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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17
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1answer
2k views

On a trick to derive the Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
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2answers
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Why don't people use Hamilton's equations for a relativistic free charged particle?

A charged relativistic free particle has the Hamiltonian in general: $$ \mathcal{H} = \sqrt{{\bf p}^2c^2+m^2c^4}.$$ I read somewhere that says, it is possible to go further and say that the EoM are ...
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5answers
1k views

Euclidean geometry in non-inertial frame

Refer, "The classical theory of Fields" by Landau&Lifshitz (Chap 3). Consider a disk of radius R, then circumference is $2 \pi R$. Now, make this disk rotate at velocity of the order of c(speed of ...
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3answers
1k views

What is the actual form of Noether current in field theory?

Let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a $D$-...
3
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4answers
428 views

Transformation of $d^4x$ under translation disregarded?

Under a translation in spacetime i.e., $$x\mapsto x^\prime=x+a,\tag{a}$$ a scalar field $\phi(x)$ $$\phi(x)\mapsto\phi^\prime(x)=\phi(x-a).\tag{b}$$ My aim is to verify the invariance of an action of ...
41
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5answers
7k views

Tree level QFT and classical fields/particles

It is well known that scattering cross-sections computed at tree level correspond to cross-sections in the classical theory. For example the tree level cross-section for electron-electron scattering ...
11
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4answers
4k views

Counting Degrees of Freedom in Field Theories

I'm somewhat unsure about how we go about counting degrees of freedom in classical field theory (CFT), and in QFT. Often people talk about field theories as having 'infinite degrees of freedom'. My ...
7
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1answer
874 views

Question about the Noether charge algebra

I'm reading these notes - page 8 and 9 - and I'm a bit confused. If we consider a field $\phi$ (which can be either bosonic or fermionic) transforming as: \begin{equation} \phi(x) \rightarrow \phi(x) ...
5
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2answers
1k views

Why my 4-divergence term added to a Lagrangian modifies the equation of motion?

I take this Lagrangian: $$\mathcal{L}=\mathcal{L}_0+\partial_\alpha f(\phi, \partial_\mu \phi).$$ In this topic Does a four-divergence extra term in a Lagrangian density matter to the field ...
4
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2answers
546 views

What spinor field corresponds to a forwards moving positron?

When we search for spinor solutions to the Dirac equation, we consider the 'positive' and 'negative' frequency ansatzes $$ u(p)\, e^{-ip\cdot x} \quad \text{and} \quad v(p)\, e^{ip\cdot x} \,,$$ ...
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2answers
847 views

Why can't $p^0$ change sign under a proper orthochronous Lorentz transformation?

If in an inertial frame $S$, $p^0$ is positive, then it is claimed that under a proper orthochronous Lorentz transformation ${\rm SO^+(3,1)}$ (i.e., those Lorentz transformations which are ...
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2answers
3k views

Need for a side book for E. Soper's Classical Theory Of Fields

I am reading now E. Soper, Classical Theory Of Fields, now and sometimes it is very hard to follow the equations. So I need a side book on classical field theory to read it comfortably. Landau & ...
5
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1answer
283 views

Theory invariance after substitution of theory's field equations back into theory's action functional?

Suppose I have a theory $A$ concerning the evolution of a set of fields $T_1, \dots, T_n$. Let the action functional for this theory be $S[T_1, \dots, T_n]$. Suppose in the action, in addition to ...
7
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1answer
626 views

Significance of symplectic form in classical field theory

I'm trying to understand the significance of construction presented to me in field theory class. Let me first briefly describe it and then ask questions. Given two solutions $\phi_1$, $\phi_2$ of the ...
5
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1answer
876 views

Big puzzle about Noether's theorem of coordinate transformation (spacetime symmetry)

For Noether theorem with only internal symmetry, I've found there has been a very clear proof. But I still struggle with the proof of coordinate transformation. Because there are so many different ...
11
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6answers
786 views

Charged particle as observed from an inertial and a non-inertial frame of reference

