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Matrix models and the quantum structure ofspace, time and matter
Harold Steinacker
Department of Physics, University of Vienna
February 2, 2018
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
outline:
status of “fundamental theory”
quantum space-time & geometry
matrix models
4D quantum spaces & cosmological space-times
towards particle physics, outlook
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
present understanding of fundamental matter & interactions:
standard model of elementary particle physics= quantum field theory
governs fundamental constituents of matter & interactionsexcept for gravity!
general relativity (GR)= classical geometrical theory of gravity
however: class. Einstein equations inconsistent with QM:
Rµν −12
gµνR+ Λgµν︸ ︷︷ ︸classical
?=
8πGc4 Tµν︸︷︷︸
quantum
GR resists quantization, not renormalizable
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
present understanding of fundamental matter & interactions:
standard model of elementary particle physics= quantum field theory
governs fundamental constituents of matter & interactionsexcept for gravity!
general relativity (GR)= classical geometrical theory of gravity
however: class. Einstein equations inconsistent with QM:
Rµν −12
gµνR+ Λgµν︸ ︷︷ ︸classical
?=
8πGc4 Tµν︸︷︷︸
quantum
GR resists quantization, not renormalizable
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
underlying issue: superposition principle of quantum mechanics
superposition of massive objects
⇓
superposition of “gravitational potentials”
⇒ need quantum theory of geometry & gravity
requires new framework for quantum space-time
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
Observational issuepresent cosmological concordance model: ΛCDM model
95 % of the Universe is not understood !!either
General Relativity is wrong, or
particle physics is incomplete, or
both (most likely)
→ expect major change in fundamental physics !H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
approaches to quantum & gravity
direct quantization of (pure) gravity:
loop quantum gravity, canonical QGcausal dynamical triangulationsasymptotic safety... (all have problems!)
quantization of very special models which include gravity
string theory (landscape? def?)Matrix Models (this talk!)
(holography)
“quantum” (noncommutative) geometrydescribed by Matrix Modelsphysics emerges from fluctuations
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
fuzzyness of space-time
Q.M. & G.R.⇒ break-down of classical space-time
measure object of size ∆x
Q.M. ⇒ energy E ≥ ~kc ∼ ~c∆x
G.R. ⇒ ∆x ≥ RSchwarzschild ∼ GMc2 ≥ ~G
c3∆x
⇒ (∆x)2 ≥ G~c3 = L2
Pl , LPl = 10−33cm
classical geometry breaks down, need pre-geometric framework
“fuzzy”, “foam-like” structure of space-time J. Wheeler 1955
“Unanwendbarkeit der Geometrie im Kleinen” Schrodinger 1934
why worry? because
virtual quantum effects in QFT probe UV scale, very significant!!(cf. UV divergences in 4D QFT)
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
fuzzyness of space-time
Q.M. & G.R.⇒ break-down of classical space-time
measure object of size ∆x
Q.M. ⇒ energy E ≥ ~kc ∼ ~c∆x
G.R. ⇒ ∆x ≥ RSchwarzschild ∼ GMc2 ≥ ~G
c3∆x
⇒ (∆x)2 ≥ G~c3 = L2
Pl , LPl = 10−33cm
classical geometry breaks down, need pre-geometric framework
“fuzzy”, “foam-like” structure of space-time J. Wheeler 1955
“Unanwendbarkeit der Geometrie im Kleinen” Schrodinger 1934
why worry? because
virtual quantum effects in QFT probe UV scale, very significant!!(cf. UV divergences in 4D QFT)
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
what is the quantum structure of geometry?
... lots of possibilities, generic problem:
modifications of geometry in UV → uncontrollable, unexpectedeffects in IR (cf. UV divergences in 4D QFT)
most approaches will fail, need protection mechanism
more precise statement: Doplicher Fredenhagen Roberts 1995
∑∆xµ∆xν ≥ L2
Pl
... space-time uncertainty relations, follows from
[Xµ,X ν ] = iθµν (cf. Q.M. !)
