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Results tagged with lagrangian-formalism
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user 185558
For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.
0
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Does an electromagnetic gauge transform induce a $U(1)$ transform on the field?
The Lagrangian of a scalar QED is given by
\begin{align}
\mathcal{L}&=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\phi)^{\ast}(D^{\mu}\phi)-m^{2}|\phi|^{2} \\
&=\frac{1}{2}(|\mathbf{E}|^{2}-|\mathbf{B}|^ …
- 2,975
2
votes
0
answers
153
views
Integrating Out Auxiliary Field of point-particle Polyakov Action
The Polyakov action of a point-particle is
$$S[X,e]=\frac{1}{2}\int d\tau\left(\frac{\dot{X}^{2}}{e}-m^{2}e\right)$$
with the $(−,+,+,+)$ Minkowski sign convention. How to perform the path-integral …
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1
vote
Gauge invariance of the Hamiltonian
First of all, the canonical Hamiltonian in classical mechanics and (or the canonical stress-energy tensor in classical field theory) is usually not necessarily gauge invariant.
For example, the Hamilt …
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0
votes
0
answers
88
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Lagrangian of Charged Particle Evaluated On-Shell
I am trying to calculate the Lagrangian of a charged particle in background gauge field evaluaed on-shell.
Let $A^{\mu}(x)$ be a gauge field. The action of a charged particle in this background gauge …
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0
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Derivative of the Lagrangian with respect to the metric tensor
I want to summarize the answer given by @peek-a-boo in a physicists friendly way:
The identity $g_{\alpha\beta}g^{\beta\delta}=\delta_{\alpha}^{\delta}$ is actually a relation between the metric tenso …
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1
vote
1
answer
269
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Spinning Particles in Background Gauge Fields
A simple model for a spinning particle is
$$L=m\int dt\left(\dot{x}^{2}-\frac{i}{2}\psi\dot{\psi}\right)$$
with SUSY algebra $\delta x=-i\epsilon\psi$ and $\delta\psi=-\epsilon\dot{x}$, where $\ep …
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7
votes
2
answers
956
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Global $U(1)$ symmetry of 2+1D Abelian-Higgs Model
In the Abelian-Higgs model,
$$S=\int d^{3}x\left\{-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}+|D\phi|^{2}-a|\phi|^{2}-b|\phi|^{4}\right\}\tag{5.34}$$
there is a $U(1)$ gauge symmetry. In David Tongs' lec …
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2
votes
Proof that the effective/proper action is the generating functional of one-particle-irreduci...
This has a mathematically rigorous proof using graph-related group theory. You can find it from the MIT lecture notes MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
On page 13, theorem 3.4 ha …
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2
votes
Local gauge transformations and Noether Current
Noether's theorem is about global symmetries. Gauge "symmetries" are not physical symmetries. They are local redundancies. In quantum field theory, a gauge transformation is a do-nothing transformatio …
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4
votes
Accepted
Gauge invariance of the Abelian Chern-Simons term
To explain the problem in details, I will start from the most generic (non-Abelian) case of the Chern-Simons theory.
Attention: if you are only interested in the answer for Abelian Chern-Simons theor …
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7
votes
Accepted
Is my expression of the Noether current $J^\mu$ for a local $\rm U(1)$ symmetry correct? If ...
Your expression is correct, but the result is not gauge invariant. Noether's theorem can only be applied to global symmetry, and cannot be applied to local invariance (aka gauge redundancy).
The physi …
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