Sobolev Spaces (Crash Course)

A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum system, if we can modify the Hilbert space of state vectors of the system so that it turns into a Hermitian operator. If this operator is time-dependent, the modified Hilbert space is generally time-dependent. This in turn leads to a generic conflict between the condition that the Hamiltonian is an observable of the system and that it generates a unitary time-evolution via the standard Schr\"odinger equation. We propose a geometric framework for addressing this problem. In particular we show that the Hamiltonian operator consists of a geometric part, which is determined by a metric-compatible connection on an underlying Hermitian vector bundle, and a non-geometric part which we identify with the energy observable. The same quantum system can be locally described using a time-dependent Hamiltonian that acts in a time-independent state space and is the sum of a geometric part and the energy operator. The full global description of the system is achieved within the framework of a moderate geometric extension of quantum mechanics where the role of the Hilbert space of state vectors is played by a Hermitian vector bundle $\mathcal{E}$ endowed with a metric compatible connection, and observables are given by global sections of a real vector bundle that is determined by $\mathcal{E}$. We examine the utility of our proposal to describe a class of two-level systems where $\mathcal{E}$ is a Hermitian vector bundle over a two-dimensional sphere.

We consider the problem of identifying a unitary Yang–Mills connection \nabla on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian \nabla^*\nabla over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods. Moreover, by using a Runge-type approximation argument along curves to recover holonomy, we are able to uniquely determine both the bundle structure and the connection. Also, we prove that the DNmap is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.

Beginning with an intuitive consideration of sequences of measurements, we define a time-ordered event space representing the collection of all imaginable outcomes for measurement sequences. We then postulate the generalized distributive relation on the event space and examine the class of measurements for which this relation can be experimentally validated. The generalized distributive relation is shown to lead to a σ-additive conditional probability on the event space and to a predictive and retrodictive formalism for stochastic processes. We then show that this formalism has a predictive and a retrodictive representation in a separable Hilbert space H, which has no counterpart in unitary quantum dynamics.

This paper discusses the local and global geometry of coisotropic foliations and complex polarizations on symplectic manifolds and draws attention to an analogy between coisotropic foliations and Hermitian vector bundles, in which connections and characteristic classes are modelled by objects in symplectic geometry.

In this chapter, we gather some basic results concerning Hermitian vector bundles over arithmetic curves, namely affine schemes defined by rings of integers of number fields.

where λ ( ξ) = (|ξ − ( )|2 + 1)1/2. For simplicity we suppose here that the mass of the particle is equal to 1. All the differential and pseudodifferential operators considered in this paper are, possibly unbounded, operators in 2(R ), defined on the Schwartz space S of rapidly decreasing smooth functions on R . In [14] it was proved that if the derivatives of any positive order of ∈ ∞(R ; R ) are bounded, then (λ ) is essentially selfadjoint on S. Let be its unique selfadjoint extension. In [16] the authors proved that if itself is bounded and if all its derivatives converge to zero at infinity, then the essential spectrum of is equal to the essential spectrum of √ + 1, where is the quantum nonrelativistic magnetic Hamiltonian with vector potential , i.e. the selfadjoint operator generated by the differential operator ( − ( ))2. We shall prove in this paper that the essential spectra of and of √ + 1 are still equal if we drop the condition of boundedness of . Thus, vector potentials which behave at infinity as | |1−e, e positive and arbitrary small, are allowed. More precisely, the main result of the paper is the following theorem.

Heat kernel expansions in vector bundles

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

SynopsisThis paper considers properties of the spectrum of differential operators derived from differential expressions of the second order. The object is to link the spectral properties of these differential operators with the analytic, function-theoretic properties of the solutions of the differential equation. This provides an alternative approach to the spectral theory of these differential operators but one which is consistent with the standard definitions used in Hilbert space theory. In this way the approach may be of interest to applied mathematicians and theoretical physicists.

In the present paper we develop a framework in which questions of quantum ergodicity for operators acting on sections of Hermitian vector bundles over Riemannian manifolds can be studied. We are particularly interested in the case of locally symmetric spaces. For locally symmetric spaces, we extend the recent construction of Silberman and Venkatesh [7] of representation theoretic lifts to vector bundles.

Methods of -representations in Hilbert space are applied to study of systems of n subspaces in a linear space. It is proved that the problem of description of n-transitive subspaces in a finite-dimensional linear space is -wild for n 5.

Abstract. Let G be a compact connected semisimple Lie group, g theLie algebra of G, g the canonical metric (the biinvariant Riemannianmetric which is induced from the Killing form of g), and r be the Levi-Civita connection for the metric g. Then, we get the fact that the Levi-Civita connection r in the tangent bundle TG over (G;g) is a Yang-Millsconnection. 1. IntroductionThe problem of nding metrics and connections which are critical pointsof some functional plays an important role in global analysis and Riemann-ian geometry. A Yang-Mills connection is a critical point of the Yang-MillsfunctionalYM(D) =12Z M kR D k 2 v g (1:1)on the space C E of all connections in a smooth vector bundle Eover a closed(compact and connected) Riemannian manifold (M;g), where R D is the cur-vature of D2C E . Equivalently, Dis a Yang-Mills connection if it satis es theYang-Mills equation (cf. [1, 5, 6]) D R D = 0; (1:2)(the Euler-Lagrange equations of the variational principle associated with (1.1)).If Dis a connection in a vector bundle Ewith bundle metric hover a Riemann-ian manifold (M;g), then the connection D given byh(D

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

The calculus of classical pseudo-differential operators has been used in a fundamental way in the study of boudary value problems associated to (systems) of elliptic differential and pseudo-differential operators. This calculus was used by Hörmander in [7] to construct parametrices for the elliptic operators, using which the boundary value problem is reduced to a system of pseudo-differential operators on the boundary. In order to treat the boundary value problems for parabolic operators in bounded cylindrical domains Piriou introduced in [15] and [16] a class of operators which called pseudo-differential operators of Volterra type. The basic idea consists in developping a calculus of an appropriate class of anisotropic pseudo-differential operators as in an earlier note of Hunt and Piriou [8].KeywordsVector BundleParabolic SystemPrincipal SymbolCylindrical DomainVolterra TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Semi-bounded restrictions of Dirac type operators and the unique continuation property