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REVIEW 2 major objections 4 minor 23 references

Clifford geometric algebra: Real and complex spinor data tables

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper presents correct algebraic spinor data for real and complex Clifford algebras: primitive idempotents, two-sided ideals, spinor bases, matrix representations, general spinors, and their squared Hermitian norms.

desk verdict Useful extension of Ablamowicz's spinor tables, but Table IV lists E12 where the paper's own left-ideal convention requires E21, so the printed tables need a correction pass before they are safe to use as reference data. read the letter →

arxiv 2412.14677 v1 pith:LPRLG4N5 submitted 2024-12-19 math-ph math.MP

A. Acus , A. Dargys This is my paper · ORCID
classification math-phmath.MP MSC 15A6615A7516D2581R25
keywords CliffordalgebraalgebraicspinorsminimalleftidealsprimitiveidempotentsspinorbasismatrixrepresentationsHermitiannormrealandcomplexalgebras
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish a complete set of algebraic spinor data for real and complex Clifford algebras $\mathrm{Cl}_{p,q}$, presented as tables: for each algebra it lists a primitive idempotent, the two-sided ideal it generates, a left-ideal (spinor) basis, matrix representations of the basis vectors in that spinor basis, the general spinor, its matrix form, and the square of its Hermitian norm. If the tables are correct, any of these expressions can be used directly in computations, without rederiving spinor bases or representations. The paper extends an earlier 1998 computation that covered only the first four items, by adding general spinors, their matrix forms, and norms, and by treating complex idempotents as well as real ones.

What carries the argument

The construction is carried by the minimal left ideal: a primitive idempotent $P$ built from $k$ commuting blades that square to $+1$ generates the left ideal $\mathrm{Cl}_{p,q}P$ that serves as the spinor space. The two-sided ideal $P\,\mathrm{Cl}_{p,q}P$ determines the division ring $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ for the matrix representation, and the left-ideal basis is ordered in 'RevLex' order so that the spinor's matrix form has only its leftmost column nonzero. The squared Hermitian norm is computed from the universal formula $\|\Psi\|^2 = \langle\Psi^\dagger\Psi\rangle$ with the reciprocal basis, and the tables list equivalent involutive forms for each algebra.

What would settle it

Recompute one table entry (for example, the matrix representation of $e_1$ in $\mathrm{Cl}_{2,2}$ or the general spinor norm in $\mathrm{Cl}_{3,1}$) with an independent implementation and compare; if any entry fails to satisfy the defining Clifford relations, the idempotent property $P^2=P$, or the stated norm value, the tables' correctness claim is falsified.

Watch

Extended reading notes

Core claim

The central claim is that for every real and complex Clifford algebra $\mathrm{Cl}_{p,q}$ with dimension $n=p+q$ up to $n=6$, the listed algebraic data are correct: the primitive idempotents are mutually annihilating and sum to unity, the two-sided ideals are isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, the left ideal bases give the spinor basis whose dimension matches the Bott periodicity table, the matrix representations of basis vectors satisfy the defining Clifford relations, and the squared Hermitian norm of a general spinor equals a positive scalar proportional to $\sum_i |s_i|^2$ with a normalization factor $2^{-k}$ where $k$ is the number of idempotent factors. The spinor norm is expressed in an algebra-independent way as $\|\Psi\|^2 = \langle \Psi^\dagger \Psi \rangle$ using the reciprocal basis, and the tables list equivalent involutive forms for each algebra.

Load-bearing premise

The tables are generated by the authors' Mathematica package, and that implementation is assumed to be free of bugs and to have applied the stated ordering rules consistently; no archived code version with a commit hash is provided, so a single software error could propagate into many table entries.

