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
0 votes
0 answers
22 views

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 ...
user avatar
1 vote
1 answer
34 views

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 ...
user avatar
0 votes
2 answers
43 views

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 \...
user avatar
0 votes
0 answers
28 views

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. ...
user avatar
  • 1
1 vote
1 answer
114 views

"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 ...
user avatar
2 votes
0 answers
44 views

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 ...
3 votes
2 answers
140 views

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 ...
user avatar
0 votes
0 answers
22 views

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 $\eta = (1, -1, -1, -1)$...
user avatar
  • 6,696
1 vote
0 answers
50 views

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 ...
3 votes
1 answer
144 views

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 $A_a$ ...
user avatar
  • 6,696
3 votes
2 answers
118 views

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 ...
user avatar
  • 6,696
2 votes
1 answer
59 views

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 ...
user avatar
  • 6,696
5 votes
1 answer
134 views

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 ...
user avatar
  • 6,696
2 votes
1 answer
81 views

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}^{...
user avatar
0 votes
1 answer
88 views

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'...
1 vote
0 answers
58 views

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 $...
user avatar
1 vote
2 answers
71 views

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 ...
user avatar
  • 2,428
5 votes
2 answers
726 views

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 $(+,-)$, ...
user avatar
  • 514
5 votes
0 answers
91 views

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 ...
user avatar
2 votes
1 answer
110 views

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^{*} \...
user avatar
0 votes
1 answer
84 views

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 \...
user avatar
  • 601
4 votes
0 answers
107 views

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). ...
user avatar
  • 2,895
1 vote
1 answer
63 views

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 ...
user avatar
  • 1,405
0 votes
2 answers
88 views

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 ...
user avatar
0 votes
0 answers
41 views

Quantum field theory not related to classical field theory [duplicate]

Overvation 1: Whatever quantization process is used, it is common to define a QFT from a classical field theory. Observation 2: On the other hand, given a lagrangian QFT, one could try to "...
user avatar
  • 883
1 vote
0 answers
77 views

Field Momentum and Canonical Momentum in the Classical Field Theory

From the definition of energy-stress tensor, $$T_\mu^\nu=\frac{\partial \mathcal{L}}{\partial(d_\nu \psi)}d_\mu\psi-\mathcal{L}\delta_\mu^\nu$$ the $i$-th component of momentum carried by "field&...
user avatar
1 vote
0 answers
82 views

Conformal invariance of the massless scalar field action

The massless scalar field action on Minkowski background is given by \begin{equation} S[\phi]=\int_{\mathbb{R}^D}d^Dx~\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi. \end{equation} This action is often ...
user avatar
  • 883
3 votes
0 answers
71 views

Can convergent perturbation series be incorrect for an action linear in the perturbation?

Non-perturbative effects are common in mathematics. For example, consider the function $$f(g) = e^{-1/g}+ g + \frac{1}{10} g^2$$ and suppose this function is the answer to some math problem. ...
user avatar
  • 1,160
1 vote
1 answer
36 views

Consistency of substitution of a canonical variable from EoM back into (momentum-less) action

I was reading this answer, where the issue of substituting equations of motion (eoms) into the action is addressed. I am fine with the basic idea that the action principle is destroyed when the eoms ...
user avatar
  • 53
0 votes
2 answers
59 views

For free fields, is there a one-to-one correspondence between probability distribution of classical field configurations, and states?

If I'm given the field operator of free fields (for example $\phi(x)$) as a function of space time, and a state (for example $\langle 0 | $, I can calculate the expectation value for every point in ...
user avatar
  • 5,499
0 votes
0 answers
68 views

When does a Lagrangian exist for arbitrary equations of motion? [duplicate]

Let's say I have some equations of motion for an arbitrary system, i.e. some implicitly or explicitly defined ODE involving $q = (q_1, q_2, q_3, \dots)$ and $\dot q = (\dot q_1, \dot q_2, \dot q_3, \...
user avatar
0 votes
0 answers
32 views

Confusion regarding a step in Schwartz's book (equation 3.44) [duplicate]

I have just started learning QFT, and am working my way through Schwartz's book. I am not able to justify to myself something he does. I have even started suspecting if this is a typo. I have attached ...
user avatar
  • 73
0 votes
0 answers
71 views

Can someone explain the steps? [duplicate]

Can anybody expand the equation .What is ω and is i power of ω in equation 2.2. And how $f_λ(r_1,r_2,...,r_n,t)=0$;λ=1,2,...,Λ gives the last equation and what $∇_k$ means?
user avatar
1 vote
0 answers
69 views

Relation between quantum and classical mass gaps

We say a QFT has a mass gap if the spectrum of the mass operator $M:=\sqrt{P_\mu P^\mu}$ is bounded below by some $\Delta >0$. I will define a $\textit{classical}$ field theory to have a mass gap ...
user avatar
8 votes
3 answers
845 views

Hamiltonian Field Theory in Peskin & Schroeder

In Section 2.2 of their QFT textbook, Peskin & Schroeder introduce the Lagrangian and Hamiltonian field theories of a classical scalar field. While defining the action $S[\phi]$ and deriving the ...
user avatar
  • 2,711
2 votes
1 answer
143 views

Confused with 4-vector notation and 4-derivative

I have a lot of trouble finding out what the rules are for doing algebra and calculus with 4-vectors. This example shall illustrate one of my problems: The Lagrangian for a real scalar field is $$\...
user avatar
  • 1,135
3 votes
1 answer
123 views

Why is Hamilton's principle (or principle of least action) still valid in a relativistic field theory?

