U
NIVERSITEIT VAN
A
MSTERDAM
MASTER
T
HESISSpontaneous Unitary Violations in
Quantum Mechanics
&
Thermalization in the Lieb-Mattis Model
Author:
Niels VERCAUTEREN
Supervisor: Prof. dr. J. van WEZEL Examinor: Prof. dr. J.S. CAUX
A thesis submitted in fulfillment of the requirements for the degree of Master of Science Physics and Astronomy
at the
60 EC April 30, 2021
iii
“It is often stated that of all the theories proposed in this[20th]century, the silliest is quantum theory. Some say that the only thing that quantum theory has going for it, in fact, is that it is unquestionably correct.”
v
Abstract
Faculteit der Natuurwetenschappen Wiskunde en Informatica (FNWI) Institute for Theoretical Physics Amsterdam (ITFA)
Master of Science Physics and Astronomy by Niels VERCAUTEREN
PartI: Spontaneous Unitary Violations in Quantum Mechanics
The first part of this report addresses the quantum measurement problem. The goal is to identify which properties of objective collapse theories are necessary and which are to be excluded in modelling quantum state reduction (QSR). A critical review of existing traditional interpretations of quantum mechanics - that do not alter the Schrödinger Equation (SE) - shows these are unable to fully solve the measurement problem. Their reliance on reduced density matrix descriptions entail inherent av-eraging through the trace operation. This can only produce a consistent theory of quantum mechanics for ensembles, not individual measurements. Modern non-traditional quantum mechanical theories that add a non-unitary component to the SE - such as GRW and CSL - are found to axiomatically assume Born’s rule and are inherently non-linear. Van Wezel’s Spontaneous Unitary Violations (SUV) model linearly alters the SE, spontaneously breaking the time translational symmetry, as is required during measurement. Energy conservation and state normalization in this model are addressed and not found to be problematic. However, linear models in general are demonstrated to be unable to produce Born’s rule. All aforementioned non-unitary models rely on the envariance based derivation of Born’s rule by Zurek, which is proven insufficient. Concluding that QSR models have to be in some way non-linear - without assuming Born’s rule - make mean-field approaches promising.
PartII: Thermalization in the Lieb-Mattis Model
Temperature is a mostly classically defined property. This work investigates the quantum origin of temperature. The reason for choosing the Lieb-Mattis (LM) model, is that it describes the collective behaviour of crystals. This made it a promising model to describe the emergence of temperature in anti-ferro magnets (AFMs). To first find an appropriate definition of temperature, classical and quantum descrip-tions of composites (system + bath) are compared. Composites which have many (in the thermodynamic limit degenerate) bath states per system state are predicted to thermalize and thus to be considered canonically typical. These general consider-ations are then applied to the LM model for AFMs. This entailed a thorough consid-eration of system and bath definitions and their associated energy levels. Neither the somewhat artificial LM model of an AFM for which the rotational symmetry is un-broken, nor the more realistic (spontaneously) symmetry broken LM model is shown to produce thermalization. The most fundamental reason for this unexpected result is found to be the inherent presence of dispersionless magnons in the LM model. They incur overwhelming placement degeneracy statistics, incongruent with those of a thermal distribution.
vii
Acknowledgements
It is with immense gratitude that I acknowledge the support and help of my super-visor Prof. dr. J. van Wezel. His flexibility and encouragement are a big part of the reason I was able to finish this thesis to my own satisfaction. I aspire to treat others with the same positivity, kindness and enthusiasm as he treats the people around him. I am also grateful for all the understanding and support I received from my family and friends during this project, foremost of which my colleagues and now close friends Alonso Corrales Salazar, Lotte Mertens and Matthijs Wesseling. The discussions we had on these topics were the most intellectually stimulating I have ever had. Lastly I would like to thank myself for the hard work, determination and perseverance I showed myself to be capable of, resulting in this thesis I am proud to present to the reader.
Contents
Page
Abstract v
Acknowledgements vii
Contents viii
I Spontaneous Unitary Violations in Quantum Mechanics 1
1 Introduction 3
2 Interpretations of Quantum Mechanics 5
2.1 The measurement problem. . . 5
2.1.1 Entropy and probabilities . . . 7
2.2 Traditional quantum mechanics . . . 9
2.2.1 The Copenhagen interpretation . . . 9
2.2.2 Decoherence. . . 10
Envariance. . . 12
2.2.3 The Many Worlds interpretation . . . 16
2.3 Non-traditional quantum mechanics . . . 17
2.3.1 GRW model . . . 17
2.3.2 CSL model . . . 19
2.3.3 Stable collapse and the problem with non-linearity . . . 23
3 Schrödinger instability 25 3.1 Spontaneous Symmetry Breaking . . . 25
3.1.1 Classical SSB . . . 26
3.1.2 Quantum SSB in the Lieb-Mattis Model . . . 27
Overview . . . 27
Symmetric energies and excitations . . . 29
Symmetry broken. . . 30
Spontaneous Symmetry Breaking . . . 31
3.2 Non-unitary perturbations . . . 32
3.2.1 Energy conservation and Normalization . . . 32
3.2.2 Spontaneous Unitary Violations in the Lieb-Mattis model . . . 33
3.3 General linear time evolution on the Bloch sphere . . . 36
ix
Page
II Thermalization in the Lieb-Mattis Model 41
5 Introduction 43
6 Thermalization 45
6.1 Microcanonical ensemble . . . 45
6.1.1 Reduced density matrix . . . 47
6.1.2 Thermalization . . . 49
6.1.3 Equilibrium, Ergodicity and Temperature . . . 51
6.2 Quantum Mechanical ensemble . . . 52
6.2.1 Canonical Typicality . . . 52
7 The Lieb-Mattis Model 57 7.1 Symmetry unbroken LM model . . . 57
7.1.1 Energy definitions . . . 59
7.1.2 Probability distribution system states . . . 61
Degeneracy corrections . . . 63
7.2 Symmetry broken LM model . . . 68
7.2.1 Energy definitions . . . 70
7.2.2 Probability distribution system states . . . 72
8 Conclusion 75
Appendix A: Defining Collapse 79
Appendix B: Differential expressions general time evolution 81
Appendix C: Independence fsof s 85
1
Part I
Spontaneous Unitary Violations in
Quantum Mechanics
3
1 Introduction
The indeterminate nature of quantum mechanics has fascinated physicists for over a century. Remarkably initial experiments hinting towards quantum mechanics - such as the first version of the double split experiment by Thomas Young in 1803 [20] -were introduced a century before the name "Quantenmechanik" was even coined by Max Born in 1924. The famous quote1 by Albert Einstein "God doesn’t play dice" in his letter to Max Born in 1926 [7], was in response to the now more widely ac-cepted interpretation of quantum mechanics called the Copenhagen interpretation, developed by Niels Bohr and Werner Heisenberg.
While the Copenhagen interpretation (described in subsection2.2.1) was and still is widely accepted among physicists, it does not address some inherent shortcom-ings of the traditional quantum mechanical framework collectively known as "the measurement problem". These shortcomings have fuelled the search for other in-terpretations of the wave function, of probabilities and of quantum mechanics in general, which is still an active field within physics today [4,16,19,28].
This work seeks to make new inroads into the measurement problem. The goal is to identify properties of objective collapse theories that are necessary and those that are to be excluded in modelling Quantum State Reduction (QSR). The reason specific properties should be excluded is that they may inherently make the model unable to fully solve the measurement problem.
