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Exact background scalar plane wave and derived current

Example

On $R^{4}$ with $x^{0} = c t$ , Minkowski metric $η = diag (- 1, 1, 1, 1)$ , and trivial bundle, set $A = 0$ , $V (u) = m_{s}^{2} u$ with $m_{s} > 0$ in $m^{- 1}$ , and

$φ (t, x) = C \exp (- i ω t + i k \cdot x),$

where $C \neq 0$ is dimensionless, $k \in R^{3}$ has units $m^{- 1}$ , and $ω : = c \sqrt{∣ k ∣^{2} + m_{s}^{2}} > 0$ has units $s^{- 1}$ . For the scalar action of Classical complex-scalar connection derivative and action, this is an exact solution of the local background scalar equation, equivalently stationary under compactly supported scalar variations. The infinite-volume action value need not be finite; only the local equation and compactly supported variations are used. If the coupling $λ \neq 0$ , the same configuration has a nonzero derived current although $F = d A = 0$ , so it fails the sourced Maxwell equation of Sourced Maxwell sign and current conservation. It is therefore a background scalar solution, not a solution of the coupled scalar–Maxwell system. No relation to $ℏ$ or empirical data is used.

Facts & Assumptions

Given: The flat metric and trivial connection above, the scalar model of Classical complex-scalar connection derivative and action, the scalar Euler–Lagrange equation and current formula of Classical charged-scalar action, current and stress variation, and the planned sourced Maxwell sign convention of Sourced Maxwell sign and current conservation.

[F1]

For this model, $E_{φ} = D_{a} D^{a} φ - V^{'} (∣ φ ∣^{2}) φ$ and $j^{a} = K_{φ} λ Im (\overline{φ} D^{a} φ)$ . (Classical charged-scalar action, current and stress variation)

[F2]

The sourced Maxwell equation is $\partial_{a} F^{a b} = - μ_{0} j^{b}$ in this inertial chart, with $μ_{0} > 0$ . (Sourced Maxwell sign and current conservation)

[A1]

The constants satisfy $c > 0$ , $K_{φ} > 0$ , $m_{s} > 0$ , $C \neq 0$ , and $ω = c \sqrt{∣ k ∣^{2} + m_{s}^{2}}$ . The example is a mathematical background-field calculation.

[A2]

The in-run geometric supplier chain carries the ${A C}_{ω}$ convention stated in The Axiom of Countable Choice ( ${A C}_{ω}$ ); this calculation makes no further choice.

Verification

1.1A1A2F1algebra

Since $A = 0$ , $D_{a} = \partial_{a}$ and $D_{a} D^{a} = - c^{- 2} \partial_{t}^{2} + Δ$ . The plane wave has $\partial_{t}^{2} φ = - ω^{2} φ$ and $Δ φ = - ∣ k ∣^{2} φ$ , so $E_{φ} = (ω^{2} / c^{2} - ∣ k ∣^{2} - m_{s}^{2}) φ = 0$ by [A1].

1.2A1F1algebra

The current formula [F1] gives $j^{0} = K_{φ} λ (ω / c) ∣ C ∣^{2}$ and $j^{i} = K_{φ} λ k_{i} ∣ C ∣^{2}$ . Thus $j = 0$ when $λ = 0$ , while for $λ \neq 0$ its temporal component is nonzero because $K_{φ}, ω, ∣ C ∣^{2}$ are positive.

2.1A1F2step 1.2algebra∎

In either case $F = d A = 0$ and hence $\partial_{a} F^{a 0} = 0$ . If $λ \neq 0$ , [F2] would require $0 = - μ_{0} j^{0}$ although $μ_{0} j^{0} \neq 0$ by step 1.2. Therefore the background scalar solution with nonzero coupling does not solve the sourced Maxwell equation; when $λ = 0$ , this particular current obstruction is absent.

Remarks

The M02/M03 suppliers are current draft carriers used only at their inspected statements. The seven typed foundational replacement carriers in the dependency closure are now authored on emg-mg02; no independent acceptance is asserted: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas.

Exact background scalar plane wave and derived current

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources