Why de Broglie-Bohm theory is probably wrong

Why de Broglie-Bohm theory is probably wrog Sha Gao Uit for History ad Philosophy of Sciece & Cetre for Time, SOPHI, Uiversity of Sydey Email: sgao7319@ui.sydey.edu.au 010-6-5, Versio 1.4 We ivestigate the validity of the field explaatio of the wave fuctio by aalyzig the mass ad charge desity distributios of a quatum system. It is argued that 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. This is also a cosequece of protective measuremet. 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 a remarkable electrostatic self-iteractio of its wave fuctio, though the gravitatioal self-iteractio is too weak to be detected presetly. This ot oly violates the superpositio priciple of quatum mechaics but also cotradicts experimetal observatios. Thus we coclude that the wave fuctio caot be a descriptio of a physical field. I the secod part of this paper, we further aalyze the implicatios of these results for the mai realistic iterpretatios of quatum mechaics, especially for de Broglie-Bohm theory. It has bee argued that de Broglie-Bohm theory gives the same predictios as quatum mechaics by meas of quatum equilibrium hypothesis. However, this equivalece is based o the premise that the wave fuctio, regarded as a Ψ-field, has o mass ad charge desity distributios, which turs out to be wrog accordig to the above results. For a charged quatum system, both Ψ-field ad Bohmia particle have charge desity distributio. This the results i the existece of a electrostatic self-iteractio of the field ad a electromagetic iteractio betwee the field ad Bohmia particle, which is cosistet with either the predictios of quatum mechaics or experimetal observatios. Therefore, de Broglie-Bohm theory as a realistic iterpretatio of quatum mechaics is probably wrog. Lastly, we suggest that the wave fuctio is a descriptio of some sort of ergodic motio (e.g. radom discotiuous motio) of particles, ad we also briefly aalyze the implicatios of this suggestio for other realistic iterpretatios of quatum mechaics icludig may-worlds iterpretatio ad dyamical collapse theories. Key words: wave fuctio; de Broglie-Bohm theory; Ψ-field; mass ad charge desity; protective measuremet; ergodic motio of particles; may-worlds iterpretatio; dyamical collapse theories 1. Itroductio.... How do mass ad charge distribute for a sigle quatum system?...3 3. Protective measuremet ad its aswer...5 4. Why the wave fuctio is ot a physical field...8 5. Neither Ψ-field or Bohmia particle is real...10 6. Further discussios...11 Ackowledgmets...13 Refereces...13

1. Itroductio De Broglie-Bohm theory is a otological iterpretatio of quatum mechaics iitially proposed by de Broglie ad later developed by Bohm (de Broglie 198; Bohm 195) 1. Accordig to the theory, a complete realistic descriptio of a quatum system is provided by the cofiguratio defied by the positios of its particles together with its wave fuctio. Although the de Broglie-Bohm theory is mathematically equivalet to stadard quatum theory, there is o clear cosesus with regard to its physical iterpretatio. I particular, the iterpretatio of the wave fuctio i this theory is still i hot debate eve today. The wave fuctio is geerally take as a objective physical field called Ψ-field. As stressed by Bell (1981): No oe ca uderstad this theory util he is willig to thik of Ψ as a real objective field rather tha just a probability amplitude. However, there are various views o exactly what field the wave fuctio is. It has bee regarded as a field similar to electromagetic field (Bohm 195), a active iformatio field (Bohm ad Hiley 1993), a field carryig eergy ad mometum (Hollad 1993), ad a causal aget more abstract tha ordiary fields (Valetii 1997) etc. I this paper, we will examie the validity of the field explaatio of the wave fuctio by aalyzig the mass ad charge desity distributio of a quatum system. First, it is argued that a quatum system with mass m ad charge Q, which is described by the wave fuctio ψ ( x,, has effective mass ad charge desity distributios mψ ( x, ad Qψ ( x, i space respectively. This result is also a cosequece of protective measuremet. Moreover, we argue that the field explaatio of the wave fuctio etails the existece of electrostatic self-iteractio for the wave fuctio of a charged quatum system, as the charge desity will be distributed i space simultaeously for a physical field. This ot oly violates the superpositio priciple of quatum