Motivation of considering instantons

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

In quantum mechanics, we solve the Schrödinger equation

${\displaystyle -{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial x^{2}}\psi (x)+V(x)\psi (x)=E\psi (x),}$

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 ${\displaystyle x=\pm 1}$ 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 ${\displaystyle \hbar }$ to be small. The time independent Schrödinger equation for the particle reads

${\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac {2m(V(x)-E)}{\hbar ^{2}}}\psi .}$

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

${\displaystyle \psi =\exp(-\mathrm {i} kx)\,}$

with

${\displaystyle k={\frac {\sqrt {2m(E-V)}}{\hbar }}.}$

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

${\displaystyle e^{-{\frac {1}{\hbar }}\int _{a}^{b}{\sqrt {2m(V(x)-E)}}\,dx},}$

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

Path integral interpretation via instantons

Alternatively, the use of 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

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

Following the process of Wick rotation (analytic continuation) to Euclidean spacetime (${\displaystyle it\rightarrow \tau }$), one gets

${\displaystyle K_{E}(a,b;\tau )=\langle x=a|e^{-{\frac {\mathbb {H} \tau }{\hbar }}}|x=b\rangle =\int d[x(\tau )]e^{-{\frac {S_{E}[x(\tau )]}{\hbar }}},}$

with the Euclidean action

${\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}\left({\frac {1}{2}}m\left({\frac {dx}{d\tau }}\right)^{2}+V(x)\right)d\tau .}$

The potential energy changes sign ${\displaystyle V(x)\rightarrow -V(x)}$ under the Wick rotation and the minima transform into maxima, thereby ${\displaystyle V(x)}$ exhibits two "hills" of maximal energy.

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

${\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+{\sqrt {2}}\int _{\tau _{a}}^{\tau _{b}}d\tau {dx \over d\tau }{\sqrt {V(x)}}}$

${\displaystyle \quad =\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+\int _{-1}^{1}dx{1 \over {\sqrt {2}}}(1-x^{2}).}$

${\displaystyle \quad \geq {2{\sqrt {2}} \over 3}.}$

The above inequality is saturated by the solution of ${\displaystyle {dx \over d\tau }={\sqrt {2V(x)}}}$ with the condition ${\displaystyle x(\tau _{a})=-1}$ and ${\displaystyle x(\tau _{b})=1}$. Such solutions exist, and the solution takes the simple form when ${\displaystyle \tau _{a}=-\infty }$ and ${\displaystyle \tau _{b}=\infty }$. The explicit formula for the instanton solution is given by

${\displaystyle x(\tau )=\tanh \left({1 \over {\sqrt {2}}}(\tau -\tau _{0})\right).}$

Here ${\displaystyle \tau _{0}}$ is an arbitrary constant. Since this solution jumps from one classical vacuum ${\displaystyle x=-1}$ to another classical vacuum ${\displaystyle x=1}$ instantaneously around ${\displaystyle \tau =\tau _{0}}$, 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^[7] 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

${\displaystyle {\frac {d^{2}y(z)}{dz^{2}}}+[E-V(z)]y(z)=0,}$

and

${\displaystyle V(z)=-{\frac {1}{4}}z^{2}h^{4}+{\frac {1}{2}}c^{2}z^{4},\;\;\;c^{2}>0,\;h^{4}>0,}$

the eigenvalues for ${\displaystyle q_{0}=1,3,5,...}$ are found to be:

${\displaystyle E_{\pm }(q_{0},h^{2})=-{\frac {h^{8}}{2^{5}c^{2}}}+{\frac {1}{\sqrt {2}}}q_{0}h^{2}-{\frac {c^{2}(3q_{0}^{2}+1)}{2h^{4}}}-{\frac {{\sqrt {2}}c^{4}q_{0}}{8h^{10}}}(17q_{0}^{2}+19)+O({\frac {1}{h^{16}}})}$

${\displaystyle \mp {\frac {2^{q_{0}+1}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{{\sqrt {\pi }}2^{q_{0}/4}[(q_{0}-1)/2]!}}e^{-h^{6}/6{\sqrt {2}}c^{2}}.}$

Clearly these eigenvalues are asymptotically (${\displaystyle h^{2}\rightarrow \infty }$) degenerate as expected as a consequence of the harmonic part of the potential.

