nLab
kinematics and dynamics

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

In the context of mechanics (broadly construed), one distinguishes between kinematics and dynamics:

  1. Kinematics concerns (only) the physical fields, states and observables, including the spaces and algebras (such as a phase space or Hilbert space of (pure) states and an appropriate algebra of observables) into which they are organised;

  2. Dynamics additionally treats the evolution of the system in time or even spacetime: as given by a Lagrangian and action functional and as given by the action of Hamiltonian quantum observables on physical states.

    In the Schrödinger picture, we think of the states as evolving, while the observables evolve in the Heisenberg picture. In the interaction picture we think of the states as evolving with respect to a given time evolution and the observables to evolve, too, with respect to a perturbation of this time evolution.

Formalization

The notions of kinematics and dynamics may be formally defined in the two formalizations of quantum field theory: FQFT and AQFT.

In FQFT

Consider a quantum field theory as given by a strong monoidal functor

Z:Bord n S𝒞 Z : Bord_n^S \to \mathcal{C}

from a category of cobordisms with SS-structure (for instance conformal structure or Riemannian structure) to some symmetric monoidal category 𝒞\mathcal{C}.

Then:

  • the kinematics of ZZ is Z 0:Ob(Bord n S)Ob(𝒞)Z_0 : Ob(Bord_n^S) \to Ob(\mathcal{C}), the action of the functor ZZ on objects.

  • with that givenm the dynamics of ZZ is Z 1:Mor(Bord n S)Mor(𝒞)Z_1 : Mor(Bord_n^S) \to Mor(\mathcal{C}), the action of the functor ZZ on morphisms.

This means that as we regard an nn-dimensional QFT as an extended QFT given by an n-functor

Z:Bord n S𝒞 Z : Bord_n^S \to \mathcal{C}

to from a (infinity,n)-category of cobordisms with SS-structure to some symmetric monoidal (infinity,n)-category 𝒞\mathcal{C} the dichotomy between kinematics and dynamics may be regarded as being blurred a bit: we can regard the action Z 0Z_0 on objects as the genuine kinematics and the action Z nZ_n on n-morphisms as the genuine dynamics, and then the actions Z 1kn1Z_{1 \leq k \leq n-1} as interpolating between these two notions.

In AQFT

(…)

Examples

In σ\sigma-models

We discuss the notions of kinematics and dynamics for sigma-model QFTs.

Charged particle

Given a Riemannian target space (X,g)(X,g) and a background gauge fields given by a circle bundle with connection (PX,)(P \to X, \nabla), the corresponding sigma-model quantum field theory is, as an FQFT, a functor

Z:Bord 1 RiemnHilb. Z : Bord_1^{Riemn} \to Hilb \,.

It sends

So in terms of the background field data we have:

Revised on January 9, 2013 02:24:41 by Urs Schreiber (82.113.98.110)