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    Mystic Doctrines
    The Reluctant Messenger

    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.




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    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.
    
    I have attempted, in the answers, to translate the precise mathematics
    of quantum theory into woolly and ambiguous English - I would appreciate
    any corrections.  In one or two instances I couldn't avoid using some
    mathematical (Dirac) notation, in particular in describing the Einstein-
    Podolsky-Rosen (EPR) experiment and Bell's Inequality and in showing how
    probabilities are derived, so I've included an appendix on the Dirac
    notation.
    
    [A] See "Does the EPR experiment prohibit locality?", "What about Bell's
    Inequality?"  and "When does Schrodinger's cat split?" 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" about the "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 and Richard Feynman.  Gell-Mann and
    Hawking recorded reservations with the name "many-worlds", but not with
    the theory's content.  Nobel Laureate Steven Weinberg is also mentioned
    as a many-worlder, although the suggestion is not when the poll was
    conducted, presumably before 1988 (when Feynman died).  The only "No,
    I don't accept MWI" named is Penrose.
    
    The findings of this poll are in accord with other polls, that many-
    worlds is most popular amongst scientists who may rather loosely be
    described as string theorists or quantum gravitists/cosmologists.  It
    is less popular amongst the wider scientific community who mostly remain
    in ignorance of it.
    
    More detail on Weinberg's views can be found in _Dreams of a Final
    Theory_ or _Life in the Universe_ Scientific American (October 1994),
    the latter where Weinberg says about quantum theory:
         "The final approach is to take the Schrodinger equation seriously
         [..description of the measurement process..] In this way, a
         measurement causes the history of the universe for practical
         purposes to diverge into different non-interfering tracks, one for
         each possible value of the measured quantity. [...] I prefer this
         last approach"
    
    In the _The Quark and the Jaguar_ and _Quantum Mechanics in the Light
    of Quantum Cosmology_ [10] Gell-Mann describes himself as an adherent
    to the (post-)Everett interpretation, although his exact meaning is
    sometimes left ambiguous.
    
    Steven Hawking is well known as a many-worlds fan and says, in an
    article on quantum gravity [H], that measurement of the gravitational
    metric tells you which branch of the wavefunction you're in and
    references Everett.
    
    Feynman, apart from the evidence of the Raub poll, directly favouring
    the Everett interpretation, always emphasized to his lecture students
    [F] that the "collapse" process could only be modelled by the
    Schrodinger wave equation (Everett's approach).
    
    [F]  Jagdish Mehra _The Beat of a Different Drum: The Life and Science
         Richard Feynman_
    [H]  Stephen W Hawking _Black Holes and Thermodynamics_ Physical Review
         D Vol 13 #2 191-197 (1976)
    [T]  Frank J Tipler _The Physics of Immortality_ 170-171
    
    Q2   What is many-worlds?
         --------------------
    AKA as the Everett, relative-state, many-histories or many-universes
    interpretation or metatheory of quantum theory.  Dr Hugh Everett, III,
    its originator, called it the "relative-state metatheory" or the "theory
    of the universal wavefunction" [1], but it is generally called "many-
    worlds" nowadays, after DeWitt [4a],[5].
    
    Many-worlds comprises of two assumptions and some consequences.  The
    assumptions are quite modest:
    1)   The metaphysical assumption: That the wavefunction does not merely
         encode the all the information about an object, but has an
         observer-independent objective existence and actually *is* the
         object.  For a non-relativistic N-particle system the wavefunction
         is a complex-valued field in a 3-N dimensional space.
    
    2)   The physical assumption:  The wavefunction obeys the empirically
         derived 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-relativistic
         systems the Schrodinger wave equation is a good approximation to
         reality.  (See "Is many-worlds a relativistic theory?" for how the
         more 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 a subject on an object
    are modelled by applying the wave equation to the joint subject-object
    system.  Some consequences are:
    1)   That each measurement causes a decomposition or decoherence of the
         universal wavefunction into non-interacting and mostly non-
         interfering branches, histories or worlds.  (See "What is
         decoherence?")  The histories form a branching tree which
         encompasses all the possible outcomes of each interaction.  (See
         "Why do worlds split?" and "When do worlds split?")  Every
         historical what-if compatible with the initial conditions and
         physical law is realised.
    
    2)   That the conventional statistical Born interpretation of the
         amplitudes in quantum theory is *derived* from within the theory
         rather than having to be *assumed* as an additional axiom.  (See
         "How do probabilities emerge within many-worlds?")
    
    Many-worlds is a re-formulation of quantum theory [1], published in 1957
    by Dr Hugh Everett III [2], which treats the process of observation or
    measurement entirely within the wave-mechanics of quantum theory, rather
    than an input as additional assumption, as in the Copenhagen
    interpretation.  Everett considered the wavefunction a real object. 
    Many-worlds is a return to the classical, pre-quantum view of the
    universe in which all the mathematical entities of a physical theory are
    real.  For example the electromagnetic fields of James Clark Maxwell or
    the atoms of Dalton were considered as real objects in classical
    physics.  Everett treats the wavefunction in a similar fashion.  Everett
    also assumed that the wavefunction obeyed the same wave equation during
    observation or measurement as at all other times.  This is the central
    assumption of many-worlds: that the wave equation is obeyed universally
    and at all times.
    
    Everett discovered that the new, simpler theory - which he named the
    "relative state" formulation - predicts that interactions between two
    (or more) macrosystems typically split the joint system into a
    superposition of products of relative states.  The states of the
    macrosystems are, after the subsystems have jointly interacted,
    henceforth correlated with, or dependent upon, each other.  Each element
    of the superposition - each a product of subsystem states - evolves
    independently of the other elements in the superposition.  The states
    of the macrosystems are, by becoming correlated or entangled with each
    other, impossible to understand in isolation from each other and must
    be viewed as one composite system.  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 one subsystem leads to a unique
    specification of the state (the "relative state") of the other
    subsystems.  (See "What is a relative state?")
    
    If one of the systems is an observer and the interaction an observation
    then the effect of the observation is to split the observer into a
    number of copies, each copy observing just one of the possible results
    of a measurement and unaware of the other results and all its observer-
    copies.  Interactions between systems and their environments, including
    communication 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, into a host of mutually unobservable but
    equally real worlds.
    
    According to many-worlds all the possible outcomes of a quantum
    interaction are realised.  The wavefunction, instead of collapsing at
    the moment of observation, carries on evolving in a deterministic
    fashion, embracing all possibilities embedded within it.  All outcomes
    exist simultaneously but do not interfere further with each other, each
    single prior world having 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 of reality,
    and free of internal inconsistencies, that I am aware of.  Briefly here
    are the defects of the most popular alternatives:
    
    1)   Copenhagen Interpretation.  Postulates that the observer obeys
         different physical laws than the non-observer, which is a return
         to vitalism.  The definition of an observer varies from one
         adherent to another, if present at all.  The status of the
         wavefunction is also ambiguous.  If the wavefunction is real the
         theory is non-local (not fatal, but unpleasant).  If the
         wavefunction is not real then the theory supplies no model of
         reality.  (See "What are the problems with quantum theory?")
    
    2)   Hidden Variables [B].  Explicitly non-local.  Bohm accepts that all
         the branches of the universal wavefunction exist.  Like Everett
         Bohm held that the wavefunction is real complex-valued field which
         never collapses.  In addition Bohm postulated that there were
         particles that move under the influence of a non-local "quantum-
         potential" derived from the wavefunction (in addition to the
         classical 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 the wavefunction.  (Bohm derives what is
         essentially a decoherence argument to show this, see section 7,#I
         [B]).
    
         The implicit, unstated assumption made by Bohm is that only the
         single branch of wavefunction associated with particles can contain
         self-aware observers, whereas Everett makes no such assumption. 
         Most of Bohm's adherents do not seem to understand (or even be
         aware of) Everett's criticism, section VI [1], that the hidden-
         variable particles are not observable since the wavefunction alone
         is sufficient to account for all observations and hence a model of
         reality.  The hidden variable particles can be discarded, along
         with the guiding quantum-potential, yielding a theory isomorphic
         to many-worlds, without affecting any experimental results.
    
         [B]  David J Bohm _A suggested interpretation of the quantum theory
              in terms of "hidden variables" I and II_ Physical Review Vol
              85 #2 166-193 (1952)
    
    3)   Quantum Logic.  Undoubtedly the most extreme of all attempts to
         solve the QM measurement problem.  Apart from abandoning one or
         other of the classical tenets of logic these theories are all
         unfinished (presumably because of internal inconsistencies).  Also
         it is unclear 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 of probability, in
         which case it suffers from the same criticism as quantum logic. 
         Also it is unclear, to me anyway, how the resultant notion of
         "complex probability" differs from the quantum "probability
         amplitude" and thus why we are justified in collapsing the complex-
         valued probability as if it were a classical, real-valued
         probability.
    
         [M]  W Muckenheim _A review of extended probabilities_ Physics
              Reports 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" to form the Born probability density. 
         It requires that the input and output states, as defined by an
         observer, act as emitters and absorbers respectively, but not any
         internal states (inside the "black box"), and, consequently,
         suffers from the familiar measurement problem of the Copenhagen
         interpretation.
    
         If the internal states *did* act as emitters/absorbers then the
         wavefunction would collapse, for example, around one of the double
         slits (an internal state) in the double slit experiment, destroying
         the observed interference fringes.  In transaction terminology a
         transaction would form between the first single slit and one of the
         double slits and another transaction would form between the same
         double slit and the point on the screen where the photon lands. 
         This never observed.
    
         [C]  John G Cramer _The transactional interpretation of quantum
              mechanics_ Reviews of Modern Physics Vol 58 #3 647-687 (1986)
    
    6)   Many-minds.  Despite its superficial similarities with many-worlds
         this is actually a very unphysical, non-operational theory.  (See
         "What is many-minds?")
    
    7)   Non-linear theories in general.  So far no non-linear theory has
         any accepted experimental support, whereas many have failed
         experiment.  (See "Is physics linear?")  Many-worlds predicts that
         non-linear theories will always fail experiment.  (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-systems which don't
    significantly interfere with other, more remote, elements in the
    superposition.  Any complex system and its coupled environment, with a
    large number of internal degrees of freedom, qualifies as a world.  An
    observer, with internal irreversible processes, counts as a complex
    system.  In terms of the wavefunction, a world is a decohered branch of
    the universal wavefunction, which represents a single macrostate.  (See
    "What is decoherence?")  The worlds all exist simultaneously in a non-
    interacting linear superposition.
    
    Sometimes "worlds" are called "universes", but more usually the latter
    is reserved the totality of worlds implied by the universal
    wavefunction.  Sometimes the term "history" is used instead of "world". 
    (Gell-Mann/Hartle's phrase, see "What is many-histories?").
    
