Importance of "euclidization" of time for instanton solutions

Question

Suppose we have theory with $SU(N)$ local gauge invariance in $3 + 1$ dimensions:
$$
S = \int d^{4}x \left(-\frac{1}{4}\text{Tr}\left[ F_{\mu \nu}F^{\mu \nu}\right] + \frac{\theta}{32 \pi^{2}}\text{Tr}\left[F_{\mu \nu}\tilde{F}^{\mu \nu}\right]\right) \qquad (1)
$$

When we look for nontrivial solutions (called instantons) which leave $(1)$ finite, we perform Wick rotation and then work in euclidean time. Then, by the definition, instantons are nontrivial solutions $(1)$ in euclidean time signature which leaves $(1)$ finite.

Why do we need euclidean time? I naively think that it is necessary since we may represent the full set of such solutions only in euclidean signature, because only in euclidean time we may identify the manifold $M$ which is mapped on $SU(N)$ group (with euclidean time we may work with simple topology of $d-$dimensional spheres). If no, what is the true reason? If yes, why can't we identify corresponding manifold in Minkowski signature?

Answer 1

The shortest answer I can give:

There are no instantons in Minkowski signature!

But let me give a bit more detail. (First off, let me point out that distinction is not due to purely topological considerations (as you asked), since changing the signature only changes the metric, but not the topology.)

Based on your comment to Neumaier, I think it's best to first clarify the difference between instantons and (theta) vacua:

A Yang-Mills (theta) vacuum: this is for a given time-slice! More exactly a vacuum is a flat gauge field (i.e. purely gauge) on space -- usually $S^3$. Physically speaking there is only one such field, since they are all related by gauge transformation. Nevertheless, sometimes they can only be related by large gauge transformations. This is a gauge transformation $g^{-1} d g$ that cannot be continuously connected to zero. Any gauge transformation is defined by a function $g: S^3 \to SU(N)$, and whether it defines a large gauge transformation or not depends on its topological class in $\pi_3(SU(N))$. It turns out that the winding number is measured by $\int \textrm{Tr} (g^{-1} d g)^3$, also called a Wess-Zumino term. A more familiar notation is perhaps $S_\textrm{CS} = \int \textrm{Tr} (A dA + \frac{2}{3} A\wedge A\wedge A)$ (with $A = g^{-1} d g$). One can indeed prove that this Chern-Simons action is an integer for flat gauge fields. Note that this concept of vacuum is perfectly well-defined in Minkowski signature (since it is for a fixed time-slice).

A Yang-Mills instanton: this is a classical solution to the Yang-Mills action $S_\textrm{YM} = \int  \textrm{Tr} (F \wedge \star F)$, where we usually take spacetime to be $S^4$ (such that the integrals are well-defined). By classical solution we mean an extremum of the action. But we know that for classical solutions in Minkowski signature, we conserve the energy, yet since our field is defined on $S^4$ it means that far in the past our curvature is exactly zero. So if we start out with zero energy/zero curvature, curvature has to stay zero and the only solution can be $F = 0$. So we see there are no (non-trivial) instantons in Minkowski signature. There are however in Euclidean signature, since our usual notion of energy is not conserved there. (This is completely analogous to how instantons appear in the usual one-dimensional tunneling example: there are no classical solutions that connect the wells, however there are once we go to Euclidean signature.)

Now, it turns out the instantons and vacua are simply related: any instanton can be interpreted as a tunneling process between two vacua ⁽¹⁾. This is again analogous to tunneling between two one-dimensional potential wells: vacua are physical concepts (i.e. defined for Minkowksi signature), and there are no classical paths connecting them, however if we want to calculate tunneling rates it can be helpful to go to the unphysical Euclidean signature, in which case `classical' tunneling solutions appear, called instantons.

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^{(1) This follows from the fact that one can prove $\int \textrm{Tr} (F \wedge \star F) \geq |\int \textrm{Tr} (F \wedge F)|$ (and in fact a stronger statement at every point), which means the instantons are defined by $\star F = \pm F$. This means the Yang-Mills action for instantons is given by the second Chern number $ C_2 = \textrm{Tr} (F \wedge F)|$, which is known to be an integer (on closed manifolds), also called the instanton number. If we now take our spactime to be $S^3 \times \mathbb R$ instead of $S^4$, then since we know our curvature has to go to zero far in the past and the future, we know that for $t\to \pm \infty$ we have vacua with a certain topological number $S_\textrm{CS}^\pm$. Now since the total derivative of the Chern-Simons form gives the Chern class, by Stokes we get $S_\textrm{CS}^+ - S_\textrm{CS}^- = C_2$. So we see the instanton number specifies between which vacua we tunnel :)}

Comments

Thank you! Now I understand what was my mispoint in the understanding of instantons: I thought that they represent the vacua, and hence the energy of instanton is zero; but in fact, they defines vacuum state only at infinity, and thus only at infinity they represent the vacuum.

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Exactly!

Answer 2

Instantons in Euclidean (imaginary) time describe tunnelling phenomena in real time through analytic continuation. The simplest instances of this can be explained in terms of quantum mechanical models.See, e.g., the 1978 paper Real-time approach to instanton phenomena: (I) by de Vega et al., which shows via WKB approximation how (in the QM case) the Euclidean description, the real time description, and the imaginary time path integral are related. 

This is then extended heuristically to QFT. I don't know of any reasoanbly rigorous treatment in the quantum field case. But you can see the arguments used in Section II of the survey article Instantons in QCD.

Comments

Thank you for the answer! But some is not clear for me. There is $\theta$-vacua in gauge theories, which is the superposition of topological $|n\rangle$ vacuum states. This state is defined by instanton solution with winding number $n$. If we want only to find the solution which corresponds to the given winding number, it is not (am I right?) important that instantons describe tunneling phenomena, and I am stuck with importance of Wick rotation.

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@NAME_XXX: At fixed winding number one doesn't need instantons. Winding numbers are a purely topological feature of $SU(3)$, independent of space and time. 

But physics is instead at fixed $\theta$, and one needs them.