Special relativity, originating from Einstein’s seminal 1905 paper "On the Electrodynamics of Moving Bodies" [1], is a theory that fundamentally reshapes our understanding of space and time. Its development was spurred by empirical observations and Maxwell’s equations of electromagnetism [2,3,4,5], which were refined by Oliver Heaviside in the 19th century [6]. Maxwell’s equations notably imply that electromagnetic signals propagate at the constant speed of light c in vacuum, suggesting that light behaves as an electromagnetic wave. Building on this, Albert Einstein [1,4,5] formulated special relativity, positing that c represents the maximum achievable velocity in the universe. According to this theory, nothing—whether an object, message, signal (even non-electromagnetic), or field—can surpass the speed of light in a vacuum. This principle introduces the concept of retardation: any change made by someone at a distance R from an observer will take at least a retardation time of $\frac{R}{c}$ for the observer to become aware of it. Consequently, actions and their reactions cannot occur simultaneously due to the finite propagation speed of signals.

Newton’s laws of motion comprise three fundamental principles that form the cornerstone of classical mechanics. These laws delineate the connection between a body, the forces acting upon it, and its resulting motion while introducing the concept of force into physics. Initially formulated by Isaac Newton in his work "Philosophiae Naturalis Principia Mathematica" (Mathematical Principles of Natural Philosophy) [7,8], first published in 1687, these laws have enduring significance in physics. Of particular interest here is Newton’s third law, which asserts that when one body exerts a force on the other body, the other body concurrently exerts a force of equal magnitude but opposite direction on the first body. This concurrent force cannot be accommodated with the principle of retardation, because a change done in one subsystem cannot be possibly affect another subsystem which is at a distant R before a duration of $\frac{R}{c}$ passes.

In the beginning of the second quarter of the twentieth century the "new" quantum mechanics was introduced [9]. Schrödinger suggested his famous quantum wave equation and soon Pauli [10] and Dirac [11] suggested corrections, motivated by experiment and the desire to accommodate quantum and relativistic theories which imply symmetry under the Lorentz transformations. This new formalism of mechanics rejected the concept of force and embraces the concept of potential instead (scalar and vector electromagnetic potentials). The gauge freedom in potential definition was thus connected to the (total) phase freedom of the wave function. Of course the potentials themselves are derived from electromagnetic theory and thus must be retarded.

It was clear that there must a connection between the classical and quantum levels of reality, and indeed such a connection was suggested by Ehrenfest [12] formalism which reintroduce the concept of force as an expectation value (to be discussed below) and also by Bohm [13,14,15] and the realistic school of quantum mechanics which suggested that one can write a Newton’s second law equation for a quantum particle that has a trajectory (regardless if measure or not). However, one needs to add a nonlocal quantum force to the electromagnetic forces in the Bohm formalism. As the quantum forces do not obey Newton’s third law and moreover they do not depend on the separation distance between particles and allow action at a distance in a very counterintuitive fashion. It follows that the Bohmian approach to quantum mechanics leads to an even stronger violation of Newton’s third law than retardation alone.

Newton’s third law stipulates that in a system unaffected by external forces, the total sum of forces is zero. This principle has garnered numerous experimental confirmations, solidifying its status as a fundamental tenet of the physical sciences. However, it’s evident that action and reaction cannot occur simultaneously due to the finite speed of signal propagation. Moreover, a quantum potential generates a force that does satisfy the classical "third law". Therefore, while Newton’s third law holds true in many practical scenarios owing to the high velocity of signal propagation, and the macroscopic dimensions of classical objects, it cannot be deemed entirely accurate in an exact sense. Consequently, the total sum of forces within a system cannot remain zero at all times. And thus the Newton’s first law demanding that a body will continue with the same velocity unless affected by an external force is also not strictly satisfied.

Current locomotive systems typically rely on coupled material components, wherein each component gains momentum equal and opposite to that gained by the other. A classic example is a rocket, which expels gas to propel itself forward. However, relativistic effects suggest an alternative type of propulsion system that involves both matter and field, rather than two distinct material elements. Initially, it may seem that the material body gains momentum, thus violating momentum conservation. However, it can be demonstrated [16] that an equivalent amount of momentum is imparted to the field, ensuring total momentum conservation. This phenomenon arises from Noether’s theorem, which asserts that systems possessing translational symmetry conserve momentum. While individual components of the system (matter or field) may not exhibit translational symmetry, the entire physical system composed of both matter and field remains invariant under translations. Feynman [5] elucidates a scenario involving two charges moving orthogonally, seemingly challenging Newton’s third law as the forces induced by the charges do not cancel each other. However, this apparent contradiction is resolved by recognizing that the momentum gained by the two-charge system is transferred to the field.

A relativistic engine is characterized by a system where the material center of mass is set in motion through the interaction of its constituent material components. These components may either move relative to each other or be fixed within a rigid framework. However, our focus lies solely on the motion of the center of mass. It’s important to note that a relativistic engine enables motion along all three axes, including vertical movement. Unlike conventional engines, it lacks moving parts and doesn’t require traditional fuel consumption, thereby eliminating carbon emissions. Instead, it operates by consuming electromagnetic energy, which can be conveniently supplied by sources like solar panels. This makes the relativistic engine particularly well-suited for space travel, where a significant portion of the spacecraft’s volume is typically allocated for fuel storage.

In our present study, we make the assumption that the medium’s magnetization and polarization are negligible. Consequently, we do not take into account corrections to the Lorentz force as proposed by Mansuripur [17]. Griffiths and Heald [18] highlighted that Coulomb’s and Biot-Savart’s laws govern the configurations of electric and magnetic fields exclusively for static sources. To extend the applicability beyond static scenarios, time-dependent generalizations of these laws, as described by Jefimenko, have been utilized. These generalized laws enable the investigation of Coulomb and Biot-Savart formulas in dynamic contexts, stepping beyond the constraints of static conditions. This is true when the sources (charge & current densities) are either classical or quantum.

In a previous study, we employed Jefimenko’s equation [4,19], to investigate the force interaction between two current loops [20]. This research was subsequently expanded to explore the forces between a current-carrying loop and a permanent magnet [21]. Since the device operates for a finite duration, it acquires mechanical momentum and energy. Consequently, the question arises whether we must relinquish the principles of momentum and energy conservation. The issue of momentum conservation was addressed in prior research [16]. Additionally, discussions in other studies delved into the exchange of energy between the mechanical components of the relativistic engine and the electromagnetic field [22]. Notably, it was demonstrated that the total electromagnetic energy expenditure exceeds the kinetic energy gained by the relativistic motor by a factor of six. Furthermore, these studies examined how energy might be radiated from the relativistic engine device if the coils are improperly configured.

In previous analyses, the assumption was made that the bodies under study were macroscopically natural, implying an equal number of electrons and ions in every volume element. However, later we relaxed this assumption and considered charged bodies [23]. Therefore, we investigated the implications of charge on the feasibility and behavior of a potential electric relativistic engine. This yielded much higher momentum and force than a non charged motor, however, the limitations of dielectric breakdown resulted in still impractical devices.

The aforementioned limitations have led us to consider utilizing the high charge densities found at the microscopic scale [24], such as those present in ionic crystals. This concept was explored in a prior publication, where we calculated the remarkably high charge densities and current densities at the atomic level. Our findings indicated that an isolated hydrogen atom, whether in a ground or excited state, does not yield momentum for a relativistic motor. However, this changes when the atom interacts with other atoms or particles, or when the electron within the atom is in a non-eigenstate state. As a result, we proposed two simplistic forms for a wave function that could potentially lead to advantageous gains for a relativistic engine: a wave packet within a hydrogen atom and an eigenstate within a simple molecule that introduces a static electric field with broken spherical symmetry.

Thus, to obtain a practical relativistic engine it is needed to manipulate matter at subatomic levels. In a previous paper [25] we investigated two ways of doing so one that is related to free electrons and the other to confined electrons. While we started with a classical description of the problem we could not and did not ignore the fact that on the atomic level a quantum description is necessary. It was also shown that the quantum effects are more significant for confined electrons in comparison with free electrons.

Previous papers relied on a classical description of a two body system, in which the two bodies interacted through retarded electromagnetic fields. When quantum effects were considered this was only done in the case of a classical system (the hydrogen nucleus) interacting with a quantum system (the electron), in this case classical formulae were use in which case the electron charge density and current density were derived from its wave function (they were taken to be proportional to the probability density and probability current density without any justification) and plugged into a classical force equation. In the current paper we would like to discuss the case in which the two bodies interacting are both quantum and wether it is justified to compare probability densities and current densities.

We will start by reminding the readers on the source of retardation effects (Maxwell equations). We shall briefly discuss the classical description of a system of particles which will be followed by a quantum description. We shall attempt two possible connections of the classical and quantum worlds, one through the Ehrenfest theorem the other through the formalism of Bohm. Finally we shall discuss the conservation of momentum showing that in either case the linear momentum of a closed system will not be conserved, an effect which can be rectified in the macroscopic level only through the concept of field momentum.

Electromagnetism is described by a set of the four Maxwell equations (in MKS units): in which $\overrightarrow{\nabla }$ has a standard meaning in vector analysis, $\overrightarrow{E}$ is the electric field, $\overrightarrow{H}$ is the magnetic field, $\overrightarrow{D}$ is the displacement field and $\overrightarrow{B}$ the magnetic flux density. Also $\rho$ is the charge density, and $\overrightarrow{J}$ is the current density. In electromagnetic theory those quantities are assumed to be given and cannot be determined from the theory itself. In the following sections we shall try to derive them from both classical and quantum mechanics.

In vacuum we have simple relations between the electric and displacement fields: in which ${\epsilon }_0$ is the vacuum susceptibility. And also: in which ${\mu }_0$ is the vacuum permeability. Thus in vacuum we may write the last two equations in a form that depends only on $\overrightarrow{E}$ and $\overrightarrow{B}$: in which the velocity of light in vacuum is defined as: $c\equiv \frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$. The first two equations (equation (1) and equation (2)) that do not depend on the charge and current densities (or the medium) can be easily solved in terms of scalar $\Phi$ and vector $\overrightarrow{A}$ potentials:

We can now solve equation (7) and equation (8) for the scalar and vector potentials and obtain the results [4] in terms of the retarded expressions:

To obtain the above solutions we must demand that the scalar and vector potentials satisfy the Lorentz gauge conditions:

This is possible since we have gauge freedom in the choice of vector and scalar potentials. The Lorentz gauge condition is guaranteed to be satisfied for all times due to equation (11) and equation (12) and since charge is conserved:

(See appendix A of [23]). The solutions for the electric field and magnetic flux density are thus given by Jefimenko’s retarded fields [4,19,23] by inserting equation (11) and equation (12) into equation (9) and equation (10):

The potentials can also be used to formulate the electromagnetic problem in a variational form in which the action is given as:

In the above the Lagrangian density of the field is:

And the Lagrangian density of the interaction is: if $\rho$ and $\overrightarrow{J}$ do not depend on the electromagnetic potentials, this is true for a system of classical particles but not in the quantum to be discussed in the following sections.

We emphasize that electromagnetic theory does not suffice to determine what $\overrightarrow{J}$ and $\rho$ are, but is certainly sufficient to determine that the fields must be retarded.

The classical picture of the world, describes it as the arena in which a very large number (say N) of point particles interact. Modern physics recognizes only four possible types of interactions: electromagnetic, gravitational, strong nuclear and weak nuclear. The nuclear interactions are only important for processes which happen to occur in the nucleus or in high energy processes, hence it is not important for chemistry, biology and most of our daily life. Nevertheless the description of those interactions is Lorentz invariant hence they suffer retardation non the less. We shall not consider this any further in the current paper.

Hence we are left with two possible types of interaction electromagnetic and gravitational, the second is much weaker than the first and needs to be considered only for astronomical bodies, thus we will also neglect gravity. We are left with only the electromagnetic interaction. The effect of an electromagnetic field on the classical particle i is given through the Lorentz force and Newton’s second law (we assume that the particle is not relativistic that is its velocity is small with respect to c):

In the above $m_i$ is the particles mass and $q_i$ is the particles charge. ${\overrightarrow{x}}_i$ designates the location of the particle with respect to the origin of an inertial frame, ${\overrightarrow{v}}_i$ is the particle velocity and ${\overrightarrow{a}}_i$ is the particles acceleration. The (retarded) fields $\overrightarrow{E}$ and $\overrightarrow{B}$ are evaluated at the particles momentary location. A variational description of the above system can be derived from the action:

In the above we have a sum of kinetic Lagrangian: and an interaction Lagrangian:

Notice, however, that according to Maxwell’s theory (equation (17) and equation (19)) the interaction Lagrangian should be written as:

To accommodate simultaneously those two forms, it seem that one should adopt the following charge density and current density definitions: in which ${\delta }^3$ is a three dimensional Dirac delta function. Thus the classical world view resolves the sources of the electromagnetic field. However, the price to be paid is introducing unphysical charge and current densities which become infinite at various points of space in which the particles happen to be. This defies physical intuition that demands that every physical quantity must be finite in every point of space. It also shows through equation (15) and equation (16) that the interaction of classical particles with each other is not immediate but retarded. We conclude this section by reminding the reader that the total classical action of field and particles is:

And thus the canonical momentum for each particle: and the classical linear momentum for each particle: are not the same unless $\overrightarrow{A}=0$.

A quantum description of an N particles system involves a complex wave function $\Psi ({\overrightarrow{x}}_1,{\overrightarrow{x}}_2\dots ,{\overrightarrow{x}}_N,t)$ and solves the Schrödinger equation: we will consider as in the previous section low velocity particles without spin. In the above $i\equiv \sqrt{-1}$ is the imaginary number, and ℏ is Planck’s constant divided by $2\pi$. Generally speaking $\Psi$ cannot be written as a multiplication of functions of each ${\overrightarrow{x}}_i$ separately (the particles are generically correlated). Moreover, it is known that for an identical set of particles the function must be either symmetric to interchange of particles (bosonic case) or antisymmetric to the interchange of particles (fermionic case). Some authors write the hamiltonian $\widehat{H}$ ([14] equation 7.1.1) in the form:

However, this form (which is rather abstract) hides the source of the interaction which as we saw in the previous section is mainly electromagnetic. It also devoids the particle of the possibility to interact with the magnetic field. Thus we shall write the Hamiltonian in the form: which overcomes those objections.

In terms of the quantum Hamiltonian operator we may write a Lagrangian density in the form:

However, this density is a density in the $3N$ dimensional configuration space (not in the standard three dimensional space). Defining the configuration space coordinate:

We may write:

As we saw in the previous section the key to identifying the charge and current densities is identifying the interaction terms in the Lagrangian, this is also true in the quantum case. For this it will be useful to write the wave function in a "polar" form:

Using this form we may write the quantum Lagrangian equation (35) in the form: in which we omitted some boundary terms that do not effect the equations. From the above Lagrangian it is easy to see that the interaction term with the scalar potential takes the form:

Introducing the reduced square amplitude:

In which $d^{3N-3}{X}_i$ is a volume element including all coordinates of configuration space except the $i^{\mathrm{th}}$ coordinate. The Born propensity rule associates with $a^2$ a probability density function such that $\int d^{3N}X{a}^2=1$, thus in terminology of random variables theory $a_i^2$ is a marginal probability density function satisfying also: $\int d^3{x}_i{a}_i^2=1$.

Now we may write:

In which the second equation one is due a change of the name of the integration variable. Comparing equation (40) with equation (24) it follows that the quantum charge density should be of the form:

For a single body this is just ${\rho }_q=q{a}^2$ which justifies the quantum charge density used in [24,25]. For a classical body one needs to take the limit: the right limit follows only if all particles are classical. We notice that while the quantum charge density is finite (and hence physical), the classical charge density is just a useful mathematical construct that is used to simplify the calculations when one is far away from the support of the reduced amplitude ("the location" of the particle).

