matrix field theory

Question

I am studying a field theory where the field is a matrix. The problem is that I have to calculate some functional derivative. How could we define functional derivative when the field is a matrix ?

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Comments

By writing the matrix in components and using the formalism of QFT with many fields.

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Answer 1

The least error-prone way for computing the functional derivative $df(M)/dM(x)$ by hand is the use of the formula

$\int dx \frac{\partial f(M)}{\partial M(x)} N(x) = \frac{d}{dt} f(M+tN) |_{t=0}$,

where $N$ is of the same type as $M$ (but c-valued if $M$ is an operator).

The right hand side is easy to work out, and the result is a linear functional in $N$, hence can always be written in the form on the left side, giving the desired functional derivative.

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