Questions tagged [instantons]
An instanton (pseudoparticle) is a classical solution to the equations of motion, usually of Yang-Mills theory, with a finite, non-zero action, on a Euclidean spacetime. Instantons normally dominate the path integral of Y-M theory and determine its quantum tunneling behavior. May use for merons and other topological field-theoretic configurations.
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Why the tunneling amplitude between two quantum states is $\left<n\right|H\left|n'\right>$?
In Srednicki's QFT textbook eq.(93.5), the tunneling amplitude between two different quantum states $\left|n\right>$ and $\left|n'\right>$ is $\left<n\right|H\left|n'\right> \sim {\rm e}^{-...
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Trying to clarify the definition of an instanton
The definition of an instanton, in [this Wikipedia article], is: a
"classical solution to equations of motion with a finite, non-zero action" and further specifies that it is "on a ...
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Why does the imaginary time Euler-Lagrange equation imply the potential goes to zero at infinite imaginary time?
On reading up about the bounce solution for false vacuum decay in S. Coleman's The Fate of the False Vacuum I it was stated that the equation
$$0 = \frac{1}{2} \frac{\partial q}{\partial \tau} . \frac{...
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Instantons in the Global $O(2)$ Model (Compact scalar field) - Polyakov textbook
This question is related with Polyakov, "Gauge Fields and Strings" section 4.2
In section 4.2, partition function is
\begin{equation}
Z=\sum_{n_{x,\delta}}\int_{-\pi}^{\pi}\prod_x\frac{d\...
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How important are purely imaginary finite action solutions for first-order instanton contributions?
I am working on a physics problem where I have to calculate instanton contributions for a non-relativistic Hamiltonian
$$H=-\frac{1}{2}\frac{d^2}{dx^2}+\frac{1}{2}x^2+\frac{1}{6}g^2x^6 \tag 1$$
for ...
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Motivation for the shape of the theta vacua
I understand that the reason why we construct the theta vacua is because instantons allow tunnelling between different vacuum states, $\left|n\right>$. This means that we have to consider a real ...
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Instantons and Spontaneous Symmetry Breaking
I'm following an introductory lecture on instantons by Hilmar Forkel. In a non-relativistic quantum mechanical setting we have the potential $$ V(x) = \dfrac{\alpha^2 m}{2 x_0^2} (x^2 - x_0^2)^2 \tag{...
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Nekrasov partition function
In the celebrated paper Seiberg-Witten prepotential from instanton counting by N. Nekrasov I can't quite understand some parts of section (2.3). The Nekrasov partition function is defined via
\begin{...
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Are loops allowed for paths in the path integral formulation?
In Wikipedia is stated that the quantum-mechanical paths are not allowed to selfintersect (create loops). (It’s stated in the description of the first figure of five paths.) On the other hand, loops ...
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Why is it justified to focus on gauge transformations constant at spatial infinity in QCD instantons?
In the context of Yang-Mills theories and QCD instantons, much of the literature and conventional treatment hinges on the consideration of gauge transformations that remain constant at spatial ...
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How can we use saddle point approximation for a bounce solution which is not even a strict local minimum of the Euclidean action?
In calculating the false vacuum decay, the main contribution to the imaginary energy part of the Euclidean path integral comes from the bounce solution. And we somehow apply saddle point approximation ...
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Double-well potential and non-perturbative energy splitting
(A reference for the topic is a QFT note (chapter 2
Instantons in Quantum Mechanics) here by Yoichi Kazama at University of Tokyo, see page 30)
Consider the double well potential in quantum mechanics,
...
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Transition fuctions only depend on homotopy
In the nLab article about Instantons in the section about the clutching construction, it is stated that one can construct $SU(2)$-bundles over $S^4$ via Chech cohomology choosing appropriate ...
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Compactification and gauge choice for instanton solutions
I have several doubts regarding topological solutions in pure YM -- these are related both to less trivial topological misunderstandings as to rudimentary gauge fixing confusions of mine.
What is the ...
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How to understand confinement of 2+1d $U(1)$ gauge theory from instanton effects?
In Section 6.3.2 of XG Wen's book Theory of Quantum Many-body System, the confinement of $2 + 1 d$ compact $U \left( 1 \right)$ gauge theory was explained by the instanton effect. I want to know how ...
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Theta vacua eigenstates
I have been trying to prove the very simple result that the eigenstates of an operator with matrix elements
$$
\langle n^\prime | H | n \rangle \sim g(|n^\prime-n|),
$$
in a basis $\{|n\rangle\}^{+\...
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Counting of zero-modes in conifold theory
I was reading Klebanov and Witten's paper on the conifold theory and at page 11 they state that
[...] In an instanton field of the first $U(N)$ with instanton number $k$, the gluinos of the first $U(...
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Topologically interesting non-abelian instantons
Consider a Yang-Mills type theory, with algebra $\mathfrak{g}$, defined over a manifold $M$. The action functional$\newcommand{\tr}{\operatorname{tr}}$ is
$$S[a] = \frac{1}{2g^2}\int_M\tr_\mathfrak{g}(...
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What is the dimension of an Instanton?
My thought process about this question was this:
A point particle couples to a $p=1$-Form field and is itself $p-1=0$-dimensional.
A string couples to a $p=2$-Form field an is $p-1=1$ dimensional.
An ...
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Wormholes as instantons?
Are all wormholes gravitational instantons in the context of General Relativity?
My question concerns also the topology of spacetime in such case.
A full Wick rotation of the metric, seems to change ...
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Arbitrary heat kernel coefficients of covariant Laplacian with instanton
The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is ...
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What do monopoles have to do with strong coupling?
My understanding is that strong coupling effects arise from instantons in the path integral.
But I sometimes read that monopoles (see the electric-magnetic duality) can allow one to calculate strong ...
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Multiple saddles in path integral
I am reading the Jerusalem Lectures by Harlow. On page 44 he calculates the thermal partition function using the path integral with no matter fields,
$$
Z(\beta) = \int \mathcal{D}[g] e^{-I_E[g]}.
$$
...
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Why are there no instantons in the gauge group $U(1)$?
I am working my way through Srednicki's QFT book and am in chapter 93. Near the end, Srednicki says "If the gauge group is $U(1)$, there are no instantons, and hence no vacuum angle."
I'm ...
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How to formulate the braiding of instanton?
I'm reading the paper https://arxiv.org/abs/2108.08835, after imposing the $\mathbb{Z}_2$ on-site symmetry, in the $J=0$ symmetric phase of the 1d Ising chain, the topological sectors of operators ...
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Barrier Penetration in Spontaneous Symmetry Breaking
In spontaneous symmetry breaking discussion in Weinberg Chapter 19 section 19.1, he says that the off-diagonal elements between two vacua $$|VAC, +> \pm |VAC->$$ is suppressed by the factor of $$...
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Topological charge change in QFT
Is it possible for the topological charge to change in quantum field theory?
The proofs in the following paper: Quantum soliton operators for vortices and superselection rules
are all based on the ...
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Why 't Hooft says: field configuration in Euclidean space that have the vacuum (or a gauge transformation thereof) at the boundary
In Symmetry Breaking through Bell-Jackiw Anomalies G. 't Hooft, Phys. Rev. Lett. 37, 8 – Published 5 July 1976,
't Hooft said that the topological quantum number $n$
$n$ is an integer for all field ...
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Why are non-perturbative solutions important and how to take them into account?
I am guilty of studying physics with an almost complete focus on the mathematical constructions (together with the motivating physical premisses) and ignoring the semantic physical intuition, which I'...
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Perturbation theory amongst supersymmetry transformations of perturbative ground states
Consider a supersymmetric one dimensional sigma-model whose target is a Riemannian manifold $M$. Moreover, assume there is a Morse function $h$ on $M$. In Hori, Kentaro, Cumrun Vafa, Sheldon Katz, ...
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What is the correct domain of integration for the index of instantons? - $\mathbb{R}^4$ or $S^4$?
I posted the original question on Math SE but it seems like a more appropriate question for Physics SE:
https://math.stackexchange.com/q/4417225/
In calculating the instanton solutions for $SU(2)$ ...
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Can a kink in a finite one dimensional box tunnel into a trivial solution?
Given a simple kink solution of the Sine Gordon equation, is it possible for such a solution in a finite volume to tunnel into a trivial vacuum solution, given that such tunneling demands a finite ...
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Why can an analytic continued Hamiltonian have squared integrable eigenfunctions?
In 1D quantum mechanics, there are no bound states and there are resonant states for the following potentials:
$$
W(q)=\frac{1}{2}q^2-gq^3,\tag{1.3.2}
$$
$$
W(q)=\frac{1}{2}q^2+\frac{g}{4}q^4,\; g<...
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Where are the multi-instantons in Supersymmetric QM?
Instantons can be used to find non perturbative corrections to ground state energies. However, the way in which they are used seems to me to be very different between the two common toy models of the ...
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What does the Pontryagin index do in BPST instanton (solution to Yang-Mills theory)?
$$
\mathcal L = -\frac12\mathrm{Tr}\ F_{\mu\nu}F^{\mu\nu}+i\bar\psi\gamma^\mu D_\mu\psi
$$
We take this Lagrangian for QCD, after this I need to calculate BPST instanton with topological Pontryagin ...
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What is the "instanton configurations of the gauge field"?
In the study of the abelian chiral anomaly, one finds that it can be written as the total derivative of a vector operator:
$$\int \mathcal{A}(x)d^4x\propto\int\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\...
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The theta term and triviality of principal bundles
Apologies if this question is trivial or has been answered before.
If we consider a Yang-Mills theory (with a simple, compact Lie group $G$) on $\mathbb{R}^4$, it is well-known that all the finite-...
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WKB application on symmetric potential well
I am a little confused how one can find a wave function by using WKB approximation? I do know the oscillation frequency $$\Omega ~=~ {2E\over h}{\rm Re} \langle L|R \rangle~=~ {E\over \pi\hbar}{\rm Re}...
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Instantons in Minkowski spacetime? or only valid in Euclidean spacetime?
In the usual description of the instanton of nonabelian gauge theory in $D=4$ spacetime, we always (or just usually?) choose the $D=4$ Euclidean spacetime see for example https://en.wikipedia.org/wiki/...
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How to Wick rotate the Yang-Mills instanton winding number?
How to Wick rotate the instanton number of Yang-Mills theory?
(Related to the earlier question Wick rotate the Yang-Mills $SU(N)$ gauge theory's field strength?)
My question is particularly about ...
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The sign of axion $F$ $F$ dual term in Weinberg
Related to the earlier question $\gamma^5$ rotation of chiral fermion in (1) Peskin&Schroeder, (2) Weinberg, or (3) Srednicki.
The sign of axion $F$ $F$ dual term in Weinberg (23.6.16) appears to ...
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$\gamma^5$ rotation of chiral fermion in (1) Peskin&Schroeder, (2) Weinberg, or (3) Srednicki
The theta angle due to the chiral gamma^5 rotation of chiral fermion results in the phase alpha(x) that has different + or - sign for
(1) Peskin&Schroeder, (2) Weinberg or (3) Srednicki.
Here
...
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What is instanton effect in QCD?
In the context of the Peccei-Quinn Symmetry in solving the Strong CP problem, it is said that the Axion develops a mass due to QCD instanton effects below $\Lambda_{QCD}$, which is very much ...
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An instanton in $d$ dimensions is often a soliton in $d + 1$ dimensions?
The title of this questions is a "folklore" I've heard from a lot of researchers, but I never understood why this is the case. I know what an instanton and soliton is, respectively in the ...
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Target space of boundary CFT dual to a bulk string theory ($AdS_3/CFT_2$)
I was reading the Maldacena Ooguri paper where they mention that for the string theory living on $AdS_3\times S_3 \times M_4$ (where $M_4$ is $K3$ or $T^4$), the boundary CFT is the supersymmetric ...
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The argument for a mass gap for the $O(3)$ Heisenberg ferromagnet
One possible argument for asymptotic freedom in the 2D $O(3)$ ferromagnetic Heisenberg model is the existence of so-called instantons, discovered in the 1975 paper of Belavin and Polyakov. This is ...
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What is a bubble from nothing?
Recently I was reading a collection of lectures on Swampland Conjecture and came across an interesting subtitle about a bubble from nothing.
A bubble from nothing is a non-perturbative instability ...
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How is the Theta angle of $SU(5)$ grand unified theory related to the three Theta angles in the $U(1) \times SU(2) \times SU(3)$ standard model?
There are three Theta angles in the $U(1) \times SU(2) \times SU(3)$ standard model:
call them
$$U(1): \theta' F \tilde{F}$$
$$SU(2):\theta'' F \tilde{F}$$
$$SU(3):\theta''' F \tilde{F}$$
But there is ...
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Holomorphic instantons in target torus
For computing instantons contributions from worldsheet torus to target torus, one can evaluate zero modes contribution of genus 1 partition function given by following expression:
$$Tr(-1)^FF_LF_Rq^{...
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How "hedgehog" or instanton event changes the configuration of neel vector in 2D antiferromagnet?
The definition of "hedgehog" or instanton event here is "a space-time event where the skyrmion number Q changes by $\pm1$ is called a hedgehog" (ref.1 & 2).
A nice figure ...
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