Chapter 33
Complex Amplitudes, and Where the Coherence Is
This chapter is the least finished thing in this part of the book, and the reader should treat it accordingly. An earlier version presented the complex instance of the decoration as delivering quantum dynamics outright. It does not, and the reason it does not is structural rather than technical — it will not be repaired by working harder at the same construction. What has changed since that version is that the structural reason is now known precisely, that it turns out to have two independent sources rather than one, and that locating it exposes a place the coherence can come from which is not among the routes the chapter previously offered. That place is the equational theory. This is progress and it is not a result: what follows states a construction, states an obstruction, and states where the construction puts what the obstruction takes away.
33.1 The complex instance
Take \(\Semi = \CC\) in Definition 30.2. The amplitude of a path is the product of its step amplitudes and the transition amplitude from \(P\) to \(Q\) is the sum over paths, \[\langle Q \mid P \rangle \;=\; \sum_{\gamma : P \to Q} W(\gamma),\] with \(|\langle Q \mid P\rangle|^2\) the candidate transition probability. This much is a construction and is not in doubt. It is also, so far, only bookkeeping: complex numbers have been attached to steps and multiplied along paths.
What would make it quantum is cancellation. Two paths \(\gamma_1, \gamma_2\) from \(P\) to \(Q\) with \(W(\gamma_1) = -W(\gamma_2)\) should annihilate, so that the transition has zero probability despite having two distinct causal histories. That is the phenomenon; everything else is notation.
33.2 Two obstructions, of different kinds
Cancellation between paths is what this framework, as built, forbids. There are two reasons and they are worth keeping apart, because one is about the semantics we chose and the other is about the construction we built.
33.2.1 The semantic obstruction
The equivalence on which this entire development rests is bisimulation, and bisimulation is a branching-time notion: it distinguishes systems by the shape of their tree of possibilities, not merely by the set of runs they admit. Two distinct paths to the same term are, for bisimulation, two distinct pieces of structure, and no assignment of weights to steps can make a branching-time equivalence identify them — still less make them annul one another. Interference adds and cancels contributions to a shared outcome, which is a statement about runs; bisimulation refuses to pool runs in the first place [62, \S13].
The obstruction is worth stating sharply because it is easy to disguise. One can write down \(\sum_\gamma W(\gamma)\) and observe that terms cancel; this is arithmetic and always available. What one cannot do, without changing the semantics, is claim that the cancellation is observed by the theory — that the resulting system is behaviourally indistinguishable from one in which the transition never occurs. Under bisimulation it is not. The two paths remain visible.
33.2.2 The constructive obstruction
The second reason is independent of the choice of equivalence, and it was not available when this chapter was first written. It comes from asking what the complex instance has to do in order to be a lift of the real one.
Chapter 13 builds, for \(\Semi = \CC\), one jump operator per rule and refinement class. Its coefficients have to be fixed somehow, and the natural-looking choice — assign an amplitude to each derivation and sum the amplitudes over derivations reaching a common target — has a consequence that is easy to miss. If \(m\) distinct redexes of one class carry a configuration to the same successor, summing amplitudes and squaring at the end gives that channel weight \(m^2\lambda\), where the real instance gives it \(m\lambda\). The complex construction would then fail to agree with the real one on a quantity both of them compute.
A complex weighting is conservative over the corresponding real one when, at zero Hamiltonian, the populations it evolves are exactly those the real weighting evolves.
Conservativity forces the other choice: sum the rates over derivations and take one square root, which is Definition 13.11. And that choice removes cross-derivation coherence by construction. The two paths do not cancel — not because a branching-time equivalence declines to pool them, but because the only normalisation under which the complex reading lifts the classical one adds them underneath the square root, where nothing can cancel.
They agree in their verdict and disagree in everything else. Remark 33.2 is a claim about observation: the cancellation could be computed but would not be seen. The constructive obstruction is a claim about arithmetic: under the only normalisation that lifts the classical semantics, there is no cancellation to compute. The second is the stronger, and it is worth noticing that it survives a change of equivalence. A reader who takes the first route of Section 33.4 — move to linear time — has removed the semantic obstruction and still has this one. The principle producing it is conservativity, not physics, and a construction declining to be conservative over its own classical instance would be buying interference by giving up the claim that these are two instances of one thing.
An earlier version of this chapter proposed letting the vectorial account \(\vect{A}\) become a positive semidefinite matrix \(\rho\), with \(\mathrm{tr}(\rho)=1\) read as the quantum analogue of resource conservation. This should not be retained. A resource account is an element of an ordered monoid, tracking what may be spent; a density matrix is a state, tracking what may be observed. They are not the same kind of thing, and identifying them conflates the conservation law of Proposition 31.3 — which is about value under conversion — with the normalisation of a probability distribution. Whatever the right quantum statement is, it is not that one.
33.3 Where the coherence is
Both obstructions are about coherence between rewrites. Neither says anything about coherence from any other source, and a presentation has another source.
A GSLT is a triple: a signature, a set of equations, and a graph of rewrite rules. The rewrites are directed and costed, so they are naturally dissipative. The equations are symmetric and cost-free, so they are naturally unitary. A quantum reading has exactly two slots to fill, and the presentation hands over exactly two kinds of relation to fill them with. Definition 13.12 does the filling: the rewrites supply the jump operators, and equations withheld from the quotient supply a Hermitian hopping term.
