aa r X i v : . [ qu a n t - ph ] F e b On configuration space, Born’s rule andontological states
Hans-Thomas Elze
Abstract
It is shown how configuration space, possibly encompassing ordi-nary spatial structures, Born’s rule, and ontological states aiming to addressan underlying reality beyond Quantum Mechanics relate to each other inmodels of Hamiltonian cellular automata.This paper is dedicated to the memory of Walter Greiner. – Always Wal-ter shared freely his wise suggestions concerning topics worth exploringin physics and beyond, all the way to ‘career moves’ to be done by hisstudents. Yet, rarely did I follow his fatherly advice. – It happened nowand then during the weekly “Palaver” seminars that Walter quickly dis-missed foundational or interpretational issues as irrelevant for physics,no matter who dared to mention them. – Yet, quite recently, followingmy talk at Walter’s F ranfurt I nstitute of A dvanced S tudy, about newapproaches to understand Quantum Mechanics as emerging from some-thing ‘intelligible’ beneath, he encouraged me strongly to pursue theseideas . . . Hans-Thomas ElzeDipartimento di Fisica “Enrico Fermi”,Universit´a di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italiae-mail: [email protected]
It shall not be our concern to derive Quantum Mechanics (QM) from somehowphysically motivated and more or less parsimonious sets of axioms, such asrecent information theoretical reconstructions of QM [2, 8, 9, 10], nor to pro-pose yet another interpretation of QM, of which there are already too many(Copenhagen, Many Worlds, Qbism, . . . ) to sort out seemingly incompatibleaspects [15]. We will explore deformations of quantum mechanical models ,due to the presence of a finite discreteness scale l (of length or time, choosing c = ¯ h = 1, henceforth). And what can be learnt from them regarding basicconcepts of QM.Presently, we shall reconsider configuration space and the Born rule in re-lation to the foundational hypotheses introduced recently by Gerard ’t Hooft,in particular, the existence and relevance of ontological states of cellular au-tomata that may give rise to the reality described by physics [11].Our discussion will refer to the class of
Hamiltonian Cellular Automata (CA) [4]. See Ref. [6] for a wider perspective on striving for the understandingof QM as an emergent structure, the raison d’ˆetre for Hamiltonian CA, andnumerous references to earlier work.
The
Hamiltonian CA describe discrete linear dynamical systems that showquantum features – especially, but not only in the continuum limit [4, 5].These are deterministic classical CA with denumerable degrees of freedom(“bit processors”). The state of such a CA is described by integer valued coordinates x αn and momenta p αn , where α ∈ N labels different degrees offreedom and n ∈ Z successive states. – We mention that a generalizationstudied earlier consists in introducing also a dynamical discrete time coor-dinate together with its associated momentum, which must be distinguishedfrom the ‘CA clock’ time n [4].Only finite differences of variables can play a role in the dynamics here,where no infinitesimals or ordinary derivatives are available!Dynamics and symmetries of such systems are all contained in a suitable action principle [4], the consistency of which places severe constraints on itsdetailed form. Ultimately, this is responsible for the essential linearity of theQM models, which result in the continuum limit within this class of CA, aswe have shown earlier.This action principle yields finite difference equations of motion:˙ x αn = S αβ p βn + A αβ x βn , (1)˙ p αn = − S αβ x βn + A αβ p βn , (2) n configuration space, Born’s rule and ontological states 3 with given integer-valued symmetric and antisymmetric matrices, ˆ S ≡ { S αβ } and ˆ A ≡ { A αβ } , respectively, defining the model under consideration (sum-mation over repeated Greek indices applies, except where stated otherwise).Note that we introduced the notation ˙ O n := O n +1 − O n − . – This may guidethe eye and indicate an analogy with Hamilton’s equations in the continuum,suggesting the name Hamiltonian CA ; however, in distinction to usually first-order derivatives, we encounter second-order finite difference operators here.This leads to discrete time reversal invariance of the Eqs. (1)-(2); i.e. theequations are invariant under reversal of the updating direction. The state ofsuch a CA can be updated in both directions, ( n ∓ , n ) → ( n ± ψ αn = − iH αβ ψ βn , (3)together with its adjoint, by introducing the self-adjoint “Hamiltonian” ma-trix, ˆ H := ˆ S + i ˆ A , and complex integer-valued ( Gaussian integer ) state vari-ables, ψ αn := x αn + ip αn . This obviously looks like the Schr¨odinger equation,despite involving only Gaussian integer quantities.The resemblance between Eq. (3) and the Schr¨odinger equation in the con-tinuum, for a whole class of models, can hardly be accidental. Indeed, wehave constructed an invertible map between the discrete equation describingHamiltonian CA in terms of the variables ψ αn and a continuous time equationdescribing the same CA in terms of a complex “wave function” ψ α ( t ) [4]. Thishas been achieved with the help of Sampling Theory [12, 16], introducing thefinite discreteness scale l in terms of a bandwidth limit or high-frequencycut-off for all such wave functions.Thus, we obtain the Schr¨odinger equation, yet modified to incorporate apower series in higher-order time derivatives with l -dependent coefficients:2 sinh( l∂ t ) ψ α ( t ) = − i ˆ H αβ ψ β ( t ) . (4)This implies corresponding modifications of other standard results of QMand may have interesting phaenomenological consequences.Naturally, known features of QM are reproduced in the continuum limit, l →
0, of the deformation of QM implied here [5, 6].
There are l -dependent conservation laws in one-to-one correspondence withthose of the corresponding quantum mechanical model obtained for l → Hans-Thomas Elze • For any matrix ˆ G that commutes with ˆ H , [ ˆ G, ˆ H ] = 0, there is a discreteconservation law : ψ ∗ αn G αβ ˙ ψ βn + ˙ ψ ∗ αn G αβ ψ βn = 0 . (5)For self-adjoint ˆ G , defined in terms of Gaussian integers, this statementconcerns real integer quantities. • Rearrangement of Eq. (5) gives the related conserved quantity q ˆ G : q ˆ G := ψ ∗ αn ˆ G αβ ψ βn − + ψ ∗ αn − ˆ G αβ ψ βn = ψ ∗ αn +1 ˆ G αβ ψ βn + ψ ∗ αn ˆ G αβ ψ βn +1 , (6) i.e. a complex (real for ˆ G = ˆ G † ) integer-valued correlation function whichis invariant under a shift n → n + m , m ∈ Z . • For ˆ G := ˆ1, the conservation law states a constraint on the state variables: q ˆ1 = 2Re ψ ∗ αn ψ αn − = 2Re ψ ∗ αn +1 ψ αn = const . (7)The latter replaces for discrete CA the familiar normalization of state vec-tors in QM, to which it reduces in the limit l →
0. This follows from mappingthe discrete to the continuum version by applying Shannon’s reconstructiontheorem, to which we alluded above [4]. Here, we obtain:const = q ˆ1 = Re ψ ∗ ( t ) cosh (cid:2) l dd t (cid:3) ψ ( t ) (8)= ψ ∗ α ( t ) ψ α ( t ) + l ψ ∗ α ( t ) d d t ψ α ( t ) + O( l ) , (9)displaying the l -dependent corrections to the continuum limit, namely thenormalization ψ ∗ α ψ α = const, which is usually conserved but not in thepresent case of CA.In order to interpret the conserved ‘two-time’ correlation function q ˆ1 ofEq. (7), we recall that the greek indices abstractly label different CA degreesof freedom . These could relate to an internal space or to localized sites of aspatial structure. However, the entire formalism describing CA and demon-strating their quantum features, so far, is independent of attaching such atraditional meaning to the degrees of freedom. This observation is valid formultipartite CA as well [5, 6]. In this sense, the primacy of configurationspace appears naturally here. This will not lead us to immediate practical consequences or new theoret-ical insights, however, we speculate that the physics of matter in space or Mathematically speaking, the Gaussian