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2014 The iP lot-Wave Perspective on Spin Travis Norsen Smith College, [email protected]
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Recommended Citation Norsen, Travis, "The iP lot-Wave Perspective on Spin" (2014). Physics: Faculty Publications. 19. https://scholarworks.smith.edu/phy_facpubs/19
This Article has been accepted for inclusion in Physics: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact [email protected] The pilot-wave perspective on spin Travis Norsena) Smith College, Northampton, Massachusetts 01060 (Received 7 May 2013; accepted 22 November 2013) The alternative pilot-wave theory of quantum phenomena—associated especially with Louis de Broglie, David Bohm, and John Bell—reproduces the statistical predictions of ordinary quantum mechanics but without recourse to special ad hoc axioms pertaining to measurement. That (and how) it does so is relatively straightforward to understand in the case of position measurements and, more generally, measurements, whose outcome is ultimately registered by the position of a pointer. Despite a widespread belief to the contrary among physicists, the theory can also account successfully for phenomena involving spin. The main goal of this paper is to explain how the pilot- wave theory’s account of spin works. Along the way, we provide illuminating comparisons between the orthodox and pilot-wave accounts of spin and address some puzzles about how the pilot-wave theory relates to the important theorems of Kochen and Specker and Bell. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4848217]
I. INTRODUCTION alone. What the critics fail to appreciate, however, is that adding the particle positions allows something to be sub- In an earlier paper, which readers are encouraged to exam- tracted elsewhere in the system. In particular, the dynamical ine first and which I refer to hereafter as “the earlier paper,” laws sketched above—namely, Eqs. (1) and (2)—constitute I attempted to give a physicists’ introduction to the alterna- the entirety of the dynamical postulates of the theory. No tive, de Broglie-Bohm pilot-wave theory of quantum 1 additional axioms or special exceptions to the usual rules— phenomena. As a so-called “hidden variable theory,” the such as the collapse postulate of ordinary QM—need to be pilot-wave theory adds something to the state descriptions of introduced in order to understand measurement or, more gen- ordinary quantum mechanics: in addition to the usual wave erally, the emergence of the familiar everyday classical function W obeying the usual Schr€odinger equation world. In the earlier paper, this point was developed in the con- @W i"h H^W; (1) text of some simple scattering experiments involving single @t ¼ particles. In such situations, the fact that the scattered parti- cle is found, whole, at some particular place at the end of the one also has definite positions for each particle in the system. experiment is explained in the simplest imaginable way: For example, for a system of N spinless nonrelativistic par- there really is a particle following a definite trajectory and ticles, the position X~n of the nth particle will evolve accord- ing to hence possessing a perfectly definite position at all times. The detector finds it at a particular place (even when its wave function is spread out across several different places) dX~ t ~j ~x1;~x2; …;~x ; t nð Þ nð N Þ ; (2) because it is already at a particular place before interacting dt ¼ q ~x1;~x2; …;~xN; t ~ ~xk Xk t k with the detector. And equivariance, in light of the QEH, ð Þ ¼ ð Þ8 guarantees that the probability for the particle to be at a cer- where ~j "h=2mi Wà ~nW W ~nWà is (what in ordi- tain place at the end of the experiment perfectly matches n ¼ð Þð r À r Þ nary QM is termed) the probability current associated with what ordinary QM would instead describe as the probability particle n and q WÃW is (what in ordinary QM is termed) that the measurement intervention triggers a collapse that ¼ the probability density. makes the particle suddenly materialize at that place. It is As explained in the earlier paper, the pilot-wave theory thus clear, in pattern, how the pilot-wave theory reproduces involves the quantum equilibrium hypothesis (QEH), accord- the statistical predictions of ordinary QM for experiments ing to which the particle positions are assumed to be random, that end with the measurement of the position of a particle. with initial (t 0) probability distribution That the pilot-wave theory makes the same predictions as ¼ ordinary QM for other kinds of measurements as well can be 2 P X~1 ~x1; …; X~N ~xN W ~x1; …;~xN; 0 : (3) understood by including the particles constituting the meas- ½ ¼ ¼ ¼j ð Þj uring device in the system under study. Consider for example It is then a purely mathematical consequence of Eqs. (1) and a simple schematic treatment of the measurement of, say, the 2 3 (2) that the probability distribution will remain W -distributed energy of a particle with initial wave function w0 x . The for all times. The family of possible particle trajectoriesj j thus measurement apparatus is imagined to have a pointerð Þ with “flows along with” q,apropertythathasbeendubbedthe spatial coordinate y and initial wave function / y /0 y , 2 2 ð Þ¼ ð Þ “equivariance” of the W probability distribution. where /0 is a narrow packet centered at the origin, corre- Critics of the theoryj oftenj argue that the addition of these sponding to a “ready” state for the apparatus. definite particle positions is pointless (or “metaphysical”) An interaction between the particle and the pointer is because at the end of the day the theory’s empirical predic- regarded as a “measurement of the energy of the particle” if tions match those of ordinary quantum mechanics, which the Schr€odinger equation (for the particle pointer system) latter predictions are of course made using the wave function produces the following sort of time-evolutionþ in the case
337 Am. J. Phys. 82 (4), April 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 337
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 that the particle is initially in an “energy eigenstate” w x known to be impossible to do, for all of the non-commuting 0ð Þ wi x with eigenvalue Ei: components of spin, what the pilot-wave theory, as sketched ¼ ð Þ above, does for position. wi x /0 y wi x /o y kEi : (4) And yet, in fact, it is entirely false that the pilot-wave ð Þ ð Þ! ð Þ ð À Þ theory cannot deal with spin. Indeed, the truth is that the pilot- That is, at the end of the experiment, the wave function for wave theory deals with spin in an almost shockingly natural— the pointer is now sharply peaked around some new point certainly a shockingly trivial—way. The goal of the rest of y kEi proportional to the initial energy Ei of the particle. this paper is to resolve this paradox and explain how. ¼ The pointer, in short, indicates the energy of the particle. The main ideas of the paper are not new. The pilot-wave But the linearity of the Schr€odinger equation then immedi- theory was applied to spin already in 1955 by Bohm et al.,8 ately implies that, in the general case in which the particle is and numerical simulations of the theory’s account of spin initially in an arbitrary superposition of energy eigenstates, measurements