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The Philosophy of Vacuum Part 10

The Philosophy of Vacuum Part 10. 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 83 extended over both halves of the mass shell 18 we obtain the point field A x X wXp Mp e J 0 r ip xlh àfl f s JE 0 r wr p br p e ip x f dp . Here the summation is over the two linearly independent spin states and p is the invariant measure on the mass shells which includes constants in h and it . It is usual to rewrite this by letting r run over four values one pair for each sign of the energy r 1 2 for the positive-mass shell and r 3 4 for the negative-mass shell i.e. for positive-energy solutions wr p wr p2 m2c2 1 2 p and for negative-energy solutions wr 2 p wr - p2 m2c2 1 2 p and similarly for b . One can then carry out the integral over the energy p0 obtaining the familiar form iA x 27rft -3 2 f X MpMp e Jp3 _ 1.2 d3 _ V2 o X M P r -P eip x r 3 4 ip xjh p here p0 p2 m2c2 1 2 the normalization is wrws 2p0 cbrs . We now consider the canonical second quantized operators. By formal application of the transformation theory using improper space of states . In that case p includes the bispinor. This is what we shall do in Section 7. If one treats instead the quantities br p as annihilation operators then they must act on the Fock space over the s J spinor representations constructed by Wigner 1939 and not over the solution space of the Dirac equation . This feature of the standard formalism is perhaps not a technicality when it comes to perturbation theory in kinematics however it is well understood. 18 For the time being we can consider the Fourier transform an application of the transformation theory in the canonical theory viewed in this way we must expand over a complete set of states therefore over both positive- and negative-energy states. 84 S . Saunders position eigenstates19 any operator A x which is local in the 1-particle theory a multiplicative function or finite derivative in configuration space coordinates can be written in the form jEi x x i x d3x. Applied to the 1-particle Hamiltonian H one obtains after some manipulation dr 7