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8/7/2019 Many Worlds Theory of Infinite Parallel Universes http://slidepdf.com/reader/full/many-worlds-theory-of-infinite-parallel-universes 1/25 Many Worlds Theory of Infinite Parallel Universes The Many Worlds Theory of Infinite Parallel Universes is the most satisfying scientific theory to explain the paradoxes inherent in Quantum Reality. Quantum Reality is the most successful scientific theory to ever explain the experimental data gathered by over a century of physics research. However, the conclusions are mind boggling to scientists because they want a nice logical explanation for the universe and quantum reality gives them a world of impreciseness and probability with data pointing to a single non-local field of energy composed of waveforms. This includes making the assumption that objects and observers are not independent but somehow linked. It is the act of observing that causes all the paradoxes. To solve all of the paradoxes of observers and object being linked somehow, the theory expresses that for every quantum event observed, the universe splits into each and every possible observable outcome and each universe continues separate, and in parallel, unaware of the other universes. In effect, a universe without observers would exist as a superimposed set of possible outcomes, with each outcome in a suspended state of unmanifested existence. It is the act of observation that makes possibilities transend from probability to reality. Archive-name: many-worlds-faq Last-modified: 17 February 1995 Posting-Frequency: in full: 3-monthly, abridged: monthly (ex *.answers) (C) Michael Clive Price, February 1995 Permission to copy in its entirety granted for non-commercial purposes. Contents: Q0 Why this FAQ? Q1 Who believes in many-worlds? Q2 What is many-worlds? Q3 What are the alternatives to many-worlds? Q4 What is a "world"? Q5 What is a measurement? Q6 Why do worlds split? What is decoherence? Q7 When do worlds split? Q8 When does Schrodinger's cat split? Q9 What is sum-over-histories? Q10 What is many-histories? What is the environment basis? Q11 How many worlds are there? Q12 Is many-worlds a local theory? Q13 Is many-worlds a deterministic theory? Q14 Is many-worlds a relativistic theory? What about quantum field theory? What about quantum gravity? Q15 Where are the other worlds? Q16 Is many-worlds (just) an interpretation? Q17 Why don't worlds fuse, as well as split? Do splitting worlds imply irreversible physics? Q18 What retrodictions does many-worlds make? Q19 Do worlds differentiate or split? Q20 What is many-minds? Q21 Does many-worlds violate Ockham's Razor? Q22 Does many-worlds violate conservation of energy? Q23 How do probabilities emerge within many-worlds? Q24 Does many-worlds allow free-will? Q25 Why am I in this world and not another? Why does the universe appear random? Q26 Can wavefunctions collapse? Q27 Is physics linear? Could we ever communicate with the other worlds? Why do I only ever experience one world? Why am I not aware of the world (and myself) splitting? Q28 Can we determine what other worlds there are? Is the form of the Universal Wavefunction knowable? Q29 Who was Everett? Q30 What are the problems with quantum theory? Q31 What is the Copenhagen interpretation? Q32 Does the EPR experiment prohibit locality? What about Bell's Inequality? Q33 Is Everett's relative state formulation the same as many-worlds? Q34 What is a relative state? Q35 Was Everett a "splitter"? Q36 What unique predictions does many-worlds make? Q37 Could we detect other Everett-worlds? Q38 Why *quantum* gravity? Q39 Is linearity exact? Q41 Why can't the boundary conditions be updated to reflect my observations in this one world? A1 References and further reading A2 Quantum mechanics and Dirac notation 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.

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  • 8/7/2019 Many Worlds Theory of Infinite Parallel Universes

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    Many Worlds Theory of Infinite Parallel Universes

    The Many Worlds Theory of Infinite Parallel Universes is the most satisfying scientific theory toexplain the paradoxes inherent in Quantum Reality. Quantum Reality is the most successful scientific

    theory to ever explain the experimental data gathered by over a century of physics research.However, the conclusions are mind boggling to scientists because they want a nice logicalexplanation for the universe and quantum reality gives them a world of impreciseness and probabilitywith data pointing to a single non-local field of energy composed of waveforms. This includes makingthe assumption that objects and observers are not independent but somehow linked. It is the act ofobserving that causes all the paradoxes. To solve all of the paradoxes of observers and object beinglinked somehow, the theory expresses that for every quantum event observed, the universe splits intoeach and every possible observable outcome and each universe continues separate, and in parallel,unaware of the other universes. In effect, a universe without observers would exist as asuperimposed set of possible outcomes, with each outcome in a suspended state of unmanifestedexistence. It is the act of observation that makes possibilities transend from probability to reality.

    Archive-name: many-worlds-faqLast-modified: 17 February 1995Posting-Frequency: in full: 3-monthly, abridged: monthly (ex *.answers)

    (C) Michael Clive Price, February 1995Permission to copy in its entirety granted for non-commercial purposes.

    Contents:

    Q0 Why this FAQ?Q1 Who believes in many-worlds?Q2 What is many-worlds?Q3 What are the alternatives to many-worlds?Q4 What is a "world"?Q5 What is a measurement?Q6 Why do worlds split?

    What is decoherence?Q7 When do worlds split?Q8 When does Schrodinger's cat split?Q9 What is sum-over-histories?

    Q10 What is many-histories?What is the environment basis?Q11 How many worlds are there?Q12 Is many-worlds a local theory?Q13 Is many-worlds a deterministic theory?Q14 Is many-worlds a relativistic theory?

    What about quantum field theory?What about quantum gravity?

    Q15 Where are the other worlds?Q16 Is many-worlds (just) an interpretation?Q17 Why don't worlds fuse, as well as split?

    Do splitting worlds imply irreversible physics?Q18 What retrodictions does many-worlds make?Q19 Do worlds differentiate or split?Q20 What is many-minds?Q21 Does many-worlds violate Ockham's Razor?Q22 Does many-worlds violate conservation of energy?

    Q23 How do probabilities emerge within many-worlds?

    Q24 Does many-worlds allow free-will?Q25 Why am I in this world and not another?

    Why does the universe appear random?Q26 Can wavefunctions collapse?Q27 Is physics linear?

    Could we ever communicate with the other worlds?Why do I only ever experience one world?Why am I not aware of the world (and myself) splitting?

    Q28 Can we determine what other worlds there are?Is the form of the Universal Wavefunction knowable?

    Q29 Who was Everett?

    Q30 What are the problems with quantum theory?Q31 What is the Copenhagen interpretation?Q32 Does the EPR experiment prohibit locality?