integer wave functions ψ , with components ψ α , α ∈ N , can be seen as elements of a linear space endowed with an integer-valued scalar product, ψ ∗ ψ := ψ ∗ α ψ α , i.e. a unitary space. Taking its incompletenessinto account, the space of states can be classified as a pre-Hilbert module over thecommutative ring of Gaussian integers [5].n configuration space, Born’s rule and ontological states 5 in spacetime and the physics of space or of spacetime itself emerge togetherfrom underlying discrete structures and CA like processes (possibly of a kindtotally unexpected). Loosely speaking, some evolving complex aggregates ofCA degrees of freedom will show up as matter and some as space(time) andmust be linked in intricate ways.Experiencing effects of these phenomena only at scales some nineteen ormore orders of magnitude away from the Planck scale, say, QM plays the roleof an effective description that has been found most successful when appliedto atomic or subatomic particles and the forces that influence their motion.This should include, in principle, the description of macroscopic amounts ofsolid, liquid, or gaseous matter.However, since the early days of QM doubts shroud the beauty of thispicture where it is supposed to reflect, within QM, the whole process of ameasurement undertaken in a laboratory or elsewhere (let alone the alivephysicist involved). There has been a puzzling gap in understanding this measurement problem , unless one is willing to undersign one of the available“interpretations” of QM which eliminate this puzzle, at the expense of intro-ducing others [15] (for an overview and proposal of a solution, see [1]). Weshall come back to this issue when discussing ontological CA states.Besides that Eq. (7) replaces the normalization of state vectors, a necessaryingredient of the Born rule in QM, we can rephrase its content by relating itto a counting procedure as follows.Recall that ψ αn := x αn + ip αn . Thus, for a pair of successive states labelledby n and n + 1 and for each degree of freedom α , there is a pair of integercomponents x αn and p αn that may be called the (numbers of) x α - and p α - initials , respectively. Correspondingly, the integer components x αn +1 and p αn +1 are called the (numbers of) x α - and p α - finals , respectively. This includesthe possibility of zero or missing (ı.e. negative) numbers of initials or finals.Summing the product of x α -initials and -finals and the product of p α -initialsand -finals defines the number of α - links , L α : L α := x ∗ αn +1 x αn + p ∗ αn +1 p αn ∈ Z , (10)dropping the summation convention henceforth. Then, the total number oflinks is given by L := P α L α and Eq. (7) states: L = q ˆ1 / , (11) i.e. , the link number is conserved . – The effect of the CA evolution accordingto the equations of motion (1) and (2) is to change or redistribute the numbersof initials and finals over the available degrees of freedom, while keeping thetotal number of links constant. One may ponder the possibility that the counting of links and their conservationlaw could be generalized such that α -initials are linked to a finite set of α ′ -finals,including the α -finals. This could be pictured as a forward lighcone -like structure, Hans-Thomas Elze The relative weights , w α := L α /L ( L = 0), with P α w α = 1, generally, arenot confined to [0 , p α := ψ ∗ α ( t ) ψ α ( t ) /ψ ∗ ψ , since w α → p α for l →
0, in accordancewith Eqs. (7)–(9). – In distinction, the case L = 0 does not allow a mean-ingful continuum limit. We will encounter an example and its interpretationin Sec.4, introducing ontological states .Thus, we recover attributes of the Born rule . Of course, our descriptionof deterministic Hamiltonian CA, so far, says as little as quantum theoryabout the origin of the randomness of experimental outcomes . However, thehypothesis of ontological states and their generally statistical relation to the(pre-)quantum states of CA will add a new element changing this situation.
