finite.tex
Physics
finite.tex
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1 \section{Finite unit registers and protected coordinates}
2 \label{a-finite-units}
3
4 Adic refinement first changes the address of a state. Its amplitudes remain
5 complex numbers, and its measurement probabilities remain ordinary Born
6 probabilities. For a finite address set $X$ we use the counting inner product
7 on $\ell^2(X;\C)$, so the address vectors $|x\rangle$ are orthonormal.
8 A bijection of address sets therefore gives a permutation unitary without
9 changing the quantum probability rule. Arithmetic determines the address
10 factorization. A Hamiltonian and a measurement prescription supply the
11 additional physical choices.
12
13 The first coordinate issue is specific to units. For an odd prime $p$ and
14 an integer $h\geq0$, set
15 \[
16 U_{p,h}=(\Z/p^{h+1}\Z)^\times.
17 \]
18 For $p=2$ the useful indexing starts one digit later:
19 \[
20 U_{2,h}=(\Z/2^{h+2}\Z)^\times.
21 \]
22 The shifted dyadic index retains the sign even when the principal phase
23 register has dimension one. It prevents the two units modulo four from
24 being collapsed into one phase coordinate.
25
26 \subsection{Torsion and principal units}
27
28 \begin{proposition}[Finite unit coordinates]
29 \label{a-finite-unit-coordinates}
30 For odd $p$, choose a unit $\zeta_{p,h}$ of order $p-1$ whose reduction
31 generates $(\Z/p\Z)^\times$. The maps
32 \begin{align}
33 (t,a)&\longmapsto\zeta_{p,h}^{\,t}(1+p)^a
34 \pmod{p^{h+1}},
35 & (t,a)&\in\Z/(p-1)\Z\times\Z/p^h\Z,
36 \label{a-finite-odd-coordinate}\\
37 (s,a)&\longmapsto(-1)^s5^a\pmod{2^{h+2}},
38 & (s,a)&\in\Z/2\Z\times\Z/2^h\Z
39 \label{a-finite-dyadic-coordinate}
40 \end{align}
41 are group isomorphisms onto $U_{p,h}$ and $U_{2,h}$ respectively.
42 \end{proposition}
43
44 \begin{proof}
45 For odd $p$ the valuation identity
46 \[
47 v_p\bigl((1+p)^{p^k}-1\bigr)=k+1
48 \qquad(k\geq0)
49 \]
50 shows that $1+p$ has order $p^h$ modulo $p^{h+1}$. To check the
51 identity, the case $k=0$ is immediate. If $u=1+c p^{k+1}$ with
52 $p\nmid c$, the binomial expansion of $u^p-1$ has first term
53 $c p^{k+2}$. Every remaining term has strictly larger $p$-valuation
54 when $p$ is odd. This proves the induction. The subgroup generated by
55 $1+p$ is contained in the kernel of reduction modulo $p$ and has the
56 same cardinality $p^h$, so it is that entire kernel.
57
58 For completeness, an order-$p-1$ lift of any nonzero residue can be chosen
59 coherently. The root equation $x^{p-1}=1$ has nonzero derivative
60 $(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting
61 $x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$
62 that cancels the error. Starting with a generator of
63 $(\Z/p\Z)^\times$ produces $\zeta_{p,h}$ with the required order.
64 The cyclicity of this residue group follows from the elementary fact that
65 a degree-$d$ polynomial over a field has at most $d$ roots: in a finite
66 subgroup of the multiplicative group, the least common multiple $m$ of
67 the element orders is realized by an element. To see this, take for each
68 prime divisor of $m$ a component having the maximal prime-power order
69 and multiply those commuting components of coprime orders. All elements solve
70 $x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's
71 theorem gives the reverse bound. The subgroup is cyclic.
72
73 If $\zeta_{p,h}^{\,t}(1+p)^a=1$, reduction modulo $p$ gives
74 $t=0$ in $\Z/(p-1)\Z$. The principal-unit order then gives
75 $a=0$ in $\Z/p^h\Z$. Thus the displayed map is injective, and its
76 domain has size $(p-1)p^h=|U_{p,h}|$, which proves surjectivity.
77
78 For the dyadic case, successive squaring gives
79 \[
80 v_2\bigl(5^{2^k}-1\bigr)=k+2.
81 \]
82 Indeed $5-1$ has valuation two, and for every such power the factor
83 $5^{2^k}+1$ has valuation one. Thus $5$ has order $2^h$ modulo
84 $2^{h+2}$. Its powers are exactly the $2^h$ units congruent to one
85 modulo four. Multiplication by $-1$ supplies the units congruent to
86 three modulo four. These two classes are disjoint, giving the claimed
87 unique coordinates.
88 \end{proof}
89
90 In the odd-prime register $t$ is a torsion coordinate and $a$ is a
91 principal phase coordinate. In the dyadic register $s$ is a sign bit and
92 $a$ is again the principal phase. The Hilbert-space factorizations are
93 therefore
94 \begin{align}
95 \ell^2(U_{p,h})&\cong
96 \C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^h\Z),
97 &&p\text{ odd},\nonumber\\
98 \ell^2(U_{2,h})&\cong
99 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^h\Z).
100 \label{a-finite-unit-hilbert}
101 \end{align}
102 These are unitary changes of basis. They do not assert that a generic
103 state is a product state, or that a generic Hamiltonian is a sum of
104 independent register Hamiltonians. Entanglement and interactions are
105 preserved under the change of coordinates.
106
107 The lifting construction in the proof permits the coherent choice
108 $\zeta_{p,h+1}\equiv\zeta_{p,h}\pmod{p^{h+1}}$. With that choice,
109 reduction of unit addresses retains $t$ and reduces $a$ modulo $p^h$.
