Variation of operator valued functions

Consider some operator valued function

F (\hat{A})

, and its variation with respect to

\hat{A}

δ F (\hat{A}) = F (\hat{A} + δ \hat{A}) - F (\hat{A}) .

It is implicit that we work to linear order in

δ \hat{A}

. This quantity is nontrivial because arbitrary variations

δ \hat{A}

in general won't commute with

\hat{A}

. Supposing

F

is analytic, we can express it as a power series:

F (\hat{A} + δ \hat{A}) = \sum_{n = 0}^{\infty} a_{n} (\hat{A} + δ \hat{A})^{n} = \sum_{n = 0}^{\infty} ({\hat{A}}^{n} + {\hat{A}}^{n - 1} (δ \hat{A}) + {\hat{A}}^{n - 2} (δ \hat{A}) \hat{A} + \dots + (δ \hat{A}) {\hat{A}}^{n - 1}) .

We can read off the variation:

δ F (\hat{A}) = \sum_{n = 0}^{\infty} ({\hat{A}}^{n - 1} (δ \hat{A}) + {\hat{A}}^{n - 2} (δ \hat{A}) \hat{A} + \dots + (δ \hat{A}) {\hat{A}}^{n - 1}) .

This is a complicated expression, however life becomes much simpler if one considers instead the variation of the trace of

F (\hat{A})

Tr (δ F (\hat{A})) = Tr (\sum_{n = 0}^{\infty} a_{n} n {\hat{A}}^{n - 1} δ \hat{A}) = Tr (F^{'} (\hat{A}) δ \hat{A}),

where we used the cyclic property of the trace.

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