# A new approach towards quantum foundation and some consequences

Inge S. Helland
Department of Mathematics, University of Oslo
P.O.Box 1053, N-0316 Oslo, Norway
<EMAIL_ADDRESS>
###### Abstract

A general theory based upon 6 postulates is introduced. The basical notions
are theoretical variables that are associated with an observer or with a group
of communicating observers. These variables may be accessible or inaccessible.
From these postulates, the ordinary formalism of quantum theory are derived.
The mathematical derivations are not given in this article, but I refer to the
recent articles [9, 10]. Three possible applications of the general theory can
be given; 1) The variables may decision variables connected to the decisions
of a person or of a group of persons. 2) The variables may be statistical
parameters or future data, But most importantly here: 3) The variables are
physical variables in some context. This last application gives a completely
new foundation of quantum mechanics, a foundation which in my opinion is much
more easy to understand than the ordinary formalism.The other applications
seem also to give interesting consequences of the approach. Socalled paradoxes
like that of Schrödinger’s cat can be clarified under the theory. Explanations
of the outcomes of David Bohm’s version of the EPR experiment and of the Bell
experiment are provided. Finally, references to links towards relativity
theory and to quantum field theory are given.

Keywords: Acessible theoretical variables; Bell experiment; Born’s formula;
complementarity; inaccessible theoretical variables; quantum foundation

##  1 Introduction

In two books [1, 2] and in a series of articles [3-10] this author has
proposed a completely new foundation of quantum theory, and also, in this
connection, discussed the interpretation of the theory. This foundation is
based upon the general notion of theoretical variables connected to some
person or jointly to a communicating group of persons in some given situation.
These variables may be accessible or inaccessible, again primitive notions.
From a mathematical point of view it is only required that if $\lambda$ is a
theoretical variable and $\theta=f(\lambda)$ for some function $f$, then
$\theta$ is a theoretical variable. And if $\lambda$ is accessible to the(se)
person(s), then $\theta$ is accessible. Some postulates on these variables are
stated, and as consequences several quantum-like conclusions are derived.

Depending upon the situation, several different applications of this theory
may be given. In a physical context the theoretical variables may be physical
variables like time, space, momentum, spin componet, energy, mass and charge.
A variable is seen to be accessible if it can be measured with arbitrary
accuracy. In a statistical context, the variables may be statistical
parameters that are seen to be accessible if they can be estimated invariantly
with respect to a certain group. Finally, in connection to human decisions,
the variables may be decision variables, which are seen to be accessible if
the corresponding decision really can be made.

The postulates of the theory will be formulated in the next Section, but I
will here mention three of them.

FIrst, it is assumed that there exists an inaccessible variable $\phi$ such
that all the accessible ones can be seen as functions of $\phi$. Group actions
are defined on the space upon which $\phi$ varies. In simple physical and
statistical situations this can easily be made concrete.

The first consequence of my postulates are given in Theorem 1 of the next
Section, Roughly speaking, this Theorem goes as follows:

Make some symmetry assumptions; in the discrete case, these seem to be
unnecessary. Then assume that there, in some context and relative to a person
or to a communicating group of persons, exist two really different maximal
accessible theoretical variables. (Using the words of Niels Bohr, these
variables may be called complementary.) The term ‘really different’ means that
there is no one-to-one function between the variables.

Then the exists a Hilbert space $\mathcal{H}$ connected to the situation, and
every accessible varable is connected to a self-adjoint operator in
$\mathcal{H}$. It is crucial that the essential assumption behind this is the
existence of two complementary variables.

In the discrete case, the eigenvalues of the operator are the possible values
of the associated variable. An accessible variable is maximal if and only if
all the eigenspaces of the operator are one-dimensional.

The Born rule is, in the physical context, a part of the conclusion. For a
detailed derivation of this rule from a few postulates, see [1,10]. In the
case where the actual accessible variables are maximal as accessible
variables, two postulates seem to be necessary in my derivation.

First, the likelihood principle from statistics is assumed (see any textbook
in theoretical statistics, or for the origin of this principle, see Berger and
Wolpert [11]). To be applicable in a quantum theory situation, a focused
version of this principle is derived in [1, 10].

The next postulate for the Born rule seems to be important. It is assumed that
the relevant observer(s) has/ have ideals, ideals that can be modeled in terms
of some abstract being which is seen to be perfectly rational. The rationality
is made precise in terms of the Dutch book principle. This postulate may be
interpreted in many different directions.

