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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 [tag:field-theory] instead.

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Does it make sense to speak in a total derivative of a functional? Part III

In this third part of the series, I will continue the deduction of Noether's theorem initiated in the previous post - Does it make sense to speak in a total derivative of a functional? Part II. ...
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Energy momentum tensor of EM field written in symmetric form

I'm reading A. Zee's book, Einstein Gravity in a Nutshell. In problem 7 of chapter IV.2, it is said that the energy momentum tensor of the electromagnetic field \begin{align} T^{\mu\nu}=\eta_{\lambda\...
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Non-existence of double time-derivative of fields in the Lagrangian and violation of equal footing of space and time

In classical field theory, we consider the Lagrangians with single time-derivative of fields whereas double derivative of the field w.r.t. space is allowed sometimes. I understand that the reason of ...
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Space translation of coordinates, classical field theory

Consider the Lagrangian density $L = -\frac{1}{4}F_{\mu\nu}F^{\mu \nu}$ with $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} $. After deriving the Euler-Lagrange equations for this ...
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Hamiltonian directly expressed in $(q,\dot{q})$ : how to find what is $p$?

I am reading a book about non relativistic quantization of E.M field. But first we do classical field theory. We directly wrote the Hamiltonian of our study, and a part of our Hamiltonian is the ...
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1answer
135 views

Simplest model in field theory which leads to a pseudo-Goldstone boson

What can be a simple (if not simplest) continuum field theory model that gives rise to a pseudo Goldstone boson (doesn't matter if it is a toy model)? For example, I would be very happy if one can ...
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Why a field theory containing only fermions does not show spontaneous symmetry breaking?

For a real scalar field $\phi$, a theory as simple as the $\phi^4$ theory, can exhibit the phenomenon of Spontaneous Symmetry Breaking (SSB). For a complex scalar field $\phi$, a theory as simple as ...
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Classical field theory with fields on different base spaces

Keeping things at a "basic level", a field is a function from a base manifold (of dimension D) to some other space. Usually the base manifold is the spacetime but may be something different (a lattice,...
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Questions about Euler-Lagrange derivation in Classical Field Theory

I'm new to classical field theory, so I have a few basic questions: From the derivation of the Euler-Lagrange equations, we have the following: \begin{align} \delta S[\phi]&=\int d^4x\delta L(...
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1answer
93 views

Problems of Klein Gordon equation

Consider the Klein-Gordon equation $$(\square+m^2)\varphi=0.$$ People usually claim that $\varphi^* \varphi$ cannot be interpreted as a probability density because $\int d^3\vec{x}\varphi(t,\vec{x})^*...
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Positive frequency definition in general spacetime for general fields

In Quantum Field Theory the positive frequency solutions to the classical field equations are quite important since they are the basis of the definition of particles. In Minkowski spacetime we have a ...
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114 views

Is every classical field theory with dimensionless couplings conformally invariant?

I'm trying to learn conformal field theory and getting rather frustrated, because I can't find any source that gives decent examples or straightforward logic. In most sources I have found, conformal ...
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Field equations for Yukawa force

I am curious if classical field equations, in the vein of Maxwell's equations and the Lorentz force law for electromagnetism, or Einstein's equations and the geodesic equation for general relativity, ...
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Intuitive/Physical reason why fields are distributions

I read in Urs Schreiber's notes on mathematical QFT that the infinities in the standard approach to QFT appear because the product between operator-valued field distributions is not always well ...
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Intuition behind the use of the Principle of Stationary Action in Classical Field Theory [duplicate]

Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the ...
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Magnetic field $\vec{A}$ as momentum potential

I was reviewing some topics on electromagnetic field theory and I came across the following interesting assertion: the electromagnetic moment $P_{EM}$, which is defined in vacuum as: $$P_{EM}=\frac{1}...
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179 views

Unification of gravity and electromagnetism

Have there been any attempts at unifying gravity and electromagnetism at least at classical level since Hermann Weyl's idea of gauge principle (1918)? We now have Standard Model which is very ...
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1answer
102 views

Gauge-invariance of Lagrangians

I am rereading David Bleecker's Gauge Theory and Variational Principles, and I have realized I don't understand something. The offending part is in 3.3 (page 50-52), however I am reproducing the ...
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246 views

How does canonical quantization work with Grassmann variables?

