Alternative action formulation — Examples

Assume Countable Choice, inherited from the field-action measure suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no additional choice is used. Work in Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional with the coordinate box $U=(-1,1{)}^4$ lying compactly inside the chosen laboratory chart, constants $c,{\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values) and $j=0$.

(a) [Coordinate and physical forms] With $x^0=ct$, set $U_t:=\{(t,x):(ct,x)\in U\}=(-1\mathrm{/}c,1\mathrm{/}c)\times (-1,1{)}^3$. The exact field dictionary gives

$S[A]=\frac{1}{2c{\mu }_0}{\int }_U(\mathrm{\mid }E{\mathrm{\mid }}^2\mathrm{/}{c}^2-\mathrm{\mid }B{\mathrm{\mid }}^2)d\lambda =\frac{1}{2}{\int }_{U_t}({\epsilon }_0\mathrm{\mid }E{\mathrm{\mid }}^2-\mathrm{\mid }B{\mathrm{\mid }}^2\mathrm{/}{\mu }_0)dt{d}^{3}x\mathrm{.}$

Also $S[\lambda A]={\lambda }^{2}S[A]$ for every real $\lambda$.

(b) [Stationarity] $A=0$ is stationary: the actual first-variation coefficient of Compact variations of the quadratic field functional: first variation and the coefficient equation is $-{\mu }_0^{-1}{F}^{\mu \nu }{\mathrm{\partial }}_{\mu }{h}_{\nu }+j^{\nu }{h}_{\nu }$, and the action variation is $c^{-1}$ times its integral. Both terms vanish at $F=0$, $j=0$.

(c) [Positive direction] Choose $\theta \in C_c^{\mathrm{\infty }}(U)$ with ${\mathrm{\nabla }}_x\theta \equiv ̸0$. Set the covariant components $h_0=\theta$, $h_i=0$, equivalently the contravariant variation $h^0=-\theta$, $h^i=0$. Then $E=c{\mathrm{\nabla }}_x\theta$, $B=0$, and

$S[h]=\frac{1}{2c{\mu }_0}{\int }_U\mathrm{\mid }{\mathrm{\nabla }}_x\theta {\mathrm{\mid }}^{2}d\lambda >0.$

(d) [Negative directions with a fixed compact support] Choose nonzero nonnegative smooth bumps $\eta ,\phi ,{\psi }_2,{\psi }_3\in C_c^{\mathrm{\infty }}((-1\mathrm{/}2,1\mathrm{/}2))$ using Test function cutoffs and euclidean localization. Extend them by zero to $\mathbb{R}$. For integers $N\ge 1$ define the contravariant variation with only its second spatial component nonzero:

$h^{(N)2}(x)=\eta (x^0)\phi (N{x}^1){\psi }_2(x^2){\psi }_3(x^3)\mathrm{.}$

All supports lie in one fixed compact subset of $U$. Writing $f_N=h^{(N)2}$, its fields are $E=-c({\mathrm{\partial }}_0{f}_N)e_2$ and $B=-({\mathrm{\partial }}_3{f}_N)e_1+({\mathrm{\partial }}_1{f}_N)e_3$. Define finite constants, with all one-dimensional integrals over $(-1,1)$,

$C_E=(\int {\eta }^{\mathrm{\prime }2})(\int {\phi }^2)(\int {\psi }_2^2)(\int {\psi }_3^2),C_B=(\int {\eta }^2)(\int {\phi }^{\mathrm{\prime }2})(\int {\psi }_2^2)(\int {\psi }_3^2)>0,$

$C_3=(\int {\eta }^2)(\int {\phi }^2)(\int {\psi }_2^2)(\int {\psi }_3^{\mathrm{\prime }2})\ge 0.$

Exact substitution and Fubini give

${\int }_U\mathrm{\mid }E{\mathrm{\mid }}^2\mathrm{/}{c}^{2}d\lambda =C_E\mathrm{/}N,{\int }_U\mathrm{\mid }B{\mathrm{\mid }}^{2}d\lambda =C_{B}N+C_3\mathrm{/}N,$

$S[h^{(N)}]=\frac{1}{2c{\mu }_0}(\frac{C_E-C_3}{N}-C_{B}N)⟶-\mathrm{\infty }\mathrm{.}$

Thus a fixed sufficiently large $N$ supplies one negative direction $h_{-}$; the direction $h_{+}$ of (c) is positive. For nonzero $\lambda$, their rescalings keep their respective signs, and each action tends to zero in absolute value as $\lambda \to 0$. A single fixed negative direction is not claimed to change sign under rescaling.

