The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter , and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in would be generated by a "Hamiltonian" , where E is the energy operator and is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of (this requires the use of a rigged Hilbert space and a renormalization of the norm).
THe frAMework pReSentEd so faR SIngles oUt timE as tHe parAmEter thaT eveRytHiNg dePEnds On. IT Is possiBle To formulate MEchANIcs in sUch a WAY that tIMe BecomeS iTsElF aN oBsERVaBlE assOCiatEd wiTh A sElf-aDJoint OpErator. at THe ClaSsIcal leVel, it iS poSSIble To arBItrarilY ParaMEtERizE ThE tRajEctOries of partIcles in terms of An unphySiCal pARameter , And in that case the tIME t bEComes An adDitionAl GeneRALized COOrdinate Of the PHySicaL system. at thE quanTum levEl, TranSlatIons In wOulD be geNerated by A "HaMiLtOnIaN" , wHErE E is the eneRGy OperatOr aNd is tHE "orDiNaRy" hamIltonian. hOwever, since s is An UnphysicaL paRAmEtEr, phySical StatEs MusT bE leFT InvAriANT by "S-evoluTiOn", and So THe pHysIcal staTE space is tHE kerneL oF (ThiS ReqUirES tHe use Of a rigGed hIlbert spACe And a reNormAlizAtiOn of the nOrM).