On Mar 8, 10:27 am, "Juan R. González-Álvarez"

So what does psibar alpha psi represent? It's a group velocity
resulting from interference of the positive and negative energy
components of the spinor field. It is not vectorial, as a true
velocity must be, rather, the time-space part of a bivector, like the
electric field. That the expected value of it is always c is
physically interesting, but not in any way connected to the ponderable
velocity of the massive Dirac particle.

Juan R. Gonzalez-Alvarez wrote
(in the thread: What is the velocity of a relativistic electron?):
Arnold Neumaier wrote on Wed, 27 Feb 2008 12:33:20 +0000:

{BLOCKQUOTE [1]
The closed time path formalism proposed by Schwinger and refined by
many others, notable Keldysh, has been frequently used to study
nonequilibrium situations. But a close inspection of the formalism
reveals that it is more of a steady state theory than a truly
time-dependent theory. }

This ''close inspection'' cannot have a deep basis. How would it then be
possible that the paper
E. Calzetta and B. L. Hu,
Nonequilibrium quantum fields: Closed-time-path effective action,
Wigner function, and Boltzmann equation, Phys. Rev. D 37 (1988),
2878-2900.
derives finite-time Boltzmann-type kinetic equations from quantum field
theory using the CTP formalism?

Moreover, the SK formalism is based in inconsistent mixture of elements
from contradictory theories, e.g. Feynman diagrams for S-matrix (only
defined for infinite times) are mixed with a LvN equation for finite
times. See also for extra criticism on SK [1].

Any current formalism for interacting quantum field theory is
inconsistent and ill-defined when viewed from a logical point of view.

Nevertheless, people derive from it predictions, often very accurate
ones.

But I haven't seen a single article that gives meaning (i.e., infrared
and ultraviolet finite, renormalization scheme independent properties)
to, say, quantum electrodynamics states at finite t and their
propagation in time.

Nevertheless, there are very useful approximations towards that goal,
which I had mentioned in the part that follows, but which you didn't
quote.

I can see you do not cite any of the classical results over which
relativistic QFT was developed. For instance, the Peirls-Landau
inequality is the reason S = U(-infinity, +infinity) and that QFT lacks
dynamical description at finite times.

There is little point in citing old results which no longer reflect the
current state of the art.

One of main points of Eugene (and others, including me) affirms there
is *not* quantum dynamical relativistic theory based in Minkowski
space-time unification: "The Einstein-Minkowski 4-dimensional spacetime
is an approximate concept as well."

I find this view mistaken and not supported by the existing research
literature.

What velocities are measuring? Instantaneous velocities or averaged
velocities?

The answer is simple, and is founded on the theory of invariants as
expressed in the Dirac algebra.

The alphas are spacetime bivectors - surface elements in spacetime, thus

alpha_i = gamma_0 gamma_i

Thus, covariants formed from a spinor and the alphas cannot possibly
represent the velocity of a material body, which is vectorial, not
bivectorial (e.g. a free electromagnetic wave).

I would advise a study of the Foldy transformation.

On 2008-03-07, Juan R. González-Álvarez <juan@canonicalscience.com
wrote:

{BLOCKQUOTE [1]
The closed time path formalism proposed by Schwinger and refined by
many others, notable Keldysh, has been frequently used to study
nonequilibrium situations. But a close inspection of the formalism
reveals that it is more of a steady state theory than a truly
time-dependent theory. }

"steady state theory" = "time-independent theory"

[1] A unified formalism of thermal quantum field theory. 1994. Int. J.
of Mod. Phys A 9(14), 2363. Chu, H; Umezawa, H.

This reference is from the authors of a formalism called "thermo field
dynamics" (many papers and at least one textbook has been written on the
subject). From your quote it appears that the authors perceive the
Schwinger-Keldysh formalism as a direct competitor and alternative,
distinct from theirs.

The point of both, BTW, is to capture time dependent phenomena in field
theory. Unfortunately, the authors of [1] are mistaken. A very clear
side by side comparison of "thermo field dynamics" and the
Schwinger-Keldysh formalism reveals that the latter completely reproduce
and is more general than the former.

