SCALING BEHAVIOR IN THE BOSE-GAS

The scaling behaviour of fluctuations of the Bose fields phi(f) in the ergodic infinite volume equilibrium states of a d-dimensional Bose ga s at temperature T and density <(rho)over bar>, can be classified in t erms of the testfunctions f. In the low density regime, the space of t estfunctions splits up in two subspaces, leading to two different type s of non-commuting macroscopic field fluctuation observables. Testfunc tions f with Fourier transform (f) over cap(0)not equal 0 yield normal fluctuation observables. The local fluctuations of the field operator s phi(f) must be scaled subnormally (i.e. with a negative scaling inde x) if the testfunction f has (f) over cap(0) = 0. The macroscopic fluc tuations of these fields can then again be described by a Bose field. The situation changes when the density of the gas exceeds the critical density. The field operators which have normal fluctuations in the lo w density regime need to be scaled abnormally in the high density regi me, yielding classical macroscopic fluctuation observables. Another di fference with the low density regime is that the space of testfunction s with (f) over cap(0) = 0 splits up in two subspaces when the critica l density is reached: for a first subspace the algebraic character of the macroscopic field fluctuation observables is also classical becaus e it is necessary to scale the fluctuations of the field operators nor mally, while for the remaining subclass, the same negative scaling ind ex is required as in the low density regime and hence also the algebra ic character of these macroscopic fluctuations is again CCR.