The formal structure

States in time

A semigroup runs forward only

A mode coupled to a bath is described by states at a time (how a time comes to be picked out is Part III). In the Markov regime its reduced state evolves under a generator of Lindblad form:

The maps $e^{t\mathcal G}$ form a semigroup: they compose forward in time, and except in the purely unitary case none has an inverse that is itself a physical, completely positive map (Lindblad 1976; Gorini, Kossakowski & Sudarshan 1976). The direction picked out at the Legendre point is now a property of the dynamics. The operators $L_r$ are the bath’s own: the operators that generate the decoherence also build the event law below.

What counts as irreversible

Paper I defines irreversibility through recoverable phase distinguishability (Def. 1, §9). Fix alternatives $|j\rangle,|k\rangle$ and the phase pair

Here $\rho_\theta(t)$ is the global state evolved from $|\psi_\theta\rangle$ and the bath, and $\mathcal E_{\rm acc}$ restricts it to the accessible algebra $\mathcal A_{\rm acc}$. That algebra is fixed by the detector architecture, causal locality and the environmental partition, not by anyone’s choice; quantum-trajectory theory needs the same specification (Wiseman & Gambetta 2012). Because the two states differ only in relative phase, $\mathcal D$ cannot be inflated by rotating populations. $\Phi_{\rm irr}$ is nondecreasing when the accessible dynamics is CP-divisible — when it can be cut into legitimate quantum steps at every time, which is the condition under which lost phase cannot come back — and the event law is stated in that scope.

The event law

For records that count events, each with jump operator $L_r$, the event intensity is

where $\rho_c$ is the state conditioned on the record so far. In general the law is the path measure of the instrument the records support, and the conditioned state is the quantum filter, which reproduces the unconditioned dynamics in the ensemble (Paper II; Belavkin 2002; Bouten, van Handel & James 2007). For an excited atom emitting into open space, $L=\sqrt{\gamma}\,\sigma_-$ and the click rate is $\gamma\langle e|\rho_c|e\rangle$. For electrons captured at a pixel, $L_x=\sqrt{\kappa}\,c_x$: the records carry no particle labels, and two electrons of the same spin are never registered at the same pixel at the same moment, because $c_xc_x=0$ (Paper II, Proposition 5).

For a record that leaves the recorded variable undisturbed (the quantum-nondemolition case), $L_\pm=\sqrt{\Lambda}\,P_\pm$ with $\sum_\pm L_\pm^{\dagger}L_\pm=\Lambda\,\mathbb I$. The total intensity is then $\Lambda=\dot\Phi_{\rm irr}$, and the evolution before the event is outcome-neutral: it does not depend on which outcome will occur. This is Paper I’s law (§§9–10):

The stopped process reproduces the reduced dephasing exactly. No additional dephasing term is imposed by the event ontology, and nothing is double-counted: decoherence is anchoring seen in the ensemble.

Which outcome

Within an outcome-neutral first-event process, agreement with the exact ensemble forces the probability of outcome $k$, given an event, to be its Born weight $p_k=\mathrm{Tr}[P_k\rho]$ (Paper I, Prop. 2). With $h_k=h\,p_k$, the first-event distribution is

If $\Phi_{\rm irr}$ stays bounded, as for a bath with a gap, a residual probability $e^{-\Phi_{\rm irr}(\infty)}$ remains that no event occurs. For single histories the conditioned Born weights form a martingale, so the single-history law and the master equation agree. The weights themselves are forced, not postulated: by one covariant chance law of the one wave, one history, sharp records with no record where the wave has no component, no signalling for every division of the world into here and elsewhere, refinable records, and continuity in the wave (Zurek’s argument in Barnum’s no-signalling form; the flagship, Appendix A; the proof in full in Paper II, Appendix A). What remains is why there is chance at all.

Release: when an event is anchored

An event is anchored when its record is released. A record $R$ formed at time $t$ is released when it commutes with every observable in the system’s accessible future, including through any return path the network provides:

Nothing the system can do afterwards can then recombine the phase the record carried away. A proved bound ties this to Paper I’s measure: the phase distinguishability any return coupling could recover is limited by the commutator, so a released record leaves no recoverable phase in any state. For a free output field every record is released as it forms, and release is the nondemolition property of the output, a theorem of quantum filtering (Belavkin). Release extends it to environments that return or evolve internally: in front of a mirror, the record that finally leaves is the one that anchors (Paper II).

Released records commute with one another, so they form a commutative algebra, and the points of that algebra are the possible record histories. Exclusivity of outcomes is therefore a theorem in this class, which leaves one primitive: one history is actual.

