Models of Quantum Spacetime, and Quantum Geometry

Models of Quantum Spacetime, and Quantum Geometry

Gherardo Piacitelli¹

1Mathematics Area SISSA- Trieste piacitel@sissa.it

Workshop on "Perspectives of Fundamental Cosmology"

Nordita, Stockholm, November 5-30, 2012

Outline

Introduction

The DFR Model in brief Heurystics

The relations

No Relations whithout Representations!

Weyl quantisation and ?-product

Optimal localisation and large scale limit Independent events

Quantum Field Theory on Quantum Space Time Universal Differential Calculus

The Universal Calculus of Dubois-Violette Volume operators

Spectrum of the 4-volume

A bound on 3-volume’s euclidean length Back to Calculus

Connection and Parallel Transport Conclusions and Outlook

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 2/34

Outline

Introduction

The DFR Model in brief Heurystics

The relations

No Relations whithout Representations!

Weyl quantisation and ?-product

Optimal localisation and large scale limit Independent events

Quantum Field Theory on Quantum Space Time Universal Differential Calculus

The Universal Calculus of Dubois-Violette Volume operators

Spectrum of the 4-volume

A bound on 3-volume’s euclidean length Back to Calculus

Connection and Parallel Transport Conclusions and Outlook

Introduction

Two reasons to review the structure of spacetime at small scale:

I Ultraviolet catastrophe and “failure of renormalisation” (no known interacting models in 4d);

I Stability of spacetime under localisation alone (localisation in small region∼high energity density∼black hole. [Bronstein, Mead, de Witt,. . . ]

N.B. The latter only is meant to prevent non dynamical black hole formation, namely only as an effect of localisation. It would result in the paradox of a measurement of position with empty output, the information being trapped in the closed surface.

Relevant scale: λ_C(m) ∼ λ_S(m) ⇒ m ∼ mP, in which case scale∼ λP ∼ 10⁻³³cm.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 4/34

Introduction

Two reasons to review the structure of spacetime at small scale:

I Ultraviolet catastrophe and “failure of renormalisation” (no known interacting models in 4d);

I Stability of spacetime under localisation alone (localisation in small region∼high energity density∼black hole. [Bronstein, Mead, de Witt,. . . ]

N.B. The latter only is meant to prevent non dynamical black hole formation, namely only as an effect of localisation. It would result in the paradox of a measurement of position with empty output, the information being trapped in the closed surface.

Relevant scale: λ_C(m) ∼ λ_S(m) ⇒ m ∼ mP, in which case scale∼ λP ∼ 10⁻³³cm.

Introduction

Two reasons to review the structure of spacetime at small scale:

I Ultraviolet catastrophe and “failure of renormalisation” (no known interacting models in 4d);

I Stability of spacetime under localisation alone (localisation in small region∼high energity density∼black hole. [Bronstein, Mead, de Witt,. . . ]

N.B. The latter only is meant to prevent non dynamical black hole formation, namely only as an effect of localisation. It would result in the paradox of a measurement of position with empty output, the information being trapped in the closed surface.

Relevant scale: λ_C(m) ∼ λ_S(m) ⇒ m ∼ mP, in which case scale∼ λP ∼ 10⁻³³cm.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 4/34

Introduction

Two reasons to review the structure of spacetime at small scale:

I Ultraviolet catastrophe and “failure of renormalisation” (no known interacting models in 4d);

I Stability of spacetime under localisation alone (localisation in small region∼high energity density∼black hole. [Bronstein, Mead, de Witt,. . . ]

N.B. The latter only is meant to prevent non dynamical black hole formation, namely only as an effect of localisation. It would result in the paradox of a measurement of position with empty output, the information being trapped in the closed surface.

Relevant scale: λ_C(m) ∼ λ_S(m) ⇒ m ∼ mP, in which case scale∼ λP ∼ 10⁻³³cm.

Introduction

Two reasons to review the structure of spacetime at small scale:

I Ultraviolet catastrophe and “failure of renormalisation” (no known interacting models in 4d);

I Stability of spacetime under localisation alone (localisation in small region∼high energity density∼black hole. [Bronstein, Mead, de Witt,. . . ]

N.B. The latter only is meant to prevent non dynamical black hole formation, namely only as an effect of localisation. It would result in the paradox of a measurement of position with empty output, the information being trapped in the closed surface.

Relevant scale: λ_C(m) ∼ λ_S(m) ⇒ m ∼ mP, in which case scale∼ λP∼ 10⁻³³cm.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 4/34

Problems:

I No direct guidance from experimental data (so far; maybe what we look for already contained in astrophysical data, but may require an already advanced theory);

I Not clear which mathematics to use, and which picture of geometry;

I What is, in the end, locality? And what is interaction?

Necessary attitude: be rigorous! Start from physically meaning basic assumptions and explore them whithout stacking “commutative expectations” on them.

One possible strategy: reason about possibly realistic, intermediate models (semiclassical quantisation) and get inspired by them.

Problems:

I No direct guidance from experimental data (so far; maybe what we look for already contained in astrophysical data, but may require an already advanced theory);

I Not clear which mathematics to use, and which picture of geometry;

I What is, in the end, locality? And what is interaction?

Necessary attitude: be rigorous! Start from physically meaning basic assumptions and explore them whithout stacking “commutative expectations” on them.

One possible strategy: reason about possibly realistic, intermediate models (semiclassical quantisation) and get inspired by them.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 5/34

Problems:

I No direct guidance from experimental data (so far; maybe what we look for already contained in astrophysical data, but may require an already advanced theory);

I Not clear which mathematics to use, and which picture of geometry;

I What is, in the end, locality? And what is interaction?

Necessary attitude: be rigorous! Start from physically meaning basic assumptions and explore them whithout stacking “commutative expectations” on them.

One possible strategy: reason about possibly realistic, intermediate models (semiclassical quantisation) and get inspired by them.

Problems:

I No direct guidance from experimental data (so far; maybe what we look for already contained in astrophysical data, but may require an already advanced theory);

I Not clear which mathematics to use, and which picture of geometry;

I What is, in the end, locality? And what is interaction?

Necessary attitude: be rigorous! Start from physically meaning basic assumptions and explore them whithout stacking “commutative expectations” on them.

One possible strategy: reason about possibly realistic, intermediate models (semiclassical quantisation) and get inspired by them.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 5/34

Problems:

I No direct guidance from experimental data (so far; maybe what we look for already contained in astrophysical data, but may require an already advanced theory);

I Not clear which mathematics to use, and which picture of geometry;

I What is, in the end, locality? And what is interaction?

Necessary attitude: be rigorous! Start from physically meaning basic assumptions and explore them whithout stacking “commutative expectations” on them.

One possible strategy: reason about possibly realistic, intermediate models (semiclassical quantisation) and get inspired by them.

Problems:

I No direct guidance from experimental data (so far; maybe what we look for already contained in astrophysical data, but may require an already advanced theory);

I Not clear which mathematics to use, and which picture of geometry;

I What is, in the end, locality? And what is interaction?

Necessary attitude: be rigorous! Start from physically meaning basic assumptions and explore them whithout stacking “commutative expectations” on them.

One possible strategy: reason about possibly realistic, intermediate models (semiclassical quantisation) and get inspired by them.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 5/34

DFR model: seek for a model of flat quantum spacetime, generated by noncommutative coordinates q^µ(selfadjoint operators on Hilbert space).

Regime of (hypothetical validity): very few processes take place at very high energies. The density of processes is too low to produce curvature, which hence is fixed to flat. The energy is sufficiently high to sense the “quantum texture” of spacetime.

General relativity takes place only in giving the stability condition of spacetime under localisation; not a model of quantum gravity, but maybe a step in this direction.

What the q^µ’s are NOT: they are not observables in the sense of quantum mechanics (they are not in contradiction with the “no time-observable” issue of QM)!

In particular, we are NOT aiming at some “(more) noncommutative quantum mechanics”!

DFR model: seek for a model of flat quantum spacetime, generated by noncommutative coordinates q^µ(selfadjoint operators on Hilbert space).

Regime of (hypothetical validity): very few processes take place at very high energies. The density of processes is too low to produce curvature, which hence is fixed to flat. The energy is sufficiently high to sense the “quantum texture” of spacetime.

General relativity takes place only in giving the stability condition of spacetime under localisation; not a model of quantum gravity, but maybe a step in this direction.

What the q^µ’s are NOT: they are not observables in the sense of quantum mechanics (they are not in contradiction with the “no time-observable” issue of QM)!

In particular, we are NOT aiming at some “(more) noncommutative quantum mechanics”!

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 6/34

DFR model: seek for a model of flat quantum spacetime, generated by noncommutative coordinates q^µ(selfadjoint operators on Hilbert space).

Regime of (hypothetical validity): very few processes take place at very high energies. The density of processes is too low to produce curvature, which hence is fixed to flat. The energy is sufficiently high to sense the “quantum texture” of spacetime.

General relativity takes place only in giving the stability condition of spacetime under localisation; not a model of quantum gravity, but maybe a step in this direction.

What the q^µ’s are NOT: they are not observables in the sense of quantum mechanics (they are not in contradiction with the “no time-observable” issue of QM)!

In particular, we are NOT aiming at some “(more) noncommutative quantum mechanics”!

DFR model: seek for a model of flat quantum spacetime, generated by noncommutative coordinates q^µ(selfadjoint operators on Hilbert space).

Regime of (hypothetical validity): very few processes take place at very high energies. The density of processes is too low to produce curvature, which hence is fixed to flat. The energy is sufficiently high to sense the “quantum texture” of spacetime.

General relativity takes place only in giving the stability condition of spacetime under localisation; not a model of quantum gravity, but maybe a step in this direction.

What the q^µ’s are NOT: they are not observables in the sense of quantum mechanics (they are not in contradiction with the “no time-observable” issue of QM)!

In particular, we are NOT aiming at some “(more) noncommutative quantum mechanics”!

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 6/34

DFR model: seek for a model of flat quantum spacetime, generated by noncommutative coordinates q^µ(selfadjoint operators on Hilbert space).

Regime of (hypothetical validity): very few processes take place at very high energies. The density of processes is too low to produce curvature, which hence is fixed to flat. The energy is sufficiently high to sense the “quantum texture” of spacetime.

General relativity takes place only in giving the stability condition of spacetime under localisation; not a model of quantum gravity, but maybe a step in this direction.

What the q^µ’s are NOT: they are not observables in the sense of quantum mechanics (they are not in contradiction with the “no time-observable” issue of QM)!

In particular, we are NOT aiming at some “(more) noncommutative quantum mechanics”!

What are the q^µ’s? Theygenerate the localisation algebra. Same rôle of the x dependence of a relativistic quantum field φ(x ) on classical spacetime.

In ordinary QFT, the label x is not an observable, but a point in the classical geometric background on which QFT is defined. In LQP

“measuring position” means: observe an event localised in a certain regionO. If we trigger the event, then we say that the resulting state is localised inO; this QFTheoretical notion of localisation is

DIFFERENT than in QM.

The slogan is “replace x by q”. We take a different, noncommutative background, and we want to do QFT on it.

