Download Three Interpretations for a Single Physical Reality

Amsterdam University College
Science
Three Interpretations for a Single
Physical Reality
Yolanda Murillo
Supervisor
Dr. Sebastian De Haro
(UvA)
Reader:
Prof. Dr. Henk De Regt
(UvA)
May 25, 2016
To Lia Den Daas
1
Contents
1 Introduction
3
2 Theories and Interpretations
2.1 What is an interpretation? . . . . . . . . . . . . . . . . . . . . . .
2.2 Comparing to Muller’s account . . . . . . . . . . . . . . . . . . .
2.3 Theoretical vs. physical equivalence . . . . . . . . . . . . . . . . .
4
4
6
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3 Three Interpreted Theories
3.1 Copenhagen Interpretation . . . .
3.1.1 States . . . . . . . . . . . .
3.1.2 Physical Quantities . . . . .
3.1.3 Dynamics . . . . . . . . . .
3.1.4 The Measurement Problem
3.1.5 The Uncertainty Principle .
3.2 Many-Worlds Interpretation . . . .
3.2.1 States . . . . . . . . . . . .
3.2.2 Physical Quantities . . . . .
3.2.3 Dynamics . . . . . . . . . .
3.2.4 The Measurement Problem
3.3 Bohmian Mechanics . . . . . . . .
3.3.1 States . . . . . . . . . . . .
3.3.2 Physical Quantities . . . . .
3.3.3 Dynamics . . . . . . . . . .
3.3.4 The Measurement Problem
3.3.5 The Uncertainty Principle .
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4 Theoretical and Physical Equivalence
22
4.1 Theoretical equivalence . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Physical equivalence . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Conclusion
25
Appendices
25
2
Abstract
This thesis studies the theoretical and physical equivalence relations
between three of the most relevant theories of quantum mechanics: the
Copenhagen, the Many-Worlds and the “Pilot-Wave” interpretations. This
is done by means of a formal description of the three theories with which
they can be, in contrast with what it is sometimes thought, formally
compared. The formal description consists of a bare theory and two interpretational maps, which connect such bare theories to what they represent in the physical reality. Contrary to most of the literature, the term
‘interpretation’ is not used to refer to the different theories, but rather
‘to interpret’ is to provide such interpretational maps. Two theories are
said to be theoretically equivalent if their bare theories are the same. Two
theories are said to be physically equivalent if they are theoretically equivalent and if they share their interpretational maps. With this framework
in mind, the Copenhagen and the Many-Worlds theories turn out to be
theoretically equivalent but not physically equivalent whereas the ‘Pilot’
wave theory does not fulfill the theoretical equivalence requirement. By
providing some insights into the different interpretations and the notion
of physical equivalence, this paper has attempted to contribute to the
quest for the best of all possible interpretations. An interpretation as
such could provide the scientific community with something it has been
restlessly searching for: a hint in quantum theory about the kind of world
(or worlds) we live in.
1
Introduction
At the beginning of the twentieth century, the founding fathers of quantum
mechanics established the mathematical formalism that would give rise to one
of the most successful devices of experimental physics. It had been originally
constructed in order to explain certain empirical inadequacies that the classical
theory could not explain. The physicists of the time wanted to find a new theory
that would explain, “what the world [was] like, intrinsically” [21]. Soon after this
formalism was developed, however, disagreements arose about how to provide
such an explanation. These disagreements were mainly over interpretation and
led to the problem of ‘the interpretations of quantum mechanics’.
In this thesis, I will study three main interpretations: the Copenhagen,
the Many-Worlds and the “Pilot Wave” interpretations. The use of the word
’interpretation’ referring to these theories, however, has often led to confusion.
One might think that choosing one interpretation of quantum mechanics over
another might partly depend on the metaphysical commitments one prefers
to adhere to. Thus the Pilot Wave quantum and the Many-Worlds quantum
theorists might be scientific realists, while the empiricist might be pre-disposed
to the Copenhagen interpretation. Without denying that this might be so; and
although the metaphysical presuppositions might differ in the different theories,
I will argue that this is not what an interpretation of quantum mechanics, in
essence, is about. As I will argue, the three interpretations may disagree, not
only about their metaphysics, but also about their physics. For this reason,
3
the aforementioned theories will not be referred to as interpretations (since
this would suggest that they are equal in all else except for ‘interpretation’)
but rather ‘interpreted theories’. They may thus differ both in ‘theory’ and in
‘interpretation’.
Applying the notions from De Haro, Teh, and Butterfield (2016) and De Haro
(2016), and because it is desirable to remain as close to a physics formalism
as possible, my thesis will offer a different notion of ‘interpretation’ in terms
of surjective maps (Section 2). Provided this, each of the three interpreted
theories mentioned above will be depicted in terms of its bare theory and its
interpretation (Section 3). All of this will be done in order to raise the question
of the physical equivalence between these different interpreted theories (Section
3). By comparing them, this thesis aims to shed light on the ongoing debate
about interpretations: more specifically, the aim is to assess to what extent the
different interpreted theories are theoretically equivalent, and to what degree
they are physically equivalent.
2
Theories and Interpretations
In section 2.1, I discuss what it will mean to interpret in my paper. Then, in
section 2.2., I will compare this notion of interpretation with the one Muller
(2015) offered. Finally, I will explain the notions of theoretical and physical
equivalence in order to reffer back to them in the discussion of the different
interpreted theories.
2.1
What is an interpretation?
What are we doing when we interpret quantum mechanics? was the question
that Muller asked himself in Circumveiloped by Obscuritads [24]. In this paper,
he provided an example of the needed clarification of what it means to interpret
in quantum mechanics. He believed that “interpretation debates about quantum mechanics are not about the meaning of words”(Section 5 [23]). Here, ‘to
give meaning’ is understood in an analogous way to how meaning is given to
literary texts (hermeneutics); assigning certain meaning to words or texts that
might not have been though of having such a meaning originally. Instead, an
interpretation of quantum mechanics should present claims that the mathematical formalism cannot offer alone. Specifically, these statements should concern
what is observed in the physical reality.
Muller was not the first one to understand the notion of interpretation in this
way 1 , but his remarks have provided insightful accounts in this debate. He explains, for example, that part of the confusion about what it means to interpret
is due to famous physicists pronouncements that quantum mechanics cannot be
understood: “we can safely conclude that no one understands quantum mechanics” (Richard Feynmann Character of the physical law 1965)[23]. The ongoing
debates about ‘the interpretation’ of quantum mechanics (which include talk
1 See
Coffey (2014) [7]
4
about what the wave-function represents, about the meaning of the terms, etc.)
suggests that one must have a serious metaphysical account, perhaps even a final
theory of the world, in order to claim that one understands quantum mechanics.
But on De Regt and Dieks’ notion of intelligibility of a theory, one can see that
this is not necessary. To understand an interpreted theory is to “recognise qualitatively its characteristic consequences without performing exact calculations”
[23]: To understand quantum mechanics cannot depend on having a complete
metaphysics of the world. The physical interpretation of a theory seems closely
connected with its intelligibility, which I take as further motivation for the idea
that the interpretation of a theory can be much something like its use. It can
be like a map that relates the theory with its characteristic consequences.
Following a different approach based on De Haro, Teh, Butterfield (2016)[8]
and De Haro (2016)[19], an interpreted theory will be comprised of two parts.
