Re: Download a new book on quantum mechanics and relativity.

From: Eugene Stefanovich (eugenev_at_synopsys.com)
Date: 10/02/04 

Date: Fri, 01 Oct 2004 17:10:14 -0700

Bilge wrote:
> Eugene Stefanovich:
> >Bilge wrote:
> >> Eugene Stefanovich:
> >> >
> >> >Dirac's equation is a semi-relativistic thing.
> >>
> >> See any textbook which derives the dirac equation (e.g.,
> >> Bjorken & Drell, Vol I, chapter 2 ``Lorentz Covariance of
> >> the Dirac Equation'', esp. sec 2.2, ``Proof of Covariance'').
> >
> >You are talking about free Dirac equation. What about Dirac
> >equation with interaction? Can you prove its covariance?
>
> Sure. How do think one could obtain the gordon reduction for
> the current and show that a charge interacts through both it's
> spin and magnetic moment if the equation wasn't covariant:
>
> \gamma^u (p_u - eA_u) = m
>
> The interaction is, e\gamma^u A_u. That interaction is part of the
> covariant derivative, D_u = d_u + ieA_u,
>
> The electromagnetic current may be obtained by multiplying through
> by \gamma^v,
>
> \gamma^v\gamma^u (p_u - eA_u) = \gamma^v m
>
> using the commutation relations for the gamma^u and evaluating
> the expression for some initial and final state, p_u and p'_u.
> You'll obtain the expression for the electromagnetic current,
> (u+)(\gamma^v) u.
>
> With the addition of the field momenta, (1/4)F^uv F_uv, that equation is
> manifestly covariant. Not only is it manifestly covariant, it's the
> qed lagrangian, which may be written in a manifestly gauge invariant
> form as,
>
> (iD/ - m) + (1/4)F^uv F_uv
>
> with D/ = d/ + ieA/ = \gamma^u (d_u + ieA_u)
>
> and F^uv = (1/ie) [D_u, D_v], which to make the invariance ridiculously
> obvious, I can write,
>
> (iD/ - m) - (1/e^2) ([D_u, D_v])([D^u, D^v])
>
> = (iD/ - m) - (1/e^2) (D_uD_v - D_vD_u)(D^uD^v - D^vD^u)
>
>
> Which now contains no term that isn't obviously a lorentz scalar
> derived from only the canonical momentum and can't not transform
> properly under a lorentz boost.
>
> >In order to satisfy my request you need to show me
> >10 generators of the representation of the Poincare group
> >which satisfy well-known commutation relations
> >(see chapter 5 in my book).
>
> When do you plan to answer me with respect to the fact that
> lorentz boosts are poincare transforms, which makes your claims
> regarding the lorentz transforms needing to be non-linear to
> obtain poincare invariance, self-contradictory? My contention
> (which so far you have deliberately tried to misconstrue) is that
> you've mistaken a gauge transformation for a physical correction
> to the lorentz transforms because you ignored the fact that your
> hamiltonian isn't covariant and not even written in terms of
> canonical invariants, yet you treat it as if it were.

I think we disagree about the definition of relativistic invariance.
My definition is given in Statement N in subsection 5.2.4.
"The theory is relativistic invariant if inertial transformations of
observers are represented by unitary operators in the Hilbert space"
In other words, there should be a unitary representation of the Poincare
group in the Hilbert space of the system. This definition follows
directly from the principle of relativity, from the group properties
of inertial transformations, and from quantum postulates, as described
in chapter 5.

It seems that your definition of relativistic invariance is different:
You think that theory is relativistically invariant if all quantities
transform as 4-tensors (or 4-vectors or 4-scalars) under boosts.
In addition, you assume that boost operators are the same for
non-interacting and interacting systems.
Could you please tell me where this definition comes from?

>
> >> Where do you think the electron spin comes from?
>
> >I think electron spin comes from Wigner's irreducible
> >representation of the Poincare group (see section 7.1 in my book
> >or 2.5 in Weinberg's book)
>
> Either write it in your post or don't bother posting it. Obviously
> the spin doesn't come from ``Wigner's irreducible representation of''
> anything. It comes from physically interpreting what he derived.
> You simply assert that what may be interpreted from wigner's derivation
> applies to you. Since my contention is that it does not, nor do
> the rest of your references, post and interpret what you assert
> as supporting evidence.

