aa r X i v : . [ phy s i c s . g e n - ph ] M a y HoloQuantum HyperNetwork Theory
Alireza Tavanfar
Instituto de Telecomunicações,Physics of Information and Quantum Technologies Group, Lisbon, [email protected]
The fundamental, general, kinematically-and-dynamically complete quantum many body theoryof the entirely-quantized HyperNetworks, namely
HoloQuantum HyperNetwork Theory , ‘ M ’, isaxiomatically defined and formulated out of a unique system of nine principles . HoloQuantumHyperNetworks are all the quantum states of a purely-information-theoretic (0+1) dimensionalclosed quantum many body system of the abstract qubits of the ‘absences-or-presences’ endowedwith a complete distinctive set of abstract many-body interactions among them. All the many-bodyinteractions and the complete total Hamiltonian of M are uniquely obtained upon realizing thekinematical-and-dynamical quantum-hypergraphical well-definedness by all the ‘cascade operators’,the quantum-hypergraphical isomorphisms, the global-phase U (1) redundancies of the multi-qubitstates, the minimally-broken equal treatment of the abstract qubits, the maximal randomness, and‘covariant completeness’. Mathematically, HoloQuantum HyperNetworks formulate in completenessall possible unitarily-evolving quantum superpositions of all the arbitrarily-chosen hypergraphs.Physically, HoloQuantum HyperNetworks formulate all the dynamical purely-information-theoreticstates of every realizable quantum many body system of the arbitrarily-chosen quantum objects andtheir arbitrarily-chosen quantum relations. Being so, HoloQuantuam HyperNetwork Theory, M , isproposed as the fundamental, complete and covariant ‘it-from-qubit’ theory of ‘All Quantum Natures’ . I. MOTIVES AND METHODOLOGY
Hypergraphs represent objects and their relations inthe most abstract and general way. The
HoloQuantumHyperNetwork Theory , ‘ M ’, which we define, constructand present here upgrades this mere representation to theform-invariant ‘it-from-qubit’ [1] theory of ‘all quantumnatures’ . That is, all the observables-and-states of all thequantum many body systems of objects-and-relationsare covariantly captured by the theory M .Mapping time-dependently to the mathematical spaceof (the maximally oriented and maximally weighted)hypergraphs, the sufficiently-refined choices of thevertices represent the arbitrarily-chosen objects, and thesufficiently-refined choices of the hyperlinks representthe arbitrarily-chosen relations between the objects.Mapping so, the hypergraphical hyperlinks representall possible physical interactions, classical statisticalcorrelations, quantum entanglement relations, geometricrelations, causality relations, structural or functionalrelations, emergent relations, multi-scale relations,and all possible observable relations. Redefining thesehypergraphical representations as HyperNetworks , oneformulates the dynamical quantum objects-and-relationsstates-and-observables from pure information .In quantum realm, the ‘fundamental informationsetting’ which assembles these objects and relationsmust be a special quantum many body system of abstractqubits . The states of every one of these abstract qubitsis spanned on a purely-information-theoretic ‘no-or-yes’,or equally ‘absence-or-presence’ binary basis. Moreover, any closed many body system of the abstractqubits for the quantum objects and their quantumrelations undergoes in its totality a unitary evolution .The total Hamiltonian of this unitary dynamics must begiven by all the many-body abstract interactions betweenall these abstract quibits which are the very fundamentalquantum degrees of freedom for all the quantum objectstogether with all their quantum relations, à la Wheeler[1]. HoloQuantum HyperNetwork Theory, M , is indeedformulated in this framework. Being so, it formulates allthe dynamical mathematically-quantized HyperNetworkswhose defining qubits and their abstract interactions allbehave as quantumly as in quantum many body systems.Being its motive, HoloQuantum HyperNetwork Theoryis ‘The Answer’ to the following Question. Question:
Let us assume that quantum physics is the exactlaw-book of the entire universe or the entire multiverse.Let us define ‘all quantum natures’ to be the entirequantum universe-or-multiverse, as well as all thearbitrarily-chosen algebraically-closed subsystems of hertotal observables probed in all the jointly-consistentscales. Equally defined, let all quantum natures be allthe theoretically-possible quantum many body systemsof quantum objects and their quantum relations. Uponthis assumption and definition, what is the ‘it-from-qubit’quantum many body theory which formulates all thosequantum natures, both fundamentally and completely,in such a ‘covariant’ way that its quantum kinematics(all the microscopic degrees of freedom) and quantumdynamics (all the many-body interactions and the totalHamiltonian) always remain form-invariant?
We will obtain the answer to this question by treatingit as an ‘equation’ and then ‘solving’ it constructively.The solution must be the unique construction in whichtwo initially-distinguished faces consistently meet oneanother and merge-in, to form an indistinguishable unitbeing a ‘quantum mathematics for all physics’.Viewed on its ‘mathematics face’ , it must be thefundamental, general and also complete theory of thedynamical entirely-quantized hypergraphs, namely‘HoloQuantum Hypergraphs’. Being so, the theorymust formulate most fundamentally, and capture mostcompletely, all the dynamical mathematically-quantizedhypergraphs which are arbitrarily structured, arbitrarilyflavored, arbitrarily weighted and arbitrarily oriented.Viewed on its ‘physics face’ , the above theory must bethe fundamental formulation of the unitarily-evolvingmany body system of a complete set of abstract qubits which covariantly code the complete time-dependentinformation of every quantum nature.Before undertaking the from-first-principle constructionof this unique theory, to be accomplished in sectionstwo to five, we first briefly characterize the operationalway in which the ‘all quantum natures’ in the originalquestion is being defined. This precise characterization,serving as a preliminary view, will secure that we willbe all clear about the defining physics-context of theresulted quantum mathematics.
First , let us consider the entire quantum universe ,or to be maximally general (as we must), the entirequantum multiverse . Physically this quantum universeor quantum multiverse is defined as the maximally-largemaximally-complete set of C ⋆ -algebra-forming quantumobservables which are based on the Hilbert space of thetotal set of independent quantum degrees of freedom.By construction, the evolving global wavefunctions of this initial all-inclusive quantum many body system contain the complete time-dependent information whichare decodable by all possible measurements on it. Second , we now consider all possible operations of ‘phenomenological reductioning’ . By definition, everysuch operation corresponds to an arbitrarily-chosen(proper) subset of the entire set of quantum observableson the Hilbert space of an arbitrarily-chosen (properor improper) subset of that total set quantum degreesof freedom, which are scale-preserved in the sense ofthe renormalization group flow, and all together form a C ⋆ -subalgebra. Every such phenomenological reduction,being a likewise-reduced quantum many body system,observationally defines a ‘phenomenologically-reducedquantum universe or multiverse’ . Clearly now, for everyreduced universe-or-multiverse, the likewise-constructedevolving global wavefunctions contain the completetime-dependent information which are decodable by allpossible measurements on it. Third , to every such collection of the C ⋆ -algebra-formingquantum observables and their base Hilbert spaces,being the complete universe-or-multiverse or each oneof the reduced universes-or-multiverses, we apply allpossible operations of ‘probes rescalings’ , by which one’sscale of probing the observables is being re-chosen in allthe arbitrarily-chosen ways (that are only constrainedby all the algebraic-and-observational consistencies).Let us call the resulted quantum many body systemsthe ‘renormalized quantum universes-or-multiverses’ .Now, all as before, the likewise-constructed evolvingglobal wavefunctions of every one of these renormalizedquantum universe-or-multiverse contains its completetime-dependent information. ‘A quantum nature’ , as we name, is every so-definedquantum universe-or-multiverse, being the entire one, areduced one, or a renormalized one. As every universeis indeed a physically-realizable quantum many bodysystem, we choose the initial system to be ‘the ultimatequantum multiverse’, defined as the set-theoretic unionof all the in-principle-realizable universal quantum manybody systems. Being so, it must be now manifest that ‘all quantum natures’ are certainly re-identifable as allthe ‘theoretically-possible’ quantum many body systemsof the arbitrarily-chosen quantum objects and theirarbitrarily-chosen quantum relations.As a theory of all physics, what this work is allabout, is to construct from first principles the covarianttotal state-space and the covariant total Hamiltonianof the most-fundamental purely-information-theoreticquantum many body theory whose unitarily-evolvingquantum states encode the complete information ofevery such-framed quantum nature. The HoloQuantumHyperNetwork Theory which we present in this workperfectly accomplishes this ‘it-from-qubit’ mission . II. HOLOQUANTUM HYPERNETWORKTHEORY: THE THREEFOLD FOUNDATIONAND THE QUANTUM KINEMATICS
HoloQuantum HyperNetwork Theory is axiomaticallydefined and formulated based upon nine principles.In constructing a theory, the choice of its principles isalmost everything, so we highlight a cardinal point aboutthese nine principles. To the best of our understanding,the chosen principles are the ones which serve the theoryas the unavoidable, and at the same time, as the verybest foundational statements to fulfill its intentionalmission, as was stated and explained in section one.Namely, these nine principles are ‘the must be’ forconsistency, and ‘the best be’ for the fundamental-ness .We now begin to answer the original question bystating the three principles which set the ‘threefoldfoundation’ of HoloQuantum HyperNetwork Theory . Principle 1:
HoloQuantum HyperNetwork Theory,‘ M ’, must be axiomatically defined and constructedto be the kinematically-and-dynamically covariantcomplete formulation of all the in-principle-realizablequantum many body systems whose defining degrees offreedom are the arbitrarily-chosen quantum objects andtheir arbitrarily-chosen quantum relations, capturingperfectly all their time-dependent states and observables.Stated equally, it must be the fundamental completetheory of ‘all quantum natures’. Mathematically, toaccomplish the above mission, the theory M must be thegeneral complete theory of HoloQuantum Hypergraphs,namely, the hypergraphs whose all-quantized definingstructures become the quantum degrees of freedomconstituting an interacting quantum many body systemwhich evolves unitarily. Explanatory Note : We must highlight that, boththe objects and their relations must be taken-in as theoriginal notions and the quantum degrees of freedom ofthe presented theory. Because all relations, by their veryconception, are to be defined between some objects, thetheory, in order to be both the most fundamental andthe most complete, does also need to have the quantumobjects included in the totality of its microscopic degreesof freedom. But hierarchically viewed, relations can alsobe ‘objectified’ and so then develop the relations betweenrelations. Being so, upon straightforward hierarchicalextensions which keep the formulation of the theoryinvariant, the defining quantum degrees of freedom willadditionally include all these ‘hierarchical relationalrelations’ . Because these hierarchical extensions are allstraightforward, they are left implicit in the present work.
Principle 2:
To accomplish the mission specifiedin principle one, M must be an entirely-pregeometricquantum field theory, defined and formulated in (0 + 1) spacetime dimensions. Namely, it must be ‘a’ theory ofquantum mechanics. Explanatory Note : HoloQuantum HyperNetworkTheory, at the level of its fundamental formulation,lives in (0 + 1) spacetime dimensions. However, itssub-theories, model extractions, solutions and phasesrealize all the in-principle-realizable many body systemsof quantum objects and their quantum relations definedin arbitrary numbers of spatial dimensions.
Principle 3:
The theory M must be the ‘it-from-qubit’fundamental theory of all quantum natures. Being so,HoloQuantum Hypergraphs must be formulated purelyinformation-theoretically as the unitarily-evolving statesof an unprecedented quantum many body system of‘abstract qubits’, namely HoloQuantum HyperNetworks.By definition, each one of these fundamental qubitsrepresents the states of the ‘absence-or-presence’ of acorresponding (sufficiently-refined) quantum vertex or a(sufficiently-refined) quantum hyperlink. Explanatory Note : ‘Formally’, the abstract qubits of M shall be formulated by means of fermionic operators.That is, the binary basis-states of each one qubit arerepresented by the eigenvalues of the number operator ofa representing fermion, being either zero or one. Clearly,as an ‘it-from-qubit’ theory, the final computations ofall the true observables of M are invariant under howone formulates its abstract qubits. Being so, these samequbits can be equally represented by spins or by anyother alternatives. But, as the ‘it-from-qubit’ theory ofall quantum natures, M embodies its simplest first-rateformulation in terms of the fermionically-representedqubits . Highlighting that this choice is simply a veryconvenient formality to represent our abstract qubits,let us name them ‘formally-fermion qubits’ .We now merge these three principles into one unit: The Threefold Foundation of M : M will be formulated as ‘a’ closed quantum many bodysystem of interacting (formally-fermion) abstract qubitsof the absences-or-presences of the arbitrarily-chosenquantum objects and their arbitrarily-chosen quantumrelations, living in (0 + 1) dimensions. The globalwavefunctions of all these qubits must correspond ina one-to-one way to the unitarily-evolving quantumsuperpositions of the arbitrarily-structured hypergraphs.To formulate HoloQuantum HyperNetwork Theory,let us choose an arbitrary M ∈ N , and let H ( M ) M bethe defining total Hilbert space of all the HoloQuantumHypergraphs whose structures can take not more than M present vertices . The degrees of freedom of H ( M ) M are atotal number of M ⋆ = M ⋆ ( M ) formally-fermion qubitsof the ‘absences or presences’, each one for a quantumvertex or for a quantum hyperlink. We take a finite M , but are also free to take M → ∞ . Moreover, letus introduce an arbitrarily-chosen all-index ‘ I ’ which even-handedly but uniquely labels all the M ⋆ qubitsas F I . With every one qubit F I come its creation andannihilation operators ( f I , f † I ) , forming the algebra, f † I = f J = 0 ; { f J , f † I } = δ JI , ∀ I , ∀ J (1)We name the qubit for a vertex a ‘ verton ’, and the qubit for a hyperlink whose identification needs ‘ m ’ number ofdefining ‘base vertices’ an ‘ m -relaton’ , where m ∈ N ≤ M .Because mathematically, a hypergraph is defined freefrom any background space, M must be fundamentallydefined in zero space dimensions. Being so, at the levelof the fundamental formulation of the theory, vertons arelabeled with one single index i ∈ N ≤ M to address theirindependent identities. We denote all the vertons of M by V i , together with their fundamental operators ( v i , v † i ) .Accordingly, the m -relatons of M are addressed uniquelyas R i ( m ) , and ( r i ( m ) , r † i ( m ) ) , with i ( m ) ≡ { i , ..., i m } beingtheir sets of base-vertons. Upon any one-to-one map, { all i ∈ N ≤ M } ∪ { all i ( m ∈ N ≤ M ) } ≡ { I } Cardinality = M ⋆ (2)Let us highlight a clarifying point. In this section, givenits mission, we only need to formulate ‘the basic class’ ofHoloQuantum HyperNetworks . By definition, these areHoloQuantum HyperNetworks for which the relatons aretotally insensitive to the ordering of their base vertons,and moreover, the relatons and the vertons are weightlessand flavorless. But, these basic-class minimalizationswill be all lifted up where in section five , on the basisof the results in sections two to four, we will perfectboth the mathematical and physical generality of M byformulating the HoloQuantum HyperNetworks whosecorresponding hypergraphs are ‘maximally flavored’,‘maximally weighted’, and also ‘maximally oriented’ .Maximal orientedness means that the base-orderingdegeneracy of the relatons are entirely lifted up, so thatevery relaton is defined by the unique ordering of all itsbase vertons.Now, the fundamental operators of the formally-fermionqubits of M are so collected, { f ( † ) I ; |{ I }| = M ⋆ ; M ⋆ = 2 M − M } ≡≡ { v ( † ) i ; r ( † ) i (1) , r ( † ) i (2) , r ( † ) i (3) , · · · , r ( † ) i ( M ) | all possible indices } (3)The simplest observables of the theory are the numberoperators of all the vertons and all the relatons, n I ≡ f † I f I ; n I = n I ; [ n I , n J ] = 0 , ∀ I , ∀ J (4)Multi-qubit states are defined as the common eigenstatesof the n I observables. Given the standard vacuum state, | i ∈ H ( M ) M ; f I | i = 0 , ∀ I (5)the ‘ m -qubit states ’ are built as follows, ≤ s ≤ m Y I s f † I s | i , ∀ { I s } ⊂{ I } Cardinality M⋆ | ≤ s ≤ M , ∀ m ∈ N ≤ M ⋆ (6)But, the defining state-space of the theory M , namely H ( M ) M , is much smaller than the space spanned by allthe multi-qubit wavefunctions constructed in (6). Thismust be clear because, as one demands, for a multi-qubitwavefunction in (6) to be a defining basis-state of H ( M ) M , it must be identifiable with a mathematically-welldefined‘classical’ hypergraph . Restated, each multi-qubit statein (6) belongs to the defining basis of H ( M ) M , if for everyrelaton R i ( m ) being present in it, all of its base-vertons { V i , ··· , V i m } are also present in it. This condition, whichdiscerns the basis of H ( M ) M , mathematically representsthe statement that the presence of a relation without thepresence of every one of its identifying objects render itscontaining wavefunctions meaningless. This point yieldsthe ‘consistency quantum constraints’ whose kinematicalversion comes in this section, but their dynamical version will come in section three. For every qubit F I , let us introduce its uniquely-defined‘qubit-presence projection operator’, being denoted by P I . By definition, it is the unique operator by which anysuperposition of the states in (6) is projected down tothe maximal part of it in which every participating statehas the presence of the qubit F I . By fermionic algebra, P I = n I ; ¯ P I ≡ − P I = 1 − n I , ∀ I (7)Given (7), let us express here the ‘kinematical quantumconstraints’ that discern which ones among the m -qubitstates in (6) form the basis of H ( M ) M , namely B ( M ) M , ∀ state (cid:12)(cid:12)(cid:12) ˆΨ E in (6) , (cid:12)(cid:12)(cid:12) ˆΨ E ∈ B ( M ) M ; if : Y i ( m ) (cid:12)(cid:12)(cid:12) ˆΨ E ≡ n i ( m ) (cid:0) − Y all i s ∈ i ( m ) n i s (cid:1) (cid:12)(cid:12)(cid:12) ˆΨ E = 0 , ∀ R i ( m ) (8)Constraints (8) are solved by replacing the r ( † ) i ( m ) s, one byone, with their ‘vertonically-dressed’ counterparts ˘ r ( † ) i ( m ) s, ˘ r ( † ) i ( m ) ≡ r ( † ) i ( m ) Y all i s ∈ i ( m ) n i s ; ˘ n i ( m ) = n i ( m ) Y all i s ∈ i ( m ) n i s (9)By definition (9), it must be clear that the operators (9)bring about only the ‘hypergraph-state relatons’ , namelythe ‘correct relatons’. Keeping vertons intact, we nowevenhandedly define the ‘hypergraph-state-qubits’ ˘ F I as, ˘ f I ≡ f I Y all the base i s for I n i s ; ˘ f † I ≡ Y all the base i s for I n i s f † I , ∀ I (10)Now, the total Hilbert space of M , namely H ( M ) M , mustbe defined by its most-complete vertonic-and-relatonictensor-product structure and its so-identified basis B ( M ) M , H ( M ) M = H ( M )(vertons) ⊗ ≤ m ≤ M H ( M )( m − relatons) B ( M ) M = (cid:8) | I · · · I m i , ∀ { I s } ⊂{ I } Cardinality M⋆ | ≤ s ≤ m , ∀ m ∈ N ≤ M ⋆ (cid:9) (where :) | I · · · I m i ≡ (cid:12)(cid:12)(cid:12) ~I k m k E ≡ ≤ s ≤ m Y I s ˘ f † I s | i (11)By the second and the third lines of (11), the basis B ( M ) M is defined to be the set of the common-eigenstatesof all the number operators n I s in which a possiblecollection of the hypergraph-state-qubits { ˘ F I · · · ˘ F I m } are present, while all the other hypergraph-state-qubits { ˘ F I } (Cardinality M ⋆ ) − { ˘ F I · · · ˘ F I m } are absent. Being so,every (cid:12)(cid:12)(cid:12) ~I k m k E ∈ B ( M ) M corresponds to a unique ‘classical’hypergraph, and so identifies the set of all (cid:12)(cid:12)(cid:12) ~I ′ k m k E ∈ B ( M ) M which as classical hypergraphs are isomorphic to it. Wewill elaborate on this classification in principle five. Result (11), concludes the quantum kinematics of M .But, let us highlight two points. First , HoloQuantumHyperNetwork Theory is, and must be, a ‘fully-internaltheory’, with no ‘external’ qubits in it whatsoever. Allits qubits correspond information-theoretically to themathematical structures of HoloQuantum Hypergraphs.Vertons are qubits for objects, the vertices, and relatonsare qubits for relations, the hyperlinks. Clearly, therecan be no other qubits in an ‘it-from-qubit’ theory of allquantum natures.
