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arXiv:0801.2026v1 [quant-ph] 14 Jan 2008
Quantum Mechanics from Focusing and
Symmetry.
Inge S. Helland
Dept. of Mathematics, Univ. of Oslo
P.O.Box 1053 Blindern, N-0316 Oslo, Norway
email: ingeh@math.uio.no
Abstract
A foundation of quantum mechanics based on the concepts of focusing
and symmetry is proposed. Focusing is connected to c-variables - inacces-
sible conceptually derived variables; several examples of such variables are
given. The focus is then on a maximal accessible parameter, a function
of the common c-variable. Symmetry is introduced via a group acting on
the c-variable. From this, the Hilbert space is constructed and state vec-
tors and operators are given a clear interpretation. The Born formula is
proved from weak assumptions, and from this the usual rules of quantum
mechanics are derived. Several paradoxes and other issues of quantum
theory are discussed.
1Introduction
Nobody doubts today that quantum mechanics is a true theory.
though the calculations devised by the standard theory give accurate predictions
that all researchers agree upon, the same theory is felt by many as being too
abstract and too formal. From the very beginning [1] leading physicists and
mathematicians have tried to find a new and better logical basis for the theory,
and this search has continued until now [2, 3, 4, 5, 6]. The discussions around the
foundation of quantum mechanics are not just theoretical; obscurities connected
to foundational issues also has as a consequence that one may be uncertain
whether certain applied statements should be classified as facts or myths [7].
Zeilinger [2] compares the situation with that of the special and general
relativity theory, which both are based on firm foundational principles, and
requests such principles for quantum theory. I agree that what one needs is not
a new formal axiomatic formulation of the mathematical foundations of quantum
mechanics, but completely new foundational conceptually defined principles. In
this paper I propose two: Maximal focusing in a situation with inaccessible
conceptually defined variables, and symmetry as given by a group acting on
these variables. These principles will be made more precise shortly, and they
But even
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will be demonstrated in the paper to lead to the essential parts of quantum
theory. What may be lacking in the present development, is a description of
more complicated quantummechanical situations without symmetry. For these
situations we can refer to the extensive new foundational recent work by Doering
and Isham [8, 9, 10, 11] based on category theory. Note that group theory may
be looked upon as a specialization of category theory, a specialization which
illustrates many basic features of that theory.
In contrast to [8, 9, 10, 11], I will here want to keep the foundation simple.
In fact, part of my aim will be to connect the discussion to daily life concepts.
Also, a connection to statistical inference will be made clear, a connection which
is crucial, Quantum theory and statistical theory are both concerned with pre-
dictions based upon observations, yet all indications of any connection between
these two worlds seem up to now to be completely absent from the literature.
Finally, the explicit use of group theory in the foundation may have interest in
itself. The physics literature is full of examples where group theory has turned
out to be useful in a quantummechanical setting.
In agreement with the statistical tradition I will regard all measurement
apparata as macroscopic, and I will use the ordinary statistical inference theory
on all measurements made. In particular, this implies that the measurement
problem in its ordinary quantum mechanical formulation where a quantum state
also is given to the measurement apparatus, is barely touched upon here. For
a recent survey paper where the measurement problem is related to various
interpretations of quantum theory, see Wallace [12].
Also, by following up the statistical way of thinking, the interpretetion of
the quantum states advocated here is epistemic. In accorance with Niels Bohr’s
words ’It is wrong to think that the task of Physics is to find out what Nature
is. Physics concerns what we can say about nature.’ [13], this can be said to be
in agreement with the classical Copenhagen school.
2Briefly on statistical theory.
A basic concept is that of a statistical parameter λ. The fundamental questions
that we ask nature, are in terms of such a parameter: What is the value of λ?
Later in the present paper, all parameters are discrete, and such a question can
be answered literally in a simple way. For continuous parameters, statistical
theory offers the tools of point estimation, interval estimation and hypothesis
testing.
In any case, what lies behind these inference procedures, is a statistical
model: A probability distribution Pλ(·) of the observed data, given the value of
the parameter λ. This leads to a very rich theory, where inference statements
about λ can be formulated in a precise way, and where one also may formulate
in various ways just how precise of these statements are. The simplest possible
statistical model is a perfect one, where each Pλis just 0 or 1 as functions of
the data, most often such that the sets where Pλis 1, are disjoint for different
λ. In this case the given observation leads to a unique estimated valueˆλ This
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simplification may be relevant in some physical applications.
In general, an estimator of a parameter λ is a functionˆλ(y) of the observa-
tions used to estimate λ. The problem of finding an estimator will in general
not have a unique solution. One aim of statistics is to find good estimators.
There are two schools of statistical inference, the frequentist and the Bayesian.
The Bayesians allow themselves to make use of, in addition to the statistical
model, prior probabilities on λ. In the present paper I will also make extensive
use of probabilities on the parameter space, but now introduced by Born’s for-
mula, and thus being transition probabilities between different states. One way
to look upon these probabilities are as priors for the next experiment, given the
result of an earlier perfect experiment.
The main idea from statistical theory that I will use in this paper, is that of
using parametersas intermediate quantities when asking questions about nature.
For those who want to read more about statistical theory, I can recommend on
the intermediate level books [14, 15] or more advanced books like [18, 19, 20].
3 Conceptually defined variables, and focusing.
There has been a long debate on hidden variables in quantum mechanics, see, e.g.
[21, 22]. Today these words are met with much scepticism, but a related concept,
hidden measurements, has been introduced successfully by Aerts [23, 24].
Hidden variables are assumed to take values. As a contrast, in this paper I
will base much of the discussion upon conceptually defined variables, abbreviated
c-variables, and denoted by φ, which as a rule do not take values at all. To some,
this may seem like a strange construction, but in fact both our daily language
and the scientific language is full of such conceptual variables.
