36 research outputs found
A Discrete Time Presentation of Quantum Dynamics
Inspired by the discrete evolution implied by the recent work on loop quantum
cosmology, we obtain a discrete time description of usual quantum mechanics
viewing it as a constrained system. This description, obtained without any
approximation or explicit discretization, mimics features of the discrete time
evolution of loop quantum cosmology. We discuss the continuum limit, physical
inner product and matrix elements of physical observables to bring out various
issues regarding viability of a discrete evolution. We also point out how a
continuous time could emerge without appealing to any continuum limit.Comment: 20 pages, RevTex, no figures. Additional Clarifications added.
Version accepted for publication in Class. Quant. Gra
Principles of Discrete Time Mechanics: IV. The Dirac Equation, Particles and Oscillons
We apply the principles of discrete time mechanics discussed in earlier
papers to the first and second quantised Dirac equation. We use the Schwinger
action principle to find the anticommutation relations of the Dirac field and
of the particle creation operators in the theory. We find new solutions to the
discrete time Dirac equation, referred to as oscillons on account of their
extraordinary behaviour. Their principal characteristic is that they oscillate
with a period twice that of the fundamental time interval T of our theory.
Although these solutions can be associated with definite charge, linear
momentum and spin, such objects should not be observable as particles in the
continuous time limit. We find that for non-zero T they correspond to states
with negative squared norm in Hilbert space. However they are an integral part
of the discrete time Dirac field and should play a role in particle
interactions analogous to the role of longitudinal photons in conventional
quantum electrodynamics.Comment: 27 pages LateX; published versio
Principles of Discrete Time Mechanics: I. Particle Systems
We discuss the principles to be used in the construction of discrete time
classical and quantum mechanics as applied to point particle systems. In the
classical theory this includes the concept of virtual path and the construction
of system functions from classical Lagrangians, Cadzow's variational principle
applied to the action sum, Maeda-Noether and Logan invariants of the motion,
elliptic and hyperbolic harmonic oscillator behaviour, gauge invariant
electrodynamics and charge conservation, and the Grassmannian oscillator. First
quantised discrete time mechanics is discussed via the concept of system
amplitude, which permits the construction of all quantities of interest such as
commutators and scattering amplitudes. We discuss stroboscopic quantum
mechanics, or the construction of discrete time quantum theory from continuous
time quantum theory and show how this works in detail for the free Newtonian
particle. We conclude with an application of the Schwinger action principle to
the important case of the quantised discrete time inhomogeneous oscillator.Comment: 35 pages, LateX, To be published in J.Phys.A: Math.Gen. Basic
principles stated: applications to field theory in subsequent papers of
series contact email address: [email protected]
Factorization and Entanglement in Quantum Systems
We discuss the question of entanglement versus separability of pure quantum
states in direct product Hilbert spaces and the relevance of this issue to
physics. Different types of separability may be possible, depending on the
particular factorization or split of the Hilbert space. A given orthonormal
basis set for a Hilbert space is defined to be of type (p,q) if p elements of
the basis are entangled and q are separable, relative to a given bi-partite
factorization of that space. We conjecture that not all basis types exist for a
given Hilbert space.Comment: 11 page
Principles of Discrete Time Mechanics: II. Classical field Theory
We apply the principles discussed in an earlier paper to the construction of
discrete time field theories. We derive the discrete time field equations of
motion and Noether's theorem and apply them to the Schrodinger equation to
illustrate the methodology. Stationary solutions to the discrete time
Schrodinger wave equation are found to be identical to standard energy
eigenvalue solutions except for a fundamental limit on the energy. Then we
apply the formalism to the free neutral Klein Gordon system, deriving the
equations of motion and conserved quantities such as the linear momentum and
angular momentum. We show that there is an upper bound on the magnitude of
linear momentum for physical particle-like solutions. We extend the formalism
to the charged scalar field coupled to Maxwell's electrodynamics in a gauge
invariant way. We apply the formalism to include the Maxwell and Dirac fields,
setting the scene for second quantisation of discrete time mechanics and
discrete time Quantum Electrodynamics.Comment: 23 pages, LateX, To be published in J.Phys.A: Math.Gen: contact email
address: [email protected]