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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28 views

Can convergent perturbation series be incorrect for an action linear in the perturbation?

Non-perturbative effects are common in mathematics. For example, consider the function $$f(g) = e^{-1/g}+ g + \frac{1}{10} g^2$$ and suppose this function is the answer to some math problem. ...
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1answer
28 views

Consistency of substitution of a canonical variable from EoM back into (momentum-less) action

I was reading this answer, where the issue of substituting equations of motion (eoms) into the action is addressed. I am fine with the basic idea that the action principle is destroyed when the eoms ...
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2answers
50 views

For free fields, is there a one-to-one correspondence between probability distribution of classical field configurations, and states?

If I'm given the field operator of free fields (for example $\phi(x)$) as a function of space time, and a state (for example $\langle 0 | $, I can calculate the expectation value for every point in ...
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2answers
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Why don't people use Hamilton's equations for a relativistic free charged particle?

A charged relativistic free particle has the Hamiltonian in general: $$ \mathcal{H} = \sqrt{{\bf p}^2c^2+m^2c^4}.$$ I read somewhere that says, it is possible to go further and say that the EoM are ...
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0answers
69 views

Can someone explain the steps? [duplicate]

Can anybody expand the equation .What is ω and is i power of ω in equation 2.2. And how $f_λ(r_1,r_2,...,r_n,t)=0$;λ=1,2,...,Λ gives the last equation and what $∇_k$ means?
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51 views

When does a Lagrangian exist for arbitrary equations of motion? [duplicate]

Let's say I have some equations of motion for an arbitrary system, i.e. some implicitly or explicitly defined ODE involving $q = (q_1, q_2, q_3, \dots)$ and $\dot q = (\dot q_1, \dot q_2, \dot q_3, \...
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0answers
30 views

Confusion regarding a step in Schwartz's book (equation 3.44) [duplicate]

I have just started learning QFT, and am working my way through Schwartz's book. I am not able to justify to myself something he does. I have even started suspecting if this is a typo. I have attached ...
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3answers
670 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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0answers
60 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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1answer
92 views

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
95 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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0answers
21 views

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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3answers
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Klein gordon field and positive/negative energy solutions

In my course we calculated the Klein-Gordon field: $$ \phi(x)= \int \frac{d^3k}{(2 \pi)^3}\frac{1}{2k_0} ~ \left[a(\vec{k})e^{-ik.x}+b^*(\vec{k})e^{i kx}\right]$$ We said that the part $ a(\vec{k})e^{-...
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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 ...
1
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1answer
228 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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0answers
51 views

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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2answers
366 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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1answer
457 views

Implication of breakdown of scale invariance for problems with intrinsic length or time scales?

According to Wikipedia article on scale invarince, the equations for electric (and magnetic) fields : $$\nabla^2\vec{E}=\frac{1}{c^2}\frac{\partial^2\vec{E}}{\partial t^2}\hspace{0.3cm}\text{and}\...
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3answers
96 views

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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1answer
194 views

Understanding the Functional Poisson Bracket

In classical field theory (for a single field $\psi$) the dynamical variables are defined to be functions of the fields $\psi$, $\pi$, $\partial_{x_{i}}\psi$ and maybe $\mathbf{r}$, where $\pi$ is the ...
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3answers
173 views

Extra term when calculating variation in Lagrangian density under infinitesimal Lorentz transform

Consider an (active) infinitesimal Lorentz transformation: $$ x^\mu \rightarrow x^\mu + {\omega^\mu}_\nu x^\nu, $$ so that any scalar field is transformed as $$ \phi(x) \rightarrow \phi'(x) = \phi(x) -...
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0answers
82 views

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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2answers
77 views

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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3answers
130 views

A question on the Poisson field equations in classical gravity and EM

I'm having a problem understanding why in the Poisson equation for gravitational potential, the term with the mass density has a positive sign, while for the electric potential, the charge density has ...
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2answers
69 views

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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3answers
7k views

Retarded and advanced Green's function and Feynman propagator

Is there a use of advanced Green's functions? If yes then when or in which context? In quantum field theory, why do we always use Feynman's prescription for finding the propagator and not the ...
2
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0answers
67 views

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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0answers
54 views

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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0answers
52 views

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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0answers
96 views

How can we prove that a non-linear equation of motion for a classical scalar field satisfies causality?

Let $\phi$ be a real-valued scalar field in $N$-dimensional spacetime with coordinates $(t,\vec x)$, and consder the equation of motion $$ (\partial_t^2-\nabla^2)\phi(t,\vec x)+V'\big(\phi(t,\vec x)\...
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2answers
53 views

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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3answers
447 views

What are Connections in physics?

This question arises from a personal misunderstanding about a conversation with a friend of mine. He asked me a question about the "truly nature" of spinors, i.e., he asked a question to me about what ...
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1answer
57 views

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 }{\...
2
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1answer
66 views

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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1answer
87 views

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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0answers
27 views

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 ...
2
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2answers
506 views

Reference request on condensed matter field theory including Classical Field Theory

I was hoping for a reference request for a book on basic/introductory condensed matter field theory. In addition to the usual topics I am looking for books with reference to classical physics (...
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4answers
2k views

What is the actual form of Noether current in field theory?

Let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a $D$-...
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0answers
50 views

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
46 views

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 ...
4
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1answer
154 views

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^...
2
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1answer
366 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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2answers
95 views

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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0answers
51 views

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
36 views

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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0answers
50 views

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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0answers
34 views

$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 ...
5
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1answer
197 views

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 ...
-1
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1answer
121 views

Are Yang-Mills Fields sections of associated bundles to the orthonormal frame bundle?

Let $\pi: P \to M$ be a principal bundle and $\omega$ a connection on it. Given a section $\sigma: M \to P$ we define Yang-Mills fields by $$A=\sigma^*\omega$$ Now since under Lorentz transformation ...
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0answers
47 views

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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