were carried out by Dewdney et al. in the the evolution goes as follows: 1980s.9 Our approach here, following Bell’s more elegant treatment in Ref. 10, instead aims at simplicity and accessi- ciwi x /0 y ciwi x /0 y kEi : (5) bility. In particular, the proposed delta-function model of a i ð Þ ð Þ! i ð Þ ð À Þ Stern-Gerlach experiment (the one genuine novelty in the pa- X X per) allows one to solve the Schr€odinger equation and deter- The resulting “Schr€odinger cat” type state is of course prob- mine the pilot-wave particle trajectories (without any lematic from the point of view of ordinary QM: instead of recourse to numerical simulations) using the “plane-wave registering some one definite outcome for the experiment, the packet” methods developed in the earlier paper. pointer itself ends up in an entangled superposition. Enter the The delta-function model is developed in the following collapse postulate to save us from the troubling implications section; the associated pilot-wave particle trajectories are of universal Schr€odinger evolution! But this is no problem at then displayed and analyzed in Sec. III. Section IV explains all in the pilot-wave theory, according to which the (directly how the pilot-wave theory deals with cases of repeated spin perceivable) final disposition of the pointer is not to be found measurements, while Secs. V and VI develop examples to in the wave function, but instead in the actual position Y of illustrate the “contextuality” and “non-locality” exhibited by the pointer particle at the end of the experiment. It follows the theory. Some concluding remarks about the relationship from the equivariance property discussed earlier that, assum- between the pilot-wave and orthodox points of view are then ing the initial particle positions are random in accordance made in Sec. VII. 2 with the QEH, there will be a probability ci that the actual value of at the end of the experiment isj (near)j . That is, Y kEi II. THE STERN-GERLACH EXPERIMENT the pointer will point to a definite place, with probabilities given by the usual quantum rules, but, here, by rigorously fol- The Stern-Gerlach experiment, first and most famously lowing the basic dynamical rules, rather than making up spe- performed in 1922, involves subjecting a beam of particles cial exceptions when they produce embarrassing results. to an inhomogeneous magnetic field so the particles experi- One can thus understand how the pilot-wave theory man- ence a force proportional to a certain component of their ages to reproduce the statistical predictions of ordinary QM, intrinsic spin angular momenta.11 Since, empirically, the at least in cases where one measures position or measures incident beam gets split into two or more discrete sub-beams some other quantity but by means of the position of a (as opposed to a continuous distribution), the experiment is pointer. But … what about spin? Of all the phenomena sur- usually understood as demonstrating the quantization of spin veyed in undergraduate quantum mechanics courses, those angular momentum. involving intrinsic spin and its measurement—which Pauli In a standard textbook treatment of the experiment, we famously described as indicating a mysterious, non-classical assume a magnetic field Zweideutigkeit or “two-valuedness”4—seem perhaps the least amenable to the pilot-wave type of analysis just B~ bz^z; (6) sketched. Indeed, it is sometimes reported that the pilot- wave theory simply cannot deal with spin phenomena in a as might be produced, for example, by the magnets indicated plausible way.5 This view was expressed very eloquently by in Fig. 1.12 For a neutral spin-1/2 particle (such as the silver one of the anonymous referees of the earlier paper: atoms used in the original S-G experiment) the interaction Hamiltonian is “It has indeed been shown that Bohmian mechanics is equivalent to non-relativistic quan- H^ l~r B~ lbzr ; (7) tum mechanics with respect to its predictions con- ¼À Á ¼À z cerning particle positions. However, the scheme where is the magnitude of the particle’s magnetic moment encounters major problems: in spite of a half- l and the components of ~ are, in the usual representation, the century of effort by its adherents, it has not been r Pauli matrices. The eigenstates of this interaction possible to incorporate spin into the theory in a Hamiltonian will obviously be (proportional to) spinors convincing way …” 1 0 v z and v z , which satisfy This type of view is perhaps motivated by the various “no þ ¼ 0 À ¼ 1 hidden variables” theorems, such as that due to Kochen and Specker, which show quite clearly that it is mathematically rzv6z 6v6z: (8) impossible to assign pre-existing values to all relevant spin ¼ components of an ensemble of particles such that the quan- Now let us consider the spatial degrees of freedom of the tum mechanical predictions are reproduced.6,7 That is, it is particle’s wave function, focusing in particular on the
338 Am. J. Phys., Vol. 82, No. 4, April 2014 Travis Norsen 338
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 Fig. 1, by imagining that the field B~ is very strong, but is nonzero only in a vanishingly small region around y 0. That is, the interaction Hamiltonian of Eq. (7) is replaced¼ with H^ lbzd y rz, so that the total Hamiltonian govern- ing the particle’s¼À ð motionÞ in the yz-plane is
"h2 @2 "h2 @2 H^ lbzd y r : (13) ¼À2m @y2 À 2m @z2 À ð Þ z
Now, the situation can be treated like an elementary 2D- scattering problem in wave mechanics. To begin with, we imagine a “plane-wave packet” like those described in the earlier paper: the initial wave func- tion should be a packet of length L along the y-direction and width w along the z-direction, with constant amplitude
Fig. 1. Neutral spin-1/2 particles with initial spin state c v z c v z move in the region where its amplitude doesn’t vanish, propagat- in the y-direction toward a Stern-Gerlach apparatus withþ magneticþ þ À fieldÀ gra- ing initially in the y-direction with (reasonably sharply dient along the z-direction. The field delivers an impulse in the 6z-direction defined) energy E þ"h2k2=2m.Letusagainstartbyassum- to the components of the beam, so downstream there arise two spatially v6z ing that the incident¼ particle’s spin degrees of freedom are non-overlapping sub-beams with amplitudes c and c . In ordinary quantum mechanics, a measurement of the position ofþ the particleÀ downstream from described by one of the eigenspinors v6z.Then,inthetime the S-G device will find the particle in (or more accurately, cause it to mate- period during which the incident packet is interacting with 2 rialize in) the upper/lower sub-beam with probability c6 . the field at y 0, the wave function can be taken to be (up j j to an overall¼ time-dependent phase which we omit for simplicity) dimension (here, z) parallel to the magnetic field gradient. The beam is initially propagating, say, in the y-direction, iky iky þ Ae v6z Bz eÀ v6z for y < 0; so let us assume that, in the z-direction, the wave function is W y; z þ ik0y 6ijz initially constant (over the range where it is nonzero) and ð Þ¼( Ce e v6z for y > 0; that the particles are in eigenstates of : rz (14) W z; 0 Av : (9) ð Þ¼ 6z in the appropriate regions of the yz-plane. (Note that the fac- Now suppose the particles experience the magnetic field for tor of z, in the term