Non-Euclidean Phase Space?

Question

In classical mechanics, the canonical equations of motion can be rendered in terms of Poisson Brackets: $$\begin{align} \left\{q_i, F(\mathbf{q},\mathbf{p})\right\} &= \frac{\partial F}{\partial p_i}, \\ \left\{p_i, F(\mathbf{q},\mathbf{p})\right\} &= -\frac{\partial F}{\partial q_i},\ \mathrm{and} \\ \left\{H, F(\mathbf{q},\mathbf{p})\right\} &= -\frac{\operatorname{d} F}{\operatorname{d} t}. \end{align}$$

This is taken to mean that the $q_i$ generates translations in the $-p_i$ direction, $p_i$ in the $q_i$ direction, and $H$ (the Hamiltonian) through time. Is there anything that can be gained by adding a Christoffel symbol like connection to the canonical equations (ie translating the phase space gradient into a covariant derivative)?

Concretely, say $V_j$ is in a vector space tangent to the phase space manifold (in some combination of $\mathbf{q}$ and $\mathbf{p}$ directions, or in an entirely unrelated vector space). Is it possible to construct a meaningful phase space by defining the Poisson brackets as: $$\begin{align} \left\{q_i, V_j(\mathbf{q},\mathbf{p})\right\} &= \frac{\partial V_j}{\partial p_i} + \left[\Gamma_p\right]_{i\hphantom{k}j}^{\hphantom{i}k} V_k, \\ \left\{p_i, V_j(\mathbf{q},\mathbf{p})\right\} &= -\frac{\partial V_j}{\partial q_i} - \left[\Gamma_q\right]_{i\hphantom{k}j}^{\hphantom{i}k} V_k,\ \mathrm{and} \\ \left\{H, V_j(\mathbf{q},\mathbf{p})\right\} &= -\frac{\operatorname{d} V_j}{\operatorname{d} t}- \left[\Gamma_t\right]_{i\hphantom{k}j}^{\hphantom{i}k} V_k, \end{align}$$ or some analogous construction?

Is the resulting curved phase space always expressible, through some transformation of coordinates and Hamiltonian, using ordinary canonical equations of motion?

Answer 1

1. Given a symplectic manifold $(M,\omega)$, it is natural to ponder what tangent bundle connection $$\nabla: \Gamma(TM)\times\Gamma(TM)\to \Gamma(TM) \tag{1}$$ to chose?

2. Generically, it is natural to choose $\nabla$ to be torsionfree $$T~=~0,\tag{2}$$ and compatible $$\nabla \omega~=~0\tag{3}$$ with the symplectic $2$-form $\omega$.

3. One may show (via partition of unity) that a torsionfree & compatible connection $\nabla$ exists on a paracompact manifold, cf. e.g. this Phys.SE post. Be aware that a such a connection $\nabla$ is far from being unique.

4. The triple $(M,\omega,\nabla)$ is called a Fedosov manifold, and it is the geometric input for the Fedosov star product $\star$ in deformation quantization.

5. Fedosov quantization can be used to define covariant derivatives and time evolution for tensor fields, cf. Refs. 1-2. The classical construction can be extracted in the $\hbar\to 0$ limit.

6. In special cases the symplectic manifold $(M,\omega)$ is endowed with a compatible metric $g$, cf. Kähler manifold. In such situations, the metric $g$ uniquely singles out the Levi-Civita connection. See also this related Phys.SE post.

7. Finally, let us mention that if the symplectic manifold $M=T^{\ast}Q$ is a cotangent bundle equipped with the tautological symplectic structure (cf. e.g. this Phys.SE post), and the base manifold $Q$ is endowed with a connection, this also leads to interesting possibilities, e.g. a super-Poisson bracket, cf. Ref. 3.

References:

1. B.V. Fedosov, A simple geometrical construction of deformation quantization, J. Diff. Geom. 40 (1994) 213.

2. B.V. Fedosov, Deformation quantization and index theory, Mathematical Topics, Vol. 9, Akademie Verlag, Berlin, 1996.

3. B. DeWitt, Supermanifolds, Cambridge Univ. Press, 1992; Section 6.7.