Editorial extensions

If this is right

  • Practitioners can copy the table entries directly into physical or numerical applications—spinor bases, matrix representations, and norms—without rederiving them, provided the chosen ordering convention matches the tables.
  • The complex idempotent tables permit the use of any blade (even one squaring to $-1$) in constructing spinors, which can simplify computations in specific signatures.
  • The explicit norm formulas provide a convenient check for numerical implementations of spinor dynamics or for verifying other software packages.
  • The block-diagonal structure for odd complex algebras, where the spinor has two independent blocks, clarifies the relation between complex and real spinor degrees of freedom.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step is to extend the tables to $n=7,8$ using the same package and the recursive periodicity of the Radon-Hurwitz numbers; the paper's ordering rules and normalization should carry over unchanged.
  • Because the paper provides only the computed tables and not an archived versioned code, users who rely on a specific sign convention might prefer to check each entry against their own implementation, particularly for anti-Euclidean signatures where reciprocal-basis signs enter.
  • The algebra-independent norm formula $\langle\Psi^\dagger\Psi\rangle$ suggests that a coordinate-free package could automate norm computations without per-algebra formulas, possibly informing an implementation that supports arbitrary signatures.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs tables of algebraic spinor data for real and complex Clifford algebras Cl(p,q) (real algebras up to dimension 6, complex up to dimension 5). For each algebra the tables list: a primitive idempotent, the two-sided ideal, a left-ideal (spinor) basis, matrix representations of the basis vectors, a general spinor, its matrix form, and the squared Hermitian norm. The constructions follow standard ideal theory, with the computations performed using the authors' Mathematica package, and a detailed worked example for Cl(2,2) is included. The central claim is that these tables are correct, directly usable reference data.

Significance. If the tables are correct, the paper provides a convenient reference resource that extends Ablamowicz's 1998 Maple-generated tables by adding the last three data items (general spinor, matrix form, norm) and by covering complex idempotents. Strengths of the paper are the explicit worked example for Cl(2,2), the reproducibility offered by the referenced Mathematica package, and the clear statement of the constructive algorithm. However, the paper's central claim is currently undermined by the concrete inconsistency in Table IV, and the lack of a versioned, archived computation means the reader cannot independently verify the remaining tables.

major comments (2)
  1. [Table IV, item 6 (rows Cl(2,0) and Cl(1,1))] The table lists the spinor matrix form for both Cl(2,0) and Cl(1,1) as s1E11 + s2E12. This is inconsistent with the table's own matrix representations. For Cl(2,0), item 4 gives e1 = E11 - E22 and e2 = E12 + E21; with P1 = ½(1+e1) represented by E11 one computes e2P1 = (E12 + E21)E11 = E21. The general spinor Ψ = (s1 + s2e2)P1 therefore has matrix form s1E11 + s2E21, not s1E11 + s2E12. For Cl(1,1), e2 = -E12 + E21 gives the same result. The printed E12 also contradicts the convention stated in Section IV, item 3, that the spinor matrix has only the leftmost nonzero column, and it contradicts the complex analogue in Table X, which correctly lists s1E11 + s2E21 for the identical spinor. A user copying Table IV obtains a transposed (row) form that is not in the left ideal generated by P1. This is a load-bearing error for a paper whose central claim is that the tables are directly usable.
  2. [Section IV A and reference [4]] The paper attributes the computations to the Mathematica notebook '10AlgebraicSpinorsOfRealCl.nb' in the package of reference [4], but no version, commit hash, or archived copy is provided, and no independent verification script is included. The detected error in Table IV shows that the package output was not checked against the paper's own left-ideal convention. For a data-tables paper, this is a reproducibility gap: a single bug in the package can propagate into many table entries, and the reader currently has no way to tell which entries are affected. The authors should supply the exact version of the code used, or an independent verification script, in addition to correcting the erroneous entries.
minor comments (4)
  1. [Abstract and references] The name 'Ablamowicz' appears as 'A b/suppress lamowicz' in the abstract and in reference [2]; the stray '/suppress' string should be removed.
  2. [Section I] The text says 'R, C and H, collectively are called fields, or just K field.' This is nonstandard terminology: quaternions are a division ring but not a field, and calling all three 'fields' may confuse readers. A brief clarification would be useful.
  3. [Section IV A, step 4] In the sentence 'the orthonormal of reciprocal generators (basis) are e1 = e1, e2 = e2, e3 = −e3, and e4 = −e4,' the notation is ambiguous; using superscripts (e^1, e^2, e^3, e^4) would make the reciprocal-basis statement clearer.
  4. [Tables III-XV] The tables are very dense, with some entries broken across lines without visible alignment; for a reference-data paper, larger spacing or splitting the tables into smaller blocks would improve usability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spinor tables are direct evaluations of standard ideal-theory constructions, not predictions fitted to their own outputs.

