Geometry and quantum field theory - 02 The steepest descent and stationary phase formulas.pdf

文档介绍：4 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
2. The steepest descent and stationary phase formulas
Now, let us forget for a moment that the integrals (1,2,3) are infinite dimensional and hence problem-
atic to define, and ask ourselves the following question: why should we expect that when the parameter
κ or
goes to zero, we recover the usual classical mechanics or field theory? The answer is that this
expectation is based on the steepest descent(respectively, stationary phase ) principle from classical
analysis: if f(x) is a function in Rd then the integrals g(x)e−f (x)/κdx, g(x)eif (x)/dx “localize” to
minima, respectively critical points, of the function f. As this classical fact is of central importance to
the whole course, let us now discuss it in some detail.
. The steepest descent f, g :[a, b] → R be smooth functions.
Theorem .(The steepest descent formula) Assumef attains that a global minimum at a unique
point c ∈(a, b), such thatf

( c) > 0. Then one has

b
(4) g(x)e −f (x)/dx =
1/2 e −f (c)/I(
),
a
√
where I(
) extends toasmoothfunction [0on, ∞) such thatI (0) = 2π√g(c) .
f

(c)
Proof. Let I(
) be defined by the equation (4).
1

Let  be a real number, such that 2 >> 0, and let I1( ) be defined by the same equation, but with
1 − 1 −
integration over [c −
2 ,c +
2 ].