13.8 The Unread Collapse Channel: Random Kicks and Energy–Momentum Cost
Conventions for continuous outcomes. From here on, an instrument with outcome space (Ω, 𝔉), a
standard Borel space such as ℝ or ℝⁿ, is a map Δ ↦ I(Δ) from 𝔉 to normal completely positive maps on
trace-class operators, countably additive in trace norm on every state, with I(Ω) trace-preserving.
Its Heisenberg duals I*(Δ) are normal, and the dual N* = I*(Ω) of the unread channel is unital. The
Gaussian collapse of Proposition 13.12 is such an instrument, with Ω = ℝ and I(Δ)(ρ) = ∫_Δ K_y ρ K_y
dy, the integral converging in trace norm. All statements about continuous outcomes below are made
in this sense.
Proposition 13.15 (The unread Gaussian collapse is a random kick). Let A be a self-adjoint operator,
possibly unbounded, σ > 0, and K_y = (2πσ²)^{−1/4} exp(−(A − y)²/(4σ²)) for y ∈ ℝ.
(a) The operators K_y are bounded, y ↦ K_y is strongly continuous, and ∫ K_y² dy = 1. The
outcome y is distributed as the Born-rule distribution of A convolved with a centred Gaussian of
variance σ².
(b) The unread channel is a Gaussian mixture of unitary kicks generated by A:
N(ρ) = ∫ K_y ρ K_y dy = ∫ γ(t) e^{−itA} ρ e^{itA} dt, γ(t) = (2σ²/π)^{1/2} exp(−2σ²t²),
so the kick parameter t is centred Gaussian with variance 1/(4σ²). In a spectral representation
of A, N multiplies the kernel ρ(a, a′) by exp(−(a − a′)²/(8σ²)), the characteristic function of
γ evaluated at a − a′.
Proof. (a) Boundedness and strong continuity follow from the functional calculus. For every vector ψ
with spectral measure μ_ψ of A, ∫ ⟨ψ, K_y² ψ⟩ dy = ∫∫ (2πσ²)^{−1/2} exp(−(a − y)²/(2σ²)) dy dμ_ψ(a)
= ‖ψ‖², and the same integrand, integrated over a only, is the convolution formula for the outcome
density. (b) By the spectral theorem in multiplication form, the Hilbert space is L²(X, ν) with A
acting as multiplication by a real function a. A trace-class ρ is Hilbert–Schmidt, with a kernel
ρ(ξ, ξ′). The kernel of K_y ρ K_y is κ_y(a(ξ)) ρ(ξ, ξ′) κ_y(a(ξ′)), with κ_y(a) = (2πσ²)^{−1/4}
exp(−(a − y)²/(4σ²)); completing the square as in the proof of Proposition 13.12 and integrating
over y gives exp(−(a(ξ) − a(ξ′))²/(8σ²)) ρ(ξ, ξ′). The kernel of e^{−itA} ρ e^{itA} is e^{−it(a(ξ) −
a(ξ′))} ρ(ξ, ξ′), and ∫ γ(t) e^{−itd} dt = exp(−d²/(8σ²)). The two trace-class operators therefore
have the same kernel and coincide. □
The selective instrument and its unread channel must be kept apart. Selectively, a collapse with
outcome y multiplies the spectral amplitude of A by a Gaussian of width σ centred at y and genuinely
narrows the distribution of A. The unread channel is a random-unitary channel, and it has a second
unravelling in which the recorded outcome is the kick parameter t. That record is independent of the
state, and each kick leaves the distribution of A unchanged. The two unravellings localize
differently and have the same channel, which is Proposition 13.2 in its sharpest form. No-signaling,
a property of the channel, can therefore be established in the random-kick picture, while
localization, a property of the unravelling, cannot. For a free field, e^{itφ(h)} W(g) e^{−itφ(h)} =
e^{−itE(h,g)} W(g), so each kick multiplies Weyl operators by phases. This is the mechanism behind
Proposition 13.12, and it is what Theorem 13.17 below abstracts.
