The emergence of action

In the latest post, we have discussed the substantial equivalence between the QM prediction of the position probability density for a free particle in 1D emitted by a perfectly localized source, via the Schrödinger equation and the Born rule (though the result is considered as non-normalizable) and the ensemble pdf of the position calculated with the proposed rules of motion and the proposed source preparation.

The particle position can be regarded as a stochastic variable defined as

where v is the momentum.

We can define another stochastic variable as

that is, the sum of momentum squared. Clearly, this variable is related to the accumulation of energy. It can take only positive values comprised between 0 and t. Its pmf is thus evaluated as

where e = (1+V²)/2 is the energy propensity, a function of the momentum propensity. This binomial distribution is approximated for large times by a Normal distribution, with the mean at e∙t. Indeed, energy × time is what often people call an action.

Now, we could be interested in computing the ensemble distribution of σ, i.e., after averaging the pmf w, or its approximating pdf, for all values of the source momentum v₀. Instead of integrating the pmf over v₀, which could not yield an easily treatable solution, we follow the second approach already used in this post. That is, we consider the relation ⟨σ⟩ = t(1+V²)/2 as a relation between two stochastic variables ⟨σ⟩ and V, and we compute

remembering that V = v₀ in this simple scenario (free 1D particle). We should aware of the fact that, while σ can take all integer values between 0 and t, ⟨σ⟩ is only defined between t/2 (when V = 0) and t (when V = ±1). Again, the pdf above is only an approximation of the true pmf.

It is maybe more interesting to consider yet another stochastic variable, that is, the action "seen" at a particular lattice node (x,t). Let us denote this vairable as σ(x,t). For example, node (1,1) always receives particles starting from (0,0) having σ = 1, and the same is true for nodes (-1,1), (1,2), and (-1,2), while node (0,1) can receive only particles with σ = 0. But node (0,2) can receive particles with σ = 2 or σ = 0. The probability of receiving a particle with σ = 0 is (1-e)², while that of receiving a particle with σ = 2 is (1-e)²/2, as it can be checked by applying the microscopic rules of motion. The total probability is w(0,V;2,0) = 3/2(1-e)². So the average action seen by that node is equal to 3/2. Computing the same quantity for all node, we obtain the general formula

The proof is contained in this old document at p. 8-9 and Appendix A.4.

Procedure to evaluate the average action seen by the node (x=0,t=2)

The function found is independent of the momentum propensity V. Consequently, it represents also the ensemble average action as seen by lattice nodes. We can compare this result with the action in Quantum Mechanics. For a free particle emitted at a perfectly localized source, the wave function in lattice units is shown in the previous post. Taking the phase of this wave function yields the quantity π(x-x₀)²/(2t), which clearly corresponds to the position-dependent part of the formula for ⟨σ(x,t)⟩ above, when it is multiplied by π to obtain a phase angle. The correspondence is almost perfect, except for the term 2t − 1 that in the proposed model replaces the term 2t predicted by QM. For large values of time, however, the two results are practically coincident.

In other terms, the phase of the wavefunction predicted by the complex Schrödinger equation is the continuum limit of the average action see by lattice nodes in the proposed model. In the same way the the square modulus of the wavefunction predicted by the Schrödinger equation is the continuum limit of the ensemble probability density of the position in the proposed model.