110 If the torsion lifts are instead chosen independently at each depth, the
111 finite factorizations remain exact, but a comparison between levels must
112 include the resulting coordinate changes. A levelwise permutation is
113 not by itself a specified refinement tower.
114
115
116
117 \subsection{CRT readdressing and the Fourier dual}
118
119 Let $\mathcal P_{\mathrm o}$ be a finite set of odd primes and put
120 \[
121 M=2^{h_2+2}\prod_{p\in\mathcal P_{\mathrm o}}p^{h_p+1}.
122 \]
123 The Chinese remainder theorem gives a bijection from the unit addresses
124 modulo $M$ to their local unit addresses. Combining it with
125 Proposition~\ref{a-finite-unit-coordinates} gives a permutation unitary
126 \begin{equation}
127 \mathcal U_{\mathrm{CRT}}:
128 \ell^2\bigl((\Z/M\Z)^\times\bigr)
129 \longrightarrow
130 \C^2_{\mathrm{sign}}\otimes\ell^2(\Z/2^{h_2}\Z)
131 \otimes\bigotimes_{p\in\mathcal P_{\mathrm o}}
132 \left(\C^{p-1}_{\mathrm{tors}}\otimes\ell^2(\Z/p^{h_p}\Z)\right).
133 \label{a-finite-crt}
134 \end{equation}
135 For each basis vector, $\mathcal U_{\mathrm{CRT}}|u\rangle$ is the basis
136 vector with its unique tuple of coordinates. Distinct units have distinct
137 tuples, so inner products are preserved. The dimensions also agree:
138 \[
139 \varphi(M)=2^{h_2+1}\prod_{p\in\mathcal P_{\mathrm o}}(p-1)p^{h_p}.
140 \]
141 Multiplication of units becomes addition in each cyclic coordinate.
142
143 We use the positive-phase Fourier convention
144 \begin{equation}
145 (F_n)_{k x}=n^{-1/2}\exp(2\pi\mathrm{i}kx/n).
146 \label{a-finite-fourier-convention}
147 \end{equation}
148 The character transform of the unit group consequently factors as
149 \begin{equation}
150 F_{U_M}=\mathcal U_{\mathrm{CRT}}^*
151 \left[F_2\otimes F_{2^{h_2}}\otimes
152 \bigotimes_{p\in\mathcal P_{\mathrm o}}(F_{p-1}\otimes F_{p^{h_p}})
153 \right]\mathcal U_{\mathrm{CRT}}.
154 \label{a-finite-unit-fourier}
155 \end{equation}
156 Here the unit-group character labels have been identified with the same
157 cyclic coordinate sets. This is a transform on the multiplicative unit
158 group, rather than the additive Fourier transform on all residues
159 modulo $M$.
160
161 The additive transform has a related factorization, with different
162 permutations on its input and its dual output. Let $a,b$ be positive
163 coprime integers and $N=ab$. Define
164 \begin{align*}
165 \mathcal P_{\mathrm{in}}|n\rangle
166 &=|n\bmod a\rangle\otimes|n\bmod b\rangle,\\
167 \mathcal P_{\mathrm{out}}|k\rangle
168 &=|k b^{-1}\bmod a\rangle\otimes|k a^{-1}\bmod b\rangle,
169 \end{align*}
170 where the inverses are taken in the indicated residue rings. Then
171 \begin{equation}
172 F_N=\mathcal P_{\mathrm{out}}^*(F_a\otimes F_b)
173 \mathcal P_{\mathrm{in}}.
174 \label{a-finite-additive-fourier}
175 \end{equation}
176 To verify the phase, let $n_a=n\bmod a$ and $n_b=n\bmod b$. At the
177 dual coordinates $k_a=kb^{-1}\bmod a$ and $k_b=ka^{-1}\bmod b$, the
178 product kernel is
179 \[
180 N^{-1/2}\exp\!\left[2\pi\mathrm{i}
181 \left(\frac{k_a n_a}{a}+\frac{k_b n_b}{b}\right)\right].
182 \]
183 The numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and
184 $b$, hence modulo $ab$. This kernel equals
185 $N^{-1/2}\exp(2\pi\mathrm{i}kn/N)$, proving
186 \eqref{a-finite-additive-fourier}. Omitting the dual-output permutation
187 would generally give a different phase labeling.
188
189 \subsection{What is lost when the sign is removed}
190
191 The linear address-forgetting map
192 \begin{equation}
193 C_{\mathrm{phase}}|s,a\rangle=|a\rangle
194 \label{a-finite-forgetting}
195 \end{equation}
196 maps a $2^{h+1}$-dimensional dyadic unit register onto a
197 $2^h$-dimensional principal phase register. Its kernel is the span of
198 the $2^h$ vectors $|0,a\rangle-|1,a\rangle$. Thus
199 \[
200 \operatorname{rank}C_{\mathrm{phase}}=2^h,
201 \qquad\dim\ker C_{\mathrm{phase}}=2^h.
202 \]
203 This is a rank-deficient linear sum of amplitudes. It is not an
204 isometry, and it is not the quantum channel that discards a subsystem.
205 For example it sends the normalized state
206 $(|0,a\rangle+|1,a\rangle)/\sqrt2$ to $\sqrt2|a\rangle$.
207 Its adjoint product therefore does not equal the identity.
208
209 Discarding the sign as a quantum operation instead takes the partial
210 trace $\rho\mapsto\operatorname{Tr}_{\mathrm{sign}}\rho$. Both pure
211 states $|0,a\rangle$ and $|1,a\rangle$ then produce the same reduced
212 phase state $|a\rangle\langle a|$. Yet the sign observable
213 \[
214 Z_{\mathrm{sign}}=(|0\rangle\langle0|-|1\rangle\langle1|)\otimes I
215 \]
216 has expectations $+1$ and $-1$ on those states. No measurement on the
217 retained phase state can reconstruct this difference. The obstruction is
218 information loss in a stated map, not an alternative probability rule.
219
220 A separate arithmetic example shows why integral residue selectors and
221 rational spectral idempotents should be distinguished. Let
222 \[
223 A=\begin{pmatrix}1&1\\0&-1\end{pmatrix},\qquad A^2=I,
224 \qquad E_\pm=\frac{I\pm A}{2}.
225 \]
226 The identities $E_\pm^2=E_\pm$, $E_+E_-=0$, and $E_++E_-=I$
227 follow directly from $A^2=I$. Both idempotents contain a denominator
228 two, and the resultant of $x-1$ and $x+1$ has dyadic valuation one.
229 Modulo two the two roots agree and $(A-I)^2=0$, while $A-I$ is
230 nonzero. Thus the residual matrix can retain a nilpotent extension even
231 though it no longer distinguishes the two rational eigenvalues.
232 These $E_\pm$ are complementary algebraic spectral idempotents. Since
233 $A$ is not self-adjoint in the standard counting inner product, they
234 are not orthogonal quantum measurement projectors in that inner product.
235 The unit-address sign projector, by contrast, is an ordinary diagonal
236 orthogonal projector.
237
238
239
240 \section{Cylinder averages, Haar details, and digit analysis}
241 \label{a-finite-haar}
242
243 The principal phase register has a canonical finite filtration. For a
244 prime $p$ and depth $h\geq0$, set
245 \[
246 G_{p,h}=\Z/p^h\Z,\qquad\mathcal H_{p,h}=\ell^2(G_{p,h};\C).
247 \]
248 At depth zero this is a one-dimensional space. A level-$r$ cylinder
249 contains all residues with a prescribed prefix modulo $p^r$, and has
250 $p^{h-r}$ members. Throughout this section depth refers to the number of
251 retained least significant base-$p$ digits.
252
253 \subsection{Orthogonal averaging and its detail resolution}
254
255 For $0\leq r\leq h$, define the cylinder average
256 \begin{equation}
257 (E_r^{(h)}f)(x)=p^{-(h-r)}
258 \sum_{y\equiv x\, (\mathrm{mod}\,p^r)}f(y).
259 \label{a-finite-cylinder-average}
260 \end{equation}
261 Its matrix is real and symmetric. Averaging twice over the same cylinder
262 has the same effect as averaging once. Thus $E_r^{(h)}$ is the
263 orthogonal projector onto functions constant on level-$r$ cylinders.
264 The range has dimension $p^r$.
265
266 \Needspace{16\baselineskip}
267 \begin{proposition}[Finite Haar resolution]
268 \label{a-finite-haar-resolution}
269 The cylinder projectors satisfy
270 \[
271 E_r^{(h)}E_s^{(h)}=E_{\min(r,s)}^{(h)},\qquad E_h^{(h)}=I.
272 \]
273 Define
274 \[
275 D_0^{(h)}=E_0^{(h)},\qquad
276 D_r^{(h)}=E_r^{(h)}-E_{r-1}^{(h)}\quad(1\leq r\leq h).
277 \]
278 Then the $D_r^{(h)}$ are mutually orthogonal projectors and
279 \begin{align}
280 \sum_{r=0}^hD_r^{(h)}&=I,
281 &D_r^{(h)}D_s^{(h)}&=\delta_{rs}D_r^{(h)},\nonumber\\
282 \operatorname{rank}D_0^{(h)}&=1,
283 &\operatorname{rank}D_r^{(h)}&=(p-1)p^{r-1}\quad(r\geq1).
284 \label{a-finite-detail-ranks}
285 \end{align}
286 \end{proposition}
287
288 \begin{proof}
289 If $r\leq s$, the level-$s$ cylinders partition each level-$r$
290 cylinder into equally sized groups. Averaging over the smaller groups
291 and then over the whole group returns the full average, giving
292 $E_rE_s=E_r$. The self-adjointness of both sides also gives
293 $E_sE_r=E_r$. This proves the minimum rule.
294
295 For $r\geq1$,
296 $(E_r-E_{r-1})^2=E_r-2E_{r-1}+E_{r-1}=D_r$.
297 For $1\leq r<s$, expansion of
298 $(E_r-E_{r-1})(E_s-E_{s-1})$ gives zero by the same minimum
299 rule. The products with $D_0$ vanish as well. Telescoping gives the
300 resolution of the identity. Finally $\Ran E_{r-1}\subset\Ran E_r$,
301 so the detail rank is $p^r-p^{r-1}$, while $E_0$ has rank one.
302 \end{proof}
303
304 The decomposition is exact at each finite depth:
305 \begin{equation}
306 \mathcal H_{p,h}=\bigoplus_{r=0}^h\Ran D_r^{(h)},\qquad
307 \norm f^2=\sum_{r=0}^h\norm{D_r^{(h)}f}^2.
308 \label{a-finite-parseval}
309 \end{equation}
310 A depth cutoff $m\leq h$ retains $E_m^{(h)}=\sum_{r=0}^mD_r^{(h)}$.
311 Its discarded norm is exactly
312 \[
313 \norm{(I-E_m^{(h)})f}^2=
314 \sum_{r=m+1}^h\norm{D_r^{(h)}f}^2.
315 \]
316 No smoothness assumption is needed for this finite Parseval identity.
317 A rate of decrease with $m$ requires an additional bound on the detail
318 weights of the state.
319
320 \subsection{The child characters form the Haar basis}
321
322 Write $\omega_p=\exp(2\pi\mathrm{i}/p)$. For $1\leq r\leq h$,
323 parent prefix $a\in\{0,\ldots,p^{r-1}-1\}$, and character
324 $q\in\{1,\ldots,p-1\}$, define
325 \begin{equation}
326 \psi_{r,a,q}^{(h)}(x)=
327 \begin{cases}
328 p^{-(h-r+1)/2}\omega_p^{q d_r(x)},
329 &x\equiv a\pmod{p^{r-1}},\\
330 0,&\text{otherwise},
331 \end{cases}
332 \label{a-finite-wavelet}
333 \end{equation}
334 where $d_r(x)$ is the digit multiplying $p^{r-1}$. Include the
335 constant scaling vector $\psi_0^{(h)}(x)=p^{-h/2}$.
336
337 Each wavelet is constant on level-$r$ cylinders and has zero average
338 on its parent level-$r-1$ cylinder, because
339 $\sum_{d=0}^{p-1}\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in
340 $\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving
341 unit norm. Distinct parents have disjoint supports. For a fixed parent
342 the inner product of the characters $q,q'$ is
343 \[
344 p^{-1}\sum_{d=0}^{p-1}\omega_p^{(q'-q)d}=\delta_{qq'}.
345 \]
346 Wavelets at different levels are orthogonal because their detail
347 projectors are orthogonal. There are exactly
348 $(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in
349 \eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence
350 \begin{equation}
351 D_r^{(h)}=\sum_{a=0}^{p^{r-1}-1}\sum_{q=1}^{p-1}
352 |\psi_{r,a,q}^{(h)}\rangle\langle\psi_{r,a,q}^{(h)}|.
353 \label{a-finite-wavelet-projector}
354 \end{equation}
355 The omitted character $q=0$ is the parent average. For $p=2$ there is
356 one average and one difference per parent. For odd $p$ there is one
357 average and $p-1$ nontrivial child characters. The dyadic sign factor of
358 \eqref{a-finite-unit-hilbert} is independent of this average/detail
359 split in the principal phase register.
360
361 \adicfigure{02_haar_channels}{For $p=3$, the normalized child characters
362 contain one average and two orthogonal complex details. Solid curves
363 show real parts and dashed curves show imaginary parts. The level
364 rank $(p-1)p^{r-1}$ grows by the branching factor. The atlas uses
365 $\mathsf F_r$ for the corresponding infinite cylinder detail; its
366 finite restriction is $D_r^{(h)}$ for $r\leq h$.}
367
368
369
370 \subsection{The refinement isometry and its adjoint}
371
372 One more digit supplies $p$ children for every address. The uniform
373 refinement map is
374 \begin{equation}
375 (V_{p,h}f)(a+d p^h)=p^{-1/2}f(a),
376 \quad 0\leq a<p^h,\quad0\leq d<p.
377 \label{a-finite-refinement}
378 \end{equation}
379 In tensor coordinates it is $f\mapsto|+_p\rangle\otimes f$,
380 where $|+_p\rangle=p^{-1/2}\sum_d|d\rangle$ and the child digit is
381 listed first. The norm calculation is
382 \[
383 \norm{V_{p,h}f}^2=\sum_a\sum_d p^{-1}|f(a)|^2=\norm f^2.
384 \]
385 The adjoint is the normalized amplitude sum
386 \begin{equation}
387 (V_{p,h}^*g)(a)=p^{-1/2}\sum_{d=0}^{p-1}g(a+d p^h).
388 \label{a-finite-refinement-adjoint}
389 \end{equation}
390 Consequently
391 \begin{equation}
392 V_{p,h}^*V_{p,h}=I,\qquad
393 V_{p,h}V_{p,h}^*=E_h^{(h+1)},\qquad
394 I-V_{p,h}V_{p,h}^*=D_{h+1}^{(h+1)}.
395 \label{a-finite-refinement-projectors}
396 \end{equation}
397 The first composition is an identity on the coarse space. The second
398 is a projector on the fine space, not a fine-space identity.
399
400 Replicated vectors are constant in the new digit. All old cylinder
401 averages commute with replication, giving
402 \begin{align}
403 E_r^{(h+1)}V_{p,h}&=V_{p,h}E_r^{(h)}\quad(0\leq r\leq h),
404 \nonumber\\
405 D_r^{(h+1)}V_{p,h}&=V_{p,h}D_r^{(h)}\quad(0\leq r\leq h),
406 &D_{h+1}^{(h+1)}V_{p,h}&=0.
407 \label{a-finite-detail-intertwining}
408 \end{align}
409 Repeated refinements compose as isometries. For $w\geq0$ and one
410 coarse address,
411 \begin{equation}
412 V_{p,h+w-1}\cdots V_{p,h}|a\rangle
413 =p^{-w/2}\sum_{\mathbf d\in\{0,\ldots,p-1\}^w}|a,\mathbf d\rangle,
414 \label{a-finite-nested-refinement}
415 \end{equation}
416 with the empty product interpreted as the identity. The factor
417 $p^{-w/2}$ is forced by the isometry and orthonormality of the
418 descendant addresses.
419
420 \subsection{Digit reversal and the polyphase split}
421 \label{a-finite-polyphase}
422
423 For $h\geq1$, put $N=p^h$ and $m=p^{h-1}$. Two permutation
424 unitaries separate different digits:
425 \begin{align}
426 \mathcal A^{\mathrm{std}}_{p,h}|pk+r\rangle
427 &=|r\rangle\otimes|k\rangle,
428 &0\leq r<p,\quad0\leq k<m,\nonumber\\
429 \mathcal A^{\mathrm{ad}}_{p,h}|a+d p^{h-1}\rangle
430 &=|d\rangle\otimes|a\rangle,
431 &0\leq d<p,\quad0\leq a<m.
432 \label{a-finite-digit-splits}
433 \end{align}
434 The standard polyphase split extracts the least significant residue.
435 The adic split extracts the newest, most significant digit at the
436 current depth. Let $\mathcal R_{p,h}$ reverse the $h$ base-$p$ digits,
437 with $\mathcal R_{p,0}=I$ on the one-dimensional space. Then
438 \begin{equation}
439 \mathcal A^{\mathrm{ad}}_{p,h}=
440 (I_p\otimes\mathcal R_{p,h-1})
441 \mathcal A^{\mathrm{std}}_{p,h}\mathcal R_{p,h}.
