International Journal of Geometric Methods in Modern Physics

Let M be an n-dimensional complex manifold endowed with a C^∞ Kähler metric g. We show that a certain Laplace-type integral ℒ_m(x), when x varies in a sufficiently small open set U ⊂ M, has an asymptotic expansion , where C_r:C^∞(U) → C^∞(U) are smooth differential operators depending on the curvature of g and its covariant derivatives. As a consequence we furnish a different proof of Lu's theorem by computing the lower order terms of Tian–Yau–Zelditch expansion in terms of the operator C_j. Finally, we compute the differential operators Q_j of the expansion Ber_m(f) = Σ_r≥0m^-rQ_r(f) of Berezin's transform in terms of the operators C_j.

Augmented variational principles are introduced in order to provide a definition of relative conservation laws. As it is physically reasonable, relative conservation laws define in turn relative conserved quantities that measure, for example, how much energy is needed in a field theory to go from one configuration (called the reference or vacuum) to another configuration (the physical state of the system). The general prescription we describe solves in a covariant way the well-known observer dependence of conserved quantities. The solution found is deeply related to the divergence ambiguity of the Lagrangian and to various formalisms that have recently appeared in literature to deal with the variation of conserved quantities (of which this is a formal integration). A number of examples relevant to fundamental physics are considered in detail, starting from classical mechanics.

We prove Wigner–Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representations is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra U_t(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of U_t(su(2)) is given.

The construction of a N_T = 3 cohomological gauge theory on the hyper-Kähler eight-fold, whose group theoretical description was given previously by Blau and Thompson [1], is performed explicitly.

To obtain the explicit form of evolution operator in the Tavis–Cummings model we must calculate the term e^-itg(S₊⊗a+S_-⊗ a† explicitly which is very hard. In this paper we try to make the quantum matrix A ≡ S₊ ⊗ a +S_- ⊗ a† diagonal to calculate e^-itgA and, moreover, to know a deep structure of the model.

For the case of one, two and three atoms we give such a diagonalization which is the first nontrivial examples as far as we know, and reproduce the calculations of e^-itgA given in quant-ph/0404034. We also give a hint to an application to non-commutative differential geometry.

However, a quantum diagonalization is not unique and is affected by some ambiguity arising from the non-commutativity of operators in quantum physics.

Our method may open a new point of view in mathematical or quantum physics.

We define a graded multiplication on the vector space of essential paths on a graph G (a tree) and show that it is associative. In most interesting applications, this tree is an ADE Dynkin diagram. The vector space of length-preserving endomorphisms of essential paths has a grading obtained from the length of paths and possesses several interesting bialgebra structures. One of these, the Double Triangle Algebra (DTA) of A. Ocneanu, is a particular kind of quantum groupoid (a weak Hopf algebra) and was studied elsewhere; its coproduct gives a filtrated convolution product on the dual vector space. Another bialgebra structure is obtained by replacing this filtered convolution product by a graded associative product. It can be obtained from the former by projection on a subspace of maximal grade, but it is interesting to define it directly, without using the DTA. What is obtained is a weak bialgebra, not a weak Hopf algebra.

Let X be strongly q-convex domain of an n-dimensional Kähler manifold M and E be a holomorphic vector bundle over M. Then, if E satisfies certain positivity conditions, we prove vanishing theorems for the -cohomology groups of X with values in E.

A general theory is given for quantizing both constrained and unconstrained systems with second-order Lagrangian, using the WKB approximation. In constrained systems, the constraints become conditions on the wave function to be satisfied in the semiclassical limit. This is illustrated with two examples.

The derivation of the standard model from a higher-dimensional action suggests a further study of the fiber bundle formulation of gauge theories to determine the variations in the choice of structure group that are allowed in this geometrical setting. The action of transformations on the projection of fibers to their submanifolds are characteristic of theories with fewer gauge vector bosons, and specific examples are given, which may have phenomenological relevance. The spinor space for the three generations of fermions in the standard model is described algebraically.