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S6c. Functional integrals, Wightman functions, and rigorous QFT
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QFT assumes the existence of interacting (operator
distribution valued) fields Phi(x) with certain properties, which
imply the existence of distributions
W(x_1,...,x_n)=<0|Phi(x_1)...Phi(x_n)|0>.
But the right hand side makes no rigorous sense in traditional QFT
as found in most text books, except for free fields. Axiomatic QFT
therefore tries to construct the W's - called the Wightman functions -
directly such that they have the properties needed to get an S-matrix
(Haag-Ruelle theory), whose perturbative expansion
can be compared with the nonrigorous mainstream computations.
This can be done successfully for many 2D theories and for some 3D
theories, but not, so far, in the physically relevant case of 4D.
To construct something means to prove its existence as a mathematically
well-defined object. Usually this is done by giving a construction
as a sort of limit, and proving that the limit is well-defined.
(This is different from solving a theory, which means computing
numerical properties, often approximately, occasionally
- for simple problems - in closed analytic form.)
To compare it to something simpler: In mathematics one constructs the
Riemann integral of a continuous function over a finite interval by
some kind of limit, and later the solution of an initial value problem
ordinary differential equations by using this and a fixed point
theorem. This shows that each (nice enough) initial value problem is
uniquely solvable. But it tells very little of its properties, and
in practice no one uses this construction to calculate anything.
But it is important as a mathematical tool since it shows that
calculus is logically consistent.
Such a logical consistence proof of any 4D interacting QFT is presently
still missing. Since logical consistency of a theory is important,
the first person who finds such a proof will become famous - it means
inventing new conceptual tools that can handle this currently
intractable problem.
Wightman functions are the moments of a linear functional on
some algebra generated by field operators, and just as linear
functionals on ordinary function spaces are treated in terms of
Lebesgue integration theory (and its generalization), so Wightman
linear functionals are naturally treated by functional integration.
The 'only' problem is that the latter behaves much more poorly from
a rigorous point of view than ordinary integration.
Wightman functions are the moments of a positive
state < . > on noncommutative polynomials in the quantum field Phi,
while time-ordered correlation functions are the moments
of a complex measure < . > on commutative
polynomials in the classical field Phi.
In both cases, we have a linear functional, and the linearity gives
rise to an interpretation in terms of a functional integral.
The exponential kernel in Feynman's path integral formula for the
time-ordered correlation functions comes from the analogy between
(analytically continued) QFT and statistical mechanics,
and the Wightman functions can also be described in a similar analogy,
though noncommutativity complicates matters. The main formal reason for
this is that a Wick theorem holds both in the commutative and the
noncommutative case.
For rigorous quantum field theory one essentially avoids the
path integral, because it is difficult to give it a rigorous
meaning when the action is not quadratic. Instead, one only keeps
the notion that an integral is a linear functional, and
constructs rigorously useful linear functionals on the relevant
algebras of functions or operators. In particular, one can define
Gaussian functionals (e.g., using the Wick theorem as
a definition, or via coherent states); these correspond exactly
to path integrals with a quadratic action.
If one looks at a Gaussian functional as a functional on the
algebra of fields appearing in the action (without derivatives
of fields), one gets - after time-ordering the fields - the
traditional path integral view and the time-ordered correlation
functions.
If one looks at it as a functional on the bigger algebra of
fields and their derivatives, one gets - after rewriting the
fields in terms of creation and annihilation operators - the
canonical quantum field theory view with Wightman functions.
The algebra is generated by the operators a(f) and a^*(f),
where f has compact support, but normally ordered
expressions of the form
S = integral dx : L(Phi(x), Nabla Phi(x)) :
make sense weakly (i.e., as quadratic forms).
The art and difficulty is to find well-defined functionals
that formally match the properties of the functionals 'defined'
loosely in terms of path integrals.
This requires a lot of functional analysis,
and has been successfully done only in dimensions d<4.
For an overview, see:
A.S. Wightman,
Hilbert's sixth problem:
Mathematical treatment of the axioms of physics,
in: Mathematical Developments Arising From Hilbert Problems,
edited by F. Browder,
(American Mathematical Society, Providence, R.I.) 1976, pp.147-240.