As an interpretation of the whole theory in its physical application, a
general epistemic interpretation of quantum mechanics is proposed: Quantum
mechanics is not a theory of the world as such, but a theory of an observer’s
or a communicating group of observers’ knowledge about this world. The term
‘knowledge’ is made precise in terms of accessible theoretical variables. In
the discrete case, a pure state represents knowledge of the form $\theta=u$
for some maximal accessible variable $\theta$, and, in general, a mixed state
represents knowledge in terms of a probability distribution over such a
variable.

One aspect of the theory should be mentioned: Since the theory begins with
constructing operators connected to variables, these are seen as the
fundamental building blocks. Pure states as vectors in $\mathcal{H}$ are only
introduced as eigenvectors of physically meaningful operators. This implies a
restriction of the superposition principle, but note that in the simplest case
at least, a qubit, linear combinations of two different state vectors
interpreted as spin components, describe new spin components in some
directions. Also, it is crucial that certain entangled states vectors also can
be given as eigenvectors of some meaningful operator.

The plan of the article is as follows: In Section 2 the theory is formulated
and made precise. As a first consequence, a discussion of Schrödinger’s cat
paradox is given in Section 3. Then the Bell experiment is briefly discussed
in Section 4. Connections to relativity theory and to field theory are
mentioned in Section 5. Sections 6 and 7 give two applications outside quantum
foundation, and Section 8 gives some concluding remarks.

##  2 The basic theory

A completely new approach towards quantum foundations is proposed in Helland
[1-10]; see in particular the most recent articles [9] and [10]. The basis can
be taken to be an observer who is in some physical situation. In this
situation there are theoretical variables, and some of these variables, say
$\theta,\lambda,\eta,...$ are related to the observer $C$. Some of these
variables are _accessible_ to him, which means roughly that it is, in some
future, in principle possible to obtain as accurate values as he wishes on the
relevant variable. Other variables are _inaccessible_. Two examples of the
latter are first the vector (position, momentum) of a particle at some time,
and secondly the full spin vector of a spin particle, an imagined vector whose
discretized projection in the direction $a$ is the spin component in that
direction.

The above characterization of accessible and inaccessible variables is related
to a purely physical implication of the theory. The theory is purely
mathematical, can be made precise in different directions, and the terms
‘accessible’ and ‘inaccessible’ are just primitive notions of the theory. Two
other ways that the theory can be made precise, are 1) quantum decision
theory, where the variables are decision variables; 2) statistical inference
theory, where the variables eiher are statistical parameters or future data.
From a mathematical point of view, it is only assumed that if $\lambda$ is a
theoretical variable and $\theta=f(\lambda)$ for some function $f$, then
$\theta$ is a theoretical variable. And if $\lambda$ is accessible, then
$\theta$ is accessible.

But the main application here is that these variables are theoretical
variables coupled to some physical situation. If necessary, an observed
variable can be modeled as a theoretical variable plus some random error.
Following Zwirn [12, 13], every description of reality must be seen from the
point of view of some observer. Hence we can assume that the variables also
exist relative to $C$.

Also, note that observers may communicate. The mathematical model developed in
the articles mentioned above is equally valid relative to a group of people
that can communicate about the physics and about the various theoretical
variables. This gives a new version of the theory, a version where all
theoretical variables are defined jointly for such a group. The only
difference here is that, for the variables to function during the
communication, they must always be possible to define them in words.

In the two examples mentioned above there are also maximal accessible
variables: In the first example this can be either position or momentum, in
the second example it can be the spin component $\theta^{a}$ in some direction
$a$. From a mathematical point of view, an accessible variable $\theta$ is
called maximal if there is no other accessible variable $\lambda$ such that
$\theta=f(\lambda)$ for some non-invertible function $f$. In other words, the
term ‘maximal’ will then be seen to be maximal with respect to the partial
ordering of variables given by $\alpha\leq\beta$ iff $\alpha=f(\beta)$ for
some function $f$.

Variables that are maximal, different and not in one-to-one correspondence,
are my interpretation of what Niels Bohr called complementary variables.

A basic assumption in my theory is the there exists an inaccessible variable
$\phi$ such that all the accessible variables can be seen as functions of
$\phi$. In the two mentioned examples, $\phi$ can be taken as the vector
(position, momentum), respectively the imagined full spin vector.

Two different acccessible variables $\theta$ and $\eta$ are defined to be
_related_ if there is a transformation $k$ in $\phi$-space and a function $f$
such that $\theta=f(\phi)$ and $\eta=f(k\phi)$.

