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.

Filter by
Sorted by
Tagged with
2
votes
1answer
48 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 ...
0
votes
1answer
37 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 ...
3
votes
0answers
37 views

Why the action should be “stationary up to terms at future and past boundaries”?

In Covariant Phase Space with Boundaries D. Harlow allows boundary terms in the variation of the action. Essentially, if we have some action $I[\Phi]$ on some spacetime $M$ with boundary $\partial M = ...
1
vote
1answer
48 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 ...
0
votes
0answers
40 views

Formalizing the action in Minkowski-Space

In classical mechanics the action is defined by \begin{align} S(\Omega) = \int_\Omega L(\dot q(t), q(t), t) dt \end{align} where $\Omega$ is the path parameterized by the time-parameter $t$. In ...
1
vote
0answers
20 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}= \...
7
votes
1answer
66 views

Why boundary terms in the variation of the action 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 ...
0
votes
1answer
60 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 ...
3
votes
1answer
38 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 ...
1
vote
1answer
61 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 ...
2
votes
1answer
84 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 ...
4
votes
0answers
57 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 ...
0
votes
0answers
31 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 ...
0
votes
0answers
61 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\...
2
votes
1answer
50 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 ...
1
vote
3answers
110 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 \...
0
votes
0answers
22 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 ...
0
votes
1answer
51 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 ...
2
votes
2answers
89 views

Why the Real Scalar Field lagrangian has this form?

The lagrangian of the Real Scalar Field $\phi$ is given by \begin{equation} \mathcal{L} = \frac{1}{2}\eta^{\mu \nu} \partial _\mu \phi \partial _{\nu} \phi - \frac{1}{2} m^2 \phi^2 \end{equation} \...
1
vote
0answers
60 views

Quantum solitons: derivation of $ \int {\phi^\prime}^2 dx = M$ using Lorentz invariance

I was reading through page 10 of this document (Chua, 2017) on quantum solitons, and came across the following statement relating to the equation for kinetic energy $$T = \left(\frac{da}{dt}\right)^2\...
3
votes
2answers
152 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 $(...
2
votes
0answers
101 views

Quantising classical harmonic oscillator like classical fields

Iam studying QFT, But Iam stucked at catching the mathematical formulae for harmonic oscillator analogue in QFT, For a classical harmonic oscillator I can write $x(t)= Ae^{i\omega t} +A^{\dagger} ...
3
votes
0answers
56 views

Initial value problem on $\mathcal{I}^-$ for Maxwell fields

In the paper "Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity" by Ashtekar and Streubel the authors state the following: Fix, as in § 2(a), a conformal completion $(...
1
vote
1answer
55 views

Why does the quantity $P^i=\int d^3x T^{0i}$ represent the momentum of a field?

For a classical field $\phi$, the space integral of $T^{0i}$ denoted by $P^i=\int d^3x T^{0i}$ where $T^{\mu\nu}$ is the energy-momentum tensor is called the momentum of the field. This is not very ...
0
votes
1answer
40 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 ...
0
votes
0answers
56 views

What's the defining equation of a retarded Green's function?

The retarded Green's function $G_R(x,x^\prime)$ is usually defined by using the Wightman function $$W(x,x^\prime) = \langle0| \varphi(x) \varphi(x^\prime) |0\rangle\,,$$ by $$ G_R(x,x^\prime) = \Theta(...
0
votes
1answer
80 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 $$ \...
0
votes
0answers
29 views

What's the physical meaning of the product of two (classical) wave packets?

For simplicity, let's focus on a classical scalar field. If we want to take interactions into account, we can use the modified Klein-Gordon equation $$ \left(\partial_\mu \partial ^{\mu} + m^2\...
0
votes
1answer
51 views

Would the Michelson-Morley Experiment have Discovered our Atmosphere?

We all know the earth is surrounded by an atmosphere, and we know that it is the medium through which sound travels. If the Michelson and Morley experiment was modified to find sound’s medium would ...
1
vote
0answers
66 views

What if the Lagrangian $\mathscr{L}$, a Lorentz scalar, is replaced by a Lorentz vector?

As an answer to this post, I made an impression that if $\mathscr{L}$ were not a Lorentz scalar in Eq.$(1)$ (see below), then Eq.$(1)$ would not be covariant. But now I think that is wrong! I state ...
1
vote
1answer
85 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 ...
0
votes
1answer
36 views

Generator of spatial translation in field theory

In classical mechanics, we know that the momentum operator is the generator of spatial translation. But it seems to me that this is no longer the case in the classical field theory. Lets first ...
0
votes
0answers
52 views

Conjugate momenta in field theory

In classical mechanics for particles, the Euler-Lagrange equations are given by: $\frac{d}{dt} \frac{\partial L}{\partial \dot q} = \frac{\partial L}{\partial q}$ and the momentum conjugate to q is ...
0
votes
1answer
97 views

Is there a superfluous statement in Schwartz's QFT book in deriving Euler-Lagrange equations?

