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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43 views

Why is it that magnetic fields(or any field)not move in space? [closed]

When I imagine a magnetic field produced by a magnet, or the electric field produced by a charge, I've learned that the fields are stationary, however, their value(across space) changes. If I placed ...
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0answers
29 views

Lagrangian of Phonon-photon

A quite interesting but also hard problem are Polaritons. As far as I have understand the concept it's about phonons coupling to light. The Lagrangian function should therefore have a term for the ...
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1answer
231 views

Relating the Yang-Mills field-strength to the Maxwell tensor in $SU(2)$ gauge theory

I'm studying topological monopoles in a $SU(2)$ Yang-Mills theory with spontaneous symmetry breaking, through the book "Topological Solitons", by Manton and Sutcliffe. In section 8.2, the authors ...
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1answer
62 views

Confused about scalar fields

A scalar field is one which is unchanged under rotation. But how do we decide whether a given field (e.g., the temperature in a room) is unchanged under rotation or not? We need to measure the ...
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2answers
58 views

Electromagnetic linear response theory in function integral language [closed]

I'm being confused with the discussion in Altland&Simons' textbook, page P391-392. How is eq(7.46) derived? Specificly, Here we have the action: $$S[\bar{\psi},\psi,A]=\int dx \bar{\psi_\sigma}\...
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2answers
79 views

How to unify the rotation matrix of $SU(2)$ operator and $(z_1, z_2)$ representation?

I am following the Xiao-Gang Wen's book: Quantum Field Theory of Many-body Systems. In Ch. 5.6 about non-linear $\sigma$ model, it use a rotation operator $U$ to change the spin quantization from $z$ ...
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0answers
41 views

Motivation for Weinberg angle in electroweak gauge interaction?

Suppose I have the following lagrangian If we only focus on the neutral current in the lagrangian: Where $L$ is defined as: And $Y_L$,$Y_{R}^{\nu}$,$Y_{R}^{e}$, are the hypercharge values of the ...
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2answers
119 views

Goldstone bosons when SSB potential has two fields

A theory consists of two complex scalar fields $φ_0$ and $φ _1$ with a symmetry-breaking potential $$V(|φ_0|^2 + |φ_1|^2).$$ How many Goldstone particles will there be in the theory?
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1answer
94 views

Theta-dependence of massive Schwinger model

I've read in Coleman's paper on the massive Schwinger model (and in other papers on the same topic, like this one) that the model's Hamiltonian contains a topological $\theta$-term. However, if I ...
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1answer
54 views

Difference between mean-field theory and large $N$ of $CP^{N-1}$

I am reading the Ch.14 of Auerbach, Interaction electrons and quantum magnetism about $CP^{N-1}$ which describes non-linear $\sigma$ model. The complex field is $\mathbf{z}=\left(z_{1}, z_{2}, \ldots,...
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0answers
17 views

Can we use discrete symmetry in order to generate neutrino mass in two higgs doublet model?

It is seen that an u(1) symmetry is generally used to explain the seesaw mechanism for neutrino mass in 2HDM.it is used because the theory then naturally predicts the existence of a right handed ...
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0answers
30 views

Help with an specific example of a higher derivative Lagrangian

I want to find the equation of motion that comes from the following Lagrangian density $$\mathscr{L}=\mathbf{E}\cdot\left(\nabla^{2}\mathbf{E}\right)$$ where $E_{i}=\partial_{i}\phi\;(i=x,y,z)$ . In ...
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1answer
58 views

Why are fields which do not transform in a certain way not fundamental?

When I was first exposed to Math physics textbooks and textbooks on vector calculus, I found: Temperature distribution in a room $T(x,y,z,t)$ or the density variation in a fluid $\rho(x,y,z,t)$ etc ...
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2answers
115 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. ...
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2answers
68 views

Computation in QFT

I'm always a mess with the upstairs and downstairs notation. To be specific, say I want to calculate the Euler-Lagrange equations of \begin{equation} \mathcal{L} = \frac{1}{2}\partial^\mu\phi \...
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2answers
183 views

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

I am trying to derive the Noether theorem from the following integral action: \begin{equation} S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}% \phi_{r},x\right) , \tag{II.1}\...
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2answers
307 views

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

I would like to consider the problem of the total derivative of a given functional \begin{equation} \mathcal{L}\bigg[\phi\big(x,y,z,t\big),\frac{\partial{\phi}}{\partial{x}}\big(x,y,z,t\big),\frac{\...
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1answer
39 views

Sum over real photon polarizations. The minus sign

Ok for real photons there is the formula when summing over the polarizations: $$ \sum_{\lambda=\pm}\epsilon^{*\mu}_\lambda\epsilon^\nu_\lambda = -\eta^{\mu\nu}$$ But if I have a matrix element of ...
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3answers
99 views

