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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Definition of scalar field in Kaku, Quantum Field Theory

I just started reading Kaku's book on QFT. In chapter 2, where he is still talking about classical fields, he defines the transformation property of a scalar field $\phi(x)$ (equation 2.23) as follows:...
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Where I should start from to study statistical field theory of electrolyte? [closed]

I recently have interest in statistical field theory (SFT) of electrolytes, mostly studied by Henri Orland and David Andelman. I watched all his video lectures in the youtube. Now, I want to study the ...
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Does the quantum (or classical) field vanish at time infinity?

The following argument is quite common in the QFT books A term like $ \int \mathrm{d}^4 x \partial_\mu M $ can be transformed to be a surface integration in the space-time infinity. $ M $ is some ...
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Doubt in classical field theory/electromagnetism

What is the basic difference between electromagnetic fields, electromagnetic waves and constant electromagnetic fields?
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When can we take the action between two fixed times in a relativistic classical field theory?

Peskin and Schroeder give a brief outline of Lagrangian field theory on page fifteen in their Quantum Field Theory book, where they write: Lagrangian Field Theory The fundamental quantity of ...
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Finding the number of independent degree of freedom of the Electromagnetic field

We know that there are only two independent degree of freedom, every point, in an Electromagnetic field. There seems to be two inequivalent ways to arrive at this conclusion. We just declare that the ...
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Question on asymptotic flatness

What is the theoretical argument for the asymptotical flatness of the four-potential? Can one assume asymptotical flatness for the scalar dilaton field as well?
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Question about dilaton monopole interaction derivation

I am trying to understand how one derives the dilaton monopole interaction. In "Black holes and membranes in higher-dimensional theories with dilaton fields", Gibbons and Maeda mentioned ...
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Green's functions co-incidentally appearing in the path integral of relativistic free particle action

When you compute the path integral of the relativistic free particle action, it's turns out to be the same as the Green's function of a classical field. This co-incidence is huge because it derives, ...
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Green's function co-incidences in Quantum Field Theory [duplicate]

Is there a deep reason for this Green's function co-incidences in Quantum Field Theory: When you compute the time ordered vaccuum expectation value of a quantum field, it turns out to be the same as ...
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Extra degrees of freedom in toy spontaneous symmetry breaking model?

Consider a Lagrangian with a real scalar field $\varphi$ and massless vector field $A_\mu$ with field strength $F_{\mu\nu}$, $$\mathcal{L} = -\frac{1}{2}\left(\partial\varphi\right)^2 - \frac{1}{4}F_{\...
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Getting the equation of state parameter for the inflaton field

So the inflaton field is taken to be a scalar field on an expanding spacetime, assuming it dominates the dynamics, yeah? $\mathcal{L}_\phi = \left( \frac{1}{2}\dot{\phi}^2 - V(\phi) \right) \sqrt{-\...
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How to perform a variational derivative of a function with two arguments?

In saddlepoint approximations of an action it often happens that you have to calculate a functional derivative of the inverse Green function which can be in position space a function of two arguments. ...
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What will happen if one used $[\phi (x), \frac{\partial L}{\partial (\partial _x\phi(x))}]=i\hbar$ to get a Quantum Field Theory?

Classical field theory does not discriminate between space and time, but canonical quantisation does. We use the relation $$[\phi (x), \frac{\partial L}{\partial (\partial _t\phi(x))}]=i\hbar$$ to get ...
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Pauli-Lubanski vector for Maxwell's equation

In the book quantum field theory by Itzykson and Zuber, page 53, the authors prove that Dirac's equation has spin 1/2 by showing that if $\psi$ is a solution to Dirac's equation, then compute that $\...
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Goldstein's derivation of Noether's theorem

This is a followup to my previoucs question: Translation invariance Noether's equation In Goldstein's derivation of the Noether's theorem in chapter 13, we have the infinitesimal transformation $$...
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Translation invariance Noether's equation

In chapter 13 of Goldstein's classical mechanics, on page 591 when talking about Noether's theorem, Goldstein says we need condition 3, which is $$\tag{13.133} \int_{\Omega'}\mathcal{L}(\eta_\rho'(x^\...
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Domain of definition of a Lagrangian in classical field theory

In classical field theory one has the action: $$S[\phi] = \int_{t_{0}}^{t_{1}}\int_{\Omega}\mathcal{L}(t,x,\phi(t,x),\dot{\phi}(t,x),\nabla\phi(t,x))dxdt$$ and we want to obtain the Euler-Lagrange ...
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Does the Newtonian gravitational field have momentum analogous to the Poynting vector?

