The Wave Fuctio ad Quatum Reality Sha Gao Uit for History ad Philosophy of Sciece & Cetre for Time, SOPHI Uiversity of Sydey, Sydey, NSW 006, Australia Abstract. We ivestigate the meaig of the wave fuctio by aalyzig the mass ad charge desity distributios of a quatum system. Accordig to protective measuremet, a charged quatum system has effective mass ad charge desity distributig i space, proportioal to the square of the absolute value of its wave fuctio. I a realistic iterpretatio, the wave fuctio of a quatum system ca be take as a descriptio of either a physical field or the ergodic motio of a particle. The essetial differece betwee a field ad the ergodic motio of a particle lies i the property of simultaeity; a field exists throughout space simultaeously, whereas the ergodic motio of a particle exists throughout space i a time-divided way. If the wave fuctio is a physical field, the the mass ad charge desity will be distributed i space simultaeously for a charged quatum system, ad thus there will exist gravitatioal ad electrostatic self-iteractios of its wave fuctio. This ot oly violates the superpositio priciple of quatum mechaics but also cotradicts experimetal observatios. Thus the wave fuctio caot be a descriptio of a physical field but be a descriptio of the ergodic motio of a particle. For the later there is oly a localized particle with mass ad charge at every istat, ad thus there will ot exist ay self-iteractio for the wave fuctio. It is further argued that the classical ergodic models, which assume cotiuous motio of particles, caot be cosistet with quatum mechaics. Based o the egative result, we suggest that the wave fuctio is a descriptio of the quatum motio of particles, which is radom ad discotiuous i ature. O this iterpretatio, the square of the absolute value of the wave fuctio ot oly gives the probability of the particle beig foud i certai locatios, but also gives the probability of the particle beig there. The suggested ew iterpretatio of the wave fuctio provides a atural realistic alterative to the orthodox iterpretatio, ad it also implies that the de Broglie-Bohm theory ad may-worlds iterpretatio are wrog ad the dyamical collapse theories are i the right directio by admittig wavefuctio collapse. Keywords: wave fuctio; charge desity; protective measuremet; ergodic motio of particles PACS: 03.65.Ta 1. INTRODUCTION The wave fuctio is the most fudametal cocept of quatum mechaics. Accordig to the stadard probability iterpretatio, the wave fuctio is a probability amplitude, ad the square of its absolute value represets the probability desity for a particle to be measured i certai locatios. However, this iterpretatio is usatisfyig whe applyig to a fudametal theory because of resortig to measuremet. I view of the problem, some alterative realistic iterpretatios of the wave fuctio have bee proposed ad widely studied [1-4]. There are i geeral two ways to iterpret the wave fuctio of a sigle quatum system i a realistic
iterpretatio 1. Oe view is to take the wave fuctio as a physical etity existig throughout space simultaeously such as a field [1,,4]. The other view is to take the wave fuctio as a descriptio of some kid of ergodic motio of a particle [3]. I this paper, we will argue that these two iterpretatios of the wave fuctio ca i fact be tested by aalyzig the mass ad charge desity distributios of a quatum system, ad the former has already bee excluded by experimetal observatios. Moreover, a further aalysis ca also determie which kid of ergodic motio of particles the wave fuctio describes. The motio turs out to be radom ad discotiuous i ature.. PROTECTIVE MEASUREMENT AND CHARGE DENSITY The mass ad charge of a charged classical system always localize i a defiite positio i space at each momet. For a charged quatum system, how do its mass ad charge distribute i space the? Although this questio seems meaigless accordig to the probability iterpretatio of the wave fuctio, it should have a physical meaig i a realistic iterpretatio of the wave fuctio. We ca measure the total charge of a quatum system by electromagetic iteractio ad fid them i some regio of space after all. It ca be reasoably guessed that a quatum system has mass ad charge desity distributig i space, proportioal to the square of the absolute value of its wave fuctio [5]. This is also a cosequece of protective measuremet; the mass ad charge desity ca be measured by protective measuremet as expectatio values of certai variables for a sigle quatum system [6,7]. Cosider a quatum system i a discrete odegeerate eergy eigestate ψ (x). A protective measuremet of a observable A, which is a ormalized projectio operator o small regios V havig volume v, will yield the followig result [7]: 1 A = ψ ( x) dv = ψ v (1) v It is the average of the desity ψ ( x) over the small regio V. Whe v 0 ad after performig measuremets i sufficietly may regios V we ca fid the whole desity distributio ψ ( x). For a charged system with charge Q, the desity ψ ( x) times the charge yields the effective charge desity Qψ (x). I particular, a appropriate adiabatic measuremet of the Gauss flux out of a certai regio will yield the value of the total charge iside this regio, amely the itegral of the effective charge desity Q ψ ( x) over this regio [7]. Similarly, we ca measure the effective mass desity of the system i priciple by a appropriate adiabatic measuremet of the flux of its gravitatioal field. Therefore, protective measuremet shows that the mass ad charge of a sigle quatum system described by the wave fuctio ψ (x) is distributed throughout space with effective mass desity m ψ ( x) ad effective charge desity Q ψ ( x) respectively. 1 For the sake of simplicity, we will maily discuss the wave fuctio of a sigle quatum system i this paper. The coclusio ca be readily exteded to may-body system, which wave fuctio is defied i cofiguratio space. A elarged versio of this paper is available olie at PhilPapers [5].
