by Michael Clive Price

Q0 Why this FAQ?

This FAQ shows how quantum paradoxes are resolved by the "many-worlds" interpretation or metatheory of quantum mechanics. This FAQ does not seek to prove that the many-worlds interpretation is the "correct" quantum metatheory, merely to correct some of the common errors and misinformation on the subject floating around.

As a physics undergraduate I was struck by the misconceptions of my tutors about many-worlds, despite that it seemed to resolve all the paradoxes of quantum theory [A]. The objections raised to many-worlds were either patently misguided [B] or beyond my ability to assess at the time [C], which made me suspect (confirmed during my graduate QFT studies) that the more sophisticated rebuttals were also invalid. I hope this FAQ will save other investigators from being lead astray by authoritative statements from mentors.

I have attempted, in the answers, to translate the precise mathematics of quantum theory into woolly and ambiguous English - I would appreciate any corrections. In one or two instances I couldn't avoid using some mathematical (Dirac) notation, in particular in describing the Einstein- Podolsky-Rosen (EPR) experiment and Bell's Inequality and in showing how probabilities are derived, so I've included an appendix on the Dirac notation.

"Political scientist" L David Raub reports a poll of 72 of the "leading cosmologists and other quantum field theorists" about the "Many-Worlds Interpretation" and gives the following response breakdown [T].

Amongst the "Yes, I think MWI is true" crowd listed are Stephen Hawking and Nobel Laureates Murray Gell-Mann and Richard Feynman. Gell-Mann and Hawking recorded reservations with the name "many-worlds", but not with the theory's content. Nobel Laureate Steven Weinberg is also mentioned as a many-worlder, although the suggestion is not when the poll was conducted, presumably before 1988 (when Feynman died). The only "No, I don't accept MWI" named is Penrose.

The findings of this poll are in accord with other polls, that many- worlds is most popular amongst scientists who may rather loosely be described as string theorists or quantum gravitists/cosmologists. It is less popular amongst the wider scientific community who mostly remain in ignorance of it.

More detail on Weinberg's views can be found in _Dreams of a Final Theory_ or _Life in the Universe_ Scientific American (October 1994), the latter where Weinberg says about quantum theory:
"The final approach is to take the Schrodinger equation seriously [..description of the measurement process..] In this way, a measurement causes the history of the universe for practical purposes to diverge into different non-interfering tracks, one for each possible value of the measured quantity. [...] I prefer this last approach"

In the The Quark and the Jaguar and Quantum Mechanics in the Light of Quantum Cosmology [10] Gell-Mann describes himself as an adherent to the (post-)Everett interpretation, although his exact meaning is sometimes left ambiguous.

Steven Hawking is well known as a many-worlds fan and says, in an article on quantum gravity [H], that measurement of the gravitational metric tells you which branch of the wavefunction you're in and references Everett.

Feynman, apart from the evidence of the Raub poll, directly favouring the Everett interpretation, always emphasized to his lecture students [F] that the "collapse" process could only be modelled by the Schrodinger wave equation (Everett's approach).

[F] Jagdish Mehra The Beat of a Different Drum: The Life and Science Richard Feynman
[H] Stephen W Hawking Black Holes and Thermodynamics Physical Review D Vol 13 #2 191-197 (1976)
[T] Frank J Tipler The Physics of Immortality 170-171

AKA as the Everett, relative-state, many-histories or many-universes interpretation or metatheory of quantum theory. Dr Hugh Everett, III, its originator, called it the "relative-state metatheory" or the "theory of the universal wavefunction" [1], but it is generally called "many- worlds" nowadays, after DeWitt [4a],[5].

Many-worlds comprises of two assumptions and some consequences. The assumptions are quite modest:
1) The metaphysical assumption: That the wavefunction does not merely encode the all the information about an object, but has an observer-independent objective existence and actually is the object. For a non-relativistic N-particle system the wavefunction is a complex-valued field in a 3-N dimensional space.

There is no other quantum theory, besides many-worlds, that is scientific, in the sense of providing a reductionist model of reality, and free of internal inconsistencies, that I am aware of. Briefly here are the defects of the most popular alternatives:

A measurement is an interaction, usually irreversible, between subsystems that correlates the value of a quantity in one subsystem with the value of a quantity in the other subsystem. The interaction may trigger an amplification process within one object or subsystem with many internal degrees of freedom, leading to an irreversible high-level change in the same object. If the course of the amplification is sensitive to the initial interaction then we can designate the system containing the amplified process as the "measuring apparatus", since the trigger is sensitive to some (often microphysical) quantity or parameter of the one of the other subsystems, which we designate the "object" system. Eg the detection of a charged particle (the object) by a Geiger counter (the measuring apparatus) leads to the generation of a "click" (high-level change). The absence of a charged particle does not generate a click. The interaction is with those elements of the charged particle's wavefunction that passes between the charged detector plates, triggering the amplification process (an irreversible electron cascade or avalanche), which is ultimately converted to a click.

A measurement, by this definition, does not require the presence of an conscious observer, only of irreversible processes.

Worlds, or branches of the universal wavefunction, split when different components of a quantum superposition "decohere" from each other [7a], [7b], [10]. Decoherence refers to the loss of coherency or absence of interference effects between the elements of the superposition. For two branches or worlds to interfere with each other all the atoms, subatomic particles, photons and other degrees of freedom in each world have to be in the same state, which usually means they all must be in the same place or significantly overlap in both worlds, simultaneously.

For small microscopic systems it is quite possible for all their atomic components to overlap at some future point. In the double slit experiment, for instance, it only requires that the divergent paths of the diffracted particle overlap again at some space-time point for an interference pattern to form, because only the single particle has been split.

Such future coincidence of positions in all the components is virtually impossible in more complex, macroscopic systems because all the constituent particles have to overlap with their counterparts simultaneously. Any system complex enough to be described by thermodynamics and exhibit irreversible behaviour is a system complex enough to exclude, for all practical purposes, any possibility of future interference between its decoherent branches. An irreversible process is one in, or linked to, a system with a large number of internal, unconstrained degrees of freedom. Once the irreversible process has started then alterations of the values of the many degrees of freedom leaves an imprint which can't be removed. If we try to intervene to restore the original status quo the intervention causes more disruption elsewhere.

In QM jargon we say that the components (or vectors in the underlying Hilbert state space) have become permanently orthogonal due to the complexity of the systems increasing the dimensionality of the vector space, where each unconstrained degree of freedom contributes a dimension to the state vector space. In a high dimension space almost all vectors are orthogonal, without any significant degree of overlap. Thus vectors for complex systems, with a large number of degrees of freedom, naturally decompose into mutually orthogonal components which, because they can never significantly interfere again, are unaware of each other. The complex system, or world, has split into different, mutually unobservable worlds.

According to thermodynamics each activated degree of freedom acquires kT energy. This works the other way around as well: the release of approximately kT of energy increases the state-space dimensionality. Even the quite small amounts of energy released by an irreversible frictive process are quite large on this scale, increasing the size of the associated Hilbert space.

The sum-over-histories or path-integral formalism of quantum mechanics was developed by Richard Feynman in the 1940s [F] as a third interpretation of quantum mechanics, alongside Schrodinger's wave picture and Heisenberg's matrix mechanics, for calculating transition amplitudes. All three approaches are mathematically equivalent, but the path-integral formalism offers some interesting additional insights into many-worlds.

In the path-integral picture the wavefunction of a single particle at (x',t') is built up of contributions of all possible paths from (x,t), where each path's contribution is weighted by a (phase) factor of exp(i*Action[path]/hbar) * wavefunction at (x,t), summed, in turn, over all values of x. The Action[path] is the time-integral of the lagrangian (roughly: the lagrangian equals kinetic minus the potential energy) along the path from (x,t) to (x',t'). The final expression is thus the sum or integral over all paths, irrespective of any classical dynamical constraints. For N-particle systems the principle is the same, except that the paths run through a 3-N space.

The simplest way to see that the many-worlds metatheory is a local theory is to note that it requires that the wavefunction obey some relativistic wave equation, the exact form of which is currently unknown, but which is presumed to be locally Lorentz invariant at all times and everywhere. This is equivalent to imposing the requirement that locality is enforced at all times and everywhere. Ergo many-worlds is a local theory.

It is trivial to relativise many-worlds, at least to the level of special relativity. All relativistic theories of physics are quantum theories with linear wave equations. There are three or more stages to developing a fully relativised quantum field theory:

First quantisation: the wavefunction of an N particle system is a complex field which evolves in 3N dimensions as the solution to either the many-particle Schrodinger, Dirac or Klein-Gordon or some other wave equation. External forces applied to the particles are represented or modelled via a potential, which appears in the wave equation as a classical, background field.

Second quantisation: AKA (relativistic) quantum field theory (QFT) handles the creation and destruction of particles by quantising the classical fields and potentials as well as the particles. Each particle corresponds to a field, in QFT, and becomes an operator. E.g. the electromagnetic field's particle is the photon. The wavefunction of a collection of particles/fields exists in a Fock space, where the number of dimensions varies from component to component, corresponding to the indeterminacy in the particle number. Many-worlds has no problems incorporating QFT, since a theory (QFT) is not altered by a metatheory (many-worlds), which makes statements about the theory.

Third quantisation: AKA quantum gravity. The gravitational metric is quantised, along with (perhaps) the topology of the space-time manifold. The role of time plays a less central role, as might be expected, but the first and second quantisation models are as applicable as ever for modelling low-energy events. The physics of this is incomplete, including some thorny, unresolved conceptual issues, with a number of proposals (strings, supersymmetry, supergravity...) for ways forward, but the extension required by many-worlds is quite trivial since the mathematics would be unchanged.

