##\Phi## seems to be the fundamental object that exists on the lattice grid. H can be either defined or not defined, being an added constraint, but the time evolution is always defined in the end in terms of how ##\Phi## interacts between adjacent cells or adjacent times. Any method of defining the time evolution locally (in terms of adjacent cells or adjacent times) will invariably result in a wave-like structure and propagation of the field grid ##\Phi##.
1. $$\Phi_x^{t+1} = \Phi_x^t$$
is a frozen field with no time evolution.
2. $$\Phi_x^{t+1} = \Phi_{x+1}^t$$
is a field where everything moves in the same direction at a constant velocity c.
3. $$\Phi_x^{t+1} = \Phi_{x+1}^t + \Phi_{x-1}^t - \Phi_x^{t-1}$$
is a classical wave equation field with 1 wave component ##\frac{d^2\Phi}{dt^2}=\frac{d^2\Phi}{dx^2}## where all waves disperse equally in all directions.
4. $$Re(\Phi_x^{t+1}) = - Im(\Phi_{x+1}^t - 2*\Phi_x^t + \Phi_{x-1}^t) + Re(\Phi_x^t)$$
$$Im(\Phi_x^{t+1}) = Re(\Phi_{x+1}^t - 2*\Phi_x^t + \Phi_{x-1}^t) + Im(\Phi_x^t)$$
is the Schrodinger wave ##i\frac{d\Phi}{dt}=-\frac{d^2\Phi}{dx^2}## with 2 components Re and Im. To get localized wave packets that move in a specific direction, a minimum of 2 wave components are needed.
5. Boson 4-component waves and fermion 8-component waves may possess other properties not achievable using fewer components.