ADW Theory: Emergent Gravity Framework

• Akama-Diakonov-Wetterich theory is a pregeometric framework where gravity emerges from fermionic bilinear tetrads acting as vacuum order parameters.
• The theory employs nonstandard dimensional assignments that render invariant observables dimensionless and reinterpret the role of the Planck constant.
• It juxtaposes Wetterich’s RG methodologies with emergent gravity ideas, proposing a speculative phase transition from QFT to quantum mechanics.

“Akama-Diakonov-Wetterich theory” is best understood as a nonstandard, not fully canonized family of pregeometric and emergent-gravity constructions rather than as a single universally fixed formalism. In the versions explicitly described in the recent literature, gravity is not fundamental: tetrads arise as composite operators or vacuum order parameters of more primitive quantum fields, the metric is built bilinearly from those tetrads, and unconventional dimensional assignments then make diffeomorphism-invariant quantities dimensionless (Volovik, 14 Feb 2026, Volovik, 2022, Volovik, 2023). The same label is also used more loosely for extensions in which Planck constants are interpreted as elements of the emergent Minkowski metric, and for partial connections to Wetterich’s renormalization-group program; however, the available literature does not present a single unified set of ADW field equations encompassing all three names in a common completed theory (Volovik, 14 Feb 2026, Bender et al., 2018, Rosaler et al., 2018).

1. Terminological status and historical scope

The expression “Akama-Diakonov-Wetterich” is not used uniformly across the supplied literature. One explicit usage appears in Volovik’s 2026 paper, which states that it is trying to extend the Akama-Diakonov-Wetterich theory by introducing the Planck constants $\hbar$ and $\not h=\hbar c$ as elements of the emergent metric (Volovik, 14 Feb 2026). By contrast, the 2023 paper on acoustic metrics is framed in terms of the Akama-Diakonov emergent tetrad idea and does not develop Wetterich’s version in detail; its connection to a broader ADW label is therefore interpretive rather than dynamical (Volovik, 2023).

A second source of ambiguity is that “Wetterich” enters the broader discussion in more than one way. In one role, it denotes the effective-average-action functional renormalization group and the exact Wetterich flow equation for scale-dependent effective actions (Bender et al., 2018). In another, it denotes a conceptual Wilsonian argument that rejects the privileging of a unique set of “fundamental” bare parameters, treating an EFT instead as a whole RG trajectory (Rosaler et al., 2018). Neither of those papers, however, develops Akama- or Diakonov-style emergent tetrads.

An unrelated use of the name “Akama” also exists in mathematical logic, where Akama et al. introduced the $E_k$ and $U_k$ hierarchy for prenex normalization; that literature concerns first-order proof theory and is unrelated to emergent gravity (Fujiwara et al., 2023). Accordingly, the physics usage of ADW must be distinguished both from unrelated Akama-authored work and from purely Wetterich-centered RG literature.

2. Pregeometric core: composite tetrads and emergent metric

The shared physical core of the ADW-style literature is the claim that spacetime geometry is not primordial. In the explicit pregeometric formulation emphasized by Volovik, the basic microscopic variables are fermionic quantum fields, while the gravitational tetrad is a fermion bilinear whose vacuum expectation value defines the classical tetrad (Volovik, 14 Feb 2026): $$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$ The metric is then emergent: $G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$

In this picture, the tetrad is not merely a convenient field redefinition of the metric; it is an order parameter of a symmetry-broken phase. The microscopic theory is more symmetric, and the vacuum with nonzero $E^a{}_\mu$ selects the low-energy geometric phase. Volovik presents this as the basic ADW thesis inherited from Diakonov’s pregeometric gravity program, where gravity with metric and tetrads arises from pre-geometric quantum fields (Volovik, 14 Feb 2026).

The same paper quotes pregeometric action terms written directly in terms of fermion-bilinear tetrads. The “cosmological term” is

$$S_{\rm ADW}=\frac{1}{24}e^{\alpha\beta\mu\nu} e_{abcd} \int d^4x \, \hat E^a_\alpha \hat E^b_\beta \hat E^c_\mu \hat E^d_\nu ,$$

while the Einstein-Cartan term is

$$S_{\rm EC}=\frac{1}{4}e^{\alpha\beta\mu\nu} e_{abcd} \int d^4x \, {\cal F}^{ab}_{\alpha\beta} \hat E^c_\mu \hat E^d_\nu .$$

These formulas encode the characteristic ADW inversion of standard gravitational ontology: one starts from spinorial or pregeometric variables and only later obtains tetrad, metric, and effective spacetime geometry.

This pregeometric orientation is also the point of closest contact with the Diakonov-centered literature on “dimensionless physics,” where the tetrad is again taken to be a fermion bilinear and the metric is treated as a composite vacuum quantity rather than a primitive background structure (Volovik, 2022).

