Three Interpretations for a Single Physical Reality

Transcription

1 Amsterdam University College Science Three Interpretations for a Single Physical Reality Supervisor Dr. Sebastian De Haro (UvA) Yolanda Murillo Reader: Prof. Dr. Henk De Regt (UvA) May 25, 2016

2 1 To Lia Den Daas

3 Contents 1 Introduction 3 2 Theories and Interpretations What is an interpretation? Comparing to Muller s account Theoretical vs. physical equivalence Three Interpreted Theories Copenhagen Interpretation States Physical Quantities Dynamics The Measurement Problem The Uncertainty Principle Many-Worlds Interpretation States Physical Quantities Dynamics The Measurement Problem Bohmian Mechanics States Physical Quantities Dynamics The Measurement Problem The Uncertainty Principle Theoretical and Physical Equivalence Theoretical equivalence Physical equivalence Conclusion 25 Appendices 25 2

4 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

5 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

6 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 c i 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 n x = 1, n y = 2 and n x = 2, n y = 1 the values for the energy are given by E = 5 π2 2 2mL : two different states share the same 2 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

7 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 (QM 0 ), 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 QM 0 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 QM 0 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 T 1 := H 1, Q 1, D 1, and the triple of theory 2 to be T 2 := H 2, Q 2, D 2 then, on the unitary notion of equivalence, T 1 = T2 iff U, such that U is a unitary operator, and ψ i H i, Q i Q i (i = 1, 2), ψ 2 = U ψ 1 Q 2 = U Q 1 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 H C, 6

8 as well as the action of operators on states, are preserved: ψ 2 := Q 2 ψ 2 = U Q 1 U U ψ 1 = U Q 1 ψ 1 = U ψ 1 ψ 2 Q 2 ψ 2 = ψ 1 U Q 2 U ψ 1 = ψ 1 Q 1 ψ 1 (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 I T. 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 D T1 be the domain of the interpreterpretation of theory 1 and let C T1 be the codomain of the same interpretation. The interpretation map I 1 which corresponds to T 1 maps its domain into its codomain I 1 : D T1 C T1. Likewise, I 2 : D T2 C T2. As described before, there exists a U such that U : D T1 D T2. Let us assume that, in addition, there is an invertible map u : C T1 C T2 between the codomains, such that: M 1 (Ux) = u M 1 (x) (3) for x being either an element of the states x S or of the physical quantities x Q M 2 (Uy) = u M 2 (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

9 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 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, separable 2 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 x = x x. (6) 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 f g = fi g i (7) For infinite dimentional vector spaces the inner product is given by 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. i 8

10 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 map, from the Hilbert space to the system: I 0 Cop : H S could be added 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 as energy, speed, etc. This 0th map I 0 Cop 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 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 ψ i, this yields a number times the same vector. In other words, if Qψ i = q i ψ i, (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 a i is given by P (A = a i Ψ) = Ψ P i Ψ (10) 9

11 where Ψ = i c iψ i and P i is the projection operator that projects Ψ onto the space of A corresponding to ψ i. P i Ψ = c i ψ i. (11) For a one dimentional eigenspace the projection operator can be expressed as P i = ψ i ψi. The probability of getting outcome a i will then be given by P (A = a i Ψ) = Ψ ψ i ψi Ψ = ψ i Ψ 2 = c i 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 S, Q, D into a theory about a physical system S, its physical magnitudes, and its dynamics, i.e., an interpreted theory. For the Copenhagen interpretation, the first map is given by I 1 Cop : Q H R, and as follows: I 1 Cop : (Q, ψ i ) a i and a i 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 degeneracy of states. The second map of the interpretation is given by I 2 Cop maps I 2 Cop : Q H H R 0, as follows: ICop(Q, 2 ψ i, Ψ) c i 2. Here, ψ i is the eigenvector of A with eigenvalue a i. Alternately, we can view this second map as the composition of two maps: I 1 Cop and then a map (a i, Ψ) to the probabilities. I 2 Cop : C Cop H R 0 where C Cop is the codomain of ICop. 1 In conclusion, the second map relates, as desired, the physical quantities that are predicted to be observed a i, with the frequency c i 2 with which they are observed in the experimental outcomes. 10

12 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 t 0, 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 t = Hψ (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 The relation between H and U is given by ψ(t) = U ψ(t 0 ) (14) U = 1 ith t2 H it3 H 3... (15) 6 3 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 t 0 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 t 0 is described by a superposition: ψ(t 0 ) = α A + β B, (17) then such system will remain superposed after a time t (Section 2 [1]) ψ(t) = U ψ(t 0 ) = Uα A + Uβ B = αu A + βu B = α A + β B (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 this question is no. A new map I 3 Cop 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 ψ i would yield the energy of such state. As it has been shown in section the energy of a system is a physical measurable quantity and thus can be explained in terms of the map I 1 Cop. 11

