nLab kinematics and dynamics

1. 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.

2. 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

$$Z : Bord_n^S \to \mathcal{C}$$

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

Then:

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

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

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

$$Z : Bord_n^S \to \mathcal{C}$$

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

In AQFT

(…)

3. Examples

In $\sigma$-models

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

Charged particle

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

$$Z : Bord_1^{Riemn} \to Hilb \,.$$

It sends

• each object – a point – to the space $\Gamma(P \times_{U(1)} \mathbb{C})$ of square integrable sections of the line bundle associated with the background gauge field;

• each morphism $(\bullet \stackrel{t}{\to} \bullet)$ to the operator $\exp(i t \nabla^* \nabla)$, where $\nabla^* \nabla$ is the Laplace-Beltrami operator of the covariant derivative $\nabla$.

So in terms of the background field data we have:

• the kinematics is encoded in a smooth principal bundle – hence a cocycle in smooth cohomology;

• the dynamics is encoded in a connection on this bundle – hence a refinement of the above cocycle to ordinary differential cohomology.