Re: Lorentz transformations are not universal and not exact

From: Bill Hobba (bhobba_at_rubbish.net.au)
Date: 07/31/04 

Date: Sat, 31 Jul 2004 08:22:50 GMT

"Eugene" <eugenev@synopsys.com> wrote in message
news:410B312E.9010306@synopsys.com...
>
>
> Bill Hobba wrote:
> > "Eugene" <eugenev@synopsys.com> wrote in message
> > news:410AA7E9.5000304@synopsys.com...
> >
> >>
> >>Perfectly Innocent wrote:
> >>
> >>>Eugene <eugenev@synopsys.com> wrote in message
> >>
> > news:<4109D313.6040800@synopsys.com>...
> >
> >>>
> >>>>A theorem was proven by Currie, Jordan, and Sudarshan (Rev. Mod.
> >>>>Phys. 35 (1963), 350) that trajectories of two interacting particles
> >>>
> > simply
> >
> >>>>cannot transform according to Lorentz.
> >>>
> >
> > And the same Sudarshan (Fields and Quanta 2 1973) showed that the
easiest
> > way around the problem is to introduce the concept of a fields - in fact
> > other ways of doing it is really the field concept in disguise. Even
more
> > directly the same conclusion was reached by Van-Dam and Wigner in 1966
and
> > well known before then. Its proof, and its resolution, can be found on
page
> > 113 - Gravitation and Space-time under the heading of Local Fields vs
Action
> > at a Distance and is based on the violation of assumed conservation
laws.
> > It is however possible to formulate EM is such a way that only action at
a
> > distance is required (Reviews of Modern Physics - Volume 21 no 3 - July
> > 1947 - Feynaman and Wheeler - Classical Electrodynamics in Terms of
Direct
> > Interparticle Action) - the out being 'the energy tensor can only be
> > regarded as a provisional means of representing matter - In reality
matter
> > consists of electrically charges particles '. In other words they
> > demonstrated if your are willing to put up with the consequences of
these
> > 'no go theorems' (ie total momentum etc in the usual sense is not
> > necessarily being conserved - you need a non intuitive many-times
> > formulation of things like energy and momentum derived directly from
their
> > lagrangian) then no problems really arise. It may be mathematically, or
> > even conceptually, less appealing but it does not have the problems of a
> > particles field acting on itself.
> >
> >
> >>>
> >>>There are many exceptions to this theorem in the literature:
> >>>
> >>>http://www.lns.cornell.edu/spr/2002-08/msg0043518.html
> >>>
> >>>Take a look in google for the two-body problem in relativistic
> >>>action-at-a-distance theories.
> >>>
> >>>Eugene Shubert
> >>>http://www.everythingimportant.org
> >>
> >>You are probably talking about "constraint dynamics" in which CJS
> >>theorem does not hold. My statement should be more specific:
> >>"In Hamiltonian dynamics, trajectories of interacting particles
> >>cannot transform according to Lorentz".
> >
> >
> > What a load of bollocks. The Langrangian formulation (which is strongly
> > related to the Hamiltonian formulation) is easily extended to relativity
and
> > is Lorentz invariant - for example see the lagrangian in Feynmans paper
> > referenced above. Also see the introductory chapters of Landau -
Classical
> > Theory of Fields. Both the Lagrangeian and Hamiltonian of the EM
> > interaction is Lorentz invariant - however the PLA applies directly to
> > Lagrangians not Hamiltonians.
> >
> >
> >>My view is that Hamiltonian
> >>dynamics is a better way to describe interactions.
> >
> >
> > The Lagrangian formulation is the usual method. But it is well known
they
> > are related.
> >
> >
> >>Quantum field theory
> >>is based on the Hamiltonian dynamics.
> >
> >
> > It is based on langrangians just as much as Hamiltonians - the exact
> > relation between them can be found on page 22 - Weinberg - Quantum
Theory of
> > Fields. Weinberg prefers Hamiltonians because it directly appears in
the
> > Schrodenger equation whereas the rational for introducing langrangians
is it
> > makes it easy 'to choose interaction Hamiltonians for which the S matrix
> > satisfies various symmetries'. Other books however take a different
view
> > and introduce Lagrangians right from the start. Indeed the modern view
is
> > to concentrate on the symmetries and lagrangians (via Noethers theorem)
> > would seem to make it the more natural choice - but Weinberg is not a
man to
> > be dismissed lightly.
> >
> >
> >>I haven't heard if
> >>"constraint dynamics" can be used for many-body quantum problems
> >>where particle creation and annihilation is permitted. Maybe I missed
> >>something.
> >>
> >>My point is that violation of Lorentz transformations by the
> >>interacting Hamiltonian dynamics is not a problem.
> >
> >
> > My point is it is not violated - the concept of a field or a willingness
to
> > introduce concepts like advanced and retarded potentials rescues it.
> >
> >
> >>Lorentz
> >>transformations in their usual form are valid only for
> >>non-interacting particles. In the presence of interactions
> >>they should be modified.
> >
> >
> > Bollocks
> >
> > Bill
> >
> >
> Thank you very much for this great summary of currently accepted views
> (and for the references!). My approach is slightly different.
> First, I think that Lagrangian formalism and QFT are rather misleading
> if we want to interpret the Lorentz transformations for time and
> position of particles. True, all equations are written in the manifestly
> covariant form of 4-vectors etc. But QFT does not allow you to predict
> time evolution of the system: All it does is to calculate the S-matrix,
> i.e. the relationship between two limits of the time evolution (infinite
> past and infinite future). I don't think it is possible to derive
> trajectories of interacting particles (in the classical limit, of
> course) in the Lagrangian version of QFT. But only having
> trajectories of particles we can speak about events (e.g.,
> intersections of trajectories) and their Lorentz transformations.
>
> To have time evolution we need Hamiltonian.

But once you have a Hamnitonian you have Lagrangian.

> This is why I truly like
> Weinberg's approach to QFT: he puts the Hamiltonian in the center.
> To make the
> Hamiltonian theory relativistically invariant one needs to treat it
> as one of the 10 generators of the Poincare group: 3 translations
generated
> by the total momentum P, 3 rotations generated by the total angular
> momentum J, 3 boosts generated by operators K, and 1 time translation
> generated by the Hamiltonian H. The relativistic invariance is ensured
> by the Poincare commutation relations between these generators (eq.
> (3.3.11) - (3.3.17) in Weinberg's book).

Same for Lagrangians - what is your point? You claimed "In Hamiltonian
dynamics, trajectories of interacting particles cannot transform according
to Lorentz". Clearly the existence of Lorentz invariant theories usually
expressed in teems of lagrangians trivially disproves your claim.

>
> The important point is that since the Hamiltonian contains interaction
> terms, we must also add interaction terms to the operator of boost K
> (see eq. (3.3.20)) in order to preserve the
> Poincare commutators and the relativistic invariance of the theory.
> What does this mean physically? The operator of boost transforms
> observables to the moving frame of reference. This transformation
> depends on interaction. Therefore, in the presence of interactions,
> the simple Lorentz formulas (derived for non-interacting systems) become
> approximate. Which proves the title of this thread.

I think your are missing the main point - Lagrangians and Hamiltonians are
intimately related. If you have Weinberg book and have a look at page 22
you will see you can not really have one without the other. So your
position is quite meaningless.

Bill

>
> Regards.
> Eugene Stefanovich.
>
>
>

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