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Metod Saniga

Publications and source records attributed to Metod Saniga.

89 records · Page 5Linked to original sources
arXiv2005

Hjelmslev Geometry of Mutually Unbiased Bases

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space H\_q, q = p^r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p^2 and rank r. The q vectors of a basis of H\_q correspond to the q points of a (so-called) neighbour class and the q+1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.

math-ph↗
arXiv2005

Abstract algebra, projective geometry and time encoding of quantum information

Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic $p$. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo 4 are also becoming fashionable for their role in time encoding and mutual unbiasedness.

quant-ph↗
arXiv2004

Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes?

This note is a short elaboration of the conjecture of Saniga et al (J. Opt. B: Quantum Semiclass. 6 (2004) L19-L20) by regarding a set of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space, d being a power of a prime, as an analogue of an arc in a (Desarguesian) projective plane of order d. Complete sets of MUBs thus correspond to (d+1)-arcs, i.e., ovals. The existence of two principally distinct kinds of ovals for d even and greater than four, viz. conics and non-conics, implies the existence of two qualitatively different groups of the complete sets of MUBs for the Hilbert spaces of corresponding dimensions.

quant-ph↗
arXiv2004

On Cremonian Dimensions Qualitatively Different from Time and Space

We examine a particular kind of six-dimensional Cremonian universe featuring one dimension of space, three dimensions of time and other two dimensions that can*not* be ranked as either time or space. One of these two, generated by a one-parametric aggregate of (straight-)lines lying on a quadratic cone, is more similar to the spatial dimension. The other, represented by a singly-parametrical set of singular space quartic curves situated on a proper ruled quadric surface, bears more resemblance to time. Yet, the two dimensions differ profoundly from both time and space because, although being macroscopic, they are not accessible to (detectable by) every Cremonian observer. This toy-model thus demonstrates that there might exist extra-dimensions that need not necessarily be compactified to remain unobservable.

physics.gen-ph↗
arXiv2004

Mutually Unbiased Bases and Finite Projective Planes

It is conjectured that the question of the existence of projective planes whose order is not a power of prime is intimately linked with the problem whether there exists a set of d+1 mutually unbiased bases in a d-dimensional Hilbert space if d differs from a power of prime.

math-ph↗
arXiv2004

On an Observer-Related Unequivalence Between Spatial Dimensions of a Generic Cremonian Universe

Given a generic Cremonian space-time, its three spatial dimensions are shown to exhibit an intriguing, "two-plus-one" partition with respect to standard observers. Such observers are found to form three distinct, disjoint groups based on which one out of the three dimensions stands away from the other two. These two subject-related properties have, to our knowledge, no analogue in any of the existing physical theories of space-time.

physics.gen-ph↗
arXiv2004

Cremonian Space-Time(s) as an Emergent Phenomenon

It is shown that the notion of fundamental elements can be extended to_any_, i.e. not necessarily homaloidal, web of rational surfaces in a three-dimensional projective space. A Cremonian space-time can then be viewed as an_emergent_ phenomenon when the condition of "homaloidity" of the corresponding web is satisfied. The point is illustrated by a couple of particular types of "almost-homaloidal" webs of quadratic surfaces. In the first case, the quadrics have a line and two distinct points in common and the corresponding pseudo-Cremonian manifold is endowed with just two spatial dimensions. In the second case, the quadrics share six distinct points, no three of them collinear, that lie in quadruples in three different planes, and the corresponding pseudo-Cremonian configuration features three time dimensions. In both the cases, the limiting process of the emergence of generic Cremonian space-times is explicitly demonstrated.

physics.gen-ph↗
arXiv2003

The Psychopathological Fabric of Time (and Space) and Its Underpinning Pencil-Borne Geometries

The paper presents, to our knowledge, a first fairly comprehensive and mathematically well-underpinned classification of the psychopathology of time (and space). After reviewing the most illustrative first-person accounts of "anomalous/peculiar" experiences of time (and, to a lesser degree, also space) we introduce and describe in detail their algebraic geometrical model. The model features six qualitatively different types of the internal structure of time dimension and four types of that of space. As for time, the most pronounced are the ordinary "past-present-future," "present-only" ("eternal/everlasting now") and "no-present" (time "standing still") patterns; the remaining types represent an intriguing superposition of the three. Concerning space, the most elementary are the ordinary, i.e. "here-and-there," mode and the "here-only" one ("omnipresence"); the remaining two cases are again a specific mixture of the former two. We then show what the admissible combinations of temporal and spatial psycho-patterns are and give a rigorous algebraic geometrical classification of them. The paper ends with a brief account of important epistemological/ontological questions stemming from the approach.

physics.gen-ph↗
arXiv2003

On an Intriguing Signature-Reversal Exhibited by Cremonian Spacetimes

It is shown that a generic quadro-quartic Cremonian spacetime, which is endowed with one spatial and three time dimensions, can continuously evolve into a signature-reversed configuration, i.e. into the classical spacetime featuring one temporal and three space dimensions. An interesting cosmological implication of this finding is mentioned.

physics.gen-ph↗
arXiv2003

Geometry of Psychological Time

The paper reviews the most illustrative cases of the "peculiar/anomalous" experiences of time (and, to a lesser extent, also space) and discusses a simple algebraic geometrical model accounting for the most pronounced of them.

physics.gen-ph↗
arXiv2003

Conics, (q+1)-Arcs, Pencil Concept of Time and Psychopathology

It is demonstrated that in the (projective plane over) Galois fields GF(q) with q=2^n and n>2 (n being a positive integer) we can define, in addition to the temporal dimensions generated by pencils of conics, also time coordinates represented by aggregates of (q+1)-arcs that are not conics. The case is illustrated by a (self-dual) pencil of conics endowed with two singular conics of which one represents a double real line and the other is a real line pair. Although this pencil does not generate the ordinary (i.e., featuring the past, present and future) arrow of time over GF(2^n), there does exist a pencil-related family of (q+1)-arcs, not conics, that closely resembles such an arrow. Some psycho(patho)logical justifications of this finding are presented, based on the "peculiar/anomalous" experiences of time by a couple of schizophrenic patients.

math.AG↗
arXiv2003

Geometry of Time and Dimensionality of Space

One of the most distinguished features of our algebraic geometrical, pencil concept of space-time is the fact that spatial dimensions and time stand, as far as their intrinsic structure is concerned, on completely different footings: the former being represented by pencils of lines, the latter by a pencil of conics. As a consequence, we argue that even at the classical (macroscopic) level there exists a much more intricate and profound coupling between space and time than that dictated by (general) relativity theory. It is surmised that this coupling can be furnished by so-called Cremona (or birational) transformations between two projective spaces of three dimensions, being fully embodied in the structure of configurations of their fundamental elements. We review properties of some of the simplest Cremona transformations and show that the corresponding "fundamental" space-times exhibit an intimate connection between the extrinsic geometry of time dimension and the dimensionality of space. Moreover, these Cremonian space-times seem to provide us with a promising conceptual basis for the possible reconciliation between two extreme concepts of (space-)time, viz. physical and psychological. Some speculative remarks in this respect are made.

physics.gen-ph↗
arXiv2001

Arithmetic of Plane Cremona Transformations and the Dimensions of Transfinite Heterotic String Space-Time

It is shown that the two sequences of characteristic dimensions of transfinite heterotic string space-time found by El Naschie can be remarkably well accounted for in terms of the arithmetic of self-conjugate homaloidal nets of plane algebraic curves of orders 3 to 20. A firm algebraic geometrical justification is thus given not only for all the relevant dimensions of the classical theory, but also for other two dimensions proposed by El Naschie, viz. the inverse of quantum gravity coupling constant (~42.36067977) and that of (one half of) fine structure constant (~68.54101967). A non-trivial coupling between the two El Naschie sequences is also revealed.

physics.gen-ph↗
arXiv2000

A Note on a Specific Pencil of Conics in the Galois Fields of Order 2^n

The pencil of conics featuring three degenerate conics each of which is a line-pair is briefly inspected in a Galois field of characteristic two. It is shown that if two degenerates are conjugate imaginary line pairs, the third must be a real line pair; this contradicts Campbell's claim (Campbell 1927) that all the three singular conics are conjugate imaginary line pairs.

math.AG↗