    Q5   What is a measurement?
         ----------------------
    A measurement is an interaction, usually irreversible, between
    subsystems that correlates the value of a quantity in one subsystem with
    the value of a quantity in the other subsystem.  The interaction may
    trigger an amplification process within one object or subsystem with
    many internal degrees of freedom, leading to an irreversible high-level
    change in the same object.  If the course of the amplification is
    sensitive to the initial interaction then we can designate the system
    containing the amplified process as the "measuring apparatus", since the
    trigger is sensitive to some (often microphysical) quantity or parameter
    of the one of the other subsystems, which we designate the "object"
    system.  Eg the detection of a charged particle (the object) by a Geiger
    counter (the measuring apparatus) leads to the generation of a "click"
    (high-level change).  The absence of a charged particle does not
    generate a click.  The interaction is with those elements of the charged
    particle's wavefunction that passes *between* the charged detector
    plates, triggering the amplification process (an irreversible electron
    cascade or avalanche), which is ultimately converted to a click.
    
    A measurement, by this definition, does not require the presence of an
    conscious observer, only of irreversible processes.
    
    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
    interference effects between the elements of the superposition.  For two
    branches or worlds to interfere with each other all the atoms, subatomic
    particles, photons and other degrees of freedom in each world have to
    be in the same state, which usually means they all must be in the same
    place or significantly overlap in both worlds, simultaneously.
    
    For small microscopic systems it is quite possible for all their atomic
    components to overlap at some future point.  In the double slit
    experiment, for instance, it only requires that the divergent paths of
    the diffracted particle overlap again at some space-time point for an
    interference pattern to form, because only the single particle has been
    split.
    
    Such future coincidence of positions in all the components is virtually
    impossible in more complex, macroscopic systems because all the
    constituent particles have to overlap with their counterparts
    simultaneously.  Any system complex enough to be described by
    thermodynamics and exhibit irreversible behaviour is a system complex
    enough to exclude, for all practical purposes, any possibility of future
    interference between its decoherent branches.  An irreversible process
    is one in, or linked to, a system with a large number of internal,
    unconstrained degrees of freedom.  Once the irreversible process has
    started then alterations of the values of the many degrees of freedom
    leaves an imprint which can't be removed.  If we try to intervene to
    restore the original status quo the intervention causes more disruption
    elsewhere.
    
    In QM jargon we say that the components (or vectors in the underlying
    Hilbert state space) have become permanently orthogonal due to the
    complexity of the systems increasing the dimensionality of the vector
    space, where each unconstrained degree of freedom contributes a
    dimension to the state vector space.  In a high dimension space almost
    all vectors are orthogonal, without any significant degree of overlap. 
    Thus vectors for complex systems, with a large number of degrees of
    freedom, naturally decompose into mutually orthogonal components which,
    because they can never significantly interfere again, are unaware of
    each other.  The complex system, or world, has split into different,
    mutually unobservable worlds.
    
    According to thermodynamics each activated degree of freedom acquires
    kT energy.  This works the other way around as well: the release of
    approximately kT of energy increases the state-space dimensionality. 
    Even the quite small amounts of energy released by an irreversible
    frictive process are quite large on this scale, increasing the size of
    the associated Hilbert space.
    
    Contact between a system and a heat sink is equivalent to increasing the
    dimensionality of the state space, because the description of the system
    has to be extended to include all parts of the environment in causal
    contact with it.  Contact with the 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 irreversible processes.  (See "What
    is a measurement?")  An irreversible process will always produce
    decoherence which splits worlds.  (See "Why do worlds split?", "What is
    decoherence?" and "When does Schrodinger's cat split?" for a concrete
    example.)
    
    In the example of a Geiger counter and a charged particle after the
    particle has passed the counter one world contains the clicked counter
    and that portion of the particle's wavefunction which passed though the
    detector.  The other world contains the unclicked counter with the
    particle's wavefunction with a "shadow" cast by the counter taken out
    of the particle's wavefunction.
    
    The Geiger counter splits when the amplification process became
    irreversible, before the click is emitted.  (See "What is a
    measurement?")  The splitting is local (originally in the region of the
    Geiger counter in our example) and is transmitted causally to more
    distant systems.  (See "Is many-worlds a local theory?" and "Does the
    EPR experiment prohibit locality?")  The precise moment/location of the
    split is not sharply defined due to the subjective nature of
    irreversibility, but can be considered complete when much more than kT
    of energy has been released in an uncontrolled fashion into the
    environment.  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 an irreversible event. 
    These events have caused the decoherence of the different branches of
    the wavefunction.  (See "What is decoherence?" and "Why do worlds
    split?")  Decoherence occurs when irreversible macro-level events take
    place and the macrostate description of an object admits no single
    description.  (A macrostate, in brief, is the description of an object
    in terms of accessible external characteristics.)
    
    The advantage of linking the definition of worlds and the splitting
    process with thermodynamics is the splitting process becomes
    irreversible and only permits forward-time-branching, following the
    increase with entropy.  (See "Why don't worlds fuse, as well as split?") 
    Like all irreversible processes, though, there are exceptions even at
    the coarse-grained level 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 a certain radioactive
    decay is detected.  For simplicity we'll imagine that the box, whilst
    closed, completely isolates the cat from its environment.  After a while
    an investigator opens the box to see if the cat is alive or dead. 
    According to the Copenhagen Interpretation the cat was neither alive nor
    dead until the box was opened, whereupon the wavefunction of the cat
    collapsed into one of the two alternatives (alive or dead cat).  The
    paradox, according to Schrodinger, is that the cat presumably knew if
    it was alive *before* the box was opened.  According to many-worlds the
    device was split into two states (cyanide released or not) by the
    radioactive decay, which is a thermodynamically irreversible process
    (See "When do worlds split?" and "Why do worlds split?").  As the
    cyanide/no-cyanide interacts with the cat the cat is split into two
    states (dead or alive).  From the surviving cat's point of view it
    occupies a different world from its deceased copy.  The onlooker 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 has no 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 split prior 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 the 1940s [F] as a third
    interpretation of quantum mechanics, alongside Schrodinger's wave
    picture and Heisenberg's matrix mechanics, for calculating transition
    amplitudes.  All three approaches are mathematically equivalent, but the
    path-integral formalism offers some interesting additional insights into
    many-worlds.
    
    In the path-integral picture the wavefunction of a single particle at
    (x',t') is built up of contributions of all possible paths from (x,t),
    where each path's contribution is weighted by a (phase) factor of
    exp(i*Action[path]/hbar) * wavefunction at (x,t), summed, in turn, over
    all values of x.  The Action[path] is the time-integral of the
    lagrangian (roughly: the lagrangian equals kinetic minus the potential
    energy) along the path from (x,t) to (x',t').  The final expression is
    thus the sum or integral over all paths, irrespective of any classical
    dynamical constraints.  For N-particle systems the principle is the
    same, except that the paths run through a 3-N space.
    
    In the path-integral approach every possible path through configuration
    space makes a contribution to the transition amplitude.  From this point
    of view the particle explores every possible intermediate configuration
    between the specified start 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 transition
    amplitudes in the future, and that each possible history has an equal
    reality.  Perhaps we shouldn't be surprised that Feynman is on record
    as believing in many-worlds.  (See "Who believes in many-worlds?")  What
    is surprising is that Everett developed his many-worlds theory entirely
    from the Schrodinger viewpoint without any detectable influence from
    Feynman's work, despite 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 with Schwinger and Tomonoga
    who had developed a less visualisable form of QED.  Dyson showed that
    these approaches were 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 a series of (Feynman) diagrams,
    which allow for the creation, exchange and annihilation of particles. 
    Each Feynman diagram represents a different contribution to the complete
    transition amplitude, provided that the external lines map onto the
    required boundary initial and final conditions (the defined in and out
    states).  QED became the prototype for all the other, later, field
    theories like electro-weak and quantum chromodynamics.
    
    [F]  Richard P Feynman _Space-time approach to non-relativistic quantum
         mechanics_ Reviews of Modern Physics, Vol 20: 267-287 (1948)
    
    Q10  What is many-histories?
         -----------------------
         What is the environment basis?
         ------------------------------
    There is considerable linkage between thermodynamics and many-worlds,
    explored in the "decoherence" views of Zurek [7a], [7b] and Gell-Mann
    and Hartle [10], Everett [1], [2] and others [4b].  (See "What is
    decoherence?")  
    
    Gell-Mann and Hartle, in particular, have extended the role of
    decoherence in defining the Everett worlds, or "histories" in their
    nomenclature.  They call their approach the "many-histories" approach,
    where each "coarse-grained or classical history" is associated with a
    unique time-ordered sequence of sets of irreversible events, including
    measurements, records, observations and the like.  (See "What is a
    measurement?")  Fine-grained histories effectively relax the
    irreversible criterion.  Mathematically the many-histories approach is
    isomorphic to Everett's many-worlds.
    
    The worlds split or "decohere" from each other when irreversible events
    occur.  (See "Why do worlds split?" and "When do worlds split?".) 
    Correspondingly many-histories defines a multiply-connected hierarchy
    of classical histories where each classical history is a "child" of any
    parent history which has only a subset of the child defining
    irreversible events and a parent of any history which has a superset of
    such events.  Climbing up the tree from child to parent moves to
    progressively coarser grained consistent histories until eventually the
    top is reached where the history has *no* defining events (and thus
    consistent with everything!).  This is Everett's universal wavefunction. 
    The bottom of the coarse-grained tree terminates with the maximally
    refined set of decohering histories.  The classical histories each have
    a probability assigned to them and probabilities are additive in the
    sense that the sum of the probabilities associated a set classical
    histories is equal to the probability associated with the unique parent
    history defined by the set.  (Below the maximally refined classical
    histories are the fine grained or quantum histories, where probabilities
    are no longer additive and different histories significantly interfere
    with each other.  The bottom level consists of complete microstates,
    which fully specified states.)
    
    The decoherence approach is useful in considering the effect of the
    environment on a system.  In many ways the environment, acting as a heat
    sink, can be regarded as performing a succession of measurement-like
    interactions upon any system, inducing associated system splits.  All
    the environment basis is is a basis chosen so as to minimise the cross-
    basis interference terms.  It makes any real-worlds calculation easy,
    since the cross terms are so small, but it does not *uniquely* select
    a basis, just eliminates a large number.
    
    Q11  How many worlds are there?
         --------------------------
    The thermodynamic Planck-Boltzmann relationship, S = k*log(W), counts
    the branches of the wavefunction at each splitting, at the lowest,
    maximally refined level of Gell-Mann's many-histories tree.  (See "What
    is many-histories?")  The bottom or maximally divided level consists of
    microstates which can be counted by the formula W = exp (S/k), where S
    = entropy, k = Boltzmann's constant (approx 10^-23 Joules/Kelvin) and
    W = number of worlds or macrostates.  The number of coarser grained
    worlds is lower, but still increasing with entropy by the same ratio,
    ie the number of worlds a single world splits into at the site of an
    irreversible event, entropy dS, is exp(dS/k).  Because k is very small
    a great many worlds split off at each macroscopic event.
    
