A Monodromy Obstruction for One-Sided Difference Dirac Operators

Balasaheb B. Waphare

doi:10.36948/ijfmr.2026.v08i05.88472

A Monodromy Obstruction for One-Sided Difference Dirac Operators

Author(s)	Dr. Balasaheb B. Waphare
Country	India
Abstract	Let ∇F be the Clifford product of the del operator on R^m with a vector F of difference operators of step h. We decide for which s a fractional power (∇F)^s exists as a Fourier multiplier, and the answer inverts the expectation suggested by lattice fermion doubling. For the central difference, whose symbol vanishes at the 2^m doubling points, the Clifford norm N arising from the identity (∇F)(F∇)=Δ∘∑_j^▒F_j^2 is real and nonnegative; no branch question can arise, and (∇F)^s exists for every s∈C on the whole cell minus the doubling points. For the one-sided differences, whose scalar symbol has no doubling zeros at all, N vanishes on a real variety of dimension m-2 inside the cell and has winding number -1 on loops linking it; consequently no continuous single-valued (∇F)^s exists on any domain linking that variety unless s∈Z, and (∇F)^s admits no absolutely convergent stencil. The obstruction is topological rather than pointwise: at each individual frequency every power exists. On the positive side we prove the invertibility threshold max_j h\|ξ_j \|<π/2, a bandwidth rule that halves the Nyquist limit and cannot be improved, together with two-sided bounds on N, an estimate for the inverse whose blow-up as the threshold is approached is of the correct order, and sectoriality on an explicit box.
Keywords	Difference Dirac Operator, Fermion Doubling, Monodromy, Fractional Power, Zero Divisor, Clifford Algebra, Functional Calculus
Field	Mathematics > Maths + Physics
Published In	Volume 8, Issue 5, September-October 2026
Published On	2026-09-30
DOI	https://doi.org/10.36948/ijfmr.2026.v08i05.88472

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A Monodromy Obstruction for One-Sided Difference Dirac Operators

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