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The Berry connection as a projection of a spinor differential operator

Miftachul Hadi

Published Aug 1, 2026• 0 citations
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Abstract

We introduce a matrix-valued spinor differential operator constructed from a normalized spinor field. We show that the Berry connection arises as the expectation value of this operator, providing an alternative operator-theoretic formulation of Berry geometry.
DOI
10.5281/zenodo.21733604

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We introduce a spinor differential operator whose expectation value reproduces the Berry connection exactly. We derive its transformation under local U(1)U(1)U(1) gauge transformations and show that the conventional Berry gauge transformation follows directly through projection onto the normalized spinor. The proposed formulation provides an operator-level formulation of Berry geometry whose projection reproduces the conventional Berry connection and its gauge transformation.
The Berry connection of a smoothly varying family of Hermitian operators is conventionally defined through the differentiation of instantaneous eigenvectors, a construction that introduces local gauge choices and becomes singular at energy degeneracies. In this paper we show that the Berry connection admits an intrinsic, gauge-independent reconstruction through the Moore-Penrose pseudoinverse of the adjoint action. By defining the connection directly on the operator level via the spectral adjoint, we eliminate the need to track eigenbases or construct local coordinates. This provides a unified operator formulation that applies equally to non-degenerate and degenerate spectral subspaces.
Influenced by Klauder’s investigations on the same subject, we study the question of correspondence principle for Dirac fields, looking for its formulation without use of Grassman algebras. We prove that with each Fermi operator (the series with respect to asymptotic free fields): Ω (ψ,ψ̄): one can associate the functional ΩC(ψC, ψ̄C) with respect to classical spinor fields. Here the projector 1F and the Hilbert (Fock) space ℱF=1FℱB are given such that the identity 1F: ΩC(ψB, ψ̄B): 1FℱF = :Ω (ψ, ψ̄):ℱF defines the mediating boson level, where coherent state expectation values of operator expressions are in order: 〈:ΩC(ψB, ψ̄B):〉=ΩC(ψC, ψ̄C). For proofs we employ functional differentiation (resp. integration) methods, especially in connection with the use of functional representations of the CCR and CAR algebras.