What is the correct terminology for a "symplectic covariant" equation?

Question

A Lorentz covariant equation is one that takes the same form even when a Lorentz transformation is applied to each variable. Lorentz covariance is generally made manifest by writing the equation with all Lorentz indices contracted together with the Minkowski metric, so that each scalar quantity in the equation is a Lorentz invariant.

$$\partial_\mu x^\mu = \eta_{\mu \nu} \partial^\mu x^\nu = 0,$$

$$x^\mu = (t,x,y,z), \quad \quad \eta_{\mu \nu} = \mathrm{diag}\{-1,1,1,1\}.$$

Question: What is the corresponding term for "symplectic covariance", where all variables with symplectic indices are contracted together using the symplectic form $\epsilon_{ab} = \{\{0,1\},\{-1,0\}\}$? Here's an example using a dynamical equations for the Wigner function in quantum mechanics.

$$\partial_t W(\alpha) = F^{ab} \partial_a \alpha_b W(\alpha) = \epsilon_{ac} \epsilon_{bd} F_{ab} \partial_c \alpha_d W(\alpha) ,$$

$$\alpha = (\alpha_x,\alpha_p) = (x,p), \quad \quad F^{ab} = \{\{F^{xx},F^{xp}\},\{F^{px},F^{pp}\}\}.$$

The reason the terminology is unclear is that this is really only covariant under linear symplectic transformations (i.e., linear transformations that preserve the symplectic form). Strictly speaking, "symplectic covariance" suggests a lot more generality, akin to diffemorphism covariance.

Answer 1

The difference you have pointed out is between the transformation rule for coordinates and the transformation rule for vector/tensor fields. For instance, if spacetime is oriented and equipped with a pseudo-Riemannian metric, then under coordinate transformations which preserve the orientation and the metric, the transformations for tangent vectors will be Lorentz transformations. If spacetime is equipped with a symplectic form, then under coordinate transformations which preserve the symplectic form, the transformations for tangent vectors will be symplectic linear transformations. While the coordinates may transform under a general diffeomorphism, tangent vectors always transform in a linear way (which can be thought of as the first derivative of the coordinate transformation). The general story is here https://en.wikipedia.org/wiki/G-structure_on_a_manifold.