Noesis

 

The Journal of the Mega Society

Number 96

August 1994

 

EditorRick Rosner

5139 Balboa Blvd #303

Encino CA  91316-3430

(818) 986-9177

 

IN THIS ISSUE

FREQUENTLY ASKED QUESTIONS ABOUT THE

MANY-WORLDS OR RELATIVE STATE FORMULATION

OF

QUANTUM MECHANICS

ANSWERS COMPILED BY MICHAEL CLIVE PRICE

 

1    What are the problems with quantum theory?

2    What is the Copenhagen interpretation?

3    What is many-worlds?

4    What is a "world"?

5    What is a measurement?

6    Why do worlds split?

7    When do worlds split?

8a   What is sum-over-histories?

8b   What is many-histories?

9    How many worlds are there?

10   Is many-worlds a local theory?

11   Is many-worlds a deterministic theory?

12   Is many-worlds a relativistic theory?

13   Is many-worlds (just) an interpretation?

14   What are the alternatives?

15   Is many-worlds testable?

16   Could previously separate worlds diverge rather than split?

17   What is many-minds?

18   Does many-worlds violate Ockham's Razor?

19   Does the multiplication of worlds violate conservation of energy?

20   How do probabilities emerge within many-worlds?

21   Does many-worlds allow free-will?

22   Why am I in this world and not another?

23   Can wavefunctions collapse?

24   Is physics linear?

25   Can we determine what other worlds there are?

26   Who was Everett?

27   Who believes in many-worlds?

30   Does the EPR experiment prohibit locality?

31   References and further reading

Q1   What are the problems with quantum theory?

Quantum theory is the most successful description of microscopic systems like atoms and molecules ever, yet often it is not applied to larger, classical systems, like observers or the entire universe.  Many scientists and philosophers are unhappy with the theory because it seems to require a fundamental quantum-classical divide.  Einstein, for example, and despite his early contributions to the subject, was never reconciled with assigning the act of observation a physical significance, which QM requires.  This contradicts the reductionist ethos that, amongst other things, observations should emerge only as a consequence of an underlying physical theory and not be present in the axioms, as they are in the Copenhagen interpretation.  Yet the Copenhagen interpretation is the most popular interpretation of quantum mechanics.  (See "What is the Copenhagen interpretation?")

Q2   What is the Copenhagen interpretation?

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, instantaneously, with the probability of any particular outcome given by the Born formula, determined by the wavefunction.  This is known as the collapse of the wavefunction.  The problems with this approach are: (1)  The collapse is an instantaneous process across an extended region ("non-local").  This is in conflict with relativity, which states that no processes can be transmitted faster than the speed of light.  (Nevertheless it has been shown that no information can be transmitted faster than light by the collapse process). (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.

It should be added that there are other, less popular, interpretations of quantum theory, but they all have their own drawbacks, which are widely reckoned more severe.  Generally speaking they try to find a mechanism that describes the collapse process or add extra physical objects to the theory, in addition to the wavefunction.  In this sense they are more complex.  (See "Is there any alternative theory?")

Q3   What is many-worlds?

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

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 information about an object, but has an observer- independent objective existence.  For an N-particle system the wavefunction is a complex-valued field in a 3-N dimensional space.  In quantum field theory the state vector spans a space of an indeterminate number of dimensions.

2)   The physical assumption:  The wavefunction obeys some standard deterministic wave equation at all times.  The observer plays no special role in the theory and, consequently, there is no collapse of the wavefunction.  Measurement and observation are modelled by applying the wave equation to the joint subject-object system.  For non-relativistic systems the Schrodinger wave equation is a good approximation to reality.  (See "Is many-worlds a relativistic theory?" for the more general case.)

The rest of the theory is working out consequences of the above assumptions.  Some consequences are: 1)   That each measurement causes a decomposition or decoherence of the universal wavefunction into non-interacting and non-interfering branches or worlds.  History forms a branching tree which encompasses all the possible outcomes of each interaction.  (See "Why do worlds split?" and "When do worlds split?")  Every historical what-if compatible with the initial conditions and physical law is realised.

2)   That the conventional statistical Born interpretation of the amplitudes in quantum theory is *derived* from within the theory rather than having to be *assumed* as an additional axiom.  (See "How do probabilities emerge within many-worlds?")

Many-worlds is a re-formulation of quantum theory [1], published in 1957 by Dr Hugh Everett III [2], which treats the process of observation or measurement entirely within the wave-mechanics of quantum theory, rather than an input an as additional assumption, as in the Copenhagen interpretation.  Everett considered the wavefunction a real object.  (Many-worlds is a return to the classical, pre-quantum view of the universe in which all the mathematical entities of a physical theory are real.  For example, the electromagnetic fields of James Clark Maxwell or the atoms of Dalton, were considered as real objects in classical physics.  Everett treats the wavefunction in a similar fashion.  Everett also assumed that the wavefunction obeyed the same wave equation during observation or measurement as at all other times.  This is the central assumption of many-worlds: that the wave equation is obeyed universally and at all times.

Everett discovered that the new, simpler theory - which he named the "relative state" formulation - predicts that interactions between two (or more) macrosystems typically split the joint system into a superposition of products of relative states.  The states of the macrosystems are henceforth correlated with each other.  Each element of the superposition - each a product of subsystem states - evolves independently of the other elements in the superposition.  The states of the macrosystems, by becoming correlated or entangled, meaning that it no longer possible to speak the state of one system in isolation from the other subsystems.  Instead we are forced to only speak of the relative states of the subsystems, with respect to the other subsystems.  Specifying the state of one subsystem leads to the state of the other subsystems.  In this sense the states of the subsystems are determined only relative to each other, hence Everett's original designation of his theory.

If one of the systems is an observer and the interaction an observation then observer has been split into a number of copies, each copy observing just one of the possible results of a measurement and unaware of the other results and its own observer-copies.  Interactions between systems and their environments, including communication between different observers in the same worlds, transmits the correlations, inducing local splitting or decoherence of branches of the universal wavefunction [7],[10].  Thus the entire world is split, quite rapidly, into a host of mutually unobservable but equally real worlds.

According to many-worlds all the possible outcomes of a quantum interaction are realised.  The wavefunction, instead of collapsing at the moment of observation, carries on evolving in a deterministic fashion, embracing all possibilities within it.  All outcomes exist simultaneously but do not interact further with each other, each world having split into mutually unobservable but equally real worlds or branches of the universal wavefunction.

Q4   What is a "world"?

