04-ilang-time / 03-relational-time
03relational timeverified
Determines whether the evolution of the rest of a system can be stated relative to clock readings instead of λ, from a state whose statistics do not depend on λ, and how rates convert.
# Question: 03-relational-time
- **Subproject:** 04-ilang-time
- **Package:** 03-relational-time
- **Equation tags:** (3.k)
- **Created:** 2026-10-10
## Goal
Determine whether, and under which conditions, the evolution of the rest of a system can be stated relative to the readings of a clock instead of $\lambda$, starting from a global state whose reading statistics do not depend on $\lambda$ at all. Determine how rates with respect to $\lambda$ then convert to rates with respect to the clock. This is criterion 2. Comparing two clocks is criterion 3 and not part of this package.
**Setting.**
- **Objects.** A clock part $K$, consisting of one object (the clock object), and its companion $R\neq\emptyset$. No object of $R$ has the type of the clock object, so a reading on $K$ keeps every state admissible (A3, A6).
- **Contracts.** $C=C_K\otimes\mathbb 1_R+\mathbb 1_K\otimes C_R+C_\partial$, as in 02-clock: $C_K$ acts on the clock object, $C_R$ on $R$, and $C_\partial=\sum_nA_n\otimes B_n$ collects the terms across the cut.
- **Clock.** The clock object has $N\ge2$ places $m\in\mathbb Z_N$, and
$$C_K=\frac{2\pi}{N\lambda_0}\sum_{k=0}^{N-1}k\,\lvert f_k\rangle\langle f_k\rvert,\qquad \lvert f_k\rangle=\frac1{\sqrt N}\sum_{m\in\mathbb Z_N}e^{2\pi ikm/N}\lvert m\rangle ,$$
with $\lambda_0>0$. By (2.9) and (2.12) of 02-clock, quoted under "Inputs", this is a uniform ring clock with tick $\lambda_0$ and $\theta_m=0$ (the case $\alpha_m=0$, $\varepsilon=0$, $\ell_k=k$), and of minimal cost spread. Its reading is the place reading.
- **Conditional states.** For a state $\Psi$ of the whole system and a clock place $m$, let $\Phi(m):=\bigl(\langle m\rvert\otimes\mathbb 1_R\bigr)\Psi\in\mathcal H_R$. By (1.3), a place reading on $K$ gives $m$ with probability $\lVert\Phi(m)\rVert^2$ and leaves the state $\lvert m\rangle\otimes\Phi(m)/\lVert\Phi(m)\rVert$.
- **Notation.** $C_R=\sum_se_sQ_s$ is the spectral decomposition of $C_R$, with distinct $e_s$.
1. **States without $\lambda$-dependence.** Determine a necessary and sufficient condition on the start $\Psi(0)$ under which the statistics of every reading, on any part and at any stated $\lambda$ (without earlier readings), do not depend on $\lambda$. For such starts, determine whether the outcome $m$ of a place reading on $K$ can change the statistics of a reading on $R$ performed right after it.
2. **Conditional states of an eigenstate.** Let $C_\partial=0$, and let $\Psi$ be an eigenvector of $C$ with eigenvalue $E$.
- (a) Determine the relation between $\Phi(m+1)$ and $\Phi(m)$ for every $m$, in terms of $C_R$, $\lambda_0$ and $E$.
- (b) Determine which components $Q_s\Phi(m)$ can be nonzero, as a condition on $e_s$, $E$, $N$ and $\lambda_0$. For a given unit vector $\chi\in\mathcal H_R$, determine a necessary and sufficient condition under which some eigenvector $\Psi$ of $C$ with eigenvalue $E$ has $\Phi(0)\propto\chi$. In that case, determine $\Psi$, all $\Phi(m)$, and the probabilities of the clock places.
- (c) For the case of (b), compare the statistics of any reading on $R$ in the state $\Phi(m)/\lVert\Phi(m)\rVert$ with the statistics of the same reading in $e^{-iC_R\lambda}\chi$. Determine the values of $\lambda$ at which they coincide for every reading on $R$.
