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Selected classical field topology and optional circle normalization — Examples

1 · Prerequisites

2 · Summary

For the punctured-space field $F_{b} = b \sin θ d θ \land d ϕ$ , the example compares the integer-normalized case $b = κ_{0} / 2$ , a noninteger normalized flux, and the zero-field case. It then distinguishes a flat connection with nontrivial holonomy from an integer flat connection removed by a single-valued $U (1)$ gauge transformation. These are exact classical calculations, not reported observations.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Classical real flux versus selected circle sector

Physical scope

Exact smooth two-form, fixed-normalization circle-patching, and flat-holonomy calculations on the specified punctured domains; no assertion of magnetic monopoles in nature or a quantum Aharonov–Bohm observation or charge-quantization rule.

Facts & Assumptions

Given: The exact punctured-domain and flat-connection cases of Local potentials and explicit global obstruction in the model, with chosen $κ_{0} > 0$ measured in webers. The example values below are constructed model parameters, not observations.

[F1]

Local potentials and explicit global obstruction in the model states that $F_{b} = b \sin θ d θ \land d ϕ$ on $M_{*} = R_{x^{0}} \times (R^{3} ∖ {0})$ has flux $4 π b$ , no global potential when $b \neq 0$ , and a fixed- $κ_{0}$ north/south connection transition exactly when $2 b / κ_{0} \in Z$ . It also states that $A = κ_{0} a d ϕ$ on $R_{x^{0}} \times S^{1} \times R^{2}$ has zero curvature and holonomy $e^{- 2 π i a}$ .

Example

Use the oriented sphere field $F_{b} = b \sin θ d θ \land d ϕ$ on the punctured spacetime of [F1]. For $b = κ_{0} / 2$ , its flux is $2 π κ_{0}$ and its transition map is $e^{- i ϕ}$ . This map is single-valued on the equatorial circle and has winding number of absolute value one. For $b = κ_{0} / (2 \sqrt{2})$ , the real field is still smooth and closed on the punctured domain, but $2 b / κ_{0} = 1 / \sqrt{2}$ is not an integer, so these north/south potentials do not define a connection with that fixed circle normalization. For $b = 0$ , the field and the chosen connection can both be zero.

Separately, on $R_{x^{0}} \times S^{1} \times R^{2}$ , the normalized flat connection $a d ϕ$ has holonomy $e^{- 2 π i a}$ . At $a = 1 / 2$ this is $- 1$ , so the connection is not gauge trivial; at $a = 1$ the single-valued map $e^{- i ϕ}$ removes it. The distinction is a classical statement about connection data and does not use a quantum wavefunction.

Verification

1.1F1algebra

For $b = κ_{0} / 2$ , the flux formula in [F1] gives $4 π (κ_{0} / 2) = 2 π κ_{0}$ , and the transition coefficient is $2 b / κ_{0} = 1$ , so $g_{N S} (ϕ) = e^{- i ϕ}$ returns to $1$ after a full $2 π$ turn. Its degree is $- 1$ for the stated north-to-south orientation, hence its absolute winding is one.

1.2F1algebra

For $b = κ_{0} / (2 \sqrt{2})$ , the normalized flux number is $2 b / κ_{0} = 1 / \sqrt{2}$ , which lies strictly between $0$ and $1$ and therefore is not an integer. The endpoint value $g_{N S} (2 π) = e^{- 2 π i / \sqrt{2}}$ is not $g_{N S} (0) = 1$ , so this patch data do not define the selected circle connection; the real closed two-form itself remains admitted in the classical sector of [F1].

1.3F1algebra

If $b = 0$ , then $F_{b} = 0$ , $A_{N} = A_{S} = 0$ , the transition is the constant map $1$ , and the global potential $A = 0$ exists. Thus the nonzero-flux obstruction in [F1] does not apply at this endpoint.

2.1F1algebraconstruct∎

For the flat case $A = κ_{0} a d ϕ$ , substituting $a = 1 / 2$ into [F1] gives holonomy $e^{- π i} = - 1$ , which differs from the zero connection's holonomy $1$ and is invariant under single-valued gauge transformations. At $a = 1$ , the holonomy is $1$ and $h (ϕ) = e^{- i ϕ}$ is single-valued because $h (ϕ + 2 π) = h (ϕ)$ ; its local lift changes the connection by $- d ϕ$ , removing $d ϕ$ . Therefore zero curvature need not mean gauge triviality, while this integer flat connection is gauge trivial.