In chapter2, the details of the measurement problem are stipulated and previ-ously proposed solutions critically reviewed. The following chapter3dives into the instability of the Schrödinger equation under linear non-unitary influences and how Spontaneous Unitary Violations (SUV) could serve as a model for QSR.
The Lieb-Mattis (LM) model is used throughout chapter3, but merely as an ex-ample of quantum Spontaneous Symmetry Breaking (SSB). PartII is wholly dedi-cated to this model, but in the context of thermalization and must therefore be seen as entirely separate from partI. PartIIhas its own introduction in chapter5.
1This is somewhat misquoted, the full quote is “Quantum theory yields much, but it hardly brings
5
2 Interpretations of Quantum
Mechanics
Before it is possible to address the proposed solutions to the measurement problem, it is important to become familiar with the essence of the problem. The measurement problem will therefore be made explicit in the following section2.1by means of an idealized (Von Neumann) measurement description. The most prominent interpre-tations of traditional quantum mechanics which "solve" the measurement problem are discussed subsequently. This serves partly as an overview for those less familiar with the problem, but also includes some original research in subsection2.2.2, which turns out to be crucial for the general conclusions of partI. Only local, causal inter-pretations are considered. Bell’s theorem excludes local hidden variable theories and global hidden variable theories such as the Pilot Wave theory are also omit-ted from this work. Finally non-traditional quantum mechanical theories - which contain some modification of the Schrödinger equation (SE) - are analysed and dis-cussed in section2.3. This functions as a setup for the next chapter3, where the SE itself is re-evaluated.
2.1
The measurement problem
The framework of quantum mechanics has a few postulates[4]:
1. Every physical system S has an associated Hilbert SpaceHS, in which the states of the system are represented by normalized state vectors|ψi. Observables O
have associated operators ˆO, whose eigenvalues onare the possible outcomes when measuring O:
ˆ
O|oni =on|oni. (2.1) The main idea - with which the reader is probably familiar - is that the quantum mechanical system’s state is fundamentally described by its time-dependent1 wavefunction|ψ(t)i. All information about the state of the system is
incorpo-rated in the wave function. However, the system state does not allow definite predictions of measurement outcomes. Only through measurement itself can definite outcomes of future measurements be established. Pre-measurement only systems which are in an exact eigenstate of an operator can be said to have definite measurement outcomes.
1Assuming the Schrödinger picture for now, in which the wavefunctions carry the time dependence.
Assuming the Heisenberg picture in which the operators become time-dependent instead of the states changes the technical interpretation somewhat, but the key ideas remain the same.
2. Given2the system state|ψ(t
0)iat time t0, its time dependence is governed by the Schrödinger equation
i¯h d
dt|ψ(t)i =Hˆ |ψ(t)i, (2.2) which for time independent Hamiltonians ˆH means that the time evolution of the quantum state is expressed as
|ψ(t)i =e−i ˆH(t−t0)|ψ(t0)i. (2.3) This familiar relation describes the quantum state through time, but does not give deterministic predictions for the outcome of experiments. An important feature to notice is that the Schrödinger equation is linear, meaning that if
|ψ1(t)iand|ψ2(t)iare solutions to2.2, so is any linear combination|ψ3(t)i =
α|ψ1(t)i +β|ψ1(t)i, as long as the linear combination is still a normalized state: α2+β2 = 1. Both these features will be reconsidered in later parts of
this work.
3. When measuring an observable O, given the wave function3 at time t,|ψ(t)i,
the outcome in general is predicted by a probability distribution, not a sin-gle value. Assuming non-degenerate eigenvalues, the probability of finding a particular eigenvalue onis given by
P[on] = |hon|ψ(t)i|2. (2.4)
4. By measuring the system, regardless of the state at the time of measurement
|ψ(t)i, it will be reduced to the eigenstate|oniassociated to the measured value onof the observable O
|ψ(t)ibefore measurement→ |oniafter measurement. (2.5) Note in particular that this means that if |ψ(t)iwas a superposition of
eigen-states, e.g. |ψ(t)i =1/
√
2|oni +1/ √
2|omi, it is reduced to only one of them. This process is called Quantum State Reduction (QSR).
In what follows, I will describe an "ideal" measurement, named after John von Neu-mann [15], which serves to clarify how the postulates above are contradictory or at the very least insufficient to form a coherent framework of quantum mechanics. As before, I start with a (microscopic) system S and an observable O. For simplicity I will assume a two-state system which has eigenvalues "up" ouand "down" od. I will now introduce a macroscopic measurement apparatus A. The apparatus has mutu-ally orthogonal pointer states, which can be thought of as literal pointer positions, pointing "up"|Aui, "down"|Adior towards its starting position denoted by|A0i.
The third (2.4) and fourth postulate (2.5) tells me that if the system is initially in an eigenstate (of the operator associated to the observable being measured), it
2A particular starting state is the unique vector in Hilbert spaceH
s, determined by the measured
values of observables, associated to the complete set of commuting operators.
3I will use the terms ’wave vector’, ’wave function’ and ’state (vector)’ interchangeably. This is
2.1. The measurement problem 7 will also be so after measurement. Because I actually read off the apparatus state to determine the outcome of the measurement, the apparatus state should be exactly correlated to the system state
|oni ⊗ |A0ibefore measurement→ |oni ⊗ |Aniafter measurement(n∈ {u, d}). (2.6) If the Schrödinger equation (2.2) describes all of nature, the measurement process should also be within its purview. Specifically using the linearity of the equation, I can now consider the measurement of the system in a superposition |u+di =
1/√2(|oui + |odi). The measurement should then produce |u+di ⊗ |A0i =1/ √ 2(|oui + |odi) ⊗ |A0i →1/ √ 2(|oui ⊗ |Aui + |odi ⊗ |Adi). (2.7) This is a macroscopic superposition of the apparatus A. As mentioned in the first postulate above, one can only assign properties to systems in an eigenstate of the operator of interest. Because the apparatus is not in a (position) eigenstate, I cannot assign a definite value to the apparatus (in this case its position). So at the end of the measurement, the apparatus is not pointing up or down. But this is exactly what is meant by measuring, that the measurement apparatus points up or down. It therefore seems that the last postulate (2.5) is at odds with the universal applicability of the Schrödinger equation (2.2).
Confronted with this disparity, one might be intuitively compelled to believe that the Schrödinger equation should only hold for microscopic (quantum) systems, not for macroscopic (classical) systems, which should be viewed as measurement apparatus. But this then invokes an immediate new question of the range of validity of quantum mechanics. When is a system too "big" and what does that mean? Is the mass the most important characteristic, or perhaps the number of degrees of freedom?
Taking the possibility of the presence of a spatial superposition to be the defining characteristic of a quantum system, it has been shown that systems up to 10−21g (around a thousand atoms) are still quantum. Definite classical behaviour in the form of absolute absence of a spatial superposition has been observed down to 10−6g ( 1018 atoms) [5]. This leave a huge 15 orders of magnitude range of meso-scopic systems for which quantum mechanics is untested, but within which it must break down.
By break down I mean that the quantum state no longer evolves deterministically governed by the Schrödinger equation but, through some form of quantum state reduction, produces measurement probabilities according to Born’s rule. This leaves three fundamental questions, which collectively are referred to as "the measurement problem":
1. When does the Schrödinger equation hold true? 2. What process describes the quantum state reduction?
3. How does this process generate probabilities, especially Born’s rule?
2.1.1 Entropy and probabilities
Classically, probabilities arise from indeterminacy related to uncertainty in our knowl-edge of the initial state. In other words, the system has a definite property with a
uniquely determinable value. Prior to measurement, one cannot know this value. Bell’s theorem excludes local hidden variable theories and thus the quantum indeter-minacy cannot be wholly related to a lack of knowledge about the state.
