Indefinite Metric Research Articles

On Representations of Groups and Algebras in Spaces with Indefinite Metric

On Representations of Groups and Algebras in Spaces with Indefinite Metric

Different Aspects of Spin in Quantum Mechanics and General Relativity

In this paper, different aspects of the concept of spin are studied. The most well-established one is, of course, the quantum mechanical aspect: spin is a broken symmetry in the sense that the solutions of the Dirac equation tend to have directional properties that cannot be seen in the equation itself. It has been clear since the early days of quantum mechanics that this has something to do with the indefinite metric in Lorentz geometry, but the mechanism behind this connection is elusive. Although spin is not the same as rotation in the usual sense, there must certainly be a close relationship between these concepts. And, a possible way to investigate this connection is to instead start from the underlying geometry in general relativity. Is there a reason why rotating motion in Lorentz geometry should be more natural than non-rotating motion? In a certain sense, the answer turns out to be yes. But, it is by no means easy to see what this should correspond to in the usual quantum mechanical picture. On the other hand, it seems very unlikely that the similarities should be just coincidental. The interpretation of the author is that this can be a golden opportunity to investigate the interplay between these two theories.

Quantum reference frames for an indefinite metric

The current theories of quantum physics and general relativity on their own do not allow us to study situations in which the gravitational source is quantum. Here, we propose a strategy to determine the dynamics of objects in the presence of mass configurations in superposition, and hence an indefinite spacetime metric, using quantum reference frame (QRF) transformations. Specifically, we show that, as long as the mass configurations in the different branches are related via relative-distance-preserving transformations, one can use an extension of the current framework of QRFs to change to a frame in which the mass configuration becomes definite. Assuming covariance of dynamical laws under quantum coordinate transformations, this allows to use known physics to determine the dynamics. We apply this procedure to find the motion of a probe particle and the behavior of clocks near the mass configuration, and thus find the time dilation caused by a gravitating object in superposition. Comparison with other models shows that semi-classical gravity and gravitational collapse models do not obey the covariance of dynamical laws under quantum coordinate transformations.

A dilation theoretic approach to approximation by inner functions

AbstractUsing results from the theory of operators on a Hilbert space, we prove approximation results for matrix‐valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction and the realization formula for functions in the unit ball of . We first prove a generalization of a result of Carathéodory. This generalization has many applications. A uniform approximation result for matrix‐valued holomorphic functions which extend continuously to the unit circle is proved using the Potapov factorization. This generalizes a theorem due to Fisher. Approximation results are proved for matrix‐valued functions for whom a naturally associated kernel has finitely many negative squares. This uses the Krein–Langer factorization. Approximation results for ‐contractive meromorphic functions where induces an indefinite metric on are proved using the Potapov–Ginzburg theorem. Moreover, approximation results for holomorphic functions on the unit disc with values in certain other domains of interest are also proved.

On an Indefinite Metric on a Four-Dimensional Riemannian Manifold

Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive definite metric, the manifold is endowed with an additional tensor structure of type (1,1), whose fourth power is minus the identity. The additional structure is skew-circulant and compatible with the metric, such that an isometry is induced in every tangent space on the manifold. Both structures define an indefinite metric. With the help of the indefinite metric, we determine circles in different two-planes in the tangent space on the manifold. We also calculate the length and area of the circles. On a smooth closed curve, such as a circle, we define a vector force field. Further, we obtain the circulation of the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analog of the well-known Green’s formula in the Euclidean space.

Coquaternions, Metric Invariants of Biologic Systems and Malignant Transformations

Different hypotheses of carcinogenesis have been proposed based on local genetic factors and physiologic mechanisms. It is assumed that changes in the metric invariants of a biologic system (BS) determine the general mechanisms of cancer development. Numerous pieces of data demonstrate the existence of three invariant feedback patterns of BS: negative feedback (NFB), positive feedback (PFB) and reciprocal links (RL). These base patterns represent basis elements of a Lie algebra sl(2,R) and an imaginary part of coquaternion. Considering coquaternion as a model of a functional core of a BS, in this work a new geometric approach has been introduced. Based on this approach, conditions of the system are identified with the points of three families of hypersurfaces in R42: hyperboloids of one sheet, hyperboloids of two sheets and double cones. The obtained results also demonstrated the correspondence of an indefinite metric of coquaternion quadratic form with negative and positive entropy contributions of the base elements to the energy level of the system. From that, it can be further concluded that the anabolic states of the system will correspond to the points of a hyperboloid of one sheet, whereas catabolic conditions correspond to the points of a hyperboloid of two sheets. Equilibrium states will lie in a double cone. Physiologically anabolic and catabolic states dominate intermittently oscillating around the equilibrium. Deterioration of base elements increases positive entropy and causes domination of catabolic states, which is the main metabolic determinant of cancer. Based on these observations and the geometric representation of a BS’s behavior, it was shown that conditions related to cancer metabolic malfunction will have a tendency to remain inside the double cone.

