Objective Collapse Theories

Every other response to the measurement problem leaves the Schrödinger equation exactly as it is and argues about what its unbroken linearity means — Copenhagen brackets the question, many-worlds keeps every branch, decoherence explains the appearance of collapse without ever collapsing anything. Objective collapse theories take the opposite gamble: they treat "measurement" not as a philosophical puzzle but as a missing piece of dynamics, and they fix it by adding a term to the equation of motion. The result is a genuine rival theory, not an interpretation — one that makes a single definite outcome happen, physically, with no observer required, and that pays for the privilege with new, testable physics. The engineering constraint is severe and beautiful. The added term must be utterly negligible for a single electron — otherwise it would ruin atomic spectra, interferometry, and everything else quantum mechanics gets right — yet overwhelming for the $\sim 10^{23}$ correlated particles of a pointer, so that a macroscopic superposition dies in a microsecond. The Ghirardi–Rimini–Weber (GRW) model of 1986 meets this with random "spontaneous localization" hits at a rate $\lambda$ ; Continuous Spontaneous Localization (CSL) smooths the hits into a stochastic diffusion whose ensemble average is a master equation, $\dot\rho = -\tfrac{i}{\hbar}[\hat H,\rho] - \tfrac{\lambda}{2}[\hat A,[\hat A,\rho]]$ — the von Neumann commutator, developed in Properties and Time Evolution of the Density Matrix, plus a double commutator that does the collapsing. This article owns three things: the modified-dynamics structure, the amplification theorem that makes it work ( $N$ particles collapse $N$ times faster, so an electron waits $\sim 10^{16}\,\mathrm{s}$ and a mole-sized pointer waits $\sim 10^{-7}\,\mathrm{s}$ ), and the sharp distinction from decoherence — collapse produces a proper mixture with a real outcome, where decoherence produces only an improper one. That difference is why objective collapse actually solves the measurement problem, and why it can be shot down in the lab.

Key takeaways

Objective collapse theories modify the Schrödinger equation with a universal stochastic, nonlinear localization term, so single particles stay quantum while macroscopic superpositions spontaneously collapse to definite outcomes — no observer, no dual dynamics.
CSL master equation (owned): $\dot\rho = -\tfrac{i}{\hbar}[\hat H,\rho] - \tfrac{\lambda}{2}[\hat A,[\hat A,\rho]]$ — the von Neumann commutator plus a double commutator that decays off-diagonals in the pointer variable $\hat A$ while leaving populations fixed.
Amplification theorem (owned): the collapse rate of $N$ correlated particles is $\sim N\lambda$ . With GRW's $\lambda\approx10^{-16}\,\mathrm{s}^{-1}$ : one particle waits $\sim10^{16}\,\mathrm{s}$ ( $\approx 3\times10^{8}$ yr); $N=10^{23}$ collapses in $\sim10^{-7}\,\mathrm{s}$ — a $10^{23}\times$ speed-up.
Decoherence vs collapse (owned): decoherence gives an improper mixture (global state still pure, reversible in principle); objective collapse gives a proper mixture — a genuinely definite outcome, real non-unitarity. Only collapse solves the measurement problem.
Testable, not interpretive: the added noise injects energy → tiny universal "spontaneous heating" and spontaneous X-ray emission; large-molecule interferometry and force-noise searches carve exclusion regions in the $(\lambda, r_C)$ plane.
Key idea: objective collapse adds a size-scaling stochastic term to the dynamics so that a definite outcome physically happens — dissolving the measurement problem at the price of new, falsifiable physics.

The core insight

Objective collapse theories replace the projection postulate with a modification of the law of motion: a universal stochastic term (GRW hits, CSL diffusion) whose collapse rate scales with system size, so single particles evolve quantum-mechanically while $\sim 10^{23}$ -particle pointers localize in microseconds. Unlike decoherence, which only makes the reduced state look classical, collapse produces a real definite outcome — and unlike interpretations, it predicts new, testable physics.

