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 $k^0 > 0$?