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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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A question on 't Hooft's paper “Computation of the quantum effects due to a four-dimensional pseudo-particle”

In the avove paper 't Hooft considers the following (rotation invarinat) operators on the Euclidean space $\mathbb{R}^4$: \begin{eqnarray} \mathcal{M}_0=-\left(\frac{\partial}{\partial r}\right)^2-\...
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Are theta vacua topologically protected?

In discussions of Yang-Mills instantons it is often stated that one should sum in the path integral over all contributions of fluctuations around all the topologically distinct vacua labelled by ...
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Are Instantons Massless?

That is, are the only field configurations which give a non-zero winding number ones in which the Fourier transform includes a factor like $\theta(k^0)\hat{D}\delta(k^2)$, where $\hat{D}$ is some ...
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How does the instanton break the $U(1)_A$ symmetry in QCD?

The $U(1)_A$ symmetry in QCD is anomalous. Its supposed to be broken by the instantons. Can anyone physically describe how does that happen?
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Reference on instantons in gauge theories

Is there a reasonably detailed and systematic exposition of the theory of instantons in non-abelian gauge theories?
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What is the difference between real and complex instantons (mathemtically, and their physical significance), and connection to Wick rotation

I am struggling to understand the difference and physical significance between real and complex instantons- I think these are also sometimes called ghost instantons? There are also anti-instantons. ...
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What is the 't Hooft determinant?

The 't Hooft vertex/determinant is somehow generated by instantons and is responsible for the generation of mass gap in pseudo-Goldstone bosons, such as an axion. For example, the complex Peccei-...
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Is there any connection between instantons and surface-interacting polymers?

Excluded volume polymers interacting with a penetrable hypersurface of variable dimension is a very interesting system to study critical behavior via perturbative renormalization. Since a penetrable ...
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Checking modularity-like transformation property

Assume $M$ is a 4 manifold. Let $Z_v$ be partition function of fixed magnetic flux $v$ with all instanton configuration summed over where $v\in H^2(M,Z/nZ)$. $\tau$ denotes complex parameter on upper ...
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Isn't there a unique vacuum of the Yang-Mills quantum theory?

The theta vacua$^1$ of a Yang-Mills quantum theory are given by $$|\theta\rangle=\sum\limits_{n=-\infty}^{\infty}e^{in\theta}|n\rangle.$$ In Srednicki's Quantum Field Theory, he claims that the ...
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What does $B+L$ anomaly have to do with a phase redefinition of the left-handed quark field?

According to this answer, the reason why $SU(2)_L$ weak theory does not have a theta vacuum is because any theta term can be reabsorbed with a phase redefinition of the left-handed quark field. ...
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How many connected components are there in the vacuum manifold of $\phi^4$ theory?

Consider the following theory in $1+1$ dimensions $$\mathcal L = \frac12(\partial\phi)^2 - \frac\lambda4 (\phi^2 - v^2)^2 \,,$$ which exhibits a $\mathbb Z_2 = \{0,1\}$ symmetry, $\phi \to -\phi$, ...
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Does Coleman-de Luccia instanton approach a Hawking-Moss instanton?

Suppose a Coleman-de Luccia instanton terminates in the basin with the true vacuum between the top of the potential barrier and the field value with potential energy equal to the potential energy of ...
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Multi-instanton contribution to path integral

Briefly, I would like to have a reference to a clear detailed exposition of the computation of the multi-instanton contribution to the path integral while computing the energy levels splitting of the ...
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What is the relationship between the JNR instantons and the BPST instanton?

The JNR instantons are related to the t'Hooft ansatz, and take the form \begin{equation} A=\sigma_{\mu\nu}\frac{\partial_\nu \rho}{\rho}dx^\mu, \end{equation} where $\rho$ takes the form \begin{...
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Why not regard all large gauge transformations as genuine ones?

A large gauge transformation is a gauge transformation that is not connected to the identity. When quantizing a gauge theory, we must take configurations related by ordinary gauge transformations to ...
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What justifies compactifying space and spacetime, in the context of instantons?