    What about Bell's Inequality?Q33 Is Everett's relative state formulation the same asmany-worlds?Q34 What is a relative state?Q35 Was Everett a "splitter"?Q36 What unique predictions does many-worlds make?Q37 Could we detect other Everett-worlds?Q38 Why *quantum* gravity?Q39 Is linearity exact?Q41 Why can't the boundary conditions be updated toreflect my

    observations in this one world?A1 References and further readingA2 Quantum mechanics and Dirac notation

    Q0 Why this FAQ?-------------

    This FAQ shows how quantum paradoxes are resolved by the "many-worlds" interpretation or metatheory of quantummechanics. 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 toresolve 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) thatthe more sophisticated rebuttals were also invalid. I hope this FAQ will save other investigators from being lead astray byauthoritative statements from mentors.

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    I have attempted, in the answers, to translate the precise mathematics of quantum theory into woolly and ambiguousEnglish - 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 showinghow probabilities are derived, so I've included an appendix on the Dirac notation.

    [A] See "Does the EPR experiment prohibit locality?", "What about Bell's Inequality?" and "When does Schrodinger's catsplit?" for how many-worlds handles the most quoted paradoxes.

    [B] Sample objection: "Creation of parallel universes violates energy conservation/Ockham's razor". (See "Does many-worlds violate conservation of energy?" and "Does many-worlds violate Ockham's Razor?")

    [C] eg "In quantum field theory the wavefunction becomes an operator". Er, what does that mean? And is this relevant?(See "What about quantum field theory?")

    Q1 Who believes in many-worlds?----------------------------

    "Political scientist" L David Raub reports a poll of 72 of the "leading cosmologists and other quantum field theorists" aboutthe "Many-Worlds Interpretation" and gives the following response breakdown [T].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 andRichard Feynman. Gell-Mann and Hawking recorded reservations with the name "many-worlds", but not with the theory'scontent. Nobel Laureate Steven Weinberg is also mentioned as a many-worlder, although the suggestion is not when thepoll was conducted, presumably before 1988 (when Feynman died). The only "No, I don't accept MWI" named isPenrose.

    The findings of this poll are in accord with other polls, that many-worlds is most popular amongst scientists who mayrather loosely be described as string theorists or quantum gravitists/cosmologists. It is less popular amongst the widerscientific 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_ ScientificAmerican (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, ameasurement causes the history of the universe for practicalpurposes to diverge into different non-interfering tracks, one foreach possible value of the measured quantity. [...] I prefer thislast approach"

    In the _The Quark and the Jaguar_ and _Quantum Mechanics in the Light of Quantum Cosmology_ [10] Gell-Manndescribes himself as an adherent to the (post-)Everett interpretation, although his exact meaning is sometimes leftambiguous.

    Steven Hawking is well known as a many-worlds fan and says, in an article on quantum gravity [H], that measurement ofthe 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 hislecture students [F] that the "collapse" process could only be modelled by the Schrodinger wave equation (Everett'sapproach).

    [F] Jagdish Mehra _The Beat of a Different Drum: The Life and ScienceRichard Feynman_

    [H] Stephen W Hawking _Black Holes and Thermodynamics_ Physical ReviewD Vol 13 #2 191-197 (1976)

    [T] Frank J Tipler _The Physics of Immortality_ 170-171

    Q2 What is many-worlds?--------------------

    AKA as the Everett, relative-state, many-histories or many-universes interpretation or metatheory of quantum theory. DrHugh Everett, III, its originator, called it the "relative-state metatheory" or the "theory of the universal wavefunction" [1], butit 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 merelyencode the all the information about an object, but has an

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    observer-independent objective existence and actually *is* theobject. For a non-relativistic N-particle system the wavefunctionis a complex-valued field in a 3-N dimensional space.

    2) The physical assumption: The wavefunction obeys the empiricallyderived standard linear deterministic wave equations at all times.The observer plays no special role in the theory and, consequently,there is no collapse of the wavefunction. For non-relativisticsystems the Schrodinger wave equation is a good approximation toreality. (See "Is many-worlds a relativistic theory?" for how themore general case is handled with quantum field theory or third quantisation.)

    The rest of the theory is just working out consequences of the above assumptions. Measurements and observations by asubject on an object are modelled by applying the wave equation to the joint subject-object system. Some consequencesare:

    1) That each measurement causes a decomposition or decoherence of theuniversal wavefunction into non-interacting and mostly non-interfering branches, histories or worlds. (See "What isdecoherence?") The histories form a branching tree whichencompasses all the possible outcomes of each interaction. (See

    "Why do worlds split?" and "When do worlds split?") Everyhistorical what-if compatible with the initial conditions andphysical law is realised.

    2) That the conventional statistical Born interpretation of theamplitudes in quantum theory is *derived* from within the theoryrather 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 theprocess of observation or measurement entirely within the wave-mechanics of quantum theory, rather than an input asadditional 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 physicaltheory are real. For example the electromagnetic fields of James Clark Maxwell or the atoms of Dalton were consideredas real objects in classical physics. Everett treats the wavefunction in a similar fashion. Everett also assumed that thewavefunction 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 thatinteractions between two (or more) macrosystems typically split the joint system into a superposition of products ofrelative states. The states of the macrosystems are, after the subsystems have jointly interacted, henceforth correlatedwith, or dependent upon, each other. Each element of the superposition - each a product of subsystem states - evolvesindependently of the other elements in the superposition. The states of the macrosystems are, by becoming correlated orentangled with each other, impossible to understand in isolation from each other and must be viewed as one compositesystem. It is no longer possible to speak the state of one (sub)system in isolation from the other (sub)systems.Instead we are forced to deal with the states of subsystems *relative* to each other. Specifying the state of onesubsystem leads to a unique specification of the state (the "relative state") of the other subsystems. (See "What is arelative state?")

    If one of the systems is an observer and the interaction an observation then the effect of the observation is to split theobserver into a number of copies, each copy observing just one of the possible results of a measurement and unaware ofthe other results and all its observer-copies. Interactions between systems and their environments, includingcommunication between different observers in the same world, transmits the correlations that induce local splitting or

    decoherence into non-interfering branches of the universal wavefunction. Thus the entire world is split, quite rapidly, intoa 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 ofcollapsing at the moment of observation, carries on evolving in a deterministic fashion, embracing all possibilitiesembedded within it. All outcomes exist simultaneously but do not interfere further with each other, each single prior worldhaving split into mutually unobservable but equally real worlds.