Let me quantify this in order to make it more concrete. To do this, I will use the concept of Von Neumann entropy
S≡ −Tr[ρLn(ρ)] (2.8)
= −
∑
iPiLn(Pi). (2.9)
In the first line ρ refers to the density matrix notation and in the second line Pi refers to the "probability" weights of the eigendecomposition ρ = ∑iPi|ii hi|. A pure, un-entangled state, can always be written in ket form, e.g.
|+i = √1
2(|oui + |odi). (2.10)
Using matrix notation for the basis vectors, |oiu ≡ 1
0 ,|oid ≡ 0 1 this can be written in density matrix notation as
ρ+= |+i h+| (2.11) =1/2(|oiuho|u+ |oiuho|d + |oidho|u+ |oidho|d) (2.12) = 1 2 1 1 1 1 . (2.13)
Because |+i is a pure state, its density matrix has (by definition) zero entropy S+ = 0. The Von Neumann entropy is the quantum mechanical extension of
Shan-non information entropy. ShanShan-non entropy can be understood as being a measure of lack of knowledge about a system. For example, if I know that a classical system is either up or down, with equal probabilities, my lack of knowledge about the sys-tem is maximal4 Sclass = Ln(2). This matches exactly with the entropy of a density matrix ρclass=1/2(|oiuho|u+ |oidho|d) (2.14) = 1 2 1 0 0 1 . (2.15)
Looking back at (2.7), one sees that measurement entangles the pure state|u+di
with the apparatus A. Defining the state after measurement as|ψi ≡ 1/
√
2(|oui ⊗ |Aui + |odi ⊗ |Adi), allows me to write the density matrix
ρψ= |ψi hψ| (2.16)
=1/2(|ou, Aui hou, Au| + |ou, Aui hod, Ad|
+ |od, Adi hou, Au| + |od, Adi hod, Ad|). (2.17) If I now look at what I know about the system S (to which the outcome on of the measurement of the observable O should correspond), von Neumann suggested [15] 4For any system the maximum possible entropy is Ln(N), where N is the dimension of the Hilbert
2.2. Traditional quantum mechanics 9 that in order to make predictions about this (sub)system, one should trace out the measuring apparatus
ρS =TrA[|ψi hψ|] (2.18)
=1/2(hAu|ψi hψ|Aui + hAd|ψi hψ|Adi) (2.19) =1/2(|oiuho|u+ |oidho|d). (2.20) Remarkably, the act of measurement seems to have made the reduced density ma-trix of the system ρsto be exactly equal to the density matrix of a classical statistical distribution ρclass! This apparently solves the origin of probabilities and even the appearance of Born’s rule. The mistake in this derivation is using the reduced den-sity matrix description. This is often regarded as a perfectly equivalent framework of quantum mechanics, but in taking the trace over all possible apparatus states one is actually averaging over multiple experiments. The decoherence framework states that this is inevitable and will treated in more detail in subsection2.2.2.
2.2
Traditional quantum mechanics
The measurement problem has been attempted to be solved in various ways. Some parts of the measurement problem can be answered through interpreting the tradi-tional (unaltered Schrödinger equation based) framework of quantum mechanics in a different way, without actually changing the equations. Below some of these inter-pretations are reviewed and their relation to the measurement problem summarized.
2.2.1 The Copenhagen interpretation
The Copenhagen interpretation is also known as the orthodox interpretation or more descriptively as wave function collapse. It makes a definite distinction between the quantum indeterminate nature of systems before measurement and the classical de-terministic nature they adhere to after measurement. The wavelike nature reduces instantaneously to a single determinate state, i.e. the wave function collapses. This clear difference between a collapsed state and a wave, without a process governing the transition, insinuates a tipping point, a scale at which the quantum-classical di-vide lies. The Copenhagen interpretation often does not concern itself with where this divide must exactly lie. The definition of an exact scale at which the divide should lie is deemed unnecessary, as the act of measurement is the fundamental cause of wave function collapse, not the scale itself. This implies an observer depen-dent description, as one experimenter might observe a particle, while another does not. This leads to untestable philosophical conundrums. Does a tree exist in a su-perposition between falling down and standing upright as long as no-one observes it? If so, who or what counts as someone? Is consciousness special and determinate in this regard? This fleeing to the fundamentally untestable realm is unsatisfactory for many physicists, who adapt a more agnostic attitude. According to them the divide must be somewhere, but it is not relevant for most experiments one would like to perform. Applying the theory to molecular, atomic and often even subatomic scales guarantees it will make accurate predictions. As a pragmatic approach, this is perfectly suitable in those regimes. New experiments are pushing the boundaries further and further however, making stable superpositions of increasingly large sys-tems [22,24,25]. These experiments require a fundamental mechanism to describe when the act of measurement - wave function collapse - will occur.
Summarizing, the Copenhagen interpretation answers the questions of the measure-ment problem as
1. The Schrödinger equation holds true as long as no measurement takes place, although what precisely constitutes a measurement is unclear.5
2. The process of quantum state reduction is postulated as wave function col-lapse, no dynamical description is given.
3. Born’s rule is a postulate.
2.2.2 Decoherence
The name decoherence refers to the loss of coherence of a quantum state. Put differ-ently, a pure state will decohere when entangling with another system, which might be the measuring apparatus, or the physical environment of the original system. A precursor to the decoherence framework was that of the statistical interpretation, succinctly defined by Ballentine: "The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain statisti-cal properties of an ensemble of similarly prepared systems, but need not provide a complete description of an individual system." [3]. In other words, this states that quantum mechanics is a complete theory, but only for ensembles. One therefore can-not know the process of single-particle wave collapse. As mentioned before, prag-matically this is sufficient, as one is often only interested in quantities depending on ensemble averages. But such quantities can be seen as the repetition of many single-particle experiments. Therefore there must be some underlying more fundamental process which is yet to be explained. Decoherence as solution to the measurement problem is a framework which claims the exact opposite, that there fundamentally do not exist single-particle experiments.
Decoherence assumes (realistically) that a measurement takes time. In the pre-vious definition of the measurement outcome (2.7), this is not into account. Specif-ically, I assumed that the pointer states were (by definition) exactly differentiable, in other words orthogonal: hAu|Adi = 0. Following Adler [1], I will repeat the de-scription of the measurement process, taking into account its finite duration and an environment E, which interacts with the measurement apparatus A, but not with the quantum system S directly. A general starting state for the combined system, apparatus and environment is then expressed as
|ψ(0)i =α|oui |A0i |E0i +β|odi |A0i |E0i. (2.21) At this point, I cannot read off the system state from the measurement apparatus, sincehA0E0|A0E0i = 1 means that the apparatus is in the same state regardless of the system. Applying unitary time evolution according to the Schrödinger equation 5For the pragmatically inclined "there is a divide somewhere" among the Copenhagen crowd,
mea-surement takes place if the apparatus used is clearly big enough to be classical. Meamea-surement does not take place if the apparatus is small enough to be itself considered a quantum system. (By a quantum system I mean a system that does not spontaneously undergo QSR when put into superposition, but instead remains in a superposition for timescales comparable to the age of the universe, as long as the system is kept isolated.) The crux is that the Copenhagen interpretation does not allow for "measure-ments" using an apparatus in the region of sizes in between these extremes.