The interaction term with the vector potential takes the following form:

Now this term is second order in $\overrightarrow{A}$ (as are also Lagrangians of classical Eulerian fluids [26]). It follows from comparison with equation (24) that the quantum current density must be dependent on $\overrightarrow{A}$. We also recall that the Maxwell field equations are obtained by taking a variational derivative with respect to $\overrightarrow{A}$ of the electromagnetic Lagrangian hence we compare the quantum and electromagnetic interaction terms when varied with respect to $\overrightarrow{A}$:

We now define the quantum velocity field:

One should notice that in the generic quantum case this velocity field depends on the entire configuration space and not only on the coordinates of particle i, which is of course connected to the inherent non-locality of quantum mechanics. Nevertheless, Bohm’s interpretation of quantum mechanics [13,14] suggests that a quantum particle has a well defined trajectory and the trajectory satisfies the differential equation: for other interpretations of quantum mechanics (Copenhagen) this is just a vector field with unit of velocity. We emphasize that in order to obtain the trajectory in configuration space one must evaluate ${\overrightarrow{v}}_{qi}$ along this trajectory. We may thus write the variation of the quantum Lagrangian with respect to $\overrightarrow{A}$ as:

We can now define a reduced quantum velocity:

And write the variation as:

Maxwell theory demands that: this form is more general than equation (24) which assumes implicitly that $\overrightarrow{J}$ does not depend on the vector potential which is not the quantum case. Comparing equation (49) and equation (50) suggests the following form of the quantum current density:

This will reduce to the classical case only if the particle is localized $a_i^2(\overrightarrow{x},t)\to {\delta }^3(\overrightarrow{x}-{\overrightarrow{x}}_i\left(t\right))$ and the reduced quantum velocity takes a classical value. Notice, however, that wether the charge and current densities are classical or quantum there electromagnetic effect on their peers is retarded as follows from equation (15) and equation (16). Moreover, we may write a total quantum Lagrangian (describing both particles and field) in the form:

This signifies the fact that the quantum wave function despite its connection to epistemological constructs such as probability (Born propensity rules) is not less real than the electromagnetic field as the field cannot be more real than its sources which are wave function derived quantities. We stress that charge and current densities are not directly related to the particles coordinates even in Bohm’s interpretation, and writing classical type of charge and current densities (containing delta functions) even with Bohm’s quantum velocity field will not lead to a correct coupling between the wave function and electromagnetic field. The wave function although determined deterministically does not deterministically determine the trajectory of a single quantum particle (only its tendency to move to specific locations) and in this sense quantum mechanics is not deterministic, this is well known result of the two slit experiment [14]. However, according to Bohm [13] this indeterminism could be removed if we knew the location of the particle at any time $t_0$, in that case equation (46) will tell us the location of the said particle in any future or past time using our knowledge of the quantum mechanical wave function phase.

The quantum Lagrangian of a single particle can be interpreted through Fisher information as wad demonstrated in [26], however, this also true for a multiple particle system. To see the rewrite $L_q$ given in equation (37) in terms of the quantum velocity defined in equation (45):

According to [26,27] the Fisher information of a $3N$ dimensional random variable is:

Hence the Fisher information of the $3N$ dimensional random can be written as sum of components each of the form:

In terms of the Fisher information components we may write the quantum Lagrangian in the form:

In the case that the quantum system is composed of particles of identical mass $m_i=m$ (but not necessarily of identical charge) the above expression can be written simply as:

Also equation (53) artificially looks like the fluid Lagrangian + Fisher information of [26] (Equation (45)), it is remarkably different due to the inherent quantum correlations encapsulated in the $\int d^{3N}X$ symbol which requires taking an integral over all the $3N$ dimensional configuration space and not over three dimensional space as the fluid analogy might suggest. However, in the special case that the phase can be partitioned to a sum of phases each dependent only on the coordinates of just one particle that is: which is the case that the system wave function can be written as a multiplication of ("independent") single particle wave functions (but is more general). If equation (58) holds we may further simplify the Lagrangian. In this case: and we obtain:

The above can be written in terms of quantum three dimensional Lagrangian densities:

In which ${\mathcal{L}}_{qi}$ can be easily seen to be the Lagrangian density of a charged potential flow by introducing the notations for mass density: and flow potential:

Thus we obtain the potential flow Lagrangian density [26] in the form:

We stress that in this case every particle is associated with its unique flow (there is no one flow describing the dynamics of all particles). The particles are aware of one another only through the electromagnetic field.

