Cosmological spinor

We build upon previous investigation of the one-dimensional conformal symmetry of the Friedman-Lemaitre-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)∼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.