Download 6 Compact quantum spaces: “fuzzy spaces”

• cotangent bundle: M ... manifold, local coords xi
T ∗ M ... bundle of 1-forms {pi dxi } over M
local coords: xi , pj
Now at a point (xi , pj ) ∈ T ∗ M, we choose the one-form θ = pi dxi . This defines a
canonical (tautological) 1-form θ on T ∗ M.
The symplectic form is defined as ω = dθ = dpi dxi
Phase space in class. mechanics!
• any orientable 2-dim. manifold
ω ... volume-form
sphere S 2 : ∃ unique SO(3) -invariant volume form
Liouville theorem, volume form. On a symplectic space, there is a natural volume
form given by
Ω = ωn =
1
ωi j ...ωin jn dxi1 dxi2 ...dxin dxjn
n! 1 1
(196)
this is analogous to (but independent of) the Riemannian volume for a Riemannian
manifold. Liouvilles theorem states that this volume form is invariant under Hamiltonian
vector fields. It follows immediately from LXH ω = 0, since LXH is a derivation and
satisfies the product rule. This is very important in statistical mechanics.
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Compact quantum spaces: “fuzzy spaces”
In this section, we discuss some examples of compact quantized symplectic (or Poisson)
spaces. These are locally “the same“ as the standard quantization of R2n as discussed
before, but they are in fact simpler: they contain only finitely many ”quantum cells“,
the corresponding Hilbert space is simply CN , and the quantized algebra of functions is
given by a simple matrix algebra AN = M at(N, C). We will start with some compact
symplectic space (M, ω), with sufficient symmetry in order to simplify life, and give a
quantization map Q : C(M) → AN which respects the quantization conditions as well
as the symmetries.
In the next section, we will then equip these spaces with an additional metric structure, in order to do non-commutative (or quantum) geometry.
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One remark on the compactness of M: the relation
�
n
Ωf
(2π) T r(Q(f )) =
(197)
M
(which we assume to hold in general...) where Ω is the Liouville volume form implies in
particular
(2π)n dim H = volΩ (M)
(198)
This means that the symplectic volume of M is quantized, which means that ω is
quantized accordingly. We will see this in the exampels below. Thus compact M
implies that H is finite-dimensional hence H = CN for some N ∈ N. This is a big
simplification compared with e.g. quantum mechanics.
6.1
Fuzzy torus TN2
classical T 2 :
We can think of the (flat!) torus T 2 as square in R2 with opposite edges identified,
or as embedded torus
xa : T 2 �→ R4 ,
a = 1, 2, ..., 4
(199)
(200)
via
x1 + ix2 = eiϕ ,
x3 + ix4 = eiψ
(201)
so that (x1 )2 + (x2 )2 = 1 and (x3 )2 + (x4 )2 = 1. This defines xa (ϕ, ψ) as functions on
T 2.
In any case, the space of (complex-valued) functions A = L2 (T 2 ) or A = C ∞ (T 2 ) is
spanned by the plane waves
φnm (ϕ, ψ) = einϕ eimψ
(202)
�
A={
cnm φnm (ϕ, ψ)}
(203)
i.e.
(Fourier series). This is an infinite-dimensional vector space and a commutative algebra.
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We equip the torus with the Poisson structure
{ϕ, ψ} = h
(204)
in (local!!) coordinates y µ = (ϕ, ψ) on T 2 , or
µ
ν
{y , y } = θ
µν
�
0 1
=h
−1 0
�
(205)
This is singled out because it respects the U (1) × U (1) symmetry of the torus. The
corresponding symplectic form is
�
�
0 1
−1
ωµν = −h
−1 0
1
ω = − h−1 (dψ ∧ dϕ − dϕ ∧ dψ) = h−1 dϕ ∧ dψ
(206)
2
To see that the Poisson structure is well-defined on T 2 , we have
{eiϕ , eiψ } = −h eiϕ eiψ
(207)
Fuzzy T 2 :
We need a corresponding quantum algebra AN and a quantization map.
We introduce the

0
0


U =

0
1
satisfy
following clock- and shift operators (Sylvester, Weyl, t’Hooft, ...)



1
1 0 ... 0
 e2πi N1

0 1 ... 0



2



...
e2πi N
(208)
, V = 




.
.


