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S6i. 2-dimensional quantum field theory
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Much of the state of the art in 2-dimensional relativistic quantum
field theories is covered in two books,
Elcio Abdalla, M. Christina Abdalla, Klaus D. Rothe
Non-Perturbative Methods in 2 Dimensional Quantum Field Theory
World Scientific, 1991, revised 2nd. ed. 2001.
and
J. Glimm and A Jaffe,
Quantum Physics: A Functional Integral Point of View,
Springer, Berlin 1987.
The first book treats exactly solvable theories, the second book
treats general polynomial interactions. The methods are completely
different in the two cases, and the two books are essentially disjoint.
Unfortunately, both books are somewhat difficult to read.
Abdallah et al. treat those (very special) 2-dimensional quantum field
theories having closed analytic expression for all S-matrix elements'.
These solvable models are to 2-dimensional quantum field theory what
the hydrogen atom is to quantum mechanics. It gives lots of details
about many solvable models, but I found it too specialized to give me
a feeling of general 2-dimensional quantum field theory.
Glimm and Jaffe assume a lot of measure theory and functional analysis.
This is summarized in Appendix A of their Part I, but working first
through Volume 3 of Thirring's Course in Mathematical Physics (which
only deals with nonrelativistic QM but in a reasonably rigorous way)
would be a good preparation for tackling Gliimm and Jaffe.
They construct - rigorously - for 2-dimensional relativistic
Lagrangian scalar field theories with polynomial interaction a Hilbert
space, a well-defined Hamiltonian, a well-defined unitary dynamics,
with well-defined bound states that are eigenstates of the Hamiltonian,
and everything is invariant under the 2D Poincare group ISO(1,1).
Chapter 3 defines a rigorous version of the path integral for ordinary
quantum mechanics, or rather for the Euclidean version of it, with the
i in the Schroedinger equation dropped. This amounts to analytic
continuation to imaginary time, where everything is easy and
respectable. In place of a hyperbolic differential equation one gets
a parabolic one (the heat equation), which makes things tractable
since the heat kernel is positive and hence the measures needed to
make the path integral rigorous are positive Wiener measures, with a
good rigorous theory.
Quantum field theory starts in Chapter 6. It is presented in a
Euclidean and a Minkowski version, the former being an analytic
continuation of the latter. Both versions are defined axiomatically,
by the Osterwalder-Schrader axioms and the Wightman axioms,
respectively. Again, the Euclidean version is the tractable one,
in which one can generalize the path integral and perform the
estimates needed for proving the existence of all the tools.
The Osterwalder-Schrader theory then guarantees that, given the
satisfaction of the Euclidean axioms, analytic continuation to
the Minkowski case is indeed possible. This is outlined in Section 6.1;
the remainder of the chapter discusses the (easy) special case of
free fields.
Chapters 7-12 and 19 then define the machinery needed to show how
to satisfy the axioms in the case of 2-dimensional relativistic
Lagrangian scalar field theories with polynomial interaction.
Chapter 7 discusses the Gaussian measures that define the Euclidean
path integral of free fields, Chapter 8 presents a rigorous theory of
perturbation theory for Euclidean path integrals, and the remaining
chapters mentioned provide the estimates needed to make sure that
everything works.