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.
484
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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 ...
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2
answers
115
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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 ...
- 544
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1
answer
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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 ...
- 13
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0
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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 ...
- 7,023
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1
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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 ...
0
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0
answers
23
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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 ...
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1
answer
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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 ...
- 13
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1
answer
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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 ...
1
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0
answers
37
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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
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1
answer
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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 ...
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1
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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:
$$ \frac{dp}{...
- 157
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0
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66
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Under what representation of the Lorentz group do scalar $\textit{fields}$ transform?
I know that if I am sitting in a spacetime $M$ at point $p$, vectors live in the tangent space $T_pM$, and tensors in the tensor product space etc. If I want to consider general tensor fields, I ...
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1
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If the solution of a field vanishes on-shell does it mean anything particular?
Let us consider an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. The solution of the field $c$ is given by the expression $f(a,b)$. On taking into account the relations ...
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Why only $\phi=\pm1$ are considered "vacuum states" in the Klein-Gordon model with $\phi^4$ potential, and not $\phi=0$?
I am reading "Kink Moduli Spaces — Collective Coordinates Reconsidered," by Manton, Oleś, Romańczukiewicz, and Wereszczyński (arXiv version), where they consider the Klein-Gordon equation,
$$...
- 113
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1
answer
58
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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\}^{+\...
- 1,623
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0
answers
53
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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
$$S = \...
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1
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Classical fermions, where are they?
Context:
Studying the path integral formulation of QFT I stumbled upon a fairly simple statement: when doing loop expansions of a partition function:
$$Z[\eta ; \bar{\eta}] = \int [d\psi][d\bar{\psi}]...
- 1,104
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votes
0
answers
60
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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
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3
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121
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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 $\mathcal{L}_M$...
- 557
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1
answer
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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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0
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65
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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 ...
3
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1
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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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3
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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 ...
- 43
2
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0
answers
66
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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 ...
- 568
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0
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37
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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 ...
- 1,345
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3
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110
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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 $$\delta\phi_a(x)\equiv {\phi^{\prime}}_a(x^{\prime})-\phi_a(x)$$ ...
- 11.1k
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87
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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 $$S=\int \mathscr{...
- 507
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0
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How does energy momentum tensor generate transformations?
I am reading https://arxiv.org/abs/hep-th/0008096 on p.20.
Since I am not very familiar with the general canonical formalism, I have some trouble interpreting the following statements:
Consider the ...
- 1,059
2
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0
answers
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Conditions for two field theories to be equivalent at one-loop level
I am asking a much shorter, more generalized version of this question in order to gather as much information as possible.
I have two field theories $A$ and $B$ which are equal at classical level: ...
- 338
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0
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43
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Ghost detection at the level of equations of motion
My question is about how to detect ghostly degrees of freedom at the level of equations of motion. It is not clear for me how does this work. Let me explain with an example:
Consider the following ...
- 567
3
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1
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Barut-Zanghi (BZ) Lagrangian derivation
I'm trying to derive the BZ Lagrangian (density) from the Dirac Hamiltonian density and some questions popped up.
BZ Lagrangian is
$$\mathcal{L} = \frac{i}{2}(\dot{\bar{\psi}}\psi - \bar{\psi}\dot{\...
- 133
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1
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102
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Current of conformal transformation
Suppose we have a theory with conformal invariance that has been extended to a diffeomorphism invariant theory in a way that the resulting energy-momentum tensor is traceless on-shell (which can ...
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0
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Lagrange multipliers not obeying classical EoMs
I will ask my question about Lagrange multipliers by using an example in string theory, as this question was inspired by my string theory course, but it applies to every theory with Lagrange ...
- 3,833
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0
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How to canonicalize a coupled scalar kinetic term?
I am working with a classical action in curved space-time that looks something like:
\begin{equation}
S = \int d^4x \frac{1}{16G\pi}\sqrt{-g} \left[R - \frac{K_\Phi}{\Phi^2} \partial_\mu \Phi \partial^...
2
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0
answers
115
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The Poisson bracket for a Dirac field on an Eddington-Finkelstein background metric
I am interested in deriving the anti-commutation relations of a Dirac field for a $(1+1)$D Eddington-Finkelstein metric given by
$$ ds^2 = f(r) dt^2 - 2 dt dr, \quad f(r) = 1 - \frac{r_s}{r}.$$
where $...