Non-relativistic quantum mechanics and quantum field theory are quite unambiguous: the other Everett-worlds occupy the same space and time as we do.

The implicit question is really, why aren't we aware of these other worlds, unless they exist "somewhere" else? To see why we aren't aware of the other worlds, despite occupying the same space-time, see "Why do I only ever experience one world?" Some popular accounts describe the other worlds as splitting off into other, orthogonal, dimensions. These dimensions are the dimensions of Hilbert space, not the more familiar space-time dimensions.

No, for four reasons:

This is really a question about why thermodynamics works and what is the origin of the "arrow of time", rather than about many-worlds.

A retrodiction occurs when already gathered data is accounted for by a later theoretical advance in a more convincing fashion. The advantage of a retrodiction over a prediction is that the already gathered data is more likely to be free of experimenter bias. An example of a retrodiction is the perihelion shift of Mercury which Newtonian mechanics plus gravity was unable, totally, to account for whilst Einstein's general relativity made short work of it.

First, the law conservation of energy is based on observations within each world. All observations within each world are consistent with conservation of energy, therefore energy is conserved.

Second, and more precisely, conservation of energy, in QM, is formulated in terms of weighted averages or expectation values. Conservation of energy is expressed by saying that the time derivative of the expected energy of a closed system vanishes. This statement can be scaled up to include the whole universe. Each world has an approximate energy, but the energy of the total wavefunction, or any subset of, involves summing over each world, weighted with its probability measure. This weighted sum is a constant. So energy is conserved within each world and also across the totality of worlds.

One way of viewing this result - that observed conserved quantities are conserved across the totality of worlds - is to note that new worlds are not created by the action of the wave equation, rather existing worlds are split into successively "thinner" and "thinner" slices, if we view the probability densities as "thickness".

Everett demonstrated [1], [2] that observations in each world obey all the usual conventional statistical laws predicted by the probabilistic Born interpretation, by showing that the Hilbert space's inner product or norm has a special property which allows us to makes statements about the worlds where quantum statistics break down. The norm of the vector of the set of worlds where experiments contradict the Born interpretation ("non-random" or "maverick" worlds) vanishes in the limit as the number of probabilistic trials goes to infinity, as is required by the frequentist definition of probability. Hilbert space vectors with zero norm don't exist (see below), thus we, as observers, only observe the familiar, probabilistic predictions of quantum theory. Everett-worlds where probability breaks down are never realised.

Strictly speaking Everett did not prove that the usual statistical laws of the Born interpretation would hold true for all observers in all worlds. He merely showed that no other statistical laws could hold true and asserted the vanishing of the Hilbert space "volume" or norm of the set of "maverick" worlds. DeWitt later published a longer derivation of Everett's assertion [4a], [4b], closely based on an earlier, independent demonstration by Hartle [H]. What Everett asserted, and DeWitt/Hartle derived, is that the collective norm of all the maverick worlds, as the number of trials goes to infinity, vanishes. Since the only vector in a Hilbert space with vanishing norm is the null vector (a defining axiom of Hilbert spaces) this is equivalent to saying that non-randomness is never realised. All the worlds obey the usual Born predictions of quantum theory. That's why we never observe the consistent violation of the usual quantum statistics, with, say, heat flowing from a colder to a hotter macroscopic object. Zero-probability events never happen.

Of course we have to assume that the wavefunction is a Hilbert space vector in the first place but, since this assumption is also made in the standard formulation, this is not a weakness of many-worlds since we are not trying to justify all the axioms of the conventional formulation of QM, merely those that relate to probabilities and collapse of the wavefunction.

In more detail the steps are:

1) Construct the tensor product of N identical systems in state |psi, according to the usual rules for Hilbert space composition (repeated indices summed):

Many-Worlds, whilst deterministic on the objective universal level, is indeterministic on the subjective level so the situation is certainly no better or worse for free-will than in the Copenhagen view. Traditional Copenhagen indeterministic quantum mechanics only slightly weakens the case for free-will. In quantum terms each neuron is an essentially classical object. Consequently quantum noise in the brain is at such a low level that it probably doesn't often alter, except very rarely, the critical mechanistic behaviour of sufficient neurons to cause a decision to be different than we might otherwise expect. The consensus view amongst experts is that free-will is the consequence of the mechanistic operation of our brains, the firing of neurons, discharging across synapses etc. and fully compatible with the determinism of classical physics. Free-will is the inability of an intelligent, self-aware mechanism to predict its own future actions due to the logical impossibility of any mechanism containing a complete internal model of itself rather than any inherent indeterminism in the mechanism's operation.