involving B that represents a reflected wave, is put in for future convenience.) This expression is a a finite period of time T during which the interaction Hamiltonian above dominates all other terms. Then, the valid solution to the time-independent Schr€odinger equation for 0 and 0 as long as Schr€odinger equation reads y < y > 2 2 2 2 2 @W "h k "h k0 j i"h lbzr W; (10) E ð þ Þ : (15) @t ¼À z ¼ 2m ¼ 2m which, for the assumed initial condition, has solution In addition, the above expression should solve the Schr€odinger equation at y 0. This will be the case if the ¼ W z; T Ae6ijzv ; (11) wave function is continuous at y 0, requiring ð Þ¼ 6z ¼ A Bz Ce6ijz; (16) where "hj lbT is the magnitude of the (upward or down- þ ¼ ward) momentum¼ that the particle acquires as a result of tra- versing the field. For a beam of particles initially propagating and if the following condition on the y-derivatives of w in the y-direction, this impulse in the 6z-direction deflects the (arrived at by integrating the time-independent Schr€odinger equation from y 0À to y 0þ) is satisfied: beam either upward or downward depending on whether the ¼ ¼ particle was initially in the spin state v z or v z. Of course, in þ À general, the initial spin might be an arbitrary linear combina- @w @w 2mlb tion of the two z-spin eigenstates: 7 zw 0; z : (17) @y À @y ¼ "h2 ð Þ y 0þ y 0À ! ¼ ¼ v0 c v z c v z: (12) ¼ þ þ þ À À Plugging in our explicit expression for w y; z converts Eq. But then, since Schr€odinger’s equation is linear, it follows (17) into ð Þ that the S-G device will split the incoming wave function into two sub-beams—one with relative amplitude c that is 6ijz 2mlb 6ijz þ ik0Ce ikA ikBz 7 zCe : (18) deflected up, and one with relative amplitude c that is À þ ¼ "h2 deflected down. The net effect is pictured in Fig. 1. À For purposes of discussing the pilot-wave account of these These two conditions, Eqs. (16) and (18), cannot be satisfied phenomena, it will be helpful to work with a simplified exactly for all z. However, they can be approximately satis- model that captures all the physics we’ve just reviewed but fied over the narrow range of z where the (width-w) wave includes also an explicit treatment of the particle’s other spa- packet has support. Thus, for example, Eq. (16) is valid to tial degrees of freedom. This can be done, in the spirit of first order in jw if A C and ¼
339 Am. J. Phys., Vol. 82, No. 4, April 2014 Travis Norsen 339
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 B 6ijC: (19) Consider, for example, the wave function W y; z in the y < 0 ¼ region from Eq. (21). We have that ð Þ Then Eq. (18) is satisfied to the same degree of approxima- tion if k k0 (which implies, in light of Eq. (15), that j k) iky c W y; z Ae þ (24) and ð Þ¼ c À mlb j : (20) so that ¼ "h2k † iky W y; z AÃeÀ cà cà : (25) Notice that there are two distinct physical assumptions here. ð Þ¼ ð þ ÀÞ First, the impulse (of magnitude "hj) imparted to the particles Plugging into Eq. (23) then gives by the Stern-Gerlach fields must be small compared to the ini- tial momentum "hk of the particles, or equivalently j=k dX~ "hk lb=2E should be small. (Notice that b has the same units as¼ a y^: (26) magnetic field even though it is not one!) That is, the magnetic dt ¼ m fields should be appropriately “gentle.” And second, the width No matter its exact location within the wave, the particle will w of the incident packet should be small compared to 1=j. simply drift in the positive y-direction with the group veloc- Then Eq. (14) provides an acceptable description of the wave ity of the incident packet.13 function in the vicinity of the Stern-Gerlach apparatus while Eventually, of course, the particle will encounter the fields the length- L packet interacts with it. Note that, in these limits, at y 0 and pass over into the overlap region on the y > 0 the (z-dependent) amplitude of the reflected wave is (every- side.¼ Here, because the and – components of W have dif- where) small. Since it also plays no significant role in the ferent dependencies on zþ, we end up with physics to be discussed, we therefore drop it. (Those interested in the details should see Ref. 13 note.) dX~ "hk0 "hj So far we have assumed that the spin degrees of freedom y^ c 2 c 2 ^z; (27) dt ¼ m þ m j þj Àj Àj are given by one of the (z-direction) eigenspinors v . But it 6z ÀÁ is now straightforward to appeal to the linearity of the which can be understood as a weighted average of the group Schr€odinger equation in order to write a solution appropriate velocities associated with the two (now differently directed) for an arbitrary initial spinor like that of Eq. (12): components of the wave. The family of possible particle trajectories thus looks iky Ae c v z c v z y < 0; something like that illustrated in Fig. 2 for the particular case W y; z ðÞþ þ þ À À ik0y ijz ijz that c is a little bigger than c . It should be clear, for ð Þ¼( Ae eþ c v z eÀ c v z y > 0; þ þ þ À À example,j þj that if c 1 and c j À0j then, while in the over- ÀÁ(21) lap region, the particlesþ ¼ will justÀ ¼ move with the group veloc- ity of the component of the wave function and all where, of course, the expression for y < 0 applies only within incoming particlesþ (no matter their lateral position within the the range w=2 < z < w=2 where the plane-wave packet has support,À and the expression for y > 0 applies only within the triangular “overlap region” (see Fig. 1) where the two (separating) components of the wave coincide.
III. PARTICLE TRAJECTORIES IN THE PILOT-WAVE THEORY Following the methods introduced in the earlier paper we can use the structure of the wave function in (especially) the “overlap region” to analyze the motion of the particles in the pilot-wave theory and to demonstrate explicitly that the theory reproduces the usual quantum-mechanical predictions for the outcome of a Stern-Gerlach spin measurement. The first question that must be addressed, though, is this: what is the analog of Eq. (2) when the wave function
W W þ (22) ¼ W À Fig. 2. Representative sample of possible particle trajectories through a is a multi-component spinor? The answer is simply that the z-oriented Stern-Gerlach apparatus for an initial wave function with spinor components c 2=3 and c 1=3. The particle will move along equation is exactly the same, except we must sum over the with the incidentþ ¼ wave until crossingÀ ¼ over into the overlap region where the spin index in both the numerator and denominator, or equiv- and – componentspffiffiffiffiffiffiffiffi of the wave arep beginningffiffiffiffiffiffiffiffi to separate. In this region, † þ alently, writing W Wà Wà , we say the particle velocity is the weighted average, given in Eq. (27), of the group ¼ð þ ÀÞ velocities associated with the two separating components of the wave. Depending on its initial lateral position within the incoming wave packet, dX~ t "h W† ~W ~W† W the particle will either be shunted into the upper ( ) or lower (–) fork. ð Þ ðr ÞÀðr Þ : (23) þ dt ¼ 2mi † Subsequent detection of the particle in the upper/lower fork will result in its W W ~x X~ t being identified as “spin up/down along z.” ¼ ð Þ
340 Am. J. Phys., Vol. 82, No. 4, April 2014 Travis Norsen 340
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 wave) will be shunted into the upward sub-beam. end up going into the upper sub-beam. In short, P repre- Alternatively, if c 0 and c 1 the particles in the over- sents the probability that the S-G spin measurementþ will lap region will moveþ ¼ with theÀ group¼ velocity of the—compo- yield the outcome “spin-up.” Setting nent and all incoming particles will be shunted downward. Dz The equivariance property mentioned in the introduction slope (29) implies that if the position of the particle within the wave is ¼ Dy random and W 2-distributed at t 0, it will remain W 2-dis- tributed forj allj subsequent times.¼ This has the immediatej j and solving for P one indeed recovers that consequence that the probability for the particle to be located þ (downstream of the Stern-Gerlach device) in the upper P c 2; (30) (lower) sub-beam, and hence counted as “spin-up” (“spin- þ ¼j þj down”) if its position there is detected, is c 2 c 2 . It is confirming explicitly that, indeed, the pilot-wave dynamics thus clear that the pilot-wave theory reproducesj þj ðj À thej Þ usual for the wave and particle are compatible in the way quantum-mechanical predictions for the statistics of such expressed by the formal equivariance property: the particle spin measurements. trajectories “bend just the right amount” to yield a final prob- One of the nice things about the way we have set things ability distribution for the particle position that is consistent up, though, is that this claim can be demonstrated explicitly with the usual QM predictions and, more importantly, what by considering the “critical trajectory,”14 which divides the is seen in experiments. trajectories emerging into the upper and lower beams. By definition, the critical trajectory arrives just at the vertex on the right of the triangular overlap region behind the magnets, IV. ADDITIONAL MEASUREMENTS as shown in Fig. 3. The slope of the critical trajectory is So far, we have concentrated on the measurement of the given by the ratio of the z- and y-components of the velocity z-component of the spin using a Stern-Gerlach device whose in the overlap region, Eq. (27): magnetic field gradient is along the z-direction. By simply rotating the device, however, we can also measure the com- c 2 c 2 j þ À ponent of the spin along (say) an arbitrary direction n^ in the slope j j2 Àj j2 : (28) ¼ c c k0 xz-plane: j þj þj Àj n^ cos h ^z sin h x^: (31) On the other hand, as explained in the figure caption, the crit- ¼ þ ical trajectory traverses a distance Dy wk0= 2j in the y- direction and a distance Dz wP w¼=2 in theð zÞ-direction, The whole measurement procedure can be analyzed in where P is the fraction of¼ the lateralþ À width w of the beam exactly the same way we’ve done above, mutatis mutandis: þ with the property that, should the particle begin there, it will the eigenspinors v6n will now be the eigenvectors of cos h sin h r ; (32) n ¼ sin h cos h À an arbitrary initial wave-packet (proportional to initial spinor v0) can be written as a linear combination of these eigenspinors
v0 c nv n c nv n; (33) ¼ þ þ þ À À and the whole analysis of Secs. II and III goes through, with z being everywhere replaced by n.15 Thus, it should be clear that the pilot wave theory reproduces the usual quantum sta- tistical predictions for spin measurements, whether it is the z- or some other component of spin that is to be measured. Perhaps the theory will fail, however, when we consider the possibility of subsequent and/or repeated measurements? Fig. 3. Particles that begin above the critical trajectory will be shunted into The pilot-wave theory is, after all, a deterministic hidden the upper (“spin-up”) wave packet downstream of the Stern-Gerlach mag- variable theory: for a given experimental setup, the outcome nets, while those that begin below the critical trajectory will be shunted into of a spin measurement is determined by the initial state of the lower (“spin-down”) packet. For initial particle positions that are (in ac- the “particle” (meaning here the particle wave combina- cordance with the QEH) uniformly distributed across the lateral extent of þ the incident packet, the probability for a particle to emerge with the “spin- tion). Standard textbook discussions might perhaps suggest up” packet will be the fraction of the lateral width (w) of the packet that is that there would be a problem. above the critical trajectory, i.e., the distance from the critical trajectory to For example, suppose that particles emerging from the the top of the incident packet is wP . The vertical displacement Dz of the “spin-up” port of a z-oriented Stern-Gerlach device (SGz) are critical trajectory as it traverses theþ triangular overlap region is then subsequently sent through an SGx device; those particles wP w=2. By considering the right triangle that is the lower half of the þ À emerging as “spin-up” along x are then subjected to a further overlap region (and whose hypotenuse is one edge of the “spin-up” packet), measurement of their spin’s z-component (see Fig. 4). The one can see that the horizontal extent Dy of the overlap region is w=2 j=k : the slope w=2 =Dy of the hypotenuse should match the ratio result, of course, is that fully half of the particles entering it ð Þð 0Þ ð Þ j=k0 of the z- and y-components of the group velocity associated with the will emerge from the final SGz device as “spin-down.” The “spin-up” part of the wave. usual interpretation is that spin-along-z and spin-along-x are
341 Am. J. Phys., Vol. 82, No. 4, April 2014 Travis Norsen 341
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 quantum theory of course exhibits this behavior: in quantum mechanics, the state of the particle, after a measurement, is postulated to “collapse” to the eigenstate of the appropriate operator corresponding to the actually-observed outcome. (It is worth noting here that the “actually-observed outcome” is yet another additional postulate of the theory—for Bohr, for Fig. 4. Schematic representation of a series of Stern-Gerlach spin measure- example, this was realized in some classical macroscopic ments, in which the devices are replaced by “black boxes” with two output ports, one for “spin-up” along the direction n indicated by the box’s label pointer somewhere, whose physical relationship to the parti- cle in question is, at best, obscure.) The point here is that the “SGn,” and one for “spin-down.” Here, particles that emerge as “spin-up” along the z-direction and then “spin-up” along the x-direction are subjected pilot-wave theory also exhibits this behavior: the physical to a further measurement of z-spin. Half of the particles exiting this final state of (in particular) the wave guiding the particle is differ- SG device are “spin-up,” with the other half being “spin-down.” As dis- z ent before, and after, the intermediate SGx device. But in the cussed in the text, both ordinary QM and the pilot-wave theory are able to pilot-wave theory, this difference—the physical influence of account for the observed statistics, because in both theories the state of a the measurement on the properties of the system in questio- particle is in general affected by subjecting it to a spin measurement. In par- n—is a natural and straightforward consequence of the usual ticular, the state of a particle exiting the first SGz device is not the same as the state of that same particle when it exits the intermediate SGx device. dynamical laws.