... noncommutative (quantum) space-time
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
what is the quantum structure of geometry?
... lots of possibilities, generic problem:
modifications of geometry in UV → uncontrollable, unexpectedeffects in IR (cf. UV divergences in 4D QFT)
most approaches will fail, need protection mechanism
more precise statement: Doplicher Fredenhagen Roberts 1995
∑∆xµ∆xν ≥ L2
Pl
... space-time uncertainty relations, follows from
[Xµ,X ν ] = iθµν (cf. Q.M. !)
... noncommutative (quantum) space-time
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
quantized space-time & geometry
old idea: Schrodinger, Heisenberg, Snyder, ...
borrow idea from QM:
QM = quantization of phase space [X ,P] = i~
quantum space = quantization [Xµ,X ν ] = iθµν of space-time
inherent uncertainty of space(time)use Q.M. techniques (coherent states, ...) for spacetime & geometry
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
basic message of GR:
cannot separate space-time↔ matter
need unified model governing space-time, matter & interaction
task:
find correct model which governs dynamical quantumspace-times
(many approaches conceivable, most will fail)
find correct space-time solutions
identify mechanisms for emergent physics(gauge theory & gravity)
how to choose model & framework ?
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
two crucial insights:
insight 1 (around 2000):
Yang-Mills gauge theory arises from fluctuations of quantum spacesin Matrix Models
[Xµ,X ν ] = iθµν , Xµ → Xµ +Aµ
[Xµ +Aµ,X ν +Aν ] = iθµν + i ∂µAν − ∂νAµ + i[Aµ,Aν ]︸ ︷︷ ︸Fµν
simpler than in classical space-time !! (use [Xµ,A] ∼ iθµν∂νA)
S = Tr [Xµ,X ν ][Xµ,Xν ] ∼∫
FµνFµν + c
what about gravity?
insight 2:
Matrix Models S = Tr [Xµ,X ν ][Xµ,Xν ] → dynamical quantum spaces
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
useful guideline: Schomerus, Chu-Ho, Seiberg-Witten, ... ∼ 1999
NC spaces & gauge theory arise on D-branes in string theory... “NonCommutative (quantum) field theory”
problem: UV/IR mixing Minwalla, van Raamsdonk, Seiberg 1999
virtual quantum effects in QFT probe shortest scalesstring-like behavior |x〉〈y | at high energies, non-localUV divergences in QFT new IR divergences
→ naive approaches will fail
avoided in one special model: IKKT matrix model2-fold origin/interpretation:
maximally SUSY Yang-Mills theorynonperturbative string theory
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
Matrix Models as candidates for a fundamental theory
IKKT or IIB model Ishibashi, Kawai, Kitazawa, Tsuchiya 1996
S[X ,Ψ] = −Tr(
[X a,X b][X a′,X b′
]ηaa′ηbb′ + Ψγa[X a,Ψ])
X a = X a† ∈ Mat(N,C) , a = 0, ...,9, N large
gauge symmetry X a → UX aU−1, SO(9,1), SUSY
proposed as non-perturbative definition of IIB string theory
maximally SUSY Yang-Mills in 3+1 dim.
