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3
votes
1answer
51 views

Is obtaining the coordinate representation of momentum operator from commutator more fundamental than generator of translation

Related post: What is the most general expression for the coordinate representation of momentum operator? There are two methods of obtaining the coordinate representation of momentum in quantum ...
7
votes
1answer
69 views

Connection beween infinite gauge symmetries and UV finiteness

In e.g., http://arxiv.org/abs/arXiv:0712.3526 the author claims: Since the massless higher-spin field theories involve infinite-dimensional gauge symmetries, one expects that such theories may be ...
6
votes
2answers
134 views

The gauge covariant derivative and it's substitution

I was wondering wether it would make a difference (in general) if I were to were introduce the gauge covariant derivative $$D_\mu=\partial_\mu+ieA_\mu$$ In the Lagrangian density and then derive the ...
3
votes
1answer
87 views

What are type system examples of local gauge transformation- and field strength-like objects?

This is essentially a follow up motivated by this answer to my question about the gauge transformation interpretation of identity types. A field $$\psi:\mathcal M\to\mathbb C^n$$ is a section of the ...
2
votes
1answer
66 views

Why gauge invariance for electromagnetic fields?

What is the physical constraint that gauge invariance is a required condition for electromagnetic fields? What would happen if the electromagnetic fields were not gauge invariant?
0
votes
0answers
29 views

How to deal with $\vec{j}\cdot\vec{A}$ or $\rho A^2$ interaction when utilizing Kubo formula? Gauge invariance?

If there exist electromagnetic fields in solids, electrons can feel interactions like $\vec{j} \cdot \vec{A}$ or $\rho A^2$ (these are not regarded as perturbations). But these are not gauge ...
0
votes
1answer
92 views

Integrating the gauge covariant derivative by parts

I was watching a set of lectures on effective field theory and the lecturer said that you can always integrate the covariant derivative by parts due to gauge symmetry. For example, if I understand ...
6
votes
1answer
98 views

What do people mean by gauge invariance of the normalization of field?

Lets have the scalar Klein-Gordon field interacting with EM field: $$ L = \partial_{\mu}\varphi \partial^{\mu}\varphi - m^2\varphi \varphi^{*} - j_{\mu}A^{\mu} + q^{2} A_{\mu}A^{\mu}\varphi ...
2
votes
1answer
69 views

Field strength vanishes iff $A_{\mu}$ is pure gauge

Is it true that the field strength $F_{\mu\nu}$ in a non-abelian YangMills case with gauge group $G$ vanishes is and only if the gauge field $A_{\mu}$ is a pure gauge? I can show one implication. ...
1
vote
1answer
83 views

Global SU(2) invariance of QED Lagrangian

I'm having problems seeing the global SU(2) invariance of the QED Lagrangian. My specific problem is seeing why \begin{equation} e^{-i a_i \sigma_i} \gamma_\mu e^{i a_i \sigma_i} = \gamma_\mu ...
11
votes
1answer
203 views

**Group structure** in Chern-Simons theory?

A non-Abelian Chern-Simons(C-S) has the action $$ S=\int L dt=\int \frac{k}{4\pi}Tr[\big( A \wedge d A + (2/3) A \wedge A \wedge A \big)] $$ We know that the common cases, $A=A^a T^a$ is the ...
5
votes
1answer
230 views

6 independent Einstein field equations?

I can't understand the comment on page 409, Gravitation, by Misner, Thorne, Wheeler It follows that the ten components $G_{\alpha\beta} =8\pi T_{\alpha\beta}$ of the field equation must not ...
4
votes
0answers
100 views

gauge invariance of the Feynman amplitudes

When we calculate the photon polarization sums over amplitudes, $$X=\sum\limits_{r=1}^{2}|\mathcal M_r|^2=\mathcal M_\alpha\mathcal M_\beta^*\sum\limits_{r=1}^{2}\epsilon_r^\alpha\epsilon_r^\beta$$ ...
2
votes
0answers
38 views

What's the necessary and sufficient condition for gauge equivalence in the projective construction?

The definition of gauge equivalence and notations used here is the same as those in my previous question. As we know, the condition $\chi_{ij}'=G_i\chi_{ij}G_j^\dagger$(where $G_i\in SU(2)$) is a ...
0
votes
0answers
77 views

What is wrong in following arguments about connection of local gauge invariance and causality?

There is a question and corresponding downvoting of my answer, so I decided to ask this question. There is my answer on it: "...The most theories of free fields are invariant under global gauge ...
4
votes
2answers
194 views

Local guage symmety implies causality

I read in a QFT book that local gauge symmetry implies causality. Could someone please explain that statement and why it's true? Thank you.
2
votes
0answers
106 views

Is the $SU(2)$ flux defined in the context of Projective Symmetry Group(PSG) an observable quantity?

The $SU(2)$ flux defined in the context of PSG is as follows: Consider the mean-field Hamiltonian $H_{MF}=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ description of a 2D lattice spin-model, the ...
2
votes
1answer
176 views

A naive question on the $U(1)$ gauge transformation of electromagnetic field?

For simplicity, in the following we set the electric charge $e=1$ and consider a lattice spinless free electron system in an external static magnetic field $\mathbf{B}=\nabla\times\mathbf{A}$ ...
1
vote
2answers
225 views

Gauge theory in classical electromagnetism

I understand gauge theory as the theory of continuous transformation group which keeps Lagrangian (or dynamics) invariant. So some integral invariants could be found. In terms of classical ...
3
votes
1answer
94 views

Are the symmetry operators well defined in the context of Projective Symmetry Group(PSG)?

Consider the Schwinger-fermion approach $\mathbf{S}_i=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$ to spin-$\frac{1}{2}$ system on 2D lattices. Just as Prof.Wen said in his seminal paper on PSG, the ...
5
votes
1answer
135 views

How do non-transverse photon polarizations cancel in Euclidean QED?

First, recall how to write scattering amplitudes in covariant fashion in Minkowskian QED. One starts by considering some process with an external photon whose momentum is chosen to be ...
8
votes
0answers
124 views

Gauge fields in Polyakov's treatment of renormalization for nonlinear sigma model

I am deriving the results of renormalization for nonlinear sigma model using Polyakov approach. I am closely following chapter 2 of Polyakov's book--- ``Gauge fields and strings''. Action for the ...
0
votes
0answers
32 views

How could I show that ${\mathcal{L}\left(e\right)}$ is ${SU\left(2\right)_{L}\times U\left(1\right)_{Y}}$ invariant?

How can I show that ${\mathcal{L}\left(e\right)}$ is ${SU\left(2\right)_{L}\times U\left(1\right)_{Y}}$ invariant by checking explicitly that the ${\chi_{L}}$, ${e_{R}}$, ${\vec{W}_{\mu}}$ and ...
1
vote
1answer
132 views

Different invariant gauge groups (IGG) on different lattices with the same form mean-filed Hamiltonian?

Suppose that we use the Schwinger-fermion ($\mathbf{S_i}=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$) mean-field theory to study the Heisenberg model on 2D lattices, and now we arrive at the mean-field ...
4
votes
3answers
244 views

Why gauge theories have such a success?

[This question was inspired by a identical question asked on a other forum] Note that we may morally include general relativity in the gauge theories. We may have several (some are deliberately ...
0
votes
1answer
102 views

How to verify gauge invariance of an amplitude

I have calculated a tree level amplitude for Compton scattering (${e\left(p\right)+\gamma\left(k\right)\to e\left(p\prime\right)+\gamma\left(k\prime\right)}$): $${ ...
1
vote
1answer
231 views

Local gauge invariance and fields

I have one question about local gauge invariance of the spinor and scalar theories. For the scalar complex field with lagrangian $L_{0}$ requirement of local gauge invariance leads us to the ...
1
vote
2answers
269 views

A simple question on $SU(2)$ gauge transformations in Wen's papers on projective symmetry group (PSG)?

Recently I am studying the projective symmetry group (PSG) and the associated concept of quantum order first proposed by prof.Wen. In Wen's paper, see the last line of Eq.(8), the local SU(2) gauge ...
4
votes
1answer
219 views

Is reparameterization invariance some kind of gauge symmetry?

On page 116 of this book it is said, that reparameterization invariance of the string action is analogous to the gauge invariance in electrodynamices. Whereas Maxwell's equations are symmetric under ...
2
votes
1answer
625 views

Lorenz and Coulomb gauge-fixing conditions

Lorenz and Coulomb gauge-fixing conditions. What is physical difference between these two gauge-fixing conditions? Mathematical expression are clear but how to we choose one of these means what they ...
3
votes
1answer
258 views

How to introduce the electromagnetic field in Quantum Field Theory?

There are many ways to introduce the electromagnrtic field in Quantum Field Theory(QFT), such as canonical quantization method which introduces the creation and annihilation oprators by treating the ...
2
votes
2answers
157 views

A question for the generalization of gauge transformation with two antisymmetric indices

I have a question about the generalization of gauge transformation with two antisymmetric indices. Starting from Eq. (3.7.6) in Polchinski's string theory book p. 108. $$S_{\sigma} = \frac{1}{4 \pi ...
4
votes
3answers
266 views

Problems with putting mass on Yang-Mills theory by hand

When Yang-Mills field theory was introduced, a problem is that the gauge invariance can not allow mass for the gauge field. Later people invented spontaneous symmetry breaking and Higgs mechanism to ...
7
votes
1answer
209 views

Invariance of Functional Integration Measure

Let us consider the functional integral: \begin{equation} \int \mathcal{D} A e^{iS[A]} \end{equation} where $S[A]$ is the action for $U(1)$ gauge field and \begin{equation} \mathcal{D}A\equiv ...
2
votes
1answer
122 views

How to justify matter-field interaction for non-gauge-invariant Hamiltonian?

I'm wondering how can one formally justify the electromagnetic response of a system which does not verify local U(1) gauge invariance. A good example of what I would like to consider is given by the ...
1
vote
3answers
160 views

Are waves on water an example of gauge invariance?

So: Is the close similarity of small waves crossing water of varying depths ("depth potentials") an example of an approximate gauge invariance? If so, do other "only the surface dynamics matter" ...
2
votes
1answer
317 views

The gauge-invariance of the probability current

It is simple to show that under the gauge transformation $$\begin{cases}\vec A\to\vec A+\nabla\chi\\ \phi\to\phi-\frac{\partial \chi}{\partial t}\\ \psi\to \psi ...
2
votes
1answer
293 views

Invariance, covariance and symmetry

Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from quantum field theory? ...
18
votes
4answers
576 views

When can a global symmetry be gauged?

Take a classical field theory described by a local Lagrangian depending on a set of fields and their derivatives. Suppose that the action possesses some global symmetry. What conditions have to be ...
0
votes
0answers
78 views

How to charge a field?

In a previous post [ Noether theorem, gauge symmetry and conservation of charge ] we were discussing the different ways to demonstrate the current conservation: via the first Noether theorem applied ...
1
vote
1answer
263 views

Local $U(1)$ gauge invariance of QED

The Lagrangian density for QED is $$ \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\bar{\psi}(i\gamma^{\mu}D_{\mu}-m)\psi $$ with $$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu} $$ $$ ...
3
votes
1answer
278 views

What conservation law corresponds to this local $U(1)$ symmetry of the CCR?

It is known that canonical commutation relations do not fix the form of momentum operator. That means that if canonical commutation relations (CCR) are given by ...
4
votes
1answer
493 views

Is spacetime an illusion?

In consistent histories, for gauge theories, can the projection operators used in the chains be not gauge invariant? In quantum gravity, for a projection operator to be gauge invariant means it has ...
2
votes
2answers
195 views

Charged quantum particle in a magnetic field - choosing a different gauge leads to different wavefunctions

Consider a charged quantum particle confined to the $xy$ plane, subject to a magnetic field $\mathbf{B}=B\hat{z}$. The Hamiltonian is: $$ H = \frac{1}{2m} \left( \mathbf{p} - \frac{e ...
2
votes
2answers
517 views

Vector Potential and Gauge Invariance in Quantum Mechanics

In classical electromagnetism, we are allowed to use gauge invariance through the argument that the only physical observable fields are the $E$-field and the $B$-field. So in that sense the scalar ...
2
votes
1answer
326 views

SU(3) gauge invariance in QCD

In QCD, the gauge-invariant lagrangian under the trasformation $ \psi \to \psi' = e^{ig T^a \theta^a(x)} \psi$ is written as: $\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - ...
4
votes
2answers
313 views

The physicality of the photon propagator

The equation for the photon propagator is straightforward $$ D_{ij} = \langle 0 |T \{ A_{i}(x')A_{j}(x) \}|0 \rangle $$ However, $A_{i}(x)$ is gauge-dependent and therefore unphysical (in the arguable ...
1
vote
0answers
124 views

Yang-Mills Coulomb Gauge

My Question is how to explicitly move into the "Coulomb gauge" in Yang-Mills theory. Using the answer provided by QMechanic, one can move into the "temporal gauge" for Yang-Mills fields: Gauge fixing ...
2
votes
1answer
197 views

What is non-Abelian about non-Abelian Chern-Simons' theory?

One is aware that in the axial gauge (say the light-cone gauge $A_{-}=0$) non-supersymmetric Chern-Simons' theory is a quadratic theory. Hence in this gauge there are no gauge-gauge interactions. Then ...
2
votes
2answers
241 views

Is the artificial gauge field a gauge field?

The so-called artificial gauge fields are actually the Berry connection. They could be $U(1)$ or $SU(N)$ which depends on the level degeneracy. For simplicity, let's focus on $U(1)$ artificial gauge ...