A Chicken-and-Egg Problem: Which Came First, the Quantum State or Spacetime?

Abstract

In this essay I will discuss the question: Is spacetime quantized, as in quantum geometry, or is it possible to derive the quantization procedure from the structure of spacetime? All proposals of quantum gravity try to quantize spacetime or derive it as an emergent phenomenon. In this essay, all major approaches are analyzed to find an alternative to a discrete structure on spacetime or to the emergence of spacetime. Here I will present the idea that spacetime defines the quantum state by using new developments in the differential topology of 3- and 4-manifolds. In particular the plethora of exotic smoothness structures in dimension 4 could be the corner stone of quantum gravity.

Appendix A: \(C^{*}\)—algebras Associated to Wild Embeddings

Let $I:K^{n}\rightarrow \mathbb {R}^{n+k}$ be a wild embedding of codimension $k$ with $k=0,1,2$. In the following we assume that the complement $\mathbb {R}^{n+k}\setminus I(K^{n})$ is non-trivial, i.e. $\pi _{1}(\mathbb {R}^{n+k}\setminus I(K^{n}))=\pi \not =1$. Now one defines the $C^{*}-$algebra $C^{*}(\mathcal {G},\pi )$ associated to the complement $\mathcal {G}=\mathbb {R}^{n+k}\setminus I(K^{n})$ with group $\pi =\pi _{1}(\mathcal {G})$. If $\pi$ is non-trivial then this group is not finitely generated. The construction of wild embeddings is given by an infinite construction³ (see Antoine’s necklace or Alexanders horned sphere). From an abstract point of view, we have a decomposition of $\mathcal {G}$ by an infinite union

$$ \mathcal {G}=\bigcup _{i=0}^{\infty }C_{i} $$

of “level sets” $C_{i}$. Then every element $\gamma \in \pi$ lies (up to homotopy) in a finite union of levels.

The basic elements of the $C^{*}-$algebra $C^{*}(\mathcal {G},\pi )$ are smooth half-densities with compact supports on $\mathcal {G}$, $f\in C_{c}^{\infty }(\mathcal {G},\Omega ^{1/2})$, where $\Omega _{\gamma }^{1/2}$ for $\gamma \in \pi$ is the one-dimensional complex vector space of maps from the exterior power $\Lambda ^{2}L$, of the union of levels $L$ representing $\gamma$, to $\mathbb {C}$ such that

$$ \rho (\lambda \nu )=|\lambda |^{1/2}\rho (\nu )\qquad \forall \nu \in \Lambda ^{2}L,\lambda \in \mathbb {R}\,. $$

For $f,g\in C_{c}^{\infty }(\mathcal {G},\Omega ^{1/2})$, the convolution product $f*g$ is given by the equality

$$ (f*g)(\gamma )=\mathop {\int }_{{\gamma _{1}\circ \gamma _{2}=\gamma }}f(\gamma _{1})g(\gamma _{2}) $$

with the group operation $\gamma _{1}\circ \gamma _{2}$ in $\pi$. Then we define via $f^{*}(\gamma )=\overline{f(\gamma ^{-1})}$ a $*$operation making $C_{c}^{\infty }(\mathcal {G},\Omega ^{1/2})$ into a $*$algebra. Each level set $C_{i}$ consists of simple pieces (for instance tubes in case of the Alexanders horned sphere) denoted by $T$. For these pieces, one has a natural representation of $C_{c}^{\infty }(\mathcal {G},\Omega ^{1/2})$ on the $L^{2}$ space over $T$. Then one defines the representation

$$ (\pi _{x}(f)\xi )(\gamma )=\mathop {\int }_{{\gamma _{1}\circ \gamma _{2}=\gamma }}f(\gamma _{1})\xi (\gamma _{2})\qquad \forall \xi \in L^{2}(T),\forall x\in \gamma . $$

The completion of $C_{c}^{\infty }(\mathcal {G},\Omega ^{1/2})$ with respect to the norm

$$ ||f||=\sup _{x\in \mathcal {G}}||\pi _{x}(f)|| $$

makes it into a $C^{*}$algebra $C_{c}^{\infty }(\mathcal {G},\pi$). The $C^{*}-$algebra $C_{c}^{\infty }(K,I)$ associated to the wild embedding $I$ is defined to be $C_{c}^{\infty }(K,j)=C_{c}^{\infty }(\mathcal {G},\pi )$. The GNS representation of this algebra is called the state space.