A realistic interpretation of quantum mechanics. Asymmetric random walks in a discrete spacetime
aa r X i v : . [ n li n . C G ] S e p A realistic interpretation of quantum mechanics.Asymmetric random walks in a discrete spacetime.
Antonio SciarrettaOctober 26, 2018
In this paper, I propose a realistic interpretation (RI) of quantum mechan-ics, that is, an interpretation according to which a particle follows a definitepath in spacetime. The path is not deterministic but it is rather a ran-dom walk. However, the probability of each step of the walk is found todepend from some average properties of the particle that can be interpretedas its propensity to have a certain macroscopic momentum and energy. Theproposed interpretation requires spacetime to be discrete. Prediction ofstandard quantum mechanics coincide with predictions of large ensemblesof particles in the RI.
Despite its general acceptance as a tool to predict the outcome of experi-ments with particles and other microscopic objects, standard quantum me-chanics (QM) is believed by many to be incomplete or, at least, not fullyunderstood. In particular, the “strange” or non-classical phenomena of QM,like self-interference and Born’s rule, are described in terms of abstract math-ematical objects. Although some proponents of the standard or Copenhagenintepretation of QM might have been ready to interpret the complex-valuedwavefunction as a real object [1], wavefunctions are essentially mathematicaltools serving to calculate probabilities from their square moduli. Contrastingto real-valued mathematics and one-to-one mapping between real variablesand observables of classical theories, the standard description is thus oftenconsidered as purely operational and thus not realistic.Consider the quantum phenomenon of particle self-interference, as illus-trated by the double-slit experiment. In the standard picture, self-interference1s due to the intrinsic superimposition of the two complex wavefunctions as-sociated to the two slits. The wavefunction is believed to describe everysingle particle, so that a particle is said to pass simultaneously from bothslits. That implies particle-wave duality, or other hardly imaginable con-cepts like diffusion in imaginary time [2].Proponents of alternative views have tried to re-conciliate QM with morerealistic assumptions concerning particle behavior. A well-known exampleis constituted by hidden-variable theories like De Broglie’s pilot wave theoryor Bohmian mechanics [3], a more recent variant of which is the determin-istic trajectory representation [4]. These theories calculate actual particletrajectories [5] that accumulate or rarefact, leading to maxima and minimaof fringes at the screen behind the slits. However, such an approach lacksa plausible mechanism for justifying non-localities, i.e., describing how thewave arises and does its guiding.The advocates of ensemble or statistical interpretation have recognizedthe standard description of QM as a description that does not apply toindividual particles but rather to ensembles of similarly prepared particles[6, 7]. However, ensemble interpretation does not provide any alternativeway to explain the behavior of single particles, nor it fills the gap betweenthe classical domain and the quantum domain [8].To explain self-interference in a realistic manner several alternative ap-proaches have been developed. A clearly non-exhaustive list includes theo-ries involving the influence of the context [9, 10], nonstandard probabilitydefinitions [11], nonstandard types of particle trajectories [12], quanta asreal particles acting as sources of real waves [13], nonlocality and discretespacetime [14]. Despite these efforts, QM is still universally believed to lacka convincing realistic interpretation.In this paper one possible realistic interpretation is presented. Althoughvery simple in its mathematical development and formalism, the proposedapproach seems to be able to predict self-interference and probability fieldsindependently from Schr¨odinger equation or wavefunctions. Unlike the stan-dard picture, only real quantities (actually, integers) are employed. In par-ticular, this approach assumes a discrete spacetime. Unlike ensemble in-terpretation, it describes single particle trajectories, however, as Markovchains, i.e., with an intrinsic randomness. Transition probabilities are simplefunctions of momentum propensity. The latter is randomly determined atthe preparation phase. Consequently, probability distributions of similarly-prepared ensembles of particles are obtained in accord to standard QM pre-dictions.The paper is organized as follows. Section 1 presents the assumptions2oncerning spacetime, which naturally lead to a realistic interpretation of un-certainty principle. Section 2 provides a realistic interpretation of Schr¨odingerequation in terms of Markov chain and transition probabilities. Section 3presents the results for a self-interference case, with only one additional as-sumption concerning the probability density of momentum propensity as de-termined at the sources. Section 4 finally presents a particle-by-particle sim-ulation of a double-source experiment, whose a posteriori distribution of ar-rival frequency practically tends to coincide with predictions of Schr¨odingerequation.
