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spacetime and H = L2 (S) is the Hilbert space of complex, square integrable
spinors ψ on M . Locally, in any coordinate neighborhood, we write the spinor
as a column vector, ψ(x) ∈ C4 , x ∈ M . The scalar product of two spinors is
de¬ned by

ψ — (x)ψ (x) dV,
(ψ, ψ ) = (78)
M

with the invariant volume form dV := | det g·· |1/2 d4 x de¬ned with the metric
tensor,
‚ ‚
gµν = g , , (79)
‚xµ ‚xν
that is the matrix of the Riemannian metric g with respect to the coordinates
xµ , µ = 0, 1, 2, 3. Note “ and this is important “ that with Euclidean signature
the Dirac action is simply a scalar product, SD = (ψ, ‚ ψ). The representation is
/
de¬ned by pointwise multiplication, (ρ(a) ψ)(x) := a(x)ψ(x), a ∈ A. For a start,
it is su¬cient to know the Dirac operator on a ¬‚at manifold M and with respect
to inertial or Cartesian coordinates xµ such that gµν = δ µ ν . Then we use Dirac™s
˜˜ ˜
˜˜ ˜ ˜
original de¬nition,

D = ‚ = iγ µ ‚/‚ xµ ,
˜
˜˜
/ (80)

with the self adjoint γ-matrices

’12
0 0 σj
γ0 = γj = 1
, , (81)
’12 ’σj
i
0 0

with the Pauli matrices
’i
0 1 0 10
σ1 = , σ2 = , σ3 = . (82)
0 ’1
1 0 i 0
306 T. Sch¨cker
u

We will construct the general curved Dirac operator later.
When the dimension of the manifold is even like in our case, the represen-
tation ρ is reducible. Its Hilbert space decomposes into left- and right-handed
spaces,
1’χ 1+χ
H = HL • HR , HL = H, HR = H. (83)
2 2
Again we make use of the unitary chirality operator,

’12 0
χ = γ5 := γ 0 γ 1 γ 2 γ 3 = . (84)
0 12

We will also need the charge conjugation or real structure, the anti-unitary
operator:
« 
’1
0 0 0
¬1 0·
0 0
J = C := γ 0 γ 2 —¦ complex conjugation =   —¦ c c, (85)
0 0 0 1
’1
0 0 0
that permutes particles and antiparticles.
The ¬ve items (A, H, D, J, χ) form what Connes calls an even, real spectral
triple [19].
A is a real, associative involution algebra with unit, represented faithfully by
bounded operators on the Hilbert space H.
D is an unbounded self adjoint operator on H.
J is an anti-unitary operator,
χ a unitary one.
They enjoy the following properties:
• J 2 = ’1 in four dimensions ( J 2 = 1 in zero dimensions).
[ρ(a), Jρ(˜)J ’1 ] = 0 for all a, a ∈ A.
• a ˜
• DJ = JD, particles and antiparticles have the same dynamics.
[D, ρ(a)] is bounded for all a ∈ A and [[D, ρ(a)], Jρ(˜)J ’1 ] = 0 for all a, a ∈
• a ˜
A. This property is called ¬rst order condition because in the calibrating
example it states that the genuine Dirac operator is a ¬rst order di¬erential
operator.
• χ2 = 1 and [χ, ρ(a)] = 0 for all a ∈ A. These properties allow the decompo-
sition H = HL • HR .
• Jχ = χJ.
• Dχ = ’χD, chirality does not change under time evolution.
• There are three more properties, that we do not spell out, orientability, which
relates the chirality to the volume form, Poincar´ duality and regularity,
e
which states that our functions a ∈ A are di¬erentiable.
Connes promotes these properties to the axioms de¬ning an even, real spectral
triple. These axioms are justi¬ed by his
Reconstruction theorem (Connes 1996 [20]): Consider an (even) spectral
Forces from Connes™ Geometry 307

