tailieunhanh - The Philosophy of Vacuum Part 9

The Philosophy of Vacuum Part 9. 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 Ti To make the connection with field theory it is essential that the particles are identical that is the states that are built up by successive applications of the creation operator cannot contain information as to which particle is in which state. This being so the set of occupation numbers a string of in tegers n2 . ni. . . is enough to parameterize these states where each subscript i determines a particular state of the 1-particle theory and each occupation number n fixes the number of particles in that state. We suppose the states form an orthonormal basis for 1 . In terms of this parameterization the action of the annihilation and creation operators is just a 0 nt . - n 1 21n . . n 1 . . . - l 1 2 nx . . n7 l . . . . The normalization constants are slightly different in the antisymmetric case. These operators are adjoints of each other as our notation suggests as a result if one is a linear map on the 1-particle space I the other must be antilinear that is for any complex scalar z a A 2a a A 2a 1 The antilinearity of the annihilation operator is so important that it is helpful to see why it holds in an intuitive way. For a state g e iVs l of the formj f2 f3 f permutations and arbitrary el we have n 1 2 i 2 3 fn permutations. 2 The antilinearity of the annihilation operator is therefore a consequence of the antilinearity of . . in its first entry the hermitean inner product on 1 . Using these operators one can write down the operator on an n- particle Hilbert space which corresponds to a 1-particle operator A on I such that this operator makes no reference to particle identity namely --------------------n-foid tensor sum-------------- A 0 0 0 0 0 . 3 H fold tensor product For an arbitrary orthonormal basis an equivalent definition is 74 S. Saunders ij The important point about this operator is that it duplicates the action of 3 on any -particle subspace . whatever the value of n therefore this expression but not 3 can be used in a .