Non-existence of double time-derivative of fields in the Lagrangian and violation of equal footing of space and time

In classical field theory, we consider the Lagrangians with single time-derivative of fields whereas double derivative of the field w.r.t. space is allowed sometimes. I understand that the reason of ...
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0answers
87 views

Two-dimensional bosonic field theory

I'm struggeling with the following question: Consider a two-dimensional bosonic field theory defined by the following action $$S =\frac{k}{2} \int dx_{1}dx_2 [(∂x_1 φ(x_1, x_2))^2 + (∂x_2 φ(x_1, ...
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1answer
24 views

Why do we need a two-Higgs doublet model and why only one extra doublet is added to the extension?

The general two higgs doublet model can neither give masses to neutrinos without considering a see-saw mechanism nor it can unify gravity with other three forces. To explain dark matter also we have ...
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1answer
57 views

Scalar product of free field and conjugate momentum

Given $[\Phi (x), \Pi(y)] = \delta^{3}(x-y)$,$ $ $\Phi|\phi\rangle = \phi(x)|\phi\rangle$ and $\Pi|\pi\rangle = \pi(x)|\pi\rangle$, I am trying to prove $\langle\phi|\pi\rangle \sim e^{i\int d^{3x}\...
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2answers
103 views

Question on energy conservation from the stress tensor of a classical scalar field

I am struggling to answer an old general relativity exam question, which is as follows: "Consider a scalar field $\phi(t,x^i)$ with potential $V(\phi)$ on a general spacetime. Its stress tensor is ...
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1answer
88 views

Is this a right approach to show that $\partial_{\mu} \phi \partial^{\mu} \phi $ is Lorentz Invariant?

When trying to convince myself that $\partial_{\mu} \phi \partial^{\mu} \phi $ is Lorentz Invariant, I stumbled upon this approach: The last equation should read - $\partial_{i} \phi \partial^{i} \...
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0answers
46 views

Magnetic field of a finite conductor cylinder

I'm trying to calculate the magnetic field outside a finite cylinder of radius R and height 2H, whose axis is the z-axis. Through the cylinder is flowing a constant current density $\vec{J} = J_0 \vec{...
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1answer
57 views

Operators of the special orthogonal group $\mathrm{SO}(3)$ in 3 dimensions

My professor taught us that when we want to rotate a 3D vector we need a $3\times 3$ matrix $R$ that is a rotation matrix. The set of all these matrices is the special orthogonal group in three ...
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0answers
34 views

Interaction Lagrangian up to quartic order

I have a Klein-Gordon Lagrangian of scalar fields and I add an interaction term that depends only on the fields (not their derivatives). The free Lagrangian is invariant under some infinitesial ...
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3answers
109 views

Why do we demand that the counterterms in $\varphi^3$ theory be $O(g^2)$?

In Srednicki's QFT book, section 9, he introduces the $\varphi^3$ lagrangian: $$\mathcal{L}= -\frac{1}{2}Z_\varphi(\partial_\mu\varphi)(\partial^\mu\varphi) -\frac{1}{2}Z_mm^2\varphi^2 +\frac{1}{6}...
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3answers
146 views

Origin of $\sqrt{-g}$ in the integral of action $S$

I have a question that might (and probably will) be stupid: I do not understand where does the factor $\sqrt{-g}$ (i.e. $\sqrt{-\det\left(g_{\mu\nu}\right)}$) come from in the action integral S when ...
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1answer
32 views

Is there any experiment going on to test the TWO HIGGS DOUBLET MODEL?

We know that the two higgs doublet model which is a beyond standard model theory predicts five higgs bosons.Is there any experiment that is going on to test this theory and if so have they found any ...
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1answer
48 views

Variation in field theory with respect to one quantity

In my QFT course we are supposed to vary the action of a for a scalar field coupled to an electromagnetic field with the following Lagrangian density: $$\mathcal{L} = [D_\mu\phi(x)]^*D^\mu\phi(x)-m^2\...
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1answer
25 views

How to see linearity of an interaction if it's lagrangian density is known?

