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Number, energy and momentum laws of a one-photon preparation

Physical scope

The outcome laws of the assigned radiation observables in a declared one-photon preparation of the model: the number observable has the sharp value one; the frequency and free-field energy laws are the push-forwards of the transverse momentum density under k -> c*|k| and k -> hbarc|k|, with means whenever the first moment is finite; and the one-photon momentum-component readouts are the push-forwards under k -> hbar*k_j, with means whenever the corresponding absolute first moment is finite. The frequency, energy and momentum-component variances are finite and strictly positive on their multiplier domains. The result is conditional on P1, P2 and the adopted radiation model, records no detector, no sampling model and no measured number, and shows that no normalisable one-photon vector has sharp frequency, energy or momentum.

Statement

Adopt the state and composition postulate P1 and the observable and probability postulate P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes), the radiation extension postulate (Radiation extension of the framework: transverse mode and photon assignments) and the radiation sector definition The radiation sector of the framework: modes and the number, energy and momentum assignments, in the state vocabulary of The state space of the framework: preparations, pure rays and density operators and the observable vocabulary of Assigned sharp observables and the outcome law of the framework. Assume Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice), including the hypotheses inherited from the radiation postulate and the cited operator and spectral items. Let $f \in h$ be a unit one-photon wave-packet vector. Under the $d^{3} k$ normalization of clause (b) of The radiation sector of the framework: modes and the number, energy and momentum assignments, its coordinate representatives $f (k)$ have units $m^{3 / 2}$ . Put $ε (k) = ℏ c ∣ k ∣$ , and let $ρ_{f}$ be the one-photon preparation of the ray of $ι_{1} (f)$ . Then:

(a) number law: the outcome law of the assigned total-number observable $(N, 1)$ in $ρ_{f}$ is $μ_{ρ_{f}}^{N} (B) = 1_{B} (1)$ for every Borel set $B \subseteq R$ : the number readout of the one-photon preparation is one with probability one;

(b) frequency and energy laws: the assigned frequency law $μ_{ρ_{f}}^{ω}$ and the outcome law of the assigned free-field energy observable $(H_{f}, J)$ in $ρ_{f}$ are the push-forwards of $∥ f (k) ∥_{C^{3}}^{2} d^{3} k$ under $k \mapsto c ∣ k ∣$ and $k \mapsto ℏ c ∣ k ∣$ , respectively: $μ_{ρ_{f}}^{ω} (B) = \int_{{k : c ∣ k ∣ \in B}} ∥ f (k) ∥_{C^{3}}^{2} d^{3} k, μ_{ρ_{f}}^{H_{f}} (B) = \int_{ε^{- 1} (B)} ∥ f (k) ∥_{C^{3}}^{2} d^{3} k (B \subseteq R Borel),$ and if $\int_{R^{3}} ∣ k ∣ ∥ f (k) ∥_{C^{3}}^{2} d^{3} k < \infty$ , their spectral-law means are $\int_{R} ω d μ_{ρ_{f}}^{ω} (ω) = c \int_{R^{3}} ∣ k ∣ ∥ f (k) ∥_{C^{3}}^{2} d^{3} k, \int_{R} E d μ_{ρ_{f}}^{H_{f}} (E) = ℏ c \int_{R^{3}} ∣ k ∣ ∥ f (k) ∥_{C^{3}}^{2} d^{3} k;$

(c) momentum readouts: for $j \in {1, 2, 3}$ the one-photon momentum-component readout of clause (d)(3) of The radiation sector of the framework: modes and the number, energy and momentum assignments has, in the identification of the one-photon sector with $h$ by $ι_{1}$ , the outcome law $μ_{ρ_{f}}^{P_{j}} (B) = \int_{{k : ℏ k_{j} \in B}} ∥ f (k) ∥_{C^{3}}^{2} d^{3} k (B \subseteq R Borel),$ the push-forward of $∥ f ∥^{2} d^{3} k$ under $k \mapsto ℏ k_{j}$ , with mean $ℏ \int k_{j} ∥ f (k) ∥^{2} d^{3} k$ whenever $\int ∣ k_{j} ∣ ∥ f (k) ∥^{2} d^{3} k < \infty$ ;

(d) no sharp frequency, energy or momentum: for every unit vector $f \in D (ε_{h})$ , the variances of the frequency and energy laws of (b) are strictly positive, and for every unit vector in $D (M_{ℏ k_{j}}^{h})$ the momentum-component variance of (c) is strictly positive; consequently the mode assignment "photon energy $E = ℏ ω$ , $ω = c ∣ k ∣$ " is a spectral assignment of the model and never an exact value of a normalisable one-photon state;

(e) all assertions are conditional on P1, P2 and the adopted radiation model; they record no detector model, no sampling model, no measured count and no empirical claim.

Facts & Assumptions

Given: the postulates P1, P2 and the adopted radiation model with transverse space $h$ , sectors $H_{n}^{+}$ , number operator $N$ , free-field energy $H_{f}$ with $ε (k) = ℏ c ∣ k ∣$ , one-photon momentum readouts $M_{ℏ k_{j}}^{h}$ , and a unit $f \in h$ .

[F1]

Sector structure. $Π_{n}$ is the orthogonal projection of $F (h)$ onto the closed subspace $ι_{n} (H_{n}^{+})$ , the ranges are pairwise orthogonal, $\sum_{n} Π_{n} = I$ in the strong operator topology, and $∥ ψ ∥^{2} = \sum_{n} ∥ Π_{n} ψ ∥^{2}$ (Fock sector projections clauses (a)-(c)).

[F2]

Number operator. $N$ is self-adjoint with $D (N) = {ψ : \sum_{n} n^{2} ∥ ψ_{n} ∥^{2} < \infty}$ , $N ι_{n} (u) = n ι_{n} (u)$ for every $u \in H_{n}^{+}$ , and $σ (N) = N$ (Number operator domain clauses (a),(b)).

[F3]

Spectral atoms. For a self-adjoint operator $A$ with spectral measure $E_{A}$ , $E_{A} ({λ})$ is the orthogonal projection onto the eigenspace $\ker (A - λ)$ ; for a unit eigenvector $u$ at $λ$ and a pure state $ρ = T_{u, u}$ one has $μ_{ρ}^{A} ({λ}) = 1$ (Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a),(b)).

[F4]

Free-field energy. $H_{f} = ⨁_{n \geq 0} T_{n}$ with $T_{n}$ self-adjoint on $ran P_{+}^{(n)}$ , $T_{n} ψ = ε_{n} ψ$ , is self-adjoint and positive with $D (H_{f}) = {ψ : ψ_{n} \in D (T_{n}) \forall n, \sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty}$ ; its one-particle restriction is $ε_{h}$ (Transverse continuum free Fock Hamiltonian clauses (a)-(c)).

[F5]

The exact free-field direct-sum spectral measure. For the actual sector operators $T_{n}$ of [F4], including $T_{0} = 0$ on $C$ , write $E_{n}$ for their regular spectral PVMs. Then $E_{H_{f}} (B) = ⨁_{n \geq 0} E_{n} (B), E_{0} (B) = 1_{B} (0) I_{C} .$ This is proved locally in Transverse continuum free Fock Hamiltonian, proof 5.1: strong countable additivity follows by finite-sector truncation and a squared-norm tail bound; scalar regularity follows by finite compact/open approximations after discarding a tail of measure; nonnegative integration gives $\int t^{2} d E_{ψ} = \sum_{n} \int t^{2} d (E_{n})_{ψ_{n}}$ . Thus the identity integral has exactly the domain ${ψ : ψ_{n} \in D (T_{n}) \forall n, \sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty}$ , and bounded spectral truncations converge there to $(T_{n} ψ_{n})_{n}$ . Uniqueness of the regular representing PVM identifies the result with $E_{H_{f}}$ . The same supplier proves that $E_{1} (B)$ is the restriction of the inverse-image indicator multiplier for $ε (k)$ to $h$ . These exact proved energy statements, rather than an unsupported generic PVM-sum assertion, are what step 1.2 uses.

[F6]

Restricted multipliers. For $m \in {ℏ c ∣ k ∣, ℏ k_{j}}$ the operator $M_{m}^{h}$ is self-adjoint on $h$ with spectral measure $E_{m} (B) = M_{1_{m^{- 1} (B)}} ∣_{h}$ , the law of a unit $g \in h$ is the push-forward of $∥ g ∥^{2} d^{3} k$ under $k \mapsto m (k)$ , and $M_{m}^{h}$ has no eigenvalues; for unit $g \in D (M_{m}^{h})$ the variance is strictly positive (Restricted momentum and energy multipliers of the transverse one-photon space clauses (a)-(d)).

[F7]

Means and variances for pure states. For a self-adjoint $A$ and a unit $u \in D (A)$ the mean of the outcome law in the pure state $T_{u, u}$ is $⟨ A u, u ⟩$ ; the variance is $∥ A u ∥^{2} - ⟨ A u, u ⟩^{2}$ when $u \in D (A)$ (Assigned sharp observables and the outcome law of the framework clause (e)).

[F8]

Monotone convergence. For nonnegative measurable functions $g_{n} ↑ g$ , their integrals against a measure increase to the integral of $g$ (Monotone convergence for the integral).

Proof

1.1F1F2F3algebra

[Number law] For $n \in N$ the eigenspace of $N$ at $n$ is ${ψ \in D (N) : N ψ = n ψ}$ ; comparing coordinates with the definition $(N ψ)_{m} = m ψ_{m}$ , this eigenspace equals $ι_{n} (H_{n}^{+})$ , the closed subspace of vectors supported in coordinate $n$ . Hence by [F3] the spectral projection is $E_{N} ({n}) = Π_{n}$ , and for the one-photon state $ρ_{f} = T_{ι_{1} f, ι_{1} f}$ we get $μ_{ρ_{f}}^{N} ({n}) = tr (ρ_{f} Π_{n}) = ⟨ Π_{n} ι_{1} f, ι_{1} f ⟩ = δ_{n, 1}$ by [F1]; since the atoms cover $N$ and the law is a measure, $μ_{ρ_{f}}^{N} (B) = 1_{B} (1)$ for every Borel set $B$ . This proves (a).

