Editing Instanton (section)

== Quantum mechanics ==
An ''instanton'' can be used to calculate the transition probability for a quantum mechanical particle tunneling through a potential barrier. One example of a system with an ''instanton'' effect is a particle in a [[double-well potential]]. In contrast to a classical particle, there is non-vanishing probability that it crosses a region of potential energy higher than its own energy.<ref name=":0" />

=== Motivation of considering instantons ===
Consider the quantum mechanics of a single particle motion inside the double-well potential <math>V(x)={1\over 4}(x^2-1)^2.</math>
The potential energy takes its minimal value at <math>x=\pm 1</math>, and these are called classical minima because the particle tends to lie in one of them in classical mechanics. There are two lowest energy states in classical mechanics.

In quantum mechanics, we solve the [[Schrödinger equation]]

:<math>-{\hbar^2\over 2m}{\partial^2\over \partial x^2}\psi(x)+V(x)\psi(x)=E\psi(x), </math>

to identify the energy eigenstates. If we do this, we will find only the unique lowest-energy state instead of two states. The ground-state wave function localizes at both of the classical minima <math>x=\pm 1</math> instead of only one of them because of the quantum interference or quantum tunneling.

Instantons are the tool to understand why this happens within the semi-classical approximation of the path-integral formulation in Euclidean time. We will first see this by using the WKB approximation that approximately computes the wave function itself, and will move on to introduce instantons by using the path integral formulation.

=== WKB approximation ===
One way to calculate this probability is by means of the semi-classical [[WKB approximation]], which requires the value of <math>\hbar</math> to be small. The [[Schrödinger equation#Time-independent equation|time independent Schrödinger equation]] for the particle reads

:<math>\frac{d^2\psi}{dx^2}=\frac{2m(V(x)-E)}{\hbar^2}\psi.</math>

If the potential were constant, the solution would be a plane wave, up to a proportionality factor,

:<math>\psi = \exp(-\mathrm{i}kx)\,</math>

with

:<math>k=\frac{\sqrt{2m(E-V)}}{\hbar}.</math>

This means that if the energy of the particle is smaller than the potential energy, one obtains an exponentially decreasing function. The associated tunneling amplitude is proportional to

:<math>e^{-\frac{1}{\hbar}\int_a^b\sqrt{2m(V(x)-E)} \, dx},</math>

where ''a'' and ''b'' are the beginning and endpoint of the tunneling trajectory.

=== Path integral interpretation via instantons ===
Alternatively, the use of [[path integral formulation|path integrals]] allows an ''instanton'' interpretation and the same result can be obtained with this approach. In path integral formulation, the transition amplitude can be expressed as

:<math>K(a,b;t)=\langle x=a|e^{-\frac{i\mathbb{H}t}{\hbar}}|x=b\rangle =\int d[x(t)]e^{\frac{iS[x(t)]}{\hbar}}.</math>

Following the process of [[Wick rotation]] (analytic continuation) to Euclidean spacetime (<math>it\rightarrow \tau</math>), one gets

:<math>K_E(a,b;\tau)=\langle x=a|e^{-\frac{\mathbb{H}\tau}{\hbar}}|x=b\rangle =\int d[x(\tau)]e^{-\frac{S_E[x(\tau)]}{\hbar}},</math>

with the Euclidean action

:<math>S_E=\int_{\tau_a}^{\tau_b}\left(\frac{1}{2}m\left(\frac{dx}{d\tau}\right)^2+V(x)\right) d\tau.</math>

The potential energy changes sign <math> V(x) \rightarrow - V(x) </math> under the Wick rotation and the minima transform into maxima, thereby <math> V(x) </math> exhibits two "hills" of maximal energy.

Let us now consider the local minimum of the Euclidean action <math>S_E</math> with the double-well potential <math>V(x)={1\over 4}(x^2-1)^2</math>, and we set <math>m=1</math> just for simplicity of computation. Since we want to know how the two classically lowest energy states <math>x=\pm1</math> are connected, let us set <math>a=-1</math> and <math>b=1</math>. 
For <math>a=-1</math> and <math> b=1</math>, we can rewrite the Euclidean action as

