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In the Multiplicative Scheme for the N-Site Hopper

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In the Multiplicative Scheme for the N-Site Hopper Page 1 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 1 2 3 4 5 6 7 8 9 10 A “problem of time” in the multiplicative scheme 11 12 for the n-site hopper 13 14 14 1,2,3 4 5 15 Fay Dowker ,Vojtˇech Havl´ıˇcek , Cyprian Lewandowski and 16 Henry Wilkes1 17 17 1 Theoretical Physics Group, Blackett Laboratory, Imperial College, London, 18 SW7 2AZ, UK 19 2 20 Institute for Quantum Computing, University of Waterloo, ON, N2L 2Y5, Canada 20 3 21 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, 22 Ontario, N2L 2Y5, Canada 23 4 Quantum Group, Department of Computer Science, University of Oxford, Wolfson 24 Building, Parks Road, Oxford, OX1 3QD, UK 25 5 Department of Physics, Massachusetts Institute of Technology, Cambridge, 26 MA 02139, USA 27 28 E-mail: [email protected] 29 30 PACS numbers: 03.65.Ta 31 32 Abstract. Quantum Measure Theory (QMT) is an approach to quantum mechanics, 33 33 based on the path integral, in which quantum theory is conceived of as a generalised 34 stochastic process. One of the postulates of QMT is that events with zero quantum 35 measure do not occur, however this is not sufficient to give a full picture of the 36 37 quantum world. Determining the other postulates is a work in progress and this 38 paper investigates a proposal called the Multiplicative Scheme for QMT in which the 39 physical world corresponds, essentially, to a set of histories from the path integral. 40 This scheme is applied to Sorkin’s n-site hopper, a discrete, unitary model of a single 41 particle on a ring of n sites, motivated by free Schr¨odinger propagation. It is shown that 42 the multiplicative scheme’s global features lead to the conclusion that no non-trivial, 43 43 time-finite event can occur. 44 45 46 Keywords: quantum foundations, histories, quantum measure theory, co-event, discrete 47 48 propagation 49 50 50 1. Introduction 51 52 53 One motivation for reformulating quantum mechanics beyond the Copenhagen 54 interpretation is to understand quantum mechanics without observers or measurements. 55 56 Another is to bring quantum theory into harmony with relativity and the four 57 dimensional world view of General Relativity. Such a framework would be relevant for 58 quantum cosmology and could contribute to the construction of a theory of quantum 59 60 gravity. A promising starting point for a realist and relativistic formulation of quantum mechanics is the path integral or sum-over-histories approach to quantum mechanics, AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 2 of 24 1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 2 4 5 initiated by P.A.M. Dirac [1] and developed by R.P. Feynman [2]. In the approach, 6 for a given system, a history is the most complete description possible of the system 7 8 throughout spacetime. For a single particle on a fixed spacetime a history would be a 9 spacetime path, for two particles it would be a pair of spacetime paths and for a quantum 10 field in a fixed spacetime it would be a field configuration on that spacetime. The path 11 12 integral shifts the focus from the state vectors, Hilbert space, Schr¨odinger evolution, and 13 state vector collapse of canonical quantum mechanics to spacetime histories, spacetime 14 14 events and their amplitudes. This allows us to consider statements about the quantum 15 16 world without appealing to the notion of collapse and measurement, and without having 17 to perform the split of the universe into quantum system and classical measuring system 18 19 necessary in Copenhagen quantum mechanics. 20 The path integral approach to quantum foundations has been championed 21 particularly by J.B. Hartle and R.D. Sorkin and there is much common ground between 22 23 Hartle’s path integral version of Decoherent Histories [3, 4, 5] and Sorkin’s Quantum 24 Measure Theory (QMT) which conceives of quantum theory as a generalised stochastic 25 process [6, 7, 8, 9]. Quantum physics, in the path integral approach, is rooted in 26 27 spacetime. The dynamics and initial condition of a quantum system are encoded 28 in a Schwinger-Keldysh double path integral for the quantum measure [2, 6], or, 29 29 equivalently, decoherence functional [3, 9, 10] on spacetime events. Some frontiers of 30 31 current knowledge in the path integral approach to quantum theory are signposted by 32 more-or-less technical questions such as how to define the path integral for quantum 33 34 mechanics as a mathematically well-defined integral over path space, including both 35 “ultraviolet” problems associated with the continuity of spacetime (see e.g. [11, 12, 13]) 36 and “infrared” problems associated with questions involving arbitrarily long times 37 38 [11, 14]. The central conceptual question in QMT is: “What, in the path integral 39 approach to quantum theory, corresponds to the physical world?” 