quantum field theory

We saw in a previous post that for a non-interacting theory (i.e. $V(\varphi) = 0$) that the generating functional can be written as $$ Z[J] = e^{\frac{1}{2} J \cdot K^{-1} \cdot J}. $$ We hinted that it is not always the case that $K$ can be naively inverted. The issue arises when we consider the Maxwell action for a $U(1)$ gauge potential $A_\mu$: $$ S(A) = \int d^4 x \left[ \frac{1}{2} A_\mu \left( \partial^2 g^{\mu \nu} - \partial^\mu \partial^\nu \right) A_\nu + A_\mu J^\mu \right]....

Quantum field theory is the study of various types of fields, the interactions between these fields, and the correlation functions describing the dynamics of the entire system. Fields are really just functions that label each spacetime coordinate with some type of mathematical object. We can talk about spinor fields, scalar fields, and even gauge fields, which can be described by vectors or tensors that transform in certain ways under an associated gauge group....