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 Schrödinger 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, Schrödinger 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. A visualisation of the 8-site hopper. The dashed lines represent the path 17 ... t 18 03552 up to time = 4. Note that the shape of the dashed lines is not physically 19 meaningful. 20 21 22 of as summarising as much of the past as is necessary to make further predictions. From 23 23 then on, at each time step the hopper transfers from site j to k with amplitude 24 (j−k)2 25 ωn 26 U(n)jk := √ , (3) 26 n 27 28 where 29 ⎧ ⎪ 1 30 ⎨⎪ e for even n 31 2n ωn := (4) 32 ⎪ 1 33 ⎩ e for odd n n 34 35 i2πx th th and e(x):=e .Soforn odd, ωn is an n root of unity and for n even, ωn is a (2n) 36 37 root of unity. The form of U(n) is motivated by the propagator of a nonrelativistic free 38 particle of mass m in continuum 1-d space from xi to xf in time T , given by [2] 39 m m x − x 2 40 (free) i ( f i) K (xf ,T; xi, 0) = exp . (5) 41 i2πT 2T 42 43 Moreover, U(n) is a unitary n × n matrix, U(n)jk only depends on the distance between 44 sites rather than their absolute position, and U(n)j(k±n) = U(n)jk reflects the periodicity 45 46 of the ring. 47 The form of the model as a process gives the space of histories a temporal structure. 48 48 For a given time t ∈ N consider a finite path consisting of the first t+1 sites of a history. 49 50 We call such a truncated history a t-path γt. The set of all such t-paths, for a given t, t+1 51 is Ωt(n) with cardinality |Ωt(n)| = n . 52 Each t-path, γt, is associated to an event called a cylinder set which corresponds 53 54 to the physical statement “the hopper’s history agrees with γt for the first t + 1 sites.” 55 55 The cylinder set, cyl(γt), is formally defined by 56 57 cyl(γt):={γ ∈ Ω(n) | γ(t )=γt(t )for0≤ t ≤ t} . (6) 58 59 For any two cylinder sets, either they are disjoint or one contains the other. The 60 cylinder set for γt can be expressed as a disjoint union of cylinder sets of t -paths where Page 5 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 5 4 5 t >tin the following way. First, for the t-path γt, consider the (t + 1)-path which is 6 6 the extension of γt,bythesitej ∈ Zn at time t + 1. We write this extension as γtj: 7 8 γ t ≤ t ≤ t 8 t( )for0 9 γtj(t ):= (7) 9 j t t 10 for = +1. 11 11 The cylinder set of γt is the disjoint union of all such extensions of γt: 12 n−1 13 14 cyl(γt)= cyl(γtj) . (8) 15 j=0 16 16 Each cylinder set in this union can be similarly expressed as a disjoint union of cylinder 17 18 sets of extended paths, and so on. 19 We will call an event E which can be expressed as a finite union of cylinder sets 20 20 time-finite because we can determine whether any given history γ is an element of E 21 22 or not by only looking at the first (t + 1) values of γ for some finite t. For a time-finite 23 event, E,andeacht ∈ N we define a corresponding set of t-paths 24 25 Et := {γt ∈ Ωt(n) | cyl(γt) ⊆ E} . (9) 26 27 we also define the defining time tE, which is the least time t such that 28 28 E γ . 29 = cyl( t) (10) 30 γt∈Et 31 For example, consider the event E corresponding to the statement “The hopper is 32 32 t t 33 at site 0 at = 0 or at site 1 at = 1” for the 2-site hopper: 34 E = cyl(0) ∪ cyl(11) = cyl(00) ∪ cyl(01) ∪ cyl(11) . (11) 35 36 36 Then, we have E0 = {0} and E1 = {00, 01, 11} and the defining time tE =1. 37 The path amplitude a[γt]ofat-path is given by the product of the sequence of 38 39 transfer amplitudes and the initial amplitude 40 40 t−1 41 a[γt]:=ψγ U(n)γ t γ t . (12) 42 t(0) t( ) t( +1) t 43 =0 44 The decoherence functional of two t-path cylinder sets is given by 45 ∗ D γ , γ a γ a γ δ , 46 (cyl( t) cyl( t)) = [ t] [ t] γt(t),γt(t) (13) 47 47 where the Kronecker delta ensures the paths meet at their end points and is a feature 48 49 of unitary systems [5]. The decoherence functional D(E,F) of two time-finite events is 50 then given by 51 52 D(E,F)= D (cyl(γt), cyl(γt)) , (14) 53 γt∈Et γt∈Ft 54 55 where t =max{tE,tF } is the greater of the two defining times. 56 Lemma 1. The sum 57 58 D(E,F)= D (cyl(γt), cyl(γ )) , (15) 59 t γ ∈E γ∈F 60 t t t t is independent of t if t ≥ max{tE,tF }. AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 6 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 6 4 5 Proof. 6 ∗ 7 D E,F a γ a γ δ 7 ( )= [ t] [ t] γt(t),γt(t) (16) 8 γt∈Et γt∈Ft 9 10 n−1 10 ∗ ∗ a γ a γ U U 11 = [ t] [ t] γt(t),f γt(t),f (17) 12 γt∈Et γt∈Ft f=0 13 13 n−1 14 ∗ ∗ = a[γt]Uγ t ,f a[γ ] Uγ t ,f (18) 15 t( ) t t( ) f γ ∈E γ∈F 16 =0 t t t t 17 n−1 18 a γ a γ ∗ = [ t+1] [ t+1] (19) 19 f=0 γt+1∈Et+1|f γ ∈Ft+1|f 20 t+1 21 = D cyl(γt+1), cyl(γt ) , (20) 22 +1 γ ∈E γ ∈F 23 t+1 t+1 t+1 t+1 24 24 where 25 26 Et|f := {γt ∈ Et | γt(t)=f} . (21) 27 28 29 30 This gives us the quantum measure of a time-finite event E with defining time tE: 31 32 32 μ(E)= D (cyl(γt), cyl(γ )) (22) 33 t 33 γt∈Et γ ∈E 34 t t 35 2 35 n−1 36 = a[γt] , (23) 37 f 38 =0 γt∈Et|f 39 39 where f is the final site of the t-paths and t ≥ tE. 40 40 A n 41 Since the quantum measure of time-finite events is defined, the event algebra ( ) 42 for the n-site hopper certainly includes all time-finite events. If μ were a classical 43 43 measure it would have an extension from the semi-ring of time-finite events to the full 44 45 sigma algebra generated by the time-finite events and A would then be this sigma 46 algebra. However, the standard extension theorems cannot be used for quantum 47 47 measures and we do not yet know to which events the measure can be extended [11, 12]. 48 49 For the purposes of this paper we will only need to consider the time-finite events so we 50 will not address this question here. 51 52 53 2.1. Preclusions 54 55 In QMT, the quantum measure μ is viewed as a generalisation of a probability measure: 56 56 quantum theories are generalisations of classical stochastic processes. Classical theories, 57 58 in this view, are special cases of quantum measure theories.