The Philosophy of Vacuum Part 14. 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 Vacuum on Null Planes 123 21a can be traced to the functional dependence of f x x x and fi x x x which renders the standard canonical commutation relations internally inconsistent on null planes. Since the canonical conjugate to the field on a null plane is determined by the field itself on the same null plane it appears that the field over all the points of the null plane comprises an irreducible set of operators. In other words every operator in the state space of the field-theoretic system can be written as a null-plane generalized functional of the field itself. In particular the null-time derivative of the field d f x x x is not required in the functionals. The dynamical circumstance that permits this is the first-order character of the equations of motion for the field in null-plane coordinates. Thus the wave equation x2 x 0 28a becomes 23 d_ 32 k2 x x x 0. 28b This equation can be integrated over x to yield arbitrary function of x and x . 29 The arbitrary function is set equal to zero on the ground that a non-zero value would contribute infinite energy to the states of the system. With this elimination we see that d j x x x is itself a functional of the field over the x null plane thus confirming the irreducibility of the null-plane field algebra. 4 Vacuum Structure on Null Planes There are two intuitively natural ways to define a vacuum state for a quantum-field-theoretic system. One is that the vacuum is the state of lowest energy which energy can conventionally be taken to be zero. The existence of such a state necessarily an energy eigenstate presumes the energy spectrum of the field system to have a lower bound. Such a lower bound seems required to prevent runaway collapse of the world modelled by the system. The second intuitive 124 G. N. Fleming definition of the vacuum state is that it is the state devoid of field quanta . the state that is transformed to the null vector 0 by application of any annihilation operator. Both of these definitions
