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Quantum field quantities and compatible SI normalizations — Examples

1 · Prerequisites

2 · Summary

Two-level measurement and formal-coefficient boundary constructs a two-dimensional Hilbert space, a normalized state, and a two-outcome projective Kraus instrument. It computes the conditional model probabilities and state updates, including the zero-probability boundary. A separate factorial-coefficient example shows that formal coefficients exist even when the evaluated power series has radius zero; the coefficient alone is not assigned a measurement probability. No observed outcome is reported.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Thought experimentStatement: Literature-sourcedProof: AI-adaptedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Two-level measurement and formal-coefficient boundary

Physical scope

Conditional ideal two-level instrument; no observed outcome and no formal coefficient identified as an exact probability.

Statement

Assume ${A C}_{ω}$ . At one fixed measurement time, take the constructed two-dimensional Hilbert space $H = C^{2}$ with inner product $⟨ x, y ⟩ = x_{0} \overline{y_{0}} + x_{1} \overline{y_{1}}$ , linear in the first variable. Let $e_{0} = (1, 0)$ , $e_{1} = (0, 1)$ and $v = α e_{0} + β e_{1}$ with $∣ α ∣^{2} + ∣ β ∣^{2} = 1$ . Prepare the rank-one density operator $ρ = T_{v, v}$ , where $T_{v, v} x = ⟨ x, v ⟩ v$ as in Rank-one operator calculus. Define the two measurement operators

$P_{0} = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}), P_{1} = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}), M_{0, 1} = P_{0}, M_{1, 1} = P_{1} .$

This is a finite normalized instrument. The postulate Physical interpretation of a supplied constructed quantum model assigns model probabilities $p_{0} = ∣ α ∣^{2}$ and $p_{1} = ∣ β ∣^{2}$ , which sum to one. When $p_{i} > 0$ its conditional state is $P_{i}$ ; when $p_{i} = 0$ the normalized update is undefined. This is a conditional model calculation, not a report of an observed result.

For the formal comparison, let $g$ be an indeterminate; the formal algebra assigns it no SI unit or numerical value. The formal series

$F (g) : = \sum_{n = 0}^{\infty} n! g^{n} \in R ⟦ g ⟧$

has a coefficient for every $n$ , while its numerical power series has radius zero after the real coefficients are embedded in $C$ for evaluation. This example separates coefficient data from numerical convergence. The Born postulate does not assign an exact probability to a coefficient in isolation: an additional map from formal data to a constructed state and measurement, together with any approximation and error assumptions, would be needed. No such map or observed outcome is claimed.

Facts & Assumptions

Given: ${A C}_{ω}$ ; the model state and finite instrument above; the usual finite-product definition of $n!$ ; and the numerical convention $g^{0} = 1$ for every $g$ , including $g = 0$ .

[A1]

Physical interpretation of a supplied constructed quantum model: as the Born interpretation postulate for the selected model, it reads $p_{a} (t) = tr (J_{a} (ρ_{t})) = tr (ρ_{t} E_{a})$ as the model probability for outcome $a$ .

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): for every family of nonempty sets indexed by $N$ , there is a function selecting one element from each set.

[A3]

For numerical evaluation, $n! : = \prod_{k = 1}^{n} k$ with $0! : = 1$ and $g^{0} : = 1$ for all $g$ . Thus $n! > 0$ in the real coefficient field for each finite $n$ .

[A4]

The numerical convergence radius used here is $R : = \sup {r \geq 0 : the series converges for every g \in C with ∣ g ∣ < r}$ .

[A5]

This thought experiment supplies no map from the formal coefficient sequence to a state, instrument, or controlled approximation.

[A6]

Physical interpretation of a supplied constructed quantum model: when $p_{a} (t) > 0$ , adopt $J_{a} (ρ_{t}) / p_{a} (t)$ as the model's conditional state; at zero probability no normalized update is assigned.

[A7]

Physical interpretation of a supplied constructed quantum model: the physical reading is adopted only for an explicitly selected model whose mathematical data have actually been constructed.

[L1]

Finite bounded Kraus instrument: normalization means $\sum_{a, r} M_{a, r}^{*} M_{a, r} = I_{H}$ , and the outcome operator and operation are $E_{a} = \sum_{r} M_{a, r}^{*} M_{a, r}$ and $J_{a} (X) = \sum_{r} M_{a, r} X M_{a, r}^{*}$ . The Hilbert adjoint satisfies $⟨ M x, y ⟩ = ⟨ x, M^{*} y ⟩$ .

