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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51 views

Relation between quantum and classical mass gaps

We say a QFT has a mass gap if the spectrum of the mass operator $M:=\sqrt{P_\mu P^\mu}$ is bounded below by some $\Delta >0$. I will define a $\textit{classical}$ field theory to have a mass gap ...
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601 views

Hamiltonian Field Theory in Peskin & Schroeder

In Section 2.2 of their QFT textbook, Peskin & Schroeder introduce the Lagrangian and Hamiltonian field theories of a classical scalar field. While defining the action $S[\phi]$ and deriving the ...
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Bohr-Kramers-Slater (BKS) theory and energy conservation only on statistically basis [migrated]

I was reading Wikipedia article on Bohr-Kramers-Slater (BKS) theory, https://en.wikipedia.org/wiki/BKS_theory. I encountered two interesting points and need your help to understand the reasons behind ...
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Confused with 4-vector notation and 4-derivative

I have a lot of trouble finding out what the rules are for doing algebra and calculus with 4-vectors. This example shall illustrate one of my problems: The Lagrangian for a real scalar field is $$\...
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1answer
90 views

Why is Hamilton's principle (or principle of least action) still valid in a relativistic field theory?

I am struggling to understand why the principle of least action which is derived in classical mechanics from d'Alembert's principle continues to be valid in a regime that treats a relativistic field. ...
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Variations in a vector field [duplicate]

When we derive Maxwell's equations from the Lagrangian that contains the Maxwell field tensor $F_{ij}$, I ran into a small confusion. With the Lagrangian being $L = F^{ij}F_{ij}$, taking variations of ...
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References for Hamiltonian field theory and Dirac Brackets [duplicate]

I'm looking for complete and detailed references on constrained Hamiltonian systems and Dirac brackets. While my main interest is electrodynamics, I would prefer a complete exposition of the theory ...
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Understanding classical massive real scalar $\phi^4$ Callan-Symanzik equation

Considering classical massive real scalar field theory by the action: $S[\phi] = \frac{1}{2}\int d^4x[\partial_\mu\phi\partial^\mu\phi-m^2\phi^2-\frac{g}{12}\phi^4] $ we assume the theory is ...
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357 views

Uniqueness of the definition of Noether current

On page 28 of Pierre Ramond Field theory - A modern primer the following is written: "we remark that a conserved current does not have a unique definition since we can always add to it the four-...
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3answers
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Confusion about vector and scalar field transformation laws

I'm a little bit confused about the transformations for scalars fields and vector fields in classical field theory. I've learned that a scalar field is a smooth function $$\phi : M \longrightarrow \...
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Proof that free scalar field is conformally invariant

So, under conformal transformations $$x\mapsto x'\\ \phi\mapsto\phi'(x')=\Omega^{(2-D)/2}\phi(x),$$ where $$\eta_{\mu\nu}\frac{\partial x^\mu}{\partial x^{'\alpha}}\frac{\partial x^\nu}{\partial x^{'\...
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Scalar field Hamiltonian $H = 0$ from parameterization independence

This question is related (but not similar) to this old one of mine: How to derive the two Friedmann-Lemaître equations from a Lagrangian? Consider the Lagrangian of an isotropic-homogeneous ...
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How show that for a free EM field, the energy of each mode is conserved in time?

For a free electromagnetic field, since different modes do not talk to each other, one would expect that the energy stored in each mode is conserved (or constant) in time. But the energy stored in a ...
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Nonlinear superposition and self-interaction in classical field theory [duplicate]

I am learning QFT (in a path integral formalism) and one thing I'm struggling with is that self-interaction is supposed to be a quantum phenomenon, not apparent in classical non-linear field theory. I ...
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Coordinate invariance in Physics

Let us consider a classical field theory on flat background spacetime. The action is $$S[\Phi] = \int d^nx \mathcal{L}(\Phi,\partial_\mu\Phi).$$ Why shouldn't this action be independent of the chosen ...
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Is classical Kaluza Klein theory stable or not?

Set Up In the original classical Kaluza Klein theory, you have a $d+1$ dimensional manifold where one space dimension is a circle $S^1$. In the "low energy limit," none of the metric ...
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Notations and Variations in the proof for Noether's theorem for fields

In the proof for Noether's theorem, as given in D.Gross's notes, there are two kind of variations used. $x'^{\mu}=x^{\mu}+X^{\mu}_\alpha(x)\omega^\alpha$ $\phi'_i(x')=\phi_i(x)+\Psi_{i\alpha}(x)\...
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Does this Lagrangian density represent anything "real"?

