Finally, let us sketch a possible generalization of general relativity.
3.10. Conformal gravity 37
4×4 Dirac gamma matrices furnish a representation of the Lie algebra so(2,4) of the conformal group. This allows us to consider a conformal holonomy group.
The conformal connection incorporates an additional vector field κµ(x) and an axial vierbein 5eαµ(x):
Bµ:= 1
2κµ(x)γ5+1
2eαµ(x)γα+1 2
5eαµ(x)γαγ5+ 1
8Aαβµ (x)[γα,γβ]. (3.190) The resulting theory involves additional dynamical variables and is no longer equivalent to Einstein gravity.
The matricesγα andγαγ5, respectively, do not correspond to ”physical”
transformations. Actually it turns out to be convenient to introduce new fields corresponding to the generators of translations and special conformal translations:
1
2eαµ(x)γα+ 1 2
5eαµ(x)γαγ5 =: ReαµγαPR+Leαµ(x)γαPL. (3.191) where the left-and righthanded vierbein are defined by
Leαµ = 1
2(eαµ− 5eαµ) (3.192)
Reαµ = 1
2(eαµ+ 5eαµ), (3.193) and PL,R denote the chiral projectors 12(1∓γ5). Note that the left-and righthanded vierbein are associated with the generators of translations and special conformal translations, respectively.
The antiautomorphism on the conformal Lie algebra is defined as in the de Sitter case. Since Φ(γ5) =−γ5 and Φ(γαγ5) = +γαγ5 it follows that
Eµ(x) = ReαµγαPR+Leαµ(x)γαPL, (3.194) Aµ(x) = 1
2κµ(x)γ5+1
8Aαβµ(x)[γα,γβ]. (3.195) Consequently, the unitary gauge group isG=Spin(1,3)×D(1), whereD(1) denotes the noncompact dilatation group which is isomorphic to the positive real numbers.
It is important to note that the appearance of two distinct vierbein fields implies a deep modification of general relativity. Now there are three different ”metrics”
LLgµν := LeαµLeαν (3.196)
RRgµν := ReαµReαν (3.197)
LRgµν := LeαµReαν (3.198)
with different behaviour under scale transformations. In particular, it fol-lows that LRg is invariant under γ5-scale transformations. It is important to note that as a result of (3.196) and (3.197) one is faced with a theory involving two different lightcones.
The odd piece of the nonunitary field strength associated with the vector potential (3.190) is
1
2(FTµν(x)−FT ∗µν(x)) = 1
2(FTαβµν(x)Mαβ+FT45µν(x)D), (3.199) whereMαβ andD denote the generators of the Lorentz group and the Di-latation group, respectively. The corresponding components of the nonuni-tary field strength are
FTαβµν =Rαβµν+ 2Leα[µReβν]−2Reα[µLeβν], (3.200) and
FT45µν =∂[µκν]+ 2Leγ[µReγν]=:fµν+ 2Leγ[µReγν]=fµν+ LRg[µν]. (3.201) κµ can be interpreted as the Weyl vector field andfµν as the length curva-ture (Weyl’s ”Streckenkr¨ummung”), [17], see also [37, 38]. Note that these components are invariant under scale transformations.
The even part ofFTµν is 1
2(FTµν(x) +FT ∗µν(x)) = 1
2(LTαµν(x)Kα+ RTαµν(x)Pα), (3.202) where Kα and Pα denote the generators of special conformal translations and translations, respectively. The components
LTαµν =∂[µLeαν]+Aαγ[µLeγν]+ Leα[µκν] (3.203)
RTαµν =∂[µReαν]+Aαγ[µReγν]− Reα[µκν] (3.204) can be considered as a ”left-and righthanded” torsion, respectively. Actually we can write
LTαµν =D[µLeαν] (3.205) and
RTαµν =D[µReαν], (3.206) whereDµ is the Weyl-covariant derivative
DµL,Reαν :=∂µL,Reαν +AαβµL,Reβν ∓ L,Reαµκν. (3.207)
3.10. Conformal gravity 39
We find that the nonunitary field strength describes generalizations of pseudo-Riemannian spaces which are known as Weyl spaces [17]. Further-more, there are two torsion tensors. Their physical interpretation is not clear. For fµν = 0 a Weyl space is equivalent to a pseudo-Riemannian space.
A natural generalization of the Einstein-Hilbert action is Z
d4x q
det(Leαµ)det(Reβν)LeµαReνβ[Rαβµν+ηαβfµν]. (3.208) For Leαµ→eαµ, Reαµ →eαµ and fµν = 0 we get back general relativity.
The field equations are not determined yet. A complete understanding of this theory with its ”exotic” features is still missing.
Discrete differential calculus
On any associative algebra over C or R a differential calculus can be de-fined, which generalizes the calculus of differential forms on a differentiable manifold. This structure has been studied for noncommutative algebras in many applications [1]. Noncommutative differential calculus was originally invented to give up the notion of a point. However, we are interested in the case, where the algebra of functions remains commutative. In this ”semi-commutative” framework conventional notions of locality and the notion of a point retain their meaning.
In this chapter we briefly review a differential calculus on arbitrary di-rected graphs which was developed by Dimakis and M¨uller-Hoissen [14, 33].
In addition we shall construct an integral calculus which is as close as pos-sible to the classical simplicial calculus on triangulated manifolds and prove Stokes law. For compatibility and proper covariance under noninjective maps, the differential calculus has to be adjusted by simplicial reduction and extended to pseudographs.
4.1 Motivation
In classical differential geometry the exterior derivative d is characterized by three properties:
• nilpotence
d2 = 0 (4.1)
• alternating Leibniz rule
d(ω∧ξ) =dω∧ξ+ (−1)nω∧dξ (4.2) for alln-forms ω and arbitrary forms ξ
• covariance
Φ?◦d=d◦Φ? (4.3)
40
4.1. Motivation 41
where Φ :M → M0 is a homeomorphic map of manifolds and Φ? : Ω(M0)→Ω(M) is the pull-back.
One fundamental difficulty in formulating a differential calculus on ar-bitrary directed graphs Γ is the breakdown of Leibniz rule. Actually it is a well-known fact that the Leibniz rule is not obeyed by the ordinary finite difference derivative on the lattice. This can be rectified by changing the commutation relations of 1-forms with functions. The algebra of functions however remains commutative.
To be more explicit, consider the D-dimensional unit lattice M:= ZD. Its sites can be labelled by integers {xµ}, µ = 1. . . D. Let A be the com-mutative algebra of real or complex functions on M, with pointwise mul-tiplication. The coordinates xµ are functions on M. The finite difference derivative is defined by1
∂µf(x) =f(x+ ˆµ)−f(x), (4.4) where x+ ˆµ denotes the nearest neighbour of x in ˆµ direction. Instead of the Leibniz rule,∂µ satisfies
∂µ(f g)(x) = (∂µf(x))g(x) +f(x+ ˆµ)∂µg(x). (4.5) However, it turns out that the exterior derivatived, which acts on functions according to
df(x) =X
µ
dxµ∂µf(x), (4.6)
will indeed satisfy the Leibniz rule if the usual commutativity of functions with 1-forms will be replaced by the relation
f(x)dxµ=dxµf(x+ ˆµ). (4.7) Using algebraic notation, eq. (4.7) can be rewritten as
fµdxµ=dxµf, (4.8)
wherefµ is the translated function defined byfµ(y) :=f(y−µ).ˆ
Now consider the valuedxµ(z) :=eyzof the 1-formdxµatzas associated with the edge of Γ fromztoy=z+ ˆµ. Specializing to the function f =ez, supported only atz, we havefµ=ey and consequently arrive at the relation
eyeyz=eyzez. (4.9)
We will see in the next section that (4.9) generalizes to arbitrary directed graphs without need for a preferred coordinate system.
1Note that heref(x) :=f◦xis a function.
Furthermore we shall discuss the remarkable fact that directed graphs behave like discrete manifolds. It is obvious that their links specify a substi-tute for topology, but it is surprising that one does not need an additional specification of a differential structure. It turns out that the corresponding exterior derivativedsatisfies (4.1)-(4.3) if Φ is assumed to be bijective. How-ever, covariance under noninjective maps requires to extend the calculus to pseudographs.