Sein oder Nichtsein als Grundfrage der Quantenphysik / To he or not to be as basic question in quantum physics

1984 ◽  
Vol 39 (2) ◽  
pp. 113-131
Author(s):  
Fritz Bopp
Keyword(s):  
Hilbert Spaces ◽  
Quantum Physics ◽  
Basic Assumption ◽  
Finite Lattice ◽  
Basic Question ◽  
Classical Physics ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Basic Ideas ◽  
Finite Dimensional Hilbert Space

The question is often asked how to interprete quantum physics. That question does not arise in classical physics, since Newton's axioms are immediately connected with basic ideas and experiences. The same is possible in quantum physics, if we remember how elementary particle physicists describe their experiments. As Helmholtz has pointed out. the basic assumption of classical physics is that of geneidentity. That means: Bodies remain the same during their motion. Obviously, that is no longer true in quantum physics. Particles can be created and annihilated. Therefore creation and annihilation must be considered as basic processes. Motion only occurs, if a particle is annihilated in a certain point, if an equal one is created in an infinitesimally neighbouring point, and if this process is continuously going on during a certain time. Motions of that kind are compatible with the existence of some manifest creation and annihilation processes. If we accept this idea, quantum physics can be derived from first principles. As in classical physics, we know therefore what happens from the very beginning. Thus questions of interpretation become dispensable. A particular mathematical method is used to exhaust continua. The theory is formulated in a finite lattice, whose point density and extension equally go to infinity. All calculations are therefore performed in a finite dimensional Hilbert space. The results are however related to an infinite dimensional one. Earlier calculations may, therefore, be essentially correct, though they must be rejected in theories which are based on manifestly infinite dimensional Hilbert spaces. Here limiting processes do not occur in the state space. They are only admissible for numerical results.

scholarly journals Dagger linear logic for categorical quantum mechanics

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Robin Cockett ◽  
Cole Comfort ◽  
Priyaa Srinivasan
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Physics ◽  
Linear Logic ◽  
Infinite Dimensional ◽  
Graphical Calculus ◽  
Finite Dimensional ◽  
Categorical Quantum Mechanics ◽  
Definition Of ◽  
Categorical Semantics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.

scholarly journals Frame for operators in finite dimensional hilbert space

2018 ◽  
Vol 49 (1) ◽  
pp. 35-48
Author(s):  
Mohammad Janfada ◽  
Vahid Reza Morshedi ◽  
Rajabali Kamyabi Gol
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Dual Frames ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Finite Dimensional Hilbert Space

In this paper, we study frames for operators ($K$-frames) in finite dimensional Hilbert spaces and express the dual of $K$-frames. Some properties of $K$-dual frames are investigated. Furthermore, the notion of their oblique $K$-dual and some properties are presented.

Orthogonality spaces of finite rank and the complex Hilbert spaces

2019 ◽  
Vol 16 (05) ◽  
pp. 1950080 ◽  
Author(s):  
Thomas Vetterlein
Keyword(s):  
Binary Relation ◽  
Hilbert Spaces ◽  
Finite Rank ◽  
Representation Theorem ◽  
Quantum Physics ◽  
Orthogonality Relation ◽  
Quadratic Space ◽  
Orthogonality Space ◽  
Finite Dimensional

An orthogonality space is a set endowed with a symmetric, irreflexive binary relation. By means of the usual orthogonality relation, each anisotropic quadratic space gives rise to such a structure. We investigate in this paper the question to which extent this strong abstraction suffices to characterize complex Hilbert spaces, which play a central role in quantum physics. To this end, we consider postulates concerning the nature and existence of symmetries. Together with a further postulate excluding the existence of nontrivial quotients, we establish a representation theorem for finite-dimensional orthomodular spaces over a dense subfield of [Formula: see text].

scholarly journals Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces

2005 ◽  
Vol 77 (4) ◽  
pp. 589-594 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Countable Family ◽  
Spectral Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Lagrangian Subspaces ◽  
Finite Dimensional Case ◽  
Adjoint Operators

We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.

scholarly journals On the Relation between States and Maps in Infinite Dimensions

2007 ◽  
Vol 14 (04) ◽  
pp. 355-370 ◽  
Author(s):  
Janusz Grabowski ◽  
Marek Kuś ◽  
Giuseppe Marmo
Keyword(s):  
Hilbert Spaces ◽  
Trace Class ◽  
Quantum Systems ◽  
Bounded Operators ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Continuous Map ◽  
Finite Dimensional ◽  
Trace Class Operators ◽  
Product Realization

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

Quantenphysikalischer Zustandsraum im Kontinuum / Quantum physical state space in continua

1984 ◽  
Vol 39 (3) ◽  
pp. 205-217
Author(s):  
Fritz Bopp
Keyword(s):  
Hilbert Space ◽  
Dirac Equation ◽  
Finite Number ◽  
State Space ◽  
Hilbert Spaces ◽  
First Principles ◽  
Quantum Physics ◽  
Physical State ◽  
Elementary Processes ◽  
Classical Physics

AbstractAs previously shown, quantum physics for single pairs of creation and annihilation processes may be derived from first principles. Quantum physics at all can be therefore considered as an interplay of such elementary processes. This is easily possible if the number of pairs of processes is finite. Difficulties arise only for infinite numbers.The difficulties are similar to those occurring in the derivation of the equation for an oscillating string from that for an oscillator chain. It is true that the spectra of both systems are not continuously connected. However, a weaker theorem is more important: The chain eigenvalue of each order converges to the string one of the same order for an infinitely growing number of oscillators of a certain kind. Therefore both systems are continuously connected in the sense of semiconvergency.Exhausting the space continuum with a sequence of lattices equably becomming infinitely large and fine, the infinitely dimensional Hilbertspace is steadily connected with the finitely dimensional one in the sense of semiconvergency. It will be shown that the Hilbert spaces in the sequence of lattices yield the suitable tool for quantum physics as an interplay in the mentioned sense. This kind of Hilbert space, the so-called rational one, must be preferred in physics rather than the real one introduced by Hilbert, since all theories in physics are based on a finite number of data.In particular, we formulate Dirac's equation in the rational Hilbert space. It is shown that, even in quantum physics, a theorem of classical physics remains true, according to which relativity results from certain principles formulating most obvious experiences. We obtain the Lorentz invariant Dirac equation mainly from a modification of Newtons definition II according to which p = Hυ/c2 (instead of p = m υ).

scholarly journals Stability Estimates for Phase Retrieval from Discrete Gabor Measurements

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Rima Alaifari ◽  
Matthias Wellershoff
Keyword(s):  
Hilbert Spaces ◽  
Phase Retrieval ◽  
Gabor Transform ◽  
Stability Estimates ◽  
Signal Process ◽  
Infinite Dimensional ◽  
Bandlimited Signals ◽  
Finite Dimensional ◽  
Order Polynomial ◽  
In Trans

AbstractPhase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016) and possibly severely ill-conditioned in finite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016). Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions (Alaifari in Found Comput Math 19(4):869–900, 2019). In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years (Bojarovska and Flinth in J Fourier Anal Appl 22(3):542–567, 2016; Eldar et al. in IEEE Signal Process Lett 22(5):638–642, 2014; Li et al. in IEEE Signal Process Lett 24(4):372–376, 2017; Nawab et al. in IEEE Trans Acoust Speech Signal Process 31(4):986–998, 1983).

scholarly journals Products of Positive Operators

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Maximiliano Contino ◽  
Michael A. Dritschel ◽  
Alejandra Maestripieri ◽  
Stefania Marcantognini
Keyword(s):  
Positive Operator ◽  
Hilbert Spaces ◽  
Spectral Properties ◽  
Compact Operators ◽  
Positive Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Special Classes

AbstractOn finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class $${\mathcal {L}^{+\,2}}$$ L + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in $${\mathcal {L}^{+\,2}}$$ L + 2 are developed, and membership in $${\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.

scholarly journals Optimization of functionals under uncertainties for Ito-Skorokhod stochastic differential equations in Hilbert spaces

2018 ◽  
pp. 65-70
Author(s):  
A. Nikitin ◽  
O. Baliasnikova
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Liquid Fuel ◽  
Degrees Of Freedom ◽  
Applied Mathematics ◽  
Optimal Controls ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Carry Over

In the article for the stochastic differential equations of Ito-Skorokhod, problems of optimization of functionals under conditions of uncertainty in Hilbert spaces are investigated. Purpose of the article is to investigate some properties of stochastic differential equations in Hilbert spaces. These objects arise in diverse areas of applied mathematics as models for various natural phenomena, in particular, the evolution of complex systems with infinitely many degrees of freedom. For instance, one may think of the liquid fuel motion in the tank of a spacecraft. Spacecraft constructors should take into account this motion, for it influences heavily the path of a spacecraft. Also, optimization of the motion is an issue of principal importance. It is not trivial to carry over the results concerning stochastic differential equations in finite-dimensional spaces to the infinite dimensional case. We give some statements, in which the existence, uniqueness is proved and the explicit form μ-optimal controls for such equations is constructed, in particular, μ-optimal control is found as a linear inverse relationship.

Entropy fluctuations to quantify mixedness

2019 ◽  
Vol 17 (08) ◽  
pp. 1941007
Author(s):  
J. A. Anaya-Contreras ◽  
A. Zúñiga-Segundo ◽  
H. M. Moya-Cessa
Keyword(s):  
Hilbert Spaces ◽  
The State ◽  
Field Interaction ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Level Atom ◽  
Final Dimension

We propose a mixedness quantifier based on entropy fluctuations. It provides information about the degree of mixedness either for finite dimensional and infinite dimensional Hilbert spaces (HS). It may be used to determine the reduction of the HS as it becomes maximum when either the state is maximally mixed, or when the HS effectively reduces its dimensions, such as in the atom field interaction where the two-level atom dictates the final dimension of the field.

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