From price@price.demon.co.uk Fri Sep 30 08:22:00 1994
Date: Wed, 28 Sep 1994 17:48:00 GMT
From: Michael Clive Price
Reply to: extropians@extropy.org
To: Extropians@extropy.org
Subject: SCI: Many-Worlds FAQ
I've had a few requests for this, so here goes:
*********************
Summary: Frequently Asked Questions about the Many-Worlds
or Relative State Formulation of Quantum Theory.
Answers compiled by Michael Clive Price
Comments to price@price.demon.co.uk
Last Modified: 28-September-1994
Contents:
1 What are the problems with quantum theory?
2 What is the Copenhagen interpretation?
3 What is many-worlds?
4 What is a "world"?
5 What is a measurement?
6 Why do worlds split?
7a When do worlds split?
7b When does Schrodinger's cat split?
8a What is sum-over-histories?
8b What is many-histories?
9 How many worlds are there?
10 Is many-worlds a local theory?
11 Is many-worlds a deterministic theory?
12 Is many-worlds a relativistic theory?
13 Is many-worlds (just) an interpretation?
14 What are the alternatives?
15 Is many-worlds testable?
16 Could previously separate worlds diverge rather than split?
17 What is many-minds?
18 Does many-worlds violate Ockham's Razor?
19 Does the multiplication of worlds violate conservation of energy?
20 How do probabilities emerge within many-worlds?
21 Does many-worlds allow free-will?
22 Why am I in this world and not another?
23 Can wavefunctions collapse?
24 Is physics linear?
25 Can we determine what other worlds there are?
26 Who was Everett?
27 Who believes in many-worlds?
28 Does the EPR experiment prohibit locality?
29 Is Everett's relative state theory the same as many-worlds?
31 References and further reading
32 Mini-guide to notation and quantum mechanics
Q1 What are the problems with quantum theory?
------------------------------------------
Quantum theory is the most successful description of microscopic systems
like atoms and molecules ever, yet often it is not applied to larger,
classical systems, like observers or the entire universe. Many
scientists and philosophers are unhappy with the theory because it seems
to require a fundamental quantum-classical divide. Einstein, for
example, and despite his early contributions to the subject, was never
reconciled with assigning the act of observation a physical
significance, which QM requires. This contradicts the reductionist
ethos that, amongst other things, observations should emerge only as a
consequence of an underlying physical theory and not be present in the
axioms, as they are in the Copenhagen interpretation. Yet the
Copenhagen interpretation is the most popular interpretation of quantum
mechanics. (See "What is the Copenhagen interpretation?")
Q2 What is the Copenhagen interpretation?
--------------------------------------
An unobserved system, according to the Copenhagen interpretation of
quantum theory, evolves in a deterministic way determined by a wave
equation. An observed system changes in a random fashion,
instantaneously, with the probability of any particular outcome given
by the Born formula, determined by the wavefunction. This is known as
the collapse of the wavefunction. The problems with this approach are:
(1) The collapse is an instantaneous process across an extended
region ("non-local"). This is in conflict with relativity,
which states that no processes can be transmitted faster than
the speed of light. (Nevertheless it has been shown that no
information can be transmitted faster than light by the
collapse process).
(2) The idea of an observer having an effect on microphysics is
repugnant to reductionism and smacks of a return to pre-scientific
notions of vitalism. Copenhagenism is a return to the old
vitalist notions that life is somehow different from other matter,
operating by different laws from inanimate matter. The collapse
is triggered by an observer, yet no definition of what an
"observer" is available, in terms of an atomic scale description,
even in principle.
For these reasons the view has generally been adopted that the
wavefunction associated with an object is not a real "thing", but merely
represents our *knowledge* of the object. This approach was developed
by Bohr and others, mainly at Copenhagen in the late 1920s. When we
perform an measurement or observation of an object we acquire new
information and so adjust the wavefunction as we would boundary
conditions in classical physics to reflect this new information. This
stance means that we can't answer questions about what's actually
happening, all we can answer is what will be the probability of a
particular result if we perform a measurement. This makes a lot of
people very unhappy since it provides no model for the object.
It should be added that there are other, less popular, interpretations
of quantum theory, but they all have their own drawbacks, which are
widely reckoned more severe. Generally speaking they try to find a
mechanism that describes the collapse process or add extra physical
objects to the theory, in addition to the wavefunction. In this sense
they are more complex. (See "Is there any alternative theory?")
Q3 What is many-worlds?
--------------------
AKA as the Everett, relative-state, many-histories or many-universes
interpretation. Dr Hugh Everett III, its originator, called it the
relative-state metatheory or the theory of the universal wavefunction
[1], but, after DeWitt [4a],[5], it is generally called many-worlds
nowadays.
Many-worlds comprises of two assumptions and some consequences. The
assumptions are quite modest:
1) The metaphysical assumption: That the wavefunction does not merely
encode the information about an object, but has an observer-
independent objective existence. For an N-particle system the
wavefunction is a complex-valued field in a 3-N dimensional space.
In quantum field theory the state vector spans a space of an
indeterminate number of dimensions.
2) The physical assumption: The wavefunction obeys some standard
linear deterministic wave equation at all times. The observer
plays no special role in the theory and, consequently, there is no
collapse of the wavefunction. Measurement and observation are
modelled by applying the wave equation to the joint subject-object
system. For non-relativistic systems the Schrodinger wave equation
is a good approximation to reality. (See "Is many-worlds a
relativistic theory?" for the more general case.)
The rest of the theory is working out consequences of the above
assumptions. Some consequences are:
1) That each measurement causes a decomposition or decoherence of the
universal wavefunction into non-interacting and non-interfering
branches or worlds. History forms a branching tree which
encompasses all the possible outcomes of each interaction. (See
"Why do worlds split?" and "When do worlds split?") Every
historical what-if compatible with the initial conditions and
physical law is realised.
2) That the conventional statistical Born interpretation of the
amplitudes in quantum theory is *derived* from within the theory
rather than having to be *assumed* as an additional axiom. (See
"How do probabilities emerge within many-worlds?")
Many-worlds is a re-formulation of quantum theory [1], published in 1957
by Dr Hugh Everett III [2], which treats the process of observation or
measurement entirely within the wave-mechanics of quantum theory, rather
than an input an as additional assumption, as in the Copenhagen
interpretation. Everett considered the wavefunction a real object.
Many-worlds is a return to the classical, pre-quantum view of the
universe in which all the mathematical entities of a physical theory are
real. For example, the electromagnetic fields of James Clark Maxwell
or the atoms of Dalton, were considered as real objects in classical
physics. Everett treats the wavefunction in a similar fashion. Everett
also assumed that the wavefunction obeyed the same wave equation during
observation or measurement as at all other times. This is the central
assumption of many-worlds: that the wave equation is obeyed universally
and at all times.
Everett discovered that the new, simpler theory - which he named the
"relative state" formulation - predicts that interactions between two
(or more) macrosystems typically split the joint system into a
superposition of products of relative states. The states of the
macrosystems are henceforth correlated with each other. Each element
of the superposition - each a product of subsystem states - evolves
independently of the other elements in the superposition. The states
of the macrosystems, by becoming correlated or entangled, meaning that
it no longer possible to speak the state of one system in isolation from
the other subsystems. Instead we are forced to only speak of the
relative states of the subsystems, with respect to the other subsystems.
Specifying the state of one subsystem leads to the state of the other
subsystems. In this sense the states of the subsystems are determined
only relative to each other, hence Everett's original designation of his
theory.
If one of the systems is an observer and the interaction an observation
then observer has been split into a number of copies, each copy
observing just one of the possible results of a measurement and unaware
of the other results and its own observer-copies. Interactions between
systems and their environments, including communication between
different observers in the same worlds, transmits the correlations,
inducing local splitting or decoherence of branches of the universal
wavefunction [7],[10]. Thus the entire world is split, quite rapidly,
into a host of mutually unobservable but equally real worlds.
According to many-worlds all the possible outcomes of a quantum
interaction are realised. The wavefunction, instead of collapsing at
the moment of observation, carries on evolving in a deterministic
fashion, embracing all possibilities within it. All outcomes exist
simultaneously but do not interact further with each other, each world
having split into mutually unobservable but equally real worlds or
branches of the universal wavefunction.
Q4 What is a "world"?
------------------
Loosely speaking a "world" is a complex, partially closed set of
interacting sub-systems which don't significantly interfere with the
other elements in a quantum superposition. Any complex system and its
coupled environment, with a large number of internal degrees of
freedom, counts as a world. An observer, with internal irreversible
processes, counts as a complex system. In terms of the wavefunction,
a world is a decohered branch of the universal wavefunction, which
represents a single macrostate. The worlds all exist simultaneously in
a non-interacting linear superposition.
Sometimes "worlds" are called "universes", but more usually this is
reserved the totality of worlds, or "histories" (Gell-Mann/Hartle's
phrase, see "What is many-histories?").
Q5 What is a measurement?
----------------------
A measurement is an interaction between subsystems that triggers an
amplification process, typically within an object (which we often
designate as the measuring apparatus) with many internal degrees of
freedom, leading to a change in the higher-level structure of the object
(which might be the recording apparatus). The trigger is sensitive to
some (often microphysical) parameter of the one of the subsystems, which
we designate the measured system. Eg the detection of a charged
particle by a Geiger counter leads to the generation of a "click". The
absence of a charged particle does not generate a click. The measured
system is the charged particle. The interaction is with those elements
of the charged particle's wavefunction that passes *between* the charged
detector plates, triggering the amplification process (an irreversible
electron cascade or avalanche), which is ultimately converted to a
click.
A measurement, by this definition, does not require the presence of an
observer.
Q6 Why do worlds split?
---------------------
Worlds, or branches of the universal wavefunction, split when different
components of a quantum superposition "decohere" from each other [7],
[10]. Decoherence refers to the loss of coherency or absence of
interference effects between the elements of the superposition. For two
components or worlds to interfere with each other all the atoms,
subatomic particles, photons etc, in each world, have to be in the same
state, usually in the same place. For microscopic, small systems it is
quite possible for all their components to match 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 point
for an interference pattern to form, because only the single particle
has been split. For more complex systems overlapping becomes harder
because all the constituent particles have to overlap with their
counterparts simultaneously. For a macroscopically sized system such
future coincidence of positions in all the components is virtually
impossible.
Irreversible processes, in particular, will always destroy any
possibility of interference effects being restored in the future. An
irreversible process is one in, or linked to, a system with a large
number of internal, unconstrained degrees of freedom. Once the 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 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 interfere again, are unaware of each other. From the
point of view of the complex systems they have 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 dimensionality available to
the system. Even the quite small amounts of energy released by a
frictive process (which is irreversible) 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. This is, therefore, a very effective destroyer of
coherency. In many ways the environment can be regarded as performing
a succession of measurement-like interactions upon the system, inducing
associated system splits.
Q7a When do worlds split?
---------------------
Worlds irrevocably "split" at the sites of measurement-like interactions
associated with thermodynamically irreversible processes. An
irreversible process will always produce decoherence which splits
worlds. (see "Why do worlds split?", [7], [10] and "When does
Schrodinger's cat split?" for a concrete example.)
In the example of a Geiger counter and a charged particle (see "What is
a measurement?") after the particle has passed the counter one world
contains the clicked counter and that portion of the particle's
wavefunction which passed though the detector. The other world contains
the unclicked counter with the particle's wavefunction with a "shadow"
cast by the counter in the particle's wavefunction. The Geiger counter
splits when the amplification process became irreversible.
The splitting is local (ie originally in the region of the Geiger
counter in our example) and is transmitted causally to more distant
systems (see "Is many-worlds a local theory?" and "Does the EPR
experiment prohibit locality?"). The precise moment/location of the
split is not sharply defined due to the subjective nature of
irreversibility, but can be considered complete when much more than kT
of energy has been released in an uncontrolled fashion into the
environment. (The event has become irreversible.)
In the language of thermodynamics the decay of the atom and the
amplification of its detection by a Geiger counter, the release of the
cyanide and the death of the cat are all irreversible events. These
events have caused the decoherence (see "Why do worlds split?") of the
different branches of the wavefunction of the cat + device + box.
Decoherence [7] occurs when irreversible macro-level events take place
and the macrostate description of an object admits no single
description. (See "Does the EPR experiment prohibit locality?" for a
fully worked out example of this.) 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 forward-time-branching, following the increase with
entropy. Like all irreversible processes, though, there are exceptions
even at the coarse-grained level and worlds will occasionally fuse. A
necessary, although not necessarily sufficient, precondition for fusing
is for all records, memories etc that discriminate between the pre-fused
worlds or histories be lost.
Q7b 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 radioactive decay is
detected. After a while a human opens the box to see if the cat is
alive or dead. According to the CI the cat was neither alive nor dead
until the box was opened, whereupon the wavefunction of the cat
collapsed into one of the two alternatives. The paradox, according to
Schrodinger, is that the cat presumably knew if it was alive *before*
the box was opened. According to many-worlds the device was split into
two states (cyanide released or not) by the radioactive decay. As the
device/cyanide interacts with the cat the cat is split into two states
(dead or alive). From the surviving cat's point of view it occupies a
different world from its unlucky and late copy. The human is split into
two copies only when the box is opened and is altered by the state of
the cat.
The cat splits when the cat (irreversibly) dies or lives. The human
splits when the human opens the box. The alive cat has no idea that
human has split, any more than it is aware that there is a dead cat in
the just split off world. The human can deduce, after the event by
examining the cyanide mechanism, that the cat split prior to opening the
box.
Q8a What is sum-over-histories?
---------------------------
The sum-over-histories or the path integral formalism was developed by
Feynman in the 1940s [F] as an alternative interpretation of quantum
mechanics, alongside Schrodinger's wave picture and Heisenberg's matrix
mechanics, for calculating transition amplitudes. All three approaches
are mathematically equivalent, but the PI formalism offers some
interesting additional insights into many-worlds.
In the PI picture the single particle wavefunction at (x',t') is built
up of contributions of all possible paths from (x,t), where each path's
contribution weighted by a (phase) factor of exp(i*Action[path]/hbar)
* wavefunction at (x,t), summed, in turn, over all values of x. The
Action[path] is the time-integral of the lagrangian (roughly: the
kinetic minus the potential energy) along the path from (x,t) to
(x',t'). The final expression is thus sum or integral over all paths,
irrespective of any classical dynamical constraints. For N-particle
systems the principle is the same, except that the paths cover a 3-N
space.
Feynman developed his PI formalism further for his work on quantum
electrodynamics, QED, with his Feynman diagrams, in parallel with
Schwinger and Tomonoga who 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.
It is quite natural, when analysing systems from the PI point of view,
to think of the particle exploring every possible intermediate
configuration between the specified start and end states. For this
reason the technique is often referred to as "sum-over-histories".
