Causal Structure for Noncommutative...

Causal Structure for Noncommutative Geometry

Micha l EcksteinJagellonian University & Copernicus Center, Krakow, Poland

Joint project with Nicolas Franco (CC, Krakow)

Marseille, 16th July 2014

Micha l Eckstein (Krakow) Causal Structure for NCG Marseille, 16th July 2014 1 / 17

Introduction & motivation

Noncommutative geometry a la Connes = spectral triples

1 Algebraisation of the Riemannian geometry2 Testing the concepts - new noncommutative horizons3 Applications - particle physics, cosmology, . . .

Drawbacks of the standard spectral approach

Relativistic physics is Lorentzian rather than RiemannianWe loose the causal structureApplications - need for a Wick rotation (t→ it)

Lorentzian spectral triples - a remedy?

1 Algebraisation of the causal structure2 Testing the concepts - almost-commutative space-times3 Applications - . . . ?


























































































Outline

1 Introduction & motivation

2 Noncommutative geometry

3 Causal structures

4 Testing the concepts – almost commutative causality

5 Summary


The axioms of noncommutative geometry

(A,H,D) - spectral triple

A - pre-C∗-algebra (unital)

H - Hilbert space

(need for indefinite products)

∃ a faithful representation π(A) ⊂ B(H)

D - the Dirac operator - selfadjoint, unbounded

(D − λ)−1 for any λ /∈ R- compact resolvent[D, π(a)] ∈ B(H) for all a ∈ A

. . .

The spectrum of Lorentzian D is way more complicated





H - Hilbert space

(need for indefinite products)

∃ a faithful representation π(A) ⊂ B(H)



. . .






H - Hilbert space (need for indefinite products)∃ a faithful representation π(A) ⊂ B(H)



. . .









. . .









. . .









. . .



Lorentzian spectral triples

(A, A,H,D, J) - Lorentzian spectral triple

A Hilbert space H.

A non-unital pre-C∗-algebra A with a faithful representation on B(H).

A preferred unitisation A of A which is a pre-C∗-algebra with a faithfulrepresentation on H and such that A is an ideal of A.

An unbounded operator D densely defined on H such that:

∀a ∈ A [D, a] extends to a bounded operator on H,

∀a ∈ A a∆−1J is compact, with ∆J :=(12 (DD∗ +D∗D) + 1

)1/2.

A bounded operator J on H - fundamental symmetry - such that:

J2 = 1, J∗ = J,[J, a] = 0 ∀a ∈ A,D∗ = −JDJ,J captures the Lorentzian signature of the metric[N. Franco, M.E. (2014b)]: J = −N [D, T ], with N ∈ A+, T ∈ L(H).




A Hilbert space H.






)1/2.






A Hilbert space H.






)1/2.






A Hilbert space H.






)1/2.






A Hilbert space H.






)1/2.






A Hilbert space H.






)1/2.






A Hilbert space H.






)1/2.






A Hilbert space H.






)1/2.






A Hilbert space H.






)1/2.




Globally hyperbolic manifold

A commutative Lorentzian spectral triple on a Lorentzian manifold M :

A ⊂ C∞0 (M) – smooth functions vanishing at ∞.

A ⊂ C∞b (M) – smooth bounded functions with all derivatives bounded.

H = L2(M,S) – space of square integrable spinor sections over M .

Non-degenerate products on H:

1 Indefinite: (f, g) =∫M(fx, gx)x

√|g|dnx.

2 Positive definite: 〈f, g〉 := (f, Jrg).

spacelike reflection r ∈ Aut(TM), r2 = 1, g(r·, r·) = g(·, ·)gr(·, ·) := g(·, r·) - positive definite metric on TM = F− ⊕ F+

Jr - fundamental symmetry associated with r

Jrc(e0)Jr = −c(re0), Jr = ic(e0) = iγ0

D = −i(c ◦ ∇S) = −iγµ∇Sµ – the Dirac operator.









√|g|dnx.














√|g|dnx.














√|g|dnx.














√|g|dnx.














√|g|dnx.














√|g|dnx.














√|g|dnx.














√|g|dnx.














√|g|dnx.







Outline



3 Causal structures


5 Summary


Causality - a reminder

Two points p, q are causally related p � q iffp = q or ∃ a future directed causal curve linking p and q.

� induces a partial order relation on the set of points of M .

global hyperbolicity =⇒ no closed causal curves

Theorem [Geroch (1967)]

Compact Lorentzian manifold always contain closed causal curves.

Theorem [Geroch (1970); Bernal, Sanchez (2005)]

M globally hyperbolic ⇔ M ' R× S (i.e. there exists global time)















































Algebraisation - “noncommutative points” (events)

Gelfand - Naimark theorem [1943]

commutative C∗-algbras1:1←→ (locally) compact Hausdorff topological spaces

States S(A) = {ϕ} on A:

positive linear functionals with ‖ϕ‖ = 1S(A) is a closed convex setP (A) - extremal points - pure states

Connes (pseudo-)distance formula: (may be infinite)

d(ϕ, χ) = sup{|ϕ(a)− χ(a)| : a ∈ A, ‖[D, a]‖ ≤ 1}.

