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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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Derivation of Equation of Motion of Graviton

In Feynman's Lectures on Graviton, Feynman tries to simplify the equation of motion of gravitation field, $$h_{\alpha\beta,\sigma}^{,\sigma} - (h_{\alpha\sigma,\beta}^{,\sigma} + h_{\beta\sigma,\alpha}...
Ting-Kai Hsu's user avatar
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32 views

Lorentz transformation of Creation and Annihilation operators for a real scalar field theory - MIT OCW QFT I Problem set 3 [closed]

I have been working through the MIT OCW's QFT lecture notes and problem sets, but I have come to realize that I have a fundamental misunderstanding of what is meant by how objects transform under ...
Nicolas Mendoza's user avatar
2 votes
0 answers
61 views

Lagrangian of a multi-dimensional scalar field

We know that the Lagrangian has to be a scalar. Would it be possible if this scalar is multi-dimensional (for example $m\times m$)? Let's say a field $\phi$ is represented with an $m\times m$ matrix ...
physics_2015's user avatar
-2 votes
3 answers
46 views

How can a non-derivative interaction involve the derivative of a scalar field?

I was reading up on the paper "The Fate of the False Vacuum" by Sidney Coleman and the claim is made that a scalar field with standard Lagrangian density: $$ L = \frac{1}{2} \delta_{\mu} \...
Adam P's user avatar
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Is my vector notation in electrostatics correct? [closed]

Let’s say we are expressing Coulomb’s law in Cartesian coordinates (not radial). Let the two charges of equal magnitude (and opposite sign) lie on x-axis. The equation for calculating the force acting ...
Alexander Djurovich's user avatar
2 votes
2 answers
56 views

Is First-Class Constraint Generator of matter Gauge Symmetry in EM example?

In EM theory, we can find first-class primary constraint, $$\Pi^{0}(x) = 0\tag{1}$$ and first-class secondary constraint, $$\partial_{i} \Pi^{i}(x) = 0\tag{2}$$ with Lagrangian $$\mathcal{L} = -(1/4)F^...
Ting-Kai Hsu's user avatar
2 votes
2 answers
140 views

When is the Lagrangian a Lorentz scalar?

The Lagrangian $\mathcal{L}$ can be defined as the Legendre transform (when it exists) of the Hamiltonian $\mathcal{H}$, a non-Lorentz scalar quantity (as $\mathcal{H} =T^{00}$). My questions are, ...
Gabriel Ybarra Marcaida's user avatar
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Field theories where the potential solves a linear Schrodinger equation

Are there physical situations/applications where the potential solves a linear time-dependent Schrodinger equation, or where the gradient of a solution to the Schrodinger equation (after somehow ...
kieransquared's user avatar
0 votes
1 answer
41 views

A nabla and surface integral transformation in Landau's book

I am reading Landau & Lifshitz's The Classical Theory of Fields. On page 182, in the solution to problem 1, the authors used a transformation:$$\int \nabla\cdot(\varphi\nabla\varphi)\mathbf rdV=\...
rioiong's user avatar
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1 answer
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How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$?

I would like to calculate the following expression: $$(D_\mu\Phi)^\dagger(D^\mu\Phi)$$ where $$D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$$ and $A_\mu^a$ are the components of a real $SU(...
Hendriksdf5's user avatar
1 vote
1 answer
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What kind of object is a function in the context of gauge theory?

