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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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Partition function computation on a Riemannian Manifold

This discussion comes from Chapter 10, Mirror Symmetry. I am given a Riemannian manifold $M$, and a classical (Euclidean) theory: $$(1)\quad S_E={\int}_0^\beta d\tau\space\Bigl(\frac{1}{2}g_{ij}{\dot{...
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One-loop diagram in scalar field theory

We see that this loop contain particle with momentum $p+k$ and anti-particle with momentum $k$. But I fall into contradiction because the Feynman integral corresponds to this loop (see the book http:...
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Why should vacuum energy be zero for LSZ formalism?

Can anyone explain why vacuum energy must be zero if we are to use LSZ formalism?
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Why should the $\phi^4$ term necessarily cause scattering while a $x^4$ term in anharmonic oscillator only causes correction of energy levels?

Consider an anharmonic oscillator in quantum mechanics, described by the Hamiltonian $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+bx^4.$$ The $bx^4$ term doesn't cause scattering. The effect of this ...
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Lagrangian with quadratic interaction

A real scalar field has Lagrangian $L=L_0+L_1$, where the free part $$L_0=-\frac{1}{2}(\partial\phi)^2-\frac{1}{2}m^2\phi^2$$ and the interaction term $$L_1 = + \frac{1}{2}g\phi^2.$$ I have two ...
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Does a proton bend spacetime?

Protons have mass and as a result of einstein's field equation dictate that the spacetime is no longer flat. But yet I find in most Quantum Field Theory books the Minkowski flat spacetime metric is ...
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The simple harmonic oscillator model relating particles and fields in QFT

In all of the introductory Quantum Field Theory texts I gave read so far, (such as Zee, Srednicki, Luke), there is an introduction to the concept of fields as operators, following the simple harmonic ...
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Problem with loop Integral (HQET)

I have come across the Integral: $$ \int_0^{\infty}dx [x^2-ixa+c]^{n-\frac{d}{2}}e^{-bx},$$ where $n = 1,2 ; a,b,c,d \in \mathbb{R}; b,d > 0$. This integral should contain some divergences for $d ...
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Expectation value of a path-ordered exponential

Let us define our path-ordered operator $\overrightarrow{U}\left(t_1,t_2\right)$: $$ \overrightarrow{U}\left(t_1,t_2\right)=\overrightarrow{\mathcal{P}}\exp\int_{t_1}^{t_2}dt\,\mathcal{O}\left(t\...
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Expectation values of states in QFT via the path integral

The path integral in QFT is usually computed only for the vacuum state, $$\langle 0 | T\{ A \} | 0 \rangle = \int \mathcal{D}\phi(x) A e^{iS[\phi]}$$ Doing it for different states is a bit trickier,...
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Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
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How does Atiyah-Singer index theorem relates instanton number to number of fermion zero modes?

I was studying this paper, where the authors consider an $SU(2)$ gauge field of instanton number 1 on a 4-sphere $M =S^4$. If $n_L$ is the number of zero modes of $\psi_L$ and $n_R$ is the number of ...
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Gauge transformation on vacuum

Suppose we have two electromagnetic vector potential operator that differ by a gauge transformation in free field theory. $A'_{\mu}= A_{\mu}-\partial_{\mu}\alpha$ is the state $A_{\mu}|0\rangle$ ...
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Generating functional for amputated connected Green functions

As I understand, generating functional for amputated connected Green functions is exatly $\mathcal{S}$-matrix of theory. One can represent this functional as (for instance, $\phi^3$ theory): $$e^{\...
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Ising model with quantum magnetic field

Hamiltonian of the Ising model with an external magnetic field is written as $$H=-J\sum_{\langle i,j \rangle} s_i s_j + h\sum_j s_j$$ where $J$ is nearest neighbor coupling constant and $h$ is the ...
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Gauge transformation on operators

Suppose we have two electromagnetic vector potential operator that differ by a gauge transformation. $A'_{\mu}= A_{\mu}-\partial_{\mu}\alpha$ Now suppose we have $\partial_{\mu}\alpha=F(x)\hat ...
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Converting a general QFT state to another inertial of light-cone reference frame