mechaics but also cotradicts experimetal observatios. Thus we coclude that the wave fuctio caot be a descriptio of a physical field. Next, we ivestigate the implicatios of these results for de Broglie-Bohm theory. To begi with, sice the wave fuctio is ot a physical field, takig it as a Ψ-field is improper. For a charged quatum system the remarkable electrostatic self-iteractio of the field cotradicts experimetal observatios. Thus the Ψ-field assumed i de Broglie-Bohm theory caot exist i the actual world. Secodly, the assumed Bohmia particles caot be real either. Iasmuch as the wave fuctio has charge desity distributio i space for a charged quatum system, there will exist a equally remarkable electromagetic iteractio betwee it ad the Bohmia particles. This also cotradicts the predictios of quatum mechaics ad experimetal observatios. As a result, de Broglie-Bohm theory as a realistic iterpretatio of quatum mechaics is probably wrog. Lastly, we further suggest that the wave fuctio is a descriptio of some sort of ergodic motio (e.g. radom discotiuous motio) of particles, ad we also briefly discuss the implicatios of this suggestio for other realistic iterpretatios of quatum mechaics icludig may-worlds iterpretatio ad dyamical collapse theories. 1 Amog other differeces, de Broglie s dyamics is first order while Bohm s dyamics is secod order. It should be poited out that the wave fuctio is also regarded as omological, e.g. a compoet of physical law rather tha of the reality described by the law (Dürr, Goldstei ad Zaghì 1997; Goldstei ad Teufel 001). We will ot discuss this view i this paper. But it might be worth otig that this o-field view may have serious drawbacks whe cosiderig the cotigecy of the wave fuctio (see, e.g. Valetii 009), ad the results obtaied i this paper seemigly disfavor this view too.

. How do mass ad charge distribute for a sigle quatum system? 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 described by the wave fuctio ψ ( x,, how do its mass ad charge distribute i space the? We ca measure the total mass ad charge of the quatum system ad fid them i some regio of space. Thus the mass ad charge of a quatum system must also exist i space with a certai distributios if assumig a realistic view. Although the mass ad charge distributios of a sigle quatum system seem meaigless accordig to the orthodox probability iterpretatio of the wave fuctio, it should have a physical meaig i a realistic iterpretatio of the wave fuctio such as de Broglie-Bohm theory 3. As we thik, the Schrödiger equatio of a charged quatum system uder a exteral electromagetic potetial already provides a importat clue. The equatio is ψ ( x, h iq ih = A + Qϕ + V ψ ( x, (1) t m hc where m ad Q is respectively the mass ad charge of the system, ϕ ad A are the electromagetic potetial, V is a exteral potetial, h is Plack s costat divided by π, c is the speed of light. The electrostatic iteractio term Qϕψ ( x, i the equatio seems to idicate that the charge of the quatum system distributes throughout the whole regio where its wave fuctio ψ ( x, is ot zero. If the charge does ot distribute i some regios where the wave fuctio is ozero, the there will ot exist ay electrostatic iteractio there. But the term Qϕψ ( x, implies that there exists a electrostatic iteractio i all regios where the wave fuctio is ozero. Thus it seems that the charge of the quatum system should distribute throughout the whole regio where its wave fuctio is ot zero. Furthermore, sice the itegral + Qψ ( x, dx is the total charge of the system, the charge desity distributio i space will be Qψ ( x,. Similarly, the mass desity ca be obtaied from the Schrödiger equatio of a quatum system with mass m uder a exteral gravitatioal potetial V G : 3 Ufortuately it seems that the orthodox probability iterpretatio of the wave fuctio still iflueces people s mid eve if they already accept a realistic iterpretatio of the wave fuctio. Oe obvious example is that few people admit that the realistic wave fuctio has eergy desity (Hollad (1993) is a otable exceptio). If the wave fuctio has o eergy, the it seems very difficult to regard it as physically real. Eve if Bohm iterpreted the Ψ-field as active iformatio, he also admitted that the field has eergy, though very little (Bohm ad Hiley 1993). Oce oe admits that the wave fuctio has eergy desity, the it seems atural to edow it with mass ad charge desity, which are two commo sources of eergy desity.

ψ ( x, h ih = + mvg + V ψ ( x, () t m The gravitatioal iteractio term mv G ψ ( x, i the equatio also idicates that the (passive gravitatioal) mass of the quatum system distributes throughout the whole regio where its wave fuctio ψ ( x, is ot zero, ad the mass desity distributio i space is mψ ( x,. The above result ca be more readily uderstood whe the wave fuctio is a complete realistic descriptio of a sigle quatum system as i may-worlds iterpretatio ad dyamical collapse theories. If the mass ad charge of a quatum system does ot distribute as above i terms of its wave fuctio ψ ( x,, the other supplemet quatities will be eeded to describe the mass ad charge distributios of the system i space, while this obviously cotradicts the premise that the wave fuctio is a complete descriptio. I fact, the dyamical collapse theories such as GRW theory already admit the existece of mass desity (Ghirardi, Grassi ad Beatti 1995). I additio, eve i de Broglie-Bohm theory, which takes the wave fuctio as a icomplete descriptio ad admits supplemet hidde variables (i.e. the trajectories of Bohmia particles accompayig the wave fuctio), there are also some argumets for the above mass ad charge desity explaatio (Hollad 1993; Brow, Dewdey ad Horto 1995). It was argued that sice the Ψ-field depeds o the parameters such as mass ad charge, it may be said to be massive ad charged (Hollad 1993, p.79). Brow, Dewdey ad Horto (1995), by examiig a series of effects i eutro iterferometry, argued that properties sometimes attributed to the particle aspect of a eutro, e.g., mass ad magetic momet, caot straightforwardly be regarded as localized at the hypothetical positio of the particle i Bohm s theory. They also argued that it is hard to uderstad how the Aharoov-Bohm effect is possible if that the charge of the electro which couples with the electromagetic vector-potetial is ot co-preset i the regios o all sides of the cofied magetic field accessible to the electro (Brow, Dewdey ad Horto 1995, p.33). Oe may object that de Broglie-Bohm theory ad may-worlds iterpretatio seemigly ever admit the above mass desity explaatio, ad o existig iterpretatio of quatum mechaics icludig dyamical collapse theories edows charge desity to the wave fuctio either. As we thik, however, protective measuremet provides a more covicig argumet for the existece of mass ad charge desity distributios 4. The wave fuctio of a sigle quatum system, especially its mass ad charge desity, ca be directly measured by protective measuremet. Therefore, a realistic iterpretatio of quatum mechaics should admit the mass ad charge desity explaatio i some way; if it caot, the it will be at least problematic cocerig its explaatio of the wave fuctio. 4 It is very strage for the author that most supporters of a realistic iterpretatio of quatum mechaics igore protective measuremet ad its implicatios. Admittedly there have bee some cotroversies about the meaig of protective measuremet, but the debate maily ceters o the reality of the wave fuctio. If oe isists o a realistic iterpretatio of quatum mechaics such as de Broglie-Bohm theory, the the debate will be mostly irrelevat ad protective measuremet will have strict restrictios o the realistic iterpretatio.

3. Protective measuremet ad its aswer I this sectio, we will give a brief itroductio of protective measuremet ad its implicatio for the existece of mass ad charge desity distributios. Differet from the covetioal measuremet, protective measuremet aims at measurig the wave fuctio of a sigle quatum system by repeated measuremets that do ot destroy its state. The geeral method is to let the measured system be i a o-degeerate eigestate of the whole Hamiltoia usig a suitable iteractio, ad the make the measuremet adiabatically so that the wave fuctio of the system either chages or becomes etagled with the measurig device appreciably. The suitable iteractio is called the protectio. As a typical example of protective measuremet (Aharoov, Aada ad Vaidma 1993; Aharoov, Aada ad Vaidma 1996), we cosider a quatum system i a discrete odegeerate eergy eigestate ψ (x). The protectio is atural for this situatio, ad o additioal protective iteractio is eeded. The iteractio Hamiltoia for measurig the value of a observable A i the state is: H I = g( PA (3) where P deotes the mometum of the poiter of the measurig device, which iitial state is take to be a Gaussia wave packet cetered aroud zero. The time-depedet couplig g ( is T ormalized to g( dt = 1, where T is the total measurig time. I covetioal vo 0 Neuma measuremets, the iteractio H I is of short duratio ad so strog that it domiates the rest of the Hamiltoia (i.e. the effect of the free Hamiltoias of the measurig device ad the system ca be eglected). As a result, the time evolutio exp( ipa/ h) will lead to a etagled state: eigestates of A with eigevalues a i are etagled with measurig device states i which the poiter is shifted by these values a i. Due to the collapse of the wave fuctio, the measuremet result ca oly be oe of the eigevalues of observable A, say a i, with a certai probability p i. The expectatio value of A is the obtaied as the statistical average of eigevalues for a esemble of idetical systems, amely < A >= i p i a i. By cotrast, protective measuremets are extremely slow measuremets. We let g ( = 1/ T for most of the time T ad assume that g ( goes to zero gradually before ad after the period T. I the limit T, we ca obtai a adiabatic process i which the system caot make a trasitio from oe eergy eigestate to aother, ad the iteractio Hamiltoia does ot chage the eergy

