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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31 views

Difference between potential and potential energy mathematically

I search google, quora, and reddit: What is the difference between potential and potential energy? Potential is the ability to do work. Potential Energy is the amount of energy it acquires. ...
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1answer
50 views

How do fields change when you transform spacetime coordinates?

In field theory, I have seen the notion that when you infinitessimaly transform your coordinates in the form $$x'_\mu = x_\mu + \delta x_\mu$$ The fields transform as $$\phi(x) \rightarrow \phi'(x') =\...
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39 views

Relationship between minima of scalar potential and mass of particles in the theory?

Consider the free scalar field theory $$ \mathcal{L}=\partial _\mu \phi \partial ^\mu \phi^* - V(|\phi|) $$ where $V(|\phi|)=-m^2|\phi|^2$. There are several answers on this site about why the ...
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41 views

Why is scalar field self-interaction quartic in the Lagrangian?

From https://www.youtube.com/watch?v=CNcTOSx2eMs at 11:34 the Higgs field is called a free field because it has no sources, no kinds of charges from which it emanates ... the fact that it is self ...
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3answers
603 views

Hamiltonian Field Theory in Peskin & Schroeder

In Section 2.2 of their QFT textbook, Peskin & Schroeder introduce the Lagrangian and Hamiltonian field theories of a classical scalar field. While defining the action $S[\phi]$ and deriving the ...
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1answer
59 views

Conserved quantity from a conserved current

In my QFT course, it was asserted that the conserved quantity associated with some conserved current is given by $Q = \int_v j^0d^3x$ where $j^0$ is the time component of the conserved current, and $d^...
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1answer
37 views

Symmetries of two Massless Fermion

I'm considering the following free Lagrangian (density): $$\mathcal{L}= \bar{\Psi} \left(i \displaystyle{\not}{\partial}\mathbb{1}_{2}\right)\Psi$$ Where $\Psi$ is a doublet of two fermion fields $\...
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10 views

Charge conjugation matrix in terms of rank-2 antisymmetric tensors

I am having trouble with the sign when computing the charge conjugation matrix in the Weyl representation, namely I am yielding an additional minus sign. Let me start with some of the conventions I ...
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1answer
53 views

Dirac equation plane wave solutions for antiparticles (Peskin QFT book overlooked?)

I am puzzled by the derivation given here about the Dirac equation plane wave solutions of Peskin QFT book (shown in (3.59) and (3.62) on the scanned image below): $$ (i \gamma^\mu \partial_\mu -m) \...
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0answers
80 views

Do fields have meaning in quantum field theory?

In classical mechanics the equation of motion of a free particle in polar coordinates is given by \begin{aligned} &m\left(\ddot{r}-r \dot{\theta}^{2}\right)=0 \\ &m(r \ddot{\theta}+2 \dot{r} \...
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2answers
152 views

Gauge Invariant terms of Lagrangian for Electromagnetism

Besides the usual EM Lagrangian $\mathcal{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu}$, we can add an additional term $\mathcal{L'} = \epsilon_{\mu \nu \rho \sigma }F^{\mu \nu}F^{\rho \sigma} = -8 \vec{E} ...
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33 views

Finding the EoM of a complex scalar field

I have a brief question about finding the field equations of a given Lagrangian. Consider the Lagrangian density given by $$\mathcal{L} = \partial_\mu \psi\partial^\mu\psi^\ast - m^2(\psi\psi^\ast) - \...
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46 views

For this simple harmonic oscillator scalar field, why are time and space producing different results?

I would expect this harmonic oscillator to oscillate in both space and time. It seems it only oscillates in time, and it disperses across space. I show these differences by making the scalar field a ...
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24 views

System of partial differential equations when solving the Dirac equation on a magnetic field

Hey I am trying to solve this system of partial differential equations that I got from the Dirac equation for a spinor on a magnetic field. $$\begin{equation} \begin{cases} k_1 + (\partial_x^2 + \...
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1answer
86 views

Representing Green function as a coherent state path integral

I am working through the problem "self-consistent T-matrix approximation" in Altland and Simons (second edition) pg 234. One of the steps involves representing the Green function as a ...
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1answer
55 views

A primer on topological solitons in scalar field theories

As the title suggests I want to learn more about topological solitons in scalar field theories. I am searching for a resource which is self-contained, in the sense that it also explains the ...
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0answers
42 views

Action of $\phi^4$ theory in lattice field theory

Euclidian action of $\phi^4$ thory is given by: \begin{equation} \int d^Dx L_E=\int d^Dx\left ( \frac{1}{2}(\partial_\mu \phi)^2 +\frac{m_0 ^2}{2}\phi^2+\frac{g_0}{4!} \phi^4\right), \end{equation} ...
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1answer
49 views

Is occupation number of a certain quantum state an observable?

I know that the number operators are projective. Can I use the number operator for measurement if it is projective? Can I use it if I want to measure the number of particles (fermions) in a certain ...
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26 views

$\phi^3$ four-point function in Collins, Feynman rules for OPE

it seems like it should be clear, but in collins book on Renormalization in chapter 6 (fig 6.1.1), he writes that for the diagram the large q-momentum expansion would be $[\frac{-g^2}{(p_1^2-m^2)(p_2^...
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2answers
40 views

Mandl & Shaw QFT chapter 1 question [closed]

Page 3 of Mandl & Shaw claims that, given a vector $\pmb{A}(\pmb{x},t)=\pmb{A}_{0}e^{i(\pmb{k}\pmb{\cdot} \pmb{x} - \omega t)}$, $\pmb{\nabla} \pmb{\cdot} \pmb{A} = 0$ (eq. 1.6) implies $\pmb{k} \...
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2answers
64 views

Finiteness of Maxwell gauge field symplectic form?

The symplectic form for a Maxwell $U(1)$ gauge field is $$ \omega = \int_\Sigma d \Sigma^\mu \delta F_{\mu \nu} \wedge \delta A^\nu $$ where $\wedge$ acts on field space. (In this notation, $\delta$ ...
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5answers
3k views

Can particles be in a superposition of times as well as positions?

We often talk about the various possible positions a particle can have upon measurement according to the probability density. But owing to the profound link of space and time in relativity, why do you ...
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1answer
61 views

Switch from $AdS$ to $dS$ in quadratic gravity using $f(R)$ trick: problem

I have some difficulties with effective quadratic gravity involving a cosmological constant with the "wrong sign". The following is the setup of my question. Let's assume one has the ...
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1answer
38 views

Weyl + Diffeomorphism invariance = Conformal symmety only in flat space?

Let $(M,g)$ be a Riemannian manifold and consider a classical field theory $\phi: M \to \mathbb{R}$ given by some action functional $S$ satisfying Weyl & diffeomorphism invariance $$ S(\phi, e^\...
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22 views

How can we write energy of lab frame of reference in centre of mass frame which equation can interlink them?

How can we write energy of lab frame of reference in centre of mass frame which equation can interlink them?
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1answer
95 views

Does electric field really exist or it is just an interpretation? [duplicate]

Did physicist create the concept of electric field to describe the interaction of charge particles at a distance? If they are real, do we have experimental evidence? Please describe some of them. And ...
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138 views

Klein–Gordon Inner Product in Reissner–Nördstrom Spacetime

We consider a massless charged scalar field $\Phi$ on the Reissner–Nördstrom black hole space: $$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta\,d\varphi^2, \tag{1}$$ with $f(r)=1-\frac{2M}{...
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1answer
69 views

Stress energy tensor and scalar field [closed]

The lagrangian for a scalar field $\phi$ is $$\mathcal{L}=\frac12 \partial_{\mu}\phi \partial^{\mu} \phi -V(\phi)$$ From Noither's theoreme we got stress energy tensor as $$T^{\mu \nu}=\partial^\mu \...
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0answers
12 views

Relationship between second moment of axial propagation constant and effective axial propagation constant?

In this paper by Feng & Winful, the authors attempt to explain the Gouy phase shift of focused beams in terms of the second moment of the axial propagation constant $\langle k_z^2\rangle$, where ...
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0answers
14 views

Four vector transformation by vector transformation and axial transformation

I want to know Why this four vector (isospin space) transform in vector transformation and axial vector transformation are like this.
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1answer
79 views

Deriving an action from a metric

I try to find out how in this paper https://arxiv.org/abs/hep-ph/9905221 the authors derived an effective action from the metric. The paper I study is related to string theory and modified gravity ...
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1answer
43 views

Equation of motion for scalar field

I am trying to derive the equation of motion for a scalar field in flat and homogenous space time where the metric is $g_{\mu \nu}=diag(-1,a^2(t),a^2(t),a^2(t))$ and the Lagrangian is given by $$\...
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2answers
49 views

Lagrangian density for flat space time

The lagrangian density of a classical scalar field is given as $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi -V(\phi).$$ For a flat and homogenous space time the FRW metric is $g_{\...
2
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1answer
83 views

Energy Momentum Tensor for a complex scalar field in GR

I want to find the EM tensor for a charged scalar field $\Psi $ with mass $m$ in curved space time. I am considering the following action: $\tag{1} S = -\frac{1}{2}\int\sqrt{-g}\text{ }d^4x\text{ }(g^{...
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2answers
61 views

How does the propagation of a complex scalar field work?

