In particular, we see that if the Riemann zeta-function has a zero at s _ 0, then the determinant of the scattering matrix has a pole at s _ 0 / 2, and hence the Selberg zeta-function has a zero at s _ 0 / 2.
32.
This means that in relativity, the product of the free real spin-0 field has a " nonzero " vacuum expectation value, because in addition to creating particles which are not annihilated and annihilating particles which are not subsequently created, it also includes a part that creates and annihilates " virtual " particles whose existence enters into interaction calculations-but never as scattering matrix indices or asymptotic states.
33.
This is the Fock space we want for " asymptotic " states of the theory; even though we will not generally succeed in choosing the particular final field configuration to which the gauge-fixed " Lagrangian " dynamics would have evolved that initial configuration, the singularity of the inner product along BRST exact degrees of freedom ensures that we will get the right entries for the physical scattering matrix.
34.
Another way of looking at the nilpotence of the BRST operator is to say that its closed forms ) . ( The " true " Lagrangian, presumed to be invariant under local gauge transformations, is in the kernel of the BRST operator but not in its image . ) The preceding argument says that we can limit our universe of initial and final conditions to asymptotic " states " field configurations at timelike infinity, where the interaction Lagrangian is " turned off " that lie in the kernel of " Q B " and still obtain a unitary scattering matrix . ( BRST closed and exact states are defined similarly to BRST closed and exact fields; closed states are annihilated by " Q B ", while exact states are those obtainable by applying " Q B " to some arbitrary field configuration .)
