path integrals

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....

Just about a year ago today, I began working on implementing an algorithm my undergrad research advisor had devised to speed up the Metropolis algorithm, in the regime where the acceptance probability is very low, which is the case in lattice simulations of quantum gravity. Quantum Gravity Physics has experienced its most rapid advancement when theories are unified: electromagnetism $\leftarrow$ electricity + magnetism + light general relativity $\leftarrow$ special relativity + curved space-time (gravity) quantum field theory $\leftarrow$ quantum mechanics + special relativity + electromagnetism + matter + nuclear forces And hopefully soon…...