Second,
HoloQuantum HyperNetworkTheory is the theory ‘of ’ the mathematically-quantizedhypergraphs, not a quantum theory ‘on’ hypergraphs.All possible backgrounds and excitons appear only assub-theories and models, solutions and phases of M .As of now, our task is developing axiomatically allthe many-body interactions, and the total Hamiltonianof HoloQuantum HyperNetworks, which determine boththeir complete unitary dynamics and phase diagram. III. HOLOQUANTUM HYPERNETWORKTHEORY: THE FUNDAMENTAL RULEAND CASCADE OPERATORS
Let us now begin to work out the dynamical side of M . In this section, based on the next five principles,principles four to eight, we formulate ‘the fundamentalrule’ of M , namely, its core many-body interactions.HoloQuantum HyperNetwork Theory, formulatedfundamentally, must be the theory of closed quantummany body system of all the vertons and all the relatons.This ‘closed-system criterion’, which is already imposedby principle one, comes on two regards. Firstly , for beingthe quantum many body theory of all quantum natures,the above criterion must be met necessarily, becauseevery quantum universe-or-multiverse must be a closedmany body system.
Secondly , this is also a prerequisiteto M to be able to formulate all forms of HoloQuantumHyperNetworks, fulfilling its mathematical completeness.Because, having got the theory of ‘closed HoloQuantumHyperNetworks’, ‘open HoloQuantum HyperNetworks’are formulated by any partitioning of the total system ofvertons-and-relatons, and deriving the master equationout of the total unitary evolution in M . Being a closedquantum system of all ˘ F I s, the total evolution of theHoloQuantum HyperNetworks must be this unitary one , U ( M ) M ( t ) ≡ e itH ( M ) M ; H ( M ) M ≡ all X κ λ κ O ( H ( M ) M ) κ [ { ˘ f J } ; { ˘ f † I } ] (12)By the right-hand side of (12), the total Hamiltonian of M , H ( M ) M , must be a sum over all the ‘HamiltonianOperators’ O ( H ( M ) M ) κ which, as composite operators madefrom the f J s and the f † I s, realize all the many-bodyinteractions between the vertons-and-relatons. We will uniquely determine (12), by realizing the nineprinciples . Going forward, we set the next four principles. Principle 4:
The ‘hypergraph-state well-definedness’ ofHoloQuantum HyperNetworks must be preserved duringtheir entire unitary evolutions whose Hamiltonian, in itsmaximally-general form, is expressed as (12).
Explanatory Note : Remember that HoloQuantumHyperNetworks are the unitarily-evolving superpositionsof the B ( M ) M -states given in (11). Being central to M ,principle four demands that, when U ( M ) M ( t ) in (12) actson each basis-state in (11) chosen as an initial state,the ‘vertonically-unsupported relatons’ should not takepresence in the wavefunctions. The resulted dynamicalquantum constraints will be solved by a whole family of ‘Cascade Operators’ . Principle 5:
In HoloQuantum HyperNetwork Theory,all the operators in S ( M ) M , which realize (the secondquantized form of) all the hypergraphical isomorphisms,must be made out of some of the exact Hamiltonianoperators O ( H ( M ) M ) κ introduced in (12). By definition,every HoloQuantum HyperNetwork is transformedby these operators into all possible states which arequantum-hypergraphically isomorphic to it, in such away that its basis-states (11) (as ‘classical’ hypergraphs)are either kept invariant automorphically or transformedisomorphically. Being so constructed, these operatorsmust then complete the purely-vertonic conversions byrealizing all the relatonic adjustments induced by them. Explanatory Note: M calls for the above principlebecause of being mathematically ‘the quantum theory ofhypergraphs’ . Let us define how the elements of S ( M ) M must be constructed. First, one defines an abstractdiscrete space V ( M ) M , the ‘vertonic field space’, for whichthe indices of all the M vertonic degrees of freedom of H ( M ) M form a ‘coordinate system’. Now, every elementof S ( M ) M is a permutational coordinate transformationon V ( M ) M , { V i } → { V P ( i ) } , being accompanied by allthe ‘induced’ adjustments on the indices of relatons, { R i ( m ) } → { R P ( i ) ( m ) } . Realized as field operations in thesecond-quantized framework, such transformations areall realized by the specific products of the so-constructed ‘order-one isomorphism operators’ , Γ ij , Γ ij ≡ ¯ P j + [ P i + ( v j G ij v † i ) ¯ P i ] P j == 1 + n j (1 − n i ) ( v j G ij v † i − , ∀ ( j, i ) (13)which act on HoloQuantum HyperNetworks | Ψ i ∈ H ( M ) M .According to (13), the permutation ‘ j → i ’ is relazied bythe purely-vertonic operator v j v † i and is simultaneouslycompleted as an isomorphism by the purely-relatonicoperator G ij , fulfilling the relaton adjustments as neededby the vertonic inductions. Principle 6:
The global U (1) transformations of thehypergraph-state-qubit operators f ( † ) I which generateglobal phases on the multi-qubit states of B ( M ) M in(11) should be all redundant observationally. Beingso, these transformations form a quantum-mechanicalglobal U (1) fundamental symmetry of M under whichall the observables, states, many-body interactions, andthe unitary evolutions of HoloQuantum HyperNetworksmust be invariant. Explanatory Note:
HoloQuantum HyperNetworkTheory is an inherently background-less quantumfield theory. As such, considering the theory at itsfundamental level, there can be no spatial-backgroundpotential or non-trivial spatial-background topology bywhich the global phases of the multi-qubit wavefunctionscan become observables. On the other hand, by theidentification of B ( M ) M in (11), these redundant phasesof the basis-states are all generated by the global U (1) transformations on the fundamental operators of thehyper-state-qubits f ( † ) I s, which so must be the global U (1) symmetry transofrmations of M . Principle 7:
All the relatons, independent of boththeir defining hyperlink degrees and their base vertons,must be treated equally in the fundamental formulationof HoloQuantum HyperNetwork Theory. Moreover, toincorporate the vertons and the relatons all together,the theory must treat all the hypergraph-state-qubits ˘ F I with the ‘maximum-possible level of equality’, atthe level of its fundamental formulation. That is,this ‘all-qubits equal-treatment’ will be explicitly butonly minimally broken by merely the kinematical anddynamical quantum constraints for the hypergraph-statedependencies of the relatons to their base vertons.Nevertheless, by this minimally-broken symmetry, thetheory M is as maximally HyperNetworkical as it can be. Explanatory Note:
Realizing this principle, the qualityof hypergraph-ness is maximally realized in M , bothkinematically and dynamically. Being one of its mostdistinctive characteristics, this principle is ‘a must’ for M to be the fundamental complete covariant theoryof all quantum natures. Specially, this cardinal featurebecomes manifested in three ways. First , In M , thehigher-degree relatons are elementary and so irreducible degrees of freedom which are not composites of anylower-degree relatons. In particular, the two-relatonsare neither the ‘building-block relatons’, nor by anyother means ‘the more special’ relatons. Second, allthe m -relatons must participate in the many-bodyinteractions and in the total Hamiltonian of M in anequal manner. Third , there indeed must be interactionsin M which convert the m -relatons and the vertons intoone another, in all possible ways which are consistentwith the dynamical hypergraphical well-definedness ofHoloQuantum HyperNetworks and with other principles. Principle 8: M , to be the fundamental and completequantum many body theory of all quantum natures,must be not only a ‘fundamentally-random theory’ butalso a ‘maximally-random theory’. Namely, at the levelof the fundamental formulation of M , absolutely allthe many-body interactions must be defined up to themaximally-random ensembles of their defining couplings. Explanatory Note: M , by its original intention ofbeing the most fundamental theory of all quantumnatures, must be constructed in ‘the maximal resonance’ with the statistical randomness which is at the centre ofthe principle of ‘Law Without Law’ by Wheeler [1]. Thisalso resembles the randomness in the matrix-and-tensormodels [2]. In HoloQuantum HyperNetwork Theory,this fundamental randomness is ‘maximally’ realized.That is, H ( M ) M receives all the many-body-interactionoperators O ( H ( M ) M ) κ by the couplings λ κ whose statisticaldistributions are ‘maximally random’. We choose hereto take all the couplings to be (the real-or-complexvalued) continuous parameters, although they may bealso taken to be parameters with any discrete spectra.By this choice, the maximally-free Gaussian distributions for all the λ k serve the formulation of M as the mostnatural ones. Principle eight supports the ‘embeddingcompleteness’ of the theory additionally, because then allthe deterministic theories are also embeddable in it, bysuitably fixing the two characteristic parameters of thecorresponding maximally-free Gaussian distributions.Already before the step-by-step construction of U ( M ) M ( t ) ,we state The Fundamental Rule of M which later in thissection is obtained by merging principles one to eight . The Fundamental Rule of M :By the first-order many-body interactions of M , namelyby its core microscopic operations, every arbitrary pairof the hypergragh-state-qubits ( ˘ F J , ˘ F I ) , either one beingany verton or any m -relaton, can become converted toone another, ˘ F J ⇆ ˘ F I . The corresponding strengthsof these core operations are set by the maximally-freeGaussian-random couplings (¯ λ IJ , λ IJ ) . If the conversionis purely relatonic, ˘ r j ( mj ) → ˘ r i ( mi ) , no accompanyingconversions are required. But, if the conversion paircontains a verton, v j ⇆ ˘ f I , a cascade of purely-relatonicconversions will be jointly triggered. These companionconversions, which secure the dynamical hypergraphicalwell-definedness of HoloQuantum HyperNetworks, arerealized by the ‘First-Order Cascade Operators’ ( C I † j , C Ij ) .Now, by the realization of all the above eight (outof nine) principles, we systematically construct thefirst-order many-body interactions of M , namely theabove fundamental rule. But also, the correct forms of the first-order cascade operators and the first-orderHamiltonian of M are determined. This shows the roadmap to construct U ( M ) M ( t ) by realizing principle nine.Let us begin this mission with realizing principle four .One must let the initial HoloQuantum HyperNetworkbe an arbitrary quantum state on the basis (11).Then the initial wavefunction (or mixed state) clearlysatisfies all the kinematical constraints (11), by beinga superposition of (or a density operator made from)the basis-states which are all the well-defined classicalhypergraphs. Now, principle four is the demand thatthe total HoloQuantum HyperNetwork at any time t , evolved by U ( M ) M ( t ) in (12) from any such initialwavefunction, must be also a superposition of the verysame basis-states (11). On one hand, this demandsimply states that the Hamiltonian whose most generalexpression is given in (12), H ( M ) M [ { ˘ f J } | ∀ J ; { ˘ f † I } | ∀ I ] ,must serve M as an operator which is consistentlydefined on the constrained state-space (11). That is, its‘basis-dependent expression’ must be of the form, H ( M ) M [ { ˘ f J } | ∀ J ; { ˘ f † I } | ∀ I ] = X ~J k s k ,~I k r k h ( ~J k s k | ~I k r k ) (cid:12)(cid:12)(cid:12) ~I k r k E D ~J k s k (cid:12)(cid:12)(cid:12) (14)On the other hand, we need to obtain the alternative formof H ( M ) M which, first of all , manifestly satisfies all thenine principles, and second of all , is basis-independentlydetermined as a composite operator which is directly,solely and explicitly made from the ‘alphabet’ operators { ˘ f J } | ∀ J ∪{ ˘ f † I } | ∀ I , as meant by the right hand side of (12).Aiming so, let us now work out the explicit statement ofthe principle four. As being stated, all the HoloQauntumHyperNetworks, evolved from the initial superpositionsof the basis-states (11) by the unitary evolution operator(12), should also satisfy (8). That is, ∀ | Ψ i satisfying Y i ( m ) | Ψ i = 0 , ∀ R i ( m ) also must have : Y i ( m ) (cid:0) U ( M ) M ( t ) | Ψ i (cid:1) = 0 , ∀ R i ( m ) , ∀ t (15)Because U ( M ) M ( t ) is generated by H ( M ) M , the conditions(15) can be equally restated as the following conditions, [ H ( M ) M , Y i ( m ) ] ∝ function of the constraints Y j ( s ) , ∀ R i ( m ) (16)Every operator O ( H ( M ) M ) κ ≡ O ( H ( M ) M ) κ [ { ˘ f J } ; { ˘ f † I } ] in (12) isan individual term of the total Hamiltonian. Being so,because of (16), it must also satisfy the constraints, [ O ( H ( M ) M ) , Y i ( m ) ] ∝ function of the constraints Y j ( s ) , ∀ R i ( m ) (17)We call the above quantum constraints (16), or equally(17), the ‘dynamical quantum constraints’ of M . Now,for the general HoloQuantum HyperNetwork to satisfyprinciple four, we directly solve the quantum constraints (17). Besides the important simplest solutions to (17), weobtain the equally-important ‘cascade-operator solutions’of the one-to-one conversions , to be later generalized bymeans of the ‘descendant cascade operators’. Clearly enough, the most straightforward family of thegeneral solutions to (17) are given by the purely-relatonicmany-body operators , namely by the following operators, O (purely relatonic) { j a ′ ( sa ′ ) } , { i a ( ra ) } ≡ (cid:0) Y ≤ a ′ ≤ m ′ ˘ r j a ′ ( sa ′ ) (cid:1)(cid:0) Y ≤ a ≤ m ˘ r † i a ( ra ) (cid:1) (18)The complementary class of solutions to (17) will begiven by the many-body operators which contain in theman arbitrary number of the verton operators . Such avertonically-involved many-body operator genericallydoes not satisfy all the dynamical constraints in (17).This must be clear because, if a vertonic annihilationoperator v j acts on a basis-state in (11), generically itturns it into a hypergraphically-incorrect multi-qubitwavefunction, as all the V j -based relatons which werepresent in the at-the-time state of the system, would benow leftover with incomplete vertonic support. Hence,every such v j -action must be accompanied by a cascadeoperator whose defining action protect the resulted stateagainst the vertonically-unsupported relatons. Here weobtain the first-order