observables - Gauge transformations at infinity

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes on Quantum Gravity: In an ordinary quantum field theory without gravity, in flat spacetime, there two types of physical observables that we most often talk about are correlation functions of gauge-invariant operators $\langle O_{1}(x_{1}) \dots O_{n}(x_{n})\rangle$, and S-matrix elements. The correlators are obviously gauge-independent. S-matrix elements are also physical, even though electrons are not gauge invariant. The reason is that the states used to de...Read more

gauge invariance - Total vs extended action on constrained dynamics

Studying the electromagnetic hamiltonian dynamics, I used the extended formalism (after finding all constraints using the primary hamiltonian, also following the Dirac's recipe) to calculate the equations of motion and eventually find the gauge transformations. With the following extended hamiltonian (where the total one is achieved by ignoring the last term) $$\mathscr{H}_{\text{ext}}=\frac{1}{4}F^{i j} F_{i j} -\frac{1}{2}\pi^{i}\pi_{i} -A_0 \partial_i \pi^{i} + j^{\mu}A_{\mu} + \lambda_1 \pi^0 +\lambda_2 \partial_{i}\pi^{i}$$I got the follow...Read more

gauge invariance - $\mathcal{N}=8$ Gauged supergravity from $d=11$ supergravity

In this lecture video on maximal supergravity, H. Nicolai mentions that dimensional reduction of $d=11$ supergravity on $T^7$ gives us $\mathcal{N}=8, d=4$ (ungauged) supergravity found by Cremer-Julia. However, compactification on $S^7$ gives us $SO(8)$ gauged supergrvavity. According to my understanding, in gauged supergravity the abelian vector fields of the original $N=8$ supergravity are made non-abelian, which then introduces minimal coupling between the vector fields and other fields of the theory. Is there an intuitive way to see how di...Read more