A design of experiment example.
given one out of two possible treatments 1 and 2. Let λ1be his expected life
time if he is given treatment 1; let λ2be his expected life time if he is given
treatment 2. Consider the vector φ = (λ1,λ2). This variable can perhaps be
given a hypothetical value, but it can never be given a real value which can
be compared to observations. More precisely, φ can never be a parameter of a
statistical model. But we can focus upon λ1, and use this to make a model for
the lifetime of the patient.
Consider a single patient who can be
Counterfactual parameters.
In very many cases at the experimental design phase it is natural to define a
conceptual variable or c-variable
As a continuation of the previous example:
φ = (λ1,λ2,...,λr)
at the outset. Only one parameter λjis realized by the experiment; the rest are
called counterfactuals. Counterfactual variables/parameters are also important
in other cases than statistical experiments. They have turned out to be essential
in causal reasoning; see the book by Pearl [25].
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A special apparatus.
is so sensitive that it is destroyed after one single measurement. Let µ be the
length which is to be measured. Assume furthermore that the measurement
uncertainty σ, defined at the outset in a Bayesian way, only can be estimated
by destroying the whole apparatus. Let φ = (µ,σ). Then it is impossible to
estimate the whole c-variable φ, only µ or σ can be estimated. In general, the
choice of experimental question is essential.
An apparatus for a very special length measurement
Too many parameters.
set of potential explanatory variables, resulting in a formal regression model
In linear prediction problems one often has a large
y = β1x1+ β2x2+ ... + βpxp+ e,
assuming the variables centered for simplicity. Here e is an independent error
term. When p is large compared to the number n of experimental units, in
particular if p is much larger than n, this can hardly be called a data model.
And φ = (β1,β2,...,βp) is hardly a parameter vector, but more a c-variable.
Again one must focus from this, but now the focusing is not necessarily related
to a choice of experiment. What one can do in this prediction situation - and
also does in practice - is to focus on a certain parameter
θ = (βi1,...,βiq)
and the corresponding model, using both the data and the conceptual setting
in the selection.
In most of the examples above, the c-variable φ is such that it is impossible
to estimate it from the available data. It is then called inaccessible.
The focusing takes place by a question and an answer: To find out something
about nature, one must not only look in all directions for facts, but often focus
upon a parameter λ, an accessible part of φ, and then look for answer to the
question: What is the value of λ?
In my interpretation of quantum mechanics, two questions are called com-
plementary if they are given by mutually exclusive functions of a common in-
accessible c-variable φ. More precisely, this means: Let the parameters of the
two questions be λ and µ. Then the vector variable θ = (µ,λ) is in itself an
inaccessible c-variable. An example might be where µ is the theoretical position
and λ is the theoretical momentum of a single particle. The word ’theoretical’
here just indicates that a measurement apparatus because of measurement un-
certainties may give different values. This latter phenomenon is a rather trivial
one, but provides the link from statistics to physics, in particular the link from
statistical parameters to theoretical physical variables.
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4 Symmetry in the parameter space.
Assume in this Section that a symmetry group G acting on a parameter space
Θ and simultaneously on the space of observations by
Pθg(A) = Pθ(Ag−1).
Some examples may be scale change, translation or rotations, It will turn out
later to be advantageous to place the group symbol to the right of the quantity
to transform.
Note that Pθ(yg ∈ B) is equal to Pθ(y ∈ Bg−1), thus Pθg(y ∈ B); so
transformation of observations is related to transformation of parameters in
simple situations. In this paper we will concentrate on the group action on
the parameters of potential statistical models or on a single model, thus on the
parameter space Θ.
4.1Orbits. Transitive group.
Fix θ0∈ Θ, a point in the parameter space. Consider the set {θ0g : g ∈ G}, the
set of all parameter values that are transforms of θ0. This is called the orbit of
G containing θ0.
Here is an example which may be illustrative: Let Θ be like the surface of
the earth, and let G act as does the rotation of the earth. Then the orbits will
be the circles of latitude.
The space Θ is always partitioned into disjoint orbits. A useful way to look
upon an orbit is that it is a minimal invariant set under the group.
If there is only one orbit, this will consist of the whole space Θ. Then we
say that the group is transitive, or more precisely: G is acting transitively upon
Θ.
4.2 Invariant measure.
Under weak conditions there exists a right-invariant measure ρ on the parameter
space:
ρ(Γg) = ρ(Γ) for g ∈ G and all Γ ⊆ Θ.
The measure ρ can be taken as a probability measure if Θ is compact. This
measure ρ is unique if and only if the parameter group is transitive. (In the
noncompact case this uniqueness is up to a multiplicative constant.)
Whenever there is a natural symmetry group acting upon Θ, in particular if
it is transitive, there are many arguments for using the right invariant measure
ρ as a noninformative prior in Bayesian data analysis.
4.3Subparameters and estimation of parameters.
A subparameter λ = λ(θ) is called permissible if
λ(θ1) = λ(θ2) implies λ(θ1g) = λ(θ2g) for all g.
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Then λ transforms under G by
λ(θ) → (λg)(θ) = λ(θg).
For any subparameter λ there is a maximal subgroup Gaof G under which
λ is permissible.
A group defined on the parameter space and conformably on the space of
observations, may be useful in finding a good estimator for λ. An estimator
which transforms under the group in the same way as the parameter, is called
equivariant. More precisely, if λ(θ) is a permissible parameter, so that (λg)(θ) =
λ(θg), then an estimatorˆλ is called equivariant ifˆλ(yg) = (ˆλg)(y) for all g and
y.
In the transitive case we have the following important result (see, e.g. [21]).
Theorem 1. The best equivariant estimator under quadratic loss is equal to
the Bayes estimator with prior equal to the invariant measure ρ (the Pitman
estimator).