The first part is the
triple T , formed by states, physical quantities, and dynamics T = H, Q, D , which constitute the description of the bare theory of the
interpreted theory (Section 1 [19]). The description of the bare theory, although
minimalistic will be assumed to exhaust the physical content of the theory.
The second part constituting an interpreted theory is its physical interpretation. For the reasons mentioned avobe, these interpretations should
define
two
surjective maps of the physical descriptions of the theory T = H, Q, D , into
the physical reality, preserving appropriate structure. The first map is one from
the subset of physical quantities in Q, into the physical quantities they represent
(e.g. position, energy..etc). The second map establishes the relation between
the values of the subset of physical quantities Q, the probability amplitudes
|ci |2 in the theory and the observed experimental outcomes. Both maps are of
course subject to appropriate restrictions, which will be discussed in due course,
such as e.g. the probabilities adding up to one. The maps need to be surjective,
as we wish every experimental outcome to correspond to at least one physical
quantity or state. However, they cannot be injective in order to account for
the cases of degeneracy. Take for example the degeneracy of energy levels of a
particle in a square box, for both nx = 1, ny = 2 and nx = 2, ny = 1 the values
π 2 ~2
for the energy are given by E = 5 2mL
2 : two different states share the same
measurable quantity.
At this point it would be insightful to offer an example of this new formalism
by means of applying it to a theory. Take Newtonian mechanics: the states
H are defined by the changes in physical quantities of massive particles. In
quantum mechanics these states are represented by eigenstates of the operators.
However, other kinds of states in quantum mechanics are simply not describable
by newtonian mechanics e.g. superpositions. For this reason, the set of states
H will be larger for the former. The physical quantities of classical mechanics Q
would be position, velocity, angular momentum, acceleration, energy...etc. The
dynamics D would be described by Newton’s second law of motion. The maps
relate the experiment outcomes with the predicted values. It was only when this
map failed, when physicists realized there was a need for theories like quantum
mechanics.
5
2.2
Comparing to Muller’s account
Even though inspired in Muller’s interest of redefining the notion of interpretation, this paper will differ with him in what this new notion is. For Muller,
an interpretation in quantum mechanics constitutes of two parts: (1) a minimal quantum mechanics (QM0 ), established by the mathematical formalism
and shared by all theories, and (2) extra postulates added by each one of the
interpretations. These added postulates, for him, constitute what it means to
interpret. He describes, for example, the Many-Worlds interpretation as QM0
plus a branching postulate (Section 5 [24]).
In contrast with Muller’s ‘minimal quantum mechanics+additional postulates’ view, in the formalism presented in this thesis, the triple of every interpret
theory describes its bare theories, and the equivalence between the triples is not
assured but rather a question to be raised.
There are certain advantages in choosing the presented notion of interpretation over the one proposed by Muller. Firstly, Muller does not make a clear distinction between the formalism and the physics—some of the postulates which
constitute QM0 would be considered to be part of the interpretation in De
Haro’s proposed formalism as it includes references to physical systems, magnitudes...etc. Secondly, if this is the case, questions like the one of physical
equivalence between theories could not be raised by means of comparing ’what
is physical in the theory’ because there is no such distinction.
2.3
Theoretical vs. physical equivalence
I take it that one natural way to approach the problem of interpretation is to
first distinguish physical equivalence from theoretical equivalence. The latter
describes the equivalence between the bare theories of the interpreted theories.
The former is in charge of the equivalence of what is physical in the theories;
their interpretation.
Theoretical equivalence between two theories will be granted for theories
whose triple describe the same states, physical quantities and dynamics. These
triples, however, need not to be the exact same—as this would be too strong a
requirement. However, since we will be dealing with quantum theories, it will
suffice that the two triples are unitary equivalent. Unitary equivalence between
two theories is a special case of isomorphism as each state from one of the triples
is mapped to a state from the other (and only to one), and each operator would
be mapped to a unitarly equivalent operator. Take the triple of theory 1 to be
T1 := hH1 , Q1 , D1 i, and the triple of theory 2 to be T2 := hH2 , Q2 , D2 i then,
on the unitary notion of equivalence, T1 ∼
= T2 iff ∃U , such that U is a unitary
operator, and ∀ψi ∈ Hi , Qi ∈ Qi (i = 1, 2),
|ψ2 i = U |ψ1 i
Q2
= U Q1 U †
(1)
This is an isomorphism because U is invertible, and because one easily checks
that the structures defined on the Hilbert space, i.e. the map h | i : H × H → C,
6
as well as the action of operators on states, are preserved:
|ψ20 i := Q2 |ψ2 i = U Q1 U † U |ψ1 i = U Q1 |ψ1 i = U |ψ10 i
hψ2 |Q2 |ψ2 i
=
hψ1 |U † Q2 U |ψ1 i = hψ1 |Q1 |ψ1 i
(2)
Two interpreted theories will be said to be physically equivalent if they
share both their triple T —or these are unitary equivalent—and their physical
interpretation IT . If two interpreted theories are not theoretically equivalent
there is no good reason to think that they will be physically equivalent. If, on
the other hand they are theoretically equivalent but not physically equivalent
then a reason to justify how their interpretations differ is needed.
Two unitary equivalent- and thus theoretically equivalent- theories will have
the same interpretation if their interpretation maps and the unitary transformation form a commutative diagram. That is, let DT1 be the domain of the
interpreterpretation of theory 1 and let CT1 be the codomain of the same interpretation. The interpretation map I1 which corresponds to T1 maps its domain
into its codomain I1 : DT1 → CT1 . Likewise, I2 : DT2 → CT2 . As described
before, there exists a U such that U : DT1 → DT2 . Let us assume that, in
addition, there is an invertible map u : CT1 → CT2 between the codomains,
such that:
M1 (U x) = u ◦ M1 (x)
(3)
for x being either an element of the states x ∈ S or of the physical quantities
x∈Q
M2 (U y) = u ◦ M2 (y)
(4)
for y ∈ S or y ∈ Q.
If one can establish a unitary relation between the two codomains of the
different interpreted theories, then they would have the same interpretation,
as they are mapping the same thing i.e. the experimental outcomes. Thus,
provided they are also theoretical equivalent, the two interpreted theories would
be physically equivalent.
3
Three Interpreted Theories
In this section both the triples and the interpretational maps of every interpreted
theory will be extensivelly described. in section 2.1 this will be done for the
Copenhagen Interpretation, in section 2.2 for the Many-Worlds, and in section
2.3 for Bohmian mechanics.
3.1
Copenhagen Interpretation
The Copenhagen interpretation was the first successful attempt to build a complete theory based on the by then recently developed mathematical formalism
of quantum mechanics. It accounted for some of the processes that could no
longer be described by classical physics. The theory is attributed to Niels Bohr,
7
Werner Heisenberg, Max Born and several other physicists of the time (Section
1 [14].
The nature of the ‘Copenhangen interpretation’, has been and is until now, a
historically controversial issue. Although the interpretation is usually attributed
to Bohr, Heisenberg, and a few others, there are, for example, differences in
what this interpretation includes according to Bohr and according to Heisenberg. Because of this lack of consensus, this paper will focus on a textbook
reconstruction of the ‘Copenhagen interpretation’ without the pretence that
this is ‘the’ Copenhagen interpretation.
3.1.1
States
For the first interpreted
theory, the Copenhagen quantum theory, the states
H in the triple H, Q, D are states in a Hilbert space. More specifically, the
pure states of a system are represented by normalized vectors in a complex,
separable2 Hilbert space. A Hilbert space is a linear vector
space over the set
of complex numbers C, endowed with an inner product .|. such that
x|y = ( y|x )∗
(5)
where ∗ represents the complex conjugate. The norm of a vector would then be
given by
q
(6)
||x|| = x|x .
The dimension of the Hilbert space can be finite or infinite, while maintaining
the vector inner product (pp. 94 [18]). For finite vector spaces the inner product
is
X ∗
f |g =
fi gi
(7)
i
For infinite dimentional vector spaces the inner product is given by
Z
f |g = f (x)∗ g(x)dx < ∞.
(8)
Hilbert spaces must also be complete: “any converging sequence of vectors in
the space converges to a vector in the space”. This will allow for the finite
(< inf ty) inner products (Section 1[20]). Some subsets of a Hilbert spaces are
thus themselves Hilbert spaces.
In the Copenhagen interpretation, a different Hilbert space can be assigned
to each physical system S. A system could constitute single physical entity or
collections of them (Section 4 [24], Section 2 [1]). The converging condition of
a Hilbert space is thus importat as the possible states of a physical quantity of
a system S (an electron being in position 1 or position 2) will be described by
a Hilbert space within the Hilbert space of the system.
2 Separable denotes that it can be divided into countable subsets, like the set of natural
numbers N, which is infinite but countable.
8
The state vectors, which have been said to live in a complex space, do not
represent actual physical realities. Niels Bohr, one of the founding fathers of
this interpretation, believed that these should be understood as a mere tools to
yield probabilities observed in experiments (Section 5 [14]).
In principle, following the previous discussion, an additional, invertible, “0th
0
: H → S could be added
map”, from the Hilbert space to the system: ICop
here. This surjective map could take certain states in the Hilbert space to
certain states of the system. But this is actually not needed, as it will map
a mathematical concept into another non-physical entity. Since the question
of study focuses on the physical equivalence between the theories, the maps
needed to compare the theories are those which map measurable quantities such
0
as energy, speed, etc. This ‘0th map’ ICop
could, however, be used, as the basis
of an ontological discussion of the theories, to answer questions like ‘does the
wave function represent something real?’ This discussion will, unfortunately,
not be part of the scope of this thesis.
3.1.2
Physical Quantities
The discusion of the triple H, Q, D might seem abstract at the moment, but
soon enough the theory will gain coherence once the interpretational maps are
be described. Hermitian operators in the state space of a system S represent
the physical quantities Q that characterize such system. If we take the product
of a Hermitian operator Q and the vector representing a state ψi , this yields a
number times the same vector. In other words, if
Qψi = qi ψi ,
(9)
then we say the state is an eigenvector of that operator. These are the kind
of states that Newtonian mechanics could still describe. This is normally understood as the system S having the measurable physical quantity Q (Section
4[24], Section 2[1]. The space for the eigenvectors representing physical quantities of the system can also be formalized into a Hilbert state space. The state
space for the spin, for example, is a two dimensional Hilbert space (Section 2
[1]). The position or the momentum state spaces, on the other hand, are constituted of square integrable functions and thus are infinite dimensional Hilbert
spaces(Section 3 [20]).
In order to connect the mathematical formalism to the measured outcomes
of the experiments, the Copenhagen interpretation offers an extra assumption:
the Born rule (Section 1 [15]). This rule constitues, in terms of the formulation
presented, the physical interpretation of the theory. For the sake of simplicity, imagine an operator Q in Q with a non-degenerate discrete spectrum; when
measuring the physical quantity the operator represents, the probability of measuring one of the eigenvalues ai is given by
P (A = ai |Ψ) = Ψ|Pi |Ψ
(10)
9
P
where Ψ = i ci ψi and Pi is the projection operator that projects Ψ onto the
space of A corresponding to ψi .
Pi |Ψ = ci |ψi .
(11)
For a one
dimentional eigenspace the projection operator can be expressed as
Pi = |ψi ψi |. The probability of getting outcome ai will then be given by
P (A = ai |Ψ) = Ψ|ψi ψi |Ψ = | ψi |Ψ |2 = |ci |2 .
(12)
This assumption was the tool to probabilistically predict the outcomes of the
processes that classical mechanics could not account for. Probability is here
construed in terms of frequencies of repeated measurements of the system S.
The system S can consist of a single subsystem, in which case the frequencies
represent the outcomes of repeated measurements on that system, prepared under identical conditions; or, more in line with experimental practices, it consists
of a multiplicity of identical subsystems, prepared under identical conditions:
in which case the probabilities represent the frequencies of the outcomes of
those subsystems. Notice, however, that the projection operator describes a
non-linear process as most of the states disappear after projection and do not
continue evolving—they are no longer considered as probabilities after measurement. The Copenhagen interpretation lacks, up until this day, from justification
for this non-linearity.
Once the connexion between the mathematics and the experimental outcomes is known, it is time to turn to the described notion of interpretation by
means of surjective maps. The interpretive maps, will turn the bare theory
hS, Q, Di into a theory about a physical system S, its physical magnitudes, and
its dynamics, i.e., an interpreted theory.
1
: Q×
For the Copenhagen interpretation, the first map is given by ICop
1
H → R, and as follows: ICop : (Q, ψi ) 7→ ai and ai is interpreted as a physical
magnitude - an outcome of a measurement of, say, energy. Notice that although
this map maps the
subset Q into the reals, what it truly maps is the entirety
of the triple T = H, Q, D : the states are included by means of Q’s cartesian
product with H. The states are determined by the Hamiltonian and thus the
dynamics D are also implicit in this mapping. In addition, this map is surjective
but in general not invertible, as discussed in section 2.1 , one must account for
2
degeneracy of states. The second map of the interpretation is given by ICop
2
2
2
maps ICop : Q × H × H → R≥0 , as follows: ICop (Q, ψi , Ψ) 7→ |ci | . Here, ψi is the
eigenvector of A with eigenvalue ai . Alternately, we can view this second map as
1
the composition of two maps: ICop
and then a map (ai , Ψ) to the probabilities.
2
1
ICop : CCop × H → R≥0 where CCop is the codomain of ICop
.
In conclusion, the second map relates, as desired, the physical quantities
that are predicted to be observed ai , with the frequency |ci |2 with which they
are observed in the experimental outcomes.
10
3.1.3
Dynamics
The last element from the triple that describes this bare theory is its dynamics
D. The dynamics of a system describe its time evolution. If a vector represents
the state of a system at time t0 , and the forces and constraints to which that
vector is subject to are available, it is possible to determine the state of the
system at any later time t (Section 3.4 [21]). In the Copenhagen interpretation,
this is done through the Schrödinger equation:
i~
∂ψ
= Hψ
∂t
(13)
where H represents the Hamiltonian operator which contains all the information
about the energy restrictions of the system. In the vector formalism, H is used
to construct an operator U , such that
|ψ(t) = U |ψ(t0 )
(14)
The relation between H and U is given by
U =1−
t2 H 2
it3 H 3
itH
−
−
−
...
~
2~2
6~3
(15)
which, since the eigenvalues of H are real and positive as it is a Hermitian
operator, converges to (Section 2.7 [20])
U = exp(
i Ht
)
~
(16)
whose eigenvalues then take values on the unit circle in the complex plane.
