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19
Quantum Mechanics beyond Hilbert space
, 1997
"... When describing a quantum mechanical system, it is convenient to consider state vectors that do not belong to the Hilbert space. In the first part of this paper, we survey the various formalisms have been introduced for giving a rigorous mathematical justification to this procedure: rigged Hilbert s ..."
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When describing a quantum mechanical system, it is convenient to consider state vectors that do not belong to the Hilbert space. In the first part of this paper, we survey the various formalisms have been introduced for giving a rigorous mathematical justification to this procedure: rigged Hilbert spaces (RHS), scales or lattices of Hilbert spaces (LHS), nested Hilbert spaces, partial inner product spaces. Then we present three types of applications in quantum mechanics, all of them involving spaces of analytic functions. First we present a LHS built around the Bargmann space, thus giving a natural frame for the FockBargmann (or phase space) representation. Then we review the RHS approach to scattering theory (resonances, Gamow vectors, etc.). Finally, we reformulate the Weinbergvan Winter integral equation approach to scattering in the LHS language, and this allows us to prove that it is in fact a particular case of the familiar complex scaling method. 1 Going beyond Hilbert space...
Quantum hidden subgroup algorithms on free groups, (in preparation
"... Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In thi ..."
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Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents
Continuous Quantum Hidden Subgroup Algorithms
, 2003
"... In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Pat ..."
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In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Paths of L 2 paths x: [0, 1] → R n in real nspace R n. Also included is a curious discrete QHS algorithm which is dual to Shor’s algorithm. Contents 1
A CONTINUOUS VARIABLE SHOR ALGORITHM
, 2004
"... Abstract. In this paper, we use the methods found in [21] to create a continuous variable analogue of Shor’s quantum factoring algorithm. By this we mean a quantum hidden subgroup algorithm that finds the period P of a function Φ: R − → R from the reals R to the reals R, where Φ belongs to a very ge ..."
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Abstract. In this paper, we use the methods found in [21] to create a continuous variable analogue of Shor’s quantum factoring algorithm. By this we mean a quantum hidden subgroup algorithm that finds the period P of a function Φ: R − → R from the reals R to the reals R, where Φ belongs to a very general class of functions, called the class of admissible functions. One objective in creating this continuous variable quantum algorithm was to make the structure of Shor’s factoring algorithm more mathematically transparent, and thereby give some insight into the inner workings of Shor’s original algorithm. This continuous quantum algorithm also gives some insight into the inner workings of Hallgren’s Pell’s equation algorithm. Two key questions remain unanswered. Is this quantum algorithm more efficient than its classical continuous variable counterpart? Is this quantum
Quantum Time Arrows, Semigroups and TimeReversal in Scattering
 ACCEPTED FOR PUBLICATION IN THE INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS
, 2004
"... 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, ..."
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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 BrusselsAustin group, is more general, involving excitations and deexcitations 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 timereversed states depending on which time arrow is chosen.
Extended Comment on “OneRange Addition Theorems for Coulomb Interaction Potential and Its Derivatives” by I
 213), Los Alamos Preprint arXiv:0704.1088v2 [mathph] (http://arXiv.org
, 2007
"... Guseinov [Chem. Phys. 309, 209 211 (2005)] derived onerange addition theorems for the Coulomb potential via the limit β → 0 in previously derived onerange addition theorems for the Yukawa potential exp ` −βr − r ′  ´ /r − r ′ . Onerange addition theorems are expansions in terms of function ..."
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Guseinov [Chem. Phys. 309, 209 211 (2005)] derived onerange addition theorems for the Coulomb potential via the limit β → 0 in previously derived onerange addition theorems for the Yukawa potential exp ` −βr − r ′  ´ /r − r ′ . Onerange addition theorems are expansions in terms of functions that are complete and orthonormal in a given Hilbert space, but Guseinov replaced the complete and orthonormal functions by nonorthogonal Slatertype functions and rearranged the resulting expansions. This is a dangerous operation whose validity must be checked. It is shown that the onecenter limit r ′ = 0 of Guseinov’s rearranged Yukawa addition theorems as well as of several other addition theorems does not exist. Moreover, the Coulomb potential does not belong to any of the Hilbert spaces implicitly used by Guseinov. Accordingly, onerange addition theorems for the Coulomb potential diverge in the mean. Instead, these onerange addition theorems have to interpreted as expansions of generalized functions that converge weakly in
On the problem of emergence of classical spacetime: The quantummechanical approach
, 2004
"... Abstract. The classical spacetime structure is derived from the structure of an abstract infinite dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H ∗ of functions of abstract parameters. Such a realization is associated with the process of measuring position o ..."
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Abstract. The classical spacetime structure is derived from the structure of an abstract infinite dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H ∗ of functions of abstract parameters. Such a realization is associated with the process of measuring position of macroscopic particles naturally occurring in the universe. The process of decoherence and collapse induced by the measurement is in return associated with the choice of a “decohered ” submanifoldM of realization H∗. The submanifoldM is then identified with the classical spacetime. The mathematical formalism is developed which permits to recover the usual Riemannian geometry on spacetime in terms of the Hilbert structure on S. The specific functional realizations of S are shown to produce spacetimes of different geometry and topology. PACS numbers: 03.65Ta, 03.65Yz, 02.40k On the problem of emergence of classical spacetime 2 1.
Partial inner product spaces of analytic functions. In Generalized functions, operator theory, and dynamical systems (Brussels
 of Chapman & Hall/CRC Res. Notes Math
, 1997
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