aa r X i v : . [ qu a n t - ph ] M a r Quantum ob jects in a sheaf framework
Antonina N. Fedorova and Michael G. Zeitlin [email protected], [email protected]
Abstract.
We consider some generalization of the theory of quantum states and demonstratethat the consideration of quantum states as sheaves can provide, in principle, more deepunderstanding of some well-known phenomena. The key ingredients of the proposedconstruction are the families of sections of sheaves with values in the proper category of thefunctional realizations of infinite-dimensional Hilbert spaces with special (multiscale) filtrationsdecomposed into the (entangled) orbits generated by actions/representations of internal hiddensymmetries. In such a way, we open a possibility for the exact description and reinterpretationof a lot of quantum phenomena.
1. Introduction. Quantum states: functions vs. sheaves
During a relative long period, it is well-known that there is a great difference between (at least)the mathematical levels of the investigation of quantum phenomena in different regions. At thesame time, even advanced modern Mathematics cannot help us in the final (at least practicallyaccepted) analysis of the long standing quantum phenomena and the final classification of a zooof interpretations [1]. The well-known incomplete list is as follows: (L) entanglement, measurement, wave function collapse, decoherence, Copenhageninterpretation, consistent histories, many-worlds interpretation/multiverse (MWI), Bohminterpretation, ensemble interpretation, (Dirac) self-interference, “instantaneous” quantuminteraction, hidden variables, etc.As a result, beyond a lot of fundamental advanced problems at planckian scales we arestill even unready to create the proper theoretical background for the reliable modeling andconstructing of quantum devices far away from planckian scales. It is very hard to believe thattrivial simple solutions, like gaussians, can exhaust all variety of possible quantum states neededfor the resolution of all contradictions, hidden inside the list (L) mentioned above. So, let uspropose the following (physical) hypothesis: (H1) the physically reasonable really existing Quantum States cannot bedescribed by means of functions. Quantum state is a complex pattern whichdemands a set/class of functions/patches instead of one function for properdescription and understanding . There is nothing unusual in (H1) for physicists sinceDirac’s description of monopole. All the more, there is nothing unusual for mathematicianswho successfully used sheaves, germs, etc in different areas. Definitely, the introducing of (H1) causes a number of standard topics, the most important of them are motivations, formal (exact)definition and (at least) particular realizations. Really, why need we to change our ideologyfter a century (since Planck) of success? The answer is trivial and related to the list (L) whichis overcompleted with contradictions and misunderstanding after many decades of discussions.
2. On the Route to Right Description: (Quantum) Patterns as Sheaves2.1 Motivations1). Arena for Quantum Evolution
First of all, we need to divide the kinematical and dynamical features of a set of QuantumStates ( QS ). From the formal point of view it means that one needs to consider some bundle( X, H, H x ) whose sections are the so-called | ψ > functions or QS . Here X is (kinematical) space-time base space with the proper kinematical symmetry group (like Galilei or Poincare ones), H is a total formal Hilbert space and H ( x ) = H x are fibers with their own internal structures andhidden symmetries. In addition, such a bundle has the corresponding structure group whichconnects different fibers. Of course, in a very particular case we have the constant bundle withthe trivial structure group but non-trivial fiber symmetry. Anyway, as we shall demonstratelater, it is very reasonable to provide the one-to-one correspondence between Quantum Statesand the proper sections | QS > : X −→ H, QS : x H x = H ( x ) . As a result, we have, at least, three different symmetries inside this construction: kinematicalone on space-time, hidden one inside each fiber and the gauge-like structure group of the bundleas a whole. It is obvious that the kinematical laws (like relativity principles) depend on theproper type of symmetry and are absolutely different in the base space and in the fibers. Itshould be noted that the functional realizations of fibers and the total space are very importantfor our aims. Roughly speaking, it can be supposed that physical effects depend on the typeof the particular functional realization of formal (infinite dimensional) Hilbert space. E.g., itis impossible to use infinite smooth approximations, like gaussians, for the reliable modeling ofchaotic/fractal phenomena. So, the part of Physics at quantum scales is encoded in the detailsof the proper functional realization.
