In free-field theory, the functional integral(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\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)[/itex]

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 [itex]d\varphi(x)[/itex], to the real and imaginary parts [itex]d\Re[\varphi(x)],d\Im[\varphi(x)][/itex]. They claim that since the transformation is unitary, they have

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

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

[itex]dx_i = U_{ij} dX_j [/itex].

Hence

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

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

[itex]\prod_{n=1} U_{n ,i_n} = \prod_{n=1}\delta_{n, i_n}[/itex].

I don't understand this?

Any help would be appreciated.

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# Functional integration

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