Algebraic Footprints of Quantum Gravity: a Stability Point of View - - PowerPoint PPT Presentation

Algebraic Footprints of Quantum Gravity: a Stability Point of View

Chryssomalis Chryssomalakos ICN - UNAM

(based on joint work with Elias Okon (ICN - UNAM)) Reference: Int. J. Mod. Phys. D 13/10 (2004) 2003–2034 (hep-th/0410212)

Contents

• 1. Motivation
• 2. Lie algebra deformations
• 3. Stable quantum relativistic kinematics
• 4. Physical implications (in progress)
• 5. To do list

Motivation

◮ Non-commutative spacetime [Xµ, Xν] = ? ◮ Modified dispersion relations E2 = p2 + m2+ ? ◮ Preferred frames — Lorentz symmetry violation ◮ Invariant length scale Quantum Gravity → ℓP ≡

• G

c3

+ Lorentz contraction ⇓

?

◮ The stability criterion Galileo Einstein [Ja, Jb] = i ǫ

c ab Jc

[Ja, Jb] = i ǫ

c ab Jc

[Ja, Kb] = i ǫ

c ab Kc stabilize

− → [Ja, Kb] = i ǫ

c ab Kc

[Ka, Kb] = 0 [Ka, Kb] = i t ǫ

c ab Jc

Newton Heisenberg [f(q, p), g(q, p)] = 0

stabilize

− → [f(q, p), g(q, p)] = i {f(q, p), g(q, p)}

Lie Algebra Deformations

• Lie algebras

Lie algebra (V, µ) V : finite-dimensional vector space (over R) µ: Lie product µ: V × V → V bilinear: µ(λx + ρy) = λµ(x) + ρµ(y) antisymmetric: µ(x, y) = −µ(y, x) Jacobi: µ(x, µ(y, z)) = µ(µ(x, y), z) + µ(y, µ(x, z)) Basis {TA}, A = 1, . . . , n of V ⇒ structure constants f

C AB

s.t. [TA, TB] ≡ i µ(TA, TB) = i fAB

CTC

Jacobi: fAR

SfBC R + fBR SfCA R + fCR SfAB R = 0

(relax)

• Ln

P PM Q ψ1

f12

1

f12

2

fn−1,n

n

GL(n) orbit of P GL(n) orbit of Q Figure 1: The space Ln of n-dimensional Lie algebras (sketch). T ′

A = MA BTB

⇒ f ′

AB C = MA RMB S(M −1)U CfRS U

Orb(P) open ⇒ GP stable (rigid), otherwise unstable

• Deformations

G0 = (V, µ0) , µ0(X, Y ) ≡ [X, Y ]0 One-parameter (formal) deformation of G0: deformed commutator : [X, Y ]t = [X, Y ]0 +

∞

• m=1

ψm(X, Y ) tm t-dependent f’s : [TA, TB]t = i f t

AB CTC

ψm : V × V → V , bilinear, antisymmetric (2-cochains over V ) Vector space of p-cochains: Cp(V ) 1-cochains: V → V linear, C1(V ) ∼ Aut(V ) 0-cochains: constant maps, C0(V ) ∼ V

• Coboundary operator

For any µ, coboundary operator sµ : Cp → Cp+1, sµ ⊲ ψ(p)(TA0, . . . , TAp) =

p

• r=0

(−1)rµ

• TAr, ψ(p)(TA1, . . . , ˆ

TAr, . . . , TAp)

• +
• r<s

(−1)r+sψ(p) µ(TAr, TAs), TA0, . . . , ˆ TAr, . . . , ˆ TAs, . . . , TAp

• Examples: (φ ∈ C1 , ψ ∈ C2)

sµ ⊲ φ(A1, A2) = [A1, φ(A2)] − [A2, φ(A1)] − φ([A1, A2]) sµ ⊲ ψ(A1, A2, A3) = [A1, ψ(A2, A3)] − [A2, ψ(A1, A3)] + [A3, ψ(A1, A2)] − ψ([A1, A2], A3) + ψ([A1, A3], A2) − ψ([A2, A3], A1) Jacobi for µ ⇒ s2

µ = 0

• Cohomology groups

Jacobi for µt ⇒ sµ0 ⊲ ψ1 = 0 ⇒ ψ1 ∈ Z2(V, sµ) (2-cocycle — similarly Zp) where µt(X, Y ) ≡ [X, Y ]t = [X, Y ]0 + ψ1(X, Y )t + . . . The deformation is trivial iff ∃φ ∈ C1(V ) s.t. ψ1 = sµ0 ⊲ φ ⇒ ψ1 ∈ B2(V, sµ) (2-coboundary, trivial 2-cocycle — similarly Bp) Non-trivial deformations generated by non-trivial 2-cocycles (tangent space interpretation, slide 6) Hp ≡ Zp/Bp p-th cohomology group of G0 H2(G0) trivial ⇒ G0 stable (converse not true) ⇒ semisimple Lie algebras stable

• The ⊼ product

⊼: Cp × Cq → Cp+q−1 α ⊼ β(X0, . . . , Xm+n) =

• σ

sgn(σ) α

• β(Xσ(0), . . . , Xσ(n)), Xσ(n+1), . . . , Xσ(m+n)
• Graded commutator:

α, β = α ⊼ β − (−1)mnβ ⊼ α (α ∈ Cm+1, β ∈ Cn+1) Jacobi for µ ⇔ µ ⊼ µ = 1

2µ, µ = 0

In general: sµ ⊲ ψ = (−1)pµ, ψ Assume µ, µ = 0 (µ Lie product). µt = µ + φt also Lie product iff µt, µt = 0 ⇒ sµ ⊲ φt − 1 2φt, φt = 0 deformation equation

• Obstructions and H3

µt = µ + φt , φt =

∞

• n=1

φntn Defomation equation ⇒ sµ ⊲ φ1 = 0 sµ ⊲ φ2 = 1 2φ1, φ1 sµ ⊲ φ3 = φ1, φ2 . . . H3(G) = 0 ⇒ φ1 ∈ H2(G) might be non-integrable Notice: φ1, φ1 = 0 ⇒ µ + φ1t Lie product

• Coboundary operator as exterior covariant derivative

ΠA: left invariant 1-forms,

• ΠA, TB
• = δ

A B

ψ(p) → ψB ⊗ TB ≡ 1 p!ψA1...Ap

B ΠA1 . . . ΠAp ⊗ TB

sµ → ∇ = d + Ω , ΩA

B = fRB AΠR

(components defined by: ψ(p)(TA1, . . . , TAp) = ψA1...Ap

BTB)

Example: Galilean kinematics [Ja, Jb] = i ǫ

c ab Jc ,

[Ja, Kb] = i ǫ

c ab Kc ,

[Ka, Kb] = 0 µ = 1 2ǫ

c ab ΠaΠb ⊗ Jc + ǫ c ab ΠaΠ ¯ b ⊗ Kc

Only non-trivial 2-cocycle: χKKJ = 1

2ǫ c ab Π¯ aΠ¯ b ⊗ Jc, with χKKJ, χKKJ = 0

⇒ [Ka, Kb]t = i tǫ

c ab Jc

Experiment says: t = − 1

c2

Heisenberg’s Route

Classical relativity GCR ( = 0) [Jµν, Jρσ] = i

• gµσJνρ + gνρJµσ − gµρJνσ − gνσJµρ
• [Jρσ, Pµ] = i
• gµσPρ − gµρPσ
• [Jρσ, Zµ] = i
• gµσZρ − gµρZσ
• ,

plus M central. Algorithm:

• 1. Most general 1-cochain: φ = φ

B A ΠA ⊗ TB (225 terms)

• 2. Most general 2-coboundary: ψ = ∇φ (1008 terms)
• 3. Most general 2-cochain: χ = χ

C AB ΠAΠB ⊗ TC (1575 terms)

• 4. Require χ a 2-cocycle, ∇χ = 0 (5672 equations in 1575 unknowns)
• 5. ⇒ 221 2-cocycles χi. For each χi, solve χi = ψ (348075 equations)

• 6. ⇒ only five χi non-trivial:

H2(GCR) = {[0], [ψH], [ψPMZ], [ψZMP], [ψPMP], [ψZMZ]} where ψH = ΠµΠ ˙

µ ⊗ M

ψPMZ = ΠµΠM ⊗ Zµ ψZMP = Π ˙

µΠM ⊗ Pµ

ψPMP = ΠµΠM ⊗ Pµ ψZMZ = Π ˙

µΠM ⊗ Zµ

Deform along ψH only → GPH(q)

Stable Quantum Relativistic Kinematics

GPH(q) (Poincar´ e plus Heisenberg): [Jµν, Jρσ] = i

• gµσJνρ + gνρJµσ − gµρJνσ − gνσJµρ
• [Jρσ, Pµ] = i
• gµσPρ − gµρPσ
• [Jρσ, Zµ] = i
• gµσZρ − gµρZσ
• [Pµ, Zν] = i q gµνM

⇒ µPH(q) = 1 2ΠαρΠ β

ρ ⊗ Jαβ + ΠαρΠρ ⊗ Pα + ΠαρΠ ˙ ρ ⊗ Zα + qΠµΠ ˙ µ ⊗ M

H2(GPH(q)) = {[0], [ζ1], [ζ2], [ζ3]} where ζ1 = ΠµΠM ⊗ Zµ + q 2ΠµΠν ⊗ Jµν ζ2 = −Π ˙

µΠM ⊗ Pµ + q

2Π ˙

µΠ ˙ ν ⊗ Jµν

ζ3 = Π ˙

µΠM ⊗ Zµ − ΠµΠM ⊗ Pµ + qΠµΠ ˙ ν ⊗ Jµν

ζi, ζj = 0 ⇒ µ(q, α) = µPH(q) + α1ζ1 + α2ζ2 + α3ζ3 Lie product. Stable quantum relativistic kinematics: [Jµν, Jρσ] = i

• gµσJνρ + gνρJµσ − gµρJνσ − gνσJµρ
• [Jρσ, Pµ] = i
• gµσPρ − gµρPσ
• [Jρσ, Zµ] = i
• gµσZρ − gµρZσ
• [Pµ, Zν] = i qgµνM + i q α3Jµν

[Pµ, Pν] = i q α1Jµν [Zµ, Zν] = i q α2Jµν [Pµ, M] = −i α3Pµ + i α1Zµ [Zµ, M] = −i α2Pµ + i α3Zµ provided α2

3 = α1α2.

When α2

3 = α1α2, χ = ζ1 + ζ2 is a non-trivial integrable 2-cocycle

GQR so(1, 5) so(2, 4) GPH(q) so(3, 3) α2 α3 χ FUTURE PAST ELSEWHERE

Figure 2: The (α1, α2, α3)-deformation space of GPH(q).

Physical Implications (in progress)

• S. Sivasubramanian, G. Castellani, N. Fabiano, A. Widom, J. Swain, Y. N.

Srivastava, G. Vitiello, “Non-commutative Geometry and Measurements of Polarized Two Photon Coincidence Counts”, Annals Phys. 311 (2004) 191–203

• D. Ahluwalia-Khalilova, “A Freely Falling Frame at the Interface of the

Gravitational and Quantum Realms”, Class. Quantum Grav. 22 (2005) 1433-1450

• D. Ahluwalia-Khalilova, “Minimal Spatio-Temporal Extent of Events,

Neutrinos, and the Cosmological Constant Problem”, hep-th/0505124 (honorable mention in the 2005 Essay Competition of the Gravity Research Foundation) Standard wisdom (q = 1): [Pµ, Pν] = i 1 R2 Jµν R = 1 √ Λ [Zµ, Zν] = iℓ2

PJµν

ℓ2

P ≡ G

⇒ noncommutative spacetime, energy- momentum space

However:

Jµν, Pµ: primitive (extensive), e.g., total angular momentum: Jtot = J1 + J2

Positions not primitive

⇒ Xµ not Lie algebra generators Newtonian limit: X12 = M1X1 + M2X2 M1 + M2 ⇒ Zµ = XµM primitive

• M =
• P µPµ
• [Zµ, Zν] = iq(XµPν − XνPµ) = iqLµν

Spinless particles: α2 = 1 (commutative spacetime!)

To Do List

◮ so(1, 5)-representations, Casimirs ◮ Wigner like particle description ◮ Relativistic Zµ? Higher spin? Zero mass? ◮ Non-commutative spacetime? ◮ Invariant length ⇒ momentum cutoff? ◮ Invariant length + Lorentz contraction = ? ◮ Supersymmetry?