Results

Results obtained from the mathematically well-defined Euclidean 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 (${\displaystyle V(x)}$) with the Minkowskian path integral corresponds to calculating the transition probability to tunnel through a classically allowed region (with potential −V(X)) in the Euclidean path integral (pictorially speaking – in the Euclidean picture – 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 + 1)-dimensional field theory – first quantized quantum mechanical description – allows to be interpreted as a tunneling effect between the two vacua (ground states – higher states require periodic instantons) of the physical (1-dimensional space + real time) Minkowskian system. In the case of the double-well potential written

${\displaystyle V(\phi )={\frac {m^{4}}{2g^{2}}}\left(1-{\frac {g^{2}\phi ^{2}}{m^{2}}}\right)^{2}}$

the instanton, i.e. solution of

${\displaystyle {\frac {d^{2}\phi }{d\tau ^{2}}}=V'(\phi ),}$

(i.e. with energy ${\displaystyle E_{cl}=0}$), is

${\displaystyle \phi _{c}(\tau )={\frac {m}{g}}\tanh \left[m(\tau -\tau _{0})\right],}$

where ${\displaystyle \tau =it}$ 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 path integral.^[8]

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 "axions" where the non-trivial QCD vacuum effects (like the instantons) spoil the Peccei–Quinn symmetry explicitly and transform massless Nambu–Goldstone bosons into massive pseudo-Nambu–Goldstone ones.

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 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 −1 respectively, but have the same Euclidean action.

"Periodic instantons" are a generalization of instantons.^[9] 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 – frequently known as "bounces", "bubbles" or the like – 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.^[10] 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.^[9]

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 ${\displaystyle k}$ can then be related to the imaginary part of the free energy ${\displaystyle F}$ by

${\displaystyle k(\beta )=-{\frac {2}{\hbar }}{\text{Im}}\mathrm {F} ={\frac {2}{\beta \hbar }}{\text{Im}}\ {\text{ln}}(Z_{k})\approx {\frac {2}{\hbar \beta }}{\frac {{\text{Im}}Z_{k}}{{\text{Re}}Z_{k}}},\ \ {\text{Re}}Z_{k}\gg {\text{Im}}Z_{k}}$

whereby ${\displaystyle Z_{k}}$is the canonical partition function which is calculated by taking the trace of the Boltzmann operator in the position representation.

${\displaystyle Z_{k}={\text{Tr}}(e^{-\beta {\hat {H}}})=\int d\mathbf {x} \left\langle \mathbf {x} \left|e^{-\beta {\hat {H}}}\right|\mathbf {x} \right\rangle }$

Using a wick rotation and identifying the Euclidean time with ${\displaystyle \hbar \beta =1/(k_{b}T)}$ one obtains a path integral representation for the partition function in mass weighted coordinates

${\displaystyle Z_{k}=\oint {\mathcal {D}}\mathbf {x} (\tau )e^{-S_{E}[\mathbf {x} (\tau )]/\hbar },\ \ \ S_{E}=\int _{0}^{\beta \hbar }\left({\frac {\dot {\mathbf {x} }}{2}}^{2}+V(\mathbf {x} (\tau ))\right)d\tau }$

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

${\displaystyle k(\beta )={\frac {2}{\beta \hbar }}\left({\frac {{\text{det}}\left[-{\frac {\partial ^{2}}{\partial \tau ^{2}}}+\mathbf {V} ''(x_{\text{RS}}(\tau ))\right]}{{\text{det}}\left[-{\frac {\partial ^{2}}{\partial \tau ^{2}}}+\mathbf {V} ''(x_{\text{Inst}}(\tau ))\right]}}\right)^{\frac {1}{2}}{\exp \left({\frac {-S_{E}[x_{\text{inst}}(\tau )+S_{E}[x_{\text{RS}}(\tau )]}{\hbar }}\right)}}$

where ${\displaystyle \mathbf {x} _{\text{Inst}}}$is a periodic instanton and ${\displaystyle \mathbf {x} _{\text{RS}}}$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

${\displaystyle {\frac {d^{2}y}{dz^{2}}}+[E-V(z)]y(z)=0,\;\;\;V(z)={\frac {1}{4}}h^{4}z^{2}-{\frac {1}{2}}c^{2}z^{4},}$

the eigenvalues as given by Müller-Kirsten are, for ${\displaystyle q_{0}=1,3,5,...,}$

${\displaystyle E={\frac {1}{2}}q_{0}h^{2}-{\frac {3c^{2}}{4h^{4}}}(q_{0}^{2}+1)-{\frac {q_{0}c^{4}}{h^{10}}}(4q_{0}^{2}+29)+O({\frac {1}{h^{16}}})\pm i{\frac {2^{q_{0}}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{(2\pi )^{1/2}[(q_{0}-1)/2]!}}e^{-h^{6}/6c^{2}}.}$

The imaginary part of this expression agrees with the well known result of Bender and Wu.^[11] In their notation ${\displaystyle \hbar =1,q_{0}=2K+1,h^{6}/2c^{2}=\epsilon .}$