TThe outcomes of measurements in the de Broglie-Bohmtheory.
G. Tastevin ∗ and F. Lalo¨e † Laboratoire Kastler Brossel, ENS-Universit´e PSL,CNRS, Sorbonne Universit´e, Coll`ege de France,24 rue Lhomond 75005 Paris, FranceFebruary 5, 2021
Abstract
Within the de Broglie-Bohm (dBB) theory, the measurement process is usuallydiscussed only in terms of the effect of the Bohmian positions of the measured systemS, while the effects of the Bohmian positions associated with the measurementsapparatus M are ignored. This article shows that the latter variables actually playan essential role in the determination of the result. Indeed, in many cases, the resultof measurement is practically independent of the initial value of a Bohmian positionassociated with S, and determined only by those of M. The measurement then doesnot reveal the value of any pre-existing variable attached to S, but just the initialstate of the measurement apparatus. Quantum contextuality then appears withparticular clarity as a consequence of the dBB dynamics for entangled systems.
A well-known feature of standard quantum mechanics is that the result of a mea-surement does not reveal the value of any pre-existing physical quantity attached to themeasured system S; in most situations, the result is actually created during the interac-tion process between S and the measurement apparatus. Jordan, as quoted by Bell [1],wrote for instance “In a measurement of position, the electron is forced .. to assumea definite position; previously, it was neither here nor there, it had not yet made itsdecision to a definite position..” Within standard quantum mechanics, quantum contex-tuality [2–4] is a well-known consequence of this property. In the de Broglie-Bohm (dBB)theory [5, 6], the result is completely determined by the initial value of the positions.The only source of randomness arises from the fact that these positions are unknown.Now, when a quantum system S interacts with a measurement apparatus M, one maywonder if the result is primarily determined by the Bohmian positions associated withS, or by those associated with the macroscopic measurement apparatus M.In most cases, the number of Bohmian positions associated with M is much largerthan that associated with S, and it seems natural that these positions should play a role. ∗ [email protected] † [email protected] a r X i v : . [ qu a n t - ph ] F e b ut, curiously, the measurement process within dBB theory is often discussed [7–11] ina model where M is treated as a classical external potential acting on S, which amountsto merely ignoring the quantum properties of M. For instance, in a Stern-Gerlach exper-iment, the effect of the magnet on the incoming atom is treated classically through theaction of given external magnetic gradient; all effects of quantum entanglement betweenS and M are then ignored. It is then clear that the initial Bohmian position of theincoming particle within its wave function is the only variable that can determine thefinal outcome (whether the particle is deviated upwards or downwards). This probablyexplains why it is often believed that, within dBB theory, the result of measurement isdetermined by the Bohmian variables attached to S.The purpose of the present article is to show that this is not the case in general. Wewill discuss the effect of all the Bohmian positions, including those of the measurementapparatus, and their influence on the final result of measurement. Indeed, we will see that,in most cases the measurement result is almost independent of the Bohmian positionsattached to S; it is rather determined by the initial values of the Bohmian position of M.In a Stern-Gerlach apparatus for instance, the direction in which the atom flies at theoutput of the magnet is primarily (if not entirely) determined by variables belonging tothe magnet itself. The measurement result is then a consequence of the initial physicalstate of the measurement apparatus, and may appear as predetermined. This ideahas some internal consistency: it seems natural that, in the interaction between a largesystem and a small system, the former dominates the process and forces the small systemto follow a given evolution. In the historical Stern-Gerlach experiment [12], what was observed was the accumulationof silver atoms onto a glass slide. This method does not seem really appropriate for themeasurement of single particles. Moreover, various recoil effects affect the atoms insidethe measurement apparatus: depending on the result of measurement, the atoms andmolecules inside in the magnet recoil upwards or downwards, various localized phononemission processes occur inside the glass plate, etc. Modelling these effects and theresulting changes of quantum states would be a complicated task. We will thereforeintroduce an optical version of the experiment that seems to be more appropriate for aquantum treatment of single measurement events.
The experiment is sketched in figure 1. The atom wave packet propagates along thedirection of the Ox axis, and then crosses two wave packets of photons having oppositedirections of propagation and circular polarizations, emitted respectively by two singlephoton sources E and E . Depending on the spin state of the atom, one of the photonsis absorbed and scattered in all directions, while the atom undergoes a recoil that pushesit either upwards of downwards. The other non absorbed photon is detected by either2etector D or D , which provides a measurement of the atomic spin along direction Oz.We assume that, initially, the atom is in a coherent spin state: α | + (cid:105) + β |−(cid:105) (1)with the normalization condition | α | + | β | = 1. O z xD D M M E E Figure 1: Schematic representation of an optical version of the Stern-Gerlach experiment.A spin 1 / | + (cid:105) spin state, it can absorb a circularly polarized photon emitted by E , but not thephoton emitted by E , since it has the opposite circular polarization. The converseis true if the particle is in the spin |−(cid:105) state. In the former case, the photon recoiltransfers a positive momentum to the wave packet along axis Oz, in the latter case anegative momentum. Either photon detector D or D then registers a clic, and itssignal is amplified sufficiently to move the position of a macroscopic pointer displayingthe results in apparatuses M or M . We assume that the spin particle is initially in acoherent superposition α | + (cid:105) + β |−(cid:105) of the two spin eigenstates.3 .2 Wave function We use a notation that is similar to that of Ref. [13]. The initial quantum state of thewhole system at time t = 0 is symbolized by the expression: | Φ( t = 0) (cid:105) = (cid:104) α | + (cid:105) + β |−(cid:105) (cid:105) (cid:12)(cid:12) ϕ z (cid:11) (cid:12)(cid:12) ϕ x (cid:11) (cid:12)(cid:12) ϕ y (cid:11) N (cid:89) p =1 (cid:12)(cid:12) χ (cid:11) p N (cid:89) n =1 (cid:12)(cid:12) χ (cid:11) n (2)where (cid:12)(cid:12) ϕ z (cid:11) (cid:12)(cid:12) ϕ x (cid:11) (cid:12)(cid:12) ϕ y (cid:11) is the ket describing the orbital state of the atom as a product ofOz, Ox and Oy states. The kets (cid:12)(cid:12) χ (cid:11) p and (cid:12)(cid:12) χ (cid:11) n respectively describe the initial statesof the pointer (or other) particles inside the first and second measurement apparatuses.For simplicity, we will treat these particules as unidimensional.First assume that each source E , emits one photon; one is scattered, and the otheris absorbed by either D or D , depending on the state of the spin. An amplificationprocess takes place in the photomultipliers, and drives the positions of the N particlesinside the pointers of both apparatuses. After the measurement has taken place, thestate of the system becomes : | Φ( t ) (cid:105) = α | + (cid:105) (cid:12)(cid:12) ϕ + z ( t ) (cid:11) (cid:12)(cid:12) ϕ x ( t ) (cid:11) (cid:12)(cid:12) ϕ y ( t ) (cid:11) N (cid:89) p =1 (cid:12)(cid:12) χ +1 ( t ) (cid:11) p N (cid:89) n =1 (cid:12)(cid:12) χ ( t ) (cid:11) n + β |−(cid:105) (cid:12)(cid:12) ϕ − z ( t ) (cid:11) (cid:12)(cid:12) ϕ x ( t ) (cid:11) (cid:12)(cid:12) ϕ y ( t ) (cid:11) N (cid:89) p =1 (cid:12)(cid:12) χ ( t ) (cid:11) p N (cid:89) n =1 (cid:12)(cid:12) χ − ( t ) (cid:11) n (3)In this expression, | ϕ ± z ( t ) (cid:105) is the ket describing the motion of the atom wave packet of theparticle along Oz when a photon has provided a positive, or negative, recoil to the atom.In both cases, we assume that the motion along Ox and Oy is not affected, and remainsdescribed by the freely propagating state (cid:12)(cid:12) ϕ x,y ( t ) (cid:11) . The wave functions associated withthese kets are: ϕ ± z ( z, t ) ∼ (cid:20) a + 4 (cid:125) t m (cid:21) − / exp (cid:40) ± i mvz (cid:125) − [ z ∓ vt ] a + i (cid:125) tm (cid:41) (4)and: ϕ x ( x, t ) ∼ (cid:20) a + 4 (cid:125) t m (cid:21) − / exp (cid:40) − x a + i (cid:125) tm (cid:41) (5)where a is the minimal width of the Gaussian wave packet (assumed to be the same alongthe three axes), m the mass of the particle, and ± v its recoil velocity after absorbing We consider that, in the final state | Φ( t ) (cid:105) , the two photons have been absorbed, either by one of thedetectors, or the environment; one should then consider that the states | χ , ( t ) (cid:105) also describes particlesin the environment. The initial and final states of the radiation in (2) and (3) are then the vacuum state,which does not have to be written explicitly. Moreover, in order the increase the recoil effect of theatom, we may assume that each source has emitted N photons instead of one. This would not changethe structure of the calculation either. or E . A similar expression gives the expression ofthe wave function along axis Oy . Since neither direction Ox nor Oy plays a role in thecalculation, we merely ignore the corresponding wave functions in what follows. Thephase S ± z ( z, t ) of ϕ ± z ( z, t ) is: S ± z ( z, t ) = ± mvz (cid:125) + 2 (cid:125) tm [ z ∓ vt ] a + (cid:126) t m (6)Similarly, the wave functions associated with the states | χ , ( t ) (cid:105) and | χ +1 , ( t ) (cid:105) are : χ ( z, t ) ∼ (cid:20) b + 4 (cid:125) t M (cid:21) − / exp (cid:40) − z b + i (cid:125) tM (cid:41) χ ± ( z, t ) ∼ (cid:20) b + 4 (cid:125) t M (cid:21) − / exp (cid:40) ± i M V z (cid:125) − [ z ∓ V t ] b + i (cid:125) tM (cid:41) (7)where M is the mass of the pointer particles, b the initial width of their wave packets,and V their velocity when the detector has counted one photon. Therefore, the phases ξ ( z, t ) and ξ + ( z, t ) of these functions are: ξ ( z, t ) = 2 (cid:125) tz M (cid:104) b + 4 (cid:126) t M (cid:105) ξ ± ( z, t ) = ± M V z (cid:125) + 2 (cid:125) tM [ z ∓ V t ] (cid:104) b + 4 (cid:126) t M (cid:105) (8) We now calculate the motion of the Bohmian positions, driven by the gradients of thephase of the wave functions.
