ψ — γ 0 [iγ µ Dµ ’ mψ 14 ] ψ

— — — —

= ψL γ 0 iγ µ Dµ ψL + ψR γ 0 iγ µ Dµ ψR ’ mψ ψL γ 0 ψR ’ mψ ψR γ 0 ψL .(60)

The relativistic energy momentum relations are quadratic in the masses. There-

fore the sign of the fermion mass mψ is conventional and merely re¬‚ects the

choice: who is particle and who is antiparticle. We can even adopt one choice for

the left-handed fermions and the opposite choice for the right-handed fermions.

Formally this can be seen by the change of ¬eld variable (chiral transformation):

ψ := exp(i±γ5 ) ψ . (61)

It leaves invariant the kinetic term and the mass term transforms as,

—

’mψ ψ γ 0 [cos(2±) 14 + i sin(2±) γ5 ]ψ . (62)

With ± = ’π/4 the Dirac Lagrangian becomes:

—

ψ γ 0 [ iγ µ Dµ + imψ γ5 ]ψ (63)

— — —0

= ψ L γ 0 iγ µ Dµ ψL + ψ R γ 0 iγ µ Dµ ψ + mψ ψ L γ iγ5 ψ R

R

—

+ mψ ψ R γ 0 iγ5 ψ L

— — — —

’ imψ ψ R γ 0 ψ L .

= ψ L γ 0 iγ µ Dµ ψL + ψ R γ 0 iγ µ Dµ ψ + imψ ψ L γ 0 ψ

R R

We have seen that gauge invariance forbids massive gauge bosons, mA = 0,

and that parity violation forbids massive fermions, mψ = 0. This is ¬xed by

spontaneous symmetry breaking, where we take the scalar mass term with wrong

sign, m• = 0, µ > 0. The shift of the scalar then induces masses for the gauge

bosons, the fermions and the physical scalars. These masses are calculable in

terms of the gauge, Yukawa, and Higgs couplings.

The other relative signs in the combined Lagrangian are ¬xed by the require-

ment that the energy density of the non-gravitational part T00 be positive (up to

a cosmological constant) and that gravity in the Newtonian limit be attractive.

In particular this implies that the Higgs potential must be bounded from below,

» > 0. The sign of the Einstein“Hilbert action may also be obtained from an

asymptotically ¬‚at space of weak curvature, where we can de¬ne gravitational

energy density. Then the requirement is that the kinetic terms of all physical

bosons, spin 0, 1, and 2, be of the same sign. Take the metric of the form

gµν = ·µν + hµν , (64)

hµν small. Then the Einstein“Hilbert Lagrangian becomes [17],

’ 16πG R |det g·· |1/2 = 16πG { 4 ‚µ h±β ‚ h ’ 1 ‚µ h± ± ‚ µ hβ β

µ ±β

1 1 1

(65)

8

’ [‚ν hµ ’ 1 ‚µ hν ][‚ν hµ ν ’ 1 ‚ µ hν ν

ν ν

] + O(h3 )}.

2 2

Here indices are raised with · ·· . After an appropriate choice of coordinates,

˜harmonic coordinats™, the bracket ‚ν hµ ν ’ 1 ‚µ hν ν vanishes and only two in-

2

dependent components of hµν remain, h11 = ’h22 and h12 . They represent the

302 T. Sch¨cker

u

two physical states of the graviton, helicity ±2. Their kinetic terms are both

positive, e.g.:

+ 16πG 1 ‚µ h12 ‚ µ h12 .

1

(66)

4

Likewise, by an appropriate gauge transformation, we can achieve ‚µ Aµ = 0,

˜Lorentz gauge™, and remain with only two ˜transverse™ components A1 , A2 of

helicity ±1. They have positive kinetic terms, e.g.:

+ 2g2 tr (‚µ A— ‚ µ A1 ).