cascade operators .First, let us consider the operator O ( H ( M ) M ) by whicha verton V j is converted to a verton V i . Because thesimplest option v j v † i can not be a Hamiltonian operator,we must enlarge it by a yet-unknown cascade operator C ij , and demand that the operator v j C ij v † i satisfies (17).We now determine the the correct form of the cascadeoperator C ij . To fulfill its intented mission, C ij must bedivisible into the following product form, C ij = M Y m =1 all choices Y y ( m ) ≡ { y ··· y m } C j → iy ( m ) (19)Every contributing y ( m ) in the right-hand side of theabove product (19), identifies an arbitrary choice of m vertons { V y , · · · , V y m } out of the M − available vertonssetting aside { V i , V j } . In (19) all such choices of y ( m ) sare incorporated impartially. Moreover, all the internaloperators C j → iy ( m ) must be expandable in this form, C j → iy ( m ) = A ⊥⊥ ¯˘ P iy ( m ) ¯˘ P y ( m ) j + A k⊥ ¯˘ P iy ( m ) ˘ P y ( m ) j + A ⊥k ˘ P iy ( m ) ¯˘ P y ( m ) j + A kk ˘ P iy ( m ) ˘ P y ( m ) j (20)In (20), the indices ‘ y ( m ) j ’ and ‘ iy ( m ) ’ identify the two ( m + 1) relatons ˘ R y ( m ) j and ˘ R iy ( m ) which relate the m vertons { V y , · · · , V y m } to V j and V i , respectively. Also, ( ˘ P y ( m ) j , ¯˘ P y ( m ) j ) and ( ˘ P iy ( m ) , ¯˘ P iy ( m ) ) are the doublets ofthe projection operators which respectively correspondto the presences and the absences of ˘ R y ( m ) j and ˘ R iy ( m ) .Furthermore, there are two sets of operator identities thatall the C ij s, and so as induced by them, each one of the C j → iy ( m ) s, must satisfy. These identities are as follows, ( C ij ) † = C ji ⇒ (cid:0) C j → iy ( m ) (cid:1) † = C i → jy ( m ) ; C jj = 1 ⇒ C j → jy ( m ) = 1 (21)By the left-side of (21), the two inversely-conversionaloperators O ( H ( M ) M ) ij ≡ v j C ij v † i and O ( H ( M ) M ) ji ≡ v i C ji v † j add up to a Hermitian unit of H ( M ) M . Right-sidecondition in (21), on the other hand, must be requiredfor the following ‘interaction-consistency criterion’ to berealized. The number operator n j = v † j v j , which surelyis a Hamiltonian operator by satisfying (17), can beregarded as the conversion of V j into the same V j , and socan be likewise recast as n j = 1 − v j C jj v † j , yielding C jj = 1 .By applying the two sets of the operator identitiesin (21) to the general expansion in (20), one obtains, C j → iy ( m ) = (cid:0) ¯˘ P iy ( m ) ¯˘ P y ( m ) j + ˘ P iy ( m ) ˘ P y ( m ) j (cid:1) + A k⊥ ¯˘ P iy ( m ) ˘ P y ( m ) j + ( A k⊥ ) † ˘ P iy ( m ) ¯˘ P y ( m ) j (22)Now, for the V j ⇆ V i conversions to realize principle four ,one demands that the enlarged operators ( v j C ij v † i , v i C ji v † j ) satisfy the dynamical constraints (17). By satisfying thisrequirement together with realizing principle five in thecorresponding form which realizes (13), and finally uponutilizing ( ˘ P I , ¯˘ P I ) = (˘ n I , − ˘ n I ) , we conclude as follows the purely-vertonic first-order cascade operators C ij , C ij = M Y m =1 all choices Y y ( m ) C j → iy ( m ) C j → iy ( m ) = (cid:0) − ˘ n iy ( m ) − ˘ n y ( m ) j + 2 ˘ n iy ( m ) ˘ n y ( m ) j (cid:1) + (1 − δ ij ) (cid:0) ˘ r † y ( m ) j ˘ r iy ( m ) + ˘ r † iy ( m ) ˘ r y ( m ) j (cid:1) (23)Now, using fermionic algebra, and because the cascadeoperator C ji always appears in H ( M ) M in the form v j C ij v † i ,one clearly sees that the cascade operators (23), can bealso operationally simplified in the following form , C ij = M Y m =1 all choices Y y ( m ) (cid:2) (cid:0) − ˘ n y ( m ) j (cid:1) + ˘ r † iy ( m ) ˘ r y ( m ) j (cid:3) (24)We now move on to formulate the first-order interactionswhich realize th conversions between vertons and relatons in HoloQuantum HyperNetwork Theory. As above, itwill be done by the Hamiltonian operator v j C Ij ˘ f † I whichconverts a verton V j to any hypergraph-state-qubit ˘ F I .By realizing principle seven , and by (24), C Ij must be so, C Ij ≡ C kj [ if index k being replaced with the index I ] , ∀ I (25)with V k = j being whatever verton. Therefore, we obtain, C Ij = M Y m =1 all choices Y y ( m ) (cid:2) (cid:0) − ˘ n y ( m ) j (cid:1) + ˘ r † Iy ( m ) ˘ r y ( m ) j (cid:3) (26)with R Iy ( m ) being the unique relaton whose indices arethe union of y ( m ) with all the vertonic indices of F I . Indeed, one can now directly see that upon the insertionof the ‘all-species cascade operator’ (26), v j C Ij ˘ f † I becomesa hypergraphically-acceptable conversion operator, asits satisfies the constraints (17). We highlight that,the inverse conversion F I → V j is conducted by itsHermitian-conjugate cascade operator, that is, by C jI = ( C Ij ) † . The result (26) completes the axiomaticconstruction of the first-order cascade operators of M .Let us now confirm that indeed principle five hasbeen correctly realized in M . By this principle, thecomplete set of the second-quantized hypergraphicalisomorphisms, S ( M ) M , must be realized in the theory asspecified both in the statement of principle five and inits explanatory note. First , the quantum-hypergraphicalisomorphisms must be identified with some compositeoperators which are constructed from a proper subset ofthe multi-qubit-conversion operators of M , that is, froma specific class of the Hamiltonian operators O H ( M ) M κ asintroduced in (12). Second , every one of these operatorsmust complete a corresponding many-body conversionin M which is purely vertonic . Being mathematicallyrepresented, the purely-vertonic conversions do realizeall the coordinate permutations on V ( M ) M , in a one-to-onemanner. Being so, to form the generators Γ ij (13), theymust be completed by the induced relatonic adjustments.Looking into C ij in (24), one confirms that, G ij = C ij , ∀ ( j, i )Γ ij = 1 + n j (1 − n i )( v j C ij v † i − (27)Being generated by the operators of the first-orderquantum-hypergraphical isomorphisms given in (27), the‘order- m ∈ N ≤ M ’ elements of S ( M ) M are now constructed, Γ i ··· i m j ··· j m ≡ Γ i j · · · Γ i m j m == ≤ s ≤ m Y s [ 1 + n j s (1 − n i s )( v j s C i s j s v † i s −
1) ]Γ i ··· i m j ··· j m | Ψ i ∈ H ( M ) M = (cid:12)(cid:12) Ψ ′ (cid:11) ∈ H ( M ) M ∼ = | Ψ i ∈ H ( M ) M , ∀ | Ψ i ∈ H ( M ) M (28) But, we highlight a very important characteristic of M .Primarily, the cascade operators C IJ are introduced in thetheory to secure the hypergraphical well-definedness ofHoloQuantum HyperNetworks during their entire timeevolutions. Being so, setting aside all the purely-vertonicones, the ‘verton-relaton-conversion cascade operators’ ,namely the pairs ( C i ( m ) j , C ji ( m ) ) in (26), do not supportany quantum-hypergraphical isometries in M . Thatis, their purely-relatonic cascades which complete theverton-relaton conversions V j ⇆ R i ( m ) , do transmute theHoloQuantum HyperNetworks ‘non-isomorphically’ . Letus then state one significant point. The immensely-largersubset of the many-body interactions in M are all thenon-isomorphic ‘transmutational interactions’ . Next, we advance to the realization of principle six . Thisprinciple defines the quantum-mechanical redundanciesof HoloQuantum HyperNetwork Theory. Re-highlightinga point of distinction, we must remind that principlefive is sourced by the mathematical character of M . Onthe other hand, principle six originates from the physicalcharacter of M , being a theory of quantum mechanics.This is simply the statement that the global phases ofthe basis-states of H ( M ) M must be unobservables. Becauseall the multi-qubit states of M are created as (5,10,11)identify, the following complete set of the global U (1) transformations on the hypergraph-state-qubits F I s, ˘ f I → exp( − iφ ) ˘ f I ; ˘ f † I → exp(+ iφ ) ˘ f † I , ∀ φ ∈ R , ∀ I (29)which develop those global phases, must be symmetrytransformations. All the observables of the theory mustbe the singlets of the global U (1) transformations (29)which assign charges ( − , +1) to the operator-doublets ( ˘ f I , ˘ f † I ) . Being so, every Hamiltonian operator O ( H ( M ) M ) must be composed from the products of equal numbersof the ˘ f J s and the ˘ f † I s, to be U (1) -chargeless. Imposingthis condition on the solutions of the quantum constraints(17) obtained so far, we conclude that all the followingoperators are among the microscopic interactions of M , { O ( H ( M ) M ) } = { O ( H ( M ) M ) [ { ˘ f J ; { ˘ f † I } ] } ⊃⊃ { (cid:0) ˘ r j s ... ˘ r j m ( sm ) (cid:1)(cid:0) ˘ r † i m ( rm ) ... ˘ r † i r (cid:1) ; v j C Ij ˘ f † I ; h.c } (30) Result (30) does summarize the fundamental rule of M . By this result, the ‘first-order’ Hamiltonian operators ofHoloQuantum HyperNetwork Theory are so concluded , { O ( H ( M ) M )(first order) } = { O ( H ) I (1) J ≡ ˘ f J C IJ ˘ f † I , ∀ J , ∀ I } (31)Let us now summarize as follows the cascade operators C IJ of the first-order Hamiltonian operators (31), C IJ ∈ { C Ij , C ˘ r i ( m ) ˘ r j ( n ) }C ˘ r i ( m ) ˘ r j ( n ) = 1 ; C jj = 1 C I = jj = M Y m =1 all choices Y w ( m ) (cid:0) − n w ( m ) j + ˘ r † Iw ( m ) ˘ r w ( m ) j (cid:1) (32)By the findings (31,32), we such obtain the first-orderHamiltonian of HoloQuantum HyperNetwork Theory , H ( M )(1 − M = X I,J λ IJ O ( H ) I (1) J = X I,J λ IJ ˘ f J C IJ ˘ f † I (33)where for the evolution to be unitary, we demand, λ JI = ¯ λ IJ (34) Now, we must realize principle eight . Being one amongthe intrinsically-Wheelerian principles of HoloQuantumHyperNetwork Theory , it does also play a central rolein fulfilling the intention of the theory M to becomethe fundamental complete quantum many body theoryof all quantum natures. Principle eight is the statementthat the couplings of all the O ( H ( M ) M ) κ operators inside thetotal fundamental Hamiltonian of the theory (12) mustbe maximally random. Applying this to the first-orderHamiltonian of M , namely to the Hamiltonian (33), allof the independent couplings λ IJ must be random anduncorrelated, defined with their maximally-free Gaussiandistributions , P ( λ IJ | ˚ λ IJ , ˆ λ IJ ) ∼ exp (cid:0) − ( λ IJ − ˚ λ IJ ) ˆ λ IJ (cid:1) (35)The characteristic parameters (˚ λ IJ , ˆ λ IJ ) in (35), by whichthe statistical mean values and mean-squared values of λ IJ s are respectively determined, take arbitrary values .We now classify all the microscopic interactions ofthe first-order Hamiltonian (33), taking care of allthe index degeneracies which are contained in thedefining summation of H ( M ) M in (33). Indeed, for everyhypergraph-state-qubit ˘ F I , there comes an independentchemical-potential µ I ≡ λ II which is Gaussian random.So, H ( M )(1 − M in (33) takes the unfolded form, H ( M )(1 − M = all X I µ I n I + all X I = J λ IJ ˘ f J C IJ ˘ f † I (36)Being microscopically discerned in terms of all thevertonic-and-relatonic interactions, the Hamiltonian (36)can be further microscopically re-expressed as follows, H ( M )(1 − M = M X i =1 µ i n i + M X m =1 | { i ( m ) } | = ( Mm ) X i ( m ) µ i ( m ) n i ( m ) ++ M X m,n =1 i ( m ) = j ( s ) X j ( s ) ,i ( m ) λ i ( m ) j ( s ) ˘ r j ( s ) ˘ r † i ( m ) + h.c ++ M X m =1 X i ( m ) X j λ i ( m ) j v j C i ( m ) j ˘ r † i ( m ) + h.c ++ i = j X j,i λ ij v j C ij v † i + h.c (37) Results (33,36,37) conclude the mission of section three .By invoking principle nine, and directly based upon thefundamental rule, we systematically develop the totalunitary evolution of the ‘flavorless theory’ in section four.In section five , we complete this constructive procedure,by finally presenting ‘the kinematically-and-dynamicallyPerfected HoloQuantum HyperNetwork Theory’ .0 IV. HOLOQUANTUM HYPERNETWORKTHEORY: COMPLETE INTERACTIONS, THETOTAL HAMILTONIAN, THE COMPACTMODEL
The mission of this section is to complete the unitaryquantum dynamics of HoloQuantum HyperNetworks byformulating all the higher-order many-body interactionsof M , and therefore, determining its total Hamiltonian.The road map to systematically get there must havealready become clear to a great extent by the results ofsection three which formulated all the core microscopicinteractions of the theory, and concluded its first-orderHamiltonian. But because to accomplish this aim, ahierarchical family of the higher-order cascade operatorsare to be correctly identified, it will be instructive for usto begin the procedure with carefully constructing thesecond-order microscopic interactions and therefore thesecond-order Hamiltonian of the theory.Based on the very same principles, the higher-ordermany-body interactions of HoloQuantum HyperNetworkTheory must be all constructed by the ‘consistentprolifications’ of its core operations as being concludedby its fundamental rule. Being so, to formulate thesecond-order interactions, we should correctly realize the‘two-to-two’ conversions between the arbitrarily-chosen‘two-sets’ { ˘ F J , ˘ F J } ↔ { ˘ F I , ˘ F I } . Beginning with thesimplest ansatz for these interactions, and albeit up tothe redundant permutations in the down-indices and inthe up-indices, we now examine the following operators, O I I J J ≡ ( ˘ f J ˘ f J )( ˘ f † I ˘ f † I ) (38)If the interaction operators (38) are purely relatonic,namely converting any two relatons to any two relatons,then by satisfying in this simplest form all the dynamicalconstraints (17), they are Hamiltonian operators as such.But if any of these four qubits is vertonic, then the ansatz(38) generically violates the dynamical hypergraphicalwell-definedness of HoloQuantum HyperNetworks for thesame reason explained in section three in the case of thecore interactions. Being so, by identifying the needfulfirst-order cascade operators for { ˘ F J , ˘ F J } ↔ { ˘ F I , ˘ F I } and combining them sequentially, we enlarge the bareoperators (38) by the corresponding assemblies of theircascade conversions. This direct simple procedure leadsus to both of the following Hamiltonian operators, O ( H )( I ,I )(2)( J ,J ) ≡ ˘ f J ˘ f J ( C I J C I J ) ˘ f † I ˘ f † I ≡ ˘ f J ˘ f J C I I J J ˘ f † I ˘ f † I O ( H )( I ,I )(2)( J ,J ) ≡ ˘ f J ˘ f J ( C I J C I J ) ˘ f † I ˘ f † I ≡ ˘ f J ˘ f J C I I J J ˘ f † I ˘ f † I (39)Given the ‘descendant cascade operators’ C II ′ JJ ′ defined in(39), now the operators O ( H )( I,I ′ )(2)( J,J ′ ) satisfy the constraints(17). Result (39) concludes all the second-order cascadeoperators and all the second-order interactions of M . Let us highlight a point here. For every