Assume, in a certain reference frame, a relativistic QFT at time $t=0$ is in the state $$ \hat{\Psi}(t=0) |vac\rangle \quad, $$ where $$ \hat{\Psi}(t) = \operatorname{e}^{-i\hat{H}t} \sum \limits_{n=1}...
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Thermal Quantum Field Theory methods

I have been reading about the various techniques that have been employed to study quantum field theories near equilibrium. It seems that the two main ways are the Schwinger-Keldysh (SK) and the ...
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Problem with converting Integral to Gamma functions (from HQET heavy quark self-energy diagram)

In the calculation of HQET radiative correction, I came across the Equation: $$\int_0^{\infty}d\lambda ~ \lambda^{-\epsilon}(\lambda+\omega)^{-\epsilon} = \frac{1}{2\sqrt{\pi}}\Gamma(\epsilon-\frac{1}{...
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One-loop correction to triple gluon vertex in QCD

I'm trying to calculate a one loop correction to the 3 gluon vertex, which is given by a circular correction, and another that's due to the 4 gluon vertex. However I'm unsure how the ghost fields ...
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Can we ever “measure” a quantum field at a given point?

In quantum field theory, all particles are "excitations" of their corresponding fields. Is it possible to somehow "measure" the "value" of such quantum fields at any point in the space (like what is ...
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Commutator of a quark current

In Quantum Chromodynamics, when we take the limit in which the u, d and s quarks have no mass, there exists a global symmetry $G \equiv SU(3)_L \otimes SU(3)_R$ in flavour space. The corresponding ...
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Singular behavior of pure gravity

Can anyone plese explain what means singular part of partition function for pure gravity? Let me specify my question. I am dealing with 2D quantum gravity and starts from path integral formulation of ...
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Should Hamiltonians in quantum field theory be linear operators?

The usual structure of quantum mechanics imposes that Hamiltonians are linear operators. I am not sure if this really holds in quantum field theory. Do non-linear Hamiltonian operators really make ...
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Closed set of operators under renormalization

While reading the article http://inspirehep.net/record/61135, I came across the concept of "closed set under renormalization". The definition they give is the following. In any renormalizable field ...
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Normalization for the overlap $\langle \phi_a | 0 \rangle$

This question is related to my previous post here. According to Wienberg's Volume I (9.2.9), we have the result $$ \langle \phi_a | 0 \rangle = \mathcal{N} \mathrm{exp}\left( - \frac{1}{2} \int d^{3}\...
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High-energy effective field theory

Usually when one speaks of effective field theories, one is looking to integrate out certain fields which are typically heavy in comparison to the regime of interest. That is one has a theory at a ...
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Mode Expansion in Klein-Gordon QFT

I have a confusion regarding the mode expansion of the Klein-Gordon field theory. I am following Peskin and Schroeder. My questions are about how we formally get to the expansion of the KG QFT in ...
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Proving equivalence of first and second quantisation (Pathria's way)

I'm trying to solve problem 11.1 form Pathria R. K. & Beale P. D. - Statistical mechanics book (the hyperlink will get you straight to the page of the problem). The point (b) is to show the ...
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Deriving Ward identity directly from a given formula for the conserved current only using the equal-time canonical commutation relation

I have a very technical question on deriving a Ward identity directly from a given explicit form of the "conserved current". Let me emphasize that I do not start with an apriori knowledge on the ...
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Couplings in the SYK model

I have read several times by now that the couplings in the SYK model are drawn randomly from a gaussian distribution. I was wondering what exactly is meant by that. To elaborate, when I compute an ...
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Can the Hermitian conjugate of a column vector still be column vector?

From what I understand, contravariant vectors are represented by column vectors, and covariant vectors are row vectors. So for a QED current, say $j ^ { \mu } = \overline { \psi } \gamma ^ { \mu } \...
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Frequency gap between neighbouring spectral lines

According to classical theory the atomic line spectra are discrete and their frequencies quantized. Have the newer quantum theories changed anything since then, giving some other expressions for the ...
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Feynman rule for 3-gluon vertex

I want to obtain the Feynman rule for the 3-gluon vertex, but looking at the result I don't really know how to tackle it. The relevant term in the Lagrangian is \begin{equation} \mathcal{L}_\text{...
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Where does Field Theory come from?

