## Tesi etd-09222015-103202 |

Thesis type

Tesi di laurea magistrale

Author

ANDREONI, ALESSIO

URN

etd-09222015-103202

Title

Relations between Hamiltonian Cellular Automata and Quantum Mechanics of composite systems

Struttura

FISICA

Corso di studi

FISICA

Supervisors

**relatore**Prof. Elze, Hans Thomas

Parole chiave

- composite
- quantum
- automata
- cellular

Data inizio appello

19/10/2015;

Consultabilità

Completa

Riassunto analitico

A Cellular Automaton (CA) is an idealization of a physical system in which space and time are discrete and the physical quantities take only a finite set of values. The concept of Cellular Automata dates back to the late 1940s and is due to John von Neumann.

They are used in many different fields, such as equilibrium and non-equilibrium statistical physics, application-oriented problems, biology, sociology, chemistry, and others.

Moreover, recently, the possibility has been explored that CAs can be used to build more or less foundamental hidden variables theories that could explain certain (if not all) features of quantum mechanics.

In this context, we want to study the properties of a class of CAs introduced by Prof. Hans-Thomas Elze, called Hamiltonian Cellular Automata (HCA). In particular, we will concentrate on apparent similarities with quantum mechanics. We will see that a HCA follows updating equations that represent a discretized analogue of Schr\"odinger's equation and obey conservation laws similar to those of quantum mechanics. We will then study the space of states of these systems finding that the degrees of freedom are doubled w.r.t. those of the corresponding quantum mechanical system. This is due to the fact that the discretized updating equations is of second kind, thus requiring twice as many initial conditions then the Schr\"odinger equation.

Once we have introduced an HCA and studied its main properties, we will consider the composition of two of them. It is possible to do this in various different ways: first, following the composition rule of classical systems and second, trying to mimick as close as possible the quantum mechanical procedure, thus, using the tensor product between their spaces of states.

Then, in the second case, we will look for entangled and non-entangled CA states, and for time-evolution operators which can give rise to entanglement starting from non-entangled states. We will see that as in quantum mechanics we can find non-interacting time-evolution operators that do not introduce entanglement between two subsystems, while the introduction of an interaction can make a non-entangled state evolve into an entangled one. We will illustrate this with some numerical evaluation.

They are used in many different fields, such as equilibrium and non-equilibrium statistical physics, application-oriented problems, biology, sociology, chemistry, and others.

Moreover, recently, the possibility has been explored that CAs can be used to build more or less foundamental hidden variables theories that could explain certain (if not all) features of quantum mechanics.

In this context, we want to study the properties of a class of CAs introduced by Prof. Hans-Thomas Elze, called Hamiltonian Cellular Automata (HCA). In particular, we will concentrate on apparent similarities with quantum mechanics. We will see that a HCA follows updating equations that represent a discretized analogue of Schr\"odinger's equation and obey conservation laws similar to those of quantum mechanics. We will then study the space of states of these systems finding that the degrees of freedom are doubled w.r.t. those of the corresponding quantum mechanical system. This is due to the fact that the discretized updating equations is of second kind, thus requiring twice as many initial conditions then the Schr\"odinger equation.

Once we have introduced an HCA and studied its main properties, we will consider the composition of two of them. It is possible to do this in various different ways: first, following the composition rule of classical systems and second, trying to mimick as close as possible the quantum mechanical procedure, thus, using the tensor product between their spaces of states.

Then, in the second case, we will look for entangled and non-entangled CA states, and for time-evolution operators which can give rise to entanglement starting from non-entangled states. We will see that as in quantum mechanics we can find non-interacting time-evolution operators that do not introduce entanglement between two subsystems, while the introduction of an interaction can make a non-entangled state evolve into an entangled one. We will illustrate this with some numerical evaluation.

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