The modern state of the Pauli Exclusion Principle (PEP) is discussed. PEP can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is the so-called spin-statistics connection (SSC). As we will discuss, the physical reasons why SSC exists are still unknown. On the other hand, according to PEP, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, both belong to one-dimensional representations of the permutation group, all other types of permutation symmetry are forbidden; whereas the solution of the Schrödinger equation may have any permutation symmetry. It is demonstrated that the proof in some widespread textbooks on quantum mechanics that only symmetric and antisymmetric states (one-dimensional representations of the permutation group) can exist is wrong. However, the scenarios, in which an arbitrary permutation symmetry (degenerate permutation states) is permitted lead to contradictions with the concepts of particle identity and their independence. Thus, the existence in our nature particles only in nondegenerate permutation states (symmetric and antisymmetric) is not accidental and so-called symmetrization postulate may not be considered as a postulate, since all other symmetry options for the total wave function may not be realized. From this an important conclusion follows: we may not expect that in future some unknown elementary particles can be discovered that are not fermions or bosons.
The Pauli exclusion principle (PEP) can be considered from two aspects. First, it asserts that particles that have half-integer spin (fermions) are described by antisymmetric wave functions, and particles that have integer spin (bosons) are described by symmetric wave functions. It is called spin-statistics connection (SSC). The physical reasons why SSC exists are still unknown. On the other hand, PEP is not reduced to SSC and can be consider from another aspect, according to it, the permutation symmetry of the total wave function can be only of two types: symmetric or antisymmetric. They both belong to one-dimensional representations of the permutation group, while other types of permutation symmetry are forbidden. However, the solution of the Schrödinger equation may have any permutation symmetry. We analyze this second aspect of PEP and demonstrate that proofs of PEP in some wide-spread textbooks on quantum mechanics, basing on the indistinguishability principle, are incorrect. The indistinguishability principle is insensitive to the permutation symmetry of wave function. So, it cannot be used as a criterion for the PEP verification. However, as follows from our analysis of possible scenarios, the permission of states with permutation symmetry more general than symmetric and antisymmetric leads to contradictions with the concepts of particle identity and their independence. Thus, the existence in our Nature particles only in symmetric and antisymmetric permutation states is not accidental, since all symmetry options for the total wave function, except the antisymmetric and symmetric, cannot be realized. From this an important conclusion follows, we may not expect that in future some unknown elementary particles that are not fermions or bosons can be discovered.
Quantum mechanics broadly classifies the particles into two categories:
(1) fermions and
(2) bosons. Fermions are half-integer spin particles, obeying Pauli's exclusion principle and Fermi-Dirac statistics. Whereas bosons are integer spin particles, not obeying Pauli's exclusion principle, and obeying Bose-Einstein statistics. However, there are two exceptions to this standard case: first, anyons, which exist in 2-dimensional systems, and secondly, paraparticles, which can exist in any dimension. Paraparticles follow their non-trivial parastatistics, obeying their generalised exclusion principle. In this paper, we provide a detailed review of the foundations of paraparticle statistics established in \cite{wang2025particle}. We extend this work further and then derive an important expression for the heat capacity of paraparticles for a specific case, which would provide a handle for the experimental detection of paraparticles in appropriate systems.
Particles that must have symmetric states are called bosons; pions, photons, and He 4+ nuclei are such particles. Particles whose states must be ant symmetric under interchange of the two particles are called fermions; electrons, protons, and neutrons are examples of fermions. Particles with integer spin are always bosons while particles with half-integer spin are always fermions. It is not at all obvious from the point of view of non-relativistic quantum mechanics why this relation, called the connection between spin and statistics, is true, and one must look within the framework of relativistic quantum field theory to understand it fully theoretically. Effects of identity can also show up in the scattering of seemingly non-identical particles. The elastic scattering of a He 4+ ion from a He 4+ atom, both are in their ground states. These are certainly distinguishable particles, since they have different charges. Particle identity also plays an important role in determining the properties of molecules made of identical atoms.
