Fixed-mode number law

[Oscillator coefficients] For $e_m$ supported in coordinate $m$, the raising formula of [F3] has its only nonzero coordinate at $m+1$, with value $\sqrt{m+1}$, so $A_{+}(1)e_m=\sqrt{m+1}{e}_{m+1}$. For $m\ge 1$ the lowering formula has its only nonzero coordinate at $m-1$, with value $\sqrt{m}$, so $A_{-}(1)e_m=\sqrt{m}{e}_{m-1}$; for $m=0$ every lowering coordinate is zero. This proves (a).

[Convergence and normalization] Let $z\in \mathbb{C}$ and $S_N:=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}{\sum }_{n\le N}{z}^n\mathrm{/}\sqrt{n!}{e}_n$. Orthogonality of the $e_n$ gives $\mathrm{\parallel }{S}_N-S_M{\mathrm{\parallel }}^2=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2}{\sum }_{M<n\le N}\mathrm{\mid }z{\mathrm{\mid }}^{2n}\mathrm{/}n!$, and ${\sum }_n\mathrm{\mid }z{\mathrm{\mid }}^{2n}\mathrm{/}n!=\exp (\mathrm{\mid }z{\mathrm{\mid }}^2)$ by [F4], so $(S_N)$ is Cauchy and converges in the complete space $F(\mathbb{C})$ to a vector ${\psi }_z$; moreover $\mathrm{\parallel }{\psi }_z{\mathrm{\parallel }}^2={\lim }_N\mathrm{\parallel }{S}_N{\mathrm{\parallel }}^2=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2}\exp (\mathrm{\mid }z{\mathrm{\mid }}^2)=1$, so $\mathrm{\parallel }{\psi }_z\mathrm{\parallel }=1$. For $z=0$ only the zeroth term survives, so convergence is absolute. For $z\mathrm{\ne }0$, convergence is absolute because the ratio of consecutive moduli of ${\sum }_n\mathrm{\mid }z{\mathrm{\mid }}^n\mathrm{/}\sqrt{n!}$ is $\mathrm{\mid }z\mathrm{\mid }\mathrm{/}\sqrt{n+1}\to 0$, so that series converges by the ratio test (Ratio test: $\mathrm{lim\hspace{0.17em}sup}\mathrm{\mid }{a}_{k+1}\mathrm{/}{a}_k\mathrm{\mid }<1$ gives absolute convergence and hence convergence, and $\mathrm{lim\hspace{0.17em}inf}\mathrm{\mid }{a}_{k+1}\mathrm{/}{a}_k\mathrm{\mid }>1$ gives divergence); and ${\sum }_n\mathrm{\mid }z{\mathrm{\mid }}^{2n}\mathrm{/}n!=\exp (\mathrm{\mid }z{\mathrm{\mid }}^2)$ is the exponential series of the real number $\mathrm{\mid }z{\mathrm{\mid }}^2\ge 0$ ([F4]).

[Annihilation eigenvalue] Every finite combination $S_N\in F_0$ lies in the domain of the closed operator $A_{-}(1)$, and $A_{-}(1)S_N=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}{\sum }_{1\le n\le N}{z}^n\mathrm{/}\sqrt{n!}\sqrt{n}{e}_{n-1}=z{e}^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}{\sum }_{0\le m\le N-1}{z}^m\mathrm{/}\sqrt{m!}{e}_m\to z{\psi }_z$ by step 1.2. Since $S_N\to {\psi }_z$ and $A_{-}(1)$ is closed, ${\psi }_z\in D(A_{-}(1))$ and $A_{-}(1){\psi }_z=z{\psi }_z$.

[Overlaps and the number law] For $z,w\in \mathbb{C}$, the series ${\psi }_z$ and ${\psi }_w$ have absolutely convergent coordinate expansions, so the inner product may be computed termwise: $⟨{\psi }_z,{\psi }_w⟩=e^{-(\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2)\mathrm{/}2}{\sum }_n{z}^n\overline{w^n}\mathrm{/}n!=e^{-(\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2)\mathrm{/}2+z\bar{w}}$, the last equality by the exponential series [F4] and its addition law [F5]. In particular $\mathrm{\parallel }{\psi }_z{\mathrm{\parallel }}^2=⟨{\psi }_z,{\psi }_z⟩=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }z{\mathrm{\mid }}^2}=1$. Since $\mathrm{\mid }⟨e_n,{\psi }_z⟩{\mathrm{\mid }}^2=\mid e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}{\bar{z}}^n\mathrm{/}\sqrt{n!}{\mid }^2=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2}\mathrm{\mid }z{\mathrm{\mid }}^{2n}\mathrm{/}n!$ and ${\sum }_n\mathrm{\mid }z{\mathrm{\mid }}^{2n}\mathrm{/}n!=\exp (\mathrm{\mid }z{\mathrm{\mid }}^2)$, the weights in (c) sum to $1$.

[Conclusion and scope] Steps 1.1-2.2 prove (a), (b) and (c); the square roots $\sqrt{n!}$ and $\sqrt{n}$ are the positive real roots. The final statement records that no measurement postulate, and hence no probability interpretation of the displayed weights, is asserted; the Fock vectors and the eigenvalue relation are purely mathematical.