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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Relation between the spin of a field and the way charges interact

When solving the problems of V. Rubakov's "Classical Theory of Gauge fields" book, I encountered the following phenomenon: For a real scalar fields (spin 0) $\phi$, if we consider the ...
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Derivative of the Metric Tensor With Respect to a Scalar Field

In Sean Caroll's Spacetime and Geometry Textbook, at page 183 (discussing scalar-tensor theories) Carroll defines a conformal metric by performing a conformal transformation as: $\tilde{g}_{\mu\nu} = ...
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Renormalization of Bending Young's Modulus and Diagrams

I'm reading through this arXiv paper and I ran into a problem when working through some of the RG calculations. In the supplemental info (p. 8), when evaluating the diagrams in Fig S1. The interaction ...
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We know the length scales of particles, but what are the length scales of bosonic and fermionic fields?

As particles are about 10-15 meters, the fields must be of inferior length scales? Then are they between Planck scales and electron scales? About which scale? Thanks in advance
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Does introducing a gauge field into the complex scalar field theory Lagrangian change its dynamics?

I've been reading Lancaster & Blundell, and in Chapter 14 they focus on the Lagrangian $$ \mathcal{L}=(\partial^\mu\psi)^\dagger(\partial_\mu\psi) - m^2\psi^\dagger\psi. $$ To impose invariance to ...
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Angular momentum in field theory

If we do an infinitesimal Lorentz transformation $\Lambda^\mu_\nu$ = $\delta^\mu_\nu$ + $\omega^\mu_\nu$ where $\omega^\mu_\nu$ is an anti-symmetric quantity, then the conserved current will be $j^\mu$...
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Symmetric energy-momentum tensor

In field theory, there's no guarantee that the energy-momentum tensor resulting from Noether's theorem is symmetric. The usual trick to construct a symmetric tensor is to add to the original energy-...
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126 views

Equations of motion from Lagrangian

I have the action $$S=\int d^4x\sqrt{-g} \Big[\frac{1}{8}\phi^2R- \frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi - \frac{1}{2}m^2\phi^2\Big]$$ where $\phi$ is a scalar field and $R$ is the ...
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48 views

Deriving equations from the Lagrangian density

I was working on problem 10.6 of the book "Problem Book in Relativity and Gravitation by A. Lightman, R. H. Price" where we derive the following equation for killing vectors: $\xi^{\nu;\...
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66 views

Lorentz invariance of the Klein-Gordon equation action

What I will say is not exclusively true for the KG equation, but let's take it as a simple example. When proving the invariance of its action under a Lorentz transformation, it suffices to show that ...
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Dirac delta and covariance [duplicate]

Is there a covariant form of the Dirac delta function? And how to build a covariant form of an identity that contains Dirac delta? To be more precise, what I am looking for is Some distribution that ...
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The action in classical field theory

In calculating the action in classical field theory, why do we integrate over all of spacetime, thus over all of time, while we don't have to do that in ordinary particle action?
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Why does Air act as a conductor in presence of a Strong Electric Field

We know that Air is a good insulator but why does it become conducting in presence of a strong electric field?.
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Lagrangian density under infinitesimal transformation

Consider the following Lagrangian density for two real scalar fields: $$\mathcal{L}=\frac{1}{2}\sum\limits^{2}_{i=1}(\partial^{\mu}\phi_i)(\partial_{\mu}\phi_i)-\frac{1}{2}\sum\limits^{2}_{i=1}m_i^2\...
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The planar limit, self-duality and their relation to two dimensions

In the lecture notes by Beisert on integrability, it is stated that integrability is a property mainly in two-dimensional field theories, with some higher-dimensional examples. As higher-dimensional ...
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Momentum commutation for boson field

Given a boson field described by $\psi(\vec{x})$, conserved momentum from the Lagrangian (which isn't relevant here) is $\vec{P} = \frac{\hbar}{2i} \int d^3 x \left( \psi^\dagger \nabla \psi - \nabla \...
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Lagrange multiplier for Robin boundary condition in variational minimisation

Consider the partition function for a scalar field $\{\phi:\mathbb{R}_{\geq 0}\to\mathbb{R}\}$, $Z=\int D\phi D\lambda\exp(-S)$ with the action $$S=\underbrace{\int_0^\infty dx \frac{1}{2}(\partial_x\...
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Writing the EL equations in the language of differential geometry

I want to explore generalised Noether currents obtained from $q$-form symmetries in an action. The regular way we obtain Noether currents is fairly straightforward: We have a 0-form symmetry $\phi \to ...
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Why in physics space cannot be discrete automata as claimed in Feynman's public Messenger lecture?

