Quantum Hilbert Space Research Articles

A unifying perspective on the Moyal and Voros products and their physical meanings

The Moyal and Voros formulations of non-commutative quantum field theory have been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular, we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective, the two formulations are therefore completely equivalent, but we also argue that only the Voros formulation admits a consistent physical interpretation. These considerations are elucidated by considering the free-particle transition amplitude in the two representations.

The Modified Bell Inequality and Its Physical Implications in the ESR Model

The extended semantic realism (ESR) model proposes a theoretical perspective which reinterprets quantum probabilities as conditional on detection rather than absolute and embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) in a noncontextual, hence local, framework. The assumptions needed to prove the Bell inequality therefore hold in the ESR model, but we show that the Bell inequality must be substituted in it by the modified Bell inequality and that the standard quantum expectation values, when reinterpreted as proposed by the ESR model, do not violate the latter inequality. Hence the long-standing conflict between “local realism” and QM is settled in the ESR model. Finally we provide an elementary example of a prediction that might be used to check whether the ESR model is correct.

Higher asymptotics of unitarity in “quantization commutes with reduction”

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. Guillemin and Sternberg (Invent Math 67:515–538, 1982, no. 3), showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M //G. This map, though, is not in general unitary, even to leading order in $${\hslash}$$ . Hall and Kirwin (Commun Math Phys 275:401–422, 2007, no. 2), showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed $${\hslash}$$ , becomes unitary in the semiclassical limit $${\hslash\rightarrow0}$$ (cf. the work of Ma and Zhang (C R Math Acad Sci Paris 341:297–302, 2005, no. 5), and (Astérisque No. 318:viii+154, 2008). The unitarity of the classical Guillemin–Sternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M //G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as $${\hslash\rightarrow0}$$ .

Generalized Observables, Bell’s Inequalities and Mixtures in the ESR Model for QM

The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) into a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide in this review an overall view on the present status of our research on this topic. We attain in a new, shortened way a mathematical representation of the generalized observables introduced by the ESR model and a generalization of the projection postulate of elementary QM. Basing on these results we prove that the Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, a modified BCHSH inequality and quantum predictions hold together in the ESR model because they refer to different parts of the picture of the physical world supplied by the model. Then we show that a new mathematical representation of mixtures must be introduced in the ESR model which does not coincide with the standard representation in QM and avoids some deep problems that arise from the representation of mixtures provided by QM. Finally we get a nontrivial generalization of the Lüders postulate, which is justified in a special case by introducing a reasonable physical assumption on the evolution of the compound system made up of the measured system and the measuring apparatus.

A Hilbert Space Representation of Generalized Observables and Measurement Processes in the ESR Model

The extended semantic realism (ESR) model recently worked out by one of the authors embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here a Hilbert space representation of the generalized observables introduced by the ESR model that satisfy a simple physical condition, propose a generalization of the projection postulate, and suggest a possible mathematical description of the measurement process in terms of evolution of the compound system made up of the measured system and the measuring apparatus.

Embedding Quantum Mechanics Into a Broader Noncontextual Theory: A Conciliatory Result

The extended semantic realism (ESR) model embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here an improved version of this model and show that it predicts that, whenever idealized measurements are performed, a modified Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality holds if one takes into account all individual systems that are prepared, standard quantum predictions hold if one considers only the individual systems that are detected, and a standard BCHSH inequality holds at a microscopic (purely theoretical) level. These results admit an intuitive explanation in terms of an unconventional kind of unfair sampling and constitute a first example of the unified perspective that can be attained by adopting the ESR model.

Matrix-valued Schrödinger operators over local fields

We consider quantum systems that have as their configuration spaces finite dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to ameasure on the Skorokhod space of paths,D[0,∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman-Kac formula in the local field setting as a consequence of the Hille-Yosida theory of semi-groups.

