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19
votes
4answers
466 views

Which exact solutions of the classical Yang-Mills equations are known?

I'm interested in the pure gauge (no matter fields) case on Minkowski spacetime with simple gauge groups. It would be nice if someone can find a review article discussing all such solutions EDIT: I ...
11
votes
1answer
378 views

How is the Dirac adjoint generalized?

I am wondering how one can generalize the Dirac adjoint to flat "spacetimes" of arbitrary dimension and signature. To be more specific, a standard situation would be to consider 4 dimensional ...
9
votes
1answer
325 views

Small oscillations of heavy string

I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points. First of all I wrote down this Lagrangian: ...
8
votes
1answer
368 views

Relativistic center of mass

Recently I realized the concept of center of mass makes sense in special relativity. Maybe it's explained in the textbooks, but I missed it. However, there's a puzzle regarding the zero mass case ...
6
votes
1answer
103 views

Question about the Noether charge algebra

I'm reading these notes - page 8 and 9 - and I'm a bit confused. If we consider a field $\phi$ (which can be either bosonic or fermionic) transforming as: \begin{equation} \phi(x) \rightarrow \phi(x) ...
6
votes
1answer
427 views

Frasca's mapping of classical Yang-Mills to $\phi^4$ theory

I recently came across an article on the arXiv 0709.2042 written by Marco Frasca, where he provides a mapping between classical Yang-Mills theory to $\phi^4$ theory. Has his idea been fruitful in ...
4
votes
2answers
128 views

Hamilton formalism for Dirac spinors

Let's have the Dirac free lagrangian: $$ L = \bar {\Psi} (i\gamma^{\mu}\partial_{\mu} - m) \Psi . $$ I can rewrite it as $$ L = i\Psi^{\dagger}\partial_{0}\Psi - H_{d}, \quad H_{d} = ...
4
votes
1answer
115 views

On a trick to derive 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 ...
4
votes
1answer
64 views

Why does the minimum energy field configuration require the fields to be constant?

I am having a hard time in understanding a well known statement always made in the context of field theory. Background Consider a classical real scalar field theory with Lagrangian density given by ...
4
votes
1answer
220 views

Can classical systems exhibit “strong coupling”?

Does the concept of strong coupling mean anything in a classical setting? If strong coupling means just an inability to apply perturbative methods to the Hamiltonian, then obviously yes, we can ...
3
votes
3answers
323 views

Relation between the determinants of metric tensors

Recently I have started to study the classical theory of gravity. In Landau, Classical Theory of Field, paragraph 84 ("Distances and time intervals") , it is written We also state that the ...
3
votes
1answer
219 views

Infinite Energy of Point Charges (in the context of classical field theories)

In the context of classical physics,is there any renormalization method to avoid infinite energy of point charges?
3
votes
0answers
123 views

Questions about classical and quantum scale invariance

This is kind of a continuation of this and this previous questions. Say one has a free "classical" field theory which is scale invariant and one develops a perturbative classical solution for an ...
2
votes
2answers
120 views

Canonical momentum density vs. energy-momentum tensor

Suppose we have a scalar field $\varphi$ with Lagrangian $$ \mathcal{L} = \frac{1}{2} \kappa \left( \frac{\partial \varphi}{\partial x} \right)^2 + \frac{1}{2} \rho \left( \frac{\partial ...
2
votes
1answer
108 views

Energy-momentum conservation without translation symmetry?

As I checked, the energy-momentum tensor defined as ${T^\mu}_\nu=\frac{\partial {\cal L}}{\partial(\partial_\mu \phi)}\partial_\nu \phi-{\cal L}{\delta^\mu}_\nu$ at the solution $\phi$ of equation of ...
2
votes
1answer
267 views

Noether First and Second Theorem

I have this question related to the the Noether's Theorems. I want to know a rigorous enough enunciation of this theorem, the context is Classical Field Theory without fancy geometrical structures ...
2
votes
2answers
392 views

Symmetry of Euler-Lagrange equations and conservation laws

Continuous symmetry of the action implies a conservation law, but what if equations of motion have a continuous symmetry? Does it imply a conservation law? Also is symmetry of equations of motion ...
2
votes
1answer
124 views

Total energy is extremal for the static solutions of equation of motions

In physics total energy is extremal for the static solutions of equation of motions. Can anyone explain this sentence to me?
1
vote
1answer
82 views

What is meant by a local Lagrangian density?

What is meant by a local Lagrangian density? How will a non-local Lagrangian look like and what is the problem that we do not consider such Lagrangian densities?
1
vote
1answer
38 views

Integrating out fields from classical systems

Has anyone ever heard of integrating out fields from classical Lagrangians if they are quadratic?
1
vote
0answers
74 views

Taylor expansion in classical 1D harmonic chain (classical field theory)

I'm struggling with something apparently really easy but not entirely straightforward. In "Condensed Matter Field Theory" of Altland and Simons the classical 1D harmonic chain is treated as ...
1
vote
1answer
74 views

Finding Hamilton's equations given a Hamiltonian

I am trying to find Hamilton's equations for a general Hamiltonian given by $$H[u]=\int_\mathbf{R} \phi(u,u_x)dx$$ Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial ...
0
votes
0answers
58 views

How to do this index notation differentiation?

I am studying classical Maxwell fields and I am stuck on this differentiating part. How can I derive the result given below ? $$\dfrac{\partial}{\partial(\partial A_{\mu}/\partial x_{\nu})} ...
0
votes
0answers
54 views

Classical systems with strong coupling

I would like to have some examples of classical systems with "strong" coupling coefficient (in their action term). Specifically, I am looking for intermediate or cross-over ranges of coupling ...
0
votes
0answers
67 views

Counting the modes of the vector potential in a coulomb gauge

With a view to quantising the EM field, consider a classical free field in the absence of charge and currents, we can take a coulomb gauge, $\phi=0, \partial_kA_k=0$. The physical fields in terms of ...