# A Mathematical Formulae for Peskin and Schroeder's Exercise 2.2 (a)

I am self studying Quantum Field Theory and I am using the book An Introduction to Quantum Field Theory by Peskin and Schroeder. Currently I am working on problem 2.2 (a). In the textbook problem, the authors provide us with a Hamiltonian, whose Hamiltonian density contains a $$\nabla \phi^*(y)\cdot \nabla \phi(y)$$. One part of the question asks us to compute the Heisenberg equation of motion, which involves computing $$[\phi(x), H]$$, and this involves computing $$[\phi(x), \nabla \phi^*(y)\cdot \nabla \phi(y)]$$. I know that for operators $$A$$, $$B$$, $$C$$ that $$[A,BC] = B[A,C] + [A,B]C.$$ This, however, involves the product of operators, but does a similar formulae hold for dot products of vector operators? I.e. is this true: $$[\phi(x), \nabla \phi^*(y)\cdot \nabla \phi(y)] = \nabla \phi^*(y)\cdot[\phi(x),\nabla \phi(y)] + [\phi(x),\nabla \phi^*(y)]\cdot \nabla \phi(y).$$

• Why don't you expand it in modes? Jun 30 at 12:08

You have the usual scalar product $$\nabla\Phi^\ast\nabla\Phi=\sum_i\partial_i\Phi^\ast\partial_i\Phi$$ and since the commutator is bilinear, your equation at the end of your post is true.