Categorical Operator Algebraic Foundations of Relational ...ffp14.cpt.univ-mrs.fr/DOCUMENTS/SLIDES/BERTOZZINI_Paolo.pdf · Physical Systems = Higher Categories of Correlations Quantum

Relational Quantum Theory - BanckgroundQuantum Correlations

(Quantum) Higher CategoriesRelational Spectral Space-Time . . .

Categorical Operator Algebraic Foundations ofRelational Quantum Theory

Paolo Bertozzini

Department of Mathematics and Statistics - Thammasat University - Bangkok.

Symposium “Foundations of Fundamental Physics 2014”Epistemology and Philosophy Section

Aix Marseille University, Marseille - 17 July 2014.

Paolo Bertozzini Categorical Operator Algebraic Foundations of RQT



Abstract 1

We provide an algebraic formulation of C.Rovelli’s relationalquantum theory1 that is based on suitable notions of“non-commutative” higher operator categories, originally developedin the study of categorical non-commutative geometry.2 3 4

1Rovelli C (1996)Relational Quantum MechanicsInt J Theor Phys 35:1637 [arXiv:quant-ph/9609002].

2B P, Conti R, Lewkeeratiyutkul W (2007)Non-commutative Geometry, Categories and Quantum PhysicsEast-West Journal of Mathematics 2007:213-259 [arXiv:0801.2826v2].

3B P, Conti R, Lewkeeratiyutkul W (2012)Categorical Non-commutative GeometryJ Phys: Conf Ser 346:012003.

4B P, Conti R, Lewkeeratiyutkul W, SuthichitranontStrict Higher C*-categories preprint(s) (to appear).




Abstract 2

As a way to implement C.Rovelli’s original intuition on therelational origin of space-time,5 in the context of our proposedalgebraic approach to quantum gravity via Tomita-Takesakimodular theory,6 we tentatively suggest to use this categoricalformalism in order to spectrally reconstruct non-commutativerelational space-time geometries from categories of correlationbimodules between operator algebras of observables.

5Rovelli C (1997)Half Way Through the WoodsThe Cosmos of Science 180-223Earman J, Norton J (eds) University of Pittsburgh Press.

6B P, Conti R, Lewkeeratiyutkul W (2010)Modular Theory, Non-commutative Geometry and Quantum GravitySIGMA 6:067 [arXiv:1007.4094v2].




Abstract 3

Part of this work is a joint collaboration with:

I Dr.Roberto Conti (Sapienza Universita di Roma),

I Assoc.Prof.Wicharn Lewkeeratiyutkul(Chulalongkorn University),

I Dr.Matti Raasakka (Paris 13 University)

I Dr.Noppakhun Suthichitranont.




Outline

I Relational Quantum Theory - Background

?? Quantum RelationsI Quantum Phase Spaces = C*-algebrasI Quantum SpectraI Morphisms of Non-commutative (Quantum) SpacesI Quantum Relations = Bimodules - Relational Networks

? Quantum Higher CategoriesI Eckmann-Hilton CollapseI Non-commutative (Quantum) ExchangeI Higher Involutions - Strict Higher C*-categoriesI Examples: Relations, Hypermatrices, Hyper C*-algebras

?? Relational Spectral Space-Time




Ideology

I covariance and dynamic laws are both described by (higher)categorical structures of “correlations” between “observers”;

I higher-C*-categories are a possible algebraic quantummathematical formalism for the study of C.Rovelli’s “relationalquantum mechanics”;

I a “physical system” is completely captured by the (higher)categorical structure of “correlations”;

I the physical geometry (space-time) of the system isdetermined by the base category of the (higher) categoricalbundle of these “interaction/correlations” between observablealgebras;

I . . . such a non-commutative space-time organization of thesystem is spectrally recovered via “relational” modular theory.




Dymanics = CorrelationsFunctions / RelationsRelational Quantum Theory

• Relational Quantum Theory - Background





Relationalism: Dynamics = Correlations

I Relationalism in physics has a long tradition: G.Leibniz,G.Berkeley, E.Mach, . . .

I Relational dynamics is a core feature of Einstein’s theory ofrelativity (special and general): the dynamics is not specifiedas an explicit functional evolution with respect to a timeparameter, but it is given by an implicit relation between theseveral variables (Rovelli’s partial observables).

I Similarly (Einstein’s hole argument), localization of events ingeneral relativity is not absolute: coordinates are gauge andpoints on a Lorentz manifold are not objective elements of thetheory (coincidences, events and correlations, that arepreserved by local diffeomorphisms, are).





Functions / RelationsIn this classical context, mathematically speaking, the transition isbetween functions and relations (more generally 1-quivers):

functionrelationor quiver

t 7→ F (t) s(τ) τ�oo � // t(τ)

tt // F (t) s(τ)

τ ′

,,τ 22 t(τ)

F : A→ B R : T → A× B





Relational Quantum Theory 1

In 1994, C.Rovelli elaborated relational quantum mechanics asan attempt to radically solve the interpretational problems ofquantum theory.7 This approach is based on two assumptions:

I relativism: all systems (necessarily quantum) have equivalentstatus, there is no difference between observers and objects.

I completeness: quantum physics is a complete andself-consistent theory of natural phenomena.

7Rovelli C (1996) Relational Quantum MechanicsInt J Theor Phys 35:1637 [arXiv:quant-ph/9609002].





Relational Quantum Theory 2

Analysis of the Schrodinger’s cat problem entails:

I states are relative to each observer: different observers cangive different (but “compatible”) accounts of the interactions.

I the only physical properties (interactions) are correlationsbetween observers.

I physics is about information exchange between agents:correlations describe the “relative information” that observersposses about each other.





Quantum and Relativistic Relationalisms

In 1996 C.Rovelli went further with the radical conjecture:8

I there is a direct connection between:I quantum relationalism via correlations of systems,I general relativistic relational status of space-time localization

determined by contiguity of events.

I This strongly suggests that it should be possible to reinterpretthe information on space-time localization (contiguity) ascorrelations (interactions) between quantum systems, openingthe way for a reconstruction of space-time “a-posteriori” frompurely quantum correlations (see also R.Haag 1990).

It is our purpose to provide some mathematical implementation insupport of this approach to quantum relativity.

8Rovelli C (1997) Half Way Through the Woods The Cosmos of Science180-223 Earman J, Norton J (eds) University of Pittsburgh Press.





Higher C*-categories in Relational Quantum Theory

F A general mathematical framework for relational physics isstill missing and we propose to formalize “correlations”(relations between quantum systems) and their“compatibility” using a higher C*-categorical environment:

F Systems and observers are represented by C*-algebraic data.F Correlations and interactions are represented by “bimodules”.F There is a modular hierarchy of systems in mutual correlation,

because we must distinguish “observers” from “observers ofobservers”, ‘observers of observers of observers” and so on . . .

F The mutual compatibility requested is encoded by thecovariance coming from the compositions operations of anhigher category.

F Systems with higher internal correlations are described byhyper-C*-algebras.




Quantum Phase Spaces = C*-algebras of ObservablesMorphisms of Quantum SpacesQuantum Correlations = BimodulesPhysical Systems = Higher Categories of Correlations

• Quantum Correlations

I Quantum systems as C*-algebras,

I Spectra of quantum spaces and their morphisms,

I Correlations as suitable bivariant bimodules,

I Higher correlations . . .





Quantum Phase Spaces = C*-algebras of Observables

Following the usual framework of algebraic quantum theory,9 weaccept as tentative assumption that:

I quantum systems can be described as C*-algebras.

I classical systems, as a special case, are described bycommutative C*-algebras.

Gel’fand-Naımark duality assures that every commutativeC*-algebra A is ∗-isomomorphic to the algebra C (Sp(A)) ofcontinuous functions over its spectrum Sp(A) that is a locallycompact Hausdorff topological space (phase space):

Classical space = spectrum of C*-algebra' locally compact Hausdorff space

9Strocchi F (2005) An Introduction to the Mathematical Structure ofQuantum Mechanics: a Short Course for Mathematicians World Scientific.





Quantum Spectra (conjectural)

Motivated by our work on spectral theory of commutative fullC*-categories10 we find inspiration in the following:

Spectral Conjecture:there is a spectral theory of non-commutative C*-algebras in termof families of Fell complex line-bundles over involutive categories.

Quantum space = spectrum of C*-algebra= Fell line-bundle over an involutive category

Quantum correlations = morphisms of quantum spaces= spectra of (higher) bimodules= (higher) quivers.

10B P, Conti R, Lewkeeratiyutkul W (2011) A Horizontal Categorificationof Gel’fand Duality Adv Math 226(1):584-607 [arXiv:0812.3601v2].





