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S5d. Why is QFT based on a classical action?
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The path integral approach to QFT begins with classical fields
that are varied to produce quantum amplitudes as a 'sum over all
possible paths'. But, with exception of the elctromagnetic field,
the classical fields one meets there are not fields occurring
in classical physics. Nevertheless they are rightfully labelled
'classical'.
Classical physics is the physics of processes slowly varying in space
and time; of course, elementary particles do not belong there.
But classical mechanics can also be considered as an abstract
mathematical framework for dynamics in a general phase space
(described by a Poisson manifold), which has much wider applicability.
The classical fields that figure in the path
integral belong in this sense to classical mechanics.
In QFT, one needs a classical action to be able to implement
unitarity of the S-matrix and the cluster decomposition.
The first is essential for a correct probabilistic interpretation of
QFT, since it amounts to preservation of probability, and the second is
necessary to account for the fact that all our experiments are done
locally, and what is far away does not contribute significantly
except through effectively classical far fields. (What happens with
the stars should be irrelevant to experiments on the earth, except for
the experiments of astronomers. This is the basis of all physics.)
In terms of microphysics, cluster decomposition means that one cannot
scatter particles (clusters of elementary particles) at very distant
particles (clusters).
The arguments why this requires a classical action expressed in terms
of creation and annihilation operators are explained in detail in
Weinberg's quantum field theory book, Volume I, Chapters 3-7.
We need cluster decomposition because it is observed. We need
local fields and microcausality, mainly because it implies
(modulo fine print involving contact terms) at least perturbatively
cluster decomposition, and there is no other known way in QFT to
ensure the latter. But there are covariant N-particle models with
cluster decomposition, discussed, e.g., in
B.D. Keister and W.N. Polyzou,
Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics,
in: Advances in Nuclear Physics, Volume 20,
(J. W. Negele and E.W. Vogt, eds.)
Plenum Press 1991.
www.physics.uiowa.edu/~wpolyzou/papers/rev.pdf
(The constructions are quite messy; they have, however, the
advantage that they do not need renormalization, and are useful
phenomenological models.)
The lack of references to cluster decomposition in standard textbooks
of QFT is explained by the fact that local QFT automatically satisfies
cluster decomposition. Most people start by taking QFT as starting
point, without asking why. Weinberg's treatise is about the only book
that asks this question and answers it in some depth.
But when you look at the literature on phenomenological covariant
multiparticle models, cluster decomposition plays an essential role
in that it is the main hurdle to overcome to get realistic models for
systems made of more than two unconfined particles. For details see
the survey by Keister and Polyzou mentioned above,
and the references there.
Cluster decomposition for field theory is also discussed from a
rigorous point of view in the book by Glimm and Jaffe, where
connections are made to multiparticle scattering.
Indeed, books on (nonrelativistic) scattering theory are the ones
where the cluster decomposition is discussed in detail, since it is
needed to describe the result of the most general multiparticle
scattering experiments, and an understanding of it is essential for
proving the asymptotic completeness of scattering states.
Nonrelativistic theory also shows that the 'correct'
cluster decomposition is always one for bound states,
as can be seen from a more detailed nonrelativistic analysis.
(This is not apparent from Weinberg's argument,
since perturbation theory breaks down in the presence of
bound states. This explains why QCD has no cluster
decomposition for isolated quarks.)
Unfortunately, most physicists tend to work in isolated fragments of the
whole edifice of physics, thus losing connections that may be important
to understanding. Cluster decomposition would perhaps be more prominent
in QFT if it were easier to calculate properties of bound states and
their scattering or breaking up, since that is where one can see the
principle at work. But such calculations are presently out of reach
without severe approximations.