We can define the total energy of the electromagnetic field as: $$\mathcal{E}_{EM}= \frac{1}{2} \int_V \left(\varepsilon_0\boldsymbol{E}^2+\frac{\boldsymbol{B}^2}{\mu_0}\right)dV$$ which satisfies the ...
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Conservation on energy in newtonian gravity

For the electromagnetic field, we can define the EM energy: $$\mathcal{E}_{EM}= \frac{1}{2} \int_V \left(\varepsilon_0\boldsymbol{E}^2+\frac{\boldsymbol{B}^2}{\mu_0}\right)dV$$ And because charged ...
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Canonical Realization of Poincare Symmetry of Dirac Spinor

I have a maybe stupid question about Noether charges and the Poisson bracket. If a classical field theory has a Poincare symmetry, then by using the Noether's theorem, one can write down its ...
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Semiclassical QCD

The classical Lagrangian density for Quantum ChromoDynamics (QCD) is given by: $$\mathcal{L}_\mathrm{QCD} = \bar{\psi}_i \left( i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}...
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Resonant cavity for sound waves

I would appreciate some assistance in understanding resonant cavities. I would like to be able to calculate the resonant frequencies of sound waves in a resonant cavity. I would think to use the ...
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Can we deduce the conservation of mass in non-relativist physics or is it just an experimental fact? [duplicate]

It is a well-known fact that mass by itself is not conserved (since, for example, a particle can annihilate with its antiparticle). However, in classical physics, and as long as there is no physical ...
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Energy of classical fields in pre-relativistic physics

In non-relativistic physics, the electromagnetic energy in all space is given by: $$E_{em} = \frac{1}{2} \int_{\mathbb{R}^3} \left(\varepsilon_0\mathbf{E}^2 + \frac{\mathbf{B}^2}{\mu_0}\right) \ \text{...
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Hamiltonian density of Abelian-Higgs Lagrangian

Given the Lagrangian density $$\mathcal{L}=-(\nabla^{\mu}\phi)^\dagger(\nabla_\mu \phi)-\frac{\lambda}{4}\left(\phi^{\dagger}\phi-v^2\right)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,\quad \nabla_\mu\phi=\...
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How are the retarded and advanced Lienard-Wiechert EM potentials interpreted?

Before QFT was developed in its current form, the Lienard-Wiechert EM potentials were mathematically interpreted as diverging/converging from/upon an electric charge q in the retarded and advanced ...
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How does the boundary term matter in scalar field and in more general cases?

People always say that boundary terms don't change the equation of motion, and some people say that boundary terms do matter in some cases. I always get confused. Here I want to consider a specific ...
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Lower vs Upper indices in stress energy tensor

In Goldstein Classical Mechanics, chapter 13 page 56, equations 13.30, the canonical stress energy tensor $T_\mu^{\,\,\,\nu}$ is defiend as: $$T_\mu^{\,\,\,\nu}=\frac{\partial\mathcal{L}}{\partial \...
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Express the power spectrum by displacement field in Lagrangian perturbation theory

Recently I'm reading this paper, Resumming Cosmological Perturbations via the Lagrangian Picture, to learn the application of Lagrangian perturbation theory in the modelling of large-scale structures. ...
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"Classical field configuration" - QFT

I often encounter the term "classical field configuration" in the scope of QFT, but I have a hard time interpreting what it really means. If I understood it correctly, then a general field ...
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Resources on Classical Field Theory [duplicate]

I have not been able to find where to ask this question on Stack Exchange; so pardon me if this is inappropriate and kindly redirect me: How may I obtain a referral to texts that cover classical field ...
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Very briefly, what is the relation/difference between classical field theory and classical thermodynamics/statistical mechanics?

This is probably not a good question, since I am at a fairly low level, but I am a little bit confused when the two concepts were described to me and it's bringing discomfort during my study. What I ...
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Proper condition for a wave of fast varying phase, relative to its amplitude-polarisation?