Lawrie, I. D.
On the relationship between thermo field dynamics and quantum
statistical mechanics
J. Phys. A: Math. Gen. 27 (1994) 1435-1452

Eugene's misconceptions regarding the content of QED have been pointed
out numerous times in this group.

On 2008-03-08, Juan R. <juan@canonicalscience.com> wrote:
Igor Khavkine wrote on Sun, 02 Mar 2008 23:24:23 +0000:

Well, then I guess that settles the question. I'm sure that people who
do electron transport measurements will be happy to hear that they
need to interpret their data so that electron velocities always come
out equal to the speed of light (to be consistent with the theoretical
definition given in your original post), instead of the centimeters
per second that their instruments have been telling them all along.

More of the same.

What velocities are measuring? Instantaneous velocities or averaged
velocities?

These measurements correspond to what has been called the "averaged
velocity" observable. Since no other kind of velocity has ever been
measured, I am completely justified in calling it "the velocity".

Each one has its own operator. This was discussed four or five times
before.

I've challenged you to come up with an experiment that has measured what
you call the "instantaneous velocity" (so far without response). Until
one can be found, the "instantaneous velocity" is merely an operator
with units of m/s and with little physical relevance.

It is ironic, you also seem completely unaware on how atomic and
molecular physicists, and quantum chemists work.

Take the classical (Darwin) expression

U_D = -1/2 (v v' + v r v' r) (e^2/r) 1/c^2

For a Dirac electron the velocity operator is

v_op = [H,x] = c alpha

substitution {v --> v_op} gives

U_B = -1/2 (alpha alpha' + alpha r alpha' r) (e^2/r)

This is the Dirac Breit operator broadly used in atomic and molecular
physics, and quantum chemistry.

Another example is the Gaunt potential, a standard in relativistic
molecular hamiltonians for bound states. Again its derivation involves

v --> c alpha.

These potentials are derived directly from the Dirac equation and remain
in the same 4-component spinor representation. These can also be
subjected to the Foldy-Wouthuysen transformation and put into
2-component Pauli-Schroedinger-like form (see [6]). Since the FW
transformation is unitary, spectral properties of the Hamiltonian remain
unchanged.

Ironically, this is also the way Feynman derived QED -now considered
the most precise theory of physics-. Take a look to his historical
paper [5].

Feynman starts from the classical interaction Lagrangian (he uses c =
1)

{ 1 - (v v') delta_+(s^2) }

and substitutes velocity by its quantum operator

{1 - (alpha alpha') delta_+(s^2) }

This gives [5] "our fundamental equation for electrodynamics."

In all those *standard* cases, the rule has been

v --> {v_op = c alpha}

Is not ironic physicists and chemists work with an velocity operator
"which quite obviously doesn't work" in your own words?

There are many equivalent formulations of quantum mechanics
(encompassing QED), including Feynmans. What is true of one formulation,
by equivalence, must be true of all other ones.

The use of the Dirac
representation, including its alpha and x operators, is not in the least
ironic.

Most of the time, only spectral properties of the Hamiltonian are of
interest. The expectation values of the x and alpha are not measured,
and hence need to be interpreted.

However, whenever high order
relativistic corrections are desired (to be computed in the
Pauli-Schroedinger representation, where position and momentum
observables have the common interpretations), these corrections are
invariably written in the FW representation (again, see [6]).

While I may not be an expert in relativistic localization, I
understand the topic well enough to point out that the question below
has nothing in particular to do with relativity.

I provided an adequate quotation "1.6 The Position operator" in a
previous message. You deleted but i reintroduce here:

{BLOCKQUOTE
The outcome, then, is that a position operator is inconsistent with
relativity.
}

I do not have access to Ticciatti's book at the moment, so I cannot
judge his statement in context. However, whatever his position is, it
doesn't change the fact that massive relativistic particles (whether
formulated on their own or in the context of field theory) possess a
well defined, frame dependent position operator

As noticed before your answer is incorrect. And your 'derivation'
inconsistent already at step 1.