40 40 In Quantum Measure Theory, the concept of a co-event has been proposed as that 41 42 which represents the physical world. A co-event is an answer to every yes-no physical 43 question that can be asked about the world, once the class of spacetime histories has 44 44 been fixed. In this paper we investigate one proposal – the multiplicative scheme – 45 46 for the physical laws governing co-events. We work in the context of a one parameter 47 family of discrete unitary quantum models, the n-site hopper [11]. The discreteness can 48 49 be viewed as a ploy to sidestep the ultraviolet problems, referred to above, that arise 50 in trying to define the path integral in the continuum, or more positively as something 51 we may actually want to keep in the end. Indeed, Sorkin’s motivation for developing 52 53 Quantum Measure Theory is to employ it to build a quantum theory of gravity in which 54 continuum spacetime emerges from a fundamentally discrete substructure. 55 55 In section 2 we introduce the n-site hopper as a quantum system within Quantum 56 57 Measure Theory. We define the decoherence functional, the quantum measure and the 58 law of preclusion. In section 3 we define the concept of a co-event and the Multiplicative 59 60 Scheme. We prove the main result of the paper, that every non-trivial finite time event is a subevent of an event of measure zero. In the Multiplicative Scheme this implies Page 3 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 3 4 5 that no non-trivial finite time event happens. We discuss the implications of this result 6 in section 4. 7 8 9 2. The n-site Hopper 10 11 n 12 The -site hopper model, proposed by Sorkin in [11], is a unitary quantum model of a 13 particle on a ring of n>1 spatial sites i ∈ Zn, with discrete times steps t =0, 1, 2 .... 14 For more details on the 2-site hopper see [12, 14] and on the 3-site hopper see [15]. 15 15 , A,D 16 In QMT [16, 17, 18] a system is defined by a triple (Ω )whereΩistheset of 17 histories, A is the event algebra and D is the decoherence functional. An event is a set 18 of histories and the event algebra is the set of all events to which the theory assigns a 19 20 measure. The event algebra is, then, a subset of the power set of Ω. The decoherence 21 functional is a function with two arguments D : A × A → C such that 22 ∗ 23 • D(A, B)=D(B,A) ∀ A, B ∈ A ; 24 24 • A ,...,A ∈ A m × m 25 For any finite collection of events 1 m ,the matrix 26 Mab := D(Aa,Ab) is positive semi-definite ; 27 27 • D , 28 (Ω Ω) = 1 ; 29 • D(A ∪ B,C)=D(A, C)+D(B,C) ∀ A, B, C ∈ A such that A ∩ B = ∅ . 30 31 The quantum measure, μ(E), of an event E ∈ A is given by the diagonal of the 32 32 decoherence functional D: 33 34 μ(E):=D(E,E) . (1) 35 36 We will now define each of these entities for the n-site hopper. First, the spacetime 37 37 γ 38 lattice is fixed so a history for the system is simply a path, , the hopper can take on 39 the lattice. Each history is an infinite sequence of spatial sites which can be conceived 40 of as a function γ : N → Zn or as an infinite string of sites. For example, here is the 41 beginning of a particular history displayed in the two ways: 42 ⎧ 43 ⎪ t 44 ⎪ 0for=0 44 ⎪ 45 ⎪ 3fort =1 46 ⎪ 46 ⎨⎪ t 47 5for=2 γ(t)= ↔ γ = 03552 ... (2) 48 ⎪ 5fort =3 49 ⎪ 49 ⎪ 50 ⎪ 2fort =4 ⎪ 51 ⎩⎪ . 52 . 53 54 Figure 1 gives a visualisation of this history for the 8-site hopper. The history space, 55 Ω(n), is the set of all possible paths γ. 56 Each path γ that starts at site i at t = 0 will have an initial amplitude of ψi and 57 n−1 |ψ |2 58 i=0 i = 1. In the canonical approach the initial amplitudes are the initial state, 59 59 |ψ = ψi|i . However, in a sum over histories approach to quantum mechanics where 60 i 60 the focus is on the paths, not the state vector, these initial amplitudes should be thought AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 4 of 24 1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 4 4 5 5 4 6 7 8 6 3 9 10 11 11 7 2 12 13 14 0 1 15 16 Figure 1.
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