Recent content by theZ

I recommend you look up the "Trotter Expansion" of a matrix exponential. The path integral can be understood as a Trotter expansion in which resolutions of the identity (alternating between Q and P) are inserted. The Trotter expansion can be derived from the Baker-Campbell-Hausdorff formula...

An interesting physical way to understand how [O_1(t), O_2(t') ] relates to causality is through the linear response (Kubo) formalism. Suppose the system begins in the vacuum state, and you perturb the Hamiltonian with an external source V(t) = J(t) O_1 . This could, for instance, be a...

As I said, you must understand the equality after summing together k, -k. Look at the creation and annihilation terms separately. For, say, the creation part, call the term to be summed, as initially written, f(k). Call the term to be summed, as the text has rewritten it, g(k). f(k) + f(-k) =...

You have to remember that this occurs under the summation over the BZ - as the sum includes -k for every k, you can take, for example, \sum_{k \in BZ} a^\dagger_k a_k \to \sum_{k \in BZ} a^\dagger_{-k} a_{-k} with impunity.