442 \label{a-finite-digit-reversal}
443 \end{equation}
444 Indeed reversing $d_0+d_1p+\cdots+d_{h-1}p^{h-1}$ makes
445 $d_{h-1}$ the least significant digit. Standard analysis removes that
446 digit, and reversing the remaining $h-1$ digits restores the prefix
447 $a=d_0+\cdots+d_{h-2}p^{h-2}$. This proves the identity on an
448 orthonormal basis.
449
450 Let the new-digit modulation be
451 $M_{p,h}|a+d p^{h-1}\rangle=\omega_p^d|a+d p^{h-1}\rangle$.
452 The selector of child digit $d$ is the exact roots-of-unity filter
453 \begin{equation}
454 R_d=\frac1p\sum_{\ell=0}^{p-1}\omega_p^{-d\ell}M_{p,h}^{\ell}.
455 \label{a-finite-root-selector}
456 \end{equation}
457 On a child with digit $e$, the coefficient is
458 $p^{-1}\sum_\ell\omega_p^{(e-d)\ell}=\delta_{de}$. Therefore
459 $R_dR_e=\delta_{de}R_d$, $R_d^*=R_d$, and $\sum_dR_d=I$.
460 These projectors select address digits. They are different from the
461 projectors onto Fourier characters of a child register.
462
463 Apply the child Fourier transform after adic analysis:
464 \begin{equation}
465 \mathcal W_{p,h}=(F_p\otimes I_m)\mathcal A^{\mathrm{ad}}_{p,h}.
466 \label{a-finite-child-fourier}
467 \end{equation}
468 The zero child-frequency row of $F_p$ is $p^{-1/2}(1,\ldots,1)$.
469 Hence
470 \begin{equation}
471 \begin{aligned}
472 (\langle0|\otimes I_m)\mathcal W_{p,h}&=V_{p,h-1}^*,\\
473 \mathcal W_{p,h}^*
474 \bigl[(I_p-|0\rangle\langle0|)\otimes I_m\bigr]
475 \mathcal W_{p,h}&=D_h^{(h)}.
476 \end{aligned}
477 \label{a-finite-child-detail}
478 \end{equation}
479 The second identity subtracts the coarse zero-frequency projector from
480 the identity using \eqref{a-finite-refinement-projectors}. Recursive
481 application to the zero-frequency channel alone constructs a unitary
482 Haar analysis map $\mathcal T_{p,h}$ whose channel projectors $\Pi_r$
483 have the ranks in \eqref{a-finite-detail-ranks} and obey
484 \begin{equation}
485 \mathcal T_{p,h}^*\Pi_r\mathcal T_{p,h}=D_r^{(h)}.
486 \label{a-finite-haar-analysis}
487 \end{equation}
488 With our positive Fourier sign, the analysis row of character $q$
489 pairs with the complex-conjugate wavelet character. This only permutes
490 the nonzero character labels $q\leftrightarrow p-q$ and leaves the
491 detail projectors unchanged.
492
493 The zero child-frequency channel retains a uniform superposition over
494 the new digit. Retaining standard residue zero instead keeps one
495 address out of each $p$ addresses. Both spaces have the same dimension,
496 but they are distinct subspaces. Their compressed dynamics and Schur
497 complements consequently need not agree. Digit reversal and child
498 Fourier analysis specify the transformations needed to compare them.
499
500
501
502 \section{Which Hamiltonians are compatible with refinement?}
503 \label{a-finite-hamiltonian}
504
505 An isometric address refinement does not automatically preserve a
506 Hamiltonian. The exact condition is a block condition on that
507 Hamiltonian. It can be checked without an approximation, a scaling
508 limit, or a physical interpretation of the new digit.
509
510 \begin{theorem}[Finite refinement classification]
511 \label{a-finite-hamiltonian-classification}
512 Let $V:\mathcal H_c\to\mathcal H_f$ be an isometry between finite
513 Hilbert spaces. Set $P=VV^*$ and $Q=I-P$. For self-adjoint operators
514 $H_c,H_f$, the following conditions are equivalent:
515 \begin{align}
516 H_fV&=VH_c,\label{a-finite-hamiltonian-intertwine}\\
517 QH_fP&=0,\qquad PH_fP=VH_cV^*.
518 \label{a-finite-hamiltonian-blocks}
519 \end{align}
520 They hold exactly when, on $\Ran P\oplus\Ran Q$,
521 \begin{equation}
522 H_f=(VH_cV^*)|_{\Ran P}\ \oplus\ H_Q,
523 \qquad H_Q=(QH_fQ)|_{\Ran Q}.
524 \label{a-finite-hamiltonian-direct-sum}
525 \end{equation}
526 The detail operator $H_Q$ may be any self-adjoint operator on
527 $\Ran Q$.
528 \end{theorem}
529
530 \begin{proof}
531 Right multiplication of \eqref{a-finite-hamiltonian-intertwine} by
532 $V^*$ gives $H_fP=VH_cV^*$. Left multiplication by $Q$ and $P$
533 gives \eqref{a-finite-hamiltonian-blocks}. Conversely $PV=V$ and
534 $V^*V=I$ imply
535 \[
536 H_fV=(PH_fP+QH_fP)V=VH_cV^*V=VH_c.
537 \]
538 Self-adjointness gives $PH_fQ=(QH_fP)^*=0$. Thus both off-diagonal
539 blocks vanish, and the remaining blocks are exactly those in
540 \eqref{a-finite-hamiltonian-direct-sum}. Conversely that direct sum
541 has the two block properties. There is no further condition on the
542 self-adjoint detail block.