In view of future generalisations: Approach with coordinates not in contradiction with GR. Even in classical GR, coordinates describe the localisation of events; clearly, this makes sense even if the

coordinates themselves are not observable quantities.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 7/34

What are the q^µ’s? Theygenerate the localisation algebra. Same rôle of the x dependence of a relativistic quantum field φ(x ) on classical spacetime.

In ordinary QFT, the label x is not an observable, but a point in the classical geometric background on which QFT is defined. In LQP

“measuring position” means: observe an event localised in a certain regionO. If we trigger the event, then we say that the resulting state is localised inO; this QFTheoretical notion of localisation is

DIFFERENT than in QM.

The slogan is “replace x by q”. We take a different, noncommutative background, and we want to do QFT on it.

In view of future generalisations: Approach with coordinates not in contradiction with GR. Even in classical GR, coordinates describe the localisation of events; clearly, this makes sense even if the

coordinates themselves are not observable quantities.

What are the q^µ’s? Theygenerate the localisation algebra. Same rôle of the x dependence of a relativistic quantum field φ(x ) on classical spacetime.

In ordinary QFT, the label x is not an observable, but a point in the classical geometric background on which QFT is defined. In LQP

“measuring position” means: observe an event localised in a certain regionO. If we trigger the event, then we say that the resulting state is localised inO; this QFTheoretical notion of localisation is

DIFFERENT than in QM.

The slogan is “replace x by q”. We take a different, noncommutative background, and we want to do QFT on it.

In view of future generalisations: Approach with coordinates not in contradiction with GR. Even in classical GR, coordinates describe the localisation of events; clearly, this makes sense even if the

coordinates themselves are not observable quantities.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 7/34

What are the q^µ’s? Theygenerate the localisation algebra. Same rôle of the x dependence of a relativistic quantum field φ(x ) on classical spacetime.

In ordinary QFT, the label x is not an observable, but a point in the classical geometric background on which QFT is defined. In LQP

“measuring position” means: observe an event localised in a certain regionO. If we trigger the event, then we say that the resulting state is localised inO; this QFTheoretical notion of localisation is

DIFFERENT than in QM.

The slogan is “replace x by q”. We take a different, noncommutative background, and we want to do QFT on it.

In view of future generalisations: Approach with coordinates not in contradiction with GR. Even in classical GR, coordinates describe the localisation of events; clearly, this makes sense even if the

coordinates themselves are not observable quantities.

Note: coordinate operators can be “measured” with abitrary precision:

for every given µ, the uncertainty ∆Ψq^µcan be made small at wish by suitable choices of Ψ.No bounds to the “measurement” of one coordinate.

But [q^µ,q^ν] 6=0 implies that they cannot be simultaneously

“measured” with arbirary precision. Relative (Heisenberg–like) bounds arise.

The Amati–Ciafaloni–Veneziano relations are not of this kind (they contain an absolute bound).

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 8/34

Note: coordinate operators can be “measured” with abitrary precision:

for every given µ, the uncertainty ∆Ψq^µcan be made small at wish by suitable choices of Ψ.No bounds to the “measurement” of one coordinate.

But [q^µ,q^ν] 6=0 implies that they cannot be simultaneously

“measured” with arbirary precision. Relative (Heisenberg–like) bounds arise.

The Amati–Ciafaloni–Veneziano relations are not of this kind (they contain an absolute bound).

Note: coordinate operators can be “measured” with abitrary precision:

for every given µ, the uncertainty ∆Ψq^µcan be made small at wish by suitable choices of Ψ.No bounds to the “measurement” of one coordinate.

But [q^µ,q^ν] 6=0 implies that they cannot be simultaneously

“measured” with arbirary precision. Relative (Heisenberg–like) bounds arise.

The Amati–Ciafaloni–Veneziano relations are not of this kind (they contain an absolute bound).

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 8/34

Outline

Introduction

The DFR Model in brief Heurystics

The relations

No Relations whithout Representations!

Weyl quantisation and ?-product

Optimal localisation and large scale limit Independent events

Quantum Field Theory on Quantum Space Time Universal Differential Calculus

The Universal Calculus of Dubois-Violette Volume operators

Spectrum of the 4-volume

A bound on 3-volume’s euclidean length Back to Calculus

Connection and Parallel Transport Conclusions and Outlook

Heurystics

With a = minj∆x^j, b = max ∆x^j, τ = ∆x⁰,

I Energy ∼ 1/τ localised in box of sides ∆x^j generates gravitational potential

|V | / 1 b min(a, τ );

I to avoid formation of trapped surface, require g⁰⁰=1 + 2V > 0;

I this gives the relations

b min(a, τ ) ' 1.

More detailed analysis with localised states construced with free fields on classical spacetime Ψ = e^{iφ(f )}Ωlocalised in box of sides

∆x^µ; estimating the corresponding energy tensor and linearising Einstein equations leads to weaker set of relations:

∆x⁰(∆x¹+ ∆x²+ ∆x³) & λ²P,

∆x¹∆x²+ ∆x²∆x³+ ∆x³∆x¹) & λ²P.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 10/34

Heurystics

With a = minj∆x^j, b = max ∆x^j, τ = ∆x⁰,

I Energy ∼ 1/τ localised in box of sides ∆x^j generates gravitational potential

|V | / 1 b min(a, τ );

I to avoid formation of trapped surface, require g⁰⁰=1 + 2V > 0;

I this gives the relations

b min(a, τ ) ' 1.

More detailed analysis with localised states construced with free fields on classical spacetime Ψ = e^{iφ(f )}Ωlocalised in box of sides

∆x^µ; estimating the corresponding energy tensor and linearising Einstein equations leads to weaker set of relations:

∆x⁰(∆x¹+ ∆x²+ ∆x³) & λ²P,

∆x¹∆x²+ ∆x²∆x³+ ∆x³∆x¹) & λ²P.