I'd like to know which parts of the following is due to convention and which part has to be a certain way. Let's assume a complex scalar field $\phi(x)$ with Lagrangian: $$ \mathcal L = \partial_\mu \...
2
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1answer
46 views

Perturbative expansion of $\varphi^4$ theory and Green's functions

I'm working in the $\varphi^4$ QFT where $$S(\varphi)=\frac{1}{2} \mu \varphi^{2}+\frac{1}{4 !} \lambda_{4} \varphi^{4}$$ and the text says that we can expand (assuming small $\lambda_4$) as $$\exp (-...
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0answers
27 views

Lorentz Invariant Space-Like Field Commutator in Diagrammatica (Cambridge Lecture Notes), Page 27

I am working my way through Diagrammatica by Martinus Veltman, and had a question about his proof for Lorentz invariance of space-like separated fields. Unfortunately, Veltman does not number his ...
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1answer
37 views

Double symmetrization in Einstein index notation

What does the following supposed to give? This appears in equation (3.12) in the following link: https://arxiv.org/abs/1312.5344 (last term) $$A^{(I(J}B^{K)L)} = $$ $$1) \quad \frac{1}{2}(A^{(IJ}B^{KL)...
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2answers
354 views

Fields as sections of fibre bundles versus functions on spacetime

Why should fields be thought of as sections on a bundle? In particular, what is the problem with thinking of them as functions (with additional conditions on target domain and smoothness) on spacetime?...
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0answers
20 views

Can the Newton field law be applied to electrostatics, gravity as well as thermal radiation in the same way?

Can the Newton field law be applied to electrostatics, gravity as well as thermal radiation in the same way?It is well known that the net gravity inside a homogenous shell is zero.Also happens to the ...
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48 views

Different Definitions of the Action Functional

So I am not able to grasp the difference between the many definitions of the Action Functional I am finding while studying CFT to get to QFT. The first one I encoutered was the following $$ S [\phi_n] ...
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5answers
64 views

Lagrangian in fields

I am wondering where this statement comes from. I've tried to find out myself and just can't seem to figure it out. But it seems to be trivial everywhere that the kinetic term in the Lagrangian of a ...
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2answers
106 views

Why change of variables in the Hamiltonian give us the same physics while changes of variables in the Lagrangian does not?

Suppose we have the Lagrangian density $$L=\partial_\mu\phi\partial^\mu\phi-m^2\phi^2\tag 1$$ With $\phi$ a scalar field and $\pi=\frac{\partial L}{\partial \dot{\phi}}=2\dot{\phi}$ we can show that ...
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1answer
94 views

What is the meaning of diffeomorphism invariance?

I have read a bunch of papers and I see "diffeomorphism invariance" and I cannot understand how it works. For instance, in asymptotic safe quantum gravity, we make 2 assumptions: ...
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0answers
27 views

Understanding some notation in one-dimensional electron gas

I am a little confused about some notation used in Altland and Simons (second edition) pg 176. In their discussion on the one-dimensional electron gas, they define bosonic operators: $$b(x) \equiv \...
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1answer
46 views

Flame changeing water temp faster when boiling as opposed to heating it up

Cooking beans I noticed that, after the beans came to a boil, they reacted much more quickly to the turn of the dial than cold. I'm wondering if the water an everything else is more sensitive and ...
2
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1answer
84 views

Why are there three charges (all with the same form) with Pauli Matrices?

I am working on problem 2.2 Part d of Peskin and Schroeder's An Introduction to Quantum Field Theory. The authors claim that there are three charges based on the three Pauli Matrices and that that ...
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0answers
13 views

References for Hamiltonian field theory and Dirac Brackets [duplicate]

I'm looking for complete and detailed references on constrained Hamiltonian systems and Dirac brackets. While my main interest is electrodynamics, I would prefer a complete exposition of the theory ...
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2answers
199 views

Rarita-Schwinger spin projection operators

Chapter 2 of the paper Symmetry of massive Rarita-Schwinger fields by T. Pilling mentions "the usual" spin projection operators. However, to me, they are not usual and I struggle with ...
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1answer
91 views

Some questions about the compact boson in David Tong's notes on Gauge Theory

The notes can be found at http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html. In Sec. 7.5.1, T-Duality, around Eq. 7.51, it says that the Bianchi identity $\partial_\mu(\epsilon^{\mu\nu}\partial_\...

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