ρ ρ (17)

We denote by ρ the Lie algebra representation of the group representation ρ.

˜

Sometimes one also wants the mixed Yang“Mills“gravitational anomalies to van-

ish:

tr ρL (X) ’ tr ρR (X) = 0, for all X ∈ g.

˜ ˜ (18)

292 T. Sch¨cker

u

3.2 Rules

It is time to open the slot machine and to see how it works. Its mechanism has

¬ve pieces:

The Yang“Mills Action. The actor in this piece is A = Aµ dxµ , called con-

nection, gauge potential, gauge boson or Yang“Mills ¬eld. It is a 1-form on

spacetime M x with values in the Lie algebra g, A ∈ „¦ 1 (M, g). We de¬ne its

curvature or ¬eld strength,

F := dA + 1 [A, A] = 1 Fµν dxµ dxν ∈ „¦ 2 (M, g), (19)

2 2

and the Yang“Mills action,

’1 —

SYM [A] = ’ 1 (F, —F ) = tr Fµν F µν dV. (20)

2 2

2gn

M M

The gauge group M G is the in¬nite dimensional group of di¬erentiable functions

g : M ’ G with pointwise multiplication. ·— is the Hermitean conjugate of

matrices, —· is the Hodge star of di¬erential forms. The space of all connections

carries an a¬ne representation (cf. Appendix) ρV of the gauge group:

ρV (g)A = gAg ’1 + gdg ’1 . (21)

Restricted to x-independent (˜rigid™) gauge transformation, the representation is

linear, the adjoint one. The ¬eld strength transforms homogeneously even under

x-dependent (˜local™) gauge transformations, g : M ’ G di¬erentiable,

ρV (g)F = gF g ’1 , (22)

and, as the scalar product (·, ·) is invariant, the Yang“Mills action is gauge

invariant,

SYM [ρV (g)A] = SYM [A] for all g ∈ M

G. (23)

Note that a mass term for the gauge bosons,

1

m2 tr A— Aµ dV,

m2 (A, —A) =

1

(24)

A A µ

2 2

gn

M M

is not gauge invariant because of the inhomogeneous term in the transformation

law of a connection (21). Gauge invariance forces the gauge bosons to be massless.

In the Abelian case G = U (1), the Yang“Mills Lagrangian is nothing but

Maxwell™s Lagrangian, the gauge boson A is the photon and its coupling con-

√

stant g is e/ 0 . Note however, that the Lie algebra of U (1) is iR and the

vector potential is purely imaginary, while conventionally, in Maxwell™s theory

it is chosen real. Its quantum version is QED, quantum electro-dynamics. For

G = SU (3) and HL = HR = C3 we have today™s theory of strong interaction,

quantum chromo-dynamics, QCD.

Forces from Connes™ Geometry 293

The Dirac Action. Schr¨dinger™s action is non-relativistic. Dirac generalized

o

it to be Lorentz invariant, e.g. [4]. The price to be paid is twofold. His gen-

eralization only works for spin 1 particles and requires that for every such

2

particle there must be an antiparticle with same mass and opposite charges.

Therefore, Dirac™s wave function ψ(x) takes values in C4 , spin up, spin down,

particle, antiparticle. antiparticles have been discovered and Dirac™s theory was

celebrated. Here it is in short for (¬‚at) Minkowski space of signature + ’ ’’,

·µν = · µν = diag(+1, ’1, ’1, ’1). De¬ne the four Dirac matrices,

’12

0 0 σj

γ0 = γj =

, , (25)

’12 ’σj

0 0

for j = 1, 2, 3 with the three Pauli matrices,

’i

0 1 0 10

σ1 = , σ2 = , σ3 = . (26)

0 ’1

1 0 i 0

They satisfy the anticommutation relations,

γ µ γ ν + γ ν γ µ = 2· µν 14 . (27)

In even spacetime dimensions, the chirality,

’12 0

γ5 := ’ 4! γ γ γ = ’iγ 0 γ 1 γ 2 γ 3 =

µνρσ

i

µνρσ γ (28)

0 12

is a natural operator and it paves the way to an understanding of parity vi-

olation in weak interactions. The chirality is a unitary matrix of unit square,

which anticommutes with all four Dirac matrices. (1 ’ γ5 )/2 projects a Dirac

spinor onto its left-handed part, (1 + γ5 )/2 projects onto the right-handed part.

The two parts are called Weyl spinors. A massless left-handed (right-handed)

spinor, has its spin parallel (anti-parallel) to its direction of propagation. The

chirality maps a left-handed spinor to a right-handed spinor. A space re¬‚ection

or parity transformation changes the sign of the velocity vector and leaves the

spin vector unchanged. It therefore has the same e¬ect on Weyl spinors as the

chirality operator. Similarly, there is the charge conjugation, an anti-unitary op-

erator (cf. Appendix) of unit square, that applied on a particle ψ produces its

antiparticle

«

0 ’1 0 0

¬1 0 0 0·

J = 1 γ 0 γ 2 —¦ complex conjugation = —¦ c c, (29)

i 00 01

0 0 ’1 0

¯ ¯

i.e. Jψ = 1 γ 0 γ 2 ψ. Attention, here and for the last time ψ stands for the complex

i

conjugate of ψ. In a few lines we will adopt a di¬erent more popular convention.

The charge conjugation commutes with all four Dirac matrices. In ¬‚at spacetime,

the free Dirac operator is simply de¬ned by,

‚ := i γ µ ‚µ .

/ (30)

294 T. Sch¨cker

u

2

It is sometimes referred to as square root of the wave operator because ‚ = ’ .

/

µ

The coupling of the Dirac spinor to the gauge potential A = Aµ dx is done via

the covariant derivative, and called Minimal coupling. In order to break parity,

we write left- and right-handed parts independently:

1 ’ γ5

¯/

ψL [ ‚ + i γ µ ρL (Aµ )]

SD [A, ψL , ψR ] = ˜ ψL dV

2

M

1 + γ5

¯/

ψR [ ‚ + i γ µ ρR (Aµ )]

+ ˜ ψR dV. (31)

2

M

The new actors in this piece are ψL and ψR , two multiplets of Dirac spinors

¯

or fermions, that is with values in HL and HR . We use the notations, ψ :=

ψ — γ 0 , where ·— denotes the Hermitean conjugate with respect to the four spinor

components and the dual with respect to the scalar product in the (internal)

Hilbert space HL or HR . The γ 0 is needed for energy reasons and for invariance

of the pseudo“scalar product of spinors under lifted Lorentz transformations. The

γ 0 is absent if spacetime is Euclidean. Then we have a genuine scalar product

and the square integrable spinors form a Hilbert space L2 (S) = L2 (R4 )—C4 , the

in¬nite dimensional brother of the internal one. The Dirac operator is then self

adjoint in this Hilbert space. We denote by ρL the Lie algebra representation in

˜

HL . The covariant derivative, Dµ := ‚µ + ρL (Aµ ), deserves its name,

˜

[‚µ + ρL (ρV (g)Aµ )] (ρL (g)ψL ) = ρL (g) [‚µ + ρL (Aµ )] ψL ,

˜ ˜ (32)

for all gauge transformations g ∈ MG. This ensures that the Dirac action (31)

is gauge invariant.

If parity is conserved, HL = HR , we may add a mass term

1 ’ γ5 1 + γ5

¯ ¯

’c ψL dV ’ c

ψ R mψ ψ L mψ ψR dV =

2 2

M M

¯

’c ψ mψ ψ dV (33)

M

to the Dirac action. It gives identical masses to all members of the multiplet. The

fermion masses are gauge invariant if all fermions in HL = HR have the same

mass. For instance QED preserves parity, HL = HR = C, the representation

being characterized by the electric charge, ’1 for both the left- and right handed

electron. Remember that gauge invariance forces gauge bosons to be massless.

For fermions, it is parity non-invariance that forces them to be massless.

Let us conclude by reviewing brie¬‚y why the Dirac equation is the Lorentz

invariant generalization of the Schr¨dinger equation. Take the free Schr¨dinger

o o

equation on (¬‚at) R . It is a linear di¬erential equation with constant coe¬cients,

4

2m ‚

’ ∆ ψ = 0. (34)

i ‚t

We compute its polynomial following Fourier and de Broglie,

p2

2m 2m

’ ω+k =’ E’

2

. (35)

2 2m

Forces from Connes™ Geometry 295

Energy conservation in Newtonian mechanics is equivalent to the vanishing of

the polynomial. Likewise, the polynomial of the free, massive Dirac equation

( ‚ ’ cmψ )ψ = 0 is

/

ωγ 0 + kj γ j ’ c m1. (36)

c

Putting it to zero implies energy conservation in special relativity,

( c )2 ω 2 ’ k2 ’ c2 m2 = 0.

2

(37)

In this sense, Dirac™s equation generalizes Schr¨dinger™s to special relativity. To

o

see that Dirac™s equation is really Lorentz invariant we must lift the Lorentz

transformations to the space of spinors. We will come back to this lift.

So far we have seen the two noble pieces by Yang“Mills and Dirac. The

remaining three pieces are cheap copies of the two noble ones with the gauge

boson A replaced by a scalar •. We need these three pieces to cure only one

problem, give masses to some gauge bosons and to some fermions. These masses

are forbidden by gauge invariance and parity violation. To simplify the notation

we will work from now on in units with c = = 1.

The Klein“Gordon Action. The Yang“Mills action contains the kinetic term

for the gauge boson. This is simply the quadratic term, (dA, dA), which by

Euler“Lagrange produces linear ¬eld equations. We copy this for our new actor,

a multiplet of scalar ¬elds or Higgs bosons,

• ∈ „¦ 0 (M, HS ), (38)

by writing the Klein“Gordon action,

(D•)— — D• = (Dµ •)— Dµ • dV,

1 1

SKG [A, •] = (39)

2 2

M M

with the covariant derivative here de¬ned with respect to the scalar representa-

tion,

D• := d• + ρS (A)•.

˜ (40)

Again we need this Minimal coupling •— A• for gauge invariance.

The Higgs Potential. The non-Abelian Yang“Mills action contains interaction

terms for the gauge bosons, an invariant, fourth order polynomial, 2(dA, [A, A])+

([A, A], [A, A]). We mimic these interactions for scalar bosons by adding the

integrated Higgs potential M —V (•) to the action.

The Yukawa Terms. We also mimic the (minimal) coupling of the gauge boson

to the fermions ψ — Aψ by writing all possible trilinear invariants,

SY [ψL , ψR , •] :=

«

n m

— gY j (ψL , ψR , •— )j .

— —

Re gY j (ψL , ψR , •)j + (41)

M j=1 j=n+1

296 T. Sch¨cker

u