Abstract An information‐theoretic degree of symmetry adaptation of a ket with respect to a given group is introduced. Then the general equivalence between spin‐free electronic kets and antisymmetrized space‐spin kets is obtained so long as our attention is restricted to spin‐independent observables. It is noted that there are spin‐free kets for which there exists no single antisymmetrized space‐spin ket giving all the same expectation values; the converse is also noted.
Abstract An information‐theoretic degree of symmetry adaptation of a ket with respect to a given group is introduced. Then the general equivalence between spin‐free electronic kets and antisymmetrized space‐spin kets is obtained so long as our attention is restricted to spin‐independent observables. It is noted that there are spin‐free kets for which there exists no single antisymmetrized space‐spin ket giving all the same expectation values; the converse is also noted.
Abstract The representation of an N‐electron Schrödinger Hamiltonian on an orthonormal, spin‐free (freon) orbital product space is exactly modeled by a second‐degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the orbital product space with respect to the symmetric group yields Gel'fand states which provide base for irreducible represntations of the unitary group. These exist in closed form as, in consequence, does the representation of the model Hamiltonian in this same basis. We retain as physical only those states characterized by tableaux with no more than two columns for which the spin labelling is one‐half the difference in the lengths of the two columns. The unitary group formulation is equivalent to standard, number‐conserving, second‐quantized, many‐body theory.
Spin-free transition density matrices are derived from spin-free kets which are symmetry-adapted to the symmetric group and its algebra. The Dirac identity establishes that these spin-free density matrices are identical to those obtained by integrating the spin from the full-spin density martices. Derivations are first given for arbitrary primitive kets which may be geminals of higher polymals, after which we consider products of orbitals, either orthonormal or nonorthonormal. Correlation in the spin-free space is discussed and we show the influence of permutational symmetry on the probability of coincidence of pairs. A special case of this correlation is the well-known Fermi hole.
It is usually assumed that a ket represents the state of an actually existing particle. But one can show there is no evidence for particles. The particle-like properties of mass, spin and charge, as well as particle-like trajectories, the photoelectric effect, and localized effects from spread-out wave functions can be explained using quantum mechanics alone. It is therefore proposed instead that kets represent particle-like solutions to a pre-representational linear partial differential equation which has Poincaré and internal symmetries. This equation underlies the completely representational character, including mass, spin, charge, internal symmetries, and symmetric and antisymmetric statistics, of current quantum mechanics.