9.3 Continuous-Time Histories as Marked Point Processes
In continuous time a history consists of event labels together with their times, and the truth
profile contains a continuum of state propositions. Both must be defined explicitly.
Definition 9.3 (Records and their law). Fix Ψ_0 ∈ H_phys and t > 0. A record on [0, t] is a finite
sequence ω = ((t_1, k_1), …, (t_m, k_m)) with m ≥ 0, 0 < t_1 < ⋯ < t_m ≤ t, and k_j ∈ 𝒦. The record
space Ω_t is the disjoint union over m of the sets of such sequences, equipped with the reference
measure given by Lebesgue measure on ordered m-tuples of times and counting measure on labels. Along
ω, the state at time s ∈ [0, t] is the normalized vector
Ψ_s(ω) ∝ e^{−iH(s − t_j)} J_{k_j} e^{−iH(t_j − t_{j−1})} ⋯ J_{k_1} e^{−iH t_1} Ψ_0
for t_j ≤ s < t_{j+1},
with t_0 = 0 and t_{m+1} = ∞, so that the state is right-continuous at each jump.
Proposition 9.4 (Density of the record). The law of the record on [0, t] has density, with respect
to the reference measure,
p_t(ω) = e^{−Λt} ‖J_{k_m} e^{−iH(t_m − t_{m−1})} ⋯ J_{k_1} e^{−iH t_1} Ψ_0‖².
The admissible records are the positive-density records, 𝒜_t = { ω ∈ Ω_t : p_t(ω) > 0 }.
Proof. The hit times of a Poisson process of rate Λ have density Λ^m e^{−Λt} on ordered m-tuples in
(0, t]. Given the hit times, the labels are drawn successively with probabilities ‖J_{k_j} φ_j‖² /
Λ, where φ_j is the normalized state just before t_j. The product of these probabilities times Λ^m
is Π_j ‖J_{k_j} φ_j‖². Because the evolution between hits is unitary, this product telescopes into
the squared norm of the unnormalized vector shown. □
Definition 9.5 (Continuous-time truth profile). The past-tensed language in continuous time consists
of state propositions Had(s, [Ψ]), read “at time s the state was [Ψ]”, and event propositions Had(s,
k), read “at time s event k occurred”. For ω ∈ 𝒜_t the truth profile at time t is
L_t(ω) = { Had(s, [Ψ_s(ω)]) : 0 ≤ s < t } ∪ { Had(t_j, k_j) : 1 ≤ j ≤ m }.
An event at time t_m = t is counted as having occurred: it belongs to the generation of the present
state, which is right-continuous.
Remark 9.6 (Pointwise and almost-sure statements). Two kinds of statement about records must be kept
apart. Pointwise statements hold for every record in a specified set: Definitions 9.3 and 9.5 assign
a state path and a truth profile to every record, Theorem 9.7 characterizes the profiles of all
admissible records, and Propositions 9.8 and 9.9 and Theorem 9.10 hold for every admissible record.
Probabilistic statements hold almost surely, that is, outside a set of records of probability zero.
Because p_t is a density, an individual record has probability zero. Admissibility means positive
density, p_t(ω) > 0; it does not mean that ω has positive probability, and positivity of a density
is not in general the same as membership in the topological support of the law. Throughout,
“admissible” is used only in the positive-density sense. In the finite-width model every Kraus
operator is invertible, so every record has positive density. In particular, records with an event
exactly at the terminal time t form a set of probability zero: for a fixed t, the Poisson process
almost surely has no point at t. Such records are admitted so that the profile at time t is defined
for every record, with right-continuous states, and so that the profiles of a single trajectory are
nested at all times; no probabilistic statement depends on them. Since every result in this section
is proved pointwise for all admissible records, it holds in particular almost surely. The
conditional law in Proposition 9.9 is likewise given by an explicit density defined for every
record, which is a version of the conditional distribution; any other version agrees with it almost
surely.
Theorem 9.7 (Admissible continuous-time profiles). A set L of propositions belongs to { L_t(ω) : ω ∈
𝒜_t } if and only if: (o) every element of L is a state proposition Had(s, [Ψ]) with s ≥ 0 or an
event proposition Had(s, k) with s > 0; (a) the event propositions are finitely many, with distinct
times in (0, t]; (b) for each s ∈ [0, t) there is exactly one state proposition with time s, and
none with time s ≥ t; (c) the state proposition at time 0 is Had(0, [Ψ_0]); and (d) if ω is the
record read off from the event propositions, then p_t(ω) > 0 and, for every s ∈ [0, t), the state
proposition at time s is Had(s, [Ψ_s(ω)]). The record is then unique, and profiles are nested: if t
≤ t′, ω′ ∈ 𝒜_{t′}, and ω is the restriction of ω′ to [0, t] (its events with times at most t), then
ω ∈ 𝒜_t and L_t(ω) ⊆ L_{t′}(ω′).