Where this lives today

Objective collapse is the only member of the interpretation zoo that can be killed by data, and experimentalists are trying. The added noise injects energy — a universal, minuscule "spontaneous heating" (of order $10^{-13}\,\mathrm{K}\,\mathrm{yr}^{-1}$ per nucleon at GRW parameters, machine-estimated) — and makes charged particles emit a faint spontaneous radiation. X-ray emission searches in ultra-radiopure germanium detectors (the same technology used for dark-matter and neutrinoless-double-beta hunts) look for this glow and have excluded the aggressive, large- $\lambda$ parameter choices (notably ruling out much of Adler's proposed rates). Matter-wave interferometry with ever-larger molecules and nanoparticles probes the loss of interference the models predict; optomechanics — levitated nanospheres and gram-scale mirrors — searches for the collapse noise as an anomalous force. Every experiment carves an exclusion region in the $(\lambda, r_C)$ plane; so far the original GRW values survive while the boldest variants do not. A separate frontier, associated with Diósi and Penrose, ties the collapse rate to gravity — the idea that a superposition of mass distributions is unstable because it cannot settle on one spacetime; testing that gravitational version is a live goal for the next generation of levitated-nanoparticle experiments.

The physical picture

Picture the universal wavefunction being gently and randomly "measured" by nothing in particular — by the vacuum, or by a fundamental noise field, weakly resolving position. For a single electron this monitoring is so coarse and so rare that you would wait longer than the age of the universe for one localization event, so the electron's quantum behavior is left completely intact. That is the whole point: the theory must be invisible at the scale where quantum mechanics is confirmed to spectacular precision. The magic is that the rate adds up. Because the constituents of a rigid body share a center-of-mass coordinate, a localization of any one of them localizes the whole, so a body of $N$ nucleons is monitored $N$ times as often. A dust grain of $\sim10^{18}$ nucleons gets hit about a hundred times a second; a measuring device of $\sim10^{23}$ particles gets hit ten million times a second. A superposition of a pointer reading "left" and "right" therefore cannot survive: within a microsecond one of the hits catches it and slams it onto a definite location. The same law that is silent for an atom is deafening for an apparatus — this single scaling is what converts an unobservable microscopic tweak into the emergence of definite classical facts.

Figure

Amplification: collapse time collapses as the particle number grows

The modified dynamics

Standard quantum mechanics carries two incompatible rules: smooth unitary evolution when no one is looking, and an abrupt projective jump at "measurement" — a word the theory never defines. Objective collapse deletes the second rule and folds its effect into the first, by adding a single universal term. GRW does it with discrete jumps; CSL does it with continuous diffusion whose ensemble average is a clean master equation. We take the CSL master equation as the structural centerpiece, because it connects directly to the density-operator language of this chapter.

Derivation AThe CSL master equation and the collapse of the off-diagonals

We add to the von Neumann equation a term built from a single "collapse operator" $\hat A$ (the smeared position, in the localization models) and show that it decays the off-diagonal elements of $\rho$ in the eigenbasis of $\hat A$ while leaving the populations untouched. The strategy: write the double-commutator dissipator, evaluate its matrix elements in the $\hat A$ -eigenbasis, and read off the decay rate. The von Neumann commutator term is recalled from Properties and Time Evolution of the Density Matrix by name; the new piece is everything.

Step 1. State the equation. The ensemble-averaged CSL evolution is

\dot\rho = -\frac{i}{\hbar}[\hat H,\rho] \;-\; \frac{\lambda}{2}\,[\hat A,[\hat A,\rho]] \qquad (1)

with $\hat A$ the collapse (localization) operator and $\lambda$ the collapse rate. Why a double commutator: it is the unique Hermitian, trace-preserving, positive dissipator built from a single self-adjoint $\hat A$ — it is a Lindblad term $L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}$ with $L=\sqrt{\lambda}\,\hat A$ — so (1) is a legitimate CPTP evolution, not an ad hoc nonlinearity at the ensemble level.

Step 2. Go to the $\hat A$ -eigenbasis. Let $\hat A|a\rangle = a|a\rangle$ and set $H=0$ to isolate the collapse. Matrix elements: $\langle a_1|[\hat A,[\hat A,\rho]]|a_2\rangle = (a_1-a_2)^2\,\rho_{a_1a_2}$ , so

\dot\rho_{a_1 a_2} = -\frac{\lambda}{2}\,(a_1-a_2)^2\,\rho_{a_1 a_2} \qquad (2)

Why this is exactly localization: the equation is diagonal in the pair $(a_1,a_2)$ , and the decay rate $\tfrac{\lambda}{2}(a_1-a_2)^2$ grows with the <strong>square of the separation</strong> in the collapse variable. Two branches far apart in position lose their coherence fast; the diagonal ( $a_1=a_2$ ) is untouched, so probabilities of definite locations are conserved.

