We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for the limiting case of the estimates. MSC(2000): 53C25, 53C27, 58B20

The first edition of these notes was written by Professor Dyson. The second edition was prepared by Michael J. Moravcsik; he is responsible for the changes made in the process of re-editing. A * : complex conjugate transposed (Hermitian conjugate) A + : complex conjugate (not transposed) A : A * β = A * γ 4 = adjoint A −1 = inverse A T = transposed I = identity matrix or operator i

The momentum operator for a particle in a box is represented by an infinite order Hermitian matrix P . Its square P 2 is well defined (and diagonal), but its cube P 3 is ill defined, because P P 2 6= P 2 P . Truncating these matrices to a finite order restores the associative law, but leads to other curious results. Classification 0260 (02.10.Sp)

We propose construction of a unique and definite metric (η +), time-reversal operator (T) and an inner product such that the pseudo-Hermitian matrix Hamiltonians are C, PT, CPT invariant and PT(CPT)-norm is indefinite (definite). Here, P and C denote the generalized symmetries : parity and charge-conjugation respectively. The limitations of the other current approaches have been brought out.

We provide a careful analysis of the generating functional in the path integral formulation of pseudo-Hermitian and in particular PT -symmetric quantum mechanics and show how the metric operator enters the expression for the generating functional. PACS number: 03.65.-w

Numerical evaluation of the overlap Dirac operator is difficult since it contains the sign function e(Hw) of the Hermitian Wilson-Dirac operator Hw with a negative mass term. The problems are due to Hw having very small eigenvalues on the equilibrium background configurations generated in current day Monte Carlo simulations. Since these are a consequence of the lattice discretization and do not...

The long standing problem of the ordering ambiguity in the definition of the Hamilton operator for a point particle in curved space is naturally resolved by using the powerful geometric calculus based on Clifford Algebra. The momentum operator is defined to be the vector derivative (the gradient) multiplied by −i; it can be expanded in terms of basis vectors γμ as p = −iγ∂μ. The product of two ...

Abstract We propose a new boson expansion method using norm operator. The small parameter expansion, in which the approximation becomes zeroth-order approximation, requires double commutation relations between phonon operators that are not closed excitation modes adopted as excitations. This results an infinite regardless of whether type is Hermitian or non-Hermitian. does hold when closed. ope...