Few basic questions about instantons

Question

For the $SU(2)$ Yang-Mill's theory, (1) how can one understand that the finite action solutions of the Euclidean equations of motion (called Instantons) exhibit tunneling effects? (2) Since, this effect is present even in a classical field theory, with no quantized particles, what does really tunnel? (3) The gauge field $A_\mu$ at $|\textbf{x}|\rightarrow \infty$ is given by $$A_\mu\rightarrow -\frac{i}{g}(\partial_\mu U) U^{-1}$$ My question is, whether this relation is true for arbitrary $U(x)\in SU(2)$. In other words, will there be an additional constraint on $U(x)$, due to the finiteness of the action, so that $\{U(x)\}$ are restricted to a subset of $SU(2)$? If yes, how is a particular homotopy class (labeled by an integer $n$) of gauge transformations, $U^{(n)}(x)$, represented by (explicitly, in terms of $2\times 2$ matrices)? (4) Hoes can one show/understand that two classes $U^{(1)}(x)$ and $U^{(2)}(x)$, labeled by two different integers (say, 1 and 2, for example) are not topologically equivalent?

Answer 1

1) Suppose that you have two configurations (here I've used Coulomb gauge with euclidean time $\tau$): $$ \tag 0 A_{i}(x) = \begin{cases} 0 = U^{(0)}\partial_{i}(U^{(0)})^{-1}, \quad \tau = -\infty \\ U^{(1)}\partial_{i}(U^{(1)})^{-1}, \quad \tau = \infty\end{cases} $$ Such situation describes tunneling between vacua with topological charges $0$ and $1$. Next you have to compute Maurer-Cartan invariant (see Eq. $(1)$), which for given configuration is equal to 1. It can be shown by gauge invariance of this quantity and by integration on the cylinder surface, in which $z$ axis denotes time, while "perpendicular" directions denotes spatial coordinates, that it is equal to $$ n = \frac{1}{24 \pi^{2}}\left(\int d^{3}\mathbf r\epsilon_{ijk}\text{Tr}A_{i}A_{j}A_{k}\right)_{\tau = -\infty}^{\tau = \infty} = |(0)| = n[U^{(1)}] - n[U^{(0)}] $$ So that you see that in the Coulomb gauge $A_{0} = 0$ the instanton solution $(0)$ really describes tunneling between vacua with topological charges $0$ and $1$. Since each instanton with arbitrary topological value can be described as set of instantons with topological value $1$ (look to 4)), and due to gauge invariance of $n$, the result stated above is true for configuration with arbitrary number $n$ and for each gauge.

2) Yes, instantons are solutions of classical equations of motion. But only the quantum system can be described as the superposition of different states. In the case of theories with nontrivial topological properties, in general the state of theory is $$ |\text{vac}\rangle = \sum_{n = -\infty}^{\infty} c(n)| n\rangle $$ Due to possibility of tunneling between vacua with different values of $n$ $c(n)$ isn't zero for all $n$. It can be shown that $c(n) = e^{-i\theta n}$, where $\theta$ is arbitrary parameter. This, of course, is impossible in classical theory.

3) In general, you have to find $U(x)$ so that it has definite topological charge. When you'll find it, you'll obtain the solution with correct asymptotics. For example, due to the fact that $U^{-1} = \tau_{\alpha}n_{\alpha}$, where $\tau_{\alpha}$ is $SU(2)$ group generator and $n_{\alpha}$ is the unit 3-vector, corresponds to the topological charge $1$, following relation is hold: $$ U^{-1}\partial_{\mu}U = -i\bar{\eta}_{\mu \alpha a}\frac{n_{\alpha}}{r}\tau_{a} \sim \frac{1}{r}, $$ which is sufficient for finiteness of action. Here $\bar{\eta}_{\mu \alpha a}$ is antiself-dual t'Hooft symbol.

The relation between finiteness of action and between configurations with different topological charges follows from the Bogomolny inequality.

4) Two different $SU(2)$ elements $U^{(1)}(x),U^{(2)}(x)$ corresponds to the different values of Maurer-Cartan invariant: $$ \tag 1 n = \frac{1}{24 \pi^{2}}\int d\sigma_{\mu}\epsilon^{\mu \nu \rho \sigma}\text{Tr}[\rho_{\nu}\rho_{\rho}\rho_{\sigma}], $$ where $$ \rho_{\alpha} \equiv U\partial_{\alpha}U^{-1} $$ This quantity is invariant under small perturbations $U \to U + \delta U$ and under coordinate replacement $x \to x{'}$. It means that $U^{(1)}$ and $U^{(2)}$ are topologically inequivalent since there is conservation law of topological charge: continuous transformation of $U^{(1)}$ which transforms it to $U^{(2)}$ doesn't exist. In principle, however, configuration with topological charge $2$ may be represented as set of configurations with summary topological charge $2$.