full rationale

The paper's derivation chain is self-contained and non-circular. Each table is produced by a fixed algebraic recipe: choose a primitive idempotent from commuting blades that square to +1 (with the Radon–Hurwitz count and an explicit InvDeg[Lex] ordering), form the left ideal S = {Cl_{p,q}}P, form the two-sided ideal K = P{Cl_{p,q}}P, obtain the ideal basis by dropping elements obtainable from earlier ones via division-ring multiplication, compute matrix representations entrywise as E_{ij}(e_k) = S(i)^# e_k S(j), and finally write the general spinor and its norm using reversion and Clifford conjugation. Nothing in this chain is fitted to the tabulated outputs, and no tabulated entry is fed back into the construction; the spinor matrix form is literally the expansion of the general spinor in the computed ideal basis, and the norm formula is the scalar part of the reversion/Clifford-conjugation expression. The external references (Lounesto's textbook, Ablamowicz's 1998 Maple data, Ablamowicz–Fauser) are independent prior computations and do not supply any load-bearing premise that is itself unverified. The only self-reference, the authors' GeometricAlgebra Mathematica package [4], is cited as the implementation tool, not as a uniqueness theorem or as evidence that the tables are correct; even if the package contained a bug, that would be a correctness or reproducibility issue, not circular reasoning. The skeptic's noted Table IV E12/E21 inconsistency, if real, is an internal data-entry or software-output error contradicting the paper's own left-ideal convention; it is not a case of the derivation reducing to its inputs. Thus the central claim that Tables III–XV present directly usable algebraic spinor data is supported by the explicit computational procedure rather than by circular definition, renamed prior results, or fitted parameters.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The computations rely on standard Clifford algebra theorems (ideal decomposition, Radon-Hurwitz periodicity, norm formulas) and on the correctness of the Mathematica package. No new axioms or empirically fitted parameters are introduced.

assumptions (5)
  • standard math Clifford algebra Cl(p,q) decomposes into minimal left ideals generated by primitive idempotents of the form (1/2)(1 + e_T).
    Invoked in Section III and used to construct spinor basis; standard result in Lounesto.
  • standard math The number of commuting positive-square blades needed for a primitive idempotent is k = q - r_{q-p}, where r are Radon-Hurwitz numbers.
    Stated in Section III without proof; follows from periodicity theorems in Clifford algebra.
  • standard math For complex Clifford algebras, any basis blade can be used in an idempotent after scaling by i, and the resulting complex idempotents are primitive.
    Used in Section V to construct complex spinor tables; standard in complexified Clifford algebras.
  • standard math The hermitian norm of a spinor is given by ⟨Ψ†Ψ⟩ where Ψ† is the reciprocal basis reverser; this matches the trace formula 2^{-k} Tr(Ψ̂†Ψ̂).
    Defined in Section IV B and used in item 7 of every table.
  • domain assumption The Mathematica package computes geometric products, reversions, and grade projections correctly.
    Underpins all tabulated data; not verified by the paper and no versioned code snapshot is provided.

