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The binomial family of the splitting pullback is multiplicative and bracket-preserving, and its logarithm is a derivation

Statement

Let $d \geq 1$ , let $(U, U_{ρ}, \tilde{A}, \tilde{B}, M, K, δ, Ψ_{h})$ be analytic splitting data, and use all notation of The truncated binomial family $T_{h}^{s, N}$ , the truncated logarithm $L_{N} (h)$ , its coefficient fields $X_{j}$ and the truncations $X^{[N]}$ : the domains $D (r)$ and $D_{k}$ , the parameters $b = ρ / (8 N)$ , $r_{N} = ρ / (C_{0} M N)$ , the binomial coefficients $(\binom{s}{k})$ , the difference operator $Δ_{h}$ , the truncated family $T_{h}^{s, N}$ and the truncated logarithm $L_{N} (h)$ . Let $f, g$ be holomorphic on $D_{0}$ , and put $E : = D (ρ / 4)$ . For every $Z \in D_{0}$ and finite $k$ , joint local holomorphy of $Ψ_{h}$ and $Ψ_{0} = i d$ give a neighborhood of $(0, Z)$ on which the first $k$ iterates are defined and stay in $D_{0}$ . Define $[h^{j}] (Δ_{h}^{k} f) (Z)$ from this local holomorphic family. Compatible flow germs make these coefficients independent of the neighborhood; coefficient extraction makes them holomorphic on $D_{0}$ . On $E$ they agree with the uniform construction at order $Q_{j} : = \max {2, j}$ for $k \leq j$ , where $∣ h ∣ < r_{Q_{j}}$ and $E \subseteq D_{k}$ . Whenever $D_{j}$ denotes a distance domain below, allocate it with order $Q_{j}$ ; thus $D_{j} \subseteq D_{0}$ even for $j = 0, 1$ . For $s \in C$ and $Z \in D_{0}$ define the coefficient polynomial

$P_{j} (s) f : = \sum_{k = 0}^{j} (\binom{s}{k}) [h^{j}] (Δ_{h}^{k} f),$

where $[h^{j}] (Δ_{h}^{k} f) : = \frac{1}{j!} \partial_{h}^{j} (Δ_{h}^{k} f) ∣_{h = 0}$ , and $P_{j} (s) f$ is a polynomial of degree at most $j$ in $s$ with coefficients in the holomorphic functions on $D_{0}$ .

Then:

Integer case. For all integers $Q \geq 2$ , $0 \leq m \leq Q$ and every $∣ h ∣ \leq r_{Q}$ , $T_{h}^{m, Q} f = f \circ Ψ_{h}^{m}$ (omit the zero summands with $k > m$ to extend the finite sum) on the domain $D_{m}$ allocated with order $Q$ .
Multiplicativity and bracket identity. For every $j \in N$ and every $s \in C$ , as identities of holomorphic functions on $D_{j}$ , $P_{j} (s) (f g) = \sum_{j_{1} + j_{2} = j} P_{j_{1}} (s) f \cdot P_{j_{2}} (s) g, P_{j} (s) {f, g} = \sum_{j_{1} + j_{2} = j} {P_{j_{1}} (s) f, P_{j_{2}} (s) g} .$
The logarithm is a derivation. Put $D_{j} f : = P_{j}^{'} (0) f \in O (D_{0})$ (the derivative in $s$ at $s = 0$ ), so that $D_{j} f = [h^{j}] (L (h) f)$ for the formal logarithm $L (h) : = \sum_{k \geq 1} \frac{(- 1)^{k + 1}}{k} Δ_{h}^{k}$ . Then $D_{0} = 0$ , each $D_{j}$ is a $C$ -linear local differential operator on holomorphic functions, and $D_{j} (f g) = D_{j} f \cdot g + f \cdot D_{j} g, D_{j} {f, g} = {D_{j} f, g} + {f, D_{j} g}$ as functions on $D_{j}$ ; equivalently the coefficient of $h^{j}$ in $L (h) (f g)$ and in $L (h) {f, g}$ obeys the product rule and the bracket derivation rule. Moreover $D_{j} f$ for $h$ -degree $j \leq N$ agrees with the $j$ -th coefficient of the concrete holomorphic function $h \mapsto L_{N} (h) f$ restricted to $∣ h ∣ \leq r_{N}$ whenever $j \leq N$ .

Moreover $D_{j} f = \sum_{i} (D_{j} z_{i}) \partial_{z_{i}} f$ for $j \geq 1$ : each coefficient derivation is the coordinate vector field used in the later Hamiltonian construction.

Facts & Assumptions

Given: The analytic splitting data and all notational conventions of The truncated binomial family $T_{h}^{s, N}$ , the truncated logarithm $L_{N} (h)$ , its coefficient fields $X_{j}$ and the truncations $X^{[N]}$ ; holomorphic $f, g$ on $D_{0}$ .

