348 research outputs found
Towards the Unification of Gravity and other Interactions: What has been Missed?
Faced with the persisting problem of the unification of gravity with other
fundamental interactions we investigate the possibility of a new paradigm,
according to which the basic space of physics is a multidimensional space
associated with matter configurations. We consider general
relativity in . In spacetime, which is a 4-dimensional subspace of
, we have not only the 4-dimensional gravity, but also other
interactions, just as in Kaluza-Klein theories. We then consider a finite
dimensional description of extended objects in terms of the center of mass,
area, and volume degrees of freedom, which altogether form a 16-dimensional
manifold whose tangent space at any point is Clifford algebra Cl(1,3). The
latter algebra is very promising for the unification, and it provides
description of fermions.Comment: 11 pages; Talk presented at "First Mediterranean Conference on
Classical and Quantum Gravity", Kolymbari, Crete, Greece, 14-18 September
200
New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
I show how there is an ambiguity in how one treats auxiliary variables in
gauge theories including general relativity cast as 3 + 1 geometrodynamics.
Auxiliary variables may be treated pre-variationally as multiplier coordinates
or as the velocities corresponding to cyclic coordinates. The latter treatment
works through the physical meaninglessness of auxiliary variables' values
applying also to the end points (or end spatial hypersurfaces) of the
variation, so that these are free rather than fixed. [This is also known as
variation with natural boundary conditions.] Further principles of dynamics
workings such as Routhian reduction and the Dirac procedure are shown to have
parallel counterparts for this new formalism. One advantage of the new scheme
is that the corresponding actions are more manifestly relational. While the
electric potential is usually regarded as a multiplier coordinate and Arnowitt,
Deser and Misner have regarded the lapse and shift likewise, this paper's
scheme considers new {\it flux}, {\it instant} and {\it grid} variables whose
corresponding velocities are, respectively, the abovementioned previously used
variables. This paper's way of thinking about gauge theory furthermore admits
interesting generalizations, which shall be provided in a second paper.Comment: 11 page
Triangleland. I. Classical dynamics with exchange of relative angular momentum
In Euclidean relational particle mechanics, only relative times, relative
angles and relative separations are meaningful. Barbour--Bertotti (1982) theory
is of this form and can be viewed as a recovery of (a portion of) Newtonian
mechanics from relational premises. This is of interest in the absolute versus
relative motion debate and also shares a number of features with the
geometrodynamical formulation of general relativity, making it suitable for
some modelling of the problem of time in quantum gravity. I also study
similarity relational particle mechanics (`dynamics of pure shape'), in which
only relative times, relative angles and {\sl ratios of} relative separations
are meaningful. This I consider firstly as it is simpler, particularly in 1 and
2 d, for which the configuration space geometry turns out to be well-known,
e.g. S^2 for the `triangleland' (3-particle) case that I consider in detail.
Secondly, the similarity model occurs as a sub-model within the Euclidean
model: that admits a shape--scale split. For harmonic oscillator like
potentials, similarity triangleland model turns out to have the same
mathematics as a family of rigid rotor problems, while the Euclidean case turns
out to have parallels with the Kepler--Coulomb problem in spherical and
parabolic coordinates. Previous work on relational mechanics covered cases
where the constituent subsystems do not exchange relative angular momentum,
which is a simplifying (but in some ways undesirable) feature paralleling
centrality in ordinary mechanics. In this paper I lift this restriction. In
each case I reduce the relational problem to a standard one, thus obtain
various exact, asymptotic and numerical solutions, and then recast these into
the original mechanical variables for physical interpretation.Comment: Journal Reference added, minor updates to References and Figure
Quantum Cosmological Relational Model of Shape and Scale in 1-d
Relational particle models are useful toy models for quantum cosmology and
the problem of time in quantum general relativity. This paper shows how to
extend existing work on concrete examples of relational particle models in 1-d
to include a notion of scale. This is useful as regards forming a tight analogy
with quantum cosmology and the emergent semiclassical time and hidden time
approaches to the problem of time. This paper shows furthermore that the
correspondence between relational particle models and classical and quantum
cosmology can be strengthened using judicious choices of the mechanical
potential. This gives relational particle mechanics models with analogues of
spatial curvature, cosmological constant, dust and radiation terms. A number of
these models are then tractable at the quantum level. These models can be used
to study important issues 1) in canonical quantum gravity: the problem of time,
the semiclassical approach to it and timeless approaches to it (such as the
naive Schrodinger interpretation and records theory). 2) In quantum cosmology,
such as in the investigation of uniform states, robustness, and the qualitative
understanding of the origin of structure formation.Comment: References and some more motivation adde
Foundations of Relational Particle Dynamics
Relational particle dynamics include the dynamics of pure shape and cases in
which absolute scale or absolute rotation are additionally meaningful. These
are interesting as regards the absolute versus relative motion debate as well
as discussion of conceptual issues connected with the problem of time in
quantum gravity. In spatial dimension 1 and 2 the relative configuration spaces
of shapes are n-spheres and complex projective spaces, from which knowledge I
construct natural mechanics on these spaces. I also show that these coincide
with Barbour's indirectly-constructed relational dynamics by performing a full
reduction on the latter. Then the identification of the configuration spaces as
n-spheres and complex projective spaces, for which spaces much mathematics is
available, significantly advances the understanding of Barbour's relational
theory in spatial dimensions 1 and 2. I also provide the parallel study of a
new theory for which positon and scale are purely relative but orientation is
absolute. The configuration space for this is an n-sphere regardless of the
spatial dimension, which renders this theory a more tractable arena for
investigation of implications of scale invariance than Barbour's theory itself.Comment: Minor typos corrected; references update
Approaching the Problem of Time with a Combined Semiclassical-Records-Histories Scheme
I approach the Problem of Time and other foundations of Quantum Cosmology
using a combined histories, timeless and semiclassical approach. This approach
is along the lines pursued by Halliwell. It involves the timeless probabilities
for dynamical trajectories entering regions of configuration space, which are
computed within the semiclassical regime. Moreover, the objects that Halliwell
uses in this approach commute with the Hamiltonian constraint, H. This approach
has not hitherto been considered for models that also possess nontrivial linear
constraints, Lin. This paper carries this out for some concrete relational
particle models (RPM's). If there is also commutation with Lin - the Kuchar
observables condition - the constructed objects are Dirac observables.
