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Questions tagged [field-theory]

For questions where the dynamical variables are fields, that is, functions of several variables (typically, one time coordinate and several space coordinates). Comprises both classical field theory and quantum field theory. Use this tag when the question applies to both classical and quantum phenomena. Otherwise, use the specific tag instead.

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23 views

How to go from $\delta(\dot{\Psi})$ to $\delta\Psi$ in variational calculations? [duplicate]

This must have been done somewhere before but I never saw a clear and rigorous explanation of why. This is possibly related to my recent post here about potential abuse of notation. Just to be ...
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2answers
98 views

Physics near null infinity

The concept of null infinity $\mathscr{I}$ is standard in general relativity, and more recently in the analysis of infrared structure of gravity (see e.g. the article by Strominger). I am curious ...
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42 views

Exact solution for (simple) Reheating Boltzmann equations?

Let's say we have the Universe which is exiting Inflation, with the inflaton field $\phi$ decaying into relativistic particles (radiation $R$). Without any other assumption, the equations describing ...
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1answer
121 views

Functional derivative and variation of action $S$ vs Lagrangian $L$ vs Lagrangian density $\mathcal{L}$ vs Lagrangian 4-form $\mathbf{L}$

I have seen many potential abuse of notation that prevents me from clearly understanding variational methods in QFT and GR that I want to get this settled once and for all. This may be a bit long but ...
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1answer
25 views

(Giamarchi) Meaning of slowly varying field in bosonization

I am currently reading Giamarchi's Quantum Physics in One Dimension. Eq. (2.30) of the book says $$ \psi_r(x)=\frac{U_r}{\sqrt{2\pi\alpha}}e^{irk_Fx}e^{-i(r\phi(x)-\theta(x))} $$ where $U_r$ is the ...
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42 views

Should the energy-momentum tensor be invariant under gauge transformations?

For example, consider the electromagnetic theory given by \begin{align} I=-\frac{1}{4}\int d^4x\, F_{\mu\nu}F^{\mu\nu}, \end{align} where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. The action ...
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1answer
47 views

Some counting of field degrees of freedom for a classical spin-1/2 Dirac field

A classical real scalar field admits a decomposition $$\phi(x)\sim a_pe^{-ip\cdot x}+a_p^*e^{+ip\cdot x}$$ which tells that at each $x$, there exists a real number i.e., one degree of freedom at each ...
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1answer
38 views

Change of variable in 4D space-time

I was reading Sidney Coleman's article "Fate of the false vacuum: Samiclassical theory" and i stumbled upon a change of variables that i can't seem to prove. The problem is this: trying to solve the ...
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1answer
57 views

Complex Scalar Fields and Killing Vectors

In a stationary and axisymmetric spacetime, there are two Killing vectors, say $\zeta^\mu$ and $\xi^\mu$, one timelike and one space like. I understand that for a real scalar field, $\phi$, that ...
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52 views

Why is this lagrangian term a surface term?

I've seen certain models add the term $J^\mu\partial_\mu \phi$ to a Lagrangian, where $J^\mu$ is the current for a $U(1)$ global charge. They always claim that, if the current is conserved $\partial_\...
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1answer
58 views

Condition for Lorentz transformation

Today I had my first class of a QFT course, and there were some things that apparently I am supposed to know, but I don't. One of them is regarding Lorentz transformations. My teacher stated that: $\...
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60 views

Reading symmetry out of Lagrangian

Consider the Lagrangian: $$\mathcal L=\mathcal L_{kin}+\frac{1}{2}m_a^2(\phi^2_1+\phi^2_2+\phi^2_3)+\frac{1}{2}m_b^2(\phi^2_4+\phi^2_5)+\lambda_a(\phi^2_1+\phi^2_2+\phi^2_3)^2+\lambda_b(\phi^2_4+\phi^...
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33 views

Correlation function calculation

From Mehran Kardar's Statistical Physics of Fields page 38 section 3.2 [...] $I_d(x,\xi)$ is the solution to the following differential equation $ \nabla^2 I_d(x) = \delta^d(x)+ I_d(x)/\xi^2$ ...
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7 views

Lagrangian formalism to create impenetration condition between two different fluids

Assume that we are given two kinds of fluids that are described by their respective Lagrangian field densities $\mathcal{L}, \mathcal{L'}$. In the case when they are not interacting we assume that the ...
2
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1answer
136 views

The equation of motion for a scalar field in curved spacetime in terms of the covariant derivative

The equation of motion for a scalar field in curved spacetime $$\frac{\partial\mathcal{L}}{\partial\phi}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}\frac{\partial\mathcal{L}}{\partial\left(\...
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2answers
122 views

Yang-Mills Bianchi identity in tensor notation vs form notation

I've seen the Yang-Mills Bianchi identity written as both $$0 = dF^a + f^{abc} A^b \wedge F^c$$ and, in tensor notation, as $$\epsilon^{\mu\nu\lambda\sigma}D_{\nu} F^a_{\lambda\sigma} = 0.$$ Here ...
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1answer
35 views

How to make a triplet out of 2 doublets in the $SU(2)$ representation?