Every quantum field theory textbook I've encountered seems to have the same logical oversight, because of the particular order they cover topics. First, the books introduce the Dirac Lagrangian, $$\...
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1answer
97 views

Poynting theorem in Landau and Lifshitz’ field theory book

In Landau & Lifshitz’s The Classical Theory of Fields, in section 31, they have proved the Poynting theorem (equation 31.6) in its integral form. In the footnote on page 76, they mention We ...
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36 views

Isometries and coordinate transformations in the context of Noether's Theorem

If I have a theory defined on some manifold, my understanding is that the dynamical objects in the theory should carry a representation of the isometry group of that manifold. Moreover, the action $S$...
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Hamilton equations of motion for matter fields coupled to general relativity in ADM formalism

Do you know what are the Hamiltonian formalism analogs of the Klein-Gordon equation and/or the Maxwell equations in general relativity? Showing how these equations of motion for matter in the ...
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1answer
76 views

Confusion about conservation of angular momentum tensor in classical field theory?

In my lectures, we considered the conserved stress energy tensor $T^{\mu \nu}$ and noted that we could always add a conserved tensor to it such that $T^{\mu \nu}$ is symmetric. As a consequence, a ...
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2answers
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Global $U(1)$ transformation properties of gauge fields

What are the Global gauge transformations of gauge bosons in Standard Model? To elaborate: Initially, we consider the global $U(1)$ transformations of scalars ($\phi$) and fermions ($\psi$) as $$\...
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1answer
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States of classical general relativity

In Classical Mechanics a state of a system is either a pair $(q,p)$ or $(q,\dot{q})$ depending if we formulate the theory on the tangent or cotangent bundle of the configuration space. The evolution ...
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Proof of the existence of the energy-momentum tensor [duplicate]

I have a problem providing or finding a general proof for this statement i found in Mussardo's statistical field theory book, section $10.3.2$: Due to the locality of the theory there exists a local ...
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Does classical physics allow a flow of electrons in vacuum to form a current?

My physics teacher today proposed this question as a homework. My view is that it does allow the current to flow classically.
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Question about the concepts of Noether charge and Noether current

I read that a noether current occurs when the lagrangian assume vector values. Well, what are noether current and noether charge in comparison to elementary classical mechanics notions of Noether's ...
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Parity Transformation on Classical Fields

I've been confused by this parity transformation in classical field theory for a long time. Let $\phi(t,\vec{x})$ be a scalar field. Then, up to some constant phase factor, it transforms to $\phi^{\...
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Noether's Theorem in Classical Field theory Confusion

Consider $N$ independent scalar fields $φ_i (x)$ in 4D space. Also consider a lagrangian density $$\mathcal{L} = \mathcal{L}(φ_i, \partial_μφ_i).$$ Suppose we perform the following infinitesimal ...
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1answer
79 views

Is it enough to assume $F_{\mu\nu}\to 0$ at infinity but not $A_\mu$ to derive the equation of motion?

Suppose the the Lagrangian $\mathscr{L}$ of the free electromagnetic field is augmented with the term $$F_{\mu\nu}\tilde{F}^{\mu\nu}=\partial_{\mu}(\epsilon^{\nu\nu\lambda\rho}A_\nu F_{\lambda\rho}).$$...
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2answers
196 views

Solution to the Klein-Gordon Equation

Most textbooks solve the Klein-Gordon equation with the ansatz $$\varphi(\mathbf{x},t) = \int \frac{\mathrm{d}^3\mathbf{k}}{f(k)}\left(a(\mathbf{k})\, \mathrm{e}^{i k x} + b(\mathbf{k})\,\mathrm{e}^{-...
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Why don't we “see” the classical Dirac field?

The electromagnetic field describes photons. If there are many photons then things become classical and we can use classical electromagnetism to describe the EM field. We can also measure the EM field ...
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101 views

Maxwell equations of motion from $S = \frac{-1}{2} \int F \wedge \ast F$

I'm trying to understand the following equation, used in the derivation of the equations of motion. Let $S = \frac{-1}{2} \int F \wedge \ast F$ and $F = dA$. Let $\delta$ denote variation. Then $$...
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Landau's 4-Accelaration Problem

I was going through Lev and Landau's 2nd volume on classical field theory. I came across a doubt. It's stated in the solution that we go to the frame having particle's velocity zero, which basically ...
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What happens when a super large charge is brought close to a conductor with limited supply of charges?

