22 research outputs found
On The Geometry of Field Theoretic Gerstenhaber Structures
Field theoretical models with first order Lagrangean can be formulated in a
covariant Hamiltonian formalism. In this article, the geometrical construction
of the Gerstenhaber structure that encodes the equations of motion is explained
for arbitrary fibre bundles. Special emphasis has been put on naturality of the
constructions. Further, the treatment of symmetries is explained. Finally, the
canonical field theoretical 2-form is obtained by pull back and integration of
the polysymplectic form over space like hypersurfaces
A general construction of Poisson brackets on exact multisymplectic manifolds
In this note the long standing problem of the definition of a Poisson bracket
in the framework of a multisymplectic formulation of classical field theory is
solved. The new bracket operation can be applied to forms of arbitary degree.
Relevant examples are discussed and important properties are stated with proofs
sketched.Comment: 8 pages LaTeX, Talk delivered at the 34th Symp. on Math. Phys.,
Torun, Poland, June 200
Geometry of Hamiltonean n-vectors in Multisymplectic Field Theory
Multisymplectic geometry - which originates from the well known de
Donder-Weyl theory - is a natural framework for the study of classical field
theories. Recently, two algebraic structures have been put forward to encode a
given theory algebraically. Those structures are formulated on finite
dimensional spaces, which seems to be surprising at first. In this article, we
investigate the correspondence of Hamiltonian functions and certain
antisymmetric tensor products of vector fields. The latter turn out to be the
proper generalisation of the Hamiltonian vector fields of classical mechanics.
Thus we clarify the algebraic description of solutions of the field equations.Comment: 22 pages, major revision of the introductio
Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory
We present a general classification of Hamiltonian multivector fields and of
Poisson forms on the extended multiphase space appearing in the geometric
formulation of first order classical field theories. This is a prerequisite for
computing explicit expressions for the Poisson bracket between two Poisson
forms.Comment: 50 page
De Donder-Weyl Equations and Multisymplectic Geometry
Multisymplectic geometry is an adequate formalism to geometrically describe
first order classical field theories. The De Donder-Weyl equations are treated
in the framework of multisymplectic geometry, solutions are identified as
integral manifolds of Hamiltonean multivectorfields. In contrast to mechanics,
solutions cannot be described by points in the multisymplectic phase space.
Foliations of the configuration space by solutions and a multisymplectic
version of Hamilton-Jacobi theory are also discussed.Comment: Talk given by H. Roemer at the 33rd Symposium on Mathematical
Physics, Torun, Poland, June 200
The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory
We present a general definition of the Poisson bracket between differential
forms on the extended multiphase space appearing in the geometric formulation
of first order classical field theories and, more generally, on exact
multisymplectic manifolds. It is well defined for a certain class of
differential forms that we propose to call Poisson forms and turns the space of
Poisson forms into a Lie superalgebra.Comment: 40 pages LaTe
Yang-Mills action from minimally coupled bosons on R^4 and on the 4D Moyal plane
We consider bosons on Euclidean R^4 that are minimally coupled to an external
Yang-Mills field. We compute the logarithmically divergent part of the cut-off
regularized quantum effective action of this system. We confirm the known
result that this term is proportional to the Yang-Mills action.
We use pseudodifferential operator methods throughout to prepare the ground
for a generalization of our calculation to the noncommutative four-dimensional
Moyal plane (also known as noncommutative flat space). We also include a
detailed comparison of our cut-off regularization to heat kernel techniques.
In the case of the noncommutative space, we complement the usual technique of
asymptotic expansion in the momentum variable with operator theoretic arguments
in order to keep separated quantum from noncommutativity effects. We show that
the result from the commutative space R^4 still holds if one replaces all
pointwise products by the noncommutative Moyal product.Comment: 37 pages, v2 contains an improved treatment of the theta function in
Appendix A.
A vertical exterior derivative in multisymplectic geometry and a graded Poisson bracket for nontrivial geometries
A vertical exterior derivative is constructed that is needed for a graded
Poisson structure on multisymplectic manifolds over nontrivial vector bundles.
In addition, the properties of the Poisson bracket are proved and first
examples are discussed.Comment: 14 pages LaTeX, needs bbm.sty; final version accepted for publ. in
Rep. on Math. Ph. 4
Singular factorizations, self-adjoint extensions, and applications to quantum many-body physics
We study self-adjoint operators defined by factorizing second order
differential operators in first order ones. We discuss examples where such
factorizations introduce singular interactions into simple quantum mechanical
models like the harmonic oscillator or the free particle on the circle. The
generalization of these examples to the many-body case yields quantum models of
distinguishable and interacting particles in one dimensions which can be solved
explicitly and by simple means. Our considerations lead us to a simple method
to construct exactly solvable quantum many-body systems of Calogero-Sutherland
type.Comment: 17 pages, LaTe