Entangled records

Two systems prepared in one entangled state $\rho_{AB}$ and recorded with settings $x$ and $y$ have the joint Born statistics

When the record $a$ is anchored at one end, the other end anchors from what is left of the one structure consistent with that record:

So the second outcome depends on the first. Averaged over the first outcome, the second end’s statistics do not depend on the first detector’s setting, and it makes no difference which end is taken first:

In Jarrett’s terms ACT violates outcome independence and keeps parameter independence, which is why nothing can be signalled; Shimony called this “passion at a distance.” For suitable settings these statistics violate the Bell–CHSH inequality, reaching $2\sqrt2$ against the local bound of 2, as loophole-free tests confirm. ACT is therefore nonlocal in Bell’s sense, by outcome dependence only. Records in spacelike-separated regions have no time order, and their instruments commute (Fewster & Verch 2020), so the record history needs no preferred frame. The Picture states the position in words.

Which alternatives: selection

Which alternatives can be recorded is fixed by what is physically coupled. In Paper I’s probe model a probe reads a system observable $\hat{\mathcal O}$ through $U=\exp(-i\hat{\mathcal O}\otimes\hat p_E)$, and the environment reads the probe’s position before free motion can turn momentum information into position information. Then every system operation obtainable from fragment records alone has the form

a classical post-processing of the localization instrument $\{M_y\}$, where $\varphi$ is the probe’s wavepacket; under mild conditions on $\varphi$, no coherent momentum-kick instrument can be realized from such records (Paper I, §12). The theorem follows from standard unitary dynamics, and its role is ontological rather than empirical: it identifies, from the record architecture, which of the mathematically available unravelings is the actual one. It holds when records form faster than the probe disperses.

Mass is a rate

Past the Legendre point the mass term sets a frequency. A massive excitation’s phase turns at its rest-energy rate,

de Broglie’s “internal periodic phenomenon” (1924), and a clock has been run from that rate (Lan et al. 2013). The rate is structure. Elapsed, recorded time exists only as anchored events: mass gives the field a clock, and anchoring gives the clock its readings.

Structure: one field and its modes

ACT works in quantum field theory because there the primitive object is the field. What is usually called a particle is, in the formalism, a quantum of the field, an excitation of one of its modes — not a thing that carries a wave around with it. ACT keeps the word for the anchored event that excitation leaves: the mark (The Picture). For a free scalar field the whole theory is fixed by one action over spacetime (in this part, natural units, $\hbar=c=1$):

The action assigns amplitudes to whole field histories. It singles out no time and no “now”: time enters $\mathcal L$ only as one of four coordinates, on the same footing as the other three. The Lagrangian has no privileged time, which is not the same as having no time variable: $\partial_t\phi$ appears in it, and the action integrates over $t$.

Superposition is the composition of modes

The field’s solutions are compositions of plane-wave modes. In the quantum theory the field operator is

This is a Fourier decomposition of the field, and $\hat a_{\mathbf k}^{\dagger}$ creates one quantum in mode $\mathbf k$. A state of the field is a composition of its modes. That is what superposition is, and what a quantum state is.

A single photon “in two places” shows it plainly. It is one excitation of one two-lobed mode:

In the mode basis $\{b,\,b_\perp\}$ it is one photon in one mode. Nothing here is two of anything: superposition is the composition of one wave.

The dynamics choose the decomposition

Which decomposition matters physically is not a human choice; the dynamics decide. For a free, translation-invariant field they select Fourier modes. For a field meeting an apparatus they select the alternatives the interaction can hold stable; spin is the reminder that not every relevant decomposition is literally Fourier. Once a mode is coupled to a detector and its environment, the state becomes

Entanglement between separate systems cannot be undone by re-choosing modes, so these terms are no longer modes in the strict sense. They are still terms in the decomposition of one state, and the dynamics fix which decomposition matters.

Mass is a scale

In $\mathcal L$ the mass $m$ is a scale, a parameter of the structure. Past the Legendre point it becomes a frequency: the rest-energy rate at which a massive excitation’s phase turns (Part I). Mass gives the field a clock; anchoring gives the clock its readings.

The bath at the Legendre point

The Legendre transform is where a time is picked out. The canonical momentum and the Hamiltonian,

cannot be defined until one coordinate has been chosen as time. For a closed system the step is only a change of variables. It is invertible, the Schrödinger equation governs the unanchored wave in Hamiltonian form, and nothing physical happens there. With a bath the step stops being optional. Three facts of the formalism show why.

1 · A damped system has no ordinary action principle

The equations of motion of a linear dissipative system with constant coefficients cannot be derived from a variational principle (Bauer 1931). The ways around this write a time direction into the variational problem itself. One puts an explicit exponential into the Lagrangian (Caldirola 1941; Kanai 1948):

Another doubles the histories and fixes them by initial data rather than by both endpoints (Bateman 1931; Galley 2013, the classical counterpart of the Schwinger–Keldysh formalism). The physical way out is the third: treat the damped system as part of a larger, conservative one that includes its bath. The damping then appears when the bath is integrated out, and so does everything else the bath does.

2 · The bath writes causal order and decoherence into the action together

Let a system quantity $F_A$ couple to a bath variable $X_A$:

A density matrix carries a ket and a bra, so on the closed time path of Schwinger and Keldysh each history is doubled, $\phi_\pm$, with $F_c=\tfrac12(F_++F_-)$ and $F_\Delta=F_+-F_-$. Integrating out a Gaussian bath (Feynman & Vernon 1963, the route Caldeira and Leggett took) gives the influence functional $\mathcal F=e^{iS_{\rm IF}/\hbar}$ with

The first kernel is the bath’s retarded response. It vanishes unless $x^0 > y^0$, so the past acts on the future and not the reverse: causal order, entering the action. The second is the noise kernel, and what it does to the influence functional is decoherence, measured by the decoherence exponent $\Phi_A$:

with $\Delta F_A=F_A[\phi_+]-F_A[\phi_-]$. When the coupled quantity is conserved by the system’s own dynamics (pure dephasing), the histories are fixed by the alternatives $a,a'$, and the reduced density matrix in that basis is

where $\Theta_A$ collects the phases. One spectral density fixes both kernels, and for a bath in equilibrium the fluctuation–dissipation relation ties them (Paper I, §8). Neither kernel can be had without the other: where the bath responds, it fluctuates. Causal order and decoherence enter the dynamics together.

The noise can only suppress coherence; it never amplifies it.

Together they make $\Phi_A$ a well-behaved measure: never negative and, where the bath keeps noise weight in the coupled channel, growing without bound, so the coherence factor $e^{-\Phi_A}$ goes to zero. A bath with a gap leaves $\Phi_A$ bounded; a single mode, or a small finite bath, gives back what it took (Paper I, §10).

3 · Without the bath the quantum step fails

Putting the damping into the Lagrangian by hand gives the Caldirola–Kanai Hamiltonian

Quantized with $[q,p]=i\hbar$, it guarantees the uncertainty relation only for the canonical momentum $p$. For the physical momentum $m\dot q=e^{-\gamma t}p$ the guaranteed bound shrinks as $\tfrac{\hbar}{2}e^{-\gamma t}$, and the uncertainty relation fails as the motion damps (Brittin 1950; Dekker 1981). The bath’s noise restores it (Senitzky 1960). Damping without fluctuation is not a consistent quantum theory; the bath supplies both.

An example: quantum Brownian motion

The Caldeira–Leggett model makes the pairing explicit (Paper I, §11). For a particle of mass $m$ coupled linearly to an ohmic bath, $J(\omega)=m\gamma\omega$, the high-temperature master equation is

The second term damps; the third decoheres; both carry the same $\gamma$. For alternatives a distance $\Delta x$ apart, the ratio of the two rates has no freedom:

For 1 g displaced by 1 cm at room temperature the ratio is about $4\times10^{40}$. This high-temperature form is not of Lindblad type; it is shown for the link between its coefficients, and Part I uses the Lindblad form. The Mechanism page sets out what the laboratory has seen of this.

The Legendre point is where the mathematics picks out a time. The bath is what happens there in the physics: it supplies causal order and decoherence together, and makes the passage into time forced and directed.

For a closed system, the passage from histories to states at a time is a choice of description. With a bath, it is forced, and it has a direction. The step needs a choice of time-slicing; its covariant form is listed among the open problems below.

A spin in a dephasing bath

The simplest case shows every step, from the influence functional to single events.

Setup. A spin starts in $|\psi\rangle=\alpha|{\uparrow}\rangle+\beta|{\downarrow}\rangle$, with $H_S=-\tfrac12\hbar\omega_0\sigma_z$, and couples to a Gaussian bath through $H_{\rm int}=\sigma_z\otimes X$.

The decoherence exponent. Because $\sigma_z$ commutes with $H_S$, the histories are fixed by the two alternatives, and $\Delta F=\pm2$. In the Markov limit, $N\approx2D\,\delta(t-t')$, so

which meets the bound of the theorem in Part III with equality.

The master equation. The same limit gives

with the populations $|\alpha|^2$ and $|\beta|^2$ unchanged. With the spin as the accessible system, $\mathcal D(t)=e^{-2\gamma_A t}$, so $\Phi_{\rm irr}=\Phi_A=2\gamma_A t$: in a Markov bath nothing comes back.

Events. The jump operators $L_\pm=\sqrt{\Lambda}\,P_\pm$ with $\Lambda=2\gamma_A$ generate exactly the same dissipator, since $\Lambda\,(P_+\rho P_++P_-\rho P_--\rho)=-\tfrac{\gamma_A}{2}[\sigma_z,[\sigma_z,\rho]]$. Events occur at rate $\Lambda$, whatever the state. Before an event only $H_S$ acts on the normalized state, which stays coherent. At the event the spin is recorded up with probability $|\alpha|^2$ and down with probability $|\beta|^2$.

