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matrix field theory

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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 ?

This post has been migrated from (A51.SE)
asked Apr 12, 2012 in Theoretical Physics by PanAkry (5 points) [ no revision ]
By writing the matrix in components and using the formalism of QFT with many fields.

This post has been migrated from (A51.SE)

1 Answer

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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.

This post has been migrated from (A51.SE)
answered Apr 13, 2012 by Arnold Neumaier (11,395 points) [ no revision ]

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