# Växjö Interpretation of Wave Function: 2012

Andrei Khrennikov
International Center for Mathematical Modelling
in Physics and Cognitive Sciences
Linnaeus University, Växjö, S-35195, Sweden

###### Abstract

We discuss the problem of interpretation of the wave function. The latest
version of the Växjö interpretation is presented. It differs essentially from
the original Växjö interpretation (2001). The main distinguishing feature of
the present Växjö interpretation is the combination of realism on the
subquantum level with nonobjectivity of quantum observables (i.e.,
impossibility to assign their values before measurements). Hence, realism is
destroyed by detectors transforming continuous subquantum reality into
discrete events, clicks of detectors. The Växjö interpretation-2012 is
fundamentally contextual in the sense that the value of an observable depends
on measurement context. This is contextuality in Bohr’s sense. It is more
general than Bell’s contextuality based on joint measurements of compatible
observables.

##  1 Introduction

During the Växjö series of conferences on quantum foundations the most
exciting spectacle started each time when the question of interpretations of
the wave function attracted the attention. Finally, I understood that the
number of different interpretations is in the best case equal to the number of
participants. If you meet two people who say that they are advocates of, e.g.,
the Copenhagen interpretation of quantum mechanics (QM), ask them about the
details. You will see immediately that their views on what is the Copenhagen
interpretation can differ very much. The same is true for other
interpretations. If two scientists tell that they are followers of Albert
Einstein’s ensemble interpretation, ask them about the details… At one of the
round tables (after two hours of debates with opinions for and against
completeness of QM) we had decided to vote on this problem. Incompleteness
advocates have won, but only because a few advocates of completeness voted for
incompleteness. The situation is really disappointing: the basic notion of QM
has not yet been properly interpreted (after 100 years of exciting, but not
very productive debates).111“When I speak with somebody and get to know their
interpretation, I understand immediately it is wrong. The main problem is that
I do not know whether my own interpretation is right.” (Theo Nieuwenhuizen)
This is the standard problem of participants of Växjö conferences. I
specifically appreciate the activity of Arkady Plotnitsky, philosopher
studying Bohr’s views, see, e.g., [1]. He teaches us (participants of Växjö
conferences) about interpretations of the wave functions a lot. First of all
we got to know that the Copenhagen interpretation cannot be rigidly coupled
with Bohr’s views. On many occasions Niels Bohr emphasized that QM is not
about physical processes in microworld, but about our measurements [2]:

“Strictly speaking, the mathematical formalism of quantum mechanics and
electrodynamics merely offers rules of calculation for the deduction of
expectations pertaining to observations obtained under well-defined
experimental conditions specified by classical physical concepts”.

The basic postulate of the Copenhagen interpretation of QM – “the wave
function describes completely the state of a quantum system” (i.e., a concrete
system, not an ensemble) – cannot be assigned to Bohr. Then we learned (again
from Plotnitsky) that Bohr’s views have been crucially changed a few times
during his life. Thus, there can be found many different Bohr’s
interpretations of QM. Bohr was definitely the father of the operational
interpretation of QM. As was already pointed out, Bohr emphasized that the
formalism of QM does not provide the intrinsic description of processes in
microworld, it describes only results of measurements. Bohr also can be
considered as one of fathers of the so-called information interpretation of
QM: the QM-formalism describes information about micro systems extracted by
means of macroscopic measurement devices. Heisenberg (and to some extent
Schrödinger) shared this viewpoint. Nowadays the information interpretation of
QM became very popular, see, e.g., [3], [4], [1]. I can mention Anton
Zeilinger [5] and Christopher Fuchs [6]–[8] among the active promoters of this
interpretation; we can also mention Mermin’s paper [9].

In this paper we discuss three main interpretations of QM: the Copenhagen,
ensemble and Växjö interpretations.

##  2 Interpretations of Wave Function

Everywhere below $H$ denotes complex Hilbert space with the scalar product
$\langle\cdot,\cdot\rangle$ and the norm $\|\cdot\|$ corresponding to the
scalar product.

Postulate IM. (The mathematical description of quantum states). Quantum (pure)
states (wave functions) are represented by normalized vectors $\psi$ (i.e.,
$\|\psi\|^{2}=\langle\psi,\psi\rangle=1$) of a complex Hilbert space $H.$
Every normalized vector $\psi\in H$ may represent a quantum state. If a vector
$\psi$ corresponding to a state is multiplied by any complex number $c,|c|=1,$
the resulting vector will correspond to the same state.222Thus states are
given by elements of the unit sphere of the Hilbert space $H.$

The physical meaning of “a quantum state” is not defined by this postulate. It
must be provided by a separate postulate; see Postulates IE, IC, IV
(respectively, the ensemble, Copenhagen and Växjö interpretations).

Postulate IE. (The ensemble interpretation). A wave function provides a
description of certain statistical properties of an ensemble of similarly
prepared quantum systems.

This interpretation is upheld, for example by Einstein, Popper, Blokhintsev,
Margenau, Ballentine, Klyshko, and recent years by, e.g., De Muynck, De Baere,
Holevo, Santos, Khrennikov, Nieuwenhuizen, Adenier, Groessing, Hofer and many
others

Postulate IC. (The Copenhagen interpretation). A wave function provides a
complete description of an individual quantum system.

This interpretation was supported by a great variety of members, from
Schrödinger’s original attempt to identify the electron with a wave function
solution of his equation to the several versions of the Copenhagen
interpretation; for example, Heisenberg, Bohr, Pauli, Dirac, von Neumann,
Landau, Fock, …, Greenberger, Mermin, Lahti, Peres.

There is an interesting story about the correspondence between Bohr and Fock
on the individual interpretation. This story was told me by one of former
students of Fock who pointed out that one of the strongest supporters of this
interpretation was Vladimir A. Fock, and that even though Bohr himself had
doubts about its consistency, he, Fock, demonstrated to Bohr inconsistency in
the Einsteinian ensemble interpretation. Thus interpretation which is commonly
known as the Copenhagen interpretation might be as well called the “Leningrad
interpretation.”

Instead of Einstein’s terminology ”ensemble interpretation”, L. Ballentine
[11], [10] used the terminology “statistical interpretation.” However,
Ballentine’s terminology is rather misleading, because the term “statistical
interpretation” was also used by J. von Neumann [12] for individual
randomness! For him “statistical interpretation” had the meaning which is
totally different from the Ballentine’s “ensemble-statistical interpretation.”
J. von Neumann wanted to emphasize the difference between deterministic
(Newtonian) classical mechanics in that the state of a system is determined by
values of two observables (position and momentum) and quantum mechanics in
that the state is determined not by values of observables, but by
probabilities. We shall follow Albert Einstein and use the terminology
“ensemble interpretation”.

##  3 Växjö interpretation of quantum mechanics

The Växjö interpretation [13] was born (in 2001) as an alternative to the
Copenhagen interpretation. The basic assumption of the latter that the wave
function describes completely the state of a quantum system is the main source
of all quantum mysteries. (We remind that Schrödinger elaborated
the“Schrödinger cat” experiment to show the absurdness of this assumption.
Nowadays the origin of the Schrödinger cat illustration of the absurdness of
the Copenhagen interpretation is practically forgotten. The presence of such
cat-states is often considered as one of natural features of QM.) Therefore
the easiest way to resolve the majority of interpretation problems of QM is to
assume that the wave function description is not the final description of
micro phenomena. By proceeding in this way one has to develop prequantum
models inducing the QM formalism as an operational formalism ignoring details
of processes in the microworld and describing only results of measurements.
Measurements are operationally described by Hermitian operators or more
generally by POVM. However, there are known many no-go statements which
prohibit any motion beyond QM (under the natural assumption of locality333We
remind that Einstein, Podolsky and Rosen considered nonlocality as the absurd
alternative to incompleteness of QM [14]. This is also practically forgotten
(ignored?). One can often find just the statement that quantum nonlocality was
originally discussed by Einstein, Podolsky and Rosen [14]. In his reply to the
EPR paper [15] Bohr did not mention nonlocality at all. He also was sure that
QM is local.). The first version [13] of the Växjö interpretation was created
as the result of analysis of the “impossibility statements”, e.g., [12], [16],
[17], playing the crucial role in the contemporary debates on quantum
foundations. Surprisingly for myself, I found that all no-go statements
contain some unphysical assumptions which are not valid for real experimental
situations, see [18] for details; cf. with Bell’s and Ballentine’s critical
analysis of assumptions of von Neumann no-go theorem [17], [10], [11].
Suddenly I understood that usage of the operational quantum formalism for
results of measurements does not contradict with the possibility of creating
prequantum classical models. These models can be both realistic and local.
Here realism (objectivity) is defined as a possibility to assign the values of
quantum observables to quantum systems before measurement. This viewpoint to
realism is common in discussions related to Bell’s inequality [17] and in
general to inter-relation of classical and quantum physics.

The Växjö interpretation-2001 was the (local) realistic interpretation [13].

In 2004 I visited Beckman Institute for Advanced Science and Technology
(University of Illinois at Urbana-Champaign). After my talk (May 3, 2004), I
discussed with A. Leggett a role of no-go theorems in QM. In particular, I
wondered why Einstein had never mentioned the von Neumann’s no-to theorem.
(After Einstein’s death, the book of von Neumann [12] was found in Einstein’s
office.) A. Leggett remarked that Einstein was mainly interested in the real
physical situation rather than in formal mathematical statements. I started to
think about the real physical situation described by QM. I understood that
even if some classical prequantum model is formally possible, in spite of all
no-go “theorems”, this does not imply that this model matches with the real
physical situation. Hence, although local realism cannot be rejected as a
consequence of, e.g., violation of Bell’s inequality, one has carefully to
analyze matching of local realism with the real physical situation. Since, as
for Einstein and Bohr, I did not take seriously quantum nonlocality, for me
the questionable point was the possibility of realistic representation of
quantum observables.

By reading Bohr’s works [15], [2] I understood the fundamental role of
measurement context in quantum measurements. Bohr stressed that the result of
measurement is the sum of impacts of the quantum system and the measurement
device. And it is impossible to distill the contribution of the system from
the integral measurement result. Thus quantum observables are not objective.
We could not assign a value of, e.g., position to a quantum system before
measurement. We remark that we discuss measurement of a single quantum
observable and not of a pair of observables. Hence, the essence of this
discussion is not in the impossibility of the joint measurement of some
quantum observables, so called incompatible observables, e.g., position and
momentum (the Bohr’s principle of complementarity [2]). We discuss
contextuality of even a single quantum observable: the properties of
measurement context cannot be separated from intrinsic properties of a quantum
system.444 May be Bohr would not accept that a quantum system has intrinsic
properties at all. (Bohr’s works are very difficult for understanding; try to
read , e.g., [15].) At least it is clear that he was not an idealist. So, he
might agree that, e.g., an electron has some intrinsic properties. However,
for Bohr, these properties were fundamentally inapproachable in measurements.

Unfortunately, in recent discussions related to Bell’s theorem [17] the notion
of contextuality is coupled to the joint measurement of compatible
observables. In this book (following N. Bohr), we always use the notion of
contextuality as the irreducible dependence of the result of measurement (of
even a single observable) on the whole experimental arrangement.

Under influence of Bohr I rejected Einstein’s realism.555I also mention an
intermediate version of the Växjö interpretation [19] in which I tried to
combine realism with contextuality, see also [18]. This interpretation might
be useful in some applications of quantum information outside of physics. Here
two circumstances of totally different nature were very important.

One of my main scientific interests is the application of the mathematical
formalism of QM outside of physics: to study statistical data from cognitive
science, psychology, finances [20]. In these applications observables, “mental
observables”, are nonobjective by their nature; contexts play a crucial role.
And this is well known for psychologists and researchers working in cognitive
science, sociology, economics. This out-physics activity improved my
understanding of the role of contextuality.

Another strong motivation to reject realism of quantum observables666We state
again that in this discussion realism is regarded as objectivity of quantum
observables, i.e., the possibility to assign the value of a quantum observable
to the quantum system before measurement. We do not reject realism in its
general philosophical meaning. In particular, quantum systems have their own
properties, properties of objects. However, these objective properties could
not be approached by the measurement devices in use. The problem is that the
class of measurement devices is not large enough to approach the subquantum
level. Of course, this theoretical discussion does not imply that it would be
possible (at least for our civilization) to elaborate novel measurement
devices to approach subquantum spatial and temporal scales. was related to the
development of a measurement theory for prequantum classical statistical field
theory, PCSFT [21]–[33]. This is the classical field theory which reproduces
quantum averages, including correlations for quantum observables (e.g., spin
or polarization) for entangled systems. Thus, opposite to claims of adherents
of the Copenhagen interpretation (starting with von Neumann [12]), quantum
randomness can be reduced to randomness of classical systems. There is,
however, a crucial proviso. While creation of PCSFT implies QM can be
interpreted as a form of classical statistical mechanics, this classical
statistical theory is not that of particles, but of fields. This means that
the mathematical formalism of QM must be translated into the mathematical
formalism of classical statistical mechanics on the infinite-dimensional phase
space. The infinite dimension of the phase space of this translation is a
price of classicality. From the mathematical viewpoint this price is very
high, because in this case the theories of measure, dynamical systems, and
distributions are essentially more complicated than in the case of the finite-
dimensional phase space found in classical statistical mechanics of
particles.777However, at the model level (similar to quantum information
theory) one can proceed with the finite-dimensional phase space by
approximating physical prequantum fields by vectors with finite number of
coordinates. On the other hand, from the physical and philosophical
viewpoints, considering QM as classical statistical mechanics of fields can
resolve the basic interpretational problems of QM. As was already pointed out,
for example and in particular, quantum correlations of entangled systems can
be reduced to correlations of classical random fields. From this perspective,
quantum entanglement is not mysterious at all, since quantum correlations are
no longer different from the classical ones.

However, usage of PCSFT reduces quantum randomness to classical only on the
level of averages and not individual events, clicks of detectors. PCSFT is the
theory of classical random fields reproducing quantum averages (including
correlations for entangled systems) as averages of quadratic forms of fields.
Since the basic model of prequantum fields is the Gaussian one, these fields
are continuous. Hence, in the same way as in the classical signal theory,
averages are calculated for variables with a continuous range of values. This
model, PCSFT, is really a prequantum model. In accordance with the Bohr’s
viewpoint, we consider QM as an operational formalism describing (predicting)
results of measurements on micro systems. QM cannot describe intrinsic
physical processes in the microworld, but only measurements performed by
macroscopic classical devices. In contrast, PCSFT describes intrinsic
processes in the microworld. However, it does not describe results of
measurements. In principle one might be satisfied with creation of a
prequantum model which reproduces only quantum averages without establishing a
direct connection with theory of measurement on the level of individual
events. This approach would match with views of Schrödinger and the Bild
concept in general. (The Bild concent was formulated by Hertz in his 1894
Prinzipien der Mechanik. Schrödinger’s views to QM were based on this
concept.) However, historical development of QM demonstrated that the Bild
concept was not attractive for the majority of physicists. Since the
experimental verification is considered as the basic counterpart of any
physical theory, Bild-like prequantum theories are considered as metaphysical.
Measurement theory connecting PCSFT with experiment on the level of individual
events was developed in my paper [34]. In this paper I elaborated a scheme of
discrete measurements of classical random signals which reproduces the basic
rule of QM, the Born’s rule. This is the scheme of threshold type detection:
such a detector clicks after it has “eaten” a special portion of energy
$\epsilon$ of a prequantum random signal. We call such measurement theory
threshold detection (TSD) model. TSD is a classical measurement model which
delivers the same predictions as QM. In contrast to QM, which is an
operational formalism888In QM measurement devices are simply black boxes which
are symbolically represented by self-adjoint operators or more generally (and
even more formally) by positive operator valued measures (POVM). , TSD
provides a detailed description of the energy balance in the process of
interaction of a classical random field (fluctuating at a fine time scale)
with a detector, as well as the conditions inducing an individual quantum
event, a click. Thus QM can be interpreted as theory of measurements of
classical random signals with the aid of threshold type detectors. They
produce discretization of continuous classical fields. This discretization is
the essence of quantum phenomena. Hence, “quantumness is created in
detectors”, there is no quantumness without measurement. In particular, this
viewpoint contradicts the views of early Einstein who claimed that the
electromagnetic field is quantized not only in the process of the energy
exchange with material bodies, but even in vacuum. In the PCSFT/TSD model
photons appear only on the level of detection, so they are nothing else than
the clicks of detectors. Surprisingly this viewpoint on the notion of photon
coincides with the views of some top level experimenters working in quantum
optics, e.g., A. Zeilinger, A. Migdal, and S. Polyakov also identify photon
with a click of a detector (their talks at Växjö conferences). Of course, the
majority of experts in quantum optics (including aforementioned) proceed under
the Bohr’s assumption on completeness of QM.

Our threshold detection model is fundamentally based on the assumption of
ergodicity of prequantum random signals, i.e., the time and ensemble averages
coincide. We stress that this ergodicity is related to the subquantum
processes, i.e., to the intrinsic physical processes in the microworld.
Another important assumption is that the prequantum random signals fluctuate
on a very fine time scale (comparing with the time scale of lab-measurements).
The duration of measurement, i.e., the interaction of a random field with a
detector, is huge on the prequantum time scale. However, it is sufficiently
small with respect to the time scale of lab-measurements. (The rigorous
mathematical justification of the derivation of the Born’s rule in the TSD-
framework is rather complicated mathematically and it is based on the theory
of ergodic stochastic processes.)

