Base class with member parameterized on type of extending class

[rcor via Digitalmars-d-learn]

I'm trying to make a game, and would like to set up the following 
hierarchy:
At any time, the game is in one Scene. Each scene has a state 
machine that manages States of type T, where T is the type of the 
scene (e.g. Overworld, Menu).

abstract class State!T {
   void update(T scene, float time, InputManager input);
   ...
}

class StateMachine!T { //manages states of type State!T }

abstract class Scene {
   alias T = // type of class extending Scene
   // methods pushState, popState, currentState access 
_stateMachine
   private StateMachine!T _stateMachine;
}

class MainMenu : Scene {
    // I want _stateMachine of type StateMachine!MainMenu
}

class Overworld : Scene {
   // I want _stateMachine of type StateMachine!Overworld
}

class MoveToLocation : State!Overworld {
   override void update(Overworld world, float time, InputManager 
input) {
     // access properties of Overworld here
   }
}

Within the Scene class, I've tried alias T = typeof(this), but 
that appears to be resolved within Scene. This means that any 
Scene, such as Overworld, have a state machine of type 
StateMachine!Scene rather than StateMachine!Overworld. Since 
States are particular to a certain scene and are designed to 
manipulate properties specific to that type of scene, this would 
involve a lot of casting if States are not parameterized. It 
feels like I need something like a class version of the (this T) 
syntax used in templates.

This all smells a bit off though, so I wouldn't be surprised if 
the answer is that I'm approaching this all wrong, but right now 
I'm not seeing it.