Gleason's theorem and completeness of inner product spaces

We generalize the result of Hamhalter and Pták showing that an inner product space whose dimension is either a nonmeasurable cardinal or an arbitrary cardinal is complete iff its lattice of strongly closed subspaces possesses at least one state or one completely additive state, respectively. Moreover, we show that this lattice of any separable space possesses many-finite measures, and we give the Gleason analogue for them.     

Test spaces,dacey spaces,and completeness of inner product spaces

We show that a test space consisting of the unit sphere of a real or complex inner product spaceS and of the set all maximal orthonormal systems inS, is algebraic iffS is Dacey or, equivalently, iffS is complete. In addition, we present another completeness criterion.This research is supported by the grant G-368 of the Slovak Academy of Sciences, Slovakia.     

Quantum logics and completeness criteria of inner product spaces

We present the survey of measure-theoretic completeness criteria for inner product spaces using methods and notions important for quantum logics. Moreover, some new criteria and open problems are given.     

State on splitting subspaces and completeness of inner product spaces

We show that an inner product spaceV is complete iff the system of all splitting subspaces, i.e., of all subspacesM for whichM + M =V, possesses at least one completely additive state.     

Regular measures and inner product spaces

We present some results concerning the properties of regular finitely additive measures on the set of all splitting subspaces of a (real or complex) inner product spaceS and their relation to completeness ofS. These results are generalized for abstract quadratic spaces to be orthomodular. Moreover, some open problems are presented.     

Semi-indefinite inner product and generalized Minkowski spaces

In this paper we develop the theories of normed linear spaces and of linear spaces with indefinite metric, for finite dimensions both of which are also called Minkowski spaces in the literature.     

Solving general gauge theories on inner product spaces

By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form ph〉 = exp [Q, ψ]ph〉₀ where Q is the nilpotent BRST operator, ψ a hermitian fermionic gauge-fixing operator, and ph〉_o BRST invariant states determined by a hermitian set of BRST doublets in involution. ph〉₀ does not belong to an inner product space although ph〉 does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for ph〉₀ and the corresponding ψ. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that ph〉₀ are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set ph〉₀ are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading.     

Completeness of inner product spaces and quantum logic of splitting subspaces

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e., of all subspaces E for which EE =S holds, is a quantum logic, that is, an orthocomplemented orthomodular -orthoposet. It is well known that the quantum logic is an important axiomatic model of quantum mechanics. This generalizes the result of G. Cattaneo and G. Marino (Lett. Math. Phys. 11, 15–20 (1986)) who required that E(S) be a lattice. Moreover, the conditions are weakened to show that S is complete whenever E(S) contains the join of any sequence of one-dimentional orthogonal subspaces.     

Completeness of inner product spaces with respect to splitting subspaces

The set E(S) of all splitting subspaces, i.e., of all subspaces M of an inner product space S for which MM =S holds, is an orthocomplemented orthomodular orthoposet and it has already been shown that the ordering property on E(S) of being a complete lattice characterizes the completeness of inner product spaces. In this work this last result is generalized proving that S is a Hilbert space under the weaker request that E(S) is a -lattice. As a marginal result, we also prove that an inner product space is complete if and only if the complete lattice (S) of all subspaces is orthomodular.     

Finitely correlated states on quantum spin chains

We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic Néel ordered states. The ergodic components have exponential decay of correlations. All states considered can be obtained as local functions of states of a special kind, so-called purely generated states, which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.     

Coherent states and symmetric spaces

Properties of system of the coherent states related to representations of the class I of principal series of the motion groups of symmetric spaces of rank 1 have been studied. It has been proved that such states are given by horospherical kernels and are the generalization of the plane waves for the case of symmetric spaces.     

Propagation of states in dilation analytic potentials and asymptotic completeness

We estimate the space-time behavior of scattering states for two-body Schrödinger operators with smooth, dilation analytic potentials. We use our estimates to give a simple proof of asymptotic completeness for a class of long-range potentials, including the Coulomb potential plus a fairly general short-range perturbation.Supported by USNSF Grant MCS-78-01885     

Existence of ground states and KMS states for approximately inner dynamics

A strongly continuous one parameter group of *-automorphisms of aC*-algebra with unit is said to be approximately inner if it can be approximated strongly by inner one parameter groups of *-automorphisms. It is shown that an approximately inner one parameter group of *-automorphisms has a ground state and, if there exists a trace state, a KMS state for all inverse temperatures. It follows that quantum lattice systems have ground states and KMS states. Conditions that a strongly continuous one parameter group of *-automorphisms of a UHF algebra be approximately inner are given in terms of the unbounded derivation which generates the automorphism group.     

On the completeness of one-particle states of a many-electron atom

A method for constructing the complete set of one-particle states of a many-electron atom is proposed in the framework of the one-configuration Hartree-Fock approximation.     

Tensor product of generalized sample spaces

The tensor product of generalized samples spacer or manuals is defined within the framework of empirical logic The requirement to accurately reflect the interaction of experimental procedures for coupled systems leads to three levels of product: the cross-product, operational product, and tensor product. The structure of the weights of these products is examined and is used to give a condition for the existence of the tensor product Categorical properties of the tensor product, including a universal mapping theorem, are given.     

Indecomposable representations with invariant inner product

Consequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space \(\mathfrak{H}\) are studied. If π has an irreducible subrepresentation π₁ on a subspace \(\mathfrak{H}_1 \) , it is shown that there exists an invariant subspace \(\mathfrak{H}_2 \) of \(\mathfrak{H}\) containing \(\mathfrak{H}_1 \) and satisfying the following conditions: (1) the representation π ₁ ^# =π mod \(\mathfrak{H}_2 \) on \(\mathfrak{H}\) mod \(\mathfrak{H}_2 \) is conjugate to the representation (π₁, \(\mathfrak{H}_1 \) ), (2) \(\mathfrak{H}_1 \) is a null space for the inner product, and (3) the induced inner product on \(\mathfrak{H}_2 \) mod \(\mathfrak{H}_1 \) is non-degenerate and invariant for the representation $$\pi _2 = (\pi _2 |_{\mathfrak{H}_2 } )\bmod \mathfrak{H}_1 ,$$ a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with \(\mathfrak{H}_1 \) =space of longitudinal photons and \(\mathfrak{H}_2 \) =the space defined by the subsidiary condition.     

Canonical quantization with indefinite inner product

In this paper an algebraic approach to canonical quantization on indefinite inner product space is presented. Concrete realization of such a quantization of a given classical system described by a symplectic space (M, σ) is obtained by means of a so-called J¹-representation of CCR algebra Δ(M, σ). So-called Fock-Krein representations of Δ(M, σ) determined by some class of complex structures on (M, σ) are studied in detail. It is shown that every Fock-Krein representation is unbounded. Starting with a fixed Fock-Krein representation of CCR sectors of non-Fock-Krein representations are constructed.     

Extremum uncertainty product and sum states

We consider the states with extremum products and sums of the uncertainties in non-commuting observables. These are illustrated by two specific examples of harmonic oscillator and the angular momentum states. It shows that the coherent states of the harmonic oscillator are characterized by the minimum uncertainty sum 〈(Δq)²〉 + 〈(Δp)²〉. The extremum values of the sums and products of the uncertainties of the components of the angular momentum are also obtained.     

IWOP积分技术证明粒子数平移态的过完备性

利用正规乘积算符内的积分技术(IWOP),导出了在相干态|z＞过完备性中,积分变量z与z*可以独立地做积分变换,而保持其完备性.利用相干态的这一特点和IWOP积分技术,进而可以方便地证明粒子数平移态D(z),|n＞具有过完备性.