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### Noether charge of local symmetries

If our Lagrangian is invariant under a local symmetry, then, by simply restricting our local symmetry to the case in which the transformation is constant over space-time, we obtain a global symmetry, ...
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### How are classical optics phenomena explained in QED (Snell's law)?

How is the following classical optics phenomenon explained in quantum electrodynamics? Reflection and Refraction Are they simply due to photons being absorbed and re-emitted? How do we get to Snell'...
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### Noether Theorem and Energy conservation in classical mechanics

I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
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### Is there a kind of Noether's theorem for the Hamiltonian formalism?

The original Noether's theorem assumes a Lagrangian formulation. Is there a kind of Noether's theorem for the Hamiltonian formalism?
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### On a trick to derive the Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
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I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: $$S=-m\... 1answer 2k views ### Infinitesimal transformations for a relativistic particle The action of a free relativistic particles can be given by$$S=\frac{1}{2}\int d\tau \left(e^{-1}(\tau)g_{\mu\nu}(X)X^\mu(\tau)X^\nu(\tau)-e(\tau)m^2\right).$$If we then make an infinitesimal ... 1answer 284 views ### Constrained Hamiltonian systems: spin 1/2 particle I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action S=-m\int ... 1answer 176 views ### Principled approach to arrive at geodesic Hamiltonian H = g^{\mu \nu} p_\mu p_\nu? The background: If we have a spacetime path x^\mu(t) parameterized by arbitrary parameter t, the proper time along the path between t_1 and t_2 is$$ \int_{t_1}^{t_2} (g_{\mu \nu} \dot x^\mu \...
The action for a string in this background $$G_{IJ}\tag{1}$$ can be written as the Nambu-Goto action $$S_{NG}=\int d\sigma^1d\sigma^2\sqrt{g}\quad\quad\Rightarrow\quad\mathcal{L}=\sqrt{g}\tag{2}$$ ...