incompatible properties, corresponding to non-commuting V. CONTEXTUALITY quantum mechanical operators, so the determination of a In the previous section I stressed that the pilot-wave definite value for spin-along-x completely erases the theory is not the “naive” sort of hidden-variable theory that previously-definite value for spin-along-z. Surely a is sometimes discussed (and, when discussed, always refuted non-quantum, deterministic hidden-variable theory could not with great pomp!) in textbooks. In these “naive” theories, reproduce this paradigmatically quantum result? measurements simply passively reveal some pre-existing But a little thought reveals that, yes, the pilot-wave theory value of the property being measured, without affecting the reproduces this prediction quite easily. The crucial point is state of the particle. In the pilot-wave theory, by contrast, that, although the particles entering the SGx device were there is a natural mechanism (which, quite literally, is noth- 1 being guided by waves proportional to , the wave guid- ing but Eqs. (1) and (2) that define the theory’s dynamics) 0 whereby the physical measurement intervention affects the ing those particles that successfully emerge from the state of the measured system. There is thus a sense in which, “spin-up” port of the SGx device is proportional to for the pilot-wave theory just as for ordinary QM, the mea- 1 1 surement cannot be thought of as passively revealing some v x p . The v x and v x components of the wave þ ¼ 2 1 þ À pre-existing quantity, but should instead be thought of as an get separated by the SGx device, whereas the particle goes active intervention which brings about the new, final state of one wayffiffi or the other. And, by definition, if the particle ends the particle corresponding to the measurement outcome. up in the “spin-up-along-x” beam, downstream of the SGx But there is an even deeper sense in which, for the pilot- device, the component of the wave now surrounding and wave theory, the measurement cannot be thought of as pas- guiding it is the “spin-up-along-x” part. So it is as if the par- sively revealing a pre-existing value. Let us discuss this in ticle’s quantum state has “collapsed,” as per the usual quan- terms of a concrete example. Recall the SGz device sketched tum mechanical rules, even though, in fact, no collapse—no in Fig. 1: the magnets produce a magnetic field near the ori- violation of the usual Schr€odinger evolution of the wave— gin that can be approximated by the expression in Eq. (6). has occurred. The other, “spin-down-along-x,” part of the The field is such that a classical particle, with magnetic wave is still present—it’s just “over there,” far away from dipole moment in the z direction, will feel a force in the z the actual location of the particle, and hence dynamically direction and hence beþ deflected up upon passing through theþ irrelevant to the present or future motion of the particle.16 SG device. Likewise, a classical particle with magnetic This separation of the different spin components of the wave dipole moment in the z direction will feel a force in the z packet of course happens in ordinary quantum mechanics as direction and will henceÀ be deflected down. This is,À in well. But in ordinary QM there is nothing like the actual essence, the basis for saying that, when a particle emerges position of the particle, in addition to the wave function, to from the SG device having been deflected up, it is “spin-up warrant talk of the particle actually being up there (i.e., talk along z,” etc. of the spin measurement having actually had the outcome But other ways of measuring the z-component of a par- “spin-up”), and so additional dynamical postulates are ticle’s spin can also be contemplated. For example, imagine required. This is an illustration of the point made in the intro- a device—let us call it here an SGz0 device—like that indi- duction, that although the pilot-wave theory adds something cated in Fig. 5. The structure is identical to the original SGz to the physical state descriptions, it is not really more (let device except that the polarity of the magnets has been alone pointlessly more) complicated than ordinary QM reversed and hence Eq. (6) is replaced with because this addition allows for the subtraction of the other- wise rather dubious measurement axioms. B~0 bz^z: (34) What is actually shown by this kind of example is not that À one cannot explain the observed statistics of spin measure- This reversal of the magnetic field gradient reverses the ments with a non-quantum, deterministic, hidden-variable effect on particles passing through the device: now, a classi- theory, but rather only that the physical state of a particle cal particle with a magnetic dipole moment in the z direc- (known, say, to be “spin-up-along-z”) must be affected by a tion will feel a force in the z direction and henceþ be measurement of (for example) its spin-along- x. Ordinary deflected down upon passing throughÀ the device, while a
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 spin of the particle” is not simply a function of the initial state of the “particle” (i.e., the particle wave complex). The exact same initial state can yield eitherþ of the two possible measure- ment outcomes, depending on which of two possible experi- mental devices is chosen for performing the experiment. This is a concrete illustration of the fact that is usually put as follows: for the pilot-wave theory, spin is “contextual.” That is, the outcome of a measurement of a certain compo- nent of a particle’s spin depends, in the pilot-wave theory, not just on the initial state of the particle but also on the overall experimental context, that is, on the particular “way” that the measurement is implemented. This “contextuality” should be contrasted to the “non- contextual” type of hidden variable theory in which for each (Hermitian) quantum mechanical operator (corresponding to Fig. 5. An alternative way to “measure the z-component of the spin of a par- some “observable” property), each particle in an ensemble ticle” is to pass the particle through the SGz0 device shown here. This is the possesses a definite value that is simply revealed by the appro- same as an SGz device, but with the polarity of the magnets—and hence the priate kind of experiment, independent of which specific ex- calibration of the device—reversed: particles that deflect up are now counted perimental implementation is used to measure the observable as “spin-down along z,” while particles that deflect down are now counted as in question. In particular, a non-contextual hidden-variable “spin-up along z.” theory would (by definition) assign a definite value to the z- component of each particle’s spin, independent of whether the classical particle with magnetic dipole moment in the z spin is measured using an SGz or an SG0 device. More gener- direction will feel a force in the z direction and will henceÀ z þ ally, non-contextuality can be understood as the requirement be deflected up. The SGz0 device is a perfectly valid one for that each observable should possess a particular definite value, measuring the z-component of the spin of a particle; it that will be revealed by a measurement of that observable, no merely has a different “calibration” (one might say) than the matter what compatible observables are perhaps being meas- original device. Whereas, with the original device, particles ured simultaneously. (The following section provides a con- deflected up are declared to be “spin-up along z,” particles crete example of this more general sort of contextuality.) deflected up by the modified device are instead declared to That the “no hidden variables” theorems of von Neumann, be “spin-down along z,” and vice versa. And indeed, a full Jauch-Piron, and Gleason tacitly assume non-contextuality quantum mechanical analysis, parallel to that undertaken for (or an even stronger and less innocent requirement) was first the original SGz device in Sec. III, leads to the conclusion clearly pointed out in Bell’s landmark 1966 