arbitrarily-chosenconversion { ˘ F J , ˘ F J } ↔ { ˘ F I , ˘ F I } , there is a freedomin enlarging the operator (38) according to either of thetwo distinguished channels of the one-to-one conversionswhich build up this process. This freedom is mirroredin the two independent second-order operators that (39)identifies. Therefore, these two independent operatorsmust contribute to the Hamiltonian of the theory withtwo independent couplings, otherwise the resulted totaldynamics of the theory would be restricted unnecessarily.This must be clear because the dimension of the completespace of the second-order interaction operators is countedby the basis-operators which, modulo all the redundantpermutations, can be so decomposed in terms of theirdirect ancestors, { O ( H ) I J O ( H ) I J = ( ˘ f J C I J ˘ f † I ) ( ˘ f J C I J ˘ f † I ) ; ∀ ~J , ∀ ~I } (40)Therefore, by the counting of (40), the Hamiltonianwhich generates the complete dynamics of the theoryshould receive an independent coupling for each one ofthe two operators identified in (39). By the result (39),and again by the realizition of principle eight, we obtainas follows the second-order Hamiltonian of HoloQuantumHyperNetwork Theory , H ( M )(2 − M = X J ,J X I ,I λ I I J J ˘ f J ˘ f J C I I J J ˘ f † I ˘ f † I (41)with three structural patterns on the λ -couplings. First ,to ensure the unitarity of the evolution, λ J J I I = ¯ λ I I J J (42) Second , to undo the permutational redundancies, λ I I J J = − λ I I J J ; λ I I J J = − λ I I J J (43) Third , absolutely all the independent couplings λ I I J J are the maximally-free Gaussian random parameters, tobe taken from the same distributions as in (35).By opening-up the hypergraph-state-qubit contentof (39), we obtain the microscopically-distinguishedsecond-order many-body interactions of HoloQuantumHyperNetwork Theory as follows, O ( H )( J ,J ) , ( I ,I ) ∈ { (˘ r j s ˘ r j s )(˘ r † i m ˘ r † i m ) ; h.c ;( v j ˘ r j s ) C i m i m j j s (˘ r † i m ˘ r † i m ) ; h.c ;( v j v j ) C i m i m j j (˘ r † i m ˘ r † i m ) ; h.c ;( v j v j ) C i m i j j ( v † i ˘ r † i m ) ; h.c ;( v j v j ) C i i j j ( v † i v † i ) ; h.c } (44)1Having manifested (44), let us further expose all themicroscopically-distinct terms in (41). By taking careof all the qubit degeneracies in the summations of (41),and by appropriately renaming some of the couplings,the microscopically-detailed form of (41) is so obtained, H ( M )(2 − M = X I ,I λ I I ˘ n I ˘ n I ++ X K J = I X J,I λ KJ,I ˘ n K ( ˘ f J C IJ ˘ f † I ) ++ { J s }∩{ I r } =0 X J ,J ,I ,I λ I ,I J J ˘ f J ˘ f J C I ,I J ,J ˘ f † I ˘ f † I (45)By the first class of interaction terms in (45), the numberoperators of all possible pairs of ‘HyperNetwork qubits’are coupled Gaussian-randomly and ‘index-globally’. Bythe interactions of the second term of (45) the one-to-oneconversions of any two distinct qubits, coupled withthe number operator of an arbitrary qubit, are turnedon Gaussian-randomly. Finally, the interactions of thethird class of Hamiltonian terms in (45) consist of all thepossible two-to-two conversions of four distinct qubits,activated by the global and Gaussian-random couplings.Next, the higher-order Hamiltonians of the theorymust be all built up in the exact same way H ( M )(1 − M and H ( M )(2 − M have been constructed. Because by nowall the steps are crystal clear, we summarize the finalresult. The ‘order- m ’ Hamiltonian of HoloQuantumHyperNetwork Theory, H ( M )( m − m ) M , is so concluded , H ( M )( m − m ) M = X J ...J m X I ...I m λ I ··· I m J ··· J m ( m Y s =1 ˘ f J s ) C I ··· I m J ··· J m ( m Y r =1 ˘ f † I r ) (46)in which the descendant ‘order-m cascade operators’ , C ~I~J ≡ C I ...I m J ...J m , being required to have the many-bodyinteraction operators of M satisfying (17), are identifiedby means of their ancestors C IJ given in (32) simply as, C I ··· I m J ··· J m ≡ ≤ s ≤ m Y s C I s J s (47)The couplings λ I ··· I m J ··· J m in (46) are antisymmetric both intheir down-indices and in their up-indices, and also forunitarity satisfy the m -body version of (42). Namely, λ I ··· I m J ··· J m = λ I ··· I m [ J ··· J m ] = λ [ I ··· I m ] J ··· J m ; λ J ··· J m I ··· I m = ¯ λ I ··· I m J ··· J m (48)Besides (48), all the independent couplings λ I ··· I m J ··· J m , atthe fundamental level of the theory , are uncorrelatedlymaximally-free Gaussian-random. Namely, P ( λ ~I~J | ˚ λ ~I~J , ˆ λ ~I~J ) ∼ exp (cid:0) − (cid:0) λ ~I~J − ˚ λ ~I~J (cid:1) ˆ λ ~I~J (cid:1) (49) Now we are at the right point to meet principle nine . Principle 9: M , for being the ‘it-from-qubit’ theoryof all quantum natures and all possible HoloQuantumHypergraphs, must be ‘ covariantly complete ’. Explanatory Note:
Principle nine, by being thestatement of ‘covariant completeness’ , is a must tosecure the fulfillment of the intention of HoloQuantumHyperNetwork Theory. By this principle, being realizedon the ‘multiverse face’ of M , every quantum (or evenclassical) many body system of the arbitrarily-chosenobjects and their arbitrarily-chosen relations, which isin-principle realizable, must be both completely andform-invariantly (= covariantly) formulable by thisvery theory. That is, the complete time-dependentinformation of all the observables and all the statesof all of these many body systems must be either extractable from the complete theory as its covariantcontextual model-specifications or consistent solutions, or be contained in its total phase diagram as thecovariant effective-or-emergent (sub)theories. Realizedon its mathematical ‘hypergraphical face’ , this principlestates that, every dynamical hypergraphical (and soalso graphical) network which is both structurally andfunctionally consistent must be covariantly formulablefrom within the this complete theory, either directly asextractions or solutions, or as its phase-specific effectiveemergences.For principle nine to come true, given the absolutegenerality of our principles and all the formulation, thereis only one condition we still need to impose. By theWilsonian renormalization group flow, the landscape ofthe low-energy fixed-points of the theory must be, bothphysics-wise and network-wise, covariantly complete.This ‘low-energy completeness’ of the theory requiresthat, all the relevant-or-marginal multi-qubits-conversionoperators which are acceptable by principles four, five,six and seven, must contribute, in the way stated byprinciple eight, to the total Hamiltonian (12). So, thetotal Hamiltonian of M , H ( M ) M , must be form-invariantunder the whole Wilsonian renormalization group flow .Let us now realize principle nine by which the completemicroscopic interactions and the total Hamiltonianof HoloQuantum HyperNetwork Theory H ( M ) M mustbe obtained. By principle two, M lives in (0 + 1) dimensions. Exactly in these dimensions, all the M ⋆ hypergraph-state ‘alphabet operators’ ( ˘ f I , ˘ f † I ) havecanonical dimension zero, by formally being fermions.As such, all the acceptable convertors, no matter how‘large’ they are as fermionically-made composite bosonicoperators, are relevant, in fact, are ‘equally relevant’.So, H ( M ) M must be formed hierarchically by receiving all O ( H )( m − m ) s. Namely, to realize principle nine, one mustsum up H ( M )( m − m ) M , ∀ m ≤ M ⋆ ≥ , to obtain H ( M ) M .2Therefore, the realization of principle nine is so fulfilled , H ( M ) M = m = M ⋆ X m =1 H ( M )( m − m ) M (50)Now, having realized all the nine principles, and giventhe m -body results (46,48,49), we conclude as follows. The complete unitary evolution operator of HoloQuantumHyperNetworks , U ( M ) M ( t ) , is generated in accordance with (12), by the following total Hamiltonian , H ( M ) M == m = M ⋆ X m =1 { X J ...J m X I ...I m λ I ··· I m J ··· J m ( m Y s =1 ˘ f J s ) ( m Y ℓ =1 C I ℓ J ℓ ) ( m Y r =1 ˘ f † I r ) } (51)Let us make explicit the microscopic content of (51). Bydiscerning all the index degeneracies in the summation,and upon renaming some of the the couplings, the totalHamiltonian (51) can be re-expressed as follows, H ( M ) M = m = M ⋆ X m =1 { ≤ r ≤ m X I r λ I ...I m ˘ n I ... ˘ n I m ++ ≤ s,r ≤ m X I r = J s λ ~I~J ( m Y s =1 ˘ f J s ) C ~I~J ( m Y r =1 ˘ f † I r ) ++ m X p =1 1 ≤ c ≤ p X K c ≤ s,r ≤ m − p X I r = J s λ ~K~J,~I ( p Y c =1 ˘ n K c ) ×× ( m − p Y s =1 ˘ f J s ) C ~I~J ( m − p Y r =1 ˘ f † I r ) } (52)That is, HoloQuantum HyperNetwork Theory realisesthree microscopically-distinguished classes of abstract m -body hypergraph-state-qubit interactions, for every m ≤ M ⋆ ≥ . By the first class, (only) the number operatorsof m hypergraph-state-qubits interact randomly. By thesecond class, m one-to-one conversions between a groupof all-distinguished hypergraph-state-qubits, togetherwith their cascade conversions, are triggered randomly.By the third class, both of the above-mentioned typesof interactions are simultaneously merged in arbitrarymixtures and by random strengths.We re-highlight three important points about H ( M ) M . First , in M , interactions between the networkical qubitsare ‘all-species-inclusive’. Namely, the multi-convertingqubits can be all vertons, can be a number of relatonsof the same-or-different degrees, or can be any arbitrarymixtures of vertons and relatons. Second , In M , all the HyperNetwork qubits withall possible indices interact, at the fundamental level.That is, the many-body interactions are index-global. Third , when one comes to develop application-specificsub-theories or models of M , all the types of necessary disciplines on the random couplings must be imposed,by simply choosing so, by imposing extra symmetries, orby the emergences. For example, in many sub-theories,solutions or phases of the theory, the ‘unfrozen’ ˆ λ ~I~J arereduced to only specific subsets of the HyperNetworkqubit spices or their indices. Moreover, the parameters ofthe Gaussian distributions (49) of the unfrozen couplingsmust be chosen, or be effectively developed, such thatthe characteristic statistical-average-measures of thosecouplings obey the necessary conditions or constraints,given the context. As a physically-significant example,in the models, solutions or phases of the theory in which geometric locality is either a built-in assumption or anemergent feature, usually a set of locality constraints on the distribution-fixing parameters of those randomcouplings should ensure that their associated statisticalmeasures do properly depond on the metric-induceddistances in the corresponding geometry.Before moving to the next section, we must nowfeature a remarkable sub-theory of the complete theory (51). We will name it the ‘Compactified HoloQuantumHyperNetwork Theory’ , and will likewise denote itby M (cid:13) . Both structurally and qualitatively, M (cid:13) must be regarded as the first ‘child’ of HoloQuantumHyperNetwork Theory M . M (cid:13) as a sub-theory of M ,is remarkable by being both ‘the maximal one’ and ‘theminimal one’ . Being qualified as a maximal sub-theory,its definition features the following characteristics. First , the entirety of the hypergraph-state-qubits ˘ F I areincluded in its quantum statics, so that its total Hilbertspace is identical to (11). Second , the complete abstractmany-body interactions of HoloQuantum HyperNetworkTheory are kept activated in its dynamics. That is, H ( M (cid:13) ) M ≡ H ( M ) M { O ( H ( M(cid:13) ) M )(order − m ) ; ∀ m ≤ M ⋆ ≥ } ≡ { O ( H ( M ) M )(order − m ) ; ∀ m ≤ M ⋆ ≥ } (53)Being qualified at the same time as a minimal sub-theory,its definition features one more characteristics as follows.The total number of independent couplings in the totalHamiltonian of M (cid:13) are kept as minimal as possible, withthe constraint that (53) still hold. To realize that, thissub-theory is dynamized by a total Hamiltonian whichexponentiates the first-order Hamiltonian (33). Namely, H ( M (cid:13) ) M ≡ exp (cid:0) H ( M )(1 − M (cid:1) = exp (cid:0) X I,J λ IJ ˘ f J C IJ ˘ f † I (cid:1) (54)The above Hamiltonian of M (cid:13) which (up to the additionof an irrelevant constant term) defines a sub-theory of(51), realizes a campactification of the moduli space ofthe Gaussian random couplings of the complete theory M to that subspace which triggers (33).3 V. HOLOQUANTUM HYPERNETWORKTHEORY: THE MAXIMALLY-FLAVOREDFORMULATION, ‘THE PERFECTED THEORY’
