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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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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How to deal with actions with interfering extrema?

How can one deal with actions with multiple extrema which difference is comparable to $\hbar$ (the extrema of the action are not widely separated) in the path integral method?
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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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What if the minimising path for the least action isn't unique? [duplicate]

Using the Lagrangian method, we find the path which minimises the action. But what if multiple paths each have the same minimum action? How do we resolve such a situation?
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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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Vacuum Degeneracy of the Standard Model

There are infinite vacua, which are degenerate, in Standard Model (people know this in the context of Sphaleron, Instanton and so on). I'd like to know why there are infinite vacua. I'm glad to ...
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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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Gelfand-Yaglom method and perturbation

I'm reading Instantons and large $N$ (https://laces.web.cern.ch/LACES10/notes/instlargen.pdf) by Marino, and I don't understand something about Gelfand–Yaglom method (from page 22), so please tell me ...
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Can an instanton be a Wilson loop?

I was wondering if one could write an $\mathrm{SU}(N)$-instanton as a Wilson loop. Since in temporal gauge, I can interpret a YM-instanton on $M\times\mathbb{R}$ as a one-parameter family of gauge ...
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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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How to visualize the $U(1)$ instanton event in (2+1)D compact lattice gauge field?

In the continuum limit of (2+1)-dimensional compact $U(1)$ gauge field, the instantons are input by hand in terms of nonconservation of magnetic flux $\int b$: \begin{eqnarray} \int dxdy [b(x,y,t+\...
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Instantons in mathematical physics [closed]

I am extremely curious about instantons in the context of mathematical physics, and I would like to learn more about the subject. Could anyone give me good references about this? Especially references ...
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Instantons in Altland/Simons

I have a question about a statement in Condensed Matter Field Theory (2nd edition) by Altland/Simons on p.124. In short, when we consider a motion in a double well we obtain a classical solution in ...
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When can I use Gaussian integration to compute a path integral?

In reading 14.4 of Gregory Moore's notes on abstract group theory, I was left with some questions on the computation he did of the path integral that may be general features. Let consider a spacetime $...
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Regularization of functional determinant over an Instanton background

I am reading the paper "ABC of instantons" and meet some problems at section 8. I simplify this problem a little bit as follows. First, we have a Euclidean path integral like \begin{equation}...
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Gauge fixing and instanton calculation

I am reading Cheng&Li's book "Gauge theory of elementary particle physics". In section 16.2, I am confused by some assumptions. Suppose we have a $SU(2)$ gauge theory in $\mathbb{R}^4$ $$...
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Boundary condition of gauge field for finite Euclidean action

I am reading the book "Gauge theory of elementary particle physics" by Cheng & Li chapter 16 and I am confused by some statements. In Euclidean 4D spacetime we have a $SU(2)$ gauge teory ...
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Is Yang-Mills Instanton number zero?

I am studying Yang-Mills instanton. Suppose we have an action in $R^4$ \begin{equation} S=\int_{M} Tr(F\wedge *F) \end{equation} where $F=dA+A\wedge A$. The instanton number $k$ is defined as \begin{...
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What is the lifetime of the deuteron with respect to sphaleron-induced decay

As far as I've understood, instantons like the sphaleron can give rise to processes that violate $B+L$ but conserve $B-L$, where the baryon and lepton number can only change by a multiple of three. ...
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Instantons in 1+1 dimensional Abelian Higgs model

Let's consider the Abelian Higgs model in 1+1 dimensions in Euclidean space-time: $$L_E=\frac{1}{4e^2}F_{\mu\nu}F_{\mu\nu}+D_\mu\phi^\dagger D_\mu\phi+ \frac{e^2}{4}(|\phi|^2-\zeta)^2$$ where $\zeta&...
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Why is the "instanton map" surjective and do we compactify the space or not?

The following line of reasoning, apart from possible misconceptions in my part, is how instantons are usually (intuitively, at least) introduced: (i) We look for minimum classical action solution for ...
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The generic form of instanton (antiinstanton) in Kahler $\sigma$-model

Suppose we have some compact Riemann surface $\Sigma$ , and scalar field $\phi$, which takes values in some Kahler manifold (target space) $M$. In other words, we have a map: $$ \phi : \Sigma \...
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Questions about the large-instanton problem

The Problem. The issue that I'm talking about is the large-instanton problem of asymptotically-free non-abelian gauge theories. You can read about it in: [1.] Section 15 of 't Hooft's 1976 paper on ...
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About the Yang-Mills equation

Perhaps this is a sick question. How do we know that Yang-Mills equation does not have a non-trivial classical solution in Minkowski spacetime in vacuum? By non-trivial solution, I mean one that ...
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Saddle point approximation and finite action configurations forming a set of zero measure

In Coleman's "Aspects of Symmetry", chapter 7, section 3.2, he makes a claim that configurations of finite action form a set of zero measure and are therefore unimportant. Further, he goes on to prove ...
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Winding number of QCD vacuum

We know that QCD vacuum has instantons, which corresponds to tunneling process. Consider $SU(2)$ gauge theory without matter. We say that in the classical configuration of vacuum state $F^a_{\mu\nu}=0$...
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Regarding instantons in 2D Abelian gauge theory

I am trying to understand the analogues of instantons in a $U(1)$ gauge theory in 2D Euclidean spacetime. If we follow the same arguments as the 4D case and say that the the gauge field must tend to ...
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Consistency condition for Yang-Mills on a Torus

So I was recently studying 't Hooft's paper on self-dual solutions to Yang-Mills on $\mathbb{T}^4$. So the basic idea is that you consider a box with periodic boundary conditions and then you impose ...
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Besides instantons and large-$N$ what are some other general non-perturbative methods for quantum field theory?

Besides large-$N$, instantons, lattice QFT, what are some other non-perturbative methods that help us better understand QFTs like the large distance dynamics of Yang Mills and QCD?
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Intuition about ADHM construction

I'm trying to understand reasons, why self-dual Yang-Mills equation can be reduced to algebraic equations. It's seem like a miracle. In article Construction of Instanton and Monopole Solutions and ...
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Why do we think that the $U(1)$ problem is solved by instantons?

It is usually thought that the $U(1)$ problem is solved when 't Hooft realized that instantons induce additional symmetry breaking of the $U(1)_A$ symmetry aside from the non-vanishing quark masses. ...
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ADHM construction of gravitational instantons

Is there a ADHM-like construction for gravitational instantons? Could somebody explain why there is not such construction?
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Instantons as classical solutions

Instantons are classical solutions of the Euclidean (i.e. imaginary time) classical equations of motion. The standard example from single particle 1D QM is taking a potential of the form $V(x)=(x^2-a^...
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Fermionic non-zero modes to standard $SU(2)$ 't Hooft instantons?

The fermionic zero modes to the "standard" $SU(2)$ 't Hooft instantons are known as follows (see page 42 in this article): For a 't Hooft instanton (self-dual field strength tensor) in the regular ...
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Topological number and gauge invariance

In QCD or other non-abelian gauge theories, we come across infinitely many vacua that are gauge equivalent but have different topological numbers. We then say that the instanton solution is tunnelling ...
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