[Paper Review] Discrete Symmetries and Clifford Algebras

В. В. Варламов|ArXiv.org|Sep 18, 2000

Algebraic and Geometric Analysis23 references5 citations

TL;DR

This paper establishes a rigorous algebraic framework linking discrete symmetries (P, T, PT) to automorphism groups of real and complex Clifford algebras, showing that the eight possible double coverings of orthogonal groups (Da̧browski groups) correspond precisely to the eight real Clifford algebra types via division ring structure. The key contribution is proving that the non-isomorphism of Pin(3,1) and Pin(1,3) arises from the signature dependence of automorphism groups, resolving a long-standing ambiguity in Lorentz group double coverings and linking it to Atiyah–Bott–Shapiro periodicity through generalized Brauer–Wall groups.

ABSTRACT

An algebraic description of basic discrete symmetries (space reversal P, time reversal T and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphisms groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dabrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah-Bott-Shapiro periodicity. Generalized Brauer-Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford-Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.

Motivation & Objective

• To resolve the ambiguity in the standard description of discrete symmetries (P, T, PT) in Lorentz groups by embedding them intrinsically within Clifford algebra automorphism groups.
• To establish a complete classification of automorphism groups of real Clifford algebras based on their division ring structure.
• To clarify the origin of the inequality Pin(p,q) ≇ Pin(q,p), particularly for (p,q) = (3,1) and (1,3), by linking it to the signature dependence of automorphism groups.
• To generalize Brauer–Wall groups over real and complex Clifford algebras and demonstrate their cyclic structure via Trautman diagrams.
• To provide a mathematically consistent framework for the eight Da̧browski Pin^{a,b,c} coverings of O(p,q) using isomorphisms between automorphism groups and finite groups.

Proposed method

• Classify automorphism groups of real Clifford algebras Cℓ_{p,q} using the Wedderburn–Artin theorem and division ring structure (R, C, H).
• Establish a one-to-one correspondence between the eight Da̧browski Pin^{a,b,c} double coverings of O(p,q) and the eight real Clifford algebra types via signature (a,b,c) determined by the algebra's division ring.
• Use the Atiyah–Bott–Shapiro periodicity (mod 8 for R, mod 2 for C) to define generalized Brauer–Wall groups BW^{a,b,c}_ℝ ≃ ℤ₂ ⊗ (ℤ₄)² ⊗ ℤ₈ and BW^{a,b,c}_ℂ ≃ ℤ₂.
• Construct Trautman diagrams to visualize the cyclic group structure of generalized Brauer–Wall groups over R and C, with labels based on p−q mod 8 and n mod 2, respectively.
• Apply the Maple V CLIFFORD package to compute and compare spinor representations and automorphism groups of Cℓ_{3,1} (Majorana algebra) and Cℓ_{1,3} (spacetime algebra).
• Analyze the Clifford–Lipschitz groups Pin(p,q) and Pin(q,p) via their automorphism groups, showing that the signature (p,q) determines the automorphism type and thus the non-isomorphism.

Experimental results

Research questions

• RQ1 How can discrete symmetries P, T, and PT be systematically embedded within the intrinsic algebraic structure of Clifford algebras?
• RQ2 What determines the choice of signature (a,b,c) for the eight Da̧browski Pin^{a,b,c} double coverings of O(p,q), and how does it relate to the signature (p,q) of the underlying space ℝ^{p,q}?
• RQ3 Why is Pin(3,1) not isomorphic to Pin(1,3), despite O(3,1) ≅ O(1,3), and what algebraic structure underlies this non-isomorphism?
• RQ4 How do the generalized Brauer–Wall groups BW^{a,b,c}_ℝ and BW^{a,b,c}_ℂ reflect the Atiyah–Bott–Shapiro periodicity in the context of Clifford algebras?
• RQ5 What is the role of the division ring structure (R, C, H) in constraining the existence and type of automorphism groups of real Clifford algebras?

Key findings

• The eight Da̧browski Pin^{a,b,c} double coverings of O(p,q) are in one-to-one correspondence with the eight real Clifford algebra types Cℓ_{p,q}, with the signature (a,b,c) determined by the division ring structure of Cℓ_{p,q}.
• The non-isomorphism of Pin(3,1) and Pin(1,3) is explained by the fact that their automorphism groups have different signatures (a,b,c), which are determined by the underlying space signature (p,q) and the algebra's division ring structure.
• The generalized Brauer–Wall group over the reals is isomorphic to ℤ₂ ⊗ (ℤ₄)² ⊗ ℤ₈, reflecting the mod 8 periodicity of real Clifford algebras and the cyclic structure of automorphism groups.
• Over the complex numbers, the generalized Brauer–Wall group is isomorphic to ℤ₂, corresponding to the mod 2 periodicity of complex Clifford algebras and the two types Cₙ and Cₙ₊₁ ≃ Cₙ ⊕ Cₙ.
• The Trautman diagrams for BW_ℝ ≃ ℤ₈ and BW_ℂ ≃ ℤ₂ visually encode the periodicity and group structure, with labels based on p−q mod 8 and n mod 2, respectively.
• The study confirms that the automorphism group of Cℓ_{3,1} (Majorana algebra) and Cℓ_{1,3} (spacetime algebra) differ in signature, explaining the non-isomorphism of their Pin groups despite isomorphic Lorentz groups.