Now, HoloQuantum HyperNetwork Theory M mustbecome ‘perfected’, based on the very nine principles . On one hand , this perfection amounts to arriving at the ‘absolute’ mathematical generality of M , by formulatingthe HoloQuantum hypergraphs-and-HyperNetworkswhich are maximally flavored, both relatonically andvertonically. By this ultimate mathematical enlargementof M , most particularly, all possible HoloQuantumHyperNetworks (and so hypergraphs) whose vertons(and so quantum vertices) and relatons (and so quantumhyperlinks) are in the most general form ‘weighted’ and ‘oriented’ are formulated. Moreover, this goal will beaccomplished in the very ‘it-from-qubit’ natural way.However we must highlight that the identifications ofthese flavors are not restricted to the hypergraphicalweights or orientations. Most-generally understood, themaximally-falvored relatons and vertons frepresent allpossible families of quantum hyperlinks and quantumvertices which, although can take up all the differentidentification (when being interpreted contextually), arenevertheless the constituting and interacting degrees offreedom of the ‘Perfected M ’. On the other hand , thismaximally-flavored formulation amounts to the ultimaterealization of principle nine in the physics-face of M .This is so because the resulted maximally-flavored M does formulate, as is intended, ‘absolutely’ all possiblequantum many body systems of quantum objects andtheir quantum relations, namely, all quantum natures .We begin to construct the total quantum kinematics,namely H ( FM ) M , and the complete quantum dynamics,namely H ( FM ) M , of the ‘Maximally-Flavored M ’ . Modulodefining the flavored cascade operators, the procedure isstraghtforward, nevertheless it perfects qualitatively M . First , we enlarge the Hilbert space H ( M ) M of section three,by replacing its relatons with the ‘upgraded relatons’ which, for every choice of the base-vertons, are flavoredin the most general form . Clearly, HyperNetworks can,and in fact do generically, develop the different ‘classes =flavors’ of hyperlinks which represent the distinct typesof relations between the same vertex-represented objects.Mathematically, for example, all possible weights ororientations of every single hyperlink can be formulatedas certain flavors, as we will soon manifest. Physically,on the other hand, the maximally-flavored relatons arequbits for the ‘distinct hyperlinks’ which represent allthe ‘distinct-type’ many-body interactions, many-bodycorrelations, or the structural-or-functinal many-bodycompositions, connections, or participatig associations.Being maximally general, we let every relaton R i ( m ) beupgraded by its most general set of the ‘purely-relatonicflavors’ = Set R i ( m ) (cid:0) α [ i ( m ) ] (cid:1) , as R α [ i ( m ) ] i ( m ) . Second , we must likewise enlarge H ( M ) M by replacing itsflavorless vertons with the ‘upgraded vertons’ which aremost-generally flavored . By definition, the such-flavoredvertons represent (as before,) information-theoreticallyall possible defining-or-refining states which can beassociated to the arbitrarily-chosen objects, either eachby each distinctly, or in the arbitrarily-chosen groups.For example, on one hand they can represent the weightsor the colors of the vertices, and on the other hand, thespins, the bosonic states of different population numbers,the ‘generations’, and any quantum numbers of particles.Being maximally general, therefore, we let every singleverton V i be now upgraded by its most general set ofthe ‘purely vertonic flavors’ = Set V i (cid:0) a i (cid:1) , as V a i i . Forconsistency, however, these purely-vertonic flavors alsoinduce their ‘vertonically-induced flavors’ on relatons.This must be clear, because relatons are defined tobe the information-theoretically-defining qubits of thequantum hyperlinks which must be uniquely identifiedby their now-flavored quantum base-vertices. Being so,aside from its purely-relatonic flavors, every relatonmsut be additionally carrying a whole sequence of thepurely-vertonic flavors which define its base-vertons, tobe identified uniquely. Collecting all the data, we denote the all-flavores-included m -relatons by R α [( i ai ) ( m ) ]( i ai ) ( m ) . Third , to have the maximally-flavored theory of M be formulated with notational compactness, assumingthat the above index-details are all remembered , themaximally-flavored hypergraph-state-qubit operatorswill be collectively denoted by (cid:0) ˘ f γ I I , ˘ f † γ I I (cid:1) , in which I ∈ N ≤ M⋆ is the all-inclusive counter of the very flavorlessqubits. Now, the total state-space of Maximally-FlavoredHoloQuantum HyperNetwork Theory , H ( FM ) M , is definedby its tensor-product structure and by its basis B ( FM ) M , H ( FM ) M = H ( FM )(vertons) ⊗ ≤ m ≤ M H ( FM )( m − relatons) B ( FM ) M = (cid:8) all choices Y I s all choices Y γ Is ˘ f † γ Is I s | i } (55)The identification (55) is the quantum-kinematicalmaximally-flavored realization of principles one to three .Reading directly from (55), the very maximally-flavoredformulation of principles seven, eight and six must beclear. Specially, the U(1) redundancy transformations ofthe maximally-flavored qubits are so formulated, ∀ ( I, γ I ) : ˘ f γ I I → e − iφ ˘ f γ I I ; ˘ f † γ I I → e + iφ ˘ f † γ I I (56)Next, we move on to formulate systematically the totalunitary dynamics of the perfected theory . Building on allthe results of the previous sections, this mission must beaccomplished as straightforwardly as possible. Being so,the major step to obtain H ( FM ) M is to realize principles four and five by defining the ‘maximally-flavored cascadeoperators’ , which generalize (32,47) correctly.4Remembering the notations { ˘ F γ I I } = { V a i i ; R α [( i ai ) ( m ) ]( i ai ) ( m ) } , the Maximally-Flavored Cascade Operators C I ηI J γJ , and alltheir order- m descendants , are so identified correctly, { C I ηI J γJ } ≡ {C I ηI j aj ( µ I ηI j aj ) ; All the purely relatonic ones = 1 }C I ηI j aj ( µ I ηI j aj ) == M Y m =1 all choices Y ( y ay ) ( m ) all choices Y α [( y ay ) ( m ) j aj ] C j aj → I ηI α [( y ay ) ( m ) j aj ] (cid:2) µ β [ I ηI ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] (cid:3) C j aj → I ηI α [( y ay ) ( m ) j aj ] (cid:2) µ β [ I ηI ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] (cid:3) == (cid:0) − ˘ n α [( y ay ) ( m ) j aj ]( y ay ) ( m ) j aj ++ all choices X β [ I ηI ( y ay ) ( m ) ] µ β [ I ηI ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] ˘ r † β [ I ηI ( y ay ) ( m ) ] I ηI ( y ay ) ( m ) ˘ r α [( y ay ) ( m ) j aj ]( y ay ) ( m ) j aj (cid:1) µ β [ I ηI ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] ≡ all choices [ β [ I ηI ( y ay ) ( m ) ] { µ β [ I ηI ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] } µ I ηI j aj ≡ M [ m =1 all choices [ ( y ay ) ( m ) all choices [ α [( y ay ) ( m ) j aj ] µ β [ I ηI ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] C I ηI ··· I ηImm J γJ ··· J γJmm = C I ηI J γJ · · · C I ηImm J γJmm (57)Here comes the identification of all the characters whichparticipate in the generalized definitions given by (57).Every index-set ( y a y ) ( m ) ≡ { y a y ... y a ym m } identifies anarbitrarily-chosen set of m ≤ M − ≥ ‘intermediate vertons’ { V a y · · · V a m y m } . Being highlighted, the second productin (57) incorporates the largest spectra of the flavorswhich must be attributed to the intermediate vertons,for them to be identified uniquely. Given the chosen ( y a y ) ( m ) , every ˘ R α [( y ay ) ( m ) j aj ]( y ay ) ( m ) j aj is the uniquely-identifiedrelaton whose base indices are given by { j a j } ∪ ( y a y ) ( m ) ,and has the purely-relatonic identity-flavor ‘ α ’. Given j a j ( y a y ) ( m ) , the third product in (57) incorporates allthe ‘such-based relatons’ which are accommodated in thelargest spectra of the purely-relatonic identity-flavors. ˘ R β [ I ηI ( y ay ) ( m ) ] I ηI ( y ay ) ( m ) is, likewise, the unique relaton whose baseis given by the union of ( y a y ) ( m ) with all the vertonicindices of I η I , and is specified by the purely-relatonicidentification-flavor ‘ β ’. By the intermediate summationin (57), all the relatons with the base I η I ( y a y ) ( m ) in thelargest spectra of their purely-relatonic identity-flavorsmust be integrated. We complete the identification of the characters in (57),by specifying the ‘ µ -couplings’ which, at the level ofthe fundamental formulation of the perfected theory,come in the definitions of its maximally-flavored cascadeoperators C I ηI j aj ( µ I ηI j aj ) . Upon realizing principle eight,for every ‘initial’ purely-relatonic flavor α [( y a y ) ( m ) j a j ] ,the ‘flavor-transition-couplings’ µ β [ I ηI ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] in thesixth line of (57) are determined by a total numberof ‘ |{ β [ I η I ( y a y ) ( m ) }| − ’ uncorrelated maximally-freeGaussian-random parameters { κ α ( j a j ; ( y a y ) ( m ) ; I η I ) } ,ensuring that they satisfy the following constraint, all choices X β [ I ηI ( y ay ) ( m ) ] µ β [ I ηI ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] = 1 (58)One can consistently restore (32) from the generalizedcascade operators. The reduction is consistent becausethe flavorless formulation developed in sections two, threeand four is equivalent with the flavored formulation inthis section, by restricting to only one purely-vertonicflavor and only one purely-relatonic flavor. Because ofthis, upon applying the summation-constraint (58) to theone-flavor case, we obtain only one coefficient µ αα = 1 ,such that (57) does consistently reduce to (32). Now, forbetter clarity , and as the simplest example, we exposehere the first-order cascade operators for the sub-theoryof HoloQuantum HyperNetworks defined with flavorlessvertons and with ‘ ± ’ flavored relatons. Here come all thenon-identity cascade operators of the model, C I ηI j = M Y m =1 all choices Y y ( m ) ≡ { y ... y m } C j → I ηI ( − ) [ y ( m ) ] C j → I ηI (+) [ y ( m ) ] C j → I ηI ( ± ) [ y ( m ) ] = ( 1 − ˘ n ± y ( m ) j ) ++ (cid:2) µ −± ( j ; y ( m ) ; I η I ) ˘ r †− I ηI y ( m ) + µ + ± ( j ; y ( m ) ; I η I ) ˘ r † + I ηI y ( m ) (cid:3) ˘ r ± y ( m ) j µ ±± ( j ; y ( m ) ; I η I ) = 12 ± κ ± ( j ; y ( m ) ; I η I ) (59)Let us confirm that principle five is indeed realized in theperfected theory. Generators of quantum-hypergraphicalisomorphisms are given by the cascade operators in (57), Γ i bi j aj ≡ n a j j (1 − n b i i )[ v a j j C i bi j aj (ˆ µ i bi j aj ) v † b i i − µ i bi j aj ≡ M [ m =1 all choices [ ( y ay ) ( m ) all choices [ α [( y ay ) ( m ) j aj ] all choices [ β [ i bi ( y ay ) ( m ) ] { δ β [ i bi ( y ay ) ( m ) ] α [( y ay ) ( m ) j aj ] } | ( ) (60)By (60), the complete set of the quantum-hypergraphicalisomorphisms in the perfected theory are realized by the C i bi j aj s whose flavor-transiotion-couplings are localized (bythe tuned Gaussian distributions of their { κ } variables)on ‘ µ βα = δ βα ’, and in addition, they do satisfy (58).5Let us highlight the wisdom for having defined thecascade operators of the perfected theory as in (57).By the definitions in (57), not only all the enlargedconversion operators ˘ f γ J J C I ηI J γJ ˘ f † η I I satisfy the flavoredgeneralizations of (17), and at the same time principlefive is realized for the maxiamlly-flavored HoloQuantumHyperNetworks, but also the unitary dynamics of theperfected theory, as we now formulate, receives its mostcomplete realization of all the nine principles.By straghtforwardly generalizing the results of sectionsthree and four, and utilizing the maximally-flavoredcascade operators given in (57), we realize the ninthprinciple to obtain as follows the total Hamiltonian ofthe Perfected HoloQuantum HyperNetwork Theory, H ( FM ) M = m = M ⋆ X m =1 H ( FM )( m − m ) M H ( FM )( m − m ) M == all X J ··· J m all X γ J ··· γ Jm all X I ··· I m all X η I ··· η Im λ I ηI ··· I ηImm J γJ ··· J γJmm Φ I ηI ··· I ηImm J γJ ··· J γJmm Φ I ηI ··· I ηImm J γJ ··· J γJmm = ( m Y s =1 ˘ f γ Js J s ) (cid:0) C I ηI J γJ · · · C I ηImm J γJmm (cid:1) ( m Y r =1 ˘ f † η Ir I r ) λ I ηI ··· I ηImm J γJ ··· J γJmm = λ I ηI ··· I ηImm [ J γJ ··· J γJmm ] = λ [ I ηI ··· I ηImm ] J γJ ··· J γJmm λ I ηI ··· I ηImm J γJ ··· J γJmm = ¯ λ J γJ ··· J γJmm I ηI ··· I ηImm P ( λ ~I ~η~I ~J ~γ ~J | ˚ λ ~I ~η~I ~J ~γ~J , ˆ λ ~I ~η~I ~J ~γ~J ) ∼ exp (cid:0) − ( λ ~I ~η~I ~J ~γ~J − ˚ λ ~I ~η~I ~J ~γ~J ) ˆ λ ~I ~η~I ~J ~γ~J (cid:1) (61)By discerning all the degeneracies in the summationsof (61), H ( FM )( m − m ) M takes a detailed form similar to(52), but now with all the flavors being incorporated.Now, knowing the complete unitary dynamics of theperfected theory, the ‘Compactified Maximally-FlavoredHoloQuantum HyperNetwork Theory’ must be defined,similar to its flavourless counterpart in section four. Thistheory is the ‘minimal-maximal child sub-theory’ of theperfected theory, with the same quantum statics as (55),but with the total hamiltonian which exponentiates thefirst-degree Hamiltonian in (61). Namely, H ( FM (cid:13) ) M = H ( FM ) M { O ( H )( FM (cid:13) )( m ) ; ∀ m ≤ M ⋆ ≥ } = { Φ I ηI ··· I ηImm J γJ ··· J γJmm ; ∀ m ≤ M ⋆ ≥ } H ( FM (cid:13) ) M = exp (cid:0) all X J , γ J all X I , η I λ I ηI J γJ ˘ f γ J J C I ηI J γJ ˘ f † η I I (cid:1) (62) The perfected theory, which we have already concluded,has been axiomatically defined and formulated based onthe nine principles. Most remarkably, its total Hilbertspace (55) and its complete unitary dynamics (61) are, bytheir axiomatic constructions, as general, fundamentaland complete as they can be. By the such-accomplishedultimate realization of principle nine, the perfectedtheory is covariantly complete. Being so, even whensheerly regarded as a theory of quantum mathematics,the Maximally-Flavored HoloQuantum HyperNetworkTheory is the kinematically-and-dynamically completetheory of ‘all possible forms’ of mathematically-quantizedhypergraphs . Now, given the extremely-generic domainof all dynamical hypergraphs which are formulated bythe flavorless theory, defined in the first four sections, wemust take in what follows one and only one more stepto work-out the ‘hypergraphical covariant completeness’of the maximally-flavored theory. That is, we mustwork-out how the perfected theory formulates all thedynamical mathematically-quantized hypergraphs whichare defined with arbitrarily-oriented hyperlinks , and with arbitrarily-weighted vertices and hyperlinks .The natural and direct ‘it-from-qubit’ way by whichone presents the general, complete formulation of themaximally-oriented and maximally-weighted quantizedhypergraphs is to have it extracted as a sub-theory of(55) and (61), upon defining all the orientations andall the weights as specific flavors . Let us state thewhole point. In HoloQuantum HyperNetwork Theory,every relaton of every arbitrarily-chosen orientation orweight, and every verton of every arbitrarily-chosenweight, must be distinctively taken to be one independentquantum degree of freedom of (55), that is, it should beidentified with one formally-fermion qubit F γ I I whichis addressed with a ‘correspondence-flavor’ . Being so,in the promised theory of the maximally-oriented andmaximally-weighted HoloQuantum HyperNetworks,one can ‘weight stuff’ and ‘orient stuff’ just with theformally-fermionic qubits. Restated, the complete set ofrelatonic orientations and weights, and their completeset of vertonic weights must be mapped in one-to-onemanners to their coresspondence sets of flavors.Let us now begin to formulate the above-featuredsub-theory of the perfected M with identifying all thehyperlink orientations in HoloQuantum HyperNetworks.By definition, a hyperlink of m base-vertices has m ! orientations, each one of them corresponding to a uniqueordering of its base-vertices. As such, to formulate themost general orientations of the quantum hyperlinks,every m -relaton, identified earlier in (11) as ˘ R i ( m ) , mustbe now ‘branched-out’ as the m ! ‘orientationally-flavoredrelatons’ ˘ R α ( m ) i ( m ) in the Hilbert space (55). Namly, themaximally-oriented HoloQuantum hypergraphs must beformulated by this family of orientation flavors , ∀ i ( m ) , ˘ R i ( m ) ֒ → { ˘ R α ( m ) i ( m ) ; α ( m ) ∈ N ≤ m ! } (63)6Like orientations (63), all the hypergraphical weights forma distinct set of the flavors, named the ‘ weight flavors ’.Stating precisely, the ‘all-inclusive