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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19 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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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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62 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
114 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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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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161 views

What's the Green's Function of the Photon's equation of motion?

The Green's function of the Klein-Gordon equation: $$\phi_s(x_\mu-y_\mu) = \int \frac{d^4k}{(2\pi)^4} \; \frac{e^{-i k^\mu (x_\mu -y_\mu)}}{-k_\mu k^\mu + m^2}$$ is the solution to the equation $$ \...
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222 views

State amplitude and field operator covariance in QFT

I'm studying QFT on Bogoliubov-Shirkov's "Introduction to the theory of quantized fields" (3d edition). In $§9.3$ they discuss transformation properties of quantum states and operators in QFT. Given ...
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49 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
68 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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1answer
134 views

What's the physical reason that a massive vector field has only three linearly-independent physical polarizations?

While a four-vector field $A_\mu$ has four components, for a massive field there are only three linearly independent combinations of these components that correspond to physical situations. This ...
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67 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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52 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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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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41 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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4answers
388 views

Confusion in Proof of Noether's theorem

This question is related to this Noether's theorem under arbitrary coordinate transformation and this Transformation of $d^4x$ under translation disregarded? To proof Noether's theorem every ...
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1answer
43 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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42 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
176 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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1answer
59 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
31 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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How to define “positive frequency” for general fields?

In Quantum Field Theory the positive frequency solutions to the classical field equations are quite important since they are the basis of the definition of particles. In Minkowski spacetime we have a ...
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2answers
76 views

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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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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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
35 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
55 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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106 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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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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2answers
95 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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88 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
73 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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1answer
96 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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1answer
73 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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468 views

Unification of gravity and electromagnetism

Have there been any attempts at unifying gravity and electromagnetism at least at classical level since Hermann Weyl's idea of gauge principle (1918)? We now have Standard Model which is very ...
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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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308 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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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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96 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
60 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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190 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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84 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
61 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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1answer
53 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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1answer
88 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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294 views

How do fields transform under special conformal transformations?

A Question in Classical Field Theory $\underline{\text{Assumption 1}}$: The definition of a transformation specifies how both the coordinates and the fields transform: They are namely $(1$-$1)$ and $(...

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