The connection between the intrinsic angular momentum (spin) of particles and quantum statistics is established by considering the response of identical particles to a common background radiation field. For this purpose, the Hamiltonian analysis previously performed in stochastic electrodynamics to derive the quantum description of a one-particle system is extended to a system of two identical bound particles subject to the same field. Depending on the relative phase of the response of the particles to a common field mode, two types of particles are distinguished by their symmetry or antisymmetry with respect to particle exchange. While any number of identical particles responding in phase can occupy the same energy state, there can only be two particles responding in antiphase. The calculation of bipartite correlations between the response functions reveals maximum entanglement as a consequence of the parallel response of the particles to the common field. The introduction of an internal rotation parameter leads to a direct link between spin and statistics and to a physical rationale for the Pauli exclusion principle.
The connection between the intrinsic angular momentum (spin) of particles and quantum statistics is established by considering the response of identical particles to a common background radiation field. For this purpose, the Hamiltonian analysis previously performed in stochastic electrodynamics to derive the quantum description of a one-particle system is extended to a system of two identical bound particles subject to the same field. Depending on the relative phase of the response of the particles to a common field mode, two types of particles are distinguished by their symmetry or antisymmetry with respect to particle exchange. While any number of identical particles responding in phase can occupy the same energy state, there can only be two particles responding in antiphase. The calculation of bipartite correlations between the response functions reveals maximum entanglement as a consequence of the parallel response of the particles to the common field. The introduction of an internal rotation parameter leads to a direct link between spin and statistics and to a physical rationale for the Pauli exclusion principle.
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.
Which is the physical agent behind the antisymmetry of the electron state vectors? With the purpose to find an answer to this key question, we analyze the stationary states of a system containing two noninteracting electrons, using the tools of stochastic electrodynamics. In previous work, the resonant response of two particles to common modes of the random zero- point field has been shown to lead to the nonfactorizability of the composite state vector. Here we extend the analysis to particles with spin. When two electrons constitute a single system, a correlation is established between their dynamical variables through the common relevant modes of the zero-point field, which acts as a mediator. An analysis of the exchange properties of the bipartite state vectors obtained is shown to lead to the connection between spin and symmetry. The conclusion is that due consideration of the vacuum field in first quantization leads to the corresponding statistics for an assembly of electrons.
The traditional Standard Quantum Mechanics (SQM) theory is unable to solve the Spin-s problem, i.e., to justify the utterly important "Pauli Exclusion Principle". A complete and straightforward solution of the Spin-Statistics problem is presented based on the "Weyl Integrable Quantum Mechanics" (WIQM) theory. This theory provides a Weyl-gauge invariant formulation of the Standard Quantum Mechanics and reproduces successfully, with no restrictions, the full set of the quantum mechanical processes, including the formulation of Dirac's or Schrödinger's equation, of Heisenberg's uncertainty relations, and of the nonlocal EPR correlations. etc. When the Weyl Integrable Quantum Mechanics is applied to a system made of many identical particles with spin, an additional constant property of all elementary particles enters naturally into play: the "intrinsic helicity", or the "intrinsic angular - momentum". This additional particle property, not considered by Standard Quantum Mechanics, determines the correct Spin-Statistics Connection (SSC) observed in Nature. All this leads to the consideration of a novel, most complete (in the EPR sense) quantum mechanical theory.
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
We study angular correlations of the radiation emitted by spatially extended sets of atoms. Such correlations result from the fact that the photons are identical quantum particles. We show that the symmetry (or the antisymmetry) alone of the wave function for identical, indistinguishable particles always leads to significant angular correlations and that in full agreement with our intuition, bosons tend to travel in the same direction while fermions tend to travel in opposite directions. In order to quantify these intuitive notions, we introduce---as a measure of angular correlations---the average value of the cosine of the angle between the directions of the momenta of two particles. The angular correlation coefficient is calculated for simple, model wave functions and the results are compared with those obtained from an exactly soluble model describing radiating harmonic oscillators. We find that even though the interaction modifies the exact values of the angular correlation coefficient, the gross features can be obtained from quantum statistics, i.e., from the symmetry of the many-photon wave functions. Moreover, the angular correlations as measured by the average cosine exhibit an almost universal character, at least for a small number of emitting atoms. Finally, we compare angular correlations obtained in quantum electrodynamics with the correlations resulting from the phenomenologically introduced randomness of the phases in the classical description of the electromagnetic field.
The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles.