<<

. 75
( 78 .)



>>

˜
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.

References
1. A. Connes, A. Lichn´rowicz and M. P. Sch¨tzenberger, Triangle de Pens´es, O. Ja-
e
e u
cob (2000), English version: Triangle of Thoughts, AMS (2001)
2. G. Amelino-Camelia, Are we at the dawn of quantum gravity phenomenology?,
Lectures given at 35th Winter School of Theoretical Physics: From Cosmology to
Quantum Gravity, Polanica, Poland, 1999, gr-qc/9910089
3. S. Weinberg, Gravitation and Cosmology, Wiley (1972)
R. Wald, General Relativity, The University of Chicago Press (1984)
4. J. D. Bjørken and S. D. Drell, Relativistic Quantum Mechanics, McGraw“Hill
(1964)
5. L. O™Raifeartaigh, Group Structure of Gauge Theories, Cambridge University Press
(1986)
346 T. Sch¨cker
u

6. M. G¨ckeler and T. Sch¨cker, Di¬erential Geometry, Gauge Theories, and Gravity,
o u
Cambridge University Press (1987)
7. R. Gilmore, Lie Groups, Lie Algebras and some of their Applications, Wiley (1974)
H. Bacry, Lectures Notes in Group Theory and Particle Theory, Gordon and Breach
(1977)
8. N. Jacobson, Basic Algebra I, II, Freeman (1974,1980)
9. J. Madore, An Introduction to Noncommutative Di¬erential Geometry and its
Physical Applications, Cambridge University Press (1995)
G. Landi, An Introduction to Noncommutative Spaces and their Geometry, hep-
th/9701078, Springer (1997)
10. J. M. Gracia-Bond´ J. C. V´rilly and H. Figueroa, Elements of Noncommutative
±a, a
Geometry, Birkh¨user (2000)
a
11. J. W. van Holten, Aspects of BRST quantization, hep-th/0201124, in this volume
12. J. Zinn-Justin, Chiral anomalies and topology, hep-th/0201220, in this volume
13. The Particle Data Group, Particle Physics Booklet and http://pdg.lbl.gov
14. G. ™t Hooft, Renormalizable Lagrangians for Massive Yang“Mills Fields, Nucl.
Phys. B35 (1971) 167
G. ™t Hooft and M. Veltman, Regularization and Renormalization of Gauge Fields,
Nucl. Phys. B44 (1972) 189
G. ™t Hooft and M. Veltman, Combinatorics of Gauge Fields, Nucl. Phys. B50
(1972) 318
B. W. Lee and J. Zinn-Justin, Spontaneously broken gauge symmetries I, II, III
and IV, Phys. Rev. D5 (1972) 3121, 3137, 3155; Phys. Rev. D7 (1973) 1049
15. S. Glashow, Partial-symmetries of weak interactions, Nucl. Phys. 22 (1961) 579
A. Salam in Elementary Particle Physics: Relativistic Groups and Analyticity,
Nobel Symposium no. 8, page 367, eds.: N. Svartholm, Almqvist and Wiksell,
Stockholm 1968
S. Weinberg, A model of leptons, Phys. Rev. Lett. 19 (1967) 1264
16. J. Iliopoulos, An introduction to gauge theories, Yellow Report, CERN (1976)
17. G. Esposito-Far`se, Th´orie de Kaluza“Klein et gravitation quantique, Th´se de
e
e e
Doctorat, Universit´ d™Aix-Marseille II, 1989
e
18. A. Connes, Noncommutative Geometry, Academic Press (1994)
19. A. Connes, Noncommutative Geometry and Reality, J. Math. Phys. 36 (1995) 6194
20. A. Connes, Gravity coupled with matter and the foundation of noncommutative
geometry, hep-th/9603053, Comm. Math. Phys. 155 (1996) 109
21. H. Rauch, A. Zeilinger, G. Badurek, A. Wil¬ng, W. Bauspiess and U. Bonse,
Veri¬cation of coherent spinor rotations of fermions, Phys. Lett. 54A (1975) 425
22. E. Cartan, Le¸ons sur la th´orie des spineurs, Hermann (1938)
c e
23. A. Connes, Brisure de sym´trie spontan´e et g´om´trie du point de vue spectral,
e e ee
S´minaire Bourbaki, 48`me ann´e, 816 (1996) 313
e e e
A. Connes, Noncommutative di¬erential geometry and the structure of space time,
Operator Algebras and Quantum Field Theory, eds.: S. Doplicher et al., Interna-
tional Press, 1997
24. T. Sch¨cker, Spin group and almost commutative geometry, hep-th/0007047
u
25. J.-P. Bourguignon and P. Gauduchon, Spineurs, op´rateurs de Dirac et variations
e
de m´triques, Comm. Math. Phys. 144 (1992) 581
e
26. U. Bonse and T. Wroblewski, Measurement of neutron quantum interference in
noninertial frames, Phys. Rev. Lett. 1 (1983) 1401
27. R. Colella, A. W. Overhauser and S. A. Warner, Observation of gravitationally
induced quantum interference, Phys. Rev. Lett. 34 (1975) 1472
Forces from Connes™ Geometry 347

28. A. Chamseddine and A. Connes, The spectral action principle, hep-th/9606001,
Comm. Math. Phys.186 (1997) 731
29. G. Landi and C. Rovelli, Gravity from Dirac eigenvalues, gr-qc/9708041, Mod.
Phys. Lett. A13 (1998) 479
30. P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah“Singer Index
Theorem, Publish or Perish (1984)
S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge
University Press (1989)
31. B. Iochum, T. Krajewski and P. Martinetti, Distances in ¬nite spaces from non-
commutative geometry, hep-th/9912217, J. Geom. Phys. 37 (2001) 100
32. M. Dubois-Violette, R. Kerner and J. Madore, Gauge bosons in a noncommutative
geometry, Phys. Lett. 217B (1989) 485
33. A. Connes, Essay on physics and noncommutative geometry, in The Interface of
Mathematics and Particle Physics, eds.: D. G. Quillen et al., Clarendon Press
(1990)
A. Connes and J. Lott, Particle models and noncommutative geometry, Nucl. Phys.
B 18B (1990) 29
A. Connes and J. Lott, The metric aspect of noncommutative geometry, in the pro-
ceedings of the 1991 Carg`se Summer Conference, eds.: J. Fr¨hlich et al., Plenum
e o
Press (1992)
34. J. Madore, Modi¬cation of Kaluza Klein theory, Phys. Rev. D 41 (1990) 3709
35. P. Martinetti and R. Wulkenhaar, Discrete Kaluza“Klein from Scalar Fluctuations
in Noncommutative Geometry, hep-th/0104108, J. Math. Phys. 43 (2002) 182
36. T. Ackermann and J. Tolksdorf, A generalized Lichnerowicz formula, the Wodzicki
residue and gravity, hep-th/9503152, J. Geom. Phys. 19 (1996) 143
T. Ackermann and J. Tolksdorf, The generalized Lichnerowicz formula and analysis
of Dirac operators, hep-th/9503153, J. reine angew. Math. 471 (1996) 23
37. R. Estrada, J. M. Gracia-Bond´ and J. C. V´rilly, On summability of distributions
±a a
and spectral geometry, funct-an/9702001, Comm. Math. Phys. 191 (1998) 219
38. B. Iochum, D. Kastler and T. Sch¨ cker, On the universal Chamseddine“Connes ac-
u
tion: details of the action computation, hep-th/9607158, J. Math. Phys. 38 (1997)
4929
L. Carminati, B. Iochum, D. Kastler and T. Sch¨ cker, On Connes™ new principle of
u
general relativity: can spinors hear the forces of space-time?, hep-th/9612228, Op-
erator Algebras and Quantum Field Theory, eds.: S. Doplicher et al., International
Press, 1997
39. M. Paschke and A. Sitarz, Discrete spectral triples and their symmetries, q-
alg/9612029, J. Math. Phys. 39 (1998) 6191
T. Krajewski, Classi¬cation of ¬nite spectral triples, hep-th/9701081, J. Geom.
Phys. 28 (1998) 1
40. S. Lazzarini and T. Sch¨cker, A farewell to unimodularity, hep-th/0104038, Phys.
u
Lett. B 510 (2001) 277
41. B. Iochum and T. Sch¨cker, A left-right symmetric model a la Connes“Lott, hep-
`
u
th/9401048, Lett. Math. Phys. 32 (1994) 153
F. Girelli, Left-right symmetric models in noncommutative geometry? hep-
th/0011123, Lett. Math. Phys. 57 (2001) 7
42. F. Lizzi, G. Mangano, G. Miele and G. Sparano, Constraints on uni¬ed gauge
theories from noncommutative geometry, hep-th/9603095, Mod. Phys. Lett. A11
(1996) 2561
43. W. Kalau and M. Walze, Supersymmetry and noncommutative geometry, hep-
th/9604146, J. Geom. Phys. 22 (1997) 77
348 T. Sch¨cker
u

44. D. Kastler, Introduction to noncommutative geometry and Yang“Mills model build-
ing, Di¬erential geometric methods in theoretical physics, Rapallo (1990), 25
” , A detailed account of Alain Connes™ version of the standard model in non-
commutative geometry, I, II and III, Rev. Math. Phys. 5 (1993) 477, Rev. Math.
Phys. 8 (1996) 103
D. Kastler and T. Sch¨ cker, Remarks on Alain Connes™ approach to the standard
u
model in non-commutative geometry, Theor. Math. Phys. 92 (1992) 522, English
version, 92 (1993) 1075, hep-th/0111234
” , A detailed account of Alain Connes™ version of the standard model in non-
commutative geometry, IV, Rev. Math. Phys. 8 (1996) 205
” , The standard model a la Connes“Lott, hep-th/9412185, J. Geom. Phys. 388
`
(1996) 1
J. C. V´rilly and J. M. Gracia-Bond´ Connes™ noncommutative di¬erential ge-
a ±a,
ometry and the standard model, J. Geom. Phys. 12 (1993) 223
T. Sch¨cker and J.-M. Zylinski, Connes™ model building kit, hep-th/9312186, J.
u
Geom. Phys. 16 (1994) 1
E. Alvarez, J. M. Gracia-Bond´ and C. P. Mart´ Anomaly cancellation and
±a ±n,
the gauge group of the Standard Model in Non-Commutative Geometry, hep-
th/9506115, Phys. Lett. B364 (1995) 33
R. Asquith, Non-commutative geometry and the strong force, hep-th/9509163,
Phys. Lett. B 366 (1996) 220
C. P. Mart´ J. M. Gracia-Bond´ and J. C. V´rilly, The standard model as a
±n, ±a a
noncommutative geometry: the low mass regime, hep-th/9605001, Phys. Rep. 294
(1998) 363
L. Carminati, B. Iochum and T. Sch¨ cker, The noncommutative constraints on the
u
standard model a la Connes, hep-th/9604169, J. Math. Phys. 38 (1997) 1269
`
R. Brout, Notes on Connes™ construction of the standard model, hep-th/9706200,
Nucl. Phys. Proc. Suppl. 65 (1998) 3
J. C. V´rilly, Introduction to noncommutative geometry, physics/9709045, EMS
a
Summer School on Noncommutative Geometry and Applications, Portugal,
september 1997, ed.: P. Almeida

<<

. 75
( 78 .)



>>