Frames associated with an operator in spaces with an indefinite metric

<abstract><p>In the present paper, we study frames associated with an operator ($ \mathcal{W} $-frames) in Krein spaces, and we give the definition of frames associated with an operator depending on the adjoint of the operator in the Krein space (Definition 4.1). We prove that the definition given in [A. Mohammed, K. Samir, N. Bounader, K-frames for Krein spaces, <italic>Ann. Funct. Anal.</italic>, <bold>14</bold> (2023), 10.], which depends on the adjoint of the operator in the associated Hilbert space, is a consequence of our definition. We prove that our definition is independent of the fundamental decomposition (Theorem 4.1) and that having $ \mathcal{W} $-frames for the Krein space necessarily gives $ \mathcal{W} $-frames for the Hilbert spaces that compose the Krein space (Theorem 4.4). We also prove that orthogonal projectors generate new operators with their respective frames (Theorem 4.2). We prove an equivalence theorem for $ \mathcal{W} $-frames (Theorem 4.3), without depending on the fundamental symmetry as usually given in Hilbert spaces.</p></abstract>

Excitation Spectrum in an Ensemble of Hubbard Bosons

An approach that makes it possible to correctly derive equations describing the Bose–Einstein condensation and the spectrum of elementary excitations in the ensemble of Hubbard bosons in the strong correlation regime (U gg {text{|}}{{t}_{{fm}}}{text{|}}) has been developed in the atomic representation using the Dyson method with the introduced indefinite metric. The kinematic Dyson interaction caused by the properties of the commutation relations of dynamic variables plays an important role in such a system. An effective Hamiltonian has been obtained using the operator form of perturbation theory at finite U values. It has been shown that the properties of the ensemble of Hubbard bosons have been determined by the kinematic interaction, correlated hopping, and the attraction between Hubbard bosons. Numerical calculations have demonstrated the effect of these interactions on the characteristics of the energy spectrum of excitations of the ensemble of Hubbard bosons and on the dependence of the density of condensate particles on the density of bosons in the system.

Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design.

Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group which is a semi-Riemannian manifold, i.e. a manifold equipped with an indefinite metric. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower-dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transformations, which are the natural isometric transformations in hyperbolic spaces. This projection is computationally efficient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it allows for weight sharing. The nested hyperbolic space representation is the core component of our network and therefore, we first compare this representation - independent of the network - with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projection. Finally, we present experiments demonstrating comparative performance of our network on several publicly available data sets.

J-states and quantum channels between indefinite metric spaces

In the present work, we introduce and study the concepts of state and quantum channel on spaces equipped with an indefinite metric. Exclusively, we will limit our analysis to the matricial framework. As it will be confirmed below, from our research it is noticed that, when passing to the spaces with indefinite metric, the use of the adjoint of a matrix with respect to the indefinite metric is required in the construction of states and quantum channels; which prevents us to consider the space of matrices of certain order \(M_{n}({\mathbb {C}})\) as a \(C^{*}\)-algebra. In our case, this adjoint is defined through a J-metric, where the matrix J is a fundamental symmetry of \(M_{n}({\mathbb {C}})\). In our paper, for quantum operators, we include the general setting in the which, these operators map \(J_{1}\)-states into \(J_{2}\)-states, where \(J_{2}\ne \pm J_{1}\) are two arbitrary fundamental symmetries. In the middle of this program, we carry out a study of the completely positive maps between two different positive matrices spaces by considering two different indefinite metrics on \({\mathbb {C}}^{n}\).

Pseudo-hermitian random matrix models: General formalism

Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite dimensional subspace. As such, they can be used, in the usual spirit of random matrix theory, to model chaotic or disordered PT-symmetric quantum systems, or their gain-loss-balanced classical analogs, in the phase of broken PT-symmetry. The eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. In this paper we introduce a family of pseudo-hermitian random matrix models, depending parametrically on their metric. We apply the diagrammatic method to obtain its averaged resolvent and density of eigenvalues as explicit functions of the metric, in the limit of large matrix size N. As a concrete example, which is essentially an ensemble of elements of the non-compact unitary Lie algebra, we choose a particularly simple set of metrics, and compute the resulting resolvent and density of eigenvalues in closed form. The spectrum consists of a finite fraction of complex eigenvalues, which occupy uniformly two two-dimensional blobs, symmetric with respect to the real axis, as well as the complimentary fraction of real eigenvalues, condensed in a finite segment, with a known non-uniform density. The numbers of complex and real eigenvalues depend on the signature of the metric, that is, the numbers of its positive and negative eigenvalues. We have also carried thorough numerical analysis of the model for these particular metrics. Our numerical results converge rapidly towards the asymptotic analytical large-N expressions.

