Re: [Phys-l] Quantum of action

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding JD's comment:

> On 03/04/2012 04:51 PM, David Bowman wrote:
>
> > I agree with JD that h happens to be the mean area occupied per
> > basis state in phase space.  This shows up quite nicely when those
> > states are given their Wigner function representation as
> > real-valued (but not necessarily positive definite) phase space
> > functions.
>
> Point taken.  I should have added a caveat that the Wigner function
> is not a complete description of the state.  Such things are great
> for making gee-whiz pictures, but you wouldn't want to calculate
> anything serious with them.
> ...

Actually, the Wigner transform from Hilbert space operators to phase space is *invertible*, and its inverse is given via a Weyl transformation.  In that sense it *is* a complete description of any legitimate quantum state.  In particular, every pure state (projection operator on Hilbert space) and every mixed state (density matrix in Hilbert space) has a unique real-valued Wigner function representation in phase space.  Also, every real-valued phase space function representing any legitimate quantum state (mixed or pure) can be Weyl-transformed back to its Hilbert space representation giving the corresponding unique density matrix, and, if the state is pure, giving back the original normalized wave function up to an irrelevant overall phase factor.  Moreover, all the physical information contained in any quantum state still exists in the Wigner function representation of that state.  In particular, the probability of any outcome of any realizable physical interrogation/measurement of the state by some ideal experimental set-up is calculable from the Wigner function, just as it is from the Hilbert Space representation of the state.

But there also exist real-valued functions on phase space that are unphysical in the sense that they can't represent any actual quantum state.  Such functions *do* represent certain self-adjoint (Hermitian) operators on Hilbert space that are not trace class. That is why they can't represent a density matrix (i.e. physical state) because any legitimate density matrix is both self-adjoint *and* trace class. For instance, a real-valued function in phase space that is more localized than allowed by the Heisenberg principle does not map back (via the Weyl transform) to any actual quantum state in Hilbert space, even if it can represent some Hermitian operator (that may be unbounded, or at least not trace class).  It is only the subspace of phase space functions called the *Schwartz space* that map back to Hilbert space as legitimate density matrices that can represent an actual quantum state.  Nevertheless, every quantum state (pure or mixed) has a well-defined unique Wigner function representation on phase space which preserves all physically knowable information about that state, and in that sense such a function is complete.

But I think I agree that I wouldn't want to use a Wigner function for some serious level QM calculations.  Nevertheless, Wigner functions have been (and are) extensively used in optics, wavelet theory, frequency-time signal processing and the like where one wants simultaneous approximate behavior of functions in both domains of a dual representation related/bridged by a Fourier transform pair.

For further information Wikipedia has a number of nice articles on the subject (see: Wigner quasi-probability distribution , Weyl quantization, negative probabilities, Wigner transform, etc.)

David Bowman