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Cite this review

Pith. "Pith review of Clifford geometric algebra: Real and complex spinor data tables." pith.science (2026). https://pith.science/paper/LPRLG4N5

@misc{pith2026241214677,
  author       = {Pith},
  title        = {Pith review of: Clifford geometric algebra: Real and complex spinor data tables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LPRLG4N5}},
  note         = {Machine review of arXiv:2412.14677}
}
read the original abstract

The modern algebra concepts are used to construct tables of algebraic spinors related to Clifford algebra multivectors with real and complex coefficients. The following data computed by Mathematica are presented in form of tables for individual Clifford geometric algebras: 1. Initial idempotent; 2. Two-sided ideal; 3. Left ideal basis (otherwise projector, or spinor basis); 4. Matrix representations (reps) for basis vectors in Clifford algebras in spinor basis; 5. General spinor; 6. Spinor in matrix form; 7. Squared hermitian norm of the spinor. Earlier in 1998, only the first four items computed by Maple were published by R. Ablamowicz.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [4]

    }P1 for the left ideal spinors I(P1)

    provides spinor basis S1 = {S1(1), S 1(2), . . . }P1 for the left ideal spinors I(P1). The individual elements S1(i) are called the projections of the ideal I(P1), or just generators. In quantum mechanics they represent spinor basis of a respective geometric algebra. The basis is sorted in RevLex ord er that ensures that the matrix rep of spinor will cont...

    work page

  2. [1]

    The idempotents that differ by signs are singled out by plus/minus signs

    defines a list of mutually annihilating idempotents Pi ∈ Cl p,q with factors sorted in InvDeg(Lex) order. The idempotents that differ by signs are singled out by plus/minus signs. F or example, P1 = 1 2 (1 + e1) ≡ (+) and P2 = 1 2 (1 − e1) ≡ (−), or P1 = 1 4 (1 + e1)(1 + e23) ≡ (+, +) and P2 = 1 4 (1 + e1)(1 − e23) ≡ (+, −)

    work page

  3. [2]

    Left ideal basis (otherwise projecto r, or spinor basis); 4

    Two-sided ideal; 3. Left ideal basis (otherwise projecto r, or spinor basis); 4. Matrix represen- tations (reps) for basis vectors in Clifford algebras in spin or basis; 5. General spinor; 6. Spinor in matrix form; 7. Squared hermitian norm of the spinor. Earlie r in 1998, only the first four items computed by Maple were published by R. Ab/suppress lamowicz...

    work page 1998

  4. [3]

    The number of elements, denoted as (1),(2) or (4), in the list of tw o-sided ideal determines a type of matrix rep: 1 ↔ R, 2 ↔ C, and 4 ↔ H, i.e

    gives a list of elements of two-sided ideal K1 = {K1(1)} , or K1 = {K1(2)}, or K1 = {K1(4)} generated by K1 = P1{Cl p,q}P1. The number of elements, denoted as (1),(2) or (4), in the list of tw o-sided ideal determines a type of matrix rep: 1 ↔ R, 2 ↔ C, and 4 ↔ H, i.e. if K1 contains single element (elements that differ by sign are considered same element)...

    work page

  5. [5]

    The matrix type and dimension is in agreement with the 8-periodicity table 1

    gives the matrix rep of basis vectors ek in the minimal left ideal basis S1. The matrix type and dimension is in agreement with the 8-periodicity table 1. The symbol Eij here denotes matrix with a single entry equal to 1 at the intersection of the i-th row and j-th column. For example, E11 − E22 = [ 1 0 0 − 1 ] , or q(−E12 + E21) = î 0 − q q 0 ó , where q...

    work page

  6. [6]

    In tables the spinor expressions for semisimple algebras includes two idempotents that are related by grade inversion (denoted by ÙPi)

    lists general spinor Ψ expanded in GA minimal left ideal basis, where si are the scalar expansion coefficients. In tables the spinor expressions for semisimple algebras includes two idempotents that are related by grade inversion (denoted by ÙPi)

    work page

  7. [7]

    gives the matrix rep ˆΨ of the GA general spinor Ψ

    work page

  8. [8]

    expresses the square of the spinor norm ‖Ψ ‖2, which is a positive scalar. The spinor hermitian norm can be written universally (in algebra independent way), as ‖Ψ ‖2 = ⟨Ψ †Ψ ⟩, where Ψ † = ∑ T c∗ T e− 1 T is the spinor in the reciprocal basis, an asterisk ∗ denotes complex conjugation of scalar coefficient and T is a multi-index, i.e. the sum is over all o...

    work page

Show all 23 references
  1. [9]

    ÙΨΨ = s2 1 + s2 2 TABLE III: Real spinor tables for GAs with vector space dimen sion n = 1 Cl 2,0[R(2)] Cl 1,1[R(2)] Cl 0,2[H] 1 {P1 = 1 2(1 + e1), P2 = (−)} {P1 = 1 2(1 + e1), P2 = (−)} {P1 = 1} 2 {P1} { P1} { 1,e1,e2,e3} 3 {P1,e2P1} { P1,e2P1} { P1} 4 {E11 − E22, E12 + E21} ...

  2. [10]

    ⟨e1‹Ψe1Ψ⟩ = 1 2(s2 1 + s2

  3. [11]

    ‹ Ù ΨΨ = s2 1 + s2 2 + s2 3 + s2 4 TABLE IV: Real spinor table for GAs with vector space dimensi on n = 2 Cl 3,0[C(2)] Cl 2,1[2R(2)] 1 {P1 = 1 2(1 + e1), P2 = (−)} { P1 = 1 4(1 + e1)(1 + e23), P2 = (+,−), P3 = (−,+), P4 = (−,−)} 2 {P11 P1, P1e23P1} { P11 P1} 3 {P1,e2P1} { P1,e...

  4. [12]

    − ⟨e3 ‹ ÙΨe3Ψ⟩ = 1 4(s2 1 + · · · + s2 4) Cl 1,2[C(2)] Cl 0,3[H(2)] 1 {P1 = 1 2(1 + e1), P2 = (−)} { P1 = 1 2(1 + e123), P2 = 1 2(1 − e123)} 2 {P11 P1, P1e23P1} { P11P1, P1e1P1, P1e2P1, P1e3P1} 3 {P1,e2P1} { e3P1} ∪ {˘e3P1} 4 {E11 − E22,−E12 + E21,−iE12 − iE21} {− q1E11 + q1E2...

  5. [13]

    ⟨‹ ÙΨΨ⟩ = 1 2(s2 1 + · · · + s2 8) TABLE V: Real spinor table for GAs with vector space dimensio n n = 3 6 Cl 4,0[H(2)] Cl 3,1[R(4)] 1 {P1 = 1 2(1 + e1), P2 = 1 2(1 − e1)} { P1 = 1 4(1 + e1)(1 + e24), P2 = (+,−), P3 = (−,+), P4 = (−,−)} 2 {P11P1, P1e23P1, P1e24P1, P1e34P1} { P...

  6. [14]

    − ⟨e123‹Ψe123Ψ⟩ = 1 4(s2 1 + · · · + s2 4) Cl 2,2[R(4)] Cl 1,3[H(2)] 1 {P1 = 1 4(1 + e1)(1 + e23), P2 = (+,−), P3 = (−,+), P4 = (−,−)} {P1 = 1 2(1 + e1), P2 = 1 2(1 − e1)} 2 {P11P1} { P11P1, P1e23P1, P1e24P1, P1e34P1} 3 {P1,e2P1,e4P1,e24P1} { P1,e2P1} 4 {E11 − E22 − E33 + E44,...

  7. [15]

    ⟨e1‹Ψe1Ψ⟩ = 1 2(s2 1 + · · · + s2 8) Cl 0,4[H(2)] 1 {P1 = 1 2(1 + e123), P2 = 1 2(1 − e123)} 2 {P11P1, P1e1P1, P1e2P1, P1e3P1} 3 {e3P1,e34P1} 4 {−q1E11 + q1E22,−q2E11 + q2E22,−q3E11 + q3E22, E12 − E21} 5 Ψ = (−s1 − s2e1 + s3e2 + s4e3 − s5e4 + s6e1,4 − s7e24 + s8e34)P1 6 (−s1 +...

  8. [16]

    Example of construction of the real spinor data table for Cl 2,2 algebra

    − ⟨e5 ‹ ÙΨe5Ψ⟩ = 1 4(s2 1 + · · · + s2 8) Cl 3,2[2R(4)] 1 {P1 = 1 8(1 + e1)(1 + e24)(1 + e35), P2 = (+,+,−), P3 = (+,−,+), P4 = (+,−,−), P5 = ıP1 = (−,+,+), P6 = (−,+,−), P7 = (−,−,+), P8 = (−,−,−)} 2 {P11P1} 3 {P1,e2P1,e3P1,e23P1} ∪ {ıP1,˘ e2P1,˘ e3P1, ˘e23P1} 4 {E11 − E22 − ...

  9. [17]

    Primitive idempotents P . In Cl 2,2 we need to choose a single pair of mutually commuting blades ( eTi , eTj ), {(e1, e23), (e1, e24), (e1, e123), (e1, e124), (e2, e13), (e2, e14), (e2, e123), (e2, e124), (e13, e24), (e13, e123), (e13, e1234), (e14, e23), (e14, e124), (e14, e1...

  10. [18]

    Find a list of two-sided ideal (division ring) K by multiplying the obtained left ideal elements by same idempotent from left, K = P S = P {Cl 2,2}P

    Two-sided ideal (division ring) . Find a list of two-sided ideal (division ring) K by multiplying the obtained left ideal elements by same idempotent from left, K = P S = P {Cl 2,2}P . The division ring of Cl 2,2 contains single element K = {K(1)} = P 1P = 1 4 (1 + e1 + e23 + ...

  11. [19]

    Ideal basis. The left ideal basis is obtained by comparing all the elements of ideal (starting from the first) and consequently dropping out all elements that can be obtained from p revious elements by multiplying them by any division ring element. Since element choice may depe...

  12. [20]

    Once the four-component ideal basis S = {S(1), S (2), S (3), S (4)} is determined one can compute a matrix rep of basis vectors in algebraic spinor bas is

    Matrix rep of GA basis vectors . Once the four-component ideal basis S = {S(1), S (2), S (3), S (4)} is determined one can compute a matrix rep of basis vectors in algebraic spinor bas is. The matrix entry ( ij) can be calculated by Eij(ek) = ~S(i)♯ekS(j), where S(i)♯ is the i...

  13. [21]

    General spinor Ψ. A general spinor is the sum of generators (item 3) multiplied by rea l coefficients si, Ψ =( s1 + s2e2 + s3e4 + s4e24)P1 = 1 4 s1(1 + e1 + e23 + e123) + 1 4 s2(e2 + e3 − e12 − e13)+ 1 4 s3(e4 − e14 + e234 − e1234) + 1 4 s4(e24 + e34 + e124 + e134)

  14. [22]

    In the matrix form the general MV spinor can be expanded in the ide al basis as ˆΨ = s1E11 + s2E21 + s3E31 + s4E41

    Spinor Ψ in matrix form . In the matrix form the general MV spinor can be expanded in the ide al basis as ˆΨ = s1E11 + s2E21 + s3E31 + s4E41

  15. [23]

    The results of the example are summarized in Table VI ( Cl 2,2 algebra)

    Square of spinor norm is ‖Ψ ‖2 = ⟨Ψ †Ψ ⟩ = ⟨‹ Ψ ♯Ψ ⟩ = −⟨e34‹ Ψ e34Ψ ⟩ = 1 4 (s2 1 + s2 2 + s2 3 + s2 4), where the tilde and brackets indicate, respectively, the reversion and scalar part projection. The results of the example are summarized in Table VI ( Cl 2,2 algebra). The...

    1998

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