Two spin components $\theta^{a}$ and $\theta^{b}$ are related, and position
and momentum are related theoretical variables. In the first case,
$\phi$-space can be taken as the plane spanned by the directions $a$ and $b$,
and $k$ can be taken as a $180^{o}$ rotation around the midline between $a$
and $b$. In the last case, $k$ is constructed by a Fourier transform.

As a summary of the above discussion, here are the first 3 postulates of the
theory:

Postulate 1: If $\eta$ is a theoretical variable and $\gamma=f(\eta)$ for some
function $f$, then $\gamma$ is also a theoretical variable.

Postulate 2: If $\theta$ is accessible to $A$ and $\lambda=f(\theta)$ for some
function $f$ , then $\lambda$ is also accessible to $A$.

Postulate 3: In the given context there exists an inaccessible variable $\phi$
such that all the accessible ones can be seen as functions of $\phi$. There is
a transitive group $K$ acting upon $\phi$.

As mentioned, in the two examples above, we may take $\phi=\mathrm{(position,\
momentum)}$, and then let $K$ be the Heisenberg-Weyl group, and $\phi=$ full
spin vector, where $K$ now is taken to be the rotation group. In the last
example, say in the spin 1/2 case, one can model the discrete spin component
$\theta^{a}$ in direction $a$ as $f(\phi)=\mathrm{sign(\ cos}(a,\phi))$.
Giving $\phi$ a reasonable distribution here, results in a correct
distribution of each $\theta^{a}$.

A definition is now needed for the fourth postulate:

Definition 1.  The accessible variable $\theta$ is called _maximal_ if
$\theta$ is maximal as an accessible variable under the partial ordering
defined by $\alpha\leq\beta$ iff $\alpha=f(\beta)$ for some function $f$.

Note that this partial ordering is consistent with accessibility: If $\beta$
is accessible and $\alpha=f(\beta)$, then $\alpha$ is accessible. Also, $\phi$
from Postulate 3 is an upper bound under this partial ordering.

Postulate 4: There exist maximal accessible variables relative to this partial
ordering. For every accessible variable $\theta$ there exists a maximal
accessible variable $\lambda$ such that $\theta$ is a function of $\lambda$.

Then, in my opinion, two different maximal accessible variables come very
close to what Bohr called complementary variables; see Plotnitsky [14] for a
thorough discussion.

It is crucial what is meant by ‘different’ here. If $\theta=f(\eta)$ where $f$
is a bijective function, there is a one-to-one correspondence between $\theta$
and $\eta$, they contain the same information, and they must be considered
‘equal’ in this sense. $\theta$ and $\eta$ are said to be ‘different’ if they
are not ‘equal’ in this meaning. This is consistent with the partial ordering
in Definition 1. The word ‘different’ is used in the same meaning in the
Theorems below.

Postulate 4 can be motivated by using Zorn’s lemma - if this lemma, which is
equivalent to the axiom of choice, is assumed to hold - and Postulate 3, but
such a motivation is not necessary if Postulate 4 is accepted. Physical
examples of maximal accessible variables are the position or the momentum of
some particle, or the spin component in some direction. In a more general
situation, the maximal accessible variable may be a vector, whose components
are simultaneously measurable.

Assuming these postulates, the main result of Helland [3, 9] is as follows:

Theorem 1 Consider a context where there are two different related maximal
accessible variables $\theta$ and $\eta$. Assume that both $\theta$ and $\eta$
are real-valued or real vectors, taking at least two values. Make the
following additional assumptions:

(i) On one of these variables, $\theta$, there can be defined a transitive
group of actions $G$ with a trivial isotropy group and with a left-invariant
measure $\rho$ on the space $\Omega_{\theta}$.

(ii) There exists a unitary multi-dimensional representation $U(\cdot)$ of the
group behind the group actions $G$ such that for some fixed
$|\theta_{0}\rangle$ the coherent states $U(g)|\theta_{0}\rangle$ are in one-
to-one correspondence with the values of $g$ and hence with the values of
$\theta$.

Then there exists a Hilbert space $\mathcal{H}$ connected to the situation,
and to every (real-valued or vector-valued) accessible variable there can be
associated a symmetric operator on $\mathcal{H}$.

The Hilbert space $\mathcal{H}$ is of course related to the representation
given in point (ii). The main result is that each accessible variable $\xi$ is
associated with an operator $A^{\xi}$. The proof goes by first constructing
$A^{\theta}$ and $A^{\eta}$, then operators associated with other accessible
variables are found by using the spectral theorem. For conditions under which
a symmetric operator is self-adjoint or Hermitian, see Hall [15].