Morphisms of Non-commutative Spaces 1

Classical space X = spectrum of Abelian C*-algebra C (X ; C)

= trivial line bundle X × C over space X

= Fell line-bundle over the space ∆X of “loops” of X

Abelian C*-algebra C (X ) = algebra Γ(X ; X × C) of sections of X × C= convolution algebra Γ(∆X ; ∆X × C)

•X

• · · · • •��

∆X

•��

· · · •��

∆X × C =

•��

•��

· · · •��





Morphisms of Non-commutative Spaces 2

Morphism of classical spaces X ,Y = map/relation/1-quiver : X → Y

= level-2 relation : ∆X → ∆Y

x � // y x ee ⇒ y99 x ∈ X , y ∈ Y .

For a relation R ⊂ X × Y (1-quiver) with reciprocal R∗ ⊂ Y × X ,the “convolution algebra” A of the Fell line-bundle with base∆X ∪ R ∪ R∗ ∪∆Y contains the C*-algebras C (X ), C (Y ), abimodule Γ(R,R × C) and its contragredient Γ(R∗; R∗ × C).

A =

[C (X ) Γ(R∗ × C)

Γ(R × C) C (Y )

]Hence the morphisms from X to Y are dually given by(Hilbert C*) bimodules over C (Y ) and C (X ).





Morphisms of Non-commutative Spaces 3Quantum space = space of points with “relations” = 1-quiver Q1

Algebra of functions on Q1 = “convolution” algebra of Q1

As a consequence we claim that:

I at the “spectral level” a morphism between two quantumspaces Q1

X and Q1Y is a 2-quiver Q2 with 2-cells like

x1

f��

//

�$

y1

g

��x2 // y2

f ∈ Q1X , g ∈ Q1

Y ,

I at the “dual level” a morphism of quantum spaces is a“level-2 bimodule” inside the convolution depth-2 hyperC*-algebra Γ(Q2) of the involutive 2-category generated bythe morphism 2-quiver Q2.

Obstacle: we need a “bivariant” notion of Hilbert C*-bimodule !Paolo Bertozzini Categorical Operator Algebraic Foundations of RQT




Quantum Correlations as Bimodules - Relational Networks

Correlations = “bimodules”:

I Inclusions of subsystems (homomorphisms) and symmetries(isomorphisms) φ : A→ B give adjoint pairs of twistedbimodules φB,Bφ.

I States ω on A, via GNS-representation (Hω, πω, ξω), givebimodules A(Hω)C.

I Conditional expectations Φ : A→ B give A-B bimodules viaKasparov GNS-representation theorem.

Rovelli’s “relational network”:“Physical system = 1-categorical structure of correlations”





Correlations from Symmetries

Two observer systems (C*-algebras) can be related by symmetries.

The usual case is given by the “geometrical symmetries” of thealgebras of localized observables in algebraic quantum field theory.

If O1,O2 are two regions in Minkowski space-time and g is anelement of the Poincare group such that g(O1) = O2, the axiomof Poincare covariance assure the existence of an isomorphismαg : A(O1)→ A(O2) between the C*-algebras localized in O1, O2.

Dynamics, as long as it is implemented via unitary evolution, canbe seen as a special case of these geometrical correlations (timetranslations) connecting observers at different times O 7→ O + t.





Symmetries as Twisted BimodulesIn algebraic quantum theory (following Wigner), symmetries aredescribed by linear isomorphisms (or conjugate-linearanti-isomorphisms) φ : A→ B between two C*-algebras ofobservables.

To every such symmetry φ, there is a naturally associated adjointpair of A-B bimodules φB and Bφ obtained by left or rightφ-twisting of the product in B:

a · x · b := φ(a)xb, ∀a ∈ A, b ∈ B, x ∈ φB,

b · x · a := bxφ(a), ∀a ∈ A, b ∈ B, x ∈ Bφ,

Composition of symmetries functorially corresponds to the internaltensor product of bimodules:

Aφ−→ B

ψ−→ C 7→ Cψ◦φ ' Cψ ⊗B Bφ.





Correlations from Localization

Any unital inclusion of C*-algebras or more generally any unital∗-homomorphism φ : A→ B between unital C*-algebras, willprovide a correlation via the φ-twisted bimodule φB.

In this way we see that “localization” can be formalized on thesame footing as covariance using correlation bimodules.

Strictly speaking we do not obtain a C*-category, because we canhave more than one correlation between the same systems, suchgeneralization of a C*-category is called a Fell bundle.





Correlations from States and Conditional Expectations

In quantum physics we have a further type of correlation betweenobservers that is responsible for the second form of dynamicalevolution, via “collapse of the wave function” as well as for the“quantum channels” of quantum information theory.

These “interactive correlations” as usually formalized in quantummechanics via completely positive maps between observablealgebras, states and conditional expectations are special cases.

Correlations via states and conditional expectations can also bereformulated using bimodules between algebras of observers.





States and Conditional Expectations as GNS-Bimodules

In algebraic quantum theory, a state is given by a normalizedpositive complex-linear functional ω : A→ C on a (unital)C*-algebra of observables A.

To every such state, we can naturally associate a A-C bimoduleHω via the usual Gel’fand-Naımark-Segal representation theorem:

I Hω is the Hilbert space (i.e. C-bimodule) obtained byseparation and completion of the vector space A under theinner product 〈x | y〉 := ω(x∗y), with x , y ∈ A,

I Hω is a left A-module via the representationπω : A→ B(Hω) obtained by linear continuous extension (ofthe quotient by the null space) of the left action of A on itself.

Conditional expectations are similary associated to a bimodule(C*-correspondence) via G.Kasparov GNS representation theorem.





Physical Systems = Categories of Correlations 1

Different observers are now mutually related by a family ofquantum correlation channels, some of them describingsymmetries, others quantum interactions.

Each observer is still equipped with a family of potential states,but now states of different observers can be compared via thefamily of binary correlations so far introduced.

The dynamic of the quantum theory has been totally codified viacorrelations and the potentially huge Cartesian product ofstate-spaces of the observers is now collapsed to a much moremanageable set of states that are compatible under the givencorrelations.





Physical Systems = Categories of Correlations 2

A a first step in the mathematical formalization of C.Rovellirelational quantum mechanics we propose the following statement:

I A physical system is totally captured by such a “category” ofbimodules of binary correlations(C.Rovelli’s “relational network”).

In principle we might also try to consider n-tuple correlationsbetween observers. Multimodules and their C*-polycategorieswould be necessary to formalize mathematically such notions.11

A physical system is for now formalized as a 1-categoricalstructure: level-1 correlations between algebras of observables ofdifferent agents. Do we need higher categories?

11C*-polycategories, work in progress.Paolo Bertozzini Categorical Operator Algebraic Foundations of RQT




Its Observers All The Way Up . . .

The “vertical categorification catastrophy” is almost inescapable:

I mathematically, the family of physical systems itself is a2-category (via functors and natural transformations);

I the ideological assumption of role interchangeability betweensystems and observers requires that such higher categoriesshould be physically relevant: the systems must themseves beobservers, object of further correlations;

I correlations between two systems could in principle bereconduced to lower level correlations between their “internalagents”, but this reductionist approach is not compatiblewith the original introduction of observers as “black buildingblocks” whose internal correlation structure is “not affected”by the (several alternative) external quantum correlations!





Higher Correlations

I Given two quantum systems A,B, a pair of observers can givea different description of their interaction correlations M,N:

A

M ''

M

ff B, A

N ''

N

ff B.

I The mutual compatibility between the two correlations isdecribed by a “higer level” morphism Φ : M→ N:

A

M ''

N

77�� Φ B.

I The morphism Φ can be seen as a (level-2) correlationbimodule between the C*-algebroids T(M), T(N) generatedrespectively by M and N. And so on . . .





Observers of Observers of . . . aka “The Hyper-Matrix” :-)

This opens the way to the scary possibility to have different levelsof “reality” for quantum properties, since observers and systems arenow not only “extensively” related, but “hierarchically” structured.

Higher categories will be necessary to formalize this situation.

I One might propose a hypercovariance principle to deal withthe invariance of the physics along the hierarchical ladder ofobservers/systems.

I Higher C*-categories and higher Fell bundles have beendeveloped from the beginning with this kind of goals in mindand can potentially deal with such a context of interacting“structured virtual realities”.




Higher Categories and (Non-commutative) ExchangeInvolutive Higher Categories - Higher C*-categoriesExamples: Hypermatrices, Hyper C*-algebras

• “Quantum” Higher Categories

I Mathematical obstacle 1:usual higher category theory cannot accommodate in anon-trivial way non-commutativity (quantum subsystems)!

I Mathematical obstacle 2:to describe higher level relational situations we need todevelop a theory of involutions for higher categories.





Globular Higher Arrows and Their Compositions

I 0-arrows (objects): •, 1-arrows: • // •

◦10-composition: A

g // Bf // C A

f ◦10g // C

I 2-arrows: •$$::

�� •


g1%%

g2

99�� Ψ B

f1 &&

f2

88�� Φ C A

f1◦10g1

((

f2◦10g2

66�� Φ◦

21Ψ C


f

�� Θ

DD

h

�� Λ

g // B A

f))

h

66�� Λ◦2

1Θ B





Cubical Higher Arrows and Their Compositions

I 2-arrows:

//

�� //

I ◦2h-composition:

A11

g1

��

f11 //

Φ

!!