To obtain a first idea of the calculation, assume first that we have a single particle withspin 1 /
2. It may be described at any time by the spinor: α | + (cid:105) (cid:12)(cid:12) ϕ + ( t ) (cid:11) + β |−(cid:105) (cid:12)(cid:12) ϕ − ( t ) (cid:11) (9)with: (cid:104) r (cid:12)(cid:12) ϕ + ( t ) (cid:11) = ϕ + ( r , t ) = R + ( r , t )e iS + ( r ,t ) (cid:104) r (cid:12)(cid:12) ϕ − ( t ) (cid:11) = ϕ − ( r , t ) = R − ( r , t )e iS − ( r ,t ) (10) For the sake of simplicity, we assume that the arbitrary axis along which all pointer particles moveis Oz; the variable z p,n gives the position of the n, p -th particle with respect to some initial referenceposition, which may differ from one particle to the other. | α | + | β | = 1; the two functions ϕ ± ( r , t ) are supposed to be normalized. Theassociated probability current is the sum of the contributions of the two spin components: J ( r ) = (cid:126) m (cid:104) | α | (cid:12)(cid:12) ϕ + ( r ) (cid:12)(cid:12) ∇ S + ( r ) + | β | (cid:12)(cid:12) ϕ − ( r ) (cid:12)(cid:12) ∇ S − ( r ) (cid:105) (11)The Bohmian velocity v ( r ) is nothing but the ratio between this current and the localprobability density: v ( r ) = (cid:126) /m | α | | ϕ + ( r ) | + | β | | ϕ − ( r ) | (cid:104) | α | (cid:12)(cid:12) ϕ + ( r ) (cid:12)(cid:12) ∇ S + ( r ) + | β | (cid:12)(cid:12) ϕ − ( r ) (cid:12)(cid:12) ∇ S − ( r ) (cid:105) (12) We now come back to the optical Stern-Gerlach experiment. From now on, and asmentioned above, we focus our calculation on the Oz component of the positions only.We call Q the Oz component of the Bohmian position of the spin particle, Z n and Z p the (one dimensional) Bohmian positions of the pointer particles. For the many particlestate (3), the Bohmian velocity of the spin particle is:dd t Q = (cid:126) mD ( Q, Z p , Z n ) (cid:34) | α | (cid:12)(cid:12) ϕ + z ( Q, t ) (cid:12)(cid:12) (cid:89) p (cid:12)(cid:12) χ + ( Z p , t ) (cid:12)(cid:12) (cid:89) n (cid:12)(cid:12) χ ( Z n , t ) (cid:12)(cid:12) ∇ S + ( Q, t )+ | β | (cid:12)(cid:12) ϕ − z ( Q, t ) (cid:12)(cid:12) (cid:89) p (cid:12)(cid:12) χ ( Z p , t ) (cid:12)(cid:12) (cid:89) n (cid:12)(cid:12) χ − ( Z n , t ) (cid:12)(cid:12) ∇ S − ( Q, t ) (cid:35) (13)with: D ( Q, Z p , Z n ) = | α | (cid:12)(cid:12) ϕ + z ( Q, t ) (cid:12)(cid:12) (cid:89) p (cid:12)(cid:12) χ + ( Z p , t ) (cid:12)(cid:12) (cid:89) n (cid:12)(cid:12) χ ( Z n , t ) (cid:12)(cid:12) + | β | (cid:12)(cid:12) ϕ − z ( Q, t ) (cid:12)(cid:12) (cid:89) p (cid:12)(cid:12) χ ( Z p , t ) (cid:12)(cid:12) (cid:89) (cid:12)(cid:12) χ − ( Z n , t ) (cid:12)(cid:12) (14)and: ∇ S ± ( Q, t ) = ± mv (cid:125) + 4 (cid:125) tm Q ∓ vt (cid:104) a + (cid:126) t m (cid:105) (15)The velocity of the spin particle is therefore a weighted average of the velocities associatedwith two wave packets at point Q , describing upwards and downwards motions resultingfrom opposite recoil effects. As expected with an entangled quantum state, the weightsdepend, not only of the position Q of the particle, but also on the positions Z p and Z n of all pointer particles. 6imilarly, the velocity of any pointer particle is given by the relation:dd t Z p = (cid:126) M D ( Q, Z p , Z n ) (cid:34) | α | (cid:12)(cid:12) ϕ + z ( Q, t ) (cid:12)(cid:12) (cid:89) p (cid:12)(cid:12) χ + ( Z p , t ) (cid:12)(cid:12) (cid:89) n (cid:12)(cid:12) χ ( Z n , t ) (cid:12)(cid:12) ∇ ξ + ( Z p , t )+ | β | (cid:12)(cid:12) ϕ − z ( Q, t ) (cid:12)(cid:12) (cid:89) p (cid:12)(cid:12) χ ( Z p , t ) (cid:12)(cid:12) (cid:89) n (cid:12)(cid:12) χ − ( Z n , t ) (cid:12)(cid:12) ∇ ξ ( Z p , t ) (cid:35) (16)with: ∇ ξ ± ( Z, t ) = ± M V (cid:125) + 4 (cid:125) tM Z ∓ V t (cid:104) b + 4 (cid:126) t M (cid:105) ∇ ξ ( Z, t ) = 4 (cid:125) tZM (cid:104) b + 4 (cid:125) t M (cid:105) (17)where Z stands for Z p or Z n . The time evolution of the positions Z n is given by anexpression similar to (16), where ∇ ξ + ( Z p , t ) is replaced by ∇ ξ ( Z n , t ) and ∇ ξ ( Z p , t ) by ∇ ξ − ( Z n , t ). Again, we see that the velocity of each pointer particle is a weighted averagebetween the velocities associated with two wave packets, but for the pointers one of thewave packets is static. The two wave packets of the spin particle separate in a time of the order of τ a (cid:39) a/v z ,those of the pointer particles separate in a time of the order of τ b (cid:39) b/V . The ratiobetween these times is: E = τ a τ b = aVbv (18)If E >
1, the pointers are fast, indicating a result before the two wave packets of thespin particle separate; if
E <
1, they are slow, and the wave packets of the spin particleno longer overlap when the pointers start to indicate a definite result.
We have assumed that all particles inside each of the two pointers have the same massand are described by Gaussian wave packets with identical initial widths. In this case,we will show that their effect on the trajectory of the spin particle is identical to that oftwo fictitious pointers, each containing a single particle with position ˆ Z , :ˆ Z = 1 √ N N (cid:88) p =1 Z p ˆ Z = 1 √ N N (cid:88) n =1 Z n (19)7n these relations, the √ N ensures that ˆ Z , have the same variance and statisticalproperties as any individual pointer position Z n,p . In this substitution of variables, wewill see that the effective velocity associated with the fictitious pointers becomes √ N V .To obtain these results, we insert relations (7) into the expression (13) of the velocityof the spin particle. We then see that several factors appear in both (cid:12)(cid:12) χ ( Z p,n , t ) (cid:12)(cid:12) and (cid:12)(cid:12) χ +2 ( Z p,n , t ) (cid:12)(cid:12) , cancelling each other in the numerator and the denominator. In (13), wecan therefore replace all the | χ ( Z p,n , t ) | by 1, and the | χ ± ( Z p,n , t ) | by the remainingfactors: (cid:12)(cid:12) χ ± ( Z p,n , t ) (cid:12)(cid:12) ⇒ exp (cid:40) b b + (cid:125) t M (cid:2) ± V tZ p,n − V t (cid:3)(cid:41) (20)Moreover, when the numbers of particles in both pointers are assumed to be equal,the terms in − V t appear N times in both components in the numerator and thedenominator, so that they also cancel each other. Taking for instance the product over p then leads to the following substitution: (cid:89) p (cid:12)(cid:12) χ ± ( Z p , t ) (cid:12)(cid:12) ⇒ exp (cid:40) ± b b + (cid:125) t M √ N V t ˆ Z (cid:41) (21)A product over n provides a similar result, where ˆ Z is replaced by ˆ Z .A similar simplification occurs with the wave function (4) of the spin particle. Thesquared moduli of two wave functions of the spin particle are: (cid:12)(cid:12) ϕ ± z ( Q, t ) (cid:12)(cid:12) ∼ exp (cid:40) − a a + (cid:126) t m [ Q ∓ vt ] (cid:41) (22)which also contain several common factors; only the terms linear in vt are relevant. Wecan therefore make the substitutions: (cid:12)(cid:12) ϕ ± z ( z, t ) (cid:12)(cid:12) ⇒ exp (cid:40) ± a a + (cid:126) t m vtQ (cid:41) (23)We then obtain the simpler relations:dd t Q = (cid:126) mD (cid:104) | α | exp { R } ∇ S + ( Q, t ) + | β | exp {− R } ∇ S − ( Q, t ) (cid:105) (24)with: D = | α | exp { R } + | β | exp {− R } (25)and: R , = 4 a a + (cid:126) t m vtQ + 4 b b + (cid:125) t M √ N V t ˆ Z , (26)These equations show that the motion of Q depends on the positions of the pointerparticles only through the variables ˆ Z and ˆ Z . One can actually even go further, and8ote that the contribution of all pointer particles reduces to that of the average position( ˆ Z + ˆ Z ) / .Physically, for the first pointer, positive initial positions of the particles tend to favorthe component of the state vector where the pointer wave packets have a positive velocity,over the other component in which they are static. If it turns out that the particles inthe pointer already tend to indicate a positive result before the measurement starts, theprobability of a positive spin result is increased. In the second pointer, positive initialpositions favor the same component of the state vector: in this component, the pointerwave packets have a zero initial velocity, while in the other they go further away witha negative velocity. So, for both pointers, positive values of the positions tend to favortrajectories of the spin particle flying upwards with a spin up. Moreover we have seenthat, in the special case where all the wave packets are gaussian, only the sum of allvalues of Z n and Z p plays a role in the process.In order to obtain the evolution of ˆ Z , we now sum relation (16) over all values of p and divide by √ N . As above, simplifications between terms in the numerator and thedenominator take place and we obtain:dd t ˆ Z = (cid:126) M D (cid:104) | α | exp { R } ∇ ˆ ξ + ( ˆ Z , t ) + | β | exp {− R } ∇ ˆ ξ ( ˆ Z , t ) (cid:105) (27)with the values obtained from (17): ∇ ˆ ξ ± ( ˆ Z, t ) = ± M √ N V (cid:125) + 4 (cid:125) tM ˆ Z ∓ √ N V t (cid:104) b + 4 (cid:126) t M (cid:105) ∇ ˆ ξ ( ˆ Z, t ) = 4 (cid:125) t ˆ ZM (cid:104) b + 4 (cid:125) t M (cid:105) (28)Similarly, we have:dd t ˆ Z = (cid:126) M D (cid:104) | α | exp { R } ∇ ˆ ξ ( ˆ Z , t ) + | β | exp {− R } ∇ ˆ ξ − ( ˆ Z , t ) (cid:105) (29)As a result, because of the particular form of the Gaussian wave packets, all Bohmianpositions inside each pointer can be replaced by the position of their center of massmultiplied by √ N , and be treated as a the position of a single fictitious pointer particle.The equations of motion (24), (27) and (29) show that the velocity of the fictitiouspointers is also multiplied by the square root √ N . If N is very large, the fictitiouspointers become very fast; we will see that this implies that the influence of the pointerson the trajectory of the spin particle is dominant. The difference ( ˆ Z − ˆ Z ) evolves separately in time, under the only effect of the spreading of thewave packets. .5 Dimensionless variables For convenience, and use in the numerical calculations, we introduce dimensionless vari-ables by setting: Q (cid:48) = Qa v (cid:48) = ma (cid:126) v z (30)All positions are then expressed in terms of the initial widths of the corresponding wavepackets. For the pointer particles, we set: Z (cid:48) p,n = Z p,n b V (cid:48) = M b (cid:126) V (31)and: ˆ Z (cid:48) , = ˆ Z , b (32)We also introduce a dimensionless time t (cid:48) by: t (cid:48) = (cid:126) ma t (33)and note that the rapidity parameter E is now given by: E = η V (cid:48) v (cid:48) (34)with: η = ma M b (35)In (24) and following equations, we can then make the substitutions:exp (cid:40) ± a a + (cid:126) t m vtQ (cid:41) ⇒ exp (cid:26) ± v (cid:48) t (cid:48) t (cid:48) ) v (cid:48) t (cid:48) Q (cid:48) (cid:27) (36)and: exp (cid:40) ± b b + (cid:125) t M √ N V t ˆ Z , (cid:41) = exp (cid:40) ± η √ N V (cid:48) t (cid:48) ˆ Z (cid:48) , η ( t (cid:48) ) (cid:41) (37)For the spin particle, the gradients of the phase are given by: ∇ S ± z ( Q ) = ± v (cid:48) a + 4 t (cid:48) a Q (cid:48) ∓ v (cid:48) t (cid:48) t (cid:48) ) (38)while, for the fictitious pointer particles, relations (28) lead to: ∇ ξ ± = ± √ N V (cid:48) b + 4 ηt (cid:48) b ˆ Z (cid:48) ∓ η √ N V (cid:48) t (cid:48) η ( t (cid:48) ) ∇ ξ = 4 ηt (cid:48) b ˆ Z (cid:48) η ( t (cid:48) ) (39)10hen these substitutions are made, we obtain dimensionless equations of evolution:dd t (cid:48) Q (cid:48) = 1 D (cid:48) (cid:20) | α | exp (cid:8) R (cid:48) (cid:9) (cid:104) v (cid:48) + 4 t (cid:48) Q t (cid:48) ) (cid:105) + | β | exp (cid:8) − R (cid:48) (cid:9) (cid:104) − v (cid:48) + 4 t (cid:48) Q t (cid:48) ) (cid:105)(cid:21) (40)with: D (cid:48) = (cid:104) | α | exp (cid:8) R (cid:48) (cid:9) + | β | exp (cid:8) − R (cid:48) (cid:9)(cid:105) (41)and: R (cid:48) , = 4 v (cid:48) t (cid:48) Q (cid:48) t (cid:48) ) + η √ N V (cid:48) t (cid:48) ˆ Z (cid:48) , η ( t (cid:48) ) (42)For the motion of the pointers, we obtain:dd t (cid:48) ˆ Z (cid:48) = ηD (cid:48) (cid:34) | α | exp (cid:8) R (cid:48) (cid:9) (cid:104) √ N V (cid:48) + 4 ηt (cid:48) ˆ Z (cid:48) − η √ N V (cid:48) t (cid:48) η ( t (cid:48) ) (cid:105) + | β | exp (cid:8) − R (cid:48) (cid:9) (cid:104) ηt (cid:48) ˆ Z (cid:48) η ( t (cid:48) ) (cid:105)(cid:35) (43)and: dd t (cid:48) ˆ Z (cid:48) = ηD (cid:48) (cid:34) | α | exp (cid:8) R (cid:48) (cid:9) (cid:104) ηt (cid:48) ˆ Z (cid:48) η ( t (cid:48) ) (cid:105) + | β | exp (cid:8) − R (cid:48) (cid:9) (cid:104) − √ N V (cid:48) + 4 ηt (cid:48) ˆ Z (cid:48) + η √ N V (cid:48) t (cid:48) η ( t (cid:48) ) (cid:105)(cid:35) (44)We notice that the Planck constant (cid:126) has disappeared from these equations. Inaddition to the initial degree of polarization of the spin: σ = | α | − | β | (45)these equations contain four dimensionless parameters: the two reduced velocities v (cid:48) and V (cid:48) , the parameter η , and the number of particles N within each pointer. Since N and V (cid:48) appear only through the product √ N V , the number of there four parameters actuallyreduces to three.
In order to understand the influence of the values of the initial positions of the spinparticle and those of the pointers, we now apply the preceding considerations to varioussituations.
We first study the case where the initial spin polarization is zero, meaning that noresult of measurement is privileged by the rules of standard quantum mechanics. Anyimbalance can then result only from the influence of the Bohmian positions, either ofthe spin particle, or of the particles in the pointers.11 .1 0.2 0.3 0.4 0.5t' - - = Q / a - - Z ' = Z / b 0.1 0.2 0.3 0.4 0.5t' - - Z ' = Z / b Figure 2: Trajectories obtained when V (cid:48) = 0, so that the motion of the pointers decouplesfrom that of the spin particle. Each pointer contains only one particle. The initial spinpolarization of the spin particle is σ = 0. In all figures of this article, we take theparameter η = ma /M b equal to unity ( η = 1) and v (cid:48) = mav z / (cid:126) = 10.The left part of the figure shows the trajectories of the spin particle starting from differentinitial positions, the central and right part the trajectories of the particles in the firstand second pointers. Since V (cid:48) = 0, the trajectories of the spin particles are exactly thesame as those usually obtained when the measurement apparatus is treated classically(through a given external potential). The initial positions of the two pointer particlesare Z (0) = 0 . Z (0) = 0 .
1, but these values are irrelevant for the trajectory of thespin particle. The pointers remain motionless, except a small drift due to the spreadingof their (otherwise static) wave packets.
If we set V (cid:48) = 0 in the equations, R (cid:48) and R (cid:48) are equal, and the motion of the spinparticle decouples from that of the pointers. Physically, when the pointer states χ ( z, t )and χ ± ( z, t ) in (7) remain identical, no quantum entanglement develops from the initialquantum state (2). Figure 2 shows the trajectories obtained in this case. They areactually exactly the same as the usual trajectories (when the effect of the measurementapparatus is treated as a classical external potential). For simplicity, we have assumedthat each pointer contains a single particle, but the value of N is actually irrelevant forthe motion of the spin particle, which remains unaffected by the pointers. The left partof the figure shows several trajectories of the spin particle starting from various initialpositions (41 different values of Q (cid:48) ( t ), equally spaced in the interval ±
3) , the centralpart the trajectory of the first pointer, and the right part the trajectory of the secondpointer; as expected in the absence of entanglement, both pointers remain still in thiscase, and therefore do not indicate any result of measurement.