.
... 0 1
N −1
2πi
0 ...
0
N
e
U V = qV U,
U N = V N = 1,
[U, V ] = (q − 1)V U
1
q = e2πi N
(209)
(210)
This generate A = M at(N, C) ... quantiz. algebra of functions on TN2
A priori, quantization is not unique. However a unique quantization is obtained if
we require Q to respect the symmetries of the manifold. The classical torus admits a
U (1) × U (1) symmetry. In the fuzzy case, only its discrete ZN × ZN subgroup survives:
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ZN × ZN action:
ZN × A → A
k
similar other ZN
k
(ω , φ) �→ U φU
−1
n m
A = ⊕N
n,m=0 U V
−k
(211)
(212)
... harmonics
quantization map:
C(T 2 ) → A = M at(N, C)
� −nm/2 n m
q
U V =: Φ̂nm , |n|, |m| < N/2
inϕ imψ
=e e
�→
0, otherwise
Q:
φn,m
(213)
(214)
satisfies
1
),
N
1
Q(i{f, g}) = [Q(f ), Q(g)] + O( 2 )
N
Q(f )† = Q(f ∗ )
Q(f g) = Q(f )Q(g) + O(
for the Poisson structure {eiϕ , eiψ } =
2π
N
eiϕ eiψ on T 2
(Q4)
(215)
(⇔ {ϕ, ψ} = − 2π
)
N
Here O( N1 ) means �.� = O( N1 ) in the operator norm.
2π
Tr(I(f ))
N
integral:
=
�
ωf,
T2
ω=
1
dϕdψ
2π
TN2 ... quantization of (T 2 , N ω)
The metric on TN2 seems “obvious”, but need extra structure, later.
sketch of proof :
It is enough to assume that f = φnm etc. First, note that
Q(φn1 m1 φn2 m2 ) = q −(n1 +n2 )(m1 +m2 )/2 U n1 +n2 V m1 +m2
Q(φn1 m1 )Q(φn2 m2 ) = q −n1 m1 /2 q −n2 m2 /2 U n2 V m2
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(216)
hence
Q(φn1 m1 φn2 m2 ) − Q(φn1 m1 )Q(φn2 m2 ) = (q −(n1 +n2 )(m1 +m2 )/2 − q −n1 m1 /2 q −n2 m2 /2 )U n2 V m2
1
(217)
= O( )
N
Note that we need to fix the indices ni , mi and then let N → ∞.
To show the second condition Q4, we make an induction on the degree deg(φnm ) =
max(n, m). For degree 1, we just verify
[Q(eiϕ ), Q(eiψ )] = [U, V ] = (q − 1)U V = i
1
2π
U V + O( 2 )
N
N
{eiϕ , eiψ } = −{ϕ, ψ}eiϕ eiψ
(218)
so clearly Q4 is satisfied for
{ϕ, ψ} = −
2π
N
(219)
Now for functions of higher degree, we simply use
[Q(f g), Q(h)] = [Q(f )Q(g) + O(
1
), Q(h)]
N
1
), Q(h)]
N
1
= Q(f )[Q(g), Q(h)] + [Q(f ), Q(h)]Q(g) + O( 2 )
N
= Q(f )[Q(g), Q(h)] + [Q(f ), Q(h)]Q(g) + [O(
(220)
and similarly
{f g, h} = f {g, h} + {f, g}h
Q({f g, h}) = Q(f )Q({g, h}) + Q({f, g})Q(h)
(221)
subtracting these two expressions and using the induction assumption establishes Q4.
(strictly speaking we use here that Q preserves the degree).
6.2
The fuzzy sphere
classical S 2 :
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xa : S 2 �→ R3
xa xa = 1
(222)
(223)
⇒ A = C ∞ (S 2 )
The sphere admits an action of SO(3)
SO(3) × A → A
(J a , φ) �→ [X a , φ]
(224)
The space of functions A then decomposes into irreducibly representations as follows
2
fuzzy sphere SN
:
(Hoppe, Madore 1992)
Again, a unique quantization is obtained if we require Q to respect the symmetries,
which is SO(3) here. This symmetry can indeed be prserved at the quantum level:
As quantized algebra we take again Aθ = M at(N, C).
Let X a ∈ A be 3 hermitian matrices
i abc c
ε X ,
CN
X a X a = 11 ,
[X a , X b ] = √
realized as X a =
√1 J a
CN
...
1
CN = (N 2 − 1)
4
(225)
(226)
N − dim irrep of su(2) on CN ,
2
generate full algebra A ∼
= M at(N, γ) of functions on SN
SO(3) action:
so(3) × A → A
(J a , φ) �→ [X a , φ]
(227)
We can then decompose A = M at(N, C) into irreps of SO(3):
A = M at(N, C) ∼
= (N ) ⊗ (N̄ ) = (1) ⊕ (3) ⊕ ... ⊕ (2N − 1)
= {Ŷ00 } ⊕ {Ŷm1 } ⊕ ... ⊕ {ŶmN −1 }.
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(228)
this defines the fuzzy spherical harmonics (polynomials in X a );
UV cutoff !
quantization map:
Q:
C(S 2 ) → A = M at(N, C)
� l
Ŷm , l < N
l
Ym �→
0, l ≥ N
(229)
(230)
satisfies
1
),
N
1
Q(i{f, g}) = [Q(f ), Q(g)] + O( 2 )
N
Q(f g) = Q(f )Q(g) + O(
Poisson structure {xa , xb } =
2
N
(231)
εabc xc on S 2
integral:
4π
Tr(I(f )) =
N
�
ωf,
S2
1
ω = εabc xa dxb dxc
2
(232)
2
is the quantization of (S 2 , N ω)
This means that SN
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Coherent states