A charged particle fixed to a frame $S^\prime$ is accelerating w.r.t an inertial frame $S$. For an observer A in the $S$ frame, the charged particle is accelerating (being attached to frame $S^\prime$)...
5
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1answer
1k views

Infinite Energy of Point Charges (in the context of classical field theories)

In the context of classical physics,is there any renormalization method to avoid infinite energy of point charges?
3
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1answer
366 views

Motivation for the Euler-Lagrange equations for fields

In Lagrangian Mechanics it is possible to motivate the Euler-Lagrange equations by means of D'alembert's principle. This is a quite more natural route to follow than to start postulating the least ...
0
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1answer
776 views

Relation between Noether's charge and the generator of a $U(1)$ symmetry

Consider a $U(1)$ symmetry of a complex scalar field realized as $$\phi\to\phi^\prime=e^{iQ\theta}\phi.$$ where $Q$ is the generator of the symmetry. The conserved Noether's charge (in $D$-spatial ...
6
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4answers
494 views

Is the description of the gravitational field as a vector field and a tensor field compatible?

By electric or magnetic fields we mean the vector fields $\vec{E}(\vec{r},t)$ and $\vec{B}(\vec{r},t)$ respectively. But a gravitational field in Newtonian theory is a vector field that $\vec{g}(\vec{...
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2answers
127 views

Does it make sense to speak in a total derivative of a functional? Part III

In this third part of the series, I will continue the deduction of Noether's theorem initiated in the previous post - Does it make sense to speak in a total derivative of a functional? Part II. ...
1
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1answer
245 views

Simplest model in field theory which leads to a pseudo-Goldstone boson

What can be a simple (if not simplest) continuum field theory model that gives rise to a pseudo Goldstone boson (doesn't matter if it is a toy model)? For example, I would be very happy if one can ...
5
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1answer
239 views

Most general Lagrangian in CFT in 0+1D

My question is about $CFT_1$. Page 18 of this says that $$L={\frac{\overset{.}{Q}^2}{2} - \frac{g}{2Q^2}}\tag{1.11}$$ is the most general Lagrangian that preserves time translation and scale ...
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0answers
91 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 ...
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3answers
441 views

Examples of non-linear field symmetries?

Consider a Lagrangian theory of fields $\phi^a(x)$. Sometime such a theory posseses a symmetry (let's talk about internal symmetries for simplicity), which means that the Lagrangian is invariant under ...
4
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2answers
291 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
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1answer
788 views

Energy momentum tensor from generalized Noether current

Proceeding like here, let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a ...
10
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2answers
188 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 ...
3
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2answers
1k views

Canonical momentum density vs. energy-momentum tensor

Suppose we have a scalar field $\varphi$ with Lagrangian $$ \mathcal{L} = \frac{1}{2} \kappa \left( \frac{\partial \varphi}{\partial x} \right)^2 + \frac{1}{2} \rho \left( \frac{\partial \varphi}{\...
2
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2answers
641 views

Can the finite dimensional irreducible $(j_+,j_-)$ representations of the Lorentz group $SO(3,1)$ be unitary?

Since the Lorentz group $SO(3,1)$ is non-compact, it doesn't have any finite dimensional unitary irreducible representation. Is this theorem really valid? One can take complex linear combinations of ...
6
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1answer
484 views

Motivation for covariant phase space

The covariant phase space idea, in one sentence, is that there is a natural symplectic structure on the space of the classical trajectories of a system and that the usual $(q,p)$ coordinates just ...
4
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1answer
1k views

Where this polarization vector is coming from?

When dealing with vector fields in his QFT book, Schwartz writes the classical field in terms of a basis which I don't know how he is getting. He first introduces the Proca Lagrangian $$\mathcal{L}=-\...
7
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2answers
726 views

Are fixed points of RG evolution really scale-invariant?

It is often stated that points in the space of quantum field theories for which all parameters are invariant under renormalisation – that is to say, fixed points of the RG evolution – are scale-...
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2answers
396 views

Is the gauge transform field in electromagnetism a Lagrange multiplier?