An important special case is when the accessible variables that take a finite
number of values, say $r$. For this case it is proved in Helland [9] that a
group $G$ and a transformation $k$ with the above properties always can be
constructed. The following Corollary then followx:

Corollary 1 Assume that there exist two different maximal accessible variables
$\theta$ and $\eta$, each taking $r$ values, and not in one-to-one
correpondence. Then, there exists an $r$-dimensional Hilbert space
$\mathcal{H}$ describing the situation, and every accessible variable in this
situation will have an associated self-adjoint operator in $\mathcal{H}$.

Theorem 1 and its Corollary constitute the first step in a new proposed
foundation of quantum theory.

The second step now is to prove the following: If $k$ is the transformation
connecting two related maximal accessible variables $\theta$ and $\eta$, and
$A^{\theta}$ and $A^{\eta}$ are the associated operators, then there is a
unitary matrix $W(k)$ such that $A^{\eta}=W(k)^{-1}A^{\theta}W(k)$. This, and
a more general related result is proved as Theorem 5 in Helland [9].

Given these results, a rich theory follows. The set of eigenvalues of the
operator $A^{\theta}$ is identical to the set of possible values of $\theta$.
The variable $\theta$ is maximal if and only if all eigenvalues of the
corresponding operator are simple. In general, the eigenspaces of $A^{\theta}$
are in one-to-one correspondence with questions ‘What is $\theta$’/ ‘What will
$\theta$ be if we measure it?’ together with sharp answers $\theta=u$.

If $\theta$ is maximal as an accessible variable, the eigenvectors of
$A^{\theta}$ have a similar interpretation. In my opinion, such eigenvectors,
where $\theta$ is some meaningful variable, should be tried to be taken as the
only possible state vectors. These have straightforward interpretations, and
from this version of the theory, also a number of so-called ‘quantum
paradoxes’ can be illuminated, see Helland [2, 9].

This version of quantum theory implies a restriction of the superposition
principle. But note the following: At least in the simplest case, a qubit,
where the states can be interpreted as spin components of a particle, a linear
combination of two pure states also can be given the same interpretation, i.e.
is an eigenvector of some spin operator. Also, the singlet state constructed
from two accessible variables, a state that is important both in David Bohm’s
version of the EPR discussion and in any discussion of the Bell experiment, is
an entangled state that is included in my theory. This will be further
discussed below.

What is lacking in the above theory, is a foundation of Born’s formula, the
means for calculating quantum probablitities. Several versions of Born’s
formula are proved from two new postulates in Helland [10]. The first
postulate is as follows:

Postulate 5: The likelihood principle from statistical theory holds.

The likelihood principle is a principle that nearly all statisticians base
their inference on, at least when the principle is restricted to a specific
context. For a basic discussion, see Berger and Wolpert [11]. It is closely
connected to the notion of a statistical model: In an experimental setting,
there is a probability model of the data $z$, given some parameter $\theta$.
Note that a statistical parameter, in an important application of my theory,
can be seen as an accessible theoretical variable. The statistical model can
be expressed by a point probability $p$ for discrete data or a probability
density $p$ for continuous data: in both cases we have $p=p(z|\theta)$.

The _likelihood_ is defined as $p$, as seen as a function of the parameter
$\theta$:

$L(\theta;z)=p(z|\theta),$ (1)

and the likelihood principle runs as follows: _Relative to any experiment, the
experimental evidence is always a function of the likelihood._ Here, the term
‘experimental evidence’ is left undefined, and cam be made precise in several
directions.

In a quantummechanical setting, a potential or actual experiment is seen in
relation to an observer $C$ or to a communicating group of observers.
Concentrate here on the first scenario. In the simplest case we assume that
$C$ knows the state $|a;i\rangle$ of a physical system, and that this state
can be interpreted as $\theta^{a}=u_{i}$ for some maximal accessible variable
$\theta^{a}$. Then assume that $C$ has focused upon a new maximal accessible
variable $\theta^{b}$, and we are interested in the probability distribution
of this variable.

The last postulate is connected to the scientific ideals of $C$, ideals that
either are given by certain conscious or unconcious principles, or are
connected to some concrete persons. These ideal are then modeled by some
higher being $D$ that $C$ considers to be perfectly rational.

Postulate 6: Consider in the context $\tau$ an epistemic setting where the the
likelihood principle from statistics is satisfied, and the whole situation is
observed by an experimentalist $C$ whose decisions can be modeled to be
influenced by a superior being $D$. Assume that one of $D$’s probabilities for
the situation is $q$, and that $D$ can be seen to be perfectly rational in
agreement with the Dutch Book Principle.