[L1]

Binomial algebra: the polynomial identity $(1 + x)^{m} = \sum_{k = 0}^{m} (\binom{m}{k}) x^{k}$ for $m \in N$ holds for the single operator $Δ_{h}$ in the algebra of operators on functions holomorphic on the appropriate domains, because all powers of $Δ_{h}$ commute; $(\binom{m}{k}) = 0$ for integers $k > m$ ; and $\frac{d}{d s} (\binom{s}{k}) ∣_{s = 0} = \frac{(- 1)^{k + 1}}{k}$ for $k \geq 1$ with value $0$ for $k = 0$ (The truncated binomial family $T_{h}^{s, N}$ , the truncated logarithm $L_{N} (h)$ , its coefficient fields $X_{j}$ and the truncations $X^{[N]}$ , The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ).

[L2]

Pullback calculus: $(f \circ Ψ_{h}^{m}) = (f \circ Ψ_{h}) \circ Ψ_{h}^{m - 1}$ , and ${f \circ Ψ_{h}^{m}, g \circ Ψ_{h}^{m}} = {f, g} \circ Ψ_{h}^{m}$ for holomorphic $f, g$ , by the chain rule and Hamiltonian flows and the exact splitting map preserve the complexified canonical form and the Poisson bracket clause 3; all maps involved are holomorphic (Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits, Complex phase domains, holomorphic maps on $C^{m}$ , and the complexified canonical form and bracket).

[L3]

Domains: $D_{j} \subseteq D_{j^{'}}$ for fixed truncation order and $j \geq j^{'}$ , and $r_{j} \leq r_{j^{'}}$ for $j \geq j^{'}$ ; $D_{h}^{k} f$ is holomorphic on $D_{k}$ for $∣ h ∣ \leq r_{k}$ (The difference operator $Δ_{h} = Ψ_{h}^{*} - i d$ loses a controlled domain margin and is bounded by $C_{d} M ∣ h ∣ / b$ ).

[L4]

Holomorphic functions of $h$ : if $h \mapsto G (h)$ is holomorphic for $∣ h ∣ \leq r$ and $A (h) = B (h)$ for all such $h$ , then all Taylor coefficients at $0$ agree; and a polynomial in $s$ vanishing at $s = 0, 1, 2, \dots$ vanishes identically (Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits clause 3, Iterated Cauchy formula, power-series expansion and Cauchy estimates for functions continuous and separately holomorphic on a polydisc).

[L5]

Chain rule as applied on this page: the pullback of a bracket and of a product of holomorphic functions is computed by differentiating the composite map, so the complexified bracket ${u, v}$ of Complex phase domains, holomorphic maps on $C^{m}$ , and the complexified canonical form and bracket is transformed by the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , A phase domain, its canonical one-form and its standard symplectic two-form).

Proof

technique · direct; identify the integer family with pullbacks, compare Taylor coefficients in $h$, then use polynomiality in $s$ and differentiate at $s=0$

1.1givenL1L2L3

Local coefficients and integer case. On any sufficiently small common neighborhood, write $Ψ_{h} (Z) = Z + h R (h, Z)$ with holomorphic $R$ . If a holomorphic family $G (h, Z)$ is $O (h^{k})$ , then $G (h, Ψ_{h} (Z)) - G (h, Z)$ is $O (h^{k + 1})$ : factor $h^{k}$ and integrate the spatial derivative of the remaining family along the short segment from $Z$ to $Ψ_{h} (Z)$ . Induction therefore gives $Δ_{h}^{k} f = O (h^{k})$ locally. We prove $T_{h}^{m, M} f = f \circ Ψ_{h}^{m}$ for $0 \leq m \leq M$ and $∣ h ∣ \leq r_{M}$ by induction on $m$ . For $m = 0$ , $T_{h}^{0, M} f = (\binom{0}{0}) f = f$ . If it holds for $m$ , then by [L1] $i d + Δ_{h} = Ψ_{h}^{*}$ on the relevant domains and $(1 + Δ_{h})^{m}$ applied to $f$ equals $f \circ Ψ_{h}^{m}$ ; applying $i d + Δ_{h}$ once more gives $(f \circ Ψ_{h}^{m}) \circ Ψ_{h} = f \circ Ψ_{h}^{m + 1}$ by [L2], while the binomial expansion over $M \geq m + 1$ reproduces $T_{h}^{m + 1, M} f$ because $(\binom{m + 1}{k}) = 0$ for $k > m + 1$ .

