Symmetrization Postulate

The symmetrization postulate states that multi-particle quantum states must be either symmetric (bosons) or antisymmetric (fermions) under particle exchange. In Logic Realism Theory, this is not an independent axiom but a consequence of Determinate Identity.

The Standard Puzzle

Why must identical particles obey specific symmetry constraints? Why are there exactly two types of quantum statistics—not one, three, or infinitely many?

The mathematical fact that the symmetric group $S_N$ has exactly two one-dimensional irreducible representations is well known. But this alone does not explain why physical states must transform according to one-dimensional representations.

The Identity Problem

Consider two electrons. In classical mechanics, we can label them “electron 1” and “electron 2” and track continuous trajectories. In quantum mechanics, trajectories are undefined.

If two electrons interact and separate, we cannot say which outgoing electron is “the same” as which incoming electron. The labels appear physically meaningless.

This is fundamentally an identity problem. Consider an arbitrary two-particle state:

This notation presupposes “particle 1” and “particle 2” refer to something. But if the particles are intrinsically indistinguishable, what grounds this distinction?

Any state that treats them differently attributes distinguishable identity to indistinguishable entities—an Id violation.

Theorem 1: Symmetrization Required

Theorem: Let N particles be identical (intrinsically indistinguishable). If Determinate Identity holds, then only states satisfying

are $L_3$-admissible, where $c: S_N \to U(1)$ is a one-dimensional unitary representation.

Proof sketch:

1. The physical content of $

   \psi\rangle$ must be independent of how labels are assigned (particles are identical)

2. By Id, a physical configuration is determinately what it is, independent of description

3. States $

   \psi\rangle$ and $P_\pi

   \psi\rangle$ (relabeled) must represent the same physical configuration

4. Same physical configuration means same state up to phase: $P_\pi

   \psi\rangle = c(\pi)

   \psi\rangle$

5. Consistency requires $c: S_N \to U(1)$ to be a group homomorphism

Theorem 2: Only Symmetric and Antisymmetric

Theorem: The symmetric group $S_N$ (for $N \geq 2$) has exactly two one-dimensional unitary representations:

1. Trivial representation: $c(\pi) = 1$ for all $\pi$ → Bosons (symmetric states)
2. Sign representation: $c(\pi) = \text{sgn}(\pi)$ → Fermions (antisymmetric states)

Proof: Every transposition $\tau$ satisfies $\tau^2 = e$, so $c(\tau)^2 = 1$, meaning $c(\tau) = \pm 1$. All transpositions are conjugate in $S_N$, so they must all have the same image. Either all $+1$ (trivial) or all $-1$ (sign).

Consequences

Pauli Exclusion

For fermions, antisymmetry immediately gives:

Two fermions cannot occupy the same state. This is not an empirical discovery but a mathematical consequence of the antisymmetry required by Id.

Bose-Einstein Condensation

For bosons, symmetric states allow (even favor) multiple particles in the same state. Bose-Einstein condensation is Id-admissible for bosons.

Parastatistics Ruled Out

Parastatistics (higher-dimensional representations of $S_N$) would attribute distinguishable identities to indistinguishable particles. Theorem 1 rules this out.

Prediction: If parastatistics were ever observed, Id (as applied here) would be falsified.

Fock Space Structure

The $L_3$-admissible sector of multi-particle Hilbert space is:

where:

• $\mathcal{F}_S$ = symmetric/bosonic Fock space
• $\mathcal{F}_A$ = antisymmetric/fermionic Fock space

This is the foundation of quantum field theory.

The Spin-Statistics Connection

The spin-statistics theorem states:

• Integer spin → bosons (symmetric)
• Half-integer spin → fermions (antisymmetric)

LRT establishes that particles must be bosons or fermions, but does not yet derive which particles are which. The connection to spin likely emerges from combining Id with Lorentz invariance—future work.

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