The Dutch Book Principle says as follows: No choice of payoffs in a series of
bets shall lead to a sure loss for the bettor.

A situation where Postulate 6 holds will be called a rational epistemic
setting. It will be seen to imply essential aspects of quantum mechanics. As
shown in [5], it also gives a foundation for probabilities in quantum decision
theory.

In Helland [1, 10], a generalized likelihood principle is proved from the
ordinary likelihood principle: Given some experiment, or more generally, some
context $\tau$ connected to an experiment, any experimental evidence will
under the above assumptions be a function of the so-called likelihood effect
$F$; a definition and a discussion is given in [1, 10]. In particular, the
probability $q$ is a function of $F$: $q(F|\tau)$.

Using these postulates and a version of Gleason’s Theorem due to Busch [16],
the followind variant of Born’s formula is proved:

Theorem 2 [Born’s formula, simple version]  Assume a rational epistemic
setting. In the above situation we have:

$P(\theta^{b}=v_{j}|\theta^{a}=u_{i})=|\langle a;i|b;j\rangle|^{2}.$ (2)

Here, $|b;j\rangle$ is the state given by $\theta^{b}=v_{j}$. In this version
of the Born formula, I have assumed _perfect measurements:_ There is no
experimental noise, so that the experiment gives a direct value of the
relevant theoretical variable.

An advantage of using the version of Gleason’s theorem due to Busch [16] in
the derivation of Born’s formula, is that this version in valid also in
dimension 2. Other derivations using the same point of departure, are Caves et
al. [17] and Wright and Weigert [18]. In Wright and Weigert [19] the class of
general probabilistic theories which also admit Gleason-type theorems is
identified. But for instance, Auffeves and Granger [20] derive the Born
formula from other postulates.

The simple Born formula formula can now be generalized to the case where the
accessible variables $\theta^{a}$ and $\theta^{b}$ are not necessarily
maximal. There is also a variant for a mixed state involving $\theta^{a}$.

For completeness, define first the mixed state associated with any accessible
variable $\theta$. If $\theta$ is not maximal, we need, for the Born formula,
the assumption that there exists a maximal accessible variable $\eta$ such
that $\theta=f(\eta)$ and such that each distribition of $\eta$, given some
$\theta=u$, is uniform. Furthermore some probability distribution of $\theta$
is assumed. Let $\Pi_{u}$ be the projection of the operator of $\theta$ upon
the eigenspace associated with $\theta=u$. Then define in the discrete case
the mixed state operator

$\rho=\sum_{j}P(\theta=u_{j})\Pi_{u_{j}}=\sum_{i}\sum_{j}P(\eta=v_{i}|\theta=u_{j}=f(v_{i}))P(\theta=u_{j})|\psi_{i}\rangle\langle\psi_{i}|.$
(3)

Here, $|\psi_{i}\rangle$ is the state vector associated with the event
$\eta=v_{i}$ for the maximal variable $\eta$.

From this definition, we can show from (2), assuming that the maximal
$\eta^{a}$ corresponding to $\theta^{a}$ also is a function of $\phi$, that in
general

$P(\theta^{b}=v|\rho^{a})=\mathrm{trace}(\rho^{a}\Pi_{v}^{b}),$ (4)

with the projection $\Pi_{v}^{b}$ projects upon the subspace of $\mathcal{H}$
given by $\theta^{b}=v$.

This result is not necessarily associated with a microscopic situation. A
macroscopic application of Born’s formula is given in [1].

The result can also be generalized to continuous theoretical variables by
first approximating them by discrete ones. For continuous variables, Born’s
formula is most easily stated on the form

$E(\theta^{b}|\rho^{a})=\mathrm{trace}(\rho^{a}A^{\theta^{b}}).$ (5)

In this formula do not assume that the accessible variable $\theta^{b}$ is
maximal. Hence a corresponding formula is also valid for any function of
$\theta^{b}$, for instance $\mathrm{exp}(i\theta^{b}x)$ for some fixed $x$.
The operator corresponding to a function of $\theta^{b}$ can be found from
$A^{\theta^{b}}$ by using the spectral theorem. From this, the probability
distribution of $\theta^{b}$, given the information in $\rho^{a}$, can in
principle be recovered.

