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 particle physics. Don’t combine with the [quantum-mechanics] tag.
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Classical Fermion and Grassmann number
In the theory of relativistic wave equations, we derive the Dirac equation and Klein-Gordon equation by using representation theory of Poincare algebra.
For example, in this paper
http://arxiv.org/abs/...
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Noether's theorem in Altland/Simmons "Condensed Matter Field Theory"
I'm reading Sec. 1.6 of "Condensed Matter Field Theory", 3e, by Altland and Simmons, where the authors derive Noether's theorem. They consider the following mapping of the spacetime ...
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Why two different spinors are Grassmann quantities?
In Rydberg Quantum Field Theory page 441 (this edition, unfortunately page 441 is not in the link) it says
If $\xi$ and $\eta$ are Majorana spinors [...] and since $\xi$ and $\eta$ are Grassmann ...
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Looking for a graph of the running of the electron mass
Is there a graph somewhere showing how the electron mass changes with energy due to renormalization up to around the Planck scale - assuming that the pure standard model is always valid (thus no GUT, ...
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Showing that the Ricci scalar equals a product of commutators
I have to compute the square of the Dirac operator, $D=\gamma^a e^\mu_a D_\mu$ , in curved space time ($D_\mu\Psi=\partial_\mu \Psi + A_\mu ^{ab}\Sigma_{ab}$ is the covariant derivative of the spinor ...
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Why are all observed particles on-shell?
I've been trying to self-learn how to do basic QFT calculations and I'm a little bit confused as to what's considered "an interaction". If I want to model an electron releasing a photon I ...
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Photon speed in a medium with negative electric susceptibility
In optical media, the real part of the electric susceptibility, Re{$\chi$}, can, in general, become negative for certain frequency ranges (near the absorption resonance). This leads to the refractive ...
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Wightman functions in quantum field theory
(Source of the 'theorem': click) Given a field $\Phi(x)$ with spin $s$ and its adjoint $\Phi^*(x)$, define the expectation values
\begin{align}
f(x-y)&:=\langle v,\Phi(x)\Phi^*(y)v\rangle, \\ ...
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Question on Wick's theorem for fermions
I have a guilty suspicion this should be obvious. What is the difference between these two expectations taken over the same measure ($\int \mathrm{d}\mu(\bar\psi,\psi)\exp{\sum \bar\psi A\psi}$ for ...
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Can I insist on having the Goldstone boson as a physical degree of freedom in scalar QED with spontaneous broken $U(1)$?
We have all learned about the Higgs mechanism in the standard QFT course. Given a complex scalar $\phi^4$ theory with global $U(1)$ symmetry, when $U(1)$ is spontaneously broken, the phase ...
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Fourier transform of $\phi$ [closed]
I was reading through David Tong's Lectures Notes on Quantum Field Theory and I was wondering how, on page 22, he derives that the Fourier transform of $\phi(\vec{x}, t), \tilde{\phi}(\vec{p}, t)$, ...
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Density of states-phase uncertainty relation
I came across this uncertainty relation for Density of states $N$ and phase $\theta$ in "Introduction to Many-Body Physics" by P Coleman on Page 15, equation (2.20).
$$\Delta N\Delta\theta &...
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Determining how a scalar field transforms considering an infinitesimal form of the Lorentz transformation
Considering the infinitesimal form of the Lorentz transformation $x^a \rightarrow x^{\prime a}=\Lambda_b^a x^b = x^a+\omega^a_b x^b$ (which preserves the Minkowski metric, such that $g_{a b} x^a x^b=...
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Accelerating frame of reference, fermions and probability conservation
I'm looking at solutions to the massless Dirac equation in an accelerating frame of reference in $(1+1)$-dimensions but the wave functions I get appear to violate probability conservation.
My ...
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Are loops counted twice in Feynman diagrams?
Consider the 2 point function in $\phi^4$ theory which is given as something proportional to
$$\int D(x-z) D(y-z) D(z-z) d^4 z,$$
where $D$ is the propagator. The corresponding Feynman diagram looks ...
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Does a quantum field theory have an effective single-particle action in the single-particle subspace?
In non-interacting quantum field theories, the particle number is conserved so we can restrict to a given subspace of fixed particle number. On the single-particle subspace, the state will evolve ...
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Quantizing Klein-Gordon Field: Sign Problem
I'm trying to re-derive the Quantization of the Klein Gordon Field but I'm running into sign problems.
My starting point is:
$$ \phi(x,t) = \frac{1}{(\sqrt{2 \pi})^3} \int \tilde{\phi}(k,t) e^{i kx}...
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Confused about square of time-reversal operator $T$
I am reading An Introduction to Quantum Field Theory by Peskin & Schroeder, and I am confused about what is the square $T^2$ of time reversal operator $T$.
My guess is that for $P^2$, $C^2$ and $T^...
Buzz♦
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Yukawa potential as the time integral of 4D retarded Green's function
I am attending an advanced QFT course, and trying to verify the instructor's claim that the retarded Green's function
$$ G_{\text{ret}}^{(4D)}(t,\mathbf{x}) = \theta(t) \left[ \frac{1}{2\pi}\delta(\...
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Renormalization in quantum field theory by discretizing space (but not time)
I'm a mathematician slowly trying to learn quantum field theory and I have a small question about renormalization, which I still have a shaky understanding of.
One common way to explain what's ...
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Understanding the basics of second quantization [duplicate]
I am new to quantum field theory and I am trying to understand how to work with quantum field operators and the notations that are used here.
Context:
Assume a hamiltonian with operator: $\hat{W} = t\...
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Wilsonian RG in QFT: what is the difference between renormalized and bare couplings?
I want to understand the relation between the Wilsonian RG and the usual QFT RG approach. Several questions have been asked, such as this and many others, yet I don't find a conceptual answer to what ...
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In QFT when performing path integral, why don’t we divide it by the volume of Poincaré group, as what we did for gauge group?
When performing path integral in gauge theory, we naively want to compute
$$
Z = \int DA \exp(iS[A])
$$
But we noticed, that because the action is the same for gauge equivalent conditions, we should ...
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What prevents photons from getting mass from higher order Feynman diagrams
The Higgs boson and gluons have no electric charge and photons couple to charge, so there is no tree level interaction between them and photons. But what prevents higher order diagrams from ...
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Fokker-Planck equation from Langevin equation in stochastic inflation
I'm reading this paper by Starobinsky and Yokoyama where they give the coarse-grained equation of motion,
$$
\dot{\bar{\phi}}({\bf x},t ) = -\frac{1}{3H}V'(\bar{\phi}) + f({\bf x},t)
$$
where $f({\bf ...
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Time ordered correlator from path integral: equation of motion?
Consider a Lagrangian $L(\phi)$ for a field $\phi$ (assume it is a free real scalar for simplicity). Then the time ordered propagator can be expressed as a path integral
$$
\langle\Omega|T\{ \phi(x) \...
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Problem with Bohr Frequency in Quantized Radiation - Matter interaction
Consider an Hydrogenic Atom (no relativistic corrections and no reduced-mass effects) in a Quantized Electromagnetic Pulse given by the wave-packet:
$$
\underline{\hat{A}}(\underline{\hat{r}}, t) = \...
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What does the Pontryagin index do in BPST instanton (solution to Yang-Mills theory)?
$$
\mathcal L = -\frac12\mathrm{Tr}\ F_{\mu\nu}F^{\mu\nu}+i\bar\psi\gamma^\mu D_\mu\psi
$$
We take this Lagrangian for QCD, after this I need to calculate BPST instanton with topological Pontryagin ...
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How does the Wave Particle Duality fit with Quantum Field Theory?
It's heard quite often that fundamental particles (photons, quarks, etc) act as both particles and waves.
Now, I'm looking at it from a Quantum Field perspective. Is this localized energy ripple ...
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Operator-state correspondence in QFT
The operator-state correspondence in CFT gives a 1-1 mapping between operators $\phi(z,\bar{z})$ and states $|\phi\rangle$,
$$
|\phi\rangle=\lim_{z,\bar{z}\mapsto 0} \phi(z,\bar{z}) |0\rangle
$$
where ...
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Calculation of Vertex factor from Lagrangian
I am studying spontaneous symmetry breaking of a complex scalar field $\phi(x)$ of a global $U(1)$ symmetry: $\phi(x)\to e^{i\alpha}\phi(x)$, where $\alpha$ is a real constant. I am considering the ...
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Doppler shift of a single photon
I'm curious to understand how the Doppler shift phenomenon occurs and is mathematically expressed for a single photon emitted by a stationary single photon emitter, such as a quantum dot, and observed ...
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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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Are Weinberg's soft theorems relevant when making predictions about collider physics?
