We know that  the time derivative of the expectation value of the momentum in any state is the negative of the expectation value of gradient of the potential energy. Dividing both sides by the mass we obtain a quantity which corresponds to the acceleration in classical mechanics.

for this observables is much better to look to the hydrodynamic approach of quantum mechanics.

Of course it's possible to define an operator for the acceleration and calculate its matrix elements, in many ways. And many thought experiments have become real experiments, using ion traps, for instance. There's no problem of principle involved in calculating the correlation functions or the matrix elements of the acceleration, defined as, either the second time derivative of the position operator, or the first time derivative of the velocity operator. 

For example: the position operator, X, satisfies the-exact-equation (hbar=1)

i (dX/dt) = [X,H] = XH-HX => i(d^2 X/dt^2) =(d/dt)(XH-HX)= i [[X,H],H] =>

(d^2 X/dt^2) = [[X,H],H]

(eventually up to signs). Here it's assumed that the Hamiltonian is independent of time-if it isn't there are additional terms in the RHS. Therefore this expression provides a consistent definition of the acceleration operator, independent of any basis and from it the matrix elements of the operator d^2 X/ dt^2 can be computed, in any basis. The Heisenberg equation, which is the consistent quantization of the classical equation in phase space, shows that the commutator of a dynamical variable with the Hamiltonian is the consistent definition of the time derivative of the variable. So it's possible to prove that for any function, f, of the phase space variables, df/dt =-i [f,H]. (one should pay attention to factors of i, here, to ensure that the operators are Hermitian.)

This isn't anything new-it's a consequence of the properties of Poisson brackets in classical mechanics and of commutators in quantum mechanics. 

Motion is an intrinsic property of physical existence. But there is a problem about concept of acceleration in theoretical physics.

First of all, what we know about acceleration? And what is the definition of acceleration? In physics, acceleration is the rate at which the velocity of an object changes over time that in classical mechanics is given by Newton's second law with a=F/m. In special relativity an accelerating particle has a world-line which is not straight. This is not difficult to handle. The 4-vector acceleration can be defined as the derivative with respect to proper time of the 4-velocity. It is possible to solve the equations of motion for a particle in electric and magnetic fields, for example. Accelerating reference frames is a different matter. In general relativity the physical equations take the same form in any co-ordinate system. In special relativity they do not but it is still possible to use co-ordinate systems corresponding to accelerating or rotating frames of reference just as it is possible to solve ordinary mechanics problems in curvilinear co-ordinate systems. This is done by introducing a metric tensor.

http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

However, Newton and Einstein define the acceleration regardless of the structure of particles (in classical mechanics and relativity). This definition belongs to Newton era or macroscopic level era. It should be noted that the interaction between large objects (e.g. collision of two bodies) under the action of the quantum layer (in fact sub quantum level) has been done. Thus, according to quantum mechanics and mass–energy equivalence E=mc^2, we must redefine acceleration. It means we should review the relativistic Newton's second law.

By using review of relativistic Newton's second law we can explain zero point energy and vacuum energy directly (without using the uncertainty principle). Also, it is a powerful tool to answer the unanswered questions and complex concepts in theoretical physics. According to the special relativity theory, nothing moves faster than light speed. Also, boundary between visible particle and invisible particles is light speed, where real space-time and virtual spacetime intertwine and interact with other.

https://www.researchgate.net/publication/279185909_Graviton_and_Newton%27s_second_law?ev=prf_pub

https://www.researchgate.net/publication/270339919_Interactions_Between_Real_and_Virtual_Spacetimes?ev=prf_pub

Article Graviton and Newton's second law

Article Interactions Between Real and Virtual Spacetimes

The problem is quite simple. In quantum mechanis there is no place for particle trajectory. This notion belongs to classical theory. Consequently, we cannot imagine velocity in QM, because it is calculated from trajectory, and evidently no place for acceleration because it is derived from velocity, and as aconsequence from trajectory. The only perceptible quantity in QM is the probability of presence, which is  calculated as the second power of wave function associated to the system.

We know that  the time derivative of the expectation value of the momentum in any state is the negative of the expectation value of gradient of the potential energy. Dividing both sides by the mass we obtain a quantity which corresponds to the acceleration in classical mechanics.

I agree with @Viazminsky's  sentiments.One can invoke Ehrenfrest's theorem to introduce acceleration.I do not think acceleration is so fundamental such that it requires a corresponding operator.

@Ghomi there are formulations of QM that model definite trajectories for particles eg   the De Broglie-Bohm formulation

In the following article dE/dt is considered as the acceleration operator. This can be understood as follows. In quantum mechanics we have E => ih_bar d/dt. Therefore, dE/dt => ih_bar d^2/dt^2. This serves the purpose of the acceleration operator. Note equations 24, 26, 27, 74, 79 and 87 in the article.