(e) [Every usual neighbourhood] Fix those two directions before rescaling. For each derivative multi-index, every compact-set supremum of a derivative of $\lambda h_{\pm }$ is $\mathrm{\mid }\lambda \mathrm{\mid }$ times the finite supremum for that fixed smooth compactly supported direction. Thus both tend to zero in the usual $C^k$, smooth compact-convergence and fixed-support and LF test-function topologies (Fixed support test function frechet space, Test function topology) (and in any usual function-space topology with continuous scalar multiplication). Every such neighbourhood of zero therefore contains configurations of positive and negative action, arbitrarily small in magnitude. The stationary zero is neither a local minimum nor a local maximum. This is a computation in the declared smooth finite-region model, not a claim about actual fields or measurements; other sources or boundary/variation restrictions require their own analysis.

Landau–Lifshitz §27 (printed pp. 68-69, with eqs. (27.4)-(27.6) on p. 69) ties the sign of $(\mathrm{\partial }A\mathrm{/}\mathrm{\partial }t{)}^2$ to a minimum requirement for $S_f$; the exact computation above shows that with the framework's adopted sign and normalisation the stationary point of the source-free box functional is a saddle rather than a minimum, so the source's minimum wording is not adopted. Tong §5.5.3 (printed p. 125) expands the action to $\int dt{d}^{3}x({\epsilon }_0{E}^2\mathrm{/}2-B^2\mathrm{/}(2{\mu }_0))$ and reads the electric term as kinetic and the magnetic term as potential; the explicit compact variations of this example are the framework's own witnesses and are not a numbered result of either source.

SI numerical values. Throughout this example, a symbol for a physical quantity denotes its real numerical value in the following fixed SI unit: $x^{\mu },u$ in m, $t$ in s, $c$ in m s${}^{-1}$, ${\mu }_0$ in N A${}^{-2}$, ${\epsilon }_0$ in F m${}^{-1}$, $A^{\mu },A_{\mu },h^{\mu },h_{\mu }$ in T m, $F_{\mu \nu },F^{\mu \nu },B$ in T, $E$ in V m${}^{-1}$, $\rho$ in C m${}^{-3}$, $j^{\mu },J$ in A m${}^{-2}$, the auxiliary gauge factors ${\phi }_{\pm }$ and gauge functions ${\chi }_{\pm },\chi$ in T m${}^2$, and $S$ in J s; $s,\zeta ,\psi$ are dimensionless. Precisely, if $q$ represents a quantity with unit $u_q$, its physical field is $q_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(X)=u_{q}q(X\mathrm{/}(1\mathrm{m}))$, with the same rule componentwise and $q_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=u_{q}q$ for constants. The box $U=(-1,1{)}^4$ thus represents the physical coordinate box $(-1\mathrm{m},1\mathrm{m}{)}^4$. All derivatives and integrals below are with respect to these numerical coordinates, so $F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(1\mathrm{T})F$ and ${\mathrm{\partial }}^{\mathrm{♯}}{\chi }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(1\mathrm{T}\mathrm{m}){\mathrm{\partial }}^{\mathrm{♯}}\chi$. The coherent SI units make the numerical action formula exactly that of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional, with $S_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}[A_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}]=(1\mathrm{J}\mathrm{s})S[A]$; the facts below use the same numerical representations of the cited quantity formulas.

Assume Countable Choice, inherited from the cited theorems. Work in the field-action setting of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional with $U=(-1,1{)}^4$, constants $c,{\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values), and the component dictionary of The electromagnetic two-form, its observer decomposition and the four-current: $F_{0i}=-E_i\mathrm{/}c$, $F^{0i}=E_i\mathrm{/}c$, $F^{10}=-E_1\mathrm{/}c$.

(a) [Source-free zero field] For $j\equiv 0$ and $A=0$ the first-variation coefficient (whose integral is multiplied by $c^{-1}$) $-{\mu }_0^{-1}{F}^{\mu \nu }{\mathrm{\partial }}_{\mu }{h}_{\nu }+j^{\nu }{h}_{\nu }$ of clause (a) of Compact variations of the quadratic field functional: first variation and the coefficient equation vanishes identically for every $h\in C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$, because $F(A)=0$ and $j=0$; hence every compact first variation of $S$ at $A=0$ is zero and $A=0$ is stationary.

