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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2answers
66 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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0answers
32 views

“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
38 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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2answers
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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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53 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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51 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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41 views

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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0answers
44 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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39 views

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
42 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
40 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
55 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
28 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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2answers
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Why is it problematic to regard the Lorentz group as ${\rm SO}(4, \mathbb{C})$? [duplicate]

If the four-vector $x^\mu$ is defined as $x^\mu\equiv(ict,{\bf x})$, instead of $x^\mu\equiv (ct,{\bf x})$, the Lorentz group will be the compact(?) ${\rm SO}(4, \mathbb{C})$ group. But the Lorentz ...
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3answers
76 views

Missing equations in Maxwells Equations

We have Maxwell's Equations (ignoring permittivity and permeability of free space) $$ \nabla\cdot E=\rho\;;\;\nabla\times E=-\frac{\partial B}{\partial t} $$ $$ \nabla\cdot B=0\;;\;\nabla\times B=\...
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0answers
59 views

Confusions on symmetry breaking and classical field theory

I am just reading some material about symmetry breaking and so-called effective action/potential Consider a lagrangian \begin{equation*} \mathcal{L}=\frac{1}{2}(\partial \phi)^2-\frac{1}{2}m^2\phi^2-...
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4answers
102 views

Why aren’t the electric and magnetic components of an EM wave complementary? [duplicate]

Every visualization of an electromagnetic wave is essentially some variation of this picture: In every one of these graphs, both the electric and magnetic components are shown as being sine waves ...
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23 views

Higher-order variation of an action

In general relativity, the first-order variation of a point particle action gives the geodesic equation while a second-order variation gives the geodesic deviation equation. Similarly, is there any ...
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1answer
41 views

What does it mean to have a function of a field?

There was a Leonard Susskind lecture on the Higgs Boson I watched the other day, and he talked about graphing the field where the domain was some sort of field space. The lecture is here at about 6 ...
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1answer
34 views

Closure of constraint algebra

In yang-mills theory , the constraint algebra closes to form a lie algebra. Even string theory has a constraint algebra which closes to form a lie algebra. I wish to know if there are other cases ...
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1answer
51 views

Involvement of a Heaviside Theta function inside an integral and its physical significance

I tried to study some scattering problem and I face the following integral- $$\int_{-a}^a \, dy_0\int_{-b}^b \, dz_0\int_0^{\infty}dt_0 \exp (-i\omega t_0)\frac{\delta (t_0-(t-\frac{s(t)}{c}))}{s(t)}$$...
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1answer
60 views

How to get the formula of the energy of EM waves?

I am trying to get the formula for energy of EM waves: $$W = \frac{E^2 + B^2}{2}$$ calculating the work done on a test charge by the force: $$\mathbf F = q(\mathbf E + v \times \mathbf B)$$ $\mathbf ...
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3answers
103 views

Lagrange formalism in field theory

I recently had a discussion with a friend of mine who is like me studying physics. And we might got used to a misconception about the Lagrange-Formalism in field theory. In common field theory books ...
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2answers
86 views

Gauge invariance for classical fields

I recently did some exercises in classical field theory and tried to think deeply about the gauge symmetry of the free electromagnetic field described by the Lagrangian $$ \mathcal L = -\frac 1 4 F^{\...
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1answer
59 views

Is the $U(1)_A$ axial vector current even under charge conjugation?

The axial current of a Dirac spinor is given by $j_A^\mu = \bar{\psi} \gamma^5 \gamma^\mu \psi$. In this book, in the paragraph under equation (2.18) it is stated that the current is even under charge ...
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2answers
81 views

Does the $U(1)$ vector current flip under charge conjugation?

The conserved $U(1)$ current of the Dirac Lagrangian is given by $j^\mu = \bar{\psi} \gamma^\mu \psi$, where $\bar{\psi} = \psi^\dagger \gamma^0$. As this is interpreted as electric current I would ...
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1answer
55 views

What is the definition of a symmetry of an action?

Symmetries of Lagrangians The definition of a symmetry of a theory is quite clear at the level of a Lagrangian. We say a Lagrangian $\mathcal{L}(\phi,\partial_\mu \phi)$ is symmetric under the ...
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1answer
91 views

Noether charge and equivalence class of Noether currents

Let some field theory be described by the Lagrangian density ${\cal L}$ on spacetime. Noether's first theorem asserts that given a quasisymmetry $\hat{\delta}\phi$ there is a class of currents $j^\mu$ ...
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3answers
85 views

Is the time evolution of physical fields unambiguous without fixing a gauge?

Context The origin of the question below stems from this lecture here by Raman Sundrum between $48.20$ to $51$ minutes. Let at some initial instant $t_0$, the electric and magnetic fields (E and B) ...
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0answers
25 views

Lorentz invariance of scalars in scalar field theory

To develop classical field theory the following model is considered: A cubic lattice of particles in which each particle is attached to its 4 closest neighbours by a spring that obeys Hooke's law. ...
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1answer
56 views

Does the Darwin Lagrangian neglect deviations from the Coulomb field?

The Darwin Lagrangian is said to describe the interaction between two charges to order $(v/c)^2$, and consists of a free part $$L_0 = \sum_{i = 1, 2} \frac12 m_i v_i^2 + \frac{1}{8c^2} m_i v_i^4$$ and ...
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3answers
284 views

When is Schwartz's method for “integrating out” a field valid?