Note however, that U is an operator but it is not Hermitian and thus it does
not describe any physical quantity, as H did. U on the other hand, describes a
unitary, linear and deterministic evolution (Section 3.4 [21]). Since U is unitary,
the states at time t0 and at time t live in the same vector space, the vector
space restricted by the constraints of the system S. As U implies the linearity
of the time evolution, if the state of a system S at time t0 is described by a
superposition:
(17)
|ψ(t0 ) = α|A + β|B ,
then such system will remain superposed after a time t (Section 2 [1])
|ψ(t) = U |ψ(t0 ) = U α|A + U β|B = αU |A + βU |B = α|A0 + β|B 0 (18)
Would we then need, an interperetation that maps what is physical in the
dynamics to the experimental oucomes we can measure of them? The answer to
3
this question is no. A new map ICop
is not needed as the dynamics are given by
the hamiltonian, which can be expresed in terms of an operator H that acting
on a state vector ψi would yield the energy of such state. As it has been shown
in section 3.1.2 the energy of a system is a physical measurable quantity and
1
thus can be explained in terms of the map ICop
.
11
As it has been mentioned, it is possible to say that the time evolution is
deterministic because the future of a state at time t0 is only determined by its
initial energy, as U depends entirely on H (Section 2.7 [20]). This does not
imply, however, that we can determine in a deterministic way, which of the
states the system S will be in, if this was initially in a superposition such as the
2
described in section 3.2 maps the
one in 17 (Section 2.7 [20]). The map ICop
theory to the experimental outcomes by means of a probabilistic approach, it
does not, however, connect the theory and the experiments in a deterministic
way.
3.1.4
The Measurement Problem
States like the one in equation (17) are the main source of trouble for both
philosophers and physicists working in quantum mechanics. The problem arises
when one wishes to measure the value of a physical quantity (e.g. the position of
a particle) on a such quantum system. The interaction between our measuring
apparatus and the system must be such that:
|ψα |φ0 (t0 ) → |ψα |φα (t)
(19)
Where ψα is the state of the system and φ0 and φα are the state of the measuring
apparatus before and after measurement. This can formalized by means of an
unitary operator U that acts on the vectors
U(|ψα |φ0 (t0 ) ) = |ψα |φα (t)
(20)
where U is given by
U=
X
|ψγ |φβ+γ φβ | ψγ |
β,γ
Proof:
U(|ψα |φ0 )
=
X
=
X
|ψγ |φβ+γ φβ |ψα ψγ |φ0
β,γ
|ψγ |φβ+γ δβ0 δγα = |ψα |φα . 2
(21)
β,γ
If the initial state ψ is in a superposition like the one in equation 17, the
final state of the measuring apparatus will be given by:
U((α|A + β|B )φ0 ) = α|A φA + β|B φB
(22)
However, in real life, pointers of measuring devices only yield one value, φ and
are never in a superposition of φA and φB . In this way, we have arrived at the
measurement problem [11]. The solution to this problem will be key for the
distinction of the different interpretations of quantum mechanics.
For the advocates of the Copenhagen interpretation, the solution lies in Von
Neumann’s projection postulate and the Born rule. In order for the mathematics to match reality, Von Neumann suggested that “the entangled state of the
12
object and the instrument collapses to a determinate state whenever a measurement takes place”(Section 7 [14]). In order words, the mere act of measuring
“collapses the wave function” into a single state: the one it is observed. The
evolution of the system becomes non-linear during measurement. How this is
done, however, “we dare not to ask” [3]. As described in the previous sections,
2
(Q, ψi , Ψ) 7→ |ci |2 does the job of “collapsing the
our interpretational map ICop
wave function” into a single state: the state that is actualiced.
In order to explain such interpretational map, Niels Bohr believed that
speaking of a separation between the system and the measuring apparatus
was the mistake, as the entanglement between these two is unavoidable due
to Plank’s constant (Section 5 [24]). He also argued that measurements do not
yield values but rather they produce them. In Bohr’s view, no physical quantity
truly belongs to the system S before measurement (Section 5 [14]).
3.1.5
The Uncertainty Principle
As a last remark it is relevant to mention Bohr’s pupil most famous contribution to quantum mechanics. In the Copenhagen interpretation, the uncertainty
principle is a consequence the formalism of non-conmuting matrix operators.
1
2
2 2
A, B
(23)
σA
σB ≥
2i
Where σX represent the standard deviation of an operator X, and [A, B] is the
canonical commutation relation between the operator A and B (Section 3 [18].
For position and momentum, for example
x̂, p̂ = i~
(24)
σx σp ≥
~
2
(25)
Where ~ is Plank’s constant divided by 2π
According to Bohr, a pair of non commuting operators forms a complementary pair. Position and momentum, for example, “are never jointly applicable
in a single experimental arrangement but only in mutually exclusive experimental arrangement”(Section 5 [24]). John Steward Bell, in his famous Six
Possible Worlds of Quantum Mechanics, thought that the word contradictoriness (instead of complementarity) would be more suitable word to describe this
characteristic of nature [3].
3.2
Many-Worlds Interpretation
In 1957, Hugh Everett III published in Reviews of Modern Physics a new formulation of quantum mechanics: the “relative state” formulation. Although
his new theory was understood and interpreted in many different ways3 , every
Everettian theory is based on the existence of many worlds, parallel to ours and
3 See
e.g. the Many Minds interpretation[22, 27]
13
thus inaccessible by us, where the outcome of a probabilistic measurements that
did not become actual in our world are actualized (Section 1 [27]).
3.2.1
States
In the Many-Worlds interpretation of quantum mechanics the states of a physical system S are also represented by normalized vectors in complex separable
Hilbert spaces. Even though not specifically differently from the Copenhagen
interpretation, in Everettian mechanics is worth mentioning the state function
of the whole universe (Section 3.3 [26]).
X
|Ψ =
αi |ψi ,
(26)
where the different ψi represent the mutually orthogonal worlds, and where the
sum represents the Cartesian sum over these worlds. In this way, indefinite
macroscopic states, like a cat in a superposition of being alive and dead at
the same time, are not described by means of a superposition of the different
states of a cat Hilbert space. Instead, they can be represented as a superposition of two worlds, one in which the cat is alive, and other where the cat is
dead. According to David Wallace, in Everett’s theory “superpositions do not
describe indefiniteness, they describe multiplicity”(Section 3 [27]): The worlds
that contain dead cats and the worlds that contain alive cats do not represent
mere probabilities but actualities, not all of which are accessible by us. For this
reason “[t]he many worlds interpretation is a natural choice for quantum cosmology, which describes the whole universe by means of a state vector”(Section
1 [28]).
In the states section of Copenhagen interpretation, I discussed the possiblity
0
0
of a ‘0th map’ ICop
: H → S. In MW this maps would be IMW
: H → S ⊗n , where
n = dim H. The tensor product is justified as every state after measurement
will lie in a different world and thus in a different dimention of the Hilbert space.
In that case, the formalism for the states resembles the one in the Copenhagen
interpretation but the interpretation is different. However, like before, this
map does not take part in the physical interpretation of the theory, as it does
not relate to what is physical i.e the measurable quantities. Again, it would
interesting to study this map in one would wish to rise a discussion about the
ontology of the theory, in particular about the nature of a state. However, as it
was mentioned before, this paper will not need to commit to such a map, since
it will not take part in the discussion of physical equivalence.
3.2.2
Physical Quantities
The physical quantities Q described by the Hilbert space formalism represent
the physical quantities in the Many-worlds interpretation in the same manner
they did in the Copenhagen interpretation.
