˜

˜ e (92)

‚ xµ

˜˜

310 T. Sch¨cker

u

give an orthonormal frame of the tangent bundle. This frame de¬nes a complete

gauge ¬xing of the Lorentz gauge group M SO(4) because it is the only orthonor-

mal frame to have symmetric coe¬cients (˜’1 )µ a with respect to the coordinate

˜

e

system xµ . We call this gauge the symmetric gauge for the coordinates xµ . Now

˜˜ ˜˜

let us perform a local change of coordinates, x = σ(˜). The holonomic frame

x

with respect to the new coordinates is related to the former holonomic one by

the inverse Jacobian matrix of σ

‚ xµ ‚

˜˜ ‚ xµ

˜˜

‚ ‚ µ

˜

µ

˜ ’1

= J ’1 J

= , (x) = . (93)

‚xµ ‚xµ ‚ xµ

˜˜ ‚ xµ

˜˜ ‚xµ

µ µ

The matrix g of the metric with respect to the new coordinates reads,

‚ ‚

= J ’1T (x)˜(σ ’1 (x))J ’1 (x)

gµν (x) := g , g , (94)

µ ‚xν µν

‚x x

and the symmetric gauge for the new coordinates x is the new orthonormal frame

µ

˜

‚ ‚ ‚

= g ’1/2 µ b J ’1 µ µ µ = J ’1

eb = e’1µ b J g ’1 J T

˜

˜ . (95)

µ ‚x˜ ‚ xµ

˜˜

‚x ˜ b

New and old orthonormal frames are related by a Lorentz transformation Λ,

eb = Λ’1 a b ea , with

˜

√

J ’1T g J ’1 g ’1 g ’1 .

J |x

Λ(σ)|x = ˜ ˜ = gJ ˜ (96)

˜ ˜

σ(˜)

x x

˜

If M is ¬‚at and xµ are ˜inertial™ coordinates, i.e. gµν = δ µ ν , and σ is a local

˜˜ ˜

˜˜ ˜ ˜

isometry then J (˜) ∈ SO(4) for all x and Λ(σ) = J . In special relativity, there-

x ˜

fore, the symmetric gauge ties together Lorentz transformations in spacetime

with Lorentz transformations in the tangent spaces.

In general, if the coordinate transformation σ is close to the identity, so is

its Lorentz transformation Λ(σ) and it can be lifted to the spin group,

S : SO(4) ’’ Spin(4)

Λ = exp ω ’’ exp 1 ωab γ ab (97)

4

with ω = ’ω T ∈ so(4) and γ ab := 1a b

2 [γ , γ ]. With our choice (81) for the γ

matrices, we have

’σj 0 σ 0

γ 0j = i γ jk = i jk 123

, , j, k = 1, 2, 3, = 1. (98)

0 σj 0 σ

We can write the local expression [24] of the lift L : Di¬(M ) ’ Di¬(M )

M

Spin(4),

(L(σ)ψ) (x) = S (Λ(σ))|σ’1 (x) ψ(σ ’1 (x)). (99)

L is a double-valued group homomorphism. For any σ close to the identity,

L(σ) is unitary, commutes with charge conjugation and chirality, satis¬es the

covariance property, and p(L(σ)) = σ. Therefore, we have locally

Spin(4) = AutL2 (S) (C ∞ (M )).

L(Di¬(M )) ‚ Di¬(M ) M

(100)

Forces from Connes™ Geometry 311

The symmetric gauge is a complete gauge ¬xing and this reduction follows Ein-

stein™s spirit in the sense that the only arbitrary choice is the one of the initial

coordinate system xµ as will be illustrated in the next section. Our computations

˜˜

are deliberately local. The global picture can be found in reference [25].

5 The Spectral Action

5.1 Repeating Einstein™s Derivation in the Commutative Case

We are ready to parallel Einstein™s derivation of general relativity in Connes™

language of spectral triples. The associative algebra C ∞ (M ) is commutative,

but this property will never be used. As a by-product, the lift L will reconcile

Einstein™s and Cartan™s formulations of general relativity and it will yield a self

contained introduction to Dirac™s equation in a gravitational ¬eld accessible to

particle physicists. For a comparison of Einstein™s and Cartan™s formulations of

general relativity see for example [6].

First Stroke: Kinematics. Instead of a point-particle, Connes takes as matter

a ¬eld, the free, massless Dirac particle ψ(˜) in the ¬‚at spacetime of special

x

µ

˜

relativity. In inertial coordinates x , its dynamics is given by the Dirac equation,

˜

˜ ψ = iδ µ a γ a ‚ ψ = 0.

˜

‚

/ (101)

‚ xµ

˜˜

We have written δ µ a γ a instead of γ µ to stress that the γ matrices are x-

˜ ˜

˜

independent. This Dirac equation is covariant under Lorentz transformations.

Indeed if σ is a local isometry then

‚

L(σ) ˜ L(σ)’1 = ‚ = iδ µ a γ a

‚

/ / . (102)

‚xµ

To prove this special relativistic covariance, one needs the identity S(Λ)γ a S(Λ)’1

= Λ’1 a b γ b for Lorentz transformations Λ ∈ SO(4) close to the identity. Take

a general coordinate transformation σ close to the identity. Now comes a long,

but straightforward calculation. It is a useful exercise requiring only matrix

multiplication and standard calculus, Leibniz and chain rules. Its result is the

Dirac operator in curved coordinates,

‚

L(σ) ˜ L(σ)’1 = ‚ = ie’1 µ a γ a

‚

/ / + s(ωµ ) , (103)

‚xµ

√

where e’1 = J J T is a symmetric matrix,

s : so(4) ’’ spin(4)

’’ 1 ωab γ ab

ω (104)

4

is the Lie algebra isomorphism corresponding to the lift (97) and

ωµ (x) = Λ|σ’1 (x) ‚µ Λ’1 . (105)

x

312 T. Sch¨cker

u

The ˜spin connection™ ω is the gauge transform of the Levi“Civita connection

“ , the latter is expressed with respect to the holonomic frame ‚µ , the former

is written with respect to the orthonormal frame ea = e’1 µ a ‚µ . The gauge

transformation passing between them is e ∈ M GL4 ,

ω = e“ e’1 + ede’1 . (106)

We recover the well known explicit expression

(‚β ea µ ) ’ (‚µ ea β ) + em µ (‚β em ± )e’1 ± a e’1 β b ’ [a ” b] (107)

ω a bµ (e) = 1

2

√

of the spin connection in terms of the ¬rst derivatives of ea µ = g a µ . Again

the spin connection has zero curvature and the equivalence principle relaxes

this constraint. But now equation (103) has an advantage over its analogue (2).

Thanks to Connes™ distance formula (86), the metric can be read explicitly in

(103) from the matrix of functions e’1 µ a , while in (2) ¬rst derivatives of the

metric are present. We are used to this nuance from electro“magnetism, where

the classical particle feels the force while the quantum particle feels the potential.

In Einstein™s approach, the zero connection ¬‚uctuates, in Connes™ approach, the

√

¬‚at metric ¬‚uctuates. This means that the constraint e’1 = J J T is relaxed

and e’1 now is an arbitrary symmetric matrix depending smoothly on x.

Let us mention two experiments with neutrons con¬rming the ˜Minimal cou-

pling™ of the Dirac operator to curved coordinates, equation (103). The ¬rst

takes place in ¬‚at spacetime. The neutron interferometer is mounted on a loud

speaker and shaken periodically [26]. The resulting pseudo forces coded in the

spin connection do shift the interference patterns observed. The second experi-

ment takes place in a true gravitational ¬eld in which the neutron interferometer

is placed [27]. Here shifts of the interference patterns are observed that do depend

on the gravitational potential, ea µ in equation (103).

Second Stroke: Dynamics. The second stroke, the covariant dynamics for

the new class of Dirac operators ‚ is due to Chamseddine & Connes [28]. It is

/

the celebrated spectral action. The beauty of their approach to general relativity

is that it works precisely because the Dirac operator ‚ plays two roles simulta-

/

neously, it de¬nes the dynamics of matter and the kinematics of gravity. For a

discussion of the transformation passing from the metric to the Dirac operator

I recommend the article [29] by Landi & Rovelli.

The starting point of Chamseddine & Connes is the simple remark that the

spectrum of the Dirac operator is invariant under di¬eomorphisms interpreted as

general coordinate transformations. From ‚ χ = ’χ ‚ we know that the spectrum

/ /

of ‚ is even. Indeed, for every eigenvector ψ of ‚ with eigenvalue E, χψ is

/ /

eigenvector with eigenvalue ’E. We may therefore consider only the spectrum of

2

the positive operator ‚ /Λ2 where we have divided by a ¬xed arbitrary energy

/

scale to make the spectrum dimensionless. If it was not divergent the trace

2

tr ‚ /Λ2 would be a general relativistic action functional. To make it convergent,

/

take a di¬erentiable function f : R+ ’ R+ of su¬ciently fast decrease such that

Forces from Connes™ Geometry 313

the action

2

SCC := tr f ( ‚ /Λ2 )

/ (108)

converges. It is still a di¬eomorphism invariant action. The following theorem,

also known as heat kernel expansion, is a local version of an index theorem [30],

that as explained in Jean Zinn-Justin™s lectures [12] is intimately related to

Feynman graphs with one fermionic loop.

Theorem: Asymptotically for high energies, the spectral action is

SCC = (109)

+ a(5 R2 ’ 8 Ricci2 ’ 7 Riemann2 )] dV + O(Λ’2 ),

[ 16πG ’

2Λc 1

16πG R

M

3π ’2

6f0 2

where the cosmological constant is Λc = f2 Λ , Newton™s constant is G = f2 Λ

f4

and a = 5760π2 . On the right-hand side of the theorem we have omitted surface

terms, that is terms that do not contribute to the Euler“Lagrange equations.

The Chamseddine“Connes action is universal in the sense that the ˜cut o¬™

∞

function f only enters through its ¬rst three ˜moments™, f0 := 0 uf (u)du,

∞

f2 := 0 f (u)du and f4 = f (0).

If we take for f a di¬erentiable approximation of the characteristic function

of the unit interval, f0 = 1/2, f2 = f4 = 1, then the spectral action just counts

the number of eigenvalues of the Dirac operator whose absolute values are below

the ˜cut o¬™ Λ. In four dimensions, the minimax example is the ¬‚at 4-torus

with all circumferences measuring 2π. Denote by ψB (x), B = 1, 2, 3, 4, the four

components of the spinor. The Dirac operator is

«

’i‚0 + ‚3 ‚1 ’ i‚2

0 0

’i‚0 ’ ‚3 ·

¬ 0 0 ‚1 + i‚2

‚= .

/ (110)

’i‚0 ’ ‚3 ’‚1 + i‚2 0 0

’‚1 ’ i‚2 ’i‚0 + ‚3 0 0

After a Fourier transform