2.1givenstep 1.1L2L4L5

Multiplicativity and bracket identities at integer $s$ . For an integer $m$ with $0 \leq m \leq M$ , both $T_{h}^{m, M} (f g)$ and $T_{h}^{m, M} f \cdot T_{h}^{m, M} g$ equal $(f \circ Ψ_{h}^{m}) (g \circ Ψ_{h}^{m})$ on $D_{m}$ by step 1.1; likewise $T_{h}^{m, M} {f, g}$ and ${T_{h}^{m, M} f, T_{h}^{m, M} g}$ both equal ${f, g} \circ Ψ_{h}^{m}$ by step 1.1 and [L2]. At any $Z \in D_{0}$ , the same finite pullback identities hold as local germs for arbitrary fixed integers $m, M$ with $M \geq m$ . Comparing Taylor coefficients in $h$ of order $j \leq M$ on that neighborhood by [L4] and using that $Δ_{h}^{k} f$ contributes only at order $\geq k$ gives, for every $j$ and every integer $m \geq 0$ (choose an integer truncation $M : = \max {2, m, j}$ , distinct from the field bound $M$ in the splitting data), $P_{j} (m) (f g) = \sum_{j_{1} + j_{2} = j} P_{j_{1}} (m) f \cdot P_{j_{2}} (m) g, P_{j} (m) {f, g} = \sum_{j_{1} + j_{2} = j} {P_{j_{1}} (m) f, P_{j_{2}} (m) g} .$

3.1givenstep 2.1L1L4

Polynomial extension. Both sides of each identity of step 2.1 are polynomials in $m$ of degree at most $j$ (for the $j$ -coefficient, products of polynomials of degrees $j_{1}, j_{2}$ with $j_{1} + j_{2} = j$ ), and they agree at every integer $m \geq 0$ ; hence by [L4] they agree for every $s \in C$ . This is clause 2.

4.1givenstep 3.1L1L3L4

Derivative at $s = 0$ . Differentiate the identities of clause 2 in $s$ at $s = 0$ and use [L1]: since $P_{0} (s) f = (\binom{s}{0}) [h^{0}] (Δ_{h}^{0} f) = f$ and $P_{j} (0) f = 0$ for $j \geq 1$ (all coefficients $(\binom{0}{k})$ vanish for $k \geq 1$ ), the product identity becomes $D_{j} (f g) = D_{j} f \cdot g + f \cdot D_{j} g$ , and the bracket identity becomes $D_{j} {f, g} = {D_{j} f, g} + {f, D_{j} g}$ . By [L1] again, $D_{j} f = \sum_{k = 1}^{j} \frac{(- 1)^{k + 1}}{k} [h^{j}] (Δ_{h}^{k} f) = [h^{j}] (L (h) f)$ . Finally, $L (h) \equiv L_{N} (h) m o d h^{N + 1}$ because the two series differ by terms of order $h^{k} \geq h^{N + 1}$ ; hence for $j \leq N$ the coefficients agree, and $h \mapsto L_{N} (h) f$ is holomorphic on $∣ h ∣ \leq r_{N}$ by The difference operator $Δ_{h} = Ψ_{h}^{*} - i d$ loses a controlled domain margin and is bounded by $C_{d} M ∣ h ∣ / b$ clause 3, so its Taylor coefficients are the $D_{j} f$ .

5.1givenstep 1.1step 4.1L4algebra

Coordinate field identification. Fix $Z_{*} \in D_{0}$ . By the integer pullback identity, each $Δ_{h}^{k} f$ is a finite linear combination of $f \circ Ψ_{h}^{m}$ with $0 \leq m \leq k$ ; for fixed $m$ , $Ψ_{h}^{m} (Z_{*})$ is holomorphic in $h$ near $0$ , equals $Z_{*}$ at $h = 0$ , and differs from $Z_{*}$ by $O (h)$ . The Taylor expansion of $f$ at $Z_{*}$ through total degree $j$ has a remainder $O (∥ W - Z_{*} ∥^{j + 1})$ , so replacing $f$ by that Taylor polynomial changes none of the coefficients $[h^{j}] (Δ_{h}^{k} f) (Z_{*})$ for $k \leq j$ . The product derivation rule of step 4.1 gives $D_{j} 1 = 0$ and annihilates at $Z_{*}$ every monomial of degree at least two in $Z - Z_{*}$ : split such a monomial into two factors vanishing at $Z_{*}$ . Only the linear Taylor terms remain, yielding $D_{j} f (Z_{*}) = \sum_{i} D_{j} z_{i} (Z_{*}) \partial_{z_{i}} f (Z_{*})$ . Since $Z_{*}$ was arbitrary, this is the claimed local differential-operator formula on $D_{0}$ .

6.1step 1.1step 3.1step 4.1step 5.1∎

Conclusion. Clause 1 is step 1.1, clause 2 is step 3.1 and clause 3 is step 4.1 together with the $C$ -linearity of $s$ -differentiation at $0$ and of the coefficient extraction.

Remarks

The key point is that the family is a polynomial in $s$ at each $h$ -degree, so the integer- $s$ identities — which are the multiplicativity and bracket naturality of the pullback — extend to all complex $s$ ; differentiating at $s = 0$ then produces the derivation identities used to build the modified Hamiltonian. No convergence of the full series $\sum_{k} (- 1)^{k + 1} Δ_{h}^{k} / k$ is asserted here: only the concrete truncated functions $L_{N} (h) f$ for $∣ h ∣ \leq r_{N}$ and their coefficients of order $\leq N$ are used.
All identities are on complex domains; no real-valuedness or physical interpretation is involved, and no choice principle is used.

The binomial family of the splitting pullback is multiplicative and bracket-preserving, and its logarithm is a derivation

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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