The solution of the Sixth Hilbert Problem: the Ultimate Galilean Revolution
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Subject Areas:Keywords: algorithmic paradigmVI Hilbert problemquantum automata
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Giacomo Mauro D’Arianoe-mail: [email protected]
The solution of the SixthHilbert Problem: the UltimateGalilean Revolution
Giacomo Mauro D’Ariano , Dipartimento di Fisica, Università degli Studi di Pavia,via Bassi 6, 27100 Pavia, INFN, Gruppo IV, Sezione di Pavia
I argue for a full mathematisation of the physicaltheory, including its axioms, which must contain nophysical primitives. In provocative words: “physicsfrom no physics”. Although this may seem anoxymoron, it is the royal road to keep completelogical coherence, hence falsifiability of the theory.For such a purely mathematical theory the physicalconnotation must pertain only the interpretation ofthe mathematics, ranging from the axioms to thefinal theorems. On the contrary, the postulates of thetwo current major physical theories either don’t havephysical interpretation (as for von Neumann’s axiomsfor quantum theory), or contain physical primitives as“clock”, “rigid rod ”, “force”, “inertial mass” (as forspecial relativity and mechanics).A purely mathematical theory as proposed here,though with limited (but relentlessly growing)domain of applicability, will have the eternal validityof mathematical truth. It will be a theory on whichnatural sciences can firmly rely. Such kind of theory iswhat I consider to be the solution of the Sixth Hilbert’sProblem.I argue that a prototype example of such amathematical theory is provided by the novelalgorithmic paradigm for physics, as in the recentinformation-theoretical derivation of quantum theoryand free quantum field theory.The present opinion paper should be regarded mainly asa manifesto, preliminary to future thorough studies. c (cid:13) The Authors. Published by the Royal Society under the terms of theCreative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author andsource are credited. a r X i v : . [ qu a n t - ph ] J a n r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A ..................................................................
1. Mathematics is eternal, physics is temporary
Who does not believe that mathematics is eternal? Will the Pitagora’s theorem continue to holdfor the next millennia?Differently from mathematical theorems, physical theories are temporary. Should they also beeternal? At first glance such possibility may look overreaching: physics has always been rivenwith contradictions, and contradictions have often played a pivotal role in the understanding ofreality–a reality that, we believe, itself possesses internal logical coherence. But, is transience anintrinsic limitation of any physical theory? Or is it just a temporary historical feature? We cannotdeny that, indeed, an impermanence phase occurred also for mathematics at its early pre-Hellenicstage, when the discipline was not so different from the “trial-and error” approach of physics (seethe case of the value of Greek-Pi [1]).Physical theories are commonly considered eternal in the sense that, whenever the theory isreplaced by a new one, nevertheless the old theory continues to hold as a limiting case of thenew theory, within a narrower phenomenological domain. Unfortunately this is not the case ofthe replacement of classical mechanics by quantum mechanics, since by no means the formercan be regarded as a limiting case of the latter. On the other hand, quantum mechanics, whichsupposedly holds for the entire physical domain, still relies on mechanical notions coming fromthe old classical theory, with the new theory built over the relics of the old one through heuristic quantisation rules , or through the mathematically undefined path integral of Feynman. Issues ofsuch extent definitely undermine the full logical coherence of the physical theory.
2. The Sixth Hilbert Problem
Was transience of physics what motivated David Hilbert to propose his Sixth Problem? Wecan imagine a positive answer from a mathematician who likely believed in permanence ofmathematical truth. Let’s examine the opening paragraph of the problem. It reads:
The investigations on the foundations of geometry suggest the problem: To treat in the same manner bymeans of axioms, those physical sciences in which mathematics plays an important part; in the first rankare the theory of probabilities and mechanics.
Benjamin Yandell writes in his book about Hilbert’s problems [2]:
Axiomatizing the theory of probabilities was a realistic goal: Kolmogorov accomplished this in 1933. Theword ’mechanics’ without a qualifier, however, is a Trojan horse.
