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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84 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 ...
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4answers
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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 ...
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0answers
73 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
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2answers
545 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 ...
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3answers
191 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}...
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0answers
64 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. $...
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2answers
92 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 ...
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0answers
50 views

Local Density of the Medium for Classical 1D Harmonic Chain

In the first chapter of Altland and Simons (2nd Edition), pg. 34, there is the following exercise: Consider the one-dimensional elastic chain discussed in Section 1.1. Convince yourself that the ...
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50 views

Do we really need to a priori assume fields, and functions of them, die off at infinity?

At the beginning of many QFT books that I've been going through, in the section on classical field theory (whose logic I presume is carried over to many other parts of those books), authors typically ...
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2answers
458 views

When can we handle a quantum field like a classical field?

I am curious that, are there any criterion to justify the use of a classical field to describe a fundamentally quantum field? To rephrase in another way, when we can take the classical limit of a ...
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1answer
431 views

Can Einstein-Hilbert action be derived from symmetry considerations?

The action of a free relativistic classical field theory can be derived from Poincare invariance, locality, and retaining terms quadratic in fields. Is there a similar set of symmetry principles which ...
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2answers
546 views

What spinor field corresponds to a forwards moving positron?

When we search for spinor solutions to the Dirac equation, we consider the 'positive' and 'negative' frequency ansatzes $$ u(p)\, e^{-ip\cdot x} \quad \text{and} \quad v(p)\, e^{ip\cdot x} \,,$$ ...
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3answers
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Why is dependence on derivatives not a problem in the definition of canonical energy-momentum tensor?

Let $\mathcal L(\phi,\partial\phi)$ be a Lagrangian for a field $\phi$. It is known that the Lagrangian $\mathcal L$ and the Lagrangian $\mathcal L+\partial_\mu K^\mu$ produces the same physics, ...
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0answers
309 views

Ambiguities in the energy-momentum tensor [duplicate]

There is a problem 3.3 in Schwartz’s QFT: Ambiguities in the energy-momentum tensor: (a) If you add a total derivative to Lagrangian ${\cal{L}} \rightarrow {\cal{L}} + \partial_\mu X^\mu$, how ...
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1answer
230 views

Why is it that in presence of a long-range force Goldstone excitations are absent?

Page 15 of this note states, If a continuous symmetry of the Lagrangian is spontaneously broken, and if there are no long-range forces, then exists a zero-frequency excitation at zero momentum....
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1answer
176 views

Contradictory solutions to the Klein-Gordon equation as an initial value problem (canonical formulation)

I'm going to begin with a prelude about constructing the solution using the action/Lagrangian formalism in order to provide context and a point of comparison for the canonical one. The main question ...
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3answers
171 views

Proving that the 3-current density corresponding to the global phase invariance vanishes at infinity

The components $j^i$ of the 3-current density $\textbf{j}$ corresponding to the global phase invariance of the action of a complex scalar field $\phi$ i.e., $\phi\to e^{-iq\theta}\phi$ is given by $$...
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0answers
257 views

What subjects are required for studying classical field theory?

I'm a undergraduate (sophomore) physics student who wants to study The Classical Theory of Fields by Landau & Lifshitz. What subject are required for studying this book/topic? I have already ...
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1answer
337 views

Classical version of tree-level QFT correlation functions

Having read this previous question (and its answers) about the relation between tree-level quantum field theory and classical field theory I can see two facts that support the (perhaps too vague) ...
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1answer
552 views

Classical Dirac equation

In QFT, electromagnetism is represented by the quantum field $\hat{A}_\mu$, and fermions (matter) by the quantum field $\hat\psi$. The same kind of formalism is used for both phenomena, even if the ...
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2answers
83 views

Violation of Derrick's theorem for finite energy, time independent solutions?

How are vortices the finite energy time independent solutions for 2+1 dimensions abelian higgs model? Doesn't it violate derricks theorem that there are no finite energy time independent solutions in ...
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0answers
368 views

Breakdown of the Legendre transform for the complex scalar field

Suppose we wish to obtain the energy density of the free complex scalar field $\varphi$ as a Legendre transform of the corresponding action. From Wikipedia, writing the action of a free complex scalar ...
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1answer
360 views

Variation of the Action under infinitesimal arbitrary transformations and Noether's Theorem

Let's consider an arbitrary infinitesimal transformation of the fields and their coordinates : $$x'^{\mu}= x^{\mu} + \delta x^{\mu} = x^{\mu} + \frac{\delta x^{\mu}}{\delta{\omega}^a}{\omega}^a\tag{1}...
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0answers
181 views

Transformation rule under translation of the Fourier transform of a scalar field

Let $\phi(x)$ be a real classical scalar field which satisfies the Klein-Gordon equation; without loss of generality it can be written as: $$ \phi(x) = \int \frac{d\vec k}{(2\pi)^{3/2} 2 \omega_k} \...
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1answer
64 views

Significance of the order of derivatives in an action

What is the significance of having higher order derivatives in an action describing some system? For example, suppose I have the following two actions \begin{align} S_1&\propto \int \text{d}^4x \...
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2answers
252 views

Noether's theorem with infinite parameters

I'm trying to understand something regarding Noether's theorem - and with the given situation, my question isn't that much of a question, I'm rather just seeking confirmation whether I'm thinking ...
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2answers
726 views

Are fixed points of RG evolution really scale-invariant?

It is often stated that points in the space of quantum field theories for which all parameters are invariant under renormalisation – that is to say, fixed points of the RG evolution – are scale-...
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2answers
1k views

Invariance of Action vs. Lagrangian in Noether's theorem?

I have recently started studying classical field theory. Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. But I find ...
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1answer
133 views

Are there linear operators and vector spaces in classical physics?