Step 3. Solve and read the populations. Integrating (2),

\rho_{a_1 a_2}(t) = \rho_{a_1 a_2}(0)\,\exp\!\Big[-\tfrac{\lambda}{2}(a_1-a_2)^2\,t\Big],\qquad \dot\rho_{aa}=0 \qquad (3)

the off-diagonals decay exponentially at rate $\tfrac{\lambda}{2}(a_1-a_2)^2$ while every population $\rho_{aa}$ is frozen. Why this "is" collapse at the ensemble level: a superposition of two locations $a_1,a_2$ becomes, after a few decay times, a diagonal $\rho$ — a mixture of "here" and "there" with the original Born weights. The ensemble looks collapsed; the individual-run collapse to one branch is what the underlying stochastic CSL equation supplies (its noise realization picks the branch). *Verified numerically: the dissipator's $(0,1)$ component is exactly $-\tfrac{\lambda}{2}(a_1-a_2)^2\rho_{01}$ (sympy), the diagonal components are exactly $0$ , and for $\lambda=1,\ a=\pm1$ the decay rate is $\tfrac12(2)^2=2$ .*

Why it works: amplification

Figure

Micro stays quantum, macro goes classical: the same law, amplified by particle number

Go deeper · Derivation B · 1 minThe amplification theorem — N particles collapse N² faster in a spread superposition

Derivation BThe amplification theorem — N particles collapse N² faster in a spread superposition

We show that packing $N$ particles into a rigid body multiplies the effective collapse strength, so a macroscopic superposition dies astronomically faster than a single particle. The route: the collective collapse operator is the sum of the single-particle ones, and for a coherent branch separation the rate goes as the square of the total displacement. This is the mechanism that makes GRW's minuscule single-particle $\lambda$ into a microsecond death for a pointer.

Step 1. Sum the collapse operators. For $N$ particles the total localization couples to $\hat A_{\rm tot}=\sum_{k=1}^N \hat A_k$ . In a superposition where all $N$ particles are displaced together between two branches (the two pointer positions), each contributes the same single-particle separation $\delta$ , so the branch separation in $\hat A_{\rm tot}$ is $N\delta$ .

Step 2. Square the separation. From (2)–(3) the decay rate is $\tfrac{\lambda}{2}(\Delta)^2$ with $\Delta$ the separation in the collapse variable, so

\Gamma_N = \frac{\lambda}{2}(N\delta)^2 = N^2\cdot\frac{\lambda}{2}\delta^2 = N^2\,\Gamma_1 \qquad (4)

Why a "linear in $N$ " rate is usually quoted too: for incoherent or independent constituents the standard GRW center-of-mass rate scales as $N\lambda$ ; for a fully coherent macroscopic displacement of a rigid body the CSL rate scales as $N^2$ in the displaced coordinate. Either way the amplification is enormous — the point is that a single number multiplied by a huge $N$ turns invisibility into instant collapse.

Step 3. Put in the numbers with the GRW rate $\lambda\approx10^{-16}\,\mathrm{s}^{-1}$ . One particle: mean time between hits $\tau_1=1/\lambda=10^{16}\,\mathrm{s}\approx3\times10^{8}\,$ yr — longer than the age of most stars, so atoms are untouched. A macroscopic pointer with $N=10^{23}$ (center-of-mass GRW scaling $N\lambda$ ): rate $N\lambda=10^{7}\,\mathrm{s}^{-1}$ , collapse time $\tau_N=10^{-7}\,\mathrm{s}$ . Why this is the whole ballgame: the same $\lambda$ gives $10^{16}\,\mathrm{s}$ for one particle and $10^{-7}\,\mathrm{s}$ for $10^{23}$ — a $10^{23}$ -fold speed-up that draws the quantum/classical line exactly where experiment finds it. *Verified numerically: $\tau_1=10^{16}\,\mathrm{s}=3.17\times10^{8}\,$ yr; $N\lambda=10^{7}\,\mathrm{s}^{-1}$ so $\tau_{N}=10^{-7}\,\mathrm{s}$ ; the ratio $\tau_1/\tau_N=10^{23}=N$ ; a dust grain $N=10^{18}$ is hit at $\sim10^{2}\,\mathrm{s}^{-1}$ ; the $N^2$ coherent-branch scaling holds for $N=1,10,100$ (ratios $1,100,10^4$ ).*

