We explore the properties of the SO(3) Majorana representation of spin. Based on its nonlocal nature, it is shown that there is an equivalence between the SO(3) Majorana representation and the Jordan-Wigner transformation in one and two dimensions. From the relation between the SO(3) Majorana representation and one-dimensional Jordan-Wigner transformation, we show that application of the SO(3) Majorana representation usually results in
Z2 gauge structure. Based on lattice Chern-Simons gauge theory, it is shown that the anticommuting link variables in the SO(3) Majorana representation make it equivalent to an operator form of compact
U(1)1 Chern-Simons Jordan-Wigner transformation in 2D. As examples of its application, we discuss two spin models, namely the quantum XY model on a honeycomb lattice and the ${90}^{\ensuremath{\circ}}$ compass model on a square lattice. It is shown that under the SO(3) Majorana representation both spin models can be exactly mapped into
Z2 gauge theory of spinons, with the standard form of
Z2 Gauss law constraint.
We present a method for studying the excitations of low-dimensional quantum spin systems based on the Jordan-Wigner transformation. Using an extended random-phase approximation (RPA) scheme we calculate the correlation function of neighboring spin flips for the one-dimensional spin-1/2 chain which well approximates the optical conductivity of
Sr2CuO3. We analyze several possible generalizations of the Jordan-Wigner approach to the two-leg spin-1/2 ladder. On the mean-field level the most accurate results are obtained when the spin operators are numbered in a meanderlike sequence. Calculation of the optical conductivity based on an extended RPA scheme for the meander-path approach yields very good agreement with a previous density matrix renormalization group evaluation. For polarization along the legs higher-order correlations are important to explain the weight of high-energy continuum excitations and we estimate the contribution of 4- and 6-fermion processes.