Quantum Mechanics from Focusing and Symmetry
ABSTRACT A foundation of quantum mechanics based on the concepts of focusing and symmetry is proposed. Focusing is connected to c-variables - inaccessible 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 vectors 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.
- [Show abstract] [Hide abstract]
ABSTRACT: A new foundation of quantum mechanics for systems symmetric under a compact symmetry group is proposed. The foundation is given by a link to classical statistics and coupled to the concept of parameter. A vector of parameters is called an inaccessible c-variable if experiments can be provided for the single parameter, but no experiment can be provided for . This is related to the concept of complementarity in quantum mechanics, but more generally to contrafactual parameters. Using these concepts and some weak assumption, the Hilbert space of quantum mechanics is constructed. The complete set of axioms of quantum mechanics is provided by proving Born's formula under weak assumptions.Journal of Physics Conference Series 06/2009; 174(1):012031. - SourceAvailable from: Inge Helland[Show abstract] [Hide abstract]
ABSTRACT: Every experiment or observational study is made in a context. This context is being explicitly considered in this book. To do so, a conceptual variable is defined as any variable which can be defined by (a group of) researchers in a given setting. Such variables are classified. Sufficiency and ancillarity are defined conditionally on the context. The conditionality principle, the sufficiency principle and the likelihood principle are generalized, and a tentative rule for when one should not condition on an ancillary is motivated by examples. The theory is illustrated by the case where a nuisance parameter is a part of the context, and for this case, model reduction is motivated. Model reduction is discussed in general from the point of view that there exists a mathematical group acting upon the parameter space. It is shown that a natural extension of this discussion also gives a conceptual basis from which essential parts of the formalism of quantum mechanics can be derived. This implies an epistemological basis for quantum theory, a kind of basis that has also been advocated by part of the quantum foundation community in recent years. Born's celebrated formula is shown to follow from a focused version of the likelihood principle together with some reasonable assumptions on rationality connected to experimental evidence. Some statistical consequences of Born's formula are sketched. The questions around Bell's inequality are approached by using the conditionality principle for each observer. The objective aspects of the world are identified with the ideal inference results upon which all observers agree (epistemological objectivity).06/2012; - SourceAvailable from: Inge Helland[Show abstract] [Hide abstract]
ABSTRACT: Compared to other physical theories, the foundation of quantum mechanics is very formal and abstract. Is it possible to find another basis which is more directly connected to what one observes in nature? The purpose of this chapter is to show that to a large extent one can give a positive answer to this question.Advances in Quantum Mechanics, 1 edited by Paul Bracken, 04/2013: chapter 15: pages 26; InTech., ISBN: 978-953-51-1089-7
Page 1
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.
1 Introduction
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
1
Page 2
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.
2 Briefly 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
2
Page 3
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].
3
Page 4
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.
4
Page 5
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.
4.1 Orbits. 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.
4.3 Subparameters 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.
5
Page 6
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]
4.4Model 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
6
Page 7
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.
6 An 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.
7
Page 8
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.
7 Conventional 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
8
Page 9
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
9
Page 10
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.
8.2 Maximally 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.
10
Page 11
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.
8.4Briefly 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.
8.5 Coupling 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
11
Page 12
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)).
12
Page 13
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
13
Page 14
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.
9.2 Auxiliary 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
View other sources
Hide other sources
- Available from Inge Helland · May 16, 2014
- Available from ArXiv