The RI assumes that the spacetime is inherently discrete. Limiting for sim-plicity the analysis to one dimension x , the RI assumes that only values x = ξX , ξ = 0 , , , . . . and t = τ T , τ = 0 , , , . . . are meaningful. Noninte-ger values of space and time are simply impossible in this picture. The twofundamental quantities X and T are the size of the lattice that constitutesthe space and the fundamental temporal resolution, respectively.Under this assumption, a particle trajectory consists of a succession ofpoints { ξ, τ } in the spacetime. Advance in time is unidirectional and unitary,that is, τ + 1 follows necessarily τ . Advance in space is still unitary butbidirectional. If at a time τ a particle resides at the location ξ of the spatiallattice, at time τ + 1 the particle can only reside at locations ξ + 1, ξ , or ξ −
1. In other words, the local velocity in lattice units β = ∆ ξ/ ∆ τ can onlytake the values +1, 0 or − v = lim N →∞ N N X τ =1 β ( τ ) XT (1)The maximum velocity that a particle can reach is the speed of light c . Light trajectory in the positive direction corresponds to β ( τ ) = 1, ∀ τ .Consequently to (1), one constraint to the fundamental lattice quantities isnecessarily XT = c (2)Another consequence of (1) is that to determine the macroscopic velocityof a particle, an observer should wait in principle a time N tending toinfinity. Every observation lasting a finite amount N of time steps will give3n approximation of v . Consider, e.g., N = 1. The observed velocity canbe +1, 0 or −
1. Thus the uncertainty of the macroscopic velocity v is 1 inabsolute value. For N = 2, the possible results for the observed velocity are1, 1 /
2, 0, − /
2, and −
1. Thus the uncertainty of the macroscopic velocityis 1/2 in absolute value. Extending these considerations, the uncertainty ofthe macroscopic velocity after an observation lasting N time steps is 1 /N in lattice units.Moreover, an observation lasting N time steps necessarily implies achange in the position of the particle. The span of the particle during theobservation ranges from N X to − N X . Thus the uncertainty of the positionof the particle at the end of the observation is obviously 2 N in lattice units.Using the two results above, and labelling ∆ v ( N ) and ∆ x ( N ) the un-certainties of velocity and position as a function of observation time N , therelationship ∆ v ( N ) · ∆ x ( N ) = cN · N X = 2 X T (3)holds.The latter equation resembles the Heisenberg uncertainty principle sinceit fixes an inverse proportionality between the uncertainty with which thevelocity of a particle can be known and the uncertainty with which its posi-tion can be known. Multiplying by the particle mass m , and comparing (3)to Heisenberg uncertainty principle, one obtains that the two fundamentallattice quantities are related to the Planck constant, m X T = ~ h π (4)The term 4 π (the solid angle of a sphere) is for three dimensional spaces. Inour example case of a 1-d space, this term reduces to Ω = π / Γ(1 / , where Γ isthe Gamma function, i.e., Ω = 2. Thus, combining (2) with the accordinglymodified (4), the values for the fundamental lattice quantities are obtainedas X = h mc (5)and T = h mc (6)These values clearly correspond to the Planck length and the Plancktime, respectively. The role of mass is not completely clear at this point.Likely, general relativity might serve to integrate it into the picture.4 Propagation: equation of Schr¨odinger
Now I describe the propagation rules of a particle on the lattice in the RI. Ateach time τ , the particle might jump to one of the nearest neighboring sitesof the lattice, or stay at rest. The actual local trajectory is not deterministic,i.e., it is not a prescribed function of previous parts of trajectory. Rather, thelocal trajectory has the characteristics of a random walk. This point is veryimportant and it implies that an intrinsic randomness affects the particlemotion. However, in contrast to pure random walks, there is a differenttransition probability for each possible jump. Label the three probabilities a , b , and c , respectively. Of course, a + b + c = 1 (7)Moreover, the RI assumes that the average or macroscopic momentum, p = v/c in lattice units (consider m = 1 henceforth), is imprinted to theparticle. This imprint is to be attributed to the preparation process. Itis thus possible to talk about momentum propensity . The quantity p canbe also reinterpreted as the probability of unitary motion in the positivedirection, so that a − c = p (8)Another average or macroscopic quantity is the energy of the particle,which in lattice units is written as e = lim N →∞ N N X τ =1 | β ( τ ) | (9)Also this energy can be reinterpreted as the propensity to unitary motionin either direction. Thus a + c = e (10)Combining (7)–(10), one obtains that a = e + p , b = 1 − e, c = e − p e must be a function of p . A well-known result of specialrelativity states that energy of a particle is the sum of the rest energy andthe kinetic energy. In lattice units, that proposition can be written as e ( p ) = 1 + p = 0 t = 1 t = 2 t = 3x = 2x = 1x = 0x = -1x = -2 bac a abac Figure 1: Particle trajectories after emission from a single source. The graphillustrates the neighboring points on the lattice and the three elementaryprobabilities a, b, c .Consequently, (11) can be rewritten as a = (cid:18) p (cid:19) , b = 1 − p , c = (cid:18) − p (cid:19) (13)The equations above, in fixing the probability of each jump at each timestep τ , determines the trajectory of the particle as a random walk. Forexample, observe Fig. 1. A particle is emitted from a source located at thesite ξ = 0 of the lattice with an intrinsic value of p and thus of e , determinedby the preparation (I will return soon on this point). After one time step, τ = 1, the particle has a probability a to be at the site ξ = 1, a probability b to be at the site ξ = 0, a probability c to be at the site ξ = −
1. Aftertwo time steps, τ = 2, the probabilities for each site from ξ = − ξ = 2are as follows: ρ p (2) = a , ρ p (1) = 2 ab , ρ p (0) = 2 ac + b , ρ p ( −
1) = 2 bc , ρ p ( −
2) = c . Notice that, since the functions a ( p ) and c ( p ) are symmetric,the probability distribution is symmetric with respect to ξ = 0. Noticealso that b = 4 ac and thus every probability ρ p is expressed as a uniquecombination of the three elementary probabilities, multiplied by a coefficient.6n general, the recursive expression for the probability of finding theparticle at time τ at site ξ is given by ρ p ( ξ, τ ) = aρ p ( ξ − , τ −
1) + bρ p ( ξ, τ −
1) + cρ p ( ξ + 1 , τ −
1) (14)Of course, the start of the recursion is ρ p (0 ,
0) = 1 for the case of a singlesource. Deriving a closed formula for the probability ρ p ( ξ, n ) is tedious butstraightforward at this point. The derivation conduces to binomials andbinomial coefficients. Instead of attempting to present here the derivation, Iwill show numerical results in the following. However, a special case is easyto calculate in closed form. The probability that a particle is at the eventhorizon, i.e., ρ p ( τ, τ ), is easily calculated as a τ . Similarly, ρ p ( − τ, τ ) = c τ .Now, a delicate passage in the theory is introduced. The RI assumes thatthe momentum propensity p is determined randomly during the preparationat the source. The probability of releasing a particle with a momentum p isuniform over the possible values of p . Since p can vary between -1 and +1,its span is 2 and thus the probability density f ( p ) = 1 / p , the probability of finding a particleat the location ξ, τ is clearly given by ρ ( ξ, τ ) = Z − f ( p ) ρ p ( ξ, τ ) dp = 12 Z − ρ p ( ξ, τ ) dp (15)Let us calculate the integral of (15) for the special case mentioned above,that is, ρ ( τ, τ ) = 12 Z − a τ dp = 12 Z − (cid:18) p τ (cid:19) τ dp (16)Using some elementary mathematics, ρ ( τ, τ ) = 12 