# What is the correct terminology for a “symplectic covariant” equation?

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.

• Although that is true regarding the question as I phrased is (thanks!), there are of course lots of cases where an object is invariant under linear symplectic, but not all symplectic, transformations, and I am left wondering whether there is terminology for this. (Invariance being distinct from the covariance using in my question.) For instance, in that example I give, the numbers $F^{ab}$ are constant over phase space, and the space of solutions to the dynamical equation has this "linear-symplectic" but not "general-symplectic" invariance. Perhaps those terms in quotes are good enough? – Jess Riedel Jul 2 at 18:31