In Y.Grossman and Y.Nir "The Standard Model" book in chapter 4 (non abelian symmetrys) they present the law of whom we can have a triplet and singlet out of 2 doublets name them $\phi_a$ and $\phi_b$, ...
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53 views

Renormalization scheme dependence

Is it possible that the QFTs at hand show dependence on the renormalization point at which renormalization conditions are introduced?
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1answer
47 views

Energy and canonical momentum conservation in non-local classical field theory

Assume we have the following Lagrangian field density where $x, x'$ both three dimensional real vectors are coordinates and $t$ represents time, field is given by $\phi$. Assume for the sake of ...
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2answers
92 views

$\delta S=0$ only for $\frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}=0$?

Condition for the variation of action is: $$0=\delta S$$ $$=\int d^4 x [\frac{\partial \mathcal{L}}{\partial \phi}\delta\phi-\partial_\mu(\frac{\partial\mathcal{L}}{\partial(\partial_\mu \phi)})\...
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2answers
60 views

Momentum density from stress energy tensor in field theory

Question is similar to one in this link 1. Let us consider very simple Lagrangian that contains only kinetic energy. Its interpretation follows from saying that the field we are varying $\mathbf{u}$ ...
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1answer
32 views

Effective action for 1D anti-ferromagnet

I'm following Fradkin's (p. 204) derivation of the effective action for a 1D anti-ferromagnet. He splits the spin field $\vec{n}$ into two pieces - a slowly varying $\vec{m}(j)$ which is the order ...
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59 views

Lagrangian for system of particles with statistical distribution $f(x_1, …, x_N)$

For system of $N$ particles it is known that it is a good model to take Lagrangian to be (ignoring electromagnetism) $$L = \sum \limits_{i=1}^N \frac{1}{2}(m_i \mathbf{v}_i^2) -U(\mathbf{x}_1, ..., \...
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1answer
76 views

How does a Lagrangian with delta potential transform to a Hamiltonian?

Suppose the Lagrangian was given as: $$L = \frac{1}{2}\int_{-\infty}^{\infty} \underbrace{\left(\dot{A(}z)^2-A(z)^2\right)+\left(\dot{Q(}z)^2\delta(z)-Q(z)^2\delta(z)\right)+2\dot{Q(}z)\cdot A(z) \...
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1answer
33 views

Variational principle if coordinate transformation depends on fields

Assume we have a Lagrangian that is given in terms of Lagrangian density. $$ L = \int \mathcal{L} (\Phi, \partial_{\mu}\Phi, x) d^N x $$ Also assume that $\Phi : \mathbb{R}^N \to \mathbb{R}^N$ and ...
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0answers
77 views

Why does light come as quanta of the harmonic oscillator?

I've recently been learning the basics of Quantum optics and it seems to be a fundamental concept that light is best described in the framework of the Quantum Harmonic Oscillator. This lead to a ...
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2answers
46 views

In QFT, are forces made out of multiple fields?

I’ve been reading about 1,5 books about quantum physics and I’ve also watched a few YouTube videos. In one book, I learnt that there are fields, such as the electromagnetic field, which carries forces ...
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1answer
159 views

$CP$ Invariance of Yang-Mills Vacua in Electroweak Theory

It is well know that quantum Yang-Mills theory has a periodic vacuum structure. Consider electroweak theory. For a single generation of fermions, the theory is CP invariant. I would like to know if ...
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1answer
90 views

Why do we have redundant degrees of freedom?

Preliminaries: Consider the homogenous Maxwell's equations $$\partial_\mu F^{\mu\nu}=0.$$ and $$\partial_{\sigma} F_{\mu \nu}+\partial_{\mu} F_{\nu \sigma}+\partial_{\nu} F_{\sigma \mu}=0$$ Since ...
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1answer
65 views

How to “transfer” indices from dot product to metric?

In this source, the author (Andrzej Pokraka, Solutions to problems from Peskin & Schroeder) is computing an integral related to scalar QED. In the step where the equation is labelled (29), the ...
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0answers
49 views

Possible Feynman diagram for $\tau^+ \rightarrow p \mu^+ \mu^-$ and $\tau^+ \rightarrow \bar{p} \mu^+ \mu^+$?

