# Question: 10-global-order

- **Subproject:** 04-ilang-time
- **Package:** 10-global-order
- **Equation tags:** (10.k)
- **Created:** 2026-10-10

## Goal

Determine which features of the ordering parameter $\lambda$ can be detected through readings: its unit, its direction, and the relation between the values of $\lambda$ at which readings on different parts are made, in particular a common "now" (equal $\lambda$) and the order of two events. This is criterion 8.

**Setting.**
- **Objects and parts.** Finitely many objects of pairwise different types, each the only instance of its type, so $\mathcal H_{\rm adm}=\mathcal H$. The objects are divided into two nonempty disjoint parts $\mathcal P_1$, $\mathcal P_2$, and $\mathcal H=\mathcal H_1\otimes\mathcal H_2$.
- **Contract.** $C$ is self-adjoint, independent of $\lambda$ and a sum of pair terms (A5). The parts are **uncoupled** if $C=C_1\otimes\mathbb 1+\mathbb 1\otimes C_2$ with self-adjoint $C_1$, $C_2$; otherwise they are **coupled**.
- **Start.** A unit vector $\Psi_0$ at $\lambda=0$; only $\lambda\ge0$ is used (A9 restricted).
- **Readings.** Readings are applied at stated values of $\lambda$ (A8, first representation); no reading is represented by a record inside the description. A local reading on a part is a partition of the joint places of that part (A4).
- **Schedule.** A finite list of **events** $e_k=(i_k,R_k,\lambda_k)$, $k=1,\dots,K$: a part index $i_k\in\{1,2\}$, a local reading $R_k$ on $\mathcal P_{i_k}$, and a value $\lambda_k\ge0$. The readings are applied by (1.3) in the order of increasing $\lambda_k$, and between them the state evolves by (1.9). Events with equal $\lambda$ are applied in any order; state why the order does not matter. The **statistics** of the schedule is the joint distribution of its $K$ outcomes.
- **Re-timing.** Two numbers $a_1,a_2>0$ and a contract $C'$ on the same objects (self-adjoint, independent of $\lambda$, a sum of pair terms). The re-timed schedule has the same events with $\lambda_k$ replaced by $a_{i_k}\lambda_k$.
- **Equivalence.** A re-timing $(a_1,a_2,C')$ is an **equivalence of $C$** if, for every start $\Psi_0$ and every schedule, the statistics of the re-timed schedule under $C'$ equal the statistics of the original schedule under $C$.
- **Detectable.** For a given $C$:
  - the **unit of $\lambda$** is detectable if every equivalence of $C$ has $a_1=a_2=1$;
  - the **common now** is detectable if every equivalence of $C$ has $a_1=a_2$. (A re-timing with $a_1\neq a_2$ changes which events on different parts have equal $\lambda$, and reverses the order of some pairs of events on different parts.)

1. **Unit and origin.**
   - (a) Determine how the statistics of a schedule behave under the re-timing $a_1=a_2=a$, $C'=C/a$. Determine whether the unit of $\lambda$ is detectable, for any $C$.
   - (b) Let the start be an eigenvector of $C$. Determine how the statistics of a schedule change when all $\lambda_k$ are increased by the same amount, and how the statistics of a schedule with a single event depend on its $\lambda$.

2. **Direction.** Take a schedule with two events on different parts, with readings $R$ at $\lambda_1$ and $R'$ at $\lambda_2>\lambda_1$.
   - Determine, for every $C$ and every start, whether the marginal distribution of the outcome of $R$ depends on whether $R'$ is made, and whether the marginal distribution of the outcome of $R'$ depends on whether $R$ is made.
   - State what this implies for the detectability of the direction of increasing $\lambda$, and which element of the formalization (the laws, (1.1)–(1.11), A1–A9) fixes that direction.

3. **Uncoupled parts.**
   - (a) Determine a family of equivalences with $a_1\neq a_2$ that holds for every start, whether or not it is a product of states of the two parts.
   - (b) Determine whether the common now is detectable. For two events on different parts with $\lambda_1<\lambda_2$, determine whether an equivalence exists after which their values of $\lambda$ are equal, and one after which their order is reversed.

4. **Coupled parts.**
   - (a) For an equivalence $(a_1,a_2,C')$ of a general $C$, determine the conditions that follow from schedules with a single event, at first and at second order in $\lambda$. Express them through commutators of $C$ and $C'$ with the place projectors of the two parts.
   - (b) Determine a sufficient condition on $C$ under which every equivalence has $a_1=a_2$. Determine a sufficient condition under which an equivalence with $a_1\neq a_2$ exists although the parts are coupled.
   - (c) For each of the following three contracts, determine all equivalences, and whether the common now is detectable. Each part is one object with two places; $\sigma^x$, $\sigma^z$ are the Pauli matrices in the place basis, with $\sigma^z$ diagonal, and $g,h,t,\kappa>0$.
     - (i) $C=g\,\sigma^x\otimes\sigma^x$.
     - (ii) $C=g\,\sigma^z\otimes\sigma^z+h\,\sigma^x\otimes\mathbb 1$.
     - (iii) $C=t\,\sigma^x\otimes\mathbb 1+\lvert2\rangle\langle2\rvert\otimes\kappa\,\sigma^x$, where $\lvert2\rangle$ is the second place of $\mathcal P_1$: a body with two places and a hopping term, whose place is recorded by a medium.

5. **The order of two events on a chain.** Setting of 07-causal-order (quoted under "Inputs"): a chain of $n$ cells, a reading $R_i$ on cell $i$ at $\lambda_1$ with projectors $\Pi_s$ and a reading $R_j$ on cell $j$ at $\lambda_2=\lambda_1+\delta$, $\delta\ge0$, with projectors $\Pi'_t$; $r=\lvert i-j\rvert\ge1$, $\Psi_1$ is the state at $\lambda_1$ and $Q_t=\Pi'_t(\delta)$. The joint statistics are $P(s,t)=\lVert Q_t\Pi_s\Psi_1\rVert^2$. Define the **order-exchanged statistics** $\tilde P(s,t):=\lVert\Pi_sQ_t\Psi_1\rVert^2$: the two readings, both carried to $\lambda_1$ by the evolution, applied in the opposite order.
   - (a) Determine the necessary and sufficient condition under which $P=\tilde P$ for every start. Determine its relation to the influence (7.1) of $R_i$ on $R_j$.
   - (b) Determine an upper bound on $\sum_{s,t}\lvert P(s,t)-\tilde P(s,t)\rvert$, valid for every start, in terms of $r$, $\delta$, $\lVert J\rVert$ and the numbers of classes of the two readings. Determine the set of $(r,\delta)$ on which this bound is below a given $\epsilon>0$, and compare it with (7.15).
   - (c) State what (a) and (b) imply for the detectability of the order of two events on different cells: at equal $\lambda$, at a small delay over a large distance, and in general.

## Inputs

From 07-causal-order@v2 (this subproject). Setting there: a chain of $n\ge2$ cells, objects $1,\dots,n$ of pairwise different types, with $C=\sum_{i=1}^nD_i+\sum_{i=1}^{n-1}J_i$, where $D_i$ acts on cell $i$ alone, $J_i$ is a pair term on cells $i$ and $i+1$, and $\lVert J\rVert:=\max_i\lVert J_i\rVert$. Notation fixed there:
- a reading $R_i$ on cell $i$ at $\lambda_1$ with outcomes $s$ and projectors $\Pi_s$ ($m$ classes), a reading $R_j$ on cell $j$ at $\lambda_2$ with outcomes $t$ and projectors $\Pi'_t$; all extended by the identity;
- $\delta:=\lambda_2-\lambda_1$, $r:=\lvert i-j\rvert$, $\Psi_1$ the state at $\lambda_1$;
- $X(\delta):=e^{iC\delta}Xe^{-iC\delta}$, $Q_t:=\Pi'_t(\delta)$ and $\mathcal M_i(X):=\sum_s\Pi_sX\Pi_s$;
- $P_j(t\mid R_i)$ is the probability of the outcome $t$ of $R_j$ averaged over the outcomes of $R_i$, and $P_j(t)$ the same probability without $R_i$; $R_i$ **influences** $R_j$ if $P_j(\cdot\mid R_i)\neq P_j(\cdot)$, and $\delta P:=\sum_t\lvert P_j(t\mid R_i)-P_j(t)\rvert$.