Points of X1:1←→ P (C(X)) ∀x∈X ϕx : A → C, ϕx(f) := f(x)

Geodesic distance: dg(x, y) = d(ϕx, ϕy).

Another option: Points of X1:1←→ maximal ideals of C(X).

Noncommutatively: Mn(C) is simple, but P (Mn(C)) ' CPn−1.








d(ϕ, χ) = sup{|ϕ(a)− χ(a)| : a ∈ A, ‖[D, a]‖ ≤ 1}.












d(ϕ, χ) = sup{|ϕ(a)− χ(a)| : a ∈ A, ‖[D, a]‖ ≤ 1}.












d(ϕ, χ) = sup{|ϕ(a)− χ(a)| : a ∈ A, ‖[D, a]‖ ≤ 1}.












d(ϕ, χ) = sup{|ϕ(a)− χ(a)| : a ∈ A, ‖[D, a]‖ ≤ 1}.












d(ϕ, χ) = sup{|ϕ(a)− χ(a)| : a ∈ A, ‖[D, a]‖ ≤ 1}.






Algebraisation - the “causal cone”

Causal functions

C(M) = {f ∈ C∞(M,R) : f − non-decreasing along future dir. causal curves}

Proposition [F. Besnard (2009)]

Let M be a globally hyperbolic Lorentzian manifold, then the set of smoothbounded causal functions C(M) ⊂ A = C∞b (M) completely determines thecausal structure on M by

∀p,q∈M , p � q iff ∀f∈C(M), f(p) ≤ f(q).

A causal cone C is a subset of elements in A such that:

(a) ∀a,b∈C a∗ = a, ∀a,b∈C a+ b ∈ C;

(c) ∀a∈C ∀λ≥0 λa ∈ C, ∀x∈R x1 ∈ C;

(e) spanC(C) = A (the closure denotes the C∗-algebra completion);

(f) ∀a∈C ∀φ∈H 〈φ, J[D, a]φ〉 ≤ 0.



Causal functions






(a) ∀a,b∈C a∗ = a, ∀a,b∈C a+ b ∈ C;

(c) ∀a∈C ∀λ≥0 λa ∈ C, ∀x∈R x1 ∈ C;


(f) ∀a∈C ∀φ∈H 〈φ, J[D, a]φ〉 ≤ 0.



Causal functions






(a) ∀a,b∈C a∗ = a, ∀a,b∈C a+ b ∈ C;

(c) ∀a∈C ∀λ≥0 λa ∈ C, ∀x∈R x1 ∈ C;


(f) ∀a∈C ∀φ∈H 〈φ, J[D, a]φ〉 ≤ 0.



Causal functions






(a) ∀a,b∈C a∗ = a, ∀a,b∈C a+ b ∈ C;

(c) ∀a∈C ∀λ≥0 λa ∈ C, ∀x∈R x1 ∈ C;


(f) ∀a∈C ∀φ∈H 〈φ, J[D, a]φ〉 ≤ 0.



Causal functions






(a) ∀a,b∈C a∗ = a, ∀a,b∈C a+ b ∈ C;

(c) ∀a∈C ∀λ≥0 λa ∈ C, ∀x∈R x1 ∈ C;


(f) ∀a∈C ∀φ∈H 〈φ, J[D, a]φ〉 ≤ 0.


Causality recovered

Proposition [N. Franco, M.E. (2013)]

Let C be a causal cone, then for every two states χ, ξ ∈ S(A) define

χ � ξ iff ∀a∈C χ(a) ≤ ξ(a).

The relation � defines a partial order relation on S(A).

Theorem [N. Franco, M.E. (2013)]

Let (A, A,H,D, J) be a commutative Lorentzian spectral triple constructed froma globally hyperbolic Lorentzian manifold M . Then,

P (A) ' Spec(A) ∼= M,

and the partial order relation � on S(A) restricted to P (A) corresponds to theusual causal relation on M .


Causality recovered

Proposition [N. Franco, M.E. (2013)]

Let C be a causal cone, then for every two states χ, ξ ∈ S(A) define

χ � ξ iff ∀a∈C χ(a) ≤ ξ(a).

The relation � defines a partial order relation on S(A).


Let (A, A,H,D, J) be a commutative Lorentzian spectral triple constructed froma globally hyperbolic Lorentzian manifold M . Then,

P (A) ' Spec(A) ∼= M,

and the partial order relation � on S(A) restricted to P (A) corresponds to theusual causal relation on M .


Outline


2 Noncommutative geometrySpectral triples - a reminderLorentzian spectral triplesCommutative examples

3 Causal structuresCausality - rudimentsAlgebraisation

4 Testing the concepts – almost commutative causalityThe “two-sheeted” space-timeThe M2(C) model

5 Summary


Almost commutative flat space-time

Theorem [N. Franco, M.E. (2014a)]

Let (AM , AM ,HM ,DM , JM ) be an even Lorentzian spectral triple withZ2-grading γM and a finite Riemannian spectral triple (AF ,HF ,DF ). Then

A = AM ⊗AF , A = AM ⊗AF , H = HM ⊗HF , D = DM ⊗ 1 + γM ⊗DF , J = JM ⊗ 1

is a Lorentzian spectral triple.

A commutative spectral triple for Minkowski space-time

AM = S(R1,n) - rapidly decreasing functions,

AM = spanC(C(M)) ⊂ Cb(R1,n),

HM = L2(R1,n,C2(n+1)/2),DM = −iγµ∂µ,JM = iγ0.

Theorem [Kadison (1986)]

If at least one of the C∗-algebras A1, A2 is commutative, thenP (A1 ⊗A2) ∼= P (A1)× P (A2), i.e. pure states on A1 ⊗A2 are separable.






































The “two-sheeted” space-time

We work on 2-dim Minkowski spacetime M .

Consider a finite spectral triple:

AF = C⊕ C, HF = C2, DF = ( 0 mm 0 ) , with m ∈ C∗.

P (AF ) ' Z2, hence M(AM ⊗AF ) ' R1,1 ∪ R1,1.

Theorem [N. Franco, M.E. (2014c)]

Let p ∈ R1,1(1) and q′ ∈ R1,1

(2) then p � q′ if and only if

1 p � q on R1,1,

No classical causality violation!

2 l(γ) ≥ π2|m| .

There is causal link between the sheets!








Let p ∈ R1,1(1) and q′ ∈ R1,1


1 p � q on R1,1,


2 l(γ) ≥ π2|m| .









Let p ∈ R1,1(1) and q′ ∈ R1,1


1 p � q on R1,1,


2 l(γ) ≥ π2|m| .









Let p ∈ R1,1(1) and q′ ∈ R1,1


1 p � q on R1,1,


2 l(γ) ≥ π2|m| .









Let p ∈ R1,1(1) and q′ ∈ R1,1


1 p � q on R1,1,


2 l(γ) ≥ π2|m| .









Let p ∈ R1,1(1) and q′ ∈ R1,1


1 p � q on R1,1,


2 l(γ) ≥ π2|m| .









Let p ∈ R1,1(1) and q′ ∈ R1,1


1 p � q on R1,1, No classical causality violation!

2 l(γ) ≥ π2|m| .









Let p ∈ R1,1(1) and q′ ∈ R1,1


1 p � q on R1,1, No classical causality violation!

2 l(γ) ≥ π2|m| . There is causal link between the sheets!


The M2(C) model

The finite spectral triple:

AF = M2(C), HF = C2, DF = diag{d1, d2}, with d1 6= d2 ∈ R∗.

P (AF ) ' CP 1, henceM(AM ⊗AF ) ' R1,1 × S2.


Two pure states ωp,ξ, ωq,ϕ are causallyrelated with ωp,ξ � ωq,ϕ if and only if:

p � q in R1,1;

No classical causalityviolation!

ξ and ϕ have the same latitude;

Agrees with Connes’ distance!

l(γ) ≥ |θϕ−θξ||d1−d2| , where l(γ)

represents the length of a causalcurve γ going from p to q on R1,1.

“Finite speed of light”


The M2(C) model






p � q in R1,1;








The M2(C) model






p � q in R1,1;








The M2(C) model






p � q in R1,1;








The M2(C) model






p � q in R1,1;








The M2(C) model






p � q in R1,1;








The M2(C) model






p � q in R1,1; No classical causalityviolation!







The M2(C) model







ξ and ϕ have the same latitude;Agrees with Connes’ distance!





The M2(C) model







ξ and ϕ have the same latitude;Agrees with Connes’ distance!


represents the length of a causalcurve γ going from p to q on R1,1.“Finite speed of light”


Outline



3 Causal structures


5 Summary


Summary and outlook

Algebraisation of Lorentzian structures is possible!

Surprising causal structure of almost commutative geometries.

Generalisation to higher dim, curved, more noncommutative, . . . –Volunteers welcome!

Is there any true physics behind the toy models?

Thank you for your attention!

N. Franco, M. Eckstein: An algebraic formulation of causality for noncommutativegeometry,Class. Quant. Grav. 30 (2013) 135007, (arXiv:1212.5171v3).

N. Franco, M. Eckstein, Exploring the Causal Structures of Almost CommutativeGeometries, SIGMA 10 (2014) 010, (arXiv:1310.8225v2).

N. Franco, M. Eckstein, Noncommutative geometry, Lorentzian structures andcausality, to appear in Mathematical Structures of the Universe, CC Press (2014).

N. Franco, M. Eckstein, The Causal Structures of two-sheeted space-time, in prep.(2014).


Summary and outlook











Summary and outlook











Summary and outlook











Summary and outlook











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