Linear operators and vector spaces form the backbone of the operator formulation of quantum mechanics. I want to ask are there operators in classical physics too? Are these operators defined on some ...
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0answers
267 views

Antisymmetry of Infinitesimal Lorentz Transformations and Noether Currents

I am going through the derivation for the six Noether currents associated with Lorentz transformations; however, I do not understand the final step in the derivation. One ends up with the following ...
5
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1answer
239 views

Most general Lagrangian in CFT in 0+1D

My question is about $CFT_1$. Page 18 of this says that $$L={\frac{\overset{.}{Q}^2}{2} - \frac{g}{2Q^2}}\tag{1.11}$$ is the most general Lagrangian that preserves time translation and scale ...
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2answers
222 views

State amplitude and field operator covariance in QFT

I'm studying QFT on Bogoliubov-Shirkov's "Introduction to the theory of quantized fields" (3d edition). In $§9.3$ they discuss transformation properties of quantum states and operators in QFT. Given ...
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1answer
162 views

Confusion with Index Notation for Fully Contracted Electromagnetic Field Tensor

I am going through the derivation of Coulomb's law from classical field theory and the very first step that is performed is re-expressing the Lagrangian in terms of derivatives of the vector potential,...
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0answers
199 views

Classical field theory: marriage of differential geometry to functional analysis?

I have always wondered (and thoroughly - but perhaps not enough - searched for literature) how the purely geometrical formulation of classical field theory (take for example a hardcore text on this ...
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0answers
157 views

A question about gauge transformations in classical electrodynamics and gauge transformations

Consider the free Maxwell's equations without being coupled to matter fields, for simplicity. The equations are invariant under $$A_\mu(x)\to A_\mu(x)+\partial_\mu\chi(x)\tag{1}$$ where $\chi(x)$ is ...
5
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1answer
834 views

Why the variation of a surface term is zero?

My original question is like: Why are the Euler-Lagrange equations invariant if we add a surface term to the action? And there is an answer by Javier: https://physics.stackexchange.com/a/...
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2answers
159 views

Why is the coupling constant $g$ relevant in the quantum theory?

I'm reading the chapter Uses of Instantons from Sidney Coleman's Aspects of Symmetry. In the introduction to the chapter, page 265, he considered the traditional $\phi^4$theory described by the ...
2
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1answer
216 views

Going from classical field to quantum field operator?

Let us say I have a classical field theory, with a field $\phi(\vec r,t)$ which satisfies the relevant Euler-Lagrange equation for the Lagrangian $\mathscr{L}$. Is the general procedure (i.e. one that ...
3
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1answer
194 views

Scalar Field Theory for Gravity

While reading the book Gravitation Foundation and Frontiers by Padmanabhan, I came across the Lagrangian for a scalar theory of gravity. But the coupling term consist of trace of the Energy Momentum ...
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1answer
78 views

How have we derived the definition of gravitational potential energy? [duplicate]

The gravitational potential energy of an object at a point above the ground is defined as the work done is raising it from the ground to that point against the gravity. How has this definition been ...
4
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1answer
589 views

Least action principle and gauge invariance

In classical electrodynamics using variational "least" action principle we arrive at equation like $$ -\int d^4x \left(j^\nu - \frac{\partial F^{\mu\nu}}{\partial x^\mu}\right) \delta A_\nu = 0 $$ ...
4
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1answer
323 views

In Hamiltonian field theory, do spatial derivatives commute with Poisson brackets?

I'm going back over some of my old notes for a current project, and I'm trying to figure out if I made an error or if I once knew something that I've now forgotten. Consider a local field theory ...
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1answer
108 views

Radiated power of a dipole in a medium

I would like to calculate the power of an electric radiating dipole inside a medium with $\varepsilon$ and $\mu$ and compare it to the power of a free dipole in vacuum. How do I go about it? My idea ...
4
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1answer
1k views

Where this polarization vector is coming from?

When dealing with vector fields in his QFT book, Schwartz writes the classical field in terms of a basis which I don't know how he is getting. He first introduces the Proca Lagrangian $$\mathcal{L}=-\...
5
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1answer
377 views

Defining a classical field corresponding to a quantum field

Why is the expectation value of the quantum field in the vacuum state $$\phi_c(x)=\langle0|\hat{\phi}(x)|0\rangle_J=\frac{\delta W}{\delta J}$$ referred to as the classical field? Why not the ...
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1answer
2k views

What is a gauge theory?

Please note that I just read about 20 forum discussions, none of which answered my question. This question is related to my earlier question Is spacetime symmetry a gauge symmetry?. I am looking for ...
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0answers
365 views

Global and local symmetries in Noether's theorem. And also Stress-Energy tensors

Noether's theorem for fields is usually given as follows: Given a field theory with action $S=\int\mathcal{L}(\phi,\partial\phi)d^4x$, and given a one-parameter variation of the fields $\phi_\epsilon$...
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1answer
432 views

Same equations of motion when the Lagrangian change by 4-divergence

In this topic Why my 4-divergence term added to a Lagrangian modifies the equation of motion? I understood that if I have : $$\mathcal{L}_1(\phi,\partial \phi)=\mathcal{L}_2(\phi,\partial \phi) + \...
2
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2answers
257 views

General question about the proof of conserved current. Example with Klein Gordon equation under $\phi \rightarrow e^{i \alpha} \phi$

There is something I never understood about Noether currents and I really want to catch it. I will ask my question with an example but it is in fact a very general question. We take the Klein Gordon ...
5
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1answer
876 views

Big puzzle about Noether's theorem of coordinate transformation (spacetime symmetry)

For Noether theorem with only internal symmetry, I've found there has been a very clear proof. But I still struggle with the proof of coordinate transformation. Because there are so many different ...

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