Cosmological spinor

We build upon previous investigation of the one-dimensional conformal symmetry of the Friedman-Lemattre-Robertson-Walker (FLRW) cosmology of a free scalar field and make it explicit through a reformulation of the theory at the classical level in terms of a manifestly SL(2, R)-invariant action principle. The new tool is a canonical transformation of the cosmological phase space to write it in terms of a spinor, i.e., a pair of complex variables that transform under the fundamental representation of SU(1, 1) similar to SL(2, R). The resulting FLRW Hamiltonian constraint is simply quadratic in the spinor and FLRW cosmology is written as a Schrodinger-like action principle. Conformal transformations can then be written as proper-time dependent SL(2, R) transformations. We conclude with possible generalizations of FLRW to arbitrary quadratic Hamiltonian and discuss the interpretation of the spinor as a gravitationally-dressed matter field or matter-dressed geometry observable.