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A nonzero pure-gauge potential with zero Yang–Mills density

Physical scope

A section-change example for the classical pure SU(2) Yang–Mills model on the trivial bundle over Minkowski spacetime. It is a classical gauge description, not an observation or quantum claim.

Example

Specialize the adopted model to the trivial bundle $P = R^{1, 3} \times S U (2)$ with canonical section $s$ . Let $T_{3} = - i σ_{3} / 2$ and choose a length $ℓ > 0$ . Change section to $s^{u} (x) = s (x) u (x)$ with $u (x) = \exp (T_{3} \sin (x^{0} / ℓ))$ . The connection whose potential in $s$ is $A = 0$ has potential in $s^{u}$ $A^{u} = u^{- 1} d u = T_{3} ℓ^{- 1} \cos (x^{0} / ℓ) d x^{0}$ , which is nonzero. Its Cartan curvature and Yang–Mills action density are zero. This is a change of gauge description of a classical connection, not a distinct physical observation.

Example references

Matthijs Vákár, Principal Bundles and Gauge Theories

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

Facts & Assumptions

Given: The adopted pure $S U (2)$ model specialized to its trivial bundle, canonical section, and length parameter $ℓ > 0$ .

[F1]

Classical pure SU(2) Yang–Mills model: each model instance may select the trivial bundle $P = R^{1, 3} \times S U (2)$ , and the local curvature convention is $F = d A + \frac{1}{2} [A \land A]$ .

[F2]

Compactness and normalized invariant form for SU(2): $T_{3}$ lies in $s u (2)$ and $[T_{3}, T_{3}] = 0$ .

[F3]

Graded bracket of Lie-algebra-valued differential forms and Principal curvature, Cartan structure equation, and Bianchi identity: if two local sections satisfy $s_{j} = s_{i} g_{i j}$ , then $A_{j} = {Ad}_{g_{i j}^{- 1}} A_{i} + g_{i j}^{- 1} d g_{i j}$ and $F_{j} = d A_{j} + \frac{1}{2} [A_{j} \land A_{j}]$ .

[A1]

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

Verification

1.1F1F2F3algebra

The section-transition law in [F3], applied to $s^{u} = s u$ and $A = 0$ , gives $A^{u} = u^{- 1} d u$ . Since $u = \exp (T_{3} \sin (x^{0} / ℓ))$ and $T_{3}$ commutes with itself, differentiation yields $u^{- 1} d u = T_{3} ℓ^{- 1} \cos (x^{0} / ℓ) d x^{0}$ . If the phase were constant, the same formula would give the zero potential; the chosen sine has nonzero derivative, so $A^{u}$ is nonzero.

2.1A1F1F2F3step 1.1algebra∎

The coefficient of $A^{u}$ depends only on $x^{0}$ and its one-form factor is $d x^{0}$ , so $d A^{u} = 0$ . Every value of $A^{u}$ is proportional to $T_{3}$ , hence $[A^{u} \land A^{u}] = 0$ by [F2]. The Cartan formula in [F3] gives $F^{u} = 0$ , and therefore $κ (F^{u} \land *_{η} F^{u}) = 0$ . The asserted zero density is exact for this classical field.

A nonzero pure-gauge potential with zero Yang–Mills density

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources