Research Interests
Prof. Dr. Michael
Forger
August 2015
Mathematical Physics
Exact methods and results in mechanics and
field theory, classical and quantum, including integrable
systems.
Current research lines and projects:
1.
General subject: Geometric Field Theory.
The general goal is the further development of
classical field theory - of the general formalism as well as of the treatment
of specific models - from a geometric viewpoint, with emphasis on structural aspects
that are important for quantization.
Project: Covariant Hamiltonian Formalism
for Classical Field Theory.
The
usual hamiltonian formalism
of field theory differs from the lagrangian formulation
in that it breaks manifest covariance (that is, Lorentz covariance in the
setting of special relativity and general covariance in the setting of general
relativity) because it presupposes the a priori choice of a Cauchy hypersurface for the initial data. There are presently two
approaches to overcome this flaw: the multisymplectic or polysymplectic
formalism – a new branch of differential geometry which plays a role
analogous to that of symplectic geometry in classical
mechanics and which in its local form goes back to the work of De Donder and Weyl in the 1930s
– and the functional formalism,
based on the concept of covariant phase
space popularized in the 1980s by Crnkovic and
Witten, which is defined to be the space of solutions of the equations of
motion, rather than the corresponding space of initial data. Best results are
obtained when these two approaches are combined.
Solved
problems: general definition of covariant Poisson brackets, in the multisymplectic and polysymplectic
approach as well as in the functional formalism (Peierls-DeWitt
bracket) [MP-P – 21,24,25,27, MP-S – 1]; general definition of multisymplectic and polysymplectic
structures encompassing the cases of interest in physics and at the same time
allowing to prove a Darboux theorem [MP-P – 30];
classification of multisymplectic and polysymplectic connections and proof of a tubular
neighborhood theorem ŕ
Problems
presently under investigation: description of symmetries in gauge theories,
minimal coupling and Utiyama’s theorem using
Lie groupoids and Lie algebroids
[MP-S – 5], generalizing [MP-P – 29]; hyperbolicity
of systems of first order partial differential equations, aiming for a proof of
existence of Green functions (retarded, advanced and causal) for the linearized DeDonder-Weyl equations
[MP-S – 6]; treatment of systems with constraints and, in particular,
gauge theories.
References: research
papers [MP-P – 21,24,25,27-31] and submitted
articles or articles in preparation [MP-S – 1,4-6].
Project: New Aspects in Noncommutative
Geometry
From
the search for a generalization of the DFR-algebra, proposed in 1995 by Doplicher, Fredenhagen and
Roberts as a model of “quantum space-time”,
from flat space-time (Minkowski space) to curved space-times
with a well-defined causal structure (globally hyperbolic lorentzian
manifolds), there has emerged a general construction of a C*-algebra, which
starts out from an arbitrary Poisson vector bundle, over an arbitrary manifold,
and reproduces the DFR-algebra as a very special case: it is the algebra of
continuous sections vanishing at infinity of a C*-bundle, over that same
manifold, whose fiber at each point is a new version of the “C*-algebra
of the canonical commutation relations”, constructed from the fiber of
the original Poisson vector bundle over that same point using techniques from Rieffel’s “strict deformation quantization”
[FM-S – 2]. The method opens new pathways for noncommutative
geometry since it provides a large class of concrete examples for investigating
the relations between noncommutative topology, described
by C*-algebras (which generalize the commutative algebra of continuous functions
on a topological space), and noncommutative geometry,
described by a certain class of Fréchet *-algebras (which
generalize the commutative algebra of smooth functions on a differentiable manifold):
the hope is that this approach may contribute to clarify one of the central
questions of noncommutative geometry, namely, what exactly
should be this “certain class”. An important and already solved partial
problem in this direction is the quest for an adequate mathematical framework for
a theory of noncommutative topological manifolds:
according to [MP-S – 3], these should be described by sheaves of locally
C*-algebras (which generalize the commutative algebra of continuous functions on
a locally compact but not compact topological space), This eliminating the need
to consider C*-algebras without unit (which generalize the commutative algebra
of continuous functions on a locally compact but not compact topological space
that vanish “at the boundary” or “at infinity”) and makes
the algebraic structure become compatible with the concepts and methods of sheaf
theory. What remains is the problem of “local regularity”, that is,
the question of how to perform the transition from the continuous world to the differentiable
world.
References: submitted
articles or articles in preparation [MP-S – 2,3].
2.
General subject: Integrable Systems.
The general goal is the further development of
the theory of integrable Hamiltonian systems, in
mechanics as well as in two-dimensional field theory, regarding general
structural properties as well as the study of specific models.
Project: New Algebraic Structures in Integrable Systems. (Presently interrupted)
The
latest example of a new algebraic structure encountered in the study of integrable systems is the concept of a dynamical R-matrix.
In contrast to the notion of an ordinary R-matrix, or numerical R-matrix,
as introduced by Yang and Baxter in the late 1960s and early 1970s, which has
become central to an entire new area of mathematics now known under the (somewhat
unfortunate) name "quantum groups", its exact mathematical status is
still not completely clear. First encountered in the mid 1980s in the analysis
of Poisson brackets between the entries of the monodromy
matrix of integrable classical non-linear sigma
models in two space-time dimensions, its investigation has been hampered by
short-distance singularities, which are typical for a field-theoretic problem
such as this one. The situation is more favorable in mechanics, where such
singularities do not appear. Therefore, we have begun a systematic
investigation of dynamical R-matrices for the Calogero
models - the most traditional integrable systems of
mechanics where this new structure arises.
References: research
papers [MP-P – 16,21,22] and conference
proceedings contribution [MP-C – 6].
Biomathematics
Symmetry and symmetry breaking – modern
concepts from physics and mathematics allowing to understand aspects of the
evolution of certain biological systems.
Current research lines and projects:
Project: Symmetry Breaking and the
Evolution of the Genetic Code.
According
to a hypothesis of Hornos and Hornos
published in 1993, the genetic code that steers the synthesis of proteins in
practically all living organisms on our planet may have evolved through a
process accompanied and guided by the phenomenon of symmetry breaking, in a
sequence of steps. The main goal of this project is to classify the possible
symmetries and symmetry breaking schemes that lead to the distribution of multiplets observed in the standard code - a purely
algebraic problem that has already largely been solved. What is still missing,
but would be a more ambitious step, apparently still out of reach, is the
formulation of a dynamical model, based on the theory of equivariant
dynamical systems (dynamical systems with symmetry), since it is well known
that equivariant bifurcations (bifurcations in the
presence of symmetries) generically lead to symmetry breaking.
References: research papers [BM-P
– 1-10] and article in preparation [BM-S – 2].
Project: Coupled Systems of Stochastic
Processes with Symmetry and Gene Expression.
Surprisingly,
an extremely simplified mathematical model for describing the regulation of gene
expression, which consists of two
adequately coupled stochastic processes, exhibits a hidden symmetry involving
the Lie algebra so(2,1), which permits some exact predictions about the
behavior of the system. This raises the question whether it is possible to
arrive at similar conclusions for more complex and, from the biological point
of view, more realistic systems.
References: research
papers [BM-P – 9,11] and submitted article [BM-S
– 1].