I want to know the possible Feynman diagram for these two lepton family, lepton and baryon number violating tau decays. These decays are forbidden in the Standard Model. But the further extension of ...
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2answers
52 views

Emergence of rotational symmetry on 2D square lattice

On page 74 of David Tong's Statistical Field Theory lecture notes, it is said that $(\partial_1\phi)^2 + (\partial_2\phi)^2 $ respects both $D_8$ (that includes discrete four-dimensional rotation ...
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1answer
38 views

PDE from Hamiltonian density

For the wave equation Hamiltonian density is $2H=\phi_t^2+\phi_x^2$ while the Lagrangian density is $2L=\phi_t^2-\phi_x^2$. I can easily compute the pde from the Lagrangian density but how does one do ...
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1answer
71 views

Transformation of the derivative of the scalar field in Ramond's book about QFT

In the book by Pierre Ramond about quantum field theory, he explores in chapter 1.4 (p.13) the behavior of fields under Poincaré transformations. He starts by explaining that infinitesimal ...
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0answers
29 views

MCS Lagrangian and Euler-Lagrange

I'm trying to solve the Euler-Lagrange equation for the MCS Lagrangian density as given by Kharzeev in this article (Eqn. 7): $$ \mathcal{L}_{\textrm{MCS}} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-A_\mu J^{...
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1answer
65 views

Expanding superfields: inconsistency of notation?

If I have a wavefunction of a fermion field $\Psi[\psi]$ I can expand it like so about some vacuum: $$\Psi[\psi] = \Psi_0[\psi]( a + \int a(x)\psi(x)dx+\int a(x,y)\psi(x)\psi(y)dxdy+...)$$ Now all ...
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2answers
192 views

Could there exist a “locality” field? [closed]

What I mean is (and I'm a layperson on the subject), can there exist a field that pervades the universe - like the Higgs field - that interacts with particles to give them "distance" or "space" ...
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1answer
60 views

Symmetry modulo total derivative term in Noether's Theorem

I came across the proof of Noether's Theorem in David Tong's notes (page 14) on QFT. He writes something like, We say that the transformation $$\delta\phi(x) = \chi (\phi) \tag{1.34}$$ is a ...
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1answer
93 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 ...
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0answers
56 views

Kosterlitz-Thouless transition and renormalisation group theory [closed]

I'm trying to understand the Kosterlitz-Thouless transition in 2d systems. There is a section in Altland and Simons' Condensed Matter Field Theory that discusses the phenomenon, but I don't really ...
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1answer
24 views

Srednicki chapter 22: continuous symmetries and conserved current

In Srednicki's book he says that: The Noether current plays a special role if we can find a set of infinitesimal field transformations that leaves the lagrangian unchanged, or invariant. In this ...
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1answer
32 views

Can you have a purely position(field) dependent Lagrangian(density)?

Let us start with the usual Euler Lagrange equations, and impose $L=L(q)$ only-$$\frac{\partial L}{\partial q}=\frac{d}{dt}(\partial L/\partial \dot{q})$$, and the RHS becomes zero, implying $$\...
3
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1answer
65 views

Why doesn't the Lagrangian depend on higher-order derivatives of position?

This isn't a duplicate of already-answered questions, but rather a follow-up of this answer. The author presents a field-theoretical argument whereby a problematic run-away particle creation is ...
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1answer
92 views

2 dimensional massless scalar field propagator in position space

I have been trying to calculate the massless scalar field propagator in position space by directly Fourier transforming the momentum space propagator. $$\int{d^2p\frac{1}{(p^0)^2-(p^1)^2}e^{-i(p^0t-p^...
3
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0answers
69 views

Confused about the gauge transformation of the amplitude tensor for gravitational waves

Far away from the field sources, where the energy-momentum tensor $$T_{mn}=0 \tag{m,n=0,1,2,3}$$ The linearized EFE becomes $$\Box \bar h_{mn}=0 \tag{1}$$ where $\bar h_{mn}$ is the trace-reverse ...
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1answer
52 views

Second quantisation for fermions

I am trying to build a model for reactions on a lattice in the Doi-Peliti formalism. Suppose there exists a lattice of $N$ sites indexed by $i$. Each site can be either occupied or unoccupied. ...
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0answers
22 views

Some questions on a certain Lagrangian related to $n$ dyons

I am trying to understand Gibbons and Manton's article. I am trying to understand the Lagrangian that they wrote down, from a physical point of view. Let me paraphrase from that article. "Consider $...
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1answer
56 views

The relation between Chern-Simons Theory and Yang-Mills Theory

So from this page, I know that there is a relation between Chern-Simons Theory and Yang-Mills Theory, but I have difficulty proving the identities in the document. I was going to prove $$\partial_\...
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0answers
44 views

Regarding notation used for infintesimal parameters of the Lorentz algebra and generators of the Lorentz group

I have a confusion regarding the notation that is used for infintesimal Lorentz transformations and the parameters that define the Lorentz transformation (used in various books such as Srednicki's and ...
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2answers
121 views

Gaussian path integral is equivalent to saddle-point?

If we have a path integral involving many fields, $$Z = \int \mathcal D \phi_1 \cdots \mathcal D \phi_n \exp(-S[\phi_1,\ldots, \phi_n]),$$ and $\phi_n$ occurs only quadratically-- i.e. the $\...