Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

4
votes
3answers
136 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 ...
2
votes
1answer
87 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 ...
8
votes
1answer
222 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, $$\...
1
vote
1answer
91 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 ...
0
votes
0answers
30 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 $S$...
1
vote
0answers
39 views

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 ...
0
votes
1answer
52 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 ...
1
vote
2answers
68 views

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 $$\...
2
votes
1answer
45 views

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 ...
1
vote
0answers
34 views

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 ...
0
votes
2answers
56 views

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.
0
votes
0answers
35 views

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 ...
1
vote
2answers
71 views

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^{\...
2
votes
2answers
116 views

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 ...
1
vote
1answer
68 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 $$F_{\mu\nu}\tilde{F}^{\mu\nu}=\partial_{\mu}(\epsilon^{\nu\nu\lambda\rho}A_\nu F_{\lambda\rho}).$$...
2
votes
2answers
124 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}^{-...
5
votes
1answer
102 views

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 ...
1
vote
1answer
94 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 $$...
0
votes
2answers
64 views

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 ...
1
vote
3answers
40 views

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 ...
1
vote
1answer
47 views

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 ...
1
vote
0answers
52 views

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, ...
3
votes
1answer
60 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 ...
3
votes
1answer
104 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+...
1
vote
1answer
116 views

Why does a the addition of a total derivative to the Lagrangian leave the equations of motion invariant?

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 ...
10
votes
2answers
978 views

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 ...
8
votes
4answers
767 views

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^{\...
0
votes
0answers
42 views

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 ...
3
votes
1answer
64 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 ...
1
vote
0answers
34 views

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}} =...
4
votes
2answers
139 views

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 ...
2
votes
2answers
113 views

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}{...
1
vote
1answer
207 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}...
1
vote
0answers
52 views

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{...
0
votes
1answer
29 views

Momentum dependence of Fourier amplitudes in the Fourier expansion of relativistic classical fields

A relativistic classical field $\phi(x)$ can be decomposed into Fourier modes as $$\phi(x)=\int\frac{d^3\textbf{p}}{(2\pi)^{3/2}\sqrt{2E_{\textbf{p}}}}\Big(a(\textbf{p})e^{-ip\cdot x}+b^*(\textbf{p})e^...
0
votes
2answers
51 views

Physical manifold with a natural linear connection on them

Of course in many situation a manifold raised from a physical situation (like spacetime or configuration manifold and so on) are really much more richer than an abstract manifold. for example phase ...
0
votes
1answer
234 views

How many degrees of freedom does an electromagnetic field have? How to correctly count them?

How many independent degrees of freedom does a most general classical electromagnetic field have in presence of sources? What is the correct way to count them? In terms of the components of the ...
2
votes
0answers
39 views

Binding energy in Euclidean Gravity

I suppose I'm asking in the context of classical gravity, but an answer in the effective GR quantum theory would be fine too. I can't seem to find much on the attractive nature of the gravitational ...
6
votes
4answers
281 views

Is the description of the gravitational field as a vector field and a tensor field compatible?

By electric or magnetic fields we mean the vector fields $\vec{E}(\vec{r},t)$ and $\vec{B}(\vec{r},t)$ respectively. But a gravitational field in Newtonian theory is a vector field that $\vec{g}(\vec{...
8
votes
1answer
154 views

Why are projective representations allowed in classical field theory?

A quantum spin $1/2$ particle does not return to itself upon a rotation by $360^\circ$, but rather itself up to a sign. This is acceptable, because this extra phase is unobservable. In general the ...
1
vote
0answers
78 views

From classical mechanics to classical field theory

Suppose I have a system of $N$ classical particles described by the Lagrangian $\mathcal{L}(\mathbf{q}_i,\dot{\mathbf{q}}_i,t)$. Similarly, we can introduce the Hamiltonian of such system via the ...
0
votes
0answers
21 views

Confusion in approxmimation method for non-linear gravity

I have been confused for the past several days on how to approach one part of a problem. This problem is from Schwartz 3.7 and is guiding the reader through solving the perihelion precession of ...
3
votes
1answer
69 views

The speed of information propagation (classical, not quantum)

It seems to be a common statement in textbooks that: "For a linear wave equation with the dispersion relation $\omega_k$, the propagation speed of information is given by the group velocity." which ...
-1
votes
2answers
58 views

Why is field inside a conductor zero? [duplicate]

No external fields penetrate the conductor as they are canceled at the outer surface by the induced charge. My question is why is field inside conductor zero when there is no other field inside the ...
3
votes
4answers
274 views

Transformation of $d^4x$ under translation disregarded?

Under a translation in spacetime i.e., $$x\mapsto x^\prime=x+a,\tag{a}$$ a scalar field $\phi(x)$ $$\phi(x)\mapsto\phi^\prime(x)=\phi(x-a).\tag{b}$$ My aim is to verify the invariance of an action of ...
1
vote
0answers
65 views

4-derivative of a vector field [closed]

I am still stuck in my homework, but this time at another point, so this is a continuation of my last question to that subject. It is given a classical field theory for a 4-vector field $A_{\mu}$ and ...
5
votes
2answers
283 views

Characteristic classes appearing in the real world?

In the 1920's, Dirac gave a wonderful proof that if there exists a magnetic monopole, then charge must be quantised (!) I remember reading that the quanisation basically comes about because the first ...
2
votes
3answers
114 views

What do the antisymmetric matrices $J_i$ represent in classical mechanics?

In physical three-dimensional space, a rotation about an arbitrary axies $\hat{\textbf{n}}$ through an angle $\phi$ can be represented by $$R(\hat{\textbf{n}},\phi)=e^{-i(\textbf{J}\cdot\hat{\textbf{n}...
2
votes
0answers
49 views

Variation of electromagnetic part of action

I've got the same problem as Gabriel Luz Almeida had out here: Variation of Maxwell action with respect to the vierbein - Einstein-Cartan Theory I try to vary the electromagnetic part of action i.e. $...
0
votes
2answers
66 views

Momentum and Higgs Fields

A photon does not have mass thus it does not interact with the Higgs Field. However, it has momentum. How can this be represented in the Higgs Field as momentum is a property exhibited by particles ...