Canonical quantization of supergravity

The canonical quantization of supergravity is developed, starting from the Hamiltonian treatment of classical supergravity. Quantum states may be represented by wave functionals $f\left({e^{A{A}^{\prime }}}_i\mathrm{}\right(x),{{\psi }^A}_i\mathrm{}(x\left)\right)$ of the spatial spinor-valued tetrad forms ${e^{A{A}^{\prime }}}_i$ and of the right-handed spatial part ${{\psi }^A}_i$ of the spin-$\frac{3}{2}$ field, or equivalently by functionals $\tilde{f}\left({e^{A{A}^{\prime }}}_i\mathrm{}\right(x),{{\overline{\psi }}^{A^{\prime }}}_i\mathrm{}(x\left)\right)$ of ${e^{A{A}^{\prime }}}_i$, and the left-handed part ${{\overline{\psi }}^{A^{\prime }}}_i$. In the first representation the momentum ${p_{A{A}^{\prime }}}^i$ classically conjugate to ${e^{A{A}^{\prime }}}_i$, together with ${{\overline{\psi }}^{A^{\prime }}}_i$, can be represented by functional differential operators such that the correct (anti) commutation relations hold; similarly for ${p_{A{A}^{\prime }}}^i$,${{\psi }^A}_i$ in the second representation. A formal inner product can be found in which ${p_{A{A}^{\prime }}}^i$ is Hermitian and ${{\psi }^A}_i$,${{\overline{\psi }}^{A^{\prime }}}_i$ are Hermitian adjoints. Physical states obey the quantum constraints $J_{\mathrm{AB}}f=0\mathrm{}$, ${\overline{J}}_{A^{\prime }{B}^{\prime }}f=0\mathrm{}$, $S_{A}f=0\mathrm{}$, ${\overline{S}}_{A^{\prime }}f=0\mathrm{}$, ${\mathcal{H}}_{A{A}^{\prime }}f=0\mathrm{}$, where $J_{\mathrm{AB}}$, ${\overline{J}}_{A^{\prime }{B}^{\prime }}$ are the quantum versions of the classical generators of local Lorentz rotations, $S_A$, ${\overline{S}}_{A^{\prime }}$ correspond to classical supersymmetry generators, and ${\mathcal{H}}_{A{A}^{\prime }}$ to generalized coordinate transformations. The constraints $J_{\mathrm{AB}}f=0\mathrm{}$, ${\overline{J}}_{A^{\prime }{B}^{\prime }}f=0\mathrm{}$ describe the invariance of $f(e,\psi )$ under local Lorentz transformations, ${\overline{S}}_{A^{\prime }}f=0\mathrm{}$ gives a simple transformation property of $f(e,\psi )$ under left-handed supersymmetry transformations applied to ${e^{A{A}^{\prime }}}_i$, ${{\psi }^A}_i$, and $S_{A}f=0\mathrm{}$ gives a corresponding property of $\tilde{f}(e,\overline{\psi })$ under right-handed transformations; these transformation properties are all that is required of a physical state. All physical wave functionals can be found by superposition from the amplitude $K$ to go from prescribed data ${({e^{A{A}^{\prime }}}_i,{{\tilde{\psi }}^{A^{\prime }}}_i)}_I$ on an initial surface to data ${({e^{A{A}^{\prime }}}_i,{{\psi }^A}_i)}_F$ on a final surface, which is given by a Feynman path integral. In a semiclassical expansion of this amplitude around a classical solution, the constraints imply that the one- and higher-loop terms $A,A_1,A_2,\dots$ are invariant under left-handed supersymmetry transformations at the final surface, and under right-handed transformations at the initial surface. An alternative approach to perturbation theory is provided by the multiple-scattering expansion, which constructs higher-order terms from the one-loop approximation $A \exp \left(\frac{i{S}_c}{\hslash }\right)$ to $K$, where $S_c$ is the classical action. This gives a resummation of the standard semiclassical expansion, which may help in improving the convergence of the theory. The invariance of $A$ under left-handed supersymmetry at the final surface is shown to limit the allowed surface divergences in $A$; there is at most one possible surface counterterm at one loop. Similar restrictions on surface counterterms in the standard expansion are expected at higher-loop order; these conditions may possibly also affect the usual volume counterterms, which must here be accompanied by surface contributions.