Kinetic Approach to Lattice Quantum Mechanics - Cellular Automata - page 114

Kinetic Approach to Lattice Quantum
Mechanics
Sauro Succi
Istituto Applicazioni Calcolo, viale del Policlinico 137, 00161, Roma, Italy,
[e-mail] ,
http://www.iac.rm.cnr.it
Abstract. We discuss some lattice discretizations of the Klein-Gordon
equation inspired by analogies withdiscrete kinetic theory.
1 Introduction
Contemporary advances in theoretical physics are clearly indicating that the
task of recounciling gravity with quantum physics may require a dramatic swing
in our current picture of space-time. In particular, the notion of space-time as a
continuum is likely to be untenable at a scale where gravitational and electro-
weak interactions become comparable in strength [1]. As observed by G.'t Hooft,
[2], it is somewhat curious that while the continuum character of space-time is
openly questioned in modern quantum field theories, many of these theories still
heavily lean on the continuum formalism of partial differential equations.
Discrete lattices are routinely used in computational field theory [3], but, with
a few notable exceptions [4], mainly as mere numerical regulators of ultraviolet
divergences. Indeed, a major point of renormalization theories is precisely to
extract lattice-independent conclusions from the numerical computations.
Given the huge gap between the Planck length (about 10 − 33 cm) and the
smallest experimentally probed scales (about 10 − 16 cm) it seems worthwile to
investigate the consequences of taking the lattice no longer as mere computa-
tional device, but as a bona-fide discrete network whose links define the only pos-
sible propagation directions for signals carrying the interactions between fields
sitting on the nodes of the network. This viewpoint has been addressed before
in a series of thorough works exploring quantum mechanics and lattice gravity
near the continuum limit [5].
Here we offer a simple view of related issues from a different perspective,
namely discrete kinetic theory, in the hope that this additional angle may help
shedding further light into this fascinating and di 4 cult topic.
In particular, we shall show that relevant quantum mechanical equations,
such as the Schroedinger and Klein-Gordon equations, can be derived within
the framework of a discrete Boltzmann formalism in which quantum fields are
bound to move like classical (although complex) distribution functions along the
light-cones of a uniform discrete space-time.