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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Intuition behind field transfomations

Consider a real field $V^{\mu}(x)$ defined on a 4-dimensional Minkowski space. Acted by a transformation $\Lambda = \Lambda^{\mu}{}_{\nu} $ it transforms like $$V^{\mu}(x) \to V^{'\mu}(x) = \Lambda^{\...
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Does the operator product expansion work for spin glasses?

The operator product expansion (OPE) is based on the assumption that the product of any two scaling operators, $\phi_i$ and $\phi_j$, can be expressed as a sum over all scaling operators $\phi_k$ as ...
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How to remove the $\exp(-i(p^0+q^0)x^0)$ term in the canonical commutation?

Using the convention A Modern Introduction to Quantum Field Theory by Michele Maggiore Eq. 4.2 or equivalently the quantum theory of fields by Steven Weinberg Eq.1.2.63. $\phi(x)=\int \frac{d^3p}{(2\...
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1answer
97 views

Confusion surrounding Noether's Theorem

As an example of Noether's Theorem, my QFT textbook gave the example of how the conservation of momentum and energy arises from symmetry in space-time translations. The book arrives at the conclusion ...
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Would quark spin or gluon spin cause a greater excitation of the surrounding field

In The Origins of Gravitational Fields, Solomon proposes that quark spin leads to the excitation of fields that becomes gravity. I am not asking about the correctness of his theory, rather I am ...
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Chaikin and Lubensky exercise: upper critical dimension of anisotropic Lifshitz potential

Exercise from Chaikin and Lubensky: Chapter 5 Calculate the upper critical dimension $d_c$ for the following critical points: b) The $(d,m)$ Lifshitz point described by the Hamiltonian $$H = \int d^...
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1answer
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Gaussian integrals with gamma matrices in their exponents

I should evaluate Gaussian integrals in the 1+1 Minkowski space, which read $$ I_{1}= \int d^{2}k \, {\rm Tr}\big[ \gamma^{5} \gamma^{\eta} \gamma^{\kappa} e^{\alpha k^{\mu}k_{\mu} + \beta \gamma^{\...
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1answer
87 views

Why there is no commutator term in the pre-sympletic density?

In this post I'm considering the Covariant Phase Space (CPS) formalism as presented by Lee & Wald in "Local symmetries and constraints ". In the CPS formalism we take the Lagrangian form ...
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1answer
48 views

Fourier transform of the energy momentum tensor

I am trying to calculate the Fourier transform of the energy momentum tensor of a scalar field, in particular the first term: \begin{equation} T_{\mu \nu}(x) = \frac{1}{2} \partial_{\mu} \phi(x) \...
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Average of Two successive momenta $m\frac{x_{k+1}-x_k}{\epsilon}m\frac{x_k-x_{k-1}}{\epsilon}$ using rules of path integral

A Problem from Feynman's Path Integral Book Let $x_i$ be coordinates at different time instances, prove that $$ \langle\chi|m\frac{x_{k+1}-x_k}{\epsilon}m\frac{x_k-x_{k-1}}{\epsilon}|\psi\rangle=\int\...
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1answer
74 views

Field exchange symmetry

I have a pheraps stupid doubt regarding the existence of a symmetry. Consider a theory such as: $$\mathcal{L}= \frac{1}{2}(\partial_\mu\phi)^2+\frac{1}{2}(\partial_\mu\psi)^2+\phi^2\psi^2$$ With some ...
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Lagrangian for Schrödinger field equation [duplicate]

I know this topic has already appeared in some posts but I think my question is uncovered. Please correct me if I'm wrong. I'm trying to find a lagrangian for the Schrödinger field equation $$i \frac{\...
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1answer
35 views

On the mass basis of the Dirac equation

The Lagrangian density used to derive the Diract equation is given as follows: $$\mathcal{L} = \bar{\Psi}(i\gamma^\mu\partial_\mu - m)\Psi$$. which can be shown to give rise to the Dirac equation: $$i\...
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52 views

Symmetry transformation for Lagrangian

I'm reading about Relativistic Quantum fields from Bjorken and Drell and i was wondering about the equation that most hold for a transformation to be a symmetry transformation. The book states: We ...
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Explicitly determining the propagator for the dual field

So I have a question which has been on my mind for some time but I never got around to asking: how can I calculate the propagator for a dual field? Let me go into some more detail. Starting off with ...
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Total divergence term in derivation of Noether's theorem [duplicate]

In the Ashok Das QFT book] pg. 212-213 (pdf), the section on Noether's theorem, the author considered general infinitesimal transformations of the form $$x^\mu\rightarrow x'^\mu,$$ $$\phi(x)\...
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1answer
69 views

Extension of Faddeev-Jackiw first-order Lagrangian formalism to fields

In this paper, Toms discusses the method that Faddeev and Jackiw proposed for quantization of constrained theories. In section III.B, he applies this method to a field theory, but I have several ...
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1answer
76 views

Intuition behind coupling and interaction terms in Lagrangian?

I'm reading through Srednicki's QFT book, and when we introduce external fields, we often multiply terms together to represent the coupling in our Lagrangian densities. My understanding of what ...
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2answers
45 views

Different definitions of configuration space

Configuration space has been mentioned in many different areas of physics and from my personal experience I have found the definition has been different from topic to topic. There are two examples to ...
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20 views

An instanton in $d$ dimensions is often a soliton in $d + 1$ dimensions?

The title of this questions is a "folklore" I've heard from a lot of researchers, but I never understood why this is the case. I know what an instanton and soliton is, respectively in the ...
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1answer
45 views

Is any continuous transformation a symmetry of action?

Consider a continuous transformation $\phi \rightarrow \phi+ \delta\phi$, where $\phi$ is a field operator and $\delta \phi$ is a infinitesmal change. If such continuous transformation is applied to ...
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3answers
575 views

Are there fields (of any kind) inside a black hole?

It is said that nothing escapes from black holes, not even light. All particles are now thought to be excitation of different fields (electric field, electromagnetic field, photon field, etc). Does it ...
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2answers
185 views

Spatial inversion and Time reversal

On the spinor field $\psi^{\mu}(x)$, I found the action of $\psi^{\mu}(x)$ on spatial inversion $P$ by postulating $\psi^{\mu}_{P}(x)=P^{\mu}_{\nu}\psi^{\nu}(P^{-1}x)=P^{\mu}_{\nu}\psi^{\nu}(t,-x)$, ...
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28 views

Can Gauss's law be generalized to use with other massless vector fields?

Can we use the integral form of Gauss's law for other types of massless vector fields? For example, if I'm thinking of the graviton field, is there any reason why it would not apply? (understanding ...
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0answers
51 views

Mathematical Charges in Classical Physics, General Relativity and QFT

I have a very easy, and naive, question: given a field $\mathbf{A}$ on some vector space $V$, we can calculate how the flux or circulation of this field behaves. For example, we have Gauss's laws for ...
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1answer
37 views

Canonical conjugate momenta of EM Field Lagrangian density

I have the EM Field Lagrangian density given as $ \mathcal{L} =- \frac{1}{4} F_{\mu \nu} F^{\mu \nu} $ where $F^{\mu \nu}$ is the Field strength tensor defined as $F^{\mu \nu} = \partial^\mu A^\nu- \...
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2answers
65 views

Dirac Hamiltonian in Peskin & Schroeder

I am currently going through the Peskin & Schroeder and have hit a snag with what seems should be an easy derivation. We see that the Dirac Lagrangian Density is given by $$\mathcal{L} = \bar{\psi}...
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1answer
47 views

Why are interacting field theories called nonlinear? Explanation for interacting EM field, in particular

The classical equation of motion for the electromagnetic field interacting with a charged fermion field $\psi$ of charge $eq$ is given by $$\Box A^\mu(x)=j^\mu(x)$$ where $j^\mu(x)=eq\bar{\psi}(x)\...
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Field of particle or particle of field? [duplicate]

Are particles consequences of fields or fields consequences of particles? Is this a useful question to ask? Or is wave-particle duality the real topic?
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53 views

Doubt of gauge covariant derivatives: how can I derive it?

In the context of general relativity (GR) it is necessary to introduce the notion of covariant derivatives. From the point of view of a basic introduction, we always start to deal with GR in a highly ...
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40 views

$SU(5)$ GUT Yukawa couplings

How exactly in $SU(5)$ GUT a coupling of fermions with different chiralities to the corresponding scalar fields is realized? The yukawa coupling in general form is $\bar5_FY_510_F5^*_H+\frac{1}{8}...
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32 views

Why is Ewald summation faster than normal summation?

I posted this on the Mathematics SE, but wasn't sure if it was purely a mathematical problem. If I should take this down because of cross-posting rules, I will do so gladly. I have been trying to ...
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5 views

Question about defining short-range and long-range potentials in Ewald summation

I am trying to learn Ewald summation for long-range forces and particles-in-grid systems, and I am following this article. On page 3 of the article, the author talks about "splitting the charge ...
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1answer
51 views

Hypercharge of the complex scalar doublet

I often see the complex scalar doublet $Φ_A$, $A=1,2$ with the opposite hypercharge arising in the Yukawa couplings as $\tildeΦ_A = iτ{_2}_{AB}Φ_B^*$ where $τ_r$ $(r=1,2,3)$ denote isospin pauli ...
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39 views

What are the momentum states in the second quantization of a free field?