set’ of all the definedweights for the structural constituents of a HoloQuantumhypergraph, being vertices or the m -degree hyperlinks,is mapped in a one-to-one manner to a complete set ofvertonic-and-relatonic weight flavors. Being maximallygeneral, the weight flavor of every structural-qubit F I comes with its distinct labeling ω ( F I ) ≡ ω I , and alsowith a distinct arbitrarily-chosen spectrum W I ≡ { ω I } .That is every hypergraph-state-qubit F I is branched-outby taking flavors as follows, ∀ I ; F I ֒ → { F ω I I ; ω I ∈ W I } ; (64)Now, the total quantum kinematics and the completeunitary quantum dynamics of all possible HoloQuantumHyperNetworks which are both maximally oriented andmaximally weighted are given by the Hilbert space (55)and the total Hamiltonian (61) whose ‘structural flavors’ are identified as follows, { F γ I I } ≡ { V ω i i ; R α ( m ) ; ω [( i ω ) ( m ) ]( i ω ) ( m ) } (65)We highlight a point about the many-body interactions(61) of HoloQuantum HyperNetwork Theory endowedwith the structural flavors (65). This point is a directimplication of the maximally-flavored version of thefundamental rule, which is equal to the very same rulestated in section three, upon the replacements F I ֒ → F γ I I and C IJ ֒ → C I γI J γI . Given this rule, at the level of thefundamental formulation of the theory, not only thequantum vertices and all the quantum hyperlinks ofdifferent degrees can freely convert into one another,but also now all their different orientations and all theirdifferent weights can convert into one another.The triplet (55,61,65) does conclude the promisedmathematical covariant-completion of the flavorless‘basic-class’ theory, by defining and formulating theHoloQuantum HyperNetworks which are endowedwith general orientations and weights. That is, thekinematically-and-dynamically complete ‘it-from-qubit’formulation of all possible mathematically-quantizedhypergraphs is the above ‘sub-theory triplet’ (55,61,65).But, now as an instructionally-good example ofthe HoloQuantum HyperNetworks whose some of flavorstake continuous spectra, and also because weights areusually taken to be real-or-complex valued numbers, let us treat the spectra of the weight-flavors to be thearbitrarily-chosen continues number-fields . By this take,the defining operators of the hypergraph-state-qubits willbe turned into formally-fermion field-function operatorsliving on the topological product of the time-dimensionand an abstract ‘wight-space’. Being so, one will obtain a ‘HoloQuantum-HyperNetworkical Continuum QuantumField Theory’ living on this abstract space, albeit at amerely formal level. To this aim, we now represent even-handedly all thehypergraph-state-qubits which are, as already specified,both maximally oriented and maximally weighted by thefield-function operators ˘ f ( † ) α I I ( ~ω I ) ≡ ˘ f ( † ) α I I ( ~x I ) , where, { ˘ f ( † ) α I I ( ~x I ) } = { v ( † ) i ( ~x i ) ; r ( † ) α ( m ) i ( m ) ( ~x i ( m ) | ~x , · · · , ~x m ) } (66)In the right hand side of (66), each finite-dimensionalvector ~x I identifies the continuously-valued weight ofthe corresponding hypergraph-state-qubit F I , whosespectrum can be independently chosen, to be maximallygeneral. For example, if the corresponding weight is areal number, ~x ≡ x ∈ R , but if it is a complex number, ~x ≡ ( z, ¯ z ) ∈ Σ with Σ being the complex plain C orany Riemannian surface whose topology one choosesarbitrarily. So, in the models where all the weightsare real numbers, or complex numbers, we obtain aformally-all-fermionic HoloQuantum-HyperNetworkicalQuantum Field Theory, with novel interactions, livingon abstract Lorentzian manifolds similar to the (1 + 1) dimensional or to the (2 + 1) dimensional spacetimes. Tohighlight, HoloQuantum HyperNetworks with multiplehyperlinks correspond to the special case in whichrelatons are integrally weighted, namely ~x = n ∈ N .The cascade operators C I βI J αJ ( ~x J ; ~x I ) and the totalHmailtonian of the complete unitary dynamics ofthis continuously-flavored sub-theory are defined as in(57,61), upon turning the summations-and-productsover all the continuous weights into integrations and theexponentials of logarithmic integrations, respectively. Inparticular, its compactified sub-theory is defined by, H ( FM (cid:13) ) M = exp (cid:0) X J,I X α J ,β I Z d~x I d~x J λ I βI J αJ ( ~x J ; ~x I ) ×× ˘ f α J J ( ~x J ) C I βI J αJ ( ~x J ; ~x I ) ˘ f † β I I ( ~x I ) (cid:1) (67)Results (55,61) conclude the ‘Perfected HoloQuantumHyperNetwork Theory’, M , axiomatically built upon allthe nine principles. As promised in section one, this isas such the fundamental and the complete form-invariant‘it-from-qubit’ theory of both all the dynamical quantumhypergraphs and all quantum natures. VI. HOLOQUANTUM HYPERNETWORKTHEORY: THE MODELS WITH EXTRASYMMETRIES, SUPERSYMMETRIC MODELS
As HoloQuantum HyperNetwork Theory is built to bethe quantum many body theory of all physically-possiblesystems of quantum objects and quantum relations, it must be necessarily maximally minimalistic in taking itsfundamental symmetries . This ‘minimalism’ is impliedby the very nine principles, so that the only symmetriestransformations in M are ‘the unavoidable ones’ .7Being maximally minimalistic in taking fundamentalsymmetries, M only has the exact U (1) symmetry ofthe global-phase redundancies, the minimally-brokensymmetry of qubits-equal-treatment, and has developedwithin it the complete set of quantum-hypergraphicalisomorohisms by specific combinations of a subset of itsmany-body-interaction operators. On the other hand,in the contextually-developed or the domain-specificsub-theories and models, solutions or phases of M ,generically additional exact-or-approximate symmetries ,either must be placed merely as the extra-structure‘constraint impositions’ , or must be emerged as the extra-structure ‘emergencies’ . We highlight here twoextremely-motivated classes of such extra symmetries.Because hypergraphs are, as fundamentally defined,pregeometric and background-less, there can not be anyspatially-local fundamental symmetry or fundamentalgauge redundancy in the perfected theory. However, inboth the ‘emergent-spacetime’ and ‘standard-model-like’ sub-theories or phases of M , a number of appropriategauge redundamcies must take place as such extrasymmetries, in either of the above ways.As an instructive and also interesting illustrationof this point, here we will incorporate supersymmetry as one such extra symmetry in a maximal sub-theory ofthe perfected theory. As it indeed must be, it will bedemonstrated that to formulate the most general familyof supersymmetric HoloQuantum HyperNetworks, oneneeds neither to introduce any extra degrees of freedominto the total quantum kinematics, nor to deform thecomplete total quantum dynamics of the perfectedtheory. That is, M stays kinematically and dynamicallyform-invariant under this extra-symmetry imposition,although indeed some of its otherwise-free structures willbecome constrained. We formulate here two differentmodel-theories of HoloQuantum HyperNetworks whichare endowed with N = 2 global supersymmetries.Moreover, to illustrate the simplest yet sufficiently-richexamples, we suffice here to the symmersymmetrizationof the HoloQuantum HyperNetworks which are flavorlessand in which only relatons are kept quantumly-active.The recipes will yield us supersymmetric HoloQuantumHyperNetworks which are formally-purely-fermionic andwhose Hamiltonians are embedded in (51).One must pick up a(ny possible) complex-conjugate pairof the formally-fermionic operators ( Q ( M ( s ) ) M , ¯ Q ( M ( s ) ) M ) on the Hilbert space H ( M ) M , which being supercharges,turn every formally-fermion hypergraph-state qubit ˘ f ( † ) I into a formally-composite-boson qubit ˘ b ( † ) I as itssupersymmetric partner, and vice versa. The algebra is, ( Q ( M ( s ) ) M ) = ( ¯ Q ( M ( s ) ) M ) = 0[ Q ( M ( s ) ) M , ˘ f † I ] = ˘ b † I ; [ Q ( M ( s ) ) M , ˘ f I ] = 0 ; h.c [ Q ( M ( s ) ) M , ˘ b I ] = ∂ t ˘ f I ; [ Q ( M ( s ) ) M , ˘ b † I ] = 0 ; h.c (68) We formulate a very general family of supersymmetrucHoloQuantum HyperNetworks . By being so, the definingsupersymmetric charges ( Q ( M ( s ) ) M , ¯ Q ( M ( s ) ) M ) must be the ‘largest-possible’ Gaussian-random composite fermionicfields , being made from the qubits { ˘ f J } ; { ˘ f † I } , in sucha way that the supersymmetric Hamiltonian, H ( M ( s ) ) M , H ( M ( s ) ) M ≡ { Q ( M ( s ) ) M , ¯ Q ( M ( s ) ) M } (69)satisfies the ‘maximal-dynamical-embedding’ criterion, H ( M ( s ) ) M ⊂ (embedded maximally) H ( M ) M (70)The first supersymmetrization recipe is inspired by theinitiative work [3], being the best method of realizing asupersymmetric sub-theory of M which is maximal bothkinematically and dynamically . By this first recipe, oneimposes an entirely-arbitrary Z partitioning on the totalHilbet space H ( M ) M . Imposing so, the formally-fermionhypergraph-state qubits ˘ F I are partitioned arbitrarilyinto two ‘chirality’ classes, { ˘ f ( † ) I } ≡ { ψ ( † )ˆ I ( − ) ; ψ ( † )ˆ I (+) } Z : ψ ( † )ˆ I ( − ) ←→ ψ ( † )ˆ I (+) (71)Let us highlight that the choice of this Z partitioningis absolutely arbitrary, and so all the different choicesfor it results in the supersymmetric models which maydiffer interpretationally, but indeed as supersymmetricquantum theories are all physically equivalent. Next, inaccordance to the chirality partitioning (71) imposed onthe hypergraph-state-qubits ˘ F I , the two supersymmetriccharges of the sub-theory M ( s ) are so defined, Q ( M ( s ) ) M ≡ all X ˆ J η ˆ J ψ ˆ J (+) + all X ˆ J ˆ J ˆ I η ˆ J ˆ J ˆ I ψ ˆ J (+) ψ ˆ J (+) ψ † ˆ I ( − ) + ... == M⋆ − X m =0 all X ˆ J ( m +1) all X ˆ I ( m ) η ˆ J ··· ˆ J m +1 ˆ I ··· ˆ I m ×× ( ψ ˆ J (+) · · · ψ ˆ J ( m +1) (+) ) ( ψ † ˆ I m ( − ) · · · ψ † ˆ I ( − ) )¯ Q ( M ( s ) ) M ≡ all X ˆ I ¯ η ˆ I ψ † ˆ I (+) + all X ˆ J ˆ I ˆ I ¯ η ˆ J ˆ I ˆ I ψ ˆ J ( − ) ψ † ˆ I (+) ψ † ˆ I (+) + ... == M⋆ − X m =0 all X ˆ J ( m ) all X ˆ I ( m +1) ¯ η ˆ J ··· ˆ J m ˆ I ··· ˆ I m +1 ×× ( ψ ˆ J ( − ) · · · ψ ˆ J m ( − ) ) ( ψ † ˆ I ( m +1) (+) · · · ψ † ˆ I (+) ) (72)In the definitions (72), all the independent defining cou-plings η ˆ J ··· ˆ J m +1 ;ˆ I ··· ˆ I m s are the statistically uncorrelatedGaussian random parameters being selected the same dis-tributions (49) for the λ -couplings of M .8By selecting the supersymmetric charges to be the onesin (72), as we can indeed compute, one now realizes themaximal sub-theory of (flavorless and purely-relatonic)supersymmetric HoloQuantum HyperNetworks, M ( s ) .That is, the resulted supersymmetric total Hamltonian(69) does satisfy the dynamical maximality criterion (70),by developing a family of ‘composite-random couplings’ , ( λ I ··· I m J ··· J m ; ¯ λ I ··· I m J ··· J m ) == some c . c functions of { η ˆ J ··· ˆ J s ˆ I ··· ˆ I s +1 , ¯ η ˆ J ··· ˆ J r ˆ I ··· ˆ I r +1 } (73)We highlight that such a ‘ ( ± ) partitioning’ of theformally-fermionic Hilbert space is very natural in manyof the sub-theories and models which are derived fromwithin M , an example of which being a Wheeleriansimplest toy model presented in the next section. Beingso, all such models are naturally amenable to thesupersymmetrization presented above, albeit whenevertheir model-defining constraints respect supersymmetry.Let us here mention an alternative recipe. Similarto the supersymmetrization in [4], the supercharges canbe also defined as follows, Q ( M ( s ) ) M ≡ M⋆ X m =1 q (2 m −
1) 1 ≤ s ≤ m X all Js η J ··· J m − ˘ f J · · · ˘ f J m − ≡≡ M⋆ X m =1 q (2 m − Q ( M ( s ) ) M (2 m − ; ¯ Q ( M ( s ) ) M ≡ (cid:0) Q ( M ( s ) ) M (cid:1) † (74)with the q (2 m − s being formal complex coefficients,by which all the ‘fixed-order supercharges’ Q M (2 m − are distinguished. By taking the supercharges givenin (74), the resulted supersymmetric Hamiltonian (69)develops, in addition to the resulted embedding of H ( M ) M [ { λ ~I~J } , c.c, } ] , the following terms ∆ H M , ∆ H M = ( n = m ) X ≤ m,n ≤ M⋆ all possible X ~J (2 m ) ,~I (2 n ) λ ~I (2 n ) ~J (2 m ) ( some Y L ˘ n L )( m Y s =0 ˘ f J s )( n Y r =0 ˘ f † I r ) λ ~I (2 n =2 m ) ~J (2 m ) ≡ q (2 m +1) ¯ q (2 n +1) M ⋆ X K η K ~J (2 m ) ¯ η K~I (2 n =2 m ) (75)However, these extra terms do violate the global U (1) symmetry of HoloQuantum Network Theory, which isnecessary for its well-definedness as a background-lessquantum theory. By demanding the preservation of the U (1) symmetry, ∆ H M = 0 , we get a set of algebraicconstraints on the η parameters, whose generic solutionallows only a single q -coefficient to be non-zero. By this,we will get a sub-theory with only a pair of ‘fixed-ordersupercharges’ ( Q ( M ( s ) ) M (2 m − ; h.c ) . As such, we favor thefirst supersymmetrization recipe in light of the principlenine of M . VII. HOLOQUANTUM HYPERNETWORKTHEORY: A SIMPLEST TOY MODEL OF‘WHEELERIAN PARTICIPATORY UNIVERSE’
In this section, we work-out from within the PerfectedHoloQuantum HyperNetwork Theory a specific ‘simplesttoy model’ on two purposes.
First , we want to exposehow concrete models of phenomenological interests canbe extracted-out from the total quantum kinematics andthe complete unitary quantum dynamics of the theory M . Second , we wan to present a minimalistic simplesttoy model of the ‘HoloQuantum-HyperNetworkicallyRealized Wheelerian Quantum Universe’ . By beingWheelerian, we mean a quantum universe which isconstructed upon, and so realizes, all of the ‘three’visions of Wheeler [1]. These three principal visionsare so stated.
First , the total quantum universe isfundamenally a gigantic quantum [Hyper-]Network ofthe ‘many-observers-participancies’ , whose ‘actions’ areher ‘elementary quantum phenomena’.
Second , there is afundamental maximal statistical randomness underlyingthis quantum HyperNetwork, sourcing the ‘law(s)without law(s)’ . Third , the principle of ‘it from [qu-]bit’ states that absolutely all the physical aspects of the world is assembled by the randomly-sourced ‘answering-qubits’of ‘no-or-yes’s to the abundantly-many binary questionsposed by all the participatory-observers. We emphasizethat the manners in which these three principles areincorporated in the toy model here are intentionally setto be the most minimalistic, the simplest, and sheerlyinstructive. But, here comes the one single significantmassage that we want to convey here.
