## Discrete Quantum Gravity

–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.

**Introduction and definitions**

We talk about the causal structure of a Lorentzian spacetime . is a partially ordered set(Poset).

For , we say if b is in the causal future of a.

In the discrete situation, the smallest length is the Planck length and the smallest time interval is

We call a finite poset a ‘* causet*‘

= All causets of cardinality m.

If , we call the ancestor and the successor.

If , we call the parent and the child if

is maximal if

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 is a string

Each such path represents a ‘universe’. Every 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

For

We have the heirarchy

This is an algebra of the subsets of

If in isomorphic ways,

we write , where is the multiplicity.

**Classical Sequential Growth Processes**

are the coupling constants.

For define

We define the* transition probability *as

where

is the number of ancestors

is the number of elements adjoined to to get .

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.

**Further readings and references-**