Heurystics

With a = minj∆x^j, b = max ∆x^j, τ = ∆x⁰,

I Energy ∼ 1/τ localised in box of sides ∆x^j generates gravitational potential

|V | / 1 b min(a, τ );

I to avoid formation of trapped surface, require g⁰⁰=1 + 2V > 0;

I this gives the relations

b min(a, τ ) ' 1.

More detailed analysis with localised states construced with free fields on classical spacetime Ψ = e^{iφ(f )}Ωlocalised in box of sides

∆x^µ; estimating the corresponding energy tensor and linearising Einstein equations leads to weaker set of relations:

∆x⁰(∆x¹+ ∆x²+ ∆x³) & λ²P,

∆x¹∆x²+ ∆x²∆x³+ ∆x³∆x¹) & λ²P.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 10/34

Heurystics

With a = minj∆x^j, b = max ∆x^j, τ = ∆x⁰,

I Energy ∼ 1/τ localised in box of sides ∆x^j generates gravitational potential

|V | / 1 b min(a, τ );

I to avoid formation of trapped surface, require g⁰⁰=1 + 2V > 0;

I this gives the relations

b min(a, τ ) ' 1.

More detailed analysis with localised states construced with free fields on classical spacetime Ψ = e^{iφ(f )}Ωlocalised in box of sides

∆x^µ; estimating the corresponding energy tensor and linearising Einstein equations leads to weaker set of relations:

∆x⁰(∆x¹+ ∆x²+ ∆x³) & λ²P,

∆x¹∆x²+ ∆x²∆x³+ ∆x³∆x¹) & λ²P.

Heurystics

With a = minj∆x^j, b = max ∆x^j, τ = ∆x⁰,

I Energy ∼ 1/τ localised in box of sides ∆x^j generates gravitational potential

|V | / 1 b min(a, τ );

I to avoid formation of trapped surface, require g⁰⁰=1 + 2V > 0;

I this gives the relations

b min(a, τ ) ' 1.

More detailed analysis with localised states construced with free fields on classical spacetime Ψ = e^{iφ(f )}Ωlocalised in box of sides

∆x^µ; estimating the corresponding energy tensor and linearising Einstein equations leads to weaker set of relations:

∆x⁰(∆x¹+ ∆x²+ ∆x³) & λ²P,

∆x¹∆x²+ ∆x²∆x³+ ∆x³∆x¹) & λ²P.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 10/34

The Relations

Q^µν= −iλ²_P[q^µ,q^ν] definition of Q^µν, [q^µ,Q^νµ] =0 (ansatz for simplicity),

Q^µνQµν =0, Q^µν(∗Q)µν = ±4I.

Note: Covariant representations must be (highly) reducible, otherwise Q^µν = θ^µνI which cannot be unitarily covariant! Q^µν must be non trivial operators!

The Uncertainty Relations (now a mathematical consequence of commutation relations):

∆(q⁰)(∆(q¹) + ∆(q²) + ∆(q³)) & λ²P,

∆(q¹)∆(q²) + ∆(q²)∆(q³) + ∆(q³)∆(q¹) & λ²P. Weakerthan those arising from heuristic analysis.

Note: ∆(·) isnotlinear, hence ∆(q^µ)isnota 4-vector. The uncertainty relations are true in any reference frame.

The Relations

Q^µν= −iλ²_P[q^µ,q^ν] definition of Q^µν, [q^µ,Q^νµ] =0 (ansatz for simplicity),

Q^µνQµν =0, Q^µν(∗Q)µν = ±4I.

Note: Covariant representations must be (highly) reducible, otherwise Q^µν = θ^µνI which cannot be unitarily covariant! Q^µν must be non trivial operators!

The Uncertainty Relations (now a mathematical consequence of commutation relations):

∆(q⁰)(∆(q¹) + ∆(q²) + ∆(q³)) & λ²P,

∆(q¹)∆(q²) + ∆(q²)∆(q³) + ∆(q³)∆(q¹) & λ²P. Weakerthan those arising from heuristic analysis.

Note: ∆(·) isnotlinear, hence ∆(q^µ)isnota 4-vector. The uncertainty relations are true in any reference frame.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 11/34

No Relations whithout Representations!

(Lorentz covariant coordinates only, for simplicity; fully Poincaré covariant coordinates may be constructed as well)

I Hilbert Space:

H=L²(L , dΛ) ⊗ L²(R²,ds1ds2), where d Λ = Haar measure ofL .

I kets:

|Λi|s₁,s2i, Λ ∈L , (s1,s2) ∈ R²,

I normalisation:

{hΛ|hs₁,s2|}{|Λ⁰i|s⁰₁,s₂⁰i} = hΛ|Λ⁰ihs₁,s2|s⁰₁,s⁰₂i =

= δ_I(Λ⁻¹Λ⁰)δ(s1− s₁⁰)δ(s2− s⁰₂), where integrals are taken with the measure d Λds1ds2.

No Relations whithout Representations!

(Lorentz covariant coordinates only, for simplicity; fully Poincaré covariant coordinates may be constructed as well)

I Hilbert Space:

H=L²(L , dΛ) ⊗ L²(R²,ds1ds2), where d Λ = Haar measure ofL .

I kets:

|Λi|s₁,s2i, Λ ∈L , (s1,s2) ∈ R²,

I normalisation:

{hΛ|hs₁,s2|}{|Λ⁰i|s⁰₁,s₂⁰i} = hΛ|Λ⁰ihs₁,s2|s⁰₁,s⁰₂i =

= δ_I(Λ⁻¹Λ⁰)δ(s1− s₁⁰)δ(s2− s⁰₂), where integrals are taken with the measure d Λds1ds2.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 12/34

No Relations whithout Representations!

