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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.

Depends on

Local potentials and explicit global obstruction in the model

Classical real flux versus selected circle sector

Facts & Assumptions

Example

Verification

Depends on

Dependency tree · 2 levels

Sources