‡ In general, quantum 59 ‡ 60 Classical and quantum theories are respectively the first and second levels in a countable, nested hierarchy of measure theories characterised by how much interference there is between histories [9]. Page 7 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 7 4 5 interference means that the quantum measure does not satisfy the Kolmogorov sum 6 rules and cannot be interpreted as a probability measure. The question then is, what is 7 8 the physical significance of the quantum measure? The full answer is yet to be worked 9 out, but at the current stage of development QMT assigns a central role to the concept of 10 preclusion [19]: an event of zero measure is precluded (“almost surely does not happen”). 11 12 It is useful to adopt the term null event to mean an event of measure zero and the Law 13 of Preclusion is then: A null event is precluded. 14 14 Null events are therefore of central importance, so we will now investigate their 15 16 structure in the case of the n-site hopper. For each t-path γt we define the phase,a ˜[γt], 17 of the path by 18 t− 19 1 t/2 a γ n U n . 20 ˜[ t]:= ( )γt(t ),γt(t +1) (24) 21 t=0 22 22 Note that this definition of the phase of a t-path does not include the phase of the initial 23 24 amplitude but only depends on the structure of the path. Then 25 2 2 2 a γ ω (γt(0)−γt(1)) ω (γt(1)−γt(2)) ...ω (γt(t−1)−γt(t)) 26 ˜[ t]= n n n (25) 27 p 27 = ωn , (26) 28 29 29 where p ∈ Zn(n) and 30 31 n n 31 2 for even 32 n (n):= (27) 33 n for odd n. 34 Now, for each time-finite event E ∈ A,eachi, f ∈ Zn, p ∈ Zn n ,andeacht ≥ tE let us 35 ( ) 36 define 37 t p 38 sifp(E):=|{γt ∈ Et | γt(0) = i, γt(t)=f, a˜[γt]=ωn }| , (28) 39 39 |·| st E t 40 where is the cardinality of a set. In other words, ifp( )isthenumberof -paths in p 41 Et which begin at i,endatf and have phase ωn . 42 Recall that the quantum measure is 43 44 2 44 n−1 45 μ(E)= a[γt] (29) 46 f 47 =0 γt∈Et|f 48 t ≥ t t μ E 49 for any E.Thesumover-path amplitudes in the formula for ( )canbere- 49 t 50 expressed in terms of sifp(E): 51 n−1 n (n)−1 52 1 t p 53 a[γt]= ψi s (E)ωn . (30) 53 nt/2 ifp 54 γt∈Et|f i=0 p=0 55 55 E f 56 is null if and only if this sum is zero for each final site . 57 We use the fact that the roots of unity sum to zero 58 n−1 59 p e = 0 (31) 60 n p=0 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 8 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 8 4 5 to give the following identities 6 6 n−1 7 p ωn =0 foroddn (32) 8 9 p=0 9 10 2n−2 n−1 p 2m 11 ωn = e =0 forevenn (33) 12 2n 12 p=0,2 m=0 13 2n−1 n−1 14 p 1 2m ωn = e e =0 forevenn. (34) 15 2n 2n 16 p=1,3 m=0 17 n E i f st E p 18 Therefore, for odd , will be null if, for each and ,the ifp( ) are equal for all . 18 t 19 When n is even, E will be null if, for each i and f,thesifp(E) are equal when the p’s 20 p+n p 20 are of the same parity. However, this is trumped by the observation that ωn = −ωn 21 t t for even n.SoE will be null if s (E)=s (E) for all i, f, p ∈ Zn. 22 ifp if(p+n) 23 The events described above will be null no matter what the initial amplitudes 24 are, but there can also be events that are null depending on the initial amplitudes. If 25 ψ cz ω pi c ∈ C z p 26 i = i n for some ﬁxed and integers i and i then 27 n−1 n (n)−1 28 t p+pi a[γt] ∝ sifp(E)ziωn (35) 29 γ ∈E | i p 30 t t f =0 =0 31 and we see that it is possible that μ(E) = 0 even when the st (E)arenotequalfor 32 ifp 32 p ψ i 33 all . A trivial example of this would be if i =0forsome , then any event that only 34 contains histories that start at i would be null. 35 35 Let us look at some examples. For the 2-site hopper, consider the event 36 37 37 A = cyl( 000 ) ∪ cyl( 010 ) . (36) 38 p=0 p=2 39 40 Then tA =2, 41 s2 A s2 A 42 000( )= 002( ) = 1 (37) 43 s2 A A 44 and all other ifp( )=0.So is null for any initial amplitudes. 45 For the 5-site hopper, the event 46 47 B = cyl(01203 ) ∪ cyl(00103 ) ∪ cyl(00203 ) ∪ cyl(00123 ) ∪ cyl(00003 ) (38) 48 p=0 p=1 p=2 p=3 p=4 49 49 t 50 has B =4and 51 51 s4 B ∀ p ∈ Z 52 03p( )=1 5 (39) 53 53 with all other s4 (B)=0.SoB is null for any initial amplitudes. 54 ifp 55 As a ﬁnal example, in the 3-site hopper the event 56 56 C ∪ ∪ 57 = cyl( 000 ) cyl( 010 ) cyl( 120 ) (40) 58 p=0 p=2 p=2 59 has tC =2and 60 s2 C s2 C s2 C , 000( )= 002( )= 102( )=1 (41) Page 9 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 9 4 2 5 with all other sifp(C) = 0. Let the initial amplitudes satisfy ψ0 = ω3ψ1. The only final 6 site for the 2-paths is f = 0. So the relevant sum is 7 8 a γ ∝ ψ s2 ωp [ 2] i i0p 3 (42) 9 γ2∈C2|0 ip 10 ψ s2 ω2 ψ s2 ψ s2 ω2 11 = 1 102 3 + 0 000 + 0 002 3 (43) 12 12 = ψ (ω2 + ω + 1) (44) 13 1 3 3 14 =0 (45) 15 16 and so C is null. 17 18 19 3. Nothing Finite Happens in the Multiplicative Scheme 20 21 3.1. Multiplicative Co-events 22 23 The set of all events, A, for a quantum measure system has a natural Boolean structure 24 25 inherited from its being a subset of the power set of Ω. In QMT, possible physical 26 worlds correspond to “answering” maps 27 28 φ : A → Z2 = {0, 1} . (46) 29 30 If φ(A)=1(φ(A)=0)wesaythateventA is affirmed (denied) by the world φ. 31 Equivalently we can say that A happens (does not happen) in the world φ. 32 32 Such a map, φ, is called a co-event [16, 18, 20] as a reminder that the map takes 33 34 events to scalars, one or zero. A co-event is a possible physical world since it gives an 35 answer to every question that can be asked about the system, once the set of histories 36 37 is fixed. One might say, “the world is everything that is the case and nothing that is 38 not the case,” echoing Wittgenstein. 39 A co-event is considered classical if it preserves the Boolean structure of A.In 40 41 particular, it must be the case that the event “A and B” is affirmed iff A is affirmed 42 and B is affirmed, that “A xor B” is affirmed iff A is affirmed or B is affirmed but 43 43 not both, and that “nothing”= ∅ is denied and “something”= Ω is affirmed. Thus the 44 45 classical co-events follow our intuitive logic. Moreover, for finite Ω, it can be shown that 46 each classical co-event can be uniquely associated with a single history γ ∈ Ω, where 47 γ 48 physically is the history that happens. For example, for a ball passing through a left 49 or a right slit, a classical co-event could be associated with the history γ =‘right slit’, 50 which would affirm the event “the ball passes through the right slit”, deny the event 51 52 “the ball passes through the left slit” and affirm the event “the ball passes through the 53 rightorleftslit”. 