In the context of special relativity (Minkowski spacetime), I define an electromagnetic wave of the following shape (I'm using units such that $c \equiv 1$ and metric signature $\eta = (1, -1, -1, -1)$...
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References for Renormalization in Classical Field Theory

Most references for texts on renormalization talk about renormalization for quantum field theories. However, I have read in some places that we can also renormalize classical field theories. So, are ...
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Deviation of light rays in a scalar gravity theory (simple modification of Nordström theory)

I'm considering a simple scalar theory of gravity in Minkowski spacetime, which isn't exactly the same as the old Nordström theory. The scalar gravity field $\phi$ and the electromagnetic field $A_a$ ...
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How to find the energy-momentum tensor of a free relativistic particle from its lagrangian?

Consider a free relativistic particle in Minkowski spacetime. Its standard action is the following, where $\sigma$ is an arbitrary parametrization ($\tau$ is the particle's proper time. I'm using ...
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Why this boundary term could be ignored for a free relativistic particle?

How can we justify that the boundary integral we get from the following could be ignored, when we want to find the equation of motion? I consider the energy-momentum of a free particle in special ...
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What’s wrong with this Nordström-like scalar theory of gravity?

I got very perplexed while reading a few papers on the old Nordström theory of relativistic scalar gravity. I would like to know what's wrong with the following, which isn't exactly the same as ...
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A Question about Diffeomorphism Invariant Action

I remember that the canonical Hamiltonian of a diffeomorphism-invariant theory, in general, is zero. For example, the geodesic equation is derived from the action of arc length $$S[g(\tau)]=\int_{a}^{...
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What are good books (or lectures) to quickly learn the classical field theory needed in quantum field theory? [duplicate]

I'm an undergrad and I'll take a graduate-level course on quantum field theory in a month or so, I have studied electromagnetism (one semester course) and a little bit of relativity (on my own), but I'...
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Is there a derivation of the classical free scalar lagrangian?

In my particle physics course notes, I see that the Lagrangian (density) for free scalars is given by $$ \mathcal{L} = \frac{1}{2} g^{\mu\nu} \partial_\mu\phi \partial_\nu \phi - \frac{1}{2}m^2\phi^2 $...
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General Relativity Subject Content

The study of electromagnetism (E&M) centers around the electric and magnetic fields, both their static configurations and dynamic, e.g waves carried by them. The Maxwell equations are essentially ...
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What is "gradient energy" in classical field theory?

For the simple theory of a single real scalar field $\phi$ in 1+1D, the Lagrange density is $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-U(\phi)\tag{1}$$ with Minkowski signature $(+,-)$, ...
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Is pair production possible in Classical Field Theory?

It is often said that quantum effects only become manifest in loops, and all tree-level calculations are classical. I am trying to figure out to what extent this claim is true. I know the claim arises ...
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How to find scaling dimensions of the scalar and gauge vector fields

The problem is "Find the scaling dimensions of the scalar and gauge vector fields." As I understand, a scalar field is a field with lagrangian: $$ \mathcal{L}=\partial_{\mu} \phi^{*} \...
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Is this a manifestation of some infinite-dimensional Cayley-Hamilton theorem?

In classical field theory, when you have a free real scalar field $\phi$ with Lagrangian (density): $$ L = \frac{1}{2} \, \eta^{\mu \nu} \, \partial_{\mu} \phi \,\partial_{\nu} \phi - \frac{1}{2} m^2 \...
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Frustrated classical field theory

The frustrated Ising model (see e.g. this answer) is an example of a system that shows no unique ground state and many metastable states (its "energy landscape" is extremely complex). ...
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Test field vs backreaction of field theory in curved spacetime

Is there a way to understand test field regime as some limit of backreaction in general relativity? Consider the Einstein-Hilbert action augmented with the standard electromagnetic field coupled ...
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Doubt about Lagrangian Density for the Electromagnetic Field [duplicate]

I have a struggle with the derivation of a term of the Electromagnetic Lagrangian. It's known that $$\mathcal{L} = -(1/4)F^{\mu \nu} F_{\mu \nu}$$ for the free Electromagnetic field. There also ...
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