The de Broglie soliton as a localized excitation of the action function

Guided by the formal analogy between the classical relativistic Hamilton–Jacobi equation and the dynamic equation for the premixed gas flame, a new class of time-dependent solutions for the relativistic quantum Hamilton–Jacobi equation, ( 1 / c 2 ) ( ∂ S / ∂ t ) 2 − ( ∇ S ) 2 = i ħ □ S + m 2 c 2 , is revealed. The equation is shown to permit solutions in the form of breathers (nondispersive oscillating/spinning solitons) displaying simultaneous particle-like and wave-like behavior adaptable to the properties of the de Broglie clock. Within this formalism the de Broglie wave acquires the meaning of a localized excitation of the action function, a complex-valued potential in configuration space. For a free non-spinning particle in the rest system the breathing action function reads, S = − m c 2 t − i ħ ln { 1 + α exp [ − i ( m c 2 / ħ ) t ] j 0 ( k r ) } , where j 0 ( k r ) = sin ( k r ) / k r , k = 3 ( m c / ħ ) , r = x 2 + y 2 + z 2 , and | α | is a parameter controlling the breather’s intensity. The problem of quantization in terms of the breathing action function and the double-slit experiment are discussed.