Re: Hawton Position Operator. (Was: EM field of photon)

On the subject of extracting information from quantum states
which are thought of as operators (density matrices) rather
than vectors from a Hilbert space, I've added a blog post
covering what is variously called "Berry phase" or
"Pancharatnam-Berry phase" or "quantum phase":

http://carlbrannen.wordpress.com/2008/02/23/Berry

Berry phase is the phase that arises when a quantum
state is sent through a sequence of states back to its
original state. It is possible that it will pick up an overall phase
change when this happens. In general, one must go through
a sequence of at least two other states for this to happen
as it is due to non commutativity.

Density matrices of qubits are particularly convenient for
this sort of calculation since the arbitrary complex phase
of the spinor is eliminated from them. The remaining phase
information is Berry phase and nothing else.

In making the calculations, it is very natural to use the
same formalism that confused Neumaier in the last few
posts. That is, while previously probabilities (real numbers)
were given as real multiples of (primitive idempotent or
quantum state) matrices, in Berry phase calculations,
it is natural to obtain complex numbers as the complex
multiples of primitive idempotents.

All these things work because if P is a primitive idempotent,
and s and t are complex numbers, one has (sP) (tP) = (st P)
and (sP + tP) = (s+t)P, etc., so in considering the complex
multiples of a primitive idempotent (or ideal), one has a
perfectly adequate copy of the complex numbers.

The kind of calculations one does in this manner are
described in the above blog post, with the example
calculation being the Berry phase of the Pauli
algebra (spin -1/2).

But this sort of phase calculation is much more
general than that. While the Pauli algebra has a
natural imaginary unit given by the product of the
three Pauli spin matrices, a more general Clifford
algebra may have no natural imaginary unit.

For example, in the complex Dirac algebra with
signature -+++, the x and y vectors (which are
usually written as gamma_1 and gamma_2 but
I will write as x and y to simplify symbols and
to promote the observation that these are analogs
of the Pauli algebra sigma matrices) satisfy
the following relations:

x x = y y = (ixy) (ixy) = 1,
x y = -y x,
x (ixy) = -(ixy) x,
y (ixy) = -(ixy) y.

These are identical to the usual relations of
the Pauli algebra:
x x = y y = z z = 1,
x y = - y x,
x z = - z x,
y z = - z y,
with the replacement z == ixy.

This is generically true for any anticommuting bilinears
taken from the Dirac algebra; such a pair of elements,
when their signatures are corrected with possible
multiplication by i so that they square to +1, give a
representation of the Pauli algebra.

Therefore, the Berry phase calculations done in the Pauli
algebra can be taken also for quantum states rotated around
the "Bloch sphere" defined by two such anticommuting Clifford
algebra elements.

And in all these cases, computations are most easily made
when one is computing with respect to the complex multiples
of matrices rather than dealing with complex numbers per se.

So the notation looks like Phase( ABCA ) where A, B, C, and
A are matrices. To compute the "phase of a matrix", one
notes that if A is a primitive idempotent, a product that begins
and ends with A, such as AMA, is a complex multiple of A.
In the usual formalism, this would be tr(AM) = tr (AMA). So
for those who are easily confused, the notation Phase(AMA)
can be thought of as an abbreviation for Phase( tr (AMA) ).
The same applies to probabilities.

.

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