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arXiv:0707.3299

Random Shooting of Entangled Particles in Vacuum

Evgeny A. Novikov

Published Jul 23, 2007• 3 citations
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Abstract

The effect of random shooting of particles is considered on the basis of solution of the Schrodinger equation and in terms of the Wigner function. Two-particles description shows, in particular, that initial correlation leads to high velocities of particles. This could be a potential mechanism for obtaining energy. Evolution of the n-particle probability distribution is descibed analytically.

DOI
10.48550/arxiv.0707.3299

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A single particle bound wavefunction may be written as a Fourier series W(x)exp(-iEt) = exp(-iEt) Sum over p a(p) exp(ipx). We argued in previous notes (1) that exp(ipx) is a statistical free particle conditional probability whose modulus of 1 corresponds to P(x)=constant. For the case of a potential V(x), we argue it is stochastic and write it as: Sum over k Vk exp(ikx). The stochastic potential gives rise to a momentum distribution: P(p/x) = a(p) exp(ipx) / W(x) at each x. A resonance, however, seems to occur because writing W(x) and V(x) in the time-independent Schrodinger equation as Fourier series, yields: pp/2m a(p) + Sum over k Vk a(p-k) = E a(p) ((1)) Thus, a(p) is really a combination of other a(p-k) or, in other words, there is a correlation between different free particle momentum states. If such is the case for single particle interactions with a stochastic potential, we argue it is possible a correlation may exist for two particles. In other words, a stochastic potential may not necessarily produce two separate resonances for two quantum particles such as ((1)), but may instead form an overall resonance involving both momenta p1 and p2 “interacting or communicating” with Vk as in the case of entanglement. We attempt to investigate these ideas in this note.

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The effect of random shooting of particles is considered on the basis of solution of the Schrodinger equation and in terms of the Wigner function. Two-particles description shows, in particular, that initial correlation leads to high velocities of particles. This could be a potential mechanism for obtaining energy. Evolution of the n-particle probability distribution is descibed analytically.
The effect of random shooting of particles by tachyonic impulses is considered on the basis of solution of the Schrödinger equation. Vortices in complex space are introduced as a representation of this effect. Two-particle description shows, in particular, that initial correlation leads to high velocities of particles. This could be a potential mechanism for obtaining energy. Evolution of the n-particle probability distribution is described analytically. 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 start with the one-dimensional case, but the 3-D generalization is simple (see below). 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) 1 where k is the wave number and ˜ Ψ(t, k) is the transform of Ψ(t, x). Solution of (4) is: ˜Ψ(t, k) = ˜ Ψ0(k)exp{−iqk 2 t} (5) where zero indicates initial function. The inverse transform of (5) gives: Ψ(t, x) = (4πiqt) −1/2
We demonstrate how the interference effects for a quantum particle in the double-slit experiment can be described by classical probabilities. We investigate a classical field theory for a complex scalar field with probabilistic initial conditions. A central element are conserved charges leading to the concept of particles. These are statistical observables which describe properties of the probability distribution for field configurations. The conserved charges define subsystems for particle excitations of a vacuum state. The classical probability distribution for the one-particle subsystem can be expressed in terms of a complex wave function. The Liouville equation for the classical probability distribution implies that the time evolution of this wave function obeys the Schrödinger equation for a quantum particle in a potential. The potential arises from space-dependent external fields in the otherwise relativistic classical field theory. It can be chosen arbitrarily, realizing the typical quantum effects of interference, tunneling or discrete energy spectra.
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 trajectories 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) = ˜ Ψ0(k)exp{−iqk 2 t} (5) 1 where zero indicates initial function. The inverse transform of (5) gives: Ψ(t, x) = (4πiqt) −1/2
A single particle bound wavefunction may be written as a Fourier series W(x)exp(-iEt) = exp(-iEt) Sum over p a(p) exp(ipx). We argued in previous notes (1) that exp(ipx) is a statistical free particle conditional probability whose modulus of 1 corresponds to P(x)=constant. For the case of a potential V(x), we argue it is stochastic and write it as: Sum over k Vk exp(ikx). The stochastic potential gives rise to a momentum distribution: P(p/x) = a(p) exp(ipx) / W(x) at each x. A resonance, however, seems to occur because writing W(x) and V(x) in the time-independent Schrodinger equation as Fourier series, yields: pp/2m a(p) + Sum over k Vk a(p-k) = E a(p) ((1)) Thus, a(p) is really a combination of other a(p-k) or, in other words, there is a correlation between different free particle momentum states. If such is the case for single particle interactions with a stochastic potential, we argue it is possible a correlation may exist for two particles. In other words, a stochastic potential may not necessarily produce two separate resonances for two quantum particles such as ((1)), but may instead form an overall resonance involving both momenta p1 and p2 “interacting or communicating” with Vk as in the case of entanglement. We attempt to investigate these ideas in this note.