Indefinite Nilsolitons and Einstein Solvmanifolds

A nilsoliton is a nilpotent Lie algebra $\mathfrak{g}$ with a metric such that $\operatorname{Ric}=\lambda \operatorname{Id}+D$, with $D$ a derivation. For indefinite metrics, this determines four different geometries, according to whether $\lambda$ and $D$ are zero or not. We illustrate with examples the greater flexibility of the indefinite case compared to the Riemannian setting. We determine the algebraic properties that $D$ must satisfy when it is nonzero. For each of the four geometries, we show that under suitable assumptions it is possible to extend the nilsoliton metric to an Einstein solvmanifold of the form $\mathfrak{g}\rtimes \mathbb{R}^k$. Conversely, we introduce a large class of indefinite Einstein solvmanifolds of the form $\mathfrak{g}\rtimes \mathbb{R}^k$ that determine a nilsoliton metric on $\mathfrak{g}$ by restriction. We show with examples that, unlike in the Riemannian case, one cannot establish a correspondence between the full classes of Einstein solvmanifolds and nilsolitons.

Reproducing Kernel in Krein Spaces

This article describes a new form to introduce a reproducing kernel for a Krein space based on orthogonal projectors enabling to describe the kernel of a Krein space as the difference between the kernel of definite positive subspace and the kernel of definite negative subspace corresponding to kernel of the associated Hilbert space. As application, the authors obtain some basic properties of both kernels for Krein spaces and exhibit that each kernel is uniquely determined by the Krein space given. The methods and results employed generalize the notion of reproducing kernel given in Hilbert spaces to the context of spaces endowed with indefinite metric.

On Representations of Groups and Algebras in Spaces with Indefinite Metric

The paper contains a survey of known results on the structure of J-symmetric operator algebras in Pontryagin and Krein spaces, as well as on representations of groups and *-algebras in these spaces.

RACP: A network with attention corrected prototype for few-shot speaker recognition using indefinite distance metric

Few-shot speaker recognition task is to identify speakers from limited support samples. We argue that query samples and support samples are both informative for classification. To help Prototypical Networks capture information from query samples, this paper proposes the relation-based indefinite distance metric attentive correction prototype network (RACP). Since the mean prototype deviates from the ideal prototype, we calculate attention scores for each query sample to customize the attention prototype. Then, to compensate for the missed query samples information, the prototype is further refined by correction data that is constructed by combining query samples with the global class attention score. Later, the indefinite distance metric of Relation Networks is introduced on Prototypical Networks, and the relation scores between the sample prototypes and the query samples are calculated for final prediction. Compare with existing methods, RACP can consider both query samples and support samples instead of ignoring the query ones. We compare RACP with strong baselines (e.g. GMM-SVM, MAML, Prototypical Networks, Res32, and VGG11). Ablation study and generalizability study of different scenarios are also conducted on different datasets. Results show that RACP achieves better performance and generalization ability.

A de Rham decomposition type theorem for contact sub-Riemannian manifolds

In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field xi is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point qin M such that the holonomy group Psi (q) acts reducibly on H(q) yielding a decomposition H(q) = H_1(q)oplus cdots oplus H_m(q) into Psi (q)-irreducible factors. Using parallel transport we obtain the decomposition H = H_1oplus cdots oplus H_m of H into sub-distributions H_i. Unlike the Riemannian case, the distributions H_i are not integrable, however they induce integrable distributions Delta _i on M/xi , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that T(U/xi )=Delta _1oplus cdots oplus Delta _m, and the latter decomposition of T(U/xi ) induces the decomposition of U/xi into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.

How to choose the signature operator such that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable?

Using a discrete version of Floquet theory in a space with indefinite metric we reconstruct periodic pseudo-Jacobi matrices from given spectral data. We use two methods to characterize the signature operator H so that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable. As a by-product of the obtained results, we precisely polish some non-rigorous assumptions in the existing literature.

Quantum measurement of space-time events

The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength.

Specific Features of Cosquares of Special Matrices in Indefinite Metric Spaces

As is known, the cosquare of an arbitrary nonsingular Hermitian matrix is the identity matrix, and the cosquare of an arbitrary unitary matrix is unitary. In a complex linear space with the symplectic metric, the symplectic and skew-Hamiltonian matrices are the counterparts of the unitary and Hermitian matrices, respectively. Specific features of cosquares for these two matrix classes are indicated.

Successions of J-bessel in Spaces with Indefinite Metric

A definition of Bessel’s sequences in spaces with an indefinite metric is introduced as a generalization of Bessel’s sequences in Hilbert spaces. Moreover, a complete characterization of Bessel’s sequences in the Hilbert space associated to a space with an indefinite metric is given. The fundamental tools of Bessel’s sequences theory are described in the formalism of spaces with an indefinite metric. It is shown how to construct a Bessel’s sequences in spaces with an indefinite metric starting from a pair of Hilbert spaces, a condition is given to decompose a Bessel’s sequences into in spaces with an indefinite metric so that this decomposition generates a pair of Bessel’s sequences for the Hilbert spaces corresponding to the fundamental decomposition. In spaces where there was no norm, it seemed impossible to construct Bessel’s sequences. The fact that in [1] frame were constructed for Krein spaces motivated us to construct Bessel’s sequences for spaces of indefinite metric.