Search Results (10)

A statistical manifold M can be defined as a Riemannian manifold each of whose points is a probability distribution on the same support. In fact, statistical manifolds possess a richer geometric structure beyond the Fisher information metric defined on the tangent bundle $TM$. Recognizing that points in M are distributions and not just generic points in a manifold, $TM$ can be extended to a Hilbert bundle $HM$. This extension proves fundamental when we generalize the classical notion of a point estimate—a single point in M—to a function on M that characterizes the relationship between observed data and each distribution in M. The log likelihood and score functions are important examples of generalized estimators. In terms of a parameterization $\theta :M\to \mathrm{\Theta }\subset {\mathbb{R}}^k$, $\widehat{\theta }$ is a distribution on $\mathrm{\Theta }$ while its generalization $g_{\widehat{\theta }}=\widehat{\theta }-E\widehat{\theta }$ as an estimate is a function over $\mathrm{\Theta }$ that indicates inconsistency between the model and data. As an estimator, $g_{\widehat{\theta }}$ is a distribution of functions. Geometric properties of these functions describe statistical properties of $g_{\widehat{\theta }}$. In particular, the expected slopes of $g_{\widehat{\theta }}$ are used to define $\mathrm{\Lambda }\left(g_{\widehat{\theta }}\right)$, the $\mathrm{\Lambda }$-information of $g_{\widehat{\theta }}$. The Fisher information I is an upper bound for the $\mathrm{\Lambda }$-information: for all g, $\mathrm{\Lambda }\left(g\right)\le I$. We demonstrate the utility of this geometric perspective using the two-sample problem. Full article

We reformulate holographic space–time models in terms of Hilbert bundles over the space of the time-like geodesics in a Lorentzian manifold. This reformulation resolves the issue of the action of non-compact isometry groups on finite-dimensional Hilbert spaces. Following Jacobson, I view the background geometry as a hydrodynamic flow, whose connection to an underlying quantum system follows from the Bekenstein–Hawking relation between area and entropy, generalized to arbitrary causal diamonds. The time-like geodesics are equivalent to the nested sequences of causal diamonds, and the area of the holoscreen (The holoscreen is the maximal $d-2$ volume (“area”) leaf of a null foliation of the diamond boundary. I use the term area to refer to its volume.) encodes the entropy of a certain density matrix on a finite-dimensional Hilbert space. I review arguments that the modular Hamiltonian of a diamond is a cutoff version of the Virasoro generator $L_0$ of a $1+1$-dimensional CFT of a large central charge, living on an interval in the longitudinal coordinate on the diamond boundary. The cutoff is chosen so that the von Neumann entropy is $\ln {\mathrm{D}}_{\diamond },$ up to subleading corrections, in the limit of a large-dimension diamond Hilbert space. I also connect those arguments to the derivation of the ’t Hooft commutation relations for horizon fluctuations. I present a tentative connection between the ’t Hooft relations and $U\left(1\right)$ currents in the CFTs on the past and future diamond boundaries. The ’t Hooft relations are related to the Schwinger term in the commutator of the vector and axial currents. The paper in can be read as evidence that the near-horizon dynamics for causal diamonds much larger than the Planck scale is equivalent to a topological field theory of the ’t Hooft CR plus small fluctuations in the transverse geometry. Connes’ demonstration that the Riemannian geometry is encoded in the Dirac operator leads one to a completely finite theory of transverse geometry fluctuations, in which the variables are fermionic generators of a superalgebra, which are the expansion coefficients of the sections of the spinor bundle in Dirac eigenfunctions. A finite cutoff on the Dirac spectrum gives rise to the area law for entropy and makes the geometry both “fuzzy” and quantum. Following the analysis of Carlip and Solodukhin, I model the expansion coefficients as two-dimensional fermionic fields. I argue that the local excitations in the interior of a diamond are constrained states where the spinor variables vanish in the regions of small area on the holoscreen. This leads to an argument that the quantum gravity in asymptotically flat space must be exactly supersymmetric. Full article

Wigner’s classification has led to the insight that projective unitary representations play a prominent role in quantum mechanics. The physics literature often states that the theory of projective unitary representations can be reduced to the theory of ordinary unitary representations by enlarging the group of physical symmetries. Nevertheless, the enlargement process is not always described explicitly: it is unclear in which cases the enlargement has to be conducted on the universal cover, a central extension, or a central extension of the universal cover. On the other hand, in the mathematical literature, projective unitary representations have been extensively studied, and famous theorems such as the theorems of Bargmann and Cassinelli have been achieved. The present article bridges the two: we provide a precise, step-by-step guide on describing projective unitary representations as unitary representations of the enlarged group. Particular focus is paid to the difference between algebraic and topological obstructions. To build the bridge mentioned above, we present a detailed review of the difference between group cohomology and Lie group cohomology. This culminates in classifying Lie group central extensions by smooth cocycles around the identity. Finally, the take-away message is a hands-on algorithm that takes the symmetry group of a given quantum theory as input and provides the enlarged group as output. This algorithm is applied to several cases of physical interest. We also briefly outline a generalization of Bargmann’s theory to time-dependent phases using Hilbert bundles. Full article

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product. Full article

In this paper, we focus on the underlying quantum structure of temporal correlations and show their peculiar nature which differentiates them from spatial quantum correlations. With a growing interest in the representation of quantum states as topological objects, we consider quantum history bundles based on the temporal manifold and show the source of the violation of monogamous temporal Bell-like inequalities. We introduce definitions for the mixture of quantum histories and consider their entanglement as sections over the Hilbert vector bundles. As a generalization of temporal Bell-like inequalities, we derive the quantum bound for multi-time Bell-like inequalities. Full article

We quantized the full Einstein equations in a globally hyperbolic spacetime $N=N^{n+1}$, $n\ge 3$, and found solutions of the resulting hyperbolic equation in a fiber bundle E which can be expressed as a product of spatial eigenfunctions (eigendistributions) and temporal eigenfunctions. The spatial eigenfunctions form a basis in an appropriate Hilbert space while the temporal eigenfunctions are solutions to a second-order ordinary differential equation in ${\mathbb{R}}_{+}^{}$. In case $n\ge 17$ and provided the cosmological constant $\Lambda$ is negative, the temporal eigenfunctions are eigenfunctions of a self-adjoint operator ${\widehat{H}}_0$ such that the eigenvalues are countable and the eigenfunctions form an orthonormal basis of a Hilbert space. Full article

Considering the case of a continual bundle of controlled dynamic processes, the authors have studied the functional-geometric conditions of existence of non-stationary coefficients-operators from the differential realization of this bundle in the class of non-autonomous bilinear second-order differential equations in the separable Hilbert space. The problem under scrutiny belongs to the type of non-stationary coefficient-operator inverse problems for the bilinear evolution equations in the Hilbert space. The solution is constructed on the basis of usage of the functional Relay-Ritz operator. Under this mathematical problem statement, the case has been studied in detail when the operators to be modeled are burdened with the condition, which provides for entire continuity of the integral representation equations of the model of realization. Proposed is the entropy interpretation of the given problem of mathematical modeling of continual bundle dynamic processes in the context of development of the qualitative theory of differential realization of nonlinear state equations of complex infinite-dimensional behavioristic dynamical system. Full article

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated. Full article

We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights. Full article

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry. Full article