From time to scale: A gentle introduction to scale dynamics

Physics is full of levels of description – atoms, molecules, fluids; spins, domains, magnets. We usually fix the level and then ask how the variables at that level evolve in time. The idea explored here is the reverse:

Let the scale flow, and ask how the Hamiltonian changes along that flow.

Doing so may give us a unified language for:

the renormalization group (RG) in quantum field theory,
effective theories in condensed-matter physics,
cross-scale phenomena such as turbulence or criticality.

The analogy: time evolution vs. scale evolution

In classical mechanics, we study how physical variables evolve with time. The Hamiltonian $H(q, p)$ governs this evolution through:

\[\frac{dq}{dt} = \frac{\partial H}{\partial p}, \quad \frac{dp}{dt} = -\frac{\partial H}{\partial q}\]

I propose a similar framework for how theories evolve with scale. Instead of treating the Hamiltonian as fixed, let’s consider the “meta-dynamics” where the Hamiltonian itself evolves as we change the scale:

\[\frac{dH}{ds} = \beta(H)\]

Here, $s$ is a scale parameter (e.g., logarithm of length scale), and $\beta(H)$ is a “scale velocity” function analogous to the familiar beta functions in quantum field theory.

Canonical transformations across scales

In usual Hamiltonian mechanics, canonical transformations preserve the symplectic structure but may change the form of the Hamiltonian. In scale dynamics, we’re interested in transformations that change the effective description while preserving certain physical outcomes.

For a system with Hamiltonian $H(q,p)$ at scale $s$, we can define a canonical transformation to new variables $(Q,P)$ at scale $s+ds$:

\[Q = Q(q,p,s), \quad P = P(q,p,s)\]

The transformation is canonical if it preserves the Poisson brackets:

\[\{Q_i, P_j\} = \delta_{ij}, \quad \{Q_i, Q_j\} = \{P_i, P_j\} = 0\]

The new Hamiltonian $K(Q,P)$ differs from $H(q,p)$ not just by variable substitution but by additional terms coming from the explicit scale dependence:

\[K(Q,P) = H(q(Q,P),p(Q,P)) + \frac{\partial F}{\partial s}\]

where $F$ is the generating function of the canonical transformation.

Scale invariance and fixed points

A particularly interesting case is when the Hamiltonian reaches a fixed point under scale transformation:

\[\beta(H^*) = 0\]

This corresponds to scale-invariant physics, where the system looks the same at different scales. Critical phenomena and phase transitions often exhibit such behavior.

For systems slightly away from a fixed point, linearizing the beta function gives:

\[\beta(H) \approx M \cdot (H - H^*)\]

The eigenvalues of the matrix $M$ determine whether perturbations grow (relevant) or shrink (irrelevant) as the scale increases.

Connection to the renormalization group

This framework makes direct contact with Wilson’s renormalization group. In that context, one integrates out short-distance degrees of freedom and rescales to obtain an effective theory at a larger length scale.

In our language, this is just a particular choice of canonical transformation corresponding to a specific flow along the scale direction. The RG equations are precisely our scale evolution equations.

The advantage of our approach is that it puts scale evolution on the same mathematical footing as time evolution, allowing us to apply the powerful machinery of Hamiltonian mechanics to understand cross-scale physics.

Beyond equilibrium: dissipative scale dynamics

What about systems far from equilibrium? For these, we can extend our framework to include dissipative terms:

\[\frac{dH}{ds} = \beta(H) + D(H)\]

where $D(H)$ represents irreversible changes in the effective description as we change scale. This is particularly relevant for systems with significant entropy production across scales, such as turbulent flows.

Experimental signatures

How might we test these ideas? Several possibilities exist:

Multi-scale correlations: Measuring correlation functions across widely separated scales might reveal signatures of the propagation of information through scale space.
Scale anomalies: Just as quantum anomalies break classical symmetries, we might find “scale anomalies” where expected scale symmetries are broken in interesting ways.
Emergent conservation laws: Our framework predicts that certain quantities should be conserved across scales even when the microscopic description changes dramatically.

Computational approaches

The practical application of these ideas requires numerical methods for tracking how Hamiltonians evolve with scale. I’m currently developing algorithms based on:

Tensor network renormalization
Neural network representations of effective Hamiltonians
Monte Carlo methods for estimating beta functions

Philosophical implications

Beyond its technical aspects, this framework suggests a profound philosophical shift. Rather than asking “what is the fundamental theory of everything?”, we might instead ask “how do theories transform across scales?”.

This perspective aligns with a growing recognition that different levels of description have their own validity and aren’t simply approximations to some more “fundamental” level.

Future directions

I’m particularly excited about applying these ideas to:

Understanding emergent spacetime in quantum gravity
Developing more efficient multi-scale numerical simulations
Exploring the connections between scale dynamics and information theory

In future work, I plan to develop more concrete examples of scale dynamics in action, particularly for strongly-coupled quantum systems where traditional perturbative methods struggle.

Conclusion

By putting scale evolution on the same mathematical footing as time evolution, we open up new ways to understand how physics works across different levels of description. This approach doesn’t replace existing techniques like the renormalization group, but rather provides a unifying language that connects them to the familiar framework of Hamiltonian mechanics.

The journey from time to scale may ultimately lead us to a deeper understanding of how nature organizes itself across its vast range of scales.