In a draft answer to another question about gauge transformations, I played around with demonstrating the action of a gauge transformation on the Lagrangian density. Beginning with the classical ...
5
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1answer
455 views

Classification of field types in QFT

In Classical Field Theory fields are sections of bundles over spacetime. In particular we almost always consider vector bundles. Some examples are: Scalar fields: these are sections of the trivial ...
4
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1answer
1k views

How to derive the two Friedmann-Lemaître equations from a Lagrangian?

Consider the Lagrangian of an isotropic-homogeneous spacetime (Robertson-Walker metric), containing a simple scalar field and a cosmological constant (this expression comes from the standard Hilbert-...
4
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2answers
535 views

Why is a nonzero VEV for a spinor field said to break Lorentz invariance?

Consider a spinor field $\psi(x)$. Its vacuum expectation value is given by $$v=\langle 0|\psi(x)|0\rangle.$$ Using the fact that the vaccum is invariant under Lorentz transformation, we get, $$v=\...
4
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1answer
846 views

Understanding instantons in pure Yang-Mills theory

Yang-Mills Instantons are defined as finite action solutions to the corresponding Euclidean equation of motion. If I understood it correct, then instantons are those classical gauge field ...
2
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1answer
113 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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1answer
297 views

Maxwell equations of motion from $S = \frac{-1}{2} \int F \wedge \ast F$

I'm trying to understand the following equation, used in the derivation of the equations of motion. Let $S = \frac{-1}{2} \int F \wedge \ast F$ and $F = dA$. Let $\delta$ denote variation. Then $$\...
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2answers
698 views

Global $U(1)$ transformation properties of gauge fields

What are the Global gauge transformations of gauge bosons in Standard Model? To elaborate: Initially, we consider the global $U(1)$ transformations of scalars ($\phi$) and fermions ($\psi$) as $$\...
11
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3answers
774 views

Classical field limit of the electron quantum field

To recover classical electromagnetic fields from the quantum electromagnetic field, we consider coherent states of the form $$\exp \left(\int d\vec{r}\, \vec{A}(\vec{r}) \vec{a}^\dagger(\vec{r}) \...
7
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1answer
462 views

Why does the minimum energy field configuration require the fields to be constant?

I am having a hard time in understanding a well known statement always made in the context of field theory. Background Consider a classical real scalar field theory with Lagrangian density given by $...
3
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0answers
256 views

Questions about classical and quantum scale invariance

This is kind of a continuation of this and this previous questions. Say one has a free "classical" field theory which is scale invariant and one develops a perturbative classical solution for an ...
3
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2answers
2k views

The variation of the Lagrangian density under an infinitesimal Lorentz transformation

I'm trying to introduce myself to QFT following these lectures by David Tong. I've started with lecture 1 (Classical Field Theory) and I'm trying to prove that under an infinitesimal Lorentz ...
3
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3answers
1k views

Klein gordon field and positive/negative energy solutions

I my course we calculated the Klein-Gordon field: $$ \phi(x)= \int \frac{d^3k}{(2 \pi)^3}\frac{1}{2k_0} ~ \left[a(\vec{k})e^{-ik.x}+b^*(\vec{k})e^{i kx}\right]$$ We said that the part $ a(\vec{k})e^{...
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2answers
252 views

Noether's theorem with infinite parameters

I'm trying to understand something regarding Noether's theorem - and with the given situation, my question isn't that much of a question, I'm rather just seeking confirmation whether I'm thinking ...
1
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1answer
454 views

Why can a Lorentz transformation not take $\Delta x^\mu$ to $-\Delta x^\mu$ when $\Delta x$ is time-like? [duplicate]

In Peskin and Schroder's Quantum field theory text (page 28, the passage below Eq. 2.53), it asserts that for a spacelike interval $(x-y)^2<0$, one can perform a Lorentz transformation taking $(x-y)...
4
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1answer
236 views

Is every classical field theory with dimensionless couplings conformally invariant?

I'm trying to learn conformal field theory and getting rather frustrated, because I can't find any source that gives decent examples or straightforward logic. In most sources I have found, conformal ...
3
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2answers
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 ...