Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

Abstract

Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic $\mathbb {R}^{4}$, and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called smooth quantum gravity by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard $\mathbb {R}^{4}$ to the exotic $R^{4}$ is a quantization of geometry.

Appendices

Appendix A Casson Handles and Labeled Trees

Let us now consider the basic construction of the Casson handle CH. Let M be a smooth, compact, simply-connected 4-manifold and $f:D^{2}\rightarrow M$ a (codimension-2) mapping. By using diffeomorphisms of $D^{2}$ and M, one can deform the mapping f to get an immersion (i.e. injective differential) generically with only double points (i.e. $\#|f^{-1}(f(x))|=2$) as singularities [62]. But to incorporate the generic location of the disk, one is rather interesting in the mapping of a 2-handle $D^{2}\times D^{2}$ induced by $f\times id:D^{2}\times D^{2}\rightarrow M$ from f. Then every double point (or self-intersection) of $f(D^{2})$ leads to self-plumbings of the 2-handle $D^{2}\times D^{2}$. A self-plumbing is an identification of $D_{0}^{2}\times D^{2}$ with $D_{1}^{2}\times D^{2}$ where $D_{0}^{2},D_{1}^{2}\subset D^{2}$ are disjoint sub-disks of the first factor disk. In complex coordinates the plumbing may be written as $(z,w)\mapsto (w,z)$ or $(z,w)\mapsto (\bar{w},\bar{z})$ creating either a positive or negative (respectively) double point on the disk $D^{2}\times 0$. Consider the pair $(D^{2}\times D^{2},\partial D^{2}\times D^{2})$ and produce finitely many self-plumbings away from the attaching region $\partial D^{2}\times D^{2}$ to get a kinky handle $(k,\partial ^{-}k)$ where $\partial ^{-}k$ denotes the attaching region of the kinky handle. A kinky handle $(k,\partial ^{-}k)$ is a one-stage tower $(T_{1},\partial ^{-}T_{1})$ and an $(n+1)$-stage tower $(T_{n+1},\partial ^{-}T_{n+1})$ is an n-stage tower union of kinky handles $\bigcup _{\ell =1}^{n}(T_{\ell },\partial ^{-}T_{\ell })$ where two towers are attached along $\partial ^{-}T_{\ell }$. Let $T_{n}^{-}$ be $(\text {interior}\,T_{n})\cup \partial ^{-}T_{n}$ and the Casson handle

$$ CH=\bigcup _{\ell =0}T_{\ell }^{-} $$

is the union of towers (with direct limit topology induced from the inclusions $T_{n}\hookrightarrow T_{n+1}$). A Casson handle is specified up to (orientation preserving) diffeomorphism (of pairs) by a labeled finitely-branching tree with base-point *, having all edge paths infinitely extendable away from *. Each edge should be given a label $+$ or − and each vertex corresponds to a kinky handle; the self-plumbing number of that kinky handle equals the number of branches leaving the vertex. The sign on each branch corresponds to the sign of the associated self plumbing. The whole process generates a tree with infinite many levels. In principle, every tree with a finite number of branches per level realizes a corresponding Casson handle. The simplest non-trivial Casson handle is represented by the tree $Tree_{+}$: each level has one branching point with positive sign $+$. The reverse construction of a Casson handle $CH_{\mathcal {T}}$ by using a labeled tree $\mathcal {T}$ can be found in the appendix A. Let $\mathcal {T}_{1}$ and $\mathcal {T}_{2}$ be two trees with $\mathcal {T}_{1}\subset \mathcal {T}_{2}$ (it is the subtree) then $CH_{\mathcal {T}_{2}}\subset CH_{\mathcal {T}_{1}}$.Given a labeled based tree Q, let us describe a subset $U_{Q}$ of $D^{2}\times D^{2}$. Now we will construct a $(U_{Q},\partial D^{2}\times D^{2})$ which is diffeomorphic to the Casson handle associated to Q. In $D^{2}\times D^{2}$ embed a ramified Whitehead link with one Whitehead link component for every edge labeled by $+$ leaving * and one mirror image Whitehead link component for every edge labeled by −(minus) leaving *. Corresponding to each first level node of Q we have already found a (normally framed) solid torus embedded in $D^{2}\times \partial D^{2}$. In each of these solid tori embed a ramified Whitehead link, ramified according to the number of $+$ and − labeled branches leaving that node. We can do that process for every level of Q. Let the disjoint union of the (closed) solid tori in the n-th family (one solid torus for each branch at level n in Q) be denoted by $X_{n}$. Q tells us how to construct an infinite chain of inclusions:

$$ \cdots \subset X_{n+1}\subset X_{n}\subset X_{n-1}\subset \cdots \subset X_{1}\subset D^{2}\times \partial D^{2} $$

and we define the Whitehead decomposition $WhC_{Q}=\bigcap _{n=1}^{\infty }X_{n}$ of Q. $WhC_{Q}$ is the Whitehead continuum [99] for the simplest unbranched tree. We define $U_{Q}$ to be

$$ U_{Q}=D^{2}\times D^{2}{\setminus }(D^{2}\times \partial D^{2}\cup \,\text {closure}(WhC_{Q})) $$

alternatively one can also write

$$\begin{aligned} U_{Q}=D^{2}\times D^{2}{\setminus }\,\text {cone}(WhC_{Q}) \end{aligned}$$

(15.33)

where $\text {cone}()$ is the cone of a space

$$ cone(A)=A\times [0, 1]/(x,0)\sim (x',0)\qquad \forall x,x'\in A $$

over the point $(0,0)\in D^{2}\times D^{2}$. As Freedman (see [53] Theorem 2.2) showed $U_{Q}$ is diffeomorphic to the Casson handle $CH_{Q}$ given by the tree Q.

Appendix B Thurston Foliation of a 3-Manifold

In [94] Thurston constructed a foliation of the 3-sphere $S^{3}$ which depends on a polygon P in the hyperbolic plane $\mathbb {H}^{2}$ so that two foliations are non-cobordant if the corresponding polygons have different areas. For later usage, we will present the main ideas of this construction only (see also the book [92] Chap. VIII for the details). Starting point is the hyperbolic plane $\mathbb {H}^{2}$ with a convex polygon $K\subset \mathbb {H}^{2}$ having k sides $s_{1},\ldots ,s_{k}$. Assuming the upper half plane model of $\mathbb {H}^{2}$ then the sides are circular arcs. The construction of the foliation depends mainly on the isometry group $PSL(2,\mathbb {R})$ of $\mathbb {H}^{2}$ realized as rational transformations (and this group can be lifted to $SL(2,\mathbb {R})$). The followings steps are needed in the construction:

• 1. The polygon K is doubled along one side, say $s_{1}$, to get a polygon $K'$. The sides are identified by (isometric) transformations $s_{i}\rightarrow s_{i}'$ (as elements of $SL(2,\mathbb {R})$).

• 2. Take $\epsilon$-neighborhoods $U_{\epsilon }(p_{i}),U_{\epsilon }(p_{i}')$ with $\epsilon >0$ sufficient small and set

  $$\begin{aligned} V^{2}&=\left( K\cup K'\right) {\setminus }\bigcup _{i=1}^{k}\left( U_{\epsilon }(p_{i})\cup U_{\epsilon }(p_{i}')\right) \\&=S^{2}{\setminus }\bigcup _{i=1}^{k}D_{i}^{2} \end{aligned}$$

  having the topology of $V^{2}=S^{2}{\setminus }\left\{ k\,\text {punctures}\right\}$ and we set $P=K\cup K'$.

• 3. Now consider the unit tangent bundle $U\mathbb {H}^{2}$, i.e. a $S^{1}$-bundle over $\mathbb {H}^{2}$ (or the tangent bundle where every vector has norm one). The restricted bundle over $V^{2}$ is trivial so that $UV^{2}=V^{2}\times S^{1}$. Let $L,L'$ be circular arcs (geodesics) in $\mathbb {H}^{2}$ (invariant w.r.t. $SL(2,\mathbb {R})$) starting at a common point which define parallel tangent vectors w.r.t. the metrics of the upper half plane model. The foliation of $V^{2}$ is given by geodesics transverse to the boundary and we obtain a foliation of $V^{2}\times S^{1}$ (as unit tangent bundle). This foliation is given by a $SL(2,\mathbb {R})$-invariant smooth 1-form $\omega$ (so that $\omega =const.$defines the leaves) which is integrable $d\omega \wedge \omega =0$. ($SL(2,\mathbb {R})$-invariant Foliation $\mathcal {F}_{SL}$)

• 4. With the relation $D^{2}=V^{2}\cup D_{1}^{2}\cup \cdots \cup D_{k-1}^{2}$, we obtain $D^{2}\times S^{1}=V^{2}\times S^{1}\cup \left( D_{1}^{2}\times S^{1}\right) \cup \cdots \cup \left( D_{k-1}^{2}\times S^{1}\right)$ or the gluing of $k-1$ solid tori to $V^{2}\times S^{1}$ gives a solid tori. Every glued solid torus will be foliated by a Reeb foliation. Finally using $S^{3}=(D^{2}\times S^{1})\cup (S^{1}\times D^{2})$ (the Heegard decomposition of the 3-sphere) again with a solid torus with Reeb foliation, we obtain a foliation on the 3-sphere.

The construction of this foliation $\mathcal {F}_{Thurston}$ (Thurston foliation) will be also work for any 3-manifold. Thurston [94] obtains for the Godbillon-Vey number

$$ GV(V^{2}\times S^{1},\mathcal {F}_{SL})=4\pi \cdot vol(P)=8\pi \cdot vol(K) $$

and

$$\begin{aligned} GV(S^{3},\mathcal {F}_{Thurston})=4\pi \cdot Area(P) \end{aligned}$$

(15.34)

so that any real number can be realized by a suitable foliation of this type. Furthermore, two cobordant foliations have the same Godbillon-Vey number (but the reverse is in general wrong). Let $[1]\in H^{3}(S^{3},\mathbb {R})$ be the dual of the fundamental class $[S^{3}]$ defined by the volume form, then the Godbillon-Vey class can be represented by

$$\begin{aligned} \varGamma _{\mathcal {F}_{a}}=4\pi \cdot Area(P)[1] \end{aligned}$$

(15.35)

The Godbillon-Vey class is an element of the deRham cohomology $H^{3}(S^{3},\mathbb {R})$. Now we will discuss the general case of a compact 3-manifold carrying a foliation of the same type like the 3-sphere above. The main idea of the construction is very simple and uses a general representation of all compact 3-manifolds by Dehn surgery. Here we will use an alternative representation of surgery by using the Dehn-Lickorish theorem ([81] Corollary 12.4 at p. 84). Let $\varSigma$ be a compact 3-manifold without boundary. There is now a natural number $k\in \mathbb {N}$ so that any orientable 3-manifold can be obtained by cutting out k solid tori from the 3-sphere $S^{3}$ and then pasting them back in, but along different diffeomorphisms of their boundaries. Moreover, it can be assumed that all these solid tori in $S^{3}$ are unknotted. Then any 3-manifold $\varSigma$ can be written as

$$ \varSigma =\left( S^{3}{\setminus }\left( \bigsqcup _{i=1}^{k}D_{i}^{2}\times S^{1}\right) \right) \cup _{\phi _{1}}\left( D_{1}^{2}\times S^{1}\right) \cup _{\phi _{2}}\cdots \cup _{\phi _{k}}\left( D_{k}^{2}\times S^{1}\right) $$

where $\phi _{i}:\partial \left( S^{3}{\setminus }\left( \bigsqcup _{i=1}^{k}D_{i}^{2}\times S^{1}\right) \right) \rightarrow \partial D_{i}^{2}\times S^{1}$ is the gluing map from each boundary component of $\left( S^{3}{\setminus }\left( \bigsqcup _{i=1}^{k}D_{i}^{2}\times S^{1}\right) \right)$ to the boundary of $\partial D_{i}^{2}\times S^{1}$. This gluing map is a diffeomorphism of tori $T^{2}\rightarrow T^{2}$ (where $T^{2}=S^{1}\times S^{1}$). The Dehn-Lickorish theorem describes all diffeomorphisms of a surface: Every diffeomorphism of a surface is the composition of Dehn twists and coordinate transformations (or small diffeomorphisms). The decomposition

$$\begin{aligned} S^{3}=\left( V^{2}\times S^{1}\right) \cup \left( D_{1}^{2}\times S^{1}\right) \cup \cdots \cup \left( D_{k-1}^{2}\times S^{1}\right) \cup \left( S^{1}\times D_{k}^{2}\right) \end{aligned}$$

(15.36)

of the 3-sphere can be used to get a decomposition of $\varSigma$ by

$$ \varSigma =\left( V^{2}\times S^{1}\right) \cup _{\phi _{1}}\left( D_{1}^{2}\times S^{1}\right) \cup _{\phi _{2}}\cdots \cup _{\phi _{k}}\left( D_{k}^{2}\times S^{1}\right) $$

which will guide us to the construction of a foliation on $\varSigma$:

• Construct a foliation $\mathcal {F}_{\varSigma ,SL}$ on $V^{2}\times S^{1}$ using a polygon P (see above) and

• Glue in k Reeb foliations of the solid tori using the diffeomorphisms $\phi _{i}$.

Finally we get a foliation $\mathcal {F}_{\varSigma ,Thurston}$ on $\varSigma$. According to the rules above, we are able to calculate the Godbillon-Vey number

$$ GV(\varSigma ,\mathcal {F}_{\varSigma ,Thurston})=4\pi \cdot vol(P) $$

Therefore for any foliation of $S^{3}$, we can construct a foliation on any compact 3-manifold $\varSigma$ with the same Godbillon-Vey number. Both foliations $\mathcal {F}_{Thurston}$ and $\mathcal {F}_{\varSigma ,Thurston}$ agree for the common submanifold $V^{2}\times S^{1}$ or there is a foliated cobordism between $V^{2}\times S^{1}\subset \varSigma$ and $V^{2}\times S^{1}\subset S^{3}$. Of course, $S^{3}$ and $\varSigma$ differ by the gluing of the solid tori but every solid torus carries a Reeb foliation which does not contribute to the Godbillon-Vey number.

Appendix C 3-Manifolds and Geometric Structures

A connected 3-manifold N is prime if it cannot be obtained as a connected sum of two manifolds $N_{1}\#N_{2}$ neither of which is the 3-sphere $S^{3}$ (or, equivalently, neither of which is the homeomorphic to N). Examples are the 3-torus $T^{3}$ and $S^{1}\times S^{2}$ but also the Poincare sphere. According to [77], any compact, oriented 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds (prime decomposition). A subset of prime manifolds are the irreducible 3-manifolds. A connected 3-manifold is irreducible if every differentiable submanifold S homeomorphic to a sphere $S^{2}$ bounds a subset D (i.e. $\partial D=S$) which is homeomorphic to the closed ball $D^{3}$. The only prime but reducible 3-manifold is $S^{1}\times S^{2}$. For the geometric properties (to meet Thurstons geometrization theorem) we need a finer decomposition induced by incompressible tori. A properly embedded connected surface $S\subset N$ is called 2-sided⁵ if its normal bundle is trivial, and 1-sided if its normal bundle is nontrivial. A 2-sided connected surface S other than $S^{2}$ or $D^{2}$ is called incompressible if for each disk $D\subset N$ with $D\cap S=\partial D$ there is a disk $D'\subset S$ with $\partial D'=\partial D$. The boundary of a 3-manifold is an incompressible surface. Most importantly, the 3-sphere $S^{3}$, $S^{2}\times S^{1}$ and the 3-manifolds $S^{3}/\varGamma$ with $\varGamma \subset SO(4)$ a finite subgroup do not contain incompressible surfaces. The class of 3-manifolds $S^{3}/\varGamma$ (the spherical 3-manifolds) include cases like the Poincare sphere ($\varGamma =I^{*}$ the binary icosaeder group) or lens spaces ($\varGamma =\mathbb {Z}_{p}$ the cyclic group). Let $K_{i}$ be irreducible 3-manifolds containing incompressible surfaces then we can N split into pieces (along embedded $S^{2}$)

$$\begin{aligned} N=K_{1}\#\cdots \#K_{n_{1}}\#_{n_{2}}S^{1}\times S^{2}\#_{n_{3}}S^{3}/\varGamma \,, \end{aligned}$$

(15.37)

where $\#_{n}$ denotes the n-fold connected sum and $\varGamma \subset SO(4)$ is a finite subgroup. The decomposition of N is unique up to the order of the factors. The irreducible 3-manifolds $K_{1},\ldots ,\, K_{n_{1}}$ are able to contain incompressible tori and one can split $K_{i}$ along the tori into simpler pieces $K=H\cup _{T^{2}}G$ [73] (called the JSJ decomposition). The two classes G and H are the graph manifold G and the hyperbolic 3-manifold H (see Fig. 15.11).

Fig. 15.11

Torus (JSJ-) decomposition, $H_{i}$ hyperbolic manifold, $S_{i}$ Graph-manifold, $T_{i}$ Tori

The hyperbolic 3-manifold H has a torus boundary $T^{2}=\partial H$, i.e. H admits a hyperbolic structure in the interior only. In this paper we need the splitting of the link/knot complement. As shown in [33], the Whitehead double of a knot leads to JSJ decomposition of the complement into the knot complement and the complement of the Whitehead link (along one torus boundary of the Whitehead link complement).

One property of hyperbolic 3-manifolds is central: Mostow rigidity. As shown by Mostow [78], every hyperbolic n-manifold $n>2$ with finite volume has this property: Every diffeomorphism (especially every conformal transformation) of a hyperbolic n-manifold with finite volume is induced by an isometry. Therefore one cannot scale a hyperbolic 3-manifold and the volume is a topological invariant. Together with the prime and JSJ decomposition

$$ N=\left( H_{1}\cup _{T^{2}}G_{1}\right) \#\cdots \#\left( H_{n_{1}}\cup _{T^{2}}G_{n_{1}}\right) \#_{n_{2}}S^{1}\times S^{2}\#_{n_{3}}S^{3}/\varGamma \,, $$

we can discuss the geometric properties central to Thurstons geometrization theorem: Every oriented closed prime 3-manifold can be cut along tori (JSJ decomposition), so that the interior of each of the resulting manifolds has a geometric structure with finite volume. Now, we have to clarify the term geometric structure’s. A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers. A geometric structure on a manifold N is a diffeomorphism from N to $X/\varGamma$ for some model geometry X, where $\varGamma$ is a discrete subgroup of G acting freely on X. t is a surprising fact that there are also a finite number of three-dimensional model geometries, i.e. 8 geometries with the following models: spherical $(S^{3},O_{4}(\mathbb {R}))$, Euclidean $(\mathbb {E}^{3},O_{3}(\mathbb {R})\ltimes \mathbb {R}^{3})$, hyperbolic $(\mathbb {H}^{3},O_{1,3}(\mathbb {R})^{+})$, mixed spherical-Euclidean $(S^{2}\times \mathbb {R},O_{3}(\mathbb {R})\times \mathbb {R}\times \mathbb {Z}_{2})$, mixed hyperbolic-Euclidean $(\mathbb {H}^{2}\times \mathbb {R},O_{1,3}(\mathbb {R})^{+}\times \mathbb {R}\times \mathbb {Z}_{2})$ and 3 exceptional cases called $\tilde{SL}_{2}$ (twisted version of $\mathbb {H}^{2}\times \mathbb {R}$), NIL (geometry of the Heisenberg group as twisted version of $\mathbb {E}^{3}$), SOL (split extension of $\mathbb {R}^{2}$ by $\mathbb {R}$, i.e. the Lie algebra of the group of isometries of 2-dimensional Minkowski space). We refer to [84] for the details.