Bose Fermi Supersymmetry with Bogoliubov Transforms in Cosmology

X iv :h ep -t h/ 06 11 04 4v 1 4 N ov 2 00 6 Bose Fermi Supersymmetry with Bogoliubov transforms in Cosmology Ajay Patwardhan Physics Department, St Xaviers college, Mahapalika Marg, Mumbai, India ABSTRACT Field theory including Supersymmetry and Bose Fermi symmetry is an active subject of particle physics and cosmology. Recent and expected observational evidence gives indicators for the creation and destruction of normal and supersymmetric dark matter in the universe. This paper uses Bogoliubov transforms in supersymmetric and Bose Fermi form for obtaining the vaccuum expectation values at any two times in cosmological and black hole geometries. The isotropic Robertson Walker and slightly anisotropic Bianchi I geometry mode functions have a differential equation form analogous to the supersymmetric hamiltonian. The conditions for mixed and distinct representations for bosonic and fermionic fields of normal and supersymmetric partner particles are found. 1.INTRODUCTION One of the fundamental symmetries of nature is expected to be Supersymmetry(SUSY). The dark matter in the universe could be neutralinos which are likey to be discovered in LHC experiments in a few years. Supersymmetric partners and the normal particles have asymmetric production and decay , possibly from the inflation and post inflation epoch. The primordial black holes and the collapsed matter black holes in both dark matter and normal matter are active subjects of research. The Fermi Bose symmetry , or exchange symmetry is a subset of the SUSY and also occurs in nuclear physics and in condensed matter physics. This occurs in the form of an exchange Hamiltonian giving rise to hyperfine structure in nuclear spectra and as Fermi Bose joint quantisation in anyonic systems in condensed matter. The quantum group representations give a commutator and anti commutator algebra ; a “q commutation algebra; of which the Fermi and Bose algebras are special cases. Grassmanian and normal co-ordinates can be used to express the Fermionic and Bosonic systems in superspace and do SUSY Yang Mills gauge theories, while superstring and supergravity theories have SUSY built in.[Ref 1 to 22] Local Lorentz invariance, asymptotically free field theories and the CPT theorem and possibly locality are usually assumed to hold. This gives the normal particles/SUSY particles a definite spin, half integer or integer, and spin statistics theorems prohibit mixing of the spin multiplets. However