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20
votes
4answers
1k 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 ...
12
votes
2answers
671 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: $$S=\...
12
votes
1answer
495 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 ...
8
votes
3answers
209 views

Classical field limit of the electron quantum field

In order to recover classical electromagnetic fields from the quantum electromagnetic field, we consider coherent states of the form $$\exp \left(\int d\vec{r}\, \vec{A}(\vec{r}) \vec{a}^\dagger(\vec{...
8
votes
1answer
761 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 ...
7
votes
1answer
95 views

Infinitesimal rotation of classical fields: why are rotation group representations important?

I understand that $SO(3)$ representations are important in quantum physics, because eigenspaces of the Hamiltonian are irreps of $SO(3)$ if it is part of the symmetry group. But I don't see the reason ...
7
votes
1answer
795 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 ...
7
votes
1answer
287 views

effect of a simultaneous local and a global $U(1)$ symmetry breaking

EDIT : I am trying to figure out the effect of symmetry breaking in a $U(1)_Y\times U(1)_Z$ invariant lagrangian where $U(1)_Y$ is local symmetry of the Lagrangian and $U(1)_Z$ is a global symmetry of ...
6
votes
1answer
338 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) ...
5
votes
1answer
596 views

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 ...
5
votes
1answer
135 views

Is $\phi^4$ theory in 4d conformally invariant at the classial level?

I used to believe the three following statements to be true (at the classical level only): From scale invariance full conformal invariance follows. Scale invariance is present if there are no ...
5
votes
0answers
64 views

Spin-dependence of the directionality of dipole radiation

I am interested in understanding how and whether the transformation properties of a (classical or quantum) field under rotations or boosts relate in a simple way to the directional dependence of the ...
4
votes
2answers
423 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} = \Psi^{\dagger}(-i(...
4
votes
2answers
560 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 \varphi}{\...
4
votes
1answer
728 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?
4
votes
3answers
1k 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 ...
4
votes
2answers
1k 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 ...
4
votes
1answer
91 views

Significance of symplectic form in classical field theory

I'm trying to understand the significance of construction presented to me in field theory class. Let me first briefly describe it and then ask questions. Given two solutions $\phi_1$, $\phi_2$ of the ...
4
votes
1answer
237 views

What are the equations of motion for the scalar field in the tetrad formalism?

The action of a massless scalar field in curved spacetime is given by: \begin{equation} S(\phi)=\int d^{4}x \sqrt{-g}\left(g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right) \end{equation} Now the action can ...
4
votes
1answer
240 views

Renormalization in Classical Field Theory

1) The statement that general relativity (GR) is not renormalizable - is it a statement only about the quantization of GR or is it non-renormalizable also as a classical field theory? 2) More ...
4
votes
1answer
211 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
325 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 ...
4
votes
0answers
88 views

Is there a classification scheme for linear classical field theories?

Central to a mathematical understanding of the Bogolyubov transformation is the study and classification of linear lattice field theories. What follows might be familiar to many people, but I just ...
4
votes
0answers
64 views

Motivation for covariant phase space

The covariant phase space idea, in one sentence, is that there is a natural symplectic structure on the space of the classical trajectories of a system and that the usual $(q,p)$ coordinates just ...
4
votes
0answers
106 views

Axion Model Field Theory Problem

This is a homework problem for a field theory class dealing with an axion model. Originally, we are given that $$S[a]=\int_Md^4x \frac{1}{2}(\partial_{\mu}a(x))^2$$ has a continuous global ...
3
votes
5answers
582 views

Euclidean geometry in non-inertial frame

Refer, "The classical theory of Fields" by Landau lifshitz (Chap 3). Consider a disk of radius R, then circumference is $2 \pi R$. Now, make this disk rotate at velocity of the order of c(speed of ...
3
votes
2answers
1k views

Need for a side book for E. Soper's Classical Theory Of Fields

I am reading now E. Soper, Classical Theory Of Fields, now and sometimes it is very hard to follow the equations. So I need a side book on classical field theory to read it comfortably. Landau & ...
3
votes
3answers
166 views

What is the point of complex fields in classical field theory?

I see a lot of books/lectures about classical field theory making use of complex scalar fields. However why complex fields are used in the first place is often not really motivated. Sometimes one can ...
3
votes
1answer
732 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 ...
3
votes
1answer
84 views

Why can you make $V$ stationary with respect to a parameter of the field in Derrick's theorem?

