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## Discrete Quantum Gravity

October 19, 2011 1 comment

Stanley Gudder, University of Denver

Q. GR deals with smooth functions on spacetime. QM deals with self adjoint operators on a Hilbert space. How are they related?

A. GR and QM both have bad singularities. Maybe we shouldn’t be looking at the continuum but the discrete picture!

We discuss the causal set approach to unify gravity and quantum mechanics. Let us begin with light cones. We know about the future and past lightcones of an event, say a.

Source: Wikipedia

Introduction and definitions

We talk about the causal structure $(M,<)$ of a Lorentzian spacetime $(M,g)$. $M$ is a partially ordered set(Poset).

For $a, b \in M$, we say $a< b$ if b is in the causal future of a.

In the discrete situation, the smallest length is the Planck length $l_p \sim 1.6 \times 10^{-35}$ and the smallest time interval is $l_t \sim 5.4 \times 10^{-43}$

We call a finite poset a ‘causet

$\mathbb{P}_m$ = All causets of cardinality m.

$\mathbb{P} = \bigcup_m \mathbb{P}_m$

If $a, we call $a$ the ancestor and $b$ the successor.

If $a, we call $a$ the parent and $b$ the child if
$!\exists c \ni a

$a$ is maximal if $!\exists b \ni a

$X \in \mathbb{P}_m, Y \in \mathbb{P}_{m+1}$

X produces Y if Y is obtained from X by adjoining a single element to X that is maximal in Y. We call X the producer and Y the offspring.

A path in $\mathbb{P}$ is a string
$\omega= \omega_1 \omega_2 \ldots \ni \omega_i \in \mathbb{P}_i, \omega_{i+1} \in \mathbb{P}_{i+1}$

Each such path represents a ‘universe’. Every $l_t$ unit of time we have a new element in the path. Note that we have 2 notions of time here, the ‘chronological time’ and the ‘geometric time’.

An n-path is
$\omega= \omega_1 \omega_2 \ldots \omega_n$

$\Omega = \{\omega|\omega= \omega_1 \omega_2 \ldots \}$

$\Omega_n = \{\omega|\omega= \omega_1 \omega_2 \ldots \omega_n \}$

$Cylinder(\omega_0) = \{\omega \in \Omega| \omega= \omega_0 \ldots\}$

For $A \subseteq \Omega_n, cyl(A)= \bigcup_{\omega \in A} cyl(\omega)$

$a_n = \{cyl(A): A \subseteq \Omega_n\}$

We have the heirarchy
$a_1 \subseteq a_2 \subseteq a_3\subseteq \ldots$

$C(\Omega) = \bigcup a_n.$ This is an algebra of the subsets of $\Omega.$

If $X \rightarrow Y$ in $r$ isomorphic ways,
we write $m(X \rightarrow Y) = r$, where $m$ is the multiplicity.

Source: DQG by Stan Gudder

Classical Sequential Growth Processes

$C= (C_0, C_1, \ldots)$
$C_i \geq 0$ are the coupling constants.

For $r\leq s \in \mathbb{N}$ define
$\lambda_c(s,r) = \displaystyle\sum\limits_{k=0}^s \begin{pmatrix} s-r\\k-r\end{pmatrix} C_k$

$X \in \mathbb{P}_m, Y \in \mathbb{P}_{m+1}, X \rightarrow Y$

We define the transition probability as

$p_c(X \rightarrow Y) = m(X \rightarrow Y) \frac{\lambda_c(\alpha, \pi)}{\lambda_c(m, 0}$

where
$\alpha$ is the number of ancestors
$\pi$ is the number of elements adjoined to $X$ to get $Y$.

It is not obvious that this is a probability but this can be shown.

For the part of the talk from here onward, I will just sketch the outline here. We can define a corresponding quantum sequential growth process which leads to a theory of quantum gravity. I would encourage interested reader to read the original papers listed below.