# 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.

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### Local Lorentz invariance of General Relativity

Consider the gravitational action as an integral over a differential 4-form $$S = \int_{\mathcal{M}} \star F_{ab}\wedge e^a \wedge e^b$$ where $\star F_{ab} = \epsilon_{abcd} F^{cd}$ and $F$ is the ...
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### Gauge invariant Green's function for a point particle

This question is a follow up to the question (Gauge invariant Green's function for electrodynamics). It is not possible to generally solve the eqution \begin{equation} \square A^{\mu}-\partial^{\...
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### Why do we talk about $U(1)$-principal bundles rather than torus bundles?

I'm trying to understand gauge theory in terms of differential geometry, and one thing I'm confused about is why we talk about $U(1)$-principal bundles when the gauge potential $A$ usually has ...
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### What is wrong with my counting of electromagnetic field degrees of freedom?

When we go from the field variables $({\vec E},\vec{B})$ to the potentials $(\phi,{\vec A})$, the number of degrees of freedom describing any electromagnetic field is reduced from $6$ ($3$ components ...
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### Can a QFT be anomaly-free on spacetimes that are boundaries but still have an anomaly on other spacetimes?

If $D$ is the Dirac operator for some dynamic spinor fields in background gauge and gravitational fields, then the partition function is supposed to be $\mathrm{det}(D)$. For this to make sense, we ...
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### Anomalies in QFT: why do we require smooth dependence on the background fields?

If $D$ is the Dirac operator for some dynamic spinor fields in a background gauge field $A$, then the partition function is supposed to be $\mathrm{det}(D)$. But if the coupling to the gauge field is ...
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### Large gauge transformations and global symmetries

In gauge theories, states in Hilbert space that are related by local gauge transformations are identified as the same physical state. This is necessary because otherwise the Hilbert space contains ...
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### $\mathbb{Z}_2$ gauge theory and disorder

I am confused about basics of $\mathbb{Z}_2$ (and likely other) gauge theories and plain disorder. Let $$H=H_F + h\,H_{EM}$$ $$H_F = -t\sum_l (c^\dagger_l \sigma^z_{l,l+1} c_{l+1} + h.c.)$$ be (the '...
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### Rotational invariance of the the elastic deformation in spherical symmetry

I write the elastic deformation $E$ for an incompressible material in spherical symmetry (Have a look at here Eq.(5.32) ). Since we are incompressible, $\mathbf{det} \,E=1$, so in spherical ...
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### Proving that free-particle Lagrangian is not invariant under $SU(3)$ local gauge invariance?

I would like to show that the free-particle Lagrangian $\mathcal L = \bar{\psi}\left( i\gamma^{\mu}\partial_{\mu} - m\right)\psi$ is not invariant under the $SU(3)$ local gauge invariance ...
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### $U(1)$ Local Gauge Invariance: What do $q$ and $\alpha(x)$ mean?

When deriving the existence of the photon, we start with the free Lagrangian $\mathcal L_{\text{free}} = \bar{\psi}\left( i\gamma^{\mu}\partial_{\mu}-m\right)\psi$ and require $U(1)$ local gauge ...