1

(67)

1

Finally, the kinetic term of the scalar is positive:

+ 1 ‚µ •— ‚ µ •. (68)

2

An old recipe from quantum ¬eld theory, ˜Wick rotation™, amounts to re-

placing spacetime by a Riemannian manifold with Euclidean signature. Then

certain calculations become feasible or easier. One of the reasons for this is that

Euclidean quantum ¬eld theory resembles statistical mechanics, the imaginary

time playing formally the role of the inverse temperature. Only at the end of the

calculation the result is ˜rotated back™ to real time. In some cases, this recipe can

be justi¬ed rigorously. The precise formulation of the recipe is that the n-point

functions computed from the Euclidean Lagrangian be the analytic continua-

tions in the complex time plane of the Minkowskian n-point functions. We shall

indicate a hand waving formulation of the recipe, that is su¬cient for our pur-

pose: In a ¬rst stroke we pass to the signature ’ + ++. In a second stroke we

replace t by it and replace all Minkowskian scalar products by the corresponding

Euclidean ones.

The ¬rst stroke amounts simply to replacing the metric by its negative. This

leaves invariant the Christo¬el symbols, the Riemann and Ricci tensors, but

reverses the sign of the curvature scalar. Likewise, in the other terms of the

Lagrangian we get a minus sign for every contraction of indices, e.g.: ‚µ •— ‚ µ • =

‚µ •— ‚µ •g µµ becomes ‚µ •— ‚µ •(’g µµ ) = ’‚µ •— ‚ µ •. After multiplication by

a conventional overall minus sign the combined Lagrangian reads now,

{ 16πG ’ 16πG R + 2g2 tr (Fµν F µν ) + g12 m2 tr (A— Aµ )

—

2Λc 1 1

A µ

—µ

+ 2 (Dµ •) D • + 2 m• |•| ’ 2 µ |•| + »|•|

2 2 12 2 4

1 1

+ ψ — γ 0 [ iγ µ Dµ + mψ 14 ]ψ } |det g·· |1/2 . (69)

To pass to the Euclidean signature, we multiply time, energy and mass by i.

This amounts to · µν = δ µν in the scalar product. In order to have the Euclidean

anticommutation relations,

γ µ γ ν + γ ν γ µ = 2δ µν 14 , (70)

we change the Dirac matrices to the Euclidean ones,

’12

0 0 σj

γ0 = γj = 1

, , (71)

’12 ’σj

i

0 0

Forces from Connes™ Geometry 303

All four are now self adjoint. For the chirality we take

’12 0

γ5 := γ 0 γ 1 γ 2 γ 3 = . (72)

0 12

The Minkowskian scalar product for spinors has a γ 0 . This γ 0 is needed for the

correct physical interpretation of the energy of antiparticles and for invariance

under lifted Lorentz transformations, Spin(1, 3). In the Euclidean, there is no

physical interpretation and we can only retain the requirement of a Spin(4)

invariant scalar product. This scalar product has no γ 0 . But then we have a

problem if we want to write the Dirac Lagrangian in terms of chiral spinors as

—

above. For instance, for a purely left-handed neutrino, ψR = 0 and ψL iγ µ Dµ ψL

vanishes identically because γ5 anticommutes with the four γ µ . The standard

trick of Euclidean ¬eld theoreticians [12] is fermion doubling, ψL and ψR are

treated as two independent, four component spinors. They are not chiral pro-

jections of one four component spinor as in the Minkowskian, equation (59).

The spurious degrees of freedom in the Euclidean are kept all the way through

the calculation. They are projected out only after the Wick rotation back to

Minkowskian, by imposing γ5 ψL = ’ψL , γ5 ψR = ψR .

In noncommutative geometry the Dirac operator must be self adjoint, which

is not the case for the Euclidean Dirac operator iγ µ Dµ + imψ 14 we get from the

Lagrangian (69) after multiplication of the mass by i. We therefore prefer the

primed spinor variables ψ producing the self adjoint Euclidean Dirac operator

iγ µ Dµ + mψ γ5 . Dropping the prime, the combined Lagrangian in the Euclidean

then reads:

—

) + g12 m2 tr (A— Aµ )

{ 16πG ’ µν

2Λc 1 1

R+

2g 2 tr (Fµν F (73)

A µ

16πG

—µ

+ 1 (Dµ •) D • + 1 m2 |•|2 ’ 1 µ2 |•|2 + »|•|4

•

2 2 2

— — — —

+ ψL iγ Dµ ψL + ψR iγ Dµ ψR + mψ ψL γ5 ψR + mψ ψR γ5 ψL } (det g·· )1/2 .

µ µ

4 Connes™ Noncommutative Geometry

Connes equips Riemannian spaces with an uncertainty principle. As in quantum

mechanics, this uncertainty principle is derived from noncommutativity.

4.1 Motivation: Quantum Mechanics

Consider the classical harmonic oscillator. Its phase space is R2 with points la-

belled by position x and momentum p. A classical observable is a di¬erentiable

function on phase space such as the total energy p2 /(2m) + kx2 . Observables can

be added and multiplied, they form the algebra C ∞ (R2 ), which is associative and

commutative. To pass to quantum mechanics, this algebra is rendered noncom-

mutative by means of the following noncommutation relation for the generators

x and p,

[x, p] = i 1. (74)

304 T. Sch¨cker

u

p

6

r /2

-x

Fig. 4. The ¬rst example of noncommutative geometry

Let us call A the resulting algebra ˜of quantum observables™. It is still associative,

has an involution ·— (the adjoint or Hermitean conjugation) and a unit 1. Let us

brie¬‚y recall the de¬ning properties of an involution: it is a linear map from the

real algebra into itself that reverses the product, (ab)— = b— a— , respects the unit,

1— = 1, and is such that a—— = a.

Of course, there is no space anymore of which A is the algebra of functions.

Nevertheless, we talk about such a ˜quantum phase space™ as a space that has

no points or a space with an uncertainty relation. Indeed, the noncommutation

relation (74) implies

∆x∆p ≥ /2 (75)

and tells us that points in phase space lose all meaning, we can only resolve cells

in phase space of volume /2, see Fig. 4. To de¬ne the uncertainty ∆a for an

observable a ∈ A, we need a faithful representation of the algebra on a Hilbert

space, i.e. an injective homomorphism ρ : A ’ End(H) (cf. Appendix). For the

harmonic oscillator, this Hilbert space is H = L2 (R). Its elements are the wave

functions ψ(x), square integrable functions on con¬guration space. Finally, the

dynamics is de¬ned by a self adjoint observable H = H — ∈ A via Schr¨dinger™s

o

equation

‚

’ ρ(H) ψ(t, x) = 0.

i (76)

‚t

Usually the representation is not written explicitly. Since it is faithful, no confu-

sion should arise from this abuse. Here time is considered an external parameter,

in particular, time is not considered an observable. This is di¬erent in the special

relativistic setting where Schr¨dinger™s equation is replaced by Dirac™s equation,

o

‚ ψ = 0.

/ (77)

Now the wave function ψ is the four-component spinor consisting of left- and

right-handed, particle and antiparticle wave functions. The Dirac operator is

Forces from Connes™ Geometry 305

not in A anymore, but ‚ ∈ End(H). The Dirac operator is only formally self

/

adjoint because there is no positive de¬nite scalar product, whereas in Euclidean

—

spacetime it is truly self adjoint, ‚ = ‚ .

/ /

Connes™ geometries are described by these three purely algebraic items, (A,

H, ‚ ), with A a real, associative, possibly noncommutative involution algebra

/

with unit, faithfully represented on a complex Hilbert space H, and ‚ is a self

/

adjoint operator on H.

4.2 The Calibrating Example: Riemannian Spin Geometry

Connes™ geometry [18] does to spacetime what quantum mechanics does to

phase space. Of course, the ¬rst thing we have to learn is how to reconstruct

the Riemannian geometry from the algebraic data (A, H, ‚ ) in the case where

/

the algebra is commutative. We start the easy way and construct the triple

(A, H, ‚ ) given a four dimensional, compact, Euclidean spacetime M . As before

/

∞

A = C (M ) is the real algebra of complex valued di¬erentiable functions on