We extend the twistor methods developed in our earlier work on linear deformations of hyperkähler manifolds [1] to the case of quaternionic-Kähler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-Kähler manifold M are in one-to-one correspondence with deformations of its 4d+ 4-dimensional hyperkähler cone S. The latter can be encoded in variations of the comple...

Quaternionic Hilbert spaces play an important role in applied physical sciences especially quantum physics. In this paper, the operator valued frames on quaternionic are introduced and studied. terms of a class partial isometries spaces, parametrization Parseval is obtained. We extend to many properties vector process. Moreover, we show that all can be obtained from single frame. Finally, sever...

It is shown (1) that Lanczos’s quaternionic formulation of Dirac’s equation does not lead to a solution of the problems that plague the standard spin 3/2 theory based on the Rarita-Schwinger equation, but (2) that the four-component solutions to the quaternionic generalization of Dirac’s equation proposed by Lanczos in 1929 may provide a consistent theory for spin 3/2 particles, although at the...

The collection of all quaternions is denoted byH and is called the real quaternionic algebra. This algebra was first introduced by Hamilton in 1843 (see [5, 6]), and is often called the Hamilton quaternionic algebra. It is well known thatH is an associative division algebra over R. For any a= a0 + a1i+ a2 j + a3k ∈H, the conjugate of a = a0 + a1i + a2 j + a3k is defined to be a = a0 − a1i− a2 j...

In this talk I shall first make some brief remarks on quaternionic quantum mechanics, and then describe recent work with A.C. Millard in which we show that standard complex quantum field theory can arise as the statistical mechanics of an underlying noncommutative dynamics. In quaternionic quantum mechanics, the Dirac transition amplitudes 〈ψ|φ〉 are quaternion valued, that is, they have the for...