Please help me with the following confusion. Yesterday I was looking at the derivation of Euler-Lagrange equation in several QFT textbooks using stationarity of the action. At the last step one needs ...
0
votes
0answers
66 views

Gravity as part of the metric but not of a 4-vector potential

In class, I've been told that the scalar potential $V(x)$ (except the one associated to gravity) has to be introduced in the Lagrangian as the time-component of a 4-vector potential $A_\mu$, so what ...
1
vote
2answers
87 views

Does anyone have references on classical field theory that develops the differential form formalism?

I am familiar with the usual way of doing Classical Field Theory, but I am currently taking a course where the professor works with differential forms to teach the subject. I wonder if anyone knows ...
0
votes
1answer
42 views

Fields and Conservation of Energy

I am a bit confused about fields, like magnetic fields, and the conservation of energy. For example, when two magnets are attracted to each other, don't they "exhaust" the field? The moon that orbits ...
2
votes
2answers
155 views

Confusing Total Derivative and Partial Derivative in Classical Field Theory - Noether Theorem

I'm really confused about total derivatives and partial derivatives. My multivariable calculus book (Guidorizzi vol 2 Um Curso de Calculo) says that if I have a function like $f(a(u,v),b(u,v))$ then ...
0
votes
0answers
47 views

Using the Martin-Siggia-Rose (MSR) formalism for oscillator with general non-harmonicity

I am wondering if using the Martin-Siggia-Rose (MSR) formalism can be convenient/treatable for calculating correlation functions [or their spectral densities] of a linear [underdamped] oscillator with ...
1
vote
0answers
24 views

How Can the Number of Degrees of Freedom per Frequency in the Rayleigh-Jean formula be Irrational, or are they not?

I've read that the Rayleigh-Jean formula for the energy density per frequency of blackbody radiation can be expressed as $u(ν,T) = nkT$ where k is Boltzmann's constant, T is the temperature, and n is ...
1
vote
1answer
101 views

Yang-Mills Action for Non-Trivial Bundle

Suppose we have a principal $G$ bundle $(P,M,π)$ where $M$ is a 4-dimenational manifold and $G$ a Lie group (and $\mathfrak{g}$ its Lie algebra).The Yang Mills action is a functional of the gauge ...
0
votes
0answers
70 views

How to transform a lagrangian after a change of coordinates?

Let's consider a generic lagrangian density in classical field theory: $$L(\phi(x),\partial_{\mu} \phi(x))$$ Now suppose I want to find the lagrangian for the same system with respect to another field ...
1
vote
1answer
109 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 ...
1
vote
3answers
108 views

How can the first Maxwell equation be valid in non-static cases?

I am thinking in the framework of Classical Gravity, where the speed of the interaction is infinite. Now it is also known that there is a correspondence between Classical gravity and electrostatics, ...
8
votes
1answer
212 views

Does the Standard Model have texture defects?

In the standard classification of topological defects, in a theory with vacuum manifold $\mathcal{M}$, $\pi_0(\mathcal{M})$ corresponds to domain walls, $\pi_1(\mathcal{M})$ corresponds to strings/...
0
votes
0answers
19 views

Does “continuum limit” and “Legendre transformation” commute?

In classical field theory, both Lagrangian density and Hamiltonian density can be considered as the continuum limit of the Lagrangian and Hamiltonian for point masses. Also, Hamiltonian density can be ...
1
vote
1answer
38 views

Klein-Gordon equation propagators: intersection with the support of the source

Let $(M,g)$ be a globally hyperbolic. Let $P = \Box - m^2$ be the Klein-Gordon differential operator. Following Fewster's notes, we may define the retarded/advanced propagators $$E^\pm : C^\infty_0(M)\...
0
votes
0answers
63 views

Global Part of Non-Abelian Gauge Transformation

I have a perhaps stupid question about Noether's theorem. In Abelian gauge theory, say $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\Psi}(iD\!\!\!\!/-m)\Psi, \tag{1.0} $$ where $D_{\mu}=\...
3
votes
1answer
76 views

A Question about Yang-Mills Equation

The non-homogeneous part of the Yang-Mills equations is given by $$D\star F=\star J,$$ where $D=d+A$ is the covariant derivative, $\star$ is the Hodge star and $J$ is the source current. Under a ...
0
votes
0answers
24 views

From the general decomposition of electric field to one polarized along $\hat{\textbf{x}}$

In the gauge $A^0=\nabla\cdot\textbf{A}=0$, starting from the Fourier decomposition of $\textbf{A}(\textbf{x},t)$, the electric field $\textbf{E}(\textbf{x},t)$ is obtained as $$\textbf{E}(\textbf{x},...

1
2 3 4 5
7