Can time be a discrete dynamical variable?

Lee, T.D.

doi:10.1016/0370-2693(83)90687-1

Cited by 173 publications

(125 citation statements)

References 3 publications

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“…an expression that is very similar to (16). Infact as was argued in detail [5] the imaginary parts of both (16) Once such a minimum spacetime scale is invoked, then we have a non commutative geometry as shown by Snyder more than fifty years ago [8,9]:…”

Section: The Approaches Of Barut and Hestenesmentioning

confidence: 92%

“…Indeed such minimum spacetime models were studied for several decades, precisely to overcome the divergences encountered in Quantum Field Theory [5], [9]- [14], [15,16]. Before proceeding further, it may be remarked that when the square of a, which we will take to be the Compton wavelength (including the Planck scale, which is a special case of the Compton scale for a Planck mass viz., 10 −5 gm), can be neglected, then we return to point Quantum Theory and the usual commutative geometry.…”

Section: The Approaches Of Barut and Hestenesmentioning

confidence: 99%

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Revisiting Zitterbewegung

Sidharth

2008

Int J Theor Phys

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The Dirac wave equation for the electron soon lead to the recognition of the Zitterbewegung. This was studied both by Schrodinger and Dirac. Later there were further elegant and sometimes dissenting insights, from different authors. We briefly review some of these developments. However in more recent times with dark energy and noncommutative spacetime coming to centre stage, the earlier studies of Zitterbewegung become questionable.

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Section: The Approaches Of Barut and Hestenesmentioning

confidence: 92%

Section: The Approaches Of Barut and Hestenesmentioning

confidence: 99%

Revisiting Zitterbewegung

Sidharth

2008

Int J Theor Phys

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“…There have been a few attempts in the literature that developed theories with time as a discrete parameter (such as [19][20][21][22][23][24][25][26][27]). To describe the time evolution of Hamiltonian systems with time as a discrete parameter Lee introduced time as a discrete dynamical variable [22].…”

Section: Introductionmentioning

confidence: 99%

A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time

Riek

2014

Entropy

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The basic microsopic physical laws are time reversible. In contrast, the second law of thermodynamics, which is a macroscopic physical representation of the world, is able to describe irreversible processes in an isolated system through the change of entropy S > 0. It is the attempt of the present manuscript to bridge the microscopic physical world with its macrosocpic one with an alternative approach than the statistical mechanics theory of Gibbs and Boltzmann. It is proposed that time is discrete with constant step size. Its consequence is the presence of time irreversibility at the microscopic level if the present force is of complex nature (F (r) = const). In order to compare this discrete time irreversible mechamics (for simplicity a "classical", single particle in a one dimensional space is selected) with its classical Newton analog, time reversibility is reintroduced by scaling the time steps for any given time step n by the variable s n leading to the Nosé-Hoover Lagrangian. The corresponding Nosé-Hoover Hamiltonian comprises a term N df k B T ln s n (k B the Boltzmann constant, T the temperature, and N df the number of degrees of freedom) which is defined as the microscopic entropyS n at time point n multiplied by T . Upon ensemble averaging this microscopic entropy S n in equilibrium for a system which does not have fast changing forces approximates its macroscopic counterpart known from thermodynamics. The presented derivation with the resulting analogy between the ensemble averaged microscopic entropy and its thermodynamic analog suggests that the original description of the entropy by Boltzmann and Gibbs is just an ensemble averaging of the time scaling variable s n which is in equilibrium close to 1, but that the entropy term itself has its root not in statistical mechanics but rather in the discreteness of time.

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“…In usual approach (see, e.g. [7], [8], [9], [10], [11]), only the discrete dependent variables are taken as the independent variational variables, while their differences are not. However, in the discrete variational principle proposed by two of the present authors and their collaborators recently [3], [4], [5], [6], the differences of the dependent variables as the discrete counterparts of derivatives are taken as the independent variational variables together with the discrete dependent variables themselves.…”

Section: Introductionmentioning

confidence: 99%

“…The approach has been applied to the symplectic and multisymplectic algorithms in the both Lagrangian and Hamiltonian formalism. It has been also generalized to the case of variable steps in order to preserve the discrete energy in addition to the symplectic or the multisymplectic structure [6], [7], [8], [9], [12].…”

Section: Introductionmentioning

confidence: 99%

Difference discrete connection and curvature on cubic lattice

Wang

Guo

2006

SCI CHINA SER A

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In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems.

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Can time be a discrete dynamical variable?

Cited by 173 publications

References 3 publications

Revisiting Zitterbewegung

Revisiting Zitterbewegung

A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time

Difference discrete connection and curvature on cubic lattice

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