This post builds on our earlier work, “From time to scale: A gentle introduction to scale dynamics”, where we introduced the idea of treating the Hamiltonian itself as a dynamic coordinate that evolves with scale rather than time.

Over the past week, we’ve made significant progress in extending the scale dynamics framework. This post summarizes three key developments that deepen our understanding of how physical theories transform across scales. While the mathematical details can be intricate, the core ideas are accessible to anyone with undergraduate physics training.

Key Developments in Scale Dynamics

Our initial proposal was simple yet powerful: replace time $t$ with log-scale $\tau = \ln(\mu/\mu_0)$ and treat the Hamiltonian $H$ as a dynamic coordinate that evolves as we zoom in or out. This gave us a symplectic picture of renormalization group (RG) flow with the minimalist Hamiltonian:

\[H_{\text{scale}} = p_i \beta_i(g)\]

where $g_i$ are coupling constants and $p_i$ their conjugate momenta. This framework reproduced standard RG results while opening a geometric path to new insights.

Today, we’ll explore three crucial extensions, each of which is discussed in more detail in the Scale Theory Insights section:

  1. Scheme independence via canonical transformations
  2. Irreversible scale dynamics using contact geometry
  3. Beyond linear momentum: higher-order scale corrections

1. Scheme Independence as Canonical Transformations

One of the most puzzling aspects of quantum field theory is that different renormalization schemes yield different-looking β-functions, yet produce identical physical predictions. In our scale dynamics framework, this mystery evaporates: scheme changes are simply canonical transformations in $(g,p)$ phase space.

Just as in classical mechanics, where a canonical transformation preserves Hamilton’s equations while changing coordinates, a change of renormalization scheme preserves physical predictions while transforming the β-functions. The technical mechanism involves generating functions $F(g,\tilde{p})$ that produce the coordinate transformation:

\[\tilde{g}_i = f_i(g)\]

with conjugate momenta determined by:

\[\tilde{p}_i = \sum_j \frac{\partial g_j}{\partial \tilde{g}_i}p_j\]

This elegant formulation explains why certain quantities remain invariant across schemes. For instance, in $\phi^4$ theory, the one-loop coefficient is scheme-independent while higher-order terms transform in specific ways. This perspective offers practical tools for optimizing perturbative calculations and discovering scheme-independent quantities.

Read more: Scheme Independence via Canonical Transformations

2. Irreversibility Through Contact Geometry

Standard RG flows are irreversible - coarse-graining destroys microscopic information, and we can’t undo this process. Yet our original symplectic framework is time-reversible, creating a conceptual tension.

The solution? Contact geometry - a natural extension of symplectic geometry that builds in asymmetry. We enlarge phase space from $(g_i,p_i)$ to $(g_i,p_i,s)$, where $s$ measures lost information, and modify our scale Hamiltonian to:

\[K(g,p,s) = p_i\beta_i(g) + F(g) - \gamma s, \quad \gamma > 0\]

This generates the contact RG equations:

\[\begin{aligned} \dot{g}_i &= \beta_i(g)\\ \dot{p}_i &= -p_j\partial_{g_i}\beta_j - \partial_{g_i}F + \gamma p_i\\ \dot{s} &= -F(g) + \gamma s \end{aligned}\]

The friction-like term $\gamma > 0$ ensures irreversibility. A remarkable consequence is that $K$ itself serves as a strictly decreasing function along RG trajectories:

\[\dot{K} = -\gamma K\]

This behavior perfectly matches the famous $C$-theorem in 2D conformal field theory and its higher-dimensional analogs, which state that certain quantities must monotonically decrease along RG flows.

In the $\phi^4$ example, trajectories now spiral toward fixed points while entropy (measured by $s$) increases - just as expected in physical RG transformations.

Read more: Irreversible Scale Dynamics Using Contact Geometry

3. Beyond Linear Momentum: Higher-Order Corrections

Our initial Hamiltonian was linear in momenta $p_i$. But what if this is just the leading approximation? The most general analytic scale Hamiltonian would be a power series:

\[H_{\text{scale}}(g,p) = \sum_{n=1}^{\infty}\frac{1}{n!}T^{(n)}_{i_1\dots i_n}(g)p_{i_1}\dots p_{i_n}\]

Where the first term $T^{(1)}_i(g) = \beta_i(g)$ reproduces the standard β-functions, and higher terms represent corrections that become important when momentum feedback is non-negligible.

Including even just quadratic terms unlocks new phenomena:

  1. Closed orbits: Instead of always flowing to fixed points, coupling constants can oscillate with scale, potentially explaining discrete scale invariance observed in certain physical systems.

  2. Non-perturbative deformations: Near fixed points, the higher-order terms modify critical exponents in ways invisible to linear analysis.

For our $\phi^4$ example, adding a simple quadratic term $A_{gg} = c\,g$ yields the Hamiltonian:

\[H = p(-\varepsilon g + \lambda g^2) + \frac{1}{2}c\,g\,p^2\]

Depending on the magnitude of $c$, the behavior ranges from conventional flow toward fixed points to closed orbits exhibiting discrete scale invariance, to runaway trajectories.

When Do Higher-Order Terms Matter?

The linear approximation (standard RG) holds when:

  • We’re in the perturbative ultraviolet regime
  • We’re near a stable fixed point
  • Contact friction dominates momentum growth

However, at strong coupling or near irrelevant fixed points, higher-order terms can dominate, requiring a non-perturbative treatment. This explains why conventional RG techniques sometimes break down in these regimes.

Read more: Beyond Linear Momentum—Higher-Order Scale Corrections

Outlook and Open Questions

The extended scale dynamics framework raises exciting possibilities:

  1. Quantizing scale mechanics - What happens when we promote $g_i$ and $p_i$ to operators?

  2. Contact-geometry deformations - How do contact flows select which higher-order terms matter in real QFTs?

  3. Numerical experiments - Can lattice implementations detect these higher-order effects?

  4. Bridge to holography - Do closed orbits in scale dynamics parallel holographic RG cycles?

Conclusion

Scale dynamics has evolved from a pedagogical tool to a powerful framework with predictive potential. By treating renormalization geometrically through symplectic and contact structures, we gain both conceptual clarity and computational advantages.

The standard β-function sits atop a tower of higher-momentum couplings. Ignoring them works well in many familiar contexts but can erase whole classes of scale phenomena when momentum feedback becomes significant.

For students looking to explore these ideas further, try applying the framework to your favorite field theory. Calculate the β-functions, introduce higher-order momentum terms, and explore how the dynamics changes. You might discover phenomena invisible to conventional RG analysis!

This blog post summarizes ongoing research. The framework presented here is exploratory, and we welcome feedback and critiques from the community.