The Bayes estimator is computed as follows: First find the posterior parame-
ter distribution using Bayes’ formula, a formula which is also recently advocated
strongly for in a quantummechanical setting [16]. Then find the expected pa-
rameter under this distribution. Even though this in principle sounds quite
straightforward, it often involves quite heavy calculations, calculations which
in the recent statistical literature are solved by Markov Chain Monte Carlo
techniques [17]
4.4 Model reduction.
Model reduction was introduced under focusing. It is important when you have
few data. When there is a symmetry group G acting upon the parameter space,
one has the following requirement to impose:
The original parameter space is invariant under the group G. Therefore it
is natural that the reduced parameter space also should be invariant under G.
This implies that
The reduced parameter space should be an orbit/ a set of orbits for G.
Then within each orbit the Pitman estimator gives an optimal solution. This
solution is unique when the reduced space is transitive, i.e., when the model
reduction leads to a single orbit.
As another application of focusing, the group G may first be defined on a
larger c-variable space, and then from this induced on the parameter space.
This is the procedure I will use when linking all this to quantum mechanics.
Then as a summary: It is useful to have a group defined, first it is useful for
selecting a suitable prior, then in the analysis of subparameters and finally in
connection to model reduction. Group theory in statistics may introduce some
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abstract notions, but the concept of a symmetry group in itself is not abstract
at all. More about the groups in statistics can be found in [18, 26].
5 A large scale EPR example.
Let Z be a matrix of data values; let t and u be latent variables (unknown
variables), and let a and b be parameters. Consider the multivariate latent
variable statistical model
Z = ta′+ ub′(+E)
with negligible error E. We assume that Z is measured, but the rest is unknown
to begin with. Call φ = (t,a,u,b) a c-variable.
Let us now have two distant stations. At station 1 it is possible to measure
t or a, but not both. At station 2 it is possible to measure u or b, but not both.
Now assume that we measure t. Let P = t(t′t)−1t′, and let v = (I − P)u.
Then we know the product
vb′= (I − P)Z.
This implies that we can find bv′vb′=(v′v)bb′, and hence in effect all of the
unknown parameter b. (Note that in the model b is only defined modulus a
scale factor.) On the other hand, an essential part of the parameter u remains
unknown.
By the complementary experiment at station 1, namely measuring a, we
obtain all info on u, while parts of b remain unknown.
There is no direct action at a distance here, but by taking a decision on what
to measure at station 1, we determine what parameter to get information on at
station 2.
In my view this simple thought experiment, first formulated to me by Har-
ald Martens, bears some relationship with the EPR experiment, an experiment
which throughout the years has caused much discussion in the quantummechan-
ical literature.
The focus is not upon what the values of u and b are, but upon what
information we can get about u and b. The corresponding general view in
quantum theory is the epistemic view, explicitly introduced recently; see, e.g.
Fuchs [27], but related to the classical Copenhagen interpretation of quantum
mechanics.
6An approach to electron spin.
To illustrate the general approach of this paper, I will describe the quantum-
mechanical spin in a new way using a conceptually defined but inaccessible
c-variable, then focusing, symmetry and model reduction.
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As a start, model the angular momentum of a particle such as the electron
by a vector φ, a c-variable. Let the symmetry group G be the group of rotations
of the vector φ, that is, the group that fixes the norm ?φ?.
Next, choose a direction a in space, and focus upon the angular momentum
component in this direction:
θa= ?φ?cos(φ,a).
The largest subgroup Gawith respect to which θais permissible, is given by
rotations around the axis a together with a reflection in a plane perpendicular
to a. However, the action on θais just reflection.
Finally, introduce a model reduction: The orbits of Gaas acting on θaare
given by two-point sets {±κ} together with the single point 0. A maximal model
reduction is to one such orbit. In this case it does not matter which non-trival
orbit we take, but to be definite, choose the single orbit {±1}. Let λabe the
reduced parameter.
The parameter λa, taking one of the values ±1, is our parameter for ex-
periments. Measuring apparata for electron spin, Stern-Gerlach apparata, are
often described as perfect in textbooks, but from a statistical point of view
they must usually be assumed to have errors of measurements. This is of some
importance, since the state specification in our approach is connected to the
statistical parameter λa.
More specifically, the important elements in thie whole description are the
c-variable φ, the basic rotation group G, the direction of focusing a, the reduced
group Gaand the reduced parameter λa. The state of an electron spin will in
our system consist of two elements:
A focused question: Given the direction a; what is the value of λa?
An answer: λa= +1 or λa= −1.
This rather concrete state concept should be contrasted to the conventional
foundation of quantum theory. Shortly we will construct the ordinary Hilbert
space from such a framework, and we will show that unit vectors in this Hilbert
space can be put into correspondence with the states just defined.
7Conventional quantum mechanics and the sta-
tistical approach.
Quantum mechanics has had an enormous empirical success, but its foundation
is very formal. Briefly, the whole theory can be derived from 4 axioms; here
taken from Isham [28]:
Rule 1 The state of a quantum system is given by a unit vector v in a Hilbert
space H.
Rule 2 The observables of the system are represented by selfadjoint operators
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T acting upon H.
Rule 3 If the observable quantity λ is represented by the selfadjoint operator T,
and the state by v, then the expected result of a perfect measuremet is
Ev(λ) = v†Tv.
Note: This implies that a state vector v is an eigenvector for T with eigen-
value λ0if and only if a perfect measurement of λ in this state gives a certain
value λ0. Eigenvectors for operators are important states; in fact all states can
in some way be seen in this light.
Rule 4 Time development is given by the Schr¨ odinger equation:
i¯ hdvt
dt
= Hvt,
where ¯ h is Planck’s constant, and where H is the special selfadjoint operator
known as the Hamiltonian.
More extensive principles of quantum mechanics are given for instance by
Volovich [29]. Such principles can also be derived from our approach by doing
some more work. It should be pointed out that the state transformer or collapse
of the wave packet here requires a different, in fact a statistical and non-formal,
discussion; see later.
