sci.physics.research

On 2008-02-26, Juan R. wrote:
> On Feb 17, 9:01 am, Igor Khavkine wrote:
>> On 2008-02-16, Juan R. wrote:

>> Velocity is defined by experiment.
>
> Velocity is defined by theory. Experimental outcomes are always
> interpreted *inside* the framework of some theory.

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.

>> I am eagerly awaiting your publications on the measurement of the "full
>> velocity of the quantum particle". Until then, we only have the usual
>> velocity measurement experiments, which are already well reproduced by
>> the velocity operator obtained with the help of the Foldy-Wouthuysen
>> transformation.
>
> As explained in [1] those "usual velocity measurement" correspond to
>
> {BLOCKQUOTE [emphasis mine]
> the *average* motion of the particle, . that given by classical
> relativistic formulae
> }
>
> I was asking for the actual velocity of the electron.

I see, so you want to talk about a quantity that *has not* been
measured and then compare it to a quantity that *has* been measured?
A rather strange endeavour.

> And i already said this before (but you ignored twice) Feynman (and
> myself also) shows that two-times arguments are not acceptable because
> of the commutation relations between v and p.

Juan R. originally wrote:
> Feynman even writes in his well-known textbook on QED [1] that
>
> {BLOCKQUOTE
> This result is sometimes made pausible by the argument that a precise
> determination of velocity implies precise determinations of position at
> two times. Then by the uncertainty principle, the momentum is completely
> uncertain and all values are equally likely, this is seen to imply that
> velocities near the speed of light are more probable, so that in the
> limit
> the expected value of the velocity is the speed of light. }
>
> Then in a footnote, Feynman recognizes that argument is invalid since v
> commutes with p. Thus, an incorrect value is justified using an
> incorrect
> argument!

Let me get this straight. You are saying that, for an electron, v is not
equal to c. Also, both you and Feynman are saying that the argument
presented above does not actually show that v = c. I completely agree
with both these statements, as long as v is the same velocity that
measures at centimeters per second in typical copper wire.

>> > How a non-observable transform into an observable in the non-
>> > relativistic limit?
>>
>> Let x - position coordinates (parameter)
>> X - position observable (operator)
>> |x> - an eigenstate of the X operator
>> chi(x) - position-space wave function
>> Psi(x) - field operator
>> |chi> - single-particle state
>> |0> - Fock vacuum
>>
>> 1. Project onto position eigenstates: chi(x) = .
>> 2. Embed state in Fock space: |chi> = int dx chi(x) Psi(x)^* |0>.
>> 3. Promote X to operator on Fock space: X = int dx x Psi(x)^* Psi(x).
>>
>> On the LHS of step 3, we have an operator X (an observable), while on
>> the RHS of the same equality we have integration over a coordinate x (a
>> field theoretic parameter). Step 3 is the answer to your question.
>>
>> This is a generic method for promoting single-particle operators to Fock
>> operators. In any particular situation, there will be differences.
>> Again, the Foldy-Wouthuysen transformation helps you pick out the right
>> position operator. See sections 59-65 of [3], for the general
>> prescription.
>
> Unfortunately my complaints already start with your step 1.
>
> It seems you do not understand difference between classical parameters
> and quantum observables. It seems also you lack knowledge of basic
> relativistic localization issues. But let us continue.

I will let my record attest to my understanding of the matter (or lack
thereof). As to your own record, you've yet to say much concrete on the
subject at all. 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.

> Relativistic quantum field theory (supposedly a fundamental theory)
> treat position x as a classical parameter (. *not* an observable)
> whereas non-relativistic quantum mechanics treat position x as an
> *observable*.
>
> The difficulty arises in a natural way. How does something (x) is not
> observable (not even being quantum!) transform into something is
> quantum and observable by making approximations?

The answer, as I've already stated, is in the formula of step 3 above.
There is no approximation, and this method works for non-relativistic as
well as relativistic theories. It describes the transition from particle
theory to field theory. The relativistic -> non-relativistic
approximation may be made at will at either end of the transition.

> Thus Dirac complaints still hold.

Unfortunately, as I've pointed out, Dirac is not here to be convinced
otherwise. Neither are any of the other authors whose objections you
have also quoted.

I've outlined a number of steps that, through a sequence of well defined
transformations and approximations, convert operators defined on the
relativistic field-theoretic Fock space to the usual position and
velocity operators on the non-relativistic particle Hilbert space, as
well as the reverse steps. In my understanding of your question, there
should be enough information here to give you an answer. While you do
not seem to be satisfied with this answer, I've yet to see you raise
concrete objections to any of these steps.

>> >> [1] Paul Strange, _Relativistic Quantum Mechanics_, Cambridge (1998).
>> >> Chapter 7.

>> >> [3] Paul A. M. Dirac, _Principles of Quantum Mechanics_.

Igor