Since we do not occupy a privileged moment in history it is natural to
wonder if alternative histories are contributing equally to transition
amplitudes in the future, and therefore that each possible history has
an equal reality. Perhaps we shouldn't be surprised that Feynman,
therefore, is on record as believing in many-worlds. (See "Who believes
in many-worlds?") What is surprising is that Everett developed his
many-worlds theory entirely from the Schrodinger viewpoint without any
detectable influence from Feynman's work, despite sharing the same
thesis supervisor, John A Wheeler.
[F] Richard P Feynman _Space-time approach to non-relativistic quantum
mechanics_ Reviews of Modern Physics, Vol 20: 267-287 (1948)
Q8b What is many-histories?
-----------------------
There is considerable linkage between thermodynamics and many-worlds,
explored in the "decoherence" views of Zurek [7] and Gell-Mann and
Hartle [10], Everett [1] and others [4b].
Gell-Mann and Hartle have extended the role of decoherence in defining
the Everett worlds, or histories in their nomenclature. They call their
approach the "many-histories" approach, where each "coarse-grained or
classical history" is associated with a unique time-ordered sequence of
sets of irreversible events, including measurements, records,
observations and the like. (Fine-grained histories effectively relax
the irreversible criterion.) Physically the many-histories approach is
isomorphic to Everett's many-worlds, although Gell-Mann and Hartle
choose not to accept Everett's metaphysical stance that each history
corresponds to an element of reality.
The worlds split or "decohere" from each other when irreversible events
occur. (See "Why do worlds split?" and "When do worlds split?".)
Correspondingly many-histories defines a multiply-connected hierarchy
of classical histories where each classical history is a "child" of any
parent history which has only a subset of the child defining
irreversible events and a parent of any history which has a superset of
such events. Climbing up the tree from child to parent moves to
progressively coarser grained consistent histories until eventually the
top is reached where the history has *no* defining events (and thus
consistent with everything!). This is Everett's universal wavefunction.
The bottom of the coarse-grained tree terminates with the maximally
refined set of decohering histories. The classical histories each have
a probability assigned to them and probabilities are additive in the
sense that the sum of the probabilities associated a set classical
histories is equal to the probability associated with the unique parent
history defined by the set. (Below the maximally refined classical
histories are the fine grained or quantum histories, where probabilities
are no longer additive and different histories significantly interfere
with each other. The bottom level consists of complete microstates,
which fully specified states.)
Q9 How many worlds are there?
--------------------------
It so happens that we can use the thermodynamic Planck-Boltzmann
relationship to count the branches at each splitting, at the lowest,
maximally refined level of Gell-Mann's many-histories tree (See "What
is many-histories?"). The bottom level consists of microstates which
can be counted by the formula W = exp (S/k), where S = entropy, k =
Boltzmann's constant (approx 10^22 Joules/Kelvin) and W = number of
worlds or macrostates. The number of coarser grained worlds is lower,
but still increasing with entropy by the same ratio, ie the number of
worlds a single worlds splits into at the site of an irreversible event
is exp(dS/k), where dS = entropy of the defining event. Because k is
very small a great many worlds split off at each macroscopic event.
Q10 Is many-worlds a local theory?
------------------------------
The simplest way to see that the many-worlds metatheory is local is to
note that it requires that the wavefunction obey some relativistic wave
equation, the exact form of which is currently unknown, but which is
presumed to be locally Lorentz invariant at all times and everywhere.
This is equivalent to imposing the requirement that locality is enforced
at all times and everywhere. Ergo many-worlds is a local theory.
Another way of seeing this is examine how macrostates evolve.
Macrostates descriptions of objects evolve in a local fashion. Worlds
split as the macrostate description locally divides inside the light
cone of the triggering event. Thus the splitting is a local process,
transmitted causally at light or sub-light speeds. (See "Does the EPR
experiment prohibit locality?" for more details and "When do worlds
split?")
Q11 Is many-worlds a deterministic theory?
--------------------------------------
Yes, many-worlds is a deterministic theory, since the wavefunction obeys
a deterministic wave equation at all times. All possible outcomes of
a measurement or interaction are embedded within the universal
wavefunction although each observer, split by acts of observation, is
only aware of single outcomes due to the linearity of the wave equation.
The world appears indeterministic, with the usual probabilistic collapse
of the wavefunction, but at the objective level which includes all
outcomes determinism is restored.
Some people are under the impression that the only motivation for many-
worlds is a desire to return to a deterministic theory of physics. This
is not true. As Everett pointed out, the objection with the standard
Copenhagen interpretation is not the indeterminism per se, but that
indeterminism occurs only with the intervention of an observer, when the
wavefunction collapses.
Q12 Is many-worlds a relativistic theory?
-------------------------------------
It is trivial to relativise many-worlds because all relativistic
theories of physics are still quantum theories with linear
wavefunctions. There are three or more stages to developing a fully
quantum relativistic theory. Simplifying slightly gives:
First quantisation: the wavefunction is a complex field which evolves
in 3N dimensions which represent N particles. The wavefunction is a
solution of either the many-particle Schrodinger, Dirac or Klein-Gordon
equation or some other wave equation.
Second quantisation: AKA quantum field theory, which handles the
creation and destruction of particles by quantising fields as well as
particles. (Each particle type corresponds to a field, in QFT. Eg the
electromagnetic field's particle is the photon, but the number of
particles involved is indeterminate.) Again many-worlds has no problems
handling QFT. The wavefunction of a collection of particles and fields
exists in a Fock space, where the number of dimensions varies from
component to component.
Third quantisation. The gravitational metric is quantised, along with
(perhaps) the topology of space-time. The physics of this is
incomplete, but there is no reason for thinking that many-worlds can't
be extended to cover this as well. (One of the original motivations of
Everett's scheme was to provide a system for quantizing the
gravitational field within quantum cosmology to yield a complete
description of the universe.)
Q13 Is many-worlds (just) an interpretation?
----------------------------------------
No, for four reasons:
First, many-worlds has testable implications (see "Is many-worlds
testable?") and interpretations, generally, do not have testable
differences from each other.
Second, the mathematical structure of many-worlds is not isomorphic to
other formulations of quantum mechanics like the Copenhagen
interpretation or Bohm's hidden variables. The Copenhagen
interpretation does not contain those elements of the wavefunction that
correspond to the other worlds. Bohm's hidden variables contain
particles, in addition to the wavefunction. Therefore neither theory
is isomorphic to each other or many-worlds and are not, therefore,
merely rival "interpretations".
Third, there is no scientific, reductionistic alternative to many-
worlds. All the other theories fail for logical reasons. (See "Is
there any alternative theory?")
Four, the interpretative side of many-worlds, like the subjective
probabilistic elements, are derived from within the theory, rather than
added in by assumption, as in the conventional approach. (See "How do
probabilities emerge within many-worlds?")
Many-Worlds should really be described as a theory or, more precisely,
a metatheory, as Everett pointed out, since it makes statements that are
applicable across a range of theories. Many-worlds is the unavoidable
implication of any quantum theory which obeys some type of wave
equation, linear with respect to the wavefunction it operates on.
Q14 What are the alternatives?
--------------------------
There is no other quantum theory, besides many-worlds, that is
scientific and free of internal inconsistencies, that I am aware of.
Briefly here are the defects of the most popular alternatives:
1) Copenhagen Interpretation. Postulates that the observer obeys
different 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 not real then
the theory supplies no model of reality. (See "What are the
problems with quantum theory?")
