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Pauli square identity and the formal g=2 coupling

Physical scope

Formal differential-operator identity on compactly supported two-component spinors for a smooth static SI vector potential in three-dimensional nonrelativistic Pauli-model kinematics. It asserts neither a closed-domain identity nor a measured particle g-factor.

Statement

Let $m > 0$ be mass in kilograms, $q \in R$ charge in coulombs, $ℏ > 0$ in joule-seconds, and let $A \in C^{\infty} (R^{3}; R^{3})$ be a real SI vector potential in tesla-metres, with $B = \nabla \times A$ in teslas. On the two-component core $C_{c}^{\infty} (R^{3}; C^{2})$ define $Π_{j} = - i ℏ \partial_{j} - q A_{j}$ . With the spin-half matrices and $S_{j} = ℏ σ_{j} / 2$ fixed in the effective Pauli model, the following identity holds on that core: ${(\sum_{j = 1}^{3} σ_{j} Π_{j})}^{2} = \sum_{j = 1}^{3} Π_{j}^{2} - q ℏ \sum_{j = 1}^{3} σ_{j} B_{j} .$ Consequently the quadratic Pauli expression $\frac{1}{2 m} (\sum_{j} σ_{j} Π_{j})^{2}$ equals the minimally coupled kinetic expression plus the spin term $- \frac{q}{m} S \cdot B$ , which is the formal effective Pauli coupling at $g = 2$ . This is a core identity only; it makes no claim about closed operator domains or the measured magnetic moment of an electron.

Statement references

Unbounded vector potential and effective Pauli model

Magnetic quadratic-form data

Divergence and curl of a $C^{1}$ vector field

Facts & Assumptions

Given: the effective spin-half model, $m > 0$ , real $q$ , $ℏ > 0$ , smooth real $A$ , $B = \nabla \times A$ , and compactly supported smooth two-component spinors.

[A1]

The spin matrices are $σ_{1} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), σ_{2} = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array}), σ_{3} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}), S_{j} = \frac{ℏ}{2} σ_{j},$ and $g$ is a declared real dimensionless effective coupling (Unbounded vector potential and effective Pauli model).

[F1]

On the test core, $Π_{j} = - i ℏ \partial_{j} - q A_{j}$ ; the curl convention is $B_{k} = \sum_{i, j} ϵ_{k i j} \partial_{i} A_{j}$ (Magnetic quadratic-form data, Divergence and curl of a $C^{1}$ vector field).

[F2]

The ordinary product rule holds for the smooth coefficient products (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Proof

1.1A1F1F2algebra

Direct multiplication of the matrices in [A1] gives $σ_{j}^{2} = I_{2}$ for each $j$ , $σ_{1} σ_{2} = i σ_{3}$ , $σ_{2} σ_{3} = i σ_{1}$ , $σ_{3} σ_{1} = i σ_{2}$ , and the reversed products have the opposite sign. Hence $σ_{i} σ_{j} = δ_{i j} I_{2} + i \sum_{k = 1}^{3} ϵ_{i j k} σ_{k} .$ For a core spinor $u$ , the product rule [F2] and the definition of $Π_{j}$ give $[Π_{i}, Π_{j}] u = i q ℏ (\partial_{i} A_{j} - \partial_{j} A_{i}) u = i q ℏ \sum_{k = 1}^{3} ϵ_{i j k} B_{k} u .$ All terms remain compactly supported smooth spinors.

2.1step 1.1algebra

Since the Pauli matrices act only on the spin factor, they commute with each $Π_{j}$ . Expanding the square using step 1.1 gives ${(\sum_{i} σ_{i} Π_{i})}^{2} = \sum_{i} Π_{i}^{2} I_{2} + i \sum_{i, j, k} ϵ_{i j k} σ_{k} Π_{i} Π_{j} = \sum_{i} Π_{i}^{2} I_{2} + \frac{i}{2} \sum_{i, j, k} ϵ_{i j k} σ_{k} [Π_{i}, Π_{j}] .$ Substitute the commutator from step 1.1. The finite identity $\sum_{i, j} ϵ_{i j k} ϵ_{i j ℓ} = 2 δ_{k ℓ}$ follows by listing the two nonzero ordered pairs when $k = ℓ$ and none when $k \neq ℓ$ ; the resulting term is $- q ℏ \sum_{k} σ_{k} B_{k}$ . This proves the displayed Pauli square.

3.1A1step 2.1algebra∎

Dividing by $2 m$ and using $S_{j} = ℏ σ_{j} / 2$ from [A1] rewrites the spin term as $- q ℏ σ \cdot B / (2 m) = - q S \cdot B / m = - 2 q S \cdot B / (2 m)$ . Comparison with the adopted coupling $- g q S \cdot B / (2 m)$ identifies its formal coefficient as $g = 2$ . The conclusion is only the stated identity of core differential expressions; the postulate separately requires a closable semibounded form before assigning a self-adjoint Pauli Hamiltonian.

Pauli square identity and the formal g=2 coupling

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources