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The Philosophy of Vacuum Part 11

The Philosophy of Vacuum Part 11. Physicists will find it extremely interesting, covering, as it does, technical subjects in an accessible way. For those with the necessary expertise, this book will provide an illuminating and authoritative exposition of a many-sided subject." -John D. Barrow, Times Literary Supplement. | The Negative-Energy Sea 93 phase space has a natural correspondence with the 1-particle subspace of I T. In the simplest case the dynamics is simply lifted from the classical Hamiltonian flow on V so that we can regard the induced evolution as the canonical second quantization of a 1-particle evolution. In this way we can preserve a very close correspondence with the 1-particle theory or equivalently with the c-number solutions to the field equations . Indeed although we start from a field theory the relationship of the field to the particle interpretation is the same as in the canonical second quantized theory.30 For interacting theories of physical interest even linear theories one cannot put this construction on any simple basis in particular there does not exist a canonical complex structure J which is preserved under the time evolution. We see that the complex structure J which tells us what we mean by complex numbers in the Hilbert space theory also tells us what we mean by particle number or more generally a particle interpretation for a quantum field. The favourable case roughly coincides with the situation where the field is kinematic it actually includes time-independent external couplings otherwise we shall suppose that interactions lead to a change in J and the particle interpretation is shifted quanta have been created or destroyed. In perturbation theory too one defines the asymptotic states and by the assumption of completeness even the interacting states in terms of the kinematic description of the quantum field. So this is a familiar limitation. What is the kinematic description It is provided by the decomposition of the field into positive- and negativefrequency parts. Here the complex numbers that enter into the classical fields play a crucial role for a given spacelike hypersurface and solution j one finds functions and such that and when T is translated in time the complex phase of these functions on rotates in opposite directions.31 This .