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Proof of the Spin–Statistics Theorem

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Abstract

The traditional standard quantum mechanics theory is unable to solve the spin–statistics problem, i.e. to justify the utterly important “Pauli Exclusion Principle”. A complete and straightforward solution of the spin–statistics problem is presented on the basis of the “conformal quantum geometrodynamics” theory. This theory provides a Weyl-gauge invariant formulation of the standard quantum mechanics and reproduces successfully all relevant quantum processes including the formulation of Dirac’s or Schrödinger’s equation, of Heisenberg’s uncertainty relations and of the nonlocal EPR correlations. When the conformal quantum geometrodynamics 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”. This property, not considered in the Standard Quantum Mechanics, determines the correct spin–statistics connection observed in Nature.

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Notes

  1. This similarity induced Weyl to identify \(\phi _i\) with the e.m. four potential to obtain an unified theory for electromagnetism and gravitation [35]. This approach however was unsuccessful and nowadays we know that e.m. forces are unified with weak nuclear forces, instead.

  2. As shown elsewhere [11, 33], external e.m. fields can be easily introduced by considering the further invariance under the e.m. gauge change \(a_i\rightarrow a_i+\partial _i\chi \) and \(\sigma \rightarrow \sigma -\chi \) with the replacement \(D_i\sigma \rightarrow D_i\sigma -a_i\) in Eq. (3).

  3. As it is well known, Bohm considers the coupled equations for modulus and phase of the wavefunction \(\Psi \) obtained from the wave equation (Klein–Gordon, Schrödinger) of the SQM. Bohm’s coupled equations have the form of a continuity equation for the density \(\rho =|\Psi |^2\) and of a Hamilton–Jacobi equation with suitable “ quantum potential” depending on \(\rho \) added to. Now, a closer inspection into Eq. (3) shows that the second equation has the form of a continuity equation for the density \(\rho =|\Psi |^2=|A|^2e^{(n-2)\phi }\) [cfr. Eq. (5)] and the first equation has the form of the Hamilton–Jacobi equation with potential given by the Weyl scalar curvature. Now Eq. (2) with \(\phi _i=\partial _i\phi \) shows that the second term on the right has the same form of Bohm’s quantum potential. It is then not surprising that Bohm’s ansatz can be used to transform Eq. (3) into the linear wave Eq. (4) [30, 33]. However, as said in the text, Eq. (4) is not a wave equation of the SQM.

  4. In Bohm’s approach, the spin is described by the coupled equations of modulus and phase of each component of the spinor [6]. We then have four Bohm’s coupled equations for Pauli spinors and eight Bohm’s coupled equations for Dirac spinors. In this case, Bohmian quantum mechanics and CQG are mutually at variance.

  5. The same mathematical method is used, for example, in \((4+N)\)-Kaluza Klein approach to gravitational plus Yang–Mills theory [27].

  6. We may think of \(ma^2\) as the particle moment of inertia and of \(a\) as the gyration radius.

  7. It can be shown that the addition of external e.m. fields introduces no dependence on \(\gamma \) as well [11, 33].

  8. The square modulus of \(\Phi _s\) is given by

    $$\begin{aligned} |\Phi _s|^2= & {} \cos ^2\frac{\beta }{2}|\psi ^\uparrow (\varvec{r},t)|^2 + \sin ^2\frac{\beta }{2}|\psi ^\downarrow (\varvec{r},t)|^2 +\nonumber \\&+\cos \frac{\beta }{2}\sin \frac{\beta }{2}[\psi ^{\uparrow *}(\varvec{r},t)\psi ^\downarrow (\varvec{r},t)+\psi ^\uparrow (\varvec{r},t)\psi ^{\downarrow *}(\varvec{r},t)] \end{aligned}$$
    (20)

  9. The terminology is taken from the similar factorization of the Lorentz group into rotations and boosts.

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Authors and Affiliations

  1. Dipartimento di Scienze Fisiche, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo, 80126, Napoli, Italy

    Enrico Santamato

  2. CNISM - Consorzio Nazionale Interuniversitario per la Struttura della Materia, Napoli, Italy

    Enrico Santamato

  3. Accademia Nazionale dei Lincei, via della Lungara 10, 00165, Roma, Italy

    Francesco De Martini

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  1. Enrico Santamato
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Correspondence to Enrico Santamato.

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Santamato, E., De Martini, F. Proof of the Spin–Statistics Theorem. Found Phys 45, 858–873 (2015). https://doi.org/10.1007/s10701-015-9912-7

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