1
The temporal gauge is a gauge choice that sets the temporal component of the gauge field to zero. It is often used in instanton calculations because it simplifies the calculation of the instanton action and makes it easier to see that the instanton is a tunneling process between different vacua. However, it is not necessary to use the temporal gauge to see this. The same conclusion can be drawn from a general gauge. The temporal gauge is just a convenient choice of gauge that makes the calculation easier
2
The compactification of $\mathbb{R}^3$ to $\mathbb{S}^3$ is a way of imposing boundary conditions on the gauge field $A$ at spatial infinity. This is done by requiring that the gauge transformation $U$ that relates $A$ to the pure gauge configuration $A = i(dU(x))U^{-1}(x)$ converges to a constant matrix at infinity. This means that $U$ is a map from $\mathbb{R}^3 \cup \{\infty\}$ to the gauge group $G$, which is topologically equivalent to a map from $\mathbb{S}^3$ to $G$. Therefore, $U$ defines an element of the homotopy group $\pi_3(G)$, which classifies the inequivalent vacua of the theory.
The reason why this compactification is needed is that without it, there would be no non-trivial vacua in the theory. This can be seen by noting that any gauge transformation U that is well-defined on $\mathbb{R}^3$ can be continuously deformed to the identity transformation by shrinking it to a point at infinity. This means that any two gauge fields $A$ and $A'$ that are related by such a $U$ are actually equivalent, and there is no topological distinction between them. Therefore, the only vacuum of the theory would be the trivial one, where $A = 0$.
The temporal gauge is a convenient choice of gauge fixing that simplifies the analysis of the compactification. In this gauge, one sets $A_0 = 0$, which means that the gauge field $A$ depends only on the spatial coordinates $x$. This makes it easier to impose the boundary condition at infinity, since one does not have to worry about the time dependence of $U$. Moreover, in this gauge, one can define a conserved charge $Q$ that is proportional to the Chern-Simons number of each time slice $Q = \frac{1}{16\pi^2} \int d^3 x \text{Tr}(F_{0i} \tilde{F}_i)$ . This charge $Q$ is related to the homotopy group $\pi_3(G)$ by the fact that it changes by an integer when an instanton or an anti-instanton crosses a given time slice. Therefore, $Q$ labels the different vacua of the theory.
3
The condition $U(|\vec x|→∞)→1$ is not a necessity of the Hamiltonian quantization. It is a requirement that is imposed on the gauge field in order to ensure that the Gauss law constraint holds. The Gauss law constraint is a consequence of the gauge symmetry of the Yang-Mills theory. The requirement that $U(|\vec x|→∞)→1$ ensures that the Gauss law constraint holds at infinity and that the gauge field is well-defined everywhere.
4
The instanton solutions are classical solutions of the Euclidean Yang-Mills equations that have a finite and non-zero action. They can be interpreted as tunneling events between different vacua of the theory, which are labeled by the Chern-Simons number, a topological invariant. The instanton solutions are characterized by their winding number, which is an integer that measures how many times they wrap around the gauge group G.
The winding number of an instanton solution can be calculated by using the following formula:
$w=\frac{1}{24π^2}\int d^4x Tr(F_{μν}F~_{μν})$
where $F_{μν}$ is the field strength tensor and $F~_{μν}$ is its dual. This formula is valid only if the gauge field $A$ vanishes at infinity, which means that $A$ is a pure gauge configuration at infinity. This condition ensures that the integral is well-defined and independent of the choice of gauge.
The compactification of $\mathbb{R}^3$ to $\mathbb{S}^3$ is a way of imposing this condition on the gauge field $A$ at spatial infinity. This is done by requiring that the gauge transformation U that relates A to the pure gauge configuration $A = i(dU)U^{-1}$ converges to a constant matrix at infinity. This means that $U$ is a map from $\mathbb{R}^3 \cup {\infty}$ to the gauge group $G$, which is topologically equivalent to a map from $\mathbb{S}^3$ to $G$. Therefore, $U$ defines an element of the homotopy group $pi_3(G)$, which classifies the inequivalent vacua of the theory.
The compactification of $\mathbb{R}^3$ to $\mathbb{S}^3$ is not purely physical and clear motivation enough, because it does not take into account the time dependence of the gauge field $A$. In order to find the instanton solutions, one needs to consider the full four-dimensional spacetime R4, which has a different topology than $\mathbb{R}^3$. The boundary of R4 is not $\mathbb{S}^3$, but rather $\mathbb{S}^4$, which is a four-dimensional sphere. Therefore, one needs to compactify $\mathbb{R}^4$ to $\mathbb{S}^4$, which requires a different boundary condition on $A$.
The boundary condition on $A$ for compactifying $\mathbb{R}^4$ to $\mathbb{S}^4$ is that A should be a pure gauge configuration not only at spatial infinity, but also at temporal infinity. This means that $U$ should be a map from $\mathbb{R}^4 \cup {\infty}$ to $G$, which is topologically equivalent to a map from $\mathbb{S}^4$ to $G$. Therefore, $U$ defines an element of the homotopy group $\pi_4(G)$, which classifies the instanton solutions of the theory.
The compactification of $\mathbb{R}^4$ to $\mathbb{S}^4$ is more complicated than the compactification of $\mathbb{R}^3$ to $\mathbb{S}^3$, because $\pi_4(G)$ is not always trivial. For example, if $G = SU(2)$, then $\pi_4(G) = \mathbb{Z}/2\mathbb{Z}$, which means that there are two types of instanton solutions: self-dual and anti-self-dual. These solutions have opposite signs for their winding number and their action. Thus, one needs to specify the sign of the winding number when looking for instanton solutions.
