  The basis of spinors in three-dimensional Euclidean space is expressed by
differential forms. Its expression is found from the spectral decomposition of
the modified Hamiltonian describing Weyl semimetals where the wavenumber
parameters are replaced by the differential forms. The generalization of the
definitions of differential forms including not only fractional powers but also
any algebraic functions is justified by algebraic extension of the symmetric
Fock space isomorphic to multivariable polynomial ring. We furthermore define
the inner product for these fractional-order differential forms written in two
ways; the formal power series and the multiple integral. While the norms of the
powers $\mathrm{d}s^{2\nu}$ are reduced to the variant of Dyson's integral and
have the value $2\nu+1$, possibly related to the dimension of irreducible
representation, the norm of two-component spinor is given by the integral
\[I=\frac{2}{\pi}\int_{x^2+y^2\ge 1 \& 0\le x \le 1 \& 0 \le y \le
1}\frac{xy\mathrm{d}x\mathrm{d}y}{\sqrt{(1-x)(1-y)(x^2+y^2-1)}}=\frac{4\sqrt{2}}{\pi}\int_0^1\frac{(1+\sqrt{z})}{(1+z)^3}\big[2E(z)-(1-z)K(z)\big]\mathrm{d}z\simeq
1.774, \] where $K(z)$ and $E(z)$ are the complete elliptic integrals of the
first and second kind, respectively. The product can also be generalized to
include one parameter $p$ analogous to those in $L^p$ space, reducing to the
original one when $p=2$. The norm with $p=\infty$ is considered as an example
and the result for $ \nu=-\frac{1}{2} $ is written by the Watson-Iwata
integral. We also discuss the ambiguity of the definition of the spinor under
coordinate rotation originating from Berry's phase, and point out that if we
heuristically set $\mathrm{d}s=0$, the ambiguity disappears, though its
implication remains unclear.
