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S2d. Forms of relativistic dynamics
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Relativistic multiparticle mechanics is an intricate subject,
and there are no-go theorems that imply that the most plausible
possibilities cannot be realized. However, these no-go theorems
depend on assumptions that, when questioned, allow meaningful
solutions. The no-go theorems thus show that one needs to be careful
not to introduce plausible but inappropriate intuition into the
formal framework.
To pose the problem, one needs to distinguish between kinematical
and dynamical quantities in the theory. Kinematics answers the
question "What are the general form and properties of objects that
are subject to the dynamics?" Thus it tells one about conceivable
solutions, mapping out the properties of the considered representation
of the phase space (or what remains of it in the quantum case).
Thus kinematics is geometric in nature. But kinematics does not know
of equations of motions, and hence can only tell general (kinematical)
features of solutions.
In contrast, dynamics is based on an equation of motion (or an
associated variational principle) and answers the question 'What
characterizes the actual solution?', given appropriate initial or
boundary conditions. Although the actual solution may not be available
in closed form, one can discuss their detailed properties and devise
numerical approximation schemes.
The difference between kinematical and dynamical is one of convention,
and has nothing to do with the physics. By choosing the representation,
i.e., the geometric setting, one chooses what is kinematical;
everything else is dynamical.
Since something which is up to the choice of the person describing
an experiment can never be distinguished experimentally, the physics
is unaffected. However, the formulas look very different in different
descriptions, and - just as in choosing coordinate systems - choosing
a form adapted to a problem may make a huge difference for actual
computations.
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 to be 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 (let us call it an 'instant 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
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 (let us call it a 'point observer')
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.
In principle, there are many other forms of relativistic dynamics:
As Dirac mentions on p. 396 of his paper, any 3-dimensional surface
in Minkowski space works as kinematical space if it meets
every world line with time like tangents exactly once.
In general, those transformations are kinematical which
are also symmetries of the surface one treats as kinematical reference
surface. By choosing a surface without symmetries _all_
transformations become dynamical. For reasons of economy, one wants
however, a large kinematical symmetry group. The full Poincare group
is possible only for free dynamics.
This leaves as interesting large subgroups two with 6 linearly
independent generators, the Euclidean group ISO(3), leading to the
instant form, and the Lorentz group SO(1,3), leading to the point form,
and one with 7 linearly independent generators, the stabilizer of
a front (or infinite momentum plane), a 3-space with lightlike normal,
leading to the front form. This third natural form of relativistic
dynamics according to Dirac, has many uses in quantum field theory,
but here I won't discuss it further.
All 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 (see the FAQ entry on
'Is there a multiparticle relativistic quantum mechanics?'),
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, once the assumption by Currie, Jordan and Sudarshan
that all observers see the same trajectories of a system of interacting
particles is rejected, their no-go theorem no longer applies.
The question then is how to find a consistent and covariant description
without this at first sight very intuitive property. But once it is
admitted that different observers see the same world but represented
in different personal spaces, the formerly intuitive property becomes
meaningless. For objectivity, it is enough that one can consistently
translate the views of any observer into that of any other observer.
Precisely this is the role of the dynamical Poincare transformations.
Thus 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 on
their different kinematical 3-surface look different.
Classically, this looks strange on first sight, although
the Poincare group provides well-defined recipes for translating
the trajectories seen by one observer into those seen by another
observer.
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 superficially 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 FAQ entry on 'Localization and position operators'.)