On the Origin of the Quantum Rules for Identical Particles

We present a proof of the Symmetrization Postulate for the special case of noninteracting, identical particles. The proof is given in the context of the Feynman formalism of Quantum Mechanics, and builds upon the work of Goyal, Knuth and Skilling (Phys. Rev. A 81, 022109 (2010)), which shows how to derive Feynman’s rules from operational assumptions concerning experiments. Our proof is inspired by an attempt to derive this result due to Tikochinsky (Phys. Rev. A 37, 3553 (1988)), but substantially improves upon his argument, by clarifying the nature of the subject matter, by improving notation, and by avoiding strong, abstract assumptions such as analyticity.

💡 Research Summary

The paper presents a rigorous derivation of the symmetrization postulate for non‑interacting identical particles using the Feynman formalism together with a set of operational assumptions. It begins by restating the three core Feynman rules: the Product Rule (amplitudes for concatenated measurement sequences multiply), the Sum Rule (amplitudes for mutually exclusive coarse‑grained alternatives add), and Reciprocity (reversing the temporal order of two immediate measurements takes the complex conjugate of the amplitude).

For a two‑particle system, the authors note that because the particles are indistinguishable, the overall amplitude must be a function H of all four single‑particle transition amplitudes a₁₁, a₁₂, a₂₁, a₂₂. Introducing a third measurement allows the use of the Product Rule twice, leading to a functional equation relating H to a new function G that depends on eight amplitudes. By repeatedly applying the Sum Rule they obtain additive and multiplicative constraints on H. In particular, they define a reduced function f(z)=H(z,0,0,1)/H(1,0,0,1) and show that f satisfies both f(u v)=f(u)f(v) and f(u+v)=f(u)+f(v). The only continuous solutions are the identity f(z)=z and its complex‑conjugate f(z)=z*.

Consequently, H can be expressed in terms of two real constants C₁=H(1,0,0,1) and C₂=H(0,1,1,0). Reciprocity forces these constants to be real, and a deterministic special case (unit probability) forces C₁²=C₂²=1, so each constant is ±1. The final form of the two‑particle amplitude is

 H(a₁₁,a₁₂,a₂₁,a₂₂)=±(a₁₁a₂₂ ± a₁₂a₂₁)

where the overall ± is an irrelevant global phase, while the inner ± distinguishes the symmetric (bosonic) and antisymmetric (fermionic) possibilities.

The authors sketch the extension to three or more particles. For three particles, H becomes a sum of six terms, each a product of three single‑particle amplitudes, multiplied by constants that are again restricted to ±1. By examining special configurations and using the indistinguishability of particle labels, they show that all constants must share the same sign (bosons) or alternate sign under any transposition (fermions), reproducing the fully symmetric or fully antisymmetric wavefunctions.

The key insight is that the Feynman rules themselves encode exchange symmetry; once expressed as functional equations, only the elementary algebraic properties of complex numbers are needed to derive the boson/fermion dichotomy. The derivation improves on earlier work by Tikochinsky by eliminating the need for analyticity assumptions and relying solely on the operationally motivated sum, product, and reciprocity rules. This provides a minimal‑assumption foundation for the symmetrization postulate and clarifies its origin within the standard quantum formalism.