Loosely speaking a "world" is a complex, partially closed set of interacting sub-systems which don't significantly interfere with other elements in a quantum superposition.  Any complex system and its coupled  environment, with a large number of internal degrees of freedom, counts as a world.  An observer, with internal irreversible processes, counts as a complex system.  In terms of the wavefunction, a world is a decohered branch of the universal wavefunction, which represents a single macrostate.  The worlds all exist simultaneously in a non- interacting linear superposition.

Sometimes "worlds" are called "universes", but more usually this is reserved the totality of worlds, or "histories" (Gell-Mann/Hartle's phrase, see "What is many-histories?").

Q5   What is a measurement?

A measurement is an interaction between subsystems that triggers an amplification process, typically within an object (which we often designate as the measuring apparatus) with many internal degrees of freedom, leading to a change in the higher-level structure of the object (which might be the recording apparatus).  The trigger is sensitive to some (often microphysical) parameter of the one of the subsystems, which we designate the measured system.  Eg the detection of a charged particle by a Geiger counter leads to the generation of a "click".  The absence of a charged particle does not generate a click.  The measured system is the charged particle.  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 observer.

Q6   Why do worlds split?

Worlds, or branches of the universal wavefunction, split when different components of a quantum superposition "decohere" from each other [7], [10].  Decoherence refers to the loss of coherency or absence of interference effects between the elements of the superposition.  For two components or worlds to interfere with each other all the atoms, subatomic particle, photons etc in each world have to be in the same state, in the same place.  For small systems this is quite possible.  In the double slit experiment, for instance, it only requires that the divergent paths of the diffracted particle overlap again at some point, because only the single particle has been split.  For more complex systems overlapping becomes harder because all the constituents particles have to overlap with their counterparts simultaneously.

In QM jargon we say that the components (or vectors in the underlying Hilbert space) have become permanently orthogonal due to the complexity of the systems increasing the dimensionality of the Hilbert space.  In a high dimension space almost all vectors are orthogonal.  Each time a new degree of freedom is activated the dimensionality of the space which the different components move through increases.  Thus vectors for complex systems, with a large number of degrees of freedom, naturally decompose into mutually orthogonal components which, because they never interfere again, are unaware of each other.  From the point of view of the complex systems they have split into different, mutually unobservable worlds.

Q7   When do worlds split?

Worlds irrevocably "split" at the sites of measurement-like interactions associated with thermodynamically irreversible processes.  An irreversible process will always produce decoherence which splits worlds.  (see "Why do worlds split?", [7], [10])

In the example of a Geiger counter and a charged particle (see "What is a measurement?") after the particle has passed the counter one world contains the clicked counter and that portion of the particle's wavefunction which passed though the detector.  The other world contains the unclicked counter with the particle's wavefunction with a "shadow" cast by the counter in the particle's wavefunction.  The Geiger counter split when the amplification process became irreversible.

The splitting is local (ie originally in the region of the Geiger counter in our example) and is transmitted causally to more distant systems (see "Is many-worlds a local theory?" and "Does the EPR experiment prohibit locality?").  The precise moment/location of the split is not sharply defined due to the subjective nature of irreversibility, but can be considered complete when much more than kT of energy has been released in an uncontrolled fashion into the environment.  (The event has become irreversible.)

Consider Schrodinger's Cat.  A cat is placed in a sealed box with a device that releases a lethal does of cyanide if a radioactive decay is detected.  After a while an observer opens the box to see if the cat is alive or dead.  According to the CI the cat was neither alive nor dead until the box was opened, whereupon the wavefunction of the cat collapsed into one of the two alternatives.  The paradox, according to Schrodinger, is that the cat presumably knew if it was alive *before* the box was opened.  According to many-worlds the device was split into two states (cyanide released or not) by the radioactive decay.  As the device/cyanide interacts with the cat the cat is split into two states (dead or alive).  From the surviving cat's point of view it occupies a different world from its unlucky and late copy.  The external observer is split into two copies only when the box is opened and is altered by the state of the cat.

In the language of thermodynamics the decay of the atom and the amplification of its detection by a Geiger counter, the release of the cyanide and the death of the cat are all irreversible events.  These events have caused the decoherence (see "Why do worlds split?") of the different branches of the wavefunction of the cat + device + box.  Decoherence [7] occurs when irreversible macro-level events take place and the macrostate description of an object admits no single description.  A macrostate, in brief, is the description of an object in terms of accessible external characteristics.

The advantage of linking the definition of worlds and the splitting process with thermodynamics is the splitting process is irreversible and forward-time-branching, following the increase with entropy.  Like all irreversible processes, though, there are exceptions even at the coarse- grained level and worlds will occasionally fuse.  A necessary, although not necessarily sufficient, precondition for fusing is for all records, memories etc that discriminate between the pre-fused worlds or histories be lost.

Q8a  What is sum-over-histories?

The sum-over-histories or the path integral formalism was developed by Feynman in the 1940s [F] as an alternative 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 PI formalism offers some interesting insights into many-worlds.  In the PI picture the single particle wavefunction at (x',t') is built up of contributions of all possible paths from (x,t), where each path's contribution weighted by a (phase) factor of exp(i*Action[path]/hbar) * wavefunction at (x,t).  The Action[path] is the time-integral of the lagrangian (roughly: the kinetic minus the potential energy) along the path from (x,t) to (x',t').  The final expression is thus 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 are over a 3-N space.

Feynman developed his PI formalism further for his work on quantum electrodynamics, QED, with his Feynman diagrams, in parallel with Schwinger and Tomonoga who developed a less visualisable form of QED.  Dyson showed that these approaches were all equivalent.

It is quite natural when analysing systems from the PI point of view to think of the particle continually splitting apart and fusing together to explore every possible intermediate configuration between the specified start and end states.  For this reason the technique is often referred to as "sum-over-histories".  Since we do not occupy a privileged moment in history it is natural to wonder if alternative histories are contributing equally to transition amplitudes in the future, and therefore that they all have equal reality.  Perhaps we shouldn't be surprised that Feynman, therefore, is on record as believing in many-worlds.  (See "Who believes in many-worlds?")  What is surprising is that Everett developed his many-worlds theory entirely from the Schrodinger viewpoint without any detectable influence from Feynman's work, despite sharing the same thesis supervisor, John A Wheeler.

[F]  Richard P Feynman, Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics, Vol 20 267 (1948)

Q8b  What is many-histories?

There is considerable linkage between thermodynamics and many-worlds, explored in the "decoherence" views of Zurek [7] and Gell-Mann and Hartle [10], Everett [1] and others [4b].

Gell-Mann and Hartle have extended the role of decoherence in defining the Everett worlds, or histories in their nomenclature.  They call their approach the "many-histories" approach, where each "coarse-grained or classical history" is associated with a unique time-ordered sequence of sets of irreversible events, including measurements, records, observations and the like.  (Fine-grained histories effectively relax the irreversible criterion.)  Physically the many-histories approach is isomorphic to Everett's many-worlds, although Gell-Mann and Hartle choose not to accept Everett's metaphysical stance that each history corresponds to an element of reality.

The worlds split or "decohere" from each other when irreversible events occur.  (See "Why do worlds split?" and "When do worlds split?".)  Correspondingly many-histories defines a multiply-connected hierarchy of classical histories where each classical history is a "child" of any parent history which has only a subset of the child defining irreversible events and a parent of any history which has a superset of such events.  Climbing up the tree from child to parent moves to progressively coarser grained consistent histories until eventually the top is reached where the history has *no* defining events (and thus consistent with everything!).  This is Everett's universal wavefunction.  The bottom of the coarse-grained tree terminates with the maximally refined set of decohering histories.  The classical histories each have a probability assigned to them and probabilities are additive in the sense that the sum of the probabilities associated a set classical histories is equal to the probability associated with the unique parent history defined by the set.  (Below the maximally refined classical histories are the fine grained or quantum histories, where probabilities are no longer additive and different histories significantly interfere with each other.  The bottom level consists of complete microstates, which fully specified states.)

Q9   How many worlds are there?

It so happens that we can use the thermodynamic Planck-Boltzmann relationship to count the branches at each splitting, at the lowest, maximally refined level of Gell-Mann's many-histories tree (See "What is many-histories?").  The bottom level consists of microstates which can be counted by the formula W = exp (S/k), where S = entropy, k = Boltzmann's constant (approx 1022 Joules/Kelvin) and W = number of worlds or macrostates.  The number of coarser grained worlds is lower, but still increasing with entropy by the same ratio, ie the number of worlds a single worlds splits into at the site of an irreversible event is exp(dS/k), where dS = entropy of the defining event.  Because k is very small a great many worlds split off at each macroscopic event.

Q10  Is many-worlds a local theory?

The simplest way to see that the many-worlds metatheory is local 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.

Another way of seeing this is examine how macrostates evolve.  Macrostates descriptions of objects evolve in a local fashion.  Worlds split as the macrostate description locally divides inside the light cone of the triggering event.  Thus the splitting is a local process, transmitted causally at light or sub-light speeds.  (See "Does the EPR experiment prohibit locality?" for more details and "When do worlds split?")

Q11  Is many-worlds a deterministic theory?

Yes, many-worlds is a deterministic theory, since the wavefunction obeys a deterministic wave equation at all times.  All possible outcomes of a measurement or interaction are embedded within the universal wavefunction although each observer, split by acts of observation, is only aware of single outcomes due to the linearity of the wave equation.  The world appears indeterministic, with the usual probabilistic collapse of the wavefunction, but at the objective level which includes all outcomes determinism is restored.

Some people are under the impression that the only motivation for many- worlds is a desire to return to a deterministic theory of physics.  This is not true.  As Everett pointed out, the objection with the standard Copenhagen interpretation is not the indeterminism per se, but that indeterminism occurs only with the intervention of an observer, when the wavefunction collapses.

Q12  Is many-worlds a relativistic theory?

It is trivial to relativise many-worlds because all relativistic theories of physics are still quantum theories with linear wavefunctions.  There are three or more stages to developing a fully quantum relativistic theory.  Simplifying slightly gives:

First quantisation: the wavefunction is a complex field which evolves in 3N dimensions which represent N particles.  The wavefunction is a solution of either the many-particle Schrodinger, Dirac or Klein-Gordon equation or some other wave equation.

Second quantisation: AKA quantum field theory, which handles the creation and destruction of particles by quantising fields as well as particles.  (Each particle type corresponds to a field, in QFT.  Eg the electromagnetic field's particle is the photon, but the number of particles involved is indeterminate.)  Again many-worlds has no problems handling QFT.  The wavefunction of a collection of particles and fields exists in a Fock space, where the number of dimensions varies from component to component.

Third quantisation.  The gravitational metric is quantised, along with (perhaps) the topology of space-time.  The physics of this is incomplete, but there is no reason for thinking that many-worlds can't be extended to cover this as well.  (One of the original motivations of Everett's scheme was to provide a system for quantizing the gravitational field within quantum cosmology to yield a complete description of the universe.)

Q13  Is many-worlds (just) an interpretation?

No, for four reasons:

First, many-worlds has testable implications (see "Is many-worlds testable?") and interpretations, generally, do not have testable differences from each other.

Second, the mathematical structure of many-worlds is not isomorphic to other formulations of quantum mechanics like the Copenhagen interpretation or Bohm's hidden variables.  The Copenhagen interpretation does not contain those elements of the wavefunction that correspond to the other worlds.  Bohm's hidden variables contain particles, in addition to the wavefunction.  Therefore neither theory is isomorphic to each other or many-worlds and are not, therefore, merely rival "interpretations".

Third, there is no scientific, reductionistic alternative to many- worlds.  All the other theories fail for logical reasons.  (See "Is there any alternative theory?")

Four, the interpretative side of many-worlds, like the subjective probabilistic elements, are derived from within the theory, rather than added in by assumption, as in the conventional approach.  (See "How do probabilities emerge within many-worlds?")

Many-Worlds should really be described as a theory or, more precisely, a metatheory, as Everett pointed out, since it makes statements that are applicable across a range of theories.  Many-worlds is the unavoidable implication of any quantum theory which obeys some type of wave equation, linear with respect to the wavefunction it operates on.

Q14  What are the alternatives?

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

1)   Copenhagen Interpretation.  Postulates that the observer obeys different physics than the non-observer.  (A return to vitalism.)  The definition of observer varies from one adherent to another, if present at all.  The status of the wavefunction is also ambiguous.  If the wavefunction is real the theory is non-local (not fatal, but unpleasant), if not real then the theory supplies no model of reality.  (See "What are the problems with quantum theory?")