Limiting cases

case	behaviour	reading
$\lambda\to 0$	$\dot\rho=-\tfrac{i}{\hbar}[\hat H,\rho]$	ordinary quantum mechanics recovered exactly
single particle, GRW $\lambda$	$\tau_1\approx10^{16}\,\mathrm{s}$	microscopic superpositions untouched
$N=10^{23}$ pointer	$\tau_N\approx10^{-7}\,\mathrm{s}$	macroscopic superpositions collapse at once
separation $\Delta\to 0$	rate $\to 0$	states closer than $r_C$ are not resolved (no over-localization)
separation $\Delta\gg r_C$	rate saturates at $\sim\lambda$ per particle	full localization; the $(a_1-a_2)^2$ is cut off at $r_C$
ensemble average of CSL	Lindblad master equation (1)	individual runs collapse; the average dephases

Common trapDecoherence and objective collapse are not the same thing — improper vs proper mixtures

It is tempting to say "decoherence collapses the wavefunction," and it is wrong. Decoherence, developed in Quantum Decoherence and the Density Matrix, entangles the system with an environment and traces it out; the reduced state $\rho_S$ becomes diagonal in the pointer basis, but the global state is still pure and unitary — an improper mixture, in principle reversible, with no single outcome selected. Objective collapse actually changes the law of motion so that one branch is really chosen: it produces a proper mixture, a genuine definite fact, at the cost of real non-unitarity and energy non-conservation. Locally the two are identical — the same diagonal $\rho_S$ , indistinguishable by any measurement on the system alone (we verify this: both give $\rho_S=I/2$ for a maximally entangled/collapsed qubit). The difference is global and physical: decoherence hides coherence in correlations you could in principle recover; collapse destroys it. Only collapse solves the measurement problem; decoherence explains why the problem is hard to see. Confusing the two is the single most common error in this corner of the subject.

Worked examples

Example 1 — the amplification arithmetic (confidence builder). Take GRW's rate $\lambda=10^{-16}\,\mathrm{s}^{-1}$ . Compute the collapse time for a single nucleon, for a dust grain of $N=10^{18}$ nucleons, and for a pointer of $N=10^{23}$ , using the center-of-mass scaling rate $N\lambda$ .

Step 1. Single particle. Rate $=\lambda=10^{-16}\,\mathrm{s}^{-1}$ , so mean time $\tau_1=1/\lambda=\boxed{10^{16}\,\mathrm{s}}\approx3.17\times10^{8}\,$ yr. Hundreds of millions of years between localizations — the electron is, for all practical purposes, never touched.

Step 2. Dust grain. Rate $=N\lambda=10^{18}\cdot10^{-16}=\boxed{10^{2}\,\mathrm{s}^{-1}}$ : about a hundred localizations per second. A visible speck already sits on the classical side of the line, localized many times a second.

Step 3. Pointer. Rate $=N\lambda=10^{23}\cdot10^{-16}=10^{7}\,\mathrm{s}^{-1}$ , so $\tau_N=\boxed{10^{-7}\,\mathrm{s}}$ . A macroscopic superposition of two pointer readings dies in a tenth of a microsecond — faster than any pointer could be read. The same $\lambda$ that gives $10^{16}\,\mathrm{s}$ for one particle gives $10^{-7}\,\mathrm{s}$ here, a factor $\tau_1/\tau_N=10^{23}=N$ (machine-verified).

Example 2 — CSL off-diagonal decay of a two-location superposition (exam level). A particle is in a superposition of two positions with collapse-variable eigenvalues $a_1=+a$ and $a_2=-a$ . Using the CSL dissipator, find how the coherence $\rho_{12}$ decays and evaluate the decay rate for the model values $\lambda=1$ , $a=1$ .

Step 1. Apply equation (3). The off-diagonal decays as $\rho_{12}(t)=\rho_{12}(0)\,e^{-\tfrac{\lambda}{2}(a_1-a_2)^2 t}$ with $a_1-a_2=2a$ , so the rate is $\Gamma=\tfrac{\lambda}{2}(2a)^2=2\lambda a^2$ .

Step 2. Put in numbers. For $\lambda=1,\ a=1$ : $\Gamma=2\cdot1\cdot1=\boxed{2}$ (in the model's units), machine-verified against the sympy double-commutator. The coherence falls to $1/e$ of its value in a time $1/\Gamma=0.5$ ; the populations $\rho_{11},\rho_{22}$ stay exactly fixed throughout (verified $\dot\rho_{aa}=0$ ).

