# Question: 05-moving-clock

- **Subproject:** 04-ilang-time
- **Package:** 05-moving-clock
- **Equation tags:** (5.k)
- **Created:** 2026-10-10

## Goal

Determine how the readings of a clock carried by a body depend on the body's motion in the background of a recording medium:
- when they do not depend on it;
- how they depend on it for a body whose contract couples motion and clock through two anticommuting terms;
- how the rate of the carried clock is related to the body's velocity and to the maximal speed of 03-ilang-space.

This is criterion 4, without the comparison of two clocks between two meetings, which is left to a later package.

**Setting.** As in 03-ilang-space/17-motion, with places that carry a clock label (quoted under "Inputs").
- **Objects.** A body $b$, the only instance of its type, and a medium $c$, one object of another type.
- **Places.** In item 1, the places of $b$ are pairs $h=(x,a)$; in items 2–4, triples $h=(x,s,a)$.
  - $x$ is a position label: from a finite set in item 1, and $x\in\mathbb Z$ in items 2–4.
  - $s\in\{1,2\}$ is a branch label (items 2–4 only).
  - $a\in\mathbb Z_N$, $N\ge2$, is a clock label.
- **Contract.** $C=C_b\otimes\mathbb 1_c+\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$, where $K_h$ depends only on the position label: $K_{(x,a)}=K_x$, respectively $K_{(x,s,a)}=K_x$. Places with the same $x$ therefore have the same record vector $u_h$ and lie in one point of the background.
- **Clock reading.** The reading on $b$ with one class $S_a$ per clock label, the set of all places with that label. Its statistics are $q(a;\lambda)$.
- **Start.** $\lvert\phi\rangle\otimes\lvert\chi\rangle$.

1. **Decoupled body.** Let $C_b=H\otimes\mathbb 1_N+\mathbb 1\otimes D'$, with $H$ self-adjoint on the position labels and $D'$ self-adjoint on the clock labels. Let the start be $\phi=\phi_X\otimes\eta$, with any $\chi$ and any $K_x$. Determine $q(a;\lambda)$ for every $\lambda$. Determine whether it depends on $H$, on $\phi_X$ or on the $K_x$.

2. **Model with anticommuting terms: branches and velocity.** Places $(x,s,a)$, $x\in\mathbb Z$, as the limit of long finite chains with the start far from both ends (A1). State how the limit is taken. Let
   $$C_b=A+B,\qquad A=\frac\tau2\sum_x\Bigl(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert\otimes(i\sigma_1)\Bigr)\otimes\mathbb 1_N,\qquad B=\sum_x\lvert x\rangle\langle x\rvert\otimes\sigma_3\otimes(m\mathbb 1_N+D),$$
   with $\tau>0$ and $m>0$. Here $\sigma_1,\sigma_3$ are the Pauli matrices on the branch label, and $D=\frac{2\pi}{N\lambda_0}\sum_{k=0}^{N-1}k\lvert f_k\rangle\langle f_k\rvert$ is the minimal ring clock of 02-clock on the clock label, with tick $\lambda_0>0$. Let the records be $K_x=\kappa\,x\,X$, with $\kappa>0$ and $X$ as in (17.15).
   - (a) Determine $AB+BA$. Determine the eigenvalues and eigenvectors of $C_b$ in terms of the plane waves of the chain, for each eigenvalue $d_k$ of $D$.
   - (b) **Weak records.** Work to leading order in $\kappa$ at fixed $\lambda$: the view of $b$ is that of $C_b$ alone, and positions are measured in the background (17.15). Determine the velocity $v$ (17-motion) of a start concentrated near a plane-wave label $p_0$ in the branch of positive eigenvalue for the clock level $d_k$. Determine $v$ as a function of $p_0$, its largest value over $p_0$, and the bound (17.7) for this $C_b$, all in units of $\kappa\sigma_X$.

3. **The moving clock.** Same model and regime. The start is concentrated near $p_0$ in the branch of positive eigenvalue, and the clock label is in the place $\lvert0\rangle$.
   - Determine $q(a;\lambda)$ to first order in $\frac{2\pi}{N\lambda_0 m}$, the level spacing of $D$ relative to $m$, and to leading order in the spread of the start around $p_0$.
   - Determine the tick $\lambda_0(p_0)$: the value of $\lambda$ at which, to this order, the clock reading has advanced by one place with certainty.
   - Determine its ratio to the tick at $p_0=0$.

4. **Rate and velocity.** Express the ratio of item 3 as a function of the velocity $v$ of item 2(b):
   - exactly on the chain;
   - and to leading order for small $p_0$.

   Compare the result with the ratio of $v$ to the bound (17.7) and to the largest velocity of item 2(b).

## Inputs

From 02-clock@v2. Notation fixed there: $S\lvert m\rangle=\lvert m+1\rangle$, $\lvert f_k\rangle=N^{-1/2}\sum_me^{2\pi ikm/N}\lvert m\rangle$; a uniform ring clock has $e^{-iC_K\lambda_0}\lvert m\rangle=e^{i\theta_m}\lvert m+1\rangle$ with tick $\lambda_0$, and $\Delta C_K$ is the spread of $C_K$.

