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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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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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1answer
37 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{1}\...
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1answer
32 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
65 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
68 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
18 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
51 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
63 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
83 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
19 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
53 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
23 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
89 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
103 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
24 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
39 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
24 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
93 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
36 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
40 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
105 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
68 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
44 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
73 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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0answers
22 views

Non-relativistic quantum electrodynamic lagrangian: number of dynamical variables greater than 6

There is an argument I do not understand given in "Introduction to quantum electrodynamics" by Cohen Tanoudji (page 111 for the french version of the book). We are dealing with the non-relativistic ...
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4answers
88 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
30 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
48 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
38 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
37 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
30 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
29 views

Piecewise solution to Euler-Lagrange equations

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
56 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
61 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
71 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 }...
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2answers
74 views

Field momentum of Klein-Gordon Lagrangian

Given the Lagrangian $L$ of the field $\phi$ the field momentum $\Pi$ reads: $$L_{KG}=-\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m^2\phi^2$$ $$\Pi=\frac{\partial L}{\partial(\partial_\...
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0answers
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Why do we need a two higgs doublet model? [duplicate]

Why do we need a two Higgs doublet model? What are the shortcomings in the standard model one Higgs doublet model? And is this 2HDM an independent approach or it is embedded in supersymmetric theory?
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0answers
34 views

The logic behind a specific step of SUSY variation

I'm following Neil Lambert's Supersymmetry notes, and there's a step in equation 5.67 which has me stumped. He says he uses the fact that "$C\gamma_\mu$ is symmetric". I don't see how that helps, and ...
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2answers
141 views

Does the electromagnetic field have a “rest mass” that is conserved?

In an answer to this Physics SE question, @ChiralAnomaly demonstrated that, indeed, there is a minimum field energy density observable at any point in an EM field. With a bit more calculation, it's ...
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0answers
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0answers
37 views

Tunneling in quantum mechanics and domain wall in 1d Ising model

I am following David Tong's lecture notes on Statistical Field Theory. You can find it here. In page 51-52, he said the domain wall in 1d Ising model is the same as quantum tunnelling in quantum ...
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2answers
49 views

Different definitions of Functional Derivative

In studying QFT and General Relativity, I came across two different definitions of Functional Derivative, and I'd like to know if they are equivalent. Firstly, in Wald's book General Relativity, as ...
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1answer
92 views

Problems of Klein Gordon equation

Consider the Klein-Gordon equation $$(\square+m^2)\varphi=0.$$ People usually claim that $\varphi^* \varphi$ cannot be interpreted as a probability density because $\int d^3\vec{x}\varphi(t,\vec{x})^*...
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0answers
28 views

Why do we need the two higgs doublet model? [duplicate]

Why do we add an extra doublet to the two higgs doublet model?I mean what are the conditions due to which we had to add an extra doublet. What are the limitations with only one doublet and how 2HDM ...
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1answer
56 views

Chern-Simons equation of motion

How do I get the equation of motion of the Chern-Simons Lagrangian below? Is there the product rule at work? Do I have to sum over the indices?
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1answer
69 views

Covariant derivative in field theory

I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-half - spin-1 field: $$ ...
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2answers
95 views

Is there a higher dimension analogue of Noether's theorem?

So I have recently read the proof of Noether's theorem from the book variation calculus by Gelfand. Basically, what I have already seen is that for any single integral functional, if we have a ...
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1answer
46 views

UV divergence integral

Could anyone please explain how to calculate integral such as $$\frac{\Omega}{2}\int_{-\infty}^{+\infty} \frac{d^3k}{(2\pi)^3}\ln\left[{1+\frac{a^2}{k^2}}\right]=-\frac{\Omega a^3}{12\pi}+I_0~?$$ ...
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1answer
72 views

Intuition between this construction of the sympletic form for classical fields

In this paper, Wald presents a quite general construction of a sympletic form for classical fields. If I understood (which I might have not, and in that case corrections are highly appreciated), the ...
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0answers
47 views

Constructing gauge symmetry for $SU(2)\times U(1)$

I'm currently reading the book "Classical Theory of Gauge Fields" by Rubakov, therefore I will use his convention in this question. In the following we assume that: $\phi$ comprises columns ...