2.2. Traditional quantum mechanics 11 (2.2) to this starting state (2.21) would produce, analogous to (2.7)
|ψ(t)i =α|oui |AuEu(t)i +β|odi |AdEd(t)i. (2.22) During the measurement, all particles in a causal region around the apparatus would have been able to interact with it. So any environmental particle within a distance of R=ct could in theory have interacted with the measurement apparatus. Depending on the final position of the pointer state of the apparatus|Auior |Adi, this interaction would entangle the environmental particles and the apparatus in a different way. The entanglement between a single environmental particle and the apparatus need not be much different depending on the final state of the apparatus. The measure to which the ith environmental particle differs after entangling with
|Auicompared to when it entangles with|Adiis expressed as hE
(i)
u |Ed(i)i ≤ 1. The closer to 1 this inner product lies, the less the states6of the environmental particle differ. The overlap will in general not be exactly 1 (a model for how this on average becomes less than one can be found in [12]). This means that the overlap of the full environment consisting of many particles
hEu(t)|Ed(t)i =
∏
ihE(ui)|Ed(i)i 1. (2.23)
The number of particles in the environment grows more than linear7 with time, so this product would exponentially decrease untilhEu(t)|Ed(t)i ≈0. In order to know what state the apparatus is in, I now need to trace out the environment
ρOA=TrE[|ψ(t)i hψ(t)|] (2.24)
=α2|oui |Aui hou| hAu| +β2|odi |Adi hod| hAd|. (2.25) But in doing so, I have taken an average over all possible environmental states. In essence, I take the average outcome of N experiments, with N the number of parti-cles in the environment. I am left with a superposition equal to a classical mixture of maximal entropy, to which I can associate classical probabilities.
Decoherence explains the appearance of classical probabilities by appealing to an ever-present environment, which could even consist of the internal degrees of freedom of the measurement machine. This environment becomes entangled with the quantum system during "measurement"8 and - according to the ensemble de-scription - needs to be traced out to predict the quantum system outcome. How-ever, this already presupposes the ensemble description to be fundamentally cor-rect. As stated before, this adherence to ensemble averages removes the possibility of describing a single measurement outcome. For a single experiment, the combined setup is in only one state. In this case that means the quantum system + measure-ment apparatus/environmeasure-ment is in a pure state. The Schrödinger equation preserves that purity through its unitary nature. The decoherence framework as a solution 6Here I talk about "the state" of the environmental particle as if it can be seen separate from the
apparatus, which is of course not exactly true for the case of entangled particles. But since I consider the measurement apparatus to be big and almost infinitely classical, this is a valid enough description that should intuitively be more clear.
7If one takes a causal sphere around the apparatus to act as the environment and have constant
density, the rate would go as∝ R3, so∝ t3.
8By "measurement" I mean the coupling of the quantum system to the (big) measuring apparatus,
which itself has many internal degrees of freedom or in turn interacts (entangles) with an external environment.
to the measurement problem introduces non-unitarity through averaging (tracing) over all possible states of the environment (2.24). This is the only reason that the inner product (2.23) appears, which is necessary in order for the off-diagonals to vanish and thus for QSR to take place.
Summarizing, the Decoherence interpretation answers the questions of the measure-ment problem as
1. The Schrödinger equation holds true always for pure states, but not for re-duced descriptions averaged over the possible states of the environment, with which the system has become entangled.
2. There is no need for a process of quantum state reduction. The system is said to be measured when a statistical average is taken over all possible realizations of the environment.
3. Born’s rule follows as a consequence of only considering ensemble averages a priori, not allowing single measurements.
Envariance
A popular addition to the traditional decoherence framework deals with the emer-gence of Born’s rule, in a different manner to taking a trace. The basis of this deriva-tion is the concept of environment-assisted invariance dubbed by Zurek [31] as en-variance. The key idea is that part of a state can be altered locally by one observer, while another observer who only has access to another part of the same wave func-tion would not be able to tell the difference. The wave funcfunc-tion is called envariant under any operation done by one observer that can be undone by another on a dif-ferent part of the wave function. Any envariant operation corresponds to an unob-servable quantity for both observers. Assuming causality to hold9boils down to the same assumption and in what follows I will use this instead.
As a reminder: entanglement is the property that a combined systemS E (S = system;E = environment) can be fully known, without the subsystemsS andE be-ing fully known separately to local observers. In other words, the density matrix
ρ = |S E i hS E |is pure, but the reduced density matrices are not for either observer.
In what follows, I assume|saiand|sbito be a priori equiprobable states of subsys-temS, i.e. neither is preferred over the other as a consequence of the symmetry of the Hilbert space. Concretely, one could think of two different positions in space, with no overlap in the wave function, so e.g. |sai = |x= −Li,|sbi = |x =Liand hx= −L|x= Li = 0. The important part of this definition is that the states them-selves are orthogonal10. This means that immediately after a measurement in the
S basis, I can be sure the subsystem S is in only one of these orthonormal states: P(|sai ∩ |sbi) =0. Similarly, the state of the environmentE is denoted|e1ior|e2i.
9Zurek argues that causality is a stronger demand than the concept of envariance. Although I do
not agree with this, assuming causality should at least preserve the argument.
10Technically it also matters that the states are orthonormal, as otherwise the states are not
equiprob-able. Remember that I want to find that the probability of an outcome only depends on the size of its weight, but if the states are not normalized, I might haveha|ai =2, whilehb|bi =1. This gets very confusing when calculating for example P(|sai) = |hsa| (λa|sai +λb|sbi)|2.
2.2. Traditional quantum mechanics 13 Below I will reconstruct the core of the argument by Zurek [31] on how to ob-tain Born’s probability rule for the outcome of a measurement without resorting to a trace operation. After arriving at his mathematical result (2.34), I will explain the crucial flaw in the reasoning he uses to interpret this result.
Two possible configurations of the initial state|ΨS Eiare
|Ii =λa|sai |e1i +λb|sbi |e2i (2.26) and
|I Ii =λa|sai |e2i +λb|sbi |e1i. (2.27) The operation which takes (2.26) to (2.27) (and back) is called a swap operation, which can locally be performed unitarily on a subsystem11. Therefore such an
oper-ation byE will be undetectable byS. Otherwise, a local action byE could influence the probability of finding12 |s1iatS, even though the two are spacelike separated. Denoting the probability of outcome|siimeasured byS on wave functionΨS E as
PΨSS E(|sii)this means that
PIS(|sii) =PI IS(|sii). (2.28) Proof: Prepare many copies of |Ii, say a 100, whileS andE are
at the same location. Now separateS andE so they are spacelike.
E now flips a coin. If heads, E applies a swap gate (acting only on their part of the wave function) to the last 50 copies of|Ii, turn-ing them into|I Ii. NowSmeasures the first 50 copies and denotes PIS(|sai)and PIS(|sbi)by their relative frequency. ThenSmeasures the last 50 copies and denotes PnewS (|sai)and PnewS (|sbi)by their rel-ative frequency. If PnewS (|sii) 6=PIS(|sii),Scan instantly detect that Eperformed a swap operation. This would enable faster than light communication, as the result of the coin flip would be transmitted from E to S instantly. As I assumed causality as an axiom, this must mean that (2.28) always holds.