triple (A, H, D, J, (χ)) whose algebra A is commutative. Then here exists a com-
pact, Riemannian spin manifold M (of even dimensions), whose spectral triple
(C ∞ (M ), L2 (S), ‚ , C, (γ5 )) coincides with (A, H, D, J, (χ)).
/
For details on this theorem and noncommutative geometry in general, I
warmly recommend the Costa Rica book [10]. Let us try to get a feeling of the
local information contained in this theorem. Besides describing the dynamics of
the spinor ¬eld ψ, the Dirac operator ‚ encodes the dimension of spacetime, its
/
Riemannian metric, its di¬erential forms and its integration, that is all the tools
that we need to de¬ne a . In Minkowskian signature, the square of the Dirac
operator is the wave operator, which in 1+2 dimensions governs the dynamics of
a drum. The deep question: ˜Can you hear the shape of a drum?™ has been raised.
This question concerns a global property of spacetime, the boundary. Can you
reconstruct it from the spectrum of the wave operator?
The dimension of spacetime is a local property. It can be retrieved from
the asymptotic behaviour of the spectrum of the Dirac operator for large
eigenvalues. Since M is compact, the spectrum is discrete. Let us order the
eigenvalues, ...»n’1 ¤ »n ¤ »n+1 ... Then states that the eigenvalues grow
asymptotically as n1/dimM . To explore a local property of spacetime we only
need the high energy part of the spectrum. This is in nice agreement with our
intuition from quantum mechanics and motivates the name ˜spectral triple™.
The metric can be reconstructed from the commutative spectral triple by
Connes distance formula (86) below. In the commutative case a point x ∈ M
is reconstructed as the pure state. The general de¬nition of a pure state of
course does not use the commutativity. A state δ of the algebra A is a linear
form on A, that is normalized, δ(1) = 1, and positive, δ(a— a) ≥ 0 for all
a ∈ A. A state is pure if it cannot be written as a linear combination of two
states. For the calibrating example, there is a one“to“one correspondence
between points x ∈ M and pure states δx de¬ned by the Dirac distribution,
δx (a) := a(x) = M δx (y)a(y)d4 y. The geodesic distance between two points
x and y is reconstructed from the triple as:
sup {|δx (a) ’ δy (a)|; a ∈ C ∞ (M ) such that ||[ ‚ , ρ(a)]|| ¤ 1} .
/ (86)
For the calibrating example, [ ‚ , ρ(a)] is a bounded operator. Indeed, [ ‚ , ρ(a)]
/ /
ψ = iγ ‚µ (aψ) ’ iaγ ‚µ ψ = iγ (‚µ a)ψ, and ‚µ a is bounded as a di¬eren-
µ µ µ

tiable function on a compact space.
For a general spectral triple this operator is bounded by axiom. In any case,
the operator norm ||[ ‚ , ρ(a)]|| in the distance formula is ¬nite.
/
Consider the circle, M = S 1 , of circumference 2π with Dirac operator
‚ = i d/dx. A function a ∈ C ∞ (S 1 ) is represented faithfully on a wave-
/
function ψ ∈ L2 (S 1 ) by pointwise multiplication, (ρ(a)ψ)(x) = a(x)ψ(x).
The commutator [ ‚ , ρ(a)] = iρ(a ) is familiar from quantum mechanics. Its
/
operator norm is ||[ ‚ , ρ(a)]|| := supψ |[ ‚ , ρ(a)]ψ|/|ψ| = supx |a (x)|, with
/ /
2π ¯
|ψ|2 = 0 ψ(x)ψ(x) dx. Therefore, the distance between two points x and
y on the circle is
sup{|a(x) ’ a(y)|; sup |a (x)| ¤ 1} = |x ’ y|. (87)
a x
308 T. Sch¨cker
u

Note that Connes™ distance formula continues to make sense for non-con-
nected manifolds, like discrete spaces of dimension zero, i.e. collections of
points.
Di¬erential forms, for example of degree one like da for a function a ∈ A, are
reconstructed as (’i)[ ‚ , ρ(a)]. This is again motivated from quantum me-
/
chanics. Indeed in a 1+0 dimensional spacetime da is just the time derivative
of the ˜observable™ a and is associated with the commutator of the Hamilton
operator with a.
Motivated from quantum mechanics, we de¬ne a noncommutative geometry by
a real spectral triple with noncommutative algebra A.

4.3 Spin Groups
Let us go back to quantum mechanics of spin and recall how a space rotation
acts on a spin 1 particle. For this we need group homomorphisms between the
2
rotation group SO(3) and the probability preserving unitary group SU (2). We
construct ¬rst the group homomorphism

p : SU (2) ’’ SO(3)
’’ p(U ).
U

With the help of the auxiliary function

f : R3 ’’ su(2)
« 
x1
x =  x2  ’’ ’ 1 ixj σj ,
2
x3

we de¬ne the rotation p(U ) by

p(U )x := f ’1 (U f (x)U ’1 ). (88)

The conjugation by the unitary U will play an important role and we give it a
special name, iU (w) := U wU ’1 , i for inner. Since i(’U ) = iU , the projection p
is two to one, Ker(p) = {±1}. Therefore the spin lift

L : SO(3) ’’ SU (2)
R = exp(ω) ’’ exp( 1 ω jk [σj , σk ]) (89)
8

is double-valued. It is a local group homomorphism and satis¬es p(L(R)) = R.
Its double-valuedness is accessible to quantum mechanical experiments: neu-
trons have to be rotated through an angle of 720—¦ before interference patterns
repeat [21].
The lift L was generalized by Dirac to the special relativistic setting, e.g. [4],
and by E. Cartan [22] to the general relativistic setting. Connes [23] generalizes it
to noncommutative geometry, see Fig. 5. The transformations we need to lift are
Forces from Connes™ Geometry 309

M
AutH (A) ← Di¬(M ) Spin(1, 3) ← SO(1, 3) — Spin(1, 3) ← SO(3) — SU (2)
C6 C6 C6 C6
O O O O
CL CL CL CL
p p p p
C C C C
?C ?C ?C ?C
Aut(A) ← ← ←
Di¬(M ) SO(1, 3) SO(3)
Fig. 5. The nested spin lifts of Connes, Cartan, Dirac, and Pauli

Lorentz transformations in special relativity, and general coordinate transforma-
tions in general relativity, i.e. our calibrating example. The latter transformations
are the local elements of the di¬eomorphism group Di¬(M ). In the setting of
noncommutative geometry, this group is the group of algebra automorphisms
Aut(A). Indeed, in the calibrating example we have Aut(A)=Di¬(M ). In order
to generalize the spin group to spectral triples, Connes de¬nes the receptacle of
the group of ˜lifted automorphisms™,
AutH (A) := {U ∈ End(H), U U — = U — U = 1, U J = JU, U χ = χU,
iU ∈ Aut(ρ(A))}. (90)
The ¬rst three properties say that a lifted automorphism U preserves probability,
charge conjugation, and chirality. The fourth, called covariance property, allows
to de¬ne the projection p : AutH (A) ’’ Aut(A) by
p(U ) = ρ’1 iU ρ (91)
We will see that the covariance property will protect the locality of ¬eld theory.
For the calibrating example of a four dimensional spacetime, a local calculation,
i.e. in a coordinate patch, that we still denote by M , yields the semi-direct prod-
uct (cf. Appendix) of di¬eomorphisms with local or gauged spin transformations,
AutL2 (S) (C ∞ (M )) = Di¬(M ) M Spin(4). We say receptacle because already in
six dimensions, AutL2 (S) (C ∞ (M )) is larger than Di¬(M ) M Spin(6). However
we can use the lift L with p(L(σ)) = σ, σ ∈Aut(A) to correctly identify the
spin group in any dimension of M . Indeed we will see that the spin group is the
image of the spin lift L(Aut(A)), in general a proper subgroup of the receptacle
AutH (A).
Let σ be a di¬eomorphism close to the identity. We interpret σ as coordinate
transformation, all our calculations will be local, M standing for one chart, on
which the coordinate systems xµ and xµ = (σ(˜))µ are de¬ned. We will work
˜˜ x
out the local expression of a lift of σ to the Hilbert space of spinors. This lift
U = L(σ) will depend on the metric and on the initial coordinate system xµ . ˜˜
In a ¬rst step, we construct a group homomorphism Λ : Di¬(M ) ’ Di¬(M )
M
SO(4) into the group of local ˜Lorentz™ transformations, i.e. the group of dif-
ferentiable functions from spacetime into SO(4) with pointwise multiplication.
Let (˜’1 (˜))µ a = (˜’1/2 (˜))µ a be the inverse of the square root of the positive
e x˜ x˜
g
matrix g of the metric with respect to the initial coordinate system xµ . Then
˜˜
˜
the four vector ¬elds ea , a = 0, 1, 2, 3, de¬ned by
˜

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