Eq. (7.1):

$$
\Delta_t:=P_j(t\mid R_i)-P_j(t)=\langle\Psi_1\vert\,\mathcal M_i(Q_t)-Q_t\,\vert\Psi_1\rangle,\qquad \mathcal M_i(X)-X=\sum_s\Pi_s\,[X,\Pi_s].
$$

Eq. (7.2):

$$
P_j(\cdot\mid R_i)=P_j(\cdot)\qquad\text{whenever }\lambda_2\le\lambda_1,\ \text{for all cells } i,j,\ \text{all starts and all readings.}
$$

Eq. (7.11), for an operator $A$ on cell $j$ and an operator $B$ on cell $i$ at distance $r$, uniform in $n$ and independent of the $D_l$:

$$
\bigl\lVert[A(\delta),B]\bigr\rVert\le2\lVert A\rVert\lVert B\rVert\sum_{k=r}^{\infty}\frac{(2\lVert J\rVert\delta)^k}{k!}\qquad(\delta\ge0,\ r\ge1).
$$

Eq. (7.12), for every start, every pair of readings and every $\delta\ge0$:

$$
\delta P\le2\Bigl(1-\frac1m\Bigr)\sum_{k\ge r}\frac{(2\lVert J\rVert\delta)^k}{k!}\le2\sum_{k\ge r}\frac{(2\lVert J\rVert\delta)^k}{k!}\le\frac{2\,(2\lVert J\rVert\delta)^r}{r!}\,e^{2\lVert J\rVert\delta}.
$$

Eq. (7.15), the region in which $\delta P<\epsilon$ is guaranteed; for $\lVert J\rVert>0$, $\delta_\epsilon(r)$ is the root of $2\sum_{k\ge r}(2\lVert J\rVert\delta)^k/k!=\epsilon$:

$$
\mathcal S_\epsilon:=\Bigl\{(r,\delta):\ r\ge1,\ \delta\ge0,\ 2\sum_{k\ge r}\frac{(2\lVert J\rVert\delta)^k}{k!}<\epsilon\Bigr\}=\bigl\{(r,\delta):\ 0\le\delta<\delta_\epsilon(r)\bigr\}.
$$

## Assumptions

- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5).
- No two objects have the same type, so every partition of joint places is an admissible reading and every state after a reading is admissible.
- Equivalences are required for every start and every schedule. "Detectable" is meant in this sense: with respect to re-timings by constant factors $a_1,a_2$ and an arbitrary contract $C'$ of the allowed form.

## Scope

- In scope:
  - the behaviour of reading statistics under re-timings of the two parts;
  - the unit, the direction and the common now of $\lambda$;
  - uncoupled parts in general, coupled parts through general conditions and the three examples;
  - the order of two events on a chain, through the order-exchanged statistics.
- Out of scope:
  - re-timings that are not linear in $\lambda$, more than two parts, and re-timings that depend on the position;
  - readings represented by records inside the description, and events defined by the readings of clocks;
  - a complete classification of the coupled contracts for which the common now is detectable;
  - bodies of the Model D type and their motion (05, 09);
  - any physical meaning of $\lambda$ beyond the definitions (M1); no theory that contains time may be used as an input (M4).

## Depth

- Item 1: short argument.
- Item 2: short argument.
- Item 3: derive.
- Item 4: derive.
- Item 5: short argument for (a) and (c); derive (b).

## Expected result

- Item 1: a statement and a yes/no answer; two statements.
- Item 2: two yes/no answers with conditions; a statement.
- Item 3: a family of equivalences; a yes/no answer and two yes/no answers.
- Item 4: conditions in closed form; two sufficient conditions; for each example the set of equivalences and a yes/no answer.
- Item 5: an equivalence of conditions; an inequality and a set; a statement.

Give every main result a tag $(10.k)$.

Consistency checks, at most three: for example $C=0$, a contract that is diagonal in the places, and example (i) against the exact evolution of two cells.

## Code

None.

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