3. Dimensional assignments and “dimensionless physics”

A defining technical feature of this research line is its nonstandard assignment of physical dimensions. In the Diakonov-inspired formulation, tetrads carry inverse length or inverse time dimensions, rather than being dimensionless soldering forms. One version states

$$[E_0{}^\mu]\sim \frac1{[t]},\qquad [E_i{}^\mu]\sim \frac1{[L]},$$

so that

$\not h=\hbar c$0

A related simplified statement, used in the dimensionless-physics paper, is

$\not h=\hbar c$1

In both variants, the invariant interval is dimensionless: $\not h=\hbar c$2 The invariant volume element is likewise dimensionless (Volovik, 2023, Volovik, 2022).

This dimension assignment reorganizes the status of standard physical quantities. For a point particle,

$\not h=\hbar c$3

and since $\not h=\hbar c$4 is dimensionless, the mass parameter $\not h=\hbar c$5 is dimensionless as well. The same logic is extended to action $\not h=\hbar c$6, scalar curvature $\not h=\hbar c$7, cosmological constant $\not h=\hbar c$8, scalar fields, and Planck mass: all diffeomorphism-invariant quantities are dimensionless in this scheme (Volovik, 2022). Volovik treats this not as a mere unit convention but as a consequence of taking the metric itself to carry physical dimensions.

The reformulation is especially explicit in the covariant matter actions. For a scalar field,

$\not h=\hbar c$9

and with the above dimensions both $E_k$0 and $E_k$1 are dimensionless (Volovik, 2022). In the 2026 extension, the same logic is pushed further: the inverse Planck constant $E_k$2 is treated as a frequency-like quantity, so that a mass $E_k$3, having the dimension $E_k$4, is again dimensionless (Volovik, 14 Feb 2026).

A central implication is that the ADW-type program relocates dimensional structure from invariant observables to vacuum-dependent metric or tetrad components. This is the conceptual setting in which later papers reinterpret one or two Planck constants as geometric parameters.

4. Planck constants as metric parameters

One of the most distinctive developments in the recent ADW-adjacent literature is the claim that the Planck constant should be regarded as part of the emergent Minkowski metric rather than as an external conversion factor. In the two-constant version motivated from Akama-Diakonov emergent tetrads, Minkowski space is parametrized by

$E_k$5

or equivalently in contravariant form by

$E_k$6

The temporal parameter has dimension time and the spatial one dimension length: $E_k$7 In this view, the two constants are vacuum parameters associated with the temporal and spatial sectors of the metric (Volovik, 2023).

In the $E_k$8 simplification used in the 2026 extension, the distinction is collapsed and the Minkowski metric is written

$E_k$9

with $U_k$0 after specialization to $U_k$1 (Volovik, 14 Feb 2026). The symmetric pregeometric phase is then characterized by

$U_k$2

so the emergence of metric, tetrad, and ordinary quantum mechanics is identified with the same phase transition.

This geometric reinterpretation of $U_k$3 is developed further in the dimensionless-physics paper, where flat Minkowski space is written as

$U_k$4

implying

$U_k$5

The usual energy-frequency relation is then re-expressed as

$U_k$6

That paper stresses that $U_k$7 so defined is Lorentz invariant but not diffeomorphism invariant, because it belongs only to the special Minkowski vacuum and not to generally covariant equations (Volovik, 2022).

The same conceptual line is supported by an acoustic analogue. For phonons in a superfluid Bose liquid one may introduce

$U_k$8

with

$U_k$9

Using the interatomic spacing $$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$0 and the estimated equality of ultraviolet scales in liquid helium, Volovik argues that

$$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$1

This is then used as analogue support for the conjectural gravitational relation

$$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$2

while explicitly noting that the corresponding trans-Planckian microphysics of the real vacuum is unknown (Volovik, 2023).

5. Emergent quantum mechanics and phase structure

The 2026 extension proposes that ordinary quantum mechanics is itself emergent within the ADW-type framework. The paper distinguishes three regimes: a pregeometric QFT phase, an emergent QM phase, and the classical-mechanics limit (Volovik, 14 Feb 2026). In its formulation, the pregeometric phase is approached as

$$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$3

where metric, gravity, and ordinary quantum mechanics disappear. The quantum-mechanical regime corresponds to finite nonzero $$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$4, while the classical limit is

$$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$5

The proposed sequence is therefore

$$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$6

In this interpretation, the inverse Planck constants $$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$7 and $$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$8 function as order parameters of the transition from a pregeometric phase to a phase with emergent metric and quantum mechanics (Volovik, 14 Feb 2026). The paper explicitly states that in the GUT-like reading these quantities are order parameters of spontaneous symmetry breaking, while in the anti-GUT or lattice-inspired reading they play a role analogous to emergent coherence scales.