It is well-known that nobody can prove that gaussians (or even standard coherent states, etc)are an adequate and proper image for Quantum States really existing in the Nature. We cansuggest that at quantum scales other classes of functions or, more generally, other functionalspaces (not C ∞ , e.g.) with the proper bases describe the underlying physical processes. Thereare two key features we are interested in. First of all, we need the best possible localizationproperties for our trial base functions. Second, we need to take into account, in appropriateform, all contributions from all internal hidden scales, from coarse-grained to finest ones. Ofcourse, it is a hypothesis but it looks very reasonable: (H2) there is a (infinite) tower of internal scales in quantum region thatcontributes to the really existing Quantum States and their evolution .So, we may suppose that the fundamental generating physical “eigen-modes” correspond toa selected functional realization and are localized in the best way. Let us note the role of theproper hidden symmetries which are responsible for the quantum self-organization and resultingcomplexity. As a result of the description above, we may have non-trivial “interaction” inside an infinitehierarchy of modes or scales. It resembles, in some sense, a sort of turbulence or intermittency.Of course, here the generating avatar is a representation theory of hidden symmetries whichcreate the non-trivial dynamics of this ensemble of hierarchies. t is well-known that symmetries generate all things (at least) in fundamental physics. Here,we have a particular case where the generic symmetry corresponds to the internal hiddensymmetry of the underlying functional realization. Moreover, as it is proposed above, we haveeven the more complicated structure because we believe that QS is not a function but a sheaf.As a result, we have interaction between two different symmetries, namely hidden symmetryin the fiber, that corresponds to the internal symmetry of the functional realization, and thestructure “gauge” group of a sheaf, which provides multifibers transition/dynamics. Both thesealgebraic structures can be parametrized by the proper group parameters which can play therole of famous “hidden variables” introduced many decades ago. Of course, MWI or Multiverse interpretation can be covered by the structure sketched above.Quantum States are the sections of our fundamental sheaf, so we can consider them as a collectionof maps between the patches of base space and fibers. All such maps simultaneously exist and, asan equivalence class, represent the same Quantum State. We postpone the detailed description tothe next Section but here let us mention that each member of the full family can be considered asan object belonged to some fixed World. Obviously, before measurement we cannot distinguishsamples but after measurement we shall have the only copy in our hands.
The main reason to introduce sheaves as a useful instrument for the analysis of QuantumStates is related to their main property which allows to assign to every region U in space-time X some family F ( U ) of algebraic or geometric objects such as functions or differentialoperators. The family can be restricted to smaller regions, and the compatible collections offamilies can be glued to give a family over larger regions, so it provides connection betweensmall and large scales, local and global data. Informal construction is as follows. Let X bethe space-time base space (some topological space) with a system of open subsets U ⊂ X ,then for every U and map F the image F ( U ) is some object with internal structure (moregenerally, F ( U ) takes values in some category H ) such that for every two open subsets, U and V , V ⊂ U there is the so-called restriction map (more generally, morphism in the category H ), r V , U : F ( U ) → F ( V ) (restriction morphism). A map F will be a presheaf if restrictionmorphism satisfies the following properties: (a) for every open subset U ⊂ X , the restrictionmorphism r U,U : F ( U ) → F ( U ) is the identity morphism, (b) if there are three open subsets W ⊂ V ⊂ U , then r W,V r V,U = r W,U . This property provides the connection or ordering ofthe underlying scales. In other words, let O ( X ) be the category of open sets on X , whoseobjects are the open sets of X and whose morphisms are inclusions. Then a presheaf F on X with values in category H is the contravariant functor from O ( X ) to H . F ( U ) is called thesection of F over U and we consider it as some pre-image for adequate Quantum State | QS > .But our goal, in this direction, is a sheaf , so we need to add two additional properties. Let { U i } i ∈ I be some family of open subsets of X , U = ∪ i ∈ I U i . (c) If Ψ and Ψ are two elementsof F ( U ) and r U i ,U (Ψ ) = r U i ,U (Ψ ) for every U i , then Ψ = Ψ . (d) for every i let a sectionΨ i ∈ F ( U i ). { Ψ i } i ∈ I are compatible if, for all i and j , r U i ∩ U j ,U i (Ψ i ) = r U i ∩ U j ,U j (Ψ j ). For everyset { Ψ i } i ∈ I of compatible sections on { U i } i ∈ I , there exists the unique section Ψ ∈ F ( U ) such that r U i ,U (Ψ) = Ψ i for every i ∈ I . The section Ψ is called the gluing of the sections Ψ i . Definitely,we can consider this property as allusion to the hypothesis of wave function collapse. Really, Ψ looks as Multiverse Quantum State Ensemble { Ψ i } while Ψ i is the result of measurementin the patch U i . And it is unique! The next step is to specify the Quantum Category H .According to our Hypothesis H2 , we consider the category of the functional realization of(infinite-dimensional) Hilbert spaces provided with proper filtration, which allows to take intoaccount multiscale decomposition for all dynamical quantities needed for the description ofQuantum Evolution. The well-known type of such filtration is the so-called multiresolutiondecomposition. It should be noted that the whole description is much more complicated becauset demands the consideration of both structures together, namely, the fiber structure generatedby internal hidden symmetry of the chosen functional realization and the family of gluing sections Ψ in the unified framework. In the companion paper, we shall consider in details one important realization of thisconstruction based on the local nonlinear harmonic analysis which has, as the key ingredient,the so-called Multiresolution Analysis (MRA). It allows us to describe internal hidden dynamicson a tower of scales. Introducing the Fock-like space structure on the whole space of internalhidden scales, we have the following MRA decomposition: H = M i O n H ni for the set of n-partial Wigner functions (states): W i = { W i , W i ( x ; t ) , . . . , W iN ( x , . . . , x N ; t ) , . . . } . So, qualitatively,
Quantum Objects can be represented by an infinite or sufficiently largeset of coexisting and interacting subsets while (Quasi)Classical Objects can be described byone or a few only levels of resolution with (almost) suppressed interscale self-interaction. It ispossible to consider Wigner functions as some measure of the quantum character of the system:as soon as it becomes positive, we arrive to classical regime and so there is no need to consider thefull hierarchy decomposition in the MRA representation. So, Dirac’s self-interference is nothingelse than the multiscale mixture/intermittency. Certainly, the degree of this self-interactionleads to different qualitative types of behaviour, such as localized quasiclassical states, separable,entangled, chaotic etc. At the same time, the instantaneous quantum interaction or transmissionof (quantum) information from Alice to Bob takes place not in the physical kinematical space-time but in Hilbert spaces of Quantum States in their proper functional realization where thereis a different kinematic life. As a result, on the proper orbits, we have nontrivial entangleddynamics, especially in contrast with its classical counterpart.
3. Conclusions
It seems very reasonable that there are no chances for the solution of long standing problemsand novel ones if we constraint ourselves by old routines and the old zoo of simple solutionslike gaussians, coherent states and all that. Evidently, that even the mathematical backgroundof regular Quantum Physics demands new interpretations and approaches. Let us mentiononly the procedures of quantization as a generic example. In this respect, we can hope thatour sheaf extension of representation for QS , which is natural from the formal point of view,may be very productive for the more deep understanding of the underlying (Quantum) Physics,especially, if we consider it together with the category of multiscale filtered functional realizationsdecomposed into the entangled orbits generated by actions of internal hidden symmetries. Insuch a way, we open a possibility for the exact description of a lot of phenomena like entanglementand measurement, wave function collapse, self-interference, instantaneous quantum interaction,Multiverse, hidden variables, etc. [2]. In the companion paper we consider the machinery neededfor the generation of a zoo of the complex quantum patterns during Wigner-Weyl evolution.
4. Perspectives: On the Route to Categorification
Sheafification together with micrlolocalization [3] and subsequent analysis of quantum dynamicson the orbits in the sections with special, the so-called MRA-filtrations [4], considered in thispaper and in the companion one, are the starting points of our attempt of CategorificationProgram for Quantum Mechanics and/or General Local Quantum Field Theory [5]. Inome sense, we hope on the same breakthrough as in the golden era of Algebraic Topologyand Algebraic Geometry in the 50s and 60s of the 20th Century, which was concludedby Grothendieck approach [6] and provided the universal description for a variety of longstanding problems. Roughly speaking, such an approach provides useful, constructive anduniversal methods to glue the complex local data into the general picture by power machinerytaking into account the topology and geometry of the underlying hidden internal structures.Definitely, the simple linear algebra of structureless Hilbert spaces cannot describe the wholerich world of quantum phenomena. Our approach introduces Grothendieck schemes [7]instead of varieties/manifolds as generic quantum objects, naturally encoded the full zoo ofphenomenological things discussed in Quantum Mechanics. The key ingredient of such anapproach is the bridge between the von Neumann description of measurement together withthe Gelfand ideal of the state and GNS (Gelfand-Naimark-Sigal)-construction [5], [8] on oneside of the river and locally ringed space, structure scheaf and (affine) scheme on the opposite(categorificated) side. We will consider all technical details in the separate paper.
References [1] A. Connes, M. Marcolli,
Noncommutative Geometry, Quantum Fields and Motives
Sheaves on Manifolds , Springer, 1994.[4] Y. Meyer,
Wavelets and Operators , Cambridge Univ. Press, 1990; H. Triebel,
Theory of Functional Spaces
Birkhauser, 1983.[5] R. Haag,
Local Quantum Physics , Springer, 1996.[6] D. Mumford,
The Red Book of Varieties and Schemes , Springer, 1999.[7] R. Hartshorne,
Algebraic Geometry , Springer, 1997.[8] O. Bratteli, D. Robinson,