I argue that the Greek soldiers hiding inside the Trojan horse–and coming out of it muchlater in the fully developed theory–are the physical primitives contained in the axioms, i.e. inthe same laws of mechanics. In fact, being physical, the primitives require either a theoretical oran operational definition. This means that, in both cases, they are not actually primitive, implyingthat the logical feedback from their definition toward the same theory is prone to be logically notcoherent. For example, no general procedure is known to recover the classical trajectories of a set of particles from the solution of theSchrödinger equation. The canonical quantisation procedure is an extrapolation of the Ehrenfest theorem. For general functions over the phase-space one has the well known issue of operator ordering, and it is only coincidental that Hamiltonians in nature are notaffected by this problem. On curved space-time there exist no definite quantisation procedure. The geometric quantisationprogram has attempted to solve this problem, however, it never ended up with a general procedure. To date, it has succeededonly in unifying older methods of quantising finite-dimensional physical systems, but with no rule for infinite-dimensionalsystems, e.g. field theories and generally non integrable quantum systems (for a short and pedagogical review of thegeometric quantisation, see Ref. [3]). The path integral works only as a formal rule for producing Feynman-diagrams expansions. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. Let’s analyse the case of mechanics. “Mechanics” means dealing with “forces” and their effectson “bodies”. Both notions of “force” and “body” are pregnant with theory, and are not actuallyprimitive. For example, in Newtonian mechanics one needs to establish if a force is “real” or“apparent” (or, equivalently, whether the reference frame is “inertial” or not. ) However, only thetheory can tell if a force is real (i.e. electromagnetic, weak, strong, ...), and this feeds the theoryback into the axiomatic primitive in a circular way. On the other hand, the notion of “body”, orphysical “object” (on which forces act), is a seemingly innocent concept, which, however, bears itsown theoretical logical discord. Indeed, the most general notion of object–the object as a “bundleof properties”–is incompatible with quantum theory, since it lacks its mereological connotation (i.e.composability of objects to make new objects). In fact, in quantum theory it is common that theproperties of the whole are incompatible with any possible property of the parts [4]. To makethings worst, the same notion of “particle” as a localizable entity does not survive quantum fieldtheory [5], and what is left of the particle concept is a purely mathematical notion: the particle isan irreducible representation of the Poincaré a group [6].Keeping Newtonian mechanics aside, and entering Einstein’s special relativity, we encounterphysical primitives as “clock” and “rigid rod”, apparently unharmful notions entering the theoryaxiomatisation “operationally”, but which instead are pregnant of theory, e.g. in regards ofprecision of clocks pace and rods length, which are both established by a separate theory:quantum theory! As an additional argument in support to a full mathematisation program of the physical theorywe have the non accidental fact that what endures time in a theory is mainly its mathematicalcomponent. Exemplar in such respect is group theory, which provides a perfect case of purelymathematical formulation that is “physical” in the interpretation. And exemplar has been thesuccess of group theory in the grand-unification of forces, ranging from the creation of the notionof spin and isospin, up to the quark model, and in providing the already mentioned geometricalcategorisation of the notion of elementary particle as an irreducible representation of the Poincarégroup.
3. Failure of previous mathematisation programs
The route for the mathematisation of physics has been deeply explored by Streater and Wightman[10] and Haag and Kastler [11] for quantum field theory. These axiomatizations, however, stillhinge on mechanical notions from the old classical theory, as those of “particle”, “mass”, butalso “space” and “time”. On the opposite side, the mathematical framework is inherited fromquantum theory, based on the von Neumann axiomatics, which notoriously lacks physicalinterpretation. For Newton this was to a problem, since he believed in absolute space. Notice that the notion of inertial frame is itself circular if nothing assesses reality of forces. One can introduce forcesoperationally through the Hooke law which, however, far from being primitive, is also operationally not defined at themicroscopic level of particle physics. In quantum theory a property is associated to a projector operator over a subspace of the Hilbert space of the system (whichwe identify with the object). Joint properties are projectors on the tensor products of the Hilbert spaces of the componentsystems. Entangled tensor projectors do not commute with any local projector: this means that such properties of the wholeare incompatible with any property of the parts. Moreover, entangled projectors are typical among projectors on tensor spaces. Einstein himself was well aware of the primitiveness issue about the two notions of “clock” and “rigid rod”. He writes:
