2. The measurement problem

Paul Budnik paul@mtnmath.com

The formulation of QM describes the deterministic unitary evolution of

a wave function. This wave function is never observed experimentally.

The wave function allows us to compute the probability that certain

macroscopic events will be observed. There are no events and no

mechanism for creating events in the mathematical model. It is this

dichotomy between the wave function model and observed macroscopic

events that is the source of the interpretation issue in QM. In

classical physics the mathematical model talks about the things we

observe. In QM the mathematical model by itself never produces

observations. We must interpret the wave function in order to relate

it to experimental observations.It is important to understand that this is not simply a philosophical

question or a rhetorical debate. In QM one often must model systems as

the superposition of two or more possible outcomes. Superpositions can

produce interference effects and thus are experimentally

distinguishable from mixed states. How does a superposition of

different possibilities resolve itself into some particular

observation? This question (also known as the measurement problem)

affects how we analyze some experiments such as tests of Bell's

inequality and may raise the question of interpretations from a

philosophical debate to an experimentally testable question. So far

there is no evidence that it makes any difference. The wave function

evolves in such a way that there are no observable effects from

macroscopic superpositions. It is only superposition of different

possibilities at the microscopic level that leads to experimentally

detectable interference effects.Thus it would seem that there is no criterion for objective events and

perhaps no need for such a criterion. However there is at least one

small fly in the ointment. In analyzing a test of Bell's inequality

one must make some determination as to when an observation was

complete, i. e. could not be reversed. These experiments depend on the

timing of macroscopic events. The natural assumption is to use

classical thermodynamics to compute the probability that a macroscopic

event can be reversed. This however implies that there is some

objective process that produces the particular observation. Since no

such objective process exists in current models this suggests that QM

is an incomplete theory. This might be thought of as the Einstein

interpretation of QM, i. e., that there are objective physical

processes that create observations and we do not yet understand these

processes. This is the view of the compiler of this document.For more information:

Ed. J. Wheeler, W. Zurek, Quantum theory and measurement, Princeton

University Press, 1983.J. S. Bell, Speakable and unspeakable in quantum mechanics, Cambridge

University Press, 1987.R.I.G. Hughes, The Structure and Interpretation of Quantum Mechanics,

Harvard University Press, 1989.3. Schrodinger's cat

Paul Budnik paul@mtnmath.com

In 1935 Schrodinger published an essay describing the conceptual

problems in QM1. A brief paragraph in this essay described the cat

paradox.

One can even set up quite ridiculous cases. A cat is penned up

in a steel chamber, along with the following diabolical device

(which must be secured against direct interference by the cat):

in a Geiger counter there is a tiny bit of radioactive

substance, so small that perhaps in the course of one hour one

of the atoms decays, but also, with equal probability, perhaps

none; if it happens, the counter tube discharges and through a

relay releases a hammer which shatters a small flask of

hydrocyanic acid. If one has left this entire system to itself

for an hour, one would say that the cat still lives if meanwhile

no atom has decayed. The first atomic decay would have poisoned

it. The Psi function for the entire system would express this by

having in it the living and the dead cat (pardon the expression)

mixed or smeared out in equal parts.It is typical of these cases that an indeterminacy originally

restricted to the atomic domain becomes transformed into

macroscopic indeterminacy, which can then be resolved by direct

observation. That prevents us from so naively accepting as valid

a ``blurred model'' for representing reality. In itself it would

not embody anything unclear or contradictory. There is a

difference between a shaky or out-of-focus photograph and a

snapshot of clouds and fog banks.We know that superposition of possible outcomes must exist

simultaneously at a microscopic level because we can observe

interference effects from these. We know (at least most of us know)

that the cat in the box is dead, alive or dying and not in a smeared

out state between the alternatives. When and how does the model of

many microscopic possibilities resolve itself into a particular

macroscopic state? When and how does the fog bank of microscopic

possibilities transform itself to the blurred picture we have of a

definite macroscopic state. That is the measurement problem and

Schrodinger's cat is a simple and elegant explanations of that

problem.References:

1 E. Schrodinger, ``Die gegenwartige Situation in der

Quantenmechanik,'' Naturwissenschaftern. 23 : pp. 807-812; 823-823,

844-849. (1935). English translation: John D. Trimmer, Proceedings of

the American Philosophical Society, 124, 323-38 (1980), Reprinted in

Quantum Theory and Measurement, p 152 (1983).4. The Copenhagen interpretation

Paul Budnik paul@mtnmath.com

This is the oldest of the interpretations. It is based on Bohr's

notion of `complementarity'. Bohr felt that the classical and quantum

mechanical models were two complementary ways of dealing with physics

both of which were necessary. Bohr felt that an experimental

observation collapsed or ruptured (his term) the wave function to make

its future evolution consistent with what we observe experimentally.

Bohr understood that there was no precise way to define the exact

point at which collapse occurred. Any attempt to do so would yield a

different theory rather than an interpretation of the existing theory.

Nonetheless he felt it was connected to conscious observation as this

was the ultimate criterion by which we know a specific observation has

occurred.References:

N. Bohr, The quantum postulate and recent the recent development of

atomic theory, Nature, 121, 580-89 (1928), Reprinted in Quantum Theory

and Measurement, p 87, (1983).5. Is QM a complete theory?

Paul Budnik paul@mtnmath.com

Einstein did not believe that God plays dice and thought a more

complete theory would predict the actual outcome of experiments. He

argued1 that quantities that are conserved absolutely (such as

momentum or energy) must correspond to some objective element of

physical reality. Because QM does not model this he felt it must be

incomplete.It is possible that events are the result of objective physical

processes that we do not yet understand. These processes may determine

the actual outcome of experiments and not just their probabilities.

Certainly that is the natural assumption to make. Any one who does not

understand QM and many who have only a superficial understanding

naturally think that observations come about from some objective

physical process even if they think we can only predict probabilities.There have been numerous attempts to develop such alternatives. These

are often referred to as `hidden variables' theories. Bell proved that

such theories cannot deal with quantum entanglement without

introducing explicitly nonlocal mechanisms2. Quantum entanglement

refers to the way observations of two particles are correlated after

the particles interact. It comes about because the conservation laws

are exact but most observations are probabilistic. Nonlocal

operations in hidden variables theories might not seem such a drawback

since QM itself must use explicit nonlocal mechanism to deal with

entanglement. However in QM the non-locality is in a wave function

which most do not consider to be a physical entity. This makes the

non-locality less offensive or at least easier to rationalize away.It might seem that the tables have been turned on Einstein. The very

argument he used in EPR to show QM must be incomplete requires that

hidden variables models have explicit nonlocal operations. However it

is experiments and not theoretical arguments that now must decide the

issue. Although all experiments to date have produced results

consistent with the predictions of QM, there is general agreement that

the existing experiments are inconclusive3. There is no conclusive

experimental confirmation of the nonlocal predictions of QM. If these

experiments eventually confirm locality and not QM Einstein will be

largely vindicated for exactly the reasons he gave in EPR. Final

vindication will depend on the development of a more complete theory.Most physicists (including Bell before his untimely death) believe QM

is correct in predicting locality is violated. Why do they have so

much more faith in the strange formalism of QM than in basic

principles like locality or the notion that observations are produced

by objective processes? I think the reason may be that they are

viewing these problems in the wrong conceptual framework. The term

`hidden variables' suggests a theory of classical-like particles with

additional hidden variables. However quantum entanglement and the

behavior of multi-particle systems strongly suggests that whatever

underlies quantum effects it is nothing like classical particles. If

that is so then any attempt to develop a more complete theory in this

framework can only lead to frustration and failure. The fault may not

be in classical principles like locality or determinism. They failure

may only be in the imagination of those who are convinced that no more

complete theory is possible.One alternative to classical particles is to think of observations as

focal points in state space of nonlinear transformations of the wave

function. Attractors in Chaos theory provide one model of processes

like this. Perhaps there is an objective physical wave function and QM

only models the average or statistical behavior of this wave function.

Perhaps the structure of this physical wave function determines the

probability that the wave function will transform nonlinearly at a

particular location. If this is so then probability in QM combines two

very different kinds of probabilities. The first is the probability

associated with our state of ignorance about the detailed behavior of

the physical wave function. The second is the probability that the

physical wave function will transform with a particular focal point.A model of this type might be able to explain existing experimental

results and still never violate locality. I have advocated a class of

models of this type based on using a discretized finite difference

equation rather then a continuous differential equation to model the

wave function4. The nonlinearity that must be introduced to discretize

the difference equation is a source of chaotic like behavior. In this

model the enforcement of the conservation laws comes about through a

process of converging to a stable state. Information that enforces

these laws is stored holographic-like over a wide region.Most would agree that the best solution to the measurement problem

would be a more complete theory. Where people part company is in their

belief in whether such a thing is possible. All attempts to prove it

impossible (starting with von Neumann5) have been shown to be flawed6.

It is in part Bell's analysis of these proofs that led to his proof

about locality in QM. Bell has transformed a significant part of this

issue to one experimenters can address. If nature violates locality in

the way QM predicts then a local deterministic theory of the kind

Einstein was searching for is not possible. If QM is incorrect in

making these predictions then a more accurate and more complete theory

is a necessity. Such a theory is quite likely to account for events by

an objective physical process.References: 1 A. Einstein, B. Podolsky and N. Rosen, Can quantum-

mechanical descriptions of physical reality be considered complete?,

Physical Review, 47, 777 (1935). Reprinted in Quantum Theory and

Measurement, p. 139, (1987).2 J. S. Bell, On the Einstein Podolosky Rosen Paradox, Physics, 1,

195-200 (1964). Reprinted in Quantum Theory and Measurement, p. 403,

(1987).3 P. G. Kwiat, P. H. Eberhard, A. M. Steinberg, and R. Y. Chiao,

Proposal for a loophole-free Bell inequality experiment, Physical

Reviews A, 49, 3209 (1994).4 P. Budnik, Developing a local deterministic theory to account for

quantum mechanical effects, hep-th/9410153, (1995).5 J. von Neumann, The Mathematical Foundations of Quantum Mechanics,

Princeton University Press, N. J., (1955).6 J. S. Bell, On the the problem of hidden variables in quantum

mechanics, Reviews of Modern Physics, 38, 447-452, (1966). Reprinted

in Quantum Theory and Measurement, p. 397, (1987).6. The shut up and calculate interpretation

Paul Budnik paul@mtnmath.com

This is the most popular of interpretations. It recognizes that the

important content of QM is the mathematical models and the ability to

apply those models to real experiments. As long as we understand the

models and their application we do not need an interpretation.Advocates of this position like to argue that the existing framework

allows us to solve all real problems and that is all that is

important. Franson's analysis of Aspect's experiment1 shows this is

not entirely true. Because there is no objective criterion in QM for

determining when a measurement is complete (and hence irreversible)

there is no objective criterion for measuring the delays in a test of

Bell's inequality. If the demise of Schrodinger's cat may not be

determined until someone looks in the box (see item 2) how are we to

know when a measurement in tests of Bells inequality is irreversible

and thus measure the critical timing in these experiments?References:

1 J. D. Franson, Bell's Theorem and delayed determinism, Physical

Review D, 31, 2529-2532, (1985).7. Bohm's theory

Paul Budnik paul@mtnmath.com

Bohm's interpretation is an explicitly nonlocal mechanistic model.

Just as Bohr saw the philosophical principle of complementarity as

having broader implications than quantum mechanics Bohm saw a deep

relationship between locality violation and the wholeness or unity of

all that exists. Bohm was perhaps the first to truly understand the

nonlocal nature of quantum mechanics. Bell acknowledged the importance

of Bohm's work in helping develop Bell's ideas about locality in QM.References: D. Bohm, A suggested interpretation of quantum theory in

terms of "hidden" variables I and II, Physical Review,85, 155-93

(1952). Reprinted in Quantum Theory and Measurement, p. 369, (1987).D. Bohm & B.J. Hiley, The Undivided Universe: an ontological

interpretation of quantum theory (Routledge: London & New York, 1993).Recently there has been renewed interest in Bohmian mechanics. D.

D"urr, S. Goldstein, N Zanghi, Phys. Lett. A 172, 6 (1992) K. Berndl

et al., Il Nuovo Cimento Vol. 110 B, N. 5-6 (1995).Peter Holland's book The Quantum Theory of Motion (Cambridge

University Press 1993) contains many pictures of numerical simulations

of Bohmian trajectories.8. Lawrence R. Mead rmead@whale.st.usm.ed The Transactional Interpre-

tation of Quantum MechanicsThe transactional interpretation of quantum mechanics (J.G. Cramer,

Phys. Rev. D 22, 362 (1980) ) has received little attention over the

one and one half decades since its conception. It is to be emphasized

that, like the Many-Worlds and other interpretations, the

transactional interpretation (TI) makes no new physical predictions;

it merely reinterprets the physical content of the very same

mathematical formalism as used in the ``standard'' textbooks, or by

all other interpretations.The following summarizes the TI. Consider a two-body system (there are

no additional complications arising in the many-body case); the

quantum mechanical object located at space-time point (R_1,T_1) and

another with which it will interact at (R_2,T_2). A quantum mechanical

process governed by E=h0, conservation laws, etc., occurs between the

two in the following way.1) The ``emitter'' (E) at (R_1,T_1) emits a retarded ``offer wave''

(OW) \Psi. This wave (or state vector) is an actual physical wave and

not (as in the Copenhagen interpretation) just a ``probability'' wave.2) The ``absorber'' (A) at (R_2,T_2) receives the OW and is stimulated

to emit an advanced ``echo'' or ``confirmation wave'' (CW)

proportional to \Psi at R_2 backward in time; the proportionality

factor is \Psi* (R_2,T_2).3) The advanced wave which arrives at 'E' is \Psi \Psi* and is

presumed to be the probability, P, that the transaction is complete

(ie., that an interaction has taken place).4) The exchange of OW's and CW's continues until a net exchange of

energy and other conserved quantities occurs dictated by the quantum

boundary conditions of the system, at which point the ``transaction''

is complete. In effect, a standing wave in space-time is set up

between 'E' and 'A', consistent with conservation of energy and

momentum (and angular momentum). The formation of this superposition

of advanced and retarded waves is the equivalent to the Copenhagen

``collapse of the state vector''. An observer perceives only the

completed transaction, however, which he would interpret as a single,

retarded wave (photon, for example) traveling from 'E' to 'A'.Q1. When does the ``collapse'' occur?

A1. This is no longer a meaningful question. The quantum measurement

process happens ``when'' the transaction (OW sent - CW received -

standing wave formed with probability \Psi \Psi*) is finished - and

this happens over a space-time interval; thus, one cannot point to a

time of collapse, only to an interval of collapse (consistent with

relativity).Q2. Wait a moment. What you are describing is time reversal invariant.

But for a massive particle you have to use the Schrodinger equation

and if \Psi is a solution (OW), then \Psi* is not a solution. What

gives?A2. Remember that the CW must be time-reversed, and in general must be

relativistically invariant; ie., a solution of the Dirac equation.

Now (eg., see Bjorken and Drell, Relativistic QM), the nonrelativistic

limit of that is not just the Schrodinger equation, but two

Schrodinger equations: the time forward equation satisfied by \Psi,

and the time reversed Schrodinger equation (which has i --> -i) for

which \Psi* is the correct solution. Thus, \Psi* is the correct CW for

\Psi as the OW.Q3. What about other objects in other places?

A3. The whole process is three dimensional (space). The retarded OW is

sent in all spatial directions. Other objects receiving the OW are

sending back their own CW advanced waves to 'E' also. Suppose the

receivers are labeled 1 and 2, with corresponding energy changes E_1

and E_2. Then the state vector of the system could be written as a

superposition of waves in the standard fashion. In particular, two

possible transactions could form: exchange of energy E_1 with

probability P_1=\Psi_1 \Psi_1*, or E_2 with probability P_2=\Psi_2

\Psi_2*. Here, the conjugated waves are the advanced waves evaluated

at the position of R_1 or R_2 respectively according to rule 3 above.Q4. Involving as it does an entire space-time interval, isn't this a

nonlocal ``theory''?A4. Yes, indeed; it was explicitly designed that way. As you know from

Bell's theorem, no ``theory'' can agree with quantum mechanics unless

it is nonlocal in character. In effect, the TI is a hidden variables

theory as it postulates a real waves traveling in space-time.Q5. What happens to OW's that are not ``absorbed'' ?

A5. Inasmuch as they do not stimulate a responsive CW, they just

continue to travel onward until they do. This does not present any

problems since in that case no energy or momentum or any other

physical observable is transferred.Q6. How about all of the standard measurement thought experiments like

the EPR, Schrodinger's cat, Wigner's friend, and Renninger's negative-

result experiment?A6. The interpretational difficulties with the latter three are due to

the necessity of deciding when the Copenhagen state reduction occurs.

As we saw above, in the TI there is no specific time when the

transaction is complete. The EPR is a completeness argument requiring

objective reality. The TI supplies this as well; the OW and CW are

real waves, not waves of probability.Q7. I am curious about more technical details. Can you give a further

reference?A7. If you understand the theory of ``advanced'' and ``retarded''

waves (out of electromagnetism and optics), many of the details of TI

calculations can be found in: Reviews of Modern Physics, Vol. 58, July

1986, pp. 647-687 available on the WWW as:

http://mist.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html9. Complex probabilities

References; Saul Youssef Quantum Mechanics as Complex Probability

Theory, hep-th 9307019. S. Youssef, Mod.Phys.Lett.A 28(1994)2571.10. Quantum logic

References: R.I.G. Hughes, The Structure and Interpretation of Quantum

Mechanics, pp. 178-217, Harvard University Press, 1989.11. Consistent histories

References: R. B. Griffiths, Consistent Histories and the

Interpretation of Quantum Mechanics, Journal of statistical Physics.,

36(12):219-272(1984)M. Gell-Mann and J. B. Hartle, in Complexity, Entropy and the Physics

of Information, edited by W. Zurek, Santa Fe Institute Studies in the

Sciences of Complexity Vol. VIII, Addison-Wesley, Reading, 1990. Also

in Proceedings of the $3$rd International Symposion on the Foundations

of Quantum Mechanics in the Light of New Technology, edited by S.

Kobayashi, H. Ezawa, Y. Murayama and S. Nomura, Physical Society of

Japan, Tokyo, 1990R. B. Griffiths, Phys. Rev. Lett. 70, 2201 (1993)

R. Omn`es, Rev. Mod. Phys. 64, 339 (1992)

In this approach serious problems arise. This is best pointed out in:

B. d'Espagnat, J. Stat. Phys. 56, 747 (1989)F. Dowker und A. Kent, On the Consistent Histories Approach to Quantum

Mechanics, University of Cambridge Preprint DAMTP/94-48, Isaac Newton

Institute for Mathematical Sciences Preprint NI 94006, August 1994.12. Spontaneous reduction models

Reference:

G. C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34, 470 (1986).

13. What is needed?

All comments suggested and contributions are welcome. We currently

have nothing but references on Complex Probabilities, Quantum Logic,

Consistent Histories and Spontaneous Reduction Models. The entries on

the following topics are minimal and should be replaced by complete

articles.o Copenhagen interpretation

o Relative State (Everett)

o Shut up and calculate

o Bohm's theory

Alternative views on any of the topics and suggestions for additional

topics are welcome.14. Is this a real FAQ?