We remark that ergodic processes are stationary. In the process of interaction
with a detector, a prequantum field (emitted by a source and evolving in the
free space in accordance with the Schrödinger equation with a random initial
condition approaches a steady state, the field becomes stationary. This first
phase is very short. Then the stationary stochastic process transmits its
energy to the detector and this stage is very long (infinitely long in the
pure mathematical framework). We remark that this discussion is about the
prequantum time scale.

Max Born postulated the probabilistic rule coupling the wave function with
quantum detection probabilities: the probability equals to the squared wave
function. Whether it is possible to derive this rule from some natural
physical principles is still the subject of intensive debates. In [34] we
derived the Born’s rule in the PCSFT/TSD framework: threshold type detectors
interacting with random signals of a special class produce a statistics of
clicks which is (approximately) described by the Born’s rule. Of course, this
framework is based on PCSFT-coupling between the covariance operator of a
random field and the density operator of the quantum system corresponding to
this random field, see [21]–[33] for details.999 This derivation of the
fundamental rule of QM which establishes the basic relation of the theoretical
predictions of QM with experiment definitely would not match with the
expectation of those who consider QM as theory which is full of mysteries.

PCSFT/TSD also proposed very natural solution of quantum measurement problem
[12]. Continuous evolution of the prequantum random field induces discrete
jumps corresponding to approaching the detection threshold. Such jumps are
formally described by the von Neumann projection postulate. However, this
solution of the measurement problem is also far from the expectations of
people excited by quantum exotics. Moreover, since, to solve the measurement
problem, we go beyond QM, our solution would not be considered as the solution
of the problem formulated by von Neumann [12]. The latter is expected to be
solved in the conventional quantum framework.

However, the TSD-transition from (continuous) prequantum random fields to
clicks of detectors (discrete events) destroys the PCSFT-objectivity. A
prequantum field emitted by a source does not predetermine the result of
measurement. There are a few different sources of non-objectivity of quantum
observables:

BF. The presence of the irreducibly random background field. PCSFT can
reproduce quantum correlations for entangled systems only by taking into
account the random background. Its presence (everywhere in space) makes
correlations stronger than they could be expected. In particular, by PCSFT
Bell’s inequality is violated only due to the contribution of the random
background. Hence, by emitting the concrete pulse from a source we are not
able to predict its structure at the moment of arrival to a detector;
moreover, the random background is also present in detectors. Therefore we are
not able to predict the output of the detector, even if we were able to
eliminate the random background in space between the source and the detector.
However, this is simply the well known problem of classical stochasticity. As
in classical physics, we may idealize the situation and to assume that the
random background can be eliminated (although in reality this is definitely
impossible). Nevertheless, objectivity cannot be recovered even under this
assumption, see CONT and DC.

CONT. In TSD (in the total accordance with Bohr’s views) the result of a
measurement depends fundamentally on experimental context, on all parameters
of the detector in use (and, in particular, on the threshold in the detector).
For the same classical field, detector can click or not depending on its
settings. On the level of individual events such “properties” as spin and
polarization cannot be assigned to the prequantum wave. They are determined by
detection context. (We remark that a prequantum wave definitely has such
properties, but in continuous and not discrete representation). Thus PCSFT/TSD
is contextual, in general Bohr’s and not restricted Bell’s sense. Suppose now
that we fixed all parameters of a detector. Can we speak about objectivity of
the corresponding quantum observable? (Of course, under the assumption that
the random background was totally eliminated.) Still not, see DC.

DC. In QM coincidence counts of two detectors reacting to a single system
emitted by a source are interpreted as the artifacts induced by noise. In TSD
such coincidence counts are fundamentally irreducible. This is a consequence
of the threshold detection scheme. Hence, real observables are multi-valued.
Even the concrete prequantum wave interacting with the concrete detector and
in the total absence of the random background can produce clicks in two
detectors for, e.g., measurement of spin projection to two axes. It is
impossible to assign to this wave neither the value spin up nor spin down.

The new Växjö interpretation combines Einstein’s and Bohr positions. As well
as Einstein, we do not believe that the quantum state describes completely the
physical state of a quantum system. The quantum state describes only
statistical features. As well as Bohr, we do not consider quantum observables
as objective quantities (intrinsic properties of objects). Quantum observables
are fundamentally contextual.

Postulate IV. (The Växjö interpretation). A wave function describes
correlations in prequantum random fields (which are symbolically represented
as quantum systems).

This paper was written under support of the grant ”Mathematical Modeling of
Complex Hierarchic Systems” of the Faculty of Natural Science and Engineering
of Linnaeus University.

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