is derived and expanded in power series of the charge of the particle. Then, solutions of the zeroth order in the charge of the iterative equations for

Q(x,p)

are considered. For them, it follows that the space-time probability density

ρ(x)

and the function

S(x)

which gradient defines the mean value of the momentum at the space time point

x

, define a complex function

ψ(x)

which exactly satisfies the Klein-Gordon (KG) equation. It is argued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. It follows that the total number of particles conserves when

ψ(x)

composed alternatively of pure positive or negative energy modes. In addition, solutions for the joint distribution function in lowest order, satisfying the positive condition are also found. They suggests a possible link with the Couder's experiments on droplet movements over oscillating liquid surfaces.

DOI: 10.48550/arxiv.1709.00355

Related papers

A relativistic formulation of the de la Peña-Cetto stochastic quantum mechanics

Alejandro Cabo Montes de, Alfredo González Lezcano

A covariant generalization of a non-relativistic stochastic quantum mechanics introduced by de la Peña and Cetto is formulated. The analysis is done in space-time and avoids the use of a non-covariant time evolution parameter in order to search for Lorentz invariance. The covariant form of the set of iterative equations for the joint coordinate and momentum distribution function Q(x; p) is derived and expanded in power series of the coupling of the particle with the stochastic forces. Then, particular solutions of the zeroth order in the charge of the iterative equations for Q(x; p) are considered. For them, it follows that the space-time probability density ρ(x) and the function S(x) which gradient defines the mean value of the momentum at the space time point x, define a complex function ψ(x) which exactly satisfies the Klein-Gordon (KG) equation. These results for the zeroth order solution reproduce the ones formerly and independently derived in the literature. It is alsoargued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. Other solutions for the joint distribution function in lowest order, satisfying the positive condition are also presented here. The are consistent with the assumed lack of stochastic forces implied by the zeroth order equations. It is also argued that such joint distributions, after considering the action of the stochastic forces, might furnish an explanation of the quantum mechanical properties, as associated to ensembles of particles in which the vacuum makes such particles behave in a similar way as Couder’s droplets moving over oscillating liquid surfaces. Some remarks on the solutions of the positive joint distribution problem proposed in the Olavos’s analysis are also presented.A covariant generalization of a non-relativistic stochastic quantum mechanics introduced by de la Peña and Cetto is formulated. The analysis is done in space-time and avoids the use of a non-covariant time evolution parameter in order to search for Lorentz invariance. The covariant form of the set of iterative equations for the joint coordinate and momentum distribution function Q(x; p) is derived and expanded in power series of the coupling of the particle with the stochastic forces. Then, particular solutions of the zeroth order in the charge of the iterative equations for Q(x; p) are considered. For them, it follows that the space-time probability density ρ(x) and the function S(x) which gradient defines the mean value of the momentum at the space time point x, define a complex function ψ(x) which exactly satisfies the Klein-Gordon (KG) equation. These results for the zeroth order solution reproduce the ones formerly and independently derived in the literature. It is alsoargued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. Other solutions for the joint distribution function in lowest order, satisfying the positive condition are also presented here. The are consistent with the assumed lack of stochastic forces implied by the zeroth order equations. It is also argued that such joint distributions, after considering the action of the stochastic forces, might furnish an explanation of the quantum mechanical properties, as associated to ensembles of particles in which the vacuum makes such particles behave in a similar way as Couder’s droplets moving over oscillating liquid surfaces. Some remarks on the solutions of the positive joint distribution problem proposed in the Olavos’s analysis are also presented.

Revista Mexicana de FísicaMar 14, 20180 citations

On the stochastic mechanics of the free relativistic particle

Michele Pavon

Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.

arXiv:quant-ph/0108139Journal of Mathematical PhysicsOct 1, 200115 citations

Quantum mechanics without quantum potentials

Adam Brownstein

The issue of non-locality in quantum mechanics can potentially be resolved by considering relativistically covariant diffusion in four-dimensional spacetime. Stochastic particles described by the Klein-Gordon equation are shown to undergo a classical diffusion process in spacetime coordinates, which is seen by transforming the quantum Cauchy-momentum equations to a Lagrangian frame of reference. Since the quantum potential term is removed under this transformation, the equations for momentum propagation along particle trajectories assume a classical form. A local stochastic de Broglie-Bohm interpretation for the Klein-Gordon system can subsequently be derived. We also introduce the concept of momentum equivariance to replace the second-order Bohm-Newton equations of motion, which break down due to non-linear terms of the stochastic Lagrangian derivative.The issue of non-locality in quantum mechanics can potentially be resolved by considering relativistically covariant diffusion in four-dimensional spacetime. Stochastic particles described by the Klein-Gordon equation are shown to undergo a classical diffusion process in spacetime coordinates, which is seen by transforming the quantum Cauchy-momentum equations to a Lagrangian frame of reference. Since the quantum potential term is removed under this transformation, the equations for momentum propagation along particle trajectories assume a classical form. A local stochastic de Broglie-Bohm interpretation for the Klein-Gordon system can subsequently be derived. We also introduce the concept of momentum equivariance to replace the second-order Bohm-Newton equations of motion, which break down due to non-linear terms of the stochastic Lagrangian derivative.

arXiv:2401.04091Jan 8, 20240 citations

Covariant Relativistic Quantum Mechanics Analysis of a Linearly Accelerated Scalar Particle

Karol McDonald

A covariant formalism of Relativistic Quantum Mechanics is demonstrated, through it's de- velopment and application. The Relativistic Case is shown to follow a similar structure to the established Non-Relativistic formalism. Reasons for preferring the new covariant formalism over the established method are presented. Solutions to the case of a scalar particle in a one-dimensional field are presented. The Relativistic Energy Eigenfunction is derived. Results are generated from initial Gaussian states via a Green's Function method. A Green's Function for the system is derived and applied. The solution to the Quantum System is shown to follow a scaled version of the classical path. The equivalent system in the Non-Relativistic case is also analysed. The Energy Eigenfunc- tion for the time-independent Schrodinger Equation is derived. A Green's Function is derived for the time-dependent SchrÄdinger Equation and solutions to the system are found using intial Gaussian States. A short demonstration of the established method, that of the Klein-Gordon equation, is provided. Some deficiencies in the Klein-Gordon equation are shown. These deficiencies are shown to not exist in the covariant formalism.A covariant formalism of Relativistic Quantum Mechanics is demonstrated, through it's de- velopment and application. The Relativistic Case is shown to follow a similar structure to the established Non-Relativistic formalism. Reasons for preferring the new covariant formalism over the established method are presented. Solutions to the case of a scalar particle in a one-dimensional field are presented. The Relativistic Energy Eigenfunction is derived. Results are generated from initial Gaussian states via a Green's Function method. A Green's Function for the system is derived and applied. The solution to the Quantum System is shown to follow a scaled version of the classical path. The equivalent system in the Non-Relativistic case is also analysed. The Energy Eigenfunc- tion for the time-independent Schrodinger Equation is derived. A Green's Function is derived for the time-dependent SchrÄdinger Equation and solutions to the system are found using intial Gaussian States. A short demonstration of the established method, that of the Klein-Gordon equation, is provided. Some deficiencies in the Klein-Gordon equation are shown. These deficiencies are shown to not exist in the covariant formalism.

Jan 1, 20090 citations

Relativistic Quantum Brownian Motion

Roumen Tsekov

where m is the particle mass at rest, c is the speed of light, p is the particle momentum and U is a potential. The root in Eq. (1) complicates substantially the way of quantizing the relativistic motion. In the literature there are two important solutions of this problem: the Klein-Gordon and Dirac equations. Their complex mathematical structures and some problems with definition of the probability density make it difficult to extend these equations for description of relativistic quantum Brownian motion. Moreover, it is not clear if the non-relativistic energy ˆ t E i   and momentum p i   operators, used in quantizing of Eq. (1), are valid in the relativistic quantum mechanics as well. The present paper proposes an alternative approach via a relativistic Wigner-Klein-Kramers equation. The quantum Brownian motion is described by the Madelung quantum hydrodynamics [4] coupled to the Breit-Fermi Hamiltonian. According to the relativistic dynamics the stochastic Langevin equation for a classical relativistic particle moving in a classical non-relativistic environment acquires the formwhere m is the particle mass at rest, c is the speed of light, p is the particle momentum and U is a potential. The root in Eq. (1) complicates substantially the way of quantizing the relativistic motion. In the literature there are two important solutions of this problem: the Klein-Gordon and Dirac equations. Their complex mathematical structures and some problems with definition of the probability density make it difficult to extend these equations for description of relativistic quantum Brownian motion. Moreover, it is not clear if the non-relativistic energy ˆ t E i   and momentum p i   operators, used in quantizing of Eq. (1), are valid in the relativistic quantum mechanics as well. The present paper proposes an alternative approach via a relativistic Wigner-Klein-Kramers equation. The quantum Brownian motion is described by the Madelung quantum hydrodynamics [4] coupled to the Breit-Fermi Hamiltonian. According to the relativistic dynamics the stochastic Langevin equation for a classical relativistic particle moving in a classical non-relativistic environment acquires the form

Jan 1, 20122 citations

Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis

Gerald Kaiser

The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube T in complex spacetime. For Klein-Gordon particles, we interpret T as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps $e_z: f\to f(z)$ of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because the time component of the conserved "probability current" can attain negative values even for positive-energy solutions. We show that this problem is solved very naturally in complex spacetime, where $|f(x-iy)|^2$ is interpreted as a probability density on all 6D phase spaces in T which, when integrated over the "momentum" variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps $e_z$ are spinor-valued relativistic coherent states. For free quantized Klein-Gordon and Dirac fields, the above formalism extends to n-particle/antiparticle coherent states whose scalar products are Wightman functions. The 2-point function plays the role of a reproducing kernel for the one-particle and antiparticle subspaces.The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube T in complex spacetime. For Klein-Gordon particles, we interpret T as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps $e_z: f\to f(z)$ of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because the time component of the conserved "probability current" can attain negative values even for positive-energy solutions. We show that this problem is solved very naturally in complex spacetime, where $|f(x-iy)|^2$ is interpreted as a probability density on all 6D phase spaces in T which, when integrated over the "momentum" variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps $e_z$ are spinor-valued relativistic coherent states. For free quantized Klein-Gordon and Dirac fields, the above formalism extends to n-particle/antiparticle coherent states whose scalar products are Wightman functions. The 2-point function plays the role of a reproducing kernel for the one-particle and antiparticle subspaces.

arXiv:0910.0352Oct 2, 20097 citations

Q(x,p)

is derived and expanded in power series of the charge of the particle. Then, solutions of the zeroth order in the charge of the iterative equations for

Q(x,p)

are considered. For them, it follows that the space-time probability density

ρ(x)

and the function

S(x)

which gradient defines the mean value of the momentum at the space time point

x

, define a complex function

ψ(x)

ψ(x)

e_z: f\to f(z)

of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because the time component of the conserved "probability current" can attain negative values even for positive-energy solutions. We show that this problem is solved very naturally in complex spacetime, where

|f(x-iy)|^2

is interpreted as a probability density on all 6D phase spaces in T which, when integrated over the "momentum" variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps

e_z

are spinor-valued relativistic coherent states. For free quantized Klein-Gordon and Dirac fields, the above formalism extends to n-particle/antiparticle coherent states whose scalar products are Wightman functions. The 2-point function plays the role of a reproducing kernel for the one-particle and antiparticle subspaces.