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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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Dealing with more complicated tree level feynman diagrams

Suppose I have a 2 -> 4 body process of the form Where all particles are scalar $\phi$ of mass $m$, dirac fermion $\chi$ of mass $M$ with interaction lagrangian $g\overline{\chi}\chi \phi$ . How does ...
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2answers
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Faster ways of computing feynman diagrams

Obviously the machinery of QFT allows us to calculate processes, such as QED diagrams, to great precision, and whilst it is effective, it seems there are many processes that make calculations (say by ...
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Effective potential and radiation corrections

I'm a bit confused on the idea of adding corrections to the classical potential of $\phi^4$ theory in QFT. From what I understand is that one should add corrections to the potential in order to ...
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$ e^+e^- \longrightarrow \mu^+\mu^-$ probability density function of $\theta$ strange trend

I'm considering the process: $$ e^+e^- \longrightarrow \mu^+\mu^-$$ The cross section in the center-of-mass frame: \begin{equation} \left( \frac{d \sigma}{d \Omega} \right)_{CoM}= \frac{\alpha^2}{4 s}...
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When can you simplify the W boson propagator

I have seen in several sources that the propagator of the $W$ boson is: $$\frac{- i \left( g^{\mu\nu} - \frac{P^\mu P^\nu}{m_W^2} \right)}{p^2 - m_W^2} $$ But then in some calculations (usually ...
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How to grasp the limits of these two integrals? [duplicate]

I find some difficulty in understanding the limits of the two integral below (on Page 27 of Peskin & Schroeder's Quantum Field Theory): $$D(x-y)=\frac{1}{4\pi^2}\int_m^\infty d E \sqrt{E^2-m^2}e^{...
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How to solve this particular problem of Wick's theorem?

So I know the basics of Wick's theorem, but unsure about how to solve this time ordered product of a term that involves normal ordering. Is it just simply the sum of all possible contractions, but no ...
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A proof that Heisenberg's and Euler Lagrange's equations are equivalent in QFT [closed]

I asked this before (link, link) but I think people didn't understand what I was asking, so I am going to try again . Thanks for everyone that helped so far. In QFT, Heisenberg's equation is ...
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Higgs-Mechanism: Why are gauge boson masses not protected by gauge symmetry

In non-spontaneously broken QFT like QED the gauge bosons cannot have a mass due to gauge symmetry (follows from Ward identity). Also they have only 2 polarizations. However in a spontaneously broken ...
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Why is the imaginary part of the Breit-Wigner propagator given by the total decay width?

The optical theorem links the imaginary part of the forward scattering amplitude to the total decay width of a particle: $\mathrm{Im}\,M_{i\to i} = m\Gamma_{tot}$. Here $\Gamma_{tot} = \frac{1}{2m} \...
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1answer
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Explicit quantization of free fermionic field

The canonical quantization of a scalar field $\phi(x)$ can explicitly be realized in the space of functionals in fields $\phi(\vec x)$ (here $\vec x$ is spacial variable) by operators \begin{eqnarray} ...
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Is the mass of a particle determined by the extent to which it interacts with other particles? [closed]

This is more a question about the Higgs field than anything else. If you were to take, for example, a neutrino and send it out into empty space how could you determine that it has a mass in the first ...
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Touching is a Pauli exclusion principle or electrostatic force? [duplicate]

According to quantum field theory touching is an electrostatic repulsion between electrons or the Pauli exclusion principle? How can physicists distinguish these two phenomena if they give the same ...
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What substitutions are allowed within time-ordered products?

I always thought of the time-ordering in QFTs as an explicit operation. Meaning the time-ordering "operator" just takes everything I write inside it and shuffles the operators around until they are in ...
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Zero-Temperature Limit of Matsubara Sums

Consider the fermionic Matsubara sum for $$\frac{1}{\beta}\sum_n G^2(k,\omega_n) = \frac{1}{\beta}\sum_n \frac{1}{(i\omega_n - \epsilon_k)^2}$$ where $G(k,\omega_n)$ is the free fermion Green's ...
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Are there any known models with limit cycles in their RG flow?

The text-book presentation of the renormalization group (RG) leaves one with the impression that all systems will eventually flow to a fixed point. This is somewhat enforced by the phenomenological ...
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How does the generalized effective action in Wetterich's exact RG scheme relate to observables at different scales?

I am not familiar with Wetterich's exact RG paradigm, and cannot understand the main idea behind it. I understand that if one could have solved the model and obtained the all the n-point functions ...
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Comparison of two Yukawa theories

I consider vacuum polarization diagram in two different Yukawa theories: with scalar coupling $g\bar{\psi}\phi\psi$ and pseudoscalar coupling $ig\bar{\psi}\gamma^5\phi\psi$. I am interested in ...
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1answer
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How does one deal with derivative operator in quantum field theory properly?

Given creation and annihilation operators, ${a^{\dagger}(x,t)}$ and $a(x,t)$ in non-relativistic quantum field theory, respectively, which satisfy the following properties: Now, I want to prove $$[...
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Parton distribution function in terms of Fock space kets

To my understanding, I can (at least, formally) express the (unnormalized) PDF for a certain constituent of a composite state as $$ f(x)=f\left(\dfrac{k}{K}\right)=\sum_j m_j^{(k)}|\langle\psi_j^{(k)}|...
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Does the background shift affects the renormalization group equations?

In Section 21 of "Quantum Field theory" by Mark Srednicki, it is shown that there are two equivalent ways to get the quantum action of the shifted field $\phi'= \phi-\tilde{\phi}$, where $\phi$ is the ...
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1answer
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Canonical Quantisation vs the Dirac Field, why does it even work?

Using the "Dirac Prescription", we can preserve the format of a differential equation in its QM form. If we define the canonical variables s.t. they have the same commutation relations times $i$ as ...
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Two ways to calculate imaginary part of vacuum polarization in pseudoscalar Yukawa theory

I consider vacuum polarization in pseudoscalr Yukawaw theory with interaction $ig\bar{\psi}\gamma^5\phi\psi$. The expression for diagram is $$i\Pi(k^2)=-g^2\int\frac{d^4p}{(2\pi)^4}\frac{\mathrm{Tr}[i\...
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LSZ reduction formula for massive vector bosons

What is the precise form of the LSZ reduction formula for massive vector bosons? The LSZ formula for scalar bosons, fermions, and photons is given e.g. in the textbook "Quantum field theory" by ...
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Question about DGLAP evolution equation

I am reading chapter 32.2 of Schwartz's QFT book, where he defines the renormalized PDFs $f_i(x, \mu)$. This leads to an equation (32.48), which relates PDFs at different scales $\mu, \mu_1$: $f_i(x,\...
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3answers
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Why can't we have charge field?

I learnt that everything is made of field, there is the electron field that occupy the entire universe and it will produce an electron which is an excitation of the field. This electron interacts with ...
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1answer
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Wick rotation vs. Feynman $i\varepsilon$-prescription

The generating functional $Z[J]$ of some scalar field theory is \begin{equation} Z[J(t,\vec{x})]=\int \mathcal{D}\phi e^{i\int (\mathcal{L}+J\phi)d^4x} \end{equation} This integral is not well ...
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1answer
36 views

QM probability to go from a point to another (Zee)

In Zee's QFT in a Nutshell book at p.10-11 it is piecewise said that : In quantum mechanics, for a Hamiltonian $\hat{H}=\hat{p}^2/2m$, the amplitude to propagate from a point $q_j$ to a point $q_{j+1}$...
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How do we determine who eats who (or which field gets massive) in a Higgs phase transition?

Consider the following theory in which a $2$-form field $B_{\mu\nu}$ with field strength $P_{\alpha\mu\nu}=\partial_{[\alpha}B_{\mu\nu]}$ is coupled to a $3$-form gauge field $C_{\alpha\beta\gamma}$ ...
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Unit of pion-decay constant

In the natural unit system, the pion-decay constant $f_{\pi}$ is $92.4\:\rm MeV$. But I think that a decay constant should have a dimension of $[T]^{-1}$, where $[T]$ is the dimension of time. Then, ...
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How are composite hadron fields related to elementary quark fields?

(This question is related to: A pedagogical exposition of the hadron physics?) I'm a mathematician who has been trying to learn quantum field theory for a while. I've gone through large parts of ...
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1answer
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Why there's a Lorentz inner product in the unitary representations of the translation group?

Consider Minkowski spacetime. Its translation group is just the additive group $\mathbb{R}^4$. This is an abelian locally compact group. Next, consider one unitary representation $T : \mathbb{R}^4\to ...
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1answer
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Why we can't write low energy effective field theory for a gapless system?