543 \end{proof}
544
545 The distinction between invariance and compression is essential.
546 Defining $H_c=V^*H_fV$ always matches the coarse block, but leaves
547 $QH_fV$ unconstrained. Thus compression alone does not establish
548 \eqref{a-finite-hamiltonian-intertwine}. When it does hold, the coarse
549 subspace is reducing, and fine evolution introduces no new detail
550 amplitude from a coarse initial state.
551
552 \begin{corollary}[Functional-calculus intertwining]
553 \label{a-finite-functional-calculus}
554 Under Theorem~\ref{a-finite-hamiltonian-classification}, for every
555 complex function $f$ defined on the finite union of the spectra,
556 \[
557 f(H_f)V=Vf(H_c).
558 \]
559 In particular, for real $t$, real $\beta$, and
560 $z\notin\sigma(H_f)$,
561 \begin{align}
562 e^{-\mathrm{i}tH_f}V&=Ve^{-\mathrm{i}tH_c},\nonumber\\
563 e^{-\beta H_f}V&=Ve^{-\beta H_c},\nonumber\\
564 (z-H_f)^{-1}V&=V(z-H_c)^{-1}.
565 \label{a-finite-evolution-intertwining}
566 \end{align}
567 \end{corollary}
568
569 \begin{proof}
570 On $\Ran P$, $V$ is a unitary identification of $H_c$ with the
571 first block of $H_f$. The finite spectral decomposition therefore
572 maps each coarse spectral subspace to the corresponding first-block
573 spectral subspace. Applying $f$ to the eigenvalues gives the identity.
574 The three displayed functions are defined on the relevant spectra.
575 \end{proof}
576
577 The Euclidean operator $e^{-\beta H}$ in this corollary is a spectral
578 multiplier. It is not generally unitary, and a normalized state after
579 Euclidean evolution requires its own normalization. The identity does
580 not compare normalized Gibbs states of the entire fine and coarse
581 systems, since the fine partition function also counts the detail
582 block. For a compatible finite tower, the coarse propagator and its
583 embedded fine propagator agree exactly. This does not make an arbitrary
584 interacting fine Hamiltonian compatible.
585
586 \subsection{Hierarchical and translation-invariant examples}
587
588 Choose a real sequence of level energies $\epsilon_{p,r}$ and define
589 \begin{equation}
590 H_{p,h}^{\mathrm{Haar}}=\sum_{r=0}^h
591 \epsilon_{p,r}D_r^{(h)}.
592 \label{a-finite-hierarchical-hamiltonian}
593 \end{equation}
594 Every wavelet in $\Ran D_r^{(h)}$ has energy $\epsilon_{p,r}$.
595 The projectors in \eqref{a-finite-detail-intertwining} show directly
596 that $H_{p,h+1}^{\mathrm{Haar}}V_{p,h}
597 =V_{p,h}H_{p,h}^{\mathrm{Haar}}$. The new detail energy
598 $\epsilon_{p,h+1}$ does not alter the embedded older levels.
599 Moreover the recursive Haar transform diagonalizes the level blocks:
600 \[
601 \mathcal T_{p,h}H_{p,h}^{\mathrm{Haar}}\mathcal T_{p,h}^*
602 =\sum_{r=0}^h\epsilon_{p,r}\Pi_r.
603 \]
604 This is exact for each finite $h$ and every finite real choice of the
605 energies. Their growth becomes a domain question only when an
606 infinite-depth operator is constructed.
607
608 Let the cyclic translation be $(S_{p,h}f)(x)=f(x-1)$, with
609 subtraction modulo $p^h$. Since a fine replicated function depends only
610 on the residue modulo $p^h$,
611 \[
612 S_{p,h+1}V_{p,h}=V_{p,h}S_{p,h}.
613 \]
614 The adjoint translation obeys the same identity. Therefore the cyclic
615 Laplacian $L_{p,h}=2I-S_{p,h}-S_{p,h}^*$ intertwines, and for real
616 $\kappa_p$ the family
617 \begin{equation}
618 H_{p,h}=H_{p,h}^{\mathrm{Haar}}+\kappa_p L_{p,h}
619 \label{a-finite-compatible-laplacian}
620 \end{equation}
621 is self-adjoint and refinement compatible. Translation permutes the
622 level-$r$ cylinders, so it commutes with each $E_r^{(h)}$ and
623 $D_r^{(h)}$. The Laplacian thus preserves every detail subspace,
624 although a chosen spatially localized Haar wavelet need not be its
625 eigenvector.
626
627 This cyclic translation uses fixed unit displacement in the address
628 group. Multiplying it by a depth-dependent factor intended as a
629 physical inverse lattice spacing changes the tower. Such a factor is
630 not covered by \eqref{a-finite-compatible-laplacian} unless the
631 resulting operators still satisfy the stated intertwining equation.
632 Exact adic compatibility and continuum spatial consistency are
633 different requirements.
634
635 \subsection{A valuation-shell propagator}
636
637 For the pure hierarchical Hamiltonian, the finite spectral theorem gives
638 \begin{equation}
639 U_{p,h}(t)=e^{-\mathrm{i}tH_{p,h}^{\mathrm{Haar}}}
640 =\sum_{r=0}^he^{-\mathrm{i}t\epsilon_{p,r}}D_r^{(h)}.
641 \label{a-finite-shell-propagator}
642 \end{equation}
643 The cylinder kernel is
644 \[
645 E_r^{(h)}(x,y)=p^{-(h-r)}
646 \mathbf1_{\{x\equiv y\, (\mathrm{mod}\,p^r)\}}.