    Q12  Is many-worlds a local theory?
         ------------------------------
    The simplest way to see that the many-worlds metatheory is a local
    theory is to note that it requires that the wavefunction obey some
    relativistic wave equation, the exact form of which is currently
    unknown, but which is presumed to be locally Lorentz invariant at all
    times and everywhere.  This is equivalent to imposing the requirement
    that locality is enforced at all times and everywhere.  Ergo many-worlds
    is a local theory.
    
    Another way of seeing this is examine how macrostates evolve. 
    Macrostates descriptions of objects evolve in a local fashion.  Worlds
    split as the macrostate description divides inside the light cone of the
    triggering event.  Thus the splitting is a local process, transmitted
    causally at light or sub-light speeds.  (See "Does the EPR experiment
    prohibit locality?" and "When do worlds split?")
    
    Q13  Is many-worlds a deterministic theory?
         --------------------------------------
    Yes, many-worlds is a deterministic theory, since the wavefunction obeys
    a deterministic wave equation at all times.  All possible outcomes of
    a measurement or interaction (See "What is a measurement?") are embedded
    within the universal wavefunction although each observer, split by each
    observation, is only aware of single outcomes due to the linearity of
    the wave equation.  The world appears indeterministic, with the usual
    probabilistic collapse of the wavefunction, but at the objective level,
    which includes all outcomes, determinism is restored.
    
    Some people are under the impression that the only motivation for many-
    worlds is a desire to return to a deterministic theory of physics.  This
    is not true.  As Everett pointed out, the objection with the standard
    Copenhagen interpretation is not the indeterminism per se, but that
    indeterminism occurs only with the intervention of an observer, when the
    wavefunction collapses.  (See "What is the Copenhagen interpretation?")
    
    Q14  Is many-worlds a relativistic theory?
         -------------------------------------
         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 quantum
    theories with linear wave equations.  There are three or more stages to
    developing a fully relativised quantum field theory:
    
    First quantisation: the wavefunction of an N particle system is a
    complex field which evolves in 3N dimensions as the solution to either
    the many-particle Schrodinger, Dirac or Klein-Gordon or some other wave
    equation.  External forces applied to the particles are represented or
    modelled via a potential, which appears in the wave equation as a
    classical, background field.
    
    Second quantisation: AKA (relativistic) quantum field theory (QFT)
    handles the creation and destruction of particles by quantising the
    classical fields and potentials as well as the particles.  Each particle
    corresponds to a field, in QFT, and becomes an operator.  Eg the
    electromagnetic field's particle is the photon.  The wavefunction of a
    collection of particles/fields exists in a Fock space, where the number
    of dimensions varies from component to component, corresponding to the
    indeterminacy in the particle number.  Many-worlds has no problems
    incorporating QFT, since a theory (QFT) is not altered by a metatheory
    (many-worlds), which makes statements *about* the theory.
    
    Third quantisation: AKA quantum gravity.  The gravitational metric is
    quantised, along with (perhaps) the topology of the space-time manifold. 
    The role of time plays a less central role, as might be expected, but
    the first and second quantisation models are as applicable as ever for
    modelling low-energy events.  The physics of this is incomplete,
    including some thorny, unresolved conceptual issues, with a number of
    proposals (strings, supersymmetry, supergravity...) for ways forward,
    but the extension required by many-worlds is quite trivial since the
    mathematics would be unchanged.
    
    One of the original motivations of Everett's scheme was to provide a
    system for quantising the gravitational field to yield a quantum
    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 occupy the same space and time as
    we do.
    
    The implicit question is really, why aren't we aware of these other
    worlds, unless they exist "somewhere" else?  To see why we aren't aware
    of the other worlds, despite occupying the same space-time, see "Why do
    I only ever experience one world?"  Some popular accounts describe the
    other worlds as splitting off into other, orthogonal, dimensions.  These
    dimensions are the dimensions of Hilbert space, not the more familiar
    space-time dimensions.
    
    The situation is more complicated, as we might expect, in theories of
    quantum gravity (See "What about quantum gravity?"), because gravity can
    be viewed as perturbations in the space-time metric.  If we take a
    geometric interpretation of 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) similar mass distribution. 
    Whenever the amplification of a quantum-scale interaction effects the
    mass distribution and hence space-time curvature the resultant
    decoherence can be regarded as splitting the local space-time manifold
    into discrete sheets.
    
    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.  Interpretations do not make
    predictions that differ.  (See "What unique predictions does many-worlds
    make?")  In addition many-worlds retrodicts 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 the Copenhagen
    interpretation or Bohm's hidden variables.  The Copenhagen
    interpretation does not contain those elements of the wavefunction that
    correspond to the other worlds.  Bohm's hidden variables contain
    particles, in addition to the wavefunction.  Neither theory is
    isomorphic to each other or many-worlds and are not, therefore, merely
    rival "interpretations".
    
    Third, there is no scientific, reductionistic alternative to many-
    worlds.  All the other theories fail for logical reasons.  (See "Is
    there any alternative theory?")
    
    Fourth, the interpretative side of many-worlds, like the subjective
    probabilistic elements, are derived from within the theory, rather than
    added to it by assumption, as in the conventional approach.  (See "How
    do probabilities emerge within many-worlds?")
    
    Many-worlds should really be described as a theory or, more precisely,
    a metatheory, since it makes statements that are applicable about a
    range of theories.  Many-worlds is the unavoidable implication of any
    quantum theory which obeys some 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 about many-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 for the number of worlds (See "How many worlds
    are there?") implies that where worlds to fuse together then entropy
    would 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 of thermodynamics, which is solely
    due to the boundary conditions (the state of universe at some chosen
    moment) imposed by the Big Bang or whatever we chose to regard as the
    initial conditions.  (See "Why can't the boundary conditions be updated
    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 convincing fashion.  The advantage
    of a retrodiction over a prediction is that the already gathered data
    is more likely to be free of experimenter bias.  An example of a
    retrodiction is the perihelion shift of Mercury which Newtonian
    mechanics plus gravity was unable, totally, to account for whilst
    Einstein's general relativity made short work of it.
    
    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 other quantum 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?") as having previous existed
    distinctly and merely differentiated, rather than the interaction as
    having split one world into many?  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 is a physical process, grounded in the dynamical evolution of
    the wave vector, not a matter of philosophical, linguistic or mental
    convenience (see "Why do worlds split?" and "When do worlds split?") 
    If you try to treat the worlds as pre-existing and separate then the
    maths and probabilistic behaviour all comes out wrong.  Also the
    differentiation theory isn't deterministic, in contradiction to the wave
    equations which are deterministic, since many-minds says that:
    
      AAAAAAAAAAAAAAABBBBBBBBBBBBBBB         --------------> time
                                             (Worlds differentiate)
      AAAAAAAAAAAAAAACCCCCCCCCCCCCCC
    
    occurs, rather than:
                     BBBBBBBBBBBBBBB
                    B
      AAAAAAAAAAAAAA                         (Worlds split)
                    C
                     CCCCCCCCCCCCCCC
    
    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?
         ------------------
    Many-minds proposes, as an extra fundamental axiom, that an infinity of
    separate minds or mental states be associated with each single brain
    state.  When the single physical brain state is split into a quantum
    superposition by a measurement (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 the
    differentiation of pre-existing separate mental states.  There is no
    physical basis for this interpretation, which is incapable of 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 or split?")
    
    In many-minds the role of the conscious observer is accorded special
    status, with its fundamental axiom about infinities of pre-existing
    minds, and as such is philosophically opposed to many-worlds, which
    seeks to remove the observer from any privileged role in physics. 
    (Many-minds was co-invented by David Albert, who has, apparently, since
    abandoned it.  See Scientific American July 1992 page 80 and contrast
    with 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 judging theories which says that
    hypotheses should not be multiplied beyond necessity.  This is known as
    Ockham's razor and is interpreted, today, as meaning that to account for
    any set of facts the simplest theories are to be preferred over more
    complex ones.  Many-worlds is viewed as unnecessarily complex, by some,
    by requiring the existence of a multiplicity of worlds to explain what
    we see, at any time, in just one world.
    
    This is to mistake what is meant by "complex".  Here's an example. 
    Analysis of starlight reveals that starlight is very similar to faint
    sunlight, both with spectroscopic absorption and emission lines. 
    Assuming the universality of physical law we are led to conclude that
    other stars and worlds are scattered, in great numbers, across the
    cosmos.  The theory that "the stars are distant suns" is the simplest
    theory and so to be preferred by Ockham's Razor to other geocentric
    theories.
    
    Similarly many-worlds is the simplest and most economical quantum theory
    because it proposes that same laws of physics apply to animate observers
    as has been observed for inanimate objects.  The multiplicity of worlds
    predicted by the 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 older heliocentric theories
    *against* Galileo's notion of the vastness of the cosmos.  The notion
    of vast empty interstellar spaces was too uneconomical to be believable
    to the Medieval mind.  Again they were confusing the notion of vastness
    with complexity [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 are consistent with
    conservation of energy, therefore energy is conserved.
    
    Second, and more precisely, conservation of energy, in QM, is formulated
    in terms of weighted averages or expectation values.  Conservation of
    energy is expressed by saying that the time derivative of the expected
    energy of a closed system vanishes.  This statement can be scaled up to
    include the whole universe.  Each world has an approximate energy, but
    the energy of the total wavefunction, or any subset of, involves summing
    over each world, weighted with its probability measure.  This weighted
    sum is a constant.  So energy is conserved within each world and also
    across the totality of worlds.
    
    One way of viewing this result - that observed conserved quantities are
    conserved across the totality of worlds - is to note that new worlds are
    not created by the action of the wave equation, rather existing worlds
    are split into successively "thinner" and "thinner" slices, if we view
    the probability densities as "thickness".
    
    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 by the probabilistic
    Born interpretation, by showing that the Hilbert space's inner product
    or norm has a special property which allows us to makes statements about
    the worlds where quantum statistics break down.  The norm of the vector
    of the set of worlds where experiments contradict the Born
    interpretation ("non-random" or "maverick" worlds) vanishes in the limit
    as the number of probabilistic trials goes to infinity, as is required
    by the frequentist definition of probability.  Hilbert space vectors
    with zero norm don't exist (see below), thus we, as observers, only
    observe the familiar, probabilistic predictions of quantum theory. 
    Everett-worlds where probability breaks down are never realised.
    
    Strictly speaking Everett did not prove that the usual statistical laws
    of the Born interpretation would hold true for all observers in all
    worlds.  He merely showed that no other statistical laws could hold true
    and asserted the vanishing of the Hilbert space "volume" or norm of the
    set of "maverick" worlds.  DeWitt later published a longer *derivation*
    of Everett's assertion [4a], [4b], closely based on an earlier,
    independent demonstration by Hartle [H].  What Everett asserted, and
    DeWitt/Hartle derived, is that the collective norm of all the maverick
    worlds, as the number of trials goes to infinity, vanishes.  Since the
    only vector in a Hilbert space with vanishing norm is the null vector
    (a defining axiom of Hilbert spaces) this is equivalent to saying that
    non-randomness is never realised.  All the worlds obey the usual Born
    predictions of quantum theory.  That's why we never observe the
    consistent violation of the usual quantum statistics, with, say, heat
    flowing from a colder to a hotter macroscopic object.  Zero-probability
    events never happen.
    
    Of course we have to assume that the wavefunction is a Hilbert space
    vector in the first place but, since this assumption is also made in the
    standard formulation, this is not a weakness of many-worlds since we are
    not trying to justify all the axioms of the conventional formulation of
    QM, merely those that relate to probabilities and collapse of the
    wavefunction.
    
    In more detail the steps are:
    
    1)   Construct the tensor product of N identical systems in state |psi>,
         according to the usual rules for Hilbert space composition
         (repeated indices summed):
         |PSI_N> = |psi_1>*|psi_2>*...... |psi_N> where
         |psi_j> = jth system prepared in state |psi>
                 = |i_j><i_j|psi> (ie the amplitude of the ith eigenstate
                                  is independent of which system it is in)
         so that 
         |PSI_N> = |i_1>|i_2>...|i_N><i_1|psi><i_2|psi>...<i_N|psi>
    
    2)   Quantify the deviation from the "expected" Born-mean for each
         component of |PSI_N> with respect to the above |i_1>|i_2>...|i_N>
         basis by counting the number of occurrences of the ith
         eigenstate/N.  Call this number RF(i).  Define the Born-deviation
         as D = sum(i)( (RF(i) - |<i|psi>|^2)^2 ).  Thus D, loosely
         speaking, for each N length sequence, quantifies by how much the
         particular sequence differs from the Born-expectation.
    
    3)   Sort out terms in the expansion of |PSI_N> according to whether D
         is less/equal to (.LE.) or greater than (.GT.) E, where E is a
         real, positive constant.  Collecting terms together we get:
         |PSI_N> = |N,"D.GT.E"> + |N,"D.LE.E">
                   worlds       worlds
                  for which    for which
                    D > E       D <= E
    
    4)   What DeWitt showed was that:
         <N,"D.GT.E"|N,"D.GT.E"> < 1/(NE)     (proof in appendix of [4b])
         Thus as N goes to infinity the right-hand side vanishes for all
         positive values of E.  (This mirrors the classical "frequentist"
         position on probability which states that if event i occurs with
         probability p(i) then the proportion of N trials with outcome i
         approaches p(i)/N as N goes to infinity [H].  This has the
         immediate benefit that sum(i) p(i) = 1.)  The norm of |N,"D.LE.E">,
         by contrast, approaches 1 as N goes to infinity.
    
         Note: this property of D is not shared by other definitions, which
         is why we haven't investigated them.  If, say, we had defined, in
         step 2), A = sum(i)( (RF(i) - |<i|psi>|)^2 ), so that A measures
         the deviation from |psi|, rather than |psi|^2, then we find that
         <A> does not have the desired property of vanishing as N goes to
         infinity.
    
    5)   The norm of the collection of non-random worlds vanishes and
         therefore must be identified with some complex multiple of the null
         vector.
    
    6)   Since (by assumption) the state vector faithfully models reality
         then the null vector cannot represent any element of reality, since
         it can be added to (or subtracted from) any other state vector
         without altering the other state vector.
    
    7)   Ergo the non-random worlds are not realised, without making any
         additional physical assumptions, such the imposition of a measure.
    
         Note: no finite sequence of outcomes is excluded from happening,
         since the concept of probability and randomness only becomes
         precise only as N goes to infinity [H].  Thus, heat *could* be
         observed to flow from a cold to hotter object, but we might have
         to wait a very long time before observing it.  What *is* excluded
         is 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 any extra interpretative
    assumptions, is another reason why the Everett metatheory should not be
    regarded as just an interpretation.  (See "Is many-worlds (just) an
    interpretation?")  The interpretative elements are forced by the
    mathematical structure of the axioms of Hilbert space.
    
    [H]  JB Hartle _Quantum Mechanics of Individual Systems_ American
         Journal of Physics Vol 36 #8 704-712 (1968)  Hartle has
         investigated the N goes to infinity limit in more detail and more
         generally.  He shows that the relative frequency operator, RF,
         obeys RF(i) |psi_1>|psi_2>.... = |<i|psi>|^2 |psi_1>|psi_2>....,
         for a normed state.  Hartle regarded his derivation as essentially
         the 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 situation is certainly
    no better or worse for free-will than in the Copenhagen view. 
    Traditional Copenhagen indeterministic quantum mechanics only slightly
    weakens the case for free-will.  In quantum terms each neuron is an
    essentially classical object.  Consequently quantum noise in the brain
    is at such a low level that it probably doesn't often alter, except very
    rarely, the critical mechanistic behaviour of sufficient neurons to
    cause a decision to be different than we might otherwise expect.  The
    consensus view amongst experts is that free-will is the consequence of
    the mechanistic operation of our brains, the firing of neurons,
    discharging across synapses etc and fully compatible with the
    determinism of classical physics.  Free-will is the inability of an
    intelligent, self-aware mechanism to predict its own future actions due
    to the logical impossibility of any mechanism containing a complete
    internal model of itself rather than any inherent indeterminism in the
    mechanism's operation.
    
    Nevertheless, some people find that with all possible decisions being
    realised in different worlds that the prima facia situation for free-
    will looks quite difficult.  Does this multiplicity of outcomes destroy
    free-will?  If both sides of a choice are selected in different worlds
    why bother to spend time weighing the evidence before selecting?  The
    answer is that whilst all decisions are realised, some are realised more
    often than others - or to put to more precisely each branch of a
    decision has 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 or collection of worlds.  A world of zero volume is
    never realised.  Worlds in which the conventional statistical
    predictions consistently break down have zero volume and so are never
    realised.  (See "How do probabilities emerge within many-worlds?")  
    
    Thus our actions, as expressions of our will, correlate with the weights
    associated with worlds.  This, of course, matches our subjective
    experience of being able to exercise our will, form moral judgements and
    be held responsible for our actions.
    
    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 as Fred-Right?  The only answer possible is that there
    was *no* reason.  From Fred's point of view it is a subjectively
    *random* choice which individual "Fred" ends up as.  To the surgeon the
    whole process is deterministic.  To both the Freds it seems random.
    
    Same with many-worlds.  There was no reason "why" you ended up in this
    world, rather than another - you end up in all the quantum worlds.  It
    is a subjectively random choice, an artifact of your brain and
    consciousness being split, along with the rest of the world, that makes
    our experiences seem random.  The universe is, in effect, performing
    umpteen split-brain operations on us all the time.  The randomness
    apparent in nature is a consequence of the continual splitting into
    mutually unobservable worlds.
    
    (See "How do probabilities emerge within many-worlds?" for how the
    subjective randomness is moderated by the usual probabilistic laws of
    QM.)
    
    [*] Split brain experiments *were* performed on epileptic patients
    (severing the corpus callosum, one of the pathways connecting the
    cerebral hemispheres, moderated epileptic attacks).  Complete
    hemispherical separation was discontinued when testing of the patients
    revealed the presence of two distinct consciousnesses in the same skull. 
    So this analogy is only 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 called decoherence (See "What is decoherence?"), but claims
    that the wavefunction does not *actually* collapse but continues to
    evolve according to the usual wave-equation.  If a *mechanism* for
    collapse could be found then there would be no need for many-worlds. 
    The reason why we doubt that collapse takes place is because no one has
    ever been able to devise a physical mechanism that could trigger it.
    
    The Copenhagen interpretation posits that observers collapse
    wavefunctions, but is unable to define "observer".  (See "What is the
    Copenhagen interpretation?" and "Is there any alternative theory?") 
    Without a definition of observer there can be no mechanism triggered by
    their presence.
    
    Another popular view is that irreversible processes trigger collapse. 
    Certainly wavefunctions *appear* to collapse whenever irreversible
    processes are involved.  And most macroscopic, day-to-day events are
    irreversible.  The problem is, as with positing observers as a cause of
    collapse, that any irreversible process is composed of a large number
    of sub-processes that are each individually reversible.  To invoke
    irreversibility as a *mechanism* for collapse we would have to show that
    new *fundamental* physics comes into play for complex systems, which is
    quite absent at the reversible atom/molecular level.  Atoms and
    molecules are empirically observed to obey some type of wave equation. 
    We have no evidence for an extra mechanism operating on more complex
    systems.  As far as we can determine complex systems are described by
    the 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 many popularisers say.)
    
    Other people have attempted to construct non-linear theories so that
    microscopic systems are approximately linear and obey the wave equation,
    whilst macroscopic systems are grossly non-linear and generates
    collapse.  Unfortunately all these 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 propagate instantaneously, or in some
    space-like fashion, otherwise the same particle could be observed more
    than once at different locations.  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 that all 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, through the existence of
    interference effects, it is impossible travel to or communicate with
    them.  Mathematically this corresponds to an empirically verified
    property of all quantum theories called linearity.  Linearity implies
    that the worlds can interfere with each other with respect to a
    external, unsplit, observer or system but the interfering worlds can't
    influence each other in the sense that an experimenter in one of the
    worlds can arrange to communicate with their own, already split-off,
    quantum copies in other worlds.
    
    Specifically, the wave equation is linear, with respect to the
    wavefunction or state vector, which means that given any two solutions
    of the wavefunction, with identical boundary conditions, then any linear
    combination of the solutions is another solution.  Since each component
    of a linear solution evolves with complete indifference as to the
    presence or absence of the other terms/solutions then we can conclude
    that no experiment in one world can have any effect on another
    experiment in another world.  Hence no communication is possible between
    quantum worlds.  (This type of linearity mustn't be confused with the
    evident non-linearity of the equations with respect to the *fields*.)
    
    Non communication between the splitting Everett-worlds also explains why
    we are not aware of any splitting process, since such awareness needs
    communication between worlds.  To be aware of the world splitting you
    would have to be receiving sensory information from, and thereby effect
    by the reverse process, more than one world.  This would enable
    communication between worlds, which is forbidden by linearity.  Ergo,
    we are not aware of any splitting precisely because we are split into
    non-interfering copies along with the rest of the world.
    
    See also "Is linearity exact?"
    
    Q28  Can we determine what other worlds there are?
         ---------------------------------------------
         Is the form of the Universal Wavefunction knowable?
         ---------------------------------------------------
    To calculate the form of the universal wavefunction requires not only
    a knowledge of its dynamics (which we have a good approximation to, at
    the moment) but also of the boundary conditions.  To actually calculate
    the form of the universal wavefunction, and hence make inferences about
    *all* the embedded worlds, we would need to know the boundary conditions
    as well.  We are presently restricted to making inferences about those
    worlds with which have shared a common history up to some point, which
    have left traces (records, fossils, etc) still discernable today.  This
    restricts us to a subset of the extant worlds which have shared the same
    boundary conditions with us.  The further we probe back in time the less
    we know of the boundary conditions and the less we can know of the
    universal wavefunction.
    