Our aim now is to derive the 4 rules above from the statistical approach:
The background of this approach consists of a c-variable φ, a group G acting
upon the range Φ of φ, then a focused parameter λawith its reduced group Ga.
State: The state consists of two elements:
a) A focused question: Given the focus a, what is the value of λa?
b) An answer: λa= λk.
Under suitable conditions now we want to find a Hilbert space H, where the
unit vectors v represent the states above, and where the operators T correspond
to the parameters λ. This will give a starting point for the relation to quantum
theory.
8 Constructing the quantum space.
8.1 Parametric quantum space for a single choice of fo-
cusing.
Consider φ, the group G and the invariant measure ρ. Focus on a choice a and
a reduced parameter λa= λa(φ) taking a discrete set of values {λk}. Recall
that φ in general takes no values. Nevertheless it is possible to define a group
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G as acting upon the space Φ of c-variables. (Think of the example with two
possible treatments and two potential expected life times for a single patient.
It is possible to define a joint scale transformation acting on these expected life
times.) Also, one can in a meaningful way define the L2-space L2(Φ,ρ).
Let the space Laconsist of all functions in L2(Φ,ρ) which are of the form
f(φ) =˜f(λa(φ)).
It is easily verified that this is a closed subspace of L2(Φ,ρ), hence a Hilbert
space.
A natural choice of operator Saconsists of multiplying:
Saf(φ) = λa(φ)f(φ).
Then the eigenfunctions of Saare indicator functions of the sets {φ : λa(φ) =
λk}, constituting a complete orthogonal basis for La.
We will call Laa parametric quantum space . The parametric quantum
space is simple, and it depends only upon the c-variable φ and the focused
statistical parameter λa, plus the group G.
I intend to use this as the basic building stone, together with a group theo-
retic method of joining the parametric quantum spaces for different choices of
focusing. There will be some necessary mathematical theory in this develop-
ment, but mainly to bind the whole thing together with the formal quantum
theoretical axioms. During the development, other links to statistical theory
will turn out.
8.2Maximally accessible parameter.
In the parametric quantum space the parameter λashould be accessible, and
maximally so. Recall that a statistical parameter is call accessible if, given the
relevant context, it can be made estimable by doing a suitable experiment. As a
background for this definition, we may look upon the set of possible parameters
as connected to more than one single model. We are free to make decisions
both in terms of experiment to perform and model to select. In this respect
the distinction between parameter and c-varable may be not quite clear always.
The important distinction is between what is accessible or not.
Definition. Make a partial ordering on the parameters so that λ ≪ θ if there
is a function h such that λ = h(θ). We say that λ is maximally accessible if it
is maximal among the accessible parameters under this ordering.
Note that this is consistent with the fact that the c-variable φ usually is
inaccessible, while each parameter λausually is accessible. The definition itself
is also consistent: When θ is accessible, then so is λ = h(θ). And if λ should be
inaccessible with λ = h(θ), then θ is also inaccessible.
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8.3A modified space.
It is always possible to transform the abstract vector space Laby a unitary
transformation. If a quantum system is defined on a space L, and W is a
unitary operator, then a completely equivalent quantum system can be defined
on H = WL by the correspondence v → Wv, T → WTW†.
Let Labe the parametric quantum space corresponding to the choice of
focusing a, and let W be any unitary operator acting on La. Then Ha= WLa
is called a simple quantum space corresponding to the choice of focusing a.
The simple quantum spaces will shortly be joined together to form ordinary
quantum spaces. The choice of W will be made later.
8.4 Briefly on group representation theory.
Let a group G be given. A group representation V is a function to the set of
operators on some Hilbert space such that
V (gh) = V (g)V (h) for all g,h ∈ G.
A subspace H is said to be invariant under the representation V if V (g)v ∈ H
whenever g ∈ G and v ∈ H. An invariant space gives a subrepresentation of V .
The right regular representation on L2(Φ,ρ) is defined by U(g)f(φ) = f(φg).
It can be shown [30] that this always is a unitary representation.
8.5Coupling different focusings together.
As a basis for coupling together the different simple quantum spaces, we make
the following assumption.
Assumption 1. For each pair of focused experiments a and b there is a group
element gab∈ G such that
λb(φ) = λa(φgab).
This holds for the electron spin case by a straightforward verification. In
general it is an assumption to the effect that the fundamental group G is large
enough to contain transformations between the differently focused experiments.
Recall that Gais the maximal subgroup under which the parameter λais per-
missible. We will look at two consequences of Assumption 1.
Lemma 1 The groups Gaand Gbcan be connected by the group element gab:
gb= gabgag−1
ab.
Proof: We define (λagab)(φ) = λa(φgab), which is consistent with earlier def-
initions. By permissibility each element gaof the group Gais determined by
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its effect on λa. By Assumption 1 we must have that λbgb= λagagabfor some
gb∈ Gb. Hence it follows that λagabgb= λagagab, and the Lemma follows.
The next result has also been anticipated in notation used earlier.
Lemma 2 Assuming that each λatakes discrete values λa
arranged such that λa
k= λkis the same for each a (k = 1,2,...).
k, these values can be
Proof: By Assumption 1
{φ : λb(φ) = λb
k} = {φ : λa(φgab) = λb
k} = {φ : λa(φ) = λb
k}gba.
The sets in brackets on the lefthand side here are disjoint with uniom Φ. But
then the sets on the righthand side are disjoint with union Φgab= Φ, and this
implies that {λb
k} gives all possible values of λa.
Now make the following observation: If fa
and fb
right regular representation of G. A consequence of this is that Lb= U(gab)La.
And a consequence of this again is that Hb= V (gab)Hafor an element of the
unitary representation V (g) = WU(g)W†.