quantization via matrix “path integral”
Z =∫
dXdΨ eiS[X ]
cf. BFSS model 1996: matrix quantum mechanics (class. time)
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
leads to “matrix geometry” (≈ NC geometry):
SE ∼ Tr [X a,X b]2 ⇒ config’s with small [X a,X b] 6= 0 dominate
i.e. “almost-commutative” configurations
∃ quasi-coherent states |x〉, minimize∑
a〈x |∆X 2a |x〉
(X a) ≈ diag., spectrum =: M ⊂ R10
〈x |X a|x ′〉 ≈ δ(x − x ′)xa, x ∈M
“condensation” of matrices→ geometry:
NC branes embedded in target space R10
X a ∼ xa : M ↪→ R10
matrices = observables X a = quantized functions xa ( cf. Q.M)
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
examples of matrix geometries:
three 2× 2 matrices
X 3 =
(x(1)
x(2)
)= x(1)|1〉〈1|+ x(2)|2〉〈2|,
X 1 + iX 2 =
(x(12)
)= x(12)|2〉〈1|
X 1 − iX 2 =
(x(21)
)= x(21)|1〉〈2|
describe two points at x(1), x(2) ∈ RD
• • (“point branes”)off-diagonal matrices ≈ strings connecting branes
spectrum of X a ↔ location in RD
→ minimal fuzzy sphere S2 ↪→ R3
X a = σa, X 21 + X 2
2 + X 23 = 3
4
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
examples of matrix geometries:
three 2× 2 matrices
X 3 =
(x(1)
x(2)
)= x(1)|1〉〈1|+ x(2)|2〉〈2|,
X 1 + iX 2 =
(x(12)
)= x(12)|2〉〈1|
X 1 − iX 2 =
(x(21)
)= x(21)|1〉〈2|
describe two points at x(1), x(2) ∈ RD
• ((66 • (“point branes”)
off-diagonal matrices ≈ strings connecting branes
spectrum of X a ↔ location in RD
→ minimal fuzzy sphere S2 ↪→ R3
X a = σa, X 21 + X 2
2 + X 23 = 3
4
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
examples of matrix geometries:
three 2× 2 matrices
X 3 =
(x(1)
x(2)
)= x(1)|1〉〈1|+ x(2)|2〉〈2|,
X 1 + iX 2 =
(x(12)
)= x(12)|2〉〈1|
X 1 − iX 2 =
(x(21)
)= x(21)|1〉〈2|
describe two points at x(1), x(2) ∈ RD
• ((66 • (“point branes”)
off-diagonal matrices ≈ strings connecting branes
spectrum of X a ↔ location in RD
→ minimal fuzzy sphere S2 ↪→ R3
X a = σa, X 21 + X 2
2 + X 23 = 3
4
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
generic class of matrix geometries:
fuzzy spaces = quantized Poisson manifolds (cf. phase space, QM)
X a ∼ xa : M ↪→ RD
quantization map:
Q : C(M) → End(H) ∼= Mat(N,C) H = CN
such that
Q(f )Q(g) = Q(fg) + O(θ), θ ∼ 1N
[Q(f ),Q(g)] = Q(i{f ,g}) + O(θ2)
f = f (X ) = f † ∈ Mat(N,C) ... quantized algebra of functions onM(“observables”) cf. QM!
in particular:X a = Q(xa)
typically Tr Q(f ) ∼∫M f (x)
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
Example: the fuzzy sphere S2N (Hoppe, Madore)
classical S2 : xa : S2 ↪→ R3
(x1)2 + (x2)2 + (x3)2 = 1
fuzzy sphere S2N :
3 matrices X a := 1√CNπN(Ja) ... spin N−1
2 generators of su(2)
(X 1)2 + (X 2)2 + (X 3)2 = 1l,
[X a,X b] = i√CNεabc X c ∼ i
N , CN = 14 (N2 − 1)
algebra A = Mat(N,C) ... quantized functions on S2N
S2N ... quantization of S2 with Poisson bracket {xa, xb} = 2
N εabc xc
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
covariance: SO(3) rotations
so(3)×A → A(Ja, φ) 7→ [πN(Ja), φ]
→ fuzzy spherical harmonics, UV cutoff
A = Mat(N,C) ∼= (N)⊗ (N) = (1) ⊕ (3) ⊕ ...⊕ (2N − 1)
= {Y 00 } ⊕ {Y 1
m} ⊕ ...⊕ {Y N−1m }.