Since it was just shown that
PIS(|sai) =PS(λa|sai |e1i) =PS(λa|sai |e2i) = PI IS(|sai), (2.29) this proves the independence of PIS(|sii)on|eji. The notation PS(λa|sai |e1i) with-out subscript Ior I I represents the probability of finding the eigenvalue associated to the system state |sai when performing a measurement in the S system, for any wave function which includes the term λa|sai |e1i. This dropping of the indices I and I I from the middle two terms in (2.29) is important! It represents the fact that the probability of the full state collapsing to a single term can only depend on the factors making up that term (assuming normalized states). The normalized state (la-belled by I or I I) does not matter for the outcome probabilities, only the term in the
11Think e.g. of spins, in which a local swap by the second subsystem (E) acts as 1
2×2⊗ 0 11 0 on the
full system.
12By "finding a state|φi", what I really mean is measure the eigenvalue associated to|φiand
reduc-ing the wave function to the sreduc-ingle state|φi. Of course one then has to assume all eigenvectors have distinct eigenvalues.
wave function we expect to find (e.g. λa|sai |e1i) and which measurement basis one uses (indicated byS,EorS E).
Let me return to the main question I am trying to answer: Given say |ΨS Ei =
|Ii, what determines PS(|sai)? I now know PS(|sai)is independent of the external state|ejiby causality and is also independent of the state|saiitself by assumption of distinct states. So there is only one factor left in the argument of PS(|sai) = PS(λa|sai |e1i) that could be responsible for its value, you guessed it, λa. So the probabilities of outcomes|siiare fully determined by the prefactors λi in the initial wave function. This is a big step in the right direction, but not quite Born’s rule yet. What is left to prove is that only the size of the weights|λi|plays a role and lastly that the probability is indeed exactly equal to the square of the size of these weights. The independence of the probabilities on the phase of λi is very easily shown by once more invoking causality. IfE would locally change the phase of their states
|eji →eiθ|eji, this would carry over to the terms in the full wave function|ΨS Ei,
ef-fectively changing the phase of λi. Should the probability of a particular outcome be affected by this phase rotation, this would mean the action ofE would be detectable byS, even ifS andE are spacelike separated. This cannot be and so proves phase independence of the probabilities.
It is now proven that measurement outcome probabilities only depend on the sizes of the initial weights. This means that a superposition of equal weights yields equal probabilities. So if I have13
|φi ≡eiφ1λ|sai |e1i +eiφ2λ|sbi |e2i +eiφ3λ|sci |e3i, (2.30) PφS(|sai) = PφS(|sbi) =P
S
φ(|sci).
Returning to a two-state system S and environment E, assuming the weights to be positive real numbers as phases do not affect the probabilities, I can write a general initial state as
|ψi =r m
N|e1i |sai + r n
N |e2i |sbi, (2.31) with m, n, N ∈ N. By choosing N large enough, I can approximate any weight with arbitrary precision. The probabilities PψS(|sii)are again independent of the external states|eji, so I am free to write these normalized(
hej|eji
=1)states in a different basis, where they are a sum of states with equal weights
|e1i ≡ 1 √ m m
∑
k=1 |Aki (2.32) |e2i ≡ 1 √ n n∑
k=1 |Bki, (2.33)The full initial wave function then reads
|ψi = r 1 N " m
∑
k=1 |Aki |sai + n∑
k=1 |Bki |sbi # . (2.34)This is the mathematical result Zurek arrives to in [31]. Each term in (2.34) has the same coefficient (1/√N) and therefore Zurek concludes each of these terms is 13Here I expanded the basis ofSandEto each include 3 orthonormal states, but all the previously
2.2. Traditional quantum mechanics 15 equally likely as the outcome of a measurement. The orthogonality of the envi-ronment states (hAi|Aji = hBi|Bji = δi,j and hAi|Bji = 0) guarantees the possible outcomes are disjoint, meaning the probability of finding a subset of them is the addition of their individual probabilities
P(|A1i ∨ |A2i ∨...∨ |Ami) =m·p. (2.35) Where p is defined as the probability of a specific single term in (2.34) being the resulting state. If the probabilities are normalized to 1, p = 1/N as there are N components. This then seems to have produced Born’s rule, as the frequency with which a term |Aki |sai that includes |sai will be the final state is dictated by the probability of any of the |Aki states being found in the final state, namely m/N according to (2.35). This is the full reasoning of Zurek and is very convincing at first sight.
However, as you might have noticed, after the basis changes (2.32) and (2.33) I dropped the upper index indicating which basis is used for measurement. The mea-surement apparatus must be configured to use a specific pointer basis. If this pointer basis corresponds toS E, the probability to measure a specific system-environment state will be
PψS E(|Ak, sai) =PψS E(|Bk, sbi) =1/N. (2.36)
Just measuring the environment gives
PψE(|Aki |sai) =PψE(|Bki |sbi) =1/N, (2.37)
this is insufficient to find the probability of finding the state|siiwhen measuringS however. Naively one might think that
m N = m
∑
k=1 PψE(|Aki |sai)=? PψS(|e1i |sai). (2.38)The crucial difference is the nature of the probabilities PψE and PψS above. The out-come of the experiment in which the measurement apparatus uses theE basis pro-duces as a final state one outcome |Aki |sai. Taking the sum over the probabilities of getting one specific outcome∑mk=1PψE(|Aki |sai)then just produces a classical en-semble of single outcome states. On the other hand, the outcome of the experiment in which the measurement apparatus uses theS basis14produces as a final state a quantum superposition of states|e1i |sai = (∑km=1|Aki) |sai. Thus, contrary to what Zurek claims in his paper [31], equation (2.38) does not necessarily hold and the decoherence framework still fails to produce the origin of Born’s rule.
Before the introduction of this derivation of Born’s rule by Zurek, the only evi-dent way to derive Born’s rule was by means of a non-linear theory. However, this result by Zurek does in no way assume non-linearity. Therefore, with this proof in hand, one could construct a linear objective collapse theory: a model that describes Quantum State Reduction (QSR) by evolution with a linear generator. The invali-dation above prompted a closer inspection of objective linear collapse models and it can be proven that in fact, they can never produce Born’s rule. This proof relies on the description of the QSR dynamics for a two-state system. The methods and 14The same argument holds when measuring in theS E basis. Measurement in any basis different
fromS does not guarantee to return the eigenvalues of the states in theS basis and therefore cannot be seen as an equivalent measurement.
results of this proof are discussed in section3.3of this work, for a full treatment, see chapter 6 of [14].
2.2.3 The Many Worlds interpretation
The Many Worlds interpretation takes the observer dependence of the Copenhagen and Decoherence frameworks as its foundation. The problem of not having observer-independent outcomes is addressed by stating that the moment the observer inter-acts with a superposition, they become entangled with the superposition and thus themself superposed. The observer only registers the part of the wave function they are entangled with. Let us take the observer to be a human called Billie for definite-ness, but many worlds does not differentiate between observers and environmental particles. Many worlds assumes the Schrödinger equation applies at all scales and throughout a "measurement". Denoting the state of Billie by |Bi, the outcome of experiment (2.22) as observed by them becomes
|ψ(t)i =α|oui |AuEuBu(t)i +β|odi |AdEdBd(t)i. (2.39) Billie themself is put into a superposition, only they do not notice this. Billie (their consciousness) lives happily in a universe (’world’) where they measured the appa-ratus to indicate "up"|Auiand also lives in a different universe where they measured the apparatus to indicate "down"|Adi. The decoherence between the environments in the two universes (two parts of the wave function) ensures there will be no inter-ference between the universes (Billie will never "meet" themself in the other universe to compare notes on the result of the experiment).
