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
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
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.
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.
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.
We show that a backward stochastic system with an imaginary diffusion coefficient in complex space is equivalent to a real space quantum system if it obeys a Newtonian-like dynamical law and its final spatial distribution is constrained to be in real space. The ensemble averages of the position, momentum, angular momentum and energy of the stochastic processes are the quantum mechanical expectation values of the corresponding operators. The product of the mean-square deviations of the position and momentum of the stochastic process agreees with the corresponding uncertainty relation of quantum mechanics. The equation of motion is invariant under the combined transformations of changing the time t! ¡ t and taking the complex conjugate of the equation. The stochastic motion reduces to the classical motion in the limit where the de Broglie wavelengh of the particle vanishes. PACS. 03.65.Bz – Foundations, theory of measurement, miscellaneous theories. PACS. 02.50.Ga – Markov processes. PACS. 02.50.Cw – Probability theory.
Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.
The equation of motion for a particle moving in complex space is obtained and then solved. The interaction between the real and imaginary components of motion can produce what can be regarded as a wave motion in real space. The formation of interference patterns and polarization of a matter wave can be accounted for by the particle’s motion in complex space. The deterministic particle view of a free particle can then be reconciled with its dual probabilistic wave picture but the formulation is entirely in complex space.