The Lagrangian of electrodynamics is $-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+A_\mu J^\mu$ we know that electrodynamics is linear in special relativity but when we go to general relativity it becomes non-...
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1answer
158 views

Infinitesimal transformations that leave the action invariant

I have the Klein-Gordon Lagrangian for three scalar fields and I want to find three independent infinitesimal transformations that leave the action invariant. I suppose that these three ...
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1answer
47 views

Free boson Equation motion from action

So in David tongs notes we have $$S=\frac{m}{8\pi}\int d^2x\partial_i\varphi\partial^i\varphi$$ and he finds that the equation of motion is $$[\partial_{t}^2-v^2\partial_{x}^2]\varphi=0$$ now my ...
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1answer
48 views

Commutation relations in Gupta-Bleuler formalism

When quantising the EM field thanks to the Gupta-Bleuler formalism, Itzykson and Zuber assume that the canonical commutation rules are $$ [\hat{A}_\rho (t,\vec{x}), \hat{\pi}^\nu(t,\vec{y})]= i \, ...
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2answers
140 views

Conserved currents in quantum electrodynamics

A general Noether theorem in fields theory says that an infinitesimal symmetry of the action leads to a conserved current $j^\mu$, i.e. $\partial_\mu j^\mu=0$. Below I would like to consider a minor ...
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1answer
110 views

Why is gravity the weakest force if, theoretically, it is made up of all the other forces' fields?

I'm quite new to the physics world and want to get an idea as to how physicists have been able to sum all fields/forces (higgs boson, electromagnetic, weak force, strong force etc) in a said object, ...
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0answers
53 views

Covariant derivative of a composite field and the chain rule

I have a gauge theory with some rather strange covariant derivatives and I am wondering how they act on a composite field like $\psi= \phi\psi'$. In my setup, the covariant derivative acting on a ...
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2answers
84 views

Showing the form of the covariant derivative of $\phi$, if $\phi$ transforms as the adjoint representation of $SU(n)$

I want to show that if $\phi$ transforms as the adjoint representation of SU(n), its covariant derivative is given by $\textbf{D}_\mu \phi = \partial_\mu \phi + i [\textbf{A}_\mu, \phi]$. (Exercise in ...
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1answer
104 views

Non-relativistic E&M Lagrangian: number of dynamical variables greater than 6

There is an argument I do not understand given in "Introduction to quantum electrodynamics" by Cohen-Tannoudji (page 111 for the French version of the book). We are dealing with the non-relativistic ...
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4answers
95 views

Are electromagnetic waves a substance? [duplicate]

I would generally consider fields to not be substances, since substances are generally associated with matter. I know that energy is not a substance. Are electromagnetic waves a substance?
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0answers
46 views

Field degrees of freedom from equations of motion and higher spin

It is my understanding that we compute the number of degrees of freedom of a quantum field as the number of its components minus the number of non trivial equations we get by taking the divergence of ...
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0answers
54 views

Derivation of Coulomb's law from classical field theory

In the section on Coulomb's law in QFT by Schwartz, he expands $-\frac{1}{4}F_{\mu\nu}^{2}$ to get $-\frac{1}{2}(\partial_{\mu}A_{\nu})^{2} + \frac{1}{2}(\partial_{\mu}A_{\mu})^{2}$, can someone ...
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1answer
54 views

$\phi^3$ 2D 1-loop diagram disambiguation

I would like to calculate the 1-loop 1-PI correction to the propagator for $\phi^3$ scalar theory in 2 dimensions, where the integral is finite. Performing the usual procedure (Feynman trick, Wick ...
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0answers
69 views

What do we mean by scale invariance in a classical field?

First of all, I read many questions but they don't seem to answer my specific question. So, here it goes According to Francesco's Conformal Field Theory and many other books, a scale transformation \...
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0answers
34 views

Canonical commutation relations for a real scalar field

I am taking my first course in QFT and have come across this problem From the canonical commutation relations for a real scalar field $\hat{\phi}$ show that $$[\partial_i \hat{\phi} , \hat{\phi}...
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0answers
51 views

Piecewise solution to Euler-Lagrange equations in effective field theory

I would like to consider a background for a quantum field theory made up by connecting continuously two different solutions of the Euler Lagrange equations. The problem is one dimensional (let's call ...
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1answer
60 views

Formally condition and sample distribution of physical field

Is there any formal way to generate samples of physical fields (e.g. electromagnetic field, fluid flow) conditional on observations? The samples would need to satisfy conditions like being continuous, ...
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0answers
68 views

Error in proof of Wick's theorem

I am having a little trouble proving wick's theorem. I'll start from the last step that I know is correct. We define $$\left\langle \prod_{j=1}^{2m}x_{i_j}\right\rangle:=\left.\frac{\partial^{2m}}{\...
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1answer
84 views

Understanding solutions of the Dirac equation

In one of the lectures that I'm currently taking we encountered the Dirac equation. The general solution was given as $$\psi ( x ) = \sum _ { s } \int \frac { d ^ { 3 } \bf { p } } { ( 2 \pi ) ^ { 2 }...