Symmetry

Research

17 pages, 955 KiB

Open AccessArticle

Maze Solving by a Quantum Walk with Sinks and Self-Loops: Numerical Analysis

by Leo Matsuoka, Kenta Yuki, Hynek Lavička and Etsuo Segawa

Symmetry 2021, 13(12), 2263; https://doi.org/10.3390/sym13122263 - 28 Nov 2021

Cited by 1 | Viewed by 2841

Maze-solving by natural phenomena is a symbolic result of the autonomous optimization induced by a natural system. We present a method for finding the shortest path on a maze consisting of a bipartite graph using a discrete-time quantum walk, which is a toy [...] Read more.

Maze-solving by natural phenomena is a symbolic result of the autonomous optimization induced by a natural system. We present a method for finding the shortest path on a maze consisting of a bipartite graph using a discrete-time quantum walk, which is a toy model of many kinds of quantum systems. By evolving the amplitude distribution according to the quantum walk on a kind of network with sinks, which is the exit of the amplitude, the amplitude distribution remains eternally on the paths between two self-loops indicating the start and the goal of the maze. We performed a numerical analysis of some simple cases and found that the shortest paths were detected by the chain of the maximum trapped densities in most cases of bipartite graphs. The counterintuitive dependence of the convergence steps on the size of the structure of the network was observed in some cases, implying that the asymmetry of the network accelerates or decelerates the convergence process. The relation between the amplitude remaining and distance of the path is also discussed briefly. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

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13 pages, 2577 KiB

Open AccessArticle

Description of Dressed-Photon Dynamics and Extraction Process

by Suguru Sangu and Hayato Saigo

Symmetry 2021, 13(10), 1768; https://doi.org/10.3390/sym13101768 - 23 Sep 2021

Viewed by 1314

Abstract

Several interesting physical phenomena and industrial applications explained by the dressed photon have been reported in recent years. These require a novel concept in an off-shell science that deviates from the conventional optics, satisfying energy and momentum conservation laws. In this paper, starting [...] Read more.

Several interesting physical phenomena and industrial applications explained by the dressed photon have been reported in recent years. These require a novel concept in an off-shell science that deviates from the conventional optics, satisfying energy and momentum conservation laws. In this paper, starting from an original model that captures dressed-photon characteristics phenomenologically, the dynamics of the dressed photon in a nanomatter system and the mechanism for extracting internal degrees of freedom of the dressed photon to an external space have been examined by theoretical and numerical approaches. Our proposal is that basis states of the dressed photon can be transformed to the form that reflects the spatial distribution of the dressed-photon steady state in the system, and some of basis states with predetermined spatial distribution can relate to the dissipation components in the external space by means of the renormalization technique. From the results of numerical simulation, it is found that quasi-static states are regarded as the photon with light mass or massless, and the extraction of active states strongly affects the spatial distribution in a new steady state. The concept for extracting dressed-photon energy to an external space will contribute to a detailed understanding of dressed-photon physics and future industrial applications. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

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9 pages, 13485 KiB

Open AccessArticle

Investigation of Eigenmode-Based Coupled Oscillator Solver Applied to Ising Spin Problems

by Shintaro Murakami, Okuto Ikeda, Yusuke Hirukawa and Toshiharu Saiki

Symmetry 2021, 13(9), 1745; https://doi.org/10.3390/sym13091745 - 19 Sep 2021

Viewed by 1625

Abstract

We evaluate a coupled oscillator solver by applying it to square lattice (N × N) Ising spin problems for N values up to 50. The Ising problems are converted to a classical coupled oscillator model that includes both positive (ferromagnetic-like) and negative (antiferromagnetic-like) [...] Read more.

