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Questions tagged [classical-field-theory]

For questions where the dynamical variables are classical fields, that is, functions of several variables (typically, one time coordinate and several space coordinates). If the question comprises both classical and quantum fields, use the tag [field-theory] instead.

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Equation of motion of free field Lagrangian

I tried to derive the equation of motion obtained by varying Lagrangian (2) in https://arxiv.org/abs/0804.4291 wrt the metric. It is supposed to give the second equation in (5) of the paper but my ...
vyali's user avatar
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10 votes
3 answers
2k views

Gross asymmetry in Maxwell Equations

Consider the statement of the vacuum Maxwell equations in the language of differential forms. The equation of motion in terms of the field strength $2$-form $F$ is $$d \star F = 0,$$ which follows ...
Silly Goose's user avatar
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31 views

How to derive the ODE from the EOM of vortex?

In the Lagrangian mode we have the equation of motion \begin{align} \partial_\mu F^{\mu\nu}&=j^\nu. \\ D_{\mu }D^{\mu}\phi +\mu^{2}\phi-\lambda(\phi^{*}\phi)\phi &=0. \end{align} Since we ...
Qian-Sheng's user avatar
2 votes
2 answers
111 views

Solutions to Maxwell's equations with $dF=0$ but $F \neq dA$ -- can the new solutions be summarized by considering only the vacuum equations?

I am trying to learn a bit about differential forms. I saw a question and answer noting that the homogeneous Maxwell equations can be written as $dF=0$. However, as noted there, depending on the ...
user196574's user avatar
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3 votes
0 answers
144 views

How is classical Chern-Simons theory topological?

Note: I am using "global" and "topological" somewhat interchangable. This seems to be the case in texts and papers, but please point out if this is inappropriate. Classical Chern-...
Silly Goose's user avatar
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1 vote
0 answers
54 views

Class of on-shell and gauge equivalent potentials in Chern-Simons theory

Let $(P, M, \pi, G)$ be a principal bundle with three dimensional manifold $M$ and compact, connected, simply-connected, and simple structure group $G$. We define a Lie algebra valued connection $1$ ...
Silly Goose's user avatar
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0 votes
1 answer
51 views

Variation of the kinetic term wrt the metric in scalar field theory

Varying $\partial_\lambda\phi\,\partial^\lambda\phi$ wrt the metric tensor $g_{\mu\nu}$ in two different ways gives me different results. Obviously I'm doing something wrong. Where am I going wrong? ...
vyali's user avatar
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1 vote
0 answers
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"Field excursion" of a single scalar particle in QFT

In the classical (typically large occupation number) limit of QFT, the dynamics of, say, a scalar field can be approximated by its classical field equation. Let's consider the Klein-Gordon equation: $$...
Guy's user avatar
  • 1,291
2 votes
0 answers
70 views

Interpretation of "spin-1/2" in classical Dirac field

I emphasize that the proceeding is purely classical physics. Consider the Grassmann-valued field (where $\mathcal{N}$ is a Grassmann number), which is a solution to the Dirac equation, given by $$\psi(...
Silly Goose's user avatar
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2 votes
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Definition of the displacement field in classical field Lagrangian

In a BSM related paper (in appendix B), the authors use an effective Lagrangian $\mathcal{L}_{EFT}$, and define the following fields: $$ \mathbf{D} = \frac{\partial\mathcal{L}_{EFT}}{\partial\mathbf{E}...
Doron Behar's user avatar
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Why do I get nonsense when trying to solve the Euler-Lagrange equations for a fluid?

Take an inviscid, incompressible fluid, ignore external forces for the sake of simplicity. The Lagrangian density is $$ \mathcal{L} = \frac{\rho}{2} {\vec v}\cdot \vec v $$ I'm trying to solve Euler-...
user2958456's user avatar
-1 votes
1 answer
38 views

Constrained Hamiltonian problems [closed]

What happens to the poisson bracket structure of Hamiltonian phase space if We have some constraints in $p$ and $q$. What physical aspects this structure represents?
Spotless-hola's user avatar
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0 answers
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Discrepancy in Maxwell's extended Hamiltonian

In the 4D Maxwell's extended Hamiltonian action, I obtain the same expression of Fuentealba, Henneaux and Troessaert (see the picture), up to the term "$\partial^i\pi^0 A_i$", although my ...
hyriusen's user avatar
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5 votes
1 answer
501 views

How can a gauge field have physical effects if it only reflects a redundancy in our mathematical description of physical reality?

I struggle to reconcile two aspects of classical gauge field theories, stated informally (and vaguely, I admit) as follows: Changing the gauge does not have any physical effects because the freedom ...
Figaro's user avatar
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1 vote
0 answers
69 views

Are negative charges sinks and positive charges sources, or is it an arbitrary choice?

Classical Electromagnetism defines negative charges as sinks and positive charges as sources. Is it based on experimental distinction, or is it just that one kind of charge must be thought of as a ...
Blacklight MG's user avatar
2 votes
0 answers
126 views

Ambiguity of Lagrangian density in field theory [duplicate]

In classical mechanics, we know $L(q,\dot{q},t)$ and $L(q,\dot{q},t)+\frac{d}{dt}\Lambda(q,t)$ give the same Euler-Lagrange equation $\frac{d}{dt}\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}_i}=\...
watahoo's user avatar
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0 votes
1 answer
64 views

Is there an experimental set up that would produce a "macroscopic" a weak or strong nuclear force fields?

I was wondering if there is an experimental set up that would produce something equivalent to a classical electromagnetic field for the weak and strong nuclear forces. I know that the those forces are ...
Bryan D's user avatar
  • 11
0 votes
2 answers
279 views

In QFT what is the frequency of oscillation in time of a scalar field mode? Does it depend on the number of particles in the mode?

In classical field theory, for a given real scalar field $\phi$, each mode $\vec{k}$ vibrates in time at a frequency $\omega = \sqrt{\vec{k}^2 + \mu^2}$ with $\mu$ being the mass. In quantum field ...
TrentKent6's user avatar
4 votes
8 answers
490 views

For a classical scalar field, how can a mode have different energies if the energy is the mode's frequency of oscillation in time?

For a classical real scalar field $\phi(\vec{x},t)$ of the type: $$\frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi+m^2\phi=0$$ The modes $\phi(\vec{p},t)$ can be obtained by: $$\phi(\vec{x},t)=\int \...
TrentKent6's user avatar
1 vote
1 answer
112 views

Canonical transformations in the covariant phase space formalism

As the title says, I'm looking for an explanation on how to apply canonical transformations when using the covariant phase space formalism. I'm familiar with the topic, but I haven't found a good ...
P. C. Spaniel's user avatar
3 votes
0 answers
83 views

Laplace Green function on $R \times S^3$

On flat Euclidean $R^4$ the Laplace operator has the Green function $G(x,y) = \frac{1}{4\pi^2(x-y)^2}$, i.e. $$-\Delta G(x,y) = \delta^4(x-y).$$ What would be the corresponding Green function on $R \...
Fetchinson0234's user avatar
1 vote
0 answers
37 views

Stress Energy Tensor VS Moment Map

The stress-energy tensor can be a map of killing vector fields to conserved currents. To be more precise this happens by contraction $J^\mu = T^{\mu\nu}\xi_\nu$ where $\xi$ is a killing vector field. ...
panoik's user avatar
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1 vote
0 answers
69 views

Why is the Klein-Gordon equation a 2nd derivative equation?

Note: I am not asking about why time is in 2nd derivative: that makes perfect intuitive sense given the relativistic need to treat space and time in equal footing. We often hear how the Klein-Gordon ...
TrentKent6's user avatar
2 votes
0 answers
73 views

Did Democritus predict atoms using Sharp Phase Transitions? How? Couldn't a classical field theory also have Sharp Phase Transitions?

In the Wikipedia page for the Ising Model it is written without citations: One of Democritus' arguments in support of atomism was that atoms naturally explain the sharp phase boundaries observed in ...
Diana's user avatar
  • 71
3 votes
2 answers
145 views

Why is the Lorentz transformation of fields linear?

I know that the coordinate, $x^\mu = (t,\vec x)$ is a 4-vector and it transforms as $$x'^\mu={\Lambda^\mu}_\nu x^\nu.$$ The related (classical or quantized) field, $\phi_a(x)$, can be classified into ...
Luessiaw's user avatar
  • 665
1 vote
1 answer
153 views

Calculating Poisson brackets in classical non-relativistic Hamiltonian field theory

Summary of the question: How can I prove the equal-time Poisson bracket relations for the classical Hamiltonian field theory? I.e $$[q(x,t),H(t)]_\mathrm{PB}=\dot{q}(x,t)\tag{1}$$ for a field $q$ and ...
Yong's user avatar
  • 13
2 votes
0 answers
48 views

How to justify the vacuum energy density without creation/annihilation operators?

In classical field theory, a well known calculation (pretty involved) gives the following expression for the total energy of the free electromagnetic field in empty space (I'm not writing the MKSA ...
Cham's user avatar
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2 votes
1 answer
54 views

Change of action after a transformation with space-time dependent parameters

I've been following David Tong's lecture on introduction to quantum field theory. In his lecture notes page number 19 (and his video class on Youtube), he talks about global transformation that ...
Andri jauhari's user avatar
0 votes
0 answers
30 views

Do topological solitons allow modeling non-degenerate multiple vacua?

I am not well-versed in the research on topological solitons but am interested to make a good sense of its implication. The highly interesting point in this new talk by Nick Manton was where he is ...
VVM's user avatar
  • 477
1 vote
1 answer
172 views

Experimental Verification of Quantum Field Theory vs Classical Field Theory [closed]

I have heard one statement that appears, in varied form, throughout many Quantum Field Theory (QFT) books and lectures: QFT is the most accurate framework in all of Physics. and then the result such ...
new_kid's user avatar
  • 13
1 vote
1 answer
133 views

Confusion on proofing Noether's theorem in field theory

I've been following along Gleb Arutyunov and Henk Stoof's Classical Field Theory lecture notes. It basically want to proof the Noether's theorem in field theory by considering an infinitesimal ...
Andri jauhari's user avatar
1 vote
0 answers
65 views

Literature Suggestions - Classical Field Theory [duplicate]

I'm a physics graduate that is joining the master degree program of my University. One of the areas of research that i'm very interested is Quantum Field Theory over curved surfaces, but the point is ...
2 votes
1 answer
112 views

Short wavelength limit in Eikonal equation

Limting the discourse of this question only for scalar waves in optics, we have $\nabla^2 \phi - \dfrac{n^2}{c^2} \dfrac{\partial^2 \phi}{\partial t^2} =0 $. Now, when we investigate about geometric ...
curious_mind's user avatar
2 votes
1 answer
142 views

Landau & Lifshitz definition of magnetic field intensity

I am trying to read through classical fields by Landau and Lifshitz but I am struggling to understand their phrasing of the magnetic field intensity. They state the equation of motion as: $$ \frac{dp}{...
jake walsh's user avatar
1 vote
0 answers
106 views

Under what representation of the Lorentz group do scalar $\textit{fields}$ transform?

I know that if I am sitting in a spacetime $M$ at point $p$, vectors live in the tangent space $T_pM$, and tensors in the tensor product space etc. If I want to consider general tensor fields, I ...
Craig's user avatar
  • 1,109
1 vote
1 answer
47 views

If the solution of a field vanishes on-shell does it mean anything particular?

Let us consider an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. The solution of the field $c$ is given by the expression $f(a,b)$. On taking into account the relations ...
vyali's user avatar
  • 382
1 vote
1 answer
71 views

Why only $\phi=\pm1$ are considered "vacuum states" in the Klein-Gordon model with $\phi^4$ potential, and not $\phi=0$?

I am reading "Kink Moduli Spaces — Collective Coordinates Reconsidered," by Manton, Oleś, Romańczukiewicz, and Wereszczyński (arXiv version), where they consider the Klein-Gordon equation, $$...
Michael Nestor's user avatar
0 votes
1 answer
80 views

Theta vacua eigenstates

I have been trying to prove the very simple result that the eigenstates of an operator with matrix elements $$ \langle n^\prime | H | n \rangle \sim g(|n^\prime-n|), $$ in a basis $\{|n\rangle\}^{+\...
GaloisFan's user avatar
  • 1,742
2 votes
0 answers
76 views

Scalar field-particle interaction in curved spacetime [closed]

Consider the interaction of a scalar field $\phi$ with a point particle $\chi \equiv \chi(\tau)$ in curved spacetime where $\tau$ is some affine parameter. The action for this system is $$S = \...
newtothis's user avatar
  • 643
3 votes
1 answer
311 views

Classical fermions, where are they?

Context: Studying the path integral formulation of QFT I stumbled upon a fairly simple statement: when doing loop expansions of a partition function: $$Z[\eta ; \bar{\eta}] = \int [d\psi][d\bar{\psi}]...
LolloBoldo's user avatar
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0 votes
0 answers
97 views

Reference request - classical field theory and mathematics

I am looking for references (books, lecture notes etc) on mathematical classical field theory. By that, I mean classical field theory under a rigorous point of view. However, I am more interested in ...
1 vote
3 answers
212 views

Varying the Einstein-Hilbert action when matter fields are off-shell

The Einstein-Hilbert action reads \begin{equation*} S_{EH} = \int d^4 x \sqrt{-g} \bigg(\frac{1}{2}R + \mathcal{L}_M(\phi,g)\bigg) \end{equation*} where $\phi$ are the matter fields and $\mathcal{L}_M$...
Panopticon's user avatar
1 vote
1 answer
263 views

Operators in quantum field theory

I am beginning to learn quantum field theory. I have a beginner level question. Please help me with it. In quantum field theory, the operators $x$ at each point are demoted to just labels and every ...
SX849's user avatar
  • 306
0 votes
0 answers
74 views

What does it mean for a classic field to be defined in terms of stochastic parameters?

I'm writing a bachelor's thesis related to inflationary cosmology and I don't quite understand some things about a paper I've been reading called Signals of a Quantum Universe. Specifically, the paper ...
J.S.A. Frugte's user avatar
3 votes
1 answer
75 views

What does it mean when the EOM of a field is trivially satisfied if other EOMs are satisfied?

If a Lagrangian has the fields $a$, $b$ and $c$ whose equations of motion (EOM) are denoted by $E_a=0, E_b=0$ and $E_c=0$ respectively, then if \begin{align} E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c\tag{1} ...
vyali's user avatar
  • 382
5 votes
3 answers
2k views

Why do we need to make a tensor for the electromagnetic field?

I was wondering why we need the electromagnetic field tensor $F_{\mu\nu}$ to be a tensor and why can't we work with the electric and magnetic fields while dealing with the electromagnetic field ...
Anargha's user avatar
  • 53
2 votes
0 answers
75 views

Statistical Treatment of Classical Field Theories

Is there a conceptual problem in formulating of Liouville's theorem and the BBGKY Hierarchy for classical field theories? I always see treatments of Lioville's theorem only in the context of classical ...
physicscircus's user avatar
3 votes
0 answers
49 views

Can any flat-space QFT be made Weyl invariant on curved space using curvature coupling?

I seem to have an argument that any theory defined on flat spacetime can be extended to a theory on general spacetimes which is Weyl-invariant, by carefully choosing the curvature coupling. I wonder ...
nodumbquestions's user avatar
2 votes
3 answers
152 views

Why should the fields not change under translation but may change under Lorentz transformations?

In order to derive the conservation of four-momentum of a field, $P^\mu$, it is assumed that the total change in the field, defined as $$\delta\phi_a(x)\equiv {\phi^{\prime}}_a(x^{\prime})-\phi_a(x)$$ ...
Solidification's user avatar
4 votes
0 answers
88 views

Why is the action for a field a quadruple integral over spacetime? [duplicate]

I've been trying to get started on classical field theories. As I had been studying classical mechanics from Goldstein, I decided to start from there. Goldstein introduces the action $$S=\int \mathscr{...
Lourenco Entrudo's user avatar

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