13 As it has been mentioned, it is possible to say that the time evolution is deterministic because the future of a state at time t 0 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 one in 17 (Section 2.7 [20]). The map I 2 Cop described in section 3.2 maps the 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 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: ψ α φ0 (t 0 ) ψ α φα (t) (19) Where ψ α is the state of the system and φ 0 and φ α 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( ψ α φ0 (t 0 ) ) = ψ α φα (t) (20) where U is given by U = β,γ ψ γ φβ+γ φβ ψ γ Proof: U( ψ α φ0 ) = ψ γ φβ+γ φβ ψ α ψγ φ 0 β,γ = ψ γ φβ+γ δβ0 δ γα = ψ α φα. (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 )φ 0 ) = α A φ A + β B φ B (22) However, in real life, pointers of measuring devices only yield one value, φ and are never in a superposition of φ A and φ B. 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

14 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, our interpretational map I 2 Cop(Q, ψ i, Ψ) c i 2 does the job of collapsing the 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]) 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 σaσ 2 B 2 ] A, B 2i[ ) 2 (23) Where σ X 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 [ˆx, ˆp ] = i (24) σ x σ p 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 ways 3, 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

15 thus inaccessible by us, where the outcome of a probabilistic measurements that did not become actual in our world are actualized (Section 1 [27]) 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]). Ψ = α i ψ i, (26) where the different ψ i 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 of a 0th map I 0 Cop : H S. In MW this maps would be I 0 MW : 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 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

16 of the physical quantity is not probabilistic, relative to a world ψ i. A system S has the physical quantity Q with value q such that: Q ψ i = q ψ 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 W i (Section 3 [27]). This, sometimes called the branching postulate, constitutes the physical interpretation I T of this interpreted theory. The first map is now I 1 MW : 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 second map of the interpretation is given by I 2 MW maps I 2 MW : C MW H R 0 where C MW is the codomain of IMW. 1 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 C Cop = i C i W i, (28) for C i corresponding to the codomain of I 1 MW codomain is actual. and W i the world in which this 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 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 completely linear evolution of the map I 1 MW, 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

17 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 φ A + β B φb = SA + SB (30) The possibility of rewriting the S in other basis of the system, like for example S = 1 2 ( A + B )( φ A + φb )+ 1 2 ( A B )( φ A φb ) = S A + S B (31) 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 E 0 U(t)( S E0 ) SA EA (t) + S B EB (t) (32) for E A (t) E B (t) δ AB for t τ D and τ D represents the coherence time. The justification for this einselection (enviromental selection) is that S A and SB are pointer states. Pointer states are the set of states that are stable under time evolution: U(t) S i E = Si Ei (t) (33) As S 1 and S 2 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

18 3.3 Bohmian Mechanics The pilot wave interpretation was originally described by Louis De Broglie in 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] 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

19 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), Ψ c ) where Q(t) is the configuration space of such state at time t and Ψ c is the conditional wave function(section 5 [10]). The pair (Q(t), Ψ c ) has a phase space of R 3N H c. 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 τ D 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. The 0th map for this interpretation would be given by I 0 BM : H c R 3N S where H c represents the Hilbert space of the conditional wave function, R 3N is the configuration space of the states and H H c as required 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

20 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 would map I 1 BM : Q H c R R 3N. The justification of this maps is that in Bohmian mechancis the physical quantities i.e. the position of the particles live in configurational space R 3N 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 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]). dq i dt = m i Im Ψ t i Ψ t Ψ t Ψ t ( Q(t) ) where Q(t) = ( Q 1 (t),..., Q N (t) ) is the configuration of N point particles moving in a physical space R 3. 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, Ψ t can be determined by using the Schrodinger equation, and if Q(t 0 ) is specified, then the position of a particle at any later time t can be determined. Determining Q(t 0 ) 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 (37) 19

21 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(t 0, 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(t 0 ) of this system can be chosen, at random, to have probability density Ψ(t 0, 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 t = (ρv) (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(t 0 ) is chosen at random with probability density Ψ(t 0, 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 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: ψ α (x)φ 0 (y) ψ α (x)φ α (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]: Ψ o (x, y) = α c α ψ α (x) φ 0 (y) Ψ t (x, y) = α c α ψ α (x) φ α (y). (40) 20