Instantons are classical solutions of the Euclidean equations of motion (Wick-rotated, t → iτ) that interpolate between degenerate vacua.
For the double-well V(φ) = λ(φ²−v²)², the instanton solution is φ_inst(τ) = v·tanh(√(2λ)·v·(τ−τ₀)).
The tunneling amplitude per unit time: Γ ~ A · exp(−S_inst/ℏ), where S_inst = ∫dτ[(dφ/dτ)² + V(φ)] = (4/3)√(2λ)v³.
Multi-instanton configurations (instanton gas) split energy levels: ΔE = 2A·exp(−S₀/ℏ) — this is the origin of the mass gap and vacuum tunneling in QCD (θ-vacuum).