Many Worlds Quantum Theory

© Michael Clive Price, February 1995

FAQ

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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