scholarly journals QUANTIZATION OF POINTLIKE PARTICLES AND CONSISTENT RELATIVISTIC QUANTUM MECHANICS

2000 ◽  
Vol 15 (28) ◽  
pp. 4499-4538 ◽  
Author(s):  
S. P. GAVRILOV ◽  
D. M. GITMAN
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Relativistic Particle ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Quantum Field ◽  
Positive Spectrum ◽  
The One ◽  
Relativistic Particles

We revise the problem of the quantization of relativistic particle models (spinless and spinning), presenting a modified consistent canonical scheme. One of the main point of the modification is related to a principally new realization of the Hilbert space. It allows one not only to include arbitrary backgrounds in the consideration but to get in the course of the quantization consistent relativistic quantum mechanics, which reproduces literally the behavior of the one-particle sector of the corresponding quantum field. In particular, in a physical sector of the Hilbert space, a complete positive spectrum of energies of relativistic particles and antiparticles is reproduced, and all state vectors have only positive norms.

The relativistic quantum mechanics of the elementary particles

1949 ◽  
Vol 45 (2) ◽  
pp. 263-274 ◽  
Author(s):  
H. S. Green
Keyword(s):  
Quantum Mechanics ◽  
Density Matrix ◽  
Relativistic Quantum ◽  
Difficult Problem ◽  
Dynamical Variable ◽  
Elementary Particles ◽  
Relativistic Quantum Mechanics ◽  
Principle Of Relativity ◽  
The One ◽  
Real Linear

The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.

scholarly journals Quantum backflow in solutions to the Dirac equation of the spin-1 2 free particle

2018 ◽  
Vol 33 (32) ◽  
pp. 1850186 ◽  
Author(s):  
Hong-Yi Su ◽  
Jing-Ling Chen
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Particle ◽  
Relativistic Quantum ◽  
Free Particle ◽  
Negative Energy ◽  
Dirac Particle ◽  
Relativistic Quantum Mechanics ◽  
Probability Current ◽  
Negative Probability

It was known that a free, non-relativistic particle in a superposition of positive momenta can, in certain cases, bear a negative probability current — hence termed quantum backflow. Here, it is shown that more variations can be brought about for a free Dirac particle, particularly when negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise in the superposition (i) of a well-defined momentum but different signs of energies, or more remarkably (ii) of different signs of both momenta and energies. Neither of these cases has a counterpart in non-relativistic quantum mechanics. A generalization by using the field-theoretic formalism is also presented and discussed.

scholarly journals PSEUDO-HERMITIAN REPRESENTATION OF QUANTUM MECHANICS

2010 ◽  
Vol 07 (07) ◽  
pp. 1191-1306 ◽  
Author(s):  
ALI MOSTAFAZADEH
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Chaos ◽  
Relativistic Quantum ◽  
Canonical Quantization ◽  
Inner Product ◽  
Relativistic Quantum Mechanics ◽  
Real Spectrum ◽  
Quantization Scheme

A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.

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