Group theoretical quantization of isotropic loop cosmology

We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled to a massless scalar field adopting the improved dynamics of loop quantum cosmology. Deparemetrizing the system using the scalar field as internal time, we first identify a complete set of phase space observables whose Poisson algebra is isomorphic to the $\mathfrak{s}\mathfrak{u}(1,1)$ Lie algebra. It is generated by the volume observable and the Hamiltonian. These observables describe faithfully the regularized phase space underlying the loop quantization: they account for the polymerization of the variable conjugate to the volume and for the existence of a kinematical nonvanishing minimum volume. Since the Hamiltonian is an element in the $\mathfrak{s}\mathfrak{u}(1,1)$ Lie algebra, the dynamics is now implemented as SU(1, 1) transformations. At the quantum level, the system is quantized as a timelike irreducible representation of the group SU(1, 1). These representations are labeled by a half-integer spin, which gives the minimal volume. They provide superselection sectors without quantization anomalies and no factor ordering ambiguity arises when representing the Hamiltonian. We then explicitly construct SU(1, 1) coherent states to study the quantum evolution. They not only provide semiclassical states but truly dynamical coherent states. Their use further clarifies the nature of the bounce that resolves the big bang singularity.

Figure 1

Plots of the volume $v$ (on the left) and of its conjugate momentum $b$ (on the right) evolving as functions of the internal time $\tau$, for ${\tau }_o=0$, for the negative branch: the Universe starts with a big bang at $\tau \to -\infty$ to expand to infinite volume $\tau \to +\infty$.

Figure 2

Plots of the volume $v$ (on the left) and of its conjugate momentum $b$ (on the right) evolving as functions of the internal time $\tau$, for ${\tau }_o=0$, for the effective loop quantum dynamics of flat FRW cosmology: the Universe starts with an infinite volume at $\tau \to -\infty$, contracts and bounces to expand again to infinite volume at $\tau \to +\infty$.

Figure 3

Plots of the volume $v$ (on the left), of $\cos b$ (in the center) and of the conjugate momentum $b$ (on the right) evolving as functions of the internal time $\tau$, for ${\tau }_0=0$, for explicit values of the parameters: $\beta =0.5$, $v_m=1$, $v_0=3$, $b_0=0.2$, $H_{\mathrm{reg}}\sim 2.062$, $A\sim 4.375$ and $B\sim -3.201$.

Figure 4

Plots of the volume $v$ (on the left), of $\cos b$ (in the center) and of the conjugate momentum $b$ (on the right) evolving as functions of the internal time $\tau$, for ${\tau }_0=0$, for explicit values of the parameters: $\beta =3$, $v_m=1$, $v_0=3$, $b_0=0.2$, $H_{\mathrm{reg}}\sim 9.562$, $A\sim -0.586$ and $B\sim -0.980$.

Figure 5

Plots of the volume $v$ (on the left) and its conjugate momentum $b$ (on the right) evolving as functions of the internal time $\tau$, for ${\tau }_0=0$, for flat FRW cosmology with positive cosmological constant $\Lambda >0$. We have two branches: an expanding one starting with a big bang singularity and a contracting one ending with a big crunch.

Figure 6

Plots of the volume $v$ (on the left) and its conjugate momentum $b$ (on the right) evolving as functions of the internal time $\tau$, for ${\tau }_0=0$, for flat FRW cosmology with negative cosmological constant $\Lambda <0$. The Universe starts in a big bang, expands to maximal volume and collapses again.