Abstract
The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength.
Highlights
Starting with the pioneering work of Dirac [1], investigations of the Hamiltonian formulation of space-time physics have been pursued by numerous authors
We look closely at the role of probability in the course of our development of a relativistic theory of quantum measurement based on the geometry of the future tube
Since the physical state of a system is defined up to an overall phase, we deduce that the manifold of coherent states is invariant under the 15-parameter conformal group
Summary
Starting with the pioneering work of Dirac [1], investigations of the Hamiltonian formulation of space-time physics have been pursued by numerous authors. We propose an alternative approach in which the future tube of complexified Minkowski space is taken to be the phase space of a relativistic system. The points of Γ correspond to lines that lie entirely in the top half of CP3 In both twistor theory and quantum field theory, the complexification of Minkowski space, natural as it may be, is introduced primarily to enable one to exploit the tools of complex analysis in relation to the positive frequency condition on fields; and there is no direct physical significance attached as such to the imaginary components of complex space-time points. It is often the case, that the measurement postulates of nonrelativistic quantum theory are used in a relativistic setup to deduce implications of the postulates, which is unsatisfactory, for what is required is a measurement postulate in a relativistic setup, as we propose here
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