The answer "both" that the many worlds interpretation gives to the question of what the outcome of a measurement will be is unsatisfying, as it removes the ver-ifiability of any quantum mechanical prediction. This is not the only problem the many worlds interpretation has. Different theories for the origin of probability in many worlds have been proposed, most promising in the context of game theory where each branching off into a new ’world’ is seen as a game where the observer considers Born’s rule to determine in which ’world’ the observer will end up15[23]. No theory as of yet exists that explains the origin of Born’s rule in the many worlds framework though. For if the frequency with which an observer16 ends up in one world or another is to be according to Born’s rule, what mechanism ensures this? The argument used to explain this, as in the inventor of the many-worlds theory Everett wrote in his PhD thesis [8], applies the same faulty logic Zurek used in his envariance paper [31], debunked above in subsection2.2.2.
Summarizing, the Many Worlds interpretation answers the questions of the mea-surement problem as
1. The Schrödinger equation holds true always, no "measurement" ever takes place.
15By "ending up" in a certain world, being conscious in a certain world is meant. A copy of the
observer will be conscious in the other world.
16For now I’ll set aside the issue of what is meant by this, as in this interpretation there is no such
thing as "an" observer that can make multiple measurements. The moment a measurement is per-formed, the world splits and so does the observer. There is no possible way to predict "which" world or observer you will become in the future. This means the notion of probability as a likelihood of an event occurring in the future has become meaningless.
2.3. Non-traditional quantum mechanics 17
2. The process of quantum state reduction does not occur, instead the observer is put in orthogonal parts of a wave function.
3. Born’s rule is postulated, even an interpretation of its meaning in this context is lacking.
2.3
Non-traditional quantum mechanics
The failure of the interpretations of traditional quantum mechanics to resolve all as-pects of the measurement problem has led to theories that deviate from the Schrödinger equation (2.2). Such theories describing the process in time by which the off-diagonal components of the density matrix disappear are called objective collapse theories.
The construction of such theories is precarious, as the traditional quantum me-chanical framework is known to give such accurate predictions when tested in the microscopic regime. Any theory suggesting a deviation from the Schrödinger equa-tion should therefore in the microscopic limit give the same predicequa-tions. At the same time, a prerequisite of any universal theory will be to produce classical mechanics in the macroscopic limit. The experimentally differentiable part of any new theory will thus be found in the mesoscopic domain. The different behaviour at different scales means any universal theory should include an amplification mechanism.
The following subsections focus on the two most well-known of these type of theories, the GRW model and the CSL model(s). These theories add a non-linear term to the Schrödinger equation in order to reduce superpositions to a single state and keep it as such. The stochastic nature of collapse either comes from discrete random noise in the GRW model or from an ever-present noise field (random in space and time) in the CSL model.
2.3.1 GRW model
The first viable universal model of non-traditional quantum mechanics was pub-lished in 1986 and named GRW after its inventors Ghirardi, Rimini and Weber [10]. The main idea is to introduce a random noise, which amplifies or ’hits’ the wave function randomly in time and space. The theory relies on the position basis as being fundamental. A system consisting of N particles can then (suppressing in-ternal degrees of freedom for simplicity) be written as|ψ(t)i = |x1, x2, ..., xN(t)iat
time t. This is actually a product state of N non-interacting distinguishable particles
|ψ(t)i = |x1(t)i ⊗ |x2(t)i ⊗...⊗ |xN(t)i, making this a not very realistic model with
limited application. To be useful, the particles also need to be in a rigid body, so that the centre of mass uncertainty correlates to the position uncertainty of individual particles. As a very simple example, consider a 1D crystal with atoms of unit mass. The position of the ith atom is xi±σi. The centre of mass of the entire crystal can then be expressed as xcm ≡ ¯x±¯σ (2.40) = 1 N N
∑
i xi± 1 N v u u t N∑
i σi2. (2.41)From this one can see that the accuracy of the centre of mass position of the entire crystal goes up if the accuracy of the position of its constituent atoms goes up. In
other words, if one atom gets hit by the random noise, its wave function will collapse - its position uncertainty σi will exponentially tend to zero - reducing the centre of mass uncertainty ¯σ as well. If enough atoms get hit (consecutively) and collapse, the centre of mass uncertainty of the total system will have been reduced so much that it falls below the measurement accuracy and one excludes it to be in a macroscopic superposition. This will be how the amplification mechanism comes about17, but let me first focus on the localization process itself.
The discrete jumps caused by the (Gaussian) noise affect the total wave function according to |x1, x2, ..., xN(t)i → (πr2C)−3/4 e−(Xˆn−x)2/2r2C|x1, x2, ..., xN(t)i (πr 2 C)−3/4 e−( ˆ Xn−x)2/2r2C|x1, x2, ..., xN(t)i , (2.42)
for a random hit at position x [5]. Here ˆXn denotes the position operator of the nth particle and r
C is a parameter setting the width of the localization wave packet (rC ≈ 10−7m). In effect, some component of each particle wave function in|ψ(t)i
gets amplified, but any position away from the hit at site x gets effectively expo-nentially suppressed by the normalization. Since most particles will have negligible amplitude around the hit site to begin with, they will not localize and remain largely unaffected. The location of the hit site is stochastic in nature, with a probability dis-tribution depending on the wave function itself, which is precisely what makes the theory non-linear. Each particle in the system is considered distinguishable from all other particles, meaning their wave function is independent of (read: a product state with) the rest of the system. This means that for every particle labelled by n, the likelihood of the hit taking place at some position x is different, namely
pn(x) ≡ (πr 2 C)−3/4e−( ˆ Xn−x)2/2r2C|x 1, x2, ..., xN(t)i . (2.43) In between hits, the time evolution is governed by the Schrödinger equation. The occurrence of the hits in time is according to a Poisson distribution, which calls for the introduction of a second parameter λGRW ≈10−16s−1.
Although the GRW formalism uses the state vector representation for its time evolution, to see how it connects the microscopic to the macroscopic, it is useful to analyse it in terms of the density matrix. For one particle the density matrix evolves according to [10]
d
dtρ(t) = − i
¯h[H, ρˆ (t)] −λGRWT[ρ(t)]. (2.44) Here, T[·]corresponds to the collapse mechanism. This operator acts on position states according to hx|T[ρ(t)]|yi = 1−e−(x−y)2/4r2C hx|ρ(t)|yi. (2.45)
Of course, I am interested in systems containing many particles. To handle multi-particle systems, I rewrite the position operator of the multi-particles using centre of mass operator ( ˆ¯x) and relative position operator (ˆri) as ˆxi ≡ ˆ¯x+ˆri. For systems in which 17A different amplification mechanism can be construed by assuming rigidity. In that case the
lo-calization of one of the atoms fixes the location of all other atoms. The larger the material, the larger the possibility of one of them being randomly localized, so large objects localize quicker. For details of both mechanisms see the original GRW paper [10].
