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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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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1answer
45 views

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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3answers
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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
48 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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31 views

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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41 views

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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208 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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98 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
109 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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42 views

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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49 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
35 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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1answer
45 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
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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
52 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
64 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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65 views

How to numerically solve this scalar field equation?

Using Mathematica, I would like to numerically solve the following partial derivatives equation : $$\tag{1} \frac{\partial^2 \Phi}{\partial T^2} - \frac{\partial^2 \Phi}{\partial X^2}- \frac{\...
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2answers
156 views

Analytical solutions to scalar field equation with Higgs-like potential

I'm considering the classical field theory of a real scalar field $\phi$ with the Higgs "Mexican hat" potential: $$\tag{1} \mathcal{V}(\phi) = \frac{\lambda}{4} (\phi^2 - \phi_0^2)^2, $$ ...
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42 views

Higher Order Equation of Motions from Lagrangian / Action

I'd like to derive the equations of motion for a scalar field in a FLRW universe, where the metric, as well as the field, are perturbed. I think I should get scalar as well as tensorial expressions. ...
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1answer
73 views

Is spin associated with rotations or boosts?

EDIT: It seems that I made an error and it was $S^{ij}$ that was used after all. I will not delete the question though because even though it is erroneous, the answer given below is rather insightful. ...
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38 views

How to Derive the Poisson Brackets for Continuous Systems

Part of my research involves studying the classical mechanics of Electromagnetic fields. I have followed the procedure for determining the Lagrangian and Hamiltonian formulations for continuous ...
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1answer
97 views

Scalar field energy density

Considering a classical scalar field theory, I can find the canonical energy momentum tensor and if I calculate the $00$ component I get: $$T^{00}= \frac{1}{2} \dot \phi^2 + \frac{1}{2} (\partial_i) \...
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95 views

Noether charges as generators of symmetry transformation in classical Hamiltonian field theories

I would like to prove that Noether charge $Q$ is a generator of the same symmetry as the one that due to the Noether's theorem led to the current $j^\mu$ and charge $Q$ in classical field theory (I ...
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1answer
80 views

Field theory Euler-Lagrange problem term

Consider the following Lagrangian (density) $$ \mathcal{L} = (\mu/2) (\partial_t q)^2 - (Y/2) (\partial_x q)^2 -\alpha(\partial_x{}^2 q)^2 $$ $\mu, Y, \alpha, q$ are respectively mass/unit length, ...
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1answer
69 views

Transformations in classical field theory and configuration space

When transforming a field in classical field theory the transformation of the four-gradient of this field follows automatically. At least this is what i have learned in my lectures. This circumstance ...
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3answers
90 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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41 views

Condition for second order Lagrangian to produce second order equation of motion

Related to a research project I am currently doing I am studying Einstein-type Lagrangians, i.e. field Lagrangians which are second order, but whose Euler-Lagrange equations are also second order ...
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54 views

Algebra of Noether's charges and algebra of symmetry transformations

I'm trying to understand the connection of algebra of transformations under a commutator and algebra of Noether's charges under Poisson bracket. I have a problem that results I infer from theoretical ...
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2answers
28 views

What determines picking a gaussian surface?

I feel like I am missing something simple here, but how exactly are gaussian surface determined? Looking at the case of a charge outside a sphere, why don't we pick a gaussian surface not including ...
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30 views

Lagrangian density under infinitesimal transformation

Consider the following Lagrangian density for two real scalar fields: $$\mathcal{L}=\frac{1}{2}\sum\limits^{2}_{i=1}(\partial^{\mu}\phi_i)(\partial_{\mu}\phi_i)-\frac{1}{2}\sum\limits^{2}_{i=1}m_i^2\...
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1answer
115 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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1answer
165 views

Why does the subclassification of fields under parity require the quantum theory?

The fields of relativistic field theory (scalars, vectors, tensors, and spinors) are all defined via their transformation properties under the restricted Lorentz group (which excludes discrete ...
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2answers
69 views

How do you actually use fields?

Note: I'm probably using the wrong letters/notation here. I apologize. I use $\omega$ to represent an object, and $\mathcal{U}$ is the universe. I'm not sure how else to do it. $m(\omega)$ and $x(\...
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“Energy” of Poisson's equation when viewed as a dynamical system

I was recently exposed to an interesting way to solve the 1-d Poisson equation in electrostatics $$ \epsilon_0\frac{d^2\phi}{dx^2} = -\rho $$ for potential $\phi$ and charge density $\rho$. If the ...
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3answers
107 views

Solving the Euler-Lagrange equations for a complex scalar field in which the time derivatives and gradient are separate

This is found at the bottom of page 9 of David Tong's QFT lectures. The Euler-Lagrange equations for the complex scalar field: $$\mathcal L=\frac{i}{2}(\psi^*\dot\psi-\dot{\psi^*}\psi)-\nabla\psi^*\...
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3answers
95 views

Probability in classical physics

I have read lots of thing on probability in QM and the different ways of intending it. Now, I am wondering how physicists intend probability in classical physics. To be more specific, I have read some ...
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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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54 views

Why should fields in AdS spacetime vanish at infinity, but not in Minkowski spacetime?

I was watching the following lectures by Prof. Ashoke Sen. Between 39:00 and 56:00, he was solving the equation of classical field in the AdS global coordinates, and says that the values of $\omega$ ...
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What does decoupling mean when studying fields?

I just started studying field theory and general relativity, and when reading paper titles I often see the word decoupling coming up. My intuition is that there must be a cross term involving some ...
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59 views

Euler-Lagrange derivative for any Lagrangian density which is a function of Metric and its first and second derivatives

I know formalism of Euler_Lagrange derivatives: which $L$ here is a Lagrangian density. now I wonder how can I make such a form as below for any Lagrangian density which is a function of metric and ...
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How can we prove that a non-linear equation of motion for a classical scalar field satisfies causality?

Let $\phi$ be a classical scalar field in $1+D$-dimensional spacetime with coordinates $(t,\vec x)$, and consder the equation of motion $$ \newcommand{\pl}{\partial} (\pl_t^2-\nabla^2)\phi+m^2\phi+ g\...
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1answer
45 views

Functional variation problem in Classical Field Theory (Non Relativistic) [closed]

An exercise of my Homework sheet make a statement about rotational variation on a scalar field $\phi(x)$:\ "Consider a scalar field $\phi(x,t)$ in a lagrangian $\mathcal{L}(\phi, \partial_t \phi, ...
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1answer
49 views

Adjoint of a Four gradient of a scalar field

Is the term $(\partial^{\mu}\phi)^{\dagger}$ same as $\partial^{\mu}\phi^{\dagger}$ for any complex scalar field $\phi$?
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1answer
64 views

Macroscopic Limit of QED

How does one go about rigorously deriving special relativistic dynamics (both relativistic mechanics and electrodynamics) from quantum electrodynamics? Is this even possible from the mathematical ...
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1answer
33 views

Fields that lend themselves to variational principles? [duplicate]

In physics, we often describe the dynamic properties of fields using variational principles like defining an action or a Lagrangian. A field however is simply some function of space $\phi(x)$ so I ...

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