Itis . . . clear that the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, but that ofcomposite structures, which must not play any independent part in theoretical physics. But it is my conviction that in the present stageof development of theoretical physics these concepts must still be employed as independent concepts; for we are still far from possessingsuch certain knowledge of the theoretical principles of atomic structure as to be able to construct solid bodies and clocks theoretically fromelementary concepts. [7]. For a critical analysis of the notions of clock and rigid rod in the axiomatisation of special relativity,see Chapt. 7 of the book of Harvey R. Brown [8] along with the analysis of A. Hagar [9]. Remarkably, such notion of particle genuinely wears an ontological version of structural realism, with the naive physicalontology replaced by a mathematical structure, whereas the physical interpretation comes in terms of a relational structure ofreality. This shows how the process of mathematisation of physics also provides a unique reconciliation between the realistand the structuralist views. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. Irreconcilable with quantum field theory, the other main theory in physics–general relativity–uses space-time as a dynamical entity, whereas in quantum field theory it plays the role ofa background required axiomatically. The same “equivalence principle”, which comes fromexperience, should originate from a microscopic mechanism giving rise to mass, but, instead,it is simply postulated. On the other hand, the particle mass, a parameter which is unboundedin quantum field theory, becomes bounded as the result of a patch with general relativity,its maximum value, the Planck mass, corresponding to the notion of mini black-hole. In thesame way, patching gravity with quantum field theory clashes with the continuum description,as a consequence of the break down of the high-energy/short-distance correspondence, sincecolliding two particles with center-of-mass energy of the order of the Planck energy creates ablack hole. This clash with general relativity must be summed to the logical mismatch entirelyinternal to quantum field theory: the Haag’s theorem [10,11].
4. Physics as interpretation of mathematics: the pet theory
The reader will ask again how is it possible to axiomatise a physical theory without resorting tophysical notions. Physics without physics? Is it not an oxymoron? As already said, the answeris simply to restricts physics to the interpretational level. Then, if the interpretation is thoroughfor axioms, it will logically propagate throughout all theorems, up to the very last stage of theirspecific instances, where physics will emerge from the mathematics through its capability ofportraying phenomena and to logically connect experimental observations.One may now legitimately ask from where can we learn the physics for the interpretation of themathematics? A concrete possibility is the direct comparison of observations with the solutionsthat the mathematical algorithm portrays, likewise a digital screen reproduces a physical scenario.Also, we can compare instances of the mathematical theory with our previous heuristic theoreticalknowledge coming from non mathematically axiomatised theories. In the following I will providea concrete example of how the “mechanical” quantum field theory can emerge in its customarydifferential form from a purely mathematical theory, as a special regime of a mathematicalalgorithm: a quantum walk.The great bonus of having a mathematical theory endowed with physical interpretation, fromaxioms to theorems, will be the availability of thorough conceptual deductions that are physicallymeaningful at each logical step. The more universal the axioms are, the more stable and generalthe theory will be. And, contrarily to the case of derivations without physical interpretation (as forthe quantum Hilbert space formalism), reasoning with mathematics with physical interpretationbuilds up a logically grounded physical insight. (a) Quantum Theory as an information theory
In order to be purely mathematical, the axioms cannot contain words requiring a physicaldefinition, but must be meaningful for describing a physical scenario. How can this work?An example of a purely mathematical framework with physical interpretation is probabilitytheory. Here the notion of “event” in its specific instances can contain everything is needed forthe physical interpretation. And, indeed, probability theory is regularly used in physics with nointerpretation issue. And, as already said, it has been axiomatised by Kolmogorov in 1933, thussolving the easiest part of the VI Hilbert problem. The Planck mass is the value of the mass for which the particle localisation size (the Compton wavelength) becomescomparable to the Schwarzild radius. Streater and Wightman write [10]:
Haag’s theorem is very inconvenient; it means that the interaction picture exists only if there is nointeraction.