    Q3 What are the alternatives to many-worlds?-----------------------------------------

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

    1) Copenhagen Interpretation. Postulates that the observer obeys different physical laws than the non-observer, whichis a return to vitalism. The definition of an observer varies from one adherent to another, if present at all. The status of

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    the wavefunction is also ambiguous. If the wavefunction is real the theory is non-local (not fatal, but unpleasant). If thewavefunction is not real then the theory supplies no model of reality. (See "What are the problems with quantumtheory?")

    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 Bohmpostulated that there were particles that move under the influence of a non-local "quantum-potential" derived from thewavefunction (in addition to the classical potentials which are already incorporated into the structure of the wavefunction).The action of the quantum-potential is such that the particles are affected by only one of the branches of thewavefunction. (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 withparticles can contain self-aware observers, whereas Everett makes no such assumption. Most of Bohm's adherents donot seem to understand (or even be aware of) Everett's criticism, section VI [1], that the hidden-variable particles are notobservable since the wavefunction alone is sufficient to account for all observations and hence a model of reality. Thehidden variable particles can be discarded, along with the guiding quantum-potential, yielding a theory isomorphic tomany-worlds, without affecting any experimental results.

    [B] David J Bohm _A suggested interpretation of the quantum theoryin terms of "hidden variables" I and II_ Physical Review Vol85 #2 166-193 (1952)

    3) Quantum Logic. Undoubtedly the most extreme of all attempts to solve the QM measurement problem. Apart fromabandoning one or other of the classical tenets of logic these theories are all unfinished (presumably because of internalinconsistencies). Also it is unclear how and 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 ofprobability, in which case it suffers from the same criticism as quantum logic. Also it is unclear, to me anyway, how theresultant notion of "complex probability" differs from the quantum "probability amplitude" and thus why we are justified incollapsing the complex-valued probability as if it were a classical, real-valued probability.

    [M] W Muckenheim _A review of extended probabilities_ PhysicsReports Vol 133 339- (1986)

    [Y] Saul Youssef _Quantum Mechanics as Complex Probability Theory_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" toform the Born probability density. It requires that the input and output states, as defined by an observer, act as emittersand absorbers respectively, but not any internal states (inside the "black box"), and, consequently, suffers from thefamiliar measurement problem of the Copenhagen interpretation.

    If the internal states *did* act as emitters/absorbers then the wavefunction would collapse, for example, around one ofthe double slits (an internal state) in the double slit experiment, destroying the observed interference fringes. Intransaction terminology a transaction would form between the fi rst single slit and one of the double slits and anothertransaction would form between the same double slit and the point on the screen where the photon lands. This neverobserved.

    [C] John G Cramer _The transactional interpretation of quantummechanics_ 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-operationaltheory. (See "What is many-minds?")

    7) Non-linear theories in general. So far no non-linear theory has any accepted experimental support, whereas manyhave failed experiment. (See "Is physics linear?") Many-worlds predicts that non-linear theories will always failexperiment. (See "Is linearity exact?")

    Q4 What is a "world"?------------------

    Loosely speaking a "world" is a complex, causally connected, partially or completely closed set of interacting sub-systemswhich don't significantly interfere with other, more remote, elements in the superposition. Any complex system and itscoupled environment, with a large number of internal degrees of freedom, qualifies as a world. An observer, with internalirreversible processes, counts as a complex system. In terms of the wavefunction, a world is a decohered branch ofthe universal wavefunction, which represents a single macrostate. (See "What is decoherence?") The worlds all existsimultaneously in a non-interacting linear superposition.

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    Sometimes "worlds" are called "universes", but more usually the latter is reserved the totality of worlds implied by theuniversal wavefunction. Sometimes the term "history" is used instead of "world". (Gell-Mann/Hartle's phrase, see "Whatis many-histories?").

    Q5 What is a measurement?----------------------A measurement is an interaction, usually irreversible, between subsystems that correlates the value of a quantity in onesubsystem with the value of a quantity in the other subsystem. The interaction may trigger an amplification process withinone object or subsystem with many internal degrees of freedom, leading to an irreversible high-level change in the sameobject. If the course of the amplification is sensitive to the initial interaction then we can designate the systemcontaining 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 acharged particle (the object) by a Geiger counter (the measuring apparatus) leads to the generation of a "click" (high-levelchange). The absence of a charged particle does not generate a click. The interaction is with those elements of thecharged 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.

    Q6 Why do worlds split?---------------------

    What is decoherence?--------------------

    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 interferenceeffects 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 usuallymeans 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 thedouble slit experiment, for instance, it only requires that the divergent paths of the diffracted particle overlap again atsome 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 systemsbecause all the constituent particles have to overlap with their counterparts simultaneously. Any system complex enoughto be described by thermodynamics and exhibit irreversible behaviour is a system complex enough to exclude, for allpractical 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 irreversibleprocess has started then alterations of the values of the many degrees of freedom leaves an imprint which can't beremoved. 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 permanentlyorthogonal due to the complexity of the systems increasing the dimensionality of the vector space, where eachunconstrained degree of freedom contributes a dimension to the state vector space. In a high dimension space almostall vectors are orthogonal, without any significant degree of overlap. Thus vectors for complex systems, with a largenumber of degrees of freedom, naturally decompose into mutually orthogonal components which, because they can neversignificantly interfere again, are unaware of each other. The complex system, or world, has split into different, mutuallyunobservable worlds.

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

    Contact between a system and a heat sink is equivalent to increasing the dimensionality of the state space, because thedescription of the system has to be extended to include all parts of the environment in causal contact with it. Contact withthe external environment is a very effective destroyer of coherency. (See "What is the environment basis?")

    Q7 When do worlds split?---------------------

    Worlds irrevocably "split" at the sites of measurement-like interactions associated with thermodynamically irreversibleprocesses. (See "What is a measurement?") An irreversible process will always produce decoherence which splitsworlds. (See "Why do worlds split?", "What is decoherence?" and "When does Schrodinger's cat split?" for a concreteexample.)

    In the example of a Geiger counter and a charged particle after the particle has passed the counter one world contains theclicked counter and that portion of the particle's wavefunction which passed though the detector. The other worldcontains the unclicked counter with the particle's wavefunction with a "shadow" cast by the counter taken out of theparticle's wavefunction.

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    The Geiger counter splits when the amplification process became irreversible, before the click is emitted. (See "What is ameasurement?") The splitting is local (originally in the region of the Geiger counter in our example) and is transmittedcausally to more distant systems. (See "Is many-worlds a local theory?" and "Does the EPR experiment prohibitlocality?") 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 theenvironment. At this stage the event has become irreversible.