Nevertheless, some people find that with all possible decisions being realised in different worlds that the prima face situation for free- will looks quite difficult. Does this multiplicity of outcomes destroy free-will? If both sides of a choice are selected in different worlds why bother to spend time weighing the evidence before selecting? The answer is that whilst all decisions are realised, some are realised more often than others - or to put to more precisely each branch of a decision has its own weighting or measure which enforces the usual laws of quantum statistics.

These are really the same questions. Consider, for a moment, this analogy:

Suppose Fred has his brain divided in two and transplanted into two different cloned bodies (this is a gedanken operation! [*]). Let's further suppose that each half-brain regenerates to full functionality and call the resultant individuals Fred-Left and Fred-Right. Fred-Left can ask, why did I end up as Fred-Left? Similarly Fred-Right can ask, why did I end up as Fred-Right? The only answer possible is that there was no reason. From Fred's point of view it is a subjectively random choice which individual "Fred" ends up as. To the surgeon the whole process is deterministic. To both the Freds it seems random.

Same with many-worlds. There was no reason "why" you ended up in this world, rather than another - you end up in all the quantum worlds. It is a subjectively random choice, an artefact of your brain and consciousness being split, along with the rest of the world, that makes our experiences seem random. The universe is, in effect, performing umpteen split-brain operations on us all the time. The randomness apparent in nature is a consequence of the continual splitting into mutually unobservable worlds.

According to our present knowledge of physics whilst it is possible to detect the presence of other nearby worlds, through the existence of interference effects, it is impossible travel to or communicate with them. Mathematically this corresponds to an empirically verified property of all quantum theories called linearity. Linearity implies that the worlds can interfere with each other with respect to a external, unsplit, observer or system but the interfering worlds can't influence each other in the sense that an experimenter in one of the worlds can arrange to communicate with their own, already split-off, quantum copies in other worlds.

Specifically, the wave equation is linear, with respect to the wavefunction or state vector, which means that given any two solutions of the wavefunction, with identical boundary conditions, then any linear combination of the solutions is another solution. Since each component of a linear solution evolves with complete indifference as to the presence or absence of the other terms/solutions then we can conclude that no experiment in one world can have any effect on another experiment in another world. Hence no communication is possible between quantum worlds. (This type of linearity mustn't be confused with the evident non-linearity of the equations with respect to the fields.)

Non communication between the splitting Everett-worlds also explains why we are not aware of any splitting process, since such awareness needs communication between worlds. To be aware of the world splitting you would have to be receiving sensory information from, and thereby effect by the reverse process, more than one world. This would enable communication between worlds, which is forbidden by linearity. Ergo, we are not aware of any splitting precisely because we are split into non-interfering copies along with the rest of the world.

To calculate the form of the universal wavefunction requires not only a knowledge of its dynamics (which we have a good approximation to, at the moment) but also of the boundary conditions. To actually calculate the form of the universal wavefunction, and hence make inferences about all the embedded worlds, we would need to know the boundary conditions as well. We are presently restricted to making inferences about those worlds with which have shared a common history up to some point, which have left traces (records, fossils, etc.) still discernible today. This restricts us to a subset of the extant worlds which have shared the same boundary conditions with us. The further we probe back in time the less we know of the boundary conditions and the less we can know of the universal wavefunction.

This limits us to drawing conclusions about a restricted subset of the worlds - all the worlds which are consistent with our known history up to a some common moment, before we diverged. The flow of historical events is, according to chaos/complexity theory/thermodynamics, very sensitive to amplification of quantum-scale uncertainty and this sensitivity is a future-directed one-way process. We can make very reliable deductions about the past from the knowledge future/present but we can't predict the future from knowledge the past/present. Thermodynamics implies that the future is harder to predict than the past is to retrodict. Books get written about this "arrow of time" problem but, for the purposes of this discussion, we'll accept the thermodynamic origin of time's arrow is as given. The fossil and historical records say that dinosaurs and Adolf Hitler once existed but have less to say about the future.

Consider the effects of that most quantum of activities, Brownian motion, on the conception of individuals and the knock-on effects on the course of history. Mutation itself, one of the sources of evolutionary diversity, is a quantum event. For an example of the biological/evolutionary implications see Stephen Jay Gould's book Wonderful Life for an popular exploration of the thesis that the path of evolution is driven by chance. According to Gould evolutionary history forms an enormously diverse tree of possible histories - all very improbable - with our path being selected by chance. According to many-worlds all these other possibilities are realised. Thus there are worlds in which Hitler won WW-II and other worlds in which the dinosaurs never died out. We can be as certain of this as we are that Hitler and the dinosaurs once existed in our own past.

Whether or not we can ever determine the totality of the universal wavefunction is an open question. If Steven Hawking's work on the no- boundary-condition condition is ultimately successful, or it emerges from some theory of everything, and many think it will, then the actual form of the total wavefunction could, in principle, we determined from a complete knowledge of physical law itself.

Hugh Everett III (1930-1982) did his undergraduate study in chemical engineering at the Catholic University of America. Studying von Neumann's and Bohm's textbooks as part of his graduate studies, under Wheeler, in mathematical physics at Princeton University in the 1950s he became dissatisfied (like many others before and since) with the collapse of the wavefunction. He developed, during discussions with Charles Misner and Aage Peterson (Bohr' assistant, then visiting Princeton), his "relative state" formulation. Wheeler encouraged his work and preprints were circulated in January 1956 to a number of physicists. A condensed version of his thesis was published as a paper to The Role of Gravity in Physics conference held at the University of North Carolina, Chapel Hill, in January 1957.

Everett was discouraged by the lack of response from others, particularly Bohr, whom he flew to Copenhagen to meet but got the complete brush-off from. Leaving physics after completing his Ph.D., Everett worked as a defense analyst at the Weapons Systems Evaluation Group, Pentagon and later became a private contractor, apparently quite successfully for he became a multimillionaire. In 1968 Everett worked for the Lambda Corp. His published papers during this period cover things like optimising resource allocation and, in particular, maximising kill rates during nuclear-weapon campaigns.

From 1968 onwards Bryce S DeWitt, one of the 1957 Chapel Hill conference organisers, but better known as one of the founders of quantum gravity, successfully popularised Everett's relative state formulation as the "many-worlds interpretation" in a series of articles [4a],[4b],[5].

Sometime in 1976-9 Everett visited Austin, Texas, at Wheeler or DeWitt's invitation, to give some lectures on QM. The strict no-smoking rule in the auditorium was relaxed for Everett (a chain smoker); the only exception ever. Everett, apparently, had a very intense manner, speaking acutely and anticipating questions after a few words. Oh yes, a bit of trivia, he drove a Cadillac with horns.

With the steady growth of interest in many-worlds in the late 1970s Everett planned returning to physics to do more work on measurement in quantum theory, but died of a heart attack in 1982. Survived by his wife.

An unobserved system, according to the Copenhagen interpretation of quantum theory, evolves in a deterministic way determined by a wave equation. An observed system changes in a random fashion, at the moment of observation, instantaneously, with the probability of any particular outcome given by the Born formula. This is known as the "collapse" or "reduction" of the wavefunction. The problems with this approach are:
(1) The collapse is an instantaneous process across an extended region ("non-local") which is non-relativistic.
(2) The idea of an observer having an effect on microphysics is repugnant to reductionism and smacks of a return to pre-scientific notions of vitalism. Copenhagenism is a return to the old vitalist notions that life is somehow different from other matter, operating by different laws from inanimate matter. The collapse is triggered by an observer, yet no definition of what an "observer" is available, in terms of an atomic scale description, even in principle.

For these reasons the view has generally been adopted that the wavefunction associated with an object is not a real "thing", but merely represents our knowledge of the object. This approach was developed by Bohr and others, mainly at Copenhagen in the late 1920s. When we perform an measurement or observation of an object we acquire new information and so adjust the wavefunction as we would boundary conditions in classical physics to reflect this new information. This stance means that we can't answer questions about what's actually happening, all we can answer is what will be the probability of a particular result if we perform a measurement. This makes a lot of people very unhappy since it provides no model for the object.

The EPR experiment is widely regarded as the definitive gedanken experiment for demonstrating that quantum mechanics is non-local (requires faster-than-light communication) or incomplete. We shall see that it implies neither.

The EPR experiment was devised, in 1935, by Einstein, Podolsky and Rosen to demonstrate that quantum mechanics was incomplete [E]. Bell, in 1964, demonstrated that any hidden variables theory, to replicate the predictions of QM, must be non-local [B]. QM predicts strong correlations between separated systems, stronger than any local hidden variables theory can offer. Bell encoded this statistical prediction in the form of some famous inequalities that apply to any type of EPR experiment. Eberhard, in the late 1970s, extended Bell's inequalities to cover any local theory, with or without hidden variables. Thus the EPR experiment plays a central role in sorting and testing variants of QM. All the experiments attempting to test EPR/Bell's inequality to date (including Aspect's in the 1980s [As]) are in line with the predictions of standard QM - hidden variables are ruled out. Here is the paradox of the EPR experiment. It seems to imply that any physical theory must involve faster-than-light "things" going on to maintain these "spooky" action-at-a-distance correlations and yet still be compatible with relativity, which seems to forbid FTL.

Let's examine the EPR experiment in more detail.

So what did EPR propose? The original proposal was formulated in terms of correlations between the positions and momenta of two once-coupled particles. Here I shall describe it in terms of the spin (a type of angular momentum intrinsic to the particle) of two electrons. [In this treatment I shall ignore the fact that electrons always form antisymmetric combinations. This does not alter the results but does simplify the maths.] Two initially coupled electrons, with opposed spins that sum to zero, move apart from each other across a distance of perhaps many light years, before being separately detected, say, by me on Earth and you on Alpha Centauri with our respective measuring apparatuses. The EPR paradox results from noting that if we choose the same (parallel) spin axes to measure along then we will observe the two electrons' spins to be anti-parallel (i.e. when we communicate we find that the spin on our electrons are correlated and opposed). However if we choose measurement spin axes that are perpendicular to each other then there is no correlation between electron spins. Last minute alterations in a detector's alignment can create or destroy correlations across great distances. This implies, according to some theorists, that faster-than-light influences maintain correlations between separated systems in some circumstances and not others.

