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Subject: Measurement in quantum mechanics FAQ
This article was archived around: Sun, 31 Oct 1999 17:24:26 GMT
Posted-By: auto-faq 3.3 (Perl 5.004)
Measurement in quantum mechanics FAQ
Maintained by Paul Budnik, email@example.com, http://www.mtn-
This FAQ describes the measurement problem in QM and approaches to its
solution. Please help make it more complete. See ``What is needed''
for details. Web version: http://www.mtnmath.com/faq/meas-qm.html
1. About this FAQ
Last modified August 5, 1998 (section 7)
The general sci.physics FAQ does a good job of dealing with technical
questions in most areas of physics. However it has no material on
interpretations of QM which are among the most frequently discussed
topics in sci.physics. Hence there is a need for this supplemental
This document is probably out of date if you are reading it more than
30 days after the date which appears in the header.
This FAQ is on the web at: http://www.mtnmath.com/faq/meas-qm.html
You can get it by e-mail or FTP from rtfm.mit.edu.
By FTP, look for the file:
By e-mail send a message to firstname.lastname@example.org with a blank
subject line and the words:
The main sci.physics FAQ is in this same directory with file names
part1 through part4 and can be retrieved in the same way. You can put
multiple send lines in a single e-mail request.
This document, as a collection, is Copyright 1995 by Paul P. Budnik
(email@example.com). The individual articles are Copyright 1995 by the
individual authors listed. All rights are reserved. Permission to
use, copy and distribute this unmodified document by any means and for
any purpose EXCEPT PROFIT PURPOSES is hereby granted, provided that
both the above Copyright notice and this permission notice appear in
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included, but not limited to, printing, copying existing prints,
publishing by electronic or other means, implies full agreement to the
above non-profit-use clause, unless upon explicit prior written
permission of the authors.
This FAQ is provided by the authors ``as is''. with all its faults.
Any express or implied warranties, including, but not limited to, any
implied warranties of merchantability, accuracy, or fitness for any
particular purpose, are disclaimed. If you use the information in
this document, in any way, you do so at your own risk.
2. The measurement problem
Paul Budnik firstname.lastname@example.org
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 email@example.com
In 1935 Schrodinger published an essay describing the conceptual
problems in QM. A brief paragraph in this essay described the cat
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
 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 firstname.lastname@example.org
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
N. Bohr, The quantum postulate and 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 email@example.com
Einstein did not believe that God plays dice and thought a more
complete theory would predict the actual outcome of experiments. He
argued 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
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 mechanisms. 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 inconclusive. 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 function. 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
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 Neumann) have been shown to be
flawed. 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:  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).
 J. S. Bell, On the Einstein Podolosky Rosen Paradox, Physics, 1,
195-200 (1964). Reprinted in Quantum Theory and Measurement, p. 403,
 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).
 P. Budnik, Developing a local deterministic theory to account for
quantum mechanical effects, hep-th/9410153, (1995).
 J. von Neumann, The Mathematical Foundations of Quantum Mechanics,
Princeton University Press, N. J., (1955).
 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 firstname.lastname@example.org
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 experiment 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
 J. D. Franson, Bell's Theorem and delayed determinism, Physical
Review D, 31, 2529-2532, (1985).
7. Bohm's theory
Paul Budnik email@example.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.
There was a recent two part article in Physics Today based in part on
Bohm's approach. The author, Sheldon Goldstein, has published a number
of other papers on this and related subjects many of which are
available at his web site, http://math.rutgers.edu/~oldstein. S
Goldstein, Quantum Theory Without Observers, Physics Today Part 1:
March 1998, 42-46, Part 2: April 1998 38-42.
8. Lawrence R. Meadrmead@ocra.st.usm.ed The Transactional Interpreta-
tion of Quantum Mechanics
The 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=h\nu, 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''
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
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
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
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-
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
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:
9. 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.,
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, 1990
R. 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
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
+ Copenhagen interpretation
+ Relative State (Everett)
+ Shut up and calculate
+ 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 firstname.lastname@example.org
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 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
Eberhard proved that any theory that reproduces the predictions of QM
is nonlocal. 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.
It 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.
P. H. Eberhard, Bell's Theorem without Hidden Variables, Il Nuovo
Cimento, V38 B 1, p 75, Mar 1977.