2)   Hidden Variables [B].  Explicitly non-local.  Bohm accepts that all the branches of the universal wavefunction exist.  Like Everett Bohm held that the wavefunction is real complex-valued field which never collapses.  In addition he postulated that there were particles that move under the influence of a non-local "quantum- potential" derived from the wavefunction, in addition to the classical potential.  The action of the quantum-potential is such that the particles are affected by only one of the branches of the wavefunction.  (Bohm derives what is essentially a decoherence argument to show this, see section 7,#I [B]).

The implicit, unstated assumption made by Bohm is that only the single branch of wavefunction associated with particles can contain self-aware observers, whereas Everett makes no such assumption.  Most of Bohm's adherents do not seem to understand (or even be aware of) Everett's criticism, section VI [1], that the hidden- variable particles are not observable since the wavefunction alone is sufficient to account for all observations.  The particles can, therefore, be discarded, along with the guiding quantum-potential, yielding a theory isomorphic to many-worlds, without affecting any experimental results.

[B]  David J Bohm A suggested interpretation of the quantum theory in terms of "hidden variables" I and II, Physical Review Vol 85 #2 166-193 (1952)

3)   Quantum Logic.  Undoubtedly the most extreme of all attempts to solve the QM measurement problem.  Apart from abandoning one or other of the classical tenets of logic these theories are all unfinished (presumably because of internal inconsistencies).  Also it is unclear why different types of logic apply on different scales.

4)   Extended Probability [M].  A bold theory in which the concept of probability is "extended" to include complex values [Y].  Whilst quite daring, I am not sure if this is logically permissable, being in conflict with the relative frequency notion of probability, in which case it suffers from the same criticism as quantum logic.  Also it is unclear, to me anyway, how the resultant notion of "complex probability" differs from the "probability amplitude" and thus why we are justified in collapsing the complex probability as if it were a classical probability.

[M]  W Muckenheim A review of extended probabilities Physics Reports Vol 133 339- (1986)

[Y]  Saul Youssef __ hep-th 9307019

5)   Transactional model [C].  Explicitly non-local.  An imaginative theory, based on the Feynman-Wheeler absorber-emitter model of EM, in which advanced and retarded probability amplitudes combine into an atemporal "transaction" to form the Born probability density.  It requires that the input and output states, as defined by an observer, act as emitters and absorbers respectively, but not any internal states (inside the "black box"), and, consequently, suffers from the familiar measurement problem of the Copenhagen interpretation.

If the internal states *did* act as emitters/absorbers then the wavefunction would collapse, for example, around one of the double slits (an internal state) in the double slit experiment, destroying the observed interference fringes.  In transaction terminology a transaction forms between the first single slit and one of the double slits and another transaction forms between the same double slit and the point of screen where the photon lands.

[C]  John G Cramer, The transactional interpretation of quantum mechanics Reviews of Modern Physics Vol 58 #3 647-687 (1986)

6)   many-minds.  Despite its superficial similarities with many-worlds this is actually a very unphysical, non-operational theory.  (See "What is many-minds?")

7)   Non-linear theories in general.  So far no non-linear theory has any accepted experimental support, whereas many have failed experiment.  (See "Is physics linear?")

Q15  Is many-worlds testable?

Yes, it is.  There are two forms of tests: retrodictions (theory follows data) and predictions (data follows theory).

A) 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 data 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.

Many-worlds retrodicts all the peculiar properties of the (apparent) wavefunction collapse in terms of decoherence.  (See "Can wavefunctions collapse?", "When do worlds split?" & "Why do worlds split?")  No other quantum theory has yet accounted for this behaviour scientifically.  (See "What are the alternatives?")

B) A prediction occurs when a theory suggests new phenomena.  

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.  See a guide to QM.)

According to many-worlds worlds split with the operation of every thermodynamically irreversible process.  The operation of our minds are irreversible, carried along for the ride, and divide with the worlds.  Normally, therefore, 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 will doable by 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 dissipates much less than kT of energy is reversible.  (This implies that frictive or dissipative forces are absent.)  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 1017 bits/sec, based on the number of neurons (approx 1010) per human brain, average number of synapses per neuron (approx 104) and the average firing rate (approx 103 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 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 measurements 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, 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.

Q16  Could previously separate worlds diverge rather than split?

This is definitely not permissable in many-worlds.  Worlds do not exist in a quantum superposition independently of each other before they decohere or split.  The splitting is a physical process, grounded in the dynamical evolution of the wave vector, not a matter of philosophical/mental convenience (see "Why do worlds split?" and "When do worlds split?")  If you try to treat the worlds as pre-existing and separate then the maths all comes out wrong.  Also the divergence theory stops being deterministic, in contradiction to the wave equations which are deterministic, since we have a

 

AAAAAAAAAAAAAAABBBBBBBBBBBBBBB                      ===========> time

                                                                                                                Worlds diverge

AAAAAAAAAAAAAAACCCCCCCCCCCCCCC

 

situation, rather than:

 

                                                BBBBBBBBBBBBBBB

                                          B

AAAAAAAAAAAAAA                                                                 Worlds splitting

                                          C

                                                CCCCCCCCCCCCCCC

 

Additionally the divergence model has to explain why:

 

AAAAAAAAAAAAAAABBBBBBBBBBBBBBB

AAAAAAAAAAAAAAABBBBBBBBBBBBBBB

 

doesn't happen!  This false divergence model, at the mental level, seems favoured by adherents of many-minds.  (See "What is many-minds?")

Q17  What is many-minds?

Many-minds proposes, as an extra fundamental axiom, that an infinity of separate minds or mental states be associated with each single brain state.  When the single physical brain state is split into a quantum superposition by a measurement the associated minds are thought of as diverging rather than splitting.  The motivation for this brain-mind dichotomy seems purely to avoid talk of minds splitting and talk instead about the divergence of pre-existing separate mental states.  There is no physical basis for this interpretation, which is incapable of an operational definition.  Indeed the divergence model for physical systems is specifically not permitted in many-worlds.  Many-minds seems to be proposing that minds follow different rules than matter.  (See "Could previously separate worlds diverge rather than split?")

In many-minds the role of the conscious observer is accorded special status, with its fundamental axiom about infinities of minds, and as such is philosophically opposed to many-worlds, which seeks to remove the observer from any privileged role in physics.  (Many-minds was co- invented by David Albert, who has, apparently, since abandoned it.  See Scientific American July 1992 page 80 and contrast with April 94.)

The two theories should not be confused.  

Q18  Does many-worlds violate Ockham's Razor?