    This limits us to drawing conclusions about a restricted subset of the
    worlds - all the worlds which are consistent with our known history up
    to a some common moment, before we diverged.  The flow of historical
    events is, according to chaos/complexity theory/thermodynamics, very
    sensitive to amplification of quantum-scale uncertainty and this
    sensitivity is a future-directed one-way process.  We can make very
    reliable deductions about the past from the knowledge future/present but
    we can't predict the future from knowledge the past/present. 
    Thermodynamics implies that the future is harder to predict than the
    past is to retrodict.  Books get written about this "arrow of time"
    problem but, for the purposes of this discussion, we'll accept the
    thermodynamic origin of time's arrow is as given.  The fossil and
    historical records say that dinosaurs and Adolf Hitler once existed but
    have less to say about the future.
    
    Consider the effects of that most quantum of activities, Brownian
    motion, on the conception of individuals and the knock-on effects on the
    course of history.  Mutation itself, one of the sources of evolutionary
    diversity, is a quantum event.  For an example of the
    biological/evolutionary implications see Stephen Jay Gould's book
    "Wonderful Life" for an popular exploration of the thesis that the path
    of evolution is driven by chance.  According to Gould evolutionary
    history forms an enormously diverse tree of possible histories - all
    very improbable - with our path being selected by chance.  According to
    many-worlds all these other possibilities are realised.  Thus there are
    worlds in which Hitler won WW-II and other worlds in which the dinosaurs
    never died out.  We can be as certain of this as we are that Hitler and
    the dinosaurs once existed in our own past.
    
    Whether or not we can ever determine the totality of the universal
    wavefunction is an open question.  If Steven Hawking's work on the no-
    boundary-condition condition is ultimately successful, or it emerges
    from some theory of everything, and many think it will, then the actual
    form of the *total* wavefunction could, in principle, we determined from
    a complete knowledge of physical law itself.
    
    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 the wavefunction.  He developed, during discussions with
    Charles Misner and Aage Peterson (Bohr' assistant, then visiting
    Princeton), his "relative state" formulation.  Wheeler encouraged his
    work and preprints were circulated in January 1956 to a number of
    physicists.  A condensed version of his thesis was published as a paper
    to "The Role of Gravity in Physics" conference held at the University
    of North Carolina, Chapel Hill, in January 1957.
    
    Everett was discouraged by the lack of response from others,
    particularly Bohr, whom he flew to Copenhagen to meet but got the
    complete brush-off from.  Leaving physics after completing his Ph.D,
    Everett worked as a defense analyst at the Weapons Systems Evaluation
    Group, Pentagon and later became a private contractor, apparently quite
    successfully for he became a multimillionaire.  In 1968 Everett worked
    for the Lambda Corp.  His published papers during this period cover
    things like optimising resource allocation and, in particular,
    maximising kill rates during nuclear-weapon campaigns.
    
    From 1968 onwards Bryce S DeWitt, one of the 1957 Chapel Hill conference
    organisers, but better known as one of the founders of quantum gravity,
    successfully popularised Everett's relative state formulation as the
    "many-worlds interpretation" in a series of articles [4a],[4b],[5].
    
    Sometime in 1976-9 Everett visited Austin, Texas, at Wheeler or DeWitt's
    invitation, to give some lectures on QM.  The strict no-smoking rule in
    the auditorium was relaxed for Everett (a chain smoker); the only
    exception ever.  Everett, apparently, had a very intense manner,
    speaking acutely and anticipating questions after a few words.  Oh yes,
    a bit of trivia, he drove a Cadillac with horns.
    
    With the steady growth of interest in many-worlds in the late 1970s
    Everett planned returning to physics to do more work on measurement in
    quantum theory, but died of a heart attack in 1982.  Survived by his
    wife.
    
    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 is not applied to larger,
    classical systems, like observers or the entire universe.  Many
    scientists and philosophers are unhappy with the theory because it seems
    to require a fundamental quantum-classical divide.  Einstein, for
    example, despite his early contributions to the subject, was never
    reconciled with assigning to the act of observation a physical
    significance, which most interpretations of QM require.  This
    contradicts the reductionist ethos that, amongst other things,
    observations should emerge only as a consequence of an underlying
    physical theory and not be present at the axiomatic level, as they are
    in the Copenhagen interpretation.  Yet the Copenhagen interpretation
    remains the most popular interpretation of quantum mechanics amongst the
    broad scientific community.  (See "What is the Copenhagen
    interpretation?")
    
    Q31  What is the Copenhagen interpretation?
         --------------------------------------
    An unobserved system, according to the Copenhagen interpretation of
    quantum theory, evolves in a deterministic way determined by a wave
    equation.  An observed system changes in a random fashion, at the moment
    of observation, instantaneously, with the probability of any particular
    outcome given by the Born formula.  This is known as the "collapse" or
    "reduction" of the wavefunction.  The problems with this approach are:
    (1)  The collapse is an instantaneous process across an extended
         region ("non-local") which is non-relativistic.
    (2)  The idea of an observer having an effect on microphysics is
         repugnant to reductionism and smacks of a return to pre-scientific
         notions of vitalism.  Copenhagenism is a return to the old vitalist
         notions that life is somehow different from other matter, operating
         by different laws from inanimate matter.  The collapse is triggered
         by an observer, yet no definition of what an "observer" is
         available, in terms of an atomic scale description, even in
         principle.
    
    For these reasons the view has generally been adopted that the
    wavefunction associated with an object is not a real "thing", but merely
    represents our *knowledge* of the object.  This approach was developed
    by Bohr and others, mainly at Copenhagen in the late 1920s.  When we
    perform an measurement or observation of an object we acquire new
    information and so adjust the wavefunction as we would boundary
    conditions in classical physics to reflect this new information.  This
    stance means that we can't answer questions about what's actually
    happening, all we can answer is what will be the probability of a
    particular result if we perform a measurement.  This makes a lot of
    people very unhappy since it provides no model for the object.
    
    It should be added that there are other, less popular, interpretations
    of quantum theory, but they all have their own drawbacks, which are
    widely reckoned more severe.  Generally speaking they try to find a
    mechanism that describes the collapse process or add extra physical
    objects to the theory, in addition to the wavefunction.  In this sense
    they are more complex.  (See "Is there any alternative theory?")
    
    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 mechanics is non-local
    (requires faster-than-light communication) or incomplete.  We shall see
    that it implies neither.
    
    The EPR experiment was devised, in 1935, by Einstein, Podolsky and Rosen
    to demonstrate that quantum mechanics was incomplete [E].  Bell, in
    1964, demonstrated that any hidden variables theory, to replicate the
    predictions of QM, must be non-local [B].  QM predicts strong
    correlations between separated systems, stronger than any local hidden
    variables theory can offer.  Bell encoded this statistical prediction
    in the form of some famous inequalities that apply to any type of EPR
    experiment.  Eberhard, in the late 1970s, extended Bell's inequalities
    to cover any local theory, with or without hidden variables.  Thus the
    EPR experiment plays a central role in sorting and testing variants of
    QM.  All the experiments attempting to test EPR/Bell's inequality to
    date (including Aspect's in the 1980s [As]) are in line with the
    predictions of standard QM - hidden variables are ruled out.  Here is
    the paradox of the EPR experiment.  It seems to imply that any physical
    theory must involve faster-than-light "things" going on to maintain
    these "spooky" action-at-a-distance correlations and yet still be
    compatible with relativity, which seems to forbid FTL.
    
    Let's examine the EPR experiment in more detail.
    
    So what did EPR propose?  The original proposal was formulated in terms
    of correlations between the positions and momenta of two once-coupled
    particles.  Here I shall describe it in terms of the spin (a type of
    angular momentum intrinsic to the particle) of two electrons.  [In this
    treatment I shall ignore the fact that electrons always form
    antisymmetric combinations.  This does not alter the results but does
    simplify the maths.]  Two initially coupled electrons, with opposed
    spins that sum to zero, move apart from each other across a distance of
    perhaps many light years, before being separately detected, say, by me
    on Earth and you on Alpha Centauri with our respective measuring
    apparatuses.  The EPR paradox results from noting that if we choose the
    same (parallel) spin axes to measure along then we will observe the two
    electrons' spins to be anti-parallel (ie when we communicate we find
    that the spin on our electrons are correlated and opposed).  However if
    we choose measurement spin axes that are perpendicular to each other
    then there is no correlation between electron spins.  Last minute
    alterations in a detector's alignment can create or destroy correlations
    across great distances.  This implies, according to some theorists, that
    faster-than-light influences maintain correlations between separated
    systems in some circumstances and not others.
    
    Now let's see how many-worlds escapes from this dilemma.
    
    The initial state of the wavefunction of you, me and the electrons and
    the rest of the universe may be written:
    
       |psi> =  |me> |electrons> |you> |rest of universe>
                 on      in       on
                Earth   deep     Alpha
                        space   Centauri
    or more compactly, ignoring the rest of the universe, as:
       |psi> =  |me,electrons,you>  
    And
         |me> represents me on Earth with my detection apparatus.
         |electrons> = (|+,-> - |-,+>)/sqrt(2) 
            represents a pair electrons, with the first electron travelling
            towards Earth and the second electron travelling towards Alpha
            Centauri.
    
       |+> represents an electron with spin in the +z direction
       |-> represents an electron with spin in the -z direction
    
    It is an empirically established fact, which we just have to accept,
    that we can relate spin states in one direction to spin states in other
    directions like so (where "i" is the sqrt(-1)):
       |left>  = (|+> - |->)/sqrt(2)    (electron with spin in -x direction)
       |right> = (|+> + |->)/sqrt(2)    (electron with spin in +x direction)
       |up>    = (|+> + |->i)/sqrt(2)   (electron with spin in +y direction)
       |down>  = (|+> - |->i)/sqrt(2)   (electron with spin in -y direction)
    and inverting:
       |+>  = (|right> + |left>)/sqrt(2) =  (|up> + |down>)/sqrt(2)
       |->  = (|right> - |left>)/sqrt(2) =  (|down> - |up>)i/sqrt(2)
    
    (In fancy jargon we say that the spin operators in different directions
    form non-commuting observables.  I shall eschew such obfuscations.)
    
    Working through the algebra we find that for pairs of electrons:
    
       |+,-> - |-,+> =  |left,right> -  |right,left>
                     =  |up,down>i    - |down,up>i
    
    I shall assume that we are capable of either measuring spin in the x or
    y direction, which are both perpendicular the line of flight of the
    electrons.  After having measured the state of the electron my state is
    described as one of either:
       |me[l]> represents me + apparatus + records having measured 
               and recorded the x-axis spin as "left"
       |me[r]> ditto with the x-axis spin as "right"
       |me[u]> ditto with the y-axis spin as "up"
       |me[d]> ditto with the y-axis spin as "down"
    
    Similarly for |you> on Alpha Centauri.  Notice that it is irrelevant
    *how* we have measured the electron's spin.  The details of the
    measurement process are irrelevant.  (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.  (The latter assumption may be relaxed without altering the
    results.)
    
