Unification of Mixed Hilbert-Space Representations in Condensed Matter Physics and Quantum Field Theory

Abstract

We present a unification of mixed-space quantum representations in Condensed Matter Physics (CMP) and Quantum Field Theory (QFT). The unifying formalism is based on being able to expand any quantum operator, for bosons, fermions, and spin systems, using a universal basis operator Y(u,v) involving mixed Hilbert spaces of P and Q, respectively, where P and Q are momentum and position operators in CMP (which can be considered as a bozonization of free Bloch electrons which incorporates the Pauli exclusion principle and Fermi-Dirac distribution), whereas these are related to the creation and annihilation operators in QFT, where {\psi}^{{\dag}}=-iP and {\psi}=Q. The expansion coefficient is the Fourier transform of the Wigner quantum distribution function (lattice Weyl transform) otherwise known as the characteristic distribution function. Thus, in principle, fermionization via Jordan-Wigner for spin systems, as well as the Holstein--Primakoff transformation from boson to the spin operators can be performed depending on the ease of the calculations. Unitary transformation on the creation and annihilation operators themselves is also employed, as exemplified by the Bogoliubov transformation. Moreover, whenever Y(u,v) is already expressed in matrix form, M_{ij}, e.g. the Pauli spin matrices, the Jordan--Schwinger transformation is a map to bilinear expressions of creation and annihilation operators which expedites computation of representations. We show that the well-known coherent states formulation of quantum physics is a special case of the present unification. A new formulation of QFT based on Q-distribution of functional-field variables is suggested. The case of nonequilibrium quantum transport physics, which not only involves non-Hermitian operators but also time-reversal symmetry breaking, is discussed in the Appendix.