For the purpose the current work an engine is a system that is able to move itself as a whole. This motion will be described by the motion of the systems center of mass which is defined as:

For classical and Bohmian particles it is meaningful to discuss the trajectories of the particles. Hence we may write:

The motion of a the classical center of mass can be written in terms of the motion of the particles of the system:

We can suppose without loss of generality that at $t=0$, ${\overrightarrow{R}}_{cm}\left(0\right)$ is the origin of axes, that is ${\overrightarrow{R}}_{cm}\left(0\right)=\overrightarrow{0}$, and that at the same initial time the engine is at rest that is ${\overrightarrow{v}}_{cm}\left(0\right)=\overrightarrow{0}$. We shall inquire, what are the conditions to put the system in motion. Those are obviously the conditions for the acceleration of the center of mass at the same time $t=0$ to be different from zero:

For only in this case can we expect to have at future time a velocity different from zero which will cause the engine to move. Thus: in we have used Newton’s second law. As the fundamental forces come about through interactions (that is due the effect of one particle on another), it follows that:

Thus:

Now according to Newton’s third law:

And it follows that:

So a closed system cannot move as a system, it needs an external force to cause motion. However, as we require Lorentz invariance of any physical interaction, all interaction must be retarded. And thus Newton third law can only be an approximation as was shown previously [23]. The total force between two subsystems 1 and 2 in an electromagnetic system calculated to second order in $\frac{1}{c}$ ([23] Equation (81)):

Let us choose the two systems to be two classical particles with indices 1 and 2. Now according to equation (25) and equation (26) those particles are associated with charge and current densities:

Plugging the above expression into equation (74) will yield after some tedious but straightforward computations: obviously every two electromagnetically interacting particles $i,j$ in a given system will yield similar expressions, which must be summed through equation (71) to obtain the center of mass acceleration which is not null. Now according to our previous discussion for the prevalent classical charged particle the most important force is electromagnetic while other interactions (gravitational & nuclear) can be neglected (but must be also retarded). We not that the force between any particles affecting the center of mass acceleration is partitioned into a force which reduces slowly as a function of inter particle distances (that is as $\frac{1}{R_{12}}$), and a force which depends on velocities and reduces in a Coulomb way (that is as $\frac{1}{R_{12}^2}$). As we assume slow moving particles this second contribution must be small. The contribution to the total momentum of those two interacting particles is:

And again we stress that in order to obtain the correct center of mass velocity of a system contributions from all interacting particles must be considered.

Now although it obvious that every conceivable system is a "retarded engine" in the sense that its center of mass must move, in many systems this effect is rather small. In fact it takes great care to design a relativistic engine with a measurable effect [20,21,23,24,25]. We will not give further examples of such devices, and refer the reader to the original literature.

We shall now study the motion of center of mass, in quantum mechanics as interpreted by Bohm. We recall that in Bohm’s quantum mechanics each point particle has a trajectory which can be calculated in principle by integrating equation (46). Thus the center of mass defined in equation (66) is well defined and moves with a velocity: in which ${\overrightarrow{v}}_{qi}$ is defined in equation (45). To prove the existence of an engine we need to show that provided that ${\overrightarrow{R}}_{cm}\left(0\right)=\overrightarrow{0}$ and ${\overrightarrow{v}}_{Bcm}\left(0\right)=\overrightarrow{0}$, one may have: $\frac{d{\overrightarrow{v}}_{Bcm}}{dt}{|}_{t=0}\ne \overrightarrow{0}$ without affecting the system externally. However:

Hence we need to study the derivative of the quantum velocity of each particle in the system. This is done through the Bohm representation of Schrödinger’s equation (30) which relies on a phase amplitude representation of the wave function given in equation (36). Assuming the Hamiltonian to be given in equation (32), we can split the complex Schrödinger’s equation (30) into two real equations (see Holland [14] equations (7.1.2) and (7.1.3) but there without a vector potential): which according to the Born propensity rule is a probability conservation equation, and the phase equation:

