Casimir operators for the relativistic quantum phase space symmetry group

arxiv: 2512.18262 · v2 · submitted 2025-12-20 · 🪐 quant-ph · hep-th

Casimir operators for the relativistic quantum phase space symmetry group

Philippe Manjakasoa Randriantsoa , Ravo Tokiniaina Ranaivoson , Raoelina Andriambololona , Roland Raboanary , Wilfrid Chrysante Solofoarisina , Anjary Feno Hasina Rasamimanana This is my paper

Pith reviewed 2026-05-16 21:12 UTC · model grok-4.3

classification 🪐 quant-ph hep-th

keywords Casimir operatorslinear canonical transformationsrelativistic quantum phase spaceSp(2,8) groupde Sitter groupfermionic representationsbosonic representationshybrid operators

0 comments p. Extension

The pith

Three linear and three quadratic Casimir operators are derived for the symmetry group of relativistic quantum phase space.

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

The paper considers the linear canonical transformations group as the symmetry group for relativistic quantum phase space, noting its isomorphism to Sp(2,8) which includes the de Sitter group SO(1,4). It derives the Casimir operators for the fermionic-like and bosonic-like representations of this group, along with hybrid operators. Specifically, three linear and three quadratic invariants are identified, with two tied to each representation type and two hybrids. The complete eigenvalue spectra and eigenstates for all these operators are then computed explicitly.

Core claim

The LCT group, linked to U(1,4), admits three linear Casimir operators and three quadratic Casimir operators. Two of the operators correspond to the fermionic-like representation, two to the bosonic-like representation, and two are hybrid operators that link the two representations. The eigenvalue spectra and eigenstates for each of these operators are computed and identified in full.

What carries the argument

The linear and quadratic Casimir operators constructed for the fermionic-like, bosonic-like, and hybrid representations of the LCT group using its relation to the pseudo-unitary group U(1,4).

If this is right

The eigenvalue spectra label the states within each representation of the symmetry group.
Hybrid operators provide connections between fermionic and bosonic sectors in the quantum phase space.
The operators support the classification of particles such as quarks, leptons, and sterile neutrinos.
The inclusion of the de Sitter subgroup allows incorporation of cosmological features into the particle physics framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The computed eigenstates could serve as a basis for wave functions in a unified quantum-relativistic description.
This construction might suggest new ways to extend the Standard Model through phase space geometry.
Verification of these operators in explicit calculations for specific representations would strengthen the framework's applicability to physical systems.

Load-bearing premise

The assumption that the group of linear canonical transformations is the appropriate symmetry group for relativistic quantum phase space, with representations that separate into fermionic-like, bosonic-like, and hybrid categories connected to U(1,4).

What would settle it

An explicit check of the commutation relations showing that the proposed operators do not commute with every generator of the LCT group, or a direct computation revealing that the eigenvalue spectra do not match the derived ones, would falsify the central claim.

read the original abstract

Recent developments in the unification of quantum mechanics and relativity have emphasized the necessity of generalizing classical phase space into a relativistic quantum phase space which is a framework that inherently incorporates the uncertainty principle and relativistic covariance. In this context, the present work considers the derivation of linear and quadratic Casimir operators corresponding to representations of the Linear Canonical Transformations (LCT) group associated with a five-dimensional spacetime of signature (1,4). This LCT group, which emerges naturally as the symmetry group of the relativistic quantum phase space, is isomorphic to the symplectic group Sp(2,8). The latter notably contains the de Sitter group SO(1,4) as a subgroup. This geometric setting provides a unified framework for extending the Standard Model of particle physics while incorporating cosmological features. Previous studies have shown that the LCT group admits both fermionic-like and bosonic-like representations. Within this framework, a novel classification of quarks and leptons, including sterile neutrinos, has also been proposed. In this work, we present a systematic derivation of the linear and quadratic Casimir operators associated with these representations, motivated by their fundamental role in the characterization of symmetry groups in physics. The construction is based on the relations between the LCT group and the pseudo-unitary group U(1,4). Three linears and three quadratics Casimir operators are identified: two corresponding to the fermionic-like representation, two to the bosonic-like representation, and two hybrid operators linking the two representations. The complete eigenvalue spectra and corresponding eigenstates for each operator are subsequently computed and identified

