Forms of relativistic dynamics (was: Connes ...)

From: Arnold Neumaier (Arnold.Neumaier_at_univie.ac.at)
Date: 11/30/04 

Date: Tue, 30 Nov 2004 18:49:15 +0000 (UTC)

Eugene Stefanovich wrote:
> Arnold Neumaier wrote:
>
>>Eugene Stefanovich wrote:
>>
>>>This would mean that boosts are kinematical. This, in turn,
>>>is rejected by
>>>the Currie-Jordan-Sudarshan theorem. The CJS is a very important
>>>statement which is often underestimated, or even ignored, in QFT.
>>
>>The CJS theorem only forbids a dynamics with observer-independent worldlines
>>for each particle. Field theories do not make this assumption, hence are
>>not affected by the theorem. So it can be safely ignored in QED.
>
> You are right that CJS theorem has no relevance to the traditional QED,
> because this approach only cares about the S-matrix, and does not care
> about the time-dependence of particle observables (this time dependence
> is represented by trajectories in the classical limit). In particular,
> QED cannot even talk about the speed of propagation of interaction,
> because it can say only about the correlation between infinite past
> and infinite future, and doesn't know what happens in between.
>
> The CJS theorem
> becomes very important in the case of dynamical description of the time
> evolution. Such a description is built in my book, and in agreement with
> the CJS theorem,
> I found that particle trajectories (or, in general, particle
> observables) do not transform in a manifestly covariant fashion between
> different reference frames. The dynamical character of the boost
> transformations is consistent with causality and instantaneous
> propagation of interactions.

Relativistic multiparticle mechanics is an intricate subject,
and there are no-go theorems that imply that the most plausible
possibilities cannot be realized. But these no-go theorems do not
enforce the instant form.

To pose the problem, one needs to distinguish between kinematical
and dynamical quantities in the theory. Kinematics answers the
question "What is the general form of objects that are subject to
the dynamics?" Thus it only tells you about conceivable solutions.
But kinematics does not know of equations of motions, and hence can
only tell general (kinematical) features of solutions.

Dirac distinguishes in his seminal paper
     Rev. Mod. Phys. 21 (1949), 392-399
three natural forms of relativistic dynamics, the instant form,
the point form, and the fromt form. They are distinguished by
what they consider as kinematical quantities and what are the
dynamical quantities.

The familiar form of dynamics is the instant form,
which treats space (hence spatial translations and rotations)
as kinematical and time (and hence time translation and Lorentz boosts)
as dynamical. This is the dynamics from the point of view of a
hypothetical observer who has knowledge about all information at some
time t (the present), and asks how this information changes as time
proceeds.

Because of causality (the finite bound of c on the speed of material
motion and communication), the resulting differential equations
should be symmetric hyperbolic differential equations for which the
initial-value problem is well-posed.
Because of Lorentz invariance, the time axis can be
any axis along a timelike 4-vector, and (in special relativity)
space is the 3-space orthogonal to it. For a real observer,
the natural timelike vector is the momentum 4-vector of the material
system defining its reference frame (e.g., the solar system).

While very close to the Newtonian view of reality, it involves
an element of fiction in that no real observer can get all the
information needed as intial data. Indeed, causality implies that
it is impossible for a physical observer to know the present anywhere
except at its own position.

A second, natural form of relativistic dynamics is, according to Dirac,
the point form. This is the form of dynamics in which a particular
space-time point x=0 (the here and now) in Minkowski space is
distinguished, and the kinematical object replacing space is,
for fixed L, a hyperboloid x^2=L^2 (and x_0<0) in the past
(it could also be the future) of the here and now.
The Lorentz transformations, as symmetries of the hyperboloid,
are now kinematical and take the role that space translations and
rotations had in the instant form. On the other hand, _all_ space and
time translations are now dynamical, since they affect the position
of the here-and-now.

This is the form of dynamics which is manifestly
Lorentz invariant, and in which space and time appear on equal footing.
An observer in the here and now can (in principle, classically)
have arbitrarily accurate information about the particles and/or fields
on the past hyperboloid; thus causality is naturally accounted for.
Information given on the past hyperboloid of a point can be propagated
to information on any other past hyperboloid using the dynamical
equations that are defined via the momentum 4-vector P, which is a
4-dimensional analogue of the nonrelativistic Hamiltonian.
The Hamiltonian corresponding to motion in a fixed timelike
direction u is given by H=u dot P. The commutativity of the components
of P is the condition for the uniqueness of the resulting state
at a different point x independent of the path x is reached from 0.

There is also a third natural form of relativistic dynamics according
to Dirac, the front form. It has many uses in quantum field theory
but I'll not explain it here.

All three forms are equivalent, related classically by canonical
transformations preserving algebraic operations and the Poisson bracket,
and quantum mechanically by unitary transformations preserving
algebraic operations and hence the commutator. This means that any
statement about a system in one of the forms can be translated into
an equivalent statement of an equivalent system in any of the other
forms.

Preferences are therefore given to one form over the other depending
solely on the relative simplicity of the computations one wants to do.
This is completely analogous to the choice of coordinate systems
(cartesian, polar, cylindric, etc.) in classical mechanics.

For a multiparticle theory, however, the different forms and the
need to pick a particular one seem to give different pictures of
reality. This invites paradoxes if one is not careful.

This can be seen by considering trajectories of classical relativistic
many-particle systems. There is a famous theorem by Currie, Jordan
and Sudarshan (Rev. Mod. Phys. 35 (1963), 350-375) which asserts that
interacting two-particle systems cannot have Lorentz invariant
trajectories in Minkowski space. Traditionally, this was taken by
mainstream physics as an indication that the multiparticle view of
relativistic mechanics is inadequate, and a field theoretical
formulation is essential. However, as time proceeded, several
approaches to valid relativistic multi-particle (quantum) dynamics
were found, and the theorem had the same fate as von Neumann's
proof that hidden-variable theories are impossible. Both results are
now simply taken as an indication that the assumptions under which
they were made are too strong.

In particular, nothing forbids an instant observer to observe
particle trajectories in its present space, or a
point observer to observe particle trajectories in its past hyperboloid.
However, the present space (or the past hyperboloid) of two different
observers is related not by kinematical transforms but dynamically,
with the result that trajectories seen by different observers look
different. Classically, this looks strange, but quantum mechanically,
trajectories are fuzzy anyway, due to the uncertainty principle, and
as various successful multiparticle theories show, there is no
mathematical obstacle for such a description.

The mathematical reason of this paradoxical situation lies in the fact
that there is no observer-independent definition of the
center of mass of relativistic particles, and the related fact that
there is no observer-independent definition of space-time coordinates
for a multiparticle system. The best one can do is to define either
a covariant position operator whose components do not commute
(thus definig a noncommutative space-time), or a spatial position
operator, the so-called Newton-Wigner position operator, which has
three commuting coordinates but is observer-dependent.

See the entry on 'Localization and position operators' in my
theoretical physics FAQ at
      http://www.mat.univie.ac.at/~neum/physics-faq.txt

What you do is simply to single out an instant observer and call its
view of space and time the only physically valid one. In your book,
you do it by declaring the Newton-Wigner position operator to be a
basic kinematic object. But the latter is defined only with respect
to an instant observer. Thus you break the inherent symmetry, and get
it back through the back door by constructing the instant form of
a representation of the Poincare group. But no one will follow you in
your restrictive postulate that one _has_ to do this and that everything
else is nonphysical. The more freedom in the description, the better,
as long as mathematical consistency is maintained. And with respect to
the latter, all forms of relativistic dynamics are the same.

Arnold Neumaier

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