Homogeneous Differential Geometry / Based on Examples
(with Dr. Alexander Levichev)
Spring 2011: we meet twice a week (Mon at 5pm, Thu at 5pm) at MCS B23
Fall 2010: we have been meeting twice a week at MCS 149.
Fall 04 additional features: meets at the SPb University (Saint Petersburg, Russia).
Meets individually (to initiate a meeting or/and to get help with understanding Chronometry: send an e-mail to levit@math.bu.edu)
Also, find CHRONOMETRY BULLETIN (below)
Lie groups, Lie algebras. Riemannian and Lorentzian spaces. Isometries. Lie algebras of vector fields. Lie algebra of a Lie group. Representations of Lie groups and Lie algebras. Induced vector bundles. Imporant Lie group actions. Standard and Chronometric description of elementary particles (to learn more about that topic, get to the "Chronometry" file on this very website).
All are invited. Topics for original research are provided. Prereqs: MA 242 (Linear Algebra), MA 225 (Multivariate Calculus), MA 226 (Differential Equations).
For more information contact Alex Levichev It is advised that a (free) BU coursepack (see its content
below) be requested directly from A. Levichev.
Content of the Coursepack: Examples of Lie groups, Lie algebras.
Lie algebra of a Lie group. Riemannian and Lorentzian spaces. Lobachevskii
dim=2 geometry of a homogeneous Riemannian space. Action of a Lie group
and tangent representation of its Lie algebra. Lorentzian spaces ("space-
times"). Causality. Einstein equations. Calculation of curvature. Groups
which are of most importance to theoretical physics. Vector bundles, induced
vector bundles. Two dim=1 bundles over a circle.
Chronometric Bulletin # 5 [6/13/04]
The article "Russian Troika as the new spatio-temporal
paradigm" has been posted on the Moscow State University
site http://www.chronos.msu.ru where one has to go to
the library of electronic publications.
Chronometric Bulletin # 4 (9/03/03)
8/30/03 it was a Special Meeting of the Seminar, a Tribute to
Irving Segal (September 13, 1918 - August 30, 1998). Professor
Daigneault (Univ of Montreal) as the Moderator, A. Levichev as the
Key Speaker. It was the first public presentation of the
"DLF-development of Segal's Chronometric Theory" (see Bulletin # 3 below).
Here is the content of Levichev's talk:
Newton-Einstein-Segal ascent. Causal structure, conformal and
Poincare groups. Lie-algebraic methods: matrices and vector fields.
Invariant form in a Lie algebra, a bi-invariant metric on a group.
Representations, particles, wave equations. Lie algebras d,l,f; the
worlds D,L,F. These three worlds (as subgroups of the conformal group)
with dim=4 orbits in U(2). Explicit realization of K=Iso(D), of H=Iso(F).
Search for a realization of Iso(L). Parallelizations of vector bundles.
Chronometric Bulletin #3 (7/19/03).
Big news of the last few months: two more worlds
have "mathematically shown" themselves. All three worlds have
been talked about since ancient times (see http://agniyoga.org):
Dense, D (where we currently live in), L ("light", the Subtle), F (the Fiery).
These very letters are used below to denote their simplest mathematical models.
D is just the Segal's chronometric world (see Levichev's webpage). Mathematically,
it is a universal cover D of the matrix group U(2) together with
a bi-invariant (Lorentzian) inner product on it. It has a constant
positive scalar curvature. Actually, the mathematical knowledge of L, F
was already available in early 80s (published in several articles). I
was then less aware of their physical significance.
The Fiery World is a universal cover F of one other matrix group U(1,1)
(with the respective bi-invariant inner product).
It is of constant negative scalar curvature. The world L (also based on a
certain group - no other such groups exist in dimension 4) has a name
of "plane wave Einstein equations solution", it is a very special space-time
among worlds of General Relativity. The curvature tensor is not zero but the
scalar curvature is zero. No surprise that it plays a role of a mediator
between D and F (we go to L when we "die"). Roughly speaking, the three worlds
share the same causal structure. They INTERACT, and, as a result, our
mathematical description of D would not be quite adequate without L, F
(especially, without F). That is why I compare it to QM "hidden variables"
showing themselves.
Chronometry Release 1 (a *.doc file, in Russian, August 02) can be
requested from A. Levichev.
Chronometry Release 2
(October 01)
(Dirk Kreimer lectures at BU but there seems to be a more radical
approach to get rid of divergencies in QFT).
From [Connes, Kreimer(98): Hopf Algebras, Renormalization...],
p.9:
"The renormalization procedure appears as the qure for the
disease caused by the unavoidable presence of UV divergences
in QFTs which describe the physics of local quantized fields."
From [Segal, Constructive non-linear QFT in four space-time
dimensions, in Perspectives on quantization (South Hadley, MA,
1996), AMS, Providence, RI, 1998, pp.145-156]:
"The apparent divergences and conceptual problems arise from
one main reason, but there is also a secondary aspect. The
primary reason is that there is a certain degeneracy about
Minkowski space-time, - the vanishing of curvature, essentially,
- which encourages singularities and the appearance of
non-existence for nontrivial QFTs living in it. In fact, the
least bit of positive curvature can make all the difference.
This runs quite contrary to the old "conventional wisdom" that
the ultraviolet (i.e. high frequency) divergences can not be
fixed by any local, let alone global, alteration in the
structure of space-time. The secondary aspect is the
traditional limitation in mathematical investigations to the
case of particles (or fields) of a definite mass.
Experimentally, the mass shows a dispersion or "width", in
all but massless or extremely stable particles, but even the
latter may have a dispersion in mass that is simply below the
threshold for observability. The incorporation of nontrivial
mass width into the theory is mathematically interesting as
well as empirically essential.
QFT can be modified to deal with these considerations in a
way that is consistent with the general physical principles
listed above. Given theories, such as the classic prototype
of QED, can then be correlated in a natural way with rigorous
mathematical objects. Locally interacting quantized systems
of an infinite number of degrees of freedom exist
rigorously..." (A.L.: in, say, Massless QFTs and the
nontriviality of phi(4,4), Nuclear Physics B 376 (1992),
129-142, by Pedersen, Segal, Zhou).
From [Segal, Zhou; Convergence of Quantum Electrodynamics in
a Curved Deformation of Minkowski Space, Annals of Physics,
232, 61-87, 1994]: Attempts to explain or reduce the
divergences of QED have a long history... But renormalization
theory did not begin to address the foundational issue, and
Schwinger (Selected Papers on QED, 1958) wrote in his
introduction to a collection of key articles in the development
of renormalization theory, "We conclude that a convergent
theory can not be formulated consistently within the framework
of present space-time concepts."
Again by Segal, Zhou (from the above article): "We show that
QED becomes convergent when the conventional energy and mass
operators in Minkowski space are modified by the introduction
of a fundamental length R. The limiting case R --> infinity of the
modified theory coinsides formally with standard relativistic
QED... The interaction takes the usual trilinear form
corresponding to the Maxwell-Dirac equations. The interaction
and total hamiltonians then become well-defined self-adjoint
operators on the tensor product of the electron and photon
quantized field Hilbert spaces... The observable implications
of the modified theory appear formally indistinquishable from
those of conventional theory, apart from ambiguities resulting
from the divergeness of the latter, assuming that R is at
least of the cosmic distance scale as estimated from redshift
observations..."