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    Tensor calculus is not the proper mathematical instrument... — Carmelics
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    Tensor calculus is not the proper mathematical instrument for translating quantum-theoretic equations of the electron into general relativity.

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    Reasons For

    1 perspective
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    • 1.Vectors and tensors transform under the most general linear (affine) transformations, meaning their transformation laws extend from Cartesian to affine coordinate systems.
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    • 2.The quantity ψ belongs to a representation of the rotation group that cannot be extended to the affine group.
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    • 3.Because ψ does not transform under affine transformations, ψ components cannot be defined relative to an arbitrary coordinate system in general relativity the way electromagnetic potentials and field strengths can.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Cartan's formalism of moving frames (repère mobile) allows spinors to be defined on curved manifolds via local orthonormal frames (vielbeins) without requiring affine transformation laws.
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    • 2.The vielbein formalism embeds tensor calculus within a broader framework where ψ components are defined relative to local Lorentz frames, not arbitrary coordinate systems, dissolving the incompatibility Weyl identifies.
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    • 3.Fock and Ivanenko (1929) demonstrated explicitly that Dirac's equation admits generally covariant formulation using this extended calculus, empirically refuting the claim that tensor methods are categorically improper.
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    Reason against 2 of 2
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    • 1.The claim conflates the inadequacy of a specific formulation of tensor calculus with the inadequacy of tensor-based frameworks generally, committing a fallacy of hasty generalization.
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    • 2.Élie Cartan's and later Penrose's spinor calculus shows that the representation theory of the Lorentz group, not the affine group, is the operative constraint, and this constraint is manageable within generalized differential geometry.
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    • 3.If the proper criterion for mathematical instruments is empirical adequacy rather than transformation-group universality, then any calculus yielding generally covariant Dirac equations satisfies the requirement, regardless of its affine limitations.
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    Related

    Because ψ does not transform under affine transformations, ψ components cannot b...Cartan's formalism of moving frames (repère mobile) allows spinors to be defined...Fock and Ivanenko (1929) demonstrated explicitly that Dirac's equation admits ge...If the proper criterion for mathematical instruments is empirical adequacy rathe...
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    The claim conflates the inadequacy of a specific formulation of tensor calculus ...The quantity ψ belongs to a representation of the rotation group that cannot be ...The vielbein formalism embeds tensor calculus within a broader framework where ψ...Vectors and tensors transform under the most general linear (affine) transformat...Élie Cartan's and later Penrose's spinor calculus shows that the representation ...

    Similar

    Mathematics should not be conceived of as a calculus separate from oth...74%What remains from general relativity and quantum mechanics is only a c...74%A calculus must not drive us from true hypotheses to false conclusions74%Quantum gravity is the more restricted problem of reconciling gravity ...73%

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    SEP: weyl
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    The tensor calculus is not the proper mathematical instrument to use in translating the quantum-theoretic equations of the electron over into the general theory of relativity. Vectors and terms [tensors] are so constituted that the law which defines the transformation of their components from one Cartesian set of axes to another can be extended to the most general linear transformation, to an affine set of axes. That is not the case for quantity \(\psi\), however; this kind of quantity bel
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    Details

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    claim
    Perspectives
    3 (1 for, 2 against)
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    1 edit