projection operators (e.g. polarizers??)

[John Denker <jsd@AV8N.COM>]

Photon polarization is widely used as a playground for
introducing students to "matrix mechanics".

The polarization of the light can be represented by a vector,
and various physical operations such as sending the light
through a birefringent crystal can be represented by a
matrix.

This representation (this model) is remarkably faithful
until we get to polarized absorbers.  These have heretofore
conventionally been represented by projection operators such
as

   Px = [ 1 0 ]
        [ 0 0 ]

   Py = [ 0 0 ]
        [ 0 1 ]

That's really quite a nasty way to represent the operation of
a physical object, because the operator is not unitary!

One of the tenets of theoretical physics is that all physical
processes are unitary.  The proof of Liouville's theorem depends
on this.  Not coincidentally our understanding of the Heisenberg
uncertainty principle depends on this, and also our understanding
of the 2nd law of thermodynamics.  Really, really bad things
happen to the foundation of physics if people start buiding
non-unitary physical devices.

Of course there are some legitimate _mathematical_ things you
can do with projection operators, such as expanding unity as
a sum of projection operators:

   1 = Px + Py

which you are free to do at your convenience ... but still I
insist that although unity itself can be realized as a physical
operator, the two terms on the RHS cannot *separately* be
realized as physical devices.

This is a pretty big deal, since for decades people used this
sort of projection operator as the model (or even the definition)
of how to make a measuring device.  Bohr did it, von Neumann
did it, Feynman did it ... but that doesn't make it right.

In starkest terms, what happens if you take a beam represented
by some wavefunction |B> and extinguish it by running it through
crossed polarizers.  The projection-operator model says that
what you get out is Px Py |B> which is 0 or equivalently 0|B> ...
but that is impossible.  That's unnormalized and unnormalizable.
What you really get is some dark state |D>.  If the frequencies
of interest are high enough and/or the polarizer is cold
enough |D> will essentially be the ground state |0> but I
reeeeeally want to emphasize that the ground state is a
legitimate normalized state ... |0> is  not to be confused
with 0 or 0|0> or 0|B>.

It is true that the ground state "looks the same" as no state
at all IF (big IF!) you are measuring things with only a photon
counter, i.e. the number operator (N = aDagger a) where
aDagger is the creation operator (ladder operator):
   <0| N |0> = 0
   <B| 0 N 0 |B> = 0     (for any B)

BUT (!) photon counters are not the only pieces of laboratory
apparatus in the world.  If you measure the ground state |0>
with a *voltmeter* (V = (aDagger + a)/2) you will see zero-point
fluctuations.

FURTHERMORE (!) the ground state |0> is not the only possible
dark state |D>.  A warm polarizer will put out radiation of
its own, independent of the input beam |B>, and this can be
detected in many ways.

We can make a much better model for the operation of the
crossed-polarizer device, or any other absorbing device, by
including a model of a heat bath.  The device then, in a
unitary way, "rotates" the incoming mode into the heat bath,
and rotates a mode out of the heat bath to create the output
wave.

    main                               main
    input  ----------\    /--------->  output
    beam              \  /             beam
                       \/
                       /\
                      /  \
    innards of ------/    \---------> innards of
    heat bath                         heat bath


Now, to apply this to actual polarization states:  If we
take some light that is polarized along the 45 degree
direction
  input = (sqrt .5) |Bx> + (sqrt .5) |By>

and run it through an X-polarizer, we do not
get
  output? = (sqrt .5) |Bx>
but rather
  output = (sqrt .5) |Bx> + (sqrt .5) |0y>
where the last term is the y-polarized component of
the ground state.  This output is a properly normalized
wavefunction.  (For simplicity this part of the discussion
is restricted to the low-temperature limit;  if the
polarizer wer giving off significant thermal radiation,
the dark "state" would be harder to describe;  density
matrices and all that.)

The matrix that describes this is:

  beam_x:       [ 1 0 0 0 ]
  beam_y:       [ 0 0 0 1 ]
  heat_x:       [ 0 0 1 0 ]
  heat_y:       [ 0 1 0 0 ]

Note that the four elements in the upper-right corner
"look like" the bad old 2x2 projection matrix, so we
can see how the bad 2x2 formulation is related to the
good, unitary, 4x4 representation.

I don't so much object when high-school books use the 2x2
non-unitary approach as a rough phenomenological model.
But people don't know when to quit!  There are lots of
supposedly-authoritative QM books out there that try to
explain or even define QM in general and measurement theory
in particular in terms of non-unitary devices ... which is
just wrong.  It is wrong in a big way, and guaranteed to
get people into trouble.  I once saw a Prize-winning
Very Important Person give an hour-long research seminar
full of wrong conclusions because of this.