From now on we restrict ourselves to the case with a finite number of pa-
rametervalues λk. And also, without much loss of generalizations, we assume
that the underlying group G is compact. From standard mathematics [30] we
then have the following result:
kis the indicator that λaequals λk,
k= U(gab)fa
kis the indicator that λbequals λk, then fb
k, where U is the
Theorem 2. For a compact group, every irreducible unitary representation
V (g) can be written in as V (g) = WU(g)W†for some W, with U(g) being a
subrepresentation of the right regular representation.
We now introduce another assumption, to the effect that there is a sufficient
amount of focused questions to ask:
Assumption 2. The reduced groups Ga,Gb,... generate the whole group G.
Theorem 3. Fixing some preliminary W0 and hence some set of unitary re-
lations V0(gbc) = W0U(gbc)W†
the fixed space Hais an invariant space for some abstract representation V of
the whole group G.
0between simple Hilbert spaces Hband Hc, say,
Proof: First we observe that Hais an invariant space for the subgroup Ga
under the right regular representation. This follows directly from the definitions.
To extend this, we look at a product g1g2g3, where g1 ∈ Ga, g2 ∈ Gband
g3∈ Gc. We can define a map from such elements to operators on Haby
V (g1g2g3) = U(g1)(V0(gba)U(g2)V0(gab))(V0(gca)U(g3)V0(gac)).
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With a similar definition for all products of elements from the subgroup, we
verify for instance that V (g1g′
Also, V (g1g2g3) = V (g1)V (g2)V (g3). It follows that V (gh) = V (g)V (h) on
these products, since the last factor of g and the first factor of h either belong
to the same subgroup or to different subgroups. In this way we see that V is a
representation on the set of finite product of elements from the subgroups, and
since by Assumption 2 these products generate G, it is a representation of G.
In particular, one is able to take Haas an invariant space for a representation
V of this group.
2)V (g′′
2g3) = V (g1g2g3) when g2 = g′
2g′′
2∈ Gc.
Choice of W: Now keep Hafixed, but be free to change the matrix W0which
fixes the relations to the other Hilbert spaces. Concretely, there is a group
representation having Haas an invariant space. By Theorem 2, we can choose
W such that V (g) = WU(g)W†is this representation
Theorem 4. Then from Hb= V (gab)Hawe have Ha= Hb= Hc= ...., and
this can be taken as the quantum mechanical space H.
Example: SU(2) gives a twodimensional invariant space for electron spin,
coupled to the rotation group.
In the space H there are state vectors va
that λa(φ) = λk. Specifically we take
kwhich are transforms of indicators
va
k= Wfa
kwith fa
k(φ) = I(λa(φ) = λk).
In the electron spin case and in other cases all unit vectors in H are of this
form. In the electron spin case this can be verified directly by a Boch sphere
argument. In general this amounts to an assumption to the effect that the set
of focused questions is rich enough. On the mathematical side, one can argue
from the fact that every unit vector can be considered as eigenvector of some
operator.
Definition The vectors va
tion: Focused question: What is λa? Answer: λa= λk.
kare the state vectors with the statistical interpreta-
In general, the vectors va
kare eigenvectors of the selfadjoint operator
Ta= WSaW†
with eigenvalues λa
k. Hence we have the result:
Theorem 5. For each choice of focused question a there is an operator Taon
the Hilbert space H which corresponds to the perfect experiment with parameter
λa. These operators have eigenvectors va
kwith eigenvalues λa
k.
Note that by the unitary transformation, the vectors va
when the fa
kare. In fact the unit vectors in H correspond in a unique way to a
kare unit vectors
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question-and-answer pair if there is a correspondance at all:
Theorem 6. Assume that two vectors in H satisfy va
’What is λa?’ is answered by λa= λiif and only if the question: ’What is λb?’
is answered by λb= λj.
i= vb
j. Then the question:
Proof: The assumption is that Wfa
I(λb(φ) = λj). Thus the level sets coincide, and the Theorem follows.
i= Wfb
j, hence fa
i(φ) = I(λa(φ) = λi) =
8.6Operators and maximality.
The operators Tawill have the form
Ta=
?
k
λkva
kva†
k.
Up to now we have assumed that all parameters λaare maximally accessible.
This implies that Tahas nondegenerate eigenvalues.
In applied quantum mechanics one also allows operators with degenerate
eigenvalues. In our setting these can be introduced through non-maximal pa-
rameters µ = h(λ), and
T =
?
k
µkva
kva†
k.
Thus these parameters are also associated with operators in the obvious way.
The quantum mechanical Rule 1 and Rule 2 hold for these operators and for
the states va
k.
9 Auxiliary quantities for one choice of focusing.
The state vector va
nition W has an arbitrary phase factor. Therefore it is sometimes replaced by
the one-dimensional projector
va
khas an arbitrary phase factor in our setting, since by defi-
kva†
k.
The information contained in this projector is again: A focused question: ’What
is λa(φ)?’ has been asked, and the answer is: λa(φ) = λk.
Taking this as a point of departure, one can introduce other operators which
play a crucial role in many applications of quantum mechanics, and which also
can be connected to statistics.
9.1 Auxiliary quantity I: Density operator.
A density operator is defined by:
σ =
?
k
πkva
kva†
k.
14
Page 15
when a prior or posterior probability πkover λais given.
Conversely, given σ, one can reproduce the focused experiment itself, all
levels of answers k and the (prior or posterior) probabilities πkfor all these k.
In the ordinary quantum mechanical tradition, a density operator is defined
as any selfadjoint operator with trace 1.
9.2Auxiliary quantity II: Effect.
Given some data y, an effect (cf. [31]) is defined by:
E =
?
j
pb
j(y)vb
jvb†
j.
An experiment with likelihood pb
Conversely, given E, one can reproduce the focused experiment itself, all
levels of answers k and the likelihood pb
j(y) for the parameter values λj.
