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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Second quantisation for dynamical systems

The paper "Perturbative approach to an $A + B \rightarrow C$ reaction-diffusion system", (Z. Phys. B 96, 137-144 (1994)), by Conrad and Trimper, applies the Fock Space formalism for the master ...
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1answer
66 views

Why is important for the energy density to be positive definite in field theories?

Why is important that the energy density be positive definite in field theories?
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22 views

Fields and gauge transformations vanishing at infinity

I find that, in field theory, it is very often assumed that the fields (classical) vanish at infinity. The same assumption is also applied to gauge transformations, for example, when saying that the ...
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1answer
25 views

Ginzburg Criterion (Ising model)

In my statistical field theory class, we were told that we want the magnetization fluctuations in the Ising model to be smaller than their background. Specifically this was written as $$\langle\phi^2\...
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Why is the Coulomb Gauge enough to fix extra degrees of freedom?

In classical electrodynamics, we have after the Coulomb gauge is applied: $$ \Delta U = -\frac{\rho}{\epsilon_0} $$ $$ \Box \vec{A} = \mu_0 \vec{j}-\frac{1}{c^2} \vec{\nabla} \frac{\partial U}{\...
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1answer
54 views

How to calculate the conserved energy $E$ from the Lagrangian?

I am reading a PhD thesis that considers the Lagrangian $$\mathcal{L}=\partial_\mu\phi\partial^\mu\phi^\star-U(|\phi^2|)$$ where $\phi$ is a complex scalar field and $U(|\phi|^2)$ is an arbitrary ...
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42 views

Constraints vs Boundary Conditions

I have a very broad question about how the mathematical framework that classical theories of physics utilize to solve problems. The question is: What are the intrinsic differences between the ...
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2answers
147 views

$SU(2)$ symmetry of $\mathcal{L}=\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - \Phi^{\dagger}M\Phi$

I'm considering a Lagrangian of two complex scalar field: $$\mathcal{L}=\partial_{\mu}\phi_1^{*}\partial^{\mu}\phi_1-m_1^2\phi_1^{*}\phi_1+\partial_{\mu}\phi_2^{*}\partial^{\mu}\phi_2-m_2^2\phi_2^{*}\...
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132 views

Inconsistency between $d_A = d + A \wedge$ and $d_A = d(..) + [A,..]$?

I am confused by something basic stated in this https://physics.stackexchange.com/a/429947/42982 by @ACuriousMind and some fact I knew of. Here $d_A$ is covariant derivative. $d_A A=F$ --- @...
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126 views

Is there a sharp distinction between particles and fields?

The quantisation of the electromagnetic field, proposed originally by Planck, blurs the distinction between particles and fields. 'Point' particles become fuzzy and subject to a wave equation. Also, ...
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33 views

Gribov's phenomenon

In the well known textbook by Itzykson-Zuber "Quantum Field Theory" there is a discussion of the Gribov phenomenon in non-abelian gauge theories (see Section 12-2-1). To my taste, the discussion ...
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76 views

Quantum field theory with only 3-point vertexes

Given an arbitrary quantum field theory, can I always write it in terms of another (different) quantum field theory containing only operators with 3 fields? (i.e. vertexes with 3 legs) I guess that ...
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Is the derivative with respect to a fermion field Grassmann-odd?

Fermion fields anticommute because they are Grassmann numbers, that is, \begin{equation} \psi \chi = - \chi \psi. \end{equation} I was wondering whether derivatives with respect to Grassmann numbers ...
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Field Theory: Converting $\int_0^{x_0} d^dx$ to $\int_0^{x_0} dr$, where $r=^{\textrm{def}}\|x\|$

For my Statistical Field Theory class (http://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf), the prof converts integrals over each element of a vector $x$ into a single integral over the magnitude of the ...
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253 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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Adding a total derivative to the Lagrangian does not preserve $\int\mathrm{d}^3\mathbf{x}~ T^{00}$

In problem 3.3 of Schwartz's QFT, the first two questions ask us to prove that if we add a total derivative to the Lagrangian: $$ \mathcal{L}\mapsto\mathcal{L}+\partial_\mu X^\mu\tag{1} $$ then $$ \...
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Simplify Yang-Mills Equation of Motion in the 1-form gauge field $A$

We know the Yang-Mills theory Equation of Motion (eom) without source $$ * D * F = * (d (* F ) + [A, (* F )])= 0. $$ My question is that what are the most simple form we can boil down this ...
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23 views

Interpretation of vanishing Noether charge

I was told that Gauge symmetries are redundancies because the Noether charge of a gauge symmetry vanishes, i.e. that there exist no observable quantities that would allow you to distinguish two ...
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1answer
50 views

Mistake or Rewriting of Yang-Mills in Nakahara

I am familiar with Yang-Mills equation of motion E.O.M. (without matter or source fields) in differential form. $$ D * F =0 $$ and Bianchi identity $$ D F=0 $$ where $F= dA + A \wedge A$ and $D=d + [...
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51 views

Why do terms in a field theory Lagrangian that are polynomial in the fields collectively called the “potential”?

Field theory Lagrangians are often of the form of a kinetic term plus a source term minus a potential term. How do we know that the potential term is a polynomial in the fields? On a related note why ...
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88 views

Connection between gauge invariance and Lorentz invariance

This question is presented in the context of Weinberg's QFT book treatment, in particular considering the electromagnetism chapter. It begins in chapter 5 where Weinberg argues that in order to have ...
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1answer
48 views

Magnetic field due to a circular ring

In the EMFT notes of MIT Course-ware, the derivation of the magnetic field due to a circular ring at its axis, using Biot-Savart's Law and the cylindrical coordinate system is done as follows, I am ...
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Why is it that the equation of a massless scalar field *must* be conformal invariant?

I'm reading a paper [1], p.111 where it is said that: However, the equation of scalar field with zero mass must be conformal invariant while equation $\square\varphi=0$ does not satisfy this ...
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49 views

Interpreting the conserved charge in scalar QED

In scalar QED, applying Noether's theorem for internal global symmetries results in a Noether current that is dependent on the gauge because of the presence of the covariant derivative. $$j_\mu=-i(\...
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3answers
42 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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23 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
222 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
60 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
57 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
77 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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40 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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91 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
51 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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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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29 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
113 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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67 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
179 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
304 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
37 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
89 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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86 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
23 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
55 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
93 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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40 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
55 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 ...