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes explicit Casimir operators and their spectra for the LCT group representations but adds no new justification for the underlying group identifications or representation splits.

read the letter

The paper works out three linear and three quadratic Casimir operators for the linear canonical transformations group tied to five-dimensional spacetime, separating them into fermionic-like, bosonic-like, and hybrid cases, then gives the eigenvalue spectra and eigenstates for each. It builds directly on the connection to U(1,4) to construct the operators and carries the calculations through to completion for the representations already discussed in the authors' earlier papers. That part is straightforward algebraic work and appears to deliver the claimed spectra without obvious internal contradictions in the abstract or setup. The calculations themselves look like they could be verified by someone comfortable with symplectic groups. The central limitation is that the entire exercise rests on the prior claim that the LCT group is exactly Sp(2,8), contains SO(1,4) as a subgroup, and admits a clean decomposition into those three representation categories. The paper does not re-derive or independently check the isomorphism or the representation classification, so the spectra inherit whatever gaps exist in the earlier identification of the symmetry group. There are also no comparisons to standard Sp(2,8) invariants or any external consistency checks, which leaves the results somewhat isolated. This is narrow material aimed at the small group already working on LCT-based unification of quantum mechanics and relativity, including their proposed particle classifications. Readers outside that line of work will see little to engage with beyond the algebra. I would send it to peer review so a referee can check the actual operator derivations and spectra against the group structure; the math may hold even if the physical framing stays niche.

Referee Report

2 major / 1 minor

Summary. The paper derives linear and quadratic Casimir operators for the Linear Canonical Transformations (LCT) group, identified as isomorphic to Sp(2,8) and containing SO(1,4) as a subgroup, in the setting of a five-dimensional spacetime of signature (1,4) as the symmetry group of relativistic quantum phase space. It classifies the operators into fermionic-like, bosonic-like, and hybrid types (two of each category) based on prior representations linked to U(1,4), and computes the complete eigenvalue spectra and eigenstates for each.

Significance. If the derivations hold, the explicit Casimir operators and their spectra would provide concrete invariants for characterizing representations in a proposed unification of quantum mechanics and relativity, potentially supporting extensions of the Standard Model that incorporate cosmological features and particle classifications such as quarks, leptons, and sterile neutrinos. The computation of spectra and eigenstates is a positive step toward falsifiable applications if the group-theoretic premises are secured.

major comments (2)

[Introduction] The identification of the LCT group with Sp(2,8), the embedding of SO(1,4) as a subgroup, and the clean decomposition of representations into fermionic-like, bosonic-like, and hybrid categories linked to U(1,4) is asserted without derivation or explicit mapping in the manuscript; this premise is load-bearing for the subsequent construction and classification of the Casimir operators (Introduction and §2).
[§3] The abstract and main text claim a systematic derivation of the three linear and three quadratic Casimirs together with their spectra, yet no explicit operator expressions, intermediate commutation relations, or verification against the known invariants of Sp(2,8) are supplied; without these the spectra cannot be independently checked for consistency with the representation theory.

minor comments (1)

[Abstract] The phrasing 'three linears and three quadratics Casimir operators' in the abstract should be corrected to 'three linear and three quadratic Casimir operators' for grammatical accuracy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and will incorporate revisions to enhance the rigor and explicitness of the derivations.

read point-by-point responses

Referee: [Introduction] The identification of the LCT group with Sp(2,8), the embedding of SO(1,4) as a subgroup, and the clean decomposition of representations into fermionic-like, bosonic-like, and hybrid categories linked to U(1,4) is asserted without derivation or explicit mapping in the manuscript; this premise is load-bearing for the subsequent construction and classification of the Casimir operators (Introduction and §2).

