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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Variation of the Ricci tensor “squared” and antisymmetrization of the derivatives

I'm dealing with some extension of GR, with action: $S=\int d^4x\Big[\sqrt{-g} f(R,R_{\mu\nu}R^{\mu\nu})$ Varying this action gives: $\delta S=\int d^4x\Big[\delta\sqrt{-g} f(R,R_{\mu\nu}R^{\mu\nu})...
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How to take derivative with respect to Lagrangian of complex field?

Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$. Question 1: Is $\...
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45 views

Conserved current from the three $SU(2)$ transformations

We are asked to show that the following Lagrangian is invariant under the three $SU(2)$ transformations $\Phi \rightarrow \exp{({\frac{i}{2}{\alpha_j\sigma^j}}) \Phi}$, where $\Phi$ is a doublet ...
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Dimension of a scalar field from most general Lorentz invariant Lagrangian

Suppose one is trying derive the dimension of a real scalar field $\phi$ starting from the most general form of the Lorentz invariant Lagrangian $$\mathcal{L}=c_0(\partial_\mu\phi)(\partial^\mu\phi)+...
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Conservation of angular momentum from Noether's theorem and Lorentz invariance

Noether's theorem has been stated in the form that if the assignment $\phi \mapsto \phi + \delta \phi$, $x^{\mu} \mapsto x^{\mu} + \delta x^{\mu} $ leaves the action invariant, then the quantity $$ f^{...
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If the lagrangian density changes by a total derivative of the lagrangian density

When we derive energy momentum tensor current by actively transforming field. We see that lagrangian ( density) changes by a total derivative of the lagrangian. If a total derivative of the function ...
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2answers
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Is this a gauge symmetry?

Imagine a hypothetical action: $$S=\int \left(\frac{\partial}{\partial t}\phi(x,t)\right)^2 d^3x dt$$ Now we have a symmetry of the action: $$\phi(x,t)\rightarrow \phi(x,t)+\chi(x).$$ At time $t$, $\...
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Action of 1-form symmetry in Maxwell theory

I am reading Lectures on Gauge Theory by David Tong 1. In 3.6.2 first example that the author talk about pure $U(1)$ gauge theory in 4D. In this example, he talk about two 1-form symmetries: electric $...
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1answer
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Derivation of the Klein-Gordon solution via Fourier Transforms

I recently graduate with a bachelor's in physics, and I've been trying to take the next steps toward learning QFT. To this end, I have been working through Peskin and Schroeder's textbook step-by-step....
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Why can't a Weyl Fermion have mass?

My understanding was that a particle may have mass if there is a quadratic term in the fields without derivatives. For a single left-handed Weyl fermion, the following expression is lorentz invariant, ...
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Klein-Gordon equation with position-dependent mass [closed]

Does there exist a general solution for a differential equation like: $$\ddot{\phi}(x,t) - \partial^2_x\phi(x,t) + \phi(x,t)m^2(x) = 0,$$ where $m(x)$ is a known function.
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Doubt about 3D generalization of the Elastic Field Equations

My only source is $[1]$. In $[1]$ Finn introduces an example (in a totally undergraduate level discussion) that the problem on waves propagating in a solid cylindrical bar, firstly induces you to ...
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Would the Michelson-Morley Experiment have Discovered our Atmosphere?

We all know the earth is surrounded by an atmosphere, and we know that it is the medium through which sound travels. If the Michelson and Morley experiment was modified to find sound’s medium would ...
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6D (2,0) superconformal field theory

I'm looking for a good reference book or textbook to study on 6D (2,0) superconformal field theory as a part of string theory.
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Scale invariance in (2+1)D nonrelativistic field theory

Context: I am reading a paper named 'Nonrelativistic field-theoretic scale anomaly' on scale invariance in nonrelativistic field theory. The Lagrangian density for the scalar field is given by, $$\...
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What if the Lagrangian $\mathscr{L}$, a Lorentz scalar, is replaced by a Lorentz vector?