The ensemble. A run is still unanchored with probability $S(t)=e^{-2\gamma_A t}$, so

which is the solution of the master equation. Its off-diagonal element is carried entirely by the runs not yet anchored. The simulation on the Mechanism page runs this model.

Without a bath. If the spin’s phase goes into a single mode instead, $\Phi$ rises and falls, the dynamics is not CP-divisible, and the configuration lies outside the law: nothing is released, and nothing anchors. If the bath has a gap, $\Phi$ saturates, and a residual probability $e^{-\Phi(\infty)}$ remains that no event occurs.

What is postulated, and what is open

• Postulated: one record history is actual. Global unitary dynamics are not modified, and no new field, constant, threshold or noise is introduced. The outcome register is exclusive: a stated commitment in Paper I, and a theorem in the record-algebra class of Paper II, which leaves this one primitive.
• Stated: nonlocal in Bell’s sense. Without superdeterminism or retrocausality, one history with Born weights cannot be Bell-local. ACT is nonlocal by outcome dependence only, with no signalling and no time order between distant records (Part I).
• Interpretive: not yet in time. Reading the pre-anchored field as structure without a temporal causal history is ACT’s interpretive layer. No proof on this page depends on it. The physics it points to is the difference between evolution that can be undone and evolution that cannot.
• Scope. The ontology is proposed generally. The derivations are established in Markovian, CP-divisible record-forming regimes, principally approximately nondemolition and linear–Gaussian ones.

Open

• A covariant release criterion. The Hamiltonian step uses a time-slicing. Tomonaga–Schwinger evolution between arbitrary spacelike surfaces is the route to a covariant statement of release, and Kent (2014) shows that a one-history ontology over unmodified unitary dynamics can be Lorentz covariant.
• Selection without an apparatus: what fixes the accessible algebra when no detector is specified.
• Delayed return in the continuum limit, and non-Markovian release in general.
• The energy ledger of a single event. In the ensemble, energy is exchanged with a physical bath whose dissipation pairs with its noise through the fluctuation–dissipation relation. The account for one event is open.
• Modes and alternatives in general: how the field’s mode structure maps onto the alternatives a record architecture supports, beyond the localization class the selection theorem settles.

1. K. Sonderegger, From Matter Waves to Anchored Events: Anchored Causality Theory and Quantum Brownian Measurement (Paper I, 2026), §§7–15 and appendices; Released records (Paper II, 2026).
2. P. S. Bauer, Proc. Natl. Acad. Sci. USA 17, 311 (1931); H. Bateman, Phys. Rev. 38, 815 (1931).
3. P. Caldirola, Nuovo Cimento 18, 393 (1941); E. Kanai, Prog. Theor. Phys. 3, 440 (1948).
4. C. R. Galley, Phys. Rev. Lett. 110, 174301 (2013).
5. R. P. Feynman & F. L. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963); J. Schwinger, J. Math. Phys. 2, 407 (1961); L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)].
6. A. O. Caldeira & A. J. Leggett, Physica A 121, 587 (1983).
7. W. E. Brittin, Phys. Rev. 77, 396 (1950); I. R. Senitzky, Phys. Rev. 119, 670 (1960); H. Dekker, Phys. Rep. 80, 1 (1981).
8. G. Lindblad, Commun. Math. Phys. 48, 119 (1976); V. Gorini, A. Kossakowski & E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976).
9. V. P. Belavkin, Rep. Prog. Phys. 65, 353 (2002); L. Bouten, R. van Handel & M. R. James, SIAM J. Control Optim. 46, 2199 (2007).
10. H. M. Wiseman & J. M. Gambetta, Phys. Rev. Lett. 108, 220402 (2012).
11. J. S. Bell, Physics 1, 195 (1964); J. P. Jarrett, Noûs 18, 569 (1984); A. Shimony, in S. Kamefuchi et al. (eds.), Foundations of Quantum Mechanics in the Light of New Technology (Physical Society of Japan, 1984), p. 225.
12. B. Hensen et al., Nature 526, 682 (2015); M. Giustina et al., Phys. Rev. Lett. 115, 250401 (2015); L. K. Shalm et al., Phys. Rev. Lett. 115, 250402 (2015); S. Storz et al., Nature 617, 265 (2023).
13. C. J. Fewster & R. Verch, Commun. Math. Phys. 378, 851 (2020).
14. S. Tomonaga, Prog. Theor. Phys. 1, 27 (1946); J. Schwinger, Phys. Rev. 74, 1439 (1948); A. Kent, Phys. Rev. A 90, 012107 (2014).
15. L. de Broglie, Recherches sur la théorie des quanta (thesis, Paris 1924); S.-Y. Lan et al., Science 339, 554 (2013).