International Journal of Geometric Methods in Modern Physics

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing scalar invariants.

It is shown that any finite complete covering of a non-commutative algebra in the sense of Calow and Matthes [16] gives rise to a Galois coring.

We write down the local equations that characterize the submanifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of TM endowed with the tangent Dirac structure. In the Poisson case, these formulas once again prove a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure if and only if N is a Dirac submanifold. In the presymplectic case it is the isotropy of the normal bundle which characterizes the corresponding notion of a Dirac submanifold. On the way, we give a simple definition of the tangent Dirac structure, make new remarks about it and establish its characteristic, local formulas for various interesting classes of submanifolds of a Dirac manifold.

We study the regular representation ρ_ζ of the single-fermion algebra , i.e., c² = c⁺² = 0, cc⁺ + c⁺c = ζ1, for ζ ∈ [0,1]. We show that ρ₀ is a four-dimensional nonunitary representation of which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, ρ₀ is the minimal faithfully irreducible representation of in the sense that every faithful representation of has a subrepresentation that is equivalent to ρ₀. We therefore identify a classical fermion with ρ₀ and view its quantization as the deformation: ζ : 0 → 1 of ρ_ζ. The latter has the effect of mapping ρ₀ into the four-dimensional, unitary, (faithfully) reducible representation ρ₁ of that is reminiscent of a Dirac fermion.

This is a challenging paper that includes some reviews and new results.

Since the non-commutative version of the classical system based on the compact group SU(2) has been constructed in (quant-ph/0502174) by making use of Jaynes–Commings model and so-called quantum diagonalization method in (quant-ph/0502147), we construct a non-commutative version of the classical system based on the non-compact group SU(1,1) by modifying the compact case.

In this model the Hamiltonian is not hermite but pseudo hermite, which causes a big difference between the two models. For example, in the classical representation theory of SU(1,1), unitary representations are infinite dimensional from the starting point. Therefore, to develop a unitary theory of non-commutative system of SU(1,1) we need an infinite number of non-commutative systems, which means a kind of second non-commutativization. This is a very hard and interesting problem.

We develop a corresponding theory though it is not always enough, and present some challenging problems concerning how classical properties can be extended to the non-commutative case.

This paper is arranged for the convenience of readers as the first subsection is based on the standard model (SU(2) system) and the next one is based on the non-standard model (SU(1,1) system). This contrast may make the similarities and differences between the standard and non-standard models clearer.

The axiomatic approach to parallel transport theory is partially discussed. Bijective correspondences between the sets of connections, (axiomatically defined) parallel transports and transports along paths that satisfy some additional conditions are constructed. In particular, the equivalence between the concepts "connection" and "(axiomatically defined) parallel transport" is established.

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds.

The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fiber bundle (which will be identified with the first-order jet bundle and the multi-momentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied.

Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analyzed.

Given a generic Lagrangian system, its Euler–Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condition holds, one can associate to a differential operator the exact chain complex with the boundary operator whose nilpotency restarts all the Noether identities characterizing the degeneracy of an original differential operator.

Lagrangian and Hamiltonian descriptions of the dynamics of a self-gravitating matter shell in General Relativity are discussed in general. The case of a spherical shell composed of an elastic fluid is then considered, its Lagrangian function is derived from first principles and the Hamiltonian is calculated. Known results for dust shells are recovered as particular cases.

A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles ("quantum bundles") over the space–time manifold yield a quantum "formalism" along any 1-dimensional timelike submanifold (a "detector"); in the flat, inertial case this reproduces the basic results of the usual quantum field theory, while in general it could be seen as a local, "linearized" description of the actual physics.

We discuss the alternative algebraic structures on the manifold of quantum states arising from the alternative Hermitian structures associated with the quantum bi-Hamiltonian systems. We also consider the consequences at the level of the Heisenberg picture in terms of the deformations of the associative product on the space of observables.

Most of the known models describing the fundamental interactions have a gauge freedom. In the standard path integral, it is necessary to "fix the gauge" in order to avoid integrating over unphysical degrees of freedom. Gauge independence might then become a tricky issue, especially when the structure of the gauge symmetries is intricate. In the modern approach to this question, it is the BRST invariance that effectively implements the gauge invariance. This set of lectures briefly reviews some key ideas underlying the BRST-antifield formalism, which yields a systematic procedure to path-integrate any type of gauge system, while (usually) manifestly preserving space–time covariance. The quantized theory possesses a global invariance under the so-called BRST transformation, which is nilpotent of order two. The cohomology of the BRST differential is the central element that controls the physics. Its relationship with the observables is sketched and explained. How anomalies appear in the "quantum master equation" of the antifield formalism is also discussed. These notes are based on the lectures given by MH at the 10th Saalburg Summer School on Modern Theoretical Methods from the 30th of August to the 10th of September, 2004 in Wolfersdorf, Germany and were prepared by AF and AM. The exercises which were discussed at the school are also included.

Based on the ideas of Marsden–Ratiu, a reduction method for Lie algebroids is developed in such a way that the canonical projection onto the reduced Lie algebroid is a homomorphism of Lie algebroids. A relation between Poisson reduction and Lie algebroid reduction is established. Reduction of Lie algebroids with symmetry is also studied using this method.

We shall review some results about the globality of Chern–Simons theory in any odd dimension. The relation between Chern–Simons theories and some recent results about conservation laws are also considered in detail.