Imagine the following induction scenario: a super large charge +Q is brought close to a real conductor with limited supply of charges; that is, the total charge of all electrons inside this conductor ...
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Trouble understanding why fields are unnaffected by translations

The following sentence appears in my classical field theory notes Fields are Lorentz tensors and spinors, and as such unaffected by translations: $ \dfrac{\delta \phi}{\delta a^{\alpha}}=0.$ Where ...
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Local charge current for gauge field and conservation of charge

Motivation: It is a well known fact that the gravitational field (in General Relativity and direct generalizations of it) has no local energy-momentum density. Usually there are two reasons stated, ...
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63 views

Functional Poincaré's lemma and the inverse Lagrangian problem

I have only encountered the inverse Lagrangian problem in mathematics books that treat Lagrangian field theory using jet bundles and homological algebra, and while I am studying this approach, I still ...
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1answer
105 views

Why do we need these two sets of modes in the gravitational collapse?

Consider the gravitational collapse spacetime: Hawking argues in his paper$^{[1]}$ about black hole radiation that the massless scalar field $\phi$ can be decomposed as $$\phi = \sum_i \{p_i b_i+...
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1answer
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Why does a the addition of a total derivative to the Lagrangian leave the equations of motion invariant? [duplicate]

Take a Lagrangian $L \rightarrow L+\partial_{\mu}F^{\mu}$. If we can show that the total derivative $\partial_{\mu}F^{\mu}$ identically satisfies the Euler-Largrange equation, then we have shown that ...
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Are all fields in the universe we know of quantum fields?

Are all fields in the universe we know of quantum fields? Do all fields that exist must be inherently quantum in nature? How about fields that are yet to be discovered (ie. a new field like Higgs ...
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Why is velocity defined as 4-vector in relativity?

In classical field theory class my professor first defined velocity as $$v^{\mu} = \frac{dx^{\mu}}{dt}$$ where, $ \frac{dx^{0}}{dt} = 1$ Then he said it doesn't transform like $${v^{'}}^{\mu} = L^{\...
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Lagrangian of Charged Particle Evaluated On-Shell

I am trying to calculate the Lagrangian of a charged particle in background gauge field evaluaed on-shell. Let $A^{\mu}(x)$ be a gauge field. The action of a charged particle in this background gauge ...
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66 views

Theory invariance after substitution of theory's field equations back into theory's action functional?

Suppose I have a theory $A$ concerning the evolution of a set of fields $T_1, \dots, T_n$. Let the action functional for this theory be $S[T_1, \dots, T_n]$. Suppose in the action, in addition to ...
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Acceleration from scalar-matter coupling in classical field theory

I have came across a text where an interaction term in a classical Lagrangian is presented that couples a matter density $\rho$ and a scalar field $\phi$ as \begin{equation} \mathcal{L}_{\text{int}} =...
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Is the gauge transform field in electromagnetism a Lagrange multiplier?

In a draft answer to another question about gauge transformations, I played around with demonstrating the action of a gauge transformation on the Lagrangian density. Beginning with the classical ...
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Are scalar fields defined up to harmonic functions?

Disclaimer: this question may be very stupid. It looks like I am missing some fundamental point. Let's consider a massive scalar $\pi$ $$ \mathcal{L}_\pi = -\frac{1}{2}(\partial \pi)^2 -\frac{m^2}{...
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1answer
245 views

Charge conjugation transformation of complex scalar field

This is a quick and simple question. I'm studynig about a charge conjugation tranformation over a complex scalar field, $\psi\left(x\right)$, $$ \psi\left(x\right)\rightarrow C\psi\left(x\right)C^{-1}...
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Solution of wave equation not obeying maxwell equation

I was reading ch. 9 of classical electrodynamics of dr. griffith's. there he wrote that if a electric/magnetic field satisfy maxwell's equation then they must solve the wave equation which is $$ \frac{...