τ +1 (cid:20) (1 + p ) τ +1 τ + 1 (cid:21) − = 12 τ +1 τ +1 τ + 1 = 12 τ + 1 (17)This result is not casual. It can be shown numerically that the same proba-bility is valid for every other site location comprised between the two eventhorizons, i.e., ρ ( ξ, τ ) = 12 τ + 1 , ∀ ξ ∈ [ − τ, τ ] (18)It is easily verified that ξ = τ X ξ = − τ ρ ( ξ, τ ) = 1 (19)7ow, let us compare this result with the predictions of standard QM,i.e., the particular solution of the Schr¨odinger equation. The wavefunctionfor a free particle is Ψ( x, t ) = 1 √ π Z e i ( kx − ωt ) ϕ ( k, dk (20)where the wavenumber k is related to the momentum of the particle and ϕ ( k,
0) = 1 √ π Z Ψ( x, e − ikx dx (21)For a single perfectly localized source at x = 0, Ψ( x,
0) = δ ( x ). Thus (21)reads ϕ ( k,
0) = 1 √ π (22)and consequently (20) is rewritten asΨ( x, t ) = 1 √ π Z exp (cid:20) i ( kx − ~ k m t ) (cid:21) dk (23)that, integrated, yieldsΨ( x, t ) = 1 √ π r mi ~ t exp (cid:20) i mx ~ t (cid:21) (24)The probability density is easily calculated as | Ψ( x, t ) | = m π ~ t = mht (25)thus it is inversely proportional to time and it does not depend on x . Nor-malizing time to lattice units and using (4) allows reducing (25) toprob. density = 12 τ X (26)The probability P ( x, t ) = | Ψ( x, t ) | X . The result compares with (17), with2 τ replacing 2 τ + 1. The two functions of τ are very similar and, indeed,practically coincident for τ sufficiently large. That could be interpreted inthe following way:The square modulus of the wavefunction predicted by the Schr¨odingerequation is an approximation of the probability in RI. This ap-proximation is as better as the measurement is farther from thesource. 8nother interesting result arises from the process (14). Let us calculatethe action of the particle, i.e., the energy accumulated by the particle duringits walk, σ ( ξ, τ ). Let us introduce also the variable Σ( ξ, τ ) = σ ( ξ, τ ) /ρ ( ξ, τ ).The random process for σ is given by σ ( ξ, τ ) = a (Σ( ξ − , τ −
1) + 1) + b Σ( ξ, τ −
1) + c (Σ( ξ + 1 , τ −
1) + 1) (27)with the initial condition Σ(0 ,
0) = 0.For example, one easily obtains σ (1 ,
1) = a and thus Σ(1 ,
1) = 1. Theaction of every particle reaching the point { , } is obviously 1. Generalizingthis result, clearly σ ( τ, τ ) = τ a and thus Σ( τ, τ ) = τ . For a point like { , } the prediction is less trivial. The possible values of action can be 2,if the particle follows a back-and-forth path, or 0, if it stays at rest for twotime steps. Using (27), one obtains for this case σ (2 ,
0) = 4 ac and thusΣ(2 ,
0) = 2 /
3, which is a weighted mean between the two possible values ofaction.The results easily calculated for other points are listed in Table 1. Ob-serving the trend of Σ for a given τ , one discovers a clear quadratic depen-dency on ξ . Indeed, the function Σ( ξ, τ ) is calculated asΣ( ξ, τ ) = ξ + τ − τ τ − ( τ ) + ξ τ − is the action at ξ = 0. Equation (28) can be easily verified byinspection of a few points, provided that { ξ, τ } > S ( x, t ) = mx ~ t (29)which, in lattice units, is S ( x, t ) = 2 π ξ τ (30)The latter equation corresponds to the second terms in the right-hand sideof (28), that is, Σ − Σ , multiplied by 2 π to obtain a phase angle. Thecorrespondence is almost perfect, except for the term 2 τ − τ predicted by standard QM. For large values of τ ,however, the two results are practically coincident. In other terms:The phase predicted by the complex Schr¨odinger equation is anapproximation of the action in RI. This approximation is as bet-ter as the measurement is farther from the source.9able 1: Calculated values of Σ( ξ, τ ) for some small values of ξ and τ . x t
31 203210-1-2-3 3- 22/3 7/56/5321 111 7/52200 - ------ -- --