3. WHY THE WAVE FUNCTION IS NOT A PHYSICAL FIELD Although protective measuremet strogly suggests a realistic iterpretatio of the wave fuctio, it does ot directly tell us what the wave fuctio is. The wave fuctio may describe a physical filed or some kid of ergodic motio of a particle. Correspodigly, the mass ad charge desity may result from a physical field or the ergodic motio of a particle. These two explaatios are essetially differet i that a field exists throughout space simultaeously, whereas the ergodic motio of a particle exists throughout space i a time-divided way. If the wave fuctio of a quatum system is a physical field, the its mass ad charge desity will simultaeously distribute i space. As a result, differet spatial parts of the wave fuctio will have gravitatioal ad electrostatic iteractios, as these parts have mass ad charge simultaeously. The the Schrödiger equatio for a free quatum system with mass m ad charge Q will be ψ ( x, h ψ ( x, ( ψ x, 3 ih = + ( kq Gm ) d x ψ ( x, () t m x x x where k is the Coulomb costat, ad G is Newto s gravitatioal costat. It has bee show that the measure of the potetial stregth of a gravitatioal selfiteractio is ε = ( 4Gm for a free system with mass m [8]. This quatity represets the stregth of the ifluece of self-iteractio o the ormal evolutio of the wave fuctio; whe ε 1 the ifluece will be sigificat. Similarly, for a free charged system with charge Q, the measure of the potetial stregth of the electrostatic self-iteractio is ε = ( 4kQ. For example, the potetial stregth 3 of the electrostatic self-iteractio is ε = (4ke 1 10 for a free electro. This idicates that the electrostatic self-iteractio will have sigificat ifluece o the evolutio of its wave fuctio. If such a iteractio ideed exists, it should have bee detected by precise experimets. As aother example, cosider the electro i the hydroge atom. Sice the potetial of its electrostatic self-iteractio is of the same order as the Coulomb potetial produced by the ucleus, the eergy levels of hydroge atoms will be sigificatly differet from those predicted by quatum mechaics ad measured by experimets. Therefore, the electrostatic self-iteractio caot exist. Sice the field explaatio of the wave fuctio etails the existece of such electrostatic self-iteractios, it caot be right, i.e. the wave fuctio caot be a descriptio of a physical field. 4. TOWARDS QUANTUM MOTION OF PARTICLES The failure of the field iterpretatio leads us to the secod view that takes the wave fuctio as a descriptio of some sort of ergodic motio of particles. O this view, the effective mass ad charge desity are formed by time average of the motio of a charged particle, ad they distribute i differet locatios at differet momets. Thus there will ot exist ay self-iteractio for the wave fuctio. I fact, if the mass ad charge desity does ot exist i differet regios simultaeously as the field
iterpretatio holds, they ca oly exist throughout space i a time-divided way. As a result, the wave fuctio must be a descriptio of the ergodic motio of particles. It ca be further argued that the classical ergodic models that assume cotiuous motio of particles caot be cosistet with quatum mechaics [5,7] 3. These models are plagued by the problems of ifiite velocity, acceleratig radiatio ad the existece of a fiite time scale etc [5,7]. I view of this egative result, it has bee suggested that aother differet kid of motio radom discotiuous motio ca aturally geerate the effective mass ad charge desity measurable by protective measuremet, ad what the wave fuctio describes is probably such quatum motio of particles, which is essetially discotiuous ad radom [11,1]. If the motio of a particle is ot cotiuous but discotiuous ad radom, the the particle ca readily move throughout all possible regios where the wave fuctio spreads durig a arbitrarily short time iterval ear a give istat. This will solve the problems of classical ergodic models [5]. I fact, by assumig the wave fuctio is a (complete) descriptio for the actual motio of particles, we ca reach the radom discotiuous motio i a more direct way. If the wave fuctio ψ ( x, is a descriptio of the state of motio for a sigle particle, the the quatity ψ ( x, dx will ot oly give the probability of the particle beig foud i a ifiitesimal space iterval dx ear positio x at istat t (as i stadard quatum mechaics), but also give the objective probability of the particle beig there. This accords with the reasoable expectatio that the probability distributio of the measuremet outcomes of a property is the same as the actual distributio of the property i the measured state. Obviously, this kid of motio is essetially radom ad discotiuous. The strict mathematical descriptio of radom discotiuous motio (RDM heceforth) ca be obtaied by usig the measure theory. It has bee show that the positio measure desity ρ ( x, ad the positio measure flux desity j ( x, provide a complete descriptio for the RDM of a sigle particle [1]. By assumig that the orelativistic evolutio equatio of RDM is the Schrödiger equatio, the wave fuctio ψ ( x, ca be uiquely expressed by ρ ( x, ad j ( x,, ad thus it also provides a complete descriptio of the RDM of a sigle particle. The ew iterpretatio of the wave fuctio i terms of RDM of particles provides a atural realistic alterative to the orthodox view. O this iterpretatio, the square of the absolute value of the wave fuctio ot oly gives the probability of a particle beig foud i certai locatios, but also gives the objective probability of the particle beig there. Certaily, the trasitio process from beig to beig foud, which is closely related to the otorious quatum measuremet problem, also eeds to be explaied. This issue will be discussed i the ext sectio. 5. FURTHER DISCUSSIONS If the wave fuctio is really a descriptio of quatum motio of particles, which is radom ad discotiuous i ature, the the mai realistic iterpretatios of quatum mechaics will be either rejected or revised. 3 It has bee poited out that the classical stochastic iterpretatios (e.g. [3]) are icosistet with quatum mechaics [9,10].