William of Ockham, 1285-1349(?) English philosopher and one of the founders of logic, proposed a maxim for judging theories which says that hypotheses should not be multiplied beyond necessity.  This is known as Ockham's razor and is interpreted, today, as meaning that to account for any set of facts the simplest theories are to be preferred over more complex ones.  Many-worlds is viewed as unnecessarily complex, by some, by requiring the existence of a multitude of worlds to explain what we see, at any time, in just one world.

This is to mistake what is meant by "complex".  Here's an example.  Analysis of starlight reveals that starlight is very similar to faint sunlight, with spectroscopic absorption and emission lines.  Assuming the universality of physical law we are led to conclude that other stars and worlds are scattered, in great numbers, across the cosmos.  The theory that "the stars are distant suns" is the simplest theory and so to be preferred by Ockham's Razor to other geocentric theories.

Similarly many-worlds is the simplest and most economical theory because it proposes that same laws of physics apply to animate observers as inanimate objects.  The multitude of worlds predicted by the theory is not a weakness for many-worlds, any more than stars are for astronomy, since the non-interacting worlds emerge from a simpler theory.

(As an historical aside it is worth noting that Ockham's razor was also falsely used to argue in favour of the older heliocentric theories *against* Galileo's notion of the vastness of the cosmos.  The notion of vast empty interstellar spaces was too uneconomical to be believable.  Again they were confusing the notion of vastness with complexity [15].)

Q19  Does the multiplication of worlds violate conservation of energy?

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 weighted averages or of expectation values.  Conservation of energy is expressed by saying that the time derivative of the expectation of the energy operator vanishes.  This statement can be scaled up to includes the whole universe.  Each world has an approximate energy, but the energy of the total wavefunction (of 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 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 smaller and smaller slices, as measured in the Hilbert space.

Q20  How do probabilities emerge within many-worlds?

Everett demonstrated [1],[2] that observations in each world obey all conventional statistical laws predicted by the probabilistic Born interpretation by showing that the Hilbert space's inner product or norm supplies a unique measure or "volume" to each world or set of worlds.  The norm 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 the limit.  Vectors with zero norm, where probability breaks down, don't exist (see below), thus we, as observers, observe the familiar predictions of quantum theory expressed as probabilistic events.

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 would hold true and asserted the vanishing of the Hilbert space volume of the set of non-random worlds.  DeWitt (with Graham) later published a longer *derivation* of Everett's assertion [4a],[4b].  What Everett asserted and DeWitt 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 a Hilbert space) this is equivalent to saying that non-randomness is never realised.  Thus all worlds obey the usual Born predictions of quantum theory.

In more detail the steps are:

1)   Construct the tensor product of N identical systems in state |psi> (repeated indices summed):

                |PSI_N> = |psi_1>*|psi_2>*...... |psi_N> where

                |psi_j>    = jth system prepared in state |psi>

                                = |i_j><i_j|psi> (ie the amplitude of the ith eigenstate is independent of which system

                                                                it is in) so that 

                |PSI_N> = |i_1>|i_2>...|i_N><i_1|psi><i_2|psi>...<i_N|psi>

2)   Quantify the deviation from the "expected" Born-mean for each component of |PSI_N> with respect to     the above |i_1>|i_2>...|i_N> basis by counting the number of occurrences of the ith eigenstate/N.      Call this number RF(i).  Define the Born-deviation as D = sum(i)( (RF(i) - |<i|psi>|2)2 ). 

                Thus D, loosely speaking, for each N length sequence expresses how "non-random" a particular

                sequence is although, of course, no finite sequence is excluded from happening since the concept

                of non-random becomes precise only as N goes to infinity [H].

3)   Sort out terms in the expansion of |PSI_N> according to whether D is less/equal to (.LE.) or greater            than (.GT.) E, where E is a real, positive constant.  Collecting terms together we get:

                |PSI_N> = |N,"D.GT.E"> + |N,"D.LE.E">

                                                worlds                    worlds

                                                for which               for which

                                                D > E                      D <= E

4)   What DeWitt showed was that:

                <N,"D.GT.E"|N,"D.GT.E"> < 1/(NE)     (proof in appendix of 4b)

                Thus as N goes to infinity then the right-hand side vanishes for all positive values of E.

                (This mirrors the classical "frequentist" position on probability which states that if i occurs

                with probability p(i) then the proportion of N trials with success i approaches p(i)/N as N goes

                to infinity [H].  This has the immediate benefit that sum(i) = 1.)  The norm of |N,"D.LE.E">,

                by contrast, approaches 1 as N goes to infinity.

5)   The norm of the collection of non-random worlds vanishes and therefore must be identified with

                the same complex multiple of the null vector.  

6)   Since (by assumption) the state vector faithfully models reality then the null vector cannot represent

                any element of reality since it can be added to (or subtracted from) any other state vector without

                altering the other state vector.

7)   Ergo the non-random worlds are not realised, without making any additional physical assumptions.

 

The emergence of Born-style probabilities as a consequence of the mathematical formalism of the theory, without any extra interpretative assumptions, is another reason why the Everett metatheory should not be regarded as just an interpretation.  (See "Is many-worlds (just) an interpretation?")  The interpretative elements are forced my the mathematical structure inherent in the axioms.

[H]  JB Hartle, Quantum Mechanics of Individual Systems, American Journal of Physics Vol 36 #8 704-712 (1968)  Hartle has investigated the N goes to infinity in more detail and more generally.  He shows that the relative frequency operator obeys RF(i) |psi_1>|psi_2>.... = |<i|psi>|2 |psi_1>|psi_2>....

Q21  Does many-worlds allow free-will?

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 - insofar as it is not an illusion, which is a perfectly acceptable way of viewing it - of the mechanistic operation of our brains, the firing of neurons, discharging of synapses etc and fully compatible with the determinism of classical physics.

Nevertheless, some people find that with all possible decisions being realised in different worlds that the prima facia 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 has its own weighting or measure which enforces the usual laws of quantum statistics.

The measure is supplied by the mathematical structure of Hilbert spaces.  Every Hilbert space has a norm, constructed from the inner product, - which we can think of as analogous to a volume - which weights each world or collection of worlds.  A world of zero volume is never realised.  Worlds in which the conventional statistical predictions consistently break down have zero volume and so are never realised.  (See "How do probabilities emerge within many-worlds?")  

Thus our actions, as expressions of our will, correlate with the weights associated with worlds.  This, of course, matches our subjective experience of being able to exercise our will, form moral judgements and be held responsible for our actions.

Q22  Why am I in this world and not another? or Why the universe appears random, but isn't.