The abovethree Wheelerian principles can be merged-and-realizedcorrectly by ‘the theory M ’ which has been defined,formulated and finally perfected in this work. Beingso, the ‘no-or-yes’ qubits of the observer-participatoryuniverse must be identified with the very qubits (55)which construct HoloQuantum HyperNetworks and somicroscopically interact with one another as determinedby the total Hamiltonian of the perfected M theory (61).Let us now conceive and formulate step-by-stepthe simplest toy model of a Wheelerian participancyuniverse which is ‘purely-relational’. By being purelyrelational, we mean the following simplification in thisminimal toy model. We will take all of the participatoryobserves to be fixedly frozen in their states of presence,and then let the relatonic qubits of the ‘multi-observerparticipany relations’ be the only dynamical quantumdegrees of freedom. It is clear that in the ultimate theoryof the Wheelerian universe, every participatory observermust also be quantumly active as a dynamical vertonwhich can switch between the states of absence andpresence. This implies that all the nontrivial cascadeoperators should be necessarily in role in any realisticWheelerian model. However, in here we reduce thetoday model to be purely relational, on the account ofthe first purpose of this section which is instructional.9By this explanation, let us consider a total numberof M participatory observers who are all fixedly present.To formulate their Wheelerian Participatory Universefrom within M , let us employ the total Hilbert space H ( FM ) M (55) of M flavorless vertons together with alltheir ‘evenly-flavored’ m -relatons. The vertons of H ( FM ) M represent, by any one-to-one correspondence, those very M quantum participatory observers. Hence, by oursimplifying ansatz, all the vertons V i are frozen in theirpresence-states, in a way that must be consistent withthe dynamics of the system. Because they are fixedlypresent, the first set of the quantum constraints of thetoy model are the following operator identities, n i = 1 , ∀ i ∈ N ≤ M (76)Wheelerianly, the fundamental degrees of freedom whichmicroscopically assemble this whole quantum universe,are all the randomly-sourced ‘no-or-yes’ answering-qubitsto a complete set of questions posed by, and shared by,all the subsets of all the M participatory-observers . The completeness of the set of binary questions means that ifwe were given the qubit data about any proper subset ofthose questions, the entire participatory universe couldnot have been assembled only upon them. Now, we letevery complete set of these ‘no-or-yes’ answering-qubitsbe collected in the set of doublets { ( N α i ( m ) i ( m ) , Y α i ( m ) i ( m ) ) } ,in which i ( m ) identifies the corresponding choice of m ≤ M participatory questioners, while the index a i ( m ) forms acomplete spectrum of the binary questions posed by theobservers. The simplest HoloQuantum HyperNetworkiantoy-modeling of this Wheelerian scenario is ‘all relatonic’.Therefore, the hypergraph-state relatons should not onlycarry the very same question-spectrum flavors, but alsobe further doubly-flavored, as follows, { ˘ R α i ( m ) ,νi ( m ) } ≡ { ˘ R α i ( m ) , ± i ( m ) }{ ˘ R α i ( m ) , − i ( m ) } ←→ { N α i ( m ) i ( m ) }{ ˘ R α i ( m ) , + i ( m ) } ←→ { Y α i ( m ) i ( m ) } (77)By (77), every such-identified degree- m relaton whosechirality flavor is positive (negative) represents thepositive (negative) answer given to one of the binaryquestions posed by those m participatory observers whoare in correspondence with its base-vertons. Let ushighlight that, by (76), the hypergraph-state-relatons ˘ R α i ( m ) ,νi ( m ) are operationally equal to their counterprarts R α i ( m ) ,νi ( m ) , everywhere in this simplest toy model.Relatons (77) are the active quantum degrees offreedom which assemble the Wheelerian Universe.Namely, the operators (cid:0) ˘ r α i ( m ) , ± i ( m ) , ˘ r † α i ( m ) , ± i ( m ) (cid:1) are set free toannihilate and create their ‘no-or-yes’ answering-qubits,and so, every new instant to re-assemble the universe . Because of the abstract purely-information-theoretic‘binary’ nature of every pair of relatons with oppositechiralities, one must impose one more set of quantumconstraints. The defining basis-states of the total Hilbertspace must be mapped, in a one-to-one manner, to theoutcomes of the ‘all-measurements’ which correspondto the ‘complete-questions-answered’ entirely-assembledWheelerian universes of the M participatory observers .By this demand, the evolving global wavefunction ofthe Wheelerian universe must be, at any arbitrary time,a superposition of the basis-states in which, for everychoice of m ≤ M observers i ( m ) , and for every of the items α i ( m ) in their questionnaire, one of the two R α i ( m ) , ± i ( m ) relatons is ‘present’, whereas its chirality-complementone is necessarily ‘absent’. Formulated as operatoridentities, we must now demand the following second setof quantum constraints as relatonic operator identities, ˘ n α i ( m ) , − i ( m ) + ˘ n α i ( m ) , + i ( m ) = 1 , ∀ i ( m ≤ M ) , ∀ α i ( m ≤ M ) (78)By imposing (76) and (78), the total Hilbert space ofthe simplest toy model of the Wheelerian ParticipatoryUniverse of the M observers , H ( W . P . U ) M , is given by thetruncation of H ( FM ) M whose defining basis is identified asfollows, B ( W . P . U ) M = { M Y m =1 all Y i ( m ) all Y α i ( m ) (cid:2) (1 − ǫ α i ( m ) i ( m ) ) ˘ r † α i ( m ) , − i ( m ) ++ ǫ α i ( m ) i ( m ) ˘ r † α i ( m ) , + i ( m ) (cid:3) | i , for all possible choices of ǫ α i ( m ) i ( m ≤ ˆ M ) ∈ { , } } (79)Moreover, the quantum-kinamatical truncation (79),which in turn is demanded by the quantum constraints(76,78), must be dynamically consistent. That is, clearly,the above kinematical truncation must be preserved inthe entire evolution of these Wheelerian HoloQuantumHyperNetworks. For this dynamical consistency to cometrue, the complete total Hamiltonian of the WheelerianParticipatory Universe, H ( W . P . U ) M , which generates itsunitary evolution, must satisfy the following costraints, [ H ( W . P . U ) M , n i ] = 0 , ∀ i ≤ M [ H ( W . P . U ) M , ˘ n α i ( m ) , − i ( m ) + ˘ n α i ( m ) , + i ( m ) ] = 0 , ∀ i ( m ≤ M ) , ∀ α i ( m ) (80)Being promised from the very beginning, the above totalHamiltonian H ( W . P . U ) M must be directly extracted fromthe total Hamiltonian of the perfected M , namely (61).To fulfill this, we take the first-order Hamiltonian of theperfected theory M , manifested in the exponent of (62),and have it now expressed as the likewise-flavored cousinof the flavorless unfoldment (36).0By applying (80,76,78) to that, and dropping constantterms, we present as follows the first-degree Hamiltonianof the ‘most minimalistic’ toy model of the HoloQuantumHyperNetworkical Wheelerian Participatory Universe , H ( W . P . U )(1 − M = ≤ m ≤ M X all i ( m ) X all α i ( m ) (cid:0) µ α i ( m ) , + i ( m ) ˘ n α i ( m ) , + i ( m ) ++ ¯ λ α i ( m ) , − + i ( m ) ˘ r α i ( m ) , + i ( m ) ˘ r † α i ( m ) , − i ( m ) ++ λ α i ( m ) , − + i ( m ) ˘ r α i ( m ) , − i ( m ) ˘ r † α i ( m ) , + i ( m ) (cid:1) (81)Let us highlight that, because in this simplest model ofWheelerian HoloQuantum HyperNetwork, all the vertonsare kept present fixedly by the quantum constraint (76),the nontrivial cascade operators do not appears in theHamiltonian. To remind, all the independent µ α i ( m ) , + i ( m ) and the λ α i ( m ) , − + i ( m ) couplings in (81) must be taken tobe uncorrelated maximally-free Gaussian-random as in(61). The above first-degree Hamiltonian can be morecompactly re-expressed as such, H ( W . P . U )(1 − M = all X i ( m | m ≤ M ≥ all X α i ( m ) ∈{±} X ν,υ λ α i ( m ) ,νυi ( m ) T α i ( m ) (1 − i ( m ) ; νυ (82)using the so-defined ‘Elementary Wheelerian Operators’ , T α i ( m ) (1 − i ( m ) ; νυ ≡ ˘ r α i ( m ) ,νi ( m ) ˘ r † α i ( m ) ,υi ( m ) (83)As the indices manifest, the above operators T α i ( m ) (1 − i ( m ) ; νυ are defined for every subset of the participatory-observers i ( m ) , and further, for every one α i ( m ) in the completespectrum of their binary questions. By definition, forevery such identification, the elementary Wheelerianoperators either switch the two answering-chiralities,the off-diagonal ones, or simply witness-and-report thechiralities, the diagonal ones, in the present-momentglobal state of the purely ‘it-from-qubit’ universe.Now, upon utilizing the expressions (82,83), thecomplete total Hamiltonian of the simplest toy model ofthe Wheelerian Participatory Universe H ( W . P . U ) M , takesthe following form, as a sub-evolution of (61), H ( W . P . U ) M = ≤ s ≤ M X s H ( W . P . U )( s − s ) M H ( W . P . U )( s − s ) M ≡≡ ( ≤ m ≤ M X all i m X all α i m ∈{±} X ν ,υ ) · · · ( ≤ m s ≤ M X all i s ( ms ) X all α is ( ms ) ∈{±} X ν s ,υ s ) λ α i m ··· α is ( ms ) ,ν υ ··· ν s υ s i m ··· i s ( ms ) T α i m (1 − i m ; ν υ · · · T α is ( ms ) (1 − i s ( ms ) ; ν s υ s (84) Being one of the central characteristics of the unitaryevolution (84), the ‘no-or-yes’ answering-qubits ˘ R α i ( m ) , ± i ( m ≤ M ) do all interact with one another as conducted by themaximally-random many-body conversion operators ofthe theory M .To complete this special section, let us also presentthe ‘child’ sub-model of this minimal toy model of theHoloQuantum-HyperNetworkically-Realized WheelerianParticipatory Universe . As before, it is the one whosetotal Hamiltonian exponentiates (81), H ( W . P . U ) (cid:13) M = exp (cid:2) ≤ m ≤ M X all i ( m ) X all α i ( m ) ∈{±} X ν,υ λ α i ( m ) ,νυi ( m ) T α i ( m ) (1 − i ( m ) ; νυ (cid:3) (85) VIII. HOLOQUANTUM HYPERNETWORKTHEORY: A GLOBAL OVERVIEW WITHSELECTED CONNECTIONAL EXPLANATIONS
We begin this section with a global overview of thetheory M , and then bring about a number of selective connectional remarks which are important.HoloQuantum HyperNetwork Theory, which we havedefined and systematically formulated in this work, is thefundamental kinematically-and-dynamically completetheory of all the entirely-quantized HyperNetworks.From the purely physics point of view, M serves thevery job which the complete ‘it-from-qubit’ theory of allphysics should do. To the best of our understanding, aswe propose here, HoloQuantum HyperNetwork Theoryis indeed the fundamental complete ‘it-from-qubit’quantum-many-body form-invariant formulation of ‘allquantum natures’ in an all-unified way. We rememberthat, ‘all quantum nature’ is a collective term for theentire quantum universe-or-multiverse and for all her‘selective descendants’ one by one, as obtained by allpossible ‘phenomenological subsettings’ together with allpossible ‘observers-probes rescalings’. All the quantumkinematics, all the many-body interactions, and thecomplete unitary quantum dynamics of the perfectedtheory M are directly sourced from the unique nineprinciples which, to the best of our undedrstanding,are both the unavoidable, ‘the must be’, and the mostcompelling, ‘the best be’, for its original intention to beentirely fulfilled.Mathematically, the central characters of M arethe HoloQuantum Hypergraphs, the unitarily-evolvingquantum states which are made by all possible quantumsuperpositions of the arbitrarily-chosen hypergraphs.But, the microscopic degrees of freedom of M are thecomplete sets of purely-information-theoretic abstractqubits for the quantum vertices and for the quantumhyperlinks, the vertons and the relatons, respectively.1As a whole, all the vertons-and-relatons form an entirelypregeometric, ‘formally’-all-fermionic, qualitatively-novel closed quantum many body system of abstract qubits.The total unitary qauntum dynamics of all the vertonsand relatons is sourced by a complete set of randommany-body interactions which impartially occur inbetween all of them. Besides all the ‘conventional’multi-fermion interactions, HoloQuantum HyperNetworkTheory has a whole lot of novel many-body interactions,being conducted by a hierarchical family of the ‘cascadeoperators’ , as implied by the evolving hypergraphicalface of its abstract quantum many body system ofqubits. These novel and cardinal cascade operatorssafeguard the dynamical hypergraphical-weldefinednessof HoloQuantum HyperNetworks, but also realize (byonly the purely-vertonic subset of them) all the quantumhypergraphical isomorphisms.Both mathematically and physically, all that thetheory M ‘takes in’ about the nature of the vertonsand the relatons is their Wheelerian abstract ‘no-or-yes’or equivalently ‘absence-or-presence’ qubit-ness for acomplete set of the maximally-flavored vertices andtheir multi-degree hyperlinks. Being so, M is purely‘it-from-qubit’ in its fully-covariant formulation of bothmathematics and physics, all the way from ‘alpha toomega’ . Because of this central characteristic, andmoreover because of its covariant-completeness , theperfected theory yields the complete formulation ofhow precisely all the abstract ‘no-or-yes’ qubits ofthe observer-participancies interact with one anotherand evolve as an abtract quantum many-bpdy system,assembling the Wheelerian universes and multiverses.As such, as an alternative to approaches such as [5], M is proposed as the right theory to develop in full precisionthe ‘it-from-qubit’ construction of all physics .To accomplish this, HoloQuantum hypergraphs representthe arbitrary choices of the quantum objects, by theirvertons, and the arbitrary choices of the multi-objectquantum relations, by their m -relatons. Indeed, thepurely relatonic operators O [ { ˘ r α [( i ai ) ( m ) ]( i ai ) ( m ) , ˘ r † α [( i ai ) ( m ) ]( i ai ) ( m ) } ] s,being correctly identified in every given context in termsof the fundamental relatons, can be the field operators ofthe arbitrarily-chosen physical interactions between thearbitrarily-chosen ‘particles’. Likewise, all the statisticalcorrelations, the functional relations, or the relationalgeometrical quantifications of these ‘particles’ can beformulated by these operators. In fact, all possiblefundamental-or-emergent relational observables, whichone can measure for any arbitrarily-chosen ‘particles’,can be formulated by these operators. However, it is onlyby inter-relating those very ‘objects’, that the relationalobservables are even definable. By taking-in all theseobjects as the qubit vertons, one arrives at M . Oneso covariantly formulate all of physics by HoloQuantumHyperNetworks of arbitrary objects-and-relations . Having made a review of HoloQuatum HyperNetworkTheory, we now come to present, in the rest of thissection, a number of elucidating connectional remarksabout ‘some selected works’ in the literature which hassome ‘instructional’ overlaps with some aspects of theperfected theory M . In each case, we will comparativelycomment and carefully elaborate not only on the notableconceptual or technical connections and similarities, butalso on the multi-dimensional characteristic distinctionswhich HoloQuantum HyperNetwork Theory makeswith all those selected works. The works which wecomparatively discuss in here belong to very differentfields of quantum physics. This vast coverage, however,is very natural as the perfected theory is by constructionthe form-invariant formulation of all quantum natures. The following selected works from the literature , we musthighlight, are picked up very ‘subjectively’, mainly forthe ‘instructional’ purposes, and so their collection is‘far from complete’ . This highlighted selectiveness ismainly because the selected works are already enoughfor clariyfing the points. These remarks also shed lighton how M can advance all these fields.Mathematically, the absolute primitivity, maximalgenerality, and intrinsic competence which distinguishesthe defining framework of hypergraphs, suggest themto be the very primary characters of the ‘pregeometry’ [6] out of which the entire space emerges. Graphs,namely hypergraphs with only the two-vertex-links, havea well-known history of being examined as models ofpregeometry. Because the entire universe, and so alsothe fundamental ‘setting’ of the spacetime, is a quantumsystem, the hypergraphical pregeometry must be definedquantumly. Indeed, some interesting quantum models ofgraphical pregeometries have been already constructedin the past and recent literature [7–13]. These works,independent of the implemented quantum statistics ofthe degrees of freedom, differently but all partially , haveemployed some features of the total quantum manybody system of the theory M . Both of the works [7, 8]already used the framework of second quantization forthe graph-structural degrees of freedom, in [7] withsome deterministic local evolution, and in [8] withsome random interactions. The more recent and moreadvanced works of [11], looking for emergent locality andgravity, formulate initiative models of second-quantizedquantum-graphical pregeometries, also including matterquanta, whose in-particular Hubbard-model-resemblingstructures-and-interactions develop interesting phases.The works of [12], using a second-quantized formalismbeing similar to the quantum-graphical methods of [11],present models of random complexes whose Markovianstate-dependent unitary evolutions are realzied by gluingface-wise the simplices or the regular polytopes. Finally,the most recent works [13] offer simple toy models ofthe randomly-interacting graph-structural qubits whoseground-states, developed in the infrared quantum phasetransitions, feature some four-diemsnional geometries.2Let us now elaborate collectively in what follows on thesignificant points of distinctions between what we havepresented in this work and those presented in [7–13],as both the partial similarities between them and theirpartial overlaps must be already obvious to the reader. First , HoloQuantum HyperNetwork Theory is a complete‘theory’ . By intention, this theory is constructed to serveas the fundamental and complete theory by which ‘allof physics’, and so also the complete theory of quantumpregeomtry and quantum gravity, can be formulatedform-invariantly. Because it is neither a specific model,nor a collection of contextually-related models, itsdefinition and formulation could not have been donein the arbitrary manners in which ‘models’ are built.Being so, nothing in this quantum theory is ad hoc,or has been implemented as a matter of examination,simplification, or specifications. Both the total staticsand the complete unitary dynamics of the theory, as theymust, are directly deduced from the merge of the statednine principles required by the ‘must-be’-and-‘best-be’.This characteristic robust-and-complete determinationis unlike all the ad-hoc structures implemented in theinitiative models presnted in [7–13]. To highlight, all themany-body interactions of the theory are determinedboth uniquely and completely by the exact-or-almostsymmetries, all the dynamical quantum constraints, theWheelerian randomness, and finally the principle of thecovariant-completeness. Indeed, these very microscopicinteractions , upon realizing all the extra structures orthe extra symmetries which are needful to be imposedin obtaining the correct theory of quantum gravity fromwithin HoloQuantum HyperNetwork Theory, will alsoformulate form-invariantly the complete dynamics of thequantum spacetiem . We will elaborate on this crucialpoint in section nine.