Now let's see how many-worlds escapes from this dilemma.

The initial state of the wavefunction of you, me and the electrons and the rest of the universe may be written:

It is an empirically established fact, which we just have to accept, that we can relate spin states in one direction to spin states in other directions like so (where "i" is the sqrt(-1)):

(In fancy jargon we say that the spin operators in different directions form non-commuting observables. I shall eschew such obfuscations.)

Working through the algebra we find that for pairs of electrons:

I shall assume that we are capable of either measuring spin in the x or y direction, which are both perpendicular the line of flight of the electrons. After having measured the state of the electron my state is described as one of either:

Q33 Is Everett's relative state formulation the same as many-worlds?

Some people believe that Everett eschewed all talk all splitting or branching observers in his original relative state formulation [2]. This is contradicted by the following quote from [2]:
[...] Thus with each succeeding observation (or interaction), the observer state "branches" into a number of different states. Each branch represents a different outcome of the measurement and the corresponding eigenstate for the object- system state. All branches exist simultaneously in the superposition after any given sequence of observations.[#] The "trajectory" of the memory configuration of an observer performing a sequence of measurements is thus not a linear sequence of memory configurations, but a branching tree, with all possible outcomes existing simultaneously in a final superposition with various coefficients in the mathematical model. [...]

[#] Note added in proof-- In reply to a preprint of this article some correspondents have raised the question of the "transition from possible to actual," arguing that in "reality" there is-as our experience testifies-no such splitting of observers states, so that only one branch can ever actually exist. Since this point may occur to other readers the following is offered in explanation.
The whole issue of the transition from "possible" to "actual" is taken care of in the theory in a very simple way- there is no such transition, nor is such a transition necessary for the theory to be in accord with our experience. From the viewpoint of the theory all elements of a superposition (all "branches") are "actual," none are any more "real" than the rest. It is unnecessary to suppose that all but one are somehow destroyed, since all separate elements of a superposition individually obey the wave equation with complete indifference to the presence or absence ("actuality" or not) of any other elements. This total lack of effect of one branch on another also implies that no observer will ever be aware of any "splitting" process.
Arguments that the world picture presented by this theory is contradicted by experience, because we are unaware of any branching process, are like the criticism of the Copernican theory that the mobility of the earth as a real physical fact is incompatible with the common sense interpretation of nature because we feel no such motion. In both case the arguments fails when it is shown that the theory itself predicts that our experience will be what it in fact is. (In the Copernican case the addition of Newtonian physics was required to be able to show that the earth's inhabitants would be unaware of any motion of the earth.)

Many-Worlds predicts that the Everett-worlds do not interact with each other because of the presumed linearity of the wave equation. However worlds do interfere with each other, and this enables the theory to be tested. (Interfere and interact mean different things in quantum mechanics. Pictorially: Interactions occur at the vertices within Feynman diagrams. Interference occurs when you add together different Feynman diagrams with the same external lines.)

According to many-worlds model worlds split with the operation of every thermodynamically irreversible process. The operation of our minds are irreversible, carried along for the ride, so to speak, and divide with the division of worlds. Normally this splitting is undetectable to us. To detect the splitting we need to set an up experiment where a mind is split but the world isn't. We need a reversible mind.

The general consensus in the literature [11], [16] is that the experiment to detect other worlds, with reversible minds, will be doable by, perhaps, about mid-21st century. That date is predicted from two trendlines, both of which are widely accepted in their own respective fields. To detect the other worlds you need a reversible machine intelligence. This requires two things: reversible nanotechnology and AI.

1) Reversible nanoelectronics. This is an straight-line extrapolation based upon the log(energy) / logic operation figures, which are projected to drop below kT in about 2020. This trend has held good for 50 years. An operation that thermally dissipates much less than kT of energy is reversible. (This implies that frictive or dissipative forces are insignificant by comparison with other processes.) If more than kT of energy is released then, ultimately, new degrees of freedom are activated in the environment and the change becomes irreversible.

2) AI. Complexity of human brain = approx 10^17 bits/sec, based on the number of neurons (approx 10^10) per human brain, average number of synapses per neuron (approx 10^4) and the average firing rate (approx 10^3 Hz). Straight line projection of log(cost) / logic operation says that human level, self-aware machine intelligences will be commercially available by about 2030-2040. Uncertainty due to present human-level complexity, but the trend has held good for 40 years.