First, the de Broglie-Bohm theory will be wrog. The theory takes the wave fuctio as a physical field (i.e. Ψ-field) ad further adds the o-ergodic motio of Bohmia particles to iterpret quatum mechaics. This is obviously icosistet with the above result. As argued previously, takig the wave fuctio as a field will lead to the existece of electrostatic self-iteractio that cotradicts both quatum mechaics ad experimetal observatios. Moreover, iasmuch as the wave fuctio has charge desity distributio i space for a charged quatum system, there will also exist a electromagetic iteractio betwee it ad the Bohmia particles. This is also icosistet with quatum mechaics 4. Next, the otology of the may-worlds iterpretatio ad dyamical collapse theories eeds to be revised from field to particle. Besides, it ca be further argued that there is oly oe world ad quatum mechaics is also a oe-world theory. The key poit is that quatum superpositio exists i a form of time divisio by meas of the RDM of particles, ad there is oly oe observer (as well as oe quatum system ad oe measurig device) all alog i a cotiuous time flow durig quatum evolutio [5]. Thus the may-worlds iterpretatio will be wrog too. Moreover, there must exist a objective process of wavefuctio collapse, which is resposible for the trasitio from microscopic ucertaity to macroscopic (approximate) certaity. Therefore, the dyamical collapse theories will be i the right directio. It has bee argued that the discreteess of spacetime may ievitably result i the collapse of the wave fuctio, ad the compete evolutio law of RDM i discrete spacetime will aturally iclude the dyamical collapse of the wave fuctio. I particular, the motio of particles just provides the radom source to collapse the wave fuctio [11,1]. This may be a promisig start. But more study is still eeded before we ca solve the quatum measuremet problem (e.g. preferred basis problem) ad fially uderstad the meaig of quatum theory. REFERENCES 1. D. Bohm, Phys. Rev. 85, 166-193 (195).. H. Everett, Rev. Mod. Phys. 9, 454-46 (1957). 3. E. Nelso, Phys. Rev. 150, 1079 1085 (1966). 4. G. C. Ghirardi, R. Grassi, ad F. Beatti, Foud. Phys., 5, 313 38 (1995). 5. S. Gao, Meaig of the wave fuctio, http://philpapers.org/rec/gaomot. 6. Y. Aharoov, J. Aada, ad L. Vaidma, Phys. Rev. A 47, 4616 (1993). 7. Y. Aharoov ad L. Vaidma, Phys. Lett. A 178, 38 (1993). 8. P. J. Salzma, Ivestigatio of the Time Depedet Schrödiger-Newto Equatio Ph.D. Thesis, Uiversity of Califoria at Davis, 005. 9. H. Grabert, P. Häggi, ad P. Talker, Phys. Rev. A 19, 440 445 (1979). 10. T. Wallstrom, Phys. Rev. A 49, 1613 1617 (1994). 11. S. Gao, It. J. Theor. Phys. 45, 1943-1957 (006). 1. S. Gao, Quatum Motio: Uveilig the Mysterious Quatum World, Bury St Edmuds: Arima Publishig, 006. 4 Oe may wat to deprive the Ψ-field of mass ad charge desity to elimiate the electrostatic self-iteractio. But, o the oe had, the theory will break its physical coectio with quatum mechaics, as the wave fuctio i quatum mechaics has mass ad charge desity, ad o the other had, sice protective measuremet ca measure the mass ad charge desity for a sigle quatum system, the theory will be uable to explai the measuremet results either. Although de Broglie-Bohm theory ca still exist i this way as a mathematical tool for experimetal predictios, it obviously departs from the iitial expectatios of de Broglie ad Bohm, ad as we thik, it already fails as a physical theory because of losig its explaatio ability.