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arXiv:2512.12071

Proof of Spin-Statistics Theorem in Quantum Mechanics of Identical Particles

Takafumi Kita

quant-phcond-mat.quant-gascond-mat.stat-mechcond-mat.str-elcond-mat.supr-conSubmitted Dec 12, 2025• Announced Dec 16, 2025• 0 citations
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Abstract

A nonrelativistic proof of the spin-statistics theorem is given in terms of the field operators satisfying commutation and anticommutation relations, which are introduced here in the coordinate space as a means to build the permutation symmetry into the brackets of identical particles. An eigenvalue problem of a $π$-rotation for a product of two annihilation operators is combined with an analysis on its rotational property to prove the connection that the field operators for integral-spin and half-integral-spin particles obey the commutation and anticommutation relations, respectively.

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5 pages

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A nonrelativistic proof of the spin-statistics theorem is given in terms of the field operators satisfying commutation and anticommutation relations, which are introduced here in the coordinate space as a means to build the permutation symmetry into the brackets of identical particles. An eigenvalue problem of a πππ-rotation for a product of two annihilation operators is combined with an analysis on its rotational property to prove the connection that the field operators for integral-spin and half-integral-spin particles obey the commutation and anticommutation relations, respectively.
This paper establishes within SSS-matrix theory the connection between spin and statistics; namely, that the multiparticle-state vectors are symmetric or antisymmetric for permutation of identical particles according as the particle concerned has integral or half-integral spin. The proof given, which is simpler than previous SSS-matrix proofs, depends on the cluster-decomposition property, crossing symmetry, and Hermitian analyticity. A considerable part of the paper is concerned with establishing a suitable framework to formulate the first of these properties, cluster decomposition. To this end we develop from first principles the idea of the tensor product $f\ensuremath{\bigotimes}g$ which, for any two state vectors fff and ggg, represents the composite state (fff and ggg).
We define statistics of quantum mechanical particles, obtaining the Bose–Einstein and Fermi–Dirac varieties of indistinguishable particles. After finding the rotation and Lorentz matrices in different Lorentz representations, we describe the spin-statistics theorem, relating fermions with half-integer spin and bosons with integer spin. We explain two simple proofs and say some words on two others. We end by discussing symmetries in more generality, and we discuss the fact that internal symmetries must commute with spacetime ones, due to the Coleman–Mandula theorem.
We demonstrate that the exchange of two identical particles can be effected by a rotation of a coordinate system defined by the two particles through an angle π. It is then a consequence of the intimate connection between the generator of a space rotation and the total angular momentum operator, that the effect of the particle exchange on the ket corresponding to the physical system is dependent upon the integer or half-integer character of the particle spins. From this it can be concluded that the connection between spin and statistics that derives from the symmetry of the identical particle system under particle exchange is explained (in the most fundamental way) by the equivalence between the exchange of two identical particles and a rotation in configuration space.
We study the relation of the spin-statistics theorem to the geometric structures on phase space, which are introduced in quantization procedures (namely a U(1) bundle and connection). The relation can be proved in both the relativistic and the nonrelativistic domain (in fact for any symmetry group including internal symmetries) by requiring that the exchange can be implemented smoothly by a class of symmetry transformations that project in the phase space of the joint system system. We discuss the interpretation of this requirement, stressing the fact that any distinction of identical particles comes solely from the choice of coordinates — the exchange then arises from suitable change of coordinate system. We then examine our construction in the geometric and the coherent-state-path-integral quantization schemes. In the appendix we apply our results to exotic systems exhibiting continuous "spin" and "fractional statistics." This gives novel and unusual forms of the spin-statistics relation.
The treatment of identical particles in quantum mechanics rests on two (related) principles: the spin‐statistics connection and the Symmetrization Postulate. In light of recent theories (such as q‐deformed commutators) that allow for ‘small’ violations of the spin‐statistics connection and the Symmetrization Postulate, we revisit the issue of how quantum mechanics deals with identical particles and how it supports or fails to support various philosophical stances concerning individuality. As a consequence of the expanded possibilities for quantum statistics, we argue that permutation symmetry is best formulated as a formal property of the state function describing the system of particles rather than as a property of the individual particles.1 Introduction2 Philosophical background2.1 Important terminology2.1.1 Identity2.1.2 Indistinguishability2.1.3 Indiscernibility2.2 When are particles indistinguishable?2.3 The Principle of the Identity of Indiscernibles and quantum mechanics2.4 The Principle of the Identity of Indiscernibles and logic2.5 Particle history2.6 Transcendental individuality3 Some quantum formalism3.1 The Principle of Permutation Invariance and the Symmetrization Postulate3.2 The configuration‐space approach3.3 Commutators and anticommutators, and identical particle statistics3.4 Q‐mutators4 Identical particle statistics: a holistic point of view5 Conclusions