Finally, one can generalize to the case where the final measurement is not
necessarily perfect. Let us assume future data $z^{b}$ instead of a perfect
theoretical variable $\theta^{b}$. Strictly speaking, for this case the
focused likelihood principle is only valid under the following condition:
$p(z^{b}|\theta^{b}=u_{j})=p(z^{b}|\theta^{b}=u_{k})$ implies $u_{j}=u_{k}$;
see [1]. But this is not needed here; we only need the focused likelihood
principle for perfect experiments, in order to prove that (4) is valid. Then
we can define an operator corresponding to $z^{b}$ by

$A^{z^{b}}=\sum_{j}z^{b}p(z^{b}|\theta^{b}=u_{j})\Pi_{u_{j}}^{b},$ (6)

and, from the version (4) of the Born formula, we obtain

$E(z^{b}|\rho^{a})=\mathrm{trace}(\rho^{a}A^{z^{b}}).$ (7)

All this not only points at a new foundation of quantum theory, and it also
suggests a general epistemic interpretation of the theory: Quantum theory is
not directly a theory about the world, but a theory about an actor’s knowledge
of the world. Versions of such an interpretation already exist, and they are
among the very many suggested interpretations of quantum mechanics. Further
discussions of this interpretation are given in [1, 9, 10].

##  3 Schrödinger’s cat

Schrödinger’s cat is a well known so-called paradox in quantum theory, and
there is a large literature on this paradox. It assumes as a thought
experiment that a cat is closed into a sealed box, and that there in this box
also is some deadly poison that can be released by the decay of some
radioactive material. The question that troubled the originator Schrödinger
and many other physicists was: At a time $t$, to an outside observer, is the
cat dead or alive? The origin of the difficulty is the superposition
principle. In a quantummechanical state connected to this observer, in a state
which is a superpostion of a ‘dead’ state and an ‘alive’ state, the cat
seemingly is both dead and alive.

My theory does not assume a genaral validity of the superposition principle. A
pure state vector is included in the theory only if it is an eigenvector of
some meaningful physical operator.

It is enlightening to see this in connection to a recent article by Maccone
[21]. There, an observator $S$ is introduced which is complementary to the
property dead/ alive associated with the cat. In my terminology, there should
be a theoretical variable $\theta$ connected to $S$. The crucial question is:
Can $\theta$ be measured? According to a long discussion in Skitiniotis et al.
[22], this can in principle be done, but it requires a tremendous measurement
device. In practical terms I would say that $\theta$ is inaccessible to any
given human observer.

To put it simply, a state is always connected to the mind of some observer,
and in my theory I allow this observer to answer ‘I don’t know’ to certain
questions, in this case, the question ‘Is the cat alive at time $t$?’ Once the
box is opened, the observer can give a precise answer to the corresponding
question.

##  4 The Bell experiment

The Bell experiment is a well known experiment whose outcome has caused much
discussion among theoretical physicists. In 2022, the Nobel prize in physics
was given to three physicists that had performed so-called loophole-free
variants of this experiment. The conclusion, which has astonished both
scientists an laymen, stands firm. It seems like there either is a violation
of the principle of locality, or that the ordinary notion of reality has to be
abandoned.

Very briefly, the Bell experiment has to do with two observers Alice and Bob
that can not communicate. Midways between the two, a pair of entangled
particles are sent out, one towards each observer. Particles with spin 1/2 are
assumed. Entanglement means that the spin components of the pair is in a
singlet state are given by

$|\psi\rangle=\frac{|1+,2-\rangle-|1-,2+\rangle}{\sqrt{2}}.$ (8)

Here, the spin components of the particles are specified in some direction,
say the vertical, 1 and 2 refer to the two particles, and - means downwards
and + means upwards. (In the most recent experiments, polarization of phtons
is measured, but the discussion is similar.)

Alice is given two directions $a$ and $a^{\prime}$ to choose from, and
measures spin components $A$ or $A^{\prime}$ in these directions. Bob is given
two directions $b$ and $b^{\prime}$ to choose from, and measures spin
components $B$ or $B^{\prime}$ in these directions. The measured spin
components are coded as -1 and +1.

The experiment is repeated many times, and each time, Alice and Bob make their
choices. As the measured spin components can be seen as random variables in
some sense, expectation of pairs presumably exist. The so-called CHSH
inequality is

$E(AB)+E(A^{\prime}B)+E(AB^{\prime})-E(A^{\prime}B^{\prime})\leq 2.$ (9)

This inequality may be derived, seemingly using only that the assumption that
$A,B,A^{\prime}$ and $B^{\prime}$ exist simultaneously as random variables
taking the values $\pm 1$, by a very simple argument. But, by suitable choices
of $a,b,a^{\prime}$ and $b^{\prime}$ it can be shown that (9) is violated by
quantum mechanics.