647 \]
648 Consequently
649 \begin{align}
650 U_{p,h}(t;x,y)
651 &=p^{-h}e^{-\mathrm{i}t\epsilon_{p,0}}\nonumber\\
652 &\quad+\sum_{r=1}^he^{-\mathrm{i}t\epsilon_{p,r}}
653 \left[p^{-(h-r)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^r)}
654 -p^{-(h-r+1)}\mathbf1_{x\equiv y\,(\mathrm{mod}\,p^{r-1})}
655 \right].
656 \label{a-finite-shell-kernel}
657 \end{align}
658 Every indicator is determined by the truncated valuation
659 $\min\{v_p(x-y),h\}$, with the diagonal assigned value $h$.
660 The transition amplitude, and therefore the individual-address Born
661 probability, is constant on each such valuation shell. At a fixed
662 initial address $y$, the shell of valuation $k<h$ contains
663 $(p-1)p^{h-k-1}$ final addresses. Its total probability includes
664 this multiplicity. The diagonal has one address. Uniform probability
665 per address on a shell does not mean equal total probabilities of
666 different shells.
667
668
669
670 \subsection{Leakage detects failure of compatibility}
671 \label{a-finite-schur}
672
673 For a self-adjoint fine Hamiltonian with coarse projector $P$, set
674 $B=QH_fP$. If a unit vector $\psi$ lies in $\Ran P$, its initial
675 detail amplitude is
676 \[
677 Qe^{-\mathrm{i}tH_f}\psi=-\mathrm{i}tB\psi+O(t^2),
678 \qquad
679 \norm{Qe^{-\mathrm{i}tH_f}\psi}^2
680 =t^2\norm{B\psi}^2+O(|t|^3).
681 \]
682 These are finite Taylor expansions for fixed $H_f$ and $\psi$.
683 Thus a nonzero $B$ supplies a state with nonzero short-time leakage.
684 The two-dimensional example
685 \[
686 H_f=\begin{pmatrix}0&\overline\gamma\\\gamma&0\end{pmatrix},
687 \qquad P=\begin{pmatrix}1&0\\0&0\end{pmatrix}
688 \]
689 has compressed Hamiltonian zero but detail probability
690 $\sin^2(|\gamma|t)$ from the first basis state. It fails exact
691 refinement compatibility whenever $\gamma\neq0$.
692
693 For $z$ outside the spectrum of $H_Q=(QH_fQ)|_{\Ran Q}$, the
694 coarse Schur self-energy is
695 \begin{equation}
696 \Sigma_P(z)=PH_fQ(z-H_Q)^{-1}QH_fP.
697 \label{a-finite-self-energy}
698 \end{equation}
699 Compatibility makes it identically zero by setting $B=0$. This zero
700 is an algebraic statement, not a uniform bound on a resolvent near
701 its spectrum. A nonzero coupling requires the energy-dependent Schur
702 term when the detail sector is eliminated. The spectral part of this
703 monograph develops that elimination and its time-domain memory.
704
705 \adicfigure{05_projector_tower}{A finite compatible Hamiltonian has
706 zero blocks between its embedded coarse space and its new detail
707 space. In the plotted example, adding an off-diagonal coupling creates
708 detail probability from a coarse initial state. This finite example
709 illustrates the block criterion and supplies a negative control for
710 unconditional refinement claims.}
711
712 \section{Normalization and collective coupling}
713 \label{a-finite-normalization}
714
715 Refinement fixes the norm of an embedded packet. It does not fix an
716 interaction matrix element. The distinction becomes visible when one
717 coarse packet couples to many descendants rather than to one already
718 normalized aggregate packet. In this section $\hbar=1$, so the
719 Hamiltonian has units of inverse time.
720
721 \subsection{Descendant weights are fixed by the isometry}
722
723 After $w$ refinements a packet has $n=p^w$ orthonormal descendants.
724 Its normalized radial vector is
725 \begin{equation}
726 |R_{p,w}\rangle=n^{-1/2}\sum_{\mathbf d}|\mathbf d\rangle.
727 \label{a-finite-radial-vector}
728 \end{equation}
729 The norm is $n^{-1}\sum_{\mathbf d}1=1$, so each descendant has
730 amplitude modulus $p^{-w/2}$ and Born probability $p^{-w}$. This
731 reproduces \eqref{a-finite-nested-refinement}. The words ``radial'' and
732 ``angular'' here refer to the uniform vector and its orthogonal
733 complement in a finite descendant space. They do not introduce a
734 spacetime radius or a rotation group.
735
736 Introduce a visible address $|c\rangle$ orthogonal to all descendants.
737 Suppose every descendant has the same coupling from $|c\rangle$,
738 with complex matrix element $\gamma_{p,w}$. The Hermitian interaction
739 is
740 \begin{equation}
741 H_{\mathrm{int}}=
742 \sum_{\mathbf d}
743 \left(\gamma_{p,w}|\mathbf d\rangle\langle c|
744 +\overline{\gamma_{p,w}}|c\rangle\langle\mathbf d|\right).
745 \label{a-finite-leaf-interaction}
746 \end{equation}
747 The conjugate reverse coefficient is necessary when
748 $\gamma_{p,w}$ is not real. Define the collective coefficient
749 \begin{equation}
750 G_{p,w}=\sqrt n\,\gamma_{p,w}.
751 \label{a-finite-collective-coefficient}
752 \end{equation}
753 Then $H_{\mathrm{int}}|c\rangle=G_{p,w}|R_{p,w}\rangle$ and
754 $H_{\mathrm{int}}|R_{p,w}\rangle=
755 \overline{G_{p,w}}|c\rangle$. If
756 $|A\rangle=\sum_{\mathbf d}a_{\mathbf d}|\mathbf d\rangle$
757 is orthogonal to $|R_{p,w}\rangle$, its coefficients satisfy
758 $\sum_{\mathbf d}a_{\mathbf d}=0$. Hence
759 $H_{\mathrm{int}}|A\rangle=0$. Only the radial vector is bright
760 under this equal-leaf interaction.
