Supergravity has no Purely Bosonic Solutions

What this means is, in a sense, pure quantum general relativity cannot arise as a limit from quantum super gravity.

Theorem (Pure Bosonic States Cannot Exist). Suppose we have a bosonic state $\Psi[e^{AA’}_{a}]$, and suppose it is Lorentz invariant. Then it cannot describe a physical state for $N=1$ supergravity.

So, before we begin the proof, we recall that canonical $N=1$ supergravity has constraints for the supersymmetry. One of them we will use, namely

\begin{equation} \bar{S}_{A’} = \epsilon^{abc}e_{AA’a}\mathcal{D}_{b}\psi^{A}_{c} + 4\pi G_{N}\hbar\psi^{A}_{a}\frac{\delta}{\delta e^{AA’}_{a}} \end{equation}

where $\mathcal{D}_{b}$ is the torsion-free covariant derivative.

Why not the other constraint for SUSY? Because it’s of the form

\begin{equation} S_{A}\sim\mathcal{D}_{j}\frac{\delta}{\delta\psi_{jA}} + \frac{\delta}{\delta\psi_{jB}}(\mbox{some mess})^{ABj}. \end{equation}

Observe for a purely bosonic solution, the functional derivatives with respect to the gravitino variable will kill the wave functional automatically. So it’s trivially satisfied.

Proof (Scaling Argument). We consider \begin{equation} I = \int\bar{\epsilon}(\vec{x})\left(\frac{1}{\Psi[e]}\bar{S}^{A’}\Psi[e]\right)\,\mathrm{d}^{3}x \end{equation} for an arbitrary continuous spinor test function $\bar{\epsilon}(\vec{x})$. Observe the first term in $\bar{S}^{A’}$ acts as a “constant”, so we end up with \begin{equation} \frac{1}{\Psi[e]}\bar{S}^{A’}\Psi[e] = \epsilon^{abc}e_{AA’a}\mathcal{D}_{b}\psi^{A}_{c} + 4\pi G_{N}\hbar\psi^{A}_{a}\frac{\delta\ln\Psi[e]}{\delta e^{AA’}_{a}}. \end{equation} We replace $\bar{\epsilon}(\vec{x})$ with $\bar{\epsilon}(\vec{x})\exp(-\phi(\vec{x}))$, and similarly $\psi^{A}_{a}(x)$ with $\psi^{A}_{a}(x)\exp(\phi(\vec{x}))$. This changes the integral to $I\to I+\Delta I$, where \begin{equation} \Delta I = -\int\epsilon^{abc}e_{AA’a}\bar{\epsilon}^{A’}\psi^{A}_{c}\partial_{b}\phi\,\mathrm{d}^{3}x. \end{equation} Observe $\Delta I$ is independent of the wave functional $\Psi[e]$. We need $\Delta I$ to vanish, otherwise the constraint $\bar{S}^{A’}$ won’t. But we can pick arbitrary $\bar{\epsilon}$, $\psi$, $\phi$, such that $\Delta I\neq0$. This would imply $\bar{S}^{A’}\Psi[e]$ would depend on $\psi^{A}_{a}$, i.e., $\Psi[e]$ could not be purely bosonic. (End of Proof)

Remark 1. We did not use the hypothesis of Lorentz invariance for $\Psi[e]$. Is it necessary, or can we drop it?

Remark 2 ($N>1$?). Observe we did not explicitly use the $N=1$ property anywhere in the proof. Could this be generalized to other $N$? (Open research problem)

Remark 3 (Scaling Argument Strategy). The basic proof was to show a constraint $\hat{C}\Psi=0$ has its integral $I=\int f(\vec{x})\hat{C}\Psi\,\mathrm{d}^{3}x$ vanish everywhere. Then scaling some of the non-physical parameters in such a manner as to leave the physics the same, we found $I\to I’\neq I$ — a contradiction!

Remark 4 (“Method of Characteristics” Proof Sketch). We could have equally considered an infinitesimal $\bar{S}$ supersymmetry transformation $e^{A}_{a}\to e^{A}_{a}+\delta e^{A}_{a}$ which leaves $\psi$ unchanged. Then we expand $\Psi[e+\delta e,\psi]=\Psi[e]+\delta\Psi$. But we may scale $\psi$ and $\bar{\epsilon}$ again, which leaves $\delta e$ invariant, and thus produces (in general) a different $\delta\Psi$. (We interpret this as telling us $\Psi$ is not independent of $\psi$ in the full configuration space of the theory.) This disputes the “method of characteristics” D’Eath performed [3], namely that D’Eath did not perform a complete analysis.

References.

  1. Sean M. Carroll, Daniel Z. Freedman, Miguel E. Ortiz and Don N. Page, “Physical states in canonically quantized supergravity’’. Nucl. Phys. B 423 (1994) 661–687. Eprint arXiv:hep-th/9401155
  2. Sean M. Carroll, Daniel Z. Freedman, Miguel E. Ortiz and Don N. Page, “Bosonic physical states in $N=1$ supergravity?” Eprint arXiv:gr-qc/9410005.
  3. Peter D. D’Eath, “Physical states in $N=1$ supergravity’’. Phys. Lett. B 321 (1994) 368–371, arXiv:hep-th/9304084.
  4. Claus Kiefer, Quantum Gravity. Third ed., Oxford Press (2012). See the end of section 5.3.6.
  5. Paulo V. Moniz, Quantum Cosmology: The Supersymmetric Perspective. Volumes 1 and 2. Lecture notes in physics, vol 804. Springer, Berlin (2010) doi:10.1007/978-3-642-11575-2 and doi:10.1007/978-3-642-11570-7.