I'm going over Coleman's derivation of Derrick's theorem for real scalar fields in the chapter Classical lumps and their quantum descendants from Aspects of Symmetry (page 194). Theorem: Let $\...
3
votes
1answer
95 views

Energy-momentum tensor transformation [closed]

I've been trying to find how the energy-momentum tensor changes if we add a total derivative to the lagrangian: $$L\to L+\mathrm d_\mu X^\mu.\tag{1}$$ From the answer key: $$T^{\mu\nu}\to T^{\mu\nu}...
3
votes
1answer
114 views

Fermionic Poisson bracket

I'd like to understand the Poisson bracket for fermions in classical field theory defined on a cylinder (with coordinates $(t,x)$, $x$ being the compact direction) and propagating on $T^n$ with ...
3
votes
0answers
42 views

Hermiticity of the Lagrangian density [duplicate]

In quantum mechanics, the dynamical variables are replaced by hermitian operators. However, the Lagrangian density in quantum field theory is not an observable but we also make sure that it is ...
3
votes
0answers
196 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
1answer
162 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
2answers
456 views

The variation of the Lagrangian density under an infinitesimal Lorentz transformation

I'm trying to introduce myself to QFT following these lectures by David Tong. I've started with lecture 1 (Classical Field Theory) and I'm trying to prove that under an infinitesimal Lorentz ...
2
votes
1answer
90 views

What's the meaning of a field?

Sorry if the title sounds meta-sciency, allow me to clarify. In physics, our goal is to understand how the universe works. To this end, we construct a theory, which hopefully makes falsifiable ...
2
votes
1answer
159 views

Is it true that the self-force prevents a classical particle from falling into a Coulomb potential? What is the physical explanation of this result? [closed]

In 1943 CJ Eliezer published a paper claiming that the self-force prevents a zero angular momentum particle from ever reaching the center of an attractive Coulomb potential (and what's more that it ...
2
votes
2answers
157 views

Higher rank $\gamma$-matrix question

I read that the higher rank $\gamma$ matrices can be written as alternate commutators and anti-commutators. For example, the rank 3 gamma matrix can be written as $$\gamma^{123} = \frac{1}{2}\{\gamma^{...
2
votes
1answer
220 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?
2
votes
0answers
60 views

Jet bundles for physicists

In order to make Classical Field Theory rigorous we need the idea of jet bundles. I've seem some books on the subject, but most of them are aimed at mathematicians and tend to go quite deep in the ...
2
votes
0answers
108 views

Problems while doing $\dfrac{\partial}{\partial(\partial_\mu \phi)}$ and $\dfrac{\partial}{\partial(\partial_\mu A_\mu)}$

In David Tong's lectures, he gives two Lagrangians as examples to derive the equations of motion: $$\mathcal{L} = \dfrac{1}{2}\eta^{\mu\nu}\,\partial_\mu\phi\,\partial_\nu \phi-\dfrac{1}{2}m^2\phi^2, ...
2
votes
0answers
84 views

Possible Error in deriving conformal generator

My professor gave me the following derivation for the full generator of the Lorentz transformations. The starting point is to consider a subgroup of the conformal group that leaves the origin fixed (...
2
votes
0answers
113 views

Similarities between laminar-turbulence transition and others like BCS-BEC crossover, quark-hadron transition etc

From my limited readings on fluid dynamics, my understanding is that as the system changes from near-laminar flows to full turbulence, the dimensionless Reynolds number changes from $ R << 1$ to ...
1
vote
1answer
39 views

Classical Field Theory: Physical meaning of various terms in total Hamiltonian

In a classical field theory problem Lagrangian density is given as $${\cal L}=\frac{1}{2}\dot{\phi }^{2}-\frac{1}{2}\left ( \bigtriangledown \phi \right )^{2}-\frac{1}{2}m^{2}\phi ^{2}\tag{2.6}$$ ...
1
vote
3answers
181 views

Symmetry at quantum level in quantum field theory

In nonrelativistic quantum mechanics, a symmetry is a transformation on states in the Hilbert space which keeps the Hamiltonian invariant and this implies that the generator of the transformation must ...
1
vote
2answers
75 views

Why don't people use Hamilton's equations for a relativistic free charged particle?

A charged relativistic free particle has the Hamiltonian in general: $$ \mathcal{H} = \sqrt{p^2c^2+m^2c^4}.$$ I read somewhere that says, it is possible to go further and say that the EoM are ...
1
vote
1answer
69 views

Classical vacuum and quantum vacuum

How to determine the ground state of a classical field, for example an electromagnetic field? What is the difference between the the ground state of a classical field and that of a quantum field?
1
vote
1answer
121 views

Proof that Maxwell equations are Lorentz invariant

In Peskin and Schroeder page 37, it is written that Using vector and tensor fields, we can write a variety of Lorentz-invariant equations. Criteria for Lorentz invariance: In general, any equation ...
1
vote
1answer
78 views

Integrating out fields from classical systems

Has anyone ever heard of integrating out fields from classical Lagrangians if they are quadratic?