4

Your intuition is correct. One way to understand this is as follows. Let $\mathcal{C}$ be the space of all gauge connections $A$ and let $G$ be the gauge group. Then the space $\hat{\mathcal{C}} = \mathcal{C} /G$ can be understood as the space of gauge inequivalent connections. To make this construction explicit, we can think of the action of $G$ on a single ...


3

Not all unitary transformations can be written as coordinate transformations. Not sure where you heard this, but it is verifiably false. By and large symmetries can be separated into two classes: internal symmetries and coordinate symmetries. The $U(1)$ transformation you're asking about is of the former type. Something like a Lorentz transformation would be ...


2

This is my understanding of the Quantum Field Theory (QFT) as used in particle physics models, after years working in experiments in particle physics to verify or find discrepancies in the Standard Model. To the point particles in the table of the standard model, the QFT mathematical model assigns a field, an electron field, an electron neutrino field.....,...


2

A crucial piece of context is that this calculation is done in the Euclidean (after a Wick rotation). In the Euclidean, $\bar{\psi}$ is just considered an independent field from $\psi$, and $\bar{\psi} = \psi^\dagger \gamma^0$ does not hold (eg see after Eq 3.21). So you don't pick up the minus sign from $\{\gamma_0,\gamma_5\}=0$.


1

You are mostly correct. However, I will note that it is the fields which transform under the gauge transformations, not the states. Indeed, the (physical) states are required to be gauge invariant. This can be understood in terms of demanding observables (expectation values/correlators) to be gauge invariant, but is also equivalent to the following: there ...


1

Fundamental particles in quantum mechanics are defined to be the quanta of fundamental fields. For example, when we apply the laws of quantum mechanics to the electromagentic field, we find out the the energy and momentum of the field can only come in discrete bundles (quanta) which we call photons. For electrons, we say that there is an electron field. When ...


1

$\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}}$There are two mistakes in your question. You erroneously resolved one of them your edit, by introducing the variation of $F$, which is irrelevant here. The $F_{\mu\nu}^a$ is not what you claim it is in the cited paper's normalisation. In the paper the authors use $$ [T_a,T_b] = iC_{ab}^{c}\,T_c \tag{1}$$...


1

The $\theta$ variable is simply a representation of the ignorance that you have on determining a wave-function because what actually is physically observable is the state (or rather expectation values of the state). When you have a Hilbert space, your (normalized) wave-function is a vector $|\psi\rangle$. Multiplying by a phase factor $|\psi\rangle \to e^{i\...


1

As was mentioned in some of the comments, the Lagrangians $$ \mathcal{L}=(\partial^\mu\psi)^\dagger(\partial_\mu\psi)-m^2\psi^\dagger\psi $$ and $$ \mathcal{L}=(D^\mu\psi)^\dagger(D_\mu\psi)-m^2\psi^\dagger\psi+\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$ represent distinct theories each with their own properties. The usual way to motivate the transition from the &...


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