Assuming that we have a reversible machine intelligence to hand then the experiment consists of the machine making three reversible measurements of the spin of an electron (or polarisation of a photon). (1) First it measures the spin along the z-axis. It records either spin "up" or spin "down" and notes this in its memory. This measurement acts just to prepare the electron in a definite state. (2) Second it measures the spin along the x-axis and records either spin "left" or spin "right" and notes this in its memory. The machine now reverses the entire x-axis measurement - which must be possible, since physics is effectively reversible, if we can describe the measuring process physically - including reversibly erasing its memory of the second measurement. (3) Third the machine takes a spin measurement along the z-axis. Again the machine makes a note of the result.

According to the Copenhagen interpretation the original (1) and final (3) z-axis spin measurements have only a 50% chance of agreeing because the intervention of the x-axis measurement by the conscious observer (the machine) caused the collapse of the electron's wavefunction. According to many-worlds the first and third measurements will always agree, because there was no intermediate wavefunction collapse. The machine was split into two states or different worlds, by the second measurement; one where it observed the electron with spin "left"; one where it observed the electron with spin "right". Hence when the machine reversed the second measurement these two worlds merged back together, restoring the original state of the electron 100% of the time.

Only by accepting the existence of the other Everett-worlds is this 100% restoration explicable.

Many-worlds makes a very definite prediction - gravity must be quantised, rather than exist as the purely classical background field of general relativity. Indeed, no one has conclusively directly detected (classical) gravity waves (as of 1994), although their existence has been indirectly observed in the slowing of the rotation of pulsars and binary systems. Some claims have been made for the detection of gravity waves from supernova explosions in our galaxy, but these are not generally accepted. Neither has anyone has directly observed gravitons, which are predicted by quantum gravity, presumably because of the weakness of the gravitational interaction. Their existence has been, and is, the subject of much speculation. Should, in the absence of any empirical evidence, gravity be quantised at all? Why not treat gravity as a classical force, so that quantum physics in the vicinity of a mass becomes quantum physics on a curved Riemannian background? According to many-worlds there is empirical evidence for quantum gravity.

To see why many-worlds predicts that gravity must be quantised, let's suppose that gravity is not quantised, but remains a classical force. If all the other worlds that many-worlds predicts exist then their gravitational presence should be detectable -- we would all share the same background gravitational metric with our co-existing quantum worlds. Some of these effects might be undetectable. For instance if all the parallel Earths shared the same gravitational field small perturbations in one Earth's orbit from the averaged background orbit across all the Everett-worlds would damp down, eventually, and remain undetectable.

However theories of galactic evolution would need considerable revisiting if many-worlds was true and gravity was not quantised, since, according to the latest cosmological models, the original density fluctuations derive from quantum fluctuations in the early universe, during the inflationary era. These quantum fluctuations lead to the formation of clusters and super-clusters of galaxies, along with variations in the cosmic microwave background (detected by Smoots et al) which vary in location from Everett-cosmos to cosmos. Such fluctuations could not grow to match the observed pattern if all the density perturbations across all the parallel Everett-cosmoses were gravitationally interacting. Stars would bind not only to the observed galaxies, but also to the host of unobserved galaxies.

A theory of classical gravity also breaks down at the scale of objects that are not bound together gravitationally. Henry Cavendish, in 1798, measured the torque produced by the gravitational force on two separated lead spheres suspended from a torsion fibre in his laboratory to determine the value of Newton's gravitational constant. Cavendish varied the positions of other, more massive lead spheres and noted how the torsion in the suspending fibre varied. Had the suspended lead spheres been gravitationally influenced by their neighbours, placed in different positions by parallel Henry Cavendishs in the parallel Everett-worlds, then the torsion would have been the averaged sum of all these contributions, which was not observed. In retrospect Cavendish established that the Everett-worlds are not detectable gravitationally. More recent experiments where the location of attracting masses were varied by a quantum random (radioactive) source have confirmed these findings. [W]

A shared gravitational field would also screw up geo-gravimetric surveys, which have successfully detected the presence of mountains, ores and other density fluctuations at the Earth's surface. Such surveys are not sensitive to the presence of the parallel Everett-Earths with different geological structures. Ergo the other worlds are not detectable gravitationally. That gravity must be quantised emerges as a unique prediction of many-worlds.

[W] Louis Witten Gravitation: an introduction to current research New York, Wiley (1962).
Essays in honor of Louis Witten on his retirement. Topics on quantum gravity and beyond: University of Cincinnati, USA, 3-4 April 1992 / editors, Freydoon Mansouri & Joseph J. Scanio. Singapore ; River Edge, NJ : World Scientific, c1993 ISBN 981021290

What is lost by this approach is a unique past assigned to each future. If you time-evolve the world-we-now-see backwards in time you get a superposition of earlier starting worlds. Similarly if you time evolve a single (initial) world forward you get a superposition of later (final) worlds.