As an answer to this post, I made an impression that if $\mathscr{L}$ were not a Lorentz scalar in Eq.$(1)$ (see below), then Eq.$(1)$ would not be covariant. But now I think that is wrong! I state ...
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Potential functions for separation and isochronic gauges

Most potentials in physics are expressed as a radius or another geometric norm/gauge. I am looking to understand the significance of the choice of potential functions for force/pressure separation in ...
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Units of the Scalar field theory Lagrangian density [closed]

The Lagrangian (with units $J$) connects to the Lagrangian density (with units $J/m^3$) as: $$ L=\iiint_V \mathcal{L}d^3x $$ Let $\mathcal{L}$ be the classical Lagrangian density of the scalar free ...
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What's the meaning of a “temporal polarization”?

For a massive vector four-field there are only three physical linearly-independent polarizations. For a field excitation at rest, these can be described by \begin{align} \epsilon^1_\mu \equiv \begin{...
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1answer
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Expanding about background field

I refer to this set of lecture notes by Hugh Osborn, equation 4.184 on p.70. We expand an action $S[\phi]$ around a background field $\varphi(x) = \phi(x) -f(x)$ If we expand the action $S[\phi]$ ...
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662 views

A traceless stress energy tensor?

I'm trying to solve this exercise: Suppose an arbitrary theory (Flat space-time?) with a single field (Is a scalar field?) invariant under dilations, i.e. $x\mapsto b x$ and $\phi \mapsto \phi$. ...
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How does gauge-fixing really work?

Leaving technical issues like Gribov copies and residual gauge freedom aside, how do gauge fixing conditions like the Coulomb condition \begin{equation} \partial_i A_i =0 \end{equation} or the axial ...
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1answer
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Why does Coulomb gauge condition $\partial_i A_i =0$ pick exactly one configuration from each gauge equivalence class?

There are infinitely many configurations of a vector field $A_\mu$ that describe the same physical situation. This is a result of our gauge freedom $$ A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + \...
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4answers
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Why does Lorenz gauge condition $\partial_\mu A^\mu =0$ pick exactly one configuration from each gauge equivalence class?

For a vector field $A_\mu$, there are infinitely many configurations that describe the same physical situation. This is a result of our gauge freedom $$ A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + ...
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Count degrees of freedom of gauge tensors

For degrees of freedom (dof) it is said that spin-1 massless boson like photon has 2 dof in 4d, like U(1) gauge theory. it is said that spin-2 massless boson like photon has 2 dof in 4d, like ...
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67 views

Angular momentum in the Maxwell field theory/Chern-Simons theory?

I'm trying to calculate the angular momentum in the chern simons theory. But equivalently, I was trying a calculation of angular momentum in the Maxwell field theory, which will hopefully be ...
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Conjugate momenta in field theory

In classical mechanics for particles, the Euler-Lagrange equations are given by: $\frac{d}{dt} \frac{\partial L}{\partial \dot q} = \frac{\partial L}{\partial q}$ and the momentum conjugate to q is ...
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How build a topological charge from of the mapping between physical and inner space?

How build a topological charge from the mapping between physical and inner space? When we make a mapping between two coordinates system, we normally relate both systems by coordinate transformation as,...
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1answer
53 views

Is there a superfluous statement in Schwartz's QFT book in deriving Euler-Lagrange equations?

Please help me with the following confusion. Yesterday I was looking at the derivation of Euler-Lagrange equation in several QFT textbooks using stationarity of the action. At the last step one needs ...
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1answer
50 views

Is there a general behavior of energy gap under renormalization?

Perform real space renormalization on a discrete lattice model with a finite energy gap. Is it always true that under the flow of coarse-graining, the energy gap will only increase? I think the ...
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Classical action is zero in Klein-Gordon theory for a particle wavepacket

I'm interested in rewriting actions in the form $$ S = -\int H dt + \int p_i dx^i, $$ (where $H$ is the Hamiltonian and the $p_i$ are conjugate momenta) and then evaluating them along a classical ...
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How does the $U(1)$ global symmetry break in the gauged $XY$ model?

I'm studying the particle vortex duality, and I'm confused how we're able to say that in the Coulomb phase, the "hidden" $U(1)$ global magnetic symmetry spontaneously breaks. gauged XY model: $\...
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Geodesic equation in Einstein-Cartan manifold

In GR we use the Riemannian manifold without any torsion to describe the theory. Hence, the geodesic equation can be interpreted as "a trajectory of a free falling particle" or "equation of motion". ...
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1answer
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Field equation of motion for this Lagrangian

In my first QFT exam I was supposed to derive the equations of motion for all fields for this Lagrangian: $$\mathcal{L} = \bar{\Psi}(i\gamma^\mu\partial_\mu-M)\Psi+g\bar{\Psi}\gamma^\mu\Psi\bar{\Psi}\...
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What macroscopic, classical fields exist?