Ontological states that underlie quantum and, a fortiori , classical states ofphysical objects are central to the
Cellular Automaton Interpretation (CAI)of QM [11].There is ample motivation to reexamine the foundations of quantum theoryin perspective of essentially classical concepts – above all, determinism andexistence of ontological states of reality – which stems from observationsof quantum features in a large variety of deterministic and, in some sense,“classical” models [4, 5, 6]. – It is worth emphasizing that quantum stateshere are considered to form part of the mathematical language used, they are“templates” for the description of the “reality beneath”, including ontologicalstates and their deterministic dynamics.Finite and discrete CA may provide the necessary versatility to accommo-date ontological states and their evolution, besides (proper) quantum “tem-plate” states (especially in the continuum limit). The following general re-marks serve to obtain an operational understanding of what we are lookingfor in such models:
ONTOLOGICAL STATES ( OS ) are states a deterministic physical systemcan be in. They are denoted by | A i , | B i , | C i , . . . . The set of all states maybe very large, but is assumed to be denumerable, for simplicity.There exist no superpositions of OS “out there” as part of physical reality.The OS evolve by permutations among themselves, . . . → | A i → | D i →| B i → . . . , for example. Apparently this is the only possibility, besides considering spatial sites labelled by index α etc. Could there correspondingly existHamiltonians that would conform with locality in the usual special relativistic sense? The possibility of negative link numbers L α or weights w α reminds of “probabilities”falling outside of [0 ,
1] appearing in QM as discussed by Wigner, Feynman, and others;see, e.g. , the reviews by Khrennikov [13] and by M¨uckenheim et al. [14]n configuration space, Born’s rule and ontological states 7 producing a growing set of states or superpositions, which do not belong tothe initial set of OS .By declaring the OS to form an orthonormal set, fixed once for all, a Hilbertspace can be defined. – Operators which are diagonal on this set of OS are be-ables . Their eigenvalues describe physical properties of the OS , correspondingto the abstract labels A, B, C, . . . above.
QUANTUM STATES ( QS ) are superpositions of OS . These are templatesfor doing physics with the help of mathematics. – The amplitudes speci-fying superpositions need to be interpreted, when applying the formalismto describe experiments. Here the Born rule is built in , i.e. , by definition!By experience, interpreting amplitudes in terms of probabilities has been anamazingly useful invention. From here the machinery of QM can be seen to depart, incorporatingespecially the powerful techniques related to unitary transformations. Thelatter exist, in general, only in a rudimentary discrete form on the level of OS , due to the absence of superpositions.While quantum theory has been very effective in describing experiments,its linearity is the characteristic feature of the unitary dynamics embodiedin the Schr¨odinger equation. For a prospective ontological theory, it is ofinterest that QM remains notoriously indifferent to any reduction or collapseprocess one might be tempted to add on, in order to modify the collapse-freelinear evolution and, in this way, solve the measurement problem.Concluding this brief recapitulation of some essential points of CAI, inparticular how QM fits into this wider realistic picture, one more remark isin order, concerning the absence of the measurement problem : CLASSICAL STATES of a macroscopic deterministic system, including bil-liard balls, pointers of apparatus, planets, are probabilistic distributions of OS , since any kind of repeated experiments performed by physicists, withonly limited control of the circumstances, pick up different initial conditionsregarding the OS . Hence, different outcomes of apparatus readings must gen-erally be expected. Yet any reduction or collapse to a δ -peaked distribution,say, of pointer positions is only an apparent effect, induced by the intermedi-ary use of quantum mechanical templates in describing the evolution of OS during an experiment. Ontologically speaking, there were/are no superposi-tions, to begin with, which could possibly collapse [11]!This provides a strong motivation for pursuing an approach to understandreality as based on ontological states. It is indeed possible to change the proportionality between absolute values squared of complex amplitudes and probabilities into a more complicated relation, however,only at the expense of mathematical simplicity [11]. This linearity is reflected by the Superposition Principle and entails interference ef-fects and the possibility of nonclassical correlations among parts of composite objects, i.e. entanglement in multipartite systems. Hans-Thomas Elze
