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Consider a scalar field theory in $AdS_{d+1}$ spacetime

\begin{equation} S= -\frac{1}{2}\int d^{d+1} x \sqrt{-g} (\partial_\mu \phi \partial_\nu \phi g^{\nu \mu} + m^2 \phi^2 ) \end{equation}

On quantization of this theory in an AdS metric, one finds that the solution can be written as

\begin{equation} f_{\omega l \vec{m}} (r,t,\Omega) = \psi_{\omega l}(r) e^{-i \omega t} Y_{l \vec{m}}(\Omega) \end{equation}

where $\psi_{\omega l}(r)$ turn out to be hypergeometric functions. The key result here is that $\omega$ gets quantized as follows

\begin{equation} \omega = \omega_{n l} = \Delta (=\frac{d}{2}+ \frac{1}{2} \sqrt{d^2 + 4m^2}) + l + 2n \end{equation}

And this quantization somehow implies that mass is not properly defined in the theory? I don't quite understand how this follows just from the fact that the $\omega$ is quantized?

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  • $\begingroup$ Do you have a source for the idea that "mass is not properly defined"? This is an unusual claim, so the context may be helpful to understand what it means. $\endgroup$ Commented Jun 10, 2022 at 0:58
  • $\begingroup$ Oh yes, I have been looking at Ashoke Sen's online lectures titled "Entanglement and Geometry". It appears in the 3rd lecture. $\endgroup$
    – Ayush Raj
    Commented Jun 10, 2022 at 12:55

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I don't know much about $AdS/CFT$, but as far as I could understand, defining mass of a scalar particle in homogeneous spacetime of non-zero curvature should be problematic.

In general, for an $AdS$ spacetime, the isometry group is the conformal group $$O(2,n).$$

Then, one expects that the "mass" should correspond to the quadratic Casimir element of its universal enveloping algebra. However, I personally find this could be ambiguous for the following reason.

First of all, the isometry group $O(2,n)$ for $AdS$ spacetime is also a conformal group. In such a theory, defining mass would be troublesome because it is not an invariant concept under scaling. Second, consider the Lagrangian density $$\mathcal{L}=\frac{\sqrt{|\det g|}}{2}\left[g^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi)-(m^{2}+\lambda R)\phi^{2}\right],$$

one finds that, since the Ricci scalar curvature $R$ for $AdS$ spacetime is a constant, the mass term is ambiguous due to its coupling with the curvature.

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  • $\begingroup$ Can you elaborate on how the symmetry group of the theory being O(2,n) implies that the "mass" should correspond to the quadratic Casimir element of its universal enveloping algebra. It will be great if you could provide a reference, I haven't encountered this thing before. $\endgroup$
    – Ayush Raj
    Commented Jun 7, 2022 at 3:09
  • $\begingroup$ @AyushRaj It's the mass-shell equation. In Minkowski spacetime, the Poincare algebra satisfies the condition $P_{\mu}P^{\mu}=m^{2}$. $\endgroup$
    – Xenomorph
    Commented Jun 7, 2022 at 5:57

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