Natural Science Forum / Physics / Research / February 2008

Thus spake ilper@abv.bg

Indeed it is. I have given a treatment of photons from this point of
view, leading to Maxwell's equations as expectation values of the
operator A, in gr-qc/0605127.

Regards

Thus spake Oz <Oz@farmeroz.port995.com>

You ignore the fact that a locality condition is obeyed by the photon
field operator, A(x), quite separately and independently of the locality
condition for charged particles. In that sense, the photon has as much
right to be considered a particle as an electron.

it *is* an em field, by definition.

photons do not generate other photons, in the way that charged particles
generate photons. For this reason it is wrong to say that a photon has
an electromagnetic field in the sense that a charged particle has an
electromagnetic field.

It does. That is qed.

I find it better understood as a two way flow of photons.

Regards

Perhaps a suggestion is relevant to the physical meaning of this discussion
in the context of Cosmic Microwave Background Radiation CMBR

Is there a mass density associated with photons or not?

http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation
Quote:
The big bang theory suggests that the cosmic microwave background fills
all of observable space, and that most of the radiation energy in the
universe is in the cosmic microwave background, which makes up a
fraction of roughly 5E-5 of the total density of the universe.
UnQuote

The fraction 5E-5  is obtained in the context of Black Body Theory. CMBR
is the most perfect Black Body in nature, so this is appropriate.

Black Body radiation (power/area/time) = stefan's constant x T^4 T= 2.7
K for CMBR.

The mass density of the Black Body is computed as: CMBR Black Body
radiation (mass/volume) = (4/c^3) x stefan's constant x T^4

This mass density is roughly 5E-5 of the universe critical density
~1E-29 g/cm^3 or ~5E-34 g/cc

So in terms of this calculation derived from the very foundation of
quantum mechanics the photonic oscillators within the Black Body have
mass density.

This is consistent in a fundamental manner with margaret hawton's
presentation.

The factor T^4 is very powerful.

At high black body temperatures (T) significant photonic mass densities
(~T^4) can be achieved.

Richard D. Saam

...

The post is rather clear on this. Rather than take traces, a
mathematically equivalent method of defining probabilities
for pure quantum states is to take them as the real multiples
of primitive idempotents (states). By doing this, one keeps
one's formalism entirely in matrix form. Probabilities are not
matrices, but ratios of matrices.

Perhaps this isn't sufficiently explicit. Working in a finite
Hilbert space, let B be a primitive idempotent. Then
BB = B and trace(B) = 1.

Let M be some matrix. Then BMB is also a matrix, and by
the properties of primitive idempotents, it will be a complex
multiple of B. Therefore, assign <M>_B to be this complex
multiple. This allows one to define quantum averages without
need of specifying "trace". Instead, trace is built into the
definition of "primitive idempotent".

You don't have to define probabilties with "trace" in order to
have the equivalent of tr(AB) = tr(BA). However, since
the alternative definition of probabilities gives the same
numbers as the trace, one can move back and forth between
the definitions however one wishes. Similarly, when one
is working in the density matrix formalism, it is frequently
very convenient to take results well known in spinor
formalism and use them.

ABA is a complex (real for Hermitian) multiple of A,
and BAB is a complex multiple of B. The statement
that tr(AB) = tr(BA) amounts to the observation that
the "complex multiples" involved (i.e. ABA / A and
BAB / B), are the same.

To prove this, without reference to trace is easy and
I will leave it as an exercise for the reader with one
hint: It might be useful to prove that the matrix product
of two primitive idempotents AB is either zero, or a
complex multiple of a (not necessarily Hermitian)
primitive idempotent. What do you suppose that
complex multiple would be?

Somewhere in here I should mention that density matrices
are particularly handy in QFT in that virtual particles are
density matrices. When one writes the formalism in density
matrix form, one unifies the formalism for the virtual and real
states.

I think the most useful applications of density matrices is
in deep bound states. In analyzing these, one wants to
have the initial and final states both be bound states.

The usual QFT makes the initial and final states free, but
for a deeply bound state they should instead be virtual states.
In other words, one needs to analyze deeply bound states by
ignoring the state vectors that one usually sandwhiches the
virtual states between, but instead one looks at the algebra
of the virtual states among themselves.

To do all this, it helps to look at things from a point of view
where the virtual particles are at the foundation and the real
particles are just approximations of virtual particles. And
that is why density matrices are useful to understand.

On the other hand, if you are dealing in the usual formalism
where the state vectors are fundamental, then it makes perfect
sense to define average values by <a| M | a>. And that naturally
turns into traces in the density matrix formalism. From the spinor
point of view, defining probabilities as ratios of matrices is rather
contrived, but from a density matrix point of view, the trace is
also a rather contrived object.

As is usual in QM, the different formalisms give the same
result. It's just easier in one formalism or another to set
up the problem.

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.