    To establish familiarity with the notation let's take the state of the
    initial wavefunction as:
    
                 |psi>_1 =  |me,left,up,you>
                                 /     \
                               /         \
        first electron in left          second electron in up state
        state heading towards              heading towards you on
            me on Earth                        Alpha Centauri
            
    After the electrons arrive at their detectors, I measure the spin
    along the x-axis and you along the y-axis.  The wavefunction evolves
    into |psi>_2:
    
                    local     
         |psi>_1 ============> |psi>_2 = |me[l],left,up,you[u]> 
                  observation
    
    which represents me having recorded my electron on Earth with spin left
    and you having recorded your electron on Alpha Centauri with spin up. 
    The index in []s indicates the value of the record.  This may be held
    in the observer's memory, notebooks or elsewhere in the local
    environment (not necessarily in a readable form).  If we communicate our
    readings to each other the wavefunctions evolves into |psi>_3:
    
                   remote 
         |psi>_2 ============> |psi>_3 = |me[l,u],left,up,you[u,l]> 
                 communication
    
    where the second index in []s represents the remote reading communicated
    to the other observer and being recorded locally.  Notice that the
    results both agree with each other, in the sense that my record of your
    result agrees with your record 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 the terms 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) which correlate with the state
    of the electron.  If we now communicate over interstellar modem (this
    will take a few years since you and I are separated by light years, but
    no matter).  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 with spin right and knowing
    that you have seen your electron with spin left.  So we jointly agree,
    in both worlds, that spin has been conserved.
    
    Now suppose that we had prearranged to measure the spins along different
    axes.  Suppose I measure the x-direction spin and you the y-direction
    spin.  Things get a bit more complex.  To analyse what happens we need
    to decompose the two electrons along their respective spin axes.
    
      |psi>_1 =
          |me,electrons,you>
        = |me>(|+,-> - |-,+>)|you>/sqrt(2) 
        = |me> (
                (|right>+|left>)i(|down>-|up>)
              - (|right>-|left>)(|down>+|up>)
               ) |you> /2*sqrt(2) 
        = |me> (
                |right>(|down>-|up>)i
              + |left> (|down>-|up>)i
              - |right>(|down>+|up>)
              + |left> (|down>+|up>)
               ) |you> /2*sqrt(2) 
        = |me> (
                |right,down> (i-1) - |right,up> (1+i)
              + |left,up> (1-i)    + |left,down> (1+i) 
               ) |you> /2*sqrt(2) 
        =  (
           + |me,right,down,you> (i-1)
           - |me,right,up,you>   (i+1)
           + |me,left,up,you>    (1-i)
           + |me,left,down,you>  (1+i) 
           ) /2*sqrt(2) 
    
    So after you and I make our local observations we get:
    
       |psi>_2 =
           (
           + |me[r],right,down,you[d]> (i-1) 
           - |me[r],right,up,you[u]>   (i+1) 
           + |me[l],left,up,you[u]>    (1-i) 
           + |me[l],left,down,you[d]>  (1+i)
           ) /2*sqrt(2)
    
    Each term realises a possible outcome of the joint measurements.  The
    interesting thing is that whilst we can decompose it into four terms
    there are only two states for each observer.  Looking at myself, for
    instance, we can rewrite this in terms of states relative to *my*
    records/memories.
    
       |psi>_2 = 
           ( 
             |me[r],right> ( |down,you[d]> (i-1) - |up,you[u]> (i+1) )
           + |me[l],left>  ( |up,you[u]> (1-i) + |down,you[d]> (1+i) )
           ) /2*sqrt(2)
    
    And we see that there are only two copies of *me*.  Equally we can
    rewrite the expression in terms of states relative to *your*
    records/memory.
    
       |psi>_2 =
           ( 
             ( |me[l],left> (1-i) - |me[r],right> (i+1) ) |up,you[u]> 
           + ( |me[r],right> (i-1) + |me[l],left> (1+i) ) |down,you[d]>
           ) /2*sqrt(2)
    
    And see that there are only two copies of *you*.   We have each been
    split into two copies, each perceiving a different outcome for our
    electron's spin, but we have not been split by the measurement of the
    remote electron'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 two observers
    communicated their results to each other they were each unsplit by each
    others' measurements, although their own local measurements had split
    themselves.  The splitting is a local process that is causally
    transmitted from system to system at light or sub-light speeds.  (This
    is a point that Everett stressed about Einstein's remark about the
    observations of a mouse, in the Copenhagen interpretation, collapsing
    the wavefunction of the universe.  Everett observed that it is the mouse
    that's split by its observation of the rest of the universe.  The rest
    of the universe is unaffected and unsplit.)
    
    When all communication is complete the worlds have finally decomposed
    or decohered from each other.  Each world contains a consistent set of
    observers, records and electrons, in perfect agreement with the
    predictions of standard QM.  Further observations of the electrons will
    agree with the earlier ones and so each observer, in each world, can
    henceforth regard the electron's wavefunction as having collapsed to
    match the historically recorded, locally observed values.  This
    justifies our operational adoption of the collapse of the wavefunction
    upon measurement, without having to strain our credibility by believing
    that it actually happens.
    
    To recap.  Many-worlds is local and deterministic.  Local measurements
    split local systems (including observers) in a subjectively random
    fashion; distant systems are only split when the causally transmitted
    effects of the local interactions reach them.  We have not assumed any
    non-local FTL effects, yet we have reproduced the standard predictions
    of QM.
    
    So where did Bell and Eberhard go wrong?  They thought that all theories
    that reproduced the standard predictions must be non-local.  It has been
    pointed out by both Albert [A] and Cramer [C] (who both support
    different interpretations of QM) that Bell and Eberhard had implicity
    assumed that every possible measurement - even if not performed - would
    have yielded a *single* definite result.  This assumption is called
    contra-factual definiteness or CFD [S].  What Bell and Eberhard really
    proved was that every quantum theory must either violate locality *or*
    CFD.  Many-worlds with its multiplicity of results in different worlds
    violates CFD, of course, and thus can be local.
    
    Thus many-worlds is the only local quantum theory in accord with the
    standard predictions of QM and, so far, with experiment.
    
    [A]  David Z Albert, _Bohm's Alternative to Quantum Mechanics_
         Scientific American (May 1994)
    [As] Alain Aspect, J Dalibard, G Roger _Experimental test of Bell's
         inequalities using time-varying analyzers_ Physical Review Letters
         Vol 49 #25 1804 (1982).
    [C]  John G Cramer _The transactional interpretation of quantum
         mechanics_ Reviews of Modern Physics Vol 58 #3 647-687 (1986)
    [B]  John S Bell:  _On the Einstein Podolsky Rosen paradox_ Physics 1
         #3 195-200 (1964).
    [E]  Albert Einstein, Boris Podolsky, Nathan Rosen:  _Can
         quantum-mechanical description of physical reality be considered
         complete?_  Physical Review Vol 41 777-780 (15 May 1935).
    [S]  Henry P Stapp _S-matrix interpretation of quantum-theory_ Physical
         Review D Vol 3 #6 1303 (1971)
    
    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 lot from 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" and DeWitt'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 the observer being
    split by a measurement.  (See "What is a measurement?")  DeWitt
    introduced the term "world" for describing the split states of an
    observer, so that we now speak of the observer's world splitting during
    the measuring process.  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 effects after a measurement, although the topics
    of orthogonality, amplification and irreversibility were covered.  (See
    "What is a measurement?", "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 something else.  What
    the heck does that mean?  It means, amongst other things, that states
    in the same Everett-world are all states relative 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 to what? 
    Relative to you and me, in this world.  In some other worlds it will be
    "Bush", "Smith", etc .......  Each possibility is realised in some world
    and it is the relative state of the President's name, relative to the
    occupants of that world.
    
    According to Everett almost all states are relative states.  Only the
    state of the universal wavefunction is not relative but absolute.
    
    Q35  Was Everett a "splitter"?
         -------------------------
    Some people believe that Everett eschewed all talk all splitting or
    branching observers in his original relative state formulation [2]. 
    This is contradicted by the following quote from [2]:
         [...] Thus with each succeeding observation (or interaction),
         the observer state "branches" into a number of different
         states. Each branch represents a different outcome of the
         measurement and the *corresponding* eigenstate for the object-
         system state. All branches exist simultaneously in the
         superposition after any given sequence of observations.[#]
           The "trajectory" of the memory configuration of an observer
         performing a sequence of measurements is thus not a linear
         sequence of memory configurations, but a branching tree, with
         all possible outcomes existing simultaneously in a final
         superposition with various coefficients in the mathematical
         model. [...]
    
           [#] Note added in proof-- In reply to a preprint of this
         article some correspondents have raised the question of the
         "transition from possible to actual," arguing that in
         "reality" there is-as our experience testifies-no such
         splitting of observers states, so that only one branch can
         ever actually exist. Since this point may occur to other
         readers the following is offered in explanation.
           The whole issue of the transition from "possible" to
         "actual" is taken care of in the theory in a very simple way-
         there is no such transition, nor is such a transition
         necessary for the theory to be in accord with our experience.
         From the viewpoint of the theory *all* elements of a
         superposition (all "branches") are "actual," none are any more
         "real" than the rest. It is unnecessary to suppose that all
         but one are somehow destroyed, since all separate elements of
         a superposition individually obey the wave equation with
         complete indifference to the presence or absence ("actuality"
         or not) of any other elements. This total lack of effect of
         one branch on another also implies that no observer will ever
         be aware of any "splitting" process.
           Arguments that the world picture presented by this theory
         is contradicted by experience, because we are unaware of any
         branching process, are like the criticism of the Copernican
         theory that the mobility of the earth as a real physical fact
         is incompatible with the common sense interpretation of nature
         because we feel no such motion. In both case the arguments
         fails when it is shown that the theory itself predicts that
         our experience will be what it in fact is. (In the Copernican
         case the addition of Newtonian physics was required to be able
         to show that the earth's inhabitants would be unaware of any
         motion of the earth.)
    
    Q36  What unique predictions does many-worlds make?
         ----------------------------------------------
    A prediction occurs when a theory suggests new phenomena.  Many-worlds
    makes at least three predictions, two of them unique: about linearity,
    (See "Is linearity exact?"), quantum gravity (See "Why *quantum*
    gravity?") and reversible quantum computers (See "Could we detect other
    Everett-worlds?").
    
    Q37  Could we detect other Everett-worlds?
         -------------------------------------
    Many-Worlds predicts that the Everett-worlds do not interact with each
    other because of the presumed linearity of the wave equation.  However
    worlds *do* interfere with each other, and this enables the theory to
    be tested.  (Interfere and interact mean different things in quantum
    mechanics.  Pictorially: Interactions occur at the vertices within
    Feynman diagrams.  Interference occurs when you add together different
    Feynman diagrams with the same external lines.)
    
    According to many-worlds model worlds split with the operation of every
    thermodynamically irreversible process.  The operation of our minds are
    irreversible, carried along for the ride, so to speak, and divide with
    the division of worlds.  Normally this splitting is undetectable to us. 
    To detect the splitting we need to set an up experiment where a mind is
    split but the world *isn't*.  We need a reversible mind.
    
