# Dirac, Bohm and the Algebraic Approach

## TL;DR

This paper explores the algebraic foundations of quantum mechanics, linking Dirac's and Bohm's approaches, and proposes a formalism that relates quantum trajectories to classical mechanics and fractal space-time structures.

## Contribution

It introduces an algebraic formalism connecting Dirac's and Bohm's methods, showing how Bohm trajectories emerge from stochastic Feynman paths and suggesting a fractal space-time framework.

## Key findings

- Bohm trajectories are averages of stochastic Feynman paths.
- The algebraic formalism relates quantum and classical mechanics without decoherence.
- The approach indicates a possible underlying fractal space-time structure.

## Abstract

In this paper we show how Dirac, in 1947, anticipated the Bohm approach using an argument based on what is now called the Heisenberg picture. From a detailed examination of these ideas, we show that the role played by the Dirac standard ket is equivalent to the introduction and use of the idempotent in the orthogonal and symplectic Clifford algebras. This formalism is then used to show that the so-called `Bohm trajectories' are the average of an ensemble of individual stochastic Feynman paths. Since the Bohm approach can be simply reduced to classical mechanics, the algebraic formalism presented here provides a natural way to relate quantum mechanics to classical mechanics without the need for decoherence. We show that this approach suggests an underlying fractal space-time of the type discussed by Nottale.