Haag’s theorem states that there can be no interaction picture, that we cannot use the Fock space of noninteractingparticles as a Hilbert space, in the sense that we would identify Hilbert spaces via field polynomials acting on a vacuum at acertain time. While some physicists and philosophers of physics have repeatedly emphasized how seriously Haag’s theoremis shaking the foundations of the theory, practitioners simply dismiss the issue, whereas most quantum field theory texts donot even mention it. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. We want now to consider the case of quantum theory. With the term “quantum theory”–instead of “quantum mechanics”–we denote the general theory of abstract systems, stripped ofits mechanical connotation. The example of probability theory suggests rethinking the theoryas being itself an extension of probability theory, with the physical interpretation containedin the event specification. Von Neumann in his quantum-logic program [12,13] was seeking asimilar route: indeed, the calculus of propositions is a purely mathematical context, with theinterpretation contained in the specific proposition. Unfortunately, however, this program didn’treach the desired result.Instead of an “alternative” kind of logic, quantum theory can be axiomatised as an “extension”of logic. This is the case of the so-called operational probabilistic theory (OPT) [14–16]. AnOPT is an extension of probability theory, to which one adds connectivity among events,in terms of a directed acyclic graph of input-output relations–the so-called “systems” of thetheory. Mathematically this framework corresponds to a combination of category theory [17]with probability theory, and constitutes the mathematical backbone of a general “theory ofinformation” [18]. Therefore, an OPT is an extension of probability theory, which in turn is anextension of logic (see Jaynes [19] and Cox [20]), and, in its spirit, this axiomatisation programremains close to the quantum-logic one, which was missing the crucial ingredient of connectivityamong events. (A concise complete exposition of the OPT framework can be found in Ref. [18].)Within the OPT framework, the six postulates of quantum theory are: P1
Atomicity ofComposition : The sequence of two atomic operations is atomic; P2 Perfect Discriminability : Everydeterministic state that is not completely mixed is perfectly discriminable from some otherstate; P3
Ideal Compression : Every state can be compressed in a lossless and efficient way; P4
Causality : The probability of a preparation is independent of the choice of observation; P5
LocalDiscriminability : It is possible to discriminate any pair of states of composite systems using onlylocal observations; P6
Purification : Every state can be always purified with a suitable ancilla, andtwo different purifications with the same ancilla are connected by a reversible transformation.(For a thorough analysis of the postulates and a didactical complete derivation of quantum theory,see the textbook [15]: noticeably all the six postulates possess an epistemological connotation,concerning the falsifiability of propositions of the theory under local observations and control.)The potential of the purely mathematical axiomatisation as an information theory is thatphysics can emerge from the network of connections among events. The mechanical paradigmis substituted by the new algorithmic paradigm. The new paradigm brings a far reachingmethodology change, since formulating the physical law as an algorithm forces the theoryto be expressed in precise mathematical terms. And the algorithm, regarded as a network of“operations” bears also an interpretation as “physical algorithm”, e.g. an experimental protocol.Quantum theory is thus a special kind of information theory, namely a theory of processes ,describing a network of connections among events, some of which are under our control (theso-called preparations or states ), others are what we observe (the so-called observations or effects ),some others are transformations , connecting preparations with observations. (b) How “mechanics” emerges from information: free quantum field theory All what was said above pertains only to the quantum theory of abstract systems. But, how canthe “mechanics”, namely quantum field theory, emerge? The answer resides in adding rules tothe process in terms of general topological requirements for the connectivity network, where theconnectivity is given by systems interactions. Such requirements are (informally): P7
Unitarity :the evolution is unitary; P8
Locality : each system interacts with a uniformly bounded numberof systems; P9
Homogeneity : the network of interactions is indistinguishable when regarded The “operation” is an event connected to others via systems. Here the term atomic means that the operation as event is notrefinable into more elementary events. It should be noted that the same can be said of classical information theory, which indeed share five postulates withquantum theory. The distinctive axiom of quantum theory is the purification postulate P6, which provides maximal controlof randomness [15]. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. from any two quantum systems; P10 Isotropy (see [21]). All postulates P7-P10 correspond tominimise the algorithmic complexity of the quantum process. Postulate P9 requires that thegraph of interactions is the Cayley graph of a finitely presented group. Then, for the easiest caseof: a) Abelian group, b) linearity of the evolution, and c) minimal dimension s of the systemfor nontrivial evolution ( s = 2 ) one finds a quantum walk which in the limit of small wavevectors gives the Weyl equation [21]. For s > the solution of the unitarity conditions becomesincreasingly hard: two solutions can be easily provided for s = 4 in terms of the direct sum andthe tensor product of two copies of the quantum walk with s = 2 : these correspond to Dirac [21]and Maxwell [27] field theories, respectively. Noticeably the relativistic quantum field theory isobtained without using any mechanics, any relativistic covariance, and not even space-time. And being quantum ab initio , the theory does not need quantisation rules. (For a review, thereader is addressed to Ref. [29]). The “mechanics”–including space-time and Lorentz covarianceemerges from a purely mathematical algorithm, at the large scale of a theory that is inherentlydiscrete. Physics emerges in form of computational patterns, in a universe which resembles animmense quantum-digital screen. (c) How physical standards come out in a purely mathematical theory
In a mathematically formulated theory all variables with physical interpretation are necessarilyadimensional. Therefore, the physical interpretation of the theory will be complete only if thesame theory contains a way to establish the physical standards. In a purely mathematical theorythis can be done either through counting, or by comparing real numbers with special values of thetheory–typically minimal or maximal allowed values of adimensional variables pregnant withphysical interpretation. The quantum cellular automata theory of fields of the previous section[21] is exemplar in this sense, as it sets measurements of space and time to counting, whereas theinterplay of unitarity and discreteness leads to an upper bound for the particle mass [21], whichcan then plays the role of the mass standard.