Now to the conclusion of very many Bell experiments: Again by choosing
$a,b,a^{\prime}$ and $b^{\prime}$ in a suitable way, (9) is also violated in
practice. Somehow, the simple argument or its assumption must be wrong. In the
physical literature, the assumpion behind (9) is called local realism.

In Helland [4, 23], a very specific explanation of the outcome of the Bell
experiment is given.

First, it is crucial to show that the state (8) is allowed by my theory. It
should be an eigenstate for some operator which is physically meaningful,
which in my terminology means that the operator is connected to a theoretical
variable that is accessible to some observer.

Let the observer Charlie be some person who after the experiments are given
all the data from Alice and Bob, and then is trying to model the results of
the experiments. During an experiment, he may be able to observe the whole
experiment and record settings and responses. To him, both the spin vectors
$\phi_{A}$ and $\phi_{B}$ for Alice and Bob are inaccessible, but it turns out
that the dot-product $\xi=\phi_{A}\cdot\phi_{B}$ is accessible to him. In
fact, he is forced to be in the state given by $\xi=-3$. This can be seen as
follows: The eigenvalues of the operator corresponding to $\xi$ are -3 and +1,
and the eigenstate corresponding to $\xi=-3$ is just the singlet state (8).
(Exercise 6.9, page 181 in Susskind and Freedman [24].) Thus, being in this
singlet state as an observer of the whole experiment, he is forced to have
$\xi=-3$.

What does it mean that the dot product of the two vectors is -3? Expressed by
the cartesian components that are $\pm 1$, we must necessarily have
$\phi_{A}^{x}\phi_{B}^{x}=\phi_{A}^{y}\phi_{B}^{y}=\phi_{A}^{z}\phi_{B}^{z}=-1$,
that is $\phi_{B}^{x}=-\phi_{A}^{x}$ etc., which implies
$\phi_{B}^{a}=-\phi_{A}^{a}$ in any direction $a$. This is an important
conclusion, and it is true for any observer Charlie. This explains the result
of David Bohm’s version of the EPR experiment, an experiment proposed by
Einstein et al [25], and a result that has caused much confusion.

But go back to the Bell experiment. Here my discussion relies on a general
mathematical result of my postulates, a result which in the physical
application must be of relevanse to the mind of any observer, any person.
Recall the definition of related accessible variables given in Section 2.

Theorem 3 Assume two related maximal accessible variables $\theta$ and $\eta$.
Then any other variable $\lambda$ which is related to $\theta$, but unrelated
to $\eta$ can not be a maximal accessible variable.

The proof is given in [4] and [23].

With the physical/ psychological application of my theory described in Section
2, this means that $\lambda$ can not be maximally accessible to the mind(s) of
any observer(s) (at the same time; if we let time vary, the observer may have
many variables in his mind, also unrelated ones.)

Apply this to the argument around the CHSH inequality. Consider the observer
Charlie, and assume that the variables $\theta=(A,B)$ and
$\eta=(A,B^{\prime})$ are accessible to him. One can show that, at a fixed
time, each of these are maximal as accessible variables: Charlie is only able
to observe one run at the time. Also, they are related relative to a $\phi$
containing all of $A,B,A^{\prime}$ and $B^{\prime}$. (They have the outcome
$A$ in common; we can let the transformation $k$ rotate $b$ onto $b^{\prime}$
and hence $B$ onto $B^{\prime}$.) Then look in addition to the variable
$\lambda=(A^{\prime},B)$. This is related to $\theta$, but unrelated to
$\eta$. If it were accessible to Charlie, it would again be maximal as an
accessible variable, but this is impossible by Theorem 3.

This, any observer Charlie can not have the three variables $\theta,\eta$ and
$\lambda$ in his mind at the same time. _As a consequence of this, he is not
able to follow the simple argument leading to the CHSH inequality._ More
concretely, if he were statistician, he is not able to put up a valid joint
probability model for the variables $A,A^{\prime},B$ and $B^{\prime}$. Note
that Charlie can be any observer.

Thus, the result of the Bell experiment can in my opinion be explained by the
fact that we all have a limited mind. This has consequences for this
experiment, but as a general observation it also has other consequences.

This explanation is not related to non-locality, and the same can be said
about my explanation of Bohm’s version of the EPR experiment discussed above.
Neither can it be directly related to non-reality, but this is a deep issue
that can only be discussed in relation to interpertations of quantum theory.