In addition to probability of arrival and action, the source determinesthe whole state of the particle, that is, other observables. In order to cal-culate a few of them, I find useful to introduce global probability fluxes,which are the probability of each possible path (“history”, in standard QMlanguage) summed over all possible p ’s. The flux J { + , , −} ( ξ, τ ) describes theprobability of jumping to site ξ from site ξ − ξ , and ξ + 1, respectively,at time τ .In terms of global probability fluxes, the probability and the action at acertain site are also calculated as ρ ( ξ, τ ) = J + ( ξ, τ ) + J ( ξ, τ ) + J − ( ξ, τ ) (31)Σ( ξ, τ ) = ρ ( ξ,τ ) { J + ( ξ, τ )[Σ( ξ − , τ −
1) + 1]++ J ( ξ, τ )[Σ( ξ, τ − J − ( ξ, τ )[Σ( ξ + 1 , τ −
1) + 1] (cid:9) (32)Other observables of the random walk that can be calculated with this ap-proach are, for instance, the average momentum (the actual observable mo-mentum, not the momentum propensity that is a property of the particle)at a certain location of the lattice, which is given by¯ p ( ξ, τ ) = J + ( ξ, τ ) − J − ( ξ, τ ) J + ( ξ, τ ) + J ( ξ, τ ) + J − ( ξ, τ ) (33)and is calculated as ¯ p = ξτ (34)10imilarly, the average energy at a certain location is given by¯ e ( ξ, τ ) = J + ( ξ, τ ) + J − ( ξ, τ ) J + ( ξ, τ ) + J ( ξ, τ ) + J − ( ξ, τ ) (35)and it is calculated as ¯ e = ξ τ (2 τ −
1) (36)Both the latter equation have a counterpart in the variables predicted bythe Schr¨odinger equation, in particular in the Bohm interpretation. Thelocal average momentum corresponds to the momentum of Bohm hiddenvariable, ∇ S = mx ~ t (37)so that the probability flux is j = ρm ∇ S . The local average energy corre-sponds to the energy of Bohm hidden variable, that is, time derivative ofthe action − ∂S∂t = mx ~ t = 12 m ( ∇ S ) (38)To see the correspondence it is sufficient to write (37)–(38) in lattice unitsor, alternatively, to put m = 1, ~ = 1 in both of them. As for the otherobservables analyzed above, the correspondence is better as τ increases. After having reproduced the predictions of the Schr¨odinger equation fora free particle, I proceed now to a second puzzling aspect of QM: parti-cle self-interference. Double-slit experiment usually serves to visualize thisphenomenon. However, the core of self-interference is isolated and better il-lustrated by a double-source preparation. Instead of having a single source,a two-slit barrier, and a screen behind the barrier, I represent the sameprocess with two independent and mutually alternative sources of particles,separated by a certain distance 2 dX , and a screen. The sources are equiva-lent to very narrow, i.e., punctiform slits. Being in a one-dimensional space,the location of the “screen” is clearly fictitious. Pictorially, the geometryof the system can be still imagined in two dimensions. One dimension is ξ ,along which the particle move with a momentum propensity p . The seconddimension is perpendicular to ξ and is traversed by the particle with mo-mentum propensity 1 (certainty of advancing in the positive direction). The“screen” is thus located at a distance τ from the sources. Figure 2 illustratesthis equivalence. 11 t "screen"source source source slitslitscreen Figure 2: Pictorial equivalence between a double-slit experiment in twodimensions and a double-source test case in one dimension.The solution of the Schr¨odinger equation for this case is based on thelinear superimposition of the two waveforms relative to the two sources,Ψ( x, t ) = Ψ ( x, t ) + Ψ ( x, t ) √ ( x, t ) and Ψ ( x, t ) are obtained from (24) by replacing theterm x (that was valid for a source at x = 0) with a term ( x − dX ) and( x + dX ) , respectively. ThusΨ( x, t ) = 12 √ π r mi ~ t (cid:26) exp (cid:20) i m ( x − dX ) ~ t (cid:21) + exp (cid:20) i m ( x + dX ) ~ t (cid:21)(cid:27) (40)The probability density is given by | Ψ( x, t ) | = 2 mht cos (cid:18) S − S (cid:19) (41)where S and S are the two independent action values, that is, the phasesof the two exponentials in (40). Finally, the probability in lattice units is P ( ξ, τ ) = 1 τ cos π ( ξ − d ) − ( ξ + d ) τ ! (42)and thus an interference term arises due to the presence of two possiblesources. The interference is related to the phase difference between the twowaveforms. 