761
762 This statement concerns the interaction block. A separate descendant
763 Hamiltonian can mix radial and angular vectors. To obtain an invariant
764 two-dimensional radial subsystem, the full descendant Hamiltonian
765 must preserve the radial line, and the full Hamiltonian must leave its
766 angular complement uncoupled from $|c\rangle$. A degenerate hidden
767 energy $\epsilon I$ is one sufficient choice.
768
769 \subsection{The exact bright two-level problem}
770
771 Assume those invariance conditions and order the basis as
772 $(|c\rangle,|R_{p,w}\rangle)$. With visible energy $E_c$ and
773 radial energy $\epsilon$, the exact bright block is
774 \begin{equation}
775 H_{\mathrm{bright}}=
776 \begin{pmatrix}E_c&\overline G_{p,w}\\G_{p,w}&\epsilon\end{pmatrix}.
777 \label{a-finite-bright-block}
778 \end{equation}
779 Set $\overline E=(E_c+\epsilon)/2$,
780 $\delta=(E_c-\epsilon)/2$, and
781 $\Omega=(\delta^2+|G_{p,w}|^2)^{1/2}$. Then
782 $(H_{\mathrm{bright}}-\overline E I)^2=\Omega^2I$, giving
783 \begin{equation}
784 e^{-\mathrm{i}tH_{\mathrm{bright}}}
785 =e^{-\mathrm{i}t\overline E}
786 \left[I\cos(\Omega t)
787 -\mathrm{i}\frac{H_{\mathrm{bright}}-\overline E I}{\Omega}
788 \sin(\Omega t)\right].
789 \label{a-finite-bright-propagator}
790 \end{equation}
791 At $\Omega=0$ this formula is understood by continuous extension.
792 Starting in $|c\rangle$, the radial probability is therefore
793 \begin{equation}
794 \Pr(R_{p,w};t)=
795 \frac{|G_{p,w}|^2}{\delta^2+|G_{p,w}|^2}
796 \sin^2\!\left(t\sqrt{\delta^2+|G_{p,w}|^2}\right).
797 \label{a-finite-radial-probability}
798 \end{equation}
799 The probability is defined to be zero when $\Omega=0$, equivalently
800 when both $\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\epsilon$ it reduces to
801 $\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are
802 $\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.
803
804 Eliminating the hidden radial level gives
805 \begin{equation}
806 \Sigma_{p,w}(z)=\frac{|G_{p,w}|^2}{z-\epsilon},
807 \qquad z\neq\epsilon.
808 \label{a-finite-radial-self-energy}
809 \end{equation}
810 This is the exact finite Schur term. The full visible Green function,
811 when its denominator is nonzero, is
812 $[z-E_c-\Sigma_{p,w}(z)]^{-1}$. It is distinct from the self-energy
813 itself, and the locations of its poles depend on both sectors.
814
815 \subsection{The coupling exponent is a model choice}
816
817 Let $g\in\C$ be fixed and choose
818 $\gamma_{p,w}=g p^{-\alpha w}$ for a real exponent $\alpha$.
819 Then
820 \begin{equation}
821 G_{p,w}=g p^{(1/2-\alpha)w},\qquad
822 |G_{p,w}|^2=|g|^2p^{(1-2\alpha)w}.
823 \label{a-finite-coupling-exponent}
824 \end{equation}
825 The exact branching multiplicity produces the factor $p^{w/2}$.
826 Thus the collective coefficient grows with $w$ for $\alpha<1/2$,
827 is independent of $w$ for $\alpha=1/2$, and decays for
828 $\alpha>1/2$. The same cases govern the operator norm of this
829 equal-leaf interaction: its two radial eigenvalues are
830 $\pm|G_{p,w}|$, so $\norm{H_{\mathrm{int}}}=|G_{p,w}|$.
831 They do not determine the boundedness of a different infinite operator
832 with many interacting depth sectors.
833
834 \begin{center}
835 \begin{tabular}{@{}lccc@{}}
836 \toprule
837 Per-leaf choice & $\alpha$ & $|G_{p,w}|$ & $|G_{p,w}|^2$\\
838 \midrule
839 Constant matrix element & $0$ & $|g|p^{w/2}$ & $|g|^2p^w$\\
840 Constant collective coupling & $1/2$ & $|g|$ & $|g|^2$\\
841 Uniform measure coefficient & $1$ & $|g|p^{-w/2}$ & $|g|^2p^{-w}$\\
842 \bottomrule
843 \end{tabular}
844 \end{center}
845
846 The measure coefficient $\alpha=1$ assigns the number $p^{-w}$ to
847 each per-leaf matrix element after factoring out $g$. It gives a
848 collective amplitude of size $p^{-w/2}$ and a self-energy numerator of
849 size $p^{-w}$. The self-energy estimate is uniform only where
850 $|z-\epsilon|$ stays bounded below. Likewise
851 \[
852 \Pr(R_{p,w};t)\leq |g|^2t^2p^{-w}
853 \quad\text{when }\alpha=1
854 \]
855 follows from \eqref{a-finite-radial-probability} and
856 $|\sin u|\leq|u|$. It is a fixed-time bound. On resonance a time
857 of order $p^{w/2}/|g|$ can still transfer order-one probability.
858
859 A model that directly couples $|c\rangle$ to one normalized hidden
860 packet with matrix element $g p^{-w}$ instead has a collective
861 coefficient of that same size and a self-energy numerator
862 $|g|^2p^{-2w}$. It is a valid two-level model, but it is not the
863 equal-leaf model \eqref{a-finite-leaf-interaction}. The number of
864 descendant addresses alone does not select between these interaction
865 geometries. A physical refinement law must specify the coupling,
866 the retained hidden modes, their energies, and the observation scale.
867
868
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