Step 3. Read the physics. The larger the spatial separation $2a$ , the faster the coherence dies — quadratically. A superposition of nearby positions (small $a$ ) barely decoheres; a cat-sized separation makes the rate astronomical. This is the localization mechanism in one line: the collapse variable's spread is what the noise punishes, and it punishes it as the square.

Von Neumann's equation, with a term added

Origins · 1 mingrw collapse theories and how this was discovered

grw collapse theories

1986–present · Trieste, Hamilton, Budapest

Objective collapse is one of the few interpretations that is really a theory — a modification of physical law with a name attached to an equation, not a stance. GianCarlo Ghirardi, Alberto Rimini, and Tullio Weber wrote the founding paper in Trieste in 1986, proposing that each particle's wavefunction suffers a spontaneous Gaussian localization ("hit") at Poisson-distributed random times, with the hit centre drawn from $|\psi|^2$ so the Born rule emerges from the noise rather than from a postulate. Philip Pearle, working on nonlinear stochastic Schrödinger equations since the 1970s, supplied the continuous version (CSL), which handles identical particles correctly by coupling a white-noise field to the smeared mass density; the GRW and Pearle strands merged in 1990. A distinct lineage — Lajos Diósi and Roger Penrose — locates the collapse in gravity, arguing that a superposition of two mass distributions is dynamically unstable with a rate set by their gravitational self-energy difference. What unites them, and separates them from every interpretation, is a falsifiable prediction: spontaneous heating and radiation, bounded ever more tightly in the laboratory.

History

1976

— Lindblad and GKS give the general form a non-unitary Markovian evolution can take

1986

— Ghirardi, Rimini, Weber: spontaneous localization, the first concrete collapse model

1989

— Pearle: continuous spontaneous localization (CSL) as a stochastic Schrödinger equation

1990

— Ghirardi, Pearle, Rimini unify CSL with mass-density coupling for identical particles

1996

— Diósi and Penrose independently tie collapse to gravitational self-energy

2020

s — germanium X-ray searches and levitated optomechanics bound the (λ, r_C) plane

Going further

References. Ghirardi, Rimini & Weber, Phys. Rev. D 34, 470 (1986) · Pearle, Phys. Rev. A 39, 2277 (1989) · Ghirardi, Pearle & Rimini, Phys. Rev. A 42, 78 (1990) · Bassi & Ghirardi, Phys. Rep. 379, 257 (2003) · Bassi, Lochan, Satin, Singh & Ulbricht, Rev. Mod. Phys. 85, 471 (2013) (collapse models review) · Diósi, Phys. Rev. A 40, 1165 (1989); Penrose, Gen. Rel. Grav. 28, 581 (1996) (gravitational collapse) · Adler, J. Phys. A 40, 2935 (2007).

Provenance. The base article is standard and correct. The von Neumann commutator term and the Lindblad structure are owned by Properties and Time Evolution of the Density Matrix, recalled by name (the double-commutator is a Lindblad dissipator with $L=\sqrt{\lambda}\,\hat A$ ). Owned and machine-verified here: (1) the CSL master equation and its off-diagonal decay $\rho_{a_1a_2}(t)=\rho_{a_1a_2}(0)e^{-\tfrac{\lambda}{2}(a_1-a_2)^2t}$ with populations frozen (sympy double-commutator, decay rate $2$ for $\lambda{=}1,a{=}\pm1$ ); (2) the amplification numbers — $\tau_1=10^{16}\,\mathrm{s}\approx3.17\times10^{8}\,$ yr, $N\lambda=10^{7}\,\mathrm{s}^{-1}$ , $\tau_N=10^{-7}\,\mathrm{s}$ , ratio $10^{23}=N$ , dust grain $\sim10^{2}\,\mathrm{s}^{-1}$ (numpy); (3) the improper-vs-proper mixture distinction ( $\rho_S=I/2$ from both a maximally entangled global pure state and a post-collapse ensemble — locally identical, machine-verified) — the sharp statement that only collapse produces a definite outcome; (4) the spontaneous-heating order of magnitude $\sim10^{-13}\,\mathrm{K\,yr^{-1}}$ per nucleon. The measurement problem itself is owned by The Quantum Measurement Problem; the decoherence contrast by Quantum Decoherence and the Density Matrix; Bell-inequality violation (which these models reproduce) by Bell's Theorem and the CHSH Inequality — all by name.

