Quantum torsion and a Hartle-Hawking beam

In the Einstein-Cartan framework the torsion-free conditions arise within the Hamiltonian treatment as second-class constraints. The standard strategy is to solve these constraints, eliminating the torsion from the classical theory, before quantization. Here we advocate leaving the torsion inside the other constraints before quantization, leading at first to wave functions that can be called “kinematical” with regards to the torsion, but not the other constraints. The torsion-free condition can then be imposed as a condition upon the physical wave packets one constructs, satisfying the usual uncertainty relations, and so with room for quantum fluctuations in the torsion. This alternative strategy has the surprising effect of clarifying the sense in which the wave functions solving an explicitly real theory are “delta-function normalizable.” Such solutions with zero (or any fixed) torsion, should be interpreted as plane waves in torsion space. Properly constructed wave packets are therefore normalizable in the standard sense. Given that they are canonical duals, this statement applies equally well to the Chern-Simons state (connection representation) and the Hartle-Hawking wave function (metric representation). We show how, when torsion is taken into account, the Hartle-Hawking wave function is replaced by a Gauss-Airy function, with finite norm, which we call the Hartle-Hawking beam. The Chern-Simons state, instead, becomes a packet with a Gaussian probability distribution in connection space. We conclude the paper with two sections explaining how to generalize these results beyond minisuperspace.

Figure 1

Comparison of $P(\tilde{b})$ for various values of ${\tilde{\sigma }}_c$.

Figure 2

Comparison of $|\varphi (\tilde{a}){|}^2$ for various values of ${\tilde{\sigma }}_c$. For ease of illustration, plots are normalized so that $\int |\varphi {|}^{2}d{\tilde{a}}^2=1$. Formally, as ${\tilde{\sigma }}_c\to 0$, the solution $\varphi$ approaches proportionality with the Hartle-Hawking wave function ${\psi }_{\mathrm{HH}}$.