I am struggling to understand why the principle of least action which is derived in classical mechanics from d'Alembert's principle continues to be valid in a regime that treats a relativistic field. ...
user avatar
0 votes
0 answers
24 views

Variations in a vector field [duplicate]

When we derive Maxwell's equations from the Lagrangian that contains the Maxwell field tensor $F_{ij}$, I ran into a small confusion. With the Lagrangian being $L = F^{ij}F_{ij}$, taking variations of ...
user avatar
  • 582
0 votes
0 answers
15 views

References for Hamiltonian field theory and Dirac Brackets [duplicate]

I'm looking for complete and detailed references on constrained Hamiltonian systems and Dirac brackets. While my main interest is electrodynamics, I would prefer a complete exposition of the theory ...
1 vote
0 answers
78 views

Understanding classical massive real scalar $\phi^4$ Callan-Symanzik equation

Considering classical massive real scalar field theory by the action: $S[\phi] = \frac{1}{2}\int d^4x[\partial_\mu\phi\partial^\mu\phi-m^2\phi^2-\frac{g}{12}\phi^4] $ we assume the theory is ...
user avatar
6 votes
2 answers
410 views

Uniqueness of the definition of Noether current

On page 28 of Pierre Ramond Field theory - A modern primer the following is written: "we remark that a conserved current does not have a unique definition since we can always add to it the four-...
user avatar
2 votes
3 answers
137 views

Confusion about vector and scalar field transformation laws

I'm a little bit confused about the transformations for scalars fields and vector fields in classical field theory. I've learned that a scalar field is a smooth function $$\phi : M \longrightarrow \...
user avatar
2 votes
1 answer
276 views

Proof that free scalar field is conformally invariant

So, under conformal transformations $$x\mapsto x'\\ \phi\mapsto\phi'(x')=\Omega^{(2-D)/2}\phi(x),$$ where $$\eta_{\mu\nu}\frac{\partial x^\mu}{\partial x^{'\alpha}}\frac{\partial x^\nu}{\partial x^{'\...
user avatar
  • 3,354
3 votes
2 answers
92 views

Scalar field Hamiltonian $H = 0$ from parameterization independence

This question is related (but not similar) to this old one of mine: How to derive the two Friedmann-Lemaître equations from a Lagrangian? Consider the Lagrangian of an isotropic-homogeneous ...
user avatar
  • 6,696
1 vote
2 answers
73 views

How show that for a free EM field, the energy of each mode is conserved in time?

For a free electromagnetic field, since different modes do not talk to each other, one would expect that the energy stored in each mode is conserved (or constant) in time. But the energy stored in a ...
user avatar
2 votes
0 answers
68 views

Nonlinear superposition and self-interaction in classical field theory [duplicate]

I am learning QFT (in a path integral formalism) and one thing I'm struggling with is that self-interaction is supposed to be a quantum phenomenon, not apparent in classical non-linear field theory. I ...
user avatar
1 vote
0 answers
60 views

Coordinate invariance in Physics

Let us consider a classical field theory on flat background spacetime. The action is $$S[\Phi] = \int d^nx \mathcal{L}(\Phi,\partial_\mu\Phi).$$ Why shouldn't this action be independent of the chosen ...
user avatar
  • 883
6 votes
0 answers
78 views

Is classical Kaluza Klein theory stable or not?

Set Up In the original classical Kaluza Klein theory, you have a $d+1$ dimensional manifold where one space dimension is a circle $S^1$. In the "low energy limit," none of the metric ...
user avatar
  • 10.9k
2 votes
2 answers
78 views

Notations and Variations in the proof for Noether's theorem for fields

In the proof for Noether's theorem, as given in D.Gross's notes, there are two kind of variations used. $x'^{\mu}=x^{\mu}+X^{\mu}_\alpha(x)\omega^\alpha$ $\phi'_i(x')=\phi_i(x)+\Psi_{i\alpha}(x)\...
user avatar
  • 1,142
2 votes
1 answer
67 views

Does this Lagrangian density represent anything "real"?

So this lagrangian was used as an example for deriving the equations of motion using the Euler-Lagrange equations in our lecture notes. $$ L (\phi )=-\phi (x,t)^{2}+m\left(\frac{\partial \phi }{\...
user avatar
  • 25

1
2 3 4 5
9