Scale Dynamics in Context: Connecting Our Framework to Recent Literature

In our recent series of posts on scale dynamics (From Time to Scale, Canonical Transformations, Irreversible Scale Dynamics, and Beyond the Linear β), we developed a framework that rewrites renormalization-group (RG) flows in Hamiltonian and contact-geometric language. Below we place those ideas next to very recent work on geometric flows, holography, and rigorous RG.

Our Framework in Brief

  1. Hamiltonian formulation – Treat the scale parameter $\tau=\ln(\mu/\mu_0)$ as a time-like coordinate and build a phase-space of couplings $g_i$ and conjugate momenta $p_i$ with a “scale Hamiltonian” $H_{\text{scale}}(g,p)$.

  2. Scheme changes as canonical transformations – Renormalization-scheme changes appear as canonical (symplectic) transformations, explaining scheme-independence of observables.

  3. Contact geometry for irreversibility – Extending to a contact manifold by adding a coordinate $s$ that tracks information loss naturally encodes the irreversibility of RG flows.

  4. Higher-order momentum terms – Terms beyond the linear approximation give rise to phenomena such as closed orbits in scale space and altered critical behaviour.

  5. Hamilton-Jacobi approach to glassy systems – Applying the Hamilton-Jacobi formulation reproduces the Vogel–Fulcher–Tammann law, linking microscopic RG parameters to macroscopic observables.

This structure also mirrors holographic RG in AdS/CFT, where the bulk radial direction acts as the RG scale and symplectic structure arises from the Hamiltonian formulation of the bulk action.

Connections to Recent Literature

Holographic RG flows with symplectic structure

Karndumri showed that electric–magnetic (“symplectic”) deformations in four-dimensional $\mathcal N=4$ gauged supergravity generate new $\mathcal N=1$ and $\mathcal N=2$ RG solutions (Phys. Rev. D 105 086009, 2022; arXiv:2203.01766).
Our Canonical Transformations post provides a model-independent explanation: because scheme changes are symplectic, physical predictions remain intact under those holographic deformations.

Rigorous RG for symplectic-fermion models

A non-perturbative construction of RG fixed points for symplectic-fermion theories was given by Giuliani, Mastropietro & Rychkov (JHEP 01 (2021) 026; arXiv:2008.04361).
Where that paper proves existence and analyticity, our Beyond the Linear β post explores what happens when quadratic and higher momentum terms are retained—e.g. closed orbits and discrete scale invariance. Both perspectives insist on preserving the underlying symplectic structure.

Symplectic geometric-flow techniques

Fei & Phong introduced dual Ricci flow and Hitchin gradient flow on symplectic manifolds, including short-time existence via a DeTurck trick (arXiv:2111.14048).
Our Irreversible Scale Dynamics post parallels these results: the contact one-form $\eta=\mathrm d s-p_i\,\mathrm d g_i$ supplies a built-in monotone, much like Perelman-type entropies in Ricci flow.

Entropic monotones and irreversibility

Information-theoretic proofs of RG irreversibility—e.g. the defect-entropy $g$-theorem (Cuomo et al., Phys. Rev. Lett. 128 021603, 2022) and work by Castro-Alvaredo et al. (arXiv:1706.01871)—resonate with our contact-geometry monotone $C_{\text{mon}}=K$, further tying geometric structure to information loss.

Novel Contributions and Future Directions

  1. Contact geometry for irreversibility – Our contact formulation supplies a geometric c-like monotone that invites comparison with information-geometric approaches.

  2. Higher-order momentum terms – Closed orbits, scheme-independent invariants, and exotic critical exponents emerge once quadratic and higher terms are kept.

  3. Bridge to non-equilibrium statistical mechanics – The contact friction term $-\gamma s$ mirrors dissipative terms in stochastic thermodynamics, hinting at fluctuation theorems for RG.

  4. Quantization of scale space – Promoting $g_i$ and $p_i$ to operators suggests a “quantum scale mechanics,” connecting to quantum-information aspects of RG and holography.

Conclusion

By emphasizing the dynamical and geometric structure of scale transformations, our framework links holographic RG, rigorous symplectic-fermion results, and modern geometric-flow techniques. We anticipate fruitful exchanges among these approaches—particularly on irreversibility, higher-order dynamics, and quantum extensions—that will deepen our understanding of how nature organizes itself across scales.

Future posts will explore stochastic thermodynamics, quantum-information aspects of RG, and applications to strongly-coupled systems. We welcome collaborations with researchers pursuing related directions.

References

  1. P. Karndumri, “Holographic RG flows and symplectic deformations of $\mathcal N=4$ gauged supergravity,” Phys. Rev. D 105 086009 (2022). https://doi.org/10.1103/PhysRevD.105.086009
  2. A. Giuliani, V. Mastropietro & S. Rychkov, “Gentle introduction to rigorous RG: a worked fermionic example,” JHEP 01 (2021) 026. https://arxiv.org/abs/2008.04361
  3. T. Fei & D. H. Phong, “Symplectic geometric flows,” arXiv:2111.14048 (2021).
  4. O. A. Castro-Alvaredo, B. Doyon & F. Ravanini, “Irreversibility of the RG flow in non-unitary QFT,” arXiv:1706.01871 (2017).
  5. C. Cuomo et al., “Renormalization-group flows on line defects,” Phys. Rev. Lett. 128 021603 (2022).