5. Conclusions
Is mathematisation of physics still premature? Although we may never achieve a complete set ofaxioms for physics, yet we can have stable subsets of them (along with alternative equivalent sets),and derive a stable collection of theorems, whose physical interpretation will provide us with“physical” truth as evident and eternal as the mathematical one. Mathematics is an evolutionof the human language–the ultimate idealisation and synthesis of observations grouped intoequivalence classes. Physics goes beyond logic, and, as such, it is an interpretation of mathematicsin terms of our experience. A thorough mathematical axiomatisation program for physics willopen a new era, purely logical, not conjectural, focused on seeking general principles whoseresulting theory will retain in time autonomous mathematical value.Four hundred years already passed after Galileo introduced the mathematical description ofphysics: the complete mathematisation of physics will be the next, ultimate, Galilean revolution. Since every deterministic transformation can be achieved unitarily in quantum theory, P7 represents the choice of minimalcomplexity. P8 and P9 correspond to infinite reduction of complexity, and P10 is also a reduction of complexity. Thecomplexity of the quantum cellular automaton can be quantified in terms of the complexity of a minimal Margolus quantumcircuit-block [22,23] for the automaton, by counting the resources of the circuit in terms of controlled-not and Pauli unitariesaccording to Deutsch theorem [24]. Differently from the Kolmogorov-Chaitin complexity [25,26], the complexity of theautomaton is computable, and it is easy to get a very good upper bound. This will then quantify the complexity of thephysical law, toward a quantification of the Occam’s razor criterion. Lorentz covariance is a result of the Galileo principle, along with causality, locality, homogeneity, and isotropy ofinteractions of abstract quantum systems. The inertial frames are those that leave the quantum automaton or quantum walkinvariant [28]. Since at the maximum value of the mass there is no information flow in the quantum walk, the same value is interpretedas that of the mini black-hole originating from the patching of quantum field theory with general relativity. Thus the massstandard is the already mentioned Planck mass. Moreover, from the comparison with the usual free quantum field theoryemerging at large scales (small wave vectors) one finds that the discrete units of time and space are the Planck’s units. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. Disclaimer.
The fact that the author calls for a mathematisation of physics does not imply that he is a goodmathematician. Reference to author’s pet theories is only for the sake of exemplification, although thesetheories were designed and elaborated exactly within the spirit and motivations of the present essay. Thereader should judge the present mathematisation program independently on his opinions on these theories.
Data Accessibility.
For a recent review about the author’s pet theory here mentioned, see the recent openaccess review by the same author in memoriam of his friend David Finkelstein [29] doi: 10.1007/s10773-016-3172-y
Funding.
This work was made possible through the support of a grant from the John Templeton Foundation(Grant ID
A Quantum-Digital Universe , and Grant ID
Quantum Causal Structures ). The opinionsexpressed in this publication are those of the author and do not necessarily reflect the views of the JohnTempleton Foundation. The manuscript has been mostly written at the Kavli Institute for Theoretical Physics,University of California, Santa Barbara, and supported in part by the National Science Foundation underGrant No. NSF PHY11-25915.
Acknowledgements.
This paper is a largely updated version of the original essay:
The unreasonableeffectiveness of Mathematics in Physics: the Sixth Hilbert Problem, and the Ultimate Galilean Revolution , writtenfor the FQXi Contest 2015:
Trick or Truth , http://fqxi.org/community/forum/topic/2395 . References
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