paper, “On the 18 that an incident wave packet proportional to c v z c v z Problem of Hidden Variables in Quantum Mechanics.” þ þ þ À À will be split, upon passage through the SGz0 device, into two (This paper, incidentally, was written in 1964, prior to Bell’s sub-beams, one proportional to c v z that has been deflected more famous 1964 paper proving what is now called “Bell’s þ þ down and the other proportional to c v z that has been Theorem,” but remained unpublished until 1966 due to an À À deflected up. editorial accident.) The somewhat more famous Kochen- The significance of these two distinct pieces of experimen- Specker “no hidden variables” theorem, which appeared the tal apparatus—both equally qualified to “measure the z-com- year after Bell’s paper, also assumes non-contextuality, as ponent of the spin of a particle”—becomes clear when we discussed in the beautiful review paper by Mermin.6,7 consider what happens when, for example, a particle with ini- As Bell explains, these proofs all rely on relating “in a 1 tial wave function proportional to 1 is incident on the nontrivial way the results of experiments that cannot be per- p2 1 formed simultaneously.” (For example, they assume that the two devices. For this case, the particle velocity in the overlap results of an SGz-based and an SGz0 -based measurement region is proportional to y^, i.e., the particleffiffi trajectories simply should be the same.) Bell elaborates: continue in a straight line all the way through the overlap “It was tacitly assumed that measurement of an region until they finally emerge into one of the now spatially observable must yield the same value independently separated sub-beams corresponding to their initial lateral posi- of what other measurements may be made tions within the incident packet. That part is identical whether simultaneously. Thus as well as [some observable the original SG or the alternative SG device is involved. But z z0 with corresponding QM operator A^] say, one might there is a crucial difference between the two devices: if (say) measure either [B^] or [C^], where [B^ and C^ the particle happens to start in the upper half of the incident commute with A^, but not necessarily with each packet, it is destined to eventually find itself in the upward- other]. These different possibilities require different deflected sub-beam and hence be counted as “spin-up along experimental arrangements; there is no a priori z”—if the measurement is carried out using the original SG z reason to believe that the results for [A^] should be device. But if instead the measurement is carried out using the the same. The result of an observation may alternative SG device, the exact same “particle”—that is, the z0 reasonably depend not only on the state of the exact same wave function and the exact same particle location system (including hidden variables) but also on the in the upper half of the wave packet—is instead destined to complete disposition of the apparatus.”18 eventually find itself in the upward-deflected sub-beam and hence be counted as “spin-down along z.”17 Bell then references Bohr’s insistence on remembering In short, for the (deterministic, hidden variable) pilot-wave “the impossibility of any sharp distinction between the theory, the outcome of “measuring the z-component of the behavior of atomic objects and the interaction with the
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 measuring instruments which serve to define the conditions in exactly the way we have discussed previously for the under which the phenomena appear.”19 Abner Shimony has pilot-wave theory of single particles. aptly described Bell’s invocation of Bohr (the arch-opponent The CWFs have, however, several perhaps-unexpected of hidden variables) in Bell’s defense of hidden variables properties that should be acknowledged. First, it should be (against the theorems supposedly, but not actually, showing appreciated that, in general, the CWFs do not obey some sim- them to be impossible) as a “judo-like manoeuvre.”20 ple one-particle Schr€odinger equation.22 Evaluating Eq. (36) For our purposes, the upshot of all this is the following. In at ~x2 X~2 gives an overall (time-dependent) multiplicative the literature on hidden variables one often finds the implica- factor,¼ which means that (even in the case of non-entangled tion that the requirement of non-contextuality is quite reasona- particles) the overall phase and normalization of the CWF can ble in the sense that contextuality would supposedly involve change erratically. This, however, is irrelevant to the motion putting in “by hand” an obviously implausible ad hoc kind of of the associated particle because any multiplicative factor apparatus-dependence. But as the simple example of the simply cancels out in Eq. (38). In addition, the CWF of each contextuality of the pilot-wave theory (involving the distinct particle carries spin indices for both particles. In the case outcomes for SGz- and SGz0 -based measurements of z-spin) where the two-particle wave function is factorizable, as in Eq. 2 makes clear, this is not the case at all. The “context- (36), the particle-2 spinor (v0) is irrelevant to the motion of dependence” arises in a perfectly straightforward and natural particle 1, and vice versa. The “extra” spinor thus acts just like way, again as a direct consequence of the fundamental the overall multiplicative constant. It is thus clear that when dynamical postulates of the theory given by Eqs. (1) and (2). the two-particle wave function factorizes—when the two par- ticles are not entangled—the motions of the two particles will be fully independent: if the particles subsequently encounter VI. NON-LOCALITY Stern-Gerlach devices, each will (independently and sepa- rately) act exactly as described in the previous sections. Bell’s 1966 paper closes by noting that Bohm’s 1952 If, however, the spins of the two particles are initially pilot-wave theory eludes the various “no hidden variables” entangled, as for example in theorems by means of its contextuality—but that the context-dependence implies a non-local action at a distance 1 1 2 1 2 in situations involving entangled but spatially-separated par- W ~x1;~x2 w ~x1;~x2 v zv z v zv z ; (39) ð Þ¼ ð Þ p2 þ À À À þ ticles: “in this theory an explicit causal mechanism exists whereby the disposition of one piece of apparatus affects the 18 then the non-locality appears.ffiffiffi Suppose for example that both results obtained with a distant piece.” Let us explore this particles are to be subjected to SG -based measurements of type of situation, in which the pilot-wave theory’s contex- z their z-spins. And suppose that particle 1 encounters its SGz tuality implies a non-local dependence on a remote context. device first. A simple calculation shows that, in the overlap Consider then a pair of spin-1/2 particles, propagating in region behind the magnets, the velocity of particle 1 will be opposite directions along (and already widely separated proportional to y^ (i.e., it will simply continue in a straight hor- along) the y-axis. The initial spatial part of the 2-particle izontal line through the overlap region). It will then exit the wave function can be taken to be overlap region into one or the other of the two downstream 0;d;0 0; d;0 sub-beams, depending on its initial z-coordinate: if the particle w ~x1;~x2 Ukð Þ ~x1 Uð kÀ Þ ~x2 ; (35) happens to have started in the upper half of the packet it will ð Þ¼ ð Þ À ð Þ 0;d;0 1 end up deflecting up and being counted as “spin-up,” whereas where Ukð Þ ~x represents a plane-wave packet centered at if it happens to have started in the lower half of the packet it ~x 0; d; 0 withð Þ central wave vector k~ ky^. Consider first ¼ð Þ ¼ will end up deflecting down and being counted as “spin- the case in which the spin part of the initial state also factor- down.” Supposing, for notational simplicity, that the two now izes, so that the total state can be written spatially-separated sub-beams are bent (say, by additional appropriate SG type magnets) so that they again propagate in 