In a seminal paper, Weinberg has shown that one can relate a $n \to m$ scattering amplitude to the $n \to m + k$ scattering amplitude that involves the same particle content plus $k$ additional ...
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Euclidean functional Integrals: Existence of zero eigenvalue due to time translation symmetry
In the chapter "Uses of Instantons" from the book "aspects of symmetry" by Sidney Coleman I have come across the euclidean version of the path integral in semi-classical ...
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What is meant in condensed matter physics by a “gap” and why is it so important?
I come from a HEP background and moved to condensed matter physics. I keep seeing the word “gap” being thrown around a lot: this system has a gap, this is a gapless system, the spectrum is gapped, ...
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Question on Majorana Path Integral
I'm studying Shankar's Quantum Field Theory and Condensed Matter and got stuck in the issue related to changing measure in Majorana path integral.
In section 9.4, the Euclidean action for the ...
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Fourier transform of Feynman Integral
In Nastase, Introduction to AdS/CFT, the first chapter talks a little about the star-triangle duality. In fact, it was claimed that the Fourier Transform of a Feynman-like diagram in position space in ...
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Hypercharge of the complex scalar doublet
I often see the complex scalar doublet $Φ_A$, $A=1,2$ with the opposite hypercharge arising in the Yukawa couplings as $\tildeΦ_A = iτ{_2}_{AB}Φ_B^*$ where $τ_r$ $(r=1,2,3)$ denote isospin pauli ...
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Analytical continuation as regularization in Quantum Field Theory, the remaining questions
There is an old question posted (Regularization) which did not get an answer, about the validation of analytic continuation as regularization. It did get some discussion in the comments, referring to ...
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How can a graviton curve space(time) if it only couples to particles in spacetime?
In the theory of quantum gravity (insofar it's incomplete version exists) gravitons couple to particles like photons or other gauge fields. Unlike the other gauge fields, which are vector-like or ...
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What is the gravitational path integral computing?
What is the gravitational path integral (which roughly goes like $\int [dg]e^{iS_{\text{EH}}[g]}$) computing?
Usually, path integrals arise from transition amplitudes such as these: $\lim_{T\to\infty}\...
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Why do different contours give different answers in the limit $\epsilon \rightarrow 0$ when calculating propagators?
Let $\phi$ denote the Klein-Gordon field. Then its propagator $\langle 0 \mid [\phi(x), \phi(y)] \mid 0 \rangle$ can be calculated as
$$\int \frac{d^4}{(2\pi)^3} \frac{-e^{-ip(x-y)}}{p^2 -m ^2}. \tag{...
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Existence of any vacuum pressure
We know that there exists an underlying background energy in space throughout the entire Universe, called vacuum energy and this is a special case of zero-point energy that relates to the quantum ...
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Is completeness relation of polarization vector equivalent to propagator?
For the Proca Lagrangian:
$$\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}m^{2}A_{\mu}A^{\mu}$$
the equation of motion is:
$$\Box A^{\mu}-\partial^{\mu}\partial_{\nu}A^{\nu}+m^{2}A^{\mu}=0$...
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Physics of vacuum expectation values in QFT
I have some trouble understanding what is the physical meaning of the vacuum expectation value (VEV) when applied in a QFT context.
Question: What is the physical meaning of a VEV?
I understand that ...
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Dirac field charge conjugation
I struggled a bit to understand the proof of the relation $C\overline\psi\psi C=\overline\psi\psi$ in Peskin's and Schroeder's book An Introduction to Quantum Field Theory (page 70, formula 3.147):
$$...
Buzz♦
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Lagrangian of chiral superfield
Consider a number of chiral superfield $\Phi_i$ with components $A_i$, $\psi_i$, $F_i$, respectively a complex scalar, a 2-component Weyl fermion and an auxiliary complex scalar. The most general ...
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Left-handed fermion oscillating into right-handed fermion
Given a Dirac fermion $\psi$
$$\mathcal{L} = \bar{\psi} \gamma^\mu \partial_\mu \psi - m \bar{\psi}\psi \ ,$$
which can be written in terms of chiral left and right handed fields as
$$\mathcal{L} = \...
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Product of Lorentz invariant factors may be Lorentz non-invariant
I'm evaluating an integral and I have three cases to consider. The result of that integral must be Lorentz invariant and independent of center-of-mass momentum. One of the cases I'm certain is in fact ...
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