The qualification carries the whole weight. An equation quotiented into the state space contributes nothing, because the two configurations it relates have become one. Most presentations quotient everything. So most weighted theories have \(H = 0\), and by Proposition 13.3 that is exactly the case in which the complex instance is the real instance wearing different notation. Principle 13.1 states the consequence: a weighted theory is quantum exactly to the extent that its presentation withholds equations from the quotient.
Three things follow, in increasing order of how much they change what this book has already said.
Complexification is not free, and now one can see what it costs.
The earlier version of this chapter treated the complex instance as a choice of semiring and nothing more, so that a modeller who wanted quantum dynamics had only to select \(\CC\). That is not so. Selecting \(\CC\) over a fully quotiented presentation changes nothing whatever. What has to change is the presentation — one must decline to quotient something, and say with what amplitude it hops — and that is a modelling decision made in the object language, visible in the source text of a theory, rather than a choice of coefficient ring made about it.
The obstruction has been relocated, not removed.
Nothing above recovers cancellation between rewrites, and nothing above should be read as claiming that a coherent hopping term makes a rewrite theory quantum in the full sense. What it does is identify the one place in a presentation where a complex weighting has purchase, and thereby convert “we cannot get interference” into the sharper and more useful “interference lives in the equations and not in the rewrites, and here is what putting it there costs”. The honest summary is that the framework is quantum in its equations and classical in its rewrites. Whether that division is a defect or a discovery is not something this chapter can settle.
The ontology of Part Part V survives.
This is the largest change, and it is a change to what the book concedes rather than to what it claims. The first route below — move to a linear-time semantics — was previously the only route recovering interference honestly, and its cost was stated plainly: the ontology in which bisimulation classes are the real things does not survive the move. That concession was live and nothing retracted it. It can now be retracted, because coherence sited in the equational theory never asks bisimulation to pool runs. Two configurations related by a withheld equation are two configurations, distinguished exactly as branching time distinguishes them, and the hopping between them is additional structure on the state space rather than an identification of paths through it. One may have a nonzero Hamiltonian and keep the ontology.
Two things, and they should not be run together with the above. First, nothing here says which equations a physical theory would withhold, or where the hopping amplitudes come from; the construction says where the slot is and leaves it to the modeller, exactly as the choice of semiring is left to the modeller. Second, the notion of equivalence appropriate to a theory with nonzero Hamiltonian is not known. Rate bisimulation is the obvious candidate at \(H = 0\), and by Proposition 13.3 it transfers there without further work; when \(H \neq 0\) a relation on configurations must also respect coherences, and no candidate is on offer. That is the precise form in which the branching-time question returns, and it is a better form than the one this chapter started with.
33.4 What would have to change
For cancellation between rewrites, three routes remain open and we have not chosen among them. The fourth item is not one of them; it is where Section 33.3 has already gone, recorded here so that the list is not read as exhausting the options.
Move to linear time. Replace bisimulation by a trace-based equivalence and accept the loss of the branching-time distinctions the rest of this book depends on. This removes the semantic obstruction and leaves the constructive one (Remark 33.3), so it is a smaller step than it once looked and buys less: one would additionally have to abandon conservativity, which is to say abandon the claim that the real and complex readings are two instances of one construction.
Quotient late. Keep bisimulation as the semantics and introduce cancellation as a separate extraction laid on top — the Born rule as an additional map out of the model rather than a property of it. This is the categorical-quantum-mechanics pattern, where the interaction structure yields a semiring of scalars and the probability rule is applied afterwards [91]. It is the most conservative route and the least explanatory.
Attach amplitudes to derivations. Cancellation between rewrites requires that two routes to a common contractum be able to carry different phases, which means a decoration indexed by derivations rather than by refinement classes. That is a strictly larger construction and not a repair of this one; whether it can be given without destroying the locality that makes the present construction executable, and without making the decoration grow with the term, is the open question.
Withhold equations from the quotient — Section 33.3. This is not a route to cancellation between rewrites and does not claim to be. It is the observation that a presentation has a second place to put coherence, that this place is untouched by either obstruction, and that using it costs nothing the book has already spent.
33.5 Simulation, in the meantime
The stochastic instance is unobstructed, and it is what the implementation actually runs. The classical Gillespie algorithm [3] simulates a continuous-time Markov chain by computing exit rates from the current state, sampling a waiting time, and selecting the next rule proportionally to rate. With \(\Semi = \Real_{\geq 0}\) and a dynamic decoration (Definition 30.4) supplying the rates, this is precisely a decorated GSLT stepped forward, with the qualification of Theorem 13.1: the chain is over configurations and not over terms, and the simulator advances the term, updates the table, debits the account and records the trace as one step of one process.
The quantum case is no longer a loop in search of a semantics, and the earlier version of this chapter was right to refuse it and wrong about what the refusal was waiting for. It was waiting for a fixed generator. Quantum analogues of the Gillespie algorithm exist for open systems described by Lindblad master equations [4], and such an analogue requires an operator assembled once and exponentiated; a decoration whose table changes as the system runs does not obviously supply one. Definition 13.10 supplies it, by indexing the basis with configurations rather than with terms, so that a table update is an edge of the generator rather than a modification of it. With that in place the unravelling is well posed, and Theorem 13.3 says the classical algorithm is its degeneration at zero Hamiltonian — so the two simulators are one simulator, and “Gillespie-inspired” is a statement about a limit rather than a resemblance.
What remains withheld is the interpretation and not the computation. One may run the unravelling, and the numbers coming out will be the numbers of a quantum dynamical semigroup. Whether the system they describe is a quantum system depends on whether the coherences answer to anything, and that is a question about the presentation — about which equations were withheld, and why — rather than about the algorithm. The framework has made the question askable and has not answered it.