6,073 research outputs found
Wigner distributions for finite state systems without redundant phase point operators
We set up Wigner distributions for state quantum systems following a
Dirac inspired approach. In contrast to much of the work on this case,
requiring a phase space, particularly when is even, our
approach is uniformly based on an phase space grid and thereby
avoids the necessity of having to invoke a `quadrupled' phase space and hence
the attendant redundance. Both odd and even cases are analysed in detail
and it is found that there are striking differences between the two. While the
odd case permits full implementation of the marginals property, the even
case does so only in a restricted sense. This has the consequence that in the
even case one is led to several equally good definitions of the Wigner
distributions as opposed to the odd case where the choice turns out to be
unique.Comment: Latex, 14 page
Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle
It is shown that a real symmetric [complex hermitian] positive
definite matrix is congruent to a diagonal matrix modulo a
pseudo-orthogonal [pseudo-unitary] matrix in [ ], for any
choice of partition . It is further shown that the method of proof in
this context can easily be adapted to obtain a rather simple proof of
Williamson's theorem which states that if is even then is congruent
also to a diagonal matrix modulo a symplectic matrix in
[]. Applications of these results considered include a
generalization of the Schweinler-Wigner method of `orthogonalization based on
an extremum principle' to construct pseudo-orthogonal and symplectic bases from
a given set of linearly independent vectors.Comment: 7 pages, latex, no figure
Classical Light Beams and Geometric Phases
We present a study of geometric phases in classical wave and polarisation
optics using the basic mathematical framework of quantum mechanics. Important
physical situations taken from scalar wave optics, pure polarisation optics,
and the behaviour of polarisation in the eikonal or ray limit of Maxwell's
equations in a transparent medium are considered. The case of a beam of light
whose propagation direction and polarisation state are both subject to change
is dealt with, attention being paid to the validity of Maxwell's equations at
all stages. Global topological aspects of the space of all propagation
directions are discussed using elementary group theoretical ideas, and the
effects on geometric phases are elucidated.Comment: 23 pages, 1 figur
The Sampling Theorem and Coherent State Systems in Quantum Mechanics
The well known Poisson Summation Formula is analysed from the perspective of
the coherent state systems associated with the Heisenberg--Weyl group. In
particular, it is shown that the Poisson summation formula may be viewed
abstractly as a relation between two sets of bases (Zak bases) arising as
simultaneous eigenvectors of two commuting unitary operators in which geometric
phase plays a key role. The Zak bases are shown to be interpretable as
generalised coherent state systems of the Heisenberg--Weyl group and this, in
turn, prompts analysis of the sampling theorem (an important and useful
consequence of the Poisson Summation Formula) and its extension from a coherent
state point of view leading to interesting results on properties of von Neumann
and finer lattices based on standard and generalised coherent state systems.Comment: 20 pages, Late
Wigner-Weyl isomorphism for quantum mechanics on Lie groups
The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie
group is developed in detail. Several New features are shown to arise which
have no counterparts in the familiar Cartesian case. Notable among these is the
notion of a `semiquantised phase space', a structure on which the Weyl symbols
of operators turn out to be naturally defined and, figuratively speaking,
located midway between the classical phase space and the Hilbert space
of square integrable functions on . General expressions for the star product
for Weyl symbols are presented and explicitly worked out for the angle-angular
momentum case.Comment: 32 pages, Latex2
Assessing the climate impacts of Chinese dietary choices using a telecoupled global food trade and local land use framework
Global emissions trajectories developed to meet the 2⁰C temperature target are likely to rely on the widespread deployment of negative emissions technologies and/or the implementation of substantial terrestrial carbon sinks. Such technologies include afforestation, carbon capture and storage (CCS) and bioenergy with carbon capture and storage (BECCS), but mitigation options for agriculture appear limited. For example, using the Global Calculator tool (http://www.globalcalculator.org/), under a 2⁰C pathway, the ‘forests and other land use’ sector is projected to become a major carbon sink, reaching -15 GtCO2e yr-1 by 2050, compared to fossil emissions of 21 GtCO2e yr-1. At the same time, rates of agricultural emissions remain static at about 6 GtCO2e yr-1, despite increasing demands for crop and livestock production to meet the forecast dietary demands of the growing and increasingly wealthy global population. Emissions in the Global Calculator are sensitive to the assumed global diet, and particularly to the level and type of meat consumption, which in turn drive global land use patterns and agricultural emissions. Here we assess the potential to use a modified down-scaled Global Calculator methodology embedded within the telecoupled global food trade framework, to estimate the agricultural emissions and terrestrial carbon stock impacts in China and Brazil, arising from a plausible range of dietary choices in China. These dietary choices are linked via telecoupling mechanisms to Brazilian crop production (e.g. Brazilian soy for Chinese animal feed provision) and drive land and global market dynamics. ‘Spill-over’ impacts will also be assessed using the EU and Malawi as case studies
Wigner distributions for finite dimensional quantum systems: An algebraic approach
We discuss questions pertaining to the definition of `momentum', `momentum
space', `phase space', and `Wigner distributions'; for finite dimensional
quantum systems. For such systems, where traditional concepts of `momenta'
established for continuum situations offer little help, we propose a physically
reasonable and mathematically tangible definition and use it for the purpose of
setting up Wigner distributions in a purely algebraic manner. It is found that
the point of view adopted here is limited to odd dimensional systems only. The
mathematical reasons which force this situation are examined in detail.Comment: Latex, 13 page
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