This proposed emergence of quantum mechanics is not derived by a full microscopic calculation. The paper states the idea at a conceptual level: integration over field variables in the QFT phase is said to transform into the path-integral formulation of QM, which then yields classical mechanics in the $$\hat E^a_\mu = \frac{1}{2}\left( \psi^\dagger \gamma^a\partial_\mu \psi - \psi^\dagger\overleftarrow{\partial_\mu} \gamma^a\psi\right), \qquad E^a_\mu=\langle \hat E^a_\mu\rangle .$$9 limit. A representative formula used to illustrate the reinterpretation of particle action is

$G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$0

The intended claim is that the broken phase supplies the $G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$1-weighted phase structure usually taken as fundamental in quantum mechanics (Volovik, 14 Feb 2026).

A plausible implication is that ADW-style pregeometry, in this extended reading, is not only a theory of emergent spacetime but also a theory of emergent quantumness. The supplied literature, however, treats this as a speculative extension rather than a settled consequence.

6. The Wetterich strand: FRG, RG interpretation, and partial integration

The “Wetterich” component of the ADW label is presently the least unified with the Akama-Diakonov tetrad program. In the functional-RG literature, Wetterich appears through the exact flow of the effective average action,

$G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$2

and through its reductions such as the local potential approximation (Bender et al., 2018). The 2018 asymptotic analysis of the LPA flow is directly about this equation, but it does not mention Akama or Diakonov and therefore provides no explicit bridge to emergent tetrads.

That work is nevertheless relevant to the Wetterich side of the compound label because it shows that, after a cutoff-dependent shift and an asymptotic variable change, the large-cutoff correction obeys a heat equation whose orientation flips at $G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$3: for $G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$4 the asymptotic problem is a forward heat equation and is well-posed, whereas for $G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$5 it is a backward heat equation and is ill-posed (Bender et al., 2018). The same paper uses Padé extrapolation in $G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$6 to compare conventional Hermitian and $G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$7-symmetric cubic and quartic models, but again this is a study of the Wetterich equation in LPA, not of ADW pregeometry.

A different Wetterich strand concerns RG interpretation rather than FRG dynamics. Rosaler and Harlander reconstruct Wetterich’s anti-fine-tuning view by arguing that an EFT may be defined by an entire Wilsonian RG trajectory,

$G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$8

rather than by a unique fundamental bare parametrization (Rosaler et al., 2018). On that reading, fine tuning of bare parameters is a parametrization artifact tied to an arbitrary reference scale. This is conceptually adjacent to ADW-style suspicion of naive microscopic fundamentality, but the paper is about naturalness and Wilsonian EFT, not emergent gravity.

A further methodological development is the Lorentzian pAQFT reformulation of a flow equation very close to the Wetterich equation,

$G_{\mu\nu}=\eta_{ab}E^a_\mu E^b_\nu .$9

That work is explicitly about Lorentzian FRG on curved spacetime and generic states, not about Akama or Diakonov (D'Angelo et al., 2022). Accordingly, the currently documented relation between AD pregeometry and Wetterich FRG remains one of juxtaposition and possible future synthesis rather than of completed formal unification.

7. Misconceptions, limitations, and open problems

A frequent misconception is to treat ADW as the name of a finished, standard theory with a single accepted microscopic action, field content, and renormalization prescription. The supplied literature does not support that reading. Instead, it presents a cluster of related claims: composite tetrads from fermion bilinears, emergent metric, dimensionless invariant quantities, reinterpretation of $E^a{}_\mu$0 as a metric parameter, and partial links to Wetterich-style RG formalisms (Volovik, 14 Feb 2026, Volovik, 2022, Volovik, 2023).

Another misconception is to assume that the recent Planck-constant papers derive their conclusions from a completed microscopic quantum-gravity model. They do not. The acoustic paper explicitly states that the relation $E^a{}_\mu$1 is supported only by analogy, because the microscopic trans-Planckian vacuum physics of real gravity is unknown (Volovik, 2023). Likewise, the 2026 extension does not provide a full dynamical theory of varying or emergent $E^a{}_\mu$2, nor a detailed derivation of how pregeometric QFT functional integration becomes a QM path integral (Volovik, 14 Feb 2026).

The FRG side has its own limitations. The asymptotic heat-equation reduction of the Wetterich flow is a leading-order statement within the local potential approximation and does not establish an ill-posedness theorem for the full FRG in richer truncations (Bender et al., 2018). The naturalness paper explicitly acknowledges that the RG-trajectory interpretation is under-formulated as a complete definition of QFT because it does not yet specify the Hilbert space and Hamiltonian in an equally developed way (Rosaler et al., 2018).

The principal open problem is therefore structural: no paper in the supplied set provides a full derivation that starts from one pregeometric microscopic theory and yields, in a single controlled framework, emergent tetrads, dimensionless covariant physics, Planck constants as metric components, ordinary quantum mechanics, and Wetterich-style functional RG. The present state of the topic is better described as a partially overlapping research program than as a single closed theory.

A plausible implication is that “Akama-Diakonov-Wetterich theory” currently functions most accurately as a historiographical and conceptual umbrella for intersecting pregeometric, dimensionless-physics, and Wetterich-inspired lines of thought, with Diakonov-style emergent tetrads providing the clearest common core (Volovik, 14 Feb 2026, Volovik, 2022).