On the Problem of Hidden Variables in Quantum Mechanics

TL;DR: The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered in this article, and it is shown that their essential axioms are unreasonable.

Abstract: The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered. It is shown that their essential axioms are unreasonable. It is urged that in further examination of this problem an interesting axiom would be that mutually distant systems are independent of one another.

Summary

Introduction

• Youth presenting for mental health care often have multiple symptoms and diagnoses, complicating the prescription of targeted treatment interventions.
• Indeed, recent examinations of data-rich brain connectivity images reveal meaningful associations between brain function and such data-driven, symptom-based classifications.

Participants and Measures

• The sample also included subjects who had a first-degree relative with bipolar disorder (“at risk” AR, n=35), and those with no psychopathology (healthy volunteers, HV, n=46).
• The copyright holder for this preprint (which wasthis version posted November 8, 2019.
• Thus, the current study contains data in an additional 112 subjects, beyond those appearing in this prior imaging paper.
• Comparisons among samples and results of this study, Kircanski et al, (2017) and Stoddard et al, (2017) are detailed in Supplementary Materials and Methods.

Latent Profile Analysis

• As in their original LPA (Kircanski et al, 2017), manifest variables in the LPA were separate parent- and child-reported ARI and SCARED scores, but only self-reported CDI and parent-reported CPRS scores.
• To be included in the LPA, a measure must have been collected within 60 days of scanning.
• As described in the introduction, the original analysis yielded a 5-class solution.
• Neural Response by Symptom Dimensions Differences in the number of classes reflect differences in the samples included in the current, vs. the original, LPA (Supplementary Materials and Methods).

Probability of Class Assignment

• Individuals differ in the in their goodness of fit to the classes identified by the group analysis (Figure 1).
• Mixture modeling, like the LPA used here, can account for this individual variation by providing the probability of an individual’s assignment to each class.
• Also, by using the probability of class assignment in their analysis, the authors can weigh the influence of each person on the results, making the analysis dimensional and hence more powerful.
• The authors denote this as Pclass overall or, for for use under a CC0 license.
• FMRI Task During fMRI, participants labelled the gender of happy, angry, and fearful face- emotion pictures from 10 actors (Ekman and Friesen, 1976).

Imaging Procedures

• Magnetic resonance images (MRI) were acquired on a General Electric 3.0 Tesla scanner with a 32-channel head coil.
• This work used the computational resources of the NIH HPC Biowulf cluster (http://hpc.nih.gov).
• The copyright holder for this preprint (which wasthis version posted November 8, 2019.
• EPI images were processed by removing the first four pre-magnetization volumes, limiting each voxel’s BOLD signal to four standard deviations from the mean trend of its time series, correcting for slice timing, affine volume to volume and volume to anatomic registration, smoothing using a 5 mm FWHM Gaussian kernel, and scaling to a mean of 100.
• Subjects were excluded if more than 15% of their volumes were censored or their mean censored volume to volume Euclidean displacement was more than or equal to 0.2mm (Table S1).

Analyses

• The ANOVA tested the fully interactive effects of each Probability of Class Assignment with Emotion and Intensity, except that no term could include an interaction between two or more .
• It is not subject to copyright under 17 USC 105 and is also made available not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
• For further descriptive analysis of each functional ROI, mean values of connectivity at each emotion by intensity were extracted via AFNI’s 3dROIstat.
• Any result no longer significant when influential subjects were removed was not reported.

Unique Associations of Amygdala Connectivity to Symptom Classes

• For regions showing unique associations between each symptom class and amygdala connectivity, see Table 2 and Figure 2.
• Increasing PIrr/ADHD was associated with decreased connectivity in response to 150% fearful and 150% angry faces, both relative to 150% happy (all p’s <.001).
• PIrr/ADHD also interacted with intensity to influence right amygdala connectivity to the bilateral precuneus (F1,1688=9.8, p<.001).
• Post hoc contrasts showed this interaction was significant at 50% intensity only.
• Post hoc analyses assessing the effects of medication for all regions of interest are presented in Table S3.

Discussion

• In a large, transdiagnostic clinical sample of youth, the authors aimed to identify neural correlates of empirically-defined symptom profiles.
• The copyright holder for this preprint (which wasthis version posted November 8, 2019.
• Page 15 of 27 is novel in applying task-based functional connectivity to validate a person-centered clustering of symptoms, which extends (Kircanski et al, 2017).
• Altogether, these studies converge to implicate aberrant amygdala function to clinical presentations involving both irritability and ADHD symptoms.
• It is not subject to copyright under 17 USC 105 and is also made available not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

Full text

SLAC-PUB-44
August
1964
ON THE PROBLEM OF HIDDEN VARIABLES IN QUANTUM M3XANICS*
J. S. Bell
f
Stanford Linear Accelerator Center
Stanford University, Stanford, California
ABSTRACT
The demonstrations of von Neumann and others, that quantum
mechanics does not permit a hidden variable interpretation, are
reconsidered.
It is shown that their essential axioms are un-
reasonable. It is urged that in further examination of this
problem an interesting axiom would be that mutually distant
systems are independent of one another.
qork supported by U. S. Atomic Energy Commission.
f
On leave of absence from CERN.
-l-

