Protective Measurement and the Interpretation of the Wave Function

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 ?-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
• The wave function is a physical field.
• dBB theory, MWI, dynamical collapse theories
  etc.
• 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 ?-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.
• According 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

• 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
• Conventional impulse measurements
• Coupling interaction short duration and strong.
• Measurement result eigenvalues of A.
• Expectation value of A is obtained from the
  ensemble average.
• Weak measurements
• Coupling interaction short duration but weak.
• Measurement result expectation value of A.
• Individual measurement is imprecise. A normal
  ensemble is needed.
• 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 ?-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 ?-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
• PM implies the existence of mass and charge
  density.
• 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.
• 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)
• Bells 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 x1, particle 2 is in position x2, ,
  and particle N is in position xN.
• 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. ?-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
  ?-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 ?-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.
• Many worlds cannot exist at instants.
• Can 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.
• Many 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/)

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, 849853.
• 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.