Transposition Symmetry Research Articles

Transfer matrix study of the Anderson transition in non-Hermitian systems

In this paper, we present in-depth transfer matrix analyses of the Anderson transition in three non-Hermitian (NH) systems, NH Anderson, U(1) and Peierls models, that belong to NH class AI$^{\dagger}$ or NH class A. We first argue a general validity of the transfer matrix analysis, and clarify the symmetry properties of the Lyapunov exponents, scattering ($S$) matrix and two-terminal conductance in these NH models. The unitarity of the $S$ matrix is violated in NH systems, where the two-terminal conductance can take arbitrarily large values. Nonetheless, we show that the transposition symmetry of a Hamiltonian leads to the symmetric nature of the $S$ matrix as well as the reciprocal symmetries of the Lyapunov exponents and conductance in certain ways in these NH models. Finite size scaling data are fitted by polynomial functions, from which we determine the critical exponent $\nu$ at different single-particle energies and system parameters of the NH models. We conclude that the critical exponents of the NH class AI$^{\dagger}$ and the NH class A are $\nu=1.19 \pm 0.01$ and $\nu=1.00 \pm 0.04$, respectively. In the NH models, a distribution of the two-terminal conductance is not Gaussian. Instead, it contains small fractions of huge conductance values, which come from rare-event states with huge transmissions amplified by on-site NH disorders. Nonetheless, a geometric mean of the conductance enables the finite-size scaling analysis. We show that the critical exponents obtained from the conductance analysis are consistent with those from the localization length in these three NH models.

The Bose-Einstein statistics: Remarks on Debye, Natanson, and Ehrenfest contributions and the emergence of indistinguishability principle for quantum particles

The principal mathematical idea behind the statistical properties of black-body radiation (photons) was introduced already by L. Boltzmann (1877/2015) and used by M. Planck (1900; 1906) to derive the frequency distribution of radiation (Planck’s law) when its discrete (quantum) structure was additionally added to the reasoning. The fundamental physical idea – the principle of indistinguishability of the quanta (photons) – had been somewhat hidden behind the formalism and evolved slowly. Here the role of P. Debye (1910), H. Kamerlingh Onnes and P. Ehrenfest (1914) is briefly elaborated and the crucial role of W. Natanson (1911a; 1911b; 1913) is emphasized. The reintroduction of this Natanson’s statistics by S. N. Bose (1924/2009) for light quanta (called photons since the late 1920s), and its subsequent generalization to material particles by A. Einstein (1924; 1925) is regarded as the most direct and transparent, but involves the concept of grand canonical ensemble of J. W. Gibbs (1902/1981), which in a way obscures the indistinguishability of the particles involved. It was ingeniously reintroduced by P. A. M. Dirac (1926) via postulating (imposing) the transposition symmetry onto the many-particle wave function. The above statements are discussed in this paper, including the recent idea of the author (Spałek 2020) of transformation (transmutation) – under specific conditions – of the indistinguishable particles into the corresponding to them distinguishable quantum particles. The last remark may serve as a form of the author’s post scriptum to the indistinguishability principle.

Universality classes of non-Hermitian random matrices

Non-Hermitian random matrices have been utilized in such diverse fields as dissipative and stochastic processes, mesoscopic physics, nuclear physics, and neural networks. However, the only known universal level-spacing statistics is that of the Ginibre ensemble characterized by complex-conjugation symmetry. Here we report our discovery of two other distinct universality classes characterized by transposition symmetry. We find that transposition symmetry alters repulsive interactions between two neighboring eigenvalues and deforms their spacing distribution. Such alteration is not possible with other symmetries including Ginibre's complex-conjugation symmetry which can affect only nonlocal correlations. Our results complete the non-Hermitian counterpart of Wigner-Dyson's threefold universal statistics of Hermitian random matrices and serve as a basis for characterizing nonintegrability and chaos in open quantum systems with symmetry.

Musical pitch quantization as an eigenvalue problem

How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.

Quantum approaches to music cognition

Quantum cognition emerged as an important discipline of mathematical psychology during the last two decades. Using abstract analogies between mental phenomena and the formal framework of physical quantum theory, quantum cognition demonstrated its ability to resolve several puzzles from cognitive psychology. Until now, quantum cognition essentially exploited ideas from projective (Hilbert space) geometry, such as quantum probability or quantum similarity. However, many powerful tools provided by physical quantum theory, e.g., symmetry groups have not been utilized in the field of quantum cognition research so far. Inspired by seminal work by Guerino Mazzola on the symmetries of tonal music, our study aims at elucidating and reconciling static and dynamic tonal attraction phenomena in music psychology within the quantum cognition framework. Based on the fundamental principles of octave equivalence, fifth similarity and transposition symmetry of tonal music that are reflected by the structure of the circle of fifths, we develop different wave function descriptions over this underlying tonal space. We present quantum models for static and dynamic tonal attraction and compare them with traditional computational models in musicology. Our approach replicates and also improves predictions based on symbolic models of music perception.

Improved PHARM for JPEG Steganalysis: Making PHARM More Efficient and Effective

The PHase-Aware pRojection Model (PHARM) feature set, built as phase-aware histograms of quantized projections obtained by convolving residuals with random matrices, achieves competitive detection performance against modern adaptive JPEG steganography while having significant computational cost. In this paper, we propose three improvements to the original PHARM, making it more efficient and effective. First, we reduce the maximum projection matrix size to decrease the computational complexity of convolution and better capture steganographic embedding changes. Second, we select more than one phase pair per projection to compute phase-aware histograms, thus correspondingly reducing the number of projections for each residual. Third, the transposition symmetry is also taken into consideration to make our features more robust while preserving the feature dimensionality. The numerous experiments are given to demonstrate the efficiency and effectiveness of our improved PHARM.

-symmetric quantum theory defined in a Krein space

We provide a mathematical framework for -symmetric quantum theory, which is applicable irrespective of whether a system is defined on or a complex contour, whether symmetry is unbroken, and so on. The linear space in which -symmetric quantum theory is naturally defined is a Krein space constructed by introducing an indefinite metric into a Hilbert space composed of square integrable complex functions in a complex contour. We show that in this Krein space every -symmetric operator is -Hermitian if and only if it has transposition symmetry as well, from which the characteristic properties of the -symmetric Hamiltonians found in the literature follow. Some possible ways to construct physical theories are discussed within the restriction to the class K(H).

Consistency of the minimal supersymmetric GUT spectra

We show that with proper accounting of convention-dependent phases the mass spectra evaluated by us in an earlier paper satisfy the trace, $SU(5)$ reassembly and Goldstone counting consistency checks. Phase accounting shows that the transposition symmetry called Hermiticity will be manifest only if signs arising from the product of six phase factors are reinserted. This uncovers the errors in the claims of others (retracted in subsequent work) concerning the inconsistency of our results. The chiral multiplet spectra of the two calculations are equivalent. However our method also gives all gauge and gauge chiral spectra as well as a decomposition of all $SO(10)$ minimal supersymmetric grand unified theory couplings, for both tensors and spinors, which are unavailable, even in principle, using the methods of the above papers.

Nonlinear supersymmetry in quantum mechanics: algebraic properties and differential representation

We study the nonlinear (polynomial, N-fold,…) supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Hamiltonian projection on the zero-mode subspace of supercharges. We show that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a nonlinear superalgebra which can be appropriately specified as N=2 SUSY. In this case we find a nontrivial hidden symmetry operator and rephrase it as a nonlinear function of the Hamiltonian on the physical state space. The full N=2 nonlinear SUSY algebra includes “central charges” both polynomial and nonpolynomial (due to a symmetry operator) in the Hamiltonian.

Komplexe des Typs [MCl4(SEt2)2] mit M = Mo und W. Kristallstrukturen, NMR-Spektren und magnetisches Verhalten/[MCl4(SEt2)2] Complexes with M = Mo and W. Crystal Structures, NMR Spectra, and Magnetic Behaviour

The crystal structures of the thioether complexes [MCl4(SEt2)2] with M = Mo and W have been solved by X-ray methods. Both compounds crystallize isotypically in the triclinic space group P1̄ with two formula units per cell unit. The metal atoms are octahedrally coordinated by four chlorine atoms and by the two sulfur atoms of the thioether molecules in transposition (symmetry Ci) with bond lengths (average): Mo-Cl 233.1, Mo-S 253.4, W-Cl 233.1, and W-S 251.7 pm. Both complexes were also characterized by 1H and 13C NMR spectroscopy as well as by measurement of the magnetic susceptibilities in the temperature range from 1.8 to 350 K.

Uniqueness of Parisi's scheme for replica symmetry breaking

Replica symmetry breaking in spin glass models is investigated using elements of the theory of permutation groups. It is shown how the various types of symmetry breaking give rise to special algebras and that Parisi's scheme may be uniquely characterized by two simple conditions on these algebras, namely transposition symmetry and simple extensibility. An alternative to the Parisi scheme is shown to be unacceptable.

A q-analog of transposition symmetry for invariant G-functions

We give an algebraic construction of Milne's v μ [ G] m ( n) ( γ; δ; z) functions in terms of the matrix coefficients of certain linear operators B μ; z . We show that the v μ [ G] m ( n) ( γ; δ; z) functions all satisfy “transposition” symmetry if and only if z = ( t,…, t) ∈ C n , unless v μ [ G] 1 ( n) ( γ; δ; z) is identically zero. This transposition symmetry is a “ q-analog” of that for the ordinary v μ [ G] m ( n) ( γ; δ;) polynomials of Biedenharn, Gustafson, and Milne. We also define a “ q-analog” of the elementary reduced Wigner coefficients for U( n) and give both a combinatorial and representation theoretic interpretation of this definition.

A new symmetry for Biedenharn's G-functions and classical hypergeometric series

We show that the general bisymmetric polynomials m μ G ( n) q ( γ 1,…, γ n ; δ 1,…, δ m ) are a limiting case of the bisymmetric, invariant polynomials n μ G ( n) q ( γ 1,…, γ n ; δ 1,…, δ n ) which characterize U( n) tensor operators 〈 p, q,…, q, 0, …, 0〉. By taking suitable limits of a pair of difference equations for n μ G ( n) q ( γ; δ) we then deduce “transposition symmetry” for m μ G ( n) q ( γ; δ) from the same symmetry for n μ G ( n) q ( γ; δ). As an application of transposition symmetry for m μ G ( n) q ( γ; δ) we derive an elegant, new contiguous relation for classical, well-poised hypergeometric series, and also prove an identity between these series and multiple hypergeometric series well-poised in SU( n).

U( n) Wigner coefficients, the path sum formula, and invariant G-functions

We prove the path sum formula for computing the U( n) invariant denominator functions associated to stretched U( n) Wigner operators. A family of U( n) invariant polynomials G [ λ] ( n) is then defined which generalize the μ G q ( n) polynomials previously studied. The G [ λ] ( n) polynomials are shown to satisfy a number of difference equations and have symmetry properties similar to the μ G q ( n) polynomials. We also give a direct proof of the important transposition symmetry for the G [ λ] ( n) polynomials. To enable the non-specialist to understand the foundations for these remarkable polynomials, we provide an exposition of the boson calculus and the construction of the multiplicity-free U( n) Wigner operators.

A new symmetry related to SU( n) for classical basic hypergeometric series

A direct proof is given of an elegant new contiguous relation for classical, well-poised basic hypergeometric series which preserves the well-poised condition. The proof involves elementary series manipulations and does not depend upon the “transposition symmetry” of the general bisymmetric polynomials m μ G ( n) q ( γ 1,…, γ n ; δ 1,…, δ m ) which was used to establish the ordinary or “ q = 1” case of the identity. The new continguous relation can be considered a generalization of the 6 Φ 5 summation theorem.

The application of Regge symmetry to the 9-j symbol

The Regge transposition symmetry is applied to one of the six Wigner coefficients in the defining expression for the Wigner (9-j) symbol; couplings are then performed, and an expression for the (9-j) symbol is obtained which factorizes the sums of the angular momenta in two rows and their differences into different factors.