Article Periodic quantum gravity and cosmology

Answer to your question: This is because quantum mechanics falls essentially out of the blue, i.e. is based on "educated guessing". If one tries better, in other words, if one really develops quantum mechanics from Newtonian mechanics assuming an active vacuum, questions like yours do not occur. Literature: L.Fritsche and M.Haugk, arXiv:0912.3442v1, Ann. Phys. (Leipzig) 12, No.6, pp.~371-402 (2003)

Dear Arbab,

Let me summarize what we know, before attempting any answers. In special relativity the relation between momentum and velocity is simple. The momentum law says that dE=vdp and it is true in both classical and special relativity theory. Introducing a force imparts that it is not compatible with the momentum law and the energy law. The consequence is of course the tensor representation of the theory of general relativity.

Asking for an operator representation of acceleration in Dirac theory is neither well-defined nor fundamental.

Dear Theophanes,

Thank you for your interest in our work. As for the initiating basic  assumption our work is identical with Nelson´s PR-1966 work: zero point motion of atoms, the stability of atoms against collaps through electron/nucleus attraction and the occurrence of the tunneling effect suggests the existence of a universal vacuum noise (“dark energy”) that acts on stable particles (electrons, nucleons, etc.) via stochastic forces whose action is calibrated by Planck’s constant. The salient point was to develop a theory of stochastic particle motion which is akin to Brownian motion. However, one has to observe a very distinct feature of difference: the stochastic forces have to guarantee energy conservation and – for free particles – momentum conservation on average.

Nelson deals with this problem by defining a backward and forward acceleration of the particle in question and requires the arithmetic mean of these accelerations times particle mass to obey Newton’s second law. But he never explains thoroughly the definition of suddenly occurring probability densities – which should actually refer to an ensemble of particles - nor does he elucidate the meaning of his velocity fields whose connection with the original particle trajectories remains completely unclear. For that reason I doubt that anyone has really understood his derivation.

My approach goes back to work of Gebelein who in 1935 derived the Navier-Stokes equation for fluid media by incorporating stochastic extra forces that act on the elementary volumes of the fluid which otherwise obey Newton’s second law. The most important property of this equation is that the dissipative part consists of two portions: one is proportional to the kinematic viscosity, the other one scales as the square of the kinematic viscosity. Hence, if one reverses the sign of the kinematic viscosity in the Navier-Stokes equation and forms the arithmetic mean with the former, the portion linear in the kinematic viscosity drops out, and  one obtains a new Navier-Stokes-type equation of motion which is dissipationless. Amazingly, it can be transformed into the time-dependent Schrödinger equation! We have worked out a very large number of consequences, in particular the inclusion of particle spin and relativistic effects. All this is missing in Nelson’s work. He was so much dissatisfied with his work that he finally revoked his PR-1966 paper at some conference in Switzerland about 30 years ago.

So, if you are really interested in this new “quantum mechanics without observers and measurers”, have a look into our arXiv-article:

arXiv: 0912.3442

Lothar

I am surprise how easily you are using vacuum with properties in such a way that you transpose the difficulty of the quantum mechanics on vacuum which is not a real vacuum!

Dear Xavier,

It's a pity that you don't give me any hint on what you consider "the difficulty of the quantum mechanics on vacuum". Did you make a try at my arXiv-article 0912.3442 ?Have a look at the table of contents and, maybe, read the first two sections. What do you think is missing? In the meantime I have worked out three more chapters on spontaneous light emission,  van der Waals interaction, Bose- and Fermi-distributions.  They are not yet included. 

Dear Lothar, sorry if I annoy you with my remark on vacuum. In the Webster’s dictionary vacuum is: a space nothing at all in it; completely empty space. I suppose that we have to understand the Quantum Mechanics according to this approach. According to your paper: Stochastic Foundation of Quantum Mechanics and the Origin of Particle Spin with Haugk you sai:

1) Our approach is based on the assumption that the probabilistic character of quantum mechanics does not originate from uncertainties caused by the process of measurement or observation, but rather reflects the presence of objectively existing vacuum fluctuations whose action on massive particles is calibrated by Planck's constant and effects an additional irregular motion.

2) For me the difficulty in Quantum Mechanics is a question of continuity and discreteness.

