Abstract
We propose a geometric framework for the Renormalization Group (RG) by treating the RG scale τ as a time-like coordinate and the RG flow as a dynamical system on a phase space of couplings and conjugate momenta. We demonstrate that standard RG is a Symplectic flow generated by a scale Hamiltonian H = p β(g). The key insight is that renormalization scheme changes are Canonical Transformations on this phase space, thereby resolving the ambiguity of renormalization schemes: physical observables (critical exponents, fixed point locations, correlation function behaviors) are geometric invariants of the symplectic structure and hence automatically scheme-independent. This elevates scheme independence from an empirical requirement to a fundamental geometric principle. Furthermore, we extend the phase space to Contact Geometry, revealing that the intrinsic irreversibility of RG flows (manifested in C-theorems) arises naturally from a friction-like term that ensures monotonic decrease of a geometric C-function. This unified approach places scale evolution on the same mathematical footing as time evolution, offering new insights into the structure of effective field theories.
Authors
- Tomer Barak
- VScode
- opencode
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