A careful analysis of the roots of the difficulties lead Penrose [309, 313] (see also [372, 235, 377
]) to
suggest the modified definition for the kinematical twistor
These anomalies lead Penrose to modify slightly [310]. This modified form is based on Tod’s form of
the kinematical twistor:
A beautiful property of the original construction was its connection with the Hamiltonian formulation of the
theory [265]. Unfortunately, such a simple Hamiltonian interpretation is lacking for the modified
constructions. Although the form of Equation (58
) is that of the integral of the Nester-Witten 2-form, and
the spinor fields
and
could still be considered as the spinor constituents of the
‘quasi-Killing vectors’ of the 2-surface
, their structure is not so simple because the factor
itself
depends on all of the four independent solutions of the 2-surface twistor equation in a rather complicated
way.
To have a simple Hamiltonian interpretation Mason suggested further modifications [265, 266]. He
considers the four solutions
,
, of the 2-surface twistor equations, and uses these solutions
in the integral (55
) of the Nester-Witten 2-form. Since
is a Hermitian bilinear form on the space of
the spinor fields (see Section 8 below), he obtains 16 real quantities as the components of the
Hermitian matrix
. However, it is not clear how the four ‘quasi-translations’
of
should be found among the 16 vector fields
(called ‘quasi-conformal Killing
vectors’ of
) for which the corresponding quasi-local quantities could be considered as the
quasi-local energy-momentum. Nevertheless, this suggestion leads us to the next class of quasi-local
quantities.
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