OK, I can briefly repeat the logic of how spin is introduced in my
theory. First, I postulate that there should be a 3-component Hermitian
operator S in the Hilbert space of the system, which corresponds to the
intrinsic angular momentum (or spin). I postulate certain (rather
obvious) properties of this operator in 6.3.1 and guess
  the expression of S through 10 basic generators of the
Poincare group (P,J,K,H) in 6.3.2. Then, in subsection 6.3.6, I show
that this guess is the unique expression satisfying all physical
requirements. This expression is valid for any physical system
with strictly positive mass spectrum. This derivation is not
new (see, e.g., references [5] and [6] in chapter 6).

Then, in chapter 7, I consider one massive elementary particle.
For me, elementary particle is a system whose representation of
the Poincare group is irreducible. Wigner classified such irreducible
representations in 1939. He showed that they are characterized by two
parameters: mass m and spin squared s(s+1). For each value of momentum
there are 2s+1 orthogonal states corresponding to different values of
the spin projection on the z-axis -s, -s+1, ..., s-1, s.
As I show in equation on page 6 of chapter 7, these states are
  eigenstates of the spin operator S introduced in chapter 6.
All this is standard and fully agrees with chapter 2 of Weinberg books.

>
> If weinberg's text supported your assertions, weinberg would have
> derived them himself and abandoned the gauge theory for which he received
> a nobel prize. Unless you are claiming he renounces gauge theories and
> quantum field theory in his book, your reference to weinberg is
> irrelevant. Do you really believe weinberg would agree with you? If so,
> write him at the University of Texas at Austin and see if he agrees with
> the context in which you've referenced his textbook.
>

We were talking about spin, but now you switched to another subject -
gauge invariance. I never said gauge theories are useless. I said
many times that it is next to impossible to derive correct Hamiltonian
(and boost operator) of QED and standard model without involving fields
and gauge invariance. I don't know what is the physical meaning of this
invariance, but it helps a lot! After Hamiltonian and boost operator
are found and checked to satisfy Poincare commutators, we don't need
fields and gauges anymore. Everything goes downhill from this point,
and I discuss only this downhill slide in my book. I do not say a word
about how QED Hamiltonian was derived. I just expand the well-known
formula in terms of
creation and annihilation operators, perform renormalization, dressing
transformation, and calculations of
the S-matrix and other observable properties. That's what I am
interested in in the book.

You often complain that I do not allow you to use terms "fields",
"gauges", "Lagrangians" to discuss electromagnetic effects. This is the
whole point of my book that electromagnetic effects can be discussed
  without the unnecessary baggage of these notions. EM theory can be
constructed as a theory where there are particles and direct
interactions between them. That's what I did. The only thing I don't
know how to do myself is how to derive the operators of interparticle
interactions. That's what I need gauge theory for. This theory gives
me the interaction operators in H and K. The rest I do myself without
fields and gauges.

Eugene.

Relevant Pages

  • Re: Download a new book on quantum mechanics and relativity.
    ... > spin and magnetic moment if the equation wasn't covariant: ... which may be written in a manifestly gauge invariant ... > lorentz boosts are poincare transforms, ... I think we disagree about the definition of relativistic invariance. ...
    (sci.physics.relativity)
  • Re: Download a new book on quantum mechanics and relativity.
    ... >> to the lorentz transforms because you ignored the fact that your ... >I think we disagree about the definition of relativistic invariance. ... Look in weinberg. ... >> derived them himself and abandoned the gauge theory for which he received ...
    (sci.physics)
  • Re: Download a new book on quantum mechanics and relativity.
    ... >> to the lorentz transforms because you ignored the fact that your ... >I think we disagree about the definition of relativistic invariance. ... Look in weinberg. ... >> derived them himself and abandoned the gauge theory for which he received ...
    (sci.physics.relativity)
  • Re: Download a new book on quantum mechanics and relativity.
    ... spin and magnetic moment if the equation wasn't covariant: ... which may be written in a manifestly gauge invariant ... properly under a lorentz boost. ... lorentz boosts are poincare transforms, ...
    (sci.physics.relativity)
  • Re: Download a new book on quantum mechanics and relativity.
    ... spin and magnetic moment if the equation wasn't covariant: ... which may be written in a manifestly gauge invariant ... properly under a lorentz boost. ... lorentz boosts are poincare transforms, ...
    (sci.physics)
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