> example, they have the same energy spectra of bound states. Quite often
> they also have the same S-matrices. In high energy physics the energies
> of bound states and S-matrix elements exhaust all properties available
> to observation. In my opinion, this created a (wrong) impression that
> unitarily connected Hamiltonians are also physically equivalent.

That depends on your notion of physical equivalence. A mass-spring system
and a small amplitude pendulum are obviously different physical systems.
But once each is abstracted enough, it can be modeled as a standard simple
harmonic oscillator. One should not fear abstraction but embrace it.
Abstraction allows you to do a calculation once and apply it many times
over.

> These considerations apply not just to Hamiltonians, but to other
> generators of inertial transformations as well (total momentum, total
> angular momentum, and boost). It has been proven by Sokolov that these
> 10 operators in different forms of dynamics can be connected by unitary
> transformations (the transformation acts on these 10 operators only,
> physical states and other observables are not touched) preserving the
> S-matrix. Many people interpreted that that different forms of dynamics
> are totally physically equivalent. In my book I am arguing that this
> interpretation is wrong. I hope that you agree with me.

I hate to shatter your hopes. I cannot pretend that I am familiar with
different forms of relativistic dynamics enough to dispute this point with
you. But Arnold Neumaier has already discussed this at length and I defer
to his opinion. In the end, I believe the differences between your
interpretation and the conventional one are of no consequence, just as
we've discovered in the case of the QED Hamiltonian. If you do believe
that you've discovered a physical prediction that is at odds with
conventional theory, you are either mistaken in the interpretation of what
is physical or have made a calculational mistake.

Now is the time to invest effort into seeing how your view is equivalent
to the conventional instead of arguing against it.

Igor

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