Moreover, this paper shows that the problem of Kuchar observables is explicitly
resolved for 1- and 2-d RPM's. Then as a first route to Halliwell's approach
for nontrivial linear constraints that is also a construction of Dirac
observables, I consider theories for which Kuchar observables are formally
known, giving the relational triangle as an example. As a second route, I apply
an indirect method that generalizes both group-averaging and Barbour's best
matching. For conceptual clarity, my study involves the simpler case of
Halliwell 2003 sharp-edged window function. I leave the elsewise-improved
softened case of Halliwell 2009 for a subsequent Paper II. Finally, I provide
comments on Halliwell's approach and how well it fares as regards the various
facets of the Problem of Time and as an implementation of QM propositions.Comment: An improved version of the text, and with various further references.
25 pages, 4 figure
Triangleland. II. Quantum Mechanics of Pure Shape
Relational particle models are of value in the absolute versus relative
motion debate. They are also analogous to the dynamical formulation of general
relativity, and as such are useful for investigating conceptual strategies
proposed for resolving the problem of time in quantum general relativity.
Moreover, to date there are few explicit examples of these at the quantum
level. In this paper I exploit recent geometrical and classical dynamics work
to provide such a study based on reduced quantization in the case of pure shape
(no scale) in 2-d for 3 particles (triangleland) with multiple harmonic
oscillator type potentials. I explore solutions for these making use of exact,
asymptotic, perturbative and numerical methods. An analogy to the mathematics
of the linear rigid rotor in a background electric field is useful throughout.
I argue that further relational models are accessible by the methods used in
this paper, and for specific uses of the models covered by this paper in the
investigation of the problem of time (and other conceptual and technical
issues) in quantum general relativity.Comment: Journal Reference added, minor updates to References and Figure
Quantum Machian Time in Toy Models of Gravity
General Relativity on closed spatial topologies can be derived, using a
technique called "best-matching", as an evolving 3-geometry subject to
constraints. These constraints can be thought of as a way of imposing temporal
and spatial relationalism. The same type of constraints can be used in
non-relativistic particle models to produce relational theories that suffer
from the same Problem of Time as that encountered in General Relativity. As a
result, these simple toy models are well suited for studying the Problem of
Time in quantum gravity. In this paper, a version of these particle models is
studied where we "best-match" the time translational invariance of the theory.
Using insights gained from this procedure, we can move back and forth between
absolute and relational time by changing the way in which the relational fields
are varied. We then proceed to quantize this theory using Dirac and path
integral quantizations. We discover that one of the constraints of the theory,
which we call the Mach constraint, is responsible for removing the dependence
of the theory on a background structure. It is this Mach constraint that is
responsible for making the theory temporally relational. Because of the deep
relationship between these models and General Relativity, this work may shed
new light on the Problem of Time in quantum gravity and how one might expect
time to emerge on quantum subsystems of the universe.Comment: 27 pages, references added, typos fixed, additional comments added to
abs/intro/concl, journal ref adde
Emergent Semiclassical Time in Quantum Gravity. I. Mechanical Models
Strategies intended to resolve the problem of time in quantum gravity by
means of emergent or hidden timefunctions are considered in the arena of
relational particle toy models. In situations with `heavy' and `light' degrees
of freedom, two notions of emergent semiclassical WKB time emerge; these are
furthermore equivalent to two notions of emergent classical
`Leibniz--Mach--Barbour' time. I futhermore study the semiclassical approach,
in a geometric phase formalism, extended to include linear constraints, and
with particular care to make explicit those approximations and assumptions
used. I propose a new iterative scheme for this in the cosmologically-motivated
case with one heavy degree of freedom. I find that the usual semiclassical
quantum cosmology emergence of time comes hand in hand with the emergence of
other qualitatively significant terms, including back-reactions on the heavy
subsystem and second time derivatives. I illustrate my analysis by taking it
further for relational particle models with linearly-coupled harmonic
oscillator potentials. As these examples are exactly soluble by means outside
the semiclassical approach, they are additionally useful for testing the
justifiability of some of the approximations and assumptions habitually made in
the semiclassical approach to quantum cosmology. Finally, I contrast the
emergent semiclassical timefunction with its hidden dilational Euler time
counterpart.Comment: References Update
The physical gravitational degrees of freedom
When constructing general relativity (GR), Einstein required 4D general
covariance. In contrast, we derive GR (in the compact, without boundary case)
as a theory of evolving 3-dimensional conformal Riemannian geometries obtained
by imposing two general principles: 1) time is derived from change; 2) motion
and size are relative. We write down an explicit action based on them. We
obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation but
also all the equations used in York's conformal technique for solving the
initial-value problem. This shows that the independent gravitational degrees of
freedom obtained by York do not arise from a gauge fixing but from hitherto
unrecognized fundamental symmetry principles. They can therefore be identified
as the long-sought Hamiltonian physical gravitational degrees of freedom.Comment: Replaced with published version (minor changes and added references
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