Euclidean instanton actions and collective coordinates

Use positive Euclidean metric, orientation epsilon one two three four equal to plus one, and natural units with hbar equal to one. Gauge generators obey trace T a T b equal to one half delta a b. Put the coupling outside the action: F equals derivative terms minus i times the gauge commutator, with no coupling inside F. This differs from the preceding lesson’s canonical component convention and is declared anew to prevent importing an extra factor of g.

A Wick rotation is a prescription for relating suitable Lorentzian observables to Euclidean integrals, including boundary conditions and convergence information. The Euclidean action is not simply the real-time action evaluated on an arbitrary imaginary trajectory. For smooth finite-action Yang–Mills configurations on Euclidean four-space, assume the field approaches a pure gauge sufficiently rapidly at infinity. The resulting map on the three-sphere at infinity gives an integer topological sector under the usual regularity and bundle assumptions.

In four Euclidean dimensions the dual of a two-form can be applied twice to recover it. Completing the action into the square of F minus or plus its dual isolates the integral of F times its dual. The square is nonnegative, so its topological part gives a lower bound. Positive charge is saturated by a self-dual field and negative charge by an anti-self-dual field under the declared orientation. Such fields also satisfy the second-order equations through the Bianchi identity.

The path-integral sector weight contains exponential minus action and, with a theta angle, a phase proportional to Q. The sign of that phase is a convention fixed here by writing the weight explicitly. Different charge sectors can interfere in an amplitude or partition function; one must not square each factor independently and then call that the same observable. A tunneling splitting, a vacuum-energy theta dependence, and a metastable decay rate involve related methods but different boundary and contour problems.

A charge-one SU two instanton on four-space has four center coordinates, one size, and three gauge-orientation parameters in a framed convention where gauge transformations are fixed at infinity. Quotienting additional global gauge rotations changes the moduli description and its volume factors. Those directions generate zero modes rather than ordinary Gaussian fluctuations, so they must be replaced by collective-coordinate integrations with Jacobians. Counting parameters without stating the gauge quotient can give an apparent disagreement that is only conventional.

Quantum fluctuations renormalize the coupling and instanton measure, making size integration a physical issue rather than a harmless normalization. Small sizes probe short distances, while large sizes can leave the weakly coupled semiclassical regime. A formal dilute-instanton expression is not automatically a controlled description of uncompactified strongly coupled Yang–Mills theory. The saddle and its action are exact classical statements under their assumptions; dominance of that saddle and convergence of its collective-coordinate integral are additional questions.