2.3. Non-traditional quantum mechanics 19 the Hamiltonian can be split into a centre of mass Hamiltonian HCM(ˆ¯x)and a relative Hamiltonian Hr(ˆri), I can then approximate (see [4] for details) the time evolution of the centre of mass density matrix (ρCM), analogous to (2.44) as
d dtρCM(t) ≈ − i ¯h[HˆCM, ρCM(t)] − N
∑
i=1 λiTCM[ρCM(t)] (2.46) ≈ −i ¯h[HˆCM, ρCM(t)] −NλGRWTCM[ρCM(t)]. (2.47) In the last line, the assumption is that all particles on average have approximately the same collapse rate (λi ≈ λGRW). This clearly shows the amplification mechanism at play, as now the non-linear part of the time evolution (TCM[ρCM]) scales with system size (N).To summarize, the GRW model proposes randomly occurring in time Gaussian noise "hits", which localize the constituents of the system independently. This in turn localizes the collective. The non-linearity of this model is linked to the proba-bility of a hit occurring for a single particle being dependent on the wave function of that particle (squared). This is somewhat unsatisfying as it in effect assumes Born’s rule. In that regard, this model does not fully solve the measurement problem. This inherent problem of non-linearity is not solved by Continuous Spontaneous Local-ization (CSL), but it does allow for the treatment of indistinguishable particles, as can be seen in the next subsection.
The GRW model answers the questions of the measurement problem as
1. The Schrödinger equation is incomplete and needs to be modified to include a non-linear component.
2. Stochastic localization events ("hits") amplify the components of the wave function randomly in space and time, causing one of them to dominate.
3. Born’s rule is silently assumed, dictating the probability distribution of hits occurring in space.
2.3.2 CSL model
The Continuous Spontaneous Localization (CSL) model introduced by Pearl in 1989 [17] is the continuous extension of the GRW model. In the first formulation of the theory, the amplification mechanism was dependent on system size. Later this was reformulated by Pearle himself together with Squires [18], placing mass at the heart of the amplification mechanism. Although the stochastic evolution is described us-ing the state vector formalism, I will focus on the time evolution of the density matrix as before. This is more insightful in general and specifically shows the vanishing of off-diagonal elements, corresponding with the reduction to a classical mixture. The
density matrix evolves in time as18
∂
∂thx|ρ(t)|yi = −
i
¯hhx|[H, ρˆ (t)]|yi −λCSLΓ(x, y) hx|ρ(t)|yi, (2.48) where x = x1, x2, ...xN·n and |xiidenotes the position eigenstate of the ith particle. There are N·n particles in total, n nucleons per atom and N atoms in the material, see FIGURE2.1for a 1D example. The first term in (2.48) is the normal unitary time evolution according to Schrödinger. In the second termΓ(x, y)is the collapse rate, defined as
Γ(x, y) = 1
2
∑
i,j F(xi−xj) +F(yi−yj) −2F(xi−yj) , (2.49) with F(z) =e−|z|2/4r2C.For a single particle x=x1and y=y1, so that the collapse rate becomes
Γsingle = 1
2[F(x1−x1) +F(y1−y1) −2F(x1−y1)] (2.50)
=1−e−|x1−y1|2/4r2C. (2.51)
This is exactly what was found in (2.45) for one-particle collapse in the GRW model. For one particle, a superposition of greatly separated position states (|x1−y1|
2rC) gives a maximal (approximately constant) collapse rate (Γ ≈ 1). On the other hand if the superposition is made up of two nearby position states (|x1−y1| 2rC), the collapse rate becomes negligible (Γ≈ |x1−y1|2/4r2
C ≈0). This behaviour (faster collapse for a superposition of wider spatial separation) hints at a useful large-distance approximation in which the collapse due to near particles is neglected.
The large separation approximation entails the crystal to be in a mesoscopic su-perposition of two position states|xiand |yi19. Assuming the particles to be
nu-cleons grouped in atoms20 (assuming identical nucleons for simplicity and disre-garding electrons by virtue of their low mass), a simple working model can be as in FIGURE2.1.
18Here I have already made some assumptions/simplifications about the constituents of the
mea-surement machine, such as it being made up of all indistinguishable particles. This simplifies this expression to the point it can be understood intuitively, without compromising the key mechanism it describes. This is responsible for the sums over positions in (2.49), which in a full treatment would be written as integrals.
19Of course, the superposition could include a component at another location|zi, but effectively
one only regards the overlap of the wave-functions, so the extension which includes any lingering component can be dealt with in the same manner subsequently.
20This assumption is only made in order to make the mathematical description discrete and therefore
easier, but a full treatment using integrations instead of sums over position and wave packets instead of delta function approximations would give the same qualitative behaviour.
2.3. Non-traditional quantum mechanics 21
FIGURE2.1: Large distance approximation CSL model for nucleons in a (1D) crystal. Each atom consists of a group of n indistinguishable particles (nucleons), there are N atoms, so N·n particles in total. The atoms are a distance a apart. This distance a is larger than the cor-relation length rC, so that particles within an atom contribute almost
nothing to the collapse, but particles in different atoms will. The po-sition state|yiis shifted a mesoscopic distance∆L compared to|xi.
Ordering the length scales: rC a ∆L L.
Having chosen 2rC as the reference length scale, this means that
|xi−xj| ( 2rC, ∀i, j : xi, xj ∈same atom 2rC, ∀i, j : xi, xj ∈/same atom |yi−yj| ( 2rC, ∀i, j : yi, yj ∈same atom 2rC, ∀i, j : yi, yj ∈/same atom |xi−yj| ( 2rC, ∀i, j : xi, yj ∈same atom 2rC, ∀i, j : xi, yj ∈/same atom.
The difference in the collapse rate lies in the number of terms for which these con-ditions hold. For separations between particles of the same state, there will be n(n−1) ≈ n2 combinations21 of nucleons in the same atom and N atoms. For
separations between particles of different states, the nucleons in an atom in x cor-respond to nucleons in a different atom in y. In that case, there will still be≈ n2 combinations of nucleons in the corresponding atoms, but there will be less than N such corresponding atoms. In the mesoscopic length difference ∆L there will be d ≡ ∆L/a atoms. The overlap as in FIGURE 2.1 will then therefore consist of N−2d=N−2∆L/a corresponding atoms. Approximating F(z) ≈0∀z 2rCand F(z) ≈ 1 ∀z 2rC again, I can evaluate the collapse rate for the large separation 21This holds for indistinguishable particles, between distinguishable particles the wave function
fac-torizes and the collapse rate is much higher (with a collapse rate λGRW, which is an order of magnitude
larger than λCSL). Materials consisting of multiple types of particles (such as protons and neutrons)
approximation as Γf ar= 1 2
∑
i,j F(xi−xj) +F(yi−yj) −2F(xi−yj) (2.52) ≈ 1 2 n2·N+n2·N−2n2 N−2∆L a (2.53) ≈2n2∆L a . (2.54)In (2.53) I have assumed that the new position state is shifted exactly by an integer number of atom spacings a. The approximation F(z) ≈1∀z 2rC does no longer hold exactly if this shift is not an integer number of atom spacings. So I replace it by F(|xi−yj|) ≈ 1−e∀|xi−yj| 2rC, where e 1. So I replace N with N(1−e)in
the rightmost term of (2.53) and see that the collapse rate actually becomes Γf ar≈ n2(Ne+ 2∆L
a )∝ n
2N, (2.55)
meaning the collapse rate is proportional to the system size n2N. Here the amplifica-tion mechanism becomes visible: larger systems have a higher collapse rate. While this example used a rather particular 1D crystal, it is also possible to consider more general 3D materials. The result is similar in that the only contributing particles in a superposition of position states are the ones not in the overlap between the states. This might not be immediately obvious from (2.55), but even though the whole sys-tem n2N may seem to contribute, it is multiplied by e, a factor measuring the lack of overlap. For a general density function ρ(z) these effects (2n2∆L/a and en2N) are combined and expressed using the centre of mass positions of the two position states xcm and ycmas [9]
Γ(xcm, ycm) = (4πr2C)3/2
Z
dz
ρ2(z) −ρ(z)ρ(z+ |xcm−ycm|) . (2.56) For a homogeneous body of constant density ρ this collapse rate becomes
Γ= (4πr2C)ρnout, (2.57) where nout indicates the number of particles outside the overlap of the bodies cen-tered around xcmand ycm. Note that for a homogeneous body, in effect e=0. Taking this region to be of size 2∆L·L2(if the bodies are cubes) results in a scalingΓ ∝ L2. The collapse rate therefore grows quadratically with the number of particles in the body.
Although this model allows the treatment of (any combination of) distinguish-able and indistinguishdistinguish-able particles, the fundamental underlying assumption of the source of the QSR is no different from GRW. CSL still assumes the probability distri-bution in space of localization events to be equal to the square of the amplitude of the wave function. This can be seen from the exponential form of the decay function F(z)in (2.49). The reason it decays exponentially for far separated states is the un-derlying mechanism of Born distributed localization hits, just as in GRW. The main difference with GRW just comes down to making these localization events hit con-tinuously for infinitesimal timespans.
2.3. Non-traditional quantum mechanics 23
1. The Schrödinger equation is incomplete and needs to be modified to include a non-linear component.
2. Stochastic localization events ("hits") amplify the components of the wave function randomly in space, but continuously in time, causing one of them to domi-nate.
3. Born’s rule is silently assumed, dictating the probability distribution of hits occurring in space.
2.3.3 Stable collapse and the problem with non-linearity
Any non-linear model can assure stable collapse analogous to the Gambler’s ruin, with wave function amplitude being the equivalent of money. In the Gambler’s ruin, gamblers bet a fraction of their money each time step and a random gambler wins the pot. The game is over the moment all but one of the players have lost all their money and therefore no longer can bet a fraction of their money. Arguably, a wave function component’s amplitude can never become quite zero in this manner, as it is a continuous quantity. This means that no matter how close the wave function component’s amplitude comes to zero, it can still by the random outcomes of the game be amplified to become the largest component. For non-linear models this is not the case however. If all but one component come close to zero, the large compo-nent will forever be the largest compocompo-nent. This is so because the probability of the magnitude of a component to be amplified is proportional to the amplitude itself. So small amplitudes will become less likely to grow and large amplitudes more so.
For example, say there are two large, roughly equal amplitudes α ≈ β, then the
randomness of the process will ensure that at some time one of them becomes very small and the other one dominates, e.g. α β. The non-linear dependence on the
wave function component amplitudes will ensure the ratio α/β will be increasingly unlikely to diminish. If one of the components dominates in this way, with the other components less and less likely to ever grow large again, the state is assumed to be collapsed. For a more rigorous definition of a collapsed state, see appendixA.
This seems like a perfect way to ensure superpositions collapse into one state and stay there, and it is. In other words non-linearity of the QSR model ensures stable collapse. But as previously seen in GRW and CSL, the assumption of non-linearity is most often22 achieved by axiomatically assuming Born’s rule somewhere at the
foundation of the model. As Born’s rule is inherently non-linear, this also makes the model itself non-linear. This way of constructing a non-linear model can therefore never fully solve the measurement problem, as it does not solve the third fundamen-tal question (3), regarding the origin of Born’s rule.
A rigorous way to be absolutely sure that a new model of QSR does not have this flaw, is to consider an alteration to the Schrödinger equation that is linear, but still models quantum state reduction. Immediately this should give pause to the atten-tive reader, bringing to mind the Gambler’s ruin for linear models. This problem of unstable collapse (where states with vanishing amplitudes can re-emerge) is actu-ally one of time translation symmetry. The many-to-one map that QSR entails is ir-reversible and therefore defines a special moment in time (that of the measurement), which creates two time realms (before and after measurement). The model worked 22I have at the time of this writing not yet encountered any objective collapse model that does not
out in the next chapter3, introduced by Van Wezel [28], is the one I worked on for this thesis. It uses linear non-unitarity perturbations to the Schrödinger equation, which spontaneously break time translation symmetry and thus reduce quantum superpositions to a stable single state. Having the advantage of being linear makes this model promising to provide new solutions to the measurement problem.
25
3 Schrödinger instability
The root of the measurement problem lies in the time-reversible nature of the Schrödinger equation. Suppose that the Schrödinger equation always holds. After a measure-ment in the energy basis, the system is reduced from a superposition to an energy eigenstate|E0i. As postulated in the previous chapter (2.3), the time evolution of the system state can then be expressed as
|E(t)i =e−iE0t|E
0i. (3.1)
If I take a negative value for t, I should be able to write down the state before mea-surement. Taking t= −t0to indicate a time before measurement, I should be able to write
|E(−t0)i =eiE0t0|E
0i. (3.2)
However, it is immediately plain to see that this can never reproduce the super-position. The time-translation symmetry of the Schrödinger equation is broken by the measurement process. This highlights the need for a non-unitary component to be added to the Schrödinger equation in order to accurately describe the time evo-lution (during measurement). As an arbitrarily large non-unitary influence would cause large conservation of energy violations and worse collapse microscopic quan-tum systems into classical ones, I will introduce the necessary non-unitarity pertur-batively in section 3.2. An amplification mechanism will ensure that macroscopic superpositions are short-lived, similar to the amplification mechanism previously seen in the CSL model.
Macroscopically different behaviour caused by an infinitesimal influence break-ing a symmetry is not a new concept. In fact, it is commonly known as Spontaneous Symmetry Breaking (SSB). What is new is the concept of breaking time-translational symmetry spontaneously. Before diving into this, I will review the key concepts of SSB, using both a classical and a quantum mechanical context.
3.1
Spontaneous Symmetry Breaking
Any SSB is characterized by three features. In a condensed matter context these are: the singular nature of the thermodynamic limit(lim N → ∞); the existence of a "thin" spectrum of extremely low energy states, which become degenerate in the thermodynamic limit; and the presence of an infinitesimal order parameter field. The classical example in the following subsection follows [28].
3.1.1 Classical SSB
The canonical example of classical spontaneous symmetry breaking1is the one I treat here, a pencil balanced on its tip, see FIGURE3.1.
FIGURE3.1: A nearly balanced pencil. The limits of making the pen-cil infinitely sharp (b → 0) and perfectly balanced (θ → 0) do not commute, so that even the smallest deflection will tip over a sharp enough pencil. The centre of mass has coordinates(x, y, z). Adapted
with permission from [28].
In the ground state of the system, the pencil lies flat on the ground(z = 0). All the possible states which have z=0 (all directions in which the pencil might have fallen) are equivalent in energy. As the pencil is a classical object, only one of these states can be occupied at any given time. Before "collapsing" into one of these ground states, the upright pencil is in what is called a stable state. It is called meta-stable because the stability is dependent on a singular limit. What I mean by that is that the order2in which limits are taken matters.
1For many more great examples and an insightful treatment of the subject, see the short paper by
Anderson [2].
2For this classical example the order of the limits can be thought of as an ordering in time, but in
general, this is not the case. In general, the limit ordering should be thought of as a mathematical construct rather than consecutive actions.