For example consider a photon that hits a half-silvered mirror and turns into a superposition of a transmitted and a reflected photon. If we time-evolve one of these later states backwards we get not the original photon, but the original photon plus a "mirror image" of the original photon. (Try the calculation and see.) Only if we retain both the reflected and transmitted photons, with the correct relative phase, do we recover the single incoming photon when we time-reverse everything. (The mirror image contributions from both the final states have opposite signs and cancel out, when they are evolved backwards in time to before the reflection event.)

All the starting states have to have their relative phases co-ordinated or correlated just right (i.e. coherently) or else it doesn't work out. Needless to say the chances that the initial states should be arranged coherently just so that they yield the one final observed state are infinitesimal and in violation of observed thermodynamics, which states, in one form, that correlations only increase with time.

Note: this is a very inadequate guide. Read a more comprehensive text ASAP. For a more technical exposition of QM the reader is referred to the standard textbooks. Here are 3 I recommend:

Richard P Feynman QED: the strange story of light and matter ISBN 0- 14-012505-1. (Requires almost no maths and is universally regarded as outstanding, despite being about quantum electrodynamics.)

Richard P Feynman The Feynman Lectures in Physics Volume III Addison- Wesley (1965) ISBN 0-201-02118-8-P. The other volumes are worth reading too!

Daniel T Gillespie A Quantum Mechanics Primer: An Elementary Introduction to the Formal Theory of Non-relativistic Quantum Mechanics (Takes an axiomatic, geometric approach and teaches all the Hilbert space stuff entirely by analogy with Euclidean vector spaces. Not sure if it is still in print.)

Quantum theory is the most successful theory of physics and chemistry ever. It accounts for a wide range of phenomena from black body radiation, atomic structure and chemistry, which were very puzzling before quantum mechanics was first developed (c1926) in its modern form. All theories of physics are quantum physics, with whole new fields, like the semiconductor and microchip technology, based upon the quantum effects. This FAQ assumes familiarity with the basics of quantum theory and with the associated "paradoxes" of wave-particle duality. It will not explain the uncertainty principle or delve into the significance of non-commuting matrix operators. Only those elements of quantum theory necessary for an understanding of many-worlds are covered here.

Quantum theory contains, as a central object, an abstract mathematical entity called the "wavefunction" or "state vector". Determining the equations that describe its form and evolution with time is an unfinished part of fundamental theoretical physics. Presently we only have approximations to some "correct" set of equations, often referred to whimsically as the Theory of Everything.

The wavefunction, in bracket or Dirac notation, is written as |symbol, where "symbol" labels the object. A dog, for example, might be represented as |dog.

A general object, labelled "psi" by convention, is represented as |psi and called a "ket". Objects called "bra"s, written <psi|, may be formed from kets. An arbitrary bra <psi'| and ket |psi may be combined together to form the bracket, <psi'|psi, or inner product, which is just a fancy way of constructing a complex number. Amongst the properties of the inner product is:

where the a_i are arbitrary complex numbers. This is what is meant by saying that the inner product is linear on the right or ket side. It is made linear on the left-hand or bra side by defining

Any ket may be expanded as:

where the states |i form an orthonormal basis, with <i|j = 1 for i = j and = 0 otherwise, and where i labels some parameter of the object (like position or momentum).

The probability amplitudes, <i|psi, are complex numbers. It is empirically observed, first noted by Max Born and afterwards called the Born interpretation, that their magnitudes squared represent the probability that, upon observation, that the value of the parameter, labelled by i, will be observed if the system is the state represented by |psi. It is also empirically observed that after observing the system in state |i that we can henceforth replace the old value of the wavefunction, |psi, with the observed value, |i. This replacement is known as the collapse of the wavefunction and is the source of much philosophical controversy. Somehow the act of measurement has selected out one of the components. This is known as the measurement problem and it was this phenomenon that Everett addressed.

When a bra, <psi|, is formed from a ket, |psi, and both are inner productted together the result, <psi|psi, is a non-negative real number, called the norm of the vector. The norm of a vector provides a basis-independent way of measuring the "volume" of the vector.

The wavefunction for a joint system is built out of products of the components from the individual subsystems.

For example if the two systems composing the joint system are a cat and a dog, each of which may be in two states, alive or dead, and the state of the cat and the dog were independent of each other then we could write the total wavefunction as a product of terms. If

More generally, though, we states of subsystems are not independent of each other we have to use a more general formula:

This is sometimes described by saying that the states of the cat and dog have become entangled. It is fairly trivial to define the state of the cat and the dog with respect to each other. For instance we could re- express the above expansion with respect to the cat's two states as:

We term the state of the dog the relative state (Everett invented this terminology) with respect to the cat, specifying which cat state (alive or dead) we are interested in. This thus the dog's relative state with respect to the cat alive state is:

where the sqrt term has been added to normalise the relative state.