The many-worlds interpretation is, however, one free of observers. The physical quantities, differently from Copenhagen interpretation, must be inherent to
the system as they cannot depend on the observer (Section 7 [26]). The value
14
of the physical quantity is not probabilistic, relative to a world ψi . A system S
has the physical quantity Q with value q such that:
Q |ψi i = q |ψi i
(27)
as an actuality in that world. The rest of the eigenvalues that would stand
for other possibilities (with their probabilities given by Born’s rule) simply became actual in a different world Wi (Section 3 [27]). This, sometimes called
the branching postulate, constitutes the physical interpretation IT of this inter1
preted theory. The first map is now IMW
: Q × H → R ⊂ R⊗n . In the Copenhagen case, we had a single R for each observable—simply the possible values a
quantity can take, referring to the same system. Now the values the quantities
can take refer to different systems -the same system in a different world, so we
need the tensor product of the real space to represent such actualities. The
2
2
: CMW × H → R≥0
maps IMW
second map of the interpretation is given by IMW
1
where CMW is the codomain of IMW . This codomain will however, depend in the
world where the state branches into. Comparing such codomain with the one
from in Copenaguen interpretation would give the following relation
X
CCop =
Ci Wi ,
(28)
i
1
and Wi the world in which this
for Ci corresponding to the codomain of IMW
codomain is actual.
3.2.3
Dynamics
The dynamics D in the Many-Worlds interpretation resemble that in the Copenhagen interpretation as well. Schrödinger’s equation holds for every system and
every point in time (pp. 42 [28]). Some versions of the interpreted theory include the theory of Decoherence in their dynamics. In this paper, Decoherence
will be studied as a solution to the preferred basis problem.
3.2.4
The Measurement Problem
The measurement problem in the Many-Worlds interpretation is not understood the same way as it is in the other interpretations. While supporters of
the Copenhagen interpretation wonder about the non-linearity of their experience, Many-World interpreters do not conceive this as a problem. Their realist
attitude towards the unitary evolving quantum states, expressed in the now
1
completely linear evolution of the map IMW
, resolves the problem of uncertainty
about the world that will become actual: all of them will (Section 1 [27]). This
perspective avoids the problem of the “collapse of the wave function” (Section
3 [26]). The measurement question could, however, be formulated differently:
which world will I branch into after this measurement? This interpretation does
not offer an answer to this question (Section 2 [29].
Another problem faced by Everett’s interpreted theory is the problem of the
preferred basis (Section 3 [27]). If the mathematical structure allows for many
15
different divisions of the state of the universe into orthonormal basis, one can
wonder why it is only possible observe the basis representing definite objects
with definite positions. This problem was one of the main critics of the Everettian theory at the start. Decoherence, a theory developed some decades
after Everett’s time, seems to solve this problem. One of the first clear explanations of Decoherence was given by Zurek:
Macroscopic quantum systems are never isolated from their environments and therefore they should not be expected to follow Schrodinger’s
equation, which is applicable only to closed systems. Classical systems suffer (or benefit) from the natural loss of quantum coherence,
which leaks out into the environment (Section 1 [28]).
If there are no observers that can measure a system (if the system is the universe, for example) classicality must emerge from the physical systems themselves (Section 1[28]). The term Decoherence indicates the entanglement between a system and its environment. The entanglement of the environment with
wave-packets states, which present a fairly definite momentum and position, occurs quite slowly. The entanglement of the environment with superposed wave
packets, however, occurs extremely quickly (Section 4 [27]). Mathematically, if
a state is in a superposition
ψ = α|A + β|B
(29)
After measurement, the state S of the measured system M and the measuring
apparatus A is given by
|S = α|A |φA + β|B |φB = |SA + |SB
(30)
The possibility of rewriting the |S in other basis of the system, like for example
1
1
0
0
+|SB
(31)
|S = (|A +|B )(|φA +|φB )+ (|A −|B )(|φA −|φB ) = |SA
2
2
constitutes the preferred basis problem [15]. This possibility is, however, discarded if the interaction of the environment is taken into account (Section 4
[29]). The environment interacts with pointer states
|S ⊗ |E0 → U (t)(|S ⊗ |E0 ) ≈ |SA ⊗ |EA (t) + |SB ⊗ |EB (t)
(32)
for EA (t)|EB (t) ≈ δAB for t τD and τD represents the coherencetime. The
justification for this einselection (enviromental selection) is that |SA and |SB
are pointer states. Pointer states are the set of states that are stable under time
evolution:
U (t)|Si |E = |Si |Ei (t)
(33)
0
0
As |S1 and |S2 do not have these properties, and thus are not einselected
(Section 2,3 [15]) (Section 4 [29]). The classically describable states (or pointer
states) will become the preferred basis [28]. In the case of a wave packet, the
supperposition states will not be considered pointer states, the branching will
occur only for the definite wave-packet state basis: there will be no world where
Scrödinger’s cat will be alive and dead at the same time.
16
3.3
Bohmian Mechanics
The “pilot wave” interpretation was originally described by Louis De Broglie in
1927. When presenting his work in the Solvay Congress, however, De Broglie
was incapable of resolving some of Pauli’s criticisms on his new theory (Section
3 [17]). This made him abandon it and, among others, he soon became a
supporter of the Copenhagen interpretation. It was not until 1950 that David
Bohm rediscovered De Broglie’s ideas. After having written his book, Quantum
Theory, where he presented an extended explanation Bohr’s description of the
quantum world, Bohm “felt some-what dissatisfied”(Section 1 [4]). Following a
conversation with Albert Einstein, Bohm was convinced that there was a need
for a more complete theory of quantum mechanics. In 1952 he published his
Suggested Interpretation of “Hidden” Variables I-II, a paper divided in two parts
where he described the same equation of motion that De Broglie had described,
together this time with a stronger and a more complete new theory. This theory
was then renamed Bohmian mechanics [5].
3.3.1
States
The key notions to understand the states H in Bohmian mechanics are the
conditional wave function of a subsystem and its configuration. Suppose the
configuration of the universe can be described in terms of the configuration X
of the subsystem x, and the configuration of its environment i.e the rest of the
universe, Y . Q(Univ) = Q(X, Y ). Now the wave equation of the universe would
given by
Ψ(q) = Ψ(x, y),
(34)
where x and y represent configuration-space variables. The question that arises
is then, what is the wave equation of the subsystem x? Bohm answers this question by introducing his notion of conditional wave function. For the subsystem
x, the conditional wave function is given by:
ψ(x) = Ψ(x, Y ) ,
(35)
where Y is (still) the configuration of the environment of the subsystem x. As
the reader might have noticed, the conditional wave function might not satisfy
Schrödinger’s equation in many situations. The factorizations of the conditional
function which do satisfy Schödinger’s equation are referred to as the effective
function of the subsystem x (Section 5 [12]). These are the factorizations in
which the system is suitably decoupled from the rest of the universe. The wave
equation of the universe will remain being a function of both x and y so
Ψ(x, y) = ψ(x) φ(y) + Φ(x, y)
(36)
with φ and Φ having macroscopically y-supports that are disjoint and with Y
lying in the support 4 ’5 of φ (Section 9 [11]).
4 If the supports of the functions are disjoint this implies that the y component both
functions will not be zero at the same time in any case.
5 The fact that is macroscopically disjoint means that “there is a macroscopic function of
y–think, say, of the orientation of a pointer–whose values for y in the support of φ differ by
17
Note that when Φ(x, y) = 0 the subsystem is decoupled and Ψ(x, y) remains
a solution to Schrodinger’s equation, and thus will evolve as expected, yielding
the expected experimental outcomes(Section 4 [17]).
The state space H differs from the one described by both the Copenhagen
and the Many-Worlds interpretation. Firstly, a state is given by (Q(t), Ψc ) where
Q(t) is the configuration space of such state at time t and Ψc is the conditional
wave function(Section 5 [10]). The pair (Q(t), Ψc ) has a phase space of R3N ×Hc
. Secondly, the effective wave function accounts for the wave functions of a
system S in the other interpreted theories, as these are always considered to
be decoupled from the environment. The effective wave function is however,
a special case of the conditional wave function, and cannot be considered in
many situations, like for example during a measurement. The conditional wave
function of a subsystem x, during measurement, will consider contributions
to the Hamiltonian from the entanglement between the measuring apparatus
and the measured the subsystem x and the collapse into a single state after
measurement will be justified in a similar way as decoherence did. The main
difference between decohered systems and the conditional wave function is that
the environment is already included in the notion of conditional wave function
and does not need a decoherence time τD to interact with the system. The
conditional wave functions accounts, thus, for the weirdly-defined collapsed wave
functions. For these reasons, there are certain conditional wave functions that
cannot be explained in terms of states H in the two other theories. As we will
see, this will be the main problem for the unitary equivalence between these
theories. An analogy naturally arises here: the conditional wave function in
Bohmian mechanics is for the states of the Copenhagen and the Many-worlds
like the states described by quantum mechanics were for the states in Newtonian
mechanics. The latter is properly contained in the former but the former is
capable of explaining certain processes that the latter cannot.
0
The ’0th map’ for this interpretation would be given by IBM
: Hc × R3N → S
where Hc represents the Hilbert space of the conditional wave function, R3N is
the configuration space of the states and H ∈ Hc as required.
3.3.2
Physical Quantities
The Bohmian world consists of particles that have determinate positions. These
positions are the the only physical quantities that suffice to characterize all
quantum phenomena “insofar as these phenomena can be characterized by the
changes in the position of the particles”(Section 8 [6]). The basic physical
quantity described by Q is position. The changes in the positions of the particles
are caused by a guiding equation, as it will be described in the dynamics section.
However, as Bell put it :
No one can understand this theory until he is willing to think of ψ as
a macroscopic amount from its values for y in the support of Φ [12] As Y ∈ supp φ this will
only be zero once the universe is over, so for all practical purposes (fapp) we can ignore the
contribution of Φ and simply consider φ(y).
18
a real objective field rather than just a probability amplitude. Even
though it propagates not in 3-space but in 3N space (Section 8 [6]).
The wave equation is also real, although we cannot measure it, and it is in
charge of guiding the particles, deterministically, to their new position. In a
few words in Bohmian mechanics “most of what can be measured is not real
and most of what is real cannot be measured, position being the exception”
(Section 9 [13]). The map of the physical quantities Q for this interpretation
1
would map IBM
: Q × Hc → R ⊂ R3N . The justification of this maps is that in
Bohmian mechancis the physical quantities i.e. the position of the particles live
in configurational space R3N and thus once the position is measured, one can
find out what the configuration was.
Before moving into the dynamics of the theory, one last remark must be
done. If the main physical quantity to study in Bohmian mechanics is position,
and it is theoretically possible to determine this position, and the rest of the
observables supposedly follow from it, then the question arises: what happens
to Heisenberg’s uncertainty principle? In future sections it will be shown that
not only does this principle remain valid, but that Bohmian mechanics gives a
better justification than other interpreted theories for Heisenberg’s principle.
3.3.3
Dynamics
Bohmian mechanics includes the Schrodinger’s equation as a part of its dynamics
D. This equation indicates, as it did in the two other interpreted theories, the
evolution in time of the wave function (pp.1 [11]). Differently from the other
interpretations, however, this is not the only equation to take into account in
the dynamics of Bohmian mechanics. While the founding fathers were puzzled
about the question: wave or particle? Both De Broglie and Bohm offered the
answer wave and particle! [3]. In their interpretation, the trajectories of the
particles, are governed by Bohm’s law of motion: (Section 4 [17, 11]).
~
Ψ∗ ∆ i Ψt
dQi
=
Q(t)
(37)
Im t ∗
dt
mi
Ψt Ψt
where Q(t) = Q1 (t), ..., QN (t) is the configuration of N point particles moving
in a physical space R3 . The subsystem x discussed above, satisfies Bohm’s law
of motion with Q = X and ψ being the conditional wave function ψ(x).
Equation (37) implies that the wave equation must not only satisfy the
Schrodinger’s equation, but is also in charge of the choreography of the particles. As Bell put it,“the wave function generates a velocity vector field (on
configuration space) which defines the Bohmian trajectories”(Section 8[13]).
Furthermore, Ψt can be determined by using the Schrodinger equation, and
if Q(t0 ) is specified, then the position of a particle at any later time t can be
determined. Determining Q(t0 ) however, is not an easy task, as we always consider the conditional function of the system and this depends on the universal
wave function -in which, if you remember correctly, the configuration of all the
particles in the universe is indicated. The problem in predicting the position
19
of a particle thus, is not inherent to the theory, but rather, in Bell’s words, is
simply because we “cannot know everything”[3].
Imagine the wave function of the universe at the initial time Ψ(t0 , Q). It is
impossible for anyone to determine the configuration of the particles constituting the universe at this initial time. The quantum equilibrium hypothesis thus
requires that the initial configuration Q(t0 ) of this system can be chosen, at random, to have probability density |Ψ(t0 , Q)|2 . This looks very similar to the Born
rule, the reader might be thinking, and indeed the reader is absolutelly right.
The difference between Bohmian Mechanics and the Copenhagen interpretation
is that, for the latter, Born’s probability rule is added to the theory in order to
account for the interpretation, whereas for the former it is typical (Section 4 [12],
pp.4 [11], Section 11[13]). The Born rule would be a probability distribution
out of ignorance.
A good argument in favour of Born’s rule representing the initial probability
density of the particles is that |Ψ|2 satisfies the continuity equation
∂ρ
= −∇(ρv)
∂t
(38)
for ρ = |Ψ|2 and v being the Bohmian velocity vector field v Ψ that Bell used
(for the proof see Appendix A). In addition to this, in Bohmian mechanics
interpreting |Ψ|2 as the probability density implies equivariance: “ If the initial
configuration Q(t0 ) is chosen at random with probability density |Ψ(t0 , Q)|2
then the configuration Q(t) at another time t is random with probability density
|Ψ(t, Q)|2 ” (pp. 3 [11]). Assuming Born’s rule for typical systems (a classical
analogous would be systems in which typically the entropy tends to increase),
one can experimentally check that the assumption was rather a prediction.
3.3.4
The Measurement Problem
The measurement problem is said to be solved for Bohmian mechanics (Section 7
[17]). However, this theory does not offer a solution: it simply does not present
the problem. When formulating a quantum “ideal measurement”, a physical
quantity represented by the Hermitian operator Q in the Hilbert space of the
system S is needed. This physical quantity (position, or anything that can be
expressed in terms of position) will ideally yield an eigenvalue α. It would be
expected that after time t the measuring apparatus φ would yield the value α
as well:
ψα (x)φ0 (y) → ψα (x)φα (y).