54 Now, the Law of Preclusion implies that a possible co-event, φ, must satisfy the 55 56 condition 57 57 ∀ A ∈ A,μA ⇒ φ A . 58 ( )=0 ( )=0 (47) 59 59 When this condition holds we say that φ is a preclusive co-event. In classic stochastic 60 theory, since μ would be a probability measure and therefore satisfies the Kolmogorov AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 10 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 10 4 5 sum rule, it would always be possible to assign a classical co-event that is preclusive to 6 the system, provided the associated history does not have zero probability. However, 7 8 with a quantum measure it is possible to cover Ω with null events, which prevents the 9 assignment of a preclusive classical co-event to the system [16, 21]. 10 Instead, in QMT we seek a more general framework for choosing co-events using 11 12 the law of preclusion. However, this condition is not strong enough to give us all the 13 predictions we want to be able to make. In particular, in the special case that the 14 14 measure μ is classical we want to recover the usual classical co-events. To achieve this, 15 16 more conditions on the possible co-events are required. We refer to a complete set of 17 conditions to be imposed on co-events that defines the full set of possible worlds as a 18 19 scheme. 20 In the multiplicative scheme [17], the most closely studied scheme to date 21 [15, 18, 21], the possible co-events satisfy 22 23 φ(A ∩ B)=φ(A)φ(B) ∀ A, B ∈ A . (48) 24 25 The scheme is attractive for several reasons. It can be shown that the multiplicative 26 condition is, essentially, equivalent to the rule of inference known as modus ponens [22]. 27 27 It can also be shown that, in systems with finite Ω, each multiplicative co-event can be 28 29 uniquely associated with a subset of Ω, know as the support of the co-event. This gives 30 the co-events a corresponding partial order, making it natural to add to the scheme 31 32 the postulate that the allowed co-events are minimal in this order. If the measure is 33 classical, then the minimal multiplicative co-events correspond to single histories in Ω, 34 resulting in classical co-events. Further, Sorkin has shown that a minimal multiplicative 35 36 co-event will be classical when restricted to the algebra of macroscopic events with 37 permanent records. So, within the multiplicative scheme, “to the extent that one is 38 38 willing to posit the existence of sufficiently permanent records of macroscopic events, 39 40 one can [...] regard the measurement problem as solved” [18]. 41 Whilst the multiplicative scheme has enjoyed these successes, the previous history 42 42 spaces it had been applied to were small and had small fixed cut-off times. This is not 43 44 thecasewiththen-site hopper model presented in this paper, which allows us to expose 45 a problem with the multiplicative scheme. Any future scheme developed for QMT would 46 47 need to take this problem into account. 48 49 3.2. A “Problem of Time” 50 51 In a classical theory, if an event A is denied by the physical world (does not happen), 52 53 then any subevent of A is also denied. This property of the physical world also holds in 54 the multiplicative scheme: 55 56 Lemma 2. Let φ be a multiplicative co-event and F be an event. If φ(F )=0and 57 E ⊆ F , then φ(E)=0. 58 59 Proof. 60 φ(E)=φ(E ∩ F ) (49) Page 11 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 11 4 5 = φ(E)φ(F ) (50) 6 6 =0. (51) 7 8 9 10 11 In a classical theory this property is familiar and causes no difficulty. In QMT, 12 quantum interference results in many more null events than in classical theory and the 13 13 Law of Preclusion together with the above Lemma then vastly expands the set of events 14 15 that must be denied by all allowed co-events. It is useful to have a term for this: 16 if μ(A) = 0, so that A is precluded and B ⊂ A and μ(B) = 0, we say that in the 17 17 B A 18 multiplicative scheme is stymied by . The fact that in the multiplicative scheme 19 there are events of nonzero measure that must be denied by all allowed co-events turns 20 out to be problematic. 21 21 n 22 In the -site hopper, the system is extended arbitrarily in time into the future. The 23 space of t-paths, Ωt(n), grows exponentially with t and the number of time-finite events 24 24 grows as an exponential of an exponential. The store of events that could potentially 25 26 stymie any given fixed time-finite event therefore grows at a huge rate. In particular 27 the most striking result follows from this next theorem, which is a generalisation of a 28 29 theorem given in [12] for the 2-site hopper. 30 30 Theorem 1. For the n-site hopper with any choice of initial amplitudes, for every time- 31 32 finite event E where E ⊇ cyl(i) ∀ i, there exists a time-finite event F such that F ⊇ E 33 and μ(F )=0. 34 35 The consequence of Theorem 1 and Lemma 2 for the multiplicative scheme is that 36 36 all non-trivial time-finite events are stymied by a null time-finite event. The only time- 37 38 finite events that are not always denied correspond to statements of the form “the 39 hopper starts at site i or ...”. In fact, we shall see that for odd n there are certain 40 40 initial conditions for which the result can be extended to stymie these events as well. 41 42 We will now prove a number of lemmas before proving Theorem 1 and discussing 43 the implications of this result for multiplicative schemes. Recall that 44 45 n n 45 2 for even 46 n (n):= (52) 46 n n 47 for odd . 