Alternative Phase Space Formulation

• Alternative phase space formulation is a theoretical framework that modifies standard phase space constraints using a Weyl-anomaly condition to expose new physical and geometric structures.
• It employs a unique maximal volume (K=0) Cauchy slice and reinstates the conformal factor to ensure a symplectomorphic match with the ADM phase space while preserving gauge symmetries.
• This approach directly connects analytically continued CFT partition functions with candidate quantum gravity states, offering a promising nonperturbative route to background-independent formulations.

An alternative phase space formulation refers to any approach that reformulates classical or quantum systems using phase space (the space of positions and momenta or generalized coordinates and their conjugate variables), but with modified or generalized structures, constraints, or gauge choices that alter the standard formulation. Such modifications can be used to encode different symmetries, incorporate physical constraints, reveal new physical or mathematical structures, or generate candidate quantum states with specific invariance properties. Alternative phase space formalisms play essential roles in diverse domains including quantum deformation quantization, nonadiabatic quantum dynamics, black hole thermodynamics, geometric quantization, and quantum gravity.

1. Structural Redefinition of Phase Space

Alternative phase space formulations frequently arise by modifying the canonical constraints or introducing new degrees of freedom to phase space. In the standard ADM (Arnowitt–Deser–Misner) approach to gravity in asymptotically Anti–de Sitter (AAdS) spacetime, the phase space is constructed from spatial geometries (metric $g_{ab}$), canonical momenta ($\Pi^{ab}$), matter fields ($\Phi^i$), and their conjugate momenta, all on a Cauchy slice $\Sigma$. These data are subject to the Hamiltonian ($\mathcal{H}$), momentum ($D_a$), and matter-gauge ($G^A$) constraints, and evolution is generated with respect to a (gauge-redundant) time variable.

In the alternative formulation described in "Conformal Cauchy Slice Holography: An Alternative Phase Space For Gravity" (Khan, 19 Jul 2025), the Hamiltonian (or time reparametrization) constraint is replaced by a real Weyl-anomaly constraint,

$(\mathcal{W} + \mathcal{A})(x) = 0,$

where $\mathcal{W}$ generates local Weyl rescalings of the physical fields and $\mathcal{A}$ is the holographically computed real conformal anomaly. The pure-gauge variable—the conformal factor and its conjugate momentum—is explicitly reinstated, and the remaining phase space includes all metrics and matter field configurations subject to this set of modified, first-class gauge constraints.

2. Gauge Constraints and Physical Equivalence

The equivalence between the conventional and alternative phase space is established under specific geometric and field-theoretic requirements:

• A unique maximal volume (K = 0) Cauchy slice must exist within all classical solutions.
• Matter fields are subject to "suitable" boundary and regularity conditions.
• The reduced phase space, upon gauge fixing (e.g., fixing the conformal gauge so that the spatial Ricci scalar equals $2\Lambda$), is symplectomorphic to the standard ADM reduced phase space.

The Poisson algebra generated by the new Weyl-anomaly constraint $(\mathcal{W} + \mathcal{A})(x)$, the momentum constraint $D_a(x)$, and the matter constraint $G^A(x)$ closes. Explicitly,

• $$\{\mathcal{W} + \mathcal{A}, \mathcal{W} + \mathcal{A}\} = 0$$
• $$\{D[N^a], \mathcal{W} + \mathcal{A}\} = (\mathcal{W} + \mathcal{A})[\mathcal{L}_N \omega]$$, for a test function $\omega$
• $\{D_a, D_b\}$ and $\{G^A, \cdot \}$ form an appropriate algebra, reflecting the underlying gauge symmetries

After imposing $(\mathcal{W} + \mathcal{A}) = 0$, spatial diffeomorphism and matter-gauge constraints, and the conformal gauge, the physical degrees of freedom and symplectic structure coincide with those of the conventional ADM phase space.

3. Quantum Gravity State Conditions

A necessary requirement for a quantum gravity state in the alternative phase space is that its functional—constructed as $\Psi_{\mathrm{QG}}[\mathfrak{g}_{ab}, \Phi^i]$ in terms of the conformal metric and matter fields—must be annihilated by the quantum operator versions of all gauge constraints:

• Real Weyl-anomaly constraint: $$(\hat{\mathcal{W}} + \hat{\mathcal{A}}) \Psi_{\mathrm{QG}} = 0$$
• Spatial diffeomorphisms: $\hat{D}_a \Psi_{\mathrm{QG}} = 0$
• Matter gauge: $\hat{G}^A \Psi_{\mathrm{QG}} = 0$

Satisfaction of these operator constraints is the analog, in this formulation, of the Wheeler–DeWitt (WDW) equation for quantum gravity. The alternative phase space approach thus alters the "time" constraint into a local conformal anomaly condition while preserving covariance and local gauge symmetry.