Consider, for a moment, this analogy:

Suppose Fred has his brain divided in two and transplanted into different cloned bodies (this is a gedanken operation!).  Let's further suppose that each half brain is regenerates to full functionality and we name 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 Fred it seems random.

Same with many-worlds.  There was no reason "why" you ended up in this world, rather than another.  It was a subjectively random choice, an artifact of your consciousness being split.  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 splitting of worlds.

(See "How do probabilities emerge within many-worlds?" for how the subjective randomness is moderated by the usual probabilistic laws of QM.)

Q23  Can wavefunctions collapse?

Many-Worlds predicts/retrodicts that wavefunctions appear to collapse (see "The EPR experiment"), when measurement-like interactions and processes occur via a process called decoherence [7], [10], but claims that they do not *actually* collapse.   If a *mechanism* for collapse could be found then there would be no need for many-worlds.  The reason why we doubt that collapse takes place is because no one has ever been able to devise a physical mechanism that could trigger it.

The Copenhagen interpretation posits that observers collapse wavefunctions, but is unable to define "observer".  (See "What is the Copenhagen interpretation?" and "Is there any alternative theory?")  Without a definition there can be no mechanism.

Another popular view is that irreversible processes trigger collapse.  Certainly wavefunctions *appear* to collapse whenever irreversible processes are involved in measurement or amplification and most macroscopic, day-to-day events are irreversible.  The problem is, as with positing observers as a cause of collapse, that any irreversible process is composed of a large number of sub-processes that are each individually reversible.  To invoke irreversibility as a *mechanism* for collapse we would have to show that new *fundamental* physics comes into play for complex systems, which is quite absent at the atom/molecular level.  Atoms and molecules are empirically observed to obey some type of wave equation.  We have no evidence for an extra mechanism operating on more complex systems.  As far as we can determine complex systems are described by the same quantum-operation of their simpler components.  (Note:  chaos, complexity theory, etc, do not introduce new fundamental physics.  They still operate within the reductionistic paradigm - despite what many popularisers say.)

Other people have attempted to construct non-linear theories so that microscopic systems are approximately linear and obey the wave equation but macroscopic systems are grossly non-linear and generates collapse.  Unfortunately all these efforts have made additional predictions which, when tested, have failed.

(Another reason for doubting that any collapse actually takes place is that the collapse would have to propagate instantaneously or in some space-like fashion, otherwise the same particle would be observed more than one at different locations.  Not fatal, but unpleasant and difficult to reconcile with relativity.)

The simplest conclusion is that wavefunctions just *don't* collapse and that all branches of the wavefunction exist.

Q24  Is physics linear? or Could we ever communicate with the other worlds?  or Why do I only ever experience one world?

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 which says that the worlds can interfere with each other with respect to a external, unsplit, observer or system (whence they are manifest as diffraction or interference patterns) but they can't influence each other in the sense that an experimenter can arrange to communicate with their own, already split-off, quantum copies.

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 also a solution itself.  Since each component of a linear solution evolves with complete indifference as to the presence or absence of the other terms 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.

Linearity (of the wavefunction) has been verified hold true to better than 1 part in 1027 [W] and some scientists believe that it is absolutely true for various theoretical reasons.  If slight non-linear effects were ever discovered then the possibility of communication with/travel to the other worlds would be opened up.

[W]  Steven Weinberg, Annals of Physics, Vol 194 #2 336-386 (1989) and Dreams of a Final Theory (1992)

Q25  Can we determine what other worlds there are? or Is the form of the Universal Wavefunction knowable?

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 discernable 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 therefore 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 they diverging.  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 we'll just have to accept this as given.  The fossil and historical records say that dinosaurs and Adolf Hitler once existed but have little to say about 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 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 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 physical law itself.

Q26  Who was Everett?

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) 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 N Carolina, Chapel Hill, N Carolina 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.  At some point he 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 enemy kill rates during nuclear-weapon campaigns.

Later (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 talks 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 and agitated manner, and spoke with a very acute style, correctly anticipating and cutting off 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 the subject of measurement in quantum theory, but died of a heart attack in 1982.  Survived by his wife.

Q27  Who believes in many-worlds?

A poll conducted by "political scientist" L David Raub, reported by Tipler [T], of 72 of the "leading cosmologists and other quantum field theorists" about the "Many-Worlds Interpretation" gives the following breakdown. 

1) Yes, I think MWI is true                                 58%

2) No, I don't accept MWI                                                  18%

3) Maybe it's true but I'm not yet convinced                   13%

4) I have no opinion one way or the other                       11%

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 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" listed by name is Penrose.

[T]  Frank J Tipler, The Physics of Immortality, pages 170-1

Q30  Does the EPR experiment prohibit locality? or Quantum mechanics without operators

The EPR experiment is widely regarded as the definitive gedanken experiment for demonstrating that quantum mechanics is non-local 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 (ie 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:

                |psi> =  |me>          |electrons>             |you> |rest of universe>

                                on           in                             on

                                Earth       deep                       Alpha

                                                space                      Centauri

or more compactly, ignoring the rest of the universe, as:

                |psi> =  |me,electrons,you>  

And

                |electrons> = (|+,-> - |-,+>)/sqrt(2) 

                                represents a pair electrons, with the first electron travelling towards Earth and

                                the second electron travelling towards Alpha Centauri.

                |+> represents an electron with spin in the +z direction

                |-> represents an electron with spin in the -z direction

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)):

                |left>  = (|+> - |->)/sqrt(2)    (electron with spin in -x direction)

                |right> = (|+> + |->)/sqrt(2)    (electron with spin in +x direction)

                |up>    = (|+> + |->i)/sqrt(2)   (electron with spin in +y direction)

                |down>  = (|+> - |->i)/sqrt(2)   (electron with spin in -y direction)

and inverting:

                |+>  = (|right> + |left>)/sqrt(2) =  (|up> + |down>)/sqrt(2)

                |->  = (|right> - |left>)/sqrt(2) =  (|down> - |up>)i/sqrt(2)

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

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

                |+,-> - |-,+>             =  |left,right> -  |right,left>

                                                =  |up,down>i    - |down,up>i

|me> represents me on Earth with my detection apparatus.  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:

                |me[l]> represents me + apparatus + records having measured  x-axis spin and recorded the x-                                            axis spin as "left"

                |me[r]> ditto with the x-axis spin as "right"

                |me[u]> ditto with the y-axis spin as "up"

                |me[d]> ditto with the y-axis spin as "down"

Similarly for |you> on Alpha Centauri.  Notice that it is irrelevant *how* we have measured the electron's spin.  The details of the measurement process are irrelevant.  To model the process it is sufficient to assume that there is a way, which we have further assumed does not disturb the electron.  (The latter assumption may be relaxed without altering the results.)