We evaluate a coupled oscillator solver by applying it to square lattice (N × N) Ising spin problems for N values up to 50. The Ising problems are converted to a classical coupled oscillator model that includes both positive (ferromagnetic-like) and negative (antiferromagnetic-like) coupling between neighboring oscillators (i.e., they are reduced to eigenmode problems). A map of the oscillation amplitudes of lower-frequency eigenmodes enables us to visualize oscillator clusters with a low frustration density (unfrustrated clusters). We found that frustration tends to localize at the boundary between unfrustrated clusters due to the symmetric and asymmetric nature of the eigenmodes. This allows us to reduce frustration simply by flipping the sign of the amplitude of oscillators around which frustrated couplings are highly localized. For problems with N = 20 to 50, the best solutions with an accuracy of 96% (with respect to the exact ground state) can be obtained by simply checking the lowest ~N/2 candidate eigenmodes. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

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17 pages, 311 KiB

Open AccessArticle

Quantum Fields as Category Algebras

by Hayato Saigo

Symmetry 2021, 13(9), 1727; https://doi.org/10.3390/sym13091727 - 17 Sep 2021

Cited by 2 | Viewed by 2030

Abstract

In the present paper, we propose a new approach to quantum fields in terms of category algebras and states on categories. We define quantum fields and their states as category algebras and states on causal categories with partial involution structures. By utilizing category [...] Read more.

In the present paper, we propose a new approach to quantum fields in terms of category algebras and states on categories. We define quantum fields and their states as category algebras and states on causal categories with partial involution structures. By utilizing category algebras and states on categories instead of simply considering categories, we can directly integrate relativity as a category theoretic structure and quantumness as a noncommutative probabilistic structure. Conceptual relationships with conventional approaches to quantum fields, including Algebraic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT), are also be discussed. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

9 pages, 267 KiB

Open AccessFeature PaperArticle

Categorical Nonstandard Analysis

by Hayato Saigo and Juzo Nohmi

Symmetry 2021, 13(9), 1573; https://doi.org/10.3390/sym13091573 - 26 Aug 2021

Cited by 1 | Viewed by 2074

Abstract

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use [...] Read more.

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor

U

on a topos of sets S together with a natural transformation

υ

, instead of the terms as “standard”, “internal”, or “external”. Moreover, we propose a general notion of a space called

U

-space, and the category

U S p a c e

whose objects are

U

-spaces and morphisms are functions called

U

-spatial morphisms. The category

U S p a c e

, which is shown to be Cartesian closed, gives a unified viewpoint toward topological and coarse geometric structure. It will also be useful to further study symmetries/asymmetries of the systems with infinite degrees of freedom, such as quantum fields. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

8 pages, 247 KiB

Open AccessArticle

Symmetry of Dressed Photon

by Hiroyuki Ochiai

Symmetry 2021, 13(7), 1283; https://doi.org/10.3390/sym13071283 - 16 Jul 2021

Cited by 1 | Viewed by 1566

Abstract

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters

n \geq r \geq s

on dimensions. We discuss the symmetry on these [...] Read more.

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters

n \geq r \geq s

on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with

n = 4, r = 2, s = 1

. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

19 pages, 343 KiB

Open AccessArticle

Towards a Measurement Theory for Off-Shell Quantum Fields

by Kazuya Okamura

Symmetry 2021, 13(7), 1183; https://doi.org/10.3390/sym13071183 - 30 Jun 2021

Cited by 1 | Viewed by 1593

Abstract

In this study, we develop quantum measurement theory for quantum systems described by C

^{*}

-algebras. This is the first step to establish measurement theory for interacting quantum fields with off-shell momenta. Unlike quantum mechanics (i.e., quantum systems with finite degrees of freedom), [...] Read more.