The Louis de Broglie assumption of a wave associates to the electron led to the discovery of the wave equation. It extends the continuum notion at the scale of the atomic interactions. So the continuum looks to us strongly linked to discreteness. However, we must keep in mind that a point is a mathematical object without a volume, so it cannot have a physical reality. The continuum is a very comfortable utility but one has to keep in mind that material reality is discreteness.  It has been by underlying that light comes from atoms, i.e. from a finite ensemble of atoms, that Einstein proposed the photon hypothesis. Let us then come back to the symmetry between the motion of a mobile and the propagation of a wave which led de Broglie to associate a wave to the electron in his thesis, it comes then possible to associate a flow of material particles to the wave, per example to light, in this way rejoining the Einstein photon hypothesis. However, when the number of photons becomes small, the continuum (i.e. the wave) is of no help any more to explain interferences: then one has to consider the concepts of the storage «in situ » of the photons to explain them (Ann. Fondation Louis de Brolgie, 30, 397-408, (2005)). See also Ann. Fondation Louis de Brolgie, 38, 33-39, (2013)

3) I was led to this approach after already in retirement, but I was able to propose a new calculation of the statistical distribution of the thermal energy of a set of atoms, introducing that two atoms cannot occupy the same place as soon as we give them a volume (1987). This replace the exclusion principle of Paoli and give good results (see: Perturbations and Statistical Distribution of the Thermal Energy on my ResearchGate site.

Dear Xavier,

Thank you very much for your detailed answer. One of the crucial assumptions in my arXiv-article concerns exactly your point. The pertinent passage reads: We shall use the terms “point mass”  and “point charge'” with the reservation that the actual size of the particles in question might well be finite of the order 10-13 cm, but very small compared to atomic diameters of the order

10-8 cm. Occasionally ``point mass'' will stand for the centroid of an atom or some composite system. In the overwhelming majority of cases in quantum mechanics one is dealing with this kind of point-like particles. Think of the Hamiltonian of a solid where the atomic nuclei are assumed to constitute point-like charges as well as the interacting electrons. All interactions are assumed to have the form of Coulomb forces between point charges. Whenever I consider in a summary fashion interactions between neutral atoms or molecules I model them by pair-potentials equal or similar to Lennard-Jones potentials. This is also underlying my short note on the derivation of Fermi- and Bose-distributions.  I used to hand it out to my students when they complained about the unbearable “indistinguishability-talk”  in standard textbooks on Bose- or Fermion-systems. I am appending a PDF-file of this short note.

As for my concept of the physical vacuum, I believe that many of my colleagues will agree. It is not empty, or as Frank Close of Oxford University says in his book “The void“: “It is full of stuff!” I like to illustrate my view by referring to a simple example: Consider a heat bath of some temperature which has the form of a hollow sphere whose interior is completely evacuated. Place a small system, for example a piece of copper in the middle of this sphere and keep it there by non-heat conducting fibers connected to the inner sphere surface. In thermal equilibrium the copper sample has attained the temperature of the heat bath. It absorbs as many photons from the heat bath as it emits. But totally independent of the details of the energy exchange between the heat bath and the copper, the distribution over the copper-eigenvalues of its total energy is governed by just one universal function: the Boltzmann function!

In the state of equilibrium there is also a thermal energy portion of the “vacuum” between the heat bath and the copper sample. In the present context it is usually not paid any attention to. But it is just the contribution of Planck’s black body radiation! If one lowers the temperature of the heat bath to zero K, the sample drops to its lowest quantum mechanical state, the groundstate in which the electrons are still moving and stay between the atomic nuclei. What prevents them from not combining with the nuclei or getting very close to them, thereby gaining and eventually emitting a tremendous amount of energy by electromagnetic radiation?  The answer of my approach is: It is exactly the energy fluctuations of the vacuum which prevents them from doing so. They are still there in the vacuum between the heat bath and the sample. Actually the vacuum also penetrates the sample.  Similar to the previous fluctuations that were calibrated by kBT of the heat bath, the action of these zero-point  fluctuations is calibrated  by Planck’s constant.  Of course, the outside vacuum should also contain an energy portion as before at non-zero temperature when it contained the black body radiation.  I am not familiar with modern cosmology. But if cosmologists would ask me which cosmological consequences my assumption on the origin of quantum mechanical randomness and stability of atoms should have, I would say: dark energy! Clearly, as long as modern quantum mechanics blames “observers” and “measurers” for all that is different  from classical mechanics and denies the fundamental role of an active vacuum, the dark energy must appear as a foreign  phenomenon.

Our senses are not made to detect vacuum fluctuations which affect the motion of objects reversibly and only within extremely short time intervals. For that reason nature and Webster’s Dictionary makes us believe that everything between objects is just empty space. I append another synopsis on some questions related to all that.

