[Paper Review] Local generalization of Pauli's theorem

Н. Г. Марчук, Dmitry Shirokov|arXiv (Cornell University)|Jan 24, 2012

Algebraic and Geometric Analysis8 references3 citations

TL;DR

This paper extends Pauli's theorem to smooth families of Clifford algebra generators in Euclidean space, proving that two such families are locally related by a smooth similarity transformation when $ n $ is even. For odd $ n $, all possible connection types are classified. The problem reduces to solving a system of partial differential equations derived from the spin connection, with explicit solutions provided for low-dimensional cases ($ n=2 $, $ r \geq 1 $ and $ n \geq 2 $, $ r=1 $).

ABSTRACT

Generalized Pauli's theorem, proved by D. S. Shirokov for two sets of anticommuting elements of a real or complexified Clifford algebra of dimension $2^n$, is extended to the case, when both sets of elements depend smoothly on points of Euclidian space of dimension $r$. We prove that in the case of even $n$ there exists a smooth function such that two sets of Clifford algebra elements are connected by a similarity transformation. All cases of connection between two sets are considered in the case of odd $n$. Using the equation for the spin connection of general form, it is shown that the problem of the local Pauli's theorem is equivalent to the problem of existence of a solution of some special system of partial differential equations. The special cases $n=2$, $r\geq 1$ and $n\geq 2$, $r=1$ with more simpler solution of the problem are considered in detail.

Motivation & Objective

• To generalize Pauli’s theorem to the case where Clifford algebra generators depend smoothly on points in Euclidean space.
• To establish conditions under which two smoothly varying sets of anticommuting Clifford algebra elements are related by a smooth similarity transformation.
• To classify all possible connection types between such sets when $ n $ is odd.
• To show that the local Pauli problem is equivalent to solving a system of partial differential equations derived from the spin connection.
• To provide explicit solutions for special cases, including $ n=2 $, $ r \geq 1 $ and $ n \geq 2 $, $ r=1 $, using exponential maps and spinor-like transformations.

Proposed method

• Formulate the problem in terms of smooth functions $ g^a(x), h^a(x) $ taking values in a real or complexified Clifford algebra $ \mathcal{C}\!\ell^{\mathbb{F}}(p,q) $, satisfying anticommutation relations $ g^a g^b + g^b g^a = 2\eta^{ab}e $.
• Define the spin connection via $ C_\mu = \frac{1}{4}(\partial_\mu h^a) h_a $, which generates the transformation $ T(x) = \exp(C(x))K $, with $ K $ invertible.
• Use the equation $ \partial_\mu S(x) = C_\mu(x) S(x) $ to derive the solution $ S(x) = \exp(C(x))K $, ensuring $ g^a(x) = T^{-1}(x) h^a(x) T(x) $.
• For $ n $ even, prove existence of a smooth invertible $ T(x) $ in a neighborhood $ O_\varepsilon(x_0) $ such that $ g^a(x) = T^{-1}(x) h^a(x) T(x) $.
• For $ n $ odd, classify all possible connection types between the two sets, showing that the transformation may not exist globally.
• Analyze special cases: $ n=2 $, $ r \geq 1 $, and $ n \geq 2 $, $ r=1 $, using explicit parameterizations of rotation/boost matrices and computing $ C_\mu $ and $ T(x) $.

Experimental results

Research questions

• RQ1 Under what conditions can two smoothly varying sets of anticommuting Clifford algebra generators in Euclidean space be related by a smooth similarity transformation?
• RQ2 How does the local Pauli problem reduce to solving a system of partial differential equations involving the spin connection?
• RQ3 What are the complete classification and possible types of connections between two such sets when $ n $ is odd?
• RQ4 Can explicit solutions be constructed for low-dimensional cases such as $ n=2 $, $ r \geq 1 $ and $ n \geq 2 $, $ r=1 $?
• RQ5 What are the implications of non-abelian spin connections (i.e., $ [C_\mu, C_\nu] \neq 0 $) for the solvability of the local Pauli problem?

Key findings

• For even $ n $, there exists a smooth invertible transformation $ T(x) $ in a neighborhood of any point $ x_0 $ such that $ g^a(x) = T^{-1}(x) h^a(x) T(x) $, with $ T(x) $ of class $ C^k $.
• The transformation $ T(x) $ is unique up to multiplication by a non-vanishing smooth scalar function (real for $ \mathbb{R} $, complex for $ \mathbb{C} $).
• For $ n=2 $, explicit solutions are constructed using $ T(x) = \exp(C(x))K $, where $ C_\mu = \frac{1}{2} \partial_\mu \varphi \cdot e^{12} $, and $ \varphi(x) $ is a smooth function.
• In the case $ n=2 $, $ r=1 $, the solution $ T(x) $ satisfies $ e^a = T^{-1}(x) h^a(x) T(x) $ for all $ x \in V $, with $ T(x) = \cos(\varphi(x)/2)e \pm \sin(\varphi(x)/2)e^{12} $.
• For $ n=3 $, $ r=1 $, with $ h^a $ parameterized by Euler angles, the spin connection $ C_\mu $ is non-abelian ($ [C_\mu, C_\nu] \neq 0 $), and the Poincaré lemma does not apply.
• The local Pauli problem is equivalent to solving $ \partial_\mu S(x) = C_\mu(x) S(x) $, with solution $ S(x) = \exp(C(x))K $, and the transformation $ T(x) $ is invertible for all $ x \in V $.