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21 votes
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Is the magnetic vector potential "real" in classical electromagnetism?

The vector potential is gauge-dependent and unobservable in both classical and quantum mechanics. Only gauge-invariant quantities — including the electric and magnetic fields — are observable. Even in ...
d_b's user avatar
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18 votes
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Trouble reconciling these two views on gauge theory

Gauge theory resembles general relativity applied to extra dimensions beyond the big four. Whether that's the true nature of gauge fields or just a convenient mental picture isn't clear, but it's at ...
benrg's user avatar
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13 votes
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How do physicists mathematically define gravitational waves?

The most straightforward way is to simply take the transverse-traceless (TT) part of $h_{ij}$. The TT part of the metric, denoted $h^{\mathrm{TT}}_{ij}$, contains precisely the two propagating degrees ...
Vincent Thacker's user avatar
10 votes

Differential Forms and Gauge Invariance?

The exterior derivative of a $k$-form is a $k+1$-form. The square of the exterior derivative is zero, so if you add an exact one-form to $A$, that is $B= d\lambda$ where $\lambda$ is a zero-form, you ...
QuantumFieldMedalist's user avatar
10 votes

Is the magnetic vector potential "real" in classical electromagnetism?

One way of thinking about this issue of what quantities are 'real' is to compare it to the more familiar situation with position. There are quantities in physics with the property that the quantity ...
Nullius in Verba's user avatar
10 votes

What is an Intuitive example of a Gauge Symmetry?

Another example of of a Gauge Symmetry can be found in the basic $V=mgh$. Here you can have your "ground" anywhere you want. This freedom reflects the key idea in gauge theory.
TomGilbertPhysics's user avatar
9 votes

Is the magnetic vector potential "real" in classical electromagnetism?

Let's take the definition of a "real field" from the chapter in Feynman's lectures you linked to: What we mean here by a “real” field is this: a real field is a mathematical function we use ...
Andrew's user avatar
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9 votes

Why are physical states not eigenstates of BRST charge?

For starters, the BRST charge operator is Grassmann-odd, so an eigenvalue would be Grassmann-odd as well, which is unphysical.
Qmechanic's user avatar
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9 votes

How can a gauge field have physical effects if it only reflects a redundancy in our mathematical description of physical reality?

You just need to phrase both your points more carefully. That changing the gauge has no physical effect does not mean the gauge field does not have any physical effect (for one, since not all possible ...
ACuriousMind's user avatar
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8 votes

Differential Forms and Gauge Invariance?

The answer is simply that the sum $A+d\lambda$ should still be a $1$-form, just like the original $A$ itself. Why this has to be a case is clearer if we express the quantities in terms of spacetime ...
Buzz's user avatar
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8 votes
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Quantum Theory of Radiation Enrico Fermi 1932

Fermi is working in the gauge $V =0$. Suppose you have some choice of potentials $\mathbf{A}(\mathbf{x},t), V(\mathbf{x},t)$. Then define $\tilde{\mathbf{A}}(\mathbf{x},t) = \mathbf{A}(\mathbf{x},t) + ...
catalogue_number's user avatar
8 votes
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Why can Principal $G$ Bundles be Trivialized when $G = SU(N)$?

A Principal G-Bundle $\pi: P\rightarrow M$ is said to be trivial if it is isomorphic to $M \times G$, which means that a global section exists. In the case of a simply connected Lie Group G, its ...
MrDBrane's user avatar
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7 votes
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Is a QFT always an EFT coming from something deeper?

No theory of the form $U(1)\times G$ can be UV complete, since the $U(1)$ factor has a Landau pole (in 4d). UV complete gauge theories always involve semi-simple groups (although this is not enough: ...
AccidentalFourierTransform's user avatar
7 votes
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What is an Intuitive example of a Gauge Symmetry?

A Gauge Symmetry (not talking about large gauge transformations) refers to mathematical symmetries that are not physical but rather redundancies in our formulation. It is a quite rich subject, but I ...
MrDBrane's user avatar
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7 votes

Why did Peter Higgs, et al. suspect that the $W$ boson(s) had mass? All the way back in 1963, '64?

Already in 1957, Julian Schwinger, Annals of Physics 2 (5) November 1957, pp 407-434, "A theory of the fundamental interactions", had inferred from the chiral structure and puny strength ...
Cosmas Zachos's user avatar
6 votes

Constraints Generating Gauge Transformations and BRST

I) It seems that OP is already aware that Gauge transformations are usually only defined in the original$^1$ field sector. In the extended field sector [which includes ghosts, Lagrange multipliers, ...
Qmechanic's user avatar
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6 votes
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Interpretation of self-interacting terms in the expansion of a pure YM Lagrangian?

For what it's worth, if one calculates the symmetric SEM tensor $T^{\mu\nu}$ of the Yang-Mills (YM) theory with a compact gauge group $G$, one may check that the 00-component $T^{00}$ (=the energy-...
Qmechanic's user avatar
  • 205k
5 votes

How inaccurate is the following mental picture of particle interaction in QFT?

A main difference is that unlike classical fields, which don't interact with other fields, This is not true. You can absolutely define a "classical" version of the electron field, in the ...
Lenard Kasselmann's user avatar
5 votes

Quantum Theory of Radiation Enrico Fermi 1932

You can find a good description of this issue in Jackson, section 6.2 and 6.3. In short, starting with $$\mathbf B =\nabla \times \mathbf A \tag{1}\label{1}$$ $$\nabla \times \left(\mathbf{E} + \frac{...
hyportnex's user avatar
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5 votes
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Why is there only one coupling constant in Yang-Mills theory? Why are gluon self-coupling and gluon-matter coupling constants the same?

In annihilation of fermion-antifermion into gluons, there are three diagrams, the usual two that appear in the analogous QED process (the $t$ and $u$ channels). However, because of the existence of ...
Buzz's user avatar
  • 16.2k
5 votes
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When we solve the Maxwell equations for $(\phi,{\bf A})$ in a gauge, will the solution $(\phi,{\bf A})$ automatically obey the gauge condition?

This was stated in a comment, but I wish to make it really clear. It is not sufficient to look for potentials merely satisfying $$ \Box \mathbf A = -\mu_0 \mathbf J \\ \; \\ \Box \phi = -\frac{\rho}{\...
Andrew Steane's user avatar
5 votes

Trouble reconciling these two views on gauge theory

A local $U(1)$ transformation alone does not leave the original Lagrangian invariant. Since the laws of physics must apply locally, we want the Lagrangian to be invariant under a local $U(1)$ ...
joseph h's user avatar
  • 29.7k
5 votes

Introduce Ghost Field to eliminate unphysical degrees of freedom in case of Photon Field

Ghosts and the usual BRST formalism for gauge theories can be introduced in QED just as in non-abelian gauge theory. What one finds in the end is that the ghost fields $c,\bar{c}$ end up contributing ...
CStarAlgebra's user avatar
  • 2,687
4 votes

Three photon amplitude in QED

I'm extremely late. The reason is that otherwise one would have to insert a counterterm in the Lagrangian that is cubic in the electromagnetic field. This term would break gauge invariance.
Greg45thParallel's user avatar
4 votes

Hofstadter butterfly patterns in different honeycomb lattice structures

Your Hofstadter butterfly for nearest-neighbor hopping on the honeycomb lattice (first figure) is almost correct, however some aperiodicity remains in the spectrum. A common cause is using artificial ...
Bart's user avatar
  • 141
4 votes

When we solve the Maxwell equations for $(\phi,{\bf A})$ in a gauge, will the solution $(\phi,{\bf A})$ automatically obey the gauge condition?

This is a very important issue that is usually overlooked in almost all books even if it being of fundamental relevance in my view. Also with a great impact on the quantization procedure. (Also for ...
Valter Moretti's user avatar
4 votes

Why the massive spin-1 photon gets more degrees of freedom than massless case; while the massive spin-1/2 electron stays the same as massless case?

To count the on-shell DOF of a spin $s=1$ field we have to solve the EL equations for the Lagrangian density $${\cal L}~=~-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}m^2A_{\mu}A^{\mu}.$$ In the ...
Qmechanic's user avatar
  • 205k
4 votes
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Magnetic vector potential in 1+1 spacetime dimensions

In 1+1D there is no magnetic field, while the electric field $E=-\partial_x\phi-\partial_tA$ and the magnetic potential $A$ have only 1 component, cf. e.g. my Phys.SE answer here. The 2-vector gauge ...
Qmechanic's user avatar
  • 205k
4 votes

How do physicists mathematically define gravitational waves?

Without going to full mathematical GR like the very nice paper you've linked, one naive criterion in the weak field limit that physics textbooks mention is that nearby test particles should experience ...
Integral fan's user avatar
4 votes

Why the expectation value of three currents is important in the anomaly?

The expectation value of three currents is important because it's the first nontrivial expectation value in the computation of the anomaly, and it determines the whole thing. Here's an illustration ...
John Dougherty's user avatar

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