Quantum Field Theory (QFT) is the theoretical framework describing the quantisation of classical fields which allows a Lorentz-invariant formulation of quantum mechanics. QFT is used both in high energy physics as well as condensed matter physics and closely related to statistical field theory. Use ...

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Spectral Decomposition for Complex Conserved Current

What is the spectral decomposition for the vacuum expectation value of a complex conserved current $J^\mu(x)$, $\langle T\{J^\mu(x),J^\nu(y)^\dagger\}\rangle$?
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If a symmetry operator S in a QFT annihilates the vacuum, why does S preserve the space of 1-particle states?

In the paper "Supersymmetry and Morse Theory", on the third page (p. 663 in the journal version), Witten says: "Now in any quantum field theory if a symmetry operator (an operator which commutes ...
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55 views

Higher rank $\gamma$-matrix question

I read that the higher rank $\gamma$ matrices can be written as alternate commutators and anti-commutators. For example, the rank 3 gamma matrix can be written as $$\gamma^{123} = ...
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Lorentz transformation - Bjorken & Drell

I'm trying to derive (14.25) in Bjorken & Drell (B&D) QFT. This is $$\tag{14.25}U(\epsilon)A^\mu(x)U^{-1}(\epsilon) = A^\mu(x') - \epsilon^{\mu\nu}A_\nu(x') + \frac{\partial ...
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Time ordering, interaction Lagrangian calculation, QED

I am trying to compute $$ \langle 0| \, T\left\{\phi^\dagger(x_1) \phi(x_2) \exp \left[i \! \int{L_1(x) \, \mathrm{d}x} \right] \right\}|0 \rangle $$ for $$ L_1(x) = ...
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What does Weinberg–Witten theorem want to express?

Weinberg-Witten theorem states that massless particles (either composite or elementary) with spin $j > 1/2$ cannot carry a Lorentz-covariant current, while massless particles with spin $j > 1$ ...
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Fast and slow modes in renormalization group of nonlinear sigma model

A general nonlinear sigma model can be expressed as \begin{equation} S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}] \end{equation} where $g$ takes value in a matrix ...
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38 views

Can asymmetrical Lorentz forces account for Relativistic affects near the speed of light?

The underlying thought here is that at low relativistic speeds all objects are subjected to emf radiation from all directions. This is basically the sum of all the radiation (light, infra-red, ...
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19 views

Real representations of chiral fields

Why we can´t have real representations of chiral fields, i.e. why does a multiplet of chiral field (Weyl spinors) under a real representaiton of a Lie Group transforms as a "vector". It is easy to see ...
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Is gravitational Chern-Simons action “topological” or not?

Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection: $$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{a} $$ $$ ...
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52 views

Heisenberg picture with creation annihilation operators

In the Schrodinger picture, states are time dependent and operators time-independent. So expected values look like: $\langle s_1,t|\hat{A}|s_1,t\rangle$. If we go over to the Heisenberg picture the ...
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23 views

Connected diagrams in Sine-Gordon action

I consider a bosonic action of the type $$\int dx d\tau\left( \underbrace{a(\nabla\theta)^2+b(\nabla\phi)^2}_{free} +\underbrace{c\cos{4\phi}}_{interaction}\right),$$ and want to treat the cosine term ...
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73 views

Total divergence term and corresponding Feynman Diagram

A total divergence term added to the Lagrangian doesn’t affect the action because the integral of a total divergence vanishes. But if one attempts to derive the Feynman rules from the Lagrangian with ...
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53 views

Getting the electric field using Green's function [closed]

Let the Green's function for the gauge field be given (after gauge fixing) as $$G_{\mu \nu}(x,y) = \delta_{\mu \nu}G(x-y) \tag{1}$$ where $$G(x-y)= \int \frac{d^dk}{(2\pi)^d} \frac{e^{ik \cdot ...
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39 views

Understaning Euclidean Green's function

Consider a scalar field coupled to a source $$(\Box - m^2)\phi(x) = -J(x)\tag{1}.$$ Then, the response of the source is determined by the Green's function $G(x-y)$, which satisfies $$(\Box - ...
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Symmetries in QM and QFT — operator transformation laws

In quantum mechanics, we implement transformations by operators $U$ that map the state $|\psi\rangle$ to the state $U|\psi\rangle$. Alternatively, we could transfer the action of $U$ onto our ...
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Is the weak interaction Lagrangian invariant under parity transformations?

