1,613 research outputs found
Relativity in Introductory Physics
A century after its formulation by Einstein, it is time to incorporate
special relativity early in the physics curriculum. The approach advocated here
employs a simple algebraic extension of vector formalism that generates
Minkowski spacetime, displays covariant symmetries, and enables calculations of
boosts and spatial rotations without matrices or tensors. The approach is part
of a comprehensive geometric algebra with applications in many areas of
physics, but only an intuitive subset is needed at the introductory level. The
approach and some of its extensions are given here and illustrated with
insights into the geometry of spacetime.Comment: 29 pages, 5 figures, several typos corrected, some discussion
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Overcoming the su(2^n) sufficient condition for the coherent control of n-qubit systems
We study quantum systems with even numbers N of levels that are completely
state-controlled by unitary transformations generated by Lie algebras
isomorphic to sp(N) of dimension N(N+1)/2. These Lie algebras are smaller than
the respective su(N) with dimension N^2-1. We show that this reduction
constrains the Hamiltonian to have symmetric energy levels. An example of such
a system is an n-qubit system. Using a geometric representation for the quantum
wave function of a finite system, we present an explicit example that shows a
two-qubit system can be controlled by the elements of the Lie algebra sp(4)
(isomorphic to spin(5) and so(5)) with dimension ten rather than su(4) with
dimension fifteen. These results enable one to envision more efficient
algorithms for the design of fields for quantum-state engineering, and they
provide more insight into the fundamental structure of quantum control.Comment: 13 pp., 2 figure
Photon position operators and localized bases
We extend a procedure for construction of the photon position operators with
transverse eigenvectors and commuting components [Phys. Rev. A 59, 954 (1999)]
to body rotations described by three Euler angles. The axial angle can be made
a function of the two polar angles, and different choices of the functional
dependence are analogous to different gauges of a magnetic field. Symmetries
broken by a choice of gauge are re-established by transformations within the
gauge group. The approach allows several previous proposals to be related.
Because of the coupling of the photon momentum and spin, our position operator,
like that proposed by Pryce, is a matrix that does not commute with the spin
operator. Unlike the Pryce operator, however, our operator has commuting
components, but the commutators of these components with the total angular
momentum require an extra term to rotate the matrices for each vector component
around the momentum direction. Several proofs of the nonexistence of a photon
position operator with commuting components are based on overly restrictive
premises that do not apply here
Relativity in Clifford's Geometric Algebras of Space and Spacetime
Of the various formalisms developed to treat relativistic phenomena, those
based on Clifford's geometric algebra are especially well adapted for clear
geometric interpretations and computational efficiency. Here we study
relationships between formulations of special relativity in the spacetime
algebra (STA) Cl{1,3} of Minkowski space, and in the algebra of physical space
(APS)Cl{3}. STA lends itself to an absolute formulation of relativity, in which
paths, fields, and other physical properties have observer-independent
representations. Descriptions in APS are related by a one-to-one mapping of
elements from APS to the even subalgebra STA+ of STA. With this mapping,
reversion in APS corresponds to hermitian conjugation in STA. The elements of
STA+ are all that is needed to calculate physically measurable quantities
because only they entail the observer dependence inherent in any physical
measurement. As a consequence, every relativistic physical process that can be
modeled in STA also has a representation in APS, and vice versa. In the
presence of two or more inertial observers, two versions of APS present
themselves. In the absolute version, both the mapping to STA+ and hermitian
conjugation are observer dependent, and the proper basis vectors are persistent
vectors that sweep out timelike planes. In the relative version, the mapping
and hermitian conjugation are then the same for all observers. Relative APS
gives a covariant representation of relativistic physics with spacetime
multivectors represented by multiparavectors. We relate the two versions of APS
as consistent models within the same algebra.Comment: 22 pages, no figure
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