:<math> S_E=\int_{\tau_a}^{\tau_b}d \tau {1\over 2}\left({d x\over d \tau}-\sqrt{2V(x)}\right)^2 + \sqrt{2}\int_{\tau_a}^{\tau_b}d \tau{d x\over d \tau}\sqrt{V(x)} </math>
:<math> \quad =\int_{\tau_a}^{\tau_b}d \tau {1\over 2}\left({d x\over d \tau}-\sqrt{2V(x)}\right)^2 + \int_{-1}^{1}d x {1\over \sqrt{2}}(1-x^2). </math>
:<math> \quad \ge {2\sqrt{2}\over 3}. </math>

The above inequality is saturated by the solution of <math> {d x\over d \tau}=\sqrt{2V(x)}</math> with the condition <math>x(\tau_a)=-1</math> and <math>x(\tau_b)=1</math>. Such solutions exist, and the solution takes the simple form when <math>\tau_a=-\infty</math> and <math>\tau_b=\infty</math>. The explicit formula for the instanton solution is given by

:<math> x(\tau)=\tanh\left({1\over \sqrt{2}}(\tau-\tau_0)\right).  </math>

Here <math>\tau_0</math> is an arbitrary constant. Since this solution jumps from one classical vacuum <math>x=-1</math> to another classical vacuum <math>x=1</math> instantaneously around <math>\tau=\tau_0</math>, it is called an instanton.

=== Explicit formula for double-well potential===

The explicit formula for the eigenenergies of the Schrödinger equation with [[double-well potential]] has been given by Müller–Kirsten<ref>H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (World Scientific, 2012), {{ISBN|978-981-4397-73-5}}; formula (18.175b), p. 525.</ref> with derivation by both a perturbation method (plus boundary conditions) applied to the Schrödinger equation, and explicit derivation from the path integral (and WKB). The result is the following. Defining parameters of the Schrödinger equation and the potential by the equations

:<math> \frac{d^2y(z)}{dz^2} + [E-V(z)]y(z) = 0, </math>

and

:<math> V(z) = -\frac{1}{4}z^2h^4 + \frac{1}{2}c^2z^4, \;\;\; c^2>0, \; h^4>0, </math>

the eigenvalues for <math>q_0=1,3,5,...</math> are found to be:

:<math>E_{\pm}(q_0,h^2) = -\frac{h^8}{2^5c^2} + \frac{1}{\sqrt{2}}q_0h^2
- \frac{c^2(3q^2_0+1)}{2h^4} - \frac{\sqrt{2}c^4q_0}{8h^{10}}(17q^2_0+19) +O(\frac{1}{h^{16}})
</math>
:<math> \mp \frac{2^{q_0+1}h^2(h^6/2c^2)^{q_0/2}}{\sqrt{\pi}2^{q_0/4}[(q_0-1)/2]!}
e^{-h^6/6\sqrt{2}c^2}.
</math>

Clearly these eigenvalues are asymptotically (<math>h^2\rightarrow\infty</math>) degenerate as expected as a consequence of the harmonic part of the potential.

=== Results ===
Results obtained from the mathematically well-defined Euclidean [[line integral|path integral]] may be Wick-rotated back and give the same physical results as would be obtained by appropriate treatment of the (potentially divergent) Minkowskian path integral. As can be seen from this example, calculating the transition probability for the particle to tunnel through a classically forbidden region (<math>V(x)</math>) with the Minkowskian path integral corresponds to calculating the transition probability to tunnel through a classically allowed region (with potential &minus;''V''(''X'')) in the Euclidean path integral (pictorially speaking &ndash; in the Euclidean picture &ndash; this transition corresponds to a particle rolling from one hill of a double-well potential standing on its head to the other hill). This classical solution of the Euclidean equations of motion is often named "kink solution" and is an example of an ''instanton''. In this example, the two "vacua" (i.e. ground states) of the [[double-well potential]], turn into hills in the Euclideanized version of the problem.

Thus, the ''instanton'' field solution of the (Euclidean, i. e., with imaginary time) (1&nbsp;+&nbsp;1)-dimensional field theory &ndash; first quantized quantum mechanical description &ndash; allows to be interpreted as a tunneling effect between the two vacua (ground states &ndash; higher states require periodic instantons) of the physical (1-dimensional space + real time) Minkowskian system. In the case of the double-well potential written

:<math> V(\phi) = \frac{m^4}{2g^2}\left(1 - \frac{g^2\phi^2}{m^2}\right)^2 </math>

the instanton, i.e. solution of

:<math> \frac{d^2\phi}{d\tau^2} = V'(\phi), </math>

(i.e. with energy <math>E_{cl} = 0</math>), is

:<math> \phi_c(\tau) = \frac{m}{g}\tanh\left[m(\tau - \tau_0)\right],</math>

where <math>\tau = it</math> is the Euclidean time.

''Note'' that a naïve perturbation theory around one of those two vacua alone (of the Minkowskian description) would never show this ''non-perturbative tunneling effect'', dramatically changing the picture of the vacuum structure of this quantum mechanical system. In fact the naive perturbation theory has to be supplemented by boundary conditions, and these supply the nonperturbative effect, as is evident from the above explicit formula and analogous calculations for other potentials such as a cosine potential (cf. [[Mathieu function]]) or other periodic potentials (cf. e.g. [[Lamé function]] and [[spheroidal wave function]]) and irrespective of whether one uses the Schrödinger equation or the [[Functional integration|path integral]].<ref>H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific, 2012, {{ISBN|978-981-4397-73-5}}.</ref>

Therefore, the perturbative approach may not completely describe the vacuum structure of a physical system. This may have important consequences, for example, in the theory of [[axion|"axions"]] where the non-trivial QCD vacuum effects (like the ''instantons'') spoil the [[Peccei–Quinn theory|Peccei–Quinn symmetry]] explicitly and transform massless [[Nambu–Goldstone boson]]s into massive [[Chiral symmetry breaking|pseudo-Nambu–Goldstone ones]].

===Periodic instantons===
{{main|Periodic instantons}}
In one-dimensional field theory or quantum mechanics one defines as "instanton" a field configuration which is a solution of the classical (Newton-like) equation of motion with Euclidean time and finite Euclidean action. In the context of [[soliton]] theory the corresponding solution is known as a [[Sine-Gordon equation#Soliton solutions|kink]]. In view of their analogy with the behaviour of classical particles such configurations or solutions, as well as others, are collectively known as [[pseudoparticles]] or pseudoclassical configurations. The "instanton" (kink) solution is accompanied by another solution known as "anti-instanton" (anti-kink), and instanton and anti-instanton are distinguished by "topological charges" +1 and &minus;1 respectively, but have the same Euclidean action.

"Periodic instantons" are a generalization of instantons.<ref name="Harald J.W. Müller-Kirsten 2012">Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012).</ref> In explicit form they are expressible in terms of [[Jacobian elliptic functions]] which are periodic functions (effectively generalisations of trigonometrical functions). In the limit of infinite period these periodic instantons &ndash; frequently known as "bounces", "bubbles" or the like &ndash; reduce to instantons.

The stability of these pseudoclassical configurations can be investigated by expanding the Lagrangian defining the theory around the pseudoparticle configuration and then investigating the equation of small fluctuations around it. For all versions of quartic potentials (double-well, inverted double-well) and periodic (Mathieu) potentials these equations were discovered to be Lamé equations, see [[Lamé function]].<ref>{{cite journal | last1=Liang | first1=Jiu-Qing | last2=Müller-Kirsten | first2=H.J.W. | last3=Tchrakian | first3=D.H. | title=Solitons, bounces and sphalerons on a circle | journal=Physics Letters B | publisher=Elsevier BV | volume=282 | issue=1–2 | year=1992 | issn=0370-2693 | doi=10.1016/0370-2693(92)90486-n | pages=105–110| bibcode=1992PhLB..282..105L }}</ref> The eigenvalues of these equations are known and permit in the case of instability the calculation of decay rates by evaluation of the path integral.<ref name="Harald J.W. Müller-Kirsten 2012"/>

=== Instantons in reaction rate theory ===
In the context of reaction rate theory, periodic instantons are used to calculate the rate of tunneling of atoms in chemical reactions. The progress of a chemical reaction can be described as the movement of a pseudoparticle on a high dimensional [[potential energy surface]] (PES). The thermal rate constant <math>k</math> can then be related to the imaginary part of the free energy <math>F</math> by<ref name=":inst_chapter">
{{cite book | last1 = Zaverkin | first1 = Viktor
	| last2 = Kästner
	| first2 = Johannes
	| author-link2 = Johannes Kästner
	| year = 2020 
	| title = Tunnelling in Molecules: Nuclear Quantum Effects from Bio to Physical Chemistry
	| chapter = Instanton Theory to Calculate Tunnelling Rates and Tunnelling Splittings
	| url = https://doi.org/10.1039/9781839160370
	| location = London
	| publisher = Royal Society of Chemistry 
	| page = 245-260
	| isbn = 978-1-83916-037-0
}}
</ref>

<math>k(\beta) = -\frac{2}{\hbar} \text{Im} \mathrm{F} = \frac{2}{\beta \hbar} \text{Im} \ \text{ln}(Z_k) \approx \frac{2}{\hbar \beta} \frac{\text{Im} Z_k }{\text{Re} Z_k } ,\ \ \text{Re} Z_k \gg \text{Im} Z_k </math>

whereby <math>Z_k</math> is the canonical partition function, which is calculated by taking the trace of the Boltzmann operator in the position representation.

