Halliday & Matthiessen (1999: 153-4):
The model that generalises across these various domains of experience is different from any one of these particular submodels. It sets up one central participant that is common to all processes. This is the participant through which the process comes to be actualised. We refer to it as the Medium. […]
The participant functioning as Medium may be affected in various different ways, depending on the particular domain — the 'trace' may be physical, mental, and so on; but the status of Medium generalises across these domains.