12he representation of the same process in the RI, i.e., in terms of discretespacetime and random walk, would give just the superimposition of twoprobability densities of the type (18), if the same initial conditions are takenas in the single-source case. No interference term would arise in this case.The key factor to represent self-interference resides in the choice of thecorrect probability density of the momentum propensity p . For a two sourceprocess, f ( p ) = 1 /
2. Some values of momentum are more probably thanothers. This fact may seem strange, but actually it is already contained inthe standard picture of QM [15]. To verify it, it is sufficient to apply (21)with Ψ( x,
0) = ( δ ( x − dX ) + δ ( x + dX )) / √
2. The result is ϕ ( k,
0) = 1 √ π (cid:16) e − ikd + e ikd (cid:17) (43)from which the probability function is f ( k ) = 1 + cos(2 dk )2 π (44)Recalling that k = p/ ~ , (44) is transformed in lattice units as f ( p ) = 1 + cos(4 πdp )2 (45)which is a function oscillating between 1 and 0 with a mean value of 1/2.Again, the fact that some momenta are more probable than others shouldnot surprise. It is exactly what happens when the two sources are approxi-mated by a barrier with two slits at a sufficiently large distance from a singlesource. The two slits filter the momenta spectrum, e.g., favoring the valuescorresponding to the directions of the axes that rely the source to the slits.The same behavior is apparently contained in the assumption (45).Now, simply introducing (45) into (15), while still using the same process(14) leading to ρ p , describes the insurgence of self-interference. This is shownnumerically in the next section. In the last two sections, the predictions of the RI were shown in closed form,using mathematical equations in terms of a priori probabilities and proba-bility fluxes. The probability of a number of observable were calculated andfound to be in accord to the predictions of standard quantum mechanics.13able 2: Pseudocode used for the simulations of Fig. 5. i = 0 for particle 1 to N P i = i + 1 p ( i ) = random value beween -1 and +1 with prob. den. given by (45) ξ (0) = random value +1 or -1 with probability 1/2 s (0) = 0 n = 0 for time 1 to N T n = n + 1 v = random value +1, 0 or -1 with prob. given by (11) ξ ( n ) = ξ ( n −
1) + vs ( n ) = s ( n −
1) + | v | end for ν ( ξ ( N T )) = ν ( ξ ( N T )) + 1 end for ν ( ξ ( N T )) = ν ( ξ ( N T )) /N P Now, I will present numerical simulations of the random walk of single par-ticles and I will calculate the a posteriori probabilities as frequencies overa large number of emissions. Thus, this section is aimed at reproducingnumerically a true experiment.Table 2 shows the pseudocode used for such simulations. The two for-cycles are for the successively released N P particles, and for time up to N T ,which corresponds to the distance between the sources and the screen in thefictitious dimension perpendicular to ξ (see discussion above). Each particleexperiences the choice of three randomly-selected values: (i) the momentumpropensity according to f ( p ), (ii) the source from which it is emitted, and(iii) at each time step, its local velocity according to p . The final code linerepresents the counting of the particle that arrive at a certain location attime N T . From this number of arrivals, an a posteriori frequency ν ( ξ ) iscalculated as the ratio to the total number of particles emitted.First, I present the case with only one source at ξ = 0. Figure 3 showsthe frequency ν ( ξ ) after a time N T = 300 for different values of N P . Asthe the number of particles emitted in the ensemble increases, a frequencydistribution builds up. For large N P , the frequency clearly tends to the apriori probability ρ ( ξ, τ = N T ), that is, a constant value given by (18).The case with two sources at ξ = ±