0;d;0 1 0; d;0 2 W ~x1;~x2 Uð Þ ~x1 v Uð À Þ ~x2 v : (36) the y-direction but with displacements 6D in the z-direc- ð Þ¼ k ð Þ 0 k ð Þ 0 þ21 tion, the two-particle wave function after particle 1 has We may introduce the so-called “conditional wave function” passed through its SGz device will take the form (CWF) for particle 1 as follows: 1 0;d0;D 0; d0;0 1 2 W Ukð Þ ~x1 Uð kÀ Þ ~x2 v zv z ~x ~x;~x ; (37) ¼ p2 ð Þ À ð Þ þ À W1 W 2 ~x2 X~2 ð Þ¼ ð Þj ¼ h 0;d0; D 0; d0;0 1 2 ffiffiffi Ukð À Þ ~x1 Uð kÀ Þ ~x2 v zv z : (40) where X~2 is, of course, the actual position of particle 2. The À ð Þ À ð Þ À þ CWF for particle 2 is defined in a parallel way. That is, the overall wave function is a superposition of two These CWFs are convenient because the dynamical law 1 2 terms: one, proportional to v v , in which the particle 1 for the particle positions, Eq. (2), can be reformulated to z z wave packet has been displacedþ aÀ distance D in the positive give, for each particle, an expression for the particle velocity 1 2 z-direction, and the other, proportional to v v , in which in terms of its own CWF: z z the particle 1 wave packet has been displacedÀ a distanceþ D in † † the negative z-direction. dX~n "h W ~Wn ~W Wn n n The actual location X~1 of particle 1 will be either near ðr ÞÀð† r Þ : (38) dt ¼ 2mi W W ~ z or near z (again, depending on its random initial n n ~x Xn D D ¼ position).¼ It is thus¼À meaningful already at this stage to speak
The CWFs can thus be thought of as “one-particle wave of the actual outcome of the SGz-based measurement of the functions,” each of which guides its corresponding particle, z-spin of particle 1. The crucial point is now that the CWF
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 for particle 2—the thing that will determine how particle 2 pilot-wave theory accounts for the empirically observed corre- behaves when it subsequently encounters its SGz device— lations in the more general sort of case in which arbitrary com- depends on where particle 1 ended up. If particle 1 went up ponents of spin are measured. One of the particles will ~ 0;d0;D (that is, if X1 is now in the support of Ukð Þ) then the CWF encounter its measuring device first; the outcome of this first of particle 2 is (proportional to) measurement will be determined by the random initial position of this measured particle within its wave packet; the comple- 0; d0;0 tion of this first measurement induces a collapse in the distant W2 ~x Uð kÀ Þ ~x v z (41) ð Þ¼ À ð Þ À particle’s CWF; and this in turn determines the statistics for a subsequent measurement on the distant particle. (Showing and its subsequent interaction with an SGz device will, inde- ~ that, in addition, the expected statistical results are reproduced pendent of the exact position X2, result in its being deflected even if one or both of the measuring devices are replaced with down. Whereas if instead particle 1 ended up going down alternative SGn0 devices is left as an exercise for the reader.) ~ 0;d0; D (that is, if X2 is now in the support of Ukð À Þ) then the That so much hangs on which measurement happens first CWF of particle 2 is instead (proportional to) (or more mathematically, that the CWF of each particle is the two-particle wave function evaluated at the actual cur- 0; d0;0 W2 ~x Uð kÀ Þ ~x v z (42) rent position of the other particle, no matter how distant) ð Þ¼ À ð Þ þ makes it clear that the non-locality of the pilot-wave theory and its subsequent interaction with an SGz device will, inde- will be difficult to reconcile with relativity. This is, however, pendent of the exact position X~2, instead result in its being in principle no different from the non-locality implied by the deflected up. That is, the CWF for particle 2 collapses, as a collapse postulate of ordinary QM. (Indeed, the collapse of result of the measurement on particle 1, to a state of definite the pilot-wave theory’s CWFs can and should be understood z-spin. This happens, however, despite the fact that the as a mathematically precise derivation of the usual textbook two-particle wave function obeys the (unitary) Schr€odinger approach to measurement, in a theory where imprecise equation without exception! And the usual quantum mechan- notions like “measurement” play no fundamental role.) And ical statistics are reproduced: the two possible joint outcomes of course, the “grossly non-local structure” of the pilot-wave (particle 1 is spin-up and particle 2 is spin-down, or particle theory and ordinary QM turns out to be “characteristic… of 1 is spin-down and particle 2 is spin-up) each occur with any such theory which reproduces exactly the quantum me- 50% probability. Of course, the pilot-wave theory being after chanical predictions”—as shown in Bell’s other landmark all deterministic, these probabilities are quite reducible. In paper, of 1964.24 particular, which of the two joint outcomes occurs depends on the random initial position of particle 1 (specifically, its initial z-coordinate). In order to make the non-local character of the theory absolutely clear, it is helpful to now consider an alternative scenario in which particle 1 is not subjected to any measure- ment. For example, perhaps Alice, who is stationed there next to it, decides (at the last possible second before particle 1 arrives) to yank the SGz device out of the way. Then, the two-particle wave function remains in a state described by Eq. (39) until particle 2 arrives at its SGz device. But then particle 2 will behave in just exactly the way we previously described for particle 1 (when it was the first to encounter an SGz device): its velocity in the overlap region behind the magnets will be proportional to y^ and the outcome of the spin measurement will depend on the initial position (in par- ticular, the z-coordinate) of particle 2. The non-locality is thus clear: even for the same initial state—two-particle wave function given by Eq. (39) and, say, both particle positions, X~1 and X~2, in the upper halves of their respective packets—the outcome of the measurement on particle 2 depends on what Alice chooses to do in the vi- cinity of particle 1. If Alice subjects particle 1 to an SGz- Fig. 6. Illustration of the two scenarios discussed in the text. In the top frame, based measurement, that measurement will have outcome particle 1 (on the right) encounters its SGz device first; the particle is found to “spin-up,” the CWF of particle 2 will collapse to a state pro- be spin-up (solid trajectories) or spin-down (dashed trajectories) depending on the initial z-coordinate of the particle. If particle 1 goes up, the collapse suffered portional to v z, and the subsequent measurement of particle À by the CWF of particle 2 (on the left) causes it to go down (solid trajectories) 2’s z-spin will have outcome “spin-down.” On the other regardless of its initial z-coordinate. On the other hand, if particle 1 goes down, hand, if Alice removes the SGz device, the measurement of the collapse causes particle 2 instead to go up (dashed trajectories) regardless particle 2’s z-spin will instead have the outcome spin-up.23 It of its initial z-coordinate. In the lower frame, particle 1 is not subjected to any is clear that the non-locality is a form of contextuality in measurement. The result of measuring the z-spin of particle 2 is then deter- which the part of the experimental “context” affecting the mined by the initial z-coordinate of particle 2. The difference between the two realized outcome of the experiment is (Fig. 6). scenarios exemplifies the contextuality of the pilot-wave theory (since the result remote of measuring the spin of particle 2 depends not only on the initial state but on Although we have focused here on the concrete example in whether or not the spin of particle 1 is measured jointly) and also its non- which the z-components of the spins of both particles are (per- locality (since the choice of whether or not to measure particle 1 could be haps) measured, it should now be clear that, and how, the made at space-like relativistic separation from the measurement of particle 2).
