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

˜

‚