When studying Yang-Mills instantons, there are two instances where one compactifies a space. When classifying vacuum states, one demands $A_\mu(\mathbf{x})$ to become a constant as $\mathbf{x} \to \...
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Vacuum Stability

I am studying one of the paper of Sidney Coleman, "Fate of the False Vacuum. II. First quantum corrections". Just before eq. (2.18) he says "Because of time translation invariance, this equation ...
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Simple explanation of the QCD VEV in terms of instantons

I've heard that instantons in QCD generate quark bilinear condensate $\langle \bar{q}_{L}q_{R}\rangle$ which is responsible for spontaneous symmetry breaking. Is there any clear and simple way to ...
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Comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of: Chern class (1st, 2nd), and ...
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Does the ABJ anomaly for the Abelian gauge field have a topological argument?

We known that the ABJ anomaly for non-abelian gauge fields with gauge group containing $SU(2)$ as a subgroup has a topological argument from the Euclidean path integral. Through studying the Euclidean ...
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Validity of the thin wall approximation

Inspired by the question How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue?, I decided to come back to the first paper by ...
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Instanton Counting (Tachikawa's Review) and “Hilbert Space”

I'm reading this review on instanton counting. I cannot understand in section 3 how supersymmetric Hilbert spaces come about. For instance in section 3.1.1, covering the supersymmetric particle on $\...
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Instanton contributions in quantum gravity

Suppose a low-energetic System, i.e. a System, where the presence of "classical" gravitational fields can be assumed to be Zero. Classically we would have e.g. the ordinary Minkowski metric or more ...
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Which vacuum is the Universe really in?

There ate two types of vacuum of the Standard model-the vacuum of the Higgs potential and that of the vacuum of the Yang-Mills fields labelled by the Chern-Simons number. See the figure 5 here. The ...
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Quark pair superconductor: Even parity is favorred than odd parity

It seems that the quark pair superconductor can be odd or even parity pairing respect to the parity $P$. Say that the even parity has the form: $$ \langle\psi C \gamma^5 \psi\rangle $$ the odd ...
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Gauged Linear Sigma Model (GLSM) with target space $E^8$ gauge group

I just read a few reviews (and also Witten's original paper http://arxiv.org/abs/hep-th/9301042) about the GLSM (Gauged Linear Sigma Model) in (2,2) and (0,2) formulations. I have several natural ...
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Experimentally distinguishing between topologically inequivalent physical states in gauge theory

In gauge theory, physical states are often said to be characterized by equivalence classes of gauge field configurations that differ by gauge transformations. But according to Large and small gauge ...
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Is it the chiral anomaly which is solely responsible for having instanton effects (and therefore, the $\theta-$term) in the QCD action?

$\textbf{Fact 1}$ In principle, the QCD Lagrangian should contain a Lorentz invariant, gauge invariant, dimension-4 term $\sim\theta \text{Tr}[F^{\mu\nu}\tilde{F}_{\mu\nu}]$. This term, however, is ...
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Finding classical action in tunneling problem

In QM: I am trying to show that the minimum action for a classical path going between two potential wells (centered at $\pm L$) in a dbl-well potential is $$S_{classical} = \int_{-L}^{L} dx' \sqrt{...
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Non-Perturbative effects QCD and the Standard Model?

I read in an article that the Standard Model leaves unanswered questions about the non-perturbative effects of the QCD. I have basic knowledge about the perturbative and non-perturbative QCD. Could ...
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Reference request for semiclassical approximations for Schwinger-Keldysh path integrals

Can some one provide some resources for understanding semi-classical approximations for Schwinger-Keldysh path integrals. Is there any discussion about instanton (and multi-instanton) (for even single ...
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Physical origin of Nekrasov Partition Function

I've seen a few papers [1,2,3,4] which defined Nekrasov Partition Function as (in particular [2,3,4]) \begin{equation} Z(\mathbf{a}, \epsilon_1,\epsilon_2,\Lambda) := \sum_{n = 0}^\infty \Lambda^n\...
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Resources for self-dual solutions to Maxwell's equation on Euclidean or pseudo-Euclidean space

I am attempting to understand a question posed to me by an acquaintance, who asked me if I could refer him to literature on "self-dual solutions to Maxwell's equations on Euclidean space, or pseudo-...
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Are Instantons Quantum or Classical?