In the ordinary quantum mechanical tradition, an effect is defined as any
selfadjoint operator with eigenvalues between 0 and 1.
Using these quantities, we now turn to the probability link between different
focusings.
j(y) and parameter λbis the basis for E.
10 Born’s formula.
Born’s formula is fundamental in linking differently focused experiment:
P(λb= λj|λa= λk) = |vb†
jva
k|2.
It has been proved in a standard quantum mechanical setting under various
assumptions by Deutsch [32], Wallace [33], Saunders [34], Aerts [35] and Zurek
[36].
The crucial assumption by Zurek in his proof of Born’s formula is what he
called envariance.
Couple the quantum system S to an environment E, and then make the
following assumption:
Every unitary transformation U = uS⊗ 1E acting solely on S:
UvSE= (uS⊗ 1E)vSE= v′
SE
can be undone by a transformation V acting solely on the environment E:
V v′
SE= (1S⊗ uE)v′
SE= vSE
It was argued in various ways in [36] that this is a relatively weak assumption.
We want to prove the formula in our setting from a somewhat different, but also
weak, assumption.
15
Page 16
10.1 Proof of Born’s formula via the Busch-Gleason the-
orem.
The proof will depend upon a recent variant (Busch [31]; Caves et al. [37]) of a
well known mathematical result by Gleason [38]. One advantage of this result is
that it is also valid for dimension 2, when the ordinary Gleason Theorem fails.
Theorem 7. Assume that there is a generalized probability measure π on the
effects in a Hilbert space H; i.e., a set function satisfying
π(E) ≥ 0 for all E,
π(I) = 1,
?
Then π is necessarily of the form π(E) = tr(σE) for some density operator
σ.
iπ(Ei) = π(E) for effects Eiwhose sum is an effect E.
Now define π(vb
effects. Such an effect will then correspond to a parameter λb, an orthonormal
set of Hilbert space vectors vb
the effect will correspond to a single data point y and for each j the probability
pb
jvb†
j) = P(λb= λj|λa= λk), and extend by linearity to all
j, each giving one value λj for λb. Furthermore,
j(y) of observing y, given that λb= λj. We then have
π(Eb) =
?
j
pb
j(y)P(λb= λj|λa= λk) = P(obs. y through experim. b|λa= λk).
Imagine now n potential such experiments b1....,bn, corresponding to n dif-
ferent questions and n different sets of likelihood pbr
one of these randomly, each with probability 1/n. He has available a measure-
ment apparatus through which each of the single experiments above can be
performed, that is, experiments with parameters λbrand each with the sin-
gle observation y as one possibility. The randomized experiment above can be
imagined to correspond to a parameter λ0with this measurement apparatus. It
is important that the randomization is performed blindly, i.e, the experimenter
does not know which experiment is carried out. His likelihood in the random-
ized experiment for observing y is p0
equals λbrwith probability 1/n. Then this experiment also has an effect E0and
a generalized probability measure
j(y). An experimenter selects
j(y) when λ0= λj. And, by definition, λ0
π(E0) =
?
j
p0
j(y)P(λ0= λj|λa= λk).
This gives
π(E0) =1
n
?
r
P(obs. y through experim. br|λa= λk) =1
n
?
r
π(Er)
with Er= Ebr.
16
Page 17
On the other hand, E0is a randomization at the measurement apparatus of
the experiments with effects Er. This implies that any hypothetical observer
knowing the result of the randomization will have the corresponding result:
E0′=
?
r
IrEr,
where Ir= 1 if experiment r was selected, otherwise Ir= 0. A similar formula
should hold for partial knowledge. For instance, if the observer does not know
whether experiment 1 or 2 was selected, the indicators I1 and I2 should both
be replaced by1
2I, where I is the indicator of experiment 1 or 2.
Since it should be a high degree of consistency between the observersknowing
the randomization and a single observer not knowing the randomization at all,
we expect that E0should be close to, if not equal to,?
condition is that these two effects should have the same generalized probability
measure.
rE(Ir)Er. A weak
Assumption 3.
π(E0) = π(
?
r
E(Ir)Er).
With the simple randomization used here, this means
π(E0) = π(1
n
?
r
Er).
This implies that π(1
?
n
?
rEr) =
1
n
?
rπ(Er), or, by changing the likelihood, if
rEris an effect:
π(
?
r
Er) =
?
r
π(Er).
Thus follows the most important premise of the Busch-Gleason theorem from
our assumption. The other premises are easily proved, and it follows that there
exists a density operator σ such that π(vv†) = v†σv for all v ∈ H. For this den-
sity operator we have that σ =?
adding to 1.
Inserting this gives π(vv†) =?
case given by v = va
kfor some k. For this case one must have?
and thus?
or |va†
phase factor), and this is a condition which can only be true for one j, it follows
that cj= 0 for all other j than this one, and that cj= 1 for this particular j.
Summarizing this, we get σ = va
k, and Born’s formula follows.
jcjuju†
j, where cj are nonnegative constants
jcj|v†uj|2. Specialize now to the particular
jcj|va†
kuj|2= 1
jcj(1 − |va†
kuj|2) = 0. This implies for each j that either cj = 0
kuj| = 1. Since the last condition implies uj= va
k(modulus an irrelevant
kva†
10.2Consequence of Born’s formula I.
Let the state of a system be given by v = va
parameter with corresponding operator T =?
k, and let λ = λbbe an arbitrary
jλjvb
jvb†
j.
17
Page 18
Then the expected result of a perfect measurement will be:
Ev(λ) =
?
j
λjP(λb= λj|λa= λk)
=
?
j
λj(va†
kvb
j)(vb†
jva
k) = va†
k(
?
j
λjvb
jvb†
j)va
k= v†Tv.
This is Rule 3 in the basis for conventional quantum mechanics.
10.3Consequence of Born’s formula II.
Let an experiment have focus on a parameter λband likelihood pj(y) for λb= λj.