(b) [A non-trivial compact test variation] Let $A$ be the smooth potential with $A_0(x)=x^1$ and $A_1=A_2=A_3=0$, and let $j\equiv 0$. Then $F_{01}=-{\mathrm{\partial }}_1{A}_0=-1$, $F^{01}=+1$, $F^{10}=-1$ and all other components vanish, so the represented field is $E=(c,0,0)$, $B=0$ (clause (a) of The electromagnetic two-form, its observer decomposition and the four-current) and ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=0$ on $U$. In quantities this is $A_{0,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(X)=(1\mathrm{T})X^1$ and $E_{1,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=c_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(1\mathrm{T})$, in T m and V m${}^{-1}$, respectively. By the equivalence of Action stationarity implies the inhomogeneous Maxwell equations, $A$ is stationary; directly, for every $h\in C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$ the first variation of $S$ at $A$ is

$\frac{d}{ds}{\mid }_{s=0}S[A+sh]=-\frac{1}{c{\mu }_0}{\int }_U({\mathrm{\partial }}_0{h}_1-{\mathrm{\partial }}_1{h}_0)d\lambda =\frac{1}{c{\mu }_0}{\int }_U({\mathrm{\partial }}_1{h}_0-{\mathrm{\partial }}_0{h}_1)d\lambda =0,$

where each of ${\int }_U{\mathrm{\partial }}_0{h}_{1}d\lambda$ and ${\int }_U{\mathrm{\partial }}_1{h}_{0}d\lambda$ vanishes by the fundamental theorem in one variable combined with Fubini, and no boundary term appears because $h$ is compactly supported in $U$ (Newton–Leibniz needs only continuity on $[a,b]$, differentiability on $(a,b)$, and a Riemann-integrable extension of the interior derivative, Fubini's theorem for L^1 functions on a sigma-finite product, The Lebesgue integral is linear on $L^1(\mu )$).

(c) [Gauge-source sign check with actual overlap] Independently choose the smooth prescribed source $j^0=c\rho (x^0)$, $j^i=0$, with nonzero nonnegative $\rho \in C_c^{\mathrm{\infty }}((-1\mathrm{/}4,1\mathrm{/}4))$ extended by zero. Choose $0\le \zeta \in C_c^{\mathrm{\infty }}((-1,1))$ with $\zeta =1$ near $[-1\mathrm{/}2,1\mathrm{/}2]$, and nonzero nonnegative $\psi \in C_c^{\mathrm{\infty }}((-1,1{)}^3)$, supplied by Test function cutoffs and euclidean localization. Put ${\phi }_{\pm }(u)=(2\pm u)\zeta (u)$ and ${\chi }_{\pm }(x)={\phi }_{\pm }(x^0)\psi (x^1,x^2,x^3)$. Both ${\phi }_{\pm }$ are nonnegative compact bumps, but they are not globally monotone: on the entire support of $\rho$ their derivatives equal $+1$ and $-1$, respectively. Use the contravariant gauge variations $({\mathrm{\partial }}^{\mathrm{♯}}{\chi }_{\pm }{)}^{\mu }={\eta }^{\mu \nu }{\mathrm{\partial }}_{\nu }{\chi }_{\pm }$ from Compact gauge variation is boundary-free for conserved source. Then, for every smooth potential $A$ on a neighbourhood of $\bar{U}$,

$S[A+{\mathrm{\partial }}^{\mathrm{♯}}{\chi }_{\pm }]-S[A]=\left({\int }_{-1}^1\rho {\phi }_{\pm }^{\mathrm{\prime }}d{x}^0\right)\left({\int }_{(-1,1{)}^3}\psi d^{3}x\right)=\pm (\int \rho )(\int \psi ),$

which is strictly positive for $+$ and strictly negative for $-$, since both factors are positive. Integration by parts gives exactly the same value as $-c^{-1}{\int }_U({\mathrm{\partial }}_{\mu }{j}^{\mu }){\chi }_{\pm }d\lambda$ with ${\mathrm{\partial }}_{\mu }{j}^{\mu }=c{\rho }^{\mathrm{\prime }}$. This verifies the relative sign of the $+j^{\mu }{A}_{\mu }$ coupling, the raised-vector/covariant-lowered gauge shift and the conservation condition, using explicitly overlapping supports. If instead $\rho$ is constant on the whole neighbourhood, the source is conserved and every compact gauge difference is zero. No compact nonzero globally increasing or decreasing function is used.

(d) [Scope] The example is an exact computation in the declared smooth finite-region model; it asserts no existence of sources or fields, no numerical value of ${\mu }_0$, and no global gauge statement.