We recall that OS evolve by permutations among themselves. This is quitedifferent from the behaviour commonly found in QM, namely the dynamicalformation of superposition states (except for stationary states).In order to assess the formation, or not, of superpositions by an evolvingHamiltonian CA, we first remind ourselves that the Schr¨odinger equation isformally solved by: ∂ t ψ ( t ) = − i ˆ Hψ ( t ) ⇒ ψ ( t ) = e − i ˆ Ht ψ (0) , given the initial state ψ (0), and consider the analogous formal solution of theCA equation of motion (3), ˙ ψ n = ψ n +1 − ψ n − = − i ˆ Hψ n . In terms of anauxiliary operator ˆ φ , defined by 2 sin ˆ φ := ˆ H , one finds indeed: ψ n = 12 cos ˆ φ (cid:0) e − in ˆ φ [e i ˆ φ ψ + ψ ] + ( − n e in ˆ φ [e − i ˆ φ ψ − ψ ] (cid:1) . (12)where two initial states are required, ψ and ψ , corresponding to the fact thatHamiltonian CA are described by a second-order finite difference equation.With the help of the general solution (12), one obtains: ψ n = ˆ T ( n − m + 1) ψ m +1 + ˆ T ( n − m ) ψ m , (13)where ˆ T is a transfer operator that can be read off by comparing with theexplicit form of this relation. This generalizes the composition law for the uni-tary time evolution operator in QM. – Furthermore, the simple exponentialexpression for the solutions in QM can be recovered from Eq. (12) by takingthe appropriate limits n → ∞ and l →
0, keeping n · l fixed, and choosinginitial conditions such that ψ ≡ ψ . In this case, we have: ψ n = (cid:2) ˆ T ( n + 1) + ˆ T ( n ) (cid:3) ψ . (14)The Eq. (13) and especially Eq. (14) tell us to generally expect the formationof superposition states and, therefore, not ontological states which evolve bypermutations among themselves.While this seems to severely obstruct the search for OS from the outset, wenow present a first simple example illustrating that evolving OS are possiblein a two-state CA. – Consider the CA described by ψ αn , α = 1 , ψ n = ψ n − − i ˆ H ψ n − , ψ n ≡ (cid:18) ψ n ψ n (cid:19) , ˆ H := (cid:18) (cid:19) ≡ ˆ σ . (15)Furthermore, we choose two orthogonal initial states, ψ = (1 , t and ψ =(0 , t . By solving the equation of motion most simply by iteration, we obtain n configuration space, Born’s rule and ontological states 9 the following sequence of states: ψ , ψ , ψ = (1 − i ) ψ , ψ = − iψ ,ψ = − iψ , ψ = − (1 + i ) ψ , ψ = − ψ , ψ = − ψ , . . . , (16)which after four more steps begins to reproduce the initial pair of states.Here the normalization of the states, considered as if of vectors embeddedin a Hilbert space for a moment, changes dynamically. This would be a disas-ter in QM! However, for Hamiltonian CA this norm is not conserved. Instead,it is replaced by a conserved ‘two-time’ correlation function , cf. Eqs. (6)–(7),reproducing the norm conservation only in the continuum limit [4].Thus, apart from the change of normalization, the evolution here essen-tially swaps two orthogonal input states, once per updating step. This pro-vides a very simple example of a CA evolving OS , in agreement with CAI. OS Following the primitive example just given, the question arises, whether thereexists any generalization describing something more interesting.Besides systems with block diagonal ˆ H for multiple two-state components,one may try a higher-dimensional state space for generalizations of the modelof Eqs. (15)–(16). Indeed, we find easily that the Hamiltonians:ˆ H := − i i − i i , ˆ H := − i i − i i − i i , (17)for three- and four-state CA, respectively, lead to analogous evolution-by-permutation of OS as the previous example, ˆ H of Eq. (15). We may generallyconsider the m -dimensional state space with Hamiltonian:ˆ H m := − i · · · i − i · · ·