##  5 Some consequences for relativity theory and quantum field theory

The issues of this Section are discussed in the book [2]. The notions of
accessible and inaccessible variables are also applicable in special and
general relativity theory, and much of the theory generalizes. In particular,
it is natural to assume that variables related to the interior of a black hole
are inaccessible to any observer. Finally, one may extend the notions to
accessible and inaccessible fields. Details are given in the book, some of the
discussion is given in [6].

##  6 Applications to statistical inference

As stated above, statistical inference is based upon a statistical model
specified by a probalility function $p(z|\theta)$ for data $z$ and parameter
$\theta$. There is a large literature on statistical inference, but in my
opinion, something related to possible structures in the parameterspace is
missing. For instance, there may be groups acting on the parameters, and one
should try to work towards a systematic theory of model reduction.

My approach involving accessible and inaccessible theoretical variables may
have something to contribute here. In Theorem 1 I need a transitive group
acting on $\theta$ and in Postulate 3 I assume a transitive group acting upon
the large and inaccessible $\phi$. If one can find nontransitive groups acting
upon parameters, a reasonable policy for model reduction is _to reduce to a
suitable orbit of the group._ At least in the case of partial least squares
regression [26] this works well. A systematic application of this principle to
quantum mechanics has not been discussed yet.

The case of partial least squares (PLS) regression and its possible relation
to the theory of Section 2 here, has been systematically treated in Helland
[27]. The total model parameter is called $\phi$, and $\theta$ is the reduced
regression parameter corresponding to a PLS model with $m$ steps. The
conditions of Theorem 1 is shown to hold in the case where $\eta$ is another
reduced parameter, so, in particular a Hilbert space exists with an operator
$A^{\theta}$ corresponding to $\theta$. Optimality of the PLS model compared
to other model reductions is discussed.

Another application is the following: Soppose that we have just done a PLS
regression based on some data. Let the estimated regression parameter be
$\widehat{\theta_{1}}$. Assume then that we will do a similar investigation
later, and call the new estimator $\widehat{\theta_{2}}$. Then we may look
upon these two estimators as maximal accessible variables, use Theorem 1
again, and by the Born formula (7) in principle find a distribution of
$\widehat{\theta_{2}}$, given a posterior distribution of
$\widehat{\theta_{1}}$, taking into account that the same parametric model is
used in the two cases.

Similar applications should be investigated.

##  7 A foundation of quantum decision theory

The simplest model of a decision process connected to some person $C$ or to a
group of communicating persons is as follows: Assume that the choice is
between $n$ possible actions $a_{1},a_{2},...,a_{n}$. Define a decision
variable $\theta$ as being equal to $u$ if action $a_{u}$ is chosen. As stated
in the Introduction, such a decision variable can be taken as a theoretical
variable, accessible if the decision really can be done. If there are two
complementary decision processes at the same time, the conditions of Corollary
1 are satisfied, and this can be taken as a new theoretical foundation of
quantum decision theory, a discipline where most of the literature is based
upon empirical investigations, see for instance Busemeyer and Bruza [28]. This
foundation is further discussed in [5].

##  8 Conclusions

The ordinary formal basis for quantum mechanics is very difficult to
understand. It has lead to very many interpretations, and for people outside
the physical community it seems to be nothing but pure formalism.

By contrast, the foundation discussed in this article is given by 6
postulates, in my opininion fairly easy to understand, also for outsiders.
Still, perhaps, interpretation can be discussed, but a general epistemic
interpretation seems to be natural. The basis is just theoretical variables,
accessible or inaccessible, and connected to an observer or to a group of
communicating observers in some physical context. Since basic quantum theory
may be derived from these postulates, everything in the literature which is
derived from this formalism can in principle also be deduced from the
postulates. An exception is that I restrict the superposition somewhat, as
discussed above, and also in [9].

The mathematics behind these derivations is not given here; it is all
contained in the recent articles [9] and [10].

Socalled paradoxes like Schrödinger’s cat can be understood using this theory.
The results of the EPR type experiments and the Bell experiments can be
explained using the theory.

It is also important that this theory has consequences outside basic quantum
theory. A foundation of quantum decision theory may be found. Links to
statistical inference theory are beginning to appear. Finally, links to
special and general relativity and to quantum field theory can be discussed. A
thorough discussion will be given in the forthcoming book [2]; a beginning can
be found in [6].

I am open to any criticism of this theory, but I also have a hope that further
research, using my theory as a basis, can be carried out.

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