To establish familiarity with the notation let's take the state of the initial wavefunction as:

                |psi>_1 =                |me,left,up,you>

                                                          /           \

                                                        /              \

                first electron in left                               second electron in up state

                state heading towards                         heading towards you on

                me on Earth                                           Alpha Centauri

After the electrons arrive at their detectors, I measure the spin along the x-axis and you along the y-axis.  The wavefunction evolves into |psi>_2:

                                       local

                |psi>_1 ============> |psi>_2 = |me[l],left,up,you[u]>

                                  observation

which represents me having recorded my electron on Earth with spin left and you having recorded your electron on Alpha Centauri with spin up.  The index in []s indicates the value of the record.  This may be held in the observer's memory, notebooks or elsewhere in the local environment (not necessarily in a readable form).  If we communicate our readings to each other the wavefunctions evolves into |psi>_3:

                                     remote 

                |psi>_2 ============> |psi>_3 = |me[l,u],left,up,you[u,l]>

                                 communication

where the second index in []s represents the remote reading entering the observers' local records.  Notice that the results both agree with each other.  Eg my record of your reading agrees with your reading.

That's the notation established.  Now let's see what happens in the more general case where, again,:

                |electrons> = (|+,-> - |-,+>)/sqrt(2).

First we'll consider the case where you and I have previously arranged to measure the our respective electron spins along the same x-axis.

Initially the wavefunction of the system of electrons and two experimenters is:

                |psi>_1   =  |me,electrons,you>

                                = |me>(|left,right> - |right,left>)|you> /sqrt(2)

                                =  |me,left,right,you> /sqrt(2) - |me,right,left,you> /sqrt(2)

After the we each perform our measurements we get:

                |psi>_2   =  |me[l],left,right,you[r]> /sqrt(2) - |me[r],right,left,you[l]> /sqrt(2)

The observers (you and me) have been split (on Earth and Alpha Centauri) into relative states (or local worlds) which correlate with the state of the electron.  If we now communicate over interstellar modem (this will take a few years since you and I are separated by light years, but no matter) then, for example, the world corresponding to the 2nd term in the above expansion contains me having seen my electron with spin right also contains you having seen your electron with spin left.  So we jointly agree, in both worlds, that spin has been conserved.

Now suppose that we had prearranged to measure the spins along different axes.  Suppose I measure the x-direction spin and you the y-direction spin.  Now things get a bit more complex.  To analyse what happens we need to decompose the two electrons along their respective spin axes.

                |psi>_1   = |me,electrons,you>

                                = |me>(|+,-> - |-,+>)|you>/sqrt(2) 

                                = |me> (

                                                  (|right>+|left>)i(|down>-|up>)

                                                - (|right>-|left>)(|down>+|up>)

                                                ) |you> /2*sqrt(2)

                                = |me> (

                                                  |right>(|down>-|up>)i

                                                + |left> (|down>-|up>)i

                                                - |right>(|down>+|up>)

                                                + |left> (|down>+|up>)

                                                ) |you> /2*sqrt(2) 

                                = |me> (

                                                  |right,down> (i-1) - |right,up> (1+i)

                                                + |left,up> (1-i) + |left,down> (1+I)

                                                ) |you> /2*sqrt(2) 

                                =  (

                                                + |me,right,down,you>        (i-1)

                                                - |me,right,up,you>               (i+1)

                                                + |me,left,up,you>                (1-i)

                                                + |me,left,down,you>           (1+I)

                                ) /2*sqrt(2) 

So after you and I make our local observations we get:

                |psi>_2 =

                                (

                                + |me[r],right,down,you[d]>               (i-1) 

                                - |me[r],right,up,you[u]>                      (I+1)

                                + |me[l],left,up,you[u]>                        (1-I)

                                + |me[l],left,down,you[d]>                  (1+I)

                                ) /2*sqrt(2)

Each term realises a possible outcome of the joint measurements.  The interesting thing is that whilst we can decompose it into four terms there are only two states for each observer.  Looking at myself, for instance, we can rewrite this in terms of states relative to *my* records/memories.

                |psi>_2 = 

                                (

                                  |me[r],right> ( |down,you[d]> (i-1) - |up,you[u]> (i+1) )

                                + |me[l],left>  ( |up,you[u]> (1-i) + |down,you[d]> (1+i) )

                                ) /2*sqrt(2)

And we see that there are only two copies of *me*.  Equally we can rewrite the expression in terms of states relative to *your* records/memory.

                |psi>_2 =

                                (

                                  ( |me[l],left> (1-i) - |me[r],right> (i+1) ) |up,you[u]> 

                                + ( |me[r],right> (i-1) + |me[l],left> (1+i) ) |down,you[d]>

                                ) /2*sqrt(2)

And see that there are only two copies of *you*.   We have each been split into two copies, each perceiving a different outcome for our electron's spin, but we have not been split by the measurement of the remote electron.  

*After* you and I communicate our readings to each other, > four years later, we get:

                |psi>_3 =

                                (

                                + |me[r,d],right,down,you[d,r]>          (i-1) 

                                - |me[r,u],right,up,you[u,r]>                (i+1) 

                                + |me[l,u],left,up,you[u,l]>  (1-I)

                                + |me[l,d],left,down,you[d,l]>             (1+i)

                                ) /2*sqrt(2)

The decomposition into four worlds is only forced and unique after communication between the remote systems.  Until the two observers communicated their results to each other they were each unsplit by each others' remote measurements, although their own local measurements had split themselves.  The splitting is a local process that is causally transmitted from system to system at light or sub-light speeds.  (This is a point that Everett stressed about Einstein's remark about the observations of a mouse, in the Copenhagen interpretation, collapsing the wavefunction of the universe.  Everett observed that it is the mouse that's split by its observation of the rest of the universe.  The rest of the universe is unaffected and unsplit.)

When all communication is complete the worlds have finally decomposed or decohered from each other.  Each world contains a consistent set of observers, records and electrons, in perfect agreement with the predictions of standard QM.  Further observations of the electrons will agree with the earlier ones and so each observer, in each world, can henceforth regard the electron's wavefunction as having collapsed to match the historically recorded, locally observed values.  This justifies our operational adoption of the collapse of the wavefunction upon measurement, without having to strain our credibility by believing that it actually happens.