2) Hidden Variables [B]. Explicitly non-local. Bohm accepts that all
the branches of the universal wavefunction exist. Like Everett
Bohm held that the wavefunction is real complex-valued field which
never collapses. In addition he postulated that there were
particles that move under the influence of a non-local "quantum-
potential" derived from the wavefunction, in addition to the
classical potential. The action of the quantum-potential is such
that the particles are affected by only one of the branches of the
wavefunction. (Bohm derives what is essentially a decoherence
argument to show this, see section 7,#I [B]).
The implicit, unstated assumption made by Bohm is that only the
single branch of wavefunction associated with particles can contain
self-aware observers, whereas Everett makes no such assumption.
Most of Bohm's adherents do not seem to understand (or even be
aware of) Everett's criticism, section VI [1], that the hidden-
variable particles are not observable since the wavefunction alone
is sufficient to account for all observations. The particles can,
therefore, be discarded, along with the guiding quantum-potential,
yielding a theory isomorphic to many-worlds, without affecting any
experimental results.
[B] David J Bohm _A suggested interpretation of the quantum theory
in terms of "hidden variables" I and II_ Physical Review Vol
85 #2 166-193 (1952)
3) Quantum Logic. Undoubtedly the most extreme of all attempts to
solve the QM measurement problem. Apart from abandoning one or
other of the classical tenets of logic these theories are all
unfinished (presumably because of internal inconsistencies). Also
it is unclear why different types of logic apply on different
scales.
4) Extended Probability [M]. A bold theory in which the concept of
probability is "extended" to include complex values [Y]. Whilst
quite daring, I am not sure if this is logically permissable, being
in conflict with the relative frequency notion of probability, in
which case it suffers from the same criticism as quantum logic.
Also it is unclear, to me anyway, how the resultant notion of
"complex probability" differs from the "probability amplitude" and
thus why we are justified in collapsing the complex probability as
if it were a classical probability.
[M] W Muckenheim _A review of extended probabilities_ Physics
Reports Vol 133 339- (1986)
[Y] Saul Youssef _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 forms between the first single slit and one of the
double slits and another transaction forms between the same double
slit and the point of screen where the photon lands.
[C] John G Cramer _The transactional interpretation of quantum
mechanics_ Reviews of Modern Physics Vol 58 #3 647-687 (1986)
6) many-minds. Despite its superficial similarities with many-worlds
this is actually a very unphysical, non-operational theory. (See
"What is many-minds?")
7) Non-linear theories in general. So far no non-linear theory has
any accepted experimental support, whereas many have failed
experiment. (See "Is physics linear?")
Q15 Is many-worlds testable?
------------------------
Yes, it is. There are two forms of tests: retrodictions (theory follows
data) and predictions (data follows theory).
A) A retrodiction occurs when already gathered data is accounted for by
a later theoretical advance in a more convincing fashion. The advantage
of a retrodiction over a prediction is that the data more likely to be
free of experimenter bias. An example of a retrodiction is the
perihelion shift of Mercury which Newtonian mechanics plus gravity was
unable, totally, to account for whilst Einstein's general relativity
made short work of it.
Many-worlds retrodicts all the peculiar properties of the (apparent)
wavefunction collapse in terms of decoherence. (See "Can wavefunctions
collapse?", "When do worlds split?" & "Why do worlds split?") No other
quantum theory has yet accounted for this behaviour scientifically.
(See "What are the alternatives?")
B) A prediction occurs when a theory suggests new phenomena.
Many-Worlds predicts that the Everett-worlds do not interact with each
other, because of the presumed linearity of the wave equation. However
worlds *do* interfere with each other, and this enables the theory to
be tested. (Interfere and interact mean different things in quantum
mechanics. See a guide to QM.)
According to many-worlds worlds split with the operation of every
thermodynamically irreversible process. The operation of our minds are
irreversible, carried along for the ride, and divide with the worlds.
Normally, therefore, this splitting is undetectable to us. To detect
the splitting we need to set an up experiment where a mind is split but
the world *isn't*. We need a reversible mind.
The general consensus in the literature [11], [16] is that the
experiment to detect other worlds will doable by about mid-21st century.
That date is predicted from two trendlines, both of which are widely
accepted in their own respective fields. To detect the other worlds you
need a reversible machine intelligence. This requires two things:
reversible nanotechnology and AI.
1) Reversible nanoelectronics. This is an straight-line extrapolation
based upon the log(energy) / logic operation figures, which are
projected to drop below kT in about 2020. This trend has held good for
50 years. An operation that dissipates much less than kT of energy is
reversible. (This implies that frictive or dissipative forces are
absent.) If more than kT of energy is released then, ultimately, new
degrees of freedom are activated in the environment and the change
becomes irreversible.
2) AI. Complexity of human brain = approx 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 measurements of the spin
of an electron (or polarisation of a photon). (1) First it measures the
spin along the z-axis. It records either spin "up" or spin "down" and
notes this in its memory. This measurements acts just to prepare the
electron in a definite state. (2) Second it measures the spin along the
x-axis and records either spin "left" or spin "right" and notes *this*
in its memory. The machine now reverses the entire x-axis measurement,
including reversibly erasing its memory of the second measurement. (3)
Third the machine takes a spin measurement along the z-axis. Again the
machine makes a note of the result.
According to the Copenhagen interpretation the original (1) and final
(3) z-axis spin measurements have only a 50% chance of agreeing because
the intervention of the x-axis measurement by the conscious observer
(the machine) caused the collapse of the electron's wavefunction.
According to many-worlds the first and third measurements will *always*
agree, because there was no intermediate wavefunction collapse. The
machine was split into two states or different worlds, by the second
measurement; one where it observed the electron with spin "left"; one
where it observed the electron with spin "right". Hence when the
machine reversed the second measurement these two worlds merged back
together, restoring the original state of the electron 100% of the time.
Q16 Could previously separate worlds diverge rather than split?
-----------------------------------------------------------
This is definitely not permissable in many-worlds. Worlds do not exist
in a quantum superposition independently of each other before they
decohere or split. The splitting is a physical process, grounded in the
dynamical evolution of the wave vector, not a matter of
philosophical/mental convenience (see "Why do worlds split?" and "When
do worlds split?") If you try to treat the worlds as pre-existing and
separate then the maths all comes out wrong. Also the divergence theory
stops being deterministic, in contradiction to the wave equations which
are deterministic, since we have a
AAAAAAAAAAAAAAABBBBBBBBBBBBBBB --------------> time
Worlds diverge
AAAAAAAAAAAAAAACCCCCCCCCCCCCCC
situation, rather than:
BBBBBBBBBBBBBBB
B
AAAAAAAAAAAAAA Worlds splitting
C
CCCCCCCCCCCCCCC
Additionally the divergence model has to explain why:
AAAAAAAAAAAAAAABBBBBBBBBBBBBBB
AAAAAAAAAAAAAAABBBBBBBBBBBBBBB
doesn't happen! This false divergence model, at the mental level, seems
favoured by adherents of many-minds. (See "What is many-minds?")
Q17 What is many-minds?
------------------
many-minds proposes, as an extra fundamental axiom, that an infinity of
separate minds or mental states be associated with each single brain
state. When the single physical brain state is split into a quantum
superposition by a measurement the associated minds are thought of as
diverging rather than splitting. The motivation for this brain-mind
dichotomy seems purely to avoid talk of minds splitting and talk instead
about the divergence of pre-existing separate mental states. There is
no physical basis for this interpretation, which is incapable of an
operational definition. Indeed the divergence model for physical
systems is specifically not permitted in many-worlds. Many-minds seems
to be proposing that minds follow different rules than matter. (See
"Could previously separate worlds diverge rather than split?")
In many-minds the role of the conscious observer is accorded special
status, with its fundamental axiom about infinities of minds, and as
such is philosophically opposed to many-worlds, which seeks to remove
the observer from any privileged role in physics. (Many-minds was co-
invented by David Albert, who has, apparently, since abandoned it. See
Scientific American July 1992 page 80 and contrast with April 94.)
The two theories should not be confused.
Q18 Does many-worlds violate Ockham's Razor?
---------------------------------------
William of Ockham, 1285-1349(?) English philosopher and one of the
founders of logic, proposed a maxim for judging theories which says that
hypotheses should not be multiplied beyond necessity. This is known as
Ockham's razor and is interpreted, today, as meaning that to account for
any set of facts the simplest theories are to be preferred over more
complex ones. Many-worlds is viewed as unnecessarily complex, by some,
by requiring the existence of a multitude of worlds to explain what we
see, at any time, in just one world.
This is to mistake what is meant by "complex". Here's an example.
Analysis of starlight reveals that starlight is very similar to faint
sunlight, with spectroscopic absorption and emission lines. Assuming
the universality of physical law we are led to conclude that other stars
and worlds are scattered, in great numbers, across the cosmos. The
theory that "the stars are distant suns" is the simplest theory and so
to be preferred by Ockham's Razor to other geocentric theories.
Similarly many-worlds is the simplest and most economical theory because
it proposes that same laws of physics apply to animate observers as
inanimate objects. The multitude of worlds predicted by the theory is
not a weakness for many-worlds, any more than stars are for astronomy,
since the non-interacting worlds emerge from a simpler theory.
(As an historical aside it is worth noting that Ockham's razor was also
falsely used to argue in favour of the older heliocentric theories
*against* Galileo's notion of the vastness of the cosmos. The notion
of vast empty interstellar spaces was too uneconomical to be believable.
Again they were confusing the notion of vastness with complexity [15].)
Q19 Does the multiplication of worlds violate conservation of energy?
-----------------------------------------------------------------
First, the law conservation of energy is based on observations within
each world. All observations within each world are consistent with
conservation of energy, therefore energy is conserved.
Second, and more precisely, conservation of energy, in QM, is formulated
in terms weighted averages or of expectation values. Conservation of
energy is expressed by saying that the time derivative of the
expectation of the energy operator vanishes. This statement can be
scaled up to include the whole universe. Each world has an approximate
energy, but the energy of (any subset of) the total wavefunction
involves summing over each world, weighted with its probability measure.
This weighted sum is a constant. So energy is conserved within each
world and across the totality of worlds.
One way of viewing this result - that observed conserved quantities are
conserved across the totality of worlds - is to note that new worlds are
not created by the action of the wave equation, rather existing worlds
are split into successively "thinner" and "thinner" slices, as measured
in the Hilbert space.
Q20 How do probabilities emerge within many-worlds?
-----------------------------------------------
Everett demonstrated [1],[2] that observations in each world obey all
conventional statistical laws predicted by the probabilistic Born
interpretation by showing that the Hilbert space's inner product or norm
supplies a unique measure or "volume" to each world or set of worlds.
The norm of the set of worlds where experiments contradict the Born
interpretation (non-random or maverick worlds) vanishes in the limit as
the number of probabilistic trials goes to the limit. Vectors with zero
norm, where probability breaks down, don't exist (see below), thus we,
as observers, observe the familiar predictions of quantum theory
expressed as probabilistic events.
Strictly speaking Everett did not prove that the usual statistical laws
of the Born interpretation would hold true for all observers in all
worlds. He merely showed that no other statistical laws would hold true
and asserted the vanishing of the Hilbert space volume of the set of
non-random worlds. DeWitt (with Graham) later published a longer
*derivation* of Everett's assertion [4a],[4b]. What Everett asserted
and DeWitt derived is that the collective norm of all the maverick
worlds, as the number of trials goes to infinity, vanishes. Since the
only vector in a Hilbert space with vanishing norm is the null vector
(a defining axiom of a Hilbert space) this is equivalent to saying that
non-randomness is never realised. Thus all worlds obey the usual Born
predictions of quantum theory.
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 in the conventional formulation of
QM, merely those that relate to probabilities and collapse.
In more detail the steps are:
1) Construct the tensor product of N identical systems in state |psi>,
according to the usual rules for Hilbert space composition
(repeated indices summed):
|PSI_N> = |psi_1>*|psi_2>*...... |psi_N> where
|psi_j> = jth system prepared in state |psi>
= |i_j> (ie the amplitude of the ith eigenstate
is independent of which system it is in)
so that
|PSI_N> = |i_1>|i_2>...|i_N>...
2) Quantify the deviation from the "expected" Born-mean for 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) - ||^2)^2 ). Thus D, loosely
speaking, for each N length sequence expresses how "non-random" a
particular sequence is although, of course, no finite sequence is
excluded from happening since the concept of non-random becomes
precise only as N goes to infinity [H].
3) Sort out terms in the expansion of |PSI_N> according to whether D
is less/equal to (.LE.) or greater than (.GT.) E, where E is a
real, positive constant. Collecting terms together we get:
|PSI_N> = |N,"D.GT.E"> + |N,"D.LE.E">
worlds worlds
for which for which
D > E D <= E
4) What DeWitt showed was that:
< 1/(NE) (proof in appendix of 4b)
Thus as N goes to infinity then the right-hand side vanishes for
all positive values of E. (This mirrors the classical
"frequentist" position on probability which states that if i occurs
with probability p(i) then the proportion of N trials with success
i approaches p(i)/N as N goes to infinity [H]. This has the
immediate benefit that sum(i) = 1.) The norm of |N,"D.LE.E">, by
contrast, approaches 1 as N goes to infinity.
5) The norm of the collection of non-random worlds vanishes and
therefore must be identified with the 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.
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.
[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 obeys
RF(i) |psi_1>|psi_2>.... = ||^2 |psi_1>|psi_2>....
Q21 Does many-worlds allow free-will?
---------------------------------
Many-Worlds, whilst deterministic on the objective universal level, is
indeterministic on the subjective level so the situation is certainly
no better or worse for free-will than in the Copenhagen view.
Traditional Copenhagen indeterministic quantum mechanics only slightly
weakens the case for free-will. In quantum terms each neuron is an
essentially classical object. Consequently quantum noise in the brain
is at such a low level that it probably doesn't often alter, except very
rarely, the critical mechanistic behaviour of sufficient neurons to
cause a decision to be different than we might otherwise expect. The
consensus view amongst experts is that free-will is the consequence of
the mechanistic operation of our brains, the firing of neurons,
discharging of synapses etc and fully compatible with the determinism
of classical physics. Free-will is the inability of a mechanism to
predict its own future actions due to the logical impossibility of any
mechanism containing a complete 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 has its own
weighting or measure which enforces the usual laws of quantum
statistics.
The measure is supplied by the mathematical structure of Hilbert spaces.
Every Hilbert space has a norm, constructed from the inner product, -
which we can think of as analogous to a volume - which weights each
world or collection of worlds. A world of zero volume is never
realised. Worlds in which the conventional statistical predictions
consistently break down have zero volume and so are never realised.
(See "How do probabilities emerge within many-worlds?")
Thus our actions, as expressions of our will, correlate with the weights
associated with worlds. This, of course, matches our subjective
experience of being able to exercise our will, form moral judgements and
be held responsible for our actions.
Q22 Why am I in this world and not another?
---------------------------------------
or
Why the universe appears random, but isn't.
-------------------------------------------
Consider, for a moment, this analogy:
Suppose Fred has his brain divided in two and transplanted into
different cloned bodies (this is a gedanken operation!). Let's further
suppose that each half brain is regenerates to full functionality and
we name the resultant individuals Fred-left and Fred-right. Fred-left
can ask, why did I end up as Fred-left? Similarly Fred-right can ask,
why did I end up as Fred-right? The only answer possible is that there
was *no* reason. From Fred's point of view it is a subjectively
*random* choice which individual Fred ends up as. To the surgeon the
whole process is deterministic. To Fred it seems random.
Same with many-worlds. There was no reason "why" you ended up in this
world, rather than another. It was a subjectively random choice, an
artifact of your consciousness being split. The universe is, in effect,
performing umpteen split-brain operations on us all the time. The
randomness apparent in nature is a consequence of the splitting of
worlds.
(See "How do probabilities emerge within many-worlds?" for how the
subjective randomness is moderated by the usual probabilistic laws of
QM.)
Q23 Can wavefunctions collapse?
---------------------------
Many-Worlds predicts/retrodicts that wavefunctions appear to collapse
(see "The EPR experiment"), when measurement-like interactions and
processes occur via a process called decoherence [7], [10], but claims
that they do not *actually* collapse but continue 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 there can be no mechanism.
Another popular view is that irreversible processes trigger collapse.
Certainly wavefunctions *appear* to collapse whenever irreversible
processes are involved in measurement or amplification and most
macroscopic, day-to-day events are irreversible. The problem is, as
with positing observers as a cause of collapse, that any irreversible
process is composed of a large number of sub-processes that are each
individually reversible. To invoke irreversibility as a *mechanism* for
collapse we would have to show that new *fundamental* physics comes into
play for complex systems, which is quite absent at the atom/molecular
level. Atoms and molecules are empirically observed to obey some type
of wave equation. We have no evidence for an extra mechanism operating
on more complex systems. As far as we can determine complex systems are
described by the same quantum-operation of their simpler components.
(Note: chaos, complexity theory, etc, do not introduce new fundamental
physics. They still operate within the reductionistic paradigm -
despite what many popularisers say.)
Other people have attempted to construct non-linear theories so that
microscopic systems are approximately linear and obey the wave equation
but macroscopic systems are grossly non-linear and generates collapse.
Unfortunately all these efforts have made additional predictions which,
when tested, have failed.
(Another reason for doubting that any collapse actually takes place is
that the collapse would have to propagate instantaneously or in some
space-like fashion, otherwise the same particle would be observed more
than one at different locations. Not fatal, but unpleasant and
difficult to reconcile with relativity.)
The simplest conclusion is that wavefunctions just *don't* collapse and
that all branches of the wavefunction exist.
Q24 Is physics linear?
------------------
or
Could we ever communicate with the other worlds?
------------------------------------------------
or
Why do I only ever experience one world?
----------------------------------------
According to our present knowledge of physics whilst it is possible to
detect the presence of other nearby worlds, through the existence of
interference effects, it is impossible travel to or communicate with
them. Mathematically this corresponds to an empirically verified
property of all quantum theories called linearity which says that the
worlds can interfere with each other with respect to a external,
unsplit, observer or system (whence they are manifest as diffraction or
interference patterns) but they can't influence each other in the sense
that an experimenter can arrange to communicate with their own, already
split-off, quantum copies.
Specifically the wave equation is linear, with respect to the
wavefunction or state vector, which means that given any two solutions
of the wavefunction with identical boundary conditions then any linear
combination of the solutions is also a solution itself. Since each
component of a linear solution evolves with complete indifference as to
the presence or absence of the other terms then we can conclude that no
experiment in one world can have any effect on another experiment in
another world. Hence no communication is possible between quantum
worlds.
Linearity (of the wavefunction) has been verified hold true to better
than 1 part in 10^27 [W] and some scientists believe that it is
absolutely true for various theoretical reasons. If slight non-linear
effects were ever discovered then the possibility of communication
with/travel to the other worlds would be opened up.
[W] Steven Weinberg __ Annals of Physics Vol 194 #2 336-386 (1989) and
_Dreams of a Final Theory_ (1992)
Q25 Can we determine what other worlds there are?
---------------------------------------------
or
Is the form of the Universal Wavefunction knowable?
---------------------------------------------------
To calculate the form of the universal wavefunction requires not only
a knowledge of its dynamics (which we have a good approximation to, at
the moment) but also of the boundary conditions. To actually calculate
the form of the universal wavefunction, and hence make inferences about
*all* the embedded worlds, we would need to know the boundary conditions
as well. We are presently restricted to making inferences about those
worlds with which have shared a common history up to some point, which
have left traces (records, fossils, etc) still discernable today. This
restricts us to a subset of the extant worlds which have shared the same
boundary conditions with us. The further we probe back in time the less
we know of the boundary conditions and therefore the less we can know
of the universal wavefunction.
This limits us to drawing conclusions about a restricted subset of the
worlds - all the worlds which are consistent with our known history up
to a some common moment, before they diverging. The flow of historical
events is, according to chaos/complexity theory/thermodynamics, very
sensitive to amplification of quantum-scale uncertainty and this
sensitivity is a future-directed one-way process. We can make very
reliable deductions about the past from the knowledge future/present but
we can't predict the future from knowledge the past/present.
Thermodynamics implies that the future is harder to predict than the
past is to retrodict. Books get written about this "arrow of time"
problem but we'll just have to accept this as given. The fossil and
historical records say that dinosaurs and Adolf Hitler once existed but
have little to say about future.
Consider the effects of that most quantum of activities, Brownian
motion, on the conception of individuals and the knock-on effects on the
course of history. Mutation itself, one of the sources of evolutionary
diversity, is a quantum event. For an example of the
biological/evolutionary implications see Stephen Jay Gould's book
"Wonderful Life" for an exploration of the thesis that the path of
evolution is driven by chance. According to Gould evolutionary history
forms an enormously diverse tree of possible histories - all very
improbable - with our path being selected by chance. According to many-
worlds all these other possibilities are realised. Thus there are
worlds in which Hitler won WW-II and worlds in which the dinosaurs never
died out. We can be as certain of this as we are that Hitler and the
dinosaurs once existed in our own past.
Whether or not we can ever determine the totality of the universal
wavefunction is an open question. If Steven Hawking's work on the no-
boundary-condition condition is ultimately successful, or it emerges
from some theory of everything, and many think it will, then the actual
form of the *total* wavefunction could, in principle, we determined from
a complete knowledge physical law itself.
Q26 Who was Everett?
----------------
Hugh Everett III (1930-1982) did his undergraduate study in chemical
engineering at the Catholic University of America. Studying von
Neumann's and Bohm's textbooks as part of his graduate studies, under
Wheeler, in mathematical physics at Princeton University in the 1950s
he became dissatisfied (like many others) with the collapse of the
wavefunction. He developed, during discussions with Charles Misner and
Aage Peterson (Bohr' assistant, then visiting Princeton), his "relative
state" formulation. Wheeler encouraged his work and preprints were
circulated in January 1956 to a number of physicists. A condensed
version of his thesis was published as a paper to "The Role of Gravity
in Physics" conference held at the University of N Carolina, Chapel
Hill, N Carolina in January 1957.
Everett was discouraged by the lack of response from others,
particularly Bohr, whom he flew to Copenhagen to meet but got the
complete brush-off from. Leaving physics after completing his Ph.D,
Everett worked as a defense analyst at the Weapons Systems Evaluation
Group, Pentagon. At some point he became a private contractor,
apparently quite successfully for he became a multimillionaire. In 1968
Everett worked for the Lambda Corp. His published papers during this
period cover things like optimising resource allocation and, in
particular, maximising enemy kill rates during nuclear-weapon campaigns.
Later (from 1968 onwards) Bryce S DeWitt, one of the 1957 Chapel Hill
conference organisers, but better known as one of the founders of
quantum gravity, successfully popularised Everett's relative state
formulation as the "many-worlds interpretation" in a series of articles
[4a],[4b],[5].
Sometime in 1976-9 Everett visited Austin, Texas, at Wheeler or DeWitt's
invitation, to give some talks on QM. The strict no-smoking rule in the
auditorium was relaxed for Everett (a chain smoker); the only exception
ever. Everett, apparently, had a very intense and agitated manner, and
spoke with a very acute style, correctly anticipating and cutting off
questions after a few words. Oh yes, a bit of trivia, he drove a
Cadillac with horns.
With the steady growth of interest in many-worlds in the late 1970s
Everett planned returning to physics to do more work on the subject of
measurement in quantum theory, but died of a heart attack in 1982.
Survived by his wife.
Q27 Who believes in many-worlds?
----------------------------
"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 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 content. Nobel Laureate Steven Weinberg is also mentioned as a
many-worlder, although the suggestion is, not when the poll was
conducted, presumably before 1988 (when Feynman died). The only "No,
I don't accept MWI" listed by name is Penrose.
[T] FJ Tipler _The Physics of Immortality_, pages 170-1
Q28 Does the EPR experiment prohibit locality?
------------------------------------------
The EPR experiment is widely regarded as the definitive gedanken
experiment for demonstrating that quantum mechanics is non-local or
incomplete. We shall see that it implies neither.
The EPR experiment was devised, in 1935, by Einstein, Podolsky and Rosen
to demonstrate that quantum mechanics was incomplete [E]. Bell, in
1964, demonstrated that any hidden variables theory, to replicate the
predictions of QM, must be non-local [B]. QM predicts strong
correlations between separated systems, stronger than any local hidden
variables theory can offer. Bell encoded this statistical prediction
in the form of some famous inequalities that apply to any type of EPR
experiment. Eberhard, in the late 1970s, extended Bell's inequalities
to cover any local theory, with or without hidden variables. Thus the
EPR experiment plays a central role in sorting and testing variants of
QM. All the experiments attempting to test EPR/Bell's inequality to
date (including Aspect's in the 1980s [As]) are in line with the
predictions of standard QM - hidden variables are ruled out. Here is
the paradox of the EPR experiment. It seems to imply that any physical
theory must involve faster-than-light "things" going on to maintain
these "spooky" action-at-a-distance correlations and yet still be
compatible with relativity, which seems to forbid FTL.
Let's examine the EPR experiment in more detail.
So what did EPR propose? The original proposal was formulated in terms
of correlations between the positions and momenta of two once-coupled
particles. Here I shall describe it in terms of the spin (a type of
angular momentum intrinsic to the particle) of two electrons. [In this
treatment I shall ignore the fact that electrons always form
antisymmetric combinations. This does not alter the results but does
simplify the maths.] Two initially coupled electrons, with opposed
spins that sum to zero, move apart from each other across a distance of
perhaps many light years, before being separately detected, say, by me
on Earth and you on Alpha Centauri with our respective measuring
apparatuses. The EPR paradox results from noting that if we choose the
same (parallel) spin axes to measure along then we will observe the two
electrons' spins to be anti-parallel (ie when we communicate we find
that the spin on our electrons are correlated and opposed). However if
we choose measurement spin axes that are perpendicular to each other
then there is no correlation between electron spins. Last minute
alterations in a detector's alignment can create or destroy correlations
across great distances. This implies, according to some theorists, that
faster-than-light influences maintain correlations between separated
systems in some circumstances and not others.
Now let's see how many-worlds escapes from this dilemma.
The initial state of the wavefunction of you, me and the electrons and
the rest of the universe may be written:
|psi> = |me> |electrons> |you> |rest of universe>
on in on
Earth deep Alpha
space Centauri
or more compactly, ignoring the rest of the universe, as:
|psi> = |me,electrons,you>
And
|electrons> = (|+,-> - |-,+>)/sqrt(2)
represents a pair electrons, with the first electron travelling
towards Earth and the second electron travelling towards Alpha
Centauri.
|+> represents an electron with spin in the +z direction
|-> represents an electron with spin in the -z direction
It is an empirically established fact, which we just have to accept,
that we can relate spin states in one direction to spin states in other
directions like so (where "i" is the sqrt(-1)):
|left> = (|+> - |->)/sqrt(2) (electron with spin in -x direction)
|right> = (|+> + |->)/sqrt(2) (electron with spin in +x direction)
|up> = (|+> + |->i)/sqrt(2) (electron with spin in +y direction)
|down> = (|+> - |->i)/sqrt(2) (electron with spin in -y direction)
and inverting:
|+> = (|right> + |left>)/sqrt(2) = (|up> + |down>)/sqrt(2)
|-> = (|right> - |left>)/sqrt(2) = (|down> - |up>)i/sqrt(2)
(In fancy jargon we say that the spin operator in different directions
form non-commuting observables. I shall eschew such obfuscations.)
Working through the algebra we find that for pairs of electrons:
|+,-> - |-,+> = |left,right> - |right,left>
= |up,down>i - |down,up>i
|me> represents me on Earth with my detection apparatus. I shall assume
that we are capable of either measuring spin in the x or y direction,
which are both perpendicular the line of flight of the electrons. After
having measured the state of the electron my state is described as one
of either:
|me[l]> represents me + apparatus + records having measured
x-axis spin and recorded the x-axis spin as "left"
|me[r]> ditto with the x-axis spin as "right"
|me[u]> ditto with the y-axis spin as "up"
|me[d]> ditto with the y-axis spin as "down"
Similarly for |you> on Alpha Centauri. Notice that it is irrelevant
*how* we have measured the electron's spin. The details of the
measurement process are irrelevant. To model the process it is
sufficient to assume that there is a way, which we have further assumed
does not disturb the electron. (The latter assumption may be relaxed
without altering the results.)
To establish familiarity with the notation let's take the state of the
initial wavefunction as:
|psi>_1 = |me,left,up,you>
/ \
/ \
first electron in left second electron in up state
state heading towards heading towards you on
me on Earth Alpha Centauri
After the electrons arrive at their detectors, I measure the spin
along the x-axis and you along the y-axis. The wavefunction evolves
into |psi>_2:
local
|psi>_1 ============> |psi>_2 = |me[l],left,up,you[u]>
observation
which represents me having recorded my electron on Earth with spin left
and you having recorded your electron on Alpha Centauri with spin up.
The index in []s indicates the value of the record. This may be held
in the observer's memory, notebooks or elsewhere in the local
environment (not necessarily in a readable form). If we communicate our
readings to each other the wavefunctions evolves into |psi>_3:
remote
|psi>_2 ============> |psi>_3 = |me[l,u],left,up,you[u,l]>
communication
where the second index in []s represents the remote reading communicated
to the other observer and being recorded locally. Notice that the
results both agree with each other, in the sense 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.
*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 between the remote systems. Until the two observers
communicated their results to each other they were each unsplit by each
others' remote measurements, although their own local measurements had
split themselves. The splitting is a local process that is causally
transmitted from system to system at light or sub-light speeds. (This
is a point that Everett stressed about Einstein's remark about the
observations of a mouse, in the Copenhagen interpretation, collapsing
the wavefunction of the universe. Everett observed that it is the mouse
that's split by its observation of the rest of the universe. The rest
of the universe is unaffected and unsplit.)
When all communication is complete the worlds have finally decomposed
or decohered from each other. Each world contains a consistent set of
observers, records and electrons, in perfect agreement with the
predictions of standard QM. Further observations of the electrons will
agree with the earlier ones and so each observer, in each world, can
henceforth regard the electron's wavefunction as having collapsed to
match the historically recorded, locally observed values. This
justifies our operational adoption of the collapse of the wavefunction
upon measurement, without having to strain our credibility by believing
that it actually happens.
To recap. Many-worlds is local and deterministic. Local measurements
split local systems (including observers) in a subjectively random
fashion; distant systems are only split when the causally transmitted
effects of the local interactions reach them. We have not assumed any
non-local FTL effects, yet we have reproduced the standard predictions
of QM.
So where did Bell and Eberhard go wrong? They thought that all theories
that reproduced the standard predictions must be non-local. It has been
pointed out by both Albert [A] and Cramer [C] (who both support
different interpretations of QM) that Bell and Eberhard had implicity
assumed that every possible measurement - even if not performed - would
have yielded a *single* definite result. This assumption is called
contra-factual definiteness or CFD [S]. What Bell and Eberhard really
proved was that every quantum theory must either violate locality *or*
CFD. Many-worlds with its multiplicity of results in different worlds
violates CFD, of course, and thus can be local.
Thus many-worlds is the only local quantum theory in accord with the
standard predictions of QM and, so far, with experiment.
[A] David Z Albert, _Bohm's Alternative to Quantum Mechanics_
Scientific American (May 1994)
[As] Alain Aspect, J Dalibard, G Roger _Experimental test of Bell's
inequalities using time-varying analyzers_ Physical Review Letters
Vol 49 #25 1804 (1982).
[C] John G Cramer _The transactional interpretation of quantum
mechanics_ Reviews of Modern Physics Vol 58 #3 647-687 (1986)
[B] John S Bell: _On the Einstein Podolsky Rosen paradox_ Physics 1
#3 195-200 (1964).
[E] Albert Einstein, Boris Podolsky, Nathan Rosen: _Can
quantum-mechanical description of physical reality be considered
complete?_ Physical Review Vol 41, 777-780 (15 May 1935).
[S] Henry P Stapp _S-matrix interpretation of quantum-theory_ Physical
Review D Vol 3 #6 1303 (1971)
Q29 Is Everett's relative state theory 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.
Everett talks more about the observer's memory sequences splitting to
form a branching structure or the state of the observer being split.
DeWitt introduced the term "world" for describing the split states of
an observer, so that we can 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 measuring apparatus being split by
the measurement into non-interfering states, without presenting any
detailed analysis of *why* a measuring apparatus was so effective at
destroying interference effects after a measurement, except to cover the
notion of amplification and irreversibility. See "Why do worlds split?"
and "When do worlds split?" DeWitt, Gell-Mann, Hartle, Zurek and others
have introduced the terminology of "decoherence" to describe the role
of thermodynamics and irreversibility much further in the amplification
process.
31 References and further reading
------------------------------
[1] Hugh Everett III _The Theory of the Universal Wavefunction,
Princeton thesis_ (1956?)
The original and most comprehensive paper on many-worlds.
Investigates and recasts the foundations of quantum theory in
information theoretic terms, before moving on to consider the
nature of interactions, observation, entropy, irreversible
processes, classical objects etc. 138 pages. Only published in
[5].
[2] Hugh Everett III _"Relative State" Formulation of Quantum
Mechanics_ Reviews of Modern Physics Vol 29 #3 454-462, (July
1957) A condensation of [1] focusing on observation.
[3] John A Wheeler _Assessment of Everett's "Relative State"
Formulation of Quantum Theory_, Reviews of Modern Physics Vol
29 #3 463-465 (July 1957) Wheeler was Everett's PhD
supervisor.
[4a] Bryce S DeWitt _Quantum Mechanics and Reality_ Physics Today,
Vol 23 #9 30-40 (September 1970) One of the earlier, and more
accurate, popularisations of Everett's work. The April 1971
issue has reader feedback and DeWitt's responses.
[4b] Bryce S DeWitt _The Many-Universes Interpretation of Quantum
Mechanics_ in _Proceedings of the International School of Physics
"Enrico Fermi" Course IL: Foundations of Quantum Mechanics_
Academic Press (1972)
[5] Bryce S DeWitt, R Neill Graham eds _The many-worlds
Interpretation of Quantum Mechanics_, Contains
[1],[2],[3],[4a],[4b] plus other material. Princeton Series
in Physics, Princeton University Press (1973) ISBN 0-691-
08126-3 (hard cover), 0-691-88131-X (paper back) The
definitive guide to many-worlds, if you can get hold of a
copy, but now (1994) only available xeroxed from microfilm
(ISBN 0-7837-1942-6) from Books On Demand, 300 N Zeeb Road,
Ann Arbor, MI 48106-1346, USA. Tel: +01-313 761 4700 or 800
521 0600.
[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] Marian O Scully, Bethold-Georg Englert, Herbert Walther _Quantum
optical tests of complementarity_ Nature, Vol 351, 111-116 (9 May
1991). Demonstrates that quantum interference effects are destroyed
by irreversible object-apparatus correlations, not by the
measurement process itself.
[10] Murray Gell-Mann, James B Hartle _Quantum Mechanics in the Light
of Quantum Cosmology_ Proceedings of the 3rd International
Symposium on the Foundations of Quantum Mechanics (1989) 321-343.
They accept the Everett's decoherence analysis, and have extended
it further, but reject many-worlds' metaphysical stance.
Tests of the Everett metatheory:
[11] David Deutsch _Quantum theory as a universal physical theory_
International Journal of Theoretical Physics, Vol 24 #1
(1985). Describes an experiment which tests for the existence
of superpositions of *consciousness (in an AI).
[16] David Deutsch _Three connections between Everett's interpretation
and experiment_ Quantum Concepts of Space and Time, eds Roger
Penrose and Chris Isham, Oxford University Press (1986). Discusses
a testable split observer experiment and quantum computing.
On quantum computers:
[12] David Deutsch _Quantum theory, the Church-Turing principle and the
universal quantum computer_ Proceedings of the Royal Society of
London, Vol. A400, 96-117 (1985).
[13] David Deutsch _Quantum computational networks_ Proceedings of
the Royal Society of London, Vol. A425, 73-90 (1989).
[14] David Deutsch and R. Jozsa _Rapid solution of problems by
quantum computation_ Proceedings of the Royal Society of
London, Vol. A439, 553-558 (1992).
[] Julian Brown _A Quantum Revolution for Computing_ New Scientist,
pages 21-24, 24-September-1994
32 Mini-guide to notation and quantum mechanics
--------------------------------------------
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.
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>.
An general object, labelled "psi" by convention, is represented as |psi>
and called a "ket". Objects called "bra"s, written may be combined
together to form the bracket, , or inner product, which is
just a fancy way of constructing a complex number. Amongst the
properties of the inner product is:
*a_1 + |psi2>*a_2) = *a_1 + *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
= complex conjugate of
Any ket may be expanded as:
|psi> = sum |i>*
i
= |1>*<1|psi> + |2>*<2|psi> + ...
where the states |i> form an orthonormal basis, with = 1 for i =
j and = 0 otherwise, and where i labels some parameter of the object
(like position or momentum).
The probability amplitudes, , 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 consider that 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, , and both are inner
productted together the result, , 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.
**************************
Mike Price price@price.demon.co.uk
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