Quantum ${{\mathbb{F}}_{{\rm un}}}$: theq= 1 limit of Galois field quantum mechanics, projective geometry and the field with one element

Abstract

We argue that the q = 1 limit of Galois Field Quantum Mechanics, which was constructed on a vector space over the Galois Field Fq = GF (q), corresponds to its 'classical limit,' where superposition of states is disallowed. The limit preserves the projective geometry nature of the state space, and can be understood as being constructed on an appropriately defined analogue of a 'vector' space over the 'field with one element' F 1 .

Interference and Oscillation in Nambu Quantum Mechanics

“…Various proposals for alternative QM theories can be found in the literature. The field/division algebra over which the state space is constructed has been modified from C to R [10], H [11], O [12][13][14], and the finite fields F q [15][16][17][18][19][20][21]. Non-linear corrections to the Schrödinger equation have been considered by Weinberg [22,23].…”

Information Theoretic Characterization of Physical Theories with Projective State Space

“…We have then showed that any theory satisfying the assumptions is a probabilistic theory in which state spaces of physical systems are generalizations of quantum state spaces where complex numbers can be substituted with a generic field of numbers (more precisely with a generic division ring). In this family of theories classical theory is a degenerate or limit case in which the field of numbers is trivial and contains only one element [29]. A straightforward interpretation of this result is that superposition principle is not so uncommon among probabilistic theories; any probabilistic theory not displaying superposition principle of states (with coefficients pertaining to some division ring) either is classical or generate systems that cannot store information in a reliable way.…”

“…Hence physical systems of a probabilistic theory satisfying Assumptions 1,2, constitute a generalization of quantum systems in which superposition principle holds with coefficients not necessarily complex but belonging to a generic (skew) field of numbers K. In this family of theories we have a notable degenerate or limit case, the one in which the division ring is trivial and contains only one element. It is showed in [29] that this case corresponds to classical theory. It is interesting to note that in the framework of probabilistic theories, starting from assumptions inspired to information theoretic principles, we obtain a family of theories in which superposition principle is very common while its absence is an exceptional case.…”

“…Indeed given a set of pure states that forms a base for the vector space, constituting a set of perfectly distinguishable states of the system, any linear combination of these states represents an allowed pure state. Exploiting the work in [29] we will point out that classical theory can be seen as a degenerate or limit case among probabilistic theories satisfying the two assumptions with superposition principle being disabled.…”

Spekkens’ Toy Model, Finite Field Quantum Mechanics, and the Role of Linearity

“…Linearity, or the superposition principle, is what leads to the various mysteries of canonical QM, so it is a property one would like to impart on any model of QM. We argue that this can be done to Spekkens' model by mapping it to a QM defined over the finite field F 5 [3,4,5,6,7].…”