(39)
Notice the difference between equation (19) and this equation: the system S is
given by the subsystem X and the environment Y . If the system is not in an
eigenstate of the physical quantity Q but rather in a mix state, the measurement
will rather look [11, 13]:
X
X
Ψo (x, y) =
cα ψα (x) φ0 (y) → Ψt (x, y) =
cα ψα (x) φα (y).
(40)
α
α
20
However, I stated previously that in Bohmian mechanics there was no measurement problem, the particle’s position is not in a superposition, the particle has
a single determined position and so will the pointer of the measuring apparatus. Taking into account the conditional wave function of the subsystem x the
measuring apparatus will yield a single value:
X
ψt (x) = Ψt (x, Y ) =
cα ψα (x)φα (Yt ) = cβ ψβ (x)φβ (Yt ) = N ψβ (x) , (41)
α
where N depends on the configuration of the environment at time t but not in
x. The rest of the possible outcomes were cancelled out by the environment
initially when considering the subsystem x, as this had a determinate initial
and a final position.
3.3.5
The Uncertainty Principle
In Bohmian mechanics, the position of a particle as well as its trajectory, are
determined by the previous state of the universe (Section 1 [13]). However,
the knowledge of the observer of this trajectory (and therefore the position and
momentum) is limited. The uncertainty relation remains valid for what we can
observe and predict and is not inherent to the system. As it was mentioned in
section 3.1.5, Heisenberg’s uncertainty principle can be derived as a consequence
of the non-commuting matrix operator formalism. The same relation can be
derived, however, from the wave mechanics interpretation and the Born rule.
Born’s probability rule is typical in Bohmian mechanics and therefore, despite
the determinism of the theory, the uncertainty principle still holds.
In Bohmian mechanics, the probability density for position of a system is
given by |ψ(x)|2 and correspondingly the probability density for momentum of
a system is given by |φ(p)|2 . The standard deviation for position would then be
Z ∞
Z ∞
2
σx2 =
x2 |ψ(x)|2 dx −
x|ψ(x)|2 .
(42)
−∞
−∞
Not surprisingly, the momentum standard deviation is given by
Z ∞
Z ∞
2
2
2
2
σx =
p |φ(p)| dp −
p|ψ(p)|2 .
−∞
(43)
−∞
For simplicity, the means will vanish -as these only represent the shift of the
coordinates
origin, thisR approximation is fairly accurate (Section 4 [9]). σx2 =
R∞ 2
∞
x |ψ(x)|2 dx σx2 = −∞ p2 |φ(p)|2 dp Now, recall Schwarz’s inequality
−∞
Z ∞
hZ ∞
i1/2 h Z ∞
i1/2
0
2
xψ(x)ψ (x)dx ≤
|xψ(x)|
|ψ 0 (x)|2 dx
.
(44)
−∞
−∞
−∞
As is well-known, position and momentum are non-commuting pairs so that one
can be written as a Fourier transform of the other (Section 12 [25]
hZ ∞
i1/2 h Z ∞
dp i1/2
,
(45)
|ψ 0 (x)|2 dx
=
|ihpφ(p)|2
2π
−∞
−∞
21
So the right hand of the inequality will contain:
2π
σx σp .
h
(46)
For the left hand of the equation, using integration by parts one can find, equals
1
2 . The Schwarz’s inequality turns into
σx σp ≥
h
~
=
4π
2
(47)
which is precisely, Heisenberg’s uncertainty principle (Section 12 [25]).
4
Theoretical and Physical Equivalence
As mentioned in Section 2.3, in order for two interpreted theories to be physically
equivalent they must be theoretically equivalent and their maps, constituting
their physical interpretations, must map the same phenomena.
4.1
Theoretical equivalence
When two interpreted theories share their triples T they are said to be theoretically equivalent. This is because, in a quantum theory, the states, the quantities,
and the dynamics are the formal structures which are assumed to exhaust the
physical content of the theory. To share a triple, however and as aforementioned,
does not imply that all the items of the triple, must be exactly the same. If this
were to be the case, theoretical equivalence would be too strong of a requirement
-the Hilbert space formalism and wave mechanics, for example, would not fulfil
this requirement (Section 3 [2]). For theoretical equivalence, it suffices that the
two triples are isomorphic T1 ∼
= T2 , in other words, it suffices for them to be
unitary equivalent. As explained in section 2.3, unitary equivalence maps every
state or physical quantity from an interpreted theory into the states and the
physical quantities of the other interpreted theory.
At this point the reader may have guessed the status of the equivalence
between the three theories that have been studied. In the Copenhagen interpretation, normalized vectors in a complex, separable Hilbert space map to the
states H. The physical quantities Q of a system S are mapped by all the self adjoint operators of the Hilbert space of the system. Finally, the dynamics D are
described by means of Scrödinger’s equation for specific Hamiltonian conditions
—which will depend on the system S. The Many-worlds interpretation may
have a different basis, but unitary equivalence discussed above allows different
choices of these. Hence, this theory shares each of the elements of its triple,
with the Copenhagen interpretation.
Two of the three studied interpreted theories are theoretically equivalent.
When defining the triple for Bohmian Mechanics, however, one encounters a
number of differences. The states H in Bohm’s theory are described by the
conditional wave function Hc . As explained in section 3.3.1, the states described
22
in both the Copenhagen and the Many-Worlds interpretation are contained
inside the notion of conditional wave function H ∈ Hc but there are certain
states defined in Bohmian mechanics that do not have a corresponding state
in the other two theories that they could be mapped to. Unitary equivalence
between the theories by means of a unitary operator U cannot be fulfilled if
the number of states in the sets are different. In this case, the set of states in
Bohmian mechanics has a larger amount of states, described by the conditional
function, than the numer of states in the sets of the other two theories.
The physical quantities depicted by Bohmian mechanics seems to be radically
different from the one described by the other interpreted theories. Although not
illustrated in this paper, however, there exist some versions of Bohm’s theory
which include the description of other physical quantities besides from position
(Section 8[6]). The dynamics D in the last interpreted theory are constituted by
two equations of motion: Schödinger’s equation 13 and Bohm’s law of motion
37. The latter one describes the movement of particles in configuration space
guided by the wave equation. Although not very often emphasized, Bohm’s law
of motion can take part in other interpreted theories of quantum mechanics.
The dynamics of this theory could be mapped into the other two, in the same
way the physical quantities could. However, as he have seen, it is not possible to
define an unitary map for the states. Hence, the theories cannot be isomorphic
TBM TCop ∼
= TMW . The theoretical inequivalence of Bohmian mechanics to
the other two interpreted theories leaves Bohm’s theory outside of the debate
of physical equivalence as the first requirement is not fulfilled.
Before moving on into the notion of physical equivalence between the two remaining interpreted theories a remark must be made: theoretical inequivalence
between these interpreted theories does not imply the falsehood of Bohmian
mechanics. It simply implies that in a world where theoretical accuracy could
be fundamentally checked6 , if one of the theories would be correct, its corresponding inequivalent pair would be considered false. In addition, as theoretical
ineqivalence also implies physical inequivalence, if one were to experimentally
falisfy Bohmian mechanics, there would still be room for taking any of the other
two theories into account.
4.2
Physical equivalence
The final point that needs to be addressed is the physical equivalence between
the Many-Worlds interpretation and the Copenhagen interpretation. If these
two interpreted theories were to share their interpretation then they would be
considered physically equivalent.
