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

(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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30 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
180 views

Boundary conditions in variational principles

In classical mechanics, the condition to fix the variation of the trajectory at the endpoints has a clear-cut meaning. We want the system to propagate from $x\in\mathcal{C}$ to $y\in\mathcal{C}$, ...
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1answer
36 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
33 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 ...
3
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1answer
621 views

Bogoliubov transformation for fermion (exercise in Piers Coleman)

I am trying to solve the exercise 3.2 in Piers Coleman's Introduction to many body physics. It's about fermionic Bogoliubov transformation with only 2 fermion operators $a_{1}^{\dagger}$, $a_{2}^{\...
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1answer
50 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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0answers
48 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
52 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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1answer
208 views

Gravitons and self-interaction

In the book quantum field theory and standard model by Schwartz, there is a problem 9.4 that says by considering lorentz invariance of Compton scattering, you can prove that for spin 1 massless field ...
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2answers
105 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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27 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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56 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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6 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 ...
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3answers
453 views

Square of the Maxwell Field Tensor

I want to prove that the square of the Maxwell field tensor $$F_{\mu\nu}F^{\mu\nu}=2(B^2-E^2),$$ but I got $F_{\mu\nu}F^{\mu\nu}=2(-B^2+E^2)$ instead. Here's what I did: $$F_{\mu\nu}F^{\mu\nu}=F_{0\nu}...
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0answers
47 views

Holography and field theory

If the holographic principle is right, and there is a FINITE but very big degrees of freedom per Planck area, would it imply that in that regime we should give up local field theory and addopt some ...
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1answer
122 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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1answer
34 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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0answers
50 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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3answers
1k views

Is String Theory a Field Theory?

Is String Theory a Field or Quantum Mechanical Theory of the String rather than a Particle? I should know this having studied this for a term, but we jumped into the deep end, without really ...
2
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1answer
41 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
150 views

How to define conserved charges in Euclidean field theory?

In a field theory with signature (1,d), conserved charges are obtained by integrating the time component of a conserved current over a spatial region. What are the corresponding equations and ...
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3answers
10k views

Differentiating Propagator, Green's function, Correlation function, etc

For the following quantities respectively, could someone write down the common definitions, their meaning, the field of study in which one would typically find these under their actual name, and most ...
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3answers
681 views

Is the shorthand $ \partial_{\mu} $ strictly a partial derivative in field theory?

The Euler-Lagrange equation for particles is given by $$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q},\tag{1}$$ and for fields it is $$ \partial_{\mu} \frac{\...
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2answers
88 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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1answer
291 views

Squaring the E&M (Maxwell) field strength tensor

In Section 3.4 of Schwartz's "Quantum Field Theory and the Standard Model", the square $F_{\mu\nu}^2$ of the field strength tensor $F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu}A_{\mu}$ is ...
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2answers
46 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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0answers
18 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 ...
4
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1answer
73 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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0answers
56 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
133 views

Why is the spacelike conserved charge due to spacetime translations the momentum?

Whilst reading several books on QFT, I have come across the derivation of the conserved charges due to the symmetry under spacetime-translations. I can follow the derivations, and have that the ...
6
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1answer
152 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 ...
3
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1answer
130 views

Force via “exchange particles” or “via field”

More or less I have come across two concepts to explain non contact forces: FIELD CONCEPT: modification of space by the source which in turn produces force on the other (That is in my classroom ...
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1answer
28 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
74 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
40 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
85 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
62 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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1answer
47 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^...
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0answers
48 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
45 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
60 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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1answer
37 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 ...
4
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1answer
59 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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2answers
188 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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0answers
1k views

Suggested reading for classical field theory [duplicate]

I am reading a marvelous book Classical Field Theory by E Soper, but it is mathematically too compact and sometimes I am unable to follow the equations. Can anyone suggest a side book for solution of ...
3
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1answer
363 views

Lagrangian description of Brownian motion?

I'm interested in the existence of a Lagrangian field theory description of Bronwnian motion, does such a thing exist? Given a particle of some spin $\sigma$, which has a Lagrangian associated with ...
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2answers
94 views

Why is effective mass a tensor?

So I came across the effective mass concept for solids the other day. It was mentioned that the effective mass is a tensor and may have different values in different directions. However, this is stark ...
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2answers
147 views

Energy contributions of Hamiltonian density

In Lancaster and Blundell, Quantum Field Theory for the Gifted Amateur, p.99, Hamiltonian density is \begin{equation} \mathcal{H}=\frac{1}{2}[\partial_0\phi(x)]^2+\frac{1}{2}[\nabla\phi(x)]^2+\frac{...