(Lorentz covariant coordinates only, for simplicity; fully Poincaré covariant coordinates may be constructed as well)

I Hilbert Space:

H=L²(L , dΛ) ⊗ L²(R²,ds1ds2), where d Λ = Haar measure ofL .

I kets:

|Λi|s₁,s2i, Λ ∈L , (s1,s2) ∈ R²,

I normalisation:

{hΛ|hs₁,s2|}{|Λ⁰i|s⁰₁,s₂⁰i} = hΛ|Λ⁰ihs₁,s2|s⁰₁,s⁰₂i =

= δ_I(Λ⁻¹Λ⁰)δ(s1− s₁⁰)δ(s2− s⁰₂), where integrals are taken with the measure d Λds1ds2.

I Position operators:

q^µ|Λi|ξi = λ_P|Λi{Λ^µ_νX^ν|ξi},

I in particular for Λ = I

X⁰|Ii|ξi = λ_P|Ii{P₁|ξi}, X¹|Ii|ξi = λ_P|Ii{P₂|ξi}, X²|Ii|ξi = λ_P|Ii{Q₁|ξi}, X³|Ii|ξi = λ_P|Ii{Q₂|ξi}. with [P_j,Q_k] = −iI, [Q_j,Q_k] = [P_j,P_k] =0 ⇐ (von Neumann “!”).

I unitary representation U ofL :

U(Λ)|Mi|s₁,s₂i = |ΛMi|s₁,s₂i;

I Lorentz covariance:

U(Λ)⁻¹q^µU(Λ) = Λ^µνq^ν.

I Commutators

Q^µν|Λi|ξi = Λ^µ_µ⁰Λ^ν_ν⁰σ₀^µ0ν0|Λi|ξi,

where [X^µ,X^ν] =iσ₀^µν They have joint spectrum (=set of common generalised eigenvalues)

Σ = {σ = −σ^t : σ^µνσµν =0, ±(∗σ)^µνσµν = ±4}.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 13/34

I Position operators:

q^µ|Λi|ξi = λ_P|Λi{Λ^µ_νX^ν|ξi},

I in particular for Λ = I

X⁰|Ii|ξi = λ_P|Ii{P₁|ξi}, X¹|Ii|ξi = λ_P|Ii{P₂|ξi}, X²|Ii|ξi = λ_P|Ii{Q₁|ξi}, X³|Ii|ξi = λ_P|Ii{Q₂|ξi}.

with [P_j,Q_k] = −iI, [Q_j,Q_k] = [P_j,P_k] =0 ⇐ (von Neumann “!”).

I unitary representation U ofL :

U(Λ)|Mi|s₁,s₂i = |ΛMi|s₁,s₂i;

I Lorentz covariance:

U(Λ)⁻¹q^µU(Λ) = Λ^µνq^ν.

I Commutators

Q^µν|Λi|ξi = Λ^µ_µ⁰Λ^ν_ν⁰σ₀^µ0ν0|Λi|ξi,

where [X^µ,X^ν] =iσ₀^µν They have joint spectrum (=set of common generalised eigenvalues)

Σ = {σ = −σ^t : σ^µνσµν =0, ±(∗σ)^µνσµν = ±4}.

I Position operators:

q^µ|Λi|ξi = λ_P|Λi{Λ^µ_νX^ν|ξi},

I in particular for Λ = I

X⁰|Ii|ξi = λ_P|Ii{P₁|ξi}, X¹|Ii|ξi = λ_P|Ii{P₂|ξi}, X²|Ii|ξi = λ_P|Ii{Q₁|ξi}, X³|Ii|ξi = λ_P|Ii{Q₂|ξi}.

with [P_j,Q_k] = −iI, [Q_j,Q_k] = [P_j,P_k] =0 ⇐ (von Neumann “!”).

I unitary representation U ofL :

U(Λ)|Mi|s₁,s₂i = |ΛMi|s₁,s₂i;

I Lorentz covariance:

U(Λ)⁻¹q^µU(Λ) = Λ^µνq^ν.

I Commutators

Q^µν|Λi|ξi = Λ^µ_µ⁰Λ^ν_ν⁰σ₀^µ0ν0|Λi|ξi,

where [X^µ,X^ν] =iσ₀^µν They have joint spectrum (=set of common generalised eigenvalues)

Σ = {σ = −σ^t : σ^µνσµν =0, ±(∗σ)^µνσµν = ±4}.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 13/34

I Position operators:

q^µ|Λi|ξi = λ_P|Λi{Λ^µ_νX^ν|ξi},

I in particular for Λ = I

X⁰|Ii|ξi = λ_P|Ii{P₁|ξi}, X¹|Ii|ξi = λ_P|Ii{P₂|ξi}, X²|Ii|ξi = λ_P|Ii{Q₁|ξi}, X³|Ii|ξi = λ_P|Ii{Q₂|ξi}.

with [P_j,Q_k] = −iI, [Q_j,Q_k] = [P_j,P_k] =0 ⇐ (von Neumann “!”).

I unitary representation U ofL :

U(Λ)|Mi|s₁,s₂i = |ΛMi|s₁,s₂i;

I Lorentz covariance:

U(Λ)⁻¹q^µU(Λ) = Λ^µνq^ν.

I Commutators

Q^µν|Λi|ξi = Λ^µ_µ⁰Λ^ν_ν⁰σ₀^µ0ν0|Λi|ξi,

where [X^µ,X^ν] =iσ₀^µν They have joint spectrum (=set of common generalised eigenvalues)

Σ = {σ = −σ^t : σ^µνσµν =0, ±(∗σ)^µνσµν = ±4}.

I Position operators:

q^µ|Λi|ξi = λ_P|Λi{Λ^µ_νX^ν|ξi},

I in particular for Λ = I

X⁰|Ii|ξi = λ_P|Ii{P₁|ξi}, X¹|Ii|ξi = λ_P|Ii{P₂|ξi}, X²|Ii|ξi = λ_P|Ii{Q₁|ξi}, X³|Ii|ξi = λ_P|Ii{Q₂|ξi}.

with [P_j,Q_k] = −iI, [Q_j,Q_k] = [P_j,P_k] =0 ⇐ (von Neumann “!”).

I unitary representation U ofL :

U(Λ)|Mi|s₁,s₂i = |ΛMi|s₁,s₂i;

I Lorentz covariance:

U(Λ)⁻¹q^µU(Λ) = Λ^µνq^ν.

I Commutators

Q^µν|Λi|ξi = Λ^µ_µ⁰Λ^ν_ν⁰σ₀^µ0ν0|Λi|ξi,

where [X^µ,X^ν] =iσ₀^µν They have joint spectrum (=set of common generalised eigenvalues)

Σ = {σ = −σ^t : σ^µνσµν =0, ±(∗σ)^µνσµν = ±4}.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 13/34

Weyl quantisation and ?-product

Given function f on R⁴, define the operator f (q) = 1

4π² Z

dk e^ikq Z

dx f (x )e^−ikx. Problem:

f (q)g(q) not of the form h(q) (some h).

Need more general symbols, i.e. functions f = f (σ, x ) of Σ × R⁴. Then DFR generalisation of Weyl quant.:

f (σ, x ) → f (Q, x )

| {z }

funct. calc.

→ f (Q, q)

| {z }

Weyl. Quant.

? :=pullback of operator product:

f (Q, q)g(Q, q) = (f ? g)(Q, q) which gives:

(f ? g)(σ, ·) = f (σ, ·) ?σg(σ, ·)

with ?σ=usual ?-product with fixed matrux σ (θ in most of literature).

The resulting algebra is E = C(Σ, K).

Weyl quantisation and ?-product

Given function f on R⁴, define the operator f (q) = 1

4π² Z

dk e^ikq Z

dx f (x )e^−ikx. Problem:

f (q)g(q) not of the form h(q) (some h).

Need more general symbols, i.e. functions f = f (σ, x ) of Σ × R⁴. Then DFR generalisation of Weyl quant.:

f (σ, x ) → f (Q, x )

| {z }

funct. calc.

→ f (Q, q)

| {z }

Weyl. Quant.

? :=pullback of operator product:

f (Q, q)g(Q, q) = (f ? g)(Q, q) which gives:

(f ? g)(σ, ·) = f (σ, ·) ?σg(σ, ·)

with ?σ=usual ?-product with fixed matrux σ (θ in most of literature).

The resulting algebra is E = C(Σ, K).

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 14/34

Optimal localisation and large scale limit

Candidate states to become points in the large scale limit? Naive answer: the pure states.

Problem 1: estimating the localisation region of pure states with the corresponding undertainties in the coordinates, one finds regions which are large compared with λP.

Problem 2: taking all the pure states, the large scale limit is R⁴× Σ, where Σ is a non compact manifold!

The only mathematically well defined possibility: states with optimal localisation, namely which minimizeP

µ(∆(q^µ))².

Of coursiˇc, this definition breaks covariance under Lorentz boosts.

Optimal localisation and large scale limit

Candidate states to become points in the large scale limit? Naive answer: the pure states.

Problem 1: estimating the localisation region of pure states with the corresponding undertainties in the coordinates, one finds regions which are large compared with λ_P.

Problem 2: taking all the pure states, the large scale limit is R⁴× Σ, where Σ is a non compact manifold!

The only mathematically well defined possibility: states with optimal localisation, namely which minimizeP

µ(∆(q^µ))².

Of coursiˇc, this definition breaks covariance under Lorentz boosts.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 15/34

Optimal localisation and large scale limit

Candidate states to become points in the large scale limit? Naive answer: the pure states.

Problem 1: estimating the localisation region of pure states with the corresponding undertainties in the coordinates, one finds regions which are large compared with λ_P.

Problem 2: taking all the pure states, the large scale limit is R⁴× Σ, where Σ is a non compact manifold!

The only mathematically well defined possibility: states with optimal localisation, namely which minimizeP

µ(∆(q^µ))².

Of coursiˇc, this definition breaks covariance under Lorentz boosts.

Optimal localisation and large scale limit

Candidate states to become points in the large scale limit? Naive answer: the pure states.

Problem 1: estimating the localisation region of pure states with the corresponding undertainties in the coordinates, one finds regions which are large compared with λ_P.

Problem 2: taking all the pure states, the large scale limit is R⁴× Σ, where Σ is a non compact manifold!

The only mathematically well defined possibility: states with optimal localisation, namely which minimizeP

µ(∆(q^µ))².

Of coursiˇc, this definition breaks covariance under Lorentz boosts.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 15/34

Optimal localisation and large scale limit

Candidate states to become points in the large scale limit? Naive answer: the pure states.

Problem 1: estimating the localisation region of pure states with the corresponding undertainties in the coordinates, one finds regions which are large compared with λ_P.

Problem 2: taking all the pure states, the large scale limit is R⁴× Σ, where Σ is a non compact manifold!

The only mathematically well defined possibility: states with optimal localisation, namely which minimizeP

µ(∆(q^µ))².

Of coursiˇc, this definition breaks covariance under Lorentz boosts.

Define the orthogonal projection E0=

Z

O(R³)

dR |RihR| ⊗ I.

We have [q^µ,E0] =0, so for every state |Ψi X

µ

(q^µ)²E₀|Ψi = λ²_P Z

dRX

µντ

R^µνR^µτ{|RihR| ⊗ X^νX^τ}|Ψi =

= λ²_P Z

dR {|RihR| ⊗X

τ

(X^τ)²}|Ψi where H0=Hamiltonian of harmonic oscillator> 1/2.

Hence, if hΨ|q^µ|Ψi = 0 and Ψ = E₀Ψ, X

µ

∆_Ψ(q^µ)²=2λ²_PhΨ|I ⊗ H₀|Ψi > 2λ²_P,

saturated by the states which are coherent on the second tensor factor= states with optimal localisation(a frame dependent definition). Note that the breakdown of covariance “only” means that relatively boosted observers do not agree on the set of states with optimal localisation; the bound stays true for every observer!

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 16/34

Define the orthogonal projection E0=

Z

O(R³)

dR |RihR| ⊗ I.

We have [q^µ,E0] =0, so for every state |Ψi X

µ

(q^µ)²E₀|Ψi = λ²_P Z

dRX

µντ

R^µνR^µτ{|RihR| ⊗ X^νX^τ}|Ψi =

= λ²_P Z

dR {|RihR| ⊗X

τ

(X^τ)²}|Ψi where H0=Hamiltonian of harmonic oscillator> 1/2.

Hence, if hΨ|q^µ|Ψi = 0 and Ψ = E₀Ψ, X

µ

∆_Ψ(q^µ)²=2λ²_PhΨ|I ⊗ H₀|Ψi > 2λ²_P,

saturated by the states which are coherent on the second tensor factor= states with optimal localisation(a frame dependent definition).

Note that the breakdown of covariance “only” means that relatively boosted observers do not agree on the set of states with optimal localisation; the bound stays true for every observer!

Using the states with optimal localisation, the classical limit is R⁴× Σ₀,

where

Σ₀= {Rσ0R^t , R ∈ O(R³)} ⊂ Σ which is compact!

Analogously one relates q^µqνto with the Hamiltonian of the anharmonic oscillator, which hase spectrum R.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 17/34

Using the states with optimal localisation, the classical limit is R⁴× Σ₀,

where

Σ₀= {Rσ0R^t , R ∈ O(R³)} ⊂ Σ which is compact!

Analogously one relates q^µqνto with the Hamiltonian of the anharmonic oscillator, which hase spectrum R.

Independent events

We go one step further

q^µ₁ =q^µ⊗ I ⊗ I ⊗ · · · , q^µ₂ =I ⊗ q^µ⊗ I ⊗ · · · ,

. . .

We take ⊗= Z -module tensor product over centre of localisation algebra (generated by Q^µν’s). Then

Q^µν⊗ I = I ⊗ Q^µν (=Q^µν).

Same relations up to a factor:

[(qj− q_k)^µ, (qj− q_k)^ν)] =2iλ²_PQµν

Same relations means same bound: X

µ

(q_j^µ− q_k^µ)²≥ 4λ²_P The Euclidean quantum distance is bounded below We want now make this a bit more systematic.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 18/34

Independent events

We go one step further

q^µ₁ =q^µ⊗ I ⊗ I ⊗ · · · , q^µ₂ =I ⊗ q^µ⊗ I ⊗ · · · ,

. . .

We take ⊗= Z -module tensor product over centre of localisation algebra (generated by Q^µν’s). Then

Q^µν⊗ I = I ⊗ Q^µν (=Q^µν).

Same relations up to a factor:

[(qj− q_k)^µ, (qj− q_k)^ν)] =2iλ²_PQµν

Same relations means same bound: X

µ

(q_j^µ− q_k^µ)²≥ 4λ²_P The Euclidean quantum distance is bounded below We want now make this a bit more systematic.

Independent events

We go one step further

q^µ₁ =q^µ⊗ I ⊗ I ⊗ · · · , q^µ₂ =I ⊗ q^µ⊗ I ⊗ · · · ,

. . .

We take ⊗= Z -module tensor product over centre of localisation algebra (generated by Q^µν’s). Then

Q^µν⊗ I = I ⊗ Q^µν (=Q^µν).

Same relations up to a factor:

[(qj− q_k)^µ, (qj− q_k)^ν)] =2iλ²_PQµν

Same relations means same bound: X

µ

(q_j^µ− q_k^µ)²≥ 4λ²_P The Euclidean quantum distance is bounded below We want now make this a bit more systematic.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 18/34

Independent events

We go one step further

q^µ₁ =q^µ⊗ I ⊗ I ⊗ · · · , q^µ₂ =I ⊗ q^µ⊗ I ⊗ · · · ,

. . .

We take ⊗= Z -module tensor product over centre of localisation algebra (generated by Q^µν’s). Then

Q^µν⊗ I = I ⊗ Q^µν (=Q^µν).

Same relations up to a factor:

[(qj− q_k)^µ, (qj− q_k)^ν)] =2iλ²_PQµν

Same relations means same bound:

X

µ

(q_j^µ− q_k^µ)²≥ 4λ²_P The Euclidean quantum distance is bounded below

We want now make this a bit more systematic.

Independent events

We go one step further

q^µ₁ =q^µ⊗ I ⊗ I ⊗ · · · , q^µ₂ =I ⊗ q^µ⊗ I ⊗ · · · ,

. . .

We take ⊗= Z -module tensor product over centre of localisation algebra (generated by Q^µν’s). Then

Q^µν⊗ I = I ⊗ Q^µν (=Q^µν).

Same relations up to a factor:

[(qj− q_k)^µ, (qj− q_k)^ν)] =2iλ²_PQµν

Same relations means same bound:

X

µ

(q_j^µ− q_k^µ)²≥ 4λ²_P The Euclidean quantum distance is bounded below We want now make this a bit more systematic.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 18/34

Perturbative models

Consider φ(x ) =R dk ˇφ(k ) free scalar quantum field of mass m;

define “third quantisation” according to Weyl quantisation:

φ(q) = Z

dk ˇφ(k ) ⊗ e^ikµqµ.

It is covariant! Evaluation on a localisation state is hω, φ(q)i =

Z

dk ˇφ(k )ω(e^ikq) = φ(fω).

if omega optimally localised around x and ωa=translation of ω by a, [φ(fω), φ(fω_a]

falls off exponentially in a in any spacelike direction.

Perturbative Dyson series with effective non local Hamiltonian: based either on : φⁿ(x ) : replaced with : φⁿ(q) : = : (φ ? · · · ? φ)(q) :or on setting q_j− q_k to minimum on : φ(q₁) · · · φ(q_n) :.

Perturbative models

Consider φ(x ) =R dk ˇφ(k ) free scalar quantum field of mass m;

define “third quantisation” according to Weyl quantisation:

φ(q) = Z

dk ˇφ(k ) ⊗ e^ikµqµ.

It is covariant! Evaluation on a localisation state is hω, φ(q)i =

Z

dk ˇφ(k )ω(e^ikq) = φ(fω).

if omega optimally localised around x and ωa=translation of ω by a, [φ(fω), φ(fω_a]

falls off exponentially in a in any spacelike direction.

Perturbative Dyson series with effective non local Hamiltonian: based either on : φⁿ(x ) : replaced with : φⁿ(q) : = : (φ ? · · · ? φ)(q) :

or on setting q_j− q_k to minimum on : φ(q₁) · · · φ(q_n) :.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 19/34

Perturbative models

Consider φ(x ) =R dk ˇφ(k ) free scalar quantum field of mass m;

define “third quantisation” according to Weyl quantisation:

φ(q) = Z

dk ˇφ(k ) ⊗ e^ikµqµ.

It is covariant! Evaluation on a localisation state is hω, φ(q)i =

Z

dk ˇφ(k )ω(e^ikq) = φ(fω).

if omega optimally localised around x and ωa=translation of ω by a, [φ(fω), φ(fω_a]

falls off exponentially in a in any spacelike direction.

Perturbative Dyson series with effective non local Hamiltonian: based either on : φⁿ(x ) : replaced with : φⁿ(q) : = : (φ ? · · · ? φ)(q) :or on setting q_j− q_k to minimum on : φ(q₁) · · · φ(q_n) :.

Good regularisation; especially second prescription leads to ultraviolet regular theory.

Problem: all approaches break covariance. We also tried

Yang-Feldan equations (apparently covariant), but then covariance brken at the level of mass renormalisation (which is frame

dependent).

Apparently the problem is conceptual: we do not know which concept should replace locality in this setting, so to reproduce it in the large scale limit.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 20/34

Good regularisation; especially second prescription leads to ultraviolet regular theory.

Problem: all approaches break covariance. We also tried

Yang-Feldan equations (apparently covariant), but then covariance brken at the level of mass renormalisation (which is frame

dependent).

Apparently the problem is conceptual: we do not know which concept should replace locality in this setting, so to reproduce it in the large scale limit.

Good regularisation; especially second prescription leads to ultraviolet regular theory.

Problem: all approaches break covariance. We also tried

Yang-Feldan equations (apparently covariant), but then covariance brken at the level of mass renormalisation (which is frame

dependent).

Apparently the problem is conceptual: we do not know which concept should replace locality in this setting, so to reproduce it in the large scale limit.

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 20/34

Outline

Introduction

The DFR Model in brief Heurystics

The relations

No Relations whithout Representations!

Weyl quantisation and ?-product

Optimal localisation and large scale limit Independent events

Quantum Field Theory on Quantum Space Time Universal Differential Calculus

The Universal Calculus of Dubois-Violette Volume operators

Spectrum of the 4-volume

A bound on 3-volume’s euclidean length Back to Calculus

Connection and Parallel Transport Conclusions and Outlook

The Universal Calculus of Dubois-Violette)

Given unital algebra A, take Λ(A) =M

n

Λⁿ(A) =M

n

A^n⊗

with product and differential

(a₁⊗ · · · a_n) · (b₁⊗ . . . ⊗ b_m) =a₁⊗ · · · ⊗ a_n−1⊗ a_nb₁⊗ b₂⊗ · · · ⊗ b_m, da = a ⊗ I − I ⊗ a,

(extended as a graded differential). Define Ω(A) as the d -stable subalgebra of Λⁿ(A), generated by A.

Want to apply this to A = M(E). Keep in mind: ⊗ = ⊗_Z. dq^µ=q^µ⊗ I − I ⊗ q^µ

interpreted as separation of independent events. It “lives” in E ⊗ E. dq^µdq^ν= (q^µ⊗ I − I ⊗ q^µ)(q^ν⊗ I − I ⊗ q^ν) =

=q^µ⊗ q^ν⊗ I − q^µ⊗ I ⊗ q^ν− I ⊗ q^µq^ν⊗ I + I ⊗ q^µ⊗ q^ν

“lives” in M(E ⊗ E ⊗ E).

Gherardo Piacitelli, SISSA -Trieste Models of Quantum Spacetime, and Quantum Geometry 22/34