Hello, this question will essentially concern quantum field theory in curved spacetime, and it has two parts to it. I have recently acquired DeWitt's treatment of the formalism, which immediately discusses the role of killing vectors in the theory. Specifically, given a killing vector field K^a (forgive me, I am still learning LaTex), we may form a 'generalized momentum' given by: P = integral (Tab K^a dΣ^b)  Where Tab is the stress energy tensor, and dΣ^b is the volume form corresponding to a cauchy hypersurface. The book continues, explaining that this P has the following effect on the fields: [Φ, P] = L_p (Φ)  Where L_p denotes the Lie derivative with respect to P. Now, the questions are as follows: 1. I can see a possible way for  to be true, following from the definition of a Lie derivative in General Relativity (i.e. given vector fields X and Y, [X, Y] = L_x(Y) ), but the Φ is not a vector field (in the GR sense), but a 'scalar' quantum operator on the Fock space. Moreover, P is scalar (perhaps it is also an operator?), and thus liable to the same concern. 2. Making the substitution  for P in , I cannot seem to extrude the right hand side of . My main issue in this is that I want to discern precisely how important the killing vector field is. Or, is it possible to construct arbitrarily a scalar operator (e.g. Q), and use this on another scalar (as [Φ, Q] = L_q(Φ) )? Could I use it on a vector operator. etc? And, more importantly, how exactly do I go about calculating the Lie derivative of these quantum fields? There is a lovely formula (abstract, and index based) in General Relativity - what is it in QFT? Thank you in advance.