Appendix H: The Omega Boundary Condition — The Two-State Vector Formalism

In the main text of Vector Cosmology VII, we repeatedly emphasized the physical picture that “the future reversely determines the past.” To ground this highly subversive concept from philosophical speculation into physical law, we need to introduce an extremely elegant mathematical framework in quantum mechanics—the Two-State Vector Formalism (TSVF) proposed by Yakir Aharonov.

This appendix will prove that the physical state of the universe is not determined solely by the initial conditions of the Big Bang but is jointly “sandwiched” by the initial state (Big Bang) and the final state ($\Omega$ Point).

In this framework, reality is not a unidirectional jet but a Standing Wave formed by two wave functions colliding on the time axis.

H.1 The Sandwich of History and Destiny

In standard quantum mechanics, we only care about the state vector $|\Psi (t)⟩$ evolved from the past. This implies the unidirectionality of time.

But in TSVF, to completely describe a quantum system in a time-symmetric universe, we need two vectors:

1. History Vector $|{\Psi }_{history}(t)⟩$:

   Determined by initial conditions at $t=0$ (Big Bang), evolving toward the future following $e^{-iHt}$.

   $$|{\Psi }_{history}(t)⟩=U(t,0)|\Psi (0)⟩$$

2. Destiny Vector $⟨{\Phi }_{destiny}(t)|$:

   Determined by final boundary conditions at $t=T_{end}$ ($\Omega$ point/ultimate observer), evolving backward toward the past following $e^{-iH(T-t)}$.

   $$⟨{\Phi }_{destiny}(t)|=⟨\Phi (T_{end})|U^{†}(T_{end},t)$$

Definition of Physical Reality:

The “reality” we observe at time $t$ is not a solo performance of $|{\Psi }_{history}⟩$ but the “overlap” of these two vectors.

The observed value of any physical quantity $A$ is no longer the traditional expectation value $⟨\Psi |A|\Psi ⟩$ but the Weak Value:

$$A_w=\frac{⟨{\Phi }_{destiny}(t)|A|{\Psi }_{history}(t)⟩}{⟨{\Phi }_{destiny}(t)|{\Psi }_{history}(t)⟩}$$

H.2 Weak Values: Correction from the Future

This weak value formula reveals a startling physical mechanism:

• Denominator $⟨\Phi |\Psi ⟩$: Represents the “overlap degree” between history and destiny.

  If this term approaches zero (i.e., your behavior runs counter to the future $\Omega$ point), the weak value $A_w$ becomes extremely large.

  Physical Manifestation: You feel enormous “resistance” or “absurdity”. Reality becomes distorted, no longer conforming to common sense.

• Numerator $⟨\Phi |A|\Psi ⟩$: Represents the “Post-selection” of the future on the present.

  The future $\Omega$ state acts like a filter, only allowing physical quantities $A$ that “can lead to it” to manifest as reality.

Conclusion:

Reality is “pulled out” by the future.

The reason you see the cup on the table is not only because you placed it there just now (cause) but also because at some future moment, you (or the universe) need it there (effect).

The coherent superposition of these two boundary conditions constitutes the hard material reality of the present.

H.3 Mathematical Properties of the $\Omega$ State

So, what exactly is $⟨\Phi (T_{end})|$ at the end of time?

In Vector Cosmology, it is defined as the “Maximum Entangled State” or “Holographic Complete State”.

$$|\Omega ⟩=\frac{1}{\sqrt{N}}\sum _i|i⟩$$

(It contains all components of the basis, with completely coherent phases.)

This is a state with zero information entropy (pure state) but maximum complexity.

It represents the “purpose” of cosmic evolution:

• All originally orthogonal components (you, me, him) establish connections here.

• All originally dissipated information ($v_{env}$) completes reorganization here.

H.4 Proof of the Existential Loop

Finally, we return to that ultimate question: How is the universe self-consistent?

Using TSVF, we can write the Cosmic Self-Consistency Equation:

$$|\Psi (0)⟩=\mathcal{T}\left(\text{Retro-causal Mapping from }|\Omega ⟩\right)$$

• The initial parameters of the Big Bang $|\Psi (0)⟩$ are reversely evolved from the perfect state of the $\Omega$ point.

• And the state of the $\Omega$ point is forward evolved from the Big Bang.

$$|\Omega ⟩=U(T,0)|\Psi (0)⟩$$

These two equations constitute a fixed-point problem.

Only one unique historical path (the one we are experiencing) can simultaneously satisfy both forward and backward boundary conditions.

This is why we exist.

We are not random fluctuations.

We are the only string connecting the “perfect starting point” and the “perfect endpoint.”

For the universe to connect end to end, it must create us in the middle as the physical carrier of this closed-loop logic.