1.10 Instantons

The vacuum-to-vacuum amplitude in the Euclidean theory is

$$Z\equiv ⟨0|0⟩\propto \int \mathcal{D}\varphi e^{-S[\varphi ,{\partial }_{\mu }\varphi ]}\text{\hspace{1em}(1.289)}$$

where $S$ is the Euclidean action. Equation (1.289) shows that the principal contribution to $Z$ comes from the values of $\varphi (x)$ which give the local minima of $S[\varphi ,{\partial }_{\mu }\varphi ]$ . In many theories there exist a number of local minima in addition to the absolute minimum. In the case of non-Abelian gauge theories these minima are called instantons.

1.10.1 Introduction

Let us consider the SU(2) gauge theory defined in the four-dimensional Euclidean space ${\mathbb{R}}^4$ . The action is

$$S=\int {\mathrm{d}}^{4}x\mathcal{L}(x)=\int {\mathrm{d}}^{4}x\left[-\frac{1}{2}\mathrm{tr}{\mathcal{F}}_{\mu v}{\mathcal{F}}^{\mu v}\right]\text{\hspace{1em}(1.290)}$$

where the field strength is

$${\mathcal{F}}_{\mu v}={\partial }_{\mu }{\mathcal{A}}_v-{\partial }_v{\mathcal{A}}_{\mu }+g\left[{\mathcal{A}}_{\mu },{\mathcal{A}}_v\right]\text{\hspace{1em}(1.291)}$$

with

$${\mathcal{A}}_{\mu }\equiv A_{\mu }^{\alpha }\frac{{\sigma }_{\alpha }}{2\mathrm{i}}{\mathcal{F}}_{\mu \nu }\equiv F_{\mu \nu }^{\alpha }\frac{{\sigma }_{\alpha }}{2\mathrm{i}}.$$

The field equation is

$${\mathcal{D}}_{\mu }{\mathcal{F}}_{\mu v}={\partial }_{\mu }{\mathcal{F}}_{\mu v}+g\left[{\mathcal{A}}_{\mu },{\mathcal{F}}_{\mu v}\right]=0.\text{\hspace{1em}(1.292)}$$

In the path integral only those field configurations with finite action contribute. Suppose ${\mathcal{A}}_{\mu }$ satisfies

$${\mathcal{A}}_{\mu }\to \mathrm{i}U(x{)}^{-1}{\partial }_{\mu }U(x)\text{a s}|x|\to \infty \text{\hspace{1em}(1.293)}$$

where $U(x)$ is an element of $\mathrm{SU}(2)$ . We easily find that ${\mathcal{F}}_{\mu \nu }$ vanishes for the ${\mathcal{A}}_{\mu }$ of (1.293). We require that on sphere $S^3$ of large radius, the gauge potential be given by (1.293).

Later we show that this configuration is characterized by the way in which $S^3$ is mapped to the gauge group SU(2). Non-trivial configurations are those that cannot be deformed continuously to a uniform configuration. They were proposed by Belavin et al (1975) and are called instantons.

1.10.2 The (anti-)self-dual solution

In general, solving a second-order differential equation is more difficult than solving a first-order one. It is nice if a second-order differential equation can be replaced by a first-order one which is equivalent to the original problem. Let us consider the inequality

$$\int {\mathrm{d}}^{4}x\mathrm{tr}{\left({\mathcal{F}}_{\mu v}\pm \ast {\mathcal{F}}_{\mu v}\right)}^2\ge 0.\text{\hspace{1em}(1.294)}$$

Clearly (1.294) is saturated if

$${\mathcal{F}}_{\mu v}=\pm \ast {\mathcal{F}}_{\mu v}.\text{\hspace{1em}(1.295)}$$

If the positive sign is chosen, $\mathcal{F}$ is said to be self-dual while the negative sign gives an anti-self-dual solution. If (1.295) is satisfied, the field equation is automatically satisfied since

$${\mathcal{D}}_{\mu }{\mathfrak{F}}_{\mu v}=\pm {\mathcal{D}}_{\mu }\ast {\mathfrak{F}}_{\mu v}=0(\text{B i a n c h i i d e n t i t y}).\text{\hspace{1em}(1.296)}$$

As we will show in section 10.5, the integral

$$Q\equiv \frac{-1}{16{\pi }^2}\int {\mathrm{d}}^{4}x\mathrm{tr}{\mathcal{F}}_{\mu \nu }\ast {\mathcal{F}}^{\mu \nu }\text{\hspace{1em}(1.297)}$$

is an integer characterizing the way $S^3$ is mapped to SU(2). If $\mathcal{F}$ is self-dual then $Q$ is positive, and if $\mathcal{F}$ is anti-self-dual then $Q$ is negative. From (1.294), we find (note that $\ast {\mathcal{F}}_{\mu \nu }\ast {\mathcal{F}}^{\mu \nu }={\mathcal{F}}_{\mu \nu }{\mathcal{F}}^{\mu \nu }$ ) that

$$\int {\mathrm{d}}^{4}x\left(2{\mathcal{F}}_{\mu v}{\mathcal{F}}^{\mu v}\pm 2\ast {\mathcal{F}}_{\mu v}\ast {\mathcal{F}}^{\mu v}\right)\ge 0.\text{\hspace{1em}(1.298)}$$

From this inequality and the definition of the action, we find that

$$S\ge 8{\pi }^2|Q|\text{\hspace{1em}(1.299)}$$

where the inequality is saturated for (1.295). Let us concentrate on the self-dual solution $\mathcal{F}=\ast \mathcal{F}$ . We look for an instanton solution of the form

$${\mathcal{A}}_{\mu }=\mathrm{i}f(r)U(x{)}^{-1}{\partial }_{\mu }U(x)\text{\hspace{1em}(1.300)}$$

where $r\equiv |x|$ and

$$f(r)\to 1\text{a s}r\to \infty \text{\hspace{1em}(1.301a)}$$

$$U(x)=\frac{1}{r}\left(x_4-\mathrm{i}{x}_i{\sigma }_i\right).\text{\hspace{1em}(1.301b)}$$

Substituting (1.300) into (1.295), we find that $f$ satisfies

$$r\frac{\mathrm{d}f(r)}{\mathrm{d}r}=2f(1-f).\text{\hspace{1em}(1.302)}$$

The solution that satisfies the boundary condition (1.301a) is

$$f(r)=\frac{r^2}{r^2+{\lambda }^2}\text{\hspace{1em}(1.303)}$$

where $\lambda$ is a parameter that specifies the size of the instanton. Substituting this into (1.300) we find that

$${\mathcal{A}}_{\mu }(x)=\frac{\mathrm{i}{r}^2}{r^2+{\lambda }^2}U(x{)}^{-1}{\partial }_{\mu }U(x)\text{\hspace{1em}(1.304)}$$

and the corresponding field strength

$${\mathcal{F}}_{\mu \nu }(x)=\frac{4{\lambda }^2}{r^2+{\lambda }^2}{\sigma }_{\mu \nu }\text{\hspace{1em}(1.305)}$$

where

$${\sigma }_{ij}\equiv \frac{1}{4\mathrm{i}}[{\sigma }_i,{\sigma }_j]{\sigma }_{i0}\equiv \frac{1}{2}{\sigma }_i=-{\sigma }_{0i}.\text{\hspace{1em}(1.306)}$$

This solution gives $Q=+1$ and $S=8{\pi }^2$ .

1.1 Consider a Hamiltonian of the form

$$H=\int {\mathrm{d}}^{n}x\left[\frac{1}{2}{\left(\frac{\partial \varphi }{\partial t}\right)}^2+\frac{1}{2}(\nabla \varphi {)}^2+V(\varphi )\right]$$

where $V(\varphi )$ ( $\ge 0$ ) is a potential. If $\varphi$ is a time-independent classical solution, we may drop the first term and write $H[\varphi ]=H_1[\varphi ]+H_2[\varphi ]$ , where

$$H_1[\varphi ]\equiv \frac{1}{2}\int {\mathrm{d}}^{n}x(\nabla \varphi {)}^2{H}_2[\varphi ]\equiv \int {\mathrm{d}}^{n}xV(\varphi ).$$

(1) Consider a scale transformation $\varphi (x)\to \varphi (\lambda x)$ . Show that $H_i[\varphi ]$ transforms as

$$H_1[\varphi ]\to H_1^{\lambda }[\varphi ]={\lambda }^{(n-2)}{H}_1[\varphi ]H_2[\varphi ]\to H_2^{\lambda }[\varphi ]={\lambda }^{-n}{H}_2[\varphi ].$$

(2) Suppose $\varphi$ satisfies the field equation. Show that

$$(2-n)H_1[\varphi ]-n{H}_2[\varphi ]=0.$$

[Hint: Take the $\lambda$ -derivative of $H_1^{\lambda }[\varphi ]+H_2^{\lambda }[\varphi ]$ and put $\lambda =1$ .]

(3) Show that time-independent topological excitations of $H[\varphi ]$ exist if and only if $n=1$ (Derrick’s theorem). Consider ways out of this restriction.

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