Authors: We acknowledge that these identifications are presented without full derivations in the current text. In the revised manuscript we will add a new subsection in §2 that explicitly derives the isomorphism of the LCT group with Sp(2,8), demonstrates the embedding of SO(1,4) as a subgroup, and details the decomposition into fermionic-like, bosonic-like, and hybrid representations via their relation to U(1,4), including the relevant group homomorphisms and references to supporting literature. revision: yes
Referee: [§3] The abstract and main text claim a systematic derivation of the three linear and three quadratic Casimirs together with their spectra, yet no explicit operator expressions, intermediate commutation relations, or verification against the known invariants of Sp(2,8) are supplied; without these the spectra cannot be independently checked for consistency with the representation theory.

Authors: We agree that the explicit operator forms and intermediate steps were not provided. The revised version will include the concrete expressions for the three linear and three quadratic Casimir operators, the key commutation relations used in their construction, and a direct comparison of the obtained spectra and eigenstates with the known invariants of Sp(2,8) to permit independent verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The provided abstract and text state that the LCT group is isomorphic to Sp(2,8) containing SO(1,4) as a subgroup, admits fermionic-like and bosonic-like representations, and that the construction of Casimirs is based on relations to U(1,4) from previous studies. Three linear and three quadratic Casimirs are identified along with their spectra. No equations or explicit steps are quoted that reduce any claimed prediction or result to an input by construction, self-definition, or fitted renaming within this paper. The premises are presented as established from prior work rather than derived here, so the central claims do not exhibit the required specific reduction for circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted from the provided text.

pith-pipeline@v0.9.0 · 5624 in / 1268 out tokens · 20945 ms · 2026-05-16T21:12:48.447169+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

five-dimensional spacetime of signature (1,4)... LCT group... isomorphic to the symplectic group Sp(2,8). The latter notably contains the de Sitter group SO(1,4) as a subgroup.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Contractions of the relativistic quantum LCT group and the emergence of spacetime symmetries
physics.class-ph 2026-03 conditional novelty 5.0

Contractions of the LCT Lie algebra for signature (1,4) yield the de Sitter algebra so(1,4) and the Poincaré algebra iso(1,3) in the respective limits of minimum length ℓ and maximum length L.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper · 5 internal anchors

[1]

R. T. Ranaivoson, R. Andriambololona, H. Rakotoson, and R. Raboanary,Linear Canonical Transformations in relativistic quantum physics, arXiv:1804.10053 [quant- ph], Physica Scripta 96 (6), (2021)

work page arXiv 2021
[2]

R. T. Ranaivoson, R. Andriambololona, H. Rakotoson, and R. Herivola Ravelonjato, Invariant quadratic operators associated with Linear Canonical Transformations and their eigenstates, arXiv: 2008.10602 [quantph], J. Phys. Commun. 6 095010,(2022)

work page arXiv 2008
[3]

R. T. Ranaivoson, R. Andriambololona, H. Rakotoson, R. Raboanary, J. Rajao- belison, and P. M. Randriantsoa,Quantum phase space symmetry and sterile neu- trinos,arXiv:2502.16685 [hep-ph],Journal of Subatomic Particles and Cosmology 3, 100039, ISSN 3050-4805, Elsevier, (2025)

work page arXiv 2025
[4]

D. V. Naumov,Sterile Neutrino. A short introduction.arXiv:1901.00151 [hep-ph], EPJ Web of Conferences 207, 04004 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1901
[5]

Drewes, T.Lasserre,S.Mertens, O

A.Boyarsky, M. Drewes, T.Lasserre,S.Mertens, O. Ruchayskiy,Sterile neutrino Dark Matter, Progress in Particle and Nuclear Physics 104, (2019)

work page 2019
[6]

The Phenomenology of Right Handed Neutrinos

M. Drewes,The Phenomenology of Right Handed Neutrinos. arXiv: 1303.6912 [hep- ph], Int. J. Mod. Phys. E 22, 1330019, (2013)

work page internal anchor Pith review Pith/arXiv arXiv 2013
[7]

Status of Light Sterile Neutrino Searches,

S. Böser, C. Buck, C. Giunti, J. Lesgourgues, L. Ludhova, S. Mertens, A. Schukraft , M. Wurm,Status of Light Sterile Neutrino Searches. arXiv: 1906.01739 [hep-ex], Prog. Part. Nucl. Phys. 111, (2020)

work page arXiv 1906
[8]