    The general consensus in the literature [11], [16] is that the
    experiment to detect other worlds, with reversible minds, will be doable
    by, perhaps, about mid-21st century.  That date is predicted from two
    trendlines, both of which are widely accepted in their own respective
    fields.  To detect the other worlds you need a reversible machine
    intelligence.  This requires two things: reversible nanotechnology and
    AI.
    
    1) Reversible nanoelectronics.  This is an straight-line extrapolation
    based upon the log(energy) / logic operation figures, which are
    projected to drop below kT in about 2020.  This trend has held good for
    50 years.  An operation that thermally dissipates much less than kT of
    energy is reversible.  (This implies that frictive or dissipative forces
    are insignificant by comparison with other processes.)  If more than kT
    of energy is released then, ultimately, new degrees of freedom are
    activated in the environment and the change becomes irreversible.
    
    2) AI.  Complexity of human brain = approx 10^17 bits/sec, based on the
    number of neurons (approx 10^10) per human brain, average number of
    synapses per neuron (approx 10^4) and the average firing rate (approx
    10^3 Hz).  Straight line projection of log(cost) / logic operation says
    that human level, self-aware machine intelligences will be commercially
    available by about 2030-2040.  Uncertainty due to present human-level
    complexity, but the trend has held good for 40 years.
    
    Assuming that we have a reversible machine intelligence to hand then the
    experiment consists of the machine making three reversible measurements
    of the spin of an electron (or polarisation of a photon).  (1) First it
    measures the spin along the z-axis.  It records either spin "up" or spin
    "down" and notes this in its memory.  This measurement acts just to
    prepare the electron in a definite state.  (2) Second it measures the
    spin along the x-axis and records either spin "left" or spin "right" and
    notes *this* in its memory.  The machine now reverses the entire x-axis
    measurement - which must be possible, since physics is effectively
    reversible, if we can describe the measuring process physically -
    including reversibly erasing its memory of the second measurement.  (3)
    Third the machine takes a spin measurement along the z-axis.  Again the
    machine makes a note of the result.  
    
    According to the Copenhagen interpretation the original (1) and final
    (3) z-axis spin measurements have only a 50% chance of agreeing because
    the intervention of the x-axis measurement by the conscious observer
    (the machine) caused the collapse of the electron's wavefunction. 
    According to many-worlds the first and third measurements will *always*
    agree, because there was no intermediate wavefunction collapse.  The
    machine was split into two states or different worlds, by the second
    measurement; one where it observed the electron with spin "left"; one
    where it observed the electron with spin "right".  Hence when the
    machine reversed the second measurement these two worlds merged back
    together, restoring the original state of the electron 100% of the time.
    
    Only by accepting the existence of the other Everett-worlds is this 100%
    restoration explicable.
    
    Q38  Why *quantum* gravity?
         ----------------------
    Many-worlds makes a very definite prediction - gravity must be
    quantised, rather than exist as the purely classical background field
    of general relativity.  Indeed, no one has conclusively directly
    detected (classical) gravity waves (as of 1994), although their
    existence has been indirectly observed in the slowing of the rotation
    of pulsars and binary systems.  Some claims have been made for the
    detection of gravity waves from supernova explosions in our galaxy, but
    these are not generally accepted.  Neither has anyone has directly
    observed gravitons, which are predicted by quantum gravity, presumably
    because of the weakness of the gravitational interaction.  Their
    existence has been, and is, the subject of much speculation.  Should,
    in the absence of any empirical evidence, gravity be quantised at all? 
    Why not treat gravity as a classical force, so that quantum physics in
    the vicinity of a mass becomes quantum physics on a curved Riemannian
    background?  According to many-worlds there *is* empirical evidence for
    quantum gravity.
    
    To see why many-worlds predicts that gravity must be quantised, let's
    suppose that gravity is not quantised, but remains a classical force. 
    If all the other worlds that many-worlds predicts exist then their
    gravitational presence should be detectable -- we would all share the
    same background gravitational metric with our co-existing quantum
    worlds.  Some of these effects might be undetectable.  For instance if
    all the parallel Earths shared the same gravitational field small
    perturbations in one Earth's orbit from the averaged background orbit
    across all the Everett-worlds would damp down, eventually, and remain
    undetectable.
    
    However theories of galactic evolution would need considerable
    revisiting if many-worlds was true and gravity was not quantised, since,
    according to the latest cosmological models, the original density
    fluctuations derive from quantum fluctuations in the early universe,
    during the inflationary era.  These quantum fluctuations lead to the
    formation of clusters and super-clusters of galaxies, along with
    variations in the cosmic microwave background (detected by Smoots et al)
    which vary in location from Everett-cosmos to cosmos.  Such fluctuations
    could not grow to match the observed pattern if all the density
    perturbations across all the parallel Everett-cosmoses were
    gravitationally interacting.  Stars would bind not only to the observed
    galaxies, but also to the host of unobserved galaxies.
    
    A theory of classical gravity also breaks down at the scale of objects
    that are not bound together gravitationally.  Henry Cavendish, in 1798,
    measured the torque produced by the gravitational force on two separated
    lead spheres suspended from a torsion fibre in his laboratory to
    determine the value of Newton's gravitational constant.  Cavendish
    varied the positions of other, more massive lead spheres and noted how
    the torsion in the suspending fibre varied.  Had the suspended lead
    spheres been gravitationally influenced by their neighbours, placed in
    different positions by parallel Henry Cavendishs in the parallel
    Everett-worlds, then the torsion would have been the averaged sum of all
    these contributions, which was not observed.  In retrospect Cavendish
    established that the Everett-worlds are not detectable gravitationally. 
    More recent experiments where the location of attracting masses were
    varied by a quantum random (radioactive) source have confirmed these
    findings. [W]
    
    A shared gravitational field would also screw up geo-gravimetric
    surveys, which have successfully detected the presence of mountains,
    ores and other density fluctuations at the Earth's surface.  Such
    surveys are not sensitive to the presence of the parallel Everett-Earths
    with different geological structures.  Ergo the other worlds are not
    detectable gravitationally.  That gravity must be quantised emerges as
    a unique prediction of many-worlds.
    
    [W]  Louis Witten _Gravitation: an introduction to current research_ 
         New York, Wiley (1962).
         _Essays in honor of Louis Witten on his retirement.  Topics on
         quantum gravity and beyond_: University of Cincinnati, USA, 3-4
         April 1992 / editors, Freydoon Mansouri & Joseph J. Scanio. 
         Singapore ; River Edge, NJ : World Scientific, c1993 ISBN 981021290
    
    Q39  Is linearity exact?
         -------------------
    Linearity (of the wavefunction) has been verified to hold true to better
    than 1 part in 10^27 [W].  If slight non-linear effects were ever
    discovered then the possibility of communication with, or travel to, the
    other worlds would be opened up.  The existence of parallel Everett-
    worlds can be used to argue that physics must be *exactly* linear, that
    non-linear effects will never be detected.  (See "Is physics linear" for
    more about linearity.)
    
    The argument for exactness uses a version of the weak anthropic
    principle and proceeds thus: the exploitation of slight non-linear
    quantum effects could permit communication with and travel to the other
    Everett-worlds.  A sufficiently advanced "early" civilisation [F] might
    colonise uninhabited other worlds, presumably in an exponentially
    spreading fashion.  Since the course of evolution is dictated by random
    quantum events (mutations, genetic recombination) and environmental
    effects (asteroidal induced mass extinctions, etc) it seems inevitable
    that in a minority, although still a great many, of these parallel
    worlds life on Earth has already evolved sapient-level intelligence and
    developed an advanced technology millions or even billions of years ago. 
    Such early arrivals, under the usual Darwinian pressure to expand, would
    spread across the parallel time tracks, if they had the ability,
    displacing their less-evolved quantum neighbours.
    
    The fossil record indicates that evolution, in our ancestral lineage,
    has proceeded at varying rates at different times.  Periods of rapid
    development in complexity (eg the Cambrian explosion of 530 millions
    years ago or the quadrupling of brain size during the recent Ice Ages)
    are interspersed with long periods of much slower development.  This
    indicates that we are not in the fast lane of evolution, where all the
    lucky breaks turned out just right for the early development of
    intelligence and technology.  Ergo none of the more advanced
    civilisations that exist in other worlds have ever been able to cross
    from one quantum world to another and interrupt our long, slow
    biological evolution.
    
    The simplest explanation is that physics is sufficiently linear to
    prevent travel between Everett worlds.  If technology is only bounded
    by physical law (the Feinberg principle [F]) then linearity would have
    to be exact.
    
    [F]  Gerald Feinberg.  _Physics and Life Prolongation_ Physics Today Vol
         19 #11 45 (1966). "A good approximation for such [technological]
         predictions is to assume that everything will be accomplished that
         does not violate known fundamental laws of science as well as  many
         things that do violate these laws."
    
    [W]  Steven Weinberg _Testing Quantum Mechanics_ Annals of Physics Vol
         194 #2 336-386 (1989) and _Dreams of a Final Theory_ (1992)
    
    Q40  Why can't the boundary conditions be updated to reflect my
         ----------------------------------------------------------
         observations in this one world?
         -------------------------------
    What is lost by this approach is a unique past assigned to each future. 
    If you time-evolve the world-we-now-see backwards in time you get a
    superposition of earlier starting worlds.  Similarly if you time evolve
    a single (initial) world forward you get a superposition of later
    (final) worlds.
    
    For example consider a photon that hits a half-silvered mirror and turns
    into a superposition of a transmitted and a reflected photon.  If we
    time-evolve one of these later states backwards we get not the original
    photon, but the original photon plus a "mirror image" of the original
    photon.  (Try the calculation and see.)  Only if we retain both the
    reflected and transmitted photons, with the correct relative phase, do
    we recover the single incoming photon when we time-reverse everything. 
    (The mirror image contributions from both the final states have opposite
    signs and cancel out, when they are evolved backwards in time to before
    the reflection event.)
    
    All the starting states have to have their relative phases coordinated
    or correlated just right (ie coherently) or else it doesn't work out. 
    Needless to say the chances that the initial states should be arranged
    coherently just so that they yield the one final observed state are
    infinitesimal and in violation of observed thermodynamics, which states,
    in one form, that correlations only increase with time.
    