In Schwartz's QFT book, heavy fields are often "integrated out" by simply solving their equations of motion formally (i.e. allowing things like $\Box^{-1}$) and plugging them back into the Lagrangian. ...
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1answer
60 views

Why a symmetry, in Lagrangian field theory, should allow a boundary term?

Following the discussion in this paper (discussion around Eq. (3) in Page 4) and these lecture notes (discussion in Section 1.2.1 in page 10) given a field theory in some spacetime $(M,g)$ described ...
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2answers
82 views

What causes the evasion of the Goldstone theorem here?

For simplicity, I'll consider perhaps the simplest possible example of a gauge theory. Consider a spontaneously broken ${\rm U(1)}$ gauge theory of a charged scalar field coupled to the ...
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1answer
52 views

Pertubational approach of a scalar field EOM

I am trying to understand a calculation in a QFT textbook. Given the equation of motion of a scalar field $h$ \begin{align} \Box h - \lambda h^2 - J = 0 \end{align} I now want to solve this equation ...
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0answers
94 views

Variational principle with $\delta I \neq 0$

In Covariant Phase Space with Boundaries D. Harlow allows boundary terms in the variation of the action. If we have some action $I[\Phi]$ on some spacetime $M$ with boundary $\partial M = \Gamma \cup \...
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1answer
77 views

Calculating the Equations of motion for a scalar field

I am recently trying to get some understanding of Quantum Field Theory, therefore I am reading Quantum Field Theory and the Standard Model by M.D. Schwartz. The author takes for an example the ...
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0answers
28 views

Equations of motion of classical chromodynamics with Yang-Mills theory

I am currently reading a paper about classical chromodynamics: https://arxiv.org/abs/hep-th/0607203 However I have problems understanding equation (2) and (4) (2): \begin{equation} F_{\mu \nu}= \...
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2answers
173 views

Why boundary terms make the variational principle ill-defined?

Let me start with the definitions I'm used to. Let $I[\Phi^i]$ be the action for some collection of fields. A variation of the fields about the field configuration $\Phi^i_0(x)$ is a one-parameter ...
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1answer
70 views

High order self-interacting potential terms in QFT

In this answer on another Physics StackExchange thread, the self-interaction potential terms of a classical field theory ($ \phi^n $ terms with $ n>2 $) are said to correspond to higher-order ...
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1answer
42 views

How are the authors obtaining the asymptotic form of the sympletic form for the Maxwell + massive field system?

I've been studying the paper "Asymptotic symmetries of QED and Weinberg’s soft photon theorem" by Campiglia & Laddha and there is one step in their analysis I'm being unable to understand. I shall ...
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1answer
100 views

Noether's theorem derivation for fields

I've been trying to understand (from several sources) how Noether's theorem for fields is derived, and reading the Wikipedia page about Noether's theorem I encountered the following: say we have the ...
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1answer
87 views

High dimensional wave equation

In 3 dimensions, the wave equation $$\Box\psi=\delta(t)\delta(\vec{x})$$ has the retarded and advanced solutions $$\psi=A_R \frac{\delta(t-x)}{4\pi x} + A_A \frac{\delta(t+x)}{4\pi x}.$$ How does this ...
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0answers
99 views

Conformal Invariance of the Scalar Field

Consider a scalar field with action $$S(\phi)=\int_Md^Dx\partial_\mu\phi\partial^\mu\phi.$$ Following the book on Conformal Field Theory of Di Francesco, Mathieu and Sénéchanl, they claim that under a ...
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0answers
39 views

Difference between Functional Variation and Total Variation of a Relativistic Classical Field

I'm studyig Classical Field Theory and I've got confused with these differente kinds of field variations. The book I'm reading states Variation of coordinates $x^\mu$ \begin{equation} x'^\mu = x^\mu ...
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67 views

Scalar field Lagrangian with extra term

I am a bit of stuck on this problem: Consider a Classical field theory in the semi-infinite space $x^1, x^2 \in \mathbb{R}$, but $x^3 \in (0, \infty)$. The Lagrangian is $$L = \int_0^{\infty}dz\...
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1answer
64 views

Green's function for the screened Poisson equation

Assuming we are given a Lagrangian \begin{equation} \mathcal{L}(\phi(r),\partial^i\phi(r)) = \frac{1}{2} \partial_i\phi \partial^i\phi + \frac{m^2}{2} \phi^2 + \lambda \phi, \end{equation} the ...
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3answers
324 views

Is the Four-gradient of a scalar field a four-vector?

Consider a scalar field $\phi$ as a function of spacetime coordinates $x^\mu$. The four-gradient of $\phi$ is given by \begin{equation} \frac{\partial \phi}{\partial x^\mu} = \left( \frac{\partial \...
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0answers
27 views

Measurement of $n$-point functions

The $n$-point correlation functions appear frequently in physics. I think I can understand how the concepts can be exploited to understand physical processes. What I struggle to understand is how they ...
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1answer
53 views

Why can we minimize the potential as just a function of a single numeric variable in spontaneous symmetry breaking?

In the discussion of spontaneous symmetry breaking one starts with considerations about the classical vacuum, defined as the field configuration $\Phi(x)$ which minimizes the energy encoded in the ...

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