Now assume that V (cid:48) = v (cid:48) , so that the pointers are entangled with the spin particle. Eachpointer still contains only one particle, with the same initial position as in figure 2. Thetrajectories are shown in figure 3. The major change is that the pointers now move and12 .1 0.2 0.3 0.4 0.5t' - - = Q / a - - Z ' = Z / b 0.1 0.2 0.3 0.4 0.5t' - - Z ' = Z / b Figure 3: The only change of the input parameters with respect to those of figure 2 isthat V (cid:48) /v (cid:48) = 1, so that the motions of the spin particle and of the pointers are coupled.The positions of the two pointer particles now significantly evolve in time: if the spinparticle flies upwards at the end of the measurement, the first pointer (figure in thecenter) moves upwards, while the second pointer (figure in the right) remains still; if thespin particle flies downwards, the first pointer remains still and the second pointer movesdownwards. The trajectories of the spin particle remain similar to those in figure 2.indicate in which direction the spin particle flies at the end of the measurement process.If the spin particle flies upwards, the trajectory of the first pointer moves upwards,while that of the second pointer remains almost motionless; if the spin particle fliesdownwards, the first pointer remains still while and the second pointer moves downwards.Nevertheless, with only two pointer particles coupled to the spin particle, and with theinitial positions Z (0) = 0 . Z (0) = 0 .
1, the trajectories of the spin particle are notstrongly changed with respect to those of figure 2.Now, if the initial positions of the pointers take larger positive values, which favoursa spin up result, figure 4 shows that more trajectories of the spin particle go in the up - - = Q / a - - Z ' = Z / b 0.1 0.2 0.3 0.4 0.5t' - - Z ' = Z / b Figure 4: Same figure as figure 3 (no spin polarization, two pointer particles), butassuming that the initial positions of the pointer particles have larger positive Z (0) = 0 . Z (0) = 0 .
4. This favors an upwards deviation of the spin particle. Even if thequantum state is exactly the same as in figure 3, the positive initial values of the positionsof the two pointer particles have a significant effect on the trajectories of the spin particle.13 .1 0.2 0.3 0.4 0.5t' - - = Q / a - - Z ' = Z / b 0.1 0.2 0.3 0.4 0.5t' - - Z ' = Z / b Figure 5: Same situation as in figure 3 (no spin polarization, ˆ Z (0) = 0 . Z (0) =0 . N increases from 1 to 25. We seethat this significantly enhances the influence of the pointer particles on the Bohmiantrajectory of the spin particle (and spin direction at the end of the measurement). In a realistic measurement apparatus, the pointers are macroscopic objects, and thenumber of particles they contain is some fraction of the Avogadro number.Figure 6 is obtained in the same conditions as figure 5, but with a much largernumber of pointer particles ( N = 10 ) and only a very small positive offset of the initialBohmian positions ˆ Z , (0) = 0 .
02. A striking effect is that this small offset is sufficient toforce all trajectories of the spin particle to fly upwards at the output of the measurementapparatus, whatever the initial position of this particle. In this case, we see that it isreally the pointers that “decide” what the measurement result should be and, so to say,force the particle to “obey to this decision” and to take a spin up value with a trajectoryflying upwards.Needless to say, and as above, the standard quantum results are recovered whenan average is taken over all initial positions of the pointers. But this “all or nothing”14 .1 0.2 0.3 0.4 0.5t' - - = Q / a - - Z ' = Z / b 0.1 0.2 0.3 0.4 0.5t' - - Z ' = Z / b Figure 6: Same as figures 3 and 5 ( σ = 0), but with N = 10 pointers and smalleroffset values for the initial positions of the pointers ( ˆ Z (0) = ˆ Z (0) = 0 . When the initial spin polarization does not vanish, it introduces a preferred result ofmeasurement, and therefore a preferred deviation of the particle trajectory. In thelimiting case where the spin is fully polarized in one direction, one of the two componentsof the many body wave function (3) vanishes, and the final direction of motion of all theBohmian positions is fixed; the initial value of the position of the spin particle plays norole in the result of measurement.It is more interesting to study intermediate situations, where a compromize has tooccur between the quantum mechanical preference for one of the results and the influenceof the initial values of the Bohmian positions.
Figure 7 shows the trajectories when the spin polarization is σ = +0 .
5, assuming that theaverage initial positions of the 25 pointer particles are positive. In this case, all particletrajectories move upwards, as if the spin polarization were 100%. Figure 8 shows whathappens when the average initial positions of the pointers are negative. In this case,some trajectories go downwards. Again, the standard quantum average is recoveredonly when a statistical average over the initial positions of the pointer is applied.15 .1 0.2 0.3 0.4 0.5t' - - = Q / a - - Z ' = Z / b 0.1 0.2 0.3 0.4 0.5t' - - Z ' = Z / b Figure 7: Bohmian trajectories of the spin particle and the particles of the pointersin a case where the initial polarization of the spin is σ = 0 .
5. Each pointer contains25 particles. If the Bohmian positions of the pointers favor a spin up result ( ˆ Z (0) =ˆ Z (0) = 0 . If the number of pointers is macroscopic, as in § σ = 0 .
01, but N is very large, N = 10 . The initial positions of the pointer particles aresupposed to vanish ( ˆ Z , (0) = 0), so that they do not favour any result of measurement(they are “neutral”). In the absence of entanglement between the spin particle and thepointer particles, the numbers of trajectories going upwards and downwards should bealmost equal, as illustrated in the right part of the figure (obtained by setting V (cid:48) = 0).One could then naively expect that neutral pointer particles cannot change this situation - - = Q / a - - Z ' = Z / b 0.1 0.2 0.3 0.4 0.5t' - - Z ' = Z / b Figure 8: As in figure 7, the initial spin polarization is σ = 0 .
5, but the initial positions ofthe 25 pointer particles now favor negative results: ˆ Z (0) = ˆ Z (0) = − .
2. A significantproportion of the trajectories of the spin particle now go downwards. An average overall possible values of ˆ Z , with a weight equal to the quantum probabilities (quantumequilibrium) would show that 3 / / .1 0.2 0.3 0.4 0.5t' - - = Q / a 0.1 0.2 0.3 0.4 0.5t' - - = Q / a Figure 9: The left part of the figure shows the trajectories obtained when the initial spinpolarization is positive but low, σ = 0 .
01; the number of the particles in the pointersis high, N = 10 , while their initial positions are exactly neutral, ˆ Z , (0) = 0. As apoint of comparison, the right part shows the same situation, but in the absence ofany entanglement with the pointers ( V (cid:48) = 0); as expected, almost the same number oftrajectories then fly up or down. This figure illustrates that, when the number of particlesin the pointers is high, even if their initial positions are exactly neutral, they completelychanges the trajectories of the spin particles, forcing all of them to fly upwards and witha spin up. The initial Bohmian position of the spin particle is then completely irrelevantconcerning the measurement result.drastically. We see that this far from being true: actually, the Bohmian variables of themeasurement apparatus can completely dominate the determination of the measurementresult. They force all spin trajectories to remain grouped, in a sort of “all or nothing”situation where all spins must take the same final positive value. The mechanism ofthis effect is as follows. Every particle in the first pointer has a slightly larger chance tohave a positive initial velocity, and to move upwards. At subsequent times, its positioncorresponds to slightly larger values of | χ ± ( Z p , t ) | than (cid:12)(cid:12) χ ( Z p , t ) (cid:12)(cid:12) , which favours thespin up component of the total wave function; a similar effect takes place in the secondpointer. Then the multiplicative effect of 10 particles transforms this small effect intoa big unbalance, which affects the conditional wave function of the spin particle andmakes it fly upwards.Figure 10 shows what happens when the initial values of the positions of the pointerparticles are slightly negative, ˆ Z , (0) = − .
01. This very small change is sufficientto completely reverse the results of figure 9, since now all trajectories fly downwards.More generally, exploring many values of the parameters but keeping N = 10 constant,we find that the trajectories of the spin particles (almost) always remain completelygrouped, whatever the value of σ is. This is of course in complete opposition with whathappens in the absence of entanglement between S and M (right part of figures 9 and10). Our main conclusion is that a large number of particles of the pointers rendersthe initial position of the spin particle completely irrelevant in the determination of the17 .1 0.2 0.3 0.4 0.5t' - - = Q / a 0.1 0.2 0.3 0.4 0.5t' - - = Q / a Figure 10: Same conditions as in figure 9 except that, here, the initial values of thepositions of the center of mass of the pointer particles are slightly negative, correspondingto ˆ Z , (0) = − .
01. This is sufficient to completely change the situation and reverse thedeviation of all trajectories of the spin particle (left part of the figure; the right part is ofcourse unchanged). Again, the initial Bohmian position of the spin particle is completelyirrelevant.result of measurement. For each run of the experiment, this result is pre-determined bythe initial value of the positions of the particles inside the pointers.
In their 1993 book, Bohm and Hiley [14] emphasize that the explicit context depen-dence of experimental results within dBB theory is well in line with the central idea ofthe Copenhagen interpretation, illustrated by a famous quotation by Bohr [15] “the ne-cessity of considering the whole experimental arrangement, the specification of which isimperative for an well-defined application of the quantum mechanical formalism”. Bohmand Hiley write “The context dependence of measurements ... also embodies, in a certainsense, Bohr’s notion of the indivisibility of the combined system of observing apparatusand observed object. Indeed, it can be said that our approach (the dBB interpretation)provides a kind of intuitive understanding of what Bohr was saying”.Our discussion is in full agreement with these remarks. It actually gives them a moreprecise content, since it relates a general statement to a specific mechanism involving theposition of Bohmian variables in the overlap region of wave functions and the consecutivedisappearance of empty waves. As we have seen, the collaborative effect of the wholeensemble of Bohmian positions within the measurement apparatus M renders the initialposition of the spin particle irrelevant for the determination of the outcome of measure-ment. This may seem counterintuitive but, after all, since the measurement apparatusis in general much larger than the measured quantum system S it seems natural that,during their interaction, the larger system dominates the process and forces the smallsystem to follow a given evolution. Then the many positions inside the measurement18ointer can anticipate a final result of the measurement performed on the spin particle,at the end the spin particle is forced to move to a state that follows this anticipation.There are nevertheless a few exceptions to this rule: in the special case where S is ini-tially in an eigenstate of measurement, the result of measurement is certain and theeffect of the Bohmian variables associated with M (and actually also S) is switched off.Our discussion also illustrates the effects of quantum entanglement within dBB the-ory and the notion of the “conditional wave function” [16,17]. The Bohmian positions ofM may determine the value of the conditional wave function of the atom, which drivesthe position of the atom when it exits the output of the magnet, and forces it to goupwards or downwards with a given spin component. The corresponding effect is ingeneral nonlocal and very similar to that discussed in Ref. [13].This also has implications concerning the calculation of correlation functions at dif-ferent times. Within standard quantum mechanics, these functions are obtained bytaking into account the first measurement and the effect of the measurement apparatus.This is done by applying a projection operator, after which one calculates the subsequentevolution of the wave function from this new state. Similarly, in dBB quantum theory, itis essential to take into account the effect of the Bohmian variables of the measurementapparatus. If this step is omitted, incorrect correlation functions