Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [gauge-invariance]

Invariance of a physical system (its action) under a continuous group of local transformations underlain by a global symmetry whose group parameters fixed in space-time have now been extended to vary in space-time instead. Use for buildup of the invariance, fixing the gauge, and accounting for the corresponding changes in the functional measure of the system.

3
votes
1answer
60 views

Why do we impose de Donder gauge?

In the field language, a massless particle corresponds to irreducible representations of the Lorentz group. In particular, given a spin-2 massless particle, we can embed the creation and annihilation ...
5
votes
0answers
39 views

Coordinate Systems vs Reference Frames vs States of Motion [closed]

This is a broad conceptual question. I am really trying to understand symmetry as deeply as possible. Please let that guide your responses. In particular, I'm looking for help finding clarity on the ...
5
votes
1answer
67 views

QED lagrangian: gauge fixing term

I have a question about the structure of the QED lagrangian, in particular the free photon lagrangian which is contained in it. My premise is: I only know how to exploit canonical quantization in ...
0
votes
0answers
65 views

Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5)

In Weinberg's QFT textbook, using a gauge transformation $$A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu}\epsilon(x)\tag{8.1.3},$$ it has: $$\delta I_{M} = \int d^4 x \frac{\delta I_{M}}{\delta A_{...
3
votes
1answer
48 views

Why is the projective symmetry group a group?

I am reading the paper from X. Wen about quantum orders and symmetric spin liquids. It can be found here: https://arxiv.org/abs/cond-mat/0107071 The Hamiltonian he is writing about looks like this: \...
3
votes
1answer
109 views

Faddeev-Popov determinant and topology of the wordline

I am studying the path integral quantization of relativistic particles, using the BRST quantization method. I have to compute the integral \begin{equation} Z\sim \int Dx \det(\partial_\tau)e^{-\int_0^...
4
votes
1answer
141 views

Non-local field redefinition and effects on path-integral measure

Consider the partition function $$ Z[0] = \int \left[\mathcal{D}A_\mu\right]\left[\mathcal{D}\pi\right] e^{-i \int d^4x \left(-\frac{1}{2}(\partial\pi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{a}{M^2}...
1
vote
1answer
37 views

Is the Dirac Lagrangian locally gauge invariant without gauge field $A$?

When it comes to the check of the invariance of the Lagrangian of the Dirac equation under local $U(1)$-transformations I have made the following observation: $$L = \bar{\psi} (i\gamma^{\mu}\...
1
vote
0answers
40 views

A puzzle about Green's function and S-matrix

From the discussion in some posts example1, example2, we know that the S-matrix is the residue of the corresponding Green's function. On the other hand, S-matix is a physical observable in QFT, but ...
5
votes
2answers
217 views

Comparison of covariant form of Maxwell equations with Einstein's GR

We know, the the vector form of Maxwell equations \begin{align} \vec\nabla\cdot\vec{E} &= 4\pi\rho \label{Diff I}\\ \vec\nabla\times\vec{B} &= \dfrac{4\pi}{c} \vec{j}+\dfrac{1}{c}\dfrac{\...
2
votes
1answer
60 views

Why Coulomb gauge is a possible gauge choice?

In classical field theory we can get, that adding gradient of some scalar field to magnetic vector potential does not change the physics at all. So, we have such a symmetry: $\boldsymbol{A}\...
1
vote
2answers
53 views

Global $U(1)$ transformation properties of gauge fields

What are the Global gauge transformations of gauge bosons in Standard Model? To elaborate: Initially, we consider the global $U(1)$ transformations of scalars ($\phi$) and fermions ($\psi$) as $$\...
0
votes
0answers
16 views

Why demand local gauge invariance? [duplicate]

Ok, so I see how demanding local gauge invariance leads to gauge fields, and how quantizing those fields then leads to gauge bosons. But why do we take that step in the first place? What leads us to ...
0
votes
1answer
33 views

Algebraic trouble in gauge invariance of Schrodinger equation

I've been trying to prove a component of a proof the gauge invariance of the schrodinger equation. Specifically the part in the first answer here where this is stated: $$\big(\frac{\nabla}{i}-q(\vec{...
1
vote
0answers
34 views

How does the photon arise from the $U(1)$ symmetry in QED?

I'm trying to study a bit of QED and I really don't get where the photon does arise. I know that we want our field theory to be gauge invariant under a $U(1)$ phase transformation, because that is ...
1
vote
3answers
86 views

Is $F^{\mu\nu}F_{\mu\nu}$ equivalent to $A^{\mu}\nabla^{\alpha}\nabla_{\alpha}A_{\mu}$ for $U(1)$ gauge field lagrangian?

