Functional Integral in Free-Field Theory: Understanding the Derivation

[jdstokes]

In free-field theory, the functional integral

\int \mathcal{D}\varphi \exp\left(i \frac{1}{2} \int d^4 x (\partial_\mu \varphi \partial^\mu \varphi - m^2 \varphi^2)\right)

can be done exactly (see e.g., Peskin and Schroeder p. 285).

I'm tyring to understand the step in their derivation where they change integration variables from the field d\varphi(x), to the real and imaginary parts d\Re[\varphi(x)],d\Im[\varphi(x)]. They claim that since the transformation is unitary, they have

\prod_i d\varphi(x_i) = \prod_i d\Re[\varphi(x_i)]d\Im[\varphi(x_i)].

I don't understand this claim. Suppose the unitary xfm relating x_i to X_i is U. Then inEinstein notation,

dx_i = U_{ij} dX_j.

Hence

\prod_i dx_i = \prod_i U_{ij} dX_j = (U_{1i}U_{2j}U_{3k}\cdots)(dX_i dX_j dX_k \cdots).

Thus P&S's claim amounts to the assertion that

\prod_{n=1} U_{n ,i_n} = \prod_{n=1}\delta_{n, i_n}.

I don't understand this?

Any help would be appreciated.

 

[Avodyne]

First of all, it's just a definition. The measure for ordinary integration over a complex variable z=x+iy is defined to be dx\,dy.

More generally, a change of variable involves the determinant of the jacobian matrix of the transformation.

 

[jdstokes]

That's interesting, I desperately need to take a course in complex analysis.

I also forgot that the change of variables involves the Jacobian determinant, which is unity for a unitary matrix.

I still don't understand why P&S go through a long argument involving integrating only over the wavevectors k such that k^0 > 0?