eigestate. As a result, the correspodig time evolutio exp( ip < A > / h) shifts the poiter by the expectatio value < A >. This result strogly cotrasts with the covetioal measuremet i which the poiter shifts by oe of the eigevalues of A. It should be stressed that T is oly a ideal situatio 5, ad a protective measuremet ca ever be performed o a sigle quatum system with absolute certaity because of the tiy uavoidable etaglemet (see also Dass ad Qureshi 1999) 6. For example, for ay give values of P ad T, the eergy shift of the above eigestate, give by first-order perturbatio theory, is < A > P δ E =< H I >= (4) T Correspodigly, we ca oly obtai the exact expectatio value < A > with a probability very close to oe, ad the measuremet result ca also be the expectatio value < A >, with a probability proportioal to 1/ T, where refers to the ormalized state i the subspace ormal to the iitial state ψ (x) as picked out by first-order perturbatio theory (Dass ad Qureshi 1999). Therefore, a esemble, which may be cosiderably small, is still eeded for protective measuremets. Although a protective measuremet ca ever be performed o a sigle quatum system with absolute certaity, the measuremet is distict from the stadard oe: i o stage of the measuremet we obtai the eigevalues of the measured variable. Each system i the small esemble cotributes the shift of the poiter proportioal ot to oe of the eigevalues, but to the expectatio value. This essetial ovel poit has bee repeatedly stressed by the ivetors of protective measuremet (see, e.g. Aharoov, Aada ad Vaidma 1996). As we kow, i the orthodox iterpretatio of quatum mechaics, the expectatio values of variables are ot cosidered as physical properties of a sigle system, as oly oe of the eigevalues is observed i the outcome of the stadard measurig procedure ad the expectatio value ca oly be defied as a statistical average of the eigevalues. However, for protective measuremets, we obtai the expectatio value directly for a sigle system ad ot as a statistical average of eigevalues for a esemble. Sice the expectatio value of a variable ca be directly measured for a sigle system, it must be a physical characteristic of a sigle system, ot of a esemble (e.g. as a statistical average of eigevalues). This is a defiite coclusio we ca reach by the aalysis of protective measuremet. I the followig we will show that the mass ad charge desity ca be measured by protective measuremet as expectatio values of certai variable for a sigle quatum system, ad thus it is the physical property of the system (Aharoov ad Vaidma 1993). Cosider agai a quatum system i a discrete odegeerate eergy eigestate ψ (x). The iteractio Hamiltoia for measurig the value of a observable A i the state assumes the same form as 5 Note that the spreadig of the wave packet of the poiter also puts a limit o the time of the iteractio (Dass ad Qureshi 1999). 6 It ca be argued that oly observables that commute with the system s Hamiltoia ca be protectively measured with absolute certaity for a sigle system (see e.g. Rovelli 1994; Uffik 1999).

Eq. (3): H = g( (5) I PA where A is a ormalized projectio operator o small regios V havig volume v, which ca be writte as follows: A 1, = v 0, x V x V (6) The a protective measuremet of A will yield the followig result: 1 ψ = ψ A = ( x) dv v (7) 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) 7. 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 (Aharoov ad Vaidma 1993; Aharoov, Aada ad Vaidma 1996). 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 ideed distributed throughout space with effective mass desity m ψ ( x) ad effective charge desity Q ψ ( x) respectively. Although protective measuremet strogly suggests a realistic iterpretatio of the wave fuctio, it does ot directly tell us what the wave fuctio is. There are two possible ways to explai the wave fuctio i a realistic way. Oe view is to take the wave fuctio of a sigle quatum system as a physical etity simultaeously distributig i space like a field. This view is assumed by de Broglie-Bohm theory, may-worlds iterpretatio ad dyamical collapse theories 7 This meas that protective measuremet ca measure the desity of the Ψ-field or pilot wave i the cotext of Bohm s theory, ad it is a way of measurig the effect of the pilot wave without ivolvig the Bohmia particle itself (see also Drezet 006).

etc 8, ad it is also supported by the ivetors of protective measuremet (Aharoov ad Vaidma 1993) 9. The other view is to take the wave fuctio of a sigle quatum system as a descriptio of some kid of ergodic motio of a particle (or corpuscle). This view is assumed by stochastic iterpretatio etc, ad it was also discussed but rejected by Aharoov ad Vaidma (1993). The essetial differece betwee a field ad the ergodic motio of a particle lies i the property of simultaeity. The field exists throughout space simultaeously, whereas the ergodic motio of a particle exists throughout space i a time-divided way. The particle is still i oe positio at ay istat, ad it is oly durig a time iterval that the ergodic motio of the particle spreads throughout space. As we will see i the ext sectio, these two explaatios of the wave fuctio ca be distiguished by further aalyzig the mass ad charge desity distributios of a sigle quatum system, ad the former has already bee refuted by experimetal observatios. 4. Why the wave fuctio is ot a physical field Now we will ivestigate the implicatios of the existece of mass ad charge desity for the field explaatio of the wave fuctio 10. For the sake of simplicity, we will restrict our discussios to the wave fuctio of a sigle quatum system. The coclusio ca be readily exteded to may-body systems 11. If the wave fuctio is a physical field, the its mass ad charge desity will simultaeously distribute i space. This has two disaster results at least. Oe is that charge will ot be quatized; the total charge iside a very small regio ca be much smaller tha a basic charge for a sigle quatum system. This obviously cotradicts the commo expectatio that charge should be quatized. But maybe our expectatio eeds to be revised. So this result is ot fatal for the field explaatio of the wave fuctio. The other is that the wave fuctio will ot satisfy the superpositio priciple. For example, for the wave fuctio of a sigle electro, differet spatial parts of the wave fuctio will have gravitatioal ad electrostatic iteractios, as these parts have mass ad charge simultaeously. Let s aalyze the secod result i more detail. Iterestigly, the so-called Schrödiger-Newto equatio, which was proposed for other purposes (Diosi 1984; Perose 1998), just describes the gravitatioal self-iteractio of the wave fuctio. The equatio for a sigle quatum system ca be writte as ψ ( x, h ψ ( x, ψ ( x, 3 ih = Gm d x ψ ( x, Vψ ( x, t m x + (8) x x where m is the mass of the quatum system, V is a exteral potetial, ad G is Newto s gravitatioal costat. Much work has bee doe to study the mathematical properties of this iterestig equatio (see, e.g. Harriso, Moroz ad Tod 003; Moroz ad Tod 1999; Salzma 8 Accordig to Everett (1957), the wave fuctio is take as the basic physical etity with o a priori iterpretatio, ad observers ad object systems They all are represeted i a sigle structure, the field. 9 Note that protective measuremet itself does ot etail the field explaatio, ad it just shows that there is some sort of mass ad charge desity distributig i space. The desity may result from a physical field or the ergodic motio of a particle. As we thik, it seems that the existece of some observables such as positio i quatum mechaics already suggests the particle explaatio. A field has o positio property. Thus the expectatio value of a variable must be a physical characteristic of the motio of a particle, ot that of a field. 10 For recet objectios to the wave fuctio otology, see Moto (00, 006) ad Wallace ad Timpso (009). 11 It has bee widely ackowledged that for may-body systems the wave fuctios livig o cofiguratio space ca hardly be cosidered as real physical fields. Here we will show that eve the wave fuctio of a sigle quatum system, which lives o real space, caot be regarded as a physical field either.

005). Some experimetal schemes have bee also proposed to test its physical validity (Salzma ad Carlip 006). As we will see, although such gravitatioal self-iteractios caot yet be excluded by experimets 1, the existece of electrostatic self-iteractio already cotradicts experimetal observatios. If there is also a electrostatic self-iteractio, the the 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, (9) t m x x x where k is the Coulomb costat. Note that the gravitatioal self-iteractio is a attractive force, while the electrostatic self-iteractio is a repulsive force. It has bee show that the 4Gm measure of the potetial stregth of a gravitatioal self-iteractio is ε = for a free hc particle with mass m (Salzma 005). 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 particle with charge Q, the measure of the potetial 4kQ stregth of the electrostatic self-iteractio is ε =. As a typical example, for a free hc electro with charge e, the potetial stregth of the electrostatic self-iteractio will be 4 ke 3 ε = 1 10. This idicates that the electrostatic self-iteractio will have hc sigificat ifluece o the evolutio of the wave fuctio of a free electro. If such a iteractio ideed exists, it should have bee detected by precise experimets o charged microscopic particles. 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 cofirmed by experimetal observatios. Therefore, the electrostatic self-iteractio caot exist for the wave fuctio of a charged quatum system. 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. Oe may object to the above argumet with the example of classical electromagetic field. Electromagetic field is a field, but it has o self-iteractio. Thus a field does ot require the 1 It has bee argued that the existece of a self-iteractio term i the Schrödiger-Newto equatio does ot have a cosistet Bor rule iterpretatio (Adler 007). The reaso is that the probability of simultaeously fidig a particle i differet positios is zero. However, i a realistic iterpretatio of quatum mechaics where the wave fuctio is regarded as a real physical etity rather tha as a mere probability amplitude, the existece of gravitatioal self-iteractio term seems quite atural. For example, the field iterpretatio ca be cosistet with covetioal quatum measuremet via a dyamical collapse process. As we thik, oe covicig objectio is that if there is a self-gravitatioal iteractio for the wave fuctio of a charged particle, the there will also exist a electrostatic self-iteractio because the charge desity always accompaies the mass desity, while the existece of electrostatic self-iteractio is already icosistet with experimetal observatios (see below). If this objectio is valid, the the Schrödiger-Newto equatio will be wrog, ad moreover, the approach of semiclassical gravity will also be excluded (cf. Salzma ad Carlip 006).

existece of self-iteractio. However, this is a commo misuderstadig. The crux of the matter is that the o-existece of electromagetic self-iteractio results from the fact that electromagetic field itself has o charge. If the electromagetic field had charge, the there would also exist electromagetic self-iteractio due to the ature of field, amely the simultaeous existece of its properties i space. I fact, although electromagetic field has o electromagetic self-iteractio, it does have gravitatioal self-iteractio; the simultaeous existece of eergy desities i differet spatial locatios for a electromagetic field must geerate a gravitatioal iteractio, though the iteractio is too weak to be detected by curret techology. Oe may further object that the superpositio priciple i quatum mechaics already prohibits the existece of the above self-iteractios. But this is just the key poit we use to argue agaist the field explaatio of the wave fuctio. Let s state the argumet more explicitly. If the wave fuctio of a charged quatum system is a physical field, the the differet spatial parts of this field will have gravitatioal ad electrostatic iteractios. But the superpositio priciple i quatum mechaics, which has bee verified withi astoishig precisio, does ot permit the existece of such remarkable self-iteractios. Therefore, the field explaatio of the wave fuctio is already refuted by the superpositio priciple of quatum mechaics. Eve if the superpositio priciple may be violated whe cosiderig gravity (see, e.g. Perose 1996) 13, it does hold true for electromagetic iteractio. There is precise experimetal verificatio for the latter. Thus we coclude that the wave fuctio caot be a physical field. 5. Neither Ψ-field or Bohmia particle is real I the followig we will ivestigate the implicatios of the above results for de Broglie-Bohm theory. The theory assumes the realistic existece of both Ψ-field ad Bohmia particles. As we will argue below, however, these two kids of assumed etities caot be real accordig to the above results. To begi with, sice the wave fuctio is ot a physical field, takig it as a Ψ-field is improper. For a charged quatum system the remarkable electrostatic self-iteractio of the field cotradicts experimetal observatios. Thus the Ψ-field assumed i de Broglie-Bohm theory caot exist i the actual world. Secodly, the assumed Bohmia particles caot be real either. Iasmuch as the wave fuctio has charge desity distributio i space for a charged quatum system, there will exist a equally remarkable electromagetic iteractio betwee it ad the Bohmia particles. This is also icosistet with the predictios of quatum mechaics ad experimetal observatios. It has bee argued that de Broglie-Bohm theory gives the precisely same predictios as quatum mechaics by meas of quatum equilibrium hypothesis. Cocretely speakig, the quatum equilibrium hypothesis provides the iitial coditios for the guidace equatio which make de Broglie-Bohm theory obey Bor s rule i terms of positio distributios. Moreover, sice all measuremets ca be fially expressed i terms of positio, e.g. poiter positios, this amouts to full accordace with all predictios of quatum mechaics. However, this equivalece is based 13 Note that eve if there is such a violatio, its cause is probably ot the self-gravitatioal iteractio. The reaso is that if there is a self-gravitatioal iteractio for the wave fuctio of a charged quatum system, the there will also exist a electrostatic self-iteractio because the charge always accompaies the mass, while the existece of electrostatic self-iteractio is already icosistet with experimetal observatios.

o the premise that Ψ-field has o mass ad charge desity distributios. If the wave fuctio has mass ad charge desity distributios as we have argued above, takig it as a Ψ-field will lead to some predictios (e.g. the existece of electrostatic self-iteractio) that cotradict both quatum mechaics ad experimetal observatios. Certaily, oe ca elimiate the electromagetic iteractio betwee the Ψ-field ad Bohmia particles by deprivig the Bohmia particles of mass ad charge. But they will be ot real particles ay more. The i what sese the de Broglie-Bohm theory provides a realistic iterpretatio of quatum mechaics? Oe may also 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 accordig to our aalysis, 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 predictio (somewhat like the orthodox iterpretatio it tries to replace), it obviously departs from the iitial expectatios of de Broglie ad Bohm, ad i fact it already fails as a physical theory because of losig its explaatio ability. To sum up, the two realities assumed i de Broglie-Bohm theory, amely Ψ-field ad Bohmia particles, are ot real. Thus the theory as a realistic iterpretatio of quatum mechaics will aturally collapse. The basic reaso is that de Broglie-Bohm theory improperly iterprets the wave fuctio as a Ψ-field, ad moreover, it makes the situatio worse by addig the Bohmia particles. I the ext sectio, we will further ivestigate the physical meaig of the wave fuctio ad its implicatios for other realistic iterpretatios of quatum mechaics. 6. Further discussios If the wave fuctio is ot a descriptio of physical field as de Broglie-Bohm theory assumes, the exactly what does the wave fuctio describe? There is already a importat clue. It is that the superpositio priciple i quatum mechaics permits o existece of the self-iteractio of the wave fuctio i real space for a sigle quatum system. This idicates that the mass ad charge desity do ot exist i differet regios simultaeously. How is this possible? It aturally leads us to the secod view referred to i Sectio 3, which takes the wave fuctio as a descriptio of some kid of ergodic motio of a particle. 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. I other words, the mass ad charge desity exists i a time divisio way. At ay istat, there is oly a localized particle with mass ad charge. Thus there will ot exist ay self-iteractio for the wave fuctio. There are ideed some realistic iterpretatios of quatum mechaics that attempt to explai the wave fuctio i terms of some sort of ergodic motio of particles. A well-kow example is the stochastic iterpretatio of quatum mechaics (e.g. Nelso 1966). Nelso (1966) derived the Schrödiger equatio from Newtoia mechaics via the hypothesis that every particle of mass m is subject to a Browia motio with diffusio coefficiet h / m ad o frictio. I more techical terms, the quatum mechaical process is claimed to be equivalet to a classical Markovia diffusio process. O this iterpretatio, particles have cotiuous trajectories but o velocities, ad the wave fuctio is a statistical average descriptio of their motio. However, it

has bee poited out that the classical stochastic iterpretatios are icosistet with quatum mechaics (Glabert, Häggi ad Talker 1979; Wallstrom 1994). Glabert, Häggi ad Talker (1979) argued that the Schrödiger equatio is ot equivalet to a Markovia process, ad the various correlatio fuctios used i quatum mechaics do ot have the properties of the correlatios of a classical stochastic process. Wallstrom (1994) further showed that oe must add by had a quatizatio coditio, as i the old quatum theory, i order to recover the Schrödiger equatio, ad thus the Schrödiger equatio ad the Madelug hydrodyamic equatios are ot equivalet. I fact, Nelso (005) also showed that there is a empirical differece betwee the predictios of quatum mechaics ad his stochastic mechaics whe cosiderig quatum etaglemet ad olocality. I additio, it has bee geerally argued that the classical ergodic models that assume cotiuous motio caot be cosistet with quatum mechaics (Aharoov ad Vaidma 1993; Gao 010) 14. Classical ergodic models are plagued by the problems of ifiite velocity ad acceleratig radiatio (Aharoov ad Vaidma 1993). I particular, a particle udergoig cotiuous motio, eve if it has ifiite velocity, caot move throughout two spatially separated regios where the wave fuctio of the particle may spread. Besides, the classical ergodic models etail the existece of a fiite ergodic time, which is also icosistet with the existig quatum theory (Gao 010). Based o these results, it has bee suggested that the wave fuctio may describe radom discotiuous motio of particles (Gao 006a, 006b, 010). This ew iterpretatio of the wave fuctio ca avoid the problems of classical ergodic models, ad it also provides a atural realistic alterative to the orthodox view. O this iterpretatio, the square of the absolute value of the wave fuctio ot merely gives the probability of the particle beig foud i certai locatios, but also gives