are obtained [18].Needless to say, the model of the measurement apparatus we have used is oversimpli-fied, assuming for instance the same Gaussian wave function for all the particles insidethe measurement apparatuses that are entangled with the spin particle. This simplifica-tion does not change the structure of the entangled quantum state of the spin particleand the measurement apparatus. Since this structure is the origin of our results, webelieve that they are generic. Moreover, for simplicity, we have focussed our discussionon the particles inside the pointers, but it is clear that other parts of the measurementapparatuses also play a role. The number of particles N should not be seen as referringonly to the physical content of the pointers themselves. It should be understood asthe number of particles that get entangled with the spin particle during the amplifica-tion process that takes place in any measurement apparatus and results in the physicaldisplacement of the pointers. Curioulsy, many discussions of quantum measurements within dBB theory emphasizethe role of the Bohmian position(s) associated with the microscopic system S, ignoringall those attached to the measurement apparatus M. Under these conditions, of course,the result of a measurement performed on S can reveal nothing but the initial value ofthe Bohmian position(s) attached to S. Nevertheless, we have learnt from the historicaldiscussions between Einstein and Bohr [15], for instance the argument concerning aninterference experiment with a moving pierced screen playing the role of a which wayapparatus, that the quantum properties of the measurement apparatus cannot be ignoredwithout running into contradictions. Indeed, the same rules have to be applied to both Sand M; otherwise one misses an essential purpose of the dBB theory, which is to propose19 framework for a completely unified dynamics.During the initial stage of a measurement process, S becomes entangled with somevariables of M, so that the velocity of the Bohmian position variables attached to Sdepend on those of M. Our analysis takes into account this entanglement and how itchanges the way the wave function of the whole system drives the positions. It shows thatthe initial value of the positions attached to M play a crucial role. This also illustratesthe intrinsic contextuality [3, 4, 19] of the dBB theory: the results associated with agiven observable do not depend only on some initial value of a variables pertaining to S;actually, for a macroscopic measurement apparatus, we have seen that they may even beindependent of the positions attached to S. This is in direct line with Bohr’s ideas, wherethe results crucially depend on the entire measurement apparatus which, in a Bohmiancontext, means that they depend on all the initial positions inside this apparatus.Another way to describe the effect of entanglement is to emphasize its nonlocalaspect. When S no longer interacts with M, the wave packets of the pointers may stilloverlap, so that no specific the result of measurement has emerged yet. Later, whenthe wave packets of the pointers have significantly moved, this overlap vanishes, forcingthe ensemble of pointer positions to “choose” a component of the wave function wherethe result is determined. In such a case, it is a nonlocal effect originating from themeasurement apparatus that drives S into the wave packet associated with the resultof measurement. The microscopic system has, so to say, to “follow the decision” of themacroscopic system. The left parts of figures 9 and 10 illustrate how the initial position ofthe spin particle then plays no role whatsoever in the determination of the measurementresult. This can be seen as the macroscopic counterpart of the nonlocal effect arising ina Bell experiment performed with two spins and two Stern-Gerlach apparatuses, wherethe position of each microscopic particle drives the position of the other particle.
References [1] J.S. Bell, “Bertlmann’s socks and the nature of reality”,
J. Physique colloques C2 ,41–62 (1981). This article is reprinted in pp. 139–158 of [2].[2] J.S. Bell, Speakable and Unspeakable in Quantum Mechanics , Cambridge UniversityPress (1987); second augmented edition (2004), which contains the complete set ofJ. Bell’s articles on quantum mechanics.[3] J.S. Bell, “On the problem of hidden variables in quantum mechanics”,
Rev. Mod.Phys. , 447–452 (1966); reprinted in Quantum Theory and Measurement , J.A.Wheeler and W.H. Zurek editors, Princeton University Press (1983), 396–402 andin chapter 1 of [2].[4] S. Kochen and E.P. Specker, “The problem of hidden variables in quantum mechan-ics”,
J. Math. Mech. , 59–87 (1967).[5] L. de Broglie, “La m´ecanique ondulatoire et la structure atomique de la mati`ere etdu rayonnement”, J. Physique et le Radium , s´erie VI, tome VIII, 225–241 (1927);20Interpretation of quantum mechanics by the double solution theory”,
Ann. Fond.Louis de Broglie , Nr 4 (1987); Tentative d’Interpr´etation Causale et Non-lin´eairede la M´ecanique Ondulatoire , Gauthier-Villars, Paris (1956).[6] D. Bohm, “A suggested interpretation of the quantum theory in terms of hiddenvariables”,
Phys. Rev. , 166–179 and 180–193 (1952).[7] P.R. Holland, The Quantum Theory of Motion , Cambridge University Press (1993).[8] X. Oriols and J. Mompart, “Overview of Bohmian mechanics”, Chapter 1 of
AppliedBohmian mechanics: from nanoscale systems to cosmology , 15-147, Editorial PanStanford Publishing Pte. Ltd. (2012) ; arXiv:1206.1084v2 [quant-ph].[9] D. D¨urr, S. Glodstein and N. Zanghi,
Quantum physics without quantum philosophy ,Springer (2013).[10] J. Bricmont,
Making sense of quantum mechanics , Springer (2016).[11] M. Gondran and A. Gondran, ‘Numerical simulation of the double-slit interferencewith ultracold atoms’, Am. J.Phys. Zeitschrift f¨ur Physik , 349-352 (1922).[13] G. Tastevin and F. Lalo¨e, “Surrealistic Bohmian trajectories do not occur withmacroscopic pointers”, Eur. Phys. J.
D 72 : 183 (2018).[14] D. Bohm and B.J. Hiley,
The undivided Universe , Routledge, London (1993).[15] N. Bohr, Discussion with Einstein on epistemological roblems in atomic physics,in: P.A. Schlipp (ed.)
Albert Einstein: Philosopher-scientist , La Salle, Open court,Illinois, p. 230.[16] D. D¨urr, S. Goldstein and N. Zanghi, “Quantum equilibrium and the origin ofabsolute uncertainty”,
Journal of Stat. Phys. , 843-905 (1992)[17] T. Norsen, “Bohmian conditional wave function (and the status of the quantumstate)”, Journal of Physics: Conference series
Ann. Physics , 371–389 (2002).[19] F. Lalo¨e,
Do we really understand quantum mechanics? , Cambridge University press(2012); §§