group representation ρ. It is de¬ned on the same vector space by ρ(exp X) =

exp(˜(X)). The ρ(X)s are not necessarily invertible endomorphisms. They sat-

ρ ˜

isfy ρ([X, Y ]) = [˜(X), ρ(Y )] := ρ(X)˜(Y ) ’ ρ(Y )˜(X).

˜ ρ ˜ ˜ ρ ˜ ρ

An a¬ne representation is the same construction as above, but we allow the

ρ(g)s to be invertible a¬ne maps, i.e. linear maps plus constants.

A.3 Semi-Direct Product and Poincar´ Group

e

The direct product G — H of two groups G and H is again a group with

multiplication law: (g1 , h1 )(g2 , h2 ) := (g1 g2 , h1 h2 ). In the direct product, all

elements of the ¬rst factor commute with all elements of the second factor:

(g, 1H )(1G , h) = (1G , h)(g, 1H ). We write 1H for the neutral element of H. Warn-

ing, you sometimes see the misleading notation G — H for the direct product.

To be able to de¬ne the semi-direct product G H we must have an action

of G on H, that is a map ρ : G ’ Di¬(H) satisfying ρg (h1 h2 ) = ρg (h1 ) ρg (h2 ),

ρg (1H ) = 1H , ρg1 g2 = ρg1 —¦ ρg2 and ρ1G = 1H . If H is a vector space carrying

a representation or an a¬ne representation ρ of the group G, we can view ρ as

an action by considering H as translation group. Indeed, invertible linear maps

and a¬ne maps are di¬eomorphisms on H. As a set, the semi-direct product

G H is the direct product, but the multiplication law is modi¬ed by help of

the action:

(g1 , h1 )(g2 , h2 ) := (g1 g2 , h1 ρg1 (h2 )). (A.4)

We retrieve the direct product if the action is trivial, ρg = 1H for all g ∈ G. Our

¬rst example is the invariance group of electro“magnetism coupled to gravity

M

Di¬(M ) U (1). A di¬eomorphism σ(x) acts on a gauge function g(x) by

ρσ (g) := g —¦ σ ’1 or more explicitly (ρσ (g))(x) := g(σ ’1 (x)). Other examples

come with other gauge groups like SU (n) or spin groups.

Our second example is the Poincar´ group, O(1, 3) R4 , which is the isometry

e

group of Minkowski space. The semi-direct product is important because Lorentz

transformations do not commute with translations. Since we are talking about

the Poincar´ group, let us mention the theorem behind the de¬nition of particles

e

as orthonormal basis vectors of unitary representations: The irreducible, unitary

representations of the Poincar´ group are characterized by mass and spin. For

e

¬xed mass M ≥ 0 and spin , an orthonormal basis is labelled by the momentum

p with E 2 /c2 ’ p2 = c2 M 2 , ψ = exp(i(Et ’ p · x)/ ) and the z-component m

of the spin with |m| ¤ , ψ = Y ,m (θ, •).

A.4 Algebras

Observables can be added, multiplied and multiplied by scalars. They form nat-

urally an associative algebra A, i.e. a vector space equipped with an associative

Forces from Connes™ Geometry 345

product and neutral elements 0 and 1. Note that the multiplication does not al-

ways admit inverses, a’1 , e.g. the neutral element of addition, 0, is not invertible.

In quantum mechanics, observables are self adjoint. Therefore, we need an invo-

lution ·— in our algebra. This is an anti-linear map from the algebra into itself,

(»a + b)— = »a— + b— , » ∈ C, a, b ∈ A, that reverses the product, (ab)— = b— a— ,

¯

respects the unit, 1— = 1, and is such that a—— = a. The set of n — n matrices

with complex coe¬cients, Mn (C), is an example of such an algebra, and more

generally, the set of endomorphisms or operators on a given Hilbert space H.

The multiplication is matrix multiplication or more generally composition of op-

erators, the involution is Hermitean conjugation or more generally the adjoint

of operators.

A representation ρ of an abstract algebra A on a Hilbert space H is a way to

write A concretely as operators as in the last example, ρ : A ’ End(H). In the

group case, the representation had to reproduce the multiplication law. Now it

has to reproduce, the linear structure: ρ(»a + b) = »ρ(a) + ρ(b), ρ(0) = 0, the

multiplication: ρ(ab) = ρ(a)ρ(b), ρ(1) = 1, and the involution: ρ(a— ) = ρ(a)— .

Therefore the tensor product of two representations ρ1 and ρ2 of A on Hilbert

spaces H1 ψ1 and H2 ψ2 is not a representation: ((ρ1 — ρ2 )(»a)) (ψ1 — ψ2 ) =

(ρ1 (»a) ψ1 ) — (ρ2 (»a) ψ2 ) = »2 (ρ1 — ρ2 )(a) (ψ1 — ψ2 ).

The group of unitaries U (A) := {u ∈ A, uu— = u— u = 1} is a subset of

the algebra A. Every algebra representation induces a unitary representation of

its group of unitaries. On the other hand, only few unitary representations of

the group of unitaries extend to an algebra representation. These representa-

tions describe elementary particles. Composite particles are described by tensor

products, which are not algebra representations.

˜

An anti-linear operator J on a Hilbert space H ψ, ψ is a map from H into

˜ ¯ ˜

itself satisfying J(»ψ + ψ) = »J(ψ) + J(ψ). An anti-linear operator J is anti-

˜ ˜

unitary if it is invertible and preserves the scalar product, (Jψ, J ψ) = (ψ, ψ).

For example, on H = Cn ψ we can de¬ne an anti-unitary operator J in the

following way. The image of the column vector ψ under J is obtained by taking

the complex conjugate of ψ and then multiplying it with a unitary n — n matrix

¯

U , Jψ = U ψ or J = U —¦ complex conjugation. In fact, on a ¬nite dimensional

Hilbert space, every anti-unitary operator is of this form.

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