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Questions tagged [quantum-field-theory]

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 this tag for many-body quantum-mechanical problems and the theory of [tag:particle-physics]. Don’t combine with [tag:quantum-mechanics].

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Anti-commutation relations in annihilation operators

It is claimed that $$\{c_\alpha,c_\beta \} = c_\alpha c_\beta + c_\beta c_\alpha = 0$$ where $c_\alpha$ and $c_\beta$ are the fermionic annihilation operators in second quantization. Why is that ...
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How to write an operator in matrix form?

Say I have the following operator: $$\hat { L } =\hbar { \sum_{ \sigma ,l,p } { l } \int_{ 0 }^{ \infty }\!{ \mathrm{d}{ k }_{ 0 }\,\hat { { { a }}}_{ \sigma ,l,p }^{ \dagger } } } \left({ k }_{...
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Imaginary part of vacuum polarization tensor

I deal with vacuum polarization diagram calculation with help of QED by Landau et al. and I would like to understand one interesting statement. I know how to compute $\Pi_{\mu\nu}(k)$ via dimensional ...
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What are soft theorems in context of scalr fields

What are soft theorems ? I tried reading Weinberg’s paper but couldn’t understand it, are there any resources on this ? I am very interested in the case of scalars.
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Zero-energy universe - What is nothing?

I am a layman, so excuse me in advance for the stupidity of my questions, and I hope you can answer them in a way that I can understand. I have read, here and there, that the Universe might have a ...
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How is mass related to the Poincare group?

I have studied about Poincaré group and some QFT, read that the Casimir elements are $p^{u}p_{u} = m^2$ and Pauli-Lubanski vector (or pseudovector). How does the mass come into picture and spin come ...
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41 views

How to know what type of diagram contributes to a two-to-two process?

There are 3 types of diagrams that can contribute to a two-to-two process; the $s$-channel, $u$-channel and $t$-channel. How do I know what diagrams can contribute to a process? I know that in QED, ...
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Cutkosky's rule for vertex correction

I don't understand how to evaluate the diagram for electron-photon vertex with help of Cutosky's rule: According to the rule, one should cut diagram "in the middle". I denote final state as F and ...
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Does the massless fermion in $2+1$ dimensions suffer from gauge anomaly?

In Fermion Path Integrals And Topological Phases Witten showed that for a massless Dirac fermion in $2+1$ dimensions $$S[\bar{\psi},\psi]=\int d^{3}x\bar{\psi}iD\!\!\!\!/_{A}\psi,$$ where $A$ is a $...
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Massless $\phi^4$ theory

Most of the standard textbooks on QFT discuss in some detail the massive $\phi^4$ theory in 4d space-time. I would be interested to see a discussion of massless $\phi^4$ theory (in fact other non-...
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QCD gluon physical polarisation sum in the three gluon vertex

In the Compton scattering quark($p_1$) + gluon ($q_1$)-> quark($p_2$) + gluon($q_2$), there is three gluon vertex contribution. If we choose the physical polarisation sum $\sum_{\lambda} \epsilon^a(\...
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Computing the spin degrees of freedom for a massless particle in $D$ dimensions

According to the paper A Lagrangian formulation of the classical and quantum dynamics of spinning particles, a relativistic spinless particle in $D$ spacetime dimensions can be described by the ...
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Poincare group and free theories

How exactly is the Poincare group related to the free relativistic theories in quantum field theory? I know Poincare group is the Lorentz group along with translations but don't see any connected why ...
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Alternative particle types?

If we consider mass and charge to be excitations of independent quantum fields, do they necessarily travel together? Can we have objects with only an excitation on the mass field, and objects with ...
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Effective theory of hierarchial fractional quantum hall state

In describing the effective field theory picture of the hierarchical fractional quantum Hall states in Tong's lecture notes, page 165 he gives the expression for filling fraction, quasi-particle ...
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$\psi\psi \longrightarrow \psi\psi$ scattering in Scalar Yukawa model

In David Tong's lecture notes, Equation 3.48 In line 2, how is $|0\rangle \langle 0|$ introduced between $\psi^{\dagger}(x_1)\psi^{\dagger}(x_2)\psi(x_1)\psi(x_2)?$ Why is $\langle p_2',p_1'|\psi^{\...
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Bound on Quantum Chaos

I am currently reading the paper A Bound on Chaos. In this paper, they evaluate the quantity C(t), which is an out-of-time-order correlator (OTOC), and use very clever arguments to show that there ...
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Unstable particles as asymptotic states

In QFT are unstable particles forbidden from being used as asymptotic states in scattering calculations? I ask because the $S$ matrix has an $I$ term which can propagate the unstable state. Is this ...
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Renormalization of sine gordon theory

So assume that we have a usual sine gordon theory in the the theory we have a term in the hamiltonian $$\frac{yu}{2\pi\alpha^2}\int dx \cos(\sqrt{8}\phi_\sigma(x))$$ where $\alpha$ is cut off ...
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How does Heisenberg's uncertainty work with more than one quantum field?

How does Heisenberg's uncertainty principle work with more than one quantum field? I am specifically asking about the time-energy uncertainty: $$\Delta E \Delta t \ge \frac {\hbar}{2}\tag{1}$$ Imagine ...
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Feynman diagram with fixed and on-shell internal lines

Say I've got two scalar fields $\pi$ and $\phi$, where $M_\phi > 2M_\pi$, and the interaction $$\frac{\lambda}{2}\pi^2\phi$$ At order $\lambda^3$ the $\phi\rightarrow\pi\pi$ decay gets a ...
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Why is particle number conserved in non-relativistic limit of QED?

I am trying to recover ordinary quantum mechanics from QED. One main feature of quantum mechanics is the conservation of particle and antiparticle number separately, i.e. $[N_{e^-},H] = [N_{e^+},H] = ...
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How does canonical quantization work with Grassmann variables?

Every quantum field theory textbook I've encountered seems to have the same logical oversight, because of the particular order they cover topics. First, the books introduce the Dirac Lagrangian, $$\...
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2answers
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Grassmann's variables under integration

If $\eta$ is a Grassmann variable, due to invariance under translations we get that, $$\int d\eta\ \eta = 1 \tag1$$ Nevertheless, for being Grassmann's, $\eta$ satisfies $\eta^2 = 0$. ...
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Yukawa scattering of fermions

I've been learning about qft from Peskins Quantum Field Theory, and I can't quite figure something out. The author considers the following scattering in the Yukawa interaction theory ($H_{I}=\int d^3x\...
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Superficial degree of divergence in $\lambda\phi^4$

Ryder at the beginning of the chapter about renormalization defines the "superficial degree of divergence" of diagrams in $\lambda \phi^4$ theory. I'll recap the derivation. A diagram in $\lambda\phi^...
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2PI-effective action and functional derivatives

I'm trying to work out the 2PI-effective action for complex scalar fields. Introducing a multi field index $(a,b,c...)$ the complex conjugation and all other degrees of freedoms are suppressed, and ...
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1answer
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Why do we impose de Donder gauge?

In the field language, a massless particle corresponds to irreducible representations of the Lorentz group. In particular, given a spin-2 massless particle, we can embed the creation and annihilation ...
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Are there any open questions in theoretical quantum physics? [closed]

I am wondering if there are any open questions about the structure of quantum mechanics. If so, how do you know that this is an open question? Topics that come to mind are electron spin, probability ...
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Subtraction scheme invariance in QFT

I'm currently reading Schwartz's QFT text and I'm confused on how observables are supposed to be independent of subtraction schemes. In the text it seems that the renormalized loop diagrams are ...
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Scattering of light in Quantum Vacuum

The quantum mechanical vacuum (i.e. the vacuum of a typical QFT) viewed as the zero-point state of the system has an energy (of course, here enters the Vacuum Catastrophe and all the mess that follows)...
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Sign ambiguity when going from position to momentum space evaluating Feynman diagrams

When calculating a simple diagram I came across an ambiguity in the conservation of momentum, i.e. it seems to me that the particle could come out of the process with opposite momentum with respect to ...
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Can particles popped into existence from the vacuum have electromagnetic effects on other particles?

I know my question might have problems, but I am curious about it. In quantum field theory, particle-antiparticle pairs continuously pop in and out of existence from vacuum. These particles have a ...
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How to understand complex masses of unstable particles? The conceptual problem of calculating decay rate

If a particle has a complex mass, $p^2-m^2=0$ leads to $p^μ \notin \mathbb R^4$. What does it mean? When you want to calculate S-matrix elements of decay process $\langle p_f,\ldots\mid p_i\rangle$, ...
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Why can we add counterterms?

I'm having a hard time understanding why renormalized perturbation theory works. Why is it permissible to add counterterms to the Lagrangian? Terms which are often divergent themselves and carry ...
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1answer
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Why are we trying to fit GR in QFT when there's a chance that GR is incomplete?

GR explains most phenomena in our universe, but not everything.. Dark matter and Dark energy still don't fit in explanation of GR. QFT, on other hand, is almost complete. Shouldn't physicists go for ...
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Mass of an anti-symmetric spin-1 field

I was wondering how does one calculate the mass of an anti-symmetric spin-1 field. For a vector field one writes $m^{2}A_{\mu}A^{\mu}$ for mass term. How does one write the mass for "any" anti-...
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Utility of the time-ordered exponential

Is the time-ordered exponential $$ \mathcal{T}\exp\left\{-i\int_{t_0}^tdt'V(t')\right\}\tag{1} $$ just a mnemonic device for the series $$ \begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
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1answer
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Goldstone theorem in Weinberg vol 2

I was reading the proof of Goldstone's theorem (the operator proof starting on page 170) in Weinberg's book on QFT (Volume II) and got confused. I am able to follow each line of the proof, but as a ...
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Mass dimension of an $n$-particle scattering amplitude in 4D

For the 4-dimensional case, and using the cross-section formula, how can we show that the mass dimensions of an $n$-particle amplitude must be $$[A_n] = 4-n~?\tag{2.99}$$ My understanding is that the ...
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Lorentz transform of light cone vectors

I'm currently taking classes on Special Relativity and Minkowski space. I have one question: how would future-oriented, contravariant time-like, space-like and light-like vectors transform under a ...
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1answer
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How to “resolve a state” with respect to a spacelike hypersurface in Minkowski Spacetime QFT?

Consider usual free QFT in Minkowski spacetime. For simplicity let us consider a real scalar field $\phi$. Usually quantization is performed with respect to one inertial reference frame. This is is ...
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Peskin and Schroeder: derivation of Dirac fields commutator

I'm perplexed by the following non numbered equation at page 54 of Peskin & Schroeder, right between $(3.92)$ and $(3.93)$ $$ [\psi_a(x),\overline{\psi}_b(x)]=\int\frac{d^3p}{(2\pi)^3}\frac{1}{...
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Path Integral in Electric-Magnetic Duality

The action of electromagnetic field is $$S=\int\left(-\frac{1}{2e^{2}}F\wedge\ast F+\frac{\theta}{8\pi^{2}}F\wedge F\right),$$ where $F=dA$ is the curvature $2$-form, and $A$ is the connection $1$-...
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If the particles show uncertainity ,then why the body made of particles cant show that? [duplicate]

What i want to know is that if the properties of the quantumn nature reduses if the particles combine? Or were we missing any fundamental relation to particle or boby made up of particles?
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Euler-Lagrange eqs. use in QFT [duplicate]

It is known that in QFT the Euler-Lagrange equations are used to obtain the equations of the quantum fields. Nevertheless, from the path integral's point of view (where you integrate over all $\it{...
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$\theta$ Term In 2d for $U(1)$ and $SO(2)$

For the $U(1)$ gauge theory in 2d, there can be a theta term $$\frac{\theta}{2\pi}\int_{M} dA$$ where $A$ is the $U(1)$ gauge field and $\theta\sim \theta+2\pi$. However, it is known that $U(1)=SO(2)...
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Propagators in interaction with derivatives

Given a Lagrangian density containing an interaction with derivates, it's easy how to guess the Feynman rules for vertexes. However i was wondering about propagators: in S-matrix expansion it's ...
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Complex Gaussian integral with different source terms

Do the source terms multiplying a complex field and its conjugate need to be conjugates for the Gaussian identity to hold? E.g. is $$\int D({\phi,\psi,b}) e^{-b^\dagger A b +f(\phi, \phi^\dagger,\...
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SUSY vacuum has 0 energy?

This is related to Modern Supersymmetry: Dynamics and Duality by Terning. Consider $N=1$ SUSY. $\{Q_a,\bar{Q}_{\dot{a}}\}=2\sigma_{a\dot{a}}^\mu P_\mu$. Sum over $a=\dot{a}=1,2$ and this yields $4P^...