Lothar

Dear Manuel,

You are saying in your previous statement

"If something exists we can observe its scale, measure its weight, and determine when and where we made our observation."

I don't think that physical theories are entirely made along these lines. For example: Think of two coils parallel to each other which function as a transformer. There is an alternating current running through the first coil and another current induced in the second coil connected to a resistive load. You could calculate the energy flow from the energy source feeding the first coil to the  load on the side of the second coil, just by using Maxwell's equations. You can construct the Pointing vector everywhere. But could you observe its scale, measure its weight? Does it depend on when you make an observation? This applies similarly to quantum mechanics where one is in general dealing with a complex-valued wave function for a moving particle. How would you contrive to "observe" this function and measure its scale? Heinrich Hertz once stated in "Three Pictures of Mechanics": "We dream up pictures of the objects we are dealing with. The only criterion that has to be  satisfied in a theory that describes the properties and processes concerning these objects is that the logical consequences within this theory have to agree with the consequences dictated by nature, the ones that are accessible to measurement. " Many elements of the theory will in general not be accessible to measurement as evidenced by my wave function example .

Lothar

Dear Lothar,

I can consider the point mass that you use as the grains that I introduce to describe the interaction. See AFLB 36, 135-157, (2011) or Atom and Crystal, memories of research on my researchgate site.

Einstein in 1905 with his study of relativity showed that there is no absolute space and that physical laws should be independent of the place of observation, which made a giant leap in physics. Yet the difficulties encountered by the corpuscular approach of quantum mechanics are well related to the notion of space. Indeed there is no absolute space and one has to question how the space is built. On the macroscopic scale it is easy to define it in relation to the objects which occupy it, without forgetting the gas that fill volumes between objects, gases that transmit the sound for example. But what about the scale of the atom? Up to now electromagnetic field has allowed us to write the motion of the electron around the nucleus.

But Einstein has also taught us that the mass is equivalent to an amount of energy by establishing the relationship: E=mc2

This energy is huge so that it suggests that the electromagnetic field can be described by a discontinuous flow of grains of matter, constituting the mass, between the proton and the electron. More precisely through two fluxes: one from the proton to the electron the other from the electron to proton, these two fluxes not following everywhere the same route in such a way that the resulting action creates the movement. Grains of matter are supposed to be extremely small before the masses of the proton and the electron made of these grains. This approach leads to consider the volume of the electron and proton. This need to consider the volume of the particles is important because it gives physical meaning to the existence of an axis of rotation in the movement.

Now it seems to me that the properties attributed to the vacuum are the result of the exchanges of grains.

Now to return to the problem of Arbab it is clear that variable mass introduce force.

@Manuel,

I think you make it too easy for you in defending your standpoint:

"If something exists we can observe its scale, measure its weight..."

If you really believe that,  flowlines of the electromagnetic energy current do not exist for you. That means, from your point of view the Poynting vector and the physics which goes with it is a non-testable part of Maxwell's theory and therefore irrelevant along with many other features of electromagnetic energy transport, for example in polarizing beam splitters  to name just one example.

We grow up with the observation that a multitude of things exist, even such  sophisticated ones like our consciousness. But the existence of many of those things we cannot explain, for example: why do electrons, protons and neutrons exist? Nevertheless, we take great intellectual pleasure in explaining how simple laws of interaction between these particles explain the existence of atoms, their size, their bonding in forming molecules and solids and myriads of other things. Within this framework of our physical theories we use constantly items of imagination which do not have observable or measurable counterparts in the objective real world which we want  describe. (Just think of our idea of an infinite straight line.) What, for heavens sake, is wrong with that? All that matters is that the results of our theory agree with nature and even more: it predicts in applied physics and engineering desired outcomes with great reliability . 

Energy and momentum accumulate and are measured as discrete particle properties. They enter quantum mechanics in probabilities of what a measurement might discover. Power, force, and acceleration do not accumulate as discrete particle properties, but occur as system variables which may alter discrete properties.

If you have some purpose for using acceleration as an operator, there is no reason not to do so, as long as it is consistent with the physical system you are representing. Adding something new should fill a need that is not addressed by existing representations. I would guess that there is some opportunity to use acceleration operators in quantum field construction of general relativity. Then someone will need to fill the place of Paul Dirac and show that your new system is equivalent to conventional GR in average results of about 30 measurements.

New graduate students come along fairly often looking for research topics. People are looking for a way to formulate QFT without infinite energy in vacuum space. I've offered hints in my partition theory, viXra:1512.0277 which is written for much younger readers. So far as I know, the issue has not been resolved in graduate studies. So there is room for something new.