The Pauli Exclusion Principle and the Spin-Statistics Theorem from Worldline Non-Injectivity: Exchange Phase, Rapidity, and Topological Sheet Structure For nearly a century, the Pauli exclusion principle (PEP) and the spin-statistics theorem have stood as foundational postulates of quantum mechanics and relativistic quantum field theory. The PEP dictates that no two identical fermions can occupy the same quantum state, determining the structure of atoms, the stability of matter, and the properties of neutron stars and white dwarfs. The spin-statistics theorem connects the intrinsic spin of a particle to its statistical behavior: half-integer spin particles (fermions) obey Fermi-Dirac statistics, while integer spin particles (bosons) obey Bose-Einstein statistics. Despite their universal success, these principles have never been derived from a more primitive geometric or topological principle — until now. This paper demonstrates that both the Pauli exclusion principle and the spin-statistics theorem emerge naturally and rigorously from a single geometric fact: the **non-injectivity of a timelike worldline** for Lorentz factors exceeding a critical threshold,
γ>γcrit. In this regime, a single worldline intersects a constant-time hypersurface
Σt in
N>1 distinct spatial points, generating a multi‑sheet spacetime structure. This phenomenon, known as **worldline non-injectivity**, is the foundation of the TPST–DGQ framework. **Key results and logical flow:** 1. **Multi‑sheet structure for a single particle:** For
γ>γcrit, a particle's worldline produces
N(ϵ)∼ϵ−(d−2) topological sheets, each carrying the same physical properties (mass, charge, spin) via the Ontological Identity Principle. 2. **Composite worldline for two identical particles:** By combining two identical particles into a composite worldline in configuration space, we obtain a two‑particle sheet structure. Identical particles share isomorphic sheet structures, reducing the
N×N sheet pairs to
N independent pairs. 3. **Exchange as a permutation of sheet blocks:** The exchange operator
P12 swaps the sheet blocks associated with the two particles. This permutation is a topological operation in the two‑particle configuration space. 4. **Exchange phase from winding number:** The exchange path corresponds to half of a complete winding cycle around the fold structure of the composite worldline. The accumulated topological phase is
Φexchange=πw, where
w is the **winding number** of the worldline. Hence the exchange phase factor is
η=eiπw=(−1)w. 5. **Spin from winding number:** The winding number
w is directly related to the spin
s by the requirement that a
4π rotation returns the physical state to itself (single‑valuedness). This yields
w=2s. Therefore
η=(−1)2s: half‑integer spin gives
η=−1 (fermions, antisymmetric), integer spin gives
η=+1 (bosons, symmetric). This is the spin‑statistics theorem derived from geometry. 6. **Pauli exclusion principle:** For two identical fermions (
η=−1) at spatial coincidence, the two‑particle amplitude satisfies
A=ηA, forcing
A=0. Thus no two fermions can occupy the same quantum state. The PEP is no longer a postulate; it is a theorem. 7. **Rapidity as inter‑sheet phase measure:** The rapidity
ζ=artanh(v/c) is shown to be a monotonic function of the inter‑sheet phase difference
Φn. The non‑injectivity threshold
γ>γcrit corresponds to
ζ>ζcrit. The speed of light
c emerges as the asymptotic limit where the inter‑sheet phase diverges, making
v=c unreachable for massive particles. 8. **Universal cancellation identity:** The same algebraic identity
N(ϵ)⋅ϵd−2=O(1) that regularizes holographic entropy, Coulomb self‑energy, wavefunction normalization, electromagnetic fields, and the cosmological constant also regularizes the two‑particle exchange amplitude — zero for fermions, finite for bosons. Quantum statistics thus joins the list of phenomena unified under the principle of worldline non‑injectivity. **Implications:** This work completes the derivation of the foundations of quantum mechanics within the TPST–DGQ framework, following previous derivations of the Born rule, the Schrödinger equation, Planck’s constant, and wavefunction collapse. It also establishes a geometric origin for the spin‑statistics connection, a result that in standard physics relies on the heavy machinery of relativistic quantum field theory (unitarity, causality, positivity of energy). Here, it follows from the topology of worldline folds. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact:degiuseppealex@gmail.com The hash files that determine ownership have been created.
This chapter discusses the properties of systems of identical particles. It presents the wave functions of a system composed of N identical and independent particles. From the properties of the wave functions, the chapter shows that identical particles can be characterized by symmetric as well as antisymmetric wave functions. As a result, it distinguishes two kinds of particles: bosons and fermions. Fermions have half integer spin, whereas bosons have integer spin. Since an arbitrary number of bosons can occupy the same state, atoms with integer spin can abruptly condensate into a single ground state when the temperature of the atoms goes below a certain critical value. In atoms, a limited number of electrons can occupy the same energy level. The Pauli principle prevents the energy states being occupied by an arbitrary number of electrons. Thus, the Pauli principle prevents all atoms having the same size and the same energy.