scholarly journals Time-Dependent and/or Nonlocal Representations of Hilbert Spaces in Quantum Theory

10.14311/1195 ◽  
2010 ◽  
Vol 50 (3) ◽  
Author(s):  
M. Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Hilbert Spaces ◽  
Nuclear Physics ◽  
Bound State ◽  
Wave Functions ◽  
Time Dependent ◽  
Moving Frame ◽  
Standard Textbook ◽  
Space Formulation

A few recent innovations of the applicability of standard textbook Quantum Theory are reviewed. The three-Hilbert-space formulation of the theory (known from the interacting boson models in nuclear physics) is discussed in its slightly broadened four-Hilbert-space update. Among applications involving several new scattering and bound-state problems the central role is played by models using apparently non-Hermitian (often called “crypto-Hermitian”) Hamiltonians with real spectra. The formalism (originally inspired by the topical need for a mathematically consistent description of tobogganic quantum models) is shown to admit even certain unusual nonlocal and/or “moving-frame” representations H(S) of the standard physical Hilbert space of wave functions.

Algebraic formulation of quantum theory, particle identity and entanglement

2016 ◽  
Vol 14 (06) ◽  
pp. 1640026
Author(s):  
T. R. Govindarajan
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Hilbert Spaces ◽  
Quantum Physics ◽  
Statistical Nature ◽  
Density Matrices ◽  
Correct Formulation ◽  
Particle Identity ◽  
Algebraic Formulation

Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators [Formula: see text] algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.

SEMI-RELATIVISTIC SOLUTIONS OF THE TWO-BODY SPINLESS SALPETER EQUATION UNDER THE WOODS–SAXON POTENTIAL AND THE PEKERIS APPROXIMATION

2013 ◽  
Vol 22 (06) ◽  
pp. 1350039 ◽  
Author(s):  
H. FEIZI ◽  
M. HOSEININAVEH ◽  
A. H. RANJBAR
Keyword(s):  
Particle Physics ◽  
Nuclear Physics ◽  
Bound States ◽  
State Energy ◽  
Bound State ◽  
Wave Functions ◽  
Shape Invariance ◽  
Energy Eigenvalues ◽  
Elementary Particle Physics ◽  
Pekeris Approximation

In this paper, by applying the Pekeris approximation and in the frame of Supersymmetric Quantum Mechanics (SUSYQM), the semi-relativistic solutions of the two-body spinless Salpeter equation are obtained analytically. For an interaction of nuclear form, we obtain the approximate bound-state energy eigenvalues and the corresponding wave functions using the shape invariance concept. The solutions are reported for any l state and some energy eigenvalues are given. These results are useful in elementary-particle physics and nuclear physics to obtain the bound states spectra of relativistic systems such as fermion–antifermion systems.

scholarly journals Quantum theory based on real numbers can be experimentally falsified

Nature ◽  
2021 ◽  
Author(s):  
Marc-Olivier Renou ◽  
David Trillo ◽  
Mirjam Weilenmann ◽  
Thinh P. Le ◽  
Armin Tavakoli ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Quantum States ◽  
Complex Numbers ◽  
Real Numbers ◽  
Quantum Formalism ◽  
Physical Experiments ◽  
Independent States

AbstractAlthough complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces1,2. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural3. In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states4. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.

Quantum Theory

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Principle ◽  
Quantum Spin ◽  
Quantum States ◽  
Wave Functions ◽  
Symbolic Representations ◽  
Quantum Numbers ◽  
The Subject ◽  
Heisenberg's Uncertainty Principle

The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being representations of quantum states as well as a source of wave functions. The quantum box and quantum spin kets are specified, as are the quantum numbers that identify them. Operators are introduced and defined in part as the symbolic representations of observable quantities such as position, momentum and quantum spin. Eigenvalues and eigenkets are defined and discussed, with the former identified as the possible outcomes of a measurement. Bras, the counterpart to kets, are introduced as the means of forming probability amplitudes from kets. Products of operators are examined, as is their role underpinning Heisenberg’s Uncertainty Principle. A variety of symbol manipulations are presented. How measurements are believed to collapse linear superpositions to one term of the sum is explored.

scholarly journals Translation and modulation invariant Hilbert spaces

2021 ◽  
Author(s):  
Joachim Toft ◽  
Anupam Gumber ◽  
Ramesh Manna ◽  
P. K. Ratnakumar
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Zero Element ◽  
Multiplicative Constant ◽  
Space Of Distributions ◽  
Feichtinger Algebra

AbstractLet $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ R d which contains at least one non-zero element of the Feichtinger algebra $$S_0$$ S 0 and is continuously embedded in $$\mathscr {D}'$$ D ′ . If $$\mathcal H$$ H is translation and modulation invariant, also in the sense of its norm, then we prove that $$\mathcal H= L^2$$ H = L 2 , with the same norm apart from a multiplicative constant.

Functional Quantum Theory of Free Relativistic Fermi Fields

1970 ◽  
Vol 25 (5) ◽  
pp. 575-586
Author(s):  
H. Stumpf
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Physical Interpretation ◽  
Functional Equations ◽  
Dirac Field ◽  
Functional Hilbert Space ◽  
Free Fermi ◽  
Special Case ◽  
Relativistic Fermi

Functional quantum theory of free Fermi fields is treated for the special case of a free Dirac field. All other cases run on the same pattern. Starting with the Schwinger functionals of the free Dirac field, functional equations and corresponding many particle functionals can be derived. To establish a functional quantum theory, a physical interpretation of the functionals is required. It is provided by a mapping of the physical Hilbert space into an appropriate functional Hilbert space, which is introduced here. Mathematical details, especially the problems connected with anticommuting functional sources are treated in the appendices.

scholarly journals Clock-dependent spacetime

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Sung-Sik Lee
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Quantum Gravity ◽  
Hilbert Spaces ◽  
Coordinate Systems ◽  
Physical Laws

Abstract Einstein’s theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.

scholarly journals Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Atoms ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 53
Author(s):  
Jack C. Straton
Keyword(s):  
Quantum Theory ◽  
Arbitrary Function ◽  
Plane Waves ◽  
Wave Functions ◽  
One Dimension ◽  
Two Dimensional ◽  
Transition Amplitudes ◽  
Multidimensional Integrals ◽  
Coulomb Potentials

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.

scholarly journals Reciprocity between moduli and phases in time-dependent wave functions

1999 ◽  
Vol 60 (3) ◽  
pp. 1802-1810 ◽  
Author(s):  
R. Englman ◽  
A. Yahalom
Keyword(s):  
Wave Functions ◽  
Time Dependent
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