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A noncommuting SU(2) configuration with nonzero Yang–Mills density

Physical scope

A smooth noncommuting connection configuration in the adopted classical pure SU(2) model on the trivial bundle over Minkowski spacetime. It computes curvature and action density only and makes no field-equation or empirical claim.

Example

Assume ${A C}_{ω}$ (Countable Choice). Specialize the adopted model to the trivial bundle $P = R^{1, 3} \times S U (2)$ with its product section. Let $T_{1}, T_{2}, T_{3}$ be the normalized basis of the model, so $[T_{1}, T_{2}] = T_{3}$ and $κ (T_{3}, T_{3}) = 1$ . For a constant $a \neq 0$ with unit $m^{- 1}$ , take the smooth connection potential $A = a (T_{1} d x^{1} + T_{2} d x^{2})$ . Its curvature is $F = a^{2} T_{3} d x^{1} \land d x^{2}$ , and its Lorentzian action density is $- \frac{a^{4}}{2 g_{Y M}^{2}} d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ . This is a field configuration; no stationarity or field-equation claim is made.

Example references

David Tong, Gauge Theory, Chapter 2, Yang-Mills Theory

Matthijs Vákár, Principal Bundles and Gauge Theories

Facts & Assumptions

Given: ${A C}_{ω}$ , the trivial-bundle specialization of the adopted pure $S U (2)$ model and the constant $a \neq 0$ .

[F1]

Classical pure SU(2) Yang–Mills model: the spacetime metric is $diag (- 1, 1, 1, 1)$ with the stated orientation.

[A1]

Classical pure SU(2) Yang–Mills model: the action density is $- κ (F \land *_{η} F) / (2 g_{Y M}^{2})$ with fixed $g_{Y M} > 0$ .

[F2]

Use the explicit matrices and trace form fixed in Compactness and normalized invariant form for SU(2): $T_{1} = \frac{1}{2} (\begin{array}{cc} 0 & - i \\ - i & 0 \end{array})$ , $T_{2} = \frac{1}{2} (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ , $T_{3} = \frac{1}{2} (\begin{array}{cc} - i & 0 \\ 0 & i \end{array})$ , and $κ (X, Y) = - 2 {Tr}_{2} (X Y)$ . Direct multiplication gives $T_{1} T_{2} = T_{3} / 2$ , $T_{2} T_{1} = - T_{3} / 2$ , and $T_{3}^{2} = - I_{2} / 4$ . Consequently $[T_{1}, T_{2}] = T_{1} T_{2} - T_{2} T_{1} = T_{3}$ and $κ (T_{3}, T_{3}) = - 2 {Tr}_{2} (- I_{2} / 4) = 1$ . These finite matrix identities are established here without countable choice.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the hypothesis inherited through the adopted model and the Cartan-curvature supplier; the finite matrix identities in [F2] use no further choice.

[F3]

Under the stated ${A C}_{ω}$ , Graded bracket of Lie-algebra-valued differential forms and Principal curvature, Cartan structure equation, and Bianchi identity: local curvature is $F = d A + \frac{1}{2} [A \land A]$ with the alternating-form bracket convention.

Verification

1.1A2F2F3algebra

The coefficients of $A$ are constant, so $d A = 0$ . The degree-one bracket convention gives $\frac{1}{2} [A \land A] (\partial_{1}, \partial_{2}) = [a T_{1}, a T_{2}] = a^{2} T_{3}$ , while its other independent coordinate components vanish. Thus $F = a^{2} T_{3} d x^{1} \land d x^{2}$ . If $a = 0$ , this same formula gives zero curvature; the Statement assumes $a \neq 0$ .

2.1A1F1F2step 1.1algebra∎

For the stated orientation and metric, $*_{η} (d x^{1} \land d x^{2}) = d x^{0} \land d x^{3}$ . Therefore $κ (F \land *_{η} F) = a^{4} κ (T_{3}, T_{3}) d x^{1} \land d x^{2} \land d x^{0} \land d x^{3} = a^{4} d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ ; the last reordering is even. Multiplying by the adopted coefficient gives the stated action density. This verifies only a configuration and its nonzero density; no Euler–Lagrange equation is asserted.

A noncommuting SU(2) configuration with nonzero Yang–Mills density

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources