# 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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-...
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### 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 $1$ ...
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### 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? ...
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### 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 ...
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### 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
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### 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 $\mathcal{L}_M$...
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### 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 ...
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### 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 ...
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### 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} ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$\delta\phi_a(x)\equiv {\phi^{\prime}}_a(x^{\prime})-\phi_a(x)$$ ...
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{...