The two seem to yield the same equation of motion is why I asked. Where of course the standard notation for exterior forms applies $dA=F$. We all know how the field strength tensor plays into the ...
1
vote
1answer
48 views

Why does gauge invariance HAVE to correspond to an observable?(Or is it the other way round)

Under the line integral of the geometrical Berry phase, a close-loop integral is gauge invariant as if we were to perform a gauge transformation of the initial state, with the end point of the path in ...
3
votes
2answers
131 views

Can the mass of a hydrogen atom be calculated in a gauge-invariant way?

Please excuse the lengthy question. It involves an interesting controversy which has arisen in discussions on this site: Energy/mass of Quantum Vacuum Relative potential energy vs Absolute potential ...
-3
votes
1answer
39 views

Can fermions and bosons exist in the same representation?

Some people have made theories where they claim fermions and bosons exist in the same representation for example $E_8$. I can't see how this is possible. But say for example it is. This would imply ...
0
votes
1answer
40 views

Why can we write lagrangian for gauge theory without the traces?

I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true: $$-\frac{1}{2}...
4
votes
2answers
184 views

Help with understanding the imposition of gauge conditions

Let $s$ be a positive integer and $h_{a_1\dots a_s}$ be a traceless and totally symmetric (real) field which is defined modulo gauge transformations of the form $$\delta_{\xi}h_{a_1\dots a_s}=\...
0
votes
2answers
54 views

Why does Maxwell's equations $\partial_{\mu} F^{\mu \nu} = 0$ have 3 independent components (DOF) in $D = 4$?

And how can we generalize this to the statement that it has $D-1$ independent components in dimension $D$? I know that $F_{\mu \nu}$ has six independent components (because of antisymmetry), how do ...
6
votes
0answers
83 views

Parity Anomaly and Gauge Invariance

In Fermionic Path Integral and Topological Phases, Witten shows that in $2+1$ dimensions, the free massless Dirac fermion suffers from parity anomaly. To be specific, he shows that it is impossible to ...
2
votes
1answer
77 views

Conserved current in scalar QED

Consider a theory of a free massless complex scalar $\phi$ which undergoes global $U(1)$ transformations. The conserved current associated to this symmetry is the usual scalar current $$ J^\mu = i\...
4
votes
2answers
498 views

Does it really make sense to talk about the color of gluons?

It is my understanding that by enforcing SU(3) gauge invariance on our lagrangian of 3-colored quark fields, we are forced to accept the existence of 8 new massless vector fields, the gluons. The 8 ...
0
votes
1answer
52 views

Quick question on choosing a gauge (E.g. Lorenz gauge)

I have been quite confused when I read about choosing a gauge. For example we have the gauge transformation $$ A_\mu\longrightarrow A_{\mu\prime}= A_\mu+\partial_\mu\alpha, $$ and we can choose any $...
2
votes
1answer
119 views

Proving Gauge invariance of Schrodinger Equation

I am trying to proof explicitly that Schrodinger equation: $$ i\hbar \partial_t \psi = \big[ -\frac{1}{2m}\big(\frac{\hbar}{i}\nabla-q\vec{A}\big)^2+qV \big]\psi$$ remains the same under the ...
5
votes
1answer
107 views

Is the electroweak $SU(2)$ gauge symmetry an exact symmetry in Standard Model before spontaneous symmetry breaking?

In Standard model, components of a $SU(2)$ doublet (for example $u$ and $d$) have different masses. This means there is no $SU(2)$ symmetry, but I think it is okay because the $SU(2)$ symmetry is ...
1
vote
3answers
75 views

Why does gauge invariance in electrodynamics mean that there are redundant degrees of freedom? [closed]

It is possible to choose different gauges in electrodynamics. I am familiar with two of them: Coulomb gauge and Lorenz gauge. Let us stick to the Coulomb gauge. It sets $$\nabla\cdot\vec{A}=0.$$ The ...
0
votes
1answer
39 views

What happens to the gauge covariant derivative if the theory contains multiple fields in different representations?

I'm studying a graduate level course in QFT, and I have a potentially very stupid question that I can't find adressed anywhere (which I guess implies that I'm missing something fundamental). I'm ...
1
vote
0answers
66 views

Gauge invariance of pseudo stress-energy tensor of gravitational waves

The pseudo stress-energy tensor of gravitational waves is given by $$T_{\mu\nu}^{(\mathrm{G}\mathrm{W})} = \frac{1}{32\pi}\left\langle \partial_{\mu} \bar{h}_{\alpha\beta} \partial_{\nu}\bar{h}^{\...
1
vote
1answer
78 views

Why are two different gauge transformations of $A_\mu=0$ in $U(1)$ gauge thoery equivalent?

Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\...
2
votes
0answers
37 views

Lapse and shift inside or outside the Poisson bracket?