    In the language of thermodynamics the amplification of the charged particle's presence by the Geiger counter is anirreversible event. These events have caused the decoherence of the different branches of the wavefunction. (See "Whatis decoherence?" and "Why do worlds split?") Decoherence occurs when irreversible macro-level events take place andthe macrostate description of an object admits no single description. (A macrostate, in brief, is the description of an objectin terms of accessible external characteristics.)

    The advantage of linking the definition of worlds and the splitting process with thermodynamics is the splitting processbecomes irreversible and only permits forward-time-branching, following the increase with entropy. (See "Why don'tworlds fuse, as well as split?") Like all irreversible processes, though, there are exceptions even at the coarse-grainedlevel and worlds will occasionally fuse. A necessary, although not sufficient, precondition for fusing is for all records,memories etc that discriminate between the pre-fused worlds or histories be lost. This is not a common occurrence.

    Q8 When does Schrodinger's cat split?----------------------------------

    Consider Schrodinger's cat. A cat is placed in a sealed box with a device that releases a lethal does of cyanide if acertain radioactive decay is detected. For simplicity we'll imagine that the box, whilst closed, completely isolates the catfrom its environment. After a while an investigator opens the box to see if the cat is alive or dead. According to theCopenhagen Interpretation the cat was neither alive nor dead until the box was opened, whereupon the wavefunction ofthe cat collapsed into one of the two alternatives (alive or dead cat). The paradox, according to Schrodinger, is that thecat presumably knew if it was alive *before* the box was opened. According to many-worlds the device was split into twostates (cyanide released or not) by the radioactive decay, which is a thermodynamically irreversible process (See "Whendo worlds split?" and "Why do worlds split?"). As the cyanide/no-cyanide interacts with the cat the cat is split into twostates (dead or alive). From the surviving cat's point of view it occupies a different world from its deceased copy. Theonlooker is split into two copies only when the box is opened and they are altered by the states of the cat.

    The cat splits when the device is triggered, irreversibly. The investigator splits when they open the box. The alive cat hasno idea that investigator has split, any more than it is aware that there is a dead cat in the neighbouring split-off world.The investigator can deduce, after the event, by examining the cyanide mechanism, or the cat's memory, that the cat splitprior to opening the box.

    Q9 What is sum-over-histories?---------------------------The sum-over-histories or path-integral formalism of quantum mechanics was developed by Richard Feynman in the1940s [F] as a third interpretation of quantum mechanics, alongside Schrodinger's wave picture and Heisenberg's matrixmechanics, for calculating transition amplitudes. All three approaches are mathematically equivalent, but the path-integralformalism 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 pathsfrom (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 lagrangianequals kinetic minus the potential energy) along the path from (x,t) to (x',t'). The final expression is thus the sum orintegral over all paths, irrespective of any classical dynamical constraints. For N-particle systems the principle is thesame, except that the paths run through a 3-N space.

    In the path-integral approach every possible path through configuration space makes a contribution to the transitionamplitude. From this point of view the particle explores every possible intermediate configuration between the specifiedstart and end states. For this reason the path-integral 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 transitionamplitudes in the future, and that each possible history has an equal reality. Perhaps we shouldn't be surprised thatFeynman is on record as believing in many-worlds. (See "Who believes in many-worlds?") What is surprising is thatEverett developed his many-worlds theory entirely from the Schrodinger viewpoint without any detectable influence fromFeynman's work, despite Feynman and Everett sharing the same Princeton thesis supervisor, John A Wheeler.

    Feynman developed his path-integral formalism further during his work on quantum electrodynamics, QED, in parallel withSchwinger and Tomonoga who had developed a less visualisable form of QED. Dyson showed that these approacheswere all equivalent. Feynman, Schwinger and Tomonoga were awarded the 1965 Physics Nobel Prize for this work.Feynman's approach was to show how any process, with defined in (initial) and out (final) states, can be represented by aseries of (Feynman) diagrams, which allow for the creation, exchange and annihilation of particles. Each Feynmandiagram represents a different contribution to the complete transition amplitude, provided that the external lines map ontothe required boundary initial and final conditions (the defined in and out states). QED became the prototype for all theother, later, field theories like electro-weak and quantum chromodynamics.

    [F] Richard P Feynman _Space-time approach to non-relativistic quantum

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    -------------------------------------What about quantum field theory?--------------------------------What about quantum gravity?

    ---------------------------

    It is trivial to relativise many-worlds, at least to the level of special relativity. All relativistic theories of physics are quantumtheories 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 thesolution to either the many-particle Schrodinger, Dirac or Klein-Gordon or some other wave equation. External forcesapplied 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 byquantising the classical fields and potentials as well as the particles. Each particle corresponds to a field, in QFT, andbecomes an operator. Eg the electromagnetic field's particle is the photon. The wavefunction of a collection ofparticles/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 atheory (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 thespace-time manifold. The role of time plays a less central role, as might be expected, but the first and secondquantisation 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 wouldbe unchanged.

    One of the original motivations of Everett's scheme was to provide a system for quantising the gravitational field to yield aquantum cosmology, permitting a complete, self-contained description of the universe. Indeed many-words actually*requires* that gravity be quantised, in contrast to other interpretations which are silent about the role of gravity. (See"Why *quantum* gravity?")

    Q15 Where are the other worlds?---------------------------

    Non-relativistic quantum mechanics and quantum field theory are quite unambiguous: the other Everett-worlds occupythe 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 seewhy we aren't aware of the other worlds, despite occupying the same space-time, see "Why do I only ever experience oneworld?" Some popular accounts describe the other worlds as splitting off into other, orthogonal, dimensions. Thesedimensions are the dimensions of Hilbert space, not the more familiar space-time dimensions.

    The situation is more complicated, as we might expect, in theories of quantum gravity (See "What about quantumgravity?"), because gravity can be viewed as perturbations in the space-time metric. If we take a geometric interpretationof gravity then we can regard differently curved space-times, each with their own distinct thermodynamic history, as non-coeval. In that sense we only share the same space-time manifold with other worlds with a (macroscopically) similarmass distribution. Whenever the amplification of a quantum-scale interaction effects the mass distribution and hencespace-time curvature the resultant decoherence can be regarded as splitting the local space-time manifold into discretesheets.

    Q16 Is many-worlds (just) an interpretation?----------------------------------------

    No, for four reasons:

    First, many-worlds makes predictions that differ from the other so-called interpretations of quantum theory. Interpretationsdo not make predictions that differ. (See "What unique predictions does many-worlds make?") In addition many-worldsretrodicts a lot of data that has no other easy interpretation. (See "What retrodictions does many-worlds make?")

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

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

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

    Many-worlds should really be described as a theory or, more precisely, a metatheory, since it makes statements that areapplicable about a range of theories. Many-worlds is the unavoidable implication of any quantum theory which obeyssome type of linear wave equation. (See "Is physics linear?")

    Q17 Why don't worlds fuse, as well as split?---------------------------------------Do splitting worlds imply irreversible physics?-----------------------------------------------

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

    First, worlds almost never fuse, in the forward time direction, but often divide, because of the way we have defined them.(See "What is decoherence?", "When do worlds split?" and "When do worlds split?") The Planck-Boltzmann formula forthe number of worlds (See "How many worlds are there?") implies that where worlds to fuse together then entropywould decrease, violating the second law of thermodynamics.

    Second, this does not imply that irreversible thermodynamics is incompatible with reversible (or nearly so) microphysics.

    The laws of physics are reversible (or CPT invariant, more precisely) and fully compatible with the irreversibility ofthermodynamics, which is solely due to the boundary conditions (the state of universe at some chosen moment) imposedby the Big Bang or whatever we chose to regard as the initial conditions. (See "Why can't the boundary conditions beupdated to reflect my observations in this one world?")

    Q18 What retrodictions does many-worlds make?-----------------------------------------

    A retrodiction occurs when already gathered data is accounted for by a later theoretical advance in a more convincingfashion. The advantage of a retrodiction over a prediction is that the already gathered data is more likely to be free ofexperimenter bias. An example of a retrodiction is the perihelion shift of Mercury which Newtonian mechanics plus gravitywas 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"What is decoherence?", "Can wavefunctions collapse?", "When do worlds split?" & "Why do worlds split?") No otherquantum theory has yet accounted for this behaviour scientifically. (See "What are the alternatives to many-worlds?")

    Q19 Do worlds differentiate or split?---------------------------------Can we regard the separate worlds that result from a measurement-like interaction (See "What is a measurement?") ashaving previous existed distinctly and merely differentiated, rather than the interaction as having split one world intomany? This is definitely not permissable in many-worlds or any theory of quantum theory consistent with experiment.Worlds do not exist in a quantum superposition independently of each other before they decohere or split. The splitting isa physical process, grounded in the dynamical evolution of the wave vector, not a matter of philosophical, linguistic ormental convenience (see "Why do worlds split?" and "When do worlds split?") If you try to treat the worlds as pre-existingand separate then the maths and probabilistic behaviour all comes out wrong. Also the differentiation theory isn'tdeterministic, in contradiction to the wave equations which are deterministic, since many-minds says that:

    AAAAAAAAAAAAAAABBBBBBBBBBBBBBBB --------------> time(Worlds differentiate)

    AAAAAAAAAAAAAAACCCCCCCCCCCCCCC

    occurs, rather than:

    BBBBBBBBBBBBBBBBB

    AAAAAAAAAAAAAA (Worlds split)CCCCCCCCCCCCCCCC

    according to many-worlds.

    This false differentiation model, at the mental level, seems favoured by adherents of many-minds. (See "What is many-minds?")

    Q20 What is many-minds?

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    ------------------Many-minds proposes, as an extra fundamental axiom, that an infinity of separate minds or mental states be associatedwith each single brain state. When the single physical brain state is split into a quantum superposition by a measurement(See "What is a measurement?") the associated infinity of minds are thought of as differentiating rather than splitting. The

    motivation for this brain-mind dichotomy seems purely to avoid talk of minds splitting and talk instead about thedifferentiation of pre-existing separate mental states. There is no physical basis for this interpretation, which is incapableof an operational definition. Indeed the differentiation 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 "Do worlds differentiate orsplit?")

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

    The two theories must not be confused.

    Q21 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 judgingtheories 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 morecomplex ones. Many-worlds is viewed as unnecessarily complex, by some, by requiring the existence of a multiplicity ofworlds 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 verysimilar to faint sunlight, both with spectroscopic absorption and emission lines. Assuming the universality of physical lawwe 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 quantum theory because it proposes that same laws ofphysics apply to animate observers as has been observed for inanimate objects. The multiplicity of worlds predicted bythe theory is not a weakness of many-worlds, any more than the multiplicity of stars are for astronomers, 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 olderheliocentric theories *against* Galileo's notion of the vastness of the cosmos. The notion of vast empty interstellar spaces

    was too uneconomical to be believable to the Medieval mind. Again they were confusing the notion of vastness withcomplexity [15].)

    Q22 Does many-worlds violate conservation of energy?------------------------------------------------

    First, the law conservation of energy is based on observations within each world. All observations within each world areconsistent 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 expectationvalues. Conservation of energy is expressed by saying that the time derivative of the expected energy of a closed systemvanishes. This statement can be scaled up to include the whole universe. Each world has an approximate energy, butthe energy of the total wavefunction, or any subset of, involves summing over each world, weighted with its probabilitymeasure. This weighted sum is a constant. So energy is conserved within each world and also across the totality ofworlds.

    One way of viewing this result - that observed conserved quantities are conserved across the totality of worlds - is to notethat 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".

    Q23 How do probabilities emerge within many-worlds?-----------------------------------------------

    Everett demonstrated [1], [2] that observations in each world obey all the usual conventional statistical laws predicted bythe probabilistic Born interpretation, by showing that the Hilbert space's inner product or norm has a special propertywhich allows us to makes statements about the worlds where quantum statistics break down. The norm of the vectorof the set of worlds where experiments contradict the Born interpretation ("non-random" or "maverick" worlds) vanishes inthe 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, probabilisticpredictions 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 allobservers in all worlds. He merely showed that no other statistical laws could hold true and asserted the vanishing of theHilbert space "volume" or norm of the set of "maverick" worlds. DeWitt later published a longer *derivation* of Everett'sassertion [4a], [4b], closely based on an earlier, independent demonstration by Hartle [H]. What Everett asserted, and

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    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 Hilbertspaces) this is equivalent to saying that non-randomness is never realised. All the worlds obey the usual Bornpredictions 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 isalso made in the standard formulat ion, this is not a weakness of many-worlds since we are not trying to justify all theaxioms 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):|PSI_N> = |psi_1>*|psi_2>*...... |psi_N> where|psi_j> = jth system prepared in state |psi>

    = |i_j> (ie the amplitude of the ith eigenstateis independent of which system it is in)

    so that|PSI_N> = |i_1>|i_2>.. .|i_N>...

    2) Quantify the deviation from the "expected" Born-mean for eachcomponent of |PSI_N> with respect to the above |i_1>|i_2>...|i_N>basis by counting the number of occurrences of the itheigenstate/N. Call this number RF(i). Define the Born-deviationas D = sum(i)( (RF(i) - ||^2)^2 ). Thus D, looselyspeaking, for each N length sequence, quantifies by how much theparticular sequence differs from the Born-expectation.

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

    worlds worldsfor which for which

    D > E D ,by contrast, approaches 1 as N goes to infinity.

    Note: this property of D is not shared by other definitions, whichis why we haven't investigated them. If, say, we had defined, instep 2), A = sum(i)( (RF(i) - ||)^2 ), so that A measuresthe deviation from |psi|, rather than |psi|^2, then we find that does not have the desired property of vanishing as N goes toinfinity.

    5) The norm of the collection of non-random worlds vanishes andtherefore must be identified with some complex multiple of the nullvector.

    6) Since (by assumption) the state vector faithfully models realitythen the null vector cannot represent any element of reality, sinceit can be added to (or subtracted from) any other state vectorwithout altering the other state vector.

    7) Ergo the non-random worlds are not realised, without making anyadditional physical assumptions, such the imposition of a measure.

    Note: no finite sequence of outcomes is excluded from happening,since the concept of probability and randomness only becomesprecise only as N goes to infinity [H]. Thus, heat *could* be

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    observed to flow from a cold to hotter object, but we might haveto wait a very long time before observing it. What *is* excludedis the possibility of this process going on forever.

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

    [H] JB Hartle _Quantum Mechanics of Individual Systems_ AmericanJournal of Physics Vol 36 #8 704-712 (1968) Hartle hasinvestigated the N goes to infinity limit in more detail and moregenerally. He shows that the relative frequency operator, RF,obeys RF(i) |psi_1>|psi_2>.... = ||^2 |psi_1>|psi_2>....,for a normed state. Hartle regarded his derivation as essentiallythe same as Everett's, despite being derived independently.

    Q24 Does many-worlds allow free-will?---------------------------------

    Many-Worlds, whilst deterministic on the objective universal level, is indeterministic on the subjective level so the situationis 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, thecritical 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, thefiring 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 ofany mechanism containing a complete internal model of itself rather than any inherent indeterminism in the mechanism'soperation.

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

    This measure is supplied by the mathematical structure of the 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 orcollection of worlds. A world of zero volume is never realised. Worlds in which the conventional statistical predictionsconsistently 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, matchesour subjective experience of being able to exercise our will, form moral judgements and be held responsible for ouractions.

    Q25 Why am I in this world and not another?---------------------------------------Why does the universe appear random?------------------------------------

    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 asFred-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 theFreds 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 allthe quantum worlds. It is a subjectively random choice, an artifact of your brain and consciousness being split, along withthe rest of the world, that makes our experiences seem random. The universe is, in effect, performing umpteen split-brainoperations on us all the time. The randomness apparent in nature is a consequence of the continual splitting into mutuallyunobservable worlds.

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

    [*] Split brain experiments *were* performed on epileptic patients (severing the corpus callosum, one of the pathwaysconnecting the cerebral hemispheres, moderated epileptic attacks). Complete hemispherical separation was discontinued

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    when testing of the patients revealed the presence of two distinct consciousnesses in the same skull. So this analogy isonly partly imaginary.

    Q26 Can wavefunctions collapse?

    ---------------------------Many-worlds predicts/retrodicts that wavefunctions appear to collapse (See "Does the EPR experiment prohibit locality?"),when measurement-like interactions (See "What is a measurement?") and processes occur via a process calleddecoherence (See "What is decoherence?"), but claims that the wavefunction does not *actually* collapse but continuesto evolve according to the usual wave-equation. If a *mechanism* for collapse could be found then there would be noneed for many-worlds. The reason why we doubt that collapse takes place is because no one has ever been able todevise 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 of observer therecan be no mechanism triggered by their presence.

    Another popular view is that irreversible processes trigger collapse. Certainly wavefunctions *appear* to collapsewhenever irreversible processes are involved. And most macroscopic, day-to-day events are irreversible. The problemis, 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 toshow that new *fundamental* physics comes into play for complex systems, which is quite absent at the reversible

    atom/molecular level. Atoms and molecules are empirically observed to obey some type of wave equation. We have noevidence for an extra mechanism operating on more complex systems. As far as we can determine complex systems aredescribed by the quantum-operation of their simpler components interacting together. (Note: chaos, complexity theory,etc, do not introduce new fundamental physics. They still operate within the reductionistic paradigm - despite what manypopularisers say.)

    Other people have attempted to construct non-linear theories so that microscopic systems are approximately linear andobey the wave equation, whilst macroscopic systems are grossly non-linear and generates collapse. Unfortunately allthese efforts have made additional predictions which, when tested, have failed. (See "Is physics linear?")

    (Another reason for doubting that any collapse actually takes place is that the collapse would have to propagateinstantaneously, or in some space-like fashion, otherwise the same particle could be observed more than once at differentlocations. Not fatal, but unpleasant and difficult to reconcile with special relativity and some conservation laws.)

    The simplest conclusion, which is to be preferred by Ockham's razor, is that wavefunctions just *don't* collapse and thatall branches of the wavefunction exist.

    Q27 Is physics linear?------------------Could we ever communicate with the other worlds?------------------------------------------------Why do I only ever experience one world?----------------------------------------Why am I not aware of the world (and myself) splitting?-------------------------------------------------------

    According to our present knowledge of physics whilst it is possible to detect the presence of other nearby worlds, throughthe existence of interference effects, it is impossible travel to or communicate with them. Mathematically this correspondsto an empirically verified property of all quantum theories called linearity. Linearity implies that the worlds can interferewith each other with respect to a external, unsplit, observer or system but the interfering worlds can't influence each otherin 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 anothersolution. Since each component of a linear solution evolves with complete indifference as to the presence or absence ofthe other terms/solutions then we can conclude that no experiment in one world can have any effect on anotherexperiment in another world. Hence no communication is possible between quantum worlds. (This type of linearitymustn'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 bereceiving sensory information from, and thereby effect by the reverse process, more than one world. This would enablecommunication between worlds, which is forbidden by linearity. Ergo, we are not aware of any splitting precisely becausewe are split into non-interfering copies along with the rest of the world.

    See also "Is linearity exact?"

    Q28 Can we determine what other worlds there are?---------------------------------------------

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    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 goodapproximation 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 boundaryconditions as well. We are presently restricted to making inferences about those worlds with which have shared acommon history up to some point, which have left traces (records, fossils, etc) still discernable today. This restricts us toa subset of the extant worlds which have shared the same boundary conditions with us. The further we probe back intime 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 ourknown history up to a some common moment, before we diverged. The flow of historical events is, according tochaos/complexity theory/thermodynamics, very sensitive to amplification of quantum-scale uncertainty and this sensitivityis a future-directed one-way process. We can make very reliable deductions about the past from the knowledgefuture/present but we can't predict the future from knowledge the past/present. Thermodynamics implies that the future isharder to predict than the past is to retrodict. Books get written about this "arrow of time" problem but, for the purposes ofthis discussion, we'll accept the thermodynamic origin of time's arrow is as given. The fossil and historical records saythat 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 popularexploration of the thesis that the path of evolution is driven by chance. According to Gould evolutionary history forms anenormously diverse tree of possible histories - all very improbable - with our path being selected by chance. According tomany-worlds all these other possibilities are realised. Thus there are worlds in which Hitler won WW-II and other worldsin which the dinosaurs never died out. We can be as certain of this as we are that Hitler and the dinosaurs once existedin our own past.

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

    Q29 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 before and since) with the collapse of thewavefunction. He developed, during discussions with Charles Misner and Aage Peterson (Bohr' assistant, then visitingPrinceton), his "relative state" formulation. Wheeler encouraged his work and preprints were circulated in January 1956 toa 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 butgot the complete brush-off from. Leaving physics after completing his Ph.D, Everett worked as a defense analyst at theWeapons Systems Evaluation Group, Pentagon and later became a private contractor, apparently quite successfully forhe became a multimillionaire. In 1968 Everett worked for the Lambda Corp. His published papers during this periodcover things like optimising resource allocation and, in particular, maximising kil l 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 thefounders of quantum gravity, successfully popularised Everett's relative state formulation as the "many-worldsinterpretation" 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 oftrivia, 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 workon measurement in quantum theory, but died of a heart attack in 1982. Survived by his wife.

    Q30 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 isnot applied to larger, classical systems, like observers or the entire universe. Many scientists and philosophers areunhappy with the theory because it seems to require a fundamental quantum-classical divide. Einstein, for example,despite his early contributions to the subject, was never reconciled with assigning to the act of observation a physicalsignificance, which most interpretations of QM require. This contradicts the reductionist ethos that, amongst other things,

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    observations should emerge only as a consequence of an underlying physical theory and not be present at the axiomaticlevel, as they are in the Copenhagen interpretation. Yet the Copenhagen interpretation remains the most popularinterpretation of quantum mechanics amongst the broad scientific community. (See "What is the Copenhageninterpretation?")

    Q31 What is the Copenhagen interpretation?--------------------------------------

    An unobserved system, according to the Copenhagen interpretation of quantum theory, evolves in a deterministic waydetermined 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 extendedregion ("non-local") which is non-relativistic.

    (2) The idea of an observer having an effect on microphysics isrepugnant to reductionism and smacks of a return to pre-scientificnotions of vitalism. Copenhagenism is a return to the old vitalistnotions that life is somehow different from other matter, operatingby different laws from inanimate matter. The collapse is triggeredby an observer, yet no definition of what an "observer" isavailable, 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 atCopenhagen in the late 1920s. When we perform an measurement or observation of an object we acquire newinformation and so adjust the wavefunction as we would boundary conditions in classical physics to reflect this newinformation. This stance means that we can't answer questions about what's actually happening, all we can answer iswhat will be the probability of a particular result if we perform a measurement. This makes a lot of people very unhappysince 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 owndrawbacks, which are widely reckoned more severe. Generally speaking they try to find a mechanism that describes thecollapse process or add extra physical objects to the theory, in addition to the wavefunction. In this sense they are morecomplex. (See "Is there any alternative theory?")

    Q32 Does the EPR experiment prohibit locality?

    ------------------------------------------What about Bell's Inequality?-----------------------------

    The EPR experiment is widely regarded as the definitive gedanken experiment for demonstrating that quantum mechanicsis 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 mechanicswas incomplete [E]. Bell, in 1964, demonstrated that any hidden variables theory, to replicate the predictions of QM, mustbe non-local [B]. QM predicts strong correlations between separated systems, stronger than any local hidden variablestheory can offer. Bell encoded this statistical prediction in the form of some famous inequalities that apply to any type ofEPR experiment. Eberhard, in the late 1970s, extended Bell's inequalities to cover any local theory, with or withouthidden variables. Thus the EPR experiment plays a central role in sorting and testing variants of QM. All the experimentsattempting to test EPR/Bell's inequality to date (including Aspect's in the 1980s [As]) are in line with the predictions ofstandard QM - hidden variables are ruled out. Here is the paradox of the EPR experiment. It seems to imply that anyphysical theory must involve faster-than-light "things" going on to maintain these "spooky" action-at-a-distancecorrelations 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 andmomenta of two once-coupled particles. Here I shall describe it in terms of the spin (a type of angular momentum intrinsicto the particle) of two electrons. [In this treatment I shall ignore the fact that electrons always form antisymmetriccombinations. This does not alter the results but does simplify the maths.] Two initially coupled electrons, with opposedspins that sum to zero, move apart from each other across a distance of perhaps many light years, before beingseparately detected, say, by me on Earth and you on Alpha Centauri with our respective measuring apparatuses. TheEPR paradox results from noting that if we choose the same (parallel) spin axes to measure along then we will observethe two electrons' spins to be anti-parallel (ie when we communicate we find that the spin on our electrons are correlatedand opposed). However if we choose measurement spin axes that are perpendicular to each other then there is nocorrelation between electron spins. Last minute alterations in a detector's alignment can create or destroy correlationsacross great distances. This implies, according to some theorists, that faster-than-light influences maintain correlationsbetween separated systems in some circumstances and not others.

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

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

    |psi> = |me>

    on Earth

    |electrons>

    in deep space

    |you>

    on Alpha Centauri

    |rest of universe>

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

    |psi> = |me,electrons,you>

    And

    |me> represents me on Earth with my detection apparatus.|electrons> = (|+,-> - |-,+>)/sqrt(2)

    represents a pair electrons, with the fi rst electron travellingtowards Earth and the second electron travelling towards AlphaCentauri.

    |+> 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 spinstates 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 operators in different directions form non-commuting observables. I shall eschewsuch obfuscations.)

