I started reading Stephen Adler's book, "Quantum Theory as an Emergent phenomenon". He introduces 'Trace dynamics', and defines the following, 'Trace derivative': "Given the trace of a Polynomial $P$ constructed from a non-commuting matrix or operator variables, one can define a derivative of complex number $\mbox{Tr} P$, as :
$$\delta \mbox{Tr} P = \mbox{Tr}\frac{\delta \mbox{Tr} P}{\delta O}\delta O$$
Question 1: How to make sense out of this?, I don't understand this definition at all, and how does one usually rigorously define derivative with respect to an Operator?.
He provides, the following example, to understand this derivative. Let, $P$ is a Bosonic monomial ( that is built out of matrices having elements which are products of an even number of Grassmannian elements) containing a single factor of $O$. $P = AOB$, $A$ and $B$ generally do not commute with each other or $O$.
Then - $\delta P =A(\delta O)B$, taking trace, using cyclic permutation gives-
$\delta \mbox{Tr} P = \epsilon_{B} \mbox{Tr} BA\delta O$, ($\epsilon_{B}$ = $+1$ when Operator is Bosonic, and $-1$ when Fermionic, i.e Matrices having elements which are products of odd number of Grassmannian elements), then -
$$\frac{\delta \mbox{Tr} P}{\delta O} = \epsilon_{B} BA$$
Question 2: I don't understand, this last step here, how did the trace disappear?
Consider the matrix-valued scalar function $\phi(\mathbf{X})$. By definition, the differential $d\phi$ is related to the gradient by $$ d\phi = \frac{\partial \phi}{\partial \mathbf{X}} : d\mathbf{X} $$ where $:$ denotes the Frobenius inner product. Because $\mathbf{A}:\mathbf{B} = \mathrm{tr}(\mathbf{A}^T\mathbf{B})$, it holds
$$ d\phi = \mathrm{tr} \left[ \left(\frac{\partial \phi}{\partial \mathbf{X}} \right)^T d\mathbf{X} \right] $$ This is I think the simple reason of the expression you give.