Feynman in this famous sixties public Cornell lecture claimed it's easy to prove "physics space cannot be discrete automata", otherwise it will soon violates existing physical observations. ...
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Why amplitudes are rational functions?

In Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory there are few statements about analytical structure of amplitudes. I want to understand statement: Tree amplitudes must ...
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1answer
51 views

Interaction of charges in gauge

Let's consider Maxwell theory: $$ \mathcal{L} = -F_{\mu\nu}F^{\mu\nu} = 2 A_\mu (\Box \eta^{\mu\nu} - \partial^\mu \partial^\nu) A_\nu $$ Is it possible to fix gauge $A_0 = 0 $ and concider ...
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51 views

Why is the creation operator of a particle in the conjugate field operator?

I am learning QFT, and we discussed that to quantize a complex scalar field, we do this: $$\begin{align*} \phi(x) &= \int \frac{d^3k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_k}} \big( a(\vec{k}) e^{-ikx} ...
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Wilson line and external source

Let's consider free Maxwell theory: $$ L = -\frac{1}{4g^2} F^{\mu\nu}F_{\mu\nu} $$ As I understand, one can describe external particles with help of Wilson lines: $$ W(q,l) = e^{iq\int_l dx^\mu A_\mu} ...
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63 views

Symmetry for dipole conservation in field theory

In article The Fracton Gauge Principle complex scalar field is considered. There's statement, that for conservation of charge one needs usual U(1) global symmetry: $$ \phi \to e^{i\alpha}\phi \...
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Time reverse transformation of 4-potential and its relation to Lorentz transformation

Until now, I thought electromagnetic potential $A^{\mu}(x)$ transform like $x^{\mu}$ under the Lorentz transformation: $$A^{\mu}(x)=\Lambda^{\mu}_{\ \nu}A^{\nu}(x).$$ But according to time reversal ...
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Why does $A_\mu$ undergo an adjoint representation matrix transformation?

This question pertains to the following passage from Weinberg's second volume on QFT. It appears on page 4, section 15.1. To make the Lagrangian invariant, we need a field $A^\alpha_\mu$, whose ...
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23 views

Invariance of $ds^2$ in Randall-Sundrum models

My question is about the invariance of the space-time interval $ds^2$ under orbifold symmetries, such as in the Randall-Sundrum model. In this model, the space-time is 5-dimensional with metric $$ds^2 ...
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1answer
55 views

Is a Lagrangian term $L_{kin}=(\partial^{\mu}\phi^{*})(\partial_{\mu}\phi)$ equivalent to $L_{kin}=\phi^{*}\partial^{\mu}\partial_{\mu}\phi$?

In looking at the Lagrangian of a (free for simplicity) complex scalar field $\phi$, we have a kinetic term that goes like: $$L_{kin}=(\partial^{\mu}\phi^{*})(\partial_{\mu}\phi)$$ Given instead, a ...
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59 views

The Conservation Laws in Peskin and Schroeder (page 309)

I'm working on the Conservation Laws in Peskin (page 309), but I was confused for it. In last section, I know that Classical: the action is stationary.i.e. $\delta S =0$ when $\phi(x)\rightarrow \...
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Calculating equation of motion in gauge theories: using ordinary derivatives or covariant derivatives?

For general gauge theories, the total Lagrangian density is given as $$L=-\frac{1}{4}F^2+L_M(\psi, D\psi)$$ where $L_M(\psi, D\psi)$ is the matter field with the ordinary derivative replaced by the ...
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1answer
58 views

Path integral in Euclidean field theory

I'm very unexperienced in QFT, but I'm reading Salmhofer's book on renormalization and, at the very begining of the book, he discusses Feynamn path integral formulation on quantum mechanics to ...
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Why in the field theory, particle's motion is described by 0+1 dimensional field theory?

I started reading the lecture notes on Path integral formulation by Ashoke Das. At the very first page of the introduction chapter, he says that - "a theory describing the motion of a particle ...
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60 views

Understanding the Functional Poisson Bracket

In classical field theory (for a single field $\psi$) the dynamical variables are defined to be functions of the fields $\psi$, $\pi$, $\partial_{x_{i}}\psi$ and maybe $\mathbf{r}$, where $\pi$ is the ...
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Euler-Lagrange equations of a scalar field

Given the Robertson-Walker metric for a scalar field $\phi(t)$, how can we obtain the equation of motion for this scalar field? I took the contravariant derivative of the scalar field is which is ...
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1answer
69 views

Relativistic Bose-Einstein Condensation (BEC)

I wonder if there is a concept similar to the one of BEC but arising from Quantum Field Theory instead that from the usual one developed in non-relativistic many-body Quantum Mechanics. In non-...
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29 views

Physical meaning for the determinant of the stress energy tensor?

