Complex Trajectories of Particles and "Battle of Tachyons" in Quantum Vacuum
Random trajectories in complex space are considered in the context of the quantum theory. Wave function is represented as effective probability density for complex trajectory. Interaction between complex and complex conjugate tra-jectories is considered as tachyon-antitachyon interaction. Vortices in complex space are introduced as a representation of such interaction. The Heisenberg's uncertainty relation reads: ∆p∆x ∼ h (1) where ∆p is uncertainty of momentum (p = mv-product of mass and velocity of particle), ∆x is uncertainty of coordinate and h is Planck's constant. Relation (1) seems to preclude the existence of trajectory: the more exactly we know location the more uncertain is velocity. Probability density of particle position is given by P = ΨΨ * , where Ψ(t, x) is the wave function and star indicates complex conjugate. For simplicity, we consider the one-dimensional case, but generalization is obvious. Schrödinger's equation has the form (see, for example, Ref. 1): ih∂Ψ/∂t = −(h 2 /2m)∂ 2 Ψ/∂x 2 + U (x)Ψ (2) Here U (x) is the potential energy of the particle in the external field, i is the imaginary unit. The strange behavior of particles is not associated with particular form of potential energy. So, we put U = 0 and consider the influence of the quantum vacuum on the motion of particle. Equation (2) becomes very simple: ∂Ψ/∂t = iq∂ 2 Ψ/∂x 2 , q ≡ h/2m (3) where q is the coefficient of imaginary diffusion (ID). Equation for Ψ * has the same form as (3) with minus in the right hand side. General solution of equation (3) in unbounded space can be obtain with use of Fourier transform: ∂ Ψ/∂t = −iqk 2 Ψ (4) where k is the wave number and Ψ(t, k) is the transform of Ψ(t, x). Solution of (4) is: Ψ(t, k) = Ψ 0 (k) exp{−iqk 2 t} (5)