A12

g2

��

f12 //

Ψ

!!

A13

g3

��A21

f21

// A22f22

// A23

7→

A11

g1

��

f12◦1hf11 //

Ψ◦2hΦ

!!

A13

g3

��A21

f22◦1hf21

// A23

I ◦2v -composition

A11

g11

��

f1 //

Φ

!!

A12

g12

��A21 f2 //

g21

��Ψ

!!

A22

g22

��A31 f3 // A32

7→

A11

g21◦1v g11

��

f1 //

Ψ◦2v Φ

!!

A12

g22◦1v g12

��A31

f3

// A32





Globular Strict Higher Categories

A globular n-category (C, ◦0, · · · ◦n−1) is a family C of n-arrowsequipped with a family of partially defined binary compositions ◦p,for p := 0, . . . , n − 1, that satisfy the following list of axioms:

I for all p = 0, . . . , n − 1, (C, ◦p) is a 1-category, whose partialidentities are denoted by Cp,

I for all q < p, a ◦q-identity is also a ◦p-identity: Cq ⊂ Cp,

I for all p, q = 0, . . . n − 1, with q < p, the ◦q-composition of◦p-identities, whenever exists, is a ◦p-identity: Cp ◦q Cp ⊂ Cp,

I the exchange property holds for all q < p: whenever(x ◦p y) ◦q (w ◦p z) exists also (x ◦q w) ◦p (y ◦p z) exists andthey coincide.12

12By symmetry, the exchange property automatically holds for all q 6= p.Paolo Bertozzini Categorical Operator Algebraic Foundations of RQT




Eckmann-Hilton Collapse

For q < p < n and n-arrows with a common q-source q-target •:◦np = ◦nq and they are commutative operations!

•��

�� ι

EE�� Ψ

// •��

�� Φ

EE�� ι

//

‖

• = •��

�� Φ

EE�� Ψ

// • = • ((66

�� Ψ◦

npΦ •

• $$::

�� Ψ •

$$::

�� Φ

‖

• = • ((66�� Φ◦

nqΨ •

•��

�� Ψ

EE�� ι

// •��

�� ι

EE�� Φ

// • = •��

�� Ψ

EE�� Φ

// • = • ((66

�� Φ◦

npΨ •

where ι is •ι1• %%

ι1•

99�� ι

2• •





Non-commutative Exchange: Globular Case

A

e%%

e

99�� ιe B

f

�� Ψ

DD

h

�� Φg // C = A

e

�� ιe

DD

e

�� ιee // B

f

�� Ψ

DD

h

�� Φg // C = A

f ◦e""

�� Ψ◦ιe

<<

h◦e

�� Φ◦ιe

g◦e // C

B

f

�� Ψ

DD

h

�� Φg // C

e&&

e

88�� ιe D = B

f

�� Ψ

DD

h

�� Φg // C

e

�� ιe

DD

e

�� ιee // D = B

e◦f""

�� ιe◦Ψ

<<

e◦h

�� ιe◦Φ

e◦g // D

non-commutative exchange:for all p-identities ι, for all q < p, the partially defined mapsι ◦q − : (C, ◦p)→ (C, ◦p) and − ◦q ι : (C, ◦p)→ (C, ◦p) arefunctorial (homomorphisms of partial 1-monoids).





Duals of Globular n-arrows

I Duals of 1-arrows: Af // B 7→ A B

f ∗oo

I Duals of 2-arrows:

∗1 : A

f%%

g99

�� Φ B 7→ A

f))

g

66� �� KS Φ∗1 B

∗0 : A

f%%

g99

�� Φ B 7→ A B

g∗hh

f ∗

vv �� Φ∗0

∗0,1 : A

f%%

g99

�� Φ B 7→ A B

g∗hh

f ∗

vv � �� KS Φ∗0,1

I For n-arrows we have 2n duals ∗α (including the identity)exchanging q-sources / q-targets for q in an arbitrary setα ⊂ {0, . . . , n − 1}.





Duals of Cubical n-arrows

I ∗2h:

A11

g1

��

f1 //

Φ!!

A12

g2

��A21 f2

// A22

7→

A11

g1

��

A12

f∗1h

1oo

g2

��Φ∗

2h

}}A21 A22

f∗1h

2

oo

I ∗2v :

A11

g1

��

f1 //

Φ!!

A12

g2

��A21 f2

// A22

7→

A11f1 // A12

A21

g∗1v

1

OO

f2//

Φ∗2v

==

A22

g∗1v

2

OO

I For a cubical n-category we have (including the identity) 2n

possible duals ∗α (preserving the composability class of then-cells), one for every subset α ⊂ {d1, . . . , dn} of directions ofthe edges of the n-cubical cells.





Higher Involutions

We have an involutive (higher) category whenever:we have some duality maps ∗α, with α ⊂ {0, . . . , n − 1} that are:

I covariant functors for all ◦q, with q /∈ α,

I contravariant functors for all ◦q, with q ∈ α,

I involutive: (x∗α)∗α = x ,

I Hermitian:13 x∗α = x , for all ◦q-identities, with q = min(α),

I commuting: (x∗α)∗β = (x∗β )∗α .

The higher category (C, ◦0, . . . , ◦n−1) is fully involutive if itsfamily of involutions generates all possible 2n dualities of n-arrows.

13This is the statement for the globular case.Paolo Bertozzini Categorical Operator Algebraic Foundations of RQT




Quantum Globular Strict Higher C*-Categories

A “quantum” fully involutive strict globular n-C*-category(C, ◦0, . . . , ◦n−1, ∗0, . . . , ∗n−1,+, ·, ‖ · ‖) is a fully involutive strictn-category with non-commutative exchange such that:

I for all a, b ∈ Cn−1, the fiber Cab is Banach with norm ‖ · ‖,14

I for all p, ◦p is fiberwise bilinear and ∗p is conjugate-linear,

I for all ◦p, ‖x ◦p y‖ ≤ ‖x‖ · ‖y‖, whenever x ◦p y exists,

I for all p, ‖x∗p ◦p x‖ = ‖x‖2, for all x ∈ C,

I for all p, x∗p ◦p x is positive in the C*-algebra envelope of Cee

(E(Cee), ◦p, ∗p,+, ·, ‖ · ‖), where e is the p-source of x .

A partially involutive strict n-C*-category will be equipped withonly a subfamily of the previous involutions and will satisfy onlythose properties that can be formalized using the given involutions.

14By definition Cab := {x ∈ C | b ◦n−1 x , x ◦n−1 a both exist}.Paolo Bertozzini Categorical Operator Algebraic Foundations of RQT




Matrices = Convolution Algebras of Pair Groupoids“A square matrix [x i

j ] ∈MN×N(C) is a section of a Fell line-bundle

E over a discrete finite pair groupoid X : X1 ⇒ X0”:

1(1,1)%% (N,1) ,, N (N,N)

xx(1,N)

ll

I a finite set of objects X0 := {1, . . . ,N}I a finite set of 1-arrows (ordered pairs)

(i , j) ∈ X1 := X0 × X0, with source j and target iI for every 1-arrow (i , j) ∈ X1, a fiber Eij over (i , j) that is just

a copy of the set C of complex numbersI a section i.e. a function x : X1 → E :=

⋃(i ,j)∈X1 Eij such that

x ij := x(i , j) ∈ Eij , for all (i , j) ∈ X1

I the same construction works with an associative involutivealgebra A in place of C: MN×N(A) 'MN×N(C)⊗C A.





Hyper Convolutions Algebras

I The same construction can be generalized taking any finiteglobular n-category15 (X, ◦0, . . . , ◦n−1) in place of N × N andany associative unital complex ∗-algebra A in place of C.

I The family of sections MX(A) of the bundle E := X×A is aconvolution algebra with n operations and n involutions:(σ ◦p ρ)z :=

∑x◦py=z σx ·A ρy ,

(σ∗p )z := (σz∗p )∗A , for all p ∈ A ⊂ {0, . . . , n − 1}.I E ⊂MX(A) is a strict globular involutive n-category.

We can think of the sections σ ∈MX(A) as “hypermatrices”whose entries σx ∈ A are indexed by n-arrows in a globular strictfinite A-involutive n-category X in place of the pair groupoid{1, . . . ,N} × {1, . . . ,N}.

15With usual exchange law or with non-commutative exchange.Paolo Bertozzini Categorical Operator Algebraic Foundations of RQT




Hyper C*-algebras

DefinitionA hyper C*-algebra (A, ◦0, . . . , ◦n−1, ∗0, . . . , ∗n−1) will be acomplete topo-linear space A equipped with different pairs ofmultiplication/involution (◦k , ∗k), for k = 0, . . . n − 1, inducing onA a C*-algebra structure, via a necessarily unique C*-norm ‖ · ‖k .

Proposition

Given unital C*-algebra A and a finite globular (cubical) higher(fully) involutive n-category X, the X-convolution ∗-algebraMX(A) is a hyper C*-algebra with the operations of◦q-convolution and ∗q-involutions, for q = 0, . . . , n − 1.





Hypermatrices 1

Elementary finite dimensional examples come from hypermatrices:MX1×···×Xn(C) := MX1

(· · ·MXn(C) · · · ) 'MX1(C)⊗· · ·⊗MXn(C),

where X1, . . .Xn are here finite involutive 1-categories and inparticular when Xk := {1, . . . ,Nk} × {1, . . . ,Nk}, k = 1, . . . , n arefinite pair groupoids.

A hypermatrix of depth-n is a multimatrix [x j1...jni1...in

] ∈MN21 ...N

2n(C)

with indexes ik , jk = 1, . . .Nk , for all k = 1, . . . , n.

I on MN21 ...N

2n(C) there are 2n different multiplications: acting

at every level either as convolution or as Schur product:

[x i1...ik ...inj1...jk ...jn

] •γ [yi ′1...i

′k ...i′n

j ′1...j′k ...j′n] := [

∑k∈γ

∑Nkok=1 x i1...ik ...in

j1...ok ...jny i1...ok ...inj1...jk ...jn

]

where γ ⊂ {1, . . . , n} is the set of contracting indexes.





Hypermatrices 2

I there are 2n involutions taking the conjugate of all the entriesand, at every level, either the transposed or the identity:

[x i1...ik ...inj1...jk ...jn

]?γ := [xi1...jk1

...jkm ...inj1...ik1

...ikm ...jn],

for all γ := {k1, . . . , km} ⊂ {1, . . . , n}.I there are 2n C*-norms taking either the operator norm or the

maximum norm at every level: using the natural isomorphismMN2

1 ...N2n(C) 'MN2

1(C)⊗C · · · ⊗C MN2

n(C), ∀γ ⊂ {1, . . . , n},

‖[x i1j1

]⊗ · · · ⊗ [x injn

]‖γ :=∏

k∈γ ‖[xikjk

]‖ ·∏

k ′ /∈γ ‖[xik′jk′

]‖∞, where

‖[x ikjk

]‖ is the C*-norm on MNk(C) and ‖[x ik

jk]‖∞ := maxi ,j |x i

j |.

(MN21 ...N

2n(C), •γ , ?γ , ‖ ‖γ , γ ⊂ {1, . . . , n}) is a hyper C*-algebra.

Hypermatrices can be seen as convolution hyper C*-algebras ofproduct cubical n-categories with “extra” compositions.




References

• Relational Spectral Space-Time

The formalization of relational quantum theory via higherC*-categories is only the second intermediate step in our ongoingresearch program on modular algebraic quantum theory




References

Modular Algebraic Quantum Theory 1 (Ideology)

∗ quantum theory is a fundamental theory of physics andshould not come from a quantization;

∗ geometry should be spectrally reconstructed a posteriori from abasic operational theory of observables and states;

∗ A.Connes’ non-commutative geometry provides the naturalenvironment where to attempt an implementation of the spectralreconstruction of a “quantum” space-time;

∗ Tomita-Takesaki modular theory should be the main tool toachieve the previous goals, associating to operational data, spectralnon-commutative geometries;

∗ categories of operational data provide the general framework forthe formulation of covariance in this context . . . and ultimately forthe identification of the geometric degrees of freedom (space-time)hidden in the theory.




References

Modular Algebraic Quantum Theory 2

In our modular algebraic quantum gravity program:16

I Every state ω on a C*-algebra O of partial observables inducesa net of subalgebras A ⊂ O such that ω|A is a KMS-state.

I By Tomita-Takesaki theory, every such KMS-state ω on aC*-subalgebra A uniquely determines a modular spectralnon-commutative geometry (Aω,Hω, ξω,Kω, Jω) where:

I Hω is the Hilbert space of the GNS representation πω inducedby ω|A, with cyclic separating unit vector ξω ∈ Hω,

I Kω := log ∆ω is the generator of the one-parameter unitarygroup t 7→ ∆it

ω spatially implementing the modularone-parameter group of ∗-automorphisms σωt ∈ Aut(A),

I Jω is the conjugate-linear operator spatially implementing themodular conjugation anti-isomorphism γω : πω(A)→ πω(A)′,

I Aω := {a ∈ A | [Kω, πω(a)] ∈ πω(A)′′},

16See section 6 in arXiv:1007.4094v2.Paolo Bertozzini Categorical Operator Algebraic Foundations of RQT



References

Modular Algebraic Quantum Theory 3

I Tomita-Takesaki modular theory is here taking the role of thequantum version of Einstein’s equation associating“geometries” to “matter content” where:

I “geometries” are spectrally described by variants of modularspectral triples (see A.Carey-A.Rennie-J.Phillips-F.Sukochev),

I “matter content” is described by the set of quantumcorrelations between observables specified by the state.

I Every pair (O, ω) gives a different “net” of modular spectralgeometries (Aω,Hω, ξω,Kω, Jω)A⊂O that are:

I quantum, since A ⊂ O are non-commutative,I state-dependent on ω,I relative to observers O.




References

Modular Algebraic Quantum Theory 4: Looking Ahead

I every pair (O, ω) selects C*-categorical data inside the C*-algebra O:the family of algebras A and some of their “correlations bimodules”;

I non-commutative space-time is now constructed topologically viathe “C*-enveloping” of the base category and we guess that itsspectral non-commutative geometry can be recovered from theadditional spectral data of the modular spectral geometries living onthe total space of such bundle;

I the investigation of “(higher) categorical modular theory” is nowone of the priorities of this program :-)




References

References on C*-categoriesI Ghez P, Lima R, Roberts J E (1985)

W*-categories Pacific J Math 120(1):79-109

I Doplicher S, Roberts J (1989)A New Duality Theory for Compact Groups Invent Math 98:157-218

I Longo R, Roberts J (1997) A Theory of Dimension K-Theory11:133-159

I Mitchener P (2002)C*-categories Proceedings of the London Math Society 84:375-404

I Yamagami S (2007)Notes on Operator Categories J Math Soc Japan 59(2):541-555

I Zito P (2007)2-C*-categories with Non-simple Units Adv Math 210 (1):122-164

I B P, Conti R, Lewkeeratiyutkul W, Suthichitranont N (2012)Categorical Non-commutative Geometry J Phys: Conf Ser 346:012003

I Blute R, Comeau M Von Neumann Categories arXiv:1209.0124

I B P, Conti R, Lewkeeratiyutkul W, Suthichitranont NStrict Higher C*-categories preprint(s) to appear




References

Other References

Higher Category Theory, Morphisms of Non-commutative Spaces

I Leinster T (2004) Higher Operads, Higher Categories CambridgearXiv:math/0305049

I Connes A (1994) Noncommutative Geometry Academic Press

I Connes A, Marcolli M (2008) Noncommutative Geometry, QuantumFields and Motives Colloquium Publications 55 AMS

I B P, Conti R, Lewkeeratiyutkul W (2006) A Category of SpectralTriples and Discrete Groups with Length Function Osaka J Math43(2):327-350

I Mesland B Unbounded Biviariant K -theory and Correspondences inNoncommutative Geometry arXiv:0904.4383

I Crane L (2009) Categorical Geometry and the MathematicalFoundations of Quantum Gravity Approaches to Quantum Gravity84-98 Oriti D (ed) arXiv:gr-qc/0602120




References

Other References 2

Relational Quantum Theory

I Rovelli C (1996) Relational Quantum Mechanics Int J Theor Phys35:1637 arXiv:quant-ph/9609002

I Rovelli C (1997) Half Way Through the Woods The Cosmos of Science180-223 Earman J, Norton J (eds) University of Pittsburgh Press

I Haag R (1996) Local Quantum Physics Springer

I B P Algebraic Relational Quantum Physics preprint (in preparation)

Modular Algebraic Quantum Gravity

I B P, Conti R, Lewkeeratiyutkul W (2010) Modular Theory,Non-commutative Geometry and Quantum Gravity SymmetryIntegrability and Geometry: Methods and Applications SIGMA 6:067arXiv:1007.4094v2

I B P, Conti R, Lewkeeratiyutkul W (2007) Non-commutative Geometry,Categories and Quantum Physics arXiv:0801.2826v2




References

Thank You for Your Kind Attention!

This file has been realized using the “beamer” LATEX package ofthe TEX-live distribution and Winefish editor on Ubuntu Linux.

We acknowledge the partial support from

I the Department of Mathematics and Statistics in ThammasatUniversity and

I the Thammasat University Research Grant n. 2/15/2556:“Categorical Non-commutative Geometry”.


Documents

Categorical Operator Algebraic Foundations of Relational ...ffp14.cpt.univ-mrs.fr/DOCUMENTS/SLIDES/BERTOZZINI_Paolo.pdf · Physical Systems = Higher Categories of Correlations Quantum