1,114 research outputs found
Phase transitions in spinor quantum gravity on a lattice
We construct a well-defined lattice-regularized quantum theory formulated in
terms of fundamental fermion and gauge fields, the same type of degrees of
freedom as in the Standard Model. The theory is explicitly invariant under
local Lorentz transformations and, in the continuum limit, under
diffeomorphisms. It is suitable for describing large nonperturbative and
fast-varying fluctuations of metrics. Although the quantum curved space turns
out to be on the average flat and smooth owing to the non-compressibility of
the fundamental fermions, the low-energy Einstein limit is not automatic: one
needs to ensure that composite metrics fluctuations propagate to long distances
as compared to the lattice spacing. One way to guarantee this is to stay at a
phase transition.
We develop a lattice mean field method and find that the theory typically has
several phases in the space of the dimensionless coupling constants, separated
by the second order phase transition surface. For example, there is a phase
with a spontaneous breaking of chiral symmetry. The effective low-energy
Lagrangian for the ensuing Goldstone field is explicitly
diffeomorphism-invariant. We expect that the Einstein gravitation is achieved
at the phase transition. A bonus is that the cosmological constant is probably
automatically zero.Comment: 37 pages, 12 figures Discussion of dimensions and of the
Berezinsky--Kosterlitz--Thouless phase adde
The quantum dilogarithm and representations quantum cluster varieties
We construct, using the quantum dilogarithm, a series of *-representations of
quantized cluster varieties. This includes a construction of infinite
dimensional unitary projective representations of their discrete symmetry
groups - the cluster modular groups. The examples of the latter include the
classical mapping class groups of punctured surfaces.
One of applications is quantization of higher Teichmuller spaces.
The constructed unitary representations can be viewed as analogs of the Weil
representation. In both cases representations are given by integral operators.
Their kernels in our case are the quantum dilogarithms.
We introduce the symplectic/quantum double of cluster varieties and related
them to the representations.Comment: Dedicated to David Kazhdan for his 60th birthday. The final version.
To appear in Inventiones Math. The last Section of the previous versions was
removed, and will become a separate pape
Breaking so(4) symmetry without degeneracy lift
We argue that in the quantum motion of a scalar particle of mass "m" on S^3_R
perturbed by the trigonometric Scarf potential (Scarf I) with one internal
quantized dimensionless parameter, \ell, the 3D orbital angular momentum, and
another, an external scale introducing continuous parameter, B, a loss of the
geometric hyper-spherical so(4) symmetry of the free motion can occur that
leaves intact the unperturbed {\mathcal N}^2-fold degeneracy patterns, with
{\mathcal N}=(\ell +n+1) and n denoting the nodes number of the wave function.
Our point is that although the number of degenerate states for any {\mathcal N}
matches dimensionality of an irreducible so(4) representation space, the
corresponding set of wave functions do not transform irreducibly under any
so(4). Indeed, in expanding the Scarf I wave functions in the basis of properly
identified so(4) representation functions, we find power series in the
perturbation parameter, B, where 4D angular momenta K\in [\ell , {\mathcal
N}-1] contribute up to the order \left(\frac{2mR^2B}{\hbar^2}\right)^{{\mathcal
N}-1-K}. In this fashion, we work out an explicit example on a symmetry
breakdown by external scales that retains the degeneracy. The scheme extends to
so(d+2) for any d.Comment: Prepared for the proceedings of the conference "Lie Theory and Its
Applications In Physics", June 17-23, 2013, Varna, Bulgari
On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models
We consider two families of commuting Hamiltonians on the cotangent bundle of
the group GL(n,C), and show that upon an appropriate single symplectic
reduction they descend to the spectral invariants of the hyperbolic Sutherland
and of the rational Ruijsenaars-Schneider Lax matrices, respectively. The
duality symplectomorphism between these two integrable models, that was
constructed by Ruijsenaars using direct methods, can be then interpreted
geometrically simply as a gauge transformation connecting two cross sections of
the orbits of the reduction group.Comment: 16 pages, v2: comments and references added at the end of the tex
The two-dimensional hydrogen atom revisited
The bound state energy eigenvalues for the two-dimensional Kepler problem are
found to be degenerate. This "accidental" degeneracy is due to the existence of
a two-dimensional analogue of the quantum-mechanical Runge-Lenz vector.
Reformulating the problem in momentum space leads to an integral form of the
Schroedinger equation. This equation is solved by projecting the
two-dimensional momentum space onto the surface of a three-dimensional sphere.
The eigenfunctions are then expanded in terms of spherical harmonics, and this
leads to an integral relation in terms of special functions which has not
previously been tabulated. The dynamical symmetry of the problem is also
considered, and it is shown that the two components of the Runge-Lenz vector in
real space correspond to the generators of infinitesimal rotations about the
respective coordinate axes in momentum space.Comment: 10 pages, no figures, RevTex
The Central Correlations of Hypercharge, Isospin, Colour and Chirality in the Standard Model
The correlation of the fractionally represented hypercharge group with the
isospin and colour group in the standard model determines as faithfully
represented internal group the quotient group
{\U(1)\x\SU(2)\x\SU(3)\over\Z_2\x\Z_3}. The discrete cyclic central
abelian-nonabelian internal correlation involved is considered with respect to
its consequences for the representations by the standard model fields, the
electroweak mixing angle and the symmetry breakdown. There exists a further
discrete -correlation between chirality and Lorentz properties and also a
continuous \U(1)-external-internal one between hyperisospin and chirality.Comment: 18 pages, latex, macros include
Quantum Theory and Galois Fields
We discuss the motivation and main results of a quantum theory over a Galois
field (GFQT). The goal of the paper is to describe main ideas of GFQT in a
simplest possible way and to give clear and simple arguments that GFQT is a
more natural quantum theory than the standard one. The paper has been prepared
as a presentation to the ICSSUR' 2005 conference (Besancon, France, May 2-6,
2005).Comment: Latex, 24 pages, 1 figur
Geometric Measure of Indistinguishability for Groups of Identical Particles
The concept of p-orthogonality (1=< p =< n) between n-particle states is
introduced. It generalizes common orthogonality, which is equivalent to
n-orthogonality, and strong orthogonality between fermionic states, which is
equivalent to 1-orthogonality. Within the class of non p-orthogonal states a
finer measure of non p-orthogonality is provided by Araki's angles between
p-internal spaces. The p-orthogonality concept is a geometric measure of
indistinguishability that is independent of the representation chosen for the
quantum states. It induces a new hierarchy of approximations for group function
methods. The simplifications that occur in the calculation of matrix elements
between p-orthogonal group functions are presented
A Chiral Spin Theory in the Framework of an Invariant Evolution Parameter Formalism
We present a formulation for the construction of first order equations which
describe particles with spin, in the context of a manifestly covariant
relativistic theory governed by an invariant evolution parameter; one obtains a
consistent quantized formalism dealing with off-shell particles with spin. Our
basic requirement is that the second order equation in the theory is of the
Schr\"{o}dinger-Stueckelberg type, which exhibits features of both the
Klein-Gordon and Schr\"{o}dinger equations. This requirement restricts the
structure of the first order equation, in particular, to a chiral form. One
thus obtains, in a natural way, a theory of chiral form for massive particles,
which may contain both left and right chiralities, or just one of them. We
observe that by iterating the first order system, we are able to obtain second
order forms containing the transverse and longitudinal momentum relative to a
time-like vector used to maintain covariance of the theory.
This time-like vector coincides with the one used by Horwitz, Piron, and Reuse
to obtain an invariant positive definite space-time scalar product, which
permits the construction of an induced representation for states of a particle
with spin. We discuss the currents and continuity equations, and show that
these equations of motion and their currents are closely related to the spin
and convection parts of the Gordon decomposition of the Dirac current. The
transverse and longitudinal aspects of the particle are complementary, and can
be treated in a unified manner using a tensor product Hilbert space.
Introducing the electromagnetic field we find an equation which gives rise to
the correct gyromagnetic ratio, and is fully Hermitian under the proposed
scalar product. Finally, we show that the original structure of Dirac'sComment: Latex, 61 pages. Minor revisions. To be published in J. Math. Phy
Pseudospectral Calculation of the Wavefunction of Helium and the Negative Hydrogen Ion
We study the numerical solution of the non-relativistic Schr\"{o}dinger
equation for two-electron atoms in ground and excited S-states using
pseudospectral (PS) methods of calculation. The calculation achieves
convergence rates for the energy, Cauchy error in the wavefunction, and
variance in local energy that are exponentially fast for all practical
purposes. The method requires three separate subdomains to handle the
wavefunction's cusp-like behavior near the two-particle coalescences. The use
of three subdomains is essential to maintaining exponential convergence. A
comparison of several different treatments of the cusps and the semi-infinite
domain suggest that the simplest prescription is sufficient. For many purposes
it proves unnecessary to handle the logarithmic behavior near the
three-particle coalescence in a special way. The PS method has many virtues: no
explicit assumptions need be made about the asymptotic behavior of the
wavefunction near cusps or at large distances, the local energy is exactly
equal to the calculated global energy at all collocation points, local errors
go down everywhere with increasing resolution, the effective basis using
Chebyshev polynomials is complete and simple, and the method is easily
extensible to other bound states. This study serves as a proof-of-principle of
the method for more general two- and possibly three-electron applications.Comment: 23 pages, 20 figures, 2 tables, Final refereed version - Some
references added, some stylistic changes, added paragraph to matrix methods
section, added last sentence to abstract
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