Second , we must remark here on a characteristicdistinction regarding the inclusion of all the additionalfundamental degrees of freedom which will play therole of matter in quantum pregeometry and quantumgravity. On one hand, M is a complete ‘purely internaltheory’, without any external degrees of freedom.HoloQuantum HyperNetworks are all the quantumstates of a closed quantum many body system of theabstract qubits of vertons and relatons which formulatethe total kinematics and the complete dynamics of thefully-quantized hypergraphs. On the other hand, bypartitioning this total quantum many body system intotwo complementary subsystems, one formulates ‘openHoloQuantum Networks’. Such complementary pairs ofopen HoloQuantum Networks naturally formulate ‘thespacetime quanta’ interacting with ‘the matter quanta’in the aimed theory of quantum gravity. Being so, thevertons and m -relatons of HoloQuantum HyperNetworkscan formulate both the ‘pure quantum geometries’ andthe ‘interacting quantum spacetimes and all possiblequantum matter’, in a covariant ‘unified’ way . Third , one independent dimension in which anyHoloQuantum-HyperNetworkical theory of the quantumpregeometry and quantum gravity must be significantlydistinct statically, by interactions and dynamically fromthe models in [7–13], is the following. The theory M not only is ‘trans-graphical’, but also both kinematicallyand dynamically is ‘maximally hypergraphic’. This isindeed as it must be, and is directly sourced by therealization of principle seven, according to which all the m > - relatons, which by definition are ‘non-reducible’to the two-relatons, are as kinematically-fundamentaland also as dynamically-active as the two-relatons .Stated by principle seven, the theory must treat allthe hypergraph-state-qubits with maximum-possoibleequality. The minimalistic breaking of this symmetry isbetween the vertons and the relatons, only realized bythe ‘dressing’ of relatons and by the cascade operators.Besides this, all qubits are equal. As such, by itstotal Hamiltonian, namely H ( FM ) M , all the R i ( m ∈ N ≤ M ) sare equally activated. As we understand, this qualityof ‘ all-relatons-equality ’, being an exact fundamentalsymmetry of the theory, is instrumental for the correcttheory of quantum gravity. Let us so conclude as follows. The acceptable HoloQuantum-Networkical formulationsof quantum pregeometry and quantum gravity are theones which are ‘maximally hypergraphic’ . Comparing H ( FM ) M (61) with its counterparts in the aforementionedreferences proves the physical significance of this point. Finally , let us conclude our connectional remarkson the models of quantum spacetime with implicationsof the covariant completeness stated by principle nine.Because both the total state-pace (55) and the completeunitary dynamics (61) of the perfected theory do fullyrealize this principle, it is already guaranteed that ‘all’the network models whose kinematics and dynamicsare ‘quantumly correct’ can be consistently embeddedinside the totality of the perfected M . These consistentembeddings can be also checked concretely, exampleby example. In particular, all the models presented in [7–13] are consistently embeddable inside (55,61). Tohighlight, as explained in sections one to five and in theoverview part of this section, this ‘consistent-embeddingcriterion’ holds independent of the quantum-statisticalidentities of the objects or relations whose ‘it-from-qubit’theory is formulated by the perfected theory. That, allthe ‘independent’ models in [7–13] are embeddable insidethe Maximally-Flavored HoloQuantum HyperNetworkTheory not only interconnects them fundamentally,but also makes all their consistent deformations andgeneralizations manifest .Now, we move on to the comparative remarks inregard with the SYK models [14–16], specially with their‘complexified versions’ [15]. On one hand, an inclusivecomparison will be made briefly. On the other hand, thishighlights how one can generate the qualitatively novelclasses of quantum many body systems, inside M .3 Merely at the level of their formulations as quantummany body systems , HoloQuantum HyperNetworks haveboth a number of notable likenesses and a number ofsignificant characteristic unlikenesses with the extremelyinteresting SYK models, specially with their complexifiedformulations as given in [15]. We begin by highlighting the ‘formal’ likenesses . The total Hilbert spaces arepurely fermionic, in both cases. All the microscopicinteractions, in both cases, are fundamentally ‘all-to-all’and also random. Moreover, simply for the naturalness,but not restrictedly, the random couplings are all setto be Gaussian. Both HoloQuantum HyperNetworkTheory and the ‘complexified SYK’ as formulated in [15]respect the fundamental symmetry of the global U (1) transformations on the fundamental fermions. Finally,in both cases, the formulations are totally pregeometric.That is, the fermionic many body systems live in (0 + 1) spacetime dimensions. We must emphasize again thatthese likenesses are at a merely formal level .Let us now move on to highlight and elaborate brieflyon the characteristic unlikenesses which, even merelyformulationally , distinguish the works. We so begin with‘ the least significant ’ among these differences. Althoughin both cases the fundamental all-to-all interactions areall Gaussian-random, in HoloQuantum HyperNetworkTheory, the independent couplings must be selected fromthe Gaussian-random distributions each one of which isfixed with tuning two arbitrary parameters, one for themean values, and one for the standard deviations. Both(formally) in HoloQuantum HyperNetwork Theory andin the model [15] together with the likewise-complexifiedextended-SYK model [16], the interactions are themulti-fermionic-conversions. But, the total Hamiltonianof HoloQuantum HyperNetwork Theory H ( FM ) M in (61) ‘must necessarily’ contain all the ( m -to- m ) conversionsimpartially, to realize the covariant completeness statedby principle nine.Now, we turn to remarking the three unlikenesses between the works which are ‘ the most significant ’. First , in the SYK models [14–16] all the fermionicdegrees of freedom are treated equally, and this equaltreatment which is both statical and dynamical, isheld exactly. But, in HoloQuantum HyperNetworkTheory this global ‘equal-treatment symmetry’ must beexplicitly broken, as minimally as possible, but indeedunavoidably. The source of the explicit breaking ofthis symmetry is a fundamental one . In HoloQuantumHyperNetwork Theory, the qubits F I fundamentally sitinto two categories, vertons and relatons. Relatons canbe present or created only when their base-vertons are allpresent in the quantum state defining the HyperNetwork.Being so, relatons are ‘system-state-conditional qubits’.By principle seven, the equal-treatment symmetry isbroken only by these relatonic conditionalities. But, as (61) does manifest in comparison to [14–16], this effectstrongly deforms the unitary dynamics H ( FM ) M . Second , because by seeing on its very mathematical face,the maximally-flavored HoloQuantum HyperNetworktheory ‘becomes equivalent with’ the most completetheory of the all-structurally-quantized dynamicalhypergraphs, the quantum-hypergraphical isomorphisms do play an important role in the construction of itsdynamical side, as stated by principle five. Indeed,realizing these quantum-isomorphism transformationshas made a significant impact on the definitions andthe formulations of the microscopic interactions andthe total unitary evolution of the perfected theory, asone can trace them back in the first five sections. But,in the SYK models [14–16] the relatonic structures areall trivial, so that this dynamically-impactful featurebecomes effectively mute.
Third , the dynamical hypergraphical-welldefinedness ofHoloQuantum HyperNetworks demands the hierarchicalfamily of cascade operators . These novel operatorsplay a central role in the fundamental interactions andso in the unitary evolution of the total many bodysystem of the vertons and all the m -relatons. However,because the hypergraph structure of the SYK modelsis effectively trivial, the interactions of HoloQuantumHyperNetwork Theory and of the models [14–16] aremajorly distinct at the microscopic level , because of theboth highly-frequent and highly-impactful presences ofthe cascade operators in H ( FM ) M as concluded in (61).Beyond the above-stated distinctions, the theory M is, by construction, the fundamental completetheory in which every consistent quantum many bodysystem is covariantly contained. In regard with theSYK-type models, in fact, all these embeddings are verystraightforward . That is, not only the models [14–16]are immediately embeddable in the Maximally-FlavoredHoloQuantum HyperNetwork Theory, but also all thedescendant versions of them which are presented inthe very recent literature, whether are generalizeddimensionally, or are deformed statically or dynamically,can be easily embedded inside the very totality of(55,61). By embedding all these SYK-type modelsinto HoloQuantum HyperNetwork Theory, all theirconsistent deformations and generalizations becomesmanifest, interconnected and systematic .Finally, we highlight a point on the gravitationalaspects of the SYK models [14, 15], connecting withour related points in section nine. The SYK modelsare already dual to some two-dimensional gravitationalmodels. But, the perfectly-realized principle nine doesguarantee that there must be a sub-theory of the theory M which holographically [18] defines and formulates thecomplete realistic theory of both quantum and classicalgravity in the emergent realistic spacetimes in which,besides the single ‘renormalization-group dimension’,the other dimesnions are all emergent.4 IX. HOLOQUANTUM HYPERNETWORKTHEORY: THE VISIONS FORWARD,TODAY AND FUTURE
HoloQuantum HyperNetwork Theory, understood in its ‘minimum level’ , serves the whole quantum-grantedphysics the very same way that category theory servesthe whole mathematics. But, it does serves physics muchmore than that , once being understood in its ‘maximumlevel’ . The theory M , by its definition, formulation andperfection, given the unique choice of its defining nineprinciples, serves as the right ‘it-from-qubit’ frameworkin which every precedented-or-unprecedented specifictheory of physics can be constructed from the beginning,and then be built-up phenomenologically in a systematicway which is direct, fundamental and also optimal .Being visited in the perspective of network science,HoloQuantum HyperNetwork Theory is ‘the’ theory which formulates every possible dynamical network,assuming that it respects all the laws of quantum physicsstructurally and functionally, which it surely does ifbeing physically realizable.In a way, it is by those unique nine principles whichHoloQuantum HyperNetwork Theory has been grantedits intentional power, that is, the above ‘minimum andmaximum’ . The nine principles are characterized intotwo classes which are ‘complementary’. Six principles among them, the principles one, two, four, five, six andfinally nine are unavoidable for its minimal realizationas ‘the category theory of all physics’ . Being so, theybelong to the ‘must-be’ class of these nine principles.In particular, we must highlight that, although thesystematic construction of the perfected theory has beenimpacted, according to the principles five and six, bythe realizations of the two distinct types of symmetrytransformations, none of them is a ‘beyond-categorical’feature. Clearly, to develop any quantum-hypergraphiccategorical theory of arbitrary objects-and-relations,the most complete set of the quantum-hypergraphicalisomorphisms must be realized by a (proper) subsetof the Hamiltonian operators. Moreover, the U (1) redundancies of the global phases must be demanded forany ‘lowest-dimensional’ categorical theory of quantumobjects and their quantum relations to compute itsobservables correctly. Now, we come to the second classof the nine principles, namely the ‘best-be’. They arethe principles seven, three and eight . Principle seven,namely the principle of ‘maximal hypergraphness’ bywhich all the multi-degree quantum hyperlinks areequally-treated, and moreover all the quantum objectsand the quantum relations are also treated as equallyas it can be, is needed to make the perfected theory ‘the optimal framework’ in which all the domain-specifictheories of physics can be formulated. Finally, theWheelerian principles three and eight are the oneswhich uplift HoloQuantum HyperNetwork Theory to itsultimate ‘it-from-qubit’ fulfillment [1]. Summarized from this purely ‘it-from-qubit’ pointof view, HoloQuantum HyperNetwork Theory is thefundamental and complete dynamical interacting theoryof the abstract qubits for the absences-or-presences of‘absolutely whatever’ of the objects and their relationsin the entire quantum universe or the multiverse.Clearly, the complete time-dependent information of ‘allquantum natures’ is capturable by the total quantummany body system of a qualitatively sufficiently-diverseand quantitatively sufficiently-immense collection of theanswring-qubits to the ‘isn’t-or-is’ or equivalently, to the‘no-or-yes’ questions. This is why, by its first-principledefinition and perfected construction, the theory M is the fundamental, general, complete and covariantWheelerian theory of ‘it-from-qubit’ . This, in particular,suggests that from within the prerfected theory, onecan generate a whole families of novel more compellingsub-theories and models of quantum information andquantum computation. We will suffice to highlight threeobvious directions in this fruitful territory. Firstly , oneshould be able to reformulate and further genralize bothstatically and dynamically, the conventional quantumcomputation theory , from within the totality of (55)and (61).
Secondly , but relatedly, based on the resultsof [17] which are embeddable in, and generalizable bythe total abstract microscopic system of HoloQuantumHyperNetwork Theory, one can devise highly-novelquantum computation processors which can be superiorfunctionally . Thirdly , by hugely completing the simplesttoy-model of section seven, one derives from within M the complete Wheelirian ‘It-From-Qubit’ theory of‘Observer-Participancy-Universe’ , an alternative to [5].HoloQuantum HyperNetwork Theory must be maximallyminimalistic in taking its fundamental symmetries tosafeguard its maximal generality as the fundamentalquantum many body theory of all quantum natures.But, one can devise diverse models of HoloQuantumHyperNetworks endowed with the contextually-chosenextra symmetries. These contextual model buildingscan be done in two ways. On one hand, as in theexamples of section six, they can be obtained as theminimalizations being directly extracted from the totaltheory which has been fully perfected in section five.On the other hand, many forms of physically-significantextra symmetries, such as the spacetime symmetriesor the internal gauge symmetries, can also emerge asthe qualitatively-distinct quantum-or-classical phasesof HoloQuantum HyperNetworks in the enormous totalphase diagram of the perfected M . Also from a purelynetwork-science viewpoint, in these very two ways onecan devise all models of unprecedented-or-precedentedquantum-or-classical ‘simple’-or-complex networks. Byits clear significance for building all such sub-theoriesand models, to probe and progressively map the distinctphases, the fixed points, and the phase transitions of theMaximally-Flavored HoloQuantum HyperNetworks willbe one of the fruitful directions to explore in future.5As elaborated in section eight, the theory M , bybeing the fundamental complete ‘it-from-qubit’ theorywhich formulates the time evolutions of all possiblesuperpositions of the arbitrarily-structured hypergraphs,is the very natural framework to define, formulate andconclude the ultimate theory of quantum pregeometryand the complete quantum gravity . To accomplish thisfar-reaching goal from within the totality of (55,61)which conclude the perfected theory, is of course awhole grand project to be conducted, surely requiring anumber of totally-unprecedented ideas . To advance in thecorrect direction toward this goal, we suffice to highlighthere the one most-significant central characteristicwhich is already very well-appreciated. This distinctivefeature is nothing but ‘the correct form’ of the mostcomplete realization of Holography [18], one must comeup with. This intrinsics holographization is clearlyone more remark on models of section eight. Let ushighlight that the whole point in here is the correctway of realizing the strongest version of Holography,formulated in its purely-information-theoretic form andthen implemented as an ‘extra’ feature or structure onthe whole generality of HoloQuantum HyperNetworks .Namely, the ‘HoloQuantum-HyperNetworkian Theory ofQuantum Gravity’ will be a very specific ‘sub-theory’ of the total HoloQuantum HyperNetwork Theory which, byintrinsic construction, is immensely larger than a theoryof only quantum gravity. This unmapped dimensionmust be explored in future works.Being one context-independent direction of analysiswithin the HoloQuantum HyperNetwork Theory, onecan systematically and also precisely analyze both theemergences and the characterizations of complexities in a unique manner which is all-inclusive. By this wemean that, all the variants of complexities which aredistinct phenomenologically, or are developed out ofthe different emergent mechanisms, can be unitedlyformulated, classified and thoroughly understood inthis unique framework. Given the extremely significantroles which the complex structures do play both inthe nature and in advanced technologies, this will bea distinctly-important field of study. In particular,because the theory M is the fundamental, general andcomplete unification of (quantum and classical) manybody systems and (quantum and classical) networks,one can now attempt to systematically interconnect thedefining characteristics, the foundational principles, andthe emergent mechanisms of complexities in complexsystems in physics and in complex networks . Besides,by modelling all the computationally-distinct classes ofthe computation processors as function-specific typesof HoloQuantum HyperNetworks, one can reformulateunitedly and analyze more optimally computationalcomplexities in both mathematics and computer science .Finally, the theory M , by being scale-covariant (onwhich we will elaborate next), is ideal for the study ofthe emergences of complexities even ‘practically’. The theory M , by realizing all of its nine principles, isa quantum many body theory whose total kinematics,all microscopic interactions, and total unitary evolutiondo remain form-invariant in formulating all quantumnatures. By the operationally constructive definition of‘all quantum natures’ given in section one, and also asexplicitly stated in the explanatory note to the principlenine, this implies the following important statement.HoloQuantum HyperNetwork Theory is form-invariantunder changing the renormalization group scale all theway down from the ultraviolet fixed point. Therefore,the quantum equations of the perfected theory are all‘scale-covariant’. Being so, one knows that proposingHoloQuantum HyperNetwork Theory as the fundamentalcomplete form-invariant ‘it-from-qubit’ theory of allquantum natures is not to be meant only in a ‘scale-wise’conventional top-down manner . Namely, although wecan limit the theory to function ‘scale-by-scale’, and soonly in fixed scales, it can do much better in functioningover the renormalization-group scales. Because of itsrealized covariant-completeness, the perfected M isintrinsically the ‘Multi-Scale Theory’ in which quantumobjects-and-relations in the arbitrarily-chosen differentrenormalization-group scales can become all activatedsimultaneously and cooperate with one another . Giventhe proven significance of the multi-scale-functioningcomplex systems both in nature and in the advancedtechnologies, it remain as an important mission for thefuture works to manifestly formulate and thoroughlyanalyze, both at structural and functional levels, all the‘multi-scale organizational aspects’ of the theory M .HoloQuantum HyperNetwork Theory, we must highlight,is being proposed in the present work as the completecovariant ‘it-from-qubit’ theory of all quantum naturesat the fundamental level, but not as ‘the ultimate theory’ .By this, one brings to attention that quantum physicsis taken for granted in HoloQuantum HyperNetworkTheory, namely as a fundamental and exact inputin it. But, it may be that some defining aspects ofquantum physics, or even all of them, are emergentform a still-unknown ‘pre-quantum theory’. We suggestthat the related problem of ‘effectively undoing time’in HoloQuantum HyperNetwork Theory, which wehighlight in the conclusive paragraph of this paper, mayserve as a good theoretical laboratory to experimentwith the initiative models of the pre-quantum theory.Finally, we must come to the very notion and thevery role of ‘time’ in the perfected theory M . As aquantum theory, time plays the role of the ‘external’evolution parameter in HoloQuantum HyperNetworkTheory. But, we may be able to ‘internalize time’relatonically into the abstract total microscopic body ofHoloQuantum HyperNetworks. By this, we will be ledto even a one-level-higher parental theory .6 ACKNOWLEDGMENTS
Alireza Tavanfar would like to very happily thankYasser Omar, and moreover, Wissam Chemissany, BrunoCoutinho, Istvan A. Kovacs, Filippo Miatto, MasoudMohseni, Giuseppe Di Molfeta, Mohammad Nouri Zonoz, Ali Parvizi, Marco Pezzutto, Daniel Reitzner, AlbertoVerga and Mario Ziman for very precious discussions.Alireza Tavanfar also thanks the support from Fundaçãopara a Ciência e a Tecnologia (Portugal), namely throughthe project UID/EEA/50008/2013. [1]
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