Proof. As for Theorem 4.8. Conditions (o) and (a) determine a unique record ω; (d) states that ω is
admissible and that every state proposition is the one the law assigns along ω; (b) and (o) exclude
anything else. For nesting, the factorization in the proof of Proposition 9.9 gives p_{t′}(ω′) =
p_t(ω) · c with c ≥ 0, so p_{t′}(ω′) > 0 implies p_t(ω) > 0 and hence ω ∈ 𝒜_t. Restriction preserves
the event propositions with times up to t and the states at all times before t, so every element of
L_t(ω) belongs to L_{t′}(ω′); for t < t′ the later profile also contains the state proposition at
time t, which is why the inclusion can be strict. □
Proposition 9.8 (Reconstruction from an earlier time). For 0 ≤ s < t and ω ∈ 𝒜_t, the present state
S_t = [Ψ_t(ω)] is fixed by the state proposition Had(s, [Ψ_s(ω)]), the event propositions with times
in (s, t], and the law: it is obtained by applying to Ψ_s(ω) the unitary evolution between those
events and the corresponding jump operators.
Proof. Definition 9.3, applied from time s rather than time 0. □
Proposition 9.8 avoids any appeal to a “latest” earlier time, which does not exist in continuous
time: any earlier time s, together with the events since s, suffices.
Proposition 9.9 (Dynamical idleness in continuous time). For t < t′, the conditional law of the
record on (t, t′], given the complete state (S_t, L_t), depends only on S_t. Its density is e^{−Λ(t′
− t)} ‖J_{k_r} e^{−iH(t_r − t_{r−1})} ⋯ J_{k_{m+1}} e^{−iH(t_{m+1} − t)} Ψ_t‖², where (t_{m+1},
k_{m+1}), …, (t_r, k_r) are the events in (t, t′].
Proof. Write the unnormalized vector of Proposition 9.4 on [0, t′] as A B Ψ_0, where B carries Ψ_0
to the unnormalized state at time t and A carries the state from t to t′. Then ‖A B Ψ_0‖² = ‖A Ψ_t‖²
‖B Ψ_0‖² with Ψ_t = B Ψ_0 / ‖B Ψ_0‖, and e^{−Λt′} = e^{−Λt} e^{−Λ(t′ − t)}. The density on [0, t′]
therefore factorizes into the density on [0, t] and the stated factor, which depends on the past
only through Ψ_t. □
Theorem 9.10 (Continuous-time covariance, correspondence, and truth preservation). For a frame A,
let the frame-A description at time t be D^A_t = (t, [R_A Ψ_t], τ_A(L_t)), where τ_A maps Had(s,
[Ψ]) to Had(s, [R_A Ψ]) and Had(s, k) to Had(s, k, E^A_k). Then:
(a) For every record ω, the densities computed in frame A, in frame B, and in the neutral
description coincide, so the sets 𝒜_t of admissible records coincide in all frames.
(b) Along every ω ∈ 𝒜_t, the frame states satisfy [R_B Ψ_s(ω)] = [V_{A→B} R_A Ψ_s(ω)] for all s
∈ [0, t].
(c) Criterion 5.5, with stage replaced by time, Theorem 5.6, and Proposition 5.8 hold for
continuous-time descriptions, and if the descriptions of two frames correspond at time 0, the
descriptions generated from them correspond at every later time along every admissible record.
(d) Truth preservation (Theorem 5.16) and no revision across frames (Proposition 5.17) hold at
every time.
Proof. (a) By Lemma 3.7, R_A J_k = J^A_k R_A and R_A e^{−iHs} = e^{−iH^A s} R_A on H_phys. Hence the
frame-A unnormalized vector of Proposition 9.4, applied to R_A Ψ_0, equals R_A applied to the
neutral one, and its norm is the same because R_A is unitary; likewise for frame B. (b) The same
identities show that the frame-A state at time s is proportional to R_A Ψ_s(ω), and R_B = V_{A→B}
R_A on H_phys. (c) The proofs of Theorem 5.6 and Proposition 5.8 use only the unitarity of the
reductions and the bijectivity of the translations, both of which hold here; preservation of
correspondence follows from (a) and (b), because both frames add, at each time, the translations of
the same neutral propositions. (d) The proofs of Theorem 5.16 and Proposition 5.17 carry over
verbatim, using (c) and the nesting in Theorem 9.7. □
The numerical check (Table 1) simulated 200 trajectories over total time T = 5 in frames A and B
with common random numbers: the 1,158 events, and their times, were identical in the two frames, and
the final states were related by V_{A→B} to machine precision.
The resulting picture matches CTGP’s description of generation: continuous evolution of the present
state, punctuated by discrete primitive events, with each moment’s state and each event passing into
the past-tensed profile.