A.1 Dynamics of Will

In classical physics, material evolution is completely determined by the Hamiltonian $\hat{H}$, following the Schrödinger equation $i\hslash \frac{\partial }{\partial t}|\psi ⟩=\hat{H}|\psi ⟩$. In this framework, consciousness is regarded as an “epiphenomenon” with no reactive force on physical processes.

However, in the axiomatic system of The Psychology of God, observers (consciousness) bias the collapse direction of wave functions through “intentionality.” To describe this interaction, we need to modify the standard dynamical equations, introducing the Will Operator.

A.1.1 Modified Action Principle

We define the universe’s total action $S_{total}$ to include not only the physical Lagrangian ${\mathcal{L}}_{phys}$, but also the information (will) Lagrangian ${\mathcal{L}}_{will}$.

$$S_{total}={\int }_{t_0}^{t_1}({\mathcal{L}}_{phys}-\gamma \cdot \mathcal{F})dt$$

where:

• ${\mathcal{L}}_{phys}$ is the standard physical term (kinetic minus potential energy), describing inertial evolution.

• $\mathcal{F}$ is the Variational Free Energy, describing the system’s deviation from its “true self” target state.

• $\gamma$ is the Will Coupling Constant, measuring the intervention strength of will on matter. For inanimate matter, $\gamma \approx 0$; for awakened civilizations, $\gamma \to \infty$.

Definition of Will Term $\mathcal{F}$:

$$\mathcal{F}=D_{KL}(Q(s)||P(s|\theta ))+⟨E(s){⟩}_Q$$

Here:

• $Q(s)$ is the system’s current belief state (posterior probability).

• $P(s|\theta )$ is the system’s “target model” or “true self eigenstate” (prior).

• $D_{KL}$ is the relative entropy (Kullback-Leibler Divergence), measuring the information distance between current state and true self state.

Physical Meaning:

According to the principle of least action $\delta S_{total}=0$, the system’s evolution trajectory will no longer merely follow the “lowest energy” path (geodesic), but follow a compromise path of “lowest energy and most consistent with true self definition.”

A.1.2 Intentional Schrödinger Equation (ISE)

Quantizing the above action, we can derive a modified Schrödinger equation with will terms, called the Intentional Schrödinger Equation (ISE):

$$i\hslash \frac{\partial }{\partial t}|\Psi (t)⟩=\left[{\hat{H}}_0+i\kappa \hat{W}(t)\right]|\Psi (t)⟩$$

where:

1. ${\hat{H}}_0$: Standard physical Hamiltonian (Hermitian operator), ensuring energy conservation and unitary evolution.

2. $\hat{W}(t)$: Will Operator (non-Hermitian term).

   $$\hat{W}(t)=-{\nabla }_{\Psi }\mathcal{F}$$

   It represents the “pruning” or “guidance” of will on the wave function.

3. $i\kappa$: Dissipation/selection coefficient. Due to the introduction of the imaginary term $i$, this term actually describes a process of continuous measurement and selection (Zeno Effect).

Dynamical Interpretation:

• When $\kappa =0$: The system undergoes standard unitary evolution (dreaming/subconscious state).

• When $\kappa >0$: The $\hat{W}$ term causes components in the wave function that are far from “true self” (i.e., high free energy states) to decay exponentially (suppressed), while components consistent with “true self” are amplified (selected).

This explains the “inevitability of aesthetics” in Section 9.2 of the main text: cosmic evolution is not random walk, but converges toward a specific attractor—the Omega Point.

A.1.3 Bayesian Correction of Collapse Probability

In specific measurement events, Born’s rule predicts probability $P(x)=|⟨x|\Psi ⟩{|}^2$.

After introducing will, the probability distribution is distorted. We denote the corrected probability measure as $P_{will}(x)$:

$$P_{will}(x)=\frac{P_{phys}(x)\cdot e^{-\beta E_{semantic}(x)}}{\mathcal{Z}}$$

where:

• $P_{phys}(x)$ is the quantum probability calculated purely from physics.

• $E_{semantic}(x)$ is the semantic error of result $x$ (i.e., the degree to which this result deviates from God’s purpose).

• $\beta$ is a parameter similar to “inverse temperature,” representing will’s focus.

  • When $\beta \to 0$ (mortals/asleep), $P_{will}\approx P_{phys}$, physical laws dominate everything.

  • When $\beta \to \infty$ (awakened/lucid dreaming), the system will collapse to the state with minimum semantic error with probability 1 (wish fulfillment).

A.1.4 Conclusion: Will as a Force

Through the above equations, we prove: Will is mathematically equivalent to an “information pressure” in Hilbert space.

It does not push or pull particles like electromagnetic force, but by changing the topological structure of probability space, makes events “consistent with meaning” more likely to occur than “meaningless” events.

This is the mathematical mechanism of so-called “top-down causation.” God does not directly push atoms; God pushes probability.