So this lagrangian was used as an example for deriving the equations of motion using the Euler-Lagrange equations in our lecture notes. $$ L (\phi )=-\phi (x,t)^{2}+m\left(\frac{\partial \phi }{\...
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Building Lagrangians for Classical Field Theory

I've been studying quantum mechanics and classical field theory for quite a while now. However, I still struggle with the idea of building scalars from vectors and tensors for the Lagrangian density. ...
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Lagrange densities for spin in different dimensions

I have been looking at the Lagrange equations for spin 0 Klein-Gordon, 1 Proca, and 1/2 Dirac (are there any others?). It seems that from these equations we can find out all of the particles in the ...
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1answer
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How to tell if a system has a direct or reverse energy cascade?

We know, in 3D turbulence one observes a direct energy cascade, where the energy flows from the large scales to small scales (see wiki 1,1), usually attributed to vortex stretching. We also know that ...
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Chern-Simons Lagrangian and gauge-fixing

Main question: Consider (2+1)D Chern-Simons action $$S = \int dt d^2\mathbf r \frac{k}{4\pi} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda.$$ Assuming the Coulomb gauge $\nabla\cdot \mathbf a ...
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1answer
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Is electric field a property of a charge or is it a spatial distribution of electrostatic force?

I am a bit confused about what gives rise to an electric field. I can look at it in two different ways as follows. When two charges are separated by a distance, the electric field gives the ...
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Gauge symmetry of massive vector field

Consider a real massive vector field with lagrangian density $$\begin{align}\mathcal{L}&=-\frac{1}{4}(\partial_\mu A_\nu-\partial_\nu A_\mu)(\partial^\mu A^\nu-\partial^\nu A^\mu)+\frac{1}{2}m^2 A^...
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Carroll: Energy-momentum tensor for a scalar field theory

In Carroll's Introduction to General Relativity: Spacetime and Geometry, there is a section titled Classical Field Theory in chapter 1. There, he mentions that: "The action leads via a direct ...
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Problems in deriving the Belinfante-Rosenfeld Energy momemtum tensor through variation

I am currently following Michael Stone's lecture notes (http://people.physics.illinois.edu/stone/torsion_review.pdf) on deriving the Belinfante-Rosenfeld Energy Momentum tensor in a variational ...
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1answer
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Landau Vol. 2: On four-vectors

In Chapter 1, section 6 of The Classical Theory of Fields by Landau and Liftshitz, I didn't understand the following: "Under purely spatial rotations (i.e. transformations not affecting the time ...
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Does a traceless energy-momentum tensor really imply a massless field?

I read a paper where there was written that "a Traceless Energy-momentum Tensor implies a massless field", so I did a bit of calculations but I seems not really true, is it true? So now I'm ...
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$T^{00} ≠ $ Hamiltonian Density?

Check page 48-49 of http://walterpfeifer.ch/qft/QFT5.pdf?. It is apparent that the Hamiltonian density of the Maxwell Field is not positive definite when expressed in terms of the Four-vector ...
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Validity of Euler-Lagrange Equation in Quantum Theory [duplicate]

Lagrangian density for a single-spin 0-real-bosonic field ($\phi$) is given by, $$\mathcal{L}=-\frac{1}{2}\partial_\mu \phi \partial^\mu \phi-\frac{m^2}{2}\phi^2$$ Now if we formulate the Euler ...
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What function space fits best in describing fields in field theory?

Question: Is there a proper mathematical space in which (at least most) of the (classical) fields treated by physicists belong? In other words, when a textbook says "Let $\phi_{i}(x)$ be a field&...
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Why light was said to be faster than electromagnetic changes in ether?

According to Wikipedia on the historical development of the Lorentz ether theory: ... Contrary to Clausius, who accepted that the electrons operate by actions at a distance, the electromagnetic field ...
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Criteria to Define a (Classical) Topological Field Lagrangian? + Conjecture

I have a question concerning topological field theories. I'd rather keep the discussion at the classical level, so as to concentrate on the feature of topological evolution, which is what interests me ...
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1answer
73 views

Is any continuous transformation a symmetry of action?