The weak interaction term in the Lagrangian reads $$ \bar \Psi \gamma_\mu P_L \Psi W^\mu. $$ Under parity transformations, because of $\Psi \rightarrow \gamma_0 \Psi$ and $\gamma_5 \rightarrow ...
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Charge dependence of operators in QED renormalization

Consider a UV cutoff regulator $\Lambda$ with an effective QED lagrangian: $\mathcal{L}_{\Lambda} = \bar{\psi}_{\Lambda}(i\not \partial - m_{\Lambda})\psi_{\Lambda} - ...
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85 views

QFT and violation of Heisenberg uncertainty principle

In some QFT books is said that a free electron can emit a virtual photon as long as it reabsorbs the photon and returns to its original state within a time: $$\Delta t<\dfrac{\hbar}{2\Delta E}$$ ...
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75 views

Can a quantum field be understood as a superposition of all particles' wave functions?

Many text books emphasize that the quantum field is not wavefunction. But because of the similarity in the format, I could not stop from wondering whether they are actually the same thing.
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$d=2$ pole argument of quadratic divergences in Peskin & Schroeder's book

Q1: My question is, in the context of dimensional regularisation(DREG, in dimension $d$), why do they mention the absence of $d=2$ pole in the gauge theory cases[1], whereas the $d=2$ pole is not ...
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89 views

Reconciling two interpretations of renormalization

I know of two fascinating and perfectly reasonable explanations of renormalization. However, I'm having difficulty reconciling the two. The first is to say that when we initially write down a ...
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142 views

How do creation operators change with time in an interacting theory?

When studying the quantization of a field theory with free fields, the creation operators $a^\dagger(k)$ are independent of time. In an interacting theory, they are time-dependant, and therefore ...
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Wavefunction renormalisation in first order perturbation theory

I just read the following in the context of scattering amplitudes in QFT: Note that the wavefunction renormalisation factor $Z$ itself is of the form $1 + \mathcal{O}(\lambda)$ in perturbation ...
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58 views

Is there a field for which neutral particle and antiparticle, can be considered as positive and negative charge?

I apologize, but QFT is not my domain. What I ask is connected with the question Do the fields exist without charges? . By analogy with the electron and proton, that carry the electric charges of the ...
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An annoying question about perturbative quantum field theory

I am so sorry for posting this long question. But I've been confused and frustrated by perturbation series in Quantum Field Theory for years. I hope someone can help me. Thank you so much! In the ...
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Gordon decomposition of Dirac current for massless electron?

We know Gordon decomposition of Dirac current is applicable only for massive (nonzero mass) Dirac particles. Is there an analog for massless Dirac particles? (I have made an attempt to answer ...
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123 views

Is Parity really violated? (Even though neutrinos are massive)

The weak force couples only to left-chiral fields, which is expressed mathematically by a chiral projection operator $P_L = \frac{1-\gamma_5}{2}$ in the corresponding coupling terms in the Lagrangian. ...
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79 views

Feynman diagrams for scalar field: which particle are we drawing?

Chapter I.7 of Zee's Quantum Field Theory in a Nutshell is an introduction for Feynman diagrams in the context of a scalar field $\varphi$, with Lagrangian $\mathcal{L} = \frac12[(\partial ...
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67 views

Do Dirac field states belong to a Hilbert space with spinor coefficients?

The quantized Dirac field at a certain space-time point can be written (roughly) as a linear combination of creation operators acting on the Hilbert space of physical states, with coefficient that are ...
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1answer
50 views

Terminology of Higgs boson and Goldstone boson

I know, the from the Higgs Mechanism, or Spontaneous symmetry breaking, the massless Goldstone boson becomes massive. So in some sense Goldstone bosons are eaten by gauge "boson". Here I got ...
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Does graviton loops affect the seperately covariant conservation of energy momentum of two noninteracting sectors of matter

Consider the action $$\int \sqrt{-g}\left[R[g]+\mathcal{L}_{m1}(g,\psi_1)+\mathcal{L}_{m2}(g,\psi_2)\right]$$ Classically we have $$\nabla^\mu T^1{}_{\mu\nu}=0,\,\,\,\,\nabla^\mu T^2{}_{\mu\nu}=0$$ ...
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Why do gauge bosons/leptoquarks not mediate proton decay in the Pati-Salam model?

In the Pati-Salam $\mathrm{SU}(4)_c\times\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R$ model, I see Wikipedia and some slides mention this model doesn't predict gauge mediated proton decay without giving ...
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1answer
66 views

Treating the spinors as Grassmann numbers or as c-number objects

In the literature on supersymmetry, the following spinor summation convention is often used (eg. Wess & Bagger's book Supersymmetry and Supergravity) $$ \psi\chi = \psi^{\alpha}\chi_{\alpha} = ...
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61 views

Interpretation of the four-vector $k$ in scalar QFT

I'm studying the canonical quantization of the Klein-Gordon real scalar quantum field theory, given by the classical Lagrangian density $$\mathscr L = {1\over ...
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2answers
33 views

What are the end points in the action integral of field theory?

In the mechanics of particles when we apply the principle of the least action the two end points are two spatial coordinates. Therefore, if we consider the variation of the action with respect to the ...
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1answer
65 views

The index of a Dirac operator and its physical meaning

I recently read Witten's paper from the 1980s and he often uses the notion of the index of a Dirac operator in K-theory. What is the meaning of the index of a Dirac operator? What exactly is the ...
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42 views

Klein-Gordon propagator integral in the light-like case

In Kerson Huang's Quantum Field Theory From Operators to Path Integrals (Amazon link), pages 28 and 29, he calculates the propagator in the following case: time-like, space-like and light-like. First ...
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How to work with singular gauge transformations in QFT [closed]

I was recently considering a problem analogous to the Aharonov-Bohm (AB) effect but in the context of quantum field theory. Consider then Dirac electrons minimally coupled to an AB flux and described ...
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37 views

A question about Ising model

If $H$ is the Hamiltonian of an Ising model of $n$ spins on a lattice then is the following quantity look like something one has seen? $([uI-H]^{-1})_{ii} - \frac{1}{n}Tr[[uI - H]^{-1}]$ where $u$ ...
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95 views

How to count and 'see' the symmetry factor of Feynman diagrams?

Could somebody explain how one can derive the symmetry factor both by counting possible contractions and by looking at the symmetry of a diagram. Consider for example this diagram in $\phi^4$-theory ...
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1answer
83 views

Making sense of the canonical anti-commutation relations for Dirac spinors

When doing scalar QFT one typically imposes the famous 'canonical commutation relations' on the field and canonical momentum: $$[\phi(\vec x),\pi(\vec y)]=i\delta^3 (\vec x-\vec y)$$ at equal times ...
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1answer
46 views

Can the vacuum energy be made finite with quantized space

From what I know the reason we have infinite vacuum energy is because according to Quantum Field Theory at every point in space we something analogous to a harmonic oscillator but since the Zero Point ...
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1answer
36 views

Finding the normal ordered momentum operator for free theory

I am asked to show that the normal ordered momentum operator for free theory is $$\hat{p^\mu} = \int \frac{d^3 p}{(2 \pi)^3} p^\mu \: a_p^\dagger \:a_p.$$ The free theory Lagrangian is given by ...
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1answer
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A test for virtual particles by measuring gravity fluctuations possible?

Ok to begin I will begin by talking briefly about my discussions with my Quantum Mechanics (specializes in Particle physics) professor and my Cosmology Professor (who studies particle physics with ...
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Materials on charged black brane

everybody! Does anyone know some good materials on charged black branes in AdS/CFT and the role of chemical potential in theses cases?
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QFT propagator, time reversal and the Born rule

As far as I understand it a propagator, $D(x-y)$, gives the amplitude for a flow of positive energy-momentum from an earlier event $y$ to a later event $x$. Addendum: Instead of talking about energy ...
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Help with derivation of the Casimir Effect?

I am at the very last part of a relatively long derivation of the Casimir effect, and I just don't understand the final step D: So far, I have derived the ground state energy to be $$\langle 0| ...
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History of QFT after 1973 [closed]

Where I can read about history of development quantum field theory after 1973? I'm interested in historical reviews, like as first chapter of the Weinberg's book.
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128 views

Stimulated emission direction

Place a sub-micron clump of crystal violet molecules in front of a multipixel detector. Raise the molecules to an electronically excited state with a beam of 590 nm light, illuminating from the side ...