This motivates the definition of the quantum potential:

Taking the ${\overrightarrow{\nabla }}_i$ of equation (30) and making some tedious by straight forward manipulations one arrived at the quantum version of Newton’s second law (compare to equation (7.1.8) of [14]):

The quantum force ${\overrightarrow{F}}_{qi}$ can be partitioned into a part which is of a purely quantum origin and an electromagnetic part:

Inserting equation (83) into equation (79) will lead to:

The above equation leads to the concept of the quantum engine. It is obvious that the first term of the quantum force which is due to the quantum potential ${\overrightarrow{F}}_{qqi}$ cannot be written generally as a sum of contributions in which each originates from a specific particle j: rather the quantum potential Q will depend in a complicated way on the wave function of the entire system. In the case of a system of statistically independent (uncorrelated) particles: in this case: thus it depends only on the location of the particle i and not on the location of any other particle so equation (86) can be written as:

It follows that this force will even for a system containing a single particle. This force clearly does not confirm to Newton’s third law as even in the uncorrelated case it appears to be the action of the particle own wave function on its motion and does not take in most cases the classical interaction form. It follows that if one carefully prepares a quantum system (which means choosing initial conditions for the wave function and choosing the Hamiltonian) making sure that the quantum forces add up constructively one can obtain a quantum engine that is a self propelling system, even if retardation effects are negligible. Moreover, in the quantum case one may write the sources of the electromagnetic field in the forms given in equation (41) and equation (51). As the expressions for the fields are linear in the sources (see equation (15) and equation (16)). We may write:

It follows that the following expression for the quantum force is permissible:

Thus we may rewrite equation (85) in the form: we notice two important facts. The first fact is that the (reduced) quantum wave function of the particle may have an electromagnetic effect on the same particle, this does not happen in the classical case. The second fact is that their is no reason to think the field force generated by the sources associated with particle j will affect particle i in an opposite direction to the field force generated by the sources associated with particle j when they affect particle i. That is generically speaking:

The identity of $F_{qemij}$ and $-{\overrightarrow{F}}_{qemji}$ will reappear only in the classical limit and the neglect of retardation. Hence in the Bohm picture retardation is not needed to cause motion, one can generate motion by a purely quantum effect.

While the classical and Bohm pictures of reality ascribe a trajectory to a point particle the Copenhagen interpretation of quantum mechanics denies the existence of such a trajectory. And even if such a trajectory does exist it cannot be calculated. The it is not meaningful to discuss the trajectory of the center of mass either, because it is defined through the trajectories of the composing point particles. Thus for more conservative interpretations of quantum mechanics we can only discuss the attributes of the wave function. For examples we may ask where is the wave function centered? Or in the language of the theory of random variables, what is the expectation of the center of mass? This will be given in the form:

The above expression can be easily written in terms of the expectation values of marginal probabilities as follows:

In the classical limit: $a_i^2({\overrightarrow{x}}^{\prime },t)\to {\delta }^3({\overrightarrow{x}}^{\prime }-{\overrightarrow{x}}_i\left(t\right))$ and thus $<{\overrightarrow{R}}_{cm}>\left(t\right)\to {\overrightarrow{R}}_{cm}\left(t\right)$.

The calculation of the temporal derivative of $<{\overrightarrow{R}}_{cm}>\left(t\right)$ is most easily done using the Ehrenfest theorem [12,18]. The theorem asserts that for any quantum operator $A_o$ with an expected value:

This equality remains true:

The operators representing position and velocity of the particle i, as defined in Griffiths’ [18] work, are: in which the quantum momentum operator is associated with the canonical momentum given in equation (28) and not with the classical linear momentum defined in equation (29). Griffiths [18] derived a set of equations for a single particle without spin, his results can be trivially generalized using the multi particle Hamiltonian given in equation (32). By substituting the aforementioned operators into the Ehrenfest theorem equation (98):