To recap.  Many-worlds is local and deterministic.  Local measurements split local systems (including observers) in a subjectively random fashion; distant systems are only split when the causally transmitted effects of the local interactions reach them.  We have not assumed any non-local FTL effects, yet we have reproduced the standard predictions of QM.

So where did Bell and Eberhard go wrong?  They thought that all theories that reproduced the standard predictions must be non-local.  It has been pointed out by both Albert [A] and Cramer [C] (who both support different interpretations of QM) that Bell and Eberhard had implicity assumed that every possible measurement - even if not performed - would have yielded a *single* definite result.  This assumption is called contra-factual definiteness or CFD [S].  What Bell and Eberhard really proved was that every quantum theory must either violate locality *or* CFD.  Many-worlds with its multiplicity of results in different worlds violates CFD, of course, and thus can be local.

Thus many-worlds is the only local quantum theory in accord with the standard predictions of QM and, so far, with experiment.

[A]  David Z Albert, “Bohm's Alternative to Quantum Mechanics” Scientific American (May 1994) [As] Alain Aspect, J Dalibard, G Roger, “Experimental test of Bell's inequalities using time-varying analyzers”, Physical Review Letters Vol 49 #25 1804 (1982). [C]  John G Cramer, “The transactional interpretation of quantum mechanics”, Reviews of Modern Physics Vol 58 #3 647-687 (1986)

[B]  John S Bell:  “On the Einstein Podolsky Rosen paradox”, Physics 1 #3 195-200 (1964).

[E]  Albert Einstein, Boris Podolsky, Nathan Rosen:  “Can quantum-mechanical description of physical reality be considered complete?”  Physical Review Vol 41, 777-780 (15 May 1935).

[S]  Henry P Stapp, “S-matrix interpretation of quantum-theory”, Physical Review D Vol 3 #6 1303 (1971)

Q31   References and further reading

[1]  Hugh Everett III, The Theory of the Universal Wavefunction, Princeton thesis (1956?) The original and most comprehensive paper on many-worlds.  Investigates and recasts the foundations of quantum theory in information theoretic terms, before moving on to consider the nature of interactions, observation, entropy, irreversible processes, classical objects etc.  138 pages.  Only published in [5].

[2]  Hugh Everett III, “"Relative State" Formulation of Quantum Mechanics”, Reviews of Modern Physics Vol 29 #3 454-462, (July 1957)  A condensation of [1] focusing on observation.

[3]  John A Wheeler, “Assessment of Everett's "Relative State" Formulation of Quantum Theory”, Reviews of Modern Physics Vol 29 #3 463-465 (July 1957)  Wheeler was Everett's PhD supervisor.

[4a] Bryce S DeWitt, “Quantum Mechanics and Reality”, Physics Today, Vol 23 #9 30-40 (September 1970)  One of the earlier, and more accurate, popularisations of Everett's work.  The April 1971 issue has reader feedback and DeWitt's responses.

[4b] Bryce S DeWitt, “The Many-Universes Interpretation of Quantum Mechanics” in Proceedings of the International School of Physics "Enrico Fermi" Course IL: Foundations of Quantum Mechanics Academic Press (1972)

[5]  Bryce S DeWitt, R Neill Graham eds., The many-worlds Interpretation of Quantum Mechanics, Contains [1],[2],[3],[4a],[4b] plus other material.  Princeton Series in Physics, Princeton University Press (1973) ISBN 0-691- 08126-3 (hard cover), 0-691-88131-X (paper back)  The definitive guide to many-worlds, if you can get hold of a copy, but now (1994) only available xeroxed from microfilm (ISBN 0-7837-1942-6) from Books On Demand, 300 N Zeeb Road, Ann Arbor, MI 48106-1346, USA.  Tel: +01-313 761 4700 or 800 521 0600.

[6]  John A Wheeler, Wojciech H Zurek eds Quantum Theory and Measurement Princeton Series in Physics, Princeton University Press (1983) ISBN 0-691-08316-9.  Contains 49 classic articles, including [2], covering the history and development of interpretations of quantum theory. 

[7]  Wojciech H Zurek “Decoherence and the Transition from the Quantum to the Classical”, Physics Today, 36-44 (October 1991). The role of thermodynamics and the properties of large ergodic systems (like the environment) are related to the decoherence or loss of interference effects between superposed macrostates.

[8]  Max Jammer The Philosophy of Quantum Mechanics Wiley, New York (1974)  Almost every interpretation of quantum mechanics is covered and contrasted.  Section 11.6 contains a lucid review of many-worlds theories.

[9]  Marian O Scully, Bethold-Georg Englert, Herbert Walther “Quantum optical tests of complementarity” Nature, Vol 351, 111-116 (9 May 1991). Demonstrates that quantum interference effects are destroyed by irreversible object-apparatus correlations, not by the measurement process itself.

[10] Murray Gell-Mann, James B Hartle “Quantum Mechanics in the Light of Quantum Cosmology” Proceedings of the 3rd International Symposium on the Foundations of Quantum Mechanics (1989) 321-343.  They accept the Everett's decoherence analysis, and have extended it further, but reject many-worlds' metaphysical stance. Tests of the Everett metatheory:

[11] David Deutsch “Quantum theory as a universal physical theory” International Journal of Theoretical Physics, Vol 24 #1 (1985).  Describes an experiment which tests for the existence of superpositions of *consciousness (in an AI). [16] David Deutsch “Three connections between Everett's interpretation and experiment” Quantum Concepts of Space and Time, eds Roger Penrose and Chris Isham, Oxford University Press (1986).  Discusses a testable split observer experiment and quantum computing. On quantum computers:

[12] David Deutsch “Quantum theory, the Church-Turing principle and the universal quantum computer” Proceedings of the Royal Society of London, Vol. A400, 96-117 (1985).

[13] David Deutsch “Quantum computational networks” Proceedings of the Royal Society of London, Vol. A425, 73-90 (1989).

[14] David Deutsch and R. Jozsa “Rapid solution of problems by quantum computation” Proceedings of the Royal Society of London, Vol. A439, 553-558 (1992).

[15] Frank J Tipler “The many-worlds interpretation of quantum mechanics in quantum cosmology” in Quantum Concepts of Space and Time eds Roger Penrose and Chris Isham, Oxford University Press (1986).  Has a discussion of Ockham's razor. On quantum theory, measurement and decoherence generally:

 

 

Mike Price                         price@price.demon.co.uk