the objective probability of the particle beig there. Moreover, it seems that the theory of radom discotiuous motio ca also provide a promisig solutio to the otorious quatum measuremet problem (Gao 006a). However, the theory is still at its prelimiary stage, ad much study is still eeded before a defiite coclusio ca be reached about the true meaig of the wave fuctio. If the wave fuctio is ot a descriptio of physical field but a descriptio of some sort of ergodic motio (e.g. radom discotiuous motio) of particles, the the mai realistic iterpretatios of quatum mechaics will be either rejected or revised. We have already discussed the de Broglie-Bohm theory i the last sectio. Here we ca uderstad its wrogess more clearly. The theory takes the wave fuctio as a real physical field (i.e. Ψ-field) ad further adds the o-ergodic motio of Bohmia particles to iterpret quatum mechaics 15. But the wave fuctio i quatum mechaics is a descriptio of the ergodic motio of particles. Thus de Broglie-Bohm theory as a realistic iterpretatio of quatum mechaics should be rejected. Lastly, we will give some brief commets o the may-worlds iterpretatio ad dyamical collapse theories. These two theories both assume that the wave fuctio is a real physical etity, ad the dyamical collapse theories also explicitly assume the mass desity otology (Ghirardi, 14 Note that some variats of stochastic iterpretatio assume that the motio of particles is discrete radom jump (Bell 1986; Vik 1993; Barrett 005). But sice each radom jump is oly limited i a local regio, ad i particular, it reduces to the Bohmia trajectory i the cotiuum limit (Vik 1993), these models caot be cosistet with quatum mechaics either. I particular, they caot explai the existece of effective mass ad charge desity, which is proportioal to the square of the absolute value of the wave fuctio. 15 It has bee geerally argued that the time averages of Bohmia particle s positios typically differ markedly from the esemble averages, ad thus the motio of Bohmia particle is ot ergodic (Eglert, Scully, Süssma ad Walther 199; Aharoov ad Vaidma 1996; Aharoov, Eglert ad Scully 1999; Aharoov, Erez ad Scully 004).

Grassi ad Beatti 1995). But, as we have argued, the mass desity explaatio ca be valid oly i terms of the ergodic motio of particles due to the observatioal restrictio of electrostatic self-iteractio, ad the wave fuctio should be a descriptio of some ergodic motio of particles. Therefore, the otology of these two theories eeds to be revised from field to motio of particles. However, we may still have otology-revised may-worlds iterpretatio ad dyamical collapse theories. The left problem is to determie which is basically right: the former deies the existece of wavefuctio collapse while the latter admit its existece. If the wave fuctio is really a descriptio of the ergodic motio of particles, the it seems that quatum mechaics should be a oe-world theory, ot a may-worlds theory. The key poit is that quatum superpositio exists i a form of time divisio by meas of the ergodic motio 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. If this argumet is valid, the our defiite coscious experiece ad the defiite measuremet outcomes (e.g. positios of poiter) i the uique world will further demad that there exists a objective process of wavefuctio collapse, which is resposible for the trasitio from microscopic ucertaity to macroscopic certaity (e.g. i Schrödiger s cat thought experime 16. Therefore, it seems that the may-worlds iterpretatio might be wrog, ad the dyamical collapse theories may be i the right directio by admittig wavefuctio collapse. But the argumet here is very prelimiary ad udoubtedly eeds to be further examied. We will leave this importat issue for future study. Ackowledgmets I am deeply grateful to Atoy Valetii for his isightful criticism ad ecouragemet ad to Dea Rickles ad Huw Price for their helpful advice ad suggestios for may improvemets. This work was supported by the Postgraduate Scholarship i Quatum Foudatios provided by the Uit for History ad Philosophy of Sciece ad Cetre for Time (SOPHI) of the Uiversity of Sydey. Refereces Adler, S. L. (007). Commets o proposed gravitatioal modificatios of Schrödiger dyamics ad their experimetal implicatios. J. Phys. A 40, 755-764. Aharoov, Y., Aada, J. ad Vaidma, L. (1993). Meaig of the wave fuctio, Phys. Rev. A 47, 4616. Aharoov, Y., Aada, J. ad Vaidma, L. (1996). The meaig of protective measuremets, Foud. Phys. 6, 117. Aharoov, Y., Eglert, B. G. ad Scully M. O. (1999). Protective measuremets ad Bohm trajectories, Phys. Lett. A 63, 137. Aharoov, Y., Erez, N. ad Scully M. O. (004). Time ad Esemble Averages i Bohmia Mechaics. Physica Scripta 69,81 83. Aharoov, Y. ad Vaidma, L. (1993). Measuremet of the Schrödiger wave of a sigle particle, 16 Certaily, this coclusio is oly obtaied based o experiece, ad the physical origi of the wavefuctio collapse still eeds to be ivestigated.

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