# Question: 02-field-object

- **Subproject:** 05-ilang-free-fall
- **Package:** 02-field-object
- **Equation tags:** (2.k)
- **Created:** 2026-10-10

## Goal

Determine what it means in the language that a field acts on a body. For a general pair term between a body and a field object: determine the states of the field in which the body sees definite numbers in the place of the field's operators, what the body and the other objects see in such a state, and what they see when the field is static but not in such a state. This is criterion 1, and the static part of criterion 8.

**Setting.**
- **Objects.** A body $b$, a field $f$, and a set $r$ of further objects (for example a recording medium), possibly empty. All objects have pairwise different types and each is the only instance of its type, so $\mathcal H_{\rm adm}=\mathcal H=\mathcal H_b\otimes\mathcal H_r\otimes\mathcal H_f$.
- **Contract.**
  $$C=C_{br}\otimes\mathbb 1_f+\sum_{j\in J}B_j\otimes\mathbb 1_r\otimes\Phi_j+\mathbb 1_{br}\otimes C_f ,$$
  where:
  - $C_{br}$ is self-adjoint on $\mathcal H_b\otimes\mathcal H_r$, a sum of pair terms among $b$ and the objects of $r$; it contains the body's own term;
  - $J$ is a finite index set, the $B_j$ are self-adjoint on $\mathcal H_b$ and the $\Phi_j$ self-adjoint on $\mathcal H_f$: this is the pair term between body and field;
  - $C_f$ is self-adjoint on $\mathcal H_f$, the field's own term (booked in the pair term by A5);
  - there is no contract between $f$ and the objects of $r$.

  The lapse-type pair term of the base problem is the case $J=\{x\}$ (the position labels of the body), $B_x=\lvert x\rangle\langle x\rvert\otimes G$.
- **Start.** $\Psi_0=\Psi_{br}\otimes\varphi$ at $\lambda=0$, with $\Psi_{br}$ any unit vector of $\mathcal H_b\otimes\mathcal H_r$ (a product or not) and $\varphi$ a unit vector of $\mathcal H_f$. Only $\lambda\ge0$ is used.
- **Sharp.** $\varphi$ is **sharp** for the family $\{\Phi_j\}$ if $\Phi_j\varphi=n_j\varphi$ for all $j\in J$, with real numbers $n_j$.
- **Readings.** Local readings on $b$ and on the objects of $r$, applied directly at stated values of $\lambda$ (A8), not represented by record objects. A **schedule** is a finite list of such readings with their values of $\lambda$; its statistics is the joint distribution of the outcomes. No reading is made on the field.
- **Reduced description.** A description without the field: the objects $b$ and $r$, a contract $C_{\rm red}$ on $\mathcal H_b\otimes\mathcal H_r$, and the start $\Psi_{br}$.

1. **Sharp and stationary field.** Let $\varphi$ be sharp with the numbers $n_j$, and let $C_f\varphi=e_f\varphi$.
   - (a) Determine $\Psi(\lambda)$ for every $\Psi_{br}$ and every $\lambda\ge0$. Determine the view of the part $\{b\}\cup r$ and the view of $\{f\}$, and whether the field becomes entangled with the other objects.
   - (b) Determine whether there is a reduced description whose schedules have, for every $\Psi_{br}$, the same statistics as the schedules of the full description; if there is, determine its contract $C_{\rm red}$.

2. **Sharp field in general.** Let $\varphi$ be sharp, without the condition $C_f\varphi=e_f\varphi$. Assume that the operators $\mathbb 1_b$ and $B_j$, $j\in J$, are linearly independent; say what changes if they are not.
   - (a) Determine the necessary and sufficient condition on $(\varphi,C_f,\{\Phi_j\})$ under which the view $V$ of $\{b\}\cup r$ is pure for every $\Psi_{br}$ and every $\lambda\ge0$. Determine whether a reduced description as in item 1(b) then exists, and its contract.
   - (b) Give an example in which $\varphi$ is sharp and the condition fails. For a sharp $\varphi$ for which the condition fails, determine the lowest power of $\lambda$ at which $1-\operatorname{Tr}V^2$ can be nonzero, with its coefficient in terms of the operators and the start.

3. **Static field that is not sharp.** Let the $\Phi_j$ commute with each other and with $C_f$, and let $\varphi$ be arbitrary.
   - (a) Determine the view of $\{b\}\cup r$ at every $\lambda\ge0$, for every $\Psi_{br}$, in terms of reduced descriptions.
   - (b) Determine the statistics of every schedule in terms of the statistics of reduced descriptions. Determine whether readings on $b$ and $r$ can distinguish this start from a start in which the field is sharp with numbers drawn at random with suitable probabilities.
   - (c) Determine the necessary and sufficient condition under which the view of $\{b\}\cup r$ is pure for every $\Psi_{br}$ and every $\lambda\ge0$. When it is not, determine the purity $\operatorname{Tr}V^2$ of that view in terms of the reduced evolutions.