For general relativity in the 3+1 ADM formulation, one has $H=\int dx [N{\cal H}+N^a{\cal H}_a]$ with $N$ and $N^a$ the lapse and shift which are undetermined Lagrange multipliers. The dynamical ...
2
votes
0answers
57 views

Conserved currents in Yang-Mills theory: gluon current vs. quark current

In Yang-Mills theory there are two currents we can construct. There is the well-known quark current related to the global $SU(3)_C$ symmetry, $$j{}^{\mu\,A}_\text{quark} = \overline{\psi}{}^i \gamma{}^...
7
votes
1answer
113 views

Counting degrees of freedom in the Higgs mechanism for different gauges

I am wondering how to count the degrees of freedom (dof) for a massive gauge field in different gauges. I've been reading some other answers, but haven't found a solution yet. I am looking at the ...
3
votes
1answer
62 views

Gauge transformations with varying phase give us conservation of the charge density. Hence charged particles cannot move?

I stumbled upon the following paragraph in Quark confinement and Topology of gauge theories by Polyakov "Gauge invariance with constant phase $\Psi \to e^{i \alpha}$ lead to conservation of the ...
0
votes
1answer
45 views

Why do we require that functions which parametrize gauge transformation are smooth?

A local $U(1)$ transformation is given by \begin{equation} f(x) = e^{i\epsilon(x)} \qquad \text{with} \qquad \epsilon(x) \in C^\infty \, . \end{equation} Why do we require that the functions in ...
0
votes
1answer
34 views

Gauge covariance of the magnetic momentum operator

In the book about Schrödinger Operators by Cycon et al. there is a step in their calculations I don’t understand. When I pick two vector potentials $A_1$ and $A_2$ such that their curl (i.e. the ...
3
votes
1answer
89 views

What happens to the global $U(1)$ symmetry in alternative formulations of Quantum Mechanics?

The global $U(1)$ symmetry in Quantum Mechanics corresponds to the freedom to shift the phase of the wave function $$ \Psi \to e^{i\varphi} \Psi \, $$ and can be used to understand the conservation ...
3
votes
2answers
83 views

The meaning of gauge-fixing in covariant quantization of the electromagnetic field

I am having trouble wrapping my head around the idea behind the covariant quantization for the electromagnetic field that is usually done in textbooks (I'm currently following Mandl & Shaw and ...
3
votes
2answers
438 views

The location of an object is gauge dependent. Therefore, it's not measurable?

The location of an object $x$ depends on how we choose our coordinate system. If we move the zero point, $x$ also changes. However, since we have translational invariance, we can always do such shifts ...
2
votes
2answers
165 views

Gauge Invariance in Electrodynamics

I am studying Electrodynamics and I have been introduced to the concept of Gauge Invariance. This was introduced by noting that $E$ and $B$ amount to 6 six degrees of freedom and the Maxwell ...
1
vote
0answers
35 views

Are there more general gauge transformations than simple phase shifts?

Usually, in the context of a $U(1)$ gauge theory, we only consider gauge transformations of the form \begin{equation} \Psi(x) \to \mathrm{e}^{i\epsilon(x)} \Psi(x) \, . \end{equation} Are there ...
2
votes
1answer
109 views

Momentum eigenfunction for non vanishing vector potential

I'm reading a section in a textbook about the Aharonov-Bohm effect, which claims that the eigenfunction (outside the solenoid, so $\operatorname{curl} A=0$) for the canonical momentum operator $$\pi = ...
3
votes
1answer
37 views

Why are transformations that only change something within a finite region redundancies?

I'm trying to build some intuition for a very particular definition of the notions global and local gauge symmetries. The definition goes as follows and appears, for example, in "Quantum Field Theory -...
3
votes
2answers
82 views

What's the physical meaning of the gauge invariant quantity $\partial_\mu \varphi(x) - A_\mu(x)$?

A famous locally gauge invariant quantity is $$ F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, , $$ which is interpreted as the measurable electric and magnetic field strengths. Now, ...
2
votes
1answer
43 views

Are the nonphysical degrees of freedom in Yang-Mills theory analogous to the worldsheet metric in the Polyakov formalism?

The Polyakov string action on a flat background (in the Euclidean signature) $$S_{P}[X,\gamma]\propto\int_{\Sigma}\mathrm{d}^2\sigma\,\sqrt{\text{det}\gamma}\,\gamma^{ab}\delta_{\mu\nu}\partial_{a}X^{...
2
votes
4answers
92 views

Is fixing the gauge the same thing as performing a Lorentz transformation?

Let's say I have a moving charged particle, with constant velocity. Its electric field is given by (generally): $$ \mathbf{E} = -\nabla\phi - \frac{\partial \mathbf{A}}{\partial t}. $$ If I perform ...
3
votes
0answers
71 views

Confusion Between Associated and Principle-G-Bundles

I realize there have been similar questions on stack before, but none of them have answered what I'm after. -My question is really whether I can import wholesale everything from the principle bundle ...
4
votes
5answers
222 views

In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are isometric, or just diffeomorphic?

In Carroll's Appendix B, he says You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold $M$ with ...