1.2F4F5F6algebra

[Frequency and energy laws] By [F5] the spectral measure of $H_{f}$ is $E (B) = ⨁_{n} E_{T_{n}} (B)$ , and on the one-photon sector $ι_{1} H_{1}^{+}$ it acts as $ι_{1} E_{T_{1}} (B) ι_{1}^{- 1}$ ; since $T_{1} = ε_{h}$ by [F4], the restriction is the spectral measure $E_{ε} (B) = M_{1_{ε^{- 1} (B)}} ∣_{h}$ of [F6]. Therefore $μ_{ρ_{f}}^{H_{f}} (B) = ⟨ E (B) ι_{1} f, ι_{1} f ⟩ = ⟨ E_{ε} (B) f, f ⟩ = \int_{ε^{- 1} (B)} ∥ f (k) ∥^{2} d^{3} k .$ The assigned frequency law is the push-forward under $k \mapsto c ∣ k ∣$ by clause (e) of The radiation sector of the framework: modes and the number, energy and momentum assignments. These are the laws in (b), and they exist for every unit $f \in h$ .

2.1F6F7F8step 1.2algebra

[Push-forward integration and frequency and energy first moments] Let $ν_{f} (A) : = \int_{A} ∥ f (k) ∥^{2} d^{3} k$ , a probability measure, and, for any real measurable $m$ , let $μ_{m} (B) : = ν_{f} (m^{- 1} (B))$ . For an indicator $g = 1_{C}$ , the defining identity gives $\int g d μ_{m} = \int g (m (k)) d ν_{f} (k)$ ; linearity gives it for nonnegative simple $g$ , and increasing simple approximation with [F8] gives it for every nonnegative Borel $g$ . Applying this identity to $g (t) = t$ for the nonnegative frequency and energy maps gives the means in (b), finite under the stated first-moment condition. If also $f \in D (ε_{h})$ , the energy mean agrees with $⟨ ε_{h} f, f ⟩$ by [F7]; the first-moment condition alone need not put $f$ in that operator domain.

3.1F6F7step 2.1algebra

[Momentum readouts and means] The same sector restriction gives $μ_{ρ_{f}}^{P_{j}} (B) = ⟨ E_{m} (B) f, f ⟩ = \int_{{m \in B}} ∥ f ∥^{2} d^{3} k$ for $m (k) = ℏ k_{j}$ , the push-forward under $k \mapsto ℏ k_{j}$ . For the signed identity function, apply the push-forward integral identity proved in step 2.1 separately to its positive and negative parts; the assumed finite absolute first moment makes both integrals finite, so their difference is the mean stated in (c). If also $f \in D (M_{m}^{h})$ , this mean agrees with $⟨ M_{m}^{h} f, f ⟩$ by [F7] and the isometry of $ι_{1}$ . This proves (c).

4.1F6step 1.2step 3.1algebra

[Strict positivity of the variances] By [F6] the multiplier $M_{ℏ c ∣ k ∣}^{h}$ has no eigenvalues and every unit $f \in D (ε_{h})$ has strictly positive finite energy variance; together with step 1.2 this gives the energy variance claim. The frequency law is the positive constant rescaling $ω = E / ℏ$ , so on the same domain its variance is $ℏ^{- 2}$ times the energy variance and is strictly positive. The same clause of [F6] applied to $m = ℏ k_{j}$ gives strictly positive finite momentum-component variance whenever $f \in D (M_{ℏ k_{j}}^{h})$ . Hence no unit vector of $h$ has a sharp value of frequency, energy or any momentum component, and the assignments of the radiation postulate are spectral assignments. This proves (d).

5.1step 1.1step 1.2step 2.1step 3.1step 4.1∎

[Assembly and scope] Step 1.1 proves (a), steps 1.2 and 2.1 prove the frequency and energy claims in (b), step 3.1 proves (c), and step 4.1 proves (d). No step introduces a detector, a sampling model, a measured count or an empirical premise, which is (e).

Remarks

Reading the mean. The energy mean $ℏ c \int ∣ k ∣ ∥ f ∥^{2}$ is the first moment of the transverse momentum density weighted by $ℏ c$ ; it is a preparation mean of the model, not a measured energy. It exists whenever $\int ∣ k ∣ ∥ f ∥^{2} d^{3} k < \infty$ , which is weaker than $f \in D (ε_{h})$ ; on that operator domain it also equals $⟨ ε_{h} f, f ⟩$ .
Why "photon energy $ℏ ω$ " remains meaningful. For a model preparation concentrated near a wavevector $k_{0}$ , the energy law is concentrated near $ℏ c ∣ k_{0} ∣$ in the sense of the push-forward measure; the item claims no limit statement and no localisation, and the exact value is never attained by a normalisable vector.

Number, energy and momentum laws of a one-photon preparation

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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