Quantization-scheme-Independent Energy and Its Implications for Holographic Bounds

TL;DR

This work addresses the scheme-dependence of holographic energy in AdS/CFT, where holographic renormalization yields different total energies depending on the quantization of a scalar field in the BF window. The authors introduce a modified energy $\mathcal{H} = \mathcal{E} + \frac{d - \Delta}{d} J \langle \mathcal{O} \rangle$ that remains invariant under standard vs. alternative quantization by combining data from both schemes in a balanced, Legendre-transform-like fashion. They demonstrate, in a concrete 4D AdS example with a massive scalar, that $\mathcal{H}$ matches between quantization schemes and then verify that a set of holographic inequalities—the AdS Penrose inequality, late-time bounds on holographic entanglement entropy growth, and Lloyd-type bounds for CV and CA complexities—hold universally when $\mathcal{H}$ is used as the energy. The results propose $\mathcal{H}$ as a robust, scheme-independent notion of bulk energy, with potential implications for extending holographic bounds to dynamical spacetimes and richer matter content.

Abstract

In holographic duality, the total energy of the dual field theory is obtained from the holographic renormalization, which depends not only on the bulk geometry but also on the choice of quantization schemes. We point out that the validity of several widely studied holographic inequalities -- including the AdS Penrose inequality, the late-time bound on entanglement entropy growth, and the growth-rate limits of CV and CA complexities -- depends on the choice of quantization schemes. Motivated by this issue, we introduce a modified total energy, which is still computed via holographic renormalization but the final value is independent of the choice of quantization schemes. We verify that this new ``total energy'' restores all these bounds to universal validity in the model of generalized free scalar field theory. Our results suggest that our modified total energy provides a more robust notion of energy when we talk about above inequalities in holographic settings.

Figures (5)

• Figure 1: The total energy parameter $\tilde{f}_{3}/2$ as a function of $\phi(r_{h})$. The red and blue curves correspond to the mass density obtained by standard and alternative quantization schemes, respectively. The purple curve represents the case where the modified energy $\mathcal{H}$ is used. The dashed line represents $r_{h}/2$.
• Figure 2: Late-time growth rate of holographic entanglement entropy $G_{\text{max}}$ as a function of the source $J$ with fixed energies under different quantization schemes. The vertical axis shows the normalized growth rate $G_{\text{max}} / G_{\text{max}0}$, where $G_{\text{max}0}$ is the value for the vacuum black hole. The blue and green dashed curves correspond to the standard and alternative quantization schemes with fixed $\mathcal{E}^{\text{(sta)}}$ and $\mathcal{E}^{\text{(alt)}}$, respectively. The red and purple solid curves represent cases with fixed modified energy $\mathcal{H}$.
• Figure 3: Late-time growth rate of holographic complexity $\sigma_{\text{max}}$ under the CV conjecture as a function of the source $J$ with fixed energies under different quantization schemes. The vertical axis shows the normalized growth rate $\sigma_{\text{max}} / \sigma_{\text{max}0}$, where $\sigma_{\text{max}0}$ corresponds to the vacuum black hole. The blue and green dashed curves correspond to the standard and alternative quantization schemes with fixed $\mathcal{E}^{\text{(sta)}}$ and $\mathcal{E}^{\text{(alt)}}$, respectively. The red and purple solid curves represent cases with fixed modified energy $\mathcal{H}$.
• Figure 4: Wheeler-DeWitt patch at late time.
• Figure 5: Late-time growth rate of holographic complexity $\dot{\mathcal{C}}_A$ under the CA conjecture as a function of the source $J$ with fixed energies under different quantization schemes. The vertical axis shows the normalized growth rate $\dot{\mathcal{C}}_A / \dot{\mathcal{C}}_{A,\rm Sch}$, where $\dot{\mathcal{C}}_{A,\rm Sch}$ is the value for the vacuum black hole. The blue dashed and green dashed curves correspond to the standard and alternative quantization schemes with fixed $\mathcal{E}^{\text{(sta)}}$ and $\mathcal{E}^{\text{(alt)}}$, respectively. The red and purple solid curves represent cases with fixed modified energy $\mathcal{H}$.