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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

+ 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

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

’ [‚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-
dependent components of hµν remain, h11 = ’h22 and h12 . They represent the
302 T. Sch¨cker

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

+ 16πG 1 ‚µ h12 ‚ µ h12 .

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 ).

Finally, the kinetic term of the scalar is positive:

+ 1 ‚µ •— ‚ µ •. (68)

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
+ 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,
0 0 σj
γ0 = γj = 1
, , (71)
’12 ’σj
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
2g 2 tr (Fµν F (73)
A 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


r /2


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

’ ρ(H) ψ(t, x) = 0.
i (76)
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,
‚ 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


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( 78 .)