Define an operatorvalued measure M by
M(dy) =
?
j
pj(y)vb
jvb†
jdy.
These operators satisfy M(S) = I for the whole sample space S and are count-
ably additive.
Assume now that the initial state is given by λa= λk. Then the probability
distribution for the result of experiment b is given by
P(dy|λa= λk) = va†
kM(dy)va
k.
A more general state assumption is a Bayesian one corresponding to this
setting: Let the current state be given by the question: ’What is the value of
λa?’, and then the probabilities π(λk) for the different values λk. Then, defining
σ =?π(λk)va
kva†
k, we get
P(dy) = tr(σM(dy)).
Operator valued measures are increasingly being used in quantum statistical
inference. Here they are traced back to a likelihood based concept.
10.4 Consequence III. Collapse of the wave packet.
Note that the density matrix va
density matrix vb
giving λb= λj has just been performed. From this and from a straightforward
application of Born’s formula one gets
kva†
kis equivalent to a pure state va
jis equivalent to the statement that a perfect measurement
k. Similarly, a
jvb†
Theorem 8. a) Assume an initial state va
surement of λbhas been done without knowing that value. Then the state is
described by a density matrix?
b) After measurement λb= λjthe state then changes to the vector vb
a) this happens with probability |va†
k, and assume that a perfect mea-
j|va†
kvb
j|2vb
jvb†
j.
j. From
kvb
j|2.
18
Page 19
This is the well known and much discussed collapse of the wave packet, or
really of the density matrix. In our statistical interpretation this represents
no problem. A similar ’collapse’ occurs in Bayesian statistics each time an
observation is made. Note that the reason why this collapse is non-problematic
is that we do not assume a simple ontological interpretation of the state vector.
The state represents just a question together with an ideal answer.
11 The state interpretation and ’action at a dis-
tance’.
Consider two electrons with their spins modeled in our terminology by c-variables
φ1and φ2. Assume that these in some distant past have been in interaction
so that φ1= −φ2, what in conventional quantum mechanics is called a singlet
state. But now the two electrons are far apart. This leads to the Einstein,
Podolsky, Rosen (EPR) [39] situation as modified by Bohm.
Assume that we can make measurements on electron 1. We are then free to
focus on some chosen parameter λa. If we make a perfect measurement in that
direction and get the result λa= λk, then the spin state of this particle is fixed.
But the spin state of the electron 2 is also determined, namely as λa= −λkby
the known coupling. And the probability distribition of perfect measurements of
spin components in any other direction for electron 2 is given by Born’s formula.
All this assumes that φ1+ φ2is an accessible c-variable, while each of φ1and
φ2are inaccessible.
Like in the earlier macroscopic example: There is no action at a distance;
we only ask questions and get answers. As we do in statistical investigations.
11.1On Bell’s inequality.
Assume again the EPR situation, where spin components λaand µbare mea-
sured in the directions given by unit vectors a and b on the two particles at
distant sites A and B. The measured valuesˆλaand ˆ µbare each assumed to
take values ±1. Let this be repeated 4 times: Two settings a,a′at site A com-
bined with two settings b,b′at site B. The CHSH version of Bell’s inequality
then reads:
E(ˆλaˆ µb) ≤ E(ˆλaˆ µb′) + E(ˆλa′ˆ µb) + E(ˆλa′ˆ µb′) + 2(1)
In fact, by a combinatorial argument we can easily show the seemingly
stronger statement:
ˆλaˆ µb≤ˆλaˆ µb′+ˆλa′ˆ µb+ˆλa′ˆ µb′+ 2(2)
As is well known, the inequality (1) can be violated in the quantum me-
chanical case, and this is also well documented experimentally. There is a large
literature on Bell’s inequality, and I will not try to summarize it here. The
19
Page 20
derivation of (2) above is quite obvious, and the usual statement in the quan-
tummechanical literature is that (1) follows under what is called local realism.
My own view is that quantum theory is a statistical theory, and should be
interpreted as such. In this connection the comparison to a classical mechanical
world picture, and the term ’local realism’ inherited from this comparison, is
not necessarily of interest. I am more interested in the comparison of ordinary
statistical theory and quantum theory.
Now take a general statistical inference point of view on any situation that
may lead to statements like (2) and (1). Then one must be prepared to take
into account that there are really 4 experiments involved in these inequalities.
Going from the inaccessible c-variable ±φ to the observations there are really
three steps involved at each node: The components θ(φ) is selected, there is a
model reduction λ = λ(θ), and finally the observationˆλ. Briefly: A model is
selected, and there is an estimation within that model.
Turn now to general statistical theory: According to the conditionality prin-
ciple, a principle on which there seems to be a fair amount of concensus among
statisticians, inference in each experiment shold always be conditional upon the
experiment actually performed. Taking this into account, it may be argued
that at least under some circumstances also in the microscopic case, different
expectations should be used in complicated enough situations leading to (1),
and then the transition from (2) to (1) is not necessarily valid.
This is dependent upon one crucial point, as seen from the conditionality
principle as formulated above: When one has the choice between two experi-
ments, the same parameter should be used in both. Here is a way to achieve
this: Focus on the Stern-Gerlach apparatus which measures the spin. Make a
fixed convention on how the measurement apparatus is moved from one location
to another. Then define a parameter λ which is −1 at one end of the apparatus
and +1 at the other end. By using λ as a common parameter for all experiments
under choice, the conditionality principle can be applied, and (1) does not follow
from (2).
The crucial point here is that the violation of the Bell inequality is not by
necessity a phenomenon that makes the quantum world completely different
from the rest of the world as we know it. It has recently been pointed out
[40, 41] that variants of Bell’s inequality may be broken in macroscopic settings.
12 Briefly on ’paradoxes’ in quantum mechan-
ics.
