# 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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### 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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### 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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### 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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### 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 $D$-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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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### 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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### 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 & ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $S^\prime$)...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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### 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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### 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 ...
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### 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 ...
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### 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-...
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