I'm talking specifically about instantons on four-manifolds, but my confusion here is probably of a more general nature. So I'd also appreciate less specific answers! Okay, so I know that in physics,...
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Peculiarities of non-Abelian gauge groups: self-coupling and topology

There are two striking aspects of non-Abelian gauge groups (compared to their Abelian cousins): (1) The pure gauge parts of non-Abelian Lagrangians contain self-interaction terms that are trilinear ...
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What happens when there are different classical trajectories before and after Wick rotation? [duplicate]

Recently I read the path integral of double well tunnelling. I am puzzled about the Wick rotation calculation. For example, I choose potential like $V(x)=(x^2-1)^2$ and Lagragian $L= \frac{1}{2} \dot ...
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A proof of Gel'fand-Yaglom theorem

I am trying to understand the Gel'fand-Yaglom theorem. The Gel'fand-Yaglom theorem is the following. Let us consider the eigenvalue problem of the operator $-\partial^2+W(t)$ with the eigenfunctions ...
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How to understand “analytical continuation” in the context of instantons?

Since this is a subtle and interesting question to me. I will give a rather detailed description. I hope you can keep reading it and find it interesting too. For simplicity, in the following I will ...
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Why true -> true vacuum instantons (tunneling) not possible in QFT?

According to Tanedo in his text 't Hooft and $\eta$'ail (see paragraph 2.6), unlike the case in QM, true->true vacuum tunneling in QFT is prohibited. They do explain the reason, but I still do not ...
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Why is a superposition of vacuum states possible in QCD, but not in electroweak theory?

There are two standard stories floating around in modern particle physics: Spontaneous symmetry breaking can only happen in a QFT, like in the electroweak theory, because no tunneling between the ...
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Connection between homotopic maps from $X\to Y$ and homotopic paths in $Y$ in the context of SU(2) Yang-Mills instantons

EDIT: I was reading little bit of homotopy theory in trying to understand the difference between homotopic maps from $X\to Y$ and homotopic paths in $Y$, and their significance in the context of SU(2) ...
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Two expressions for instanton winding number integral

Suppose the winding number $n$ of the Yang-Mills instanton configuration. It ia given by the expression $$ \tag 1 n = \frac{1}{16\pi^2}\int \limits_{S^4} d^{4}x\text{tr}\big[F_{ij}\tilde{F}^{ij}\big], ...
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Boundary condition for solitons in 1+1 dimensions to have finite energy

Suppose a classical field configuration of a real scalar field $\phi(x,t)$, in $1+1$ dimensions, has the energy $$E[\phi]=\int\limits_{-\infty}^{+\infty} dx\, \left[\frac{1}{2}\left(\frac{\partial\phi}...
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Understanding typical non-perturbative calculations in QFT [closed]

Perturbative calculations in quantum field theory are based on S-matrix expansion and calculating the Feynman diagrams. These Feynman diagrams are related to the scattering cross-sections and decay ...
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How does Euclidean Quantum Field Theory describe tunneling?

We know that Euclidean QFT originates from path integral formalism of $$\langle\phi_f|e^{-\beta\hat{H}}|\phi_x\rangle.\tag{1}$$ We can understand that for $\beta\rightarrow\infty$, we can obtain the ...
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Theta Vacuum of Yang-Mills theory and Baryon number violation

Background 1. In classical SU(N) Yang-Mills theories, there are a countably infinite number of homotopically inequivalent gauge field configurations of zero energy labelled by a winding number $n\in \...
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How are the vacua of Yang-Mills theory connected by or changed by instanton effects?

Warning: This question is based on poor/immature understanding of instantons and the vacuum structure of the SU(N) Yang-Mills theory. Different vacua of the SU(2) Yang-Mills theory and the ...
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Form of $SU(N)$ gauge transformations in $SU(N)$ Yang-Mills theory

For $SU(N)$ Yang-Mills theory, instantons correspond to finite action solutions $A_\mu(x)$ of the Euclidean equation of motion. The requirement of finite action demands that $A_\mu(x)$ is a pure gauge ...
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Understanding instantons in pure Yang-Mills theory

Yang-Mills Instantons are defined as finite action solutions to the corresponding Euclidean equation of motion. If I understood it correct, then instantons are those classical gauge field ...