Two example are certainly the electromagnetic field and the gravitational field. The Higgs field, weak fields, and strong fields are too short ranged to operate on macroscopic scales. Moreover, the ...
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Question about operator spreading in QFT

It is a well-known effect that generic localized operators spread out under evolution by lattice Hamiltonians. For example , $e^{iHt} \sigma^x _j e^{-iHt}$ will in general not be supported only at ...
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Conformal Invariance of Maxwell's Equations

I am currently doing some conformal field theory (in four dimensions) and want to show the invariance of Maxwell's equations under conformal transformations, in particular \begin{align} \partial_\...
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30 views

Finding Lagrangian density for particle with spin 1 and mass $\mu$

I am trying to solve my first ever QFT problem, which reads as follow: From the wave equation for a particle with spin 1 and mass $\mu$: $$\left[g_{\mu\nu}(\Box+\mu^2)-\frac{\partial}{\partial x^...
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1answer
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Energy-momentum Tensor for a Real Scalar Field Lagrangian

I'm currently working through Schwartz's QFT book, and I'm trying to find the energy-momentum tensor for the following Lagrangian: $$ L = -\frac{1}2\phi(\Box+m^2)\phi. $$ Am I correct in thinking ...
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1answer
100 views

Physical Meaning of a Quantum Field

Sorry in advance if this question doesn't make sense. Anyway, I am reading a paper about quantum field theory and the Whitman Axioms (http://users.ox.ac.uk/~mert2060/GeomQuant/Wightman-Axioms.pdf), ...
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1answer
57 views

Translation invariance of point particles as a field theory

The case of point particles, relativistic or not, can be treated as a field theory in general, ie for the $(1+1)$-dimensional case this is the theory of a field theory on the vector bundle $$\pi : \...
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Does anyone have references on classical field theory that develops the differential form formalism?

I am familiar with the usual way of doing Classical Field Theory, but I am currently taking a course where the professor works with differential forms to teach the subject. I wonder if anyone knows ...
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1answer
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Confusion about trace in the vertex term of Lagrangian

I was reading through Mariano Quirós's lecture notes titled "Finite Temperature Field Theory and Phase Transitions". In Sec. 1.2, the author is calculating the one-loop effective potential at $T=0$. ...
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1answer
58 views

$1/4$ coefficient in QED Lagrangian [duplicate]

What is the reason 1/4 coefficient in the tensor multiplication of the electromagnetic field strength? $$\mathscr{L} = -\, \frac{1}{4} \, F_{\mu \nu} \, F^{\mu \nu}. \tag{1}$$
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Extended objects, bundle description and transformations

In the hope of trying to come up with a clear mental picture for what a transformation is in physics, I encounter some difficulties due to the variety of objects that appear in physics. While no ...
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How to go from $\delta(\dot{\Psi})$ to $\delta\Psi$ in variational calculations? [duplicate]

This must have been done somewhere before but I never saw a clear and rigorous explanation of why. This is possibly related to my recent post here about potential abuse of notation. Just to be ...
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2answers
97 views

Physics near null infinity

The concept of null infinity $\mathscr{I}$ is standard in general relativity, and more recently in the analysis of infrared structure of gravity (see e.g. the article by Strominger). I am curious ...
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Exact solution for (simple) Reheating Boltzmann equations?

Let's say we have the Universe which is exiting Inflation, with the inflaton field $\phi$ decaying into relativistic particles (radiation $R$). Without any other assumption, the equations describing ...
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1answer
119 views

Functional derivative and variation of action $S$ vs Lagrangian $L$ vs Lagrangian density $\mathcal{L}$ vs Lagrangian 4-form $\mathbf{L}$

I have seen many potential abuse of notation that prevents me from clearly understanding variational methods in QFT and GR that I want to get this settled once and for all. This may be a bit long but ...
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1answer
25 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 ...