Coherent states in noncommutative quantum mechanics

Gazeau–Klauder coherent states in noncommutative quantum mechanics are considered. We find that these states share similar properties to those of ordinary canonical coherent states in the sense that they saturate the related position uncertainty relation, obey a Poisson distribution, and possess a flat geometry. Using the natural isometry between the quantum Hilbert space of Hilbert-Schmidt operators and the tensor product of the classical configuration space and its dual, we reveal the inherent vector feature of these states.

SCHRÖDINGER OPERATORS ON LOCAL FIELDS: SELF-ADJOINTNESS AND PATH INTEGRAL REPRESENTATIONS FOR PROPAGATORS

We consider quantum systems that have as their configuration spaces finite-dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to a measure on the Skorokhod space of paths, D [0, ∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman–Kac formula in the local field setting as a consequence of the Hille–Yosida theory of semigroups.

Singular unitarity in “quantization commutes with reduction”

Let M be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group G . Let M / / G = ϕ − 1 ( 0 ) / G = M 0 be the symplectic quotient at value 0 of the moment map ϕ . The space M 0 may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over M 0 and the G -invariant subspace of the quantum Hilbert space over M . In this paper, without any regularity assumption on the quotient M 0 , we discuss the relation between the inner products of these two quantum Hilbert spaces under the above natural isomorphism; we establish asymptotic unitarity to leading order in Planck’s constant of a modified map of the above isomorphism under a “metaplectic correction” of the two quantum Hilbert spaces.

A nonstandard geometric quantization of the harmonic oscillator

In the standard geometric quantization of the harmonic oscillator in (R2,dp∧dx), using standard holomorphic coordinate z=x−ip induces a standard complex structure which Kähler polarizes the prequantum line bundle. Hence, the quantum Hilbert space maybe identified with holomorphic functions which are L2 integrable with respect to Gaussian measure, denoted by H. The corresponding quantized Hamiltonian is then given by ℏz(∂∕∂z). We propose using a different almost complex structure J (under some restrictions) compatible with dp∧dx and construct a quantum Hilbert space HJ containing nontrivial L2 sections and a quantized Hamiltonian QJ. The main result is that (HJ,QJ) is unitarily equivalent to (H,ℏz(∂∕∂z)).

Unitarity in “Quantization Commutes with Reduction”

Let M be a compact Kahler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M // G. Guillemin and Sternberg [Invent. Math. 67, 515–538 (1982)] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M //G and the G-invariant subspace of the quantum Hilbert space over M.

Probability in modal interpretations of quantum mechanics

Modal interpretations have the ambition to construe quantum mechanics as an objective, man-independent description of physical reality. Their second leading idea is probabilism: quantum mechanics does not completely fix physical reality but yields probabilities. In working out these ideas an important motif is to stay close to the standard formalism of quantum mechanics and to refrain from introducing new structure by hand. In this paper we explain how this programme can be made concrete. In particular, we show that the Born probability rule, and sets of definite-valued observables to which the Born probabilities pertain, can be uniquely defined from the quantum state and Hilbert space structure. We discuss the status of probability in modal interpretations, and to this end we make a comparison with many-worlds alternatives. An overall point that we stress is that the modal ideas define a general framework and research programme rather than one definite and finished interpretation.

The Uncertainty of Fluxes

In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3-manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We also develop the Hamilton theory of self-dual fields, noting that they are quantized by Pontrjagin self-dual cohomology theories and that the quantum Hilbert space is \({\mathbb{Z}/2\mathbb{Z}}\) -graded, so typically contains both bosonic and fermionic states. Significantly, these ideas apply to the Ramond-Ramond field in string theory, showing that its K-theory class cannot be measured.

ON THE FUNCTORIALITY OF THE CHERN–SIMONS LINE BUNDLE AND THE DETERMINANT LINE BUNDLE

We investigate functorial properties of two hermitian line bundles over the moduli space of flat SU(n)-connections on a closed oriented surface; that is, of the Chern–Simons line bundle and the determinant line bundle. We investigate actions of cyclic subgroups of the mapping class group on them. As a consequence, we show that if we modify the determinant line bundle by the Hodge bundle over the moduli space of Riemann surfaces, then these line bundles are functorially isomorphic. This implies two quantum Hilbert spaces defined by the Chern–Simons line bundle and the modified determinant line bundle are functorially isomorphic.