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 $\lvert0\rangle$:

$$
\min\Delta C_K=\frac{2\pi}{\lambda_0}\sqrt{\frac{N^2-1}{12N^2}},\quad\text{attained iff } \{\ell_k\}=\{c,c+1,\dots,c+N-1\},\ c\in\mathbb Z .
$$

Eq. (2.13), the place-reading distribution of a self-driven uniform ring clock of a minimal member, started at $\lvert0\rangle$:

$$
p(m;\lambda)=\frac{\sin^2(\pi\lambda/\lambda_0)}{N^2\sin^2\bigl(\pi(\lambda/\lambda_0-m)/N\bigr)}\qquad(\text{value }1\text{ where the denominator vanishes}).
$$

From 03-ilang-space/05-recording-contract@v1. These hold in the setting of 05: one recorded object, one medium object, the recording contract $\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$, and the start $\lvert\phi\rangle\otimes\lvert\chi\rangle$ with $\phi_h\neq0$ for all $h$. Notation fixed there: $g(v,w):=\operatorname{Re}\langle v\vert w\rangle$, and $h\sim_0h'$ iff $u_h=u_{h'}$.

Eq. (5.10):

$$
\lim_{\lambda\to0^+}\frac{\alpha(h,h';\lambda)}{\lambda}=d_0(h,h')=\bigl\lVert u_h-u_{h'}\bigr\rVert,
\qquad \lvert u_h\rangle=\bigl(K_h-\langle K_h\rangle\bigr)\lvert\chi\rangle .
$$

Eq. (5.11):

$$
\bigl(H_V/{\sim_0},\,d_0\bigr)\;\cong\;\bigl(\{u_h:h\in H_V\},\,\lVert\cdot\rVert\bigr)\subset(\mathcal H_c,g).
$$

From 03-ilang-space/17-motion@v2. Setting there: $C=C_b\otimes\mathbb 1_c+\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$, $C_b=\sum_{h,h'}t_{hh'}\lvert h\rangle\langle h'\rvert$ with $t_{hh'}=t_{h'h}^*$; the hopping graph joins $h\neq h'$ iff $t_{hh'}\neq0$; the start is $\lvert\phi\rangle\otimes\lvert\chi\rangle$; $\langle X\rangle:=\langle\chi\vert X\vert\chi\rangle$, $\lvert u_h\rangle:=(K_h-\langle K_h\rangle)\lvert\chi\rangle$, $d_0(h,h')=\lVert u_h-u_{h'}\rVert$. Definitions fixed there, with $p_h(\lambda)$ the place weights of the view of $b$:

$$\bar x(\lambda):=\sum_hp_h(\lambda)\,u_h,\qquad s(\lambda)^2:=\sum_hp_h(\lambda)\,\bigl\lVert u_h-\bar x(\lambda)\bigr\rVert^2,\qquad v(\lambda):=\frac{\mathrm d\bar x}{\mathrm d\lambda}.$$

Eq. (17.7), with $W$ the witness weight of the view of $b$, $q_h:=\sqrt{p_h}$, and $a_{hh'}:=\lvert t_{hh'}\rvert d_0(h,h')$ with largest eigenvalue $\varrho(a)$; the bound holds for every start and every $\lambda$:

$$
\lVert v(\lambda)\rVert\le\sum_{\{h,h'\}\in E}2\lvert t_{hh'}\rvert d_0(h,h')\sqrt{W(h,h';\lambda)p_hp_{h'}}\le\sum_{h,h'}a_{hh'}q_hq_{h'}\le v_{\max}:=\varrho(a)\le\max_h\sum_{h'}\lvert t_{hh'}\rvert\,d_0(h,h') .
$$

Eq. (17.15), for the chain $h_m$, $m\in\mathbb Z$, with $K_{h_m}=m\,X$, $X$ self-adjoint on $\mathcal H_c$, $\sigma_X:=(\langle X^2\rangle-\langle X\rangle^2)^{1/2}>0$ and $e:=(X-\langle X\rangle)\chi$, so that $\lVert e\rVert=\sigma_X$ and $u_{h_m}=m\,e$:

$$
d_0(h_m,h_n)=\lvert m-n\rvert\,\sigma_X,\qquad \bar x=\bar m\,e,\quad \bar m:=\sum_mm\,p_{h_m},\qquad s^2=\sigma_X^2\sum_m(m-\bar m)^2p_{h_m} .
$$

Eq. (17.20), the bound (17.7) for the chain with $C_b=t\sum_m(\lvert h_m\rangle\langle h_{m+1}\rvert+\lvert h_{m+1}\rangle\langle h_m\rvert)$, $t>0$:

$$
v_{\max}=2t\,\sigma_X .
$$

## Assumptions

- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5), and $\lambda\ge0$.
- No object other than $b$ has the type of $b$, and $c$ is the only instance of its type.
- Items 2–4 use the weak-record regime stated in item 2(b); state every place where it is used.

## Scope

- In scope:
  - the decoupled body in general (item 1);
  - the model of items 2–4;
  - the clock reading of the carried clock;
  - starts concentrated near one plane-wave label in one branch.
- Out of scope:
  - the comparison of two carried clocks between two meetings;
  - inertia (criterion 5, a separate package) and a common maximal speed for all types (criterion 6);
  - branches of negative eigenvalue, and starts that mix the two branches, beyond what item 3 needs;
  - effects of the records beyond leading order in $\kappa$;
  - any physical meaning of $\lambda$ beyond the definitions (M1).

## Depth

- Item 1: short argument.
- Item 2: derive. Standard facts on plane waves and group velocity may be used, stated.
- Item 3: derive.
- Item 4: derive.

## Expected result

- Item 1: a closed form and a yes/no answer for each dependence.
- Item 2: an identity and closed forms; a closed form, its maximum, and the value of the bound.
- Item 3: a closed form to the stated order, a closed form for $\lambda_0(p_0)$, and a ratio.
- Item 4: an exact relation, its small-$p_0$ form, and a comparison.

Give every main result a tag $(5.k)$.

Consistency checks, at most three: for example $p_0=0$, $N=2$, and the limit $m\to0$ or $\tau\to0$.

## Code

None.

Author: Ivan Honis · ivan.honis@ndot.io · License: CC BY 4.0 · Human–AI note