<math>Z_k = \text{Tr}(e^{-\beta \hat{H}}) = \int d\mathbf{x} \left\langle \mathbf{x} \left| e^{-\beta \hat{H}} \right| \mathbf{x} \right\rangle</math>

Using a Wick rotation and identifying the Euclidean time with <math>\hbar\beta = 1/(k_b T)</math>, one obtains a path integral representation for the partition function in mass-weighted coordinates:<ref name=":inst_review">
{{cite journal
	| last1 = Kästner
	| first1 = Johannes
	| date = 2014
	| title = Theory and Simulation of Atom Tunneling in Chemical Reactions
	| url = https://doi.org/10.1002/wcms.1165
	| journal = WIREs Comput. Mol. Sci.
	| volume = 4
	| pages = 158
	| doi = 10.1002/wcms.1165
| url-access = subscription
	}}
</ref>

<math>Z_k = \oint \mathcal{D} \mathbf{x}(\tau) e^{-S_E[\mathbf{x}(\tau)]/\hbar}, \ \ \ S_E = \int_0^{\beta \hbar} \left( \frac{\dot{\mathbf{x}}}{2}^2 + V(\mathbf{x}(\tau)) \right) d\tau</math>

The path integral is then approximated via a steepest descent integration, which takes into account only the contributions from the classical solutions and quadratic fluctuations around them. This yields for the rate constant expression in mass-weighted coordinates

<math>k(\beta) = \frac{2}{\beta\hbar} \left( \frac{ \text{det}\left[ -\frac{\partial^2}{\partial \tau^2} + \mathbf{V}''(x_\text{RS}(\tau)) \right] }{\text{det} \left[- \frac{\partial^2}{\partial \tau^2} + \mathbf{V}''(x_\text{Inst}(\tau)) \right] } \right)^\frac{1}{2}{\exp\left({\frac{-S_E[x_\text{inst}(\tau) + S_E[x_\text{RS}(\tau)] }{\hbar}}\right)}</math>

where <math>\mathbf{x}_\text{Inst}</math> is a periodic instanton and <math>\mathbf{x}_\text{RS}</math> is the trivial solution of the pseudoparticle at rest which represents the reactant state configuration.

===Inverted double-well formula===

As for the double-well potential one can derive the eigenvalues for the inverted double-well potential. In this case, however, the eigenvalues are complex. Defining parameters by the equations
:<math> \frac{d^2y}{dz^2} + [E - V(z)]y(z) = 0, \;\;\;
V(z) = \frac{1}{4}h^4z^2 - \frac{1}{2}c^2z^4, </math>
the eigenvalues as given by Müller-Kirsten are, for <math>q_0 = 1,3,5,...,</math>
:<math>E = \frac{1}{2}q_0h^2 - \frac{3c^2}{4h^4}(q^2_0+1) -\frac{q_0c^4}{h^{10}}(4q^2_0+29) + O(\frac{1}{h^{16}}) \pm i\frac{2^{q_0}h^2(h^6/2c^2)^{q_0/2}}{(2\pi)^{1/2}[(q_0-1)/2]!}e^{-h^6/6c^2}.</math>
The imaginary part of this expression agrees with the well known result of Bender and Wu.<ref>{{cite journal | last1=Bender | first1=Carl M. | last2=Wu | first2=Tai Tsun | title=Anharmonic Oscillator. II. A Study of Perturbation Theory in Large Order | journal=Physical Review D | publisher=American Physical Society (APS) | volume=7 | issue=6 | date=1973-03-15 | issn=0556-2821 | doi=10.1103/physrevd.7.1620 | pages=1620–1636| bibcode=1973PhRvD...7.1620B }}</ref> In their notation <math>\hbar = 1, q_0=2K+1, h^6/2c^2 = \epsilon.</math>