The physical interpretation of a theory IT was described in section 2.3 as a
pair of maps which stipulate what is physical about the theory. More preciselly,
these maps map the triples hH, Q, Di to the observed phenomena. In this way,
the physical interpretation is the responsible to establish two appropriate maps
6 Which, as Popper taught us, would never be possible as any theory can be falsified (by
giving counter-examples), but never verified(Section 2 [16].
23
from what is physical in the theory to what these represent in the physical
reality i.e the experimental outcomes. The first map is the one that relates the
the expectation values of hermitian operators to the physical quantities that can
be measured in the systems. The eigenvalues of the Hamiltonian operator, for
example, can be mapped to represent the energy of a system. As it has been
shown, this map is the same for both the Copenhagen and the Many-Worlds
interpretation:
1
ICop
:Q×H→R
(48)
1
IMW
: Q × H → R.
(49)
⊗n
With the only difference that for MW R ⊂ R .
The second map, however, seems to demolish this possibility. This map is
in charge of relating the probability amplitudes |ci |2 in Q (evaluated on states)
to the experimentally measured outcomes. Calling these amplitudes probability
amplitudes is, however, already thinking in terms of the Copenhagen interpretation. As described above, for this interpretation,P
the probability of finding
a system S described by the wave function Ψ =
i ci ψi in the state ψi is
given by |ci |2 . These probabilities would describe the results of a system (the
measurement of system S, whether it consists of a single system or a series of
equally prepared subsystems) with respect to the world where they are being
measured. The domain and codomain of this surjective map IT2 for the Copenhagen interpretation would be respectively, the subset of Q evaluated on states
(the amplitudes squared), and the probabilities of the outcomes at that world
W . As it was mentioned, for the Many-Worlds interpretation the amplitudes
squared |ci |2 do not represent probabilities as frequencies at a single world W
but rather multiplicities, i.e. the actualities at the different possible worlds. For
this reason, the domain of the second surjective map remains the same, but the
codomain of the two interpreted theories is not the same for the same world W
where the measurements were being done. The codomain CCop described by the
Copenhagen interpretation is given by
X
CCop =
Ci Wi
(50)
i
If understood that the sum over all Wi is W i.e. the world where the experimental outcomes are observed in the Copenhagen theory. For a same world Wi , the
Copenhagen interpretation would describe the codomain given by (50) whereas
the Many-Worlds interpretation’s codomain would be given by CMW = Ci . If
one recalls the maps
2
ICop
: CCop × H → R≥0
(51)
1
where CCop is the codomain of ICop
2
IMW
: CMW × H → R≥0
(52)
1
where CMW is the codomain of IMW
. The convergence condition described in
2.3 cannot be fulfilled as in mapping CCop into CMW the unitarity of U cannot be preserved. These codomains not only differ but also cannot be unitary
24
equivalent, and thus, the maps are not the same. The physical equivalence
requirement fails when it comes to what is physical with respect to the world
where the experiments are taking place.
The Copenhagen and the Many-World interpretations are theoretically equivalent but not physical equivalent due two their second surjective map. Bohmian
mechanics is not theoretically equivalent with any of the other two interpreted
theories and thus is also not physically equivalent. The three theories are physically inequivalent, which luckily will allow them to be falsibiable with respect
to one another. Someday, it will not be necessary to speak of several interpretations but rather a theory of quantum mechanics!
5
Conclusion
The main aims of this thesis were (1) to give a new meaning to the notion of
interpretation in quantum mechanics and (2) to study the physical equivalence
between three interpreted theories: the Copenhagen interpretation, the ManyWorlds interpretation and Bohmian mechanics.
Firstly, in section 2, an explanation of the new notion of interpretation was
given. To interpret is to offer a connexion between the theory and reality. A
physical interpretation thus connects what is physical in the triple T H, W, D
with the experimental outcomes observed that the theory was supposed to predict. In the light of De Haro, Teh and Butterfield (2016), this connection was
given by means of surjective maps. Secondly, and taking into account the new
notion of interpretation, an extensive explanation of each of the three interpreted theories was displayed. This explanations included the description of
their bare theory, characterized by their states H, physical quantities Q and
dynamics D and the description of their physical interpretation by means of the
interpretational maps IT1 , IT2 of their bare theory T . With this in mind, this paper addressed the question of physical equivalence. Bohmian mechanics, despite
having the same experimental predictions of the other two theories, was argued
to be theoretically inequivalent to the other two interpreted theories, leaving it
out of the debate of physical equivalence. Copenhagen and Many-Worlds were
argued to be theoretically equivalent, as they share their bare theory but physically inequivalent, as one of their interpretational maps could not relate their
two codomains in a unitary way. The findings from this study make several
contributions to the current literature in the fundaments of quantum mechanics: it offers an answer to the needed clarification of what it means to interpret
by using De Haro’s natural notions of bare theory and physical interpretation,
and it extends the knowledge of the ongoing debate about the physical equivalence between these interpreted theories allowing their falsifiability. Hopefully
allowing
[F]uture generations [to] look back, from the vantage point of a more
sophisticade theory, and wonder how we could have been so gullible.
(pp. 434 [18]).
25
Appendix A
Proof of the continuity equation from Schrödinger’s equation and the wave function in polar form
The wave function can be written in polar form: (Section 5.2 [6], pp. 169-170
[5] )
iS
(53)
Ψ = Re ~
The Scrödinger’s equation is
i~
∂Ψ
~2 2
=−
∇ Ψ+VΨ
∂t
2m
(54)
Plugging (1) into the left side of (2) we get
i~
iS
iS
i ∂S
∂R iS
∂S
∂R iS
e ~ + i~
R e ~ = i~
e~ −
Re ~
∂t
~ ∂t
∂t
∂t
(55)
Doing the same thing for the right side and setting V Ψ = 0
−
~2
∇(∇Ψ)
2m
(56)
Step by step
iS
∇ Ψ = ∇Re ~ +
iS
i
∇ S Re ~
~
iS
iS
iS
i
∇(∇Ψ) = ∇(∇R e ~ ) + ∇( ∇ S Re ~ ) = ∇2 R e ~
~
iS
iS
iS
iS
i
i
i
+ ∇ S ∇ R e ~ + [∇2 SR e ~ + ∇S(∇Re ~ + R ∇Se ~ )]
~
~
~
(57)
(58)
Taking only the imaginary part of the right side
Im(−
iS
iS
~2
~2 i
i
∇(∇Ψ)) = −
(2 ∇ S ∇ R e ~ + ∇2 SRe ~ )
2m
2m ~
~
(59)
Equaling it to the left side’s imaginary part
i~
iS
iS
~2 i
i
∂R iS
e~ =−
(2 ∇ S ∇ R e ~ + ∇2 SRe ~ )
∂t
2m ~
~
(60)
Cancelling the i’s the ~’s and the exponential term
∂R
1
=−
(2∇ S ∇ R + R∇2 S)
∂t
2m
(61)
Multiplying both sides times 2R
2R
∂R
1
= − (2R∇ S ∇ R + R2 ∇2 S)
∂t
m
26
(62)
Which rearranging can be seen equals
∂R2
∇S
= −∇ R2
∂t
m
(63)
Now, the probability density was given by ρ = R2 = |Ψ|2 and the velocity vector
field is given v = ∇mS . Substituting we arrive at the continuity equation
∂ρ
= −∇(ρv)
∂t
(64)
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