I wonder if there is any physical meaning associated to the determinant of the stress energy tensor. Or do we know at least some context in which this quantity is meaningful?
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How to calculate the invariant amplitude for a decay process using wick's theorem?

I have difficulties to apply Wick's theorem to the following problem-set: We have three free scalar fields $\phi_1, \phi_2, \phi_3$. While the field $\phi_3$ has the mass $M$ while $\phi_1$ and $\...
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Feynman rules: Difference between vector-scalar-scalar and vector-fermion-fermion vertex

I'm trying to understand the difference in cross section between top-quark pair production and pair production of scalar top quarks (top squarks) as predicted by Supersymmetry. Assume the mass of the ...
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1answer
85 views

Multiplying terms with index notation

I am trying to expand the flat-space action $$ S_{BI} = -T_p \int{d^{p+1}} \sigma \ \mathrm{Tr}\left( e^{-\phi} \sqrt{ -\det(\eta_{ab} + 4\pi^2\alpha^2 \partial_a\Phi^i\partial_b\Phi^i + 2\pi \alpha ...
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Field strength and Levi-Civita tensor in Euclidean spacetime

I am trying to formulate gauge theory in Euclidean spacetime. I have Googled a lot of thing, but I cannot find any standard way. The following is what I am doing. Suppose in Minkowski spacetime, we ...
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39 views

Computing the Euler-Lagrange PDE for a given action

I am a bit confused on how to compute the Euler Lagrange equation for the action \begin{align} S(\phi, \bar\phi) = \int d^2xdt\ \left\{|\nabla \phi(t, x)|^2-|\partial_t \phi(t, x)|^2+\phi_1(t, x)^2\...
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55 views

Does a vanishing self-energy imply a Gaussian fixed point in a $\phi^4$ theory below the upper critical dimension?

Assuming a QFT description of a second-order phase transition. From the free theory, one obtains some critical exponents and one performs an $\epsilon$-expansion below the upper critical dimension. ...
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Concept in Quantum Field Theory [duplicate]

What is Quantum Field in QFT? How can we visualize such fields?
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UV and IR cutoff in massless quantum scalar field hamiltonian

I'm writing my bachelor's thesis and working on Srednicki's discussion about area laws for a massless scalar quantum field with Hamiltonian The field is in a spherical lattice of discrete points with ...
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1answer
58 views

Solving wave equations with Fourier transform: where are the time-independent solutions?

One typically solves waves (fields) equations in Fourier space. For example, the 1D wave equation $\frac{\partial^2\phi(x,t)}{\partial t^2}-\frac{\partial^2\phi(x,t)}{\partial x^2} = 0$ in Fourier ...
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114 views

How to measure the information loss due to coarse-graining of a physical system into a graphical representation?

Let's consider a system of bead-spring with $N+1$ beads connected with $N$ springs: The Hamiltonian of such a chain is: $$ \mathcal{H} = \frac{1}{2} k \sum_{i=1} ^N (\mathbf{r}_{i+1} - \mathbf{r}_{i})...
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1answer
76 views

Comparison between formulations of Noether's theorem

Version 1: An infinitesimal variation on the fields $\phi\mapsto\phi'$ is said to be a symmetry if $\delta \mathcal{L}:=\mathcal{L}(\phi',\partial\phi')-\mathcal{L}(\phi,\partial\phi)$ is a total ...
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1answer
124 views

Yukawa potential in higher dimensions

I am trying to calculate the integral \begin{align} E_n(\mathbf{r}) = \int \frac{d^n \mathbf{k}}{(2\pi)^n} \frac{ e^{i\mathbf{k}\cdot\mathbf{r}} }{ \mathbf{k}^2 + m^2 } \end{align} for $n > 2$ (the ...
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2answers
83 views

Lagrangian of Newtonian gravity

In this wiki page we can read: The Lagrangian density for Newtonian gravity is: $$\mathcal{L}(\mathbf{x},t)= - \rho (\mathbf{x},t) \Phi (\mathbf{x},t) - {1 \over 8 \pi G} (\nabla \Phi (\mathbf{x},t))^...
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1answer
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Deriving the field operators for Quantum Field theories

I always see the form of the field operators derived by, in the case of a scalar spin 0 particle, imposing the field commutation relations on the classical field solutions of the Klein Gordon equation ...

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