S2N is fully covariant!
metric encoded in NC Laplace operator
2 : A → A,2φ = [X a, [X b, φ]]δab = 1
CNJaJaφ
compute 2Y lm = 1
CNl(l + 1)Y l
m
spectrum identical with classical case ∆gφ = 1√|g|∂µ(√|g|gµν∂νφ)
⇒ effective metric of 2 = round metric on S2
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
can measure such matrix geometries {X a}:
idea: Berenstein - Dzienkowski arXiv:1204.2788, Ishiki arXiv:1503.01230,Schneiderbauer - HS arXiv:1606.00646
measure energy E(x) of string connectingM with point at x ∈ RD
location ofM⊂ RD ↔ minima of E(x)
Mathematica package “Bprobe” DOI 10.5281/zenodo.45045Schneiderbauer - HS
examples:
squashed fuzzy CP2N ⊂ R6 fuzzy torus T 2
N
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
4D covariant quantum spaces
issue: NC spaces: [Xµ,X ν ] =: iθµν 6= 0breaks Lorentz invariance
however: ∃ fully SO(5) covariant fuzzy four-sphere S4N
Grosse-Klimcik-Presnajder 1996; Castelino-Lee-Taylor; Ramgoolam; Kimura;
Hasebe; Medina-O’Connor; Karabail-Nair; Zhang-Hu 2001 (QHE!) ...
complication/bonus: “internal structure”
fluctuations includes spin 2 modes → higher spin theory
“gravity“ naturally emergesHS arXiv:1606.00769 M. Sperling, HS arXiv:1707.00885
Euclidean case unphysical
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
∃ analogous
covariant cosmological quantum space-time solutions of IKKT modelsHS arXiv:1710.11495, arXiv:1709.10480
exactly homogeneous & isotropic
finite density of microstates
mechanism for Big Bang
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
starting point: Euclidean fuzzy hyperboloid H4n
Hasebe arXiv:1207.1968Mab ... hermitian generators of so(4,2),
[Mab,Mcd ] = i(ηacMbd − ηadMbc − ηbcMad + ηbdMac) .
ηab = diag(−1,1,1,1,1,−1)choose “short” discrete unitary irreps Hn (“minireps”, doubletons)
special properties:
irreducible under so(4,1), multiplicity-free
positive discrete spectrum
spec(M05) = {E0,E0 + 1, ...}, E0 = 1 +n2
eigenspaces ofM05 finite-dim.
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
fuzzy hyperboloid H4n :
def. 5 hermitian matrices
X a := rMa5, a = 0, ...,4
satisfy
ηabX aX b = X iX i − X 0X 0 = −R21l, R2 = r2(n2 − 4)
[X a,X b] = ir2Mab =: iΘab
one-sided hyperboloid in R1,4, covariant under SO(4,1)
note: induced metric is EuclideanH. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
oscillator construction: 4 bosonic oscillators [ψα, ψβ] = δβα
thenX a = r ψγaψ
on Fock space Hn γa ... SO(4,1) Gamma matrices
H4n is really quantized CP1,2 = S2 bundle over H4, selfdual θµν
(analogous to S4N )
spec(X 0 =M05) discrete, finite degeneracy
⇒ finite density of microstates!
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
open FRW universe from projected H4n HS arXiv:1710.11495
Yµ := Xµ, for µ = 0,1,2,3 (drop X 4 !)
= projection of H4n in R1,3 via
Yµ ∼ yµ : CP1,2 → H4 Π−→ R1,3 .
is solution of IKKT with m2 = −3r2, since
[Yµ, [Yµ,Y ν ]] = ir2[Yµ,Mµν ] (no sum)
= r2
Y ν , ν 6= µ 6= 0−Y ν , ν 6= µ = 00, ν = µ
hence
2Y Yµ = [Y ν , [Yν ,Yµ]] = 3r2Yµ (= eom of IKKT)
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
properties:
SO(3,1) manifest ⇒ foliation into SO(3,1)-invariant space-like3-hyperboloids H3
t ( homogeneous & isotropic! )
double-covered FRW space-time with hyperbolic (k = −1)spatial geometries
ds2 = dt2 − a(t)2dΣ2,
dΣ2 ... SO(3,1)-invariant metric on space-like H3
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
effective metric: H.S. arXiv:1003.4134
encoded in 2Y = [Yµ, [Yµ, .]] ∼ 1√|G|∂µ(√|G|Gµν∂ν .):
Gµν = α gµ′ν′ [θµ′µθν
′ν ]S2 , α =√|θµν ||γµν |
∼= α diag(c0(η), c(η), c(η), c(η))
where [.]S2 ... averaging over the internal S2
c(η) = 1− 13 cosh2(η)
c0(η) = cosh2(η)− 1 ≥ 0
signature change at c(η) = 0
cosh2(η0) = 3 ...Big Bang!
Euclidean for η < η0, Minkowski (+−−−) for η > η0
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
FRW metric and scale factor
ds2G = dt2 − a2(t)dΣ2
withdy0
dt=
c0(y0)12
|c(y0)| 34, a(t) = |c(y0)| 12 y2
0 .
Big Bang:
shortly after the BB :
a(t) ∝ c(t)14 ∝ t1/7
1 2 3 4 5
0.5
1.0
1.5
2.0
a(t)
conformal factor & 4-volume form |θµν | responsible for singularexpansion!
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
late times:
linear coasting cosmology
a(t) ∼ 3√
32
t .
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
a(t) ∼ t is remarkably close to observation:
age of univ. 13.9× 109y from present Hubble parametersimilar to Milne Univ;
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5
Sca
le f
acto
r -
a(t
)
t/t0
a(t) - ΛCDMa(t) - Milne Universe
artificial within GR, natural in M.M.gravity should emerge below cosm. scales (?!)
can reproduce SN1a (without acceleration)
different physics for early universe (CMB etc.)
cf. Melia Mon.Not.Roy.Astron.Soc. 446 (2015); Benoit-Levy , ChardinAstron.Astrophys. 537 (2012); Nielsen, Guffanti, Sarkar Sci.Rep. 6(2016)
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
other features:
∃ Euclidean pre-BB era
2 sheets with opposite intrinsic“chirality”
fluctuations on internal S2 → higher-spin fluctuation modes
Aµ = θµνhνρ(x)Pρ
expect (higher-spin-extended) gravity (spin 2 = gravity!)in progress, cf. HS arXiv:1606.00769 M. Sperling, HS arXiv:1707.00885
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
towards particle physics from the IKKT model
IKKT model has 9+1 “dimensions“
idea (cf. string theory):
consider solutionsM3,1 ×KN with 6 fuzzy extra dimensions
requires embedding of standard model fields in adjoint of SU(N):indeed: A. Chatzistavrakidis, H.S., G. Zoupanos arXiv:1107.0265
H. Grosse, F. Lizzi H. S. arXiv:1001.2703
Ψ =
02 0 0 lL QL
0(
0 eR0 νR
)QR
0 003
,
whereQL =
(uLdL
), lL =
(νLeL
), QR =
(dRuR
)similar to brane constructions of the S.M. in string theory
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
3 generations
from projected quantized coadjoint SU(3) orbits in extra dim.Ya = πµ(Ta), Ta ... su(3) root generators
C[µ] ↪→ R8 Π→ R6
(ya)a=1,...,8 7→ (ya)a=1,2,4,5,6,7
= solutions of IKKT model with cubic term,4- or 6-dimensional self-intersecting variety in R6
H.S., J. Zahn arxiv:1409.1440, H.S. arXiv:1504.05703chiral fermions from strings linking self-intersecting sheets
ψα,Λ = |µ′〉〈µ| , |µ〉 ... coherent statestriple self-intersection→ 3 generations!
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
no obstacle in principle to get (extended) standard model from IKKT,
no need for string compactifications
the simplest possible model might actually work !?
H. Steinacker Matrix models and the quantum structure of space, time and matter
Motivation Quantum geometry IKKT model, NC branes Cosmological space-times
Summary
understanding of fundamental physics is not complete↪→ opportunities!
matrix models: promising framework for quantum theory ofspace-time & geometry
analogous math as in Q.M.
extremely simple, good UV behavior (IKKT model)
all ingredients for gravity & partice physics
cosmological space-times & BB from fuzzy H4n
finite d.o.f. per volume
gravity expected to emerge
... exciting potential, seize the opportunity!
H. Steinacker Matrix models and the quantum structure of space, time and matter