Physics as Becoming as Computational Process

Recents posts have discussed the role of Planck's constant $h$ in Standard Quantum Mechanics StdQM presented as the smallest quantum of action as one of Nature's deepest secrets. It can thus be of interest to seek to understand the concept of action as formally energy x time or momentum x length in combinations without clear physical meaning.

Let us then ask if in physics concepts which have a more or less direct physical representation, have a special role? We thus compare concepts like mass, position, time, length, velocity, momentum,  and force, which have physical representations, with the concepts of energy and action, which are not carried the same way in physical terms.    

To seek an answer recall that in the age of the computer it is natural to view the World as evolving from one time instant to a next in processes involving exchange of forces, which can be simulated in computations involving exchange of information, as computational dynamical systems where the dynamics of the World is realised/simulated in time stepping algorithms. Stephen Wolfram has presented such a view. It is a computational form of the general idea of a World evolving in time from one time instant to the next.

This is a World of becoming with focus shifting from what the World is to what the World does, from state to process. 

The time-stepping process for the evolution of the state of a system described by $\Psi (t)$ from time $t$ to time $t+dt$ with $dt$ a small time step, takes the form 

• $\Psi (t+dt) =\Psi (t)+dt\times F(t)$   (or $\frac{d\Psi}{dt} = F$)   (P)

where $F(t)$ represents the force acting on the system at time $t$, which may also depend on the present state $\Psi (t)$. We speak here about 

• state $\Psi (t)$
• force $F(t)$ 
• process (P).

We see that the concept of state (what is) is still present, but we can bring forward the process (becoming) to be of main concern including the force $F(t)$. 

We can describe such a world as a Dynamical Newtonian World based on Newton's Law

• $\frac{dv}{dt}=\frac{f}{m}$ or $v(t+dt)=v(t)+dt\times F(t)$,

with $v(t)$ velocity, $m$ mass, $F(t)=\frac{f(t)}{m}$ and $f(t)$ force. 

This is the ever-changing world of Heracleitos based on state and force and process with physical representations. 

But there is also the world of Parmenides as a static world as Einstein's space-time block Universe. 

The idea of a space-time block Universe is present in the minds of theoretical physicists speaking about physics governed by a Principle of Stationary Action as 

• stationarity of $A(\Psi )\equiv\int_0^T L(\Psi (t))dt$      (PSA)

where $A(\Psi )$ is action, $L$ is a Lagrangian depending on $\Psi (t)$ and $t=0$ is an initial time and $T$ a final time for the dynamical system $\Psi (t)$. PSA means that the actual evolution $\bar\Psi$ is characterised by vanishing change of the total action $A(\Psi )$ under small variations of $\Psi$ of $\bar\Psi$. We note that the action $A(\Psi )$ does not have a direct physical representation but requires a counting clerk to take specific value.

PSA is not realised by computing $A(\Psi )$ for all $\Psi$ and then chosing the true $\bar\Psi$ from stationarity, since the amount of computational work is overwhelming. Instead PSA is realised by time-stepping as a form of (P) with suitable $F$.

Before the computer a problem formulation in terms of PSA was often preferred because the Lagrangian had a given analytical form allowing (P) to be formulated and the $\Psi (t)$ could be determined analytically. But the range of applications was very limited.

With the computer, the focus shifts to (P) allowing unlimited generality. The shift is from PSA where action does not have a physical representation to (P) with physical representation.

Let us now return to $h$ as the smallest quantum of action, with the experience that action is a concept in the head of a counting clerk without direct physical representation and so as the smallest quantum of action. 

This adds to the discussion in recent posts questioning the role of Planck's constant as a fundamental constant of Nature. It does not seem to be so fundamental after all. There is no smallest quantum of action in Nature.

We can compare (P) with a computational gradient method to solve to find an equilibrium state characterised by minimal energy, again with physical representation of (P) but not of energy.