Abstract
We discuss the consequences of the introduction of a quantum of time {tau}{sub 0} in the formalism of non-relativistic quantum mechanics, by referring ourselves, in particular, to the theory of the chronon as proposed by P. Caldirola. Such an interesting ``finite difference`` theory, forwards - at the classical level - a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz`s and Dirac`s approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and - at the quantum level - yields a remarkable mass spectrum for leptons. After having briefly reviewed Caldirola`s approach, our first aim is to work out, discuss, and compare to one another the new representations of Quantum Mechanics (QM) resulting from it, in the Schroedinger, Heisenberg and density-operator (Liouville-von Neumann) pictures, respectively. Moreover, for each representation, three (retarded, symmetric and advanced) formulations are possible, which refer either to times t and t-{tau}{sub 0}, or to times t-{tau}{sub 0}/2 and t+{tau}{sub 0}/2, or to times t and t+{tau}{sub 0}, respectively. It is interesting to notice that, when the chronon tends
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Farias, R H.A.;
[1]
Recami, E
[2]
- LNLS - Laboratorio Nacional de Luz Sincrotron, Campinas, S.P. (Brazil)
- Facolta di Ingegneria, Universita Statale di Bergamo, Bergamo (Italy)
Citation Formats
Farias, R H.A., and Recami, E.
Introduction of a quantum of time (``chronon``) and its consequences for quantum mechanics.
IAEA: N. p.,
1998.
Web.
Farias, R H.A., & Recami, E.
Introduction of a quantum of time (``chronon``) and its consequences for quantum mechanics.
IAEA.
Farias, R H.A., and Recami, E.
1998.
"Introduction of a quantum of time (``chronon``) and its consequences for quantum mechanics."
IAEA.
@misc{etde_308975,
title = {Introduction of a quantum of time (``chronon``) and its consequences for quantum mechanics}
author = {Farias, R H.A., and Recami, E}
abstractNote = {We discuss the consequences of the introduction of a quantum of time {tau}{sub 0} in the formalism of non-relativistic quantum mechanics, by referring ourselves, in particular, to the theory of the chronon as proposed by P. Caldirola. Such an interesting ``finite difference`` theory, forwards - at the classical level - a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz`s and Dirac`s approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and - at the quantum level - yields a remarkable mass spectrum for leptons. After having briefly reviewed Caldirola`s approach, our first aim is to work out, discuss, and compare to one another the new representations of Quantum Mechanics (QM) resulting from it, in the Schroedinger, Heisenberg and density-operator (Liouville-von Neumann) pictures, respectively. Moreover, for each representation, three (retarded, symmetric and advanced) formulations are possible, which refer either to times t and t-{tau}{sub 0}, or to times t-{tau}{sub 0}/2 and t+{tau}{sub 0}/2, or to times t and t+{tau}{sub 0}, respectively. It is interesting to notice that, when the chronon tends to zero, the ordinary QM is obtained as the limiting case of the ``symmetric`` formulation only; while the ``retarded`` one does naturally appear to describe QM with friction, i.e., to describe dissipative quantum systems (like a particle moving in an absorbing medium). In this sense, discretized QM is much richer than the ordinary one. We also obtain the (retarded) finite-difference Schroedinger equation within the Feynman path integral approach, and study some of its relevant solutions. We then derive the time-evolution operators of this discrete theory, and use them to get the finite-difference Heisenberg equations. When discussing the mutual compatibility of the various pictures listed above, we find that they can be written down in a form such that they result to be equivalent (as it happens in the ``continuous`` case of ordinary QM), even if the Heisenberg picture cannot be derived by ``discretizing`` directly the ordinary Heisenberg representation. Afterwards, some typical applications and examples are studied, as the free particle, the harmonic oscillator and the hydrogen atom; and various cases are pointed out, for which the predictions of discrete QM differ from those expected from ``continuous`` QM. At last, the density matrix formalism is applied to the solution of the measurement problem in QM, with very interesting results, as for instance a natural explication of ``decoherence``, which reveal the power of discretized (in particular, retarded) QM. (author) 86 refs, 11 figs}
place = {IAEA}
year = {1998}
month = {Jul}
}
title = {Introduction of a quantum of time (``chronon``) and its consequences for quantum mechanics}
author = {Farias, R H.A., and Recami, E}
abstractNote = {We discuss the consequences of the introduction of a quantum of time {tau}{sub 0} in the formalism of non-relativistic quantum mechanics, by referring ourselves, in particular, to the theory of the chronon as proposed by P. Caldirola. Such an interesting ``finite difference`` theory, forwards - at the classical level - a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz`s and Dirac`s approaches (and even allowing a clear answer to the question whether a free falling charged particle does or does not emit radiation), and - at the quantum level - yields a remarkable mass spectrum for leptons. After having briefly reviewed Caldirola`s approach, our first aim is to work out, discuss, and compare to one another the new representations of Quantum Mechanics (QM) resulting from it, in the Schroedinger, Heisenberg and density-operator (Liouville-von Neumann) pictures, respectively. Moreover, for each representation, three (retarded, symmetric and advanced) formulations are possible, which refer either to times t and t-{tau}{sub 0}, or to times t-{tau}{sub 0}/2 and t+{tau}{sub 0}/2, or to times t and t+{tau}{sub 0}, respectively. It is interesting to notice that, when the chronon tends to zero, the ordinary QM is obtained as the limiting case of the ``symmetric`` formulation only; while the ``retarded`` one does naturally appear to describe QM with friction, i.e., to describe dissipative quantum systems (like a particle moving in an absorbing medium). In this sense, discretized QM is much richer than the ordinary one. We also obtain the (retarded) finite-difference Schroedinger equation within the Feynman path integral approach, and study some of its relevant solutions. We then derive the time-evolution operators of this discrete theory, and use them to get the finite-difference Heisenberg equations. When discussing the mutual compatibility of the various pictures listed above, we find that they can be written down in a form such that they result to be equivalent (as it happens in the ``continuous`` case of ordinary QM), even if the Heisenberg picture cannot be derived by ``discretizing`` directly the ordinary Heisenberg representation. Afterwards, some typical applications and examples are studied, as the free particle, the harmonic oscillator and the hydrogen atom; and various cases are pointed out, for which the predictions of discrete QM differ from those expected from ``continuous`` QM. At last, the density matrix formalism is applied to the solution of the measurement problem in QM, with very interesting results, as for instance a natural explication of ``decoherence``, which reveal the power of discretized (in particular, retarded) QM. (author) 86 refs, 11 figs}
place = {IAEA}
year = {1998}
month = {Jul}
}