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
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
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
Quantum mechanics utilizes a wavefunction W(x)=Sum over p a(p)exp(ipx) (in time independent cases) with exp(ipx) being what we call a free particle complex probability. In such a case, W(x) is an OR scenario of these various probabilities. We have argued in previous notes that exp(ipx) is the probability for a p impulse hit at x, even if the center-of-mass of the particle is not at x. This means that impulse hits may occur probabilistically in space as in the 2-slit example which requires adding two probabilities as an OR case for each slit. The point then seems to be that W(x) (or even its momentum frozen form W*(x)W(x) gives a profile in space of the number of interactions (various p impulse hits) at each x. Classically, the kinetic energy (nonrelativistic) is pp/2m. In a classical bound problem this kinetic energy is considered to exist (in calculations) at the center-of-mass, but forces are also said to be rendered at this point so pp/2m, the center-of-mass and the force hit point all coincide. In quantum mechanics, a p impulse hit and the center-of-mass do not coincide. If p is said to hit at a probabilistic x, then it seems that pp/2m should also be seen to exist at this same hit point. A physical interaction point verifies in a sense the location of pp/2m in a probabilistic manner, i.e. a(p) exp(ipx)/W(x) is the probability used to calculate KEaverage and this is based on impulse hit locations. By normalizing using W(x), one removes the relative number of average hits at each x and obtains a pure energy value at x. (This may then be used in a KEave(x)+V(x)=E equation.) On average W(x) has zero points, which means that if one considers W*(x)W(x) to represent the motion-frozen-out spatial distribution of hits, then no hits are rendered at certain x points (on average). One might argue then that the particle cannot exist at x, but the center-of-mass can exist anywhere. One might next argue that there is no kinetic energy at x, but that cannot be true because 1/2m d/dx d/dx W has the same 0 points as W and the two divide. Thus, there is average kinetic energy at all bound points and this represents an average based on impulse hits (exp(ipx)). Classically, if one has many particles in a potential, one may write density(x)V(x) to find the potential energy of the particles. This implies that the particles interact with V(x), e.g. neutrinos usually do not interact. Density(x) in a classical sense means that the center-of-mass and force hit point are both at x. In quantum mechanics, exp(ipx) represents the probability for a force hit point, W(x) represents an average and W*(x)W(x) represents a motion-frozen average hit point. W*(x)W(x)V(x) seems to represent the average potential energy of the particle at x because W*(x)W(x) is linked with interaction. Even if a particle is not present, however, there is energy V(x) in space. It seems that if W*(x)W(x)=0 at some x, then there is no average particle interaction at this point, but there is still average kinetic energy Sum over p a(p)pp/2m exp(ipx) / W(x) based on the probability for average probability hits. The key idea is that one normalizes probability by dividing by W(x) in the average kinetic energy calculation. W*(x)W(x) is normalized not at x, but integrating over dx and setting this value to 1. One has relative hit at x values represented by W*(x)W(x). For exp(-ipx)exp(ipx)=1, each dx has the same average hit value (with one not being able to distinguish details in hbar/p, i.e. this is a motion frozen result. The question then becomes: Why would one write: W*(x)W(x) KEave(x)? One might argue that this term occurs if one takes the quantum mechanical equation: KEave(x) +V(x) = En and multiplies by W*(x)W(x). One may ask: Does V(x)W*(x)W(x) have physical meaning? V(x) is the potential energy at x. The average interaction probability with V(x) at x should be W(x)V(x) and not W*(x)W(x)V(x). The latter is a classical mechanical idealization. If W*(x)W(x) is 3 at x1 and 2 at x2, then it is as if the particle interacts 1.5 times more at x1 and so if V(x1)=V(x2), then this approach assumes 1.5 the interaction energy at x1. KEave(x), however, is simply free particle energy (average) at x and one does not need to take this average value and multiply it by the relative number of interactions at x. W(x) has zero points meaning that there is no interaction at such a point on average. KEave(x) on the other hand is the average energy at x based on the particle kinetic energy presence pp/2m being at the point of an impulse hit. This does not mean that an interaction is occurring with V(x) at this point in this formalism. Even a free particle has an exp(ipx) probability when it is not interacting. The sum Sum over p a(p)exp(ipx) means that all of the interactions linked with x have occurred and one is making an OR calculation. KEave(x) is free particle kinetic energy at x and V(x) is potential energy (not interaction potential energy with the particle). Multiplying by W*(x)W(x) is a classical minded approach to creating a classical equation: density(x) (KEave(x)+V(x)) = E density(x) ((1)) Here KEave(x) would represent the average kinetic energy of a single particle and density(x) would be the average number of particles at x. density(x)KE(x) has a clear meaning. W*(x)W(x) KEave(x) on the other hand is different. KEave(x) is actually the classical kinetic energy value at x and so it is not clear why one would multiply it by W*(x)W(x). W*(x)W(x) represents the probability for an impulse hit at x in a p-frozen out picture. The notion of exp(ipx) (i.e. probability for a p hit at x) is already buried within the quantum calculation of KEave(x) (which yields the classical value). In order to find the physical meaning of W*(x)W(x) multiplied by KEave(x) or V(x) it seems that one has broken dx into equally sized segments which correspond to different detection hits of the particle. This is then somehow equated with the presence of the particle per unit time in dx which is then somehow rendered as the average energy at x multiplied by the detected presence at x through a steady rate of measurement/time. Thus, multiplying by W*(x)W(x) is not multiplying by a physical density.
The existence of precise particle trajectories in any quantum state is accounted for in a consistent way by allowing delocalization of the particle charge. The relativistic mass of the particle remains within a small volume surrounding a singularity moving along the particle trajectory. The singularity is the source of an electric displacement field. The field induces a polarization charge in the vacuum and this charge is equated with the charge of the particle. Under dynamic conditions a distributed charge density rho(x,t) is induced in the vacuum. The volume integral of the charge density is equal to the charge of the particle and is rigorously conserved. The charge density is derived from a complex-valued physical field psi(x,t) such that rho(x,t) = |psi(x,t)|^2. The position probability density is equated with the mean charge density. The mean field psi(x,t) for many sample realizations with a given energy E and potential V(x) is the sum of the individual fields. In order for the sum to be non-zero, the components in the spectral decompositions of the individual fields must be spatially coherent. The particle has a spin frequency given by Planck's relation hv = T - V + mc^2, where T is the kinetic energy, determined from the momentum p and V is a quantum potential such that E = T + V is conserved. The instantaneous phase of the spin is given by the phase of exp(ikx) in the spectral decomposition a(k) of psi(x,t). It is spatially coherent, due to the dependence on x. The momentum probability distribution is given by the squared magnitude of the coefficients a(k). The Schrodinger equation is derived by requiring local conservation of mean energy.
Quantum mechanics is concerned with measurement and we suggest that in a bound problem impulses due to momentum are the form of measurement which occurs in x space. The Lorentz invariant A= -Et+px, which is also the free particle relativistic or nonrelativistic action with v=x/t, associates energy = pp/2m with time and p with x. Creating eigenvalue equations yields id/dt exp(-iEt) = E exp(-iEt) and -id/dx exp(ipx) = p exp(ipx). P (momentum) is associated with physical impulses and with its own two dimensional probability distribution exp(ipx). E is associated with t and has its own distribution in time exp(-iEt) which implies that the same E holds for each x point without any notion of x probability associated with it. In classical physics, a momentum p exists in a tiny dx (dx→0). For exp(ipx), a p impulse may occur within hbar/p with different probabilities, so one cannot restrict the measurement of p to a tiny dx unless dx is of the order of hbar/p. Nevertheless energy exists at all x points. Given that p measurements occur through specific impulses at different x points with exp(ipx) probabilities, one must calculate an average energy using {Sum over p pp/2m a(p) exp(ipx)} / {Sum over p a(p)exp(ipx)} using the probability associated with the measurement of p in space i.e. exp(ipx). This adds to V(x) to yield a constant E at each x point. We suggest that weighted exp(ipx)s are used to calculate average energy at each x, because it is through impulse measurements that one determines p at different x values and hence the average energy. One may note that one does not calculate {Sum over p a(p)exp(ipx) p } / {Sum over p a(p)exp(ipx)}. An impulse hit is associated with a certain probability and this measurement in x yields information about the corresponding energy pp/2m. Each impulse hit also yields information about a time probability exp(-i ppt/2m), but only if the particle is knocked out of the bound system i.e. only if pp/2m is the only energy present, which is not the case for a single particle bound system.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.