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\bf
{\Large Manchester Geometry Seminar 1999/2000}\medskip

18 November 1999. Maths Tower, N6. 3 p.m.\smallskip

{\large Aspects of functional integration in Mathematical
Physics}\smallskip

{\large R.K. Bullough (UMIST)}

%\href{mailto:robin.bullough@umist.ac.uk}
{\texttt{robin.bullough@umist.ac.uk}}

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In this lecture I will give a wholly \textit{heuristic} calculation (i.e. one designed to produce the
results obtained) of a large dimensional \textit{quantum mechanical} partition function $Z$. $Z$ is
defined as a functional integral taken over all classical paths in a large dimensional symplectic
manifold $M^{2N}$ ($N$ uncountable) the paths being weighted by $\exp S$, the exponential
\textit{classical} action $S$ of a chosen dynamical system. The (large) class of dynamical systems
I will (mostly) consider is the class of \textit{integrable} field theories in $1+1$ (one time ($t$) and
one space ($x$)) dimensions and the result obtained will be \textit{exact}. The motions of these
integrable systems in $M^{2N}$ are not ergodic and are confined to large dimensional tori in
$M^{2N}$. This latter fact will be exploited by canonical transformation to appropriate
action-angle variables co-ordinatising $M^{2N}$. A (considerable) complication is that the real
parameters
$t,x$ parametrising the paths in $M^{2N}$ lie in a 2-torus, $0\leq t < \beta$, $-\frac{1}{2} L \leq
x < \frac{1}{2} L$ (so that the fields $\phi(x,t)$ considered satisfy periodic boundary conditions
of period $\beta$ in $t$, and of period $L$ in $x$). $Z$ defines a ``free energy'' $F = - \beta^{-1}
\ell n Z$,  and this must be ``extensive'', $\lim_{L\rightarrow\infty} FL^{-1} =$ finite and
non-zero. Thus we need a ``finite density'' limit on $Z$ as $L\rightarrow \infty$ and the paths on
the
large dimensional torus in $M^{2N}$ must reflect this fact.

The integrable field theories in $1+1$ dimensions considered will mostly be the sine-Gordon and
sinh-Gordon models (also briefly I will consider the nonlinear Schr\"odinger models): these are
each nonlinear p.d.e.s in $(x,t)$ of rather fundamental significance. Both the attractive NLS and
the sine-Gordon equation have so-called \textit{soliton} solutions. To this extent $Z$ provides a
quantum statistical mechanics for `quantum solitons' \footnote{Note that because the class of
models we consider is a class of \textit{integrable} models, the motions are not ergodic and the
methods of statistical mechanics may seem in doubt. Indeed they might be in doubt \textit{for any
finite $N$} and any generic Hamiltonian system since in this case such generic Hamiltonian
systems are neither integrable nor ergodic -- because of the KAM theorem. For uncountable $N$'s
this aspect is (I think) wide open. However, it seems that our applications of statistical mechanics
are valid because they rest ultimately on the quantum mechanical super-position principle.
According to Dirac quantum mechanics is a theory of \textit{linear} operators on a Hilbert space (of
large dimension). This does \textit{not} mean of course that the models we consider are linear! For
the curious there is a a host of problems here requiring rigorous proof.}.

All of the integrable field theories considered are `completely integrable Hamiltonian systems' in
the sense of Liouville-Arnold (hence the large torus and the action-angle variables): $N$
independent constants of the motion commuting under a Poisson bracket on $M^{2N}$ can be
found but $N$ \textit{is} `large' (uncountable). However the extension of the Liouville-Arnold
theorem to this large dimensional case is not proved (I think). Nor is the validity of the methods
for $Z$ sketched in this lecture anywhere rigorously demonstrated\footnote{Contact with
algebraic geometry is made by the representation of the large dimensional torus as a punctured
Riemann surface with $N\sqrt{~}$ branch points. -- Genus $\infty$ since $N = \infty$??}.

It is therefore remarkable that such heuristic calculations\footnote{Even rigorous proof would
still present a problem -- see e.g. E.P. Wigner, ``The Unreasonable Effectiveness of Mathematics
in the Natural Sciences'', Comm. Pure and Appl. Math. {\bf 13}, 1 (1960)!}, as this one is, can
deliver results of physical interest. Very recently we have used not unrelated methods to calculate
correlation functions for the quantum repulsive NLS system which seem to agree with those
observed in experiments at $10^{-9}$ K (nano-Kelvin temperatures)!
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