    A1   References and further reading
         ------------------------------
    [1]  Hugh Everett III _The Theory of the Universal Wavefunction,
         Princeton thesis_ (1956?)
         The original and most comprehensive paper on many-worlds. 
         Investigates and recasts the foundations of quantum theory in
         information theoretic terms, before moving on to consider the
         nature of interactions, observation, entropy, irreversible
         processes, classical objects etc.  138 pages.  Only published in
         [5].
    [2]  Hugh Everett III _"Relative State" Formulation of Quantum
         Mechanics_ Reviews of Modern Physics Vol 29 #3 454-462, (July
         1957)  A condensation of [1] focusing on observation.
    [3]  John A Wheeler _Assessment of Everett's "Relative State"
         Formulation of Quantum Theory_, Reviews of Modern Physics Vol
         29 #3 463-465 (July 1957)  Wheeler was Everett's PhD
         supervisor.
    [4a] Bryce S DeWitt _Quantum Mechanics and Reality_ Physics Today,
         Vol 23 #9 30-40 (September 1970)  An early and accurate
         popularisations of Everett's work.  The April 1971 issue has
         reader feedback and DeWitt's responses.
    [4b] Bryce S DeWitt _The Many-Universes Interpretation of Quantum
         Mechanics_ in _Proceedings of the International School of Physics
         "Enrico Fermi" Course IL: Foundations of Quantum Mechanics_
         Academic Press (1972)
    [5]  Bryce S DeWitt, R Neill Graham eds _The many-worlds
         Interpretation of Quantum Mechanics_, Contains
         [1],[2],[3],[4a],[4b] plus other material.  Princeton Series
         in Physics, Princeton University Press (1973) ISBN 0-691-
         08126-3 (hard cover), 0-691-88131-X (paper back)  The
         definitive guide to many-worlds, if you can get hold of a
         copy, but now (1994) only available xeroxed from microfilm
         (ISBN 0-7837-1942-6) from Books On Demand, 300 N Zeeb Road,
         Ann Arbor, MI 48106-1346, USA.  Tel: +01-313 761 4700 or 800
         521 0600.
    [15] Frank J Tipler _The many-worlds interpretation of quantum mechanics
         in quantum cosmology_ in _Quantum Concepts of Space and Time_ eds
         Roger Penrose and Chris Isham, Oxford University Press (1986).  Has
         a discussion of Ockham's razor.
    On quantum theory, measurement and decoherence generally:
    [6]  John A Wheeler, Wojciech H Zurek eds _Quantum Theory and
         Measurement_ Princeton Series in Physics, Princeton University
         Press (1983) ISBN 0-691-08316-9.  Contains 49 classic
         articles, including [2], covering the history and development
         of interpretations of quantum theory. 
    [7a] Wojciech H Zurek _Decoherence and the Transition from the
         Quantum to the Classical_, Physics Today, 36-44 (October
         1991). The role of thermodynamics and the properties of large
         ergodic systems (like the environment) are related to the
         decoherence or loss of interference effects between superposed
         macrostates.
    [7b] Wojciech H Zurek _Preferred States, Predictability, Classicality,
         and the Environment-Induced Decoherence_  Progress of Theoretical
         Physics, Vol 89 #2 281-312 (1993)  A fuller expansion of [7a]
    [8]  Max Jammer _The Philosophy of Quantum Mechanics_ Wiley, New
         York (1974)  Almost every interpretation of quantum mechanics
         is covered and contrasted.  Section 11.6 contains a lucid
         review of many-worlds theories.
    [9]  Bethold-Georg Englert, Marlan O Scully, Herbert Walther _Quantum
         optical tests of complementarity_ Nature, Vol 351, 111-116 (9 May
         1991). Demonstrates that quantum interference effects are destroyed
         by irreversible object-apparatus correlations ("measurement"), not
         by Heisenberg's uncertainty principle itself.  See also _The
         Duality in Matter and Light_ Scientific American, (December 1994)
    [10] Murray Gell-Mann, James B Hartle _Quantum Mechanics in the Light
         of Quantum Cosmology_ Proceedings of the 3rd International
         Symposium on the Foundations of Quantum Mechanics (1989) 321-343. 
         They accept the Everett's decoherence analysis, and have extended
         it further.
    Tests of the Everett metatheory:
    [11] David Deutsch _Quantum theory as a universal physical theory_
         International Journal of Theoretical Physics, Vol 24 #1
         (1985).  Describes an experiment which tests for the existence
         of superpositions of *consciousness (in an AI).
    [16] David Deutsch _Three connections between Everett's interpretation
         and experiment_ Quantum Concepts of Space and Time, eds Roger
         Penrose and Chris Isham, Oxford University Press (1986).  Discusses
         a testable split observer experiment and quantum computing.
    On quantum computers:
    [12] David Deutsch _Quantum theory, the Church-Turing principle and the
         universal quantum computer_ Proceedings of the Royal Society of
         London, Vol. A400, 96-117 (1985).
    [13] David Deutsch _Quantum computational networks_ Proceedings of
         the Royal Society of London, Vol. A425, 73-90 (1989).
    [14] David Deutsch and R. Jozsa _Rapid solution of problems by
         quantum computation_ Proceedings of the Royal Society of
         London, Vol. A439, 553-558 (1992).
    [17] Julian Brown _A Quantum Revolution for Computing_ New Scientist,
         pages 21-24, 24-September-1994
    
    A2   Quantum mechanics and Dirac notation 
         ------------------------------------
    Note: this is a very inadequate guide.  Read a more comprehensive text
    ASAP.  For a more technical exposition of QM the reader is referred to
    the standard textbooks.  Here are 3 I recommend:
    
    Richard P Feynman _QED: the strange story of light and matter_ ISBN 0-
    14-012505-1.  (Requires almost no maths and is universally regarded as
    outstanding, despite being about quantum electrodynamics.)
    
    Richard P Feynman _The Feynman Lectures in Physics_ Volume III Addison-
    Wesley (1965) ISBN 0-201-02118-8-P.  The other volumes are worth reading
    too!
    
    Daniel T Gillespie _A Quantum Mechanics Primer: An Elementary
    Introduction to the Formal Theory of Non-relativistic Quantum Mechanics_ 
    (Takes an axiomatic, geometric approach and teaches all the Hilbert
    space stuff entirely by analogy with Euclidean vector spaces.  Not sure
    if it is still in print.)
    
    Quantum theory is the most successful theory of physics and chemistry
    ever.  It accounts for a wide range of phenomena from black body
    radiation, atomic structure and chemistry, which were very puzzling
    before quantum mechanics was first developed (c1926) in its modern form. 
    All theories of physics are quantum physics, with whole new fields, like
    the semiconductor and microchip technology, based upon the quantum
    effects.  This FAQ assumes familiarity with the basics of quantum theory
    and with the associated "paradoxes" of wave-particle duality.  It will
    not explain the uncertainty principle or delve into the significance of
    non-commuting matrix operators.  Only those elements of quantum theory
    necessary for an understanding of many-worlds are covered here.
    
    Quantum theory contains, as a central object, an abstract mathematical
    entity called the "wavefunction" or "state vector".  Determining the
    equations that describe its form and evolution with time is an
    unfinished part of fundamental theoretical physics.  Presently we only
    have approximations to some "correct" set of equations, often referred
    to whimsically as the Theory of Everything.
    
    The wavefunction, in bracket or Dirac notation, is written as |symbol>,
    where "symbol" labels the object.  A dog, for example, might be
    represented as |dog>.
    
    A general object, labelled "psi" by convention, is represented as |psi>
    and called a "ket".  Objects called "bra"s, written <psi|, may be formed
    from kets.  An arbitrary bra <psi'| and ket |psi> may be combined
    together to form the bracket, <psi'|psi>, or inner product, which is
    just a fancy way of constructing a complex number.  Amongst the
    properties of the inner product is:
    
       <psi'|(|psi1>*a_1 + |psi2>*a_2) = <psi'|psi1>*a_1 + <psi'|psi2>*a_2
    
    where the a_i are arbitrary complex numbers.  This is what is meant by
    saying that the inner product is linear on the right or ket side.  It
    is made linear on the left-hand or bra side by defining 
    
       <psi|psi'> = complex conjugate of <psi'|psi>
    
    Any ket may be expanded as:
    
      |psi> = sum |i>*<i|psi> 
               i
            = |1>*<1|psi> + |2>*<2|psi> + ...
    where the states |i> form an orthonormal basis, with <i|j> = 1 for i =
    j and = 0 otherwise, and where i labels some parameter of the object
    (like position or momentum).
    
    The probability amplitudes, <i|psi>, are complex numbers.  It is
    empirically observed, first noted by Max Born and afterwards called the
    Born interpretation, that their magnitudes squared represent the
    probability that, upon observation, that the value of the parameter,
    labelled by i, will be observed if the system is the state represented
    by |psi>.  It is also empirically observed that after observing the
    system in state |i> that we can henceforth replace the old value of the
    wavefunction, |psi>, with the observed value, |i>.  This replacement is
    known as the collapse of the wavefunction and is the source of much
    philosophical controversy.  Somehow the act of measurement has selected
    out one of the components.  This is known as the measurement problem and
    it was this phenomenon that Everett addressed.
    
    When a bra, <psi|, is formed from a ket, |psi>, and both are inner
    productted together the result, <psi|psi>, is a non-negative real
    number, called the norm of the vector.  The norm of a vector provides
    a basis-independent way of measuring the "volume" of the vector.
    
    The wavefunction for a joint system is built out of products of the
    components from the individual subsystems.  
    
    For example if the two systems composing the joint system are a cat and
    a dog, each of which may be in two states, alive or dead, and the state
    of the cat and the dog were *independent* of each other then we could
    write the total wavefunction as a product of terms. If
        |cat> = |cat alive> * c_a + |cat dead> * c_d
    and 
        |dog> = |dog alive> * d_a + |dog dead> * d_d
    then
        |dog+cat> = |cat>x|dog>           where x = tensor product
           =  (|cat alive> * c_a + |cat dead> * c_d)
            x (|dog alive> * d_a + |dog dead> * d_d)
           =    |cat alive> x |dog alive> * c_a * d_a 
              + |cat alive> x |dog dead> * c_a * d_d
              + |cat dead> x |dog alive> * c_d * d_a
              + |cat dead> x |dog dead> * c_d * d_d
           =    |cat alive, dog alive> * c_a * d_a 
              + |cat alive, dog dead> * c_a * d_d
              + |cat dead, dog alive> * c_d * d_a
              + |cat dead, dog dead> * c_d * d_d
    
    More generally, though, we states of subsystems are not independent of
    each other we have to use a more general formula:
    
       |dog+cat> = |cat alive, dog alive> * a_1
                 + |cat alive, dog dead> * a_2
                 + |cat dead, dog alive> * a_3
                 + |cat dead, dog dead> * a_4
    
    This is sometimes described by saying that the states of the cat and dog
    have become entangled.  It is fairly trivial to define the state of the
    cat and the dog with respect to each other.  For instance we could re-
    express the above expansion with respect to the cat's two states as:
    
       |dog+cat> = 
            |cat alive>x(|dog alive> * a_1 + |dog dead> * a_2)
          + |cat dead>x(|dog alive> * a_3 + |dog dead> * a_4)
    
    We term the state of the dog the *relative state* (Everett invented this
    terminology) with respect to the cat, specifying which cat state (alive
    or dead) we are interested in.  This thus the dog's relative state with
    respect to the cat alive state is:
    
          (|dog alive> * a_1 + |dog dead> * a_2)/sqrt(|a_1|^2 + |a_2|^2)
    
    where the sqrt term has been added to normalise the relative state.
    


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