Question 7.4: In a problem in quantum cosmology (Lemos, 1996) the action h......
In a problem in quantum cosmology (Lemos, 1996) the action has the form
S =\int_{t_{1}}^{t_{2}}{dt\left\{\dot{a}p_{a}+\dot{\phi }p_{\phi} -N\left(\frac{p^{2}_{a}}{24a}-\frac{p^{2}_{\phi}}{2a^{3}} +6ka\right) \right\} }, (7.111)
where k is a constant related to the curvature of space and t designates the evolution parameter. Given that N and a are strictly positive quantities, obtain the equations of motion in the extended phase space (a, \phi, p_{a}, p_{\phi}) and the Hamiltonian in the reduced phase space that results from the choice t = \phi.
The action (7.111) has the form (7.86) with N acting as a Lagrange multiplier. The function a(t) is the scale factor of the Universe, which determines the distances between points in space, with an increasing a corresponding to an expanding universe. In this oversimplified model \phi(t) is a homogeneous scalar field that represents the material content of the universe and is responsible for the expansion. Requiring δS = 0 with the variations of a, \phi, p_{a}, p_{\phi},N arbitrary and independent, the equations of motion in the extended phase space are
S\left[q, p,\lambda \right] =\int_{\theta _{1}}^{\theta _{2}}{d\theta \left\{\sum\limits_{i=1}^{n+1}{p_{i}q^{\prime }_{i}}-\lambda \mathcal{H} \left(q,p\right) \right\} \equiv \int_{\theta _{1}}^{\theta _{2}}{d\theta F\left(q, p,\lambda ,q^{\prime },p^{\prime },\lambda ^{\prime }\right) }}. (7.86)
\dot{a}=\frac{N p_{a}}{12a}, \dot{p}_{a}=N\left(\frac{p^{2}_{a}}{24a^{2}}-\frac{3p^{2}_{\phi}}{2a^{4}} -6k\right) , \dot{\phi}=-\frac{N p_{\phi}}{a^{3}}, \dot{p}_{\phi}=0, (7.112)
supplemented by the super-Hamiltonian constraint
\frac{p^{2}_{a}}{24a}-\frac{p^{2}_{\phi}}{2a^{3}}+6ka=0. (7.113)
In general relativity the evolution parameter t is arbitrary and, as in the case of the free relativistic particle, the fixation of the Lagrange multiplier N corresponds to a choice of time, the evolution parameter for the system. The choice t = \phi requires us to solve Eq. (7.113) for p_{\phi}, and the Hamiltonian in the reduced phase space is H = −p_{\phi}. Since now \dot{\phi} = 1, the third of Eqs. (7.112) shows that p_{\phi} < 0 because N and a are positive. Solving (7.113) for p_{\phi} and taking the negative square root, we find
p_{\phi} =-\sqrt{\frac{a^{2}p^{2}_{a}}{12}+12k a^{4}}. (7.114)
The action in the reduced phase space is obtained by substituting \dot{\phi} = 1 and the solution (7.114) for p_{\phi} into (7.111). The result is
S =\int_{t_{1}}^{t_{2}}{dt \left\{\dot{a}p_{a}- \sqrt{\frac{a^{2}p^{2}_{a}}{12}+12k a^{4}}\right\} }, (7.115)
where t = \phi. The Hamiltonian in the reduced phase space spanned by the canonical variables a, p_{a} is
H\left(a, p_{a}\right) =\sqrt{\frac{a^{2}p^{2}_{a}}{12}+12k a^{4}}. (7.116)