Quasi-Linear Quantum Field Theories for Maps to Groups and Their Quotients

I describe a functional integral for maps from \(\mathbb{R}\times \mathbb{R}^{\rm n}\) to a Lie group or its quotient which has a simple renormalization that leads to a quantum field theory for maps from \(\mathbb{R}^{\rm n}\) into the Lie group or its quotient whose Hamiltonian is the time translation generator for a unitary action of the n+1 dimensional Poincare group on the quantum Hilbert space. I also explain how the renormalization provides a functional integral for maps from a Riemann surface to a compact Lie group or its quotient that exhibits many conformal field theoretic properties.

Coherent states in geometric quantization

In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kähler and non-Kähler cases. We then study the reproducing kernels associated to the quantum Hilbert spaces and use them to define symplectic coherent states. The rest of the paper is dedicated to the properties of symplectic coherent states and the corresponding Berezin–Toeplitz quantization. Specifically, we study overcompleteness, symplectic analogues of the basic properties of Bargmann’s weighted analytic function spaces, and the ‘maximally classical’ behavior of symplectic coherent states. We also find explicit formulas for symplectic coherent states on compact Riemann surfaces.

Quantum Time Arrows, Semigroups and Time-Reversal in Scattering

Two approaches toward the arrow of time for scattering processes have been proposed in rigged Hilbert space quantum mechanics. One, due to Arno Bohm, involves preparations and registrations in laboratory operations and results in two semigroups oriented in the forward direction of time. The other, employed by the Brussels-Austin group, is more general, involving excitations and de-excitations of systems, and apparently results in two semigroups oriented in opposite directions of time. It turns out that these two time arrows can be related to each other via Wigner's extensions of the spacetime symmetry group. Furthermore, their are subtle differences in causality as well as the possibilities for the existence and creation of time-reversed states depending on which time arrow is chosen.

Forms of Quantum Nonseparability and Related Philosophical Consequences

Standard quantum mechanics unquestionably violates the separability principle that classical physics (be it point-like analytic, statistical, or field-theoretic) accustomed us to consider as valid. In this paper, quantum nonseparability is viewed as a consequence of the Hilbert-space quantum mechanical formalism, avoiding thus any direct recourse to the ramifications of Kochen-Specker’s argument or Bell’s inequality. Depending on the mode of assignment of states to physical systems – unit state vectors versus non-idempotent density operators – we distinguish between strong/relational and weak/deconstructional forms of quantum nonseparability. The origin of the latter is traced down and discussed at length, whereas its relation to the all important concept of potentiality in forming a coherent picture of the puzzling entangled interconnections among spatially separated systems is also considered. Finally, certain philosophical consequences of quantum non-separability concerning the nature of quantum objects, the question of realism in quantum mechanics, and possible limitations in revealing the actual character of physical reality in its entirety are explored.

GEOMETRICAL ASPECTS OF BRST COHOMOLOGY IN AUGMENTED SUPERFIELD FORMALISM

In the framework of augmented superfield approach, we provide the geometrical origin and interpretation for the nilpotent (anti-)BRST charges, (anti-)co-BRST charges and a non-nilpotent bosonic charge. Together, these local and conserved charges turn out to be responsible for a clear and cogent definition of the Hodge decomposition theorem in the quantum Hilbert space of states. The above charges owe their origin to the de Rham cohomological operators of differential geometry which are found to be at the heart of some of the key concepts associated with the interacting gauge theories. For our present review, we choose the two (1+1)-dimensional (2D) quantum electrodynamics (QED) as a prototype field theoretical model to derive all the nilpotent symmetries for all the fields present in this interacting gauge theory in the framework of augmented superfield formulation and show that this theory is a unique example of an interacting gauge theory which provides a tractable field theoretical model for the Hodge theory.