2 research outputs found
A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues
It is proved that a compact Kahler manifold whose Ricci tensor has two
distinct, constant, non-negative eigenvalues is locally the product of two
Kahler-Einstein manifolds. A stronger result is established for the case of
Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant
eigenvalues of the Ricci tensor are shown to exist in various situations: there
are homogeneous examples of any complex dimension n > 1, if one eigenvalue is
negative and the other positive or zero, and of any complex dimension n > 2, if
the both eigenvalues are negative; there are non-homogeneous examples of
complex dimension 2, if one of the eigenvalues is zero. The problem of
existence of Kahler metrics whose Ricci tensor has two distinct, constant
eigenvalues is related to the celebrated (still open) Goldberg conjecture.
Consequently, the irreducible homogeneous examples with negative eigenvalues
give rise to complete, Einstein, strictly almost Kahler metrics of any even
real dimension greater than 4.Comment: 18 pages; final version; accepted for publication in International
Journal of Mathematic
PT-symmetry, indefinite metric, and nonlinear quantum mechanics
If a Hamiltonian of a quantum system is symmetric under space-time reflection,
then the associated eigenvalues can be real. A conjugation operation for quantum
states can then be defined in terms of space-time reflection, but the resulting Hilbert
space inner product is not positive definite and gives rise to an interpretational
difficulty. One way of resolving this difficulty is to introduce a superselection rule that
excludes quantum states having negative norms. It is shown here that a quantum
theory arising in this way gives an example of Kibble’s nonlinear quantum mechanics,
with the property that the state space has a constant negative curvature. It then
follows from the positive curvature theorem that the resulting quantum theory is not
physically viable. This conclusion also has implications to other quantum theories
obtained from the imposition of analogous superselection rules