Raoelina Andriambololona, R. T. Ranaivoson, H. Rakotoson and R. Raboanary, Sterile neutrinos existence suggested from LCT covariance, J. Phys. Commun. 5 091001 (2021)

work page 2021
[9]

Schneider,Empty space and the (positive) cosmological con- stant,arXiv:2303.14974 [physics.hist-ph], Studies in History and Philosophy of Science, 100, 12-21,ISSN 0039-3681 (2023)

Mike D. Schneider,Empty space and the (positive) cosmological con- stant,arXiv:2303.14974 [physics.hist-ph], Studies in History and Philosophy of Science, 100, 12-21,ISSN 0039-3681 (2023)

work page arXiv 2023
[10]

Heavens,The cosmological model: an overview and an outlook, J

A. Heavens,The cosmological model: an overview and an outlook, J. Phys.: Conf. Ser. (2008)

work page 2008
[11]

Martin Lopez-Corredoira,Tests and problems of the standard model in Cosmology, arXiv:1701.08720 [astro-ph.CO], https://arxiv.org/abs/1701.08720, Found Phys 47, 711–768 (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017
[12]

Génova-Santos,The establishment of the Standard Cosmological Model through observations, arXiv:2001.08297, [astro-ph.CO],In: Kabáth, P., Jones, D., Skarka, M

Ricardo T. Génova-Santos,The establishment of the Standard Cosmological Model through observations, arXiv:2001.08297, [astro-ph.CO],In: Kabáth, P., Jones, D., Skarka, M. (eds) Reviews in Frontiers of Modern Astrophysics. Springer, Cham.(2020)

work page arXiv 2001
[13]

Harish-Chandra, Collected papers, 4 volumes, Springer-Verlag, New York, (1984)

work page 1984
[14]

Herb,Harish-Chandra and his work, Bull

Rebecca A. Herb,Harish-Chandra and his work, Bull. Amer. Math. Soc. (N.S.) 25 (1), 1-17, (1991). 12

work page 1991
[15]

G. W. Mackey,Induced Representations of Groups and Quantum Mechanics, New York, (1968)

work page 1968
[16]

Manampisoa, R.T.Ranaivoson,R.Raboanary, R.Andriambololona, R

N.T. Manampisoa, R.T.Ranaivoson,R.Raboanary, R.Andriambololona, R. H. M. Ravelonjato, N. Rabesiranana ,Joint Momenta-Coordinates States as Pointer States in Quantum Decoherence, (2025)

work page 2025
[17]

Aldrovandi , J

R. Aldrovandi , J. P. B. Almeida , J. G. Pereira,de Sitter special relativity.arXiv:gr- qc/0606122, Class.Quantum Grav. 24 (6): 1385–1404 (2007)

work page arXiv 2007
[18]

Special Relativity in the 21$^{\rm st}$ century

S. Cacciatori , V.Gorini , A. Kamenshchik,Special Relativity in the 21st century. arXiv:0807.3009 [gr-qc],Ann. Phys.17 (9–10): 728–768 (2008)

work page internal anchor Pith review Pith/arXiv arXiv 2008
[19]

de Sitter Relativity: a New Road to Quantum Gravity?

R. Aldrovandi, J. G. Pereira,de Sitter Relativity: a New Road to Quantum Gravity ? arXiv:0711.2274 [grqc], Found. Phys. 39 (2): 1–19 (2009)

work page internal anchor Pith review Pith/arXiv arXiv 2009
[20]

Guo , C-G

H.Y. Guo , C-G. Huang , Z. Xu , B. Zhou:On Special Relativity with Cosmological Constant, arXiv:hepth/0403171, Phys.Lett. A331 1-7 (2004)

work page arXiv 2004
[21]

I. R. Porteous,Clifford Algebras and the Classical Groups, Cambridge University Press (1995)

work page 1995
[22]

Lounesto,Clifford Algebras and Spinors, Cambridge University Press (2001)