Consider a continuous transformation $\phi \rightarrow \phi+ \delta\phi$, where $\phi$ is a field operator and $\delta \phi$ is a infinitesmal change. If such continuous transformation is applied to ...
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Hamilton's principle for fields

According to Goldstein, Hamilton's principle can be summerized as follows: The motion of the system from time $t_{1}$ to time $t_{2}$ is such that the line integral (called the action or the action ...
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How to evaluate the equal time Poisson bracket $\{ \phi(x), \vec{\nabla}_y\phi(y) \cdot \vec{\nabla}_y\phi(y)\}$?

I learned that for a classical scalar field theory in 4 dimensions, we can use the equal time Poisson brackets $$\{ \phi(x), \phi(y) \}_{x_0=y_0}=0$$ $$\{ \pi(x),\pi(y)\}_{x_0=y_0} =0$$ $$\{\phi(x),\...
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1answer
258 views

Understanding Hamilton's equations in classical field theory in a rigorous way

So, I'm in a quest of understanding classical field theory on my own, and I'm interested in its rigorous construction. Here's the link for a previous post of mine on mathoverflow. The interesting ...
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2answers
104 views

Energy Positivity of Classical QED Field Theory in Presence of Sources

It's well known that the classical electromagnetic field has positive definite energy, simply because: $$\mathcal{H}=\frac{1}{2}\epsilon_0\vec{E}^2+\frac{1}{2\mu_0}\vec{B}^2.$$ However, this result ...
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1answer
131 views

Spontaneous symmetry breaking, massless bosons and the equations of motion

I am currently studying spontaneous symmetry breaking, and I don't entirely understand the implications of what we are doing at certain places. Consider the standard complex scalar field with the $\...
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Classical field theory correlation function

I'm studying QFT from Schwartz's "Quantum Field Theory and the Standard Model", and in chapter 7 he derives the Schwinger-Dyson equations for the correlation functions in a scalar field ...
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54 views

Equations of motion in classical field theory

To my current understanding, the general equations of motion of charged particles are given by an action which is created from summing the field action, the particle action, and the action which ...
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1answer
37 views

Is there an equivalent term in the lagrangian of classical newtonian physics to the corresponding $\frac{1}{2}m^2\phi^2$ term of QFT?

In classical field theory and quantum field theory, the lagrangian could have a mass term in the form: $\frac{1}{2}m^2\phi^2$. Is there an equivalent term in the lagrangian of classical newtonian ...
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52 views

Additional term in the Noether current

I've seen this same question before Why is there an extra term in definition of Noether current for spacetime translations? but I didn't understand the answer that was given so I would like to ask ...
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1answer
82 views

Gravitational attraction/repulsion of cosmic strings and domain walls

It is well known that straight static cosmic strings don't produce any gravitational effects on test-particles, and that static flat domain walls are repulsive. This can be seen from the linearized ...
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2answers
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What happens if we expand out fields in terms of different functions?

When we "expand" our classical fields, for example the Dirac field, in the standard way which we later go on to "quantise": $$\psi(x,t)=\int d^3\tilde k \sum_{a=1,2}\left(b_a(k)u^a(...
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2answers
61 views

Example of a classical action changing by a nonzero boundary term under a continuous transformation

Is there an example of a continuous transformation in classical field theory under which the classical action changes by a nonzero boundary term? I'd prefer an example from field theory in flat ...
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1answer
78 views

What are the boundary conditions for an ideal fluid in a frictionless pool?

Suppose you want to numerically solve the classical 2D waves equation for an ideal incompressible fluid in a square pool. The pool's walls are frictionless, so the fluid could vertically move freely ...
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Is there a kink solution in general relativity?

In the special case of the $\phi^4$ scalar field theory in special relativity, a nice "kink" solution is very well known: $$\tag{1} \phi(z) = v \tanh \Bigl(\sqrt{\frac{\lambda}{2}} \: v \, z ...
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Is there any relativistic constraint on the rate of change of a scalar field?

Consider a scalar field $\phi(t, x, y, z)$ obeying the waves equation with an Higgs-like potential (the "mexican hat"): $$\tag{1} \mathcal{V}(\phi) = \frac{\lambda}{4} (\phi^2 - \phi_0^2)^2, ...
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Solving the action without solving for fields?

This is more of a philosophical question for a problem I'm trying to solve: are there examples in physics for which we evaluate the on-shell action without solving for the fields in the action? The ...

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