2.2 Why Unitarity? — Conservation of Probability and Logical Consistency

When we claim the universe is a unitary QCA, we are actually making a commitment about the deepest layer of existence: information conservation.

In standard quantum mechanics textbooks, unitarity is usually expressed as an abstract mathematical property: evolution operator $U$ must satisfy $U^{†}U=\mathbb{I}$. For physics students, this means the modulus squared of the wave function (sum of probabilities) is always 1. Without unitarity, particles might vanish into thin air, or the sum of probabilities might become 1.5, destroying all statistical predictive power.

But in our discrete ontology, unitarity is not merely “conservation of probability”; it is the physical embodiment of logical consistency.

2.2.1 Reversibility and the Eternity of Information

A core property of unitary transformations is reversibility. If $U$ is unitary, then the inverse operator $U^{-1}=U^{†}$ necessarily exists and is also unitary.

This means that given the current state $|\Psi (t)⟩$, we can not only uniquely predict the future $|\Psi (t+1)⟩$, but also uniquely trace back the past $|\Psi (t-1)⟩$. The current state contains all information about past and future.

What happens if the universe is not unitary?

1. Information Loss (Non-injective): If two different states $|A⟩$ and $|B⟩$ evolve to the same state $|C⟩$, then when we are in $|C⟩$, memory of the past is permanently erased. This corresponds to irreversible processes (entropy increase) in thermodynamics. But at the fundamental physics level, if we believe microscopic laws are symmetric, such erasure is forbidden.

2. Information Created from Nothing (Non-surjective): If some state $|D⟩$ has no predecessor, how did it “suddenly appear”? This violates the continuity of causality.

Therefore, the unitarity axiom is equivalent to the “law of information conservation.” In this universe, no bit is truly deleted, and no bit is created from nothing. The macroscopic “forgetting” or “dissipation” we see is merely information transferring from local degrees of freedom to environmental degrees of freedom we cannot track (entanglement diffusion). For the wave function of the entire universe, entropy is always constant (and zero, if we start from a pure state).

2.2.2 The Rigidity of Logic

Looking deeper, unitarity ensures the rigidity of physical logic.

In the derivation of the Light Path Conservation Theorem, we see that the relation $c^2=v_{ext}^2+v_{int}^2$ directly stems from unitary decomposition of operators. If non-unitary evolution were allowed, this “circle” would deform, causing physical constants (such as light speed $c$ or Planck constant $\hslash$) to fluctuate with time or state.

A non-unitary universe is a logically “soft” universe. In that universe, $1+1$ might equal 2 today and 1.9 tomorrow. This is not only a disaster for physics, but also for mathematics.

By forcing $\hat{U}$ to be unitary, we are actually saying: The underlying logic of the universe is unbreakable. No matter how violent the interactions (black hole mergers, Big Bang), the underlying Hilbert space structure remains unchanged, angles between vectors (orthogonality) remain unchanged. This geometric rigidity is the fundamental reason we can describe the physical world with mathematics.

2.2.3 Reconciliation with the Measurement Problem

Readers might ask: “If we see wave function collapse (non-unitary process), how can we say the universe is unitary?”

This is exactly what we will solve in Chapter 7. Here, we emphasize: The unitarity claimed by Axiom $\Omega$ is global, microscopic unitarity.

The so-called “non-unitary measurement” is only because the observer is also part of the system (subsystem), and can only see a tiny slice of the entire universe’s Hilbert space. When information flows out of this slice (into the environment), unitarity seems broken to the observer. But this is a perspective illusion, like feeling centrifugal force in a rotating room.

As long as we expand our view to the entire universe (or a sufficiently large closed system), unitarity is perfectly restored.

Therefore, insisting on the unitarity axiom is insisting on the many-worlds (or many-histories) perspective—all possible historical branches truly exist and evolve in parallel, together maintaining the modulus conservation of the entire universe’s wave function.

After establishing that “the universe does not forget,” the next question is: How does the universe know “where is where”? This is the theme of the next section: Locality.