In the context of differential geometry, we have the Levi-Civita connection that tells us how to take derivatives of tensors. Two examples of the covariant derivative are $$\nabla_\mu \phi = \partial_\...
dolefeast's user avatar
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1 answer
74 views

Gauge transformation rule for $dA$, where $A$ is the gauge field

Let $G$ be a non-Abelian simple compact gauge group and $\{ t^\alpha\}$ be a normalized set of generators for its Lie algebra $\mathfrak{g}$. Let $C^{\alpha \beta}_\gamma$ be the coupling constant for ...
Keith's user avatar
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1 vote
0 answers
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Deriving the Noether's theorem

I am familiar with how Noether's theorem is derived in some sources/books, the answer in 534699 is particularly clear. However, I'm reading A First Book of Quantum Field Theory by Pal, and although ...
mathemania's user avatar
2 votes
2 answers
165 views

QFT introduction: From point mechanics to the continuum

In any introductory quantum field theory course, one gets introduced with the modification of the classical Lagrangian and the conjugate momentum to the field theory lagrangian (density) and conjugate ...
Xhorxho's user avatar
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Effects of Localized Medium Changes on Field Propagation

I've studied various theories related to fields. These theories often include equations describing how the activity of a source is transmitted to other locations. The properties of the medium ...
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1 answer
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"Mass Shell" Condition on Euclidean Scalar Field

This is a basic qft question. I am looking for the condition on a free scalar $\phi$ of mass $m$ in Euclidean space such that it satisfies the Klein-Gordon equation. The Euclidean space Klein-Gordon ...
Sam's user avatar
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0 answers
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How to add a non-chiral lepton doublet to the Standard Model?

How would the Standard Model Lagrangian (before symmetry breaking) change if we were to add a non-chiral lepton doublet $\ell_{L,R}$ with weak hypercharge $y=-\frac{1}{2}$ to the $SU(2)\times U(1)$ ...
spiderhouse's user avatar
1 vote
1 answer
121 views

Lagrange Multiplier as chemical potential in Lagrangian Density

Within Matsubara formalism, we often add one chemical potential term to our Lagrangian density: $\mu\phi^\dagger\phi$, claiming that the chemical potential $\mu$ is one Lagrange multiplier. But at ...
JinH's user avatar
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1 answer
44 views

Is the Dirac adjoint in the representation dual to Dirac spinor?

As seen in this Wikipedia page, the Lorentz group is not compact and the Dirac spinor (spin $\frac{1}{2}$) representation is NOT unitary. Therefore, the complex conjugate representation does NOT ...
Keith's user avatar
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3 answers
224 views

2+1-dimensional $SU(N)$ Yang-Mills Theory

In recent years, there has been significant progress and growing interest in conducting quantum simulations of field theories using quantum devices. This typically involves formulating a Hamiltonian ...
Quantization's user avatar
2 votes
1 answer
63 views

How to find a covariant gauge derivative from a field transformation

For reference: I'm self-studying from Peskin's Particle Physics 2019, which tries to sweep all QFT under the rug. Consider an SU(3) gauge theory; I am told a $3\times 3$ scalar field $\Phi$ transforms ...
spiderhouse's user avatar
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0 answers
32 views

Decomposing physical degrees of freedom for spin-3 (Schwartz)

In section 8.7.2 of Schwartz, he decomposes the polarization vector for a massive spin-2 particle into 2 transverse and 3 longitudinal components, $$h_{\mu\nu} = h^T_{\mu\nu} + \partial_\mu (\pi^T_\nu ...
rrose's user avatar
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2 votes
1 answer
60 views

Derivation of Noether Current in Condensed Matter Field Theory by Altland and Simons

In Section 1.6 of Condensed Matter Field Theory by Altland and Simons, they prove Noether's theorem. In order to do so, they consider an infinitesimal transformation of the coordinates and the field: $...
zeroknowledgeprover's user avatar
6 votes
1 answer
182 views

Uniqueness of Maxwell Lagrangian: Why does it not include the term $c_3 (\partial_\mu j_\nu)F^{\mu \nu}$?