I.
INTRODUCTION
To know the quantum mechanical state of a system implies, in general, only
statistical restrictions on the results of measurements.
It seems interesting
t.o ask if this statistical element be thought of as arising, as in classical
statistical mechanics,
because the states in question are averages over better
defined states for which individually the results would be quite determined.
These hypothetical "dispersion free
M states would be specified not only by the
quantum mechanical state vector but also by additional "hidden variables" -
"hidden"because if states with prescribed values of these variables could
actually be prepared, quantum mechanics would be observably inadequate.
Whether this question is indeed interesting has been the subject of debate.ly2
The present paper does not contribute to that debate.
It is addressed to those
who do find the question interesting,
and more particularly to those among them
who believe that' "the question concerning,the existence of such hidden vari-
ables received an early and rather decisive answer in the form of von Neumann's
proof on the mathematical impossibility of such variables in quantum theory."
An attempt will be made to clarify what von Neumann and his successors actually
demonstrated.
This will cover, as well as von Neumann's treatment, the recent
version of the argument by Jauch and Piron, and the stronger result consequent
on the work of Gleason.*
It will be urged that these analyses leave the real
question untouched. In fact it will be seen that these demonstrations require
from the hypothetical dispersion free states,
not only that appropriate
ensembles thereof should have all measurable properties of quantum mechanical
states, but certain other properties as well.
These additional demands appear
reasonable when results of measurement are loosely identified with properties
-2-

of isolated systems.
They are seen to be quite unreasonable when one
remembers with Bohr5 "the impossibility of any sharp distinction between
the behavior of atomic objects and the interaction with the measuring in-
struments which serve to define the conditions under which the phenomena
appear."
The realization that von Neumann's proof is of limited relevance has been
gaining ground since the 1952 work of Bohm.'
However it is far from universal.
Moreover the writer has not found in the literature any adequate analysis of
what went wrong.7
Like all authors of non-commissioned reviews he thinks
that he can restate the position with such clarity and simplicity that all
previous discussions will be eclipsed.
II. ASSUMF'TIONS, AND A SIMPLE EXAMPLE
The authors of the demonstrations to be reviewed were concerned to assume
as little as possible about quantum mechanics.
This is valuable for some.
purposes, but not for ours.
We are interested only in the possibility of
hidden variables in ordinary quantum mechanics,
and will use freely all the -
usual notions.
Thereby the demonstrations will be substantially shortened.
A quantum mechanical "system" is supposed to have "observables" represented
by Hermitian operators in,a complex linear vector space.
Every "measurement"
of an observable yields one of the eigenvalues of the corresponding operator.
Observables with commuting operators can be measured simultaneously.g A
quantum mechanical "state"
is represented by a vector in the linear state
space. For a state vector
J# the statistical expectation value of an
-3-

observable with operator
0 is the normalized inner product ($,O$)/($,$).
The question at issue is whether the quantum mechanical states can be
regarded as ensemblks of states further specified by additional variable's,
such that given values of these variables together with the state vector
determine precisely the results of individual measurements.
These hypo-
thetical well-specified states are said to be "dispersion free."
In the following discussion it will be useful to keep in mind as a simple
example a system with a 2-dimensional state space.
Consider for definiteness
a spin-
2 particle without translational motion.
A quantum mechanical state
is represented by a 2-component state vector, or spinor, $r.
The observables
are represented by 2 x 2 Hermitian matrices
a+p - 5
Fsr
where a is a real number, k a real vector, and 2
Pauli matrices; Q: is understood to multiply the unit
such an observable yields one of the eigenvalues
cl*
I I
&
with relative probabilities that can be inferred from
the expectation value
(1)
has for components the
matrix.
Measurement of
(2)
For this system a hidden variable scheme can be supplied as follows: The
dispersion, free states are specified by a real number A, in the interval
1
1
--
<A<-
2 - -2'
as well as the spinor q .
To describe how A determines
which eigenvalue the measurement gives,
we note that by a rotation of co-
ordinates $
can be brought to the form

Let DxY By’ B,’
be the components of @
in the new coordinate system.
Then
A
measurement of Q: + &
* z, on the state specified by $ and h results with
certainty in the eigenvalue
a + 111 sign (h IL1 + ;I BzI) sign X
(3)
where
x=p
Z.
if p, # 0
=
PX
if p, = 0,
B, + 0
= By
if $, = 0,
and B, = 0
and
sign X = +l
if X>O
=
-1 if XC0
The quantum mechanical state specified by Jr is obtained by uniform averaging
over A. This gives the expectation value
as required.
It should be stressed that no physical significance is attributed here to
the parameter h. All that is offered is a trivial example of a possibility
which von Neumann's reasoning was for long thought to exclude.
-5-

Frequently asked questions

Q1. What are the contributions in this paper?

The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered this paper.