- 3,766
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1
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Proof that the axial current is conserved in classical QED
I am trying to use the Lagrangian of QED (without kinetic terms for photons) to prove that the axial current of QED satisfies $\partial_\mu j^\mu_5 = 2im\bar\psi\gamma^5\psi,$ where $j^\mu_5 = \bar\...
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1
answer
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Relationship between Lagrangians describing a particle interacting with a scalar field
In Susskind's Particles and Fields lecture, he considered the Lagrangian obtained by considering a particle and the effects of a scalar field $\phi(t, x)$ with coupling constant $g$ on the particle (...
4
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1
answer
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Is there a classical description of Hawking Radiation?
Before quantum theory we knew accelerating electrons radiated electric fields. This is modelled classically (even though we know it is a quantum process emitting photons)
Similarly is there a possible ...
user84158
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1
answer
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Compatibility of renormalisation with the quantum-classical correspondence principle
We know that Quantum Theories obey the Heisenberg equations of the motion, taking the expected values of which gives us the classical equations.
Also, We replace the mass and coupling parameters of a ...
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Schroedinger equation for wave functional (QFT)
As far as I'm aware you can solve for the wave functional $\Psi[\phi]$ of a field using the Schrodinger equation $$i\hbar\frac{\partial \Psi}{\partial t}=H\Psi.$$
Should $H$ here be the Hamiltonian, ...
- 107
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0
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Energy Flux Through Horizon
In Kerr spacetime, given the energy-momentum tensor $T^{ab}$ of a field, what is the energy flux (as measured at infinity)
$$
\frac{d^2E}{dt d\Omega}
$$
i.e., the amount of energy passing through the ...
- 21
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0
answers
52
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Rewriting Maxwell Lagrangian [duplicate]
I'm having some problems with rewriting the Maxwell Lagrangian. The text states, \begin{align}\mathcal{L}&=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-A_\mu J^\mu \\
&= -\dfrac{1}{2}(\partial_\mu A_\nu)^...
- 11
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0
answers
111
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Momentum conservation in classical electrodynamics
The partial of tensor is equal
$$\partial_{\mu}T^{\mu j} = \frac{\partial}{\partial t}T^{0j} + \nabla_{j} T^{ij} = -(\rho \vec{E} + \vec{j}\; \times \vec{B})^{j}$$
where does it come from?
In the ...
user256423
0
votes
1
answer
152
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Poisson brackets for a field theory
I'm performing a calculation involving Dirac constraints theory, and I need to calculate the Poisson brackets between constraints and the total Hamiltonian. The starting theory is described by a ...
- 150
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0
answers
62
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Integrating infinitesimal transformations in CFT
Consider a conformal transformation $z\mapsto f(z)$ of the complex plane described infinitesimally by the vector field $X\partial+\bar{X}\bar{\partial}$. I was wondering how one can, starting from the ...
- 3,795
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0
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Primary Fields under SCTs
Primary fields transform under global conformal transformations like
$$\tilde{\phi}(\tilde{x})=\Omega^{-\Delta}D(R)\phi(x),$$
where $R$ is the rotation associated to the conformal transformation. My ...
- 3,795
1
vote
0
answers
85
views
Lagrangian formulation of Maxwell's equations with magnetic monopole
If we set $\nabla \cdot {\bf B}=\rho_m$ where pm is the density of magnetic charges we lose the ability to write ${\bf B}=\nabla \times{\bf A}$ . Can we get a new Lagrangian that leads to the new ...
2
votes
1
answer
121
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What do we mean by "Degrees of Freedom" when we Talk about the electromagentic field?
For point-like particles, the term "degree of freedom" seems rather clear: It's the number of independent coordinate functions $q_i(t)$ that we need to specify to completely describe the ...
- 6,515
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votes
2
answers
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It is possible to create a relativistic theory of gravity more simple than General Relativity via Jefimenko's equations? [closed]
I've came across the Jefimenko's equations, which are the general solutions of Maxwell's Equations and are compatible with Special Relativity.
They are formulated in terms of the retarded potential:
$$...
- 1,213
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votes
0
answers
57
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Four-momentum elements for real scalar field in momentum representation
I'm trying to derive spatial elements of the four-momentum for the real scalar field. I'm working in momentum representation given by Fourier transform
$$\varphi^{\pm}(x)=\frac{1}{(2\pi)^{\frac{3}{2}}}...
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