About my background: I'm currently a 4th year undergrad, and planning to do a PhD in theoretical physics. I think I have a decent understanding of basic physics, and I know how to do calculations in ...
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Tricks to evaluate expectation values for operator strings in second quantisation

I am taking a course in many-body quantum mechanics. Often, I have to evaluate expectation values on strings of creation/annihilation operators. I was told that to evaluate these, I should use the (...
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Questions on how Wilson loops relate to field & charge conservation, and lattice QFT

The path-ordered exponential from which the Wilson loop is traced is, crudely, $$ \prod (I+ A_\alpha dx^\alpha) = \mathcal{P}\,\mathrm{exp}(i \oint A_\alpha dx^\alpha )$$ which returns a matrix $\...
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Composite particles and Weinberg Witten (WW) theorem

I am quite familiar with the proof of Weinberg Witten (WW) theorem. One major result which follow from WW is that the graviton cannot be a composite particle. I have 2 questions here: How do we tell (...
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Is it possible to define an energy momentum tensor for classical point particles from a QFT?

I have a question about the semi-classical limit of a QFT that so far I have never been able to solve. Let's start with a second quantized Klein-Gordon field with Lagrangian $$\mathcal{L}(\phi)=\frac{...
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Time reversal transformation of the complex scalar field

consider a complex scalar field $\phi$ $$\phi(t,x)=\int\frac{d^3k}{\sqrt{2\omega_k(2\pi)^{3}}} \big(a_ke^{i\vec{k}\cdot\vec{x}-i\omega_kt} +b^\dagger_ke^{-i\vec{k}\cdot\vec{x}+i\omega_kt}\big)$$ By ...
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3-particle phase space in $d$ dimensions

recently I came across a problem concerning the 3-particle phase space. I am trying to show, that the 3-particle phase space for massless particles with momenta $p_1$, $p_2$ and $k$ is given by $$ d\...
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Is one allowed to split path integrals in the Feynman-Vernon Influence theory

In QFT the propagator $J(t,t_0,x_f,x_i) = \langle x_f | U(t,t_0) | x_i \rangle$ fulfills the property $$ J(t,t_0,x_f,x_i) = \int_{-\infty}^{\infty}dx' J(t,t',x_f,x')J(t',t_0,x',x_i) $$ and can be ...
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Is wave function collapse the only source of 'randomness' in QM? What about field fluctuations? Are these two even distinct?

Basically I want to know the validity of the statement, "All randomness originates from wave function collapse" or maybe "The only true random event is the collapse of wavefunctions" This seemed to ...
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Why does Bell's field theory model need to be stochastic on a space-time lattice?

In "Beables for quantum field theory", John Bell has presented a realistic interpretation of any fermionic quantum field theory, along the pilot-wave ideas. This model is formulated on a spatial lattice ...
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Dimension of gamma matrices in dimensional regularization

When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form $$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$ For instance, in $d$ ...
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How to describe quantum mechanically that a particle falls into a black hole?

Consider a black hole spacetime originated by gravitational collapse, like the following Vaidya geometry $$ds^2=-\left(1-\frac{2M\theta(v)}{r}\right)dv^2+2dvdr+r^2(d\theta^2+\sin^2\theta d\phi^2).$$ ...
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Question regarding the degeneracy of vacuum state in various spacetimes

In a talk, it was mentioned that if one attempts to do quantum gravity in the following spacetimes (of any dimension), then the vacuum state has the following degeneracy. In $AdS$ - There is a ...
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Hawking said virtual particles can almost become real particles, following closed trajectories. What does that mean?

Here's hawking quote from his latest book. Brief Answers to the Big Questions "When space-time gets warped almost enough to allow travel into the past, virtual particles can almost become real ...
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What does the identity operator look like in QFT, written in momentum eigenstates? [duplicate]

This is a followup to a question I read recently: What does the identity operator look like in Quantum Field Theory? Out of curiosity, I was writing down what I figured to be the momentum basis ...
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Equivalence between $t$ and $u$ channels

Reading about QFT diagrams, I've seen examples like Bhabha scattering where the channel $u$ wasn't necessary due to the final states are distinguisable for being made of the different particles and ...