48 48 This will allow us to make statements that cover both even and odd n. 49 50 Lemma 3. For the n-site hopper with even n,ifat-path with t ≥ 1 begins at site i, 51 51 f ω p p ∈ Z 52 ends at site and has phase n where 2n, then 53 53 p + i + f ≡ 0(mod2). (53) 54 55 55 n n t γ ∈ n 56 Proof. Consider the -site hopper for any . Consider the -path t Ωt( ) with phase 57 p 57 a˜[γt]=ωn where p ∈ Zn(n).Then 58 59 t−1 59 2 60 p = [γt(k) − γt(k +1)] mod n (n) (54) k=0 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 12 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 12 4 4 t− 5 1 5 γ k 2 γ k 2 − γ k γ k n n 6 = t( ) + t( +1) 2 t( ) t( +1) mod ( ) (55) 7 k=0 8 t−1 9 2 2 9 = γt(0) + γt(t) − 2γt(0)γt(1) + 2 γt(k)[γt(k) − γt(k +1)] mod n (n) . (56) 10 10 k 11 =1 11 12 Now specialise to even n,son (n)=2n, and fix γt(0) = i and γt(t)=f.Wehave 13 13 2 2 14 p = i + f +2[...]mod2n. (57) 15 15 n 16 Noting that the mod 2 operation does not change the parity of the operand, we find 17 17 p ≡ i2 + f 2 (mod 2) (58) 18 19 and, using the fact that the parity of an integer is equal to the parity of its square 20 21 p + i + f ≡ 0(mod2). (59) 22 23 24 25 25 Corollary 1. For the n-site hopper with n even, for any time-finite event E,ifp ∈ Z n 26 2 26 i, f ∈ Z p i f st E ∀ t ≥ t 27 and n satisfy + + mod2=1, then ifp( )=0 E. 28 p 29 Given this result we can restrict the sum over in the measure calculation for even 29 t 30 n to exclude trivially zero sifp(E). Let us define 31 ⎧ 32 ⎨⎪ {0, 2,...,2n − 2} if n is even and (i − f)iseven 33 Zn,i,f := {1, 3,...,2n − 1} if n is even and (i − f)isodd (60) 34 ⎩⎪ 35 Zn if n is odd. 36 37 Then 38 n(n)−1 39 t t sifp(E) · (...)= sifp(E) · (...) . (61) 40 p p∈Z 41 =0 n,i,f 42 42 The next lemma shows that the number of paths in Et, with fixed phase and initial 43 t 44 and final sites, grows exponentially with . 45 Lemma 4. For t ≥ tE +2n − 1, i, f ∈ Zn and p ∈ Zn,i,f , 46 47 t t−tE −2n+1 sifp(E) ≥ ci(E)n , (62) 48 49 c E t E i where i( ) is the number of E-paths in tE that start at site . 50 51 p Proof. We will construct a path γt ∈ Et, with phasea ˜[γt]=ωn , that starts at site i 52 52 f 53 and ends at site , as a concatenation of three paths in three temporal regions as shown 54 in figure 2. The first region is from time 0 to tE, the second from tE to t − 2(n − 1), 55 and the third from t − 2(n − 1) to t.Now,forat-path γt to be in Et it must match 56 t γ ∈ E t 57 upwithoneofthe E-paths tE tE for the first E steps, but after this there are no 58 restrictions on it. 59 60 Page 13 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 13 4 5 i 6 ff 7 8 γ1 j γ2 γ3 9 10 0 tE t − 2(n − 1) t 11 12 t 13 Figure 2. A diagram of a -path across three temporal regions. The times are written 14 along the bottom, the height of the path specifies the site at that time. Drawn in bold 1 15 is a specific path in Et, in the first region on the left the path is given by γ from site 16 i to site j, in the middle region it is given by γ2 from site j to site f, and in the final 3 17 region it is given by γ from site f to site f. In grey are other choices for γ1 and γ2 18 18 that can be used to find a new path in Et. 19 20 21 1 Choose a site j ∈ Zn. In the first region, choose γ from among the paths in Et 22 E 23 that start at site i and end at site j.Letcij(E) be the total number of possible choices 24 for γ1. 25 2 In the second region, we choose γ to be any path starting at site j at tE and ending 26 27 at f at t − 2(n − 1). Between these two sites there are (t − tE − 2n + 1) time steps, and 28 at each one the path could go to any one of the n sites, so there are nt−tE −2n+1 such 29 30 paths to choose from. 31 In the third region γ3 starts at site f and ends at f, but we must choose it carefully 32 1 2 3 p such that γt, which is the concatenation of γ , γ and γ , will have a phase ofa ˜[γt]=ωn . 33 34 The phase is 35 35 a γ a γ1 a γ2 a γ3 . 36 ˜[ t]=˜[ ]˜[ ]˜[ ] (63) 37 37 Since we have already chosen γ1 and γ2 we have fixed 38 39 1 2 q a˜[γ ]ã[γ ]=ωn (64) 40 41 1 2 41 for some q ∈ Zn,i,f , since γ starts at i and γ ends at f. Therefore, to get a γt with the 42 42 γ3 43 right phase we need to choose the path such that its phase is 44 3 (p−q)modn(n) a˜[γ ]=ωn . (65) 45 46 For odd n,(p − q)modn could be any number in Zn. For even n, since both p and q 47 47 Z p − q n { , ,..., n − } 48 are in n,i,f ,( )mod2 could be any number in 0 2 2( 1) . 49 With this in mind, we construct γ3 in the following way. It starts by zig-zagging 50 50 between f → (f +1) → f a number of times and ends by staying at f for the remaining 51 52 time steps. The number of time steps in region 3 allows for up to a maximum of (n − 1) 53 zig-zags, the exact number of which will determine the amplitude of γ3 in the following 54 55 way 56 02+02+...+02 0 no zigzags :a ˜ [ ]=ωn = ωn (66) 57 12+12+02+...+02 2 58 one zigzag :a ˜ [ ]=ωn = ωn (67) 59 59 a ω 4 60 two zigzags : ˜ [ ]= n (68) AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 14 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 14 4 5 ... 6 2(n−2) mod n(n) n − 2 zigzags :a ˜ [ ]=ωn (69) 7 2(n−1) mod n(n) 8 n − 1 zigzags :a ˜ [ ]=ωn . (70) 9 10 Therefore, choosing such a form for γ3, we are able to obtain any phase in 11 ,ω ,...,ω n−1 n 12 2k mod n(n) 1 n n for odd ωn k ∈ Zn = (71) 13 2 2(n−1) 1,ωn ,...,ωn for even n 14 15 ω (p−q)modn(n) γ 16 including the phase n required to give t the phase we want. 16 p 17 In conclusion, the number of γt with phase ωn that can constructed in this way 18 equals 19 n−1 20 t−tE −2n+1 21 cij(E)n . (72) 22 j=0 23 c E c E st E 24 Deﬁning i( )= j ij( ) we can place the lower bound on ifp( ): the total number p 25 of t-paths in Et that start at site i,endatsitef, and have phase ωn 26 t t−tE −2n+1 27 sifp(E) ≥ ci(E)n . (73) 28 29 30 31 We will now prove a few lemmas that show that the unitary dynamics is periodic. 32 Lemma 5. The matrix U(n) for the n-site hopper has eigenvectors [vj], j ∈ Zn, with 33 34 components 35 e(jk/n) 36 [vj]k = √ ,k∈ Zn (74) 36 n 37 38 with corresponding eigenvalues 39 ⎧ 39 ⎪ 1 −j2 40 ⎪ e e for even n ⎪ n 41 ⎪ 8 2 42 ⎪ 2 42 ⎪ −j 43 ⎪ e for n ≡ 1 (mod 4) and even j ⎪ 4n 44 ⎨⎪ 45 −j2 46 λj = i e for n ≡ 1 (mod 4) and odd j (75) 46 ⎪ 4n 47 ⎪ ⎪ 2 48 ⎪ −j ⎪ i e for n ≡ 3 (mod 4) and even j 49 ⎪ 4n 50 ⎪ 50 ⎪ 2 51 ⎪ −j 51 ⎩ e for n ≡ 3 (mod 4) and odd j. 52 4n 53 54 Proof. First note that 55 (j−k)2 56 ωn 56 U(n)jk = √ (76) 57 n 58 ((j±1)−(k±1))2 ωn 59 = √ (77) 60 n

= U(n)(j±1)(k±1) . (78) Page 15 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 15 4 5 Therefore U(n) is a circulant matrix. All n × n circulant matrices have orthonormal 6 6 eigenvalues vj of the form [23] 7 8 e(jk/n) [vj]k = √ , (79) 9 n 10 11 where j, k run through Zn. Now, we will use this to derive the eigenvalues 12 12 n−1 n−1 13 1 (k−k)2 U(n)kk[vj]k = ωn e(jk/n) (80) 14 n 15 k=0 k=0 16 n−k −1 1 k2 17 = ωn e(jk/n)e(jk /n) (81) 18 n 18 k=−k 19 19 n− n−k− 20 1 1 20 1 (k−n)2 k2 = √ ωn e(j(k − n)/n)+ ωn e(jk/n) [vj]k . (82) 21 n 22 k=n−k k=0 23 23 ω n2−2nk n e −j 24 But n =1forbothevenandodd ,and ( )=1so 25 n−1 n−1 26 1 k2 26 U(n)kk[vj]k = √ ωn e(jk/n)[vj]k . (83) 27 n k=0 k=0 28 29 This implies 29 ⎧ 30 ⎪ n−1 31 ⎪ k2 +2jk ⎪ e for even n 32 ⎨ 2n 33 1 k=0 λj = √ (84) 34 n ⎪ n−1 34 ⎪ k2 + jk 35 ⎪ e for odd n. 36 ⎩ n 36 k=0 37 38 The ﬁnal sums are Gauss sums, which are number theoretic sums of the form 39 39 n−1 2 40 ak + bk 40 G(a, b, n):= e , (85) 41 n k 42 =0 43 where n is a natural number and a and b are usually integers or half-integers. Now, we 44 44 will use the Landsberg-Schaar relation for Gauss sums [24]. For any a, n ∈ N and b ∈ Z, 45 46 if an + b is even then ∗ 47 a b n 1 −b2 n b 48 G , ,n = e e G , ,a . (86) a an 49 2 2 8 8 2 2 50 For a proof to a similar relation that can be easily generalised to this see [25]. 51 Now, for even n we have 52 53 1 1 54 λj = √ G ,j,n (87) 55 n 2 55 56 1 −j2 n ∗ 57 = e e G ,j,1 (88) 57 8 2n 2 58 59 1 −j2 = e e . (89) 60 8 2n AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 16 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 16 4 5 For odd n we have 6 1 λj = √ G (1,j,n) (90) 7 n 8 2 9 1 1 −j n ∗ = √ e e G ,j,2 (91) 10 2 8 4n 2 11 ∗ 12 1 1 n j −j2 12 = √ e 1+e + e (92) 13 2 8 4 2 4n 14 ⎧ 14 ⎪ −j2 15 ⎪ e n ≡ j ⎪ for 1 (mod 4) and even 16 ⎪ 4n 17 ⎪ 17 ⎪ −j2 18 ⎨⎪ i e for n ≡ 1 (mod 4) and odd j 19 4n 19 = (93) 20 ⎪ −j2 ⎪ 21 ⎪ i e for n ≡ 3 (mod 4) and even j 22 ⎪ 4n 22 ⎪ 23 ⎪ −j2 24 ⎩⎪ e for n ≡ 3 (mod 4) and odd j. 24 4n 25 26 27 28 Lemma 6. 29 n ≡ 30 2n 1 for 0(mod4) 31 U(n) = (94) 31 −1 for n ≡ 2(mod4), 32 33 n ≡ n 1 for 1(mod4) 34 U(n) = (95) 35 −i1 for n ≡ 3(mod4). 36 37 Proof. First, notice that we can use the eigenvectors from Lemma 5 to form unitary 38 38 matrix [V ]withcomponents[V ]ij =[vi]j that diagonalises U(n). Therefore, if ∃ N ∈ N 39 39 λ N λ ∀ j U n N λ 40 such that j = ,then ( ) = 1. 41 Now, for n =4m, m ∈ N 42 42 8m 43 2n 1+i 2 λj = √ e −j (96) 44 2 45 45 =1. (97) 46 47 For n =4m +2,m ∈ N 48 m 49 8 +4 2n 1+i 2 50 λj = √ e −j (98) 51 2 52 = −1 . (99) 53 54 For n =4m +1,m ∈ N 55 ⎧ 55 ⎪ −j2 56 ⎨⎪ e for even j 57 n 4 λj = (100) 58 ⎪ −j2 ⎪ 4m+3 59 ⎩ i e for odd j 60 4 =1. (101) Page 17 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 17 4 5 For n =4m +3,m ∈ N ⎧ 6 2 ⎪ m −j 7 ⎨⎪ i4 +3 e for even j 8 n 4 λj = (102) 9 ⎪ −j2 10 ⎩ e for odd j 11 4 12 = −i . (103) 13 14 15 16 Corollary 2. For all n site hoppers 17 17 U n 4n . 18 ( ) = 1 (104) 19 19 We will now prove Theorem 1. 20 21 E n 22 Proof of Theorem 1. Let be a time-ﬁnite event for the -site hopper such that 22 ¯ ¯ 23 E ⊇ cyl(i) ∀ i.LetE := Ω(n) \ E be the complement of E.NotethatE is a time-ﬁnite 24 24 event and tE¯ = tE. We will construct a superset of E, F = E ∪ G,whereG ⊆ E¯,such 25 that μ(F ) = 0. We will do this by choosing a t>tE large enough and selecting t-paths 26 27 from E¯t with amplitudes that cancel the amplitudes in μ(E). G is then the union of 28 the cylinder sets of these selected t-paths. 29 30 We have E ∩ G = ∅ so for any t>tE 30 31 2 n−1 32 33 μ(E ∪ G)= a[γt]+ a[γt] . (105) 33 34 f=0 γt∈Et|f γt∈Gt|f 35 Notice that 36 37 a[γt]= a[γt− f] (106) 38 1 39 γt∈Et|f γt−1∈Et−1 40 n−1 41 41 = a[γt−1]U(n)jf , (107) 42 j=0 γt−1∈Et−1|j 43 44 44 where we used that Et−1 still contains all the histories needed to reproduce E because 45 (t − 1) ≥ tE. Now, applying (107) iteratively to (105) gives 46 47 2 47 n−1 n−1 48 48 μ E ∪ G U n t−tE a γ a γ . 49 ( )= ( ) jf [ tE ]+ [ t] (108) f j γ ∈E | γ ∈G | 50 =0 =0 tE tE j t t f 51 t t nm m ∈ N 52 Then we choose = E +4 for some , so we can use corollary 2 to replace 52 t−t 53 U(n) E with the identity: 54 54 2 55 n−1 56 μ E ∪ G a γ a γ 56 ( )= [ tE ]+ [ t] (109) 57 f γ ∈E | γ ∈G | =0 tE tE f t t f 58 2 59 n−1 n−1 t 1 sifp(G) 60 tE p = ψi sifp(E)+ ωn . (110) ntE n(t−tE )/2 f=0 i=0 p∈Zn,i,f AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 18 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 18 4 p 5 The number of paths in Et that start at site i,endatsitef and have phase ωn is 6 bounded from below by Lemma 4 7 8 t t−tE −2n+1 8 sifp(E¯) ≥ ci(E¯)n . (111) 9 10 Since E ⊇ cyl(i) for any initial site i, E¯ must contain paths that starts at site i,so 11 c E¯ ≥ i, f ∈ Z p ∈ Z st E¯ 12 i( ) 1. Therefore, for every n and n,i,f , ifp( ) will scale at least 12 t 13 as fast n . In particular, this means there will exist a finite m, and therefore a finite 14 t = tE +4nm,suchthat 15 16 t ¯ (t−tE )/2 tE tE sifp(E) ≥ n max sifq(E) − sifp(E) . (112) 17 q∈Zn,i,f 18 19 Given such a t, since G can be any subset of E¯, we can choose unique paths in E¯t that 20 p 20 start at site i and end at site f with phase ωn to construct a Gt such that 21 22 t (t−tE )/2 tE tE sifp(G)=n max sifq(E) − sifp(E) (113) 23 q∈Zn,i,f 24 25 for all i, f ∈ Zn and p ∈ Zn,i,f . Note that the right hand side is a non-negative integer 26 t 26 so this is allowed. Substituting sifp(G) into (110) gives 27 2 28 n− n− 1 1 1 29 μ E ∪ G ψ stE E ω p ( )= t i max ifq( ) n (114) 30 n E q∈Zn,i,f f i p∈Z 31 =0 =0 n,i,f 32 =0, (115) 33 34 where we used (32), (33) and (34) in the last step. 35 36 37 3.3. Null events dependent on the initial amplitudes 38 39 We will now show that, for certain initial amplitudes, it is even possible to stymie “initial 40 position” events, i.e. events E where E ⊇ cyl(i)forsomei for which ψi =0. 41 41 We first define the following set of initial sites 42 43 I(E):={i | E ⊇ cyl(i)} . (116) 44 i 45 For this scenario we will split our event E into the subsets E( ) defined by 46 46 E(i) {γ ∈ E | γ i} . 47 := (0) = (117) 48 (i) Note that for i ∈ I(E), E =cyl(i), and thus t (i) = 0. Therefore for all t ≥ tE 49 E 50 50 a[γt]= a[γt]+ a[γt] (118) 51 γt∈Et|f i∈I(E) γ ∈E(i)| i∈I(E) γ ∈E(i)| 52 t t f t t f 53 n−1 54 U n t a γ a γ , = ( ) jf [ 0]+ [ t] (119) 55 i∈I(E) j=0 γ ∈E(i)| i∈I(E) γ ∈E(i)| 56 0 0 f t t f 57 i 57 by applying (107) iteratively. Now, E( ) = {γ },whereγ (0) = i and so 58 0 0 0 59 n−1 60 a γ U n t ψ δ a γ . 60 [ t]= ( ) jf i ij + [ t] (120) γt∈Et|f i∈I(E) j=0 i∈I(E) (i) γt∈Et |f Page 19 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 19 4 5 And for further simplicity we can choose a t =4nm ≥ tE for some large enough m ∈ N 6 to make U(n)t = 1 via Corollary 2. Now, we want to find a null superset F ⊃ E. 7 7 F E ∪ G E ∩ G ∅ 8 Following the same method as before we split it into two parts = , = . i 9 Note that necessarily G( ) = ∅ for i ∈ I(E), therefore 10 10 n− 11 1 2 11 μ F ψ δ a γ . 12 ( )= i if + [ t] (121) 13 f=0 i∈I(E) i∈I(E) (i) 13 γt∈Ft |f 14 15 We will now restrict ourselves to an n-site hopper with odd n. Furthermore, we 16 will consider the ‘worst case scenario’ of an event E which does not contain all paths 17 17 that start at site ˇı, but does contain all paths that do not startatsiteˇı, such that 18 19 I(E)=Zn \{ˇı}. Note that if we can find a null superset for all such events then this 20 implies we can do so for every time-finite event E =Ω( n). In addition, we will choose 21 ψ cz ω qi c ∈ C z ∈ Z 22 initial amplitudes of the form i = i n for some (non-unique) , i and 23 qi ∈ Zn. Note that we could have generalised zi to be a rational number, but then we 24 could just write all the zi with the same denominator and absorbed it into c. Without 25 25 q c 26 loss of generality we will choose ˇı = 0 by changing . 27 Whilst maintaining t =4nm, we will also choose T =4nM, M>m, such that 28 28 U(n)T −t = 1. Then, as in Theorem 1’s proof, we can split the sum over path amplitudes 29 30 for Ft into two disjoint sums over Et and GT to eventually derive 31 n−1 n−1 st E sT G 2 32 ˇıfp( ) ˇıfp( ) p μ(F )= ψf (1 − δıf )+ψı + ωn (122) 33 ˇ ˇ nt/2 nT/2 34 f=0 p=0 34 35 2 35 n−1 n−1 st E sT G 36 ˇıfp( ) ˇıfp( ) p ∝ δpq (1 − δıf )zf + zı + zı ωn . (123) 37 f ˇ ˇ nt/2 ˇ nT/2 f p 38 =0 =0 39 39 If zˇı = 0 then we cannot find a G such that μ(F ) = 0, which makes sense since all the 40 G z 41 histories in would have zero amplitudes. We will now assume ˇı =0. 42 Now, as before, we must form G using the cylinder sets of paths in E¯T , and again 43 we want to choose a particular number of paths that start at ˇı,endatsitef and with 44 44 ω p sT E¯ 45 a phase of n , but this is restricted by the number ˇıfp( ). However we know from 46 Lemma 4 that this is bounded from below by 47 sT E¯ ≥ c nT −tE −2n+1 . 48 ˇıfp( ) ˇı (124) 49 50 Naively, one might use the arguments from Theorem 1’s proof to also argue that, 51 provided we choose a large enough T , we should be able to choose a G such that 52 52 z |z | 53 T (T −t)/2 t t T/2 f f sıfp(G)=n max sıfq(E) − sıfp(E) + n (1 − δıf ) (1 − δpq ) + (125) 54 ˇ q ˇ ˇ ˇ f 54 zˇı |zˇı| 55 55 f,p ∈ Z 56 for all n. However, such a choice would not be possible if the right hand side is 57 not a non-negative integer. The right hand side is non-negative but is not necessarily 58 z nT/2z f T nM 59 an integer, and so we find that ˇı must divide f for all for some =4 .Note 59 60 that if we satisfy this at some M then it remains satisfied at M >Mand so we are still free to make M larger. AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 20 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 20 4 T/2 5 Therefore, if zˇı divides n zf , we can make this choice for G, and we can use (32) 6 to show that μ(F ) = 0. Note that if we tried to apply this argument to the even n-site 7 7 sT G p ∈ Z 8 hopper we would have the further restriction that ˇıfp( ) = 0 necessarily if n,ˇı−f , 9 and so if qf ∈ Zn,ˇı−f we would not be able to satisfy the condition. 10 11 12 4. Discussion 13 14 Our result shows that the Multiplicative Scheme for the n-site hopper has a rather 15 16 severe “infrared” problem stemming from the existence of null events with arbitrarily 17 large defining times. To what extent does this indicate a problem with the Multiplicative 18 Scheme more generally? 19 20 One possibility for escaping the conclusion that no time-finite event happens is 21 to postulate that the universe is finite in extent in time. For the n-site hopper, this 22 22 translates into a time t at which the universe ends, so we are not free to look for 23 max 24 stymieing null events with defining times larger than tmax. Inspecting the proof of 25 Theorem 1 we see that to stymie an event E, we constructed a null event with defining 26 26 n t n t 27 time that was at least of order larger than E. So, for example, if max,thenitis 28 likely that this issue can be avoided. 29 Even if the “no time-finite event happens” problem can be alleviated in some 30 31 way, the stymieing of events by null events with later defining times is inherent in 32 the Multiplicative Scheme. Consider for example the 2-site hopper event, 33 34 E = cyl(001) ∪ cyl(0000) ∪ cyl(0101) (126) 35 ∪ ∪ ∪ 36 = cyl(0010) cyl(0011) cyl(0000) cyl(0101) (127) 37 3 3 3 3 with tE =3.Thenwehave,s = s = s = s = 1 which implies that E is 38 000 002 011 013 39 null. E stymies subevents with the same defining time, such as cyl(0000) ∪ cyl(0101). 40 It also stymies events of the form “E and then something”. An example is cyl(00101) ∪ 41 41 cyl(00111) ∪ 00001 ∪ cyl(01011) which corresponds to “E and then the hopper is at site 42 43 1 at time t = 4”. These are examples of physical inferences of a sort familiar from 44 classical physics. But stymieing also blocks events to E’s past, such as cyl(001). It 45 46 is this latter effect, this essentially global-in-time nature of the multiplicative scheme 47 which is disturbing, and it is not confined to the n-site hopper.§ 48 This could be an appealing result to anyone who thinks of the universe as a single, 49 50 simultaneously existing, spacetime block [26, 27], but for those who favour the concept 51 of Becoming, the stymieing of events in the past is in conflict with the idea of a 52 52 physical passage of time [28, 29]. Moreover, this feature makes physical prediction 53 54 highly impractical: to know what events are possible at early times we have to look at 55 all future events and their measures first. 56 57 § These problems are absent when the theory is classical because a classical (probability) measure 58 obeys Prob(G)=0⇒ Prob(E) = 0 for any G ⊃ E. Therefore, there is no stymieing since subevents of 59 a null event are null and precluded in their own right. 60 Page 21 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 21 4 5 One could argue that the n-site hopper is an exception because it is highly 6 6 symmetric and regular, which means that the number of t-paths in Ω(n)t grows 7 7 t n n 8 exponentially whilst the number of -path phases is fixed at ( ), resulting in many 9 null events. However, it seems plausible that there exists a larger class of models with 10 unitary transfer matrices, U, such that for every time-finite event E there exists a 11 11 F ⊃ E t 12 sequence of time-finite supersets k with increasing Fk , such that for any given 13 >0 we can find a K such that μ(Fk) < ∀ k ≥ K. The question of how to deal with 14 14 “approximate preclusion” is an open one in Quantum Measure Theory but for such a 15 16 situation, it would be hard to avoid the conclusion that E would be stymied. 17 To work towards finding such a class of models, consider a hopper on the same 18 n n 19 -spatial-sites lattice, with the same set of histories Ω( ) but with a general unitary 20 transfer matrix U of dimension n. And consider an event E.Thenfort>tE 20 21 n−1 22 a γ U t−tE a γ 23 [ t] = [ ]jf [ tE ] (128) 24 γ ∈E | j γ ∈E | 24 t t f =0 tE tE j 25 n− 26 1 ≤ U t−tE a γ 27 [ ]jf [ tE ] (129) 28 j γ ∈E | 28 =0 tE tE j 29 n− 30 1 ≤ a γ , 31 [ tE ] (130) 32 j γ ∈E | 32 =0 tE tE j 33 34 where we used that |Ujk|≤1 for a unitary matrix. So whilst in the n-site hopper 35 35 we could find times where the sum of amplitudes over t-paths in Et was unchanged, 36 37 for a more general unitary system this sum of amplitudes is bounded in magnitude 38 from above. And so one could potentially proceed along the same lines as the proof 39 for Theorem 1 by arguing that for a large enough t there are enough paths outside 40 40 E E 41 of with the right amplitudes to cancel ’s amplitude, and since the amplitudes are 42 becoming smaller in magnitude for longer paths, but not too small, we can make this 43 43 cancellation more and more precise to make the measure as small as we like. There is 44 45 reason to believe that one could show this using the fact that the quantum measure for 46 most unitary systems is not of bounded variation [12]. 47 47 There is another, related, aspect of the multiplicative scheme that seems 48 49 problematic: for unitary systems the only way that a history, γ ∈ Ω, can distinguish 50 itself dynamically from another, γ that ends at the same position, is via its amplitude. 51 51 γ γ 52 If the amplitudes of and are equal the dynamical laws that govern the allowed 53 co-events of the Multiplicative Scheme treat them the same. For example, given a null 54 event, E, we can replace any history, γ ∈ E with another history, γ ∈/ E with the same 55 55 E γ 56 amplitude to get a new null event, . And if is an element of the support of an 57 allowed multiplicative co-event, then it can be replaced by γ with the same amplitude 58 58 to give another allowed co-event. If the n-site hopper is a guide, there will always be 59 60 many more histories than there are amplitudes and there will be wildly different histories sharing the same amplitude. It seems that any scheme that treats all histories with the AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 22 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 22 4 5 same amplitude equally would struggle to provide an explanation of the emergence of 6 classical behaviour from quantum theory. 7 8 There are at least two possible avenues for revising the scheme: change the 9 preclusion law or give up multiplicativity. As a physical guide to how to make these 10 revisions we can look to the nature of the model as a temporal process and to the 11 12 heuristic of Becoming. For example, we could consider relaxing the strict Law of 13 Preclusion and instead adopt a new rule that null events that would stymie earlier 14 14 events are not precluded: “Prior Existence trumps Law.” Or, we could keep the 15 16 strict preclusion rule but drop the multiplicative condition in favour of an evolving 17 17 construction in which, for each t there is a time-finite co-event φt on the subalgebra, At 18 t 19 of events with defining time or less. This partial co-event corresponds to the physical 20 world of the hopper system up to time t. We build up the co-event step by step and 21 require that at each time t the partial co-event agrees with the previous partial co-event 22 22 A 2|Ωt| 23 on subalgebra t−1. Note that the number of potential partial co-events grows as 2 24 and there must be additional conditions on allowed co-events in order for the scheme to 25 be viable. This is under current investigation. 26 27 Finally, the n-site hopper was introduced as a discrete analogue of the free particle 28 in the continuum. We can ask to what extent this is more than an analogy: is there a 29 n 30 continuum limit for the -site hopper which gives the continuum propagator? Consider 30 t 31 the n-site hopper propagator, [U(n) ]if . Using the eigenvectors and values from Lemma 32 5, we have 33 ⎧ 33 n− 34 ⎪ t 21 i − f k − tk2 34 ⎪ 1 e e 2( ) n 35 ⎪ for even ⎪ 2n 8 2n 36 ⎪ k=0 37 ⎨⎪ n− 37 1 1 2(i − f)k − tk2 38 U n t e n ≡ ( ) if = for 1(mod4) (131) 39 ⎪ n n ⎪ k=0 40 ⎪ ⎪ n−1 41 ⎪ 1 t 2(i − f)k − tk2 ⎩⎪ e e for n ≡ 3(mod4). 42 n 4 n 43 k=0 44 These Gauss sums can be transformed to other Gauss sums over O(t) terms using the 45 46 Landsberg-Schaar relation given in Lemma 5. The values of these sums are not generally 47 known, however we can compare this to the propagator for a particle on a ring of radius 48 R 49 ∞ 50 (ring) (c) 1 (c) 2 K (φf ,t ; φi, 0) = exp i(φf − φi)k − i t k , (132) 51 2π 2mR2 52 k=−∞ 53 where φ is the angular position on the ring. This looks very similar in form to the 54 n 55 Gauss sums for the -site hopper. It remains to be shown if there is a way to take 55 t 56 the limit n →∞so that [U(n) ]if (appropriately rescaled) tends to the propagator on 57 the ring, or the free propagator on the line. If so, this would increase the relevance of 58 58 the n-site hopper. The decoherence functional or quantum measure in the continuum 59 60 is not yet able to be defined for events other than finite unions of events corresponding to finite sequences of projections onto position [5, 11, 30, 31]. If the n-site hoppers have Page 23 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 23 4 5 a continuum limit one could potentially use this to deﬁne the quantum measure of an 6 event in the continuum as the limit of a sequence of appropriate n-site hopper events as 7 7 n →∞ 8 . 9 10 10 Acknowledgments 11 12 13 We thank Rafael Sorkin for helpful discussions and the suggestion to use Gauss sums to 14 calculate powers of the transfer matrix. Research at Perimeter Institute for Theoretical 15 16 Physics is supported in part by the Government of Canada through NSERC and by 17 the Province of Ontario through MRI. FD is supported by STFC grant ST/L00044X/1. 18 HW is supported by the STFC. 19 20 21 References 22 23 23 [1] Paul Dirac. The Lagrangian in quantum mechanics. Physikalische Zeitschrift der Sowjetunion, 24 3(1), 1933. 25 26 [2] Richard Feynman. Space-time approach to non-relativistic quantum mechanics. Reviews of 27 Modern Physics, 20(2), 1948. 28 [3] James Hartle. The quantum mechanics of cosmology. In S. Coleman, J. Hartle, T. Piran, 29 and S. Weinberg, editors, Quantum Cosmology and Baby Universes: Proceedings of the 1989 30 Jerusalem Winter School for Theoretical Physics. World Scientiﬁc, Singapore, 1991. 31 [4] Jame Hartle. The spacetime approach to quantum mechanics. Vistas in Astronomy, 37, 1993. 32 32 arXiv:gr-qc/9210004. 33 33 [5] James Hartle. Spacetime quantum mechanics and the quantum mechanics of spacetime. In B. Julia 34 and J. Zinn-Justin, editors, Gravitation and Quantizations: Proceedings of the 1992 Les Houches 35 36 Summer School, North Holland, Amsterdam, 1993. arXiv:gr-qc/9304006. 37 [6] Rafael Sorkin. On the role of time in the sum over histories framework for gravity. International 38 Journal of Theoretical Physics, 33, 1994. Talk delivered at the conference, ”The History of 39 Modern Gauge Theories,” Logan, Utah, USA, July 1987 DOI:10.1007/BF00670514. 40 [7] Rafael Sorkin. Problems with Causality in the Sum Over Histories Framework for Quantum 41 Mechanics. In Conceptual Problems of Quantum Gravity. Birkh/”auser, Boston, 1991. 42 42 [8] Sukanya Sinha and Rafael Sorkin. A sum over histories account of an EPR(B) experiment. 43 43 Foundations of Physics Letters, 4, 1991. DOI:10.1007/BF00665892. 44 [9] Rafael Sorkin. Quantum mechanics as quantum measure theory. Modern Physics Letters A, 45 46 09(33), 1994. arXiv:gr-qc/9401003. 47 [10] Carlton Caves. Quantum mechanics of measurements distributed in time. a path-integral 48 formulation. Physical Review D, 33, Mar 1986. DOI:10.1103/PhysRevD.33.1643. 49 [11] Rafael Sorkin. Toward a “fundamental theorem of quantal measure theory”. Mathematical 50 Structures in Computer Science, 22(5), 2012. arXiv:1104.0997. 51 [12] Fay Dowker, Steven Johnston, and Sumati Surya. On extending the quantum measure. Journal 52 52 of Physics A: Mathematical and Theoretical, 43(50), 2010. arXiv:1007.2725. 53 53 [13] Robert Geroch. Path integrals. Unpublished notes at 54 www.perimeterinstitute.ca/personal/rsorkin/lecture.notes/geroch.pdf. 55 56 [14] Stan Gudder and Rafael Sorkin. Two-site quantum random walk. General Relativity and 57 Gravitation, 43(12), 2011. arXiv:1105.0705. 58 [15] Rafael Sorkin. Does a quantum particle know its own energy? Journal of Physics: Conference 59 Series, 442, 2013. arXiv:1304.7550. 60 [16] Rafael Sorkin. Quantum dynamics without the wave function. Journal of Physics A: Mathematical and Theoretical, 40(12), 2007. arXiv:quant-ph/0610204. AUTHOR SUBMITTED MANUSCRIPT - JPhysA-108413.R1 Page 24 of 24

1 2 2 n 3 A “problem of time” in the multiplicative scheme for the -site hopper 24 4 5 [17] Rafael Sorkin. An exercise in “anhomomorphic logic”. Journal of Physics: Conference Series, 67, 6 2007. arXiv:quant-ph/0703276. 7 [18] Rafael Sorkin. Logic is to the quantum as geometry is to gravity. In Jeff Murugan, Amanda 8 Weltman, and George Ellis, editors, Foundations of Space and Time: Reflections on Quantum 9 9 Gravity. Cambridge University Press, Cambridge, UK, 2012. arXiv:1004.1226. 10 [19] Robert Geroch. The Everett interpretation. Noûs, 18(4), 1984. 11 [20] Petros Wallden. The coevent formulation of quantum theory. Journal of Physics: Conference 12 13 Series, 442, 2013. arXiv:1301.5704. 14 [21] Fay Dowker and Yousef Ghazi-Tabatabai. The Kochen-Specker theorem revisited in quantum 15 measure theory. Journal of Physics A: Mathematical and Theoretical, 41(10), 2008. 16 arXiv:0711.0894. 17 [22] Fay Dowker, Kate Clements, and Petros Wallden. Physical logic. In S. Barry Cooper and 18 Mariya Soskova, editors, The Incomputable: Journeys Beyond the Turing Barrier. Springer 19 19 International Publishing, 2017. arXiv:1201.6266. 20 [23] Robert Gray. Toeplitz and circulant matrices: A review. Foundations and TrendsR in 21 Communications and Information Theory, 2(3), 2006. www-ee.stanford.edu/ gray/toeplitz.html. 22 23 [24] Bruce Berndt and Ronald Evans. The determination of Gauss sums. American Mathematical 24 Society, 5(3), 1981. 25 [25] William Coppel. Number Theory: An Introduction to Mathematics: Part A. Springer 26 Science+Business Media, Inc., New York, USA, 2006. 27 [26] Paul Davies. That mysterious flow. Scientific American Special Edition, A Matter of Time, 2002. 28 [27] Huw Price. The flow of time. In Craig Callender, editor, The Oxford Handbook of Philosophy of 29 29 Time. Oxford University Press, Oxford, UK, 2011. philsci-archive.pitt.edu/4829/. 30 30 [28] Rafael Sorkin. Relativity theory does not imply that the future already exists: a counterexample. 31 In Vesselin Petkov, editor, Relativity and the Dimensionality of the World, volume 153 of 32 33 Fundamental Theories of Physics. Springer, Dordrecht, The Netherlands, 2010. arXiv:gr- 34 qc/0703098. 35 [29] Fay Dowker. The birth of spacetime atoms as the passage of time. Annals of the New York 36 Academy of Sciences, 1326(1), 2014. arXiv:1405.3492. 37 [30] Fay Dowker, Steven Johnston, and Rafael Sorkin. Hilbert spaces from path integrals. Journal of 38 Physics A: Mathematical and Theoretical, 43(27), 2010. arXiv:1002.0589. 39 39 [31] Jonathan Halliwell and James Yearsley. Amplitudes for spacetime regions and the quantum Zeno 40 40 effect: pitfalls of standard path integral constructions. Journal of Physics: Conference Series, 41 442(1), 2013. arXiv:1205.3773. 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60