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

    |+,-> - |-,+> = |left,right> - |right,left>= |up,down>i - |down,up>i

    I shall assume that we are capable of either measuring spin in the x or y direction, which are both perpendicular the line offlight 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 measuredand 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 detailsof the measurement process are irrelevant. (See "What is a measurement?" if you're not convinced.) To model the

    process it is sufficient to assume that there is a way, which we have further assumed does not disturb the electron. (Thelatter 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 statestate heading towards heading towards you on

    me on Earth Alpha CentauriAfter the electrons arrive at their detectors, I measure the spin along the x-axis and you along the y-axis. Thewavefunction evolves into |psi>_2:

    local

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    |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 toeach 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 communicated to the other observer and being recordedlocally. Notice that the results both agree with each other, in the sense that my record of your result agrees with yourrecord of your result. And vice versa. Our records are consistent.

    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)

    Neither you or I are yet unambiguously split.

    Suppose I perform my measurement first (in some time frame). We get

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

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

    My measurement has split me, although you, having made no measurement, remain unsplit. In the full expansion theterms that correspond to you are identical.

    After the we each have performed our measurements we get:

    |psi>_3= |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) whichcorrelate with the state of the electron. If we now communicate over interstellar modem (this will take a few years sinceyou and I are separated by light years, but no matter). We get:

    |psi>_4

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

    The world corresponding to the 2nd term in the above expansion, for example, contains me having seen my electron withspin right and knowing that you have seen your electron with spin left. So we jointly agree, in both worlds, that spin hasbeen conserved.

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

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

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

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

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    - (|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 decomposeit into four terms there are only two states for each observer. Looking at myself, for instance, we can rewrite this in termsof 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],lef t> (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 differentoutcome for our electron's spin, but we have not been split by the measurement of the remote electron's spin.

    *After* you and I communicate our readings to each other, more than 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 forced and unambiguous after communication with the remote system. Until the twoobservers communicated their results to each other they were each unsplit by each others' measurements, although theirown local measurements had split themselves. The splitting is a local process that is causally transmitted from system tosystem at light or sub-light speeds. (This is a point that Everett stressed about Einstein's remark about the observationsof a mouse, in the Copenhagen interpretation, collapsing the wavefunction of the universe. Everett observed that it is themouse that's split by its observation of the rest of the universe. The rest of the universe is unaffected and unsplit.)

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    When all communication is complete the worlds have finally decomposed or decohered from each other. Each worldcontains 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, canhenceforth 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 tostrain our credibility by believing that it actually happens.

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

    So where did Bell and Eberhard go wrong? They thought that all theories that reproduced the standard predictions mustbe 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 haveyielded a *single* definite result. This assumption is called contra-factual definiteness or CFD [S]. What Bell andEberhard really proved was that every quantum theory must either violate locality *or* CFD. Many-worlds with itsmultiplicity 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, withexperiment.

    [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'sinequalities using time-varying analyzers_ Physical Review LettersVol 49 #25 1804 (1982).

    [C] John G Cramer _The transactional interpretation of quantummechanics_ 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: _Canquantum-mechanical description of physical reality be consideredcomplete?_ Physical Review Vol 41 777-780 (15 May 1935).

    [S] Henry P Stapp _S-matrix interpretation of quantum-theory_ PhysicalReview D Vol 3 #6 1303 (1971)

    Q33 Is Everett's relative state formulation the same as many-worlds?----------------------------------------------------------------Yes, Everett's formulation of the relative state metatheory is the same as many-worlds, but the language has evolved a lotfrom Everett's original article [2] and some of his work has been extended, especially in the area of decoherence. (See"What is decoherence?") This has confused some people into thinking that Everett's "relative state metatheory" andDeWitt's "many-worlds interpretation" are different theories.

    Everett [2] talked about the observer's memory sequences splitting to form a "branching tree" structure or the state of theobserver being split by a measurement. (See "What is a measurement?") DeWitt introduced the term "world" fordescribing the split states of an observer, so that we now speak of the observer's world splitting during the measuringprocess. The maths is the same, but the terminology is different. (See "What is a world?")

    Everett tended to speak in terms of the measuring apparatus being split by the measurement, into non-interfering states,without presenting a detailed analysis of *why* a measuring apparatus was so effective at destroying interference effectsafter a measurement, although the topics of orthogonality, amplification and irreversibility were covered. (See "What is ameasurement?", "Why do worlds split?" and "When do worlds split?") DeWitt [4b], Gell-Mann and Hartle [10], Zurek [7a]and others have introduced the terminology of "decoherence" (See "What is decoherence?") to describe the role of

    amplification and irreversibility within the framework of thermodynamics.

    Q34 What is a relative state?-------------------------

    The relative state of something is the state that something is in, *conditional* upon, or relative to, the state of somethingelse. What the heck does that mean? It means, amongst other things, that states in the same Everett-world are all statesrelative to each other. (See "Quantum mechanics and Dirac notation" for more precise details.)

    Let's take the example of Schrodinger's cat and ask what is the relative state of the observer, after looking inside the box?The relative state of the observer (either "saw cat dead" or "saw cat alive") is conditional upon the state of the cat (either"dead" or "alive").

    Another example: the relative state of the last name of the President of the Unites States, in 1995, is "Clinton". Relative towhat? Relative to you and me, in this world. In some other worlds it will be "Bush", "Smith", etc ....... Each possibility isrealised in some world and it is the relative state of the President's name, relative to the occupants of that world.

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    According to Everett almost all states are relative states. Only the state of the universal wavefunction is not relative butabsolute.

    Q35 Was Everett a "splitter"?

    -------------------------Some people believe that Everett eschewed all talk al l splitting or branching observers in his original relative stateformulation [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 differentstates. Each branch represents a different outcome of themeasurement and the *corresponding* eigenstate for the object-system state. All branches exist simultaneously in thesuperposition after any given sequence of observations.[#]

    The "trajectory" of the memory configuration of an observerperforming a sequence of measurements is thus not a linearsequence of memory configurations, but a branching tree, withall possible outcomes existing simultaneously in a finalsuperposition with various coefficients in the mathematicalmodel. [...]

    [#] Note added in proof-- In reply to a preprint of thisarticle some correspondents have raised the question of the"transition from possible to actual," arguing that in"reality" there is-as our experience testifies-no su