By V. E. Nazaĭkinskiĭ, Viktor Evgenʹevich Shatalov, Boris Yu. Sternin
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Suppose also that f(A) is defined for any f E F. Then " f (A) = f (xg for any f E Y. 7 There are no Feynman indices in the statement of this theorem. However, the proof employs Feynman ordering heavily. 5 for generalization to multivariate symbols. Proof 1°. Let q) : A —›- B be a continuous homomorphism of algebras. Set f (ço( A)) tf ki. (A)), that is, Py(A) = PIAA. Evidently, pw (A) is a continuous homomorphism (as a composition of such mappings). 1). 2°. 2. We can write 2 3 1 1 6f 3 1 23 f (A) B — B f (A) = B ( f (A) — f (A)) = B (A — A)-' (A, A) .
Hence, the autonomous brackets can safely (and almost without trace) be removed in T-exponentials. The 0 (At2 ) argument is no longer usable with functions of unbounded operators. However, under appropriate functional-analytic conditions (not to be discussed here) the conclusion remains the same. Consider the simplest case in which f (x, y, Then r t) =x ± y. 12 exp (I' [[ f (A, B , -c)]]dt = ell A+1311! , 0 and, by setting ti = tIN, i = 0, .. , N, we obtain the Trotter product formula  e liti+mr _ lim e(tIN) B e(t IN) A .
We will pose a similar question for functions of noncommuting operators, and the machinery developed to answer it will be called noncommutative differential calculus. We will see that the matter is rich in subtleties and try to explain them as clearly as possible. Thus, we should consider f (A --I-- AA), where AA is "small", AA —> 0; but what does this requirement mean? We follow the usual practice of perturbation theory and take AA = EB, where E is a small numerical parameter. This permits us to keep track of infinitesimals of different orders easily, since the orders are indicated by the powers of s.