Schwinger’s theory of angular momentum

There are a variety of formalisms that allow for a mapping of a discrete quantum system onto a continuous analog (for reviews see Refs. 218 and 219). The most prominent examples are Schwinger s theory of angular momentum [98] and the Holstein-Primakoff transformation [97], both of which allow a continuous representation of spin degrees of freedom. To discuss these two theories, we consider a spin DoF that is described by the spin operators Si,S2,Si with commutation relations... 

Consider, furthermore, a (2i- - 1)-dimensional subspace of the Hilbert space with fixed 5. Then, according to Schwinger s theory of angular momentum [98], this discrete spin DoF can be represented by two bosonic oscillators described by creation and annihilation operators with commutation relations... 

In obvious analogy to Schwinger s theory of angular momentum, this N-level system can be represented by N oscillators, whereby the mapping relations for the operator and the basis states read [99]... 

The basic idea of the mapping approach is to change from the discrete representation employed in Eq. (53) to a continuous representation. There are several ways to do so, most of them are based on the representation of spin operators by boson operators. Well-known examples of such mappings are the Holstein-Primakoff transformation, which represents a spin system by a single nonlinear boson DoF, and Schwinger s theory of angular momentum,which represents a spin system by two independent boson DoF. 

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