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S6e. Constructive field theory
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Rigorously defined Lorentz-covariant quantum field theories are known
to exist in 2 and 3 dimensions; the standard reference (for d=2)
is the book by
J. Glimm and A. Jaffe,
Quantum physics. A functional integral point of view
New York, 1981
A recent review of the achievements of constructive
quantum field theory in dimensions < 4 is
V. Rivasseau
Constructive Field Theory and Applications:
Perspectives and Open Problems,
J. Math. Phys. 41 (2000), 3764-3775.
http://lanl.arxiv.org/pdf/math-ph/0006017
The case d=4 is a famous unsolved problem; the special case of 4D
quantum Yang-Mills gauge theory with a compact simple, nonabelian
gauge group is one of the Clay Millenium problems with a 1 million
Dollar prize attached to its solution.
Let me explain some aspects of the construction given in
Glimm and Jaffe.
First one needs to understand that the construction breaks the Lorentz
symmetry. This is (although they don't draw this connection) because in
irreducible Poincare representations, one can construct only three
commuting coordinates, and their construction is observer-dependent,
i..e, dependent on singling out a preferred time. Of course, the final
theory is again Lorentz invariant.
To motivate construction, one therefore needs to choose a time
coordinate, then one makes analytical continuation to Euclidean time
(i.e. it in place of t), and shows that one gets an SO(4) symmetric
field theory in place of the Lorentz symmetry. The advantage gained is
that the functional calculus over a space with definite metric is
well-defined mathematically (via a limit approach through lattices, or
via Wiener measures) - this is just classical stochastic calculus.
Conversely, and this is the constructive part, given an SO(4) symmetric
field theory, one can choose a direction as Euclidean time and obtain
(via a fairly simple construction detailed in Chapter 7) within that
theory a well-defined Hamiltonian on a suitably constructed Hilbert
space of 3-dimensional fields. This Hamiltonian defines a time
evolution as in ordinary quantum mechanics. The nontrivial part (which
is the Osterwalder-Schrader reconstruction theorem stated in Chapter 7
but proved much later in the book - the forward references in Glimm
and Jaffe are, unfortunately, quite confusing) is to show that the
resulting theory is Lorentz invariant.
Thus the construction reduces to constructing the Euclidean field
theory. This is done via a Lattice regularization; indeed, all lattice
field theory and computation is based on the Euclidean formulation
rather than the Minkowski formulation.
In 2D and 3D, the existing analytic error estimation techniques are
sufficient to prove the existence of the limit with suitably
renormalized operators. In 4D, there are additional technical
problems that have not been overcome so far. But neither has it been
proved that any of the 4D field theories cannot exist. There are some
informal arguments suggesting this or that, but none of them is
conclusive in the sense of having paved the way towards a construction
or a no-go theorem.