1. Protective Measurement and the Interpretation of the Wave Function Shan Gao Unit for HPS & Center for time University of Sydney

2. PM & the Interpretation of the Wave Function Schrödinger asked at the fifth Solvay Conference (1927): What does the Ψ-function mean now, that is, how does the system described by it really look like in three dimensions?

3. PM & the Interpretation of the Wave Function Two views PM idea My analysis

4. Two views Two realistic views: 1.The wave function is a physical field. dBB theory, MWI, dynamical collapse theories etc. 2.The wave function is a description of some sort of ergodic motion of particles. It is assumed by stochastic interpretation etc.

5. Two views de Broglie-Bohm theory: The wave function is generally considered as an objective physical field, called Ψ-field. Various views on the nature of the field –a field similar to electromagnetic field (Bohm 1952) –active information field (Bohm and Hiley 1993) –a field carrying energy and momentum (Holland 1993) –causal agent more abstract than ordinary fields (Valentini 1997) * nomological view (Dürr, Goldstein and Zanghì 1997)

6. Two views Many-worlds interpretation: The wave function is taken as the basic physical entity with no a priori interpretation. Observers and object systems… They all are represented in a single structure, the field. (Everett 1957)

7. Two views Dynamical collapse theories: Mass density ontology (Ghirardi, Grassi and Benatti 1995) What the theory is about, what is real ‘out there’ at a given space point x, is just a field, i.e. a variable m(x,t) given by the expectation value of the mass density operator M(x) at x. (Ghirardi 2002)

8. Two views The essential difference lies in simultaneity: –A field exists throughout space simultaneously. –The ergodic motion of a particle exists throughout space in a time-divided way. A particle is still in one position at each instant, and it is only during a time interval that the ergodic motion of the particle spreads throughout space. Which view is right?

9. Two views Outline of my argument The above two views of the wave function can be tested by analyzing the mass and charge density of a quantum system. The field interpretation leads to self-interactions that contradict experimental observations. A further analysis can also determine which sort of ergodic motion of particles the wave function describes.

10. PM idea How do mass and charge distribute for a single quantum system? The mass and charge of a classical system always localize in a definite position in space. A ccording to PM, a quantum system has effective mass and charge density distributing in space, proportional to the modulus square of its wave function. (Aharonov, Anandan and Vaidman 1993)

11. PM idea 1.Suggested by the interaction terms in the Schrödinger equation ~~~ –If the charge does not distribute in some regions where the wave function is nonzero, then there will not exist any electrostatic interaction there. But the interaction does exist. –Since the integral is the total charge of the system, the charge density distribution in space will be.

12. PM idea Note that even in another theory different from SQM such as de Broglie-Bohm theory, the explanation of the measurement result of a PM should be the same. For example, the result of an appropriate adiabatic measurement of the Gauss flux out of a certain region should also be explained as the integral of the effective charge density over this region. Otherwise these measurement results can only be regarded as meaningless. But this strategy can hardly be satisfying, as we can then dismiss the measurement results of any property (by standard von Neumann procedure) as meaningless and thus deny the existence of all properties. Therefore, the above conclusion that a quantum system has an effective mass and charge density also holds true in other interpretations of QM.

13. PM idea Standard von Neumann procedure 1.Conventional impulse measurements Coupling interaction: short duration and strong. Measurement result: eigenvalues of A. Expectation value of A is obtained from the ensemble average. 2.Weak measurements Coupling interaction: short duration but weak. Measurement result: expectation value of A. Individual measurement is imprecise. A normal ensemble is needed. 3.Protective measurements Coupling interaction: long duration and weak. Measurement result: expectation value of A. Individual measurement is precise. Only a small ensemble is needed.

14. PM idea The mass and charge density can be measured by protective measurement as expectation values of certain variables for a single quantum system. –An appropriate adiabatic measurement of the Gauss flux out of a certain region will yield the value of the total charge inside this region, namely the integral of the effective charge density over this region. –Similarly, we can measure the effective mass density of the system in principle by an appropriate adiabatic measurement of the flux of its gravitational field.

15. PM idea A quantum system has effective mass and charge density distributing in space, proportional to the modulus square of its wave function. –Admittedly there have been some controversies about the meaning of protective measurement, but the debate mainly centers on the reality of the wave function (see e.g. Rovelli 1994; Uffink 1999; Dass and Qureshi 1999). –If one insists on a realistic interpretation of QM, then the debate will be mostly irrelevant and protective measurement will have strict restrictions on the realistic views of WF. Which view is consistent with this result?

16. My analysis If the mass and charge density simultaneously distributes in space (i.e. taking the wave function as a physical field), –A field by definition is a physical entity which properties are simultaneously distributed in space, no matter what type of field it is. Then the densities in different regions will have gravitational and electrostatic interactions. This not only violates the superposition principle of QM but also contradicts experimental observations.

17. My analysis The free Schrödinger equation with electrostatic and gravitational self-interactions is The measure of the strength of the electrostatic self-interaction (Salzman 2005) is The evolution of the wave function of an electron will be remarkably different from that predicted by QM and confirmed by experiments. [The energy levels of hydrogen atoms will be remarkably changed]

18. My analysis Therefore, the mass and charge density can only exist throughout space in a time-divided way. This means that at every instant there is only a localized particle with mass and charge, and only during a time interval, the time average of the ergodic motion of the particle forms the effective mass and charge density. As a result, the wave function is a description of some sort of ergodic motion of particles.

19. My analysis Objection 1: Charge density talking is nonsense. We never hear about it. Read the papers about PM. –Aharonov, Y., Anandan, J. and Vaidman, L. (1993). Meaning of the wave function, Phys. Rev. A 47, 4616. –Aharonov, Y. and Vaidman, L. (1993). Measurement of the Schrödinger wave of a single particle, Phys. Lett. A 178, 38. –Aharonov, Y., Anandan, J. and Vaidman, L. (1996). The meaning of protective measurements, Found. Phys. 26, 117. –Dass, N. D. H. and Qureshi, T. (1999). Critique of protective measurements. Phys. Rev. A 59, 2590. –Drezet, A. (2006). Comment on “Protective measurements and Bohm trajectories”, Phys. Lett. A 350, 416. –Gao, S. (2010). Meaning of the wave function. Forthcoming.

20. My analysis Objection 2: No mass and charge density exists, because SE is linear and forbids self-interactions. The existence of mass and charge density does not lead to self-interactions if the wave function is a description of some sort of ergodic motion of particles. Only the field view entails remarkable electrostatic self- interaction that contradicts experiments. In this sense, what linear SE rejects is not charge density but the field view.

21. My analysis Objection 3: Fields do not necessarily entail self- interactions. EM field is an example. The non-existence of EM self-interaction results from the fact that EM field itself has no charge. If the EM field had charge, then there would also exist EM self-interaction due to the nature of field, namely the simultaneous existence of its properties in space. Although an EM field has no EM self-interaction, it does have gravitational self-interaction ; the simultaneous existence of energy densities in different locations for an EM field must generate a gravitational interaction, though the interaction is too weak to be detected by current technology.

22. My analysis Objection 4: Ψ-field is so different from ordinary fields that it does not lead to self-interactions, even if it has mass and charge. A field by definition is a physical entity which properties are simultaneously distributed in space, no matter what type of field it is. Consider a single electron. Its wave function lives in real space and its mass and charge are simultaneously distributed in the space when taking the wave function as a field. So it is very difficult to deny the existence of an electrostatic self- interaction of the field. The fact that Ψ-field lives on configuration space does not remove the simultaneity nature of field; rather, it is in fact a common objection to the field view, and moreover, it in some sense favors the particle view. One can readily explain the multi-dimensionality of the wave function in terms of the ergodic motion of many particles.

23. My analysis Summary 1.PM implies the existence of mass and charge density. 2.In order to avoid self-interactions, the mass and charge density can only exist throughout space in a time-divided way, i.e., it is formed by time average of the motion of particles. 3.Therefore, the wave function is a description of some sort of ergodic motion of particles.

24. My analysis Which sort of motion? I only have time to tell my answer. –For details see my paper Meaning of the wave function. (philsci-archive.pitt.edu/8342/) What the WF describes is random discontinuous motion of particles.

25. My analysis Which sort of ergodic motion? The classical ergodic models that assume continuous motion of particles are not consistent with QM. –problems of stochastic interpretation –infinite velocity at the nodes of a stationary state –sudden acceleration and large radiation near these nodes –finite ergodic time

26. My analysis Double-slit experiment A single particle passes through both slits in a discontinuous way. A phenomenon which is impossible, absolutely impossible, to explain in any classical way. —R. Feynman

27. My analysis The ergodic motion must be discontinuous. If the motion of a particle is discontinuous and random, then the particle can readily move throughout all possible regions where the wave function is nonzero during an arbitrarily short time interval near a given instant. This will solve the problems plagued by the classical ergodic models. –no finite time scale –readily spreading to spatially separated regions –no infinite velocity and accelerating radiation –new definitions of energy and momentum

28. My analysis By assuming the wave function is a (complete) description for the motion of particles, we can reach this conclusion in a more direct way, independent of the above analysis. The modulus square of the wave function not only gives the probability density of finding a particle in certain locations, but also gives the objective probability density of the particle being there. (they should be the same when assuming M reflects R) Obviously, this kind of motion is essentially random and discontinuous.

29. My analysis The wavefunction gives not the density of stuff, but gives rather (on squaring its modulus) the density of probability. Probability of what exactly? Not of the electron being there, but of the electron being found there, if its position is ‘measured’. Why this aversion to ‘being’ and insistence on ‘finding’? The founding fathers were unable to form a clear picture of things on the remote atomic scale. — J. S. Bell, Against “measurement” (1990) Bell’s Everett (?) theory (1981)

30. My analysis The wave function is a description of quantum motion of particles, which is essentially discontinuous and random. (Implied by PM)

31. My analysis Description of random discontinuous motion (RDM) –position measure density and position measure flux density –Its equation of motion is the Schrödinger equation in QM. (spacetime translation invariance & relativistic invariance) The wave function is a complete description of RDM of particles.

32. My analysis It is very direct to extend the description of RDM of a single particle to RDM of many particles. –For the RDM state of N particles, we can define a joint position measure density. It represents the relative probability density of the situation in which particle 1 is in position x 1, particle 2 is in position x 2, …, and particle N is in position x N. –In a similar way, we can define the joint position measure flux density –The many-body wave function composed of them is then defined in 3N-dimensional configuration space, not in the real 3D space.

33. My analysis It seems that random discontinuous motion (RDM) provides a natural realistic alternative to the orthodox view. But the transition process from “being there” to “being found there”, which is closely related to the quantum measurement problem, needs to be further explained.

34. Implications The main realistic interpretations of QM cannot readily accommodate the result that the wave function has mass and charge density. –de Broglie-Bohm theory –Many-worlds interpretation –Dynamical collapse theories

35. Implications de Broglie-Bohm theory will be wrong –The theory takes the wave function as a physical field (i.e. Ψ-field) and further adds the non- ergodic motion of Bohmian particles to interpret QM. –This obviously runs counter to the picture of RDM of particles.

36. Implications It is often claimed that dBB theory gives the same predictions as QM by means of a quantum equilibrium hypothesis. But this equivalence is based on the wrong premise that the wave function, regarded as a Ψ-field, has no mass and charge density. Can dBB theory accommodate the result that the wave function has mass and charge density?

37. Implications Taking the wave function as a Ψ-field will lead to the existence of electrostatic self-interaction that contradicts both QM and experiments. No matter how to interpret the wave function, there will also exist an electromagnetic interaction between it and the Bohmian particle for a charged quantum system, as they both have charge. This also contradicts QM and experiments. To sum up, dBB theory is either empirically incorrect (by admitting these interactions) or logically inconsistent (by denying them).

