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We present a unification of mixed-space quantum representations in CondensedMatter Physics (CMP) and Quantum Field Theory (QFT). The unifying formalism isbased on being able to expand any quantum operator, for bosons, fermions, andspin systems, using a universal basis operator Y(u,v) involving mixed Hilbertspaces of P and Q, respectively, where P and Q are momentum and positionoperators in CMP (which can be considered as a bozonization of free Blochelectrons which incorporates the Pauli exclusion principle and Fermi-Diracdistribution), whereas these are related to the creation and annihilationoperators in QFT, where {\psi}^{{\dag}}=-iP and {\psi}=Q. The expansioncoefficient is the Fourier transform of the Wigner quantum distributionfunction (lattice Weyl transform) otherwise known as the characteristicdistribution function. Thus, in principle, fermionization via Jordan-Wigner forspin systems, as well as the Holstein--Primakoff transformation from boson tothe spin operators can be performed depending on the ease of the calculations.Unitary transformation on the creation and annihilation operators themselves isalso employed, as exemplified by the Bogoliubov transformation. Moreover,whenever Y(u,v) is already expressed in matrix form, M_{ij}, e.g. the Paulispin matrices, the Jordan--Schwinger transformation is a map to bilinearexpressions of creation and annihilation operators which expedites computationof representations. We show that the well-known coherent states formulation ofquantum physics is a special case of the present unification. A new formulationof QFT based on Q-distribution of functional-field variables is suggested. Thecase of nonequilibrium quantum transport physics, which not only involvesnon-Hermitian operators but also time-reversal symmetry breaking, is discussedin the Appendix.

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