In this study, we develop quantum measurement theory for quantum systems described by C

^{*}

-algebras. This is the first step to establish measurement theory for interacting quantum fields with off-shell momenta. Unlike quantum mechanics (i.e., quantum systems with finite degrees of freedom), measurement theory for quantum fields is still in development because of the difficulty of quantum fields that are typical quantum systems with infinite degrees of freedom. Furthermore, the mathematical theory of quantum measurement is formulated in the von Neumann algebraic setting in previous studies. In the paper, we aim to extend the applicable area of quantum measurement theory to quantum systems described by C

^{*}

-algebras from a mathematical viewpoint, referring to the sector theory that is related to symmetry and based on the theory of integral decomposition of states. In particular, we define central subspaces of the dual space of a C

^{*}

-algebra and use them to define instruments. This attempt makes the connection between measurement theory and sector theory explicit and enables us to understand the macroscopic nature and the physical meaning of measurement. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

12 pages, 259 KiB

Open AccessArticle

Category Algebras and States on Categories

by Hayato Saigo

Symmetry 2021, 13(7), 1172; https://doi.org/10.3390/sym13071172 - 29 Jun 2021

Cited by 4 | Viewed by 2225

Abstract

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional [...] Read more.

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of

^{†}

-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

21 pages, 810 KiB

Open AccessArticle

Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs

by Norio Konno, Etsuo Segawa and Martin Štefaňák

Symmetry 2021, 13(7), 1169; https://doi.org/10.3390/sym13071169 - 29 Jun 2021

Cited by 5 | Viewed by 1636

Abstract

We connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace [...] Read more.

We connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace of the Grover walk is the attractor eigenspace of the Grover walk with sinks. It is described by the persistent eigenspace of the underlying random walk whose support has no overlap to the boundaries of the graph and combinatorial flow in graph theory. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

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15 pages, 312 KiB

Open AccessArticle

A Discontinuity of the Energy of Quantum Walk in Impurities

by Kenta Higuchi, Takashi Komatsu, Norio Konno, Hisashi Morioka and Etsuo Segawa

Symmetry 2021, 13(7), 1134; https://doi.org/10.3390/sym13071134 - 24 Jun 2021

Cited by 2 | Viewed by 1694

Abstract

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region

{0, 1, \dots, M - 1}

and free at the other positions. We obtain the stationary state [...] Read more.

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region

{0, 1, \dots, M - 1}

and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow

ω^{n}

at time n

(| ω | = 1)

. From this expression, we compute the scattering on the surface of

- 1

and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

19 pages, 392 KiB

Open AccessArticle

On the Dressed Photon Constant and Its Implication for a Novel Perspective on Cosmology

by Hirofumi Sakuma and Izumi Ojima

Symmetry 2021, 13(4), 593; https://doi.org/10.3390/sym13040593 - 2 Apr 2021

Cited by 2 | Viewed by 2515

Abstract

As an important follow-up report on the latest study of the first author (H.S.) on an off-shell quantum field causing a dressed photon and dark energy, we further discuss a couple of intriguing subjects based on the new notion of simultaneous conformal symmetry [...] Read more.

As an important follow-up report on the latest study of the first author (H.S.) on an off-shell quantum field causing a dressed photon and dark energy, we further discuss a couple of intriguing subjects based on the new notion of simultaneous conformal symmetry breaking. One is the dressed photon constant. If we use it, in addition to

\bar{h}

and c, as the third component of natural units, it is defined as the geometric mean of the smallest and the largest lengths: Planck length and that relating to the cosmological constant. Interestingly, this length (≈50 nanometers) seems to give a rough measure of the Heisenberg cut for electromagnetic phenomena. The other is a new perspective on cosmology that combines two original notions, i.e., twin universes and conformal cyclic cosmology, proposed, respectively, by Petit and Penrose, into one novel picture where universes expand self-similarly. We show the possibility that twin universes having a dual structure of (matter with (dark energy and matter)) vs. corresponding anti-entities, separated by an event horizon embedded in the geometric structure of de Sitter space, undergo endless cyclic processes of birth and death, as in the case of the pair creation and annihilation of elementary particles through the intervention of a conformal light field. Full article

(This article belongs to the Special Issue Quantum Fields and Off-Shell Sciences)

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