THE EVERETT FAQ
Michael Clive Price
February 1995
Permission to copy in its entirety granted for non-commercial purposes.CONTENTS:
<br />Q0 <a href=#faq>Why this FAQ?</a><br /><br />Q1 <a href=#believes in>Who believes in many-worlds?</a><br />Q2 <a href=#what is>What is many-worlds?</a><br />Q3 <a href=#alternatives>What are the alternatives to many-worlds?</a><br />Q4 <a href=#is a>What is a "world"?</a><br />Q5 <a href=#measurement>What is a measurement?</a><br />Q6 <a href=#split>Why do worlds split?</a><br /><br /> <a href=#decoherence>What is decoherence?</a><br />Q7 <a href=#do split>When do worlds split?</a><br />Q8 <a href=#splitsh>When does Schrodinger's cat split?</a><br />Q9 <a href=#sum>What is sum-over-histories?</a><br />Q10 <a href=#many>What is many-histories?</a><br /> <a href=#environment>What is the environment basis?</a><br /><br />Q11 <a href=#how many>How many worlds are there?</a><br />Q12 <a href=#local>Is many-worlds a local theory?</a><br />Q13 <a href=#deterministic>Is many-worlds a deterministic theory?</a><br />Q14 <a href=#relativistic>Is many-worlds a relativistic theory?</a><br /> <a href=#field>What about quantum field theory?</a><br /> <a href=#gravity>What about quantum gravity?</a><br /><br />Q15 <a href=#where are>Where are the other worlds?</a><br />Q16 <a href=#interpretation>Is many-worlds (just) an interpretation?</a><br />Q17 <a href=#fuse>Why don't worlds fuse, as well as split?</a><br /> <a href=#irreversible> Do splitting worlds imply irreversible physics?</a><br />Q18 <a href=#retrodictions>What retrodictions does many-worlds make?</a><br />Q19 <a href=#differentiate>Do worlds differentiate or split?</a><br /><br />Q20 <a href=#minds>What is many-minds?</a><br />Q21 <a href=#ockham's>Does many-worlds violate Ockham