200 0 20000.0050.010.0150.02 n −200 0 20002468 x 10 −3 −200 0 2000246 x 10 −3 x n −200 0 2000246 x 10 −3 x Figure 3: Probability of arrival of a particle emitted at ξ = 0 as a functionof ξ after N T = 300. From top-left to bottom-right, N P = 500, 5000, 10000,and 50000, respectively.behavior, illustrated by Fig. 4. As the the number of particles emittedin the ensemble increases, an interference pattern clearly builds up. Thesimulations thus reproduce the outcome of famous experiments like those ofMerli et al. [16] and Tonomura et al. [17]. Notice that, despite the fact thatnone of the probability fluxes of the RI can be negative or zero, neverthelessthere are valleys in the function ν ( ξ ), that is, points where the particlesseldom arrive.Figure 5 shows how the frequency ν ( ξ ) varies with the observation time N T . The number of particles is N P = 50000 in these simulations. Thecurves used as a basis of comparison are the predictions of standard QM,namely, the probability function P ( ξ, τ = N T ) given by (42). Clearly, thelatter approximate ν ( ξ ) as better as the observation time N T is longer. Inthe case N T = 300 the function ν ( ξ ) is very well smoothed by the analyticalfunction P ( ξ, τ = N T ). 15
200 0 20000.0050.010.0150.02 n −200 0 20000.0050.010.015−200 0 20000.0020.0040.0060.0080.01 x n −200 0 20002468 x 10 −3 x Figure 4: Probability of arrival of a particle emitted at ξ = ± ξ after N T = 300. From top-left to bottom-right, N P = 500, 5000, 10000,and 50000, respectively. In contrast to the standard picture of QM, the RI does not only predictthe probability of an event over an ensemble of similiarly prepared particles.Instead, it also represents a possible behavior of every single particle in theensemble. Probability distribution of the various observables in the stateare thus obtained a posteriori by evaluating the relative frequency of occur-rence of a certain event. Probability distributions can be also determined a priori , by applying simple probability conservation equations. Moreover,while standard QM calculates the a priori probabilities as squared mod-uli of complex-valued wavefunctions, the RI directly yields the real-valued(actually, rational-valued) probabilities, without appealing to mathematicalabstractions like complex numbers.Since no other proposals to describe in a realistic way the beavior of16
200 0 20002468 x 10 −3 n , P −100 −50 0 50 10000.0050.010.0150.02−40 −20 0 20 4000.010.020.030.040.05 x n , P −10 −5 0 5 1000.050.10.150.2 x Figure 5: Probability of arrival of a particle emitted at ξ = 0 as a functionof ξ after four values of N T . From top-left to bottom-right, N T = 300,100, 40, and 10, respectively. Solid lines: predictions with standard QM, P ( ξ, τ = N T ). Squares: a posteriori probability ν ( ξ ) with the RI.single particles exist so far, at least to the knowledge of the author, wemight be tempted to believe that particles really behave like described inthe RI.The results of this paper have concerned free particles. However, the RIseems naturally capable to integrate also external forces into the picture.Each interaction of the particle with its sourrounding is indeed expected tomodify its intrinsic properties, namely, its momentum propensity. References [1] Kiefer C. On the interpretation of quantum theory - from Copenhagento the present day. In: Castell L, Ischebeck O (eds.),
Time, quantumand information , Springer, Berlin, 2003.172] Nagasawa M. Schr¨odinger equations and diffusion theory, Birkh¨auser,Basel, 1993.[3] Bohm D. A suggested interpretation of the quantum theory in terms of“hidden variables”, Phys. Rev. 85, 166(I) – 180(II), 1952.[4] Floyd E. Welcher weg? A trajectory representation of a quantumdiffraction experiment, arXiv:quant-ph/0605121v1, 2006.[5] Philippidis C, Dewdney C, Hiley B. Quantum interference and thequantum potential, Il Nuovo Cimento 52B:15, 1979.[6] Ballentine LE.