[L2]

Finite bounded instruments normalize measurement probabilities: assuming ${A C}_{ω}$ , a unit vector $ψ$ gives the rank-one density operator $T_{ψ, ψ}$ , with outcome probability $p_{a} = \sum_{r} ∥ M_{a, r} ψ ∥^{2}$ ; for positive probability the conditional update is a density operator, and at zero probability it is undefined.

[L3]

Rank-one operator calculus: the rank-one map is $T_{u, v} x = ⟨ x, v ⟩ u$ ; for a unit vector $x$ , $T_{x, x}$ is positive and trace class with trace one.

[F1]

Real and complex inner-product spaces and their induced length: the chosen complex inner product is linear in its first argument, conjugate-linear in its second, and has $∥ x ∥^{2} = ⟨ x, x ⟩$ .

[F2]

Hilbert space: a complex Hilbert space is a complex inner-product space complete in its induced norm.

[F3]

The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts: the complex plane with its Euclidean metric is complete.

[F4]

Formal power series over a commutative ring and the coefficient-extraction functional $[x^{n}]$ : for a commutative ring $R$ , a formal power series in $R ⟦ g ⟧$ is a coefficient function; $g$ is an indeterminate, and formal membership asserts no analytic convergence or chosen value of $g$ .

[F5]

The Cauchy-sequence reals are Archimedean: the Cauchy-sequence reals are Archimedean: for every $x \in R$ there is a natural number $N$ with $x < N$ .

[F6]

The reals form a totally ordered field: the repository real numbers form a totally ordered field, and hence a commutative coefficient ring for [F4].

[F7]

Real and imaginary parts, complex conjugation, and modulus: if $z = a + b i$ , then $∣ z ∣ = \sqrt{a^{2} + b^{2}}$ , so $∣ z ∣ \geq 0$ and $∣ z ∣ = 0$ only for $z = 0$ .

Proof

1.1givenF1F7algebra

The displayed pairing is linear in the first variable by coordinatewise addition and scalar multiplication, is conjugate-symmetric by complex conjugation, and satisfies $⟨ x, x ⟩ = ∣ x_{0} ∣^{2} + ∣ x_{1} ∣^{2} \geq 0$ , with equality only for $x = 0$ . Thus it is a positive-definite complex inner product with norm $∥ x ∥ = (∣ x_{0} ∣^{2} + ∣ x_{1} ∣^{2})^{1 / 2}$ .

1.2givenA3F4F6algebra

Take the coefficient ring to be the repository real field, which is a commutative ring by [F6], and define $a_{n} : = n!$ for each $n \in N_{0}$ . Each coefficient is a finite product of real integers, so $(a_{n})_{n \geq 0}$ is a coefficient function and hence $F (g) \in R ⟦ g ⟧$ by [F4]. This formal membership does not assert convergence.

1.3givenA1A5A7F4algebra

By [F4], a formal coefficient is an entry in a coefficient function and does not itself select a numerical argument or a model. By [A1], the postulate assigns outcome probabilities only through its model Born rule; by [A5] and [A7], this formal example supplies no state, instrument, or map from the coefficient sequence to such data. Therefore the postulate assigns no exact probability to an isolated coefficient; this does not exclude a separate model that supplies such a map.

2.1givenF2F3F7step 1.1algebra

Let $(x_{n})$ be Cauchy in the norm of step 1.1, with coordinates $x_{n} = (z_{n, 0}, z_{n, 1})$ . For each $j$ , the estimate $∣ z_{n, j} - z_{m, j} ∣ \leq ∥ x_{n} - x_{m} ∥$ makes $(z_{n, j})$ Cauchy in $C$ . By [F3] it converges to some $z_{j}$ . Then $∥ x_{n} - (z_{0}, z_{1}) ∥^{2} = ∣ z_{n, 0} - z_{0} ∣^{2} + ∣ z_{n, 1} - z_{1} ∣^{2} \to 0$ . So $C^{2}$ is complete, and is a complex Hilbert space by [F2]; it is nonzero since $e_{0} \neq 0$ .