Paul Budnik paul@mtnmath.com

A FAQ is generally understood to be a reasonably objective set of

answers to frequently asked questions in a news group. In cases where

an issue is controversial the FAQ should include all credible opinions

and/or the consensus view of the news group.Establishing factual accuracy is not easy. No consensus is possible on

interpretations of QM because many aspects of interpretations involve

metaphysical questions. My intention is that this be an objective

accurate FAQ that allows for the expression of all credible relevant

opinions. I did not call it a FAQ until I had significant feedback

from the `sci.physics' group. I have responded to all criticism and

have made some corrections. Nonetheless there have been a couple of

complaints about this not being a real FAQ and there is one issue that

has not been resolved.If anyone thinks there are technical errors in the FAQ please say what

you think the errors are. I will either fix the problem or try to

reach on a consensus with the help of the `sci.physics' group about

what is factually accurate. I do not feel this FAQ should be limited

to noncontroversial issues. A FAQ on measurement in quantum mechanics

should highlight and underscore the conceptual issues and problems in

the theory.The one area that has been discussed and not resolved is the status of

locality in Everett's interpretation. Here is what I believe the facts

are.Eberhard proved that any theory that reproduces the predictions of QM

is nonlocal1. This proof assumes contrafactual definiteness (CFD) or

that one could have done a different experiment and have gotten a

definite result. This assumption is widely used in statistical

arguments. Here is what Eberhard means by nonlocal:Let us consider two measuring apparata located in two different

places A and B. There is a knob a on apparatus A and a knob b on

apparatus B. Since A and B are separated in space, it is

natural to think what will happen at A is independent of the

setting of knob b and vice versa. The principles of relativity

seem to impose this point of view if the time at which the knobs

are set and the time of the measurements are so close that, in

the time laps, no light signal can travel from A to B and vice

versa. Then, no signal can inform a measurement apparatus of

what the knob setting on the other is. However, there are cases

in which the predictions of quantum theory make that

independence assumption impossible. If quantum theory is true,

there are cases in which the results of the measurements A will

depend on the setting of the knob b and/or the results of the

measurements in B will depend on the setting of the knob a.1It is logically possible to deny CFD and thus to avoid Eberhard's

proof. This assumption can be made in Everett's interpretation.

Everett's interpretation does not imply CFD is false and CFD can be

assumed false in other interpretations. I do not think it is

reasonable to deny CFD in some experiments and not others but that is

a judgment call on which intelligent people can differ.It is mathematically impossible to have a unitary relativistic wave

function from which one can compute probabilities that will violate

Bell's inequality. A unitary wave function does satisfy CFD and thus

is subject to Eberhard's proof. This is a problem for some advocates

of Everett who insist that only the wave function exists. There is no

wave function consistent with both quantum mechanics and relativity

and it is mathematically impossible to construct such a function.

Quantum field theory requires a nonlocal and thus nonrelativistic

state model. The predications of quantum field theory are the same in

any frame of reference but the mechanisms that generate nonlocal

effects must operate in an absolute frame of reference. Quantum

uncertainty makes this seemingly paradoxical situation possible. There

is a nonlocal effect but we cannot tell if the effect went from A to B

or B to A because of quantum uncertainty. As a result the predictions

are the same in any frame of reference but any mechanism that produces

these predictions must be tied to an absolute frame of reference.There is a certain Alice in Wonderland quality to arguments on these

issues. Many physicists claim that classical mathematics does not

apply to some aspects of quantum mechanics, yet there is no other

mathematics. The wave function model is a classical causal

deterministic model. The computation of probabilities from that model

is as well. The aspect of quantum mechanics that one can claim lies

outside of classical mathematics is the interpretation of those

probabilities. Most physicists believe these probabilities are

irreducible, i. e., do not come from a more fundamental deterministic

process the way probabilities do in classical physics. Because there

is no mathematical theory of irreducible probabilities one can invent

new metaphysics to interpret these probabilities and here is where the

problems and confusion rest. Some physicists claim there is new

metaphysics and within this metaphysics quantum mechanics is local.References:

P. H. Eberhard, Bell's Theorem without Hidden Variables, Il Nuovo

Cimento, V38 B 1, p 75, Mar 1977.Science & MathematicsThe Uncle Taz LibraryUncle Taz Home Page 2Site search Web search

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