4. Role of Conformal Field Theories

The construction identifies a direct link between conformal field theory (CFT) partition functions and candidate quantum gravity states in the alternative phase space. For a holographic CFT on the maximal volume slice (the K = 0 Cauchy hypersurface), the partition function $Z_{\mathrm{CFT}}[g, \Phi; \psi_{\mathrm{CFT}}]$—as a function of background metric $g$ and matter source $\Phi$—automatically satisfies spatial diffeomorphism and gauge constraints by covariance. However, it solves an "imaginary Weyl-anomaly" constraint

$$(\hat{\mathcal{W}} - i\hat{\mathcal{A}}) Z_{\mathrm{CFT}} = 0.$$

Analytically continuing the central charge from $c$ to $ic$ converts this to the required real Weyl-anomaly constraint in the alternative phase space:

$$(\hat{\mathcal{W}} + \hat{\mathcal{A}}) Z^{(ic)}_{\mathrm{CFT}} = 0.$$

Thus, every state $\psi_{\mathrm{CFT}}$ in the boundary CFT Hilbert space yields a corresponding candidate quantum gravity bulk state:

$$\Psi_{\mathrm{QG}}[\mathfrak{g}, \Phi] = Z^{(ic)}_{\mathrm{CFT}}[\mathfrak{g}, \Phi; \psi_{\mathrm{CFT}}].$$

5. Consequences and Interpretational Advances

This alternative phase space formulation confers several salient features:

• Refined Temporal Structure: By gauge fixing the Hamiltonian constraint to the maximal volume (K=0) slice, time in the bulk is fixed geometrically, resulting in a more objective handling of bulk evolution and removing the "branched" time ambiguities associated with the standard WDW approach.
• Direct Holographic Dictionary: The direct identification of CFT partition functions (analytically continued) with quantum gravity wavefunctionals (i.e., $\psi_{\mathrm{CFT}} \mapsto \Psi_{\mathrm{QG}}$) provides a nonperturbative, UV-complete scheme for constructing quantum gravity states from the dual CFT, circumventing limitations of effective field theory deformations such as $T^2$-deformed models.
• Covariance and Gauge Structure Retention: All the underlying gauge symmetries (diffeomorphisms, local Weyl, matter gauge) are maintained and imposed at the quantum level, ensuring background independence up to the conformal anomaly effects imported from the CFT.
• Applicability to Bulk Locality and Singularity Problems: By associating physical states directly to a unique maximal slice, the ambiguities associated with superpositions of different classical geometries are suppressed, supporting analysis of singularity resolution and the emergence of bulk locality.
• Potential for Higher Curvature Corrections: The framework sets a foundation for incorporating higher curvature or quantum corrections through the corresponding CFT structure.

6. Mathematical and Physical Formulation Table

Structure	Standard ADM Phase Space	Alternative (Weyl-Anomaly) Phase Space
Primary Constraints	Hamiltonian ($\mathcal{H}$), $D_a$, $G^A$	$(\mathcal{W}+\mathcal{A})$, $D_a$, $G^A$
Gauge-fixed Slice	Arbitrary “time”/foliation	Unique maximal volume ($K = 0$) slice
Quantum State Condition	$\hat{\mathcal{H}}\Psi=0,\, \hat{D}_a\Psi=0$	$(\hat{\mathcal{W}}+\hat{\mathcal{A}})\Psi=0$
Holographic Bulk Dual	Indirect, via boundary-to-bulk mapping	Direct, via $Z_{\mathrm{CFT}}^{(ic)}$
Temporal Redundancy	Generic branching/superpositions	Eliminated by geometric gauge fixing

7. Perspectives and Outlook

The alternative phase space construction is physically equivalent—modulo the specified geometric and matter field requirements—to the ADM phase space but is technically and conceptually distinct. The identification of CFT partition functions (with analytically continued central charge) as candidate quantum gravity states in this phase space not only provides a concrete realization of holography but also offers a systematic route for constructing and analyzing states with desired gauge and conformal properties. Opportunities for future investigation include the extension to higher curvature corrections, more general matter couplings, and the elucidation of detailed bulk–boundary data correspondences in this new phase space setting.