Also You seems to be not disturbed becaue i was explcitely asking for
the QM well-known result

x =/= parameter

whereas your 'answer' began with assumption

x = parameter

Aha, perhaps you missed the second line of what I wrote above, where I
introduce X (capital X) as a single particle position operator
observable. Are you perchange denying the fact that a self-adjoint
operator has a real spectrum or that any state vector can be written as
a linear superposition of its eigenstates? You may want to read over
what I wrote above somewhat more carefully.

Igor Khavkine wrote on Mon, 10 Mar 2008 19:22:05 +0000:

I've challenged you to come up with an experiment that has measured what
you call the "instantaneous velocity" (so far without response). Until
one can be found, the "instantaneous velocity" is merely an operator
with units of m/s and with little physical relevance.

My problem is that before explaining to you anything about
measurements of
*velocity*, you would first understand what velocity i am speaking on!

Feynman original formulation of QED using relativistic
quantum mechanics and particles is *not* equivalent to QED formulation
based in fields over classical spacetime.

Somewhat as Wheeler/Feynman classical electrodynamics is not
equivalent to Maxwell electrodynamics based in fields.

You wrote:

Let x - position coordinates (parameter)
X - position observable (operator)
|x> - an eigenstate of the X operator

and you add:

X|x> = x|x

In my initial reply i wrote

{BLOCKQUOTE
It seems you do not understand difference between classical parameters
and quantum observables. [...] But let us continue.
}

But thanks by confirming now. I do not need to ask you more.

[5] Space-Time Approach to Quantum Electrodynamics. 1949.
Phys. Rev. 76, 769. Feynman, R. P.

Feynman original formulation of QED using relativistic quantum
mechanics and particles is *not* equivalent to QED formulation based in
fields over classical spacetime.

Somewhat as Wheeler/Feynman classical electrodynamics is not equivalent
to Maxwell electrodynamics based in fields.

That's false. To quote from the start of Section 1 of Feynman's paper
that you referenced [5]:

"Electrodynamics can be looked upon in two equivalent and complementary
ways. One is the description of the behavior of a field (Maxwell's
equations). The other is as a description of a direct interaction at a
distance (albeit delayed in time) between charges (the solutions of
Lienard and Wiechert)."

Presuming that Feynman is referring to his own work with Wheeler, he is
only a little imprecise. The equivalence is established exactly when the
boundary condition of no incoming or outgoing radiation is imposed on
Maxwell's equations.

The equivalence of the Fock space (field description) and Feynman
diagram approaches is demonstrated in any good book on QFT.

[...]
There exist not equivalence. You may be confused by Feynman usage of the
term 'field'. Moreover, you do not need to 'presume' Feynman thinking,
just go directly to Wheeler/Feynman paper on the topic [6].

{BLOCKQUOTE
The theory of direct interparticle action is equivalent, not to the
usual field theory, but to a modified or /adjunct field theory/.

[...]

This description of nature differs from that given by the usual field
theory in [...]

}

The equivalence of the Fock space (field description) and Feynman
diagram approaches is demonstrated in any good book on QFT.

Are not equivalent approaches. See quantum part on [7].

[6] Classical Electrodynamics in Terms of Direct Interparticle Action.
1949. Rev. Mod. Phys. 21(3), 425. Wheeler, J. A; Feynman, R. P.

[7] Cosmology and action-at-distance-electrodynamics. 1995. Rev. Mod.
Phys. 67(1), 113. Hoyle F; Narlikar, J. V.

The equivalence of the Fock space (field description) and Feynman
diagram approaches is demonstrated in any good book on QFT.

Igor Khavkine wrote
(in the thread ''What is the velocity of a relativistic electron?''):

The equivalence of the Fock space (field description) and Feynman
diagram approaches is demonstrated in any good book on QFT.

The equivalence is usually demonstrated only in the range of validity
of perturbation theory. The nonperturbative interpretation of field
theory is more complicated (solitons, instantons, etc.), and
different formalisms tend to reproduce different aspects of the
nonperturbative setting.

I don't know of any clear description of the relation of the various
approaches; but it is apparent from the applications that they are
not fully equivalent.