Schr¨ odinger’s cat:
substance and a bottle of poison. Decay of the substance leads to release of the
poison.
The c-variable φ gives a complete description of the system, including the
death status of the cat. The focused parameter λ may or may not include the
latter.
A cat is contained in a box together with some radioactive
20
Page 21
Wigner’s friend:
observes everything, including Wigner.
In principle a statistical model can be formulated excluding an observer or
including an observer. There is no contradiction.
A person (Wigner) observes the world, and a friend also
Hidden variables:
only hidden conceptual c-variables φ, in general taking no value. This does not
prevent us from defining a group action on the c-variable. Also, a parametric
Hilbert space, a subspace of L2(Φ,ρ), can be defined in a meaningful way.
There are no hidden variables in our treatment; there are
13Continuous parameters.
So far, the focused parameter λ has been discrete. This has had the advantage
that the Hilbert space could be constructed in a rigorous way. Some theory of
continuous parameters can be simply developed, however, using c-variables and
the group theory approach. A straightforward example is when the c-variable
consists of the position and the momentum of a single particle. We consider the
ideal values of these quantities, i.e., the corresponding parameters.
13.1 Symmetry in space and time.
To begin with, I indicate a relativistic treatment in this subsection. As is known
from special relativity, the four-vectorof space-time positions ξ = (ξ1,ξ2,ξ3,ξ0=
cτ) and the four-vector of momentum-energy π = (π1,π2,π3,π0= c−1ǫ) trans-
form according to the extended Lorentz transformation, the Poincar´ e transfor-
mation. This is the group which fixes c2dτ2
c−2ǫ2−?3
tion group to link to the eightdimensional c-variable φ = (ξ1,ξ2,ξ3,τ,π1,π2,π3,ǫ).
associated with aparticle at time τ. A very thorough treatment of the Poincar´ e
group and its representations is given in [42]. For simplicity we will limit our-
selves here to the subgroup G of translations. Several further subgroups may
be of interest, for instance the group B1of translations in the ξ1-direction.
0= c2dτ2−?3
i=1dξ2
iand c2m2
0=
i=1π2
i, where m0is the rest mass. This may be a natural transforma-
Theorem 9. Vξ= {f : f(φ) = q(ξ1(φ)) for some q is a subspace of L2(Φ,ρ)
which is invariant under the group B1. The right regular representations have
the form U1(g)q(ξ1) = q(ξ1g) = q(ξ1+ b).
But by a Taylor expansion
q(ξ1+ b) =
∞
?
k=0
bk
k!
∂k
∂ξk
1
q(ξ1) = exp(b∂
∂ξ1)q(ξ1) = exp(ibP1
¯ h
)q(ξ1),
where P1is the familiar momentum operator
P1=¯ h
i
∂
∂ξ1.
21
Page 22
Thus the particular group formulated above has a Lie group representation
on an invariant space with a generator equal to the corresponding momentum
operator of quantum mechanics. The proportionality constant ¯ h can be argued
to be the same for all momentum components (and energy) by the conservation
of the 4-vector. By similarly considering systems of particles one can argue that
¯ h is a universal constant.
In a similar way we can show
Theorem 10. Time translation τ → τ +t has a right regular representation as
a Lie group with generator
exp(−iHt
¯ h
), (3)
where H is the Hamiltonian operator.
13.2 The Schr¨ odinger equation.
I have just shown that for a single particle, the time translation τ → τ+t has the
right regular representation found by (3). This can be generalized to systems
of several particles, assuming an additive Hamiltonian, and assuming that the
particles at some point of time were pairwise in contact, or at least so close
with respect to space and velocity that relativistic time scale differences can be
neglected. The operator (3) acts on the Hilbert space H. In this subsection I
work under the non-relativistic approximation.
Assume further that at time 0 a maximal measurement was done, so that
the system is in some state v0∈ H. This means, according to my interpretation
that some measurement with focused parameter λahas been done, resulting in
some value λ1. The parameter λais associated with an operator Ta. After time
t the parameter λawill have developed into λa(t) with an operator Ta(t) =
exp(−iHt
¯ h.Then, since after time t the state vector vtshould still be
an eigenvector of Ta(t), then with eigenvalue λa(t) = exp(−iHt
vector must be
vt= exp(−iHt
¯ h)TaexpiHt
¯ h)λa, the state
¯ h
)v0.
As is well known, the latter equation is just a formulation of the familiar
Schr¨ odinger equation
Theorem 11. The time development of the state vector can be found from
i¯ h∂
∂tvt= Hvt.
22
Page 23
14 Concluding remarks.
The starting point of this paper is partly related to another discipline than
quantum physics, namely statistics. From one point of view this may be an
advantage, giving an opportunity to see the foundational problems with new
eyes. On the other hand, the discussion here goes much further than what is
common in the literature in mathematical statistics. The language of this paper
may be seen as a synthesis of the existing languages in quantum theory on the
one hand and that of statistical inference on the other hand.
Understanding across cultural barriers is crucial. If nothing else, this paper
may point at the trivial fact - a fact that still may be difficult to accept -
that different cultures exist in what we perceive as a very objective science.
Sometimes it is useful to know that we can translate languages and learn from
each other.
Statistics and quantum theory have lived side by side for 75 years and more
without any appreciable interaction, that is, at least since the publication of
von Neumann’s monograph [43]. Still one might infer from the present paper
that deep conceptual links seem to exist.
Non-concrete concepts are crucial in almost any human discussion. Different
focusing from these concepts is also common both in the verbal exchanges of
daily life and in the literature of social sciences, say. The link from this to
statistics/ quantum theory should be developed further.
Then finally: The formal quantum theory has proved to be very powerful
in calculations. But this should not imply that the logical foundation of the
theory by necessity should be formal. In fact, the essential elements of quantum
theory - as an epistemic discipline under focusing and symmetry - may seem to
follow logically from rather simple assumptions.
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