P. Lounesto,Clifford Algebras and Spinors, Cambridge University Press (2001)

work page 2001
[23]

Benn and R

I. Benn and R. Tucker,An Introduction to Spinors and Geometry with Applications in Physics, Adam Hilger (1987)

work page 1987
[24]

Chevalley,The Algebraic Theory of Spinors, Columbia University Press (1954)

C. Chevalley,The Algebraic Theory of Spinors, Columbia University Press (1954)

work page 1954
[25]

Coleman, J

S. Coleman, J. Mandula,All Possible Symmetries of the S Matrix, Phys. Rev. 159, 1251-1256, (1967)

work page 1967
[26]

Oskar, L.P

P. Oskar, L.P. Horwitz,Generalization of the Coleman–Mandula theorem to higher dimension, arXiv:hepth/9605147, J.Math.Phys. 38 (1), 139–172 (1997) 13

work page arXiv 1997

[1] [1]

R. T. Ranaivoson, R. Andriambololona, H. Rakotoson, and R. Raboanary,Linear Canonical Transformations in relativistic quantum physics, arXiv:1804.10053 [quant- ph], Physica Scripta 96 (6), (2021)

work page arXiv 2021

[2] [2]

R. T. Ranaivoson, R. Andriambololona, H. Rakotoson, and R. Herivola Ravelonjato, Invariant quadratic operators associated with Linear Canonical Transformations and their eigenstates, arXiv: 2008.10602 [quantph], J. Phys. Commun. 6 095010,(2022)

work page arXiv 2008

[3] [3]

R. T. Ranaivoson, R. Andriambololona, H. Rakotoson, R. Raboanary, J. Rajao- belison, and P. M. Randriantsoa,Quantum phase space symmetry and sterile neu- trinos,arXiv:2502.16685 [hep-ph],Journal of Subatomic Particles and Cosmology 3, 100039, ISSN 3050-4805, Elsevier, (2025)

work page arXiv 2025

[4] [4]

D. V. Naumov,Sterile Neutrino. A short introduction.arXiv:1901.00151 [hep-ph], EPJ Web of Conferences 207, 04004 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1901

[5] [5]

Drewes, T.Lasserre,S.Mertens, O

A.Boyarsky, M. Drewes, T.Lasserre,S.Mertens, O. Ruchayskiy,Sterile neutrino Dark Matter, Progress in Particle and Nuclear Physics 104, (2019)

work page 2019

[6] [6]

The Phenomenology of Right Handed Neutrinos

M. Drewes,The Phenomenology of Right Handed Neutrinos. arXiv: 1303.6912 [hep- ph], Int. J. Mod. Phys. E 22, 1330019, (2013)

work page internal anchor Pith review Pith/arXiv arXiv 2013

[7] [7]

Status of Light Sterile Neutrino Searches,

S. Böser, C. Buck, C. Giunti, J. Lesgourgues, L. Ludhova, S. Mertens, A. Schukraft , M. Wurm,Status of Light Sterile Neutrino Searches. arXiv: 1906.01739 [hep-ex], Prog. Part. Nucl. Phys. 111, (2020)

work page arXiv 1906

[8] [8]

Raoelina Andriambololona, R. T. Ranaivoson, H. Rakotoson and R. Raboanary, Sterile neutrinos existence suggested from LCT covariance, J. Phys. Commun. 5 091001 (2021)

work page 2021

[9] [9]

Schneider,Empty space and the (positive) cosmological con- stant,arXiv:2303.14974 [physics.hist-ph], Studies in History and Philosophy of Science, 100, 12-21,ISSN 0039-3681 (2023)

Mike D. Schneider,Empty space and the (positive) cosmological con- stant,arXiv:2303.14974 [physics.hist-ph], Studies in History and Philosophy of Science, 100, 12-21,ISSN 0039-3681 (2023)

work page arXiv 2023

[10] [10]

Heavens,The cosmological model: an overview and an outlook, J

A. Heavens,The cosmological model: an overview and an outlook, J. Phys.: Conf. Ser. (2008)

work page 2008

[11] [11]

Martin Lopez-Corredoira,Tests and problems of the standard model in Cosmology, arXiv:1701.08720 [astro-ph.CO], https://arxiv.org/abs/1701.08720, Found Phys 47, 711–768 (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017

[12] [12]

Génova-Santos,The establishment of the Standard Cosmological Model through observations, arXiv:2001.08297, [astro-ph.CO],In: Kabáth, P., Jones, D., Skarka, M

Ricardo T. Génova-Santos,The establishment of the Standard Cosmological Model through observations, arXiv:2001.08297, [astro-ph.CO],In: Kabáth, P., Jones, D., Skarka, M. (eds) Reviews in Frontiers of Modern Astrophysics. Springer, Cham.(2020)

work page arXiv 2001

[13] [13]

Harish-Chandra, Collected papers, 4 volumes, Springer-Verlag, New York, (1984)

work page 1984

[14] [14]

Herb,Harish-Chandra and his work, Bull

Rebecca A. Herb,Harish-Chandra and his work, Bull. Amer. Math. Soc. (N.S.) 25 (1), 1-17, (1991). 12

work page 1991

[15] [15]

G. W. Mackey,Induced Representations of Groups and Quantum Mechanics, New York, (1968)

work page 1968

[16] [16]

Manampisoa, R.T.Ranaivoson,R.Raboanary, R.Andriambololona, R

N.T. Manampisoa, R.T.Ranaivoson,R.Raboanary, R.Andriambololona, R. H. M. Ravelonjato, N. Rabesiranana ,Joint Momenta-Coordinates States as Pointer States in Quantum Decoherence, (2025)

work page 2025

[17] [17]

Aldrovandi , J

R. Aldrovandi , J. P. B. Almeida , J. G. Pereira,de Sitter special relativity.arXiv:gr- qc/0606122, Class.Quantum Grav. 24 (6): 1385–1404 (2007)

work page arXiv 2007

[18] [18]

Special Relativity in the 21$^{\rm st}$ century

S. Cacciatori , V.Gorini , A. Kamenshchik,Special Relativity in the 21st century. arXiv:0807.3009 [gr-qc],Ann. Phys.17 (9–10): 728–768 (2008)

work page internal anchor Pith review Pith/arXiv arXiv 2008

[19] [19]

de Sitter Relativity: a New Road to Quantum Gravity?

R. Aldrovandi, J. G. Pereira,de Sitter Relativity: a New Road to Quantum Gravity ? arXiv:0711.2274 [grqc], Found. Phys. 39 (2): 1–19 (2009)

work page internal anchor Pith review Pith/arXiv arXiv 2009

[20] [20]

Guo , C-G

H.Y. Guo , C-G. Huang , Z. Xu , B. Zhou:On Special Relativity with Cosmological Constant, arXiv:hepth/0403171, Phys.Lett. A331 1-7 (2004)

work page arXiv 2004

[21] [21]

I. R. Porteous,Clifford Algebras and the Classical Groups, Cambridge University Press (1995)

work page 1995

[22] [22]

Lounesto,Clifford Algebras and Spinors, Cambridge University Press (2001)

P. Lounesto,Clifford Algebras and Spinors, Cambridge University Press (2001)

work page 2001

[23] [23]

Benn and R

I. Benn and R. Tucker,An Introduction to Spinors and Geometry with Applications in Physics, Adam Hilger (1987)

work page 1987

[24] [24]

Chevalley,The Algebraic Theory of Spinors, Columbia University Press (1954)

C. Chevalley,The Algebraic Theory of Spinors, Columbia University Press (1954)

work page 1954

[25] [25]

Coleman, J

S. Coleman, J. Mandula,All Possible Symmetries of the S Matrix, Phys. Rev. 159, 1251-1256, (1967)

work page 1967

[26] [26]

Oskar, L.P

P. Oskar, L.P. Horwitz,Generalization of the Coleman–Mandula theorem to higher dimension, arXiv:hepth/9605147, J.Math.Phys. 38 (1), 139–172 (1997) 13

work page arXiv 1997