3. **Crossing contracts.** Let $C_\partial\neq0$. Determine whether a relation $\Phi(m+1)=U\,\Phi(m)$, with one unitary $U$ on $\mathcal H_R$ independent of $m$, holds for every eigenvector of $C$. Give a sufficient condition on $C_\partial$ under which it does, with $U$, and an example of a crossing contract for which it does not.
4. **Rates.** From items 2 and 3, determine how the rate of change of the statistics of a reading on $R$ with respect to $\lambda$, under the evolution of $R$ alone, converts to its rate of change with respect to the clock reading. State the conditions under which this conversion holds.
## Inputs
From 02-clock@v2. Setting and notation fixed there:
- a uniform ring clock is a single object with $N\ge2$ places $m\in\mathbb Z_N$, with $e^{-iC_K\lambda_0}\lvert m\rangle=e^{i\theta_m}\lvert m+1\rangle$ for every $m$, where $\lambda_0>0$ is its tick;
- $S\lvert m\rangle=\lvert m+1\rangle$ is the shift, and $\lvert f_k\rangle=N^{-1/2}\sum_me^{2\pi ikm/N}\lvert m\rangle$;
- $\Delta C_K$ is the spread of $C_K$ in the view of $K$.
Eq. (2.9), "where the $\alpha_m$ and $\varepsilon$ are arbitrary reals. The phases are then $\theta_m=\alpha_{m+1}-\alpha_m-\varepsilon\lambda_0$ (mod $2\pi$, with $\alpha_N:=\alpha_0$). If $\theta_m=0$ for all $m$, then $e^{-iC_K\lambda_0}=S$. The same argument applied to $S$, whose eigenvalues $e^{-2\pi ik/N}$ on $\lvert f_k\rangle$ are distinct, gives exactly $C_K=\sum_k\frac{2\pi\ell_k}{N\lambda_0}\lvert f_k\rangle\langle f_k\rvert$ with $\ell_k\equiv k$."
$$
C_K=\sum_{k\in\mathbb Z_N}E_k\lvert v_k\rangle\langle v_k\rvert,\quad
\lvert v_k\rangle=\frac{1}{\sqrt N}\sum_{m}e^{2\pi ikm/N}e^{i\alpha_m}\lvert m\rangle,\quad
E_k=\varepsilon+\frac{2\pi\ell_k}{N\lambda_0},\quad \ell_k\in\mathbb Z,\ \ell_k\equiv k\ (\mathrm{mod}\ N),
$$
Eq. (2.12), for the start $\lvert0\rangle$:
$$
\min\Delta C_K=\frac{2\pi}{\lambda_0}\sqrt{\frac{N^2-1}{12N^2}},\quad\text{attained iff } \{\ell_k\}=\{c,c+1,\dots,c+N-1\},\ c\in\mathbb Z .
$$
## Assumptions
- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5), and $\lambda\in\mathbb R$ (A9).
- Readings are applied at stated values of $\lambda$ (A8). In item 1 they serve only to show that the value of $\lambda$ does not matter. Items 2–4 use readings on the eigenstate $\Psi$, which needs no value of $\lambda$.
## Scope
- In scope:
- one clock of the form above and the rest $R$, a general finite system;
- eigenstates of $C$ and their conditional states;
- crossing contracts only as far as item 3 needs them.
- Out of scope:
- comparison of clocks (criterion 3), moving clocks and inertia (criteria 4–5);
- clocks of other forms, and clocks whose type also occurs in $R$;
- events and causal order (criterion 7), and whether the order of $\lambda$ is detectable (criterion 8);
- any physical meaning of $\lambda$ beyond the definitions (M1).
## Depth
- Item 1: short argument.
- Item 2: derive.
- Item 3: short argument for the sufficient condition; derive the counterexample.
- Item 4: short argument.
## Expected result
- Item 1: an equivalence; a yes/no answer.
- Item 2: an identity; a condition, an equivalence and closed forms; a set of values of $\lambda$.
- Item 3: a yes/no answer, a sufficient condition with $U$, and an explicit counterexample.
- Item 4: a conversion rule and its conditions.
Give every main result a tag $(3.k)$.
Consistency checks, at most three: for example $N=2$, an $R$ whose state is an eigenvector of $C_R$, and the normalization of the place probabilities.
## Code
None.