2.2givenA3F5F6F7step 1.2algebra

At $g = 0$ , the numerical sum equals $1$ by [A3]. Fix any nonzero complex $g$ and let $u_{n} : = n! g^{n}$ . By [F5] choose a natural $N > 2 / ∣ g ∣$ . For every $n \geq N$ , $∣ u_{n + 1} ∣ / ∣ u_{n} ∣ = (n + 1) ∣ g ∣ > 2$ , so $∣ u_{n} ∣ \geq ∣ u_{N} ∣ > 0$ . Thus the terms do not tend to zero. If the partial sums converged, their consecutive differences $u_{n}$ would tend to zero, a contradiction. Therefore the numerical series diverges for every nonzero $g$ .

3.1givenA2F1F7L1step 2.1algebra

For $P_{0}$ , both inner products in the symmetry identity equal $x_{0} \overline{y_{0}}$ ; for $P_{1}$ , both equal $x_{1} \overline{y_{1}}$ . Thus $P_{i}^{*} = P_{i}$ . Multiplying the diagonal matrices gives $P_{i}^{2} = P_{i}$ and $P_{0} + P_{1} = I_{2}$ . For $x = (x_{0}, x_{1})$ , $∥ P_{0} x ∥^{2} = ∣ x_{0} ∣^{2} \leq ∥ x ∥^{2}$ and $∥ P_{1} x ∥^{2} = ∣ x_{1} ∣^{2} \leq ∥ x ∥^{2}$ , so both operators are bounded. With two outcomes and one Kraus operator each, $\sum_{i = 0}^{1} M_{i, 1}^{*} M_{i, 1} = P_{0} + P_{1} = I_{2}$ ; [L1] therefore identifies the data as a normalized finite instrument on the Hilbert space of step 2.1.

3.2givenA2F1F7L2L3step 2.1algebra

The normalization of $v$ gives $⟨ v, v ⟩ = ∣ α ∣^{2} + ∣ β ∣^{2} = 1$ . By [L3], $T_{v, v} x = ⟨ x, v ⟩ v$ ; substituting $⟨ x, v ⟩ = x_{0} \overline{α} + x_{1} \overline{β}$ gives $ρ = T_{v, v} = (\begin{array}{cc} ∣ α ∣^{2} & α \overline{β} \\ β \overline{α} & ∣ β ∣^{2} \end{array})$ . The formula in [L3] gives the displayed matrix, and [L2] identifies this unit-vector rank-one operator as a density operator under ${A C}_{ω}$ .

3.3givenA4step 2.2algebra

By the radius definition [A4], for every $r > 0$ the nonzero real number $g = r / 2$ lies inside that disk. Step 2.2 proves divergence at this $g$ , so no positive $r$ belongs to the defining set. The value $r = 0$ belongs vacuously because its disk is empty; hence $R = 0$ .

4.1givenA1A2F7L2step 3.1step 3.2algebra

The actions are $P_{0} v = α e_{0}$ and $P_{1} v = β e_{1}$ , so their squared norms are $∣ α ∣^{2}$ and $∣ β ∣^{2}$ . The pure-state formula in [L2] gives $p_{0} = ∣ α ∣^{2}$ and $p_{1} = ∣ β ∣^{2}$ , whose sum is one. By [A1] these are the adopted model probabilities; each lies in $[0, 1]$ .

5.1givenA6L1L2L3step 3.1step 3.2step 4.1algebra∎

From [L1], $J_{0} (ρ) = P_{0} ρ P_{0}$ and $J_{1} (ρ) = P_{1} ρ P_{1}$ . Using the matrix in step 3.2, $P_{0} ρ P_{0} = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}) (\begin{array}{cc} ∣ α ∣^{2} & α \overline{β} \\ β \overline{α} & ∣ β ∣^{2} \end{array}) (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}) = ∣ α ∣^{2} P_{0},$ while $P_{1} ρ P_{1} = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}) (\begin{array}{cc} ∣ α ∣^{2} & α \overline{β} \\ β \overline{α} & ∣ β ∣^{2} \end{array}) (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}) = ∣ β ∣^{2} P_{1} .$ Also $P_{i} = T_{e_{i}, e_{i}}$ by [L3], so [L2] makes $P_{i}$ a density operator. If $p_{0} > 0$ , $J_{0} (ρ) / p_{0} = P_{0}$ ; if $p_{0} = 0$ , division is undefined. The same argument gives $P_{1}$ when $p_{1} > 0$ and no normalized update when $p_{1} = 0$ , exactly as adopted in [A6].

5 · Examples, experiments and counterexamples

None yet.