Assume that we have an Hamiltonian $$H=H_0+H_{int}$$ where $H_0$ has gappless excitations and $H_{int}$ has usual two body interactions. If $H_0$ were gapped I would able to get low energy effective ...
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1answer
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Path integral and Out-of-time-ordered (OTOC) correlator

A simple observation that any insertions within the path integral are classical variables (Not operators) and hence, objects inside the path integral "commute" (is symmetric under exchange). Hence, ...
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1answer
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What are the orthogonal eigenstates of the field operator?

In Peskin & Schroeder section 9.2, they derive the two-point function in the path integral formalism: $$\langle \Omega | \mathcal{T} \left\{ \hat{\phi}(x_1)\hat{\phi}(x_2)\right\} | \Omega \...
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Calculating vacuum expectation value of graviton field

I've been reading a section in this thesis (pp 25-27) which reviews Duff's paper (pp 6-7 in particular) in which he calculates the tree-level vacuum expectation value (vev) of the graviton field (upon ...
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How to find groundstate energy of a simple Hamiltonian at $N/L$-filling using Jordan-Wigner (JW) transformation?

$\underline{\textbf{Model:}}$ Let we have the $t-V$ model for spinless fermions on a 1D lattice, which is defined in second quantization operators as follows: $$H_1 = -t\sum_i \big(c_i^\dagger c_{i+...
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1answer
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What are three-point functions?

I came across this term while I was trying to read this paper related CFT correlators. Can some please take some time out to explain what does it mean in general?
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Matrix elements of stress-energy tensor $\langle q | T^{\mu\nu} |q\rangle$ in QFT?

In many QFTs we can define a stress tensor $T^{\mu\nu}$. What is the matrix element of $T^{\mu\nu}$ in momentum eigenstates? For instance, consider $$\langle q | T^{\mu\nu} |q\rangle$$ in QCD, ...
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1answer
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Is a constant electric field CP violating?

Consider, for instance, a fundamental massless three-form field $C_{\alpha\beta\gamma}$ in the Coulomb phase: $$ \mathcal L = E_{\mu\alpha\beta\gamma}E^{\mu\alpha\beta\gamma} + C_{\alpha\beta\gamma}J^...
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Poincare group during interactions

Weinberg in Vol 1 says for non-interacting particles, we can take the state of particles to direct product of one particle states. I want to know what happens during the representations of the ...
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1answer
70 views

Localization of Electron Matter Field Excitation in Simple Electron QFT Model

I believe QFT represents a single free (stationary) electron as a an excitation of the electron matter field which then couples to the EM field to create a local 'attached' EM field - if this is ...
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Diagrammatics of a current-current correlation function $\langle 0| T\{J^{\mu}(x) J^{\nu}(0)\}|0\rangle$

Say $J^{\mu} = \bar{\psi} \gamma^{\mu} \psi$ is the QED current. While it is clear to me how to compute something like $$\langle 0 |T\{ \bar \psi(x) \psi(0)\} |0\rangle$$ using a Feynman diagram ...
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1answer
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The Majorana condition and C violation

Is the Majorana condition $$ \psi = \psi^c = C \overline{\psi}^T, $$ general? The point is often made that Majorana particles should be defined by CPT symmetry and not C as generally theories do not ...
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Field Coupling in QFT

I'm trying to understand field coupling in QFT in more detail, using a free electron as the simplest example. I understand that an excitation in the electron matter field couples to the EM field to ...
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31 views

A confusing point in linear response theory on the ground state

Information about a quantum system could be drawn from its response to a small perturbation. This is formulated in what is known as linear response theory. In second quantization, consider a ...
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computing the correction to free propagator in $\phi^4$ theory

I want to calculate the first order correction to the free propagator in $\phi^4$ theory. Srednicki "calculates" it in chapter 31, where he directly writes the answer and tells us to use the "usual ...
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Screening of the electric field in the Higgs phase

Dvali states in his paper on Three-Form Gauging of axion Symmetries and Gravity that “As usual, in the Higgs phase the electric field is completely screened in the vacuum,” and in another ...
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Furry's theorem and spontaneous parametric down-conversion

In spontaneous parametric down-conversion, a single photon interacts with a nonlinear medium to produce a pair of lower energy photons. It is observed that energy, momentum, and orbital-angular-...
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1answer
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Doubts on representations of poincare group and QFT

I am studying Poincare group and encountered the term massless representations of the Poincare group. I know Poincare group is studied by the studying the little group of various momenta, massless and ...