0. . .0 · · · i − i · · · i , (18)which works like the previous examples for ‘neighbouring’ pairs of orthogonalinitial states, ψ = ψ ( k ) := (0 , . . . , , , , . . . , t , with nonzero k -th entry(1 ≤ k ≤ m − ψ = ψ ( k +1) . The CA evolution with Hamiltonian ˆ H m , Eq. (18), does not change the nor-malization of these states, but can introduce phases ( ± , ± i ) when permutingthem. To give an explicit example, choosing ψ = ψ ( m − and ψ = ψ ( m ) ,the result of one updating step is ψ = ψ − i ˆ H m ψ = − iψ (1) . Such phasesare carried on by further updating steps until they are eventually cancelledand the initial configuration reappears, only after 4 m updates.Therefore, all the 4 m OS , which eventually differ by phases , must be con-sidered as different states here. Note that in x ′ + ip ′ ≡ ψ ′ = iψ ≡ i ( x + ip ),for example, the roles of coordinates (real parts) and momenta (imaginaryparts) are exchanged, x ′ = − p , p ′ = x . Such states cannot be seen as embed-ded in a projective space, which would be the case of normalized states inQM. Loosely speaking, they are ‘more classical’.All pairs of (initial) states of this kind have the conserved link number L = 0, cf. Eqs. (10)–(11). Whereas initial configurations with ψ ≡ ψ have L > cf. introduction of this Sec.4 and discussion of Eqs. (13)–(14). Thus,the conserved link number illustrates the ontology conservation law [11].The dynamics described by ˆ H m resembles the cogwheel model discussed inRefs. [3, 7, 11] – first introduced by ’t Hooft as a ‘particle’ making uniformjumps over fixed positions on a circle, one per fixed time interval – whichhas been shown to have surprising quantum features (providing a discreterepresentation of the quantum harmonic oscillator). However, while evolutionwas given by a unitary first-order updating rule in those models, it is a second-order process determined by a self-adjoint operator, ˆ H m , in the present case.Before closing, a remark is in order here. Namely, interacting multipartiteHamiltonian CA [6], in particular those consisting of two-state “Ising spin”subsystems, offer an alternative to look for more complex behaviour con-cerning ontological states than in the one-component examples chosen herefor simplicity. Such systems have been considered recently [3], with furtherresults to be presented elsewhere. Generalization in this direction seems nec-essary, in order to develop ontological models that possibly can serve as arealistic base from which the theory of interacting relativistic quantum fieldscan emerge in analogous ways as the QM models we described. In retrospect, one could subsume our results, cf. especially the summary givenin Sec.2, as pertaining to a particular discretization of QM, which introducesthe finite scale l , conceivably the Planck scale.However, we have reported in Sec. 3 the resulting conservation laws in suchmodels of discrete Hamiltonian CA , which are entirely described in terms ofinteger valued quantities, and illustrated the one-to-one correspondence with n configuration space, Born’s rule and ontological states 11 those of continuum models of quantum theory, which are recovered for l → counting procedure – unnoticeable in quantum theory, since thereonly the coincidence limit of the correlation matters! This seems to shed adifferent light on the
Born rule .Up to this point, though, we still did not encounter the randomness even-tually seen in experimental outcomes.As we have argued, following ’t Hooft, the probabilistic features of QM canbe understood to result from the available mathematical description of theunderlying deterministic reality. The unavoidable mismatch between the twocan be precisely traced to the nonexistence of superposition states of “stuff”that is ontologically there and the powerful use that is made of such formalsuperpositions in quantum theory.By the hypothesis of ontological states and by illustrating their existencewithin the present class of discrete models, of Hamiltonian CA kind, one hasleft standard QM, as suggested by the
Cellular Automaton Interpretation [11], cf. the introduction to Sec.4. This may provide some indication thatreality can be understood to exist “out there”, sometimes misnamed “Ein-stein’s dream”, that ontological states describe states in which a deterministicphysical system can be and how it evolves. Yet QM is neither abandoned norhas quantum theory been changed, but one begins to understand it as a mosteffective mathematical construct/language to describe the reality of what weperceive. Much more is left to be done.
Acknowledgements
It is a pleasure to thank Dirk Rischke and Horst St¨ocker for theinvitation to the
International Symposion on Discoveries at the Frontiers of Science in honour of Walter Greiner (FIAS, Frankfurt, June 26-30, 2017), for support, andespecially for kind hospitality.
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