Proposition 13.16 (Energy–momentum cost of Gaussian collapse). Let N be the unread channel of
Proposition 13.15.
(a) Position collapse of a particle. Take one degree of freedom with [Q, P] = iħ and a real
potential V that is locally integrable and bounded below, and let H be the self-adjoint operator
of the closed quadratic form q(ψ) = ‖Pψ‖²/(2m) + ⟨ψ, Vψ⟩ on the form domain D(P) ∩ D(|V|^{1/2}).
Call a state ρ = Σ_i p_i |ψ_i⟩⟨ψ_i| of finite energy if every ψ_i lies in the form domain and
ℰ(ρ) = Σ_i p_i q(ψ_i) is finite. For A = Q, if ρ has finite energy, then so does N(ρ), and
ℰ(N(ρ)) − ℰ(ρ) = ħ² / (8mσ²),
independently of ρ and of V. Collapsing all three coordinates of a particle in three dimensions
costs 3ħ²/(8mσ²).
(b) Quadratic systems. For finitely many canonical degrees of freedom R = (q_1, …, q_n, p_1, …,
p_n), let H = ½ Rᵀ M R with M real, symmetric, and positive definite, and let A be linear in R.
Then [A, [A, H]] is a multiple of the identity and, for every state of finite energy in the
sense of (a),
ℰ(N(ρ)) − ℰ(ρ) = −(1/(8σ²)) [A, [A, H]].
(c) Free fields in 3 + 1 dimensions. For a free scalar field of mass m ≥ 0 on four-dimensional
Minkowski space in its Fock representation, with energy–momentum P^μ, a real test function h
with Eh ≠ 0, and A = φ(h), every state ρ of finite energy, Tr(ρ H) < ∞, satisfies Tr(N(ρ) P^μ) =
Tr(ρ P^μ) + ΔP^μ(h) for all four generators, where
ΔP^μ(h) = (1/(4σ²)) ∫ dμ_m(k) k^μ |ĥ(k)|², dμ_m(k) = d³k / ((2π)³ 2ω_k),
with k⁰ = ω_k = (|k|² + m²)^{1/2} and ĥ(k) = ∫ h(x) e^{ik·x} d⁴x. Equivalently, ΔP^μ(h) is the
conserved energy–momentum of the classical solution Eh, divided by 4σ². It does not depend on
the state; it is unchanged when h is translated; it transforms as a four-vector, ΔP(h ∘ Λ^{−1})
= Λ ΔP(h); and it is future-directed and timelike.
(d) Curved backgrounds. Let φ be the minimally coupled free scalar field of mass m ≥ 0 on a
four-dimensional globally hyperbolic spacetime, ω a Hadamard state, and h a real test function.
Then the state ω_N = ω ∘ N* after an unread Gaussian collapse of φ(h) is Hadamard, its
symmetrized two-point function is that of ω plus (Eh ⊗ Eh)/(4σ²), and its renormalized
stress–energy tensor, defined by Hadamard point-splitting, is
⟨T_μν⟩_{ω_N} = ⟨T_μν⟩_ω + T^cl_μν[Eh] / (4σ²),
where T^cl_μν[u] = ∇_μu ∇_νu − ½ g_μν (∇^αu ∇_αu − m²u²) is the classical stress–energy tensor,
in signature (+, −, −, −).
Proof. In each case N*(B) = ∫ γ(t) e^{itA} B e^{−itA} dt by Proposition 13.15(b). (a) The unitary
e^{−itQ} preserves the form domain: it commutes with V, and the Weyl relation e^{itQ} P e^{−itQ} = P
− ħt gives P e^{−itQ} = e^{−itQ}(P − ħt) on D(P). Hence q(e^{−itQ}ψ) = q(ψ) − (ħt/m)⟨ψ, Pψ⟩ +
ħ²t²‖ψ‖²/(2m), where ⟨ψ, Pψ⟩ is finite on the form domain. The energy functional ℰ is affine, and,
being a supremum of normal functionals given by spectral truncations of the bounded-below operator
H, it satisfies ℰ(∫ γ(t) ρ_t dt) = ∫ γ(t) ℰ(ρ_t) dt by monotone convergence. Averaging over t, with
γ of mean zero and variance 1/(4σ²), gives the result. (b) The unitary e^{itA} is a Weyl operator of
the canonical system. It preserves the form domain of H and shifts R by t times a constant vector s,
so H becomes H + t L + t² H(s), with L linear in R and H(s) the value of the quadratic form on s.
Averaging gives H(s)/(4σ²), and differentiating twice at t = 0 shows that −[A, [A, H]] = 2H(s). (c)
For real t, e^{itφ(h)} φ(f) e^{−itφ(h)} = φ(f) − t E(h, f), so conjugation by e^{itφ(h)} displaces
the field by the classical solution −t Eh. Under a displacement by a classical solution u of finite
energy, the normal-ordered generators transform as P^μ ↦ P^μ + t L^μ + t² P^μ[u], with L^μ linear in
the creation and annihilation operators and P^μ[u] the classical energy–momentum of u. Because ω_k ≥
|k^i| mode by mode, −H ≤ P^i ≤ H as quadratic forms, so a state of finite energy has finite
expectations of all four generators, and N(ρ) again has finite energy. The linear terms have the
form L^μ = a(g_μ) + a(g_μ)*, with ω^{−1/2} g_μ square-integrable because ĥ decays rapidly on the
mass shell, and the bound |Tr(ρ a(g))| ≤ ‖ω^{−1/2} g‖ Tr(ρ H)^{1/2} makes their expectations finite.
Averaging over t therefore gives ΔP^μ = P^μ[Eh]/(4σ²). In the mode expansion, φ(h) = ∫ dμ_m(k)
(ĥ(k)* a_k + ĥ(k) a_k†) with [a_k, a_q†] = (2π)³ 2ω_k δ³(k − q), and the same computation gives the
stated integral. Translating h multiplies ĥ by a phase. Since ĥ ∘ Λ^{−1} is the transform of h ∘
Λ^{−1} and dμ_m is Lorentz-invariant, ΔP transforms as a four-vector. The Fourier transform of Eh is
proportional to ĥ on the mass shell, so Eh ≠ 0 exactly when ĥ is not identically zero there. Then ΔP
is an integral of future-directed causal vectors with a nonnegative weight that is not almost
everywhere zero. For m > 0 these vectors are timelike. For m = 0 they are null, but ĥ is real-
analytic and the future light cone in 3 + 1 dimensions, with its vertex removed, is connected, so ĥ
is nonzero on an open subset of the cone containing non-parallel null directions, and their sum is
timelike. (d) By Proposition 13.12, ω_N(W(f)) = exp(−E(h, f)²/(8σ²)) ω(W(f)). Differentiating
ω_N(W(sf + tg)) twice at s = t = 0 shows that the symmetrized two-point function changes by E(h, f)
E(h, g)/(4σ²), whose integral kernel is (Eh)(x)(Eh)(y)/(4σ²), a smooth function because Eh is a
smooth solution; the antisymmetric part, iE/2, is unchanged. Adding a smooth symmetric kernel
preserves the wavefront-set condition that defines Hadamard states. In Hadamard point-splitting, the
renormalized stress–energy tensor is obtained by applying a fixed bidifferential operator to the
two-point function minus a state-independent Hadamard parametrix, taking the coincidence limit, and
adding local curvature terms that are the same for every state (Wald 1994; Hollands & Wald 2015).
The difference between ω_N and ω is therefore that operator applied to (Eh ⊗ Eh)/(4σ²), whose
coincidence limit is T^cl_μν[Eh]/(4σ²). □