4. **Conventions (M6).** Let the operators $\mathbb 1_b$ and $B_j$, $j\in J$, be linearly independent.
   - (a) Determine all replacements of $C_{br}$, of the $\Phi_j$ and of $C_f$, at fixed $B_j$, that leave $C$ unchanged. Determine how the numbers $n_j$ and the contract $C_{\rm red}$ of item 1 change under them.
   - (b) Determine which of the following are well defined: the numbers $n_j$; their differences $n_j-n_{j'}$; the contract $C_{\rm red}$. For a lapse-type pair term and a set $r$ that contains a medium recording the position of the body through $\sum_x\lvert x\rangle\langle x\rvert\otimes K_x$, determine the same for the on-site block of $C_{\rm red}$ at a position $x$ and for the differences of these blocks between two positions. Here every replacement that leaves $C$ unchanged counts, including those that move a term diagonal in the positions of the body between the body's own term, its pair term with the medium and its pair term with the field (A5, M6).

5. **One object as recording medium and as field.** Let there be no separate field object: the body is recorded by a medium $c$ through $\sum_x\lvert x\rangle\langle x\rvert\otimes K_x$ with $K_x=\kappa\,x\,X$ as in (17.15), and the same pair term between $b$ and $c$ also contains the lapse-type term $\sum_x\lvert x\rangle\langle x\rvert\otimes G\otimes\Phi_x$ with $\Phi_x=\mathbb 1_c+g\,x\,Y$, where $Y$ is self-adjoint on $\mathcal H_c$, $[X,Y]=0$, $g\neq0$, and the medium has no further term. The medium starts in $\chi$.
   - (a) Determine the condition on $\chi$ under which the medium is sharp for the $\Phi_x$ and, at the same time, different positions have different record vectors ($d_0>0$ in (17.15)). Determine what the factor $\Phi_x$ then amounts to for the body.
   - (b) When the medium is not sharp for the $\Phi_x$, determine the view of the body in terms of reduced descriptions, and which profiles $n_x$ and which gradients of the records occur together in them.

## Inputs

From 03-ilang-space/17-motion@v2. Setting there: one body $b$ and a medium $c$, $C=C_b\otimes\mathbb 1_c+\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$; the start is $\lvert\phi\rangle\otimes\lvert\chi\rangle$; $\langle X\rangle:=\langle\chi\vert X\vert\chi\rangle$, record vectors $\lvert u_h\rangle:=(K_h-\langle K_h\rangle)\lvert\chi\rangle$, distances $d_0(h,h')=\lVert u_h-u_{h'}\rVert$, and $p_h$ the place weights of the view of $b$.

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 $u_{h_m}=m\,e$:

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

## 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.
- The field starts unentangled with the other objects ($\Psi_0=\Psi_{br}\otimes\varphi$). The state $\varphi$ of the field is an assumption of each item and is named in it (A11, M9).

## Scope

- **Class.** Items 1–4 are asked for every contract of the stated form (every $C_{br}$, every finite family $B_j$, $\Phi_j$, every $C_f$) and every $\Psi_{br}$; the state $\varphi$ is as each item states. Item 5 is asked for the stated pair term and every $\chi$.
- **Controls.** Item 2(b) and item 3 are the controls of item 1: field states for which the body does not see definite numbers.
- In scope:
  - the exact evolution for a sharp field, and the reduced description;
  - the condition for a pure view of the body and the other objects;
  - a static field in an arbitrary state;
  - the conventions in the split between body and field;
  - one object as recording medium and field, in the stated form.
- Out of scope:
  - a field whose operators $\Phi_j$ do not commute with its own term, beyond item 2 (a later package);
  - which operator $G$ of a body the field multiplies, and the motion and the clock of a body in a given profile (03-clock-and-fall);
  - the source of the field and any effect of the body on the field's later action on other bodies;
  - starts in which the field is entangled with the other objects.

## Depth

- Item 1: short argument.
- Item 2: derive.
- Item 3: derive.
- Item 4: short argument.
- Item 5: short argument.

## Expected result

- Item 1: a closed form for the state and the two views, and a yes/no answer; a yes/no answer with a contract.
- Item 2: an equivalence with a contract; an example, a power of $\lambda$ and a coefficient.
- Item 3: a closed form; a closed form and a yes/no answer; an equivalence and a closed form.
- Item 4: the general form of the replacements and the transformation of each quantity; a list of well-defined quantities.
- Item 5: a condition and a statement; a closed form and a statement.

Give every main result a tag $(2.k)$.

Consistency checks, at most three: for example $\Phi_j=\mathbb 1_f$ for all $j$, a field with one place, and a two-place field with one index $j$.

## Code

None.

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