Dimensionality in Field Theory and in Spin Wave Theory

The different meaning of dimensionality and universality in field theory and in spin wave theory is illustrated on account of experimental examples. In spin wave theory it is distinguished between the dimensionality of the spin and the dimensionality of the exchange interactions. According to Renormalization Group (RG) theory, these atomistic characteristics are unimportant for the critical dynamics. Instead by inter-atomic interactions the dynamics of the ordered state is determined by the excitations of the continuous magnetic medium. These excitations are bosons. Consequently, the dimensionality of ordered magnets has to be assessed to the dimensionality of the relevant boson field. The most serious consequence of RG theory is that the magnetic ordering transition also is executed by the boson field. Typical for boson dynamics is a finite width of the critical range. In the atomistic models universality applies asymptotically at $T_{c}$ only. It is evident that the critical power functions of the field dynamics are different from those of the atomistic dynamics.