The uncertainty relations state that quantized spaces consist of quantum cells with area
� or θ. There are in fact states which saturate this bound, called “coherent states”. We
2
exhibit these states both on R2� and on SN
, becasue they nicely illustrate the geometry
and the quasi-classical meaning of quantum spaces.
7.0.1
Oscillator realization of R2θ
It is useful to rewrite the canonical commutation relations in terms of the complex
generators
1
a† = (X1 + iX2 ),
2
(X1 , X2 ) ≡ (Q, P )
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1
a = (X1 − iX2 )
2
(233)
which satisfy
[a† , a] = �
(234)
This allows to introduce the following Fock basis for the (standard) Hilbert space H:
�
H=
|n�,
|n� := (a† )n |0�
n≥0
a|0� = 0
(235)
This is usually done if one wants to quantize the harmonic oscillator with Hamilton
function h = p2 + q 2 and correpsonding Hamiltonian H = P 2 + Q2 , but of course this
can always be done, irrespective of any Hamiltonian.
Note that the vacuum state |0� is an optimally localized state at the origin,
�0|Xi |0� = 0,
�
�0|X12 |0� = = �0|X22 |0�
2
(236)
with minimal uncertainty compatible with the uncertainty relation. Therefore it “occupies” a quantum cell with area �, illustrating the Bohr-Somerfeld quantization rule.
(can also realize ellipsoidal states).
Since we know how to translate functions on R2θ , we can immediately write down the
optimally localized states centered at any point (c1 , c2 ) in (quantized) R2� :
|c1 , c2 � = ei(c1 X1 +c2 X2 ) |0�
= ei(ca+c
� a† )
|0� ∼ eic
� a†
|0�
(237)
up to an (irrelevant) phase factor. This is familiar from the harmonic oscillator, where
such states - rotating around circles with h = const - are quasi-classical solutions of the
Schrödinger equations.
7.0.2
2
Coherent states on SN
2
It is now obvious how to construct the corresopnding optimally localized states on SN
:
the state localized at the north pole pn = (0, 0, R) is given by the highest weight state,
|pn � ≡ |L, L�
(238)
N = 2L + 1 and where J 3 |L, L� = L|L, L�; note that
�pn |X2,3 |pn � = 0
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(239)
We know how to move this state around the sphere using SO(3). One can show that
2
these are again the optimally localized states on SN
.
We define more generally
|pg � := πN (g)|L, L�
(240)
where πN denotes the N -dimensional irrep of SO(3). Since the stabilizer group U (1) of
p0 acts on |pg � via a complex phase, the associated one-dimensional projector
Πg = |pg ��pg |
(241)
is independent of U (1), so that there is a well-defined map
SO(3)/U (1) ∼
= S 2 → End(H)
p �→ Πg = |pg ��pg |
(242)
(2)
=: (4π)−1 δN (p − x).
(243)
The notation on the rhs should indicate that these are the optimally localized wave2
functions on SN
. To proceed, we label coherent states related by U (1) with the corresponding point on S 2 , so that |p� := |pg(p) � ∼ |pg(p)� �. One can then show the following
results [?]
�
|p��p| = c11
overcomplete
S2
�
�N −1
,
ϑ = �(p, p� ) localization p ≈ p�
|�p|p� �| = cos(ϑ/2)
pa X a |p� = |p�
�p|X a |p� = TrX a Πp = pa ∈ S 2
which provide a justification for the interpretation of X a ∼ xa : S 2 �→ R3 as quantized
embedding maps. One can also show that these states are optimally localized on S 2 ,
(ΔX 1 )2 + (ΔX 2 )2 + (ΔX 3 )2 ≥
�
N −1
1
=
∼
�p|(X a − �p|X a |p�)2 |p�.
2CN
2N
a
(244)
In principle, such considerations should apply to all matrix geometries under consideration here, although the explicit realization of such coherent states is in general not
known.
All this illustrates the fact that for scales larger than �, these quantum spaces look
like ordinary spaces. The quantum structure shows up only if one considers processes
which probe scales smaller than �. In QM of course, not only the algebra of observables
but also states play a central role, and these may be de-localized (superpositions!) leading to large-scale effects such as Bell correlations. However these will not play a central
role here.
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8
Noncommutative geometry and quantized spacetime
In this second part of the lecture, we discuss an application of the same mathematical
structures in a totally different physical context. In Quantum mechanics, the above
“quantized spaces” play the role of a quantized phase space. Now we will use the same
concept for a quantized space to model configuration spaces, which may play the role
of physical space-time (or e.g. quantized etra dimensions, or similar. In particular, this
means that we want to add a metric structure on quantum spaces.
To understand the motivation we recall some relevant background:
8.1
The present understanding of fundamental interactions and
its limitations
• gravity: described by general relativity (1910’s)
space-time described by pseudo-Riemannian manifold (M, g), dynamical metric
gµν describes gravity through the Einstein equations.
1
8πG
Rµν − gµν R + gµν Λ = 4 Tµν
2
c
(245)
• all other interactions: electroweak and strong interactions
described by standard model of fundamental interactions (1970’s)
= Quantum field theory (gauge theory) with SU (3) × SU (2) × U (1) gauge group
= Q.M. & special relativity & suitable ingredients
However, up to now there is no consistent quantum theory of all fundamental interactions including gravity.
There are many reasons to expect some sort of quantum structure of space-time at
very short distances, notably at the Planck scale. We will discuss a possible mathematical realization of such a quantum or noncommutative (NC) structure. Moreover, we will
study certain matrix models which were introduced in string theory, which lead to dynamical NC spaces along with gauge theory and (some form of) gravity. They should be
seen as candidates for a quantum theory of fundamental interactions including gravity.
The basic problem:
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Q.M. & G.R. imply a break down of classical space-time below LP l = 10−33 cm
classical concept of space-time as manifold physically not meaningful at scales
(Δx)2 ≤ L2P l = �G/c3
standard argument:
(246)
Consider an object of size Δx.
Heisenbergs uncertainty relation ⇒ momentum is uncertain by Δx · Δp ≥ �2 ,
i.e. momentum takes values up to at least Δp =
it has an energy or mass mc2 = E ≥ Δpc =
⇒
G.R.
�
.
2Δx
⇒
Δx ≥ RSchwarzschild ∼ 2G cE4 ≥
�c
2Δx
�G
c3 Δx
⇒
(Δx)2 ≥ �G/c3 = L2P l
(247)
related issues:
• singularities in GR (black holes)
• QFT ceases to make sense below LP l
• UV divergences in QFT reflect point-like singularities
can be handled using renormalization technique, but leads to fine-tuning problems
no satisfactory understanding of space-time at very short distances
Idea:
use mathematical structure of QM phase space to model physical space-times
[X µ , X ν ] = iθµν
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(248)
⇒ space-time has intrinsic minimal length, consistent with above arguments
idea is quite old ( Heisenberg 1938, Schrödinger
“universal length”)
mathematical tools: “Noncommutative Geometry”
try to formulate physical models on such a “Quantum”, “Noncommutative” space
→ NC field theory, NC QFT, NC gravity 199x
fuzzy sphere etc.
Madore, Grosse, ...
NC gauge theory → matrix models!
string theory:
• Matrix Models
BFSS, IKKT 1996
proposed as nonperturbative description of string theory
naturally lead to NC spaces
• NCG on D-branes (string theory) 1998
(Seiberg Witten etc 1998 ff)
strings end on “D-branes” (=submanifolds)
D-branes in background B–field
8.2
⇒
D-branes = NC spaces
Mathematical motivation
aim: generalize geometrical concepts into “quantized (noncommutative) spaces”
Gelfand-Naimark theorem:
every commutative C ∗ - algebra A with 1 is isomorphic to a C ∗ - algebra of continuous
functions on compact Hausdorff space M.
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