38. Implications Many-worlds interpretation will be wrong –Its ontology needs to be revised from field to particle. –It can be further argued that there is only one world and QM is also a one-world theory in terms of RDM.

39. My analysis QM is a one-world theory: –Quantum superposition exists in a form of time division by means of RDM of particles. –During quantum evolution, there is only one observer (as well as one quantum system and one measuring device) all along in a continuous time flow.

40. Implications 1st serious objection to MWI: –If there are indeed many worlds, then each world can only exist in a discontinuous dense instant set, a time sub-flow of the continuous time flow. –As a result, at every instant only one of these worlds exists, and all other worlds do not exist at all. M any worlds cannot exist at instants. C an they exist during a time interval?

41. Implications 2nd serious objection to MWI: –Since the dense instant set occupied by each world is essentially random according to RDM, many worlds can never be formed. –Each world cannot ”know” which future instants it should occupy. It has no way to select from the random discontinuous instants the definite continuous content that should belong to it. M any worlds cannot exist during a time interval either.

42. My analysis Dynamical collapse theories will be in the right direction by admitting wavefunction collapse. But existing collapse theories require major revision –Ontology-revised from field to particle –Reformulated in the framework of RDM (e.g. the random source to collapse the wave function is not a classical field but the inherent random motion of particles) RDM will help to solve the problems of existing theories. The compete evolution law of RDM in discrete spacetime will include two parts: (1) linear Schrödinger evolution; (2) nonlinear stochastic evolution describing dynamical WF collapse (Gao 2006). No doubt much work still needs to be done… I am working on the collapse models satisfying energy conservation.

43. Implications But existing collapse theories require major revision –Ontology-revised from field to particle –Reformulated in the framework of RDM The random source to collapse the wave function is not a classical field but the inherent random motion of particles. Individual collapse processes satisfy energy conservation. Finite-sized instants (or discrete Planck time) is needed ( to release the randomness and discontinuity existing at instants). The staying tendency of particles as the real cause of WFC. The compete evolution law of RDM in discrete spacetime will include two parts: (1) linear Schrödinger evolution; (2) nonlinear stochastic evolution describing dynamical WF collapse (Gao 2006).

44. PM & the Interpretation of the Wave Function Summary PM implies WF has mass and charge density. –The field view leads to self-interactions. –Classical ergodic models of particles also fail. What the wave function describes is random discontinuous motion of particles. Shan Gao (2010), Meaning of the wave function. (http://philsci-archive.pitt.edu/8342/)http://philsci-archive.pitt.edu/8342/

45. PM & the Interpretation of the Wave Function …so crowded with empty sophistication that it is extremely difficult to perceive the simple errors at the basis. It is like fighting the hydra-cut off one ugly head, and eight formalizations take its place. --- P. K. Feyerabend (1924-1994) How to Defend Society Against Science Against “quantum foundations”

46. Selected publications S. Gao (2004) Quantum collapse, consciousness and superluminal communication, Foundations of Physics Letters 17(2), 167-182. S. Gao (2006) A model of wavefunction collapse in discrete space-time, International Journal of Theoretical Physics 45, 1965. S. Gao (2006) Quantum Motion: Unveiling the Mysterious Quantum World. Bury St Edmunds: Arima Publishing. S. Gao (2008) God Does Play Dice with the Universe. Bury St Edmunds: Arima Publishing. S. Gao (2010) On Diósi-Penrose criterion of gravity-induced quantum collapse, International Journal of Theoretical Physics 49, 849–853. S. Gao (2010) Meaning of the wave function, Forthcoming in International Journal of Quantum Chemistry. S. Gao (2010) The wave function and quantum reality, Forthcoming in AIP Conference Proceedings: Advances in Quantum Theory 2010.

47. Acknowledgments This work was supported by the Postgraduate Scholarship in Quantum Foundations provided by the Unit for HPS and Centre for Time (SOPHI) of the University of Sydney. I am very grateful to Dean Rickles, Huw Price, Antony Valentini, and Hans Westman for helpful discussions.