The Foldy-Wouthuysen Velocity Operator -- Is this Correct?

Jay R. Yablon

There has been a fair amount of discussion on sci.physics.foundations in
the thread "Might Foldy-Wouthuysen Transformations contain a Hidden
Fermion Mass Generation Mechanism?" regarding the fact that the velocity
operator "alpha" of the Dirac Hamiltonian appears to suggest that a
fermion must travel at the speed of light, which some describe as making
"no sense."

While I had earlier thought that this might be the seat of a
mass-generation mechanism, I do not at this juncture believe that is the
case. In particular, as shown below, all rest masses cancel, and the
velocity then becomes amenable to distinct treatment independent of rest
mass.

In the 3-page file linked below, I have shown how this velocity operator
is (I think??) transformed into the Newton-Wigner representation using
the Foldy-Wouthuysen transformation, and how in that latter
representation, that velocity operator comes to make perfect sense.

http://jayryablon.files.wordpress.co...y-operator.pdf

Perhaps more importantly, I have shown how the position operator may
developed, based not on a time parameter t, but on the Lorentz-invariant
physical proper time tau.

I would appreciate comments in two respects. 1) Is this a correct
calculation leading to (12)? 2) If so, is the particular result in
equation (12), something that is already known?

Of course, I welcome any other comments / discussion.

Thanks,

Jay.
____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm

Arnold Neumaier

Jay R. Yablon schrieb:
> There has been a fair amount of discussion on sci.physics.foundations in
> the thread "Might Foldy-Wouthuysen Transformations contain a Hidden
> Fermion Mass Generation Mechanism?" regarding the fact that the velocity
> operator "alpha" of the Dirac Hamiltonian appears to suggest that a
> fermion must travel at the speed of light, which some describe as making
> "no sense."
>
> While I had earlier thought that this might be the seat of a
> mass-generation mechanism, I do not at this juncture believe that is the
> case. In particular, as shown below, all rest masses cancel, and the
> velocity then becomes amenable to distinct treatment independent of rest
> mass.
>
> In the 3-page file linked below, I have shown how this velocity operator
> is (I think??) transformed into the Newton-Wigner representation using
> the Foldy-Wouthuysen transformation, and how in that latter
> representation, that velocity operator comes to make perfect sense.

For this connection, see the entry
S2h. Localization and position operators
in my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt
For related information, see also
S2g. Particle positions and the position operato


Arnold Neumaier

Jay R. Yablon

"Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> wrote in message
news:48764317.9090104@univie.ac.at...
> Jay R. Yablon schrieb:
>> There has been a fair amount of discussion on sci.physics.foundations
>> in
>> the thread "Might Foldy-Wouthuysen Transformations contain a Hidden
>> Fermion Mass Generation Mechanism?" regarding the fact that the
>> velocity
>> operator "alpha" of the Dirac Hamiltonian appears to suggest that a
>> fermion must travel at the speed of light, which some describe as
>> making
>> "no sense."
>>
>> While I had earlier thought that this might be the seat of a
>> mass-generation mechanism, I do not at this juncture believe that is
>> the
>> case. In particular, as shown below, all rest masses cancel, and the
>> velocity then becomes amenable to distinct treatment independent of
>> rest
>> mass.
>>
>> In the 3-page file linked below, I have shown how this velocity
>> operator
>> is (I think??) transformed into the Newton-Wigner representation
>> using
>> the Foldy-Wouthuysen transformation, and how in that latter
>> representation, that velocity operator comes to make perfect sense.
>
> For this connection, see the entry
> S2h. Localization and position operators
> in my theoretical physics FAQ at
> http://www.mat.univie.ac.at/~neum/physics-faq.txt
> For related information, see also
> S2g. Particle positions and the position operato
>
>
> Arnold Neumaier
>
Thank you, I did look over what you referred to and it was helpful (also
didn't know FW were unaware of NW when they came up with their
transformation), so let me pinpoint my main question:

In equation (12) of
http://jayryablon.files.wordpress.co...y-operator.pdf, I
derive the position operator:

X_op(tau,x)=alpha tau + beta x(tau) (12)

Above, tau is the proper time in the fermion rest frame (a Lorentz
invariant), alpha^k=gamma^0.gamma^k, and beta=gamma^0. x is the
ordinary coordinate position. Yes, as you say, the x in the above
requires an observer-dependent choice of coordinates, and so is a
"personal 3-space."

For the moment, I simply wish to know if (12) above and in the file
linked above is a) an accurate expression and b) a known expression.

Any chance someone can give me a yea or nay, and if nay, tell we what
seems to be wrong.

Thanks,

Jay.
___________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm