# Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension

## Abstract

We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of homogeneity, parity and time-reversal invariance. The comparison between the automaton and the Dirac evolutions is rigorously set as a discrimination problem between unitary channels. We derive an exact lower bound for the probability of error in the discrimination as an explicit function of the mass, the number and the momentum of the particles, and the duration of the evolution. Computing this bound with experimentally achievable values, we see that in that regime the QCA model cannot be discriminated from the usual Dirac evolution. Finally, we show that the evolution of one-particle states with narrow-band in momentum can be efficiently simulated by a dispersive differential equation for any regime. This analysis allows for a comparison with the dynamics of wave-packets as it is described by the usual Dirac equation. This paper is a first step in exploring the idea that quantum field theory could be grounded on a more fundamental quantum cellular automaton model and that physical dynamics could emerge from quantum information processing.more »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22447603

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics; Journal Volume: 354; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIRAC EQUATION; GAUGE INVARIANCE; ONE-DIMENSIONAL CALCULATIONS; PARITY; QUANTUM FIELD THEORY; QUANTUM INFORMATION; QUANTUM SYSTEMS; T INVARIANCE

### Citation Formats

```
Bisio, Alessandro, D’Ariano, Giacomo Mauro, and Tosini, Alessandro, E-mail: alessandro.tosini@unipv.it.
```*Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension*. United States: N. p., 2015.
Web. doi:10.1016/J.AOP.2014.12.016.

```
Bisio, Alessandro, D’Ariano, Giacomo Mauro, & Tosini, Alessandro, E-mail: alessandro.tosini@unipv.it.
```*Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension*. United States. doi:10.1016/J.AOP.2014.12.016.

```
Bisio, Alessandro, D’Ariano, Giacomo Mauro, and Tosini, Alessandro, E-mail: alessandro.tosini@unipv.it. Sun .
"Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension". United States.
doi:10.1016/J.AOP.2014.12.016.
```

```
@article{osti_22447603,
```

title = {Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension},

author = {Bisio, Alessandro and D’Ariano, Giacomo Mauro and Tosini, Alessandro, E-mail: alessandro.tosini@unipv.it},

abstractNote = {We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of homogeneity, parity and time-reversal invariance. The comparison between the automaton and the Dirac evolutions is rigorously set as a discrimination problem between unitary channels. We derive an exact lower bound for the probability of error in the discrimination as an explicit function of the mass, the number and the momentum of the particles, and the duration of the evolution. Computing this bound with experimentally achievable values, we see that in that regime the QCA model cannot be discriminated from the usual Dirac evolution. Finally, we show that the evolution of one-particle states with narrow-band in momentum can be efficiently simulated by a dispersive differential equation for any regime. This analysis allows for a comparison with the dynamics of wave-packets as it is described by the usual Dirac equation. This paper is a first step in exploring the idea that quantum field theory could be grounded on a more fundamental quantum cellular automaton model and that physical dynamics could emerge from quantum information processing. In this framework, the discretization is a central ingredient and not only a tool for performing non-perturbative calculation as in lattice gauge theory. The automaton model, endowed with a precise notion of local observables and a full probabilistic interpretation, could lead to a coherent unification of a hypothetical discrete Planck scale with the usual Fermi scale of high-energy physics. - Highlights: • The free Dirac field in one space dimension as a quantum cellular automaton. • Large scale limit of the automaton and the emergence of the Dirac equation. • Dispersive differential equation for the evolution of smooth states on the automaton. • Optimal discrimination between the automaton evolution and the Dirac equation.},

doi = {10.1016/J.AOP.2014.12.016},

journal = {Annals of Physics},

number = ,

volume = 354,

place = {United States},

year = {Sun Mar 15 00:00:00 EDT 2015},

month = {Sun Mar 15 00:00:00 EDT 2015}

}