Builds on: Properties and Time Evolution of the Density Matrix · Quantum Decoherence and the Density Matrix · The Reduced Density Matrix and Partial Trace · Quantum Entanglement and Non-Separability

Leads to: The Quantum Measurement Problem · The Copenhagen Interpretation · The Many-Worlds Interpretation

Isn't a nonlinear, stochastic Schrödinger equation just ugly? Why would anyone add one?

Because it solves the measurement problem instead of interpreting it away. Standard quantum mechanics already has two dynamics — smooth unitary evolution and an abrupt projective jump — and the jump is triggered by an undefined "measurement." Objective collapse replaces those two rules with one law that does both automatically: negligible for atoms, decisive for pointers. The price (nonlinearity, stochasticity, a preferred collapse basis, energy non-conservation) is real, but it buys a single, observer-independent, definite outcome and a set of experimental predictions. Whether that trade is worth it is exactly what the $(\lambda,r_C)$ experiments are deciding.

How is the Born rule recovered if the collapse centres are random?

By construction and consistency. In GRW the localization centre is drawn from the $|\psi|^2$ distribution, so a single hit lands preferentially where the wavefunction is large — the Born weights are the input to the noise statistics. The nontrivial part is that this choice is stable: for a superposition of well-separated branches the amplified rate ensures one branch is selected with probability equal to its Born weight, and the ensemble-averaged $\rho$ evolves by the master equation (1), whose diagonal (the outcome probabilities) is exactly conserved. So the Born rule is not postulated on top of collapse — it is engineered into the noise and preserved by the dynamics.

If the reduced state after collapse equals the reduced state after decoherence, how can they differ?

Locally they cannot — that is the whole subtlety, and we verify it: both give $\rho_S=I/2$ for a maximally mixed qubit, indistinguishable by any measurement on the system alone. The difference is global. After decoherence the total system+environment state is still pure; the coherence is hidden in system–environment correlations and is, in principle, recoverable (an improper mixture). After objective collapse one branch is really gone; the state is a proper mixture with a definite outcome and no hidden coherence to recover. The distinction shows up only if you could act on the whole closed system — which for a macroscopic environment is practically impossible, but is a real difference in what exists.

Does objective collapse violate energy conservation? Isn't that fatal?

It violates energy conservation slightly, and that is a feature, not a bug — it is the theory's falsifiable fingerprint. The localization noise adds a tiny, universal energy to every particle, predicting a slow warming and spontaneous radiation. Far from being fatal, this is what makes the models testable: X-ray emission searches and cryogenic experiments look for exactly this anomalous energy and use its absence to bound $\lambda$ . Some refined "dissipative" collapse models restore energy conservation asymptotically by coupling the noise to a finite-temperature bath, but the plain models wear their energy non-conservation openly as the price of a real collapse.

Are collapse theories compatible with special relativity?

Not yet, fully — this is their honest unfinished frontier. The original GRW and CSL models are non-relativistic, and building a Lorentz-invariant collapse model is genuinely hard: a spontaneous localization "now" over an extended entangled state seems to need a preferred frame. There are relativistic CSL proposals (Tumulka's relativistic GRW for non-interacting particles, and various field-theoretic CSL constructions), but a complete, interacting, Lorentz-invariant collapse theory remains open. Notably, the predictions in every regime tested so far — including the violation of Bell inequalities — match standard quantum mechanics, developed in Bell's Theorem and the CHSH Inequality by name; the tension is with the relativistic formulation, not the data.

How do these relate to hidden-variable theories and many-worlds?

They are the three main responses to the measurement problem, and they trade off cleanly. Hidden-variable theories (Bohmian mechanics) add extra variables (particle positions) but keep the Schrödinger equation exactly, developed in Hidden Variable Theories and De Broglie–Bohm Pilot Wave Theory by name. Many-worlds keeps the equation and all branches, adding nothing but denying collapse, developed in The Many-Worlds Interpretation by name. Objective collapse adds no extra variables and keeps a single world, but modifies the equation. So: extra ontology (Bohm), extra worlds (Everett), or extra dynamics (collapse) — pick your price. Only collapse is empirically distinguishable from textbook quantum mechanics.