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Real observers solving imaginary problems

Juan Maldacena

TL;DR

The paper analyzes the sphere partition function in Euclidean de Sitter gravity and the puzzling dimension-dependent phase $i^{D+2}$ that hinders a straightforward state-count interpretation. By incorporating an observer modeled as a particle with a clock and carefully handling gauge fixing, negative modes, and contour prescriptions, Maldacena shows that most of the phase cancels in the refined quantity intended to count states, leaving a residual overall sign that is not yet fully explained. The analysis combines Polchinski's one-loop results, a toy particle-on-a-sphere model, and a Hamiltonian-constraint viewpoint to define a count-like quantity ${\cal Z}_{\rm Count}$ that encodes de Sitter entropy via $e^{S_{dS}}$ and clock entropy, while highlighting subtleties in gauge choices and analytic continuation. This work clarifies how observer degrees of freedom affect the gravity path integral and its entropic interpretation, and it points to further questions about the residual sign and its physical meaning, including subsequent refinements that remove the sign in related work. It also motivates connections to holographic perspectives and to analogues in JT gravity, suggesting a broader framework for understanding entropy and observables in semiclassical quantum gravity.

Abstract

The sphere partition function is one of the simplest euclidean gravity computations. It is usually interpreted as count of states. However, the one loop gravity correction contains a dimension dependent phase factor, $i^{D+2}$, which seems confusing for such an interpretation. We show that, after including an observer, this phase gets mostly cancelled for the quantity that should correspond to a count of states. However, an overall minus sign remains.

Real observers solving imaginary problems

TL;DR

The paper analyzes the sphere partition function in Euclidean de Sitter gravity and the puzzling dimension-dependent phase that hinders a straightforward state-count interpretation. By incorporating an observer modeled as a particle with a clock and carefully handling gauge fixing, negative modes, and contour prescriptions, Maldacena shows that most of the phase cancels in the refined quantity intended to count states, leaving a residual overall sign that is not yet fully explained. The analysis combines Polchinski's one-loop results, a toy particle-on-a-sphere model, and a Hamiltonian-constraint viewpoint to define a count-like quantity that encodes de Sitter entropy via and clock entropy, while highlighting subtleties in gauge choices and analytic continuation. This work clarifies how observer degrees of freedom affect the gravity path integral and its entropic interpretation, and it points to further questions about the residual sign and its physical meaning, including subsequent refinements that remove the sign in related work. It also motivates connections to holographic perspectives and to analogues in JT gravity, suggesting a broader framework for understanding entropy and observables in semiclassical quantum gravity.

Abstract

The sphere partition function is one of the simplest euclidean gravity computations. It is usually interpreted as count of states. However, the one loop gravity correction contains a dimension dependent phase factor, , which seems confusing for such an interpretation. We show that, after including an observer, this phase gets mostly cancelled for the quantity that should correspond to a count of states. However, an overall minus sign remains.

Paper Structure

This paper contains 13 sections, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of Polchinski's computation of the phase factor
  3. An analogy: A massive particle on a sphere
  4. Discussion
  5. The sign of the phase factor for gravity
  6. The partition function with the observer included
  7. Partition function of the clock
  8. Addressing one other factor of $i$
  9. Discussion
  10. Conformal Transformations of the sphere
  11. Result for the sphere partition function
  12. Comment on a factor of $i$ in JT gravity
  13. Comment on the zero mode counting

Figures (3)

  • Figure 1: We consider a particle of mass $m$ propagating on a Euclidean sphere. A possible classical solution is a trajectory along a maximal circle. The sum over paths also involves shorter paths which are the dominant ones.
  • Figure 2: In red we see the defining contours for the integral (\ref{['ContInt']}). Note that the ${\cal C}_-$ part of the contour is oriented differently for the $D$ even case relative to the $D$ odd case. In (a) we discuss the $D$ odd case. Here we can add and subtract the small piece ${\cal C}_s$ which give us the simple terms in (\ref{['OddD']}). Then we can shift the contour to ${\cal C}'$ and move it to the upper half plane picking up the poles through ${\cal C}_p$. In (b) we discuss the $D$ even case. In this case the orientations of the contours is such that we cannot move the whole contour to the upper half plane. But we can move it to the direction where the exponential $e^{ i \nu t}$ decreases the fastest. This is the dotted line, when the phase of $\nu$ is deformed as indicated. In this process we pick up the same poles as before, by the contours ${\cal C}_p$. If we had chosen the other sign of $\epsilon$ the dotted line would have been in the upper right quadrant and the orientation of the ${\cal C}_p$ contours would have been the opposite.
  • Figure 3: (a) Rotation of the contour for the integral over the negative modes discussed in equation (\ref{['ThetCon']}). The dotted line denotes the direction of maximum increase. We rotate the contour in such a way that we avoid this direction. The original contour is ${\cal C}$ and the new contour is ${\cal C}'$. (b) Rotation of the contour for the $\delta \beta = \beta - \beta_0$ integral discussed around equations (\ref{['IntCo']}) (\ref{['RotIntBe']}).