Instantons are saddle-point solutions of Euclidean equations of motion that mediate quantum tunneling. Visualize the instanton path in imaginary time and compute tunneling amplitudes via steepest descent.
Instanton: classical solution in Euclidean time τ=it with x_cl(τ) = x_0·tanh((τ-τ_0)/Δτ) for double-well. Tunneling amplitude ~ exp(-S_E/ℏ) where S_E = ∫dτ[½(dx/dτ)² + V(x)] is the Euclidean action. Dilute instanton gas: ground state energy splitting ΔE = C·exp(-S_E/ℏ). At one loop: prefactor from fluctuation determinant (Bender-Wu 1969). QFT applications: vacuum structure of QCD (θ-vacuum), gauge theories.