I am reading through Schwartz, and have become quite confused by his discussion of second quantization of a free field (Section 2.3, page 20-21). To be clear, I think I understand second quantization ...
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1answer
71 views

Yukawa matrices

It is known that the masses of fermions in the Standard Model are represented in the form of singular values of complex Yukawa matrices (Yukawa couplings). The question is, are the values of the ...
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1answer
53 views

Two consecutive symmetry transformation generated via Poisson brackets

Question If an infinitesimal symmetry transformation parametrized by Killing field $f^\mu(x)$ $$ \delta_f\phi=\phi'(x)-\phi(x)=f^\mu\partial_\mu\phi\tag1 $$ can be generated via Poisson bracket $$ \...
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1answer
84 views

Infinite dimensional representation of Lorentz group

Given a multicomponent field $\Phi_a$ we have the transformation law $$ \Phi_a(x) \rightarrow M_{ab}(\Lambda) \Phi_b(\Lambda^{-1} x) $$ Where $M_{ab}$ is some finite dimensional representation of the ...
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1answer
104 views

Why is the electromagnetic duality an S-duality?

One of the examples that Wikipedia gives of S-duality is the EM duality. Namely that $$ \begin{align} \mathbf{E} &\rightarrow\mathbf{B} \\ \mathbf{B} &\rightarrow -\frac{1}{c^2}\mathbf{E} \...
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1answer
53 views

Finding the free action in momentum space

I have the free action in position space $$S_0[J,\phi] = \int \! d^4x[\frac{1}{2} \phi (\partial_\mu \partial^\mu - m^2 + i\epsilon)\phi + \frac{\hbar}{i}J\phi].$$ Knowing that the Fourier transform ...
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1answer
194 views

Understanding Hamilton's equations in classical field theory in a rigorous way

So, I'm in a quest of understanding classical field theory on my own, and I'm interested in its rigorous construction. Here's the link for a previous post of mine on mathoverflow. The interesting ...
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0answers
37 views

Problem with derivation of Dirac equation

In my book on QFT (Lancaster & Blundell) while deriving the Dirac equation, they arrive at the following result: $$(\partial ^2 +m^{2})=(\not\!\partial -im)(\not\!\partial+im)$$ They then state: &...
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2answers
90 views

Energy Positivity of Classical QED Field Theory in Presence of Sources

It's well known that the classical electromagnetic field has positive definite energy, simply because: $$\mathcal{H}=\frac{1}{2}\epsilon_0\vec{E}^2+\frac{1}{2\mu_0}\vec{B}^2.$$ However, this result ...
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1answer
72 views

How these two approaches to spinors in curved spacetimes relate?

Regarding spinors in curved spacetimes I have seem basically two approaches. In a set of lecture notes by a Physicist at my department he works with spinors in a curved spacetime $(M,g)$ by picking a ...
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2answers
69 views

Why does the 3-point function of a real scalar field vanish?

$$\langle0\lvert T\hat\phi(x_1)\hat\phi(x_2)\hat\phi(x_3)\rvert0\rangle$$ I'm looking for an intuition to it, if not an actual interpretation. Otherwise, I know how to get the result 0 using Wick's ...
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0answers
118 views

Eqs of motion under dual field strength tensor

With a $SU(2)$-invariant Lagrangian of the form $$ {\cal L} = -\frac{1}{2g^2}tr\{F_{\mu\nu}F^{\mu\nu}\} - \frac{\theta}{2g^2}tr\{F_{\mu\nu}\widetilde{F}^{\mu\nu}\},\quad \widetilde{F}^{\mu\nu} = \frac{...
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0answers
44 views

What's the symmetry group $SU(N)/Z_N$?

I'm trying to understand David Tong's notes, specifically the discussion around page 92 where he's arguing that a different symmetry group may the group of QCD, namely $G'=SU(N)/Z_N$ instead of $G=SU(...
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1answer
63 views

$\phi$ a pseudoscalar field $\Rightarrow$ $\phi(x^0, \vec 0) = 0$?

On page 9 of Relativistic Quantum Mechanics and Introduction to Quantum Field Theory, A. Z. Capri defines scalar and pseudoscalar fields as follows. Under a parity transformation, $$ {x^0}' = x^0, \...
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1answer
40 views

Magnetization and Polarization in an electromagnetic field theory

I am currently reading through a paper by Hughes and Ramamurthy (ref: https://arxiv.org/abs/1508.01205), which describes the electromagnetic response of a line-node semimetal by the action $$S[A,B] = \...
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2answers
123 views

Is a field a physical object? [closed]

I read in various sources that fields in physics are some distribution of quantities at different points in space. For example, an electric field is the distribution of the value of the electric ...

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