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 VII. CONCLUSIONS as its actual z-component of spin that is supposed to be revealed in the experiment. For a general ini- Contrary to an apparently widespread belief, the pilot- tial wave function there is no such property. What wave theory has no trouble accounting for phenomena is more, the transparency of the [pilot-wave] analy- involving spin. It does so, actually, in the most straightfor- sis of this experiment makes it clear that there is ward imaginable way: the wave function (a scalar-valued nothing the least bit remarkable (or for that matter field on the configuration space in the case of spinless par- “nonclassical”) about the nonexistence of this ticles) is replaced with the appropriate spinor-valued func- property.”27 tion, evolving according to the appropriate generalization of the Schr€odinger equation, just exactly as in ordinary QM. As explained by Bell, the appearance to the contrary—that The dynamical law for the motion of the particles, Eq. (2), is, the tacit assumption that there must be some real “z-com- doesn’t change at all—one need simply recall, again from or- ponent of spin” property that the measurements unveil— dinary QM, that for particles with spin the definitions of both seems to arise from the unfortunate and inappropriate conno- ~ jn and q involve summing over the spin indices. Thus under- tations of the word “measurement”: stood, the pilot-wave theory reproduces the usual quantum “the word comes loaded with meaning from statistical predictions for spin measurements (including everyday life, meaning which is entirely sequences of measurements performed on a single particle, inappropriate in the quantum context. When it is and joint measurements performed on pairs of perhaps- said that something is “measured” it is difficult not entangled particles) but without any additional postulates or to think of the result as referring to some pre- ad hoc exceptions to the usual dynamical laws. And it does existing property of the object in question. This is this, incidentally, in a way that could have been anticipated: to disregard Bohr’s insistence that in quantum phe- spin measurements too (like position measurements and nomena the apparatus as well as the system is measurements whose results are registered by the position of essentially involved. If it were not so, how could a pointer) ultimately come down to the position of a particle we understand, for example, that measurement of a (downstream from an appropriate Stern-Gerlach device). component of “angular momentum”—in an arbi- The theory exhibits “contextuality,” as illustrated in the two trarily chosen direction—yields one of a discrete examples discussed: the result of a measurement depends not set of values? When one forgets the role of the ap- only on the state of the particle prior to measurement but also paratus, as the word measurement makes all too on the measurement’s specific experimental implementation. likely, one despairs of ordinary logic—hence In particular, the outcome of a measurement of the z-compo- “quantum logic.” When one remembers the role of nent of the spin of a particle can depend on whether the spin is the apparatus, ordinary logic is just fine.”29 measured using an SGz or an SGz0 device, and/or can depend on which (commuting) observable is also being jointly meas- Unfortunately, even some of the people in the best possi- ured. Considered in the abstract, the idea of a contextual hid- ble position to understand this important point—namely, den variable theory perhaps sounds contrived and implausible. proponents of the pilot-wave theory—have missed it and Such, at least, has apparently been the thinking behind the sug- have thus pointlessly laden the theory with additional varia- gestions that the various “no hidden variables” theorems (due bles corresponding to such actual spin components.30 One to von Neumann, Kochen-Specker, and so on) rule out the pos- goal of the present paper is thus to present, in a clear and ac- sibility of (or even an important class of) hidden variable theo- cessible way, the simplest possible pilot-wave account of ries. But the example of the pilot-wave theory shows, quite spin, in order to rectify misconceptions like that held by the simply and conclusively, that they don’t. And furthermore, anonymous referee quoted in the introduction. like it or lump it, the theory (backed up by Bell’s theorem) Finally, note that it is by no means only champions of hid- shows that there is nothing the least bit contrived or intolerable den variable theories that are sometimes seduced into thinking in the sort of contextuality that is required to eliminate the of spin components as real properties. Although the official “unprofessionally vague and ambiguous” aspects of ordinary 28 party line of the orthodox quantum theory, as expressed, for QM in favor of something clear and mathematically precise. example, in standard textbooks, is of course that there are no The idea that there should be something contrived or intol- such things, one often finds that the authors, perhaps swept erable about contextuality undoubtedly arises from the idea away by all the talk of “measuring” “observables,” are some- that, if a property really exists, measurement of it should—by what conflicted about this. Townsend, for example, whose definition—simply reveal its value. It would be hard, actually, beautiful textbook begins with five full chapters on spin, to disagree with this sentiment. The key question, though, is stresses early on that, because the operators corresponding to precisely whether any such property exists. As has been dis- distinct spin components fail to commute, “the angular mo- cussed in illuminating detail in Ref. 25, the real lesson to be mentum never really ‘points’ in any definite direction.”31 This taken away from examining the pilot-wave perspective on already implies the usual, orthodox view that there is simply spin is that so-called “contextual properties” (like the individ- no such thing as a spin angular momentum vector, pointing in ual spin components in the pilot-wave theory) are not proper- some particular direction, at all. ties at all.26 They simply do not exist and there is nothing And yet—even for Townsend, who is far more careful mysterious about this at all, just as there is nothing mysterious about this issue than most authors32—there is a tendency to in the fact that the eventual flavor of a loaf of bread (which occasionally slip into suggesting that there is such a thing, or depends not just on the ingredients but also on how it is later at least that (even though there isn’t) it is sometimes helpful to baked!) is not a pre-existing property of the raw dough: pretend that there is. Thus, for example, Townsend later “Note that one can completely understand what’s explains that “placing [a spin-1/2] particle in a magnetic field going on in [a] Stern-Gerlach experiment without in the z direction rotates the spin of the particle about the invoking any putative property of the electron such z-axis as time progresses …” We also find statements about
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 how, for example, in the weak decay of the muon, “the posi- for example, “an electron.” Of course, according to orthodox QM, an elec- tron is preferentially emitted in a direction opposite to the spin tron is not literally a particle in the sense of being a point-like object fol- direction of the muon” and even a description of x t lowing a definite trajectory. Whereas, according to the pilot-wave theory, w a single electron comprises both a quantum wave and a (literal) particle. and y w t , the amplitudes for a particle inh spinþ j stateð Þi hþ j ð Þi Readers are hereafter assumed to be able to judge, from the context, in w t to be found, respectively, spin-up along the x- and y- which sense the word “particle” is used. directions,j ð Þi as “the components of the intrinsic spin in the x- y 4W. Pauli, “Uber€ den Zusammenhang des Abschlusses der plane.” So perhaps after all particles do have definite spin vec- Elektronengruppen im Atom mit der Komplexstruktur der Spektren,” Z. Phys. 31, 765–783 (1925). tors? No, Townsend reminds us that this language and the 5 associated physical picture are not to be taken too seriously: See, for example, Lubos Motl, “David Bohm born 90 years ago,”
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53 convince oneself of this. Alternatively, one can recognize that the statistics associated with spin are reproduced. Thus the picture resulting from a of the measurement outcomes will not depend on the exact cross-sectional hidden-variable account of quantum mechanics need not very much resem- profile of the incident packet and hence imagine that the cross-sectional ble the traditional classical picture that the researcher may, secretly, have profile of the incident packet is rotated in tandem with the SG device so been keeping in mind. The electron need not turn out to be a small spin- that the analysis remains transparent. ning yellow sphere.” Gestures in the direction of this same point were 16Of course, the mere fact that the two sub-beams are spatially separated is made even earlier by M. Renninger, “Zum Wellen-Korpuskel-Dualismus,” not sufficient to prove that the empty packet is forevermore dynamically Z. Phys. 136, 251–261 (1953), translated by W. De Baere as “On Wave- irrelevant to the motion of the particle. The “empty” spin-down sub-beam Particle Duality,”
348 Am. J. Phys., Vol. 82, No. 4, April 2014 Travis Norsen 348
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.229.64.25 On: Fri, 05 Aug 2016 14:48:53