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Proof of the Spin Statistics Connection 2: Relativistic Theory

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Abstract

The traditional standard theory of quantum mechanics is unable to solve the spin–statistics problem, i.e. to justify the utterly important “Pauli Exclusion Principle” but by the adoption of the complex standard relativistic quantum field theory. In a recent paper (Santamato and De Martini in Found Phys 45(7):858–873, 2015) we presented a proof of the spin–statistics problem in the nonrelativistic approximation on the basis of the “Conformal Quantum Geometrodynamics”. In the present paper, by the same theory the proof of the spin–statistics theorem is extended to the relativistic domain in the general scenario of curved spacetime. The relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. No relativistic quantum field operators are used and the particle exchange properties are drawn from the conservation of the intrinsic helicity of elementary particles. It is therefore this property, not considered in the standard quantum mechanics, which determines the correct spin–statistics connection observed in Nature (Santamato and De Martini in Found Phys 45(7):858–873, 2015). The present proof of the spin–statistics theorem is simpler than the one presented in Santamato and De Martini (Found Phys 45(7):858–873, 2015), because it is based on symmetry group considerations only, without having recourse to frames attached to the particles. Second quantization and anticommuting operators are not necessary.

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Notes

  1. In some axiomatic approaches to the structure of spacetime, the so-called “Weyl tensor” \(W^i_{jkl}\) is used in place of the curvature tensor \(R^i_{jkl}\). The Weyl tensor is obtained from the full curvature tensor by subtracting out various traces and describes the tidal part of the gravitational forces. The Weyl tensor is unrelated to the scalar curvature \(R_W\) in Eq. (3), because all contractions of \(W^i_{jkl}\) are zero.

  2. However the positive measure \(d\mu =\rho \sqrt{g}d^nq=|{\varPsi }|^2\sqrt{g}d^nq\) is finite over any compact region of \(V^N\) and, in particular, over the region where the Weyl invariant complete figure introduced in Sect. 2.1 can be defined.

  3. The boost \(B({\tilde{y}})\) considered here differs from the boost commonly used in the relativistic mechanics because \(B({\tilde{y}})\) contains a space rotation too. An alternative form of left translation rule is \(R_2(\gamma ')=B^{-1}({\tilde{y}}'){\bar{{\varLambda }}} B({\tilde{y}})\), which represents “Wigner’s rotation” in the present case.

  4. In the case \(s=0\) we have \(D^{(0,0)}=1\) and the SST reduces to the symmetry condition (19).

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

    Enrico Santamato

  2. Accademia Nazionale dei Lincei, via della Lungara 10, 00165, Rome, Italy

    Francesco De Martini

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  1. Enrico Santamato
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  2. Francesco De Martini
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Santamato, E., De Martini, F. Proof of the Spin Statistics Connection 2: Relativistic Theory. Found Phys 47, 1609–1625 (2017). https://doi.org/10.1007/s10701-017-0114-3

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