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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Poincare invariant Lagrangian?

I only see it mentioned that we want Lorentz invariant Lagrangians in quantum field theory, but I would expect that we additionally also need translational invariance, i.e. Poincare invariance. After ...
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2 votes
1 answer
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Antifields in BV formalism - do they also have gauge transformation laws?

I am studying Weinberg Vol 2 and the BV formalism of the gauge theory. There, the antifields are introduced somewhat out of thin air. I am a little bit confused about their properties. For example, ...
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Sources to learn Gauge Theory, Groups, Lie Algebra, etc [duplicate]

As seen in previous questions, I'm interested in gauge theory, although I have no idea how to do any of the mathematics, though i'd like to start. With that in mind, Are there any good sources that ...
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Second-order Dirac equation

I'm wondering if one of you could tell me about the following equation: $$\partial_t \Psi = i \sigma_z m - \sigma_y k \partial_x \Psi + i \sigma_y k' \partial_{xx}\Psi$$ where $m, k,k'$ are real ...
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Parity in Effective Lagrangians

Given the following Lagrangian $$\mathscr{L} = c\frac{g}{m}\bar{\psi}_A\Gamma_5\gamma^\mu\psi_B (i\partial_\mu)\phi$$ where $\Gamma_5 \in \{\gamma_5, 1\}$, for two spin one-half particles $A$ and $B$ ...
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Tachyon velocity in relation to light speed [duplicate]

tachyonic particle are merely hypothetical particle that always travels faster than light., But if real, how fast would they be?
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2 votes
1 answer
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Relativistic Euler-Lagrange equations for a four-vector (or one-form) field

I think the best way to ask my question is by considering the maxwell-Lagrangian, $$\mathcal{L}=-\frac{1}{4}F^{\mu \nu}F_{\mu \nu}=-\frac{1}{2}(\partial^{\mu}A^{\nu}\partial_{\mu}A_{\nu}-\partial^{\mu}...
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Chiral Transformation and Dirac Bilinear

I need to compute the following Dirac bilinears: $$\overline{\psi} \psi \quad \text{and} \quad \overline{\psi} \gamma_\mu \psi$$ Under the following Chiral transformation: $$\psi \rightarrow \psi' = \...
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Propagators in Quantum Field Theory at Finite Temperature

While reading section 5.8.2 of Quantum Field Theory An Integrated Approach by Fradkin, I had a few questions, not able to think them though myself. The thermal propagator is given as $$G_{T}^{(0)}(\...
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Question about the parity violation of weak interaction Lagrangian

In the textbook of A. Zee, Quantum Field Theory in a Nutshell, the author states that the following Lagrangian: $$ \mathcal{L} = G (\overline{\psi}_{1L} \gamma^\mu \psi_{2L})(\overline{\psi}_{3L} \...
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What does it exactly mean by right and left functional derivatives?

In BV formalism of the gauge theory, we need to compute the right / left functional derivatives of the actions that include fermions. I do not quite see what it means by that. For example, let us ...
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What are momentum transfer rules for interacting scalar fields?

Consider the lagrangian density of the form $$L= \frac{1}{2}\partial^µφ_1 \partial_µφ_1 − \frac{1}{2}\partial_µ φ_2 \partial^\mu φ_2 -(\frac{λ_1}{4!})[(φ_1)^4+(φ_2)^4]+(\frac{λ_2}{4})(φ_1^2φ_2^2)$$ In ...
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Spherical harmonics expansion: from scalars to tensors

It is well known that a scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any ...
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Derive interaction lagrangian for KG equation in QED

The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$ By ...
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3 answers
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Problem 6 of Sheet 1 - Quantum field theory David Tong - Variation of Lagrangian density

The Problem reads: Consider the infinitesimal form of the Lorentz transformation derived in the previous question: $x^\mu \rightarrow x^\mu +\omega^{\mu}_\nu x^\nu$. Show that the scalar field ...
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Noether current associated with transformation $\delta \psi=i\alpha \psi$

I'm doing problem 3 from sheet 2 of David Tong's lecture notes. We have given the complex field $\psi(x)$ which is governed by the Lagrangian $$\mathcal{L}=\partial_\mu \psi^*\partial^\mu \psi -m^2\...
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Relative signs between interaction terms

What is the interpretation / meaning of relative signs between interaction terms in a Lagrangian density? (If there is none, are they even physically reasonable?) Example: Let $\phi$ be a scalar field,...
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Path-integral and measurements [duplicate]

I did a question a couple days ago and I didn't express it correctly. What happened is that i have a field obtained by solving the equations of motion of the $\lambda\phi^{4}$ theory. It was solved in ...
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How does the boundary term matter in scalar field and in more general cases?

People always say that boundary terms don't change the equation of motion, and some people say that boundary terms do matter in some cases. I always get confused. Here I want to consider a specific ...
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1 vote
1 answer
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Chiral symmetry of the Dirac Lagrangian

I need to show that in the mass to zero limit the lagrangian density: $$\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi$$ is invariant under the transformations: $$\psi'=e^{i\alpha\gamma^5} \psi$...
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Yet more gauge group nonsense: $D3$? $Q8$? $Z8$?

This'll probably make me look like a total idgit, but I have a new question in the same vein as mine about $SU(4)$, but this time without any guesses. I've looked a bit into groups, and it looks like ...
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What would the force arising from an $SU(4)$ gauge field operate like? (As in, how many charges, whether the boson would interact with the force, etc)

Heyo, i'm new to this all, and deadly curious what this would look like. If this isn't specific enough, lemme know.
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-2 votes
3 answers
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What is the evidence that gravitational fields don't sum up as a superposition?

Einstein's field equations are non-linear. Gravity gravitates (self-interacts). It's very complicated to solve Einstein's field equations for more than one central object. That are keystones in ...
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1 answer
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Discretization of derivative of delta function and affine Kac-Moody algebra

In equation (4.16) of https://arxiv.org/abs/1506.06601, a discretization of the (classical) affine Kac-Moody algebra is presented: $$ \frac{1}{\gamma}\left\{J_{m}^{1}, J^2_{n}\right\}=J_{m}^{1} J_{n}^{...
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1 answer
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Can a primary constraint contain spatial derivative of the field?

I am currently studying the Hamiltonian formulation of GR and I have problems understanding this definition of primary constraint. In the textbooks, primary constraint occurs when a momentum conjugate ...
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Representation of one-body operator in field theory

Section 2.1 Introduction to second quantization, Page No 47 of Condensed Matter Field Theory reads Representation of operator (one-body) Single-particle or one-body operator $\mathcal{O}_1$ acting in ...
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How is this Fourier transform done?

Given a Hermitian operator $\hat{\phi}(x,t)$, why we can write it in terms of Fourier transformations as $$\hat{\phi}(x,t)=\int^\infty_{-\infty}\frac{dk}{(2\pi)(2\omega)}[\hat{a}(k)e^{ikx-i\omega t}+\...
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3 votes
1 answer
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What does "conformally coupled scalar" mean?

"Conformally coupled scalar $\phi$" - I encounter it a lot, but I can't find what it means.
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1 answer
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What is the electric field and potential inside and outside grounded conducting and non grounded conducting sphere?

I'm taking an electromagnetic theory course, but I have trouble understanding the field and potential inside and outside of a conductive sphere when it is connected and not connected to ground. And ...
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1 vote
2 answers
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What is the correct way of looking at the Dirac field?

All quantum fields are operators in QFT. However, the Dirac field operator $\hat{\psi}$ has the following difference with the scalar field operator $\hat{\phi}$: For the $\hat{\psi}$, it makes sense ...
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Scalar theory with non-trivial boundary conditions. Green function

Let us consider the free scalar field theory $\varphi(x,t)$ in a space of dimension 1+1, with Minkowski metric $\eta_{\mu\nu}=diag(+,-)$. It is well known that it is described by the classical action ...
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Advantages of using the potentials $A$ and $\phi$ instead of the fields $E$ and $B$ [duplicate]

I'm taking a quantum mechanics course and we briefly reviewed some facts of ED, namely the Maxwell equations and their equivalent version by expressing the electric field $E$ and magnetic field $B$ ...
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Lagrangians related by field redefinition

Is there a sufficient criteria in the form of a theorem to check if two Lagrangian densities $\mathscr{L}$ are related via field redefinitions $\phi\rightarrow f(\phi)$, where $f(\phi)$ is an analytic ...
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2 answers
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Must all field theories depend on the spatial derivate of the fields?

For instance, if I have encountered \begin{equation} \label{eqq2} \frac{\partial \mathcal{L}}{\partial (\partial_i \phi)} = 0 \end{equation} This tells us that $\mathcal{L}$ cannot depend on $\...
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Addition of four-momenta in scattering amplitude of nucleon-anti-nucleon pair in scalar Yakuwa theory

Intro I'm studying QFT using David Tong's lecture notes. In section 3.5 on examples of scattering amplitudes in scalar Yukawa theory, the scattering amplitude $A$ of a nucleon-anti-nucleon scattering ...
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2 votes
2 answers
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What does it mean by spin 1/2 or spin 2 field?

I see in common discussions people simply use the terminology spin 1/2 field or spin 2 fields as if it is some common term like hamiltonian. How to think about these fields and understand what it ...
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4 votes
1 answer
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What is a propagating degree of freedom?

Given a gauge field theory, the various fields involved have (pointwise) degrees of freedom. For instance, if I consider the gauge theory of gravity in four dimensions, I have a set of tetrads $\{ e_\...
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Notation and Terminology Questions from Schwartz' QFT Book

I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing. First off, on page 34 he defines a translation of a field to first order as $$...
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Invariance of Intervals [duplicate]

I have started reading Landau & Lifshitz Vol. 2 (fields theory) and I've got confused about something I read. to prove the Lorentz transform, it defines interval: $$ds^{2} = c^{2} dt^{2} - dr^{2}\...
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2 votes
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What is the lagrangian interaction term of a charged current weak interaction?

For example, if you have an electron coupling to a $W^{-}$ and producing an electron neutrino $e\rightarrow \nu_{e}W^{-}$. Is something like $$\frac{-g_{w}}{2\sqrt{2}}\bar{\nu}_{e}\gamma^{\mu}W^{-}_{\...
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1 vote
0 answers
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How does SUSY differ from just having standard model (SM) particles in new Lorentz Group representations?

Under the standard model (SM), it is the irreducible representations of the Lorentz group which allows us to classify particles according to their spin. We can then further label particles using the ...
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5 votes
1 answer
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Is Galilean boost actually a gauge transformation?

In elementary physics, it is well-known that the Newton's law $$\vec{F}=m\vec{a}$$ is invariant under Galilean transformations. However, Galilean relativity is not introduced in details in ordinary ...
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1 vote
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Difference between a stress energy tensor and an effective stress energy tensor?

As I was working through this article by Bronnikov regarding Einstein–Cartan theory: Bronnikov, K. A., & Galiakhmetov, A. M. (2015). Wormholes without exotic matter in Einstein–Cartan theory. ...
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What is the Mathematical description of Weak Interaction at low energies?

Introduction When I started to study gauge theory the mathematical road map seemed to be quite "simple". After all the concepts and notions about principal the differential geometry of fibre ...
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Hubble Damping Intuition

When we calculate the equations of motion of, say, a scalar field in an expanding FLRW universe, the metric introduces the following `Hubble friction' to the equations of motion $$ 3 H\dot\phi\,. $$ ...
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Why do scalars and fermions have a different result in a Lagrangian?

Consider the Lagrangian for Yukawa theory: $$ \mathcal{L} =i\bar{\psi}\not{\partial}\psi- \bar{\psi}m_F \psi +\frac{1}{2} \partial_\mu \phi \partial^{\mu} \phi - \frac{1}{2}m_s^2 \phi^2 + \mathcal{L}_{...
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1 vote
1 answer
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Four-vector differentiation (E-M Euler-Lagrange eq.)

$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
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Generalization of the Hessian to field theory

For the Ostrogradski theorem to apply, in classical mechanics (finite number of degrees of freedom) the Lagrangian needs to be non-degenerate. This means that the Hessian matrix (second derivatives of ...
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Stress tensor for a real massive vector field in General Relativity

Let's consider the classical Lagrangian density for a real vector field $A_\mu$, $$ \mathcal{L}_v=\sqrt{-g}\left(-\frac{1}{2}A_{\mu;\alpha}A^{\mu; \alpha}-\frac{1}{2} R_{\mu \nu} A^\mu A^\nu+\frac{1}{...
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Integrating by parts differentiated vector fields in Lagrangian [duplicate]

when surface terms being ignored, from p.556, ch.26 QFT lectures of Sidney Coleman To get this result, do I have to integrate by parts twice? Do I have to switch the derivatives around and how? ...
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