In the textbook Condensed Matter Field Theory by Altland and Simons, it is said that the Maxwell Lagrangian $\mathcal{L}$ coupled to a four-current $j^\mu$ satisfying $\partial_\mu j^\mu = 0$ is the ...
zeroknowledgeprover's user avatar
1 vote
1 answer
53 views

Reference request: scalar $O(N)$ gauge theory

I am interested in scalar $O(N)$ gauge theory and what you can do with it. Is there a standard reference section in a textbook/monograph/paper/whatever that has a decent overview? Wikipedia has a ...
3 votes
1 answer
89 views

Equation for real/complex $\phi^4$ theory

On wikipedia (see this link), the Lagrangians of the $\phi^4$ equation for real AND complex scalar fields are given. One may derive the Klein-Gordon equation by inserting into the Euler-Lagrange-...
Octavius's user avatar
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2 votes
1 answer
93 views

Commuting/anticommuting properties of fermionic ghost fields in BRST Quantization

I was reading the paper "Batalin-Vilkovisky analysis of supersymmetric systems" (by Laurent Baulieu and others). I am struggling to understand how commutation/anticommutation relations of ...
Aravind Madhavan's user avatar
2 votes
0 answers
33 views

Partition function of harmonic oscillator using field integral

I'm currently reading Altland and Simon's Field Theory, and while trying to solve the partition function of the harmonic oscillator I ended up with a question. Using a Hamiltonian of the form $H=\hbar ...
Tiago Pinto's user avatar
1 vote
1 answer
73 views

Is the scalar field in the Yukawa interaction real or complex?

Consider a theory containing a Dirac field $\psi$ and a scalar field $\varphi$ where the only interaction is given by a Yukawa potential $$ V = -g\bar{\psi}\varphi\psi $$ I know that real scalar ...
paulina's user avatar
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2 votes
0 answers
50 views

Invariance under Lorentz transformations but not translations?

I've read here that it's not possible to construct a field theory which is invariant under boosts but not invariant under rotations. The reason is essentially that boosts aren't closed under ...
WillHallas's user avatar
2 votes
1 answer
48 views

Does Noether's theorem apply to a strict on-shell symmetry of the action that holds on every integration region?

I've worked through different proofs of Noether's theorem and read various posts about it on this site. Some important takeaways, among others from this and this post by Qmechanic were Every off-...
WillHallas's user avatar
0 votes
0 answers
38 views

Free fields in Weinberg QFT vol.1

Background: In section 5.1 Weinberg discusses free fields. He had shown that for interaction of the form, $V(t) = \int{d^3x \mathscr{H}(\mathbf{x},t)}$ if $$U_0(\Lambda,a) \mathscr{H}(x) U_0^{-1}(\...
Damo's user avatar
  • 56
0 votes
1 answer
95 views

About Lorentz Invariant

I am reading Schwartz's book, Quantum Field Theory and Standard Model. I have some questions about invariance under Lorentz transformations. In the book, the scalar fields are functions of space-time ...
MichaelS's user avatar
1 vote
1 answer
76 views

Classical open string in Polchinski -- consistency of Neumann boundary conditions with gauge choice

In Section 1.3 of String Theory, Volume 1, Polchinski derives the open string spectrum from the Polyakov action with Neumann boundary conditions, by first considering the classical open string in ...
Alex's user avatar
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0 answers
27 views

Detailed derivation of ESCK gravity and Extended Friedmann Equations with Torsion

Do you know a textbook on the Einstein-Cartan-Sciama-Kibble theory of Gravitation and its application to derive Extended Friedmann Equations with Torsion, which shows the calculations in detail?
Alexandre Masson Vicente's user avatar
1 vote
1 answer
62 views

Symmetry transformation exact meaning

In whatever text/review I happen to come across (like for example From Noether’s Theorem to Bremsstrahlung: A pedagogical introduction to Large gauge transformations and Classical soft theorems, ...
schris38's user avatar
  • 3,992
1 vote
0 answers
37 views

Fierz identity problem

I am trying to figure out whether the following has any meaningful transformation in Fierz identities. Suppose w's are either u or v spinors. Then $$ \overline{w}_1 P_L w_2 \; \overline{w}_3 P_L w_4 \...
Hubert John's user avatar
4 votes
1 answer
117 views

Interpretation of self-interacting terms in the expansion of a pure YM Lagrangian?

Let $A^{\alpha}_\mu$ be the gauge field of a Yang-Mills theory where $\alpha$ is the gauge index of generators for some Lie algebra with structure constant $C_{\alpha \beta}^\gamma$ and $\mu$ is the ...
Keith's user avatar
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0 votes
0 answers
51 views

QED without fermions? [duplicate]

Is it possible to write down a sensible analog to QED but without fermions? Or better yet, with only scalar particles? Would two scalar fields with an interaction term $\lambda \phi_1 \phi_2^2$ lead ...
user34722's user avatar
  • 2,504
1 vote
1 answer
77 views

Chain rule with functional derivative?

I posted the same question on math exchange but no answer yet, so I post it also here: "I'd like to make the functional derivative of the functional $S[\phi(x)]$ with respect to the Fourier ...
Filippo's user avatar
  • 477
2 votes
1 answer
105 views

What is the Lie derivative of Ashtekar connection and its conjugate momentum in LQG?

I am using the reference Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons for this question. I am trying to understand the two equations (30) that give the variation ...
mortimer's user avatar
3 votes
0 answers
58 views

Mode expansions of canonical quantized fields in QFT

I have been self-studying QFT Quantum Field Theory for the Gifted Amateur by Stephen Blundell and Tom Lancaster. They devote Chapters 11, 12, 13 to the subject of canonically quantizing any given ...
Ethan's user avatar
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0 answers
25 views

How to study regularity of a Green's function when solving field equations perturbatively?

Preliminaries Consider a nonlinear differential operator $\mathcal{O}$ acting on a field $\phi$, with source $\rho$ $$\mathcal{O}(\phi)=\rho$$ Let's say the charge density is small, so we can define $\...
P. C. Spaniel's user avatar
1 vote
1 answer
73 views

Geometrical interpretation of gauge fields of spin other than 2

Gravitation can be interpreted as a gauge theory with a spin 2 graviton field. This graviton field in general relativity is also interpreter as a Riemannian metric. Do other gauge theories also have ...
Andreas Christophilopoulos's user avatar
1 vote
1 answer
51 views

Invariance of general equations under Lorentz Transformation

I was reading Quantum Field Theory by Peskin and Schroeder. On page number 37 He mentions "In general, any equation in which each term has the same set of uncontracted Lorentz indices will ...
Eviciium's user avatar
2 votes
1 answer
50 views

How to calculate the Hodge dual of a two-form defined in self-dual BF theory?

I am trying to calculate the Hodge dual of the two form \begin{align}\Sigma^i=e^0\wedge e^i+\epsilon^i{}_{jk}e^j\wedge e^k,\quad (i,j,k \in \{1,2,3\}),\end{align} where Hodge dual with respect to ...
mortimer's user avatar
0 votes
0 answers
26 views

Some details about variational calculation on variational bi-complex

I am reading 1801.07064, where the covariant phase space formalism is elaborated. From what I have learnt from classical mechanics, the variation of Lagrangian for field theory $L[\Phi,\partial\Phi]$ ...
LaplaceSpell's user avatar
0 votes
1 answer
42 views

How does Diagonalizing Mass Terms Affect the Lagrangian?

One thing I don't get about mass diagonalization, is doesn't this also change the kinetic terms of your theory? You would get some off-diagonal kinetic terms. How do we deal with this? What also ...
SamuelFGC's user avatar
0 votes
1 answer
57 views

Does the Wigner little group classification of particles have consequences for classical field theory?

Does the Wigner little group classification of particles have consequences for classical field theory? In particular, I'm curious whether it can be used to predict the two propagating modes for ...
user196574's user avatar
  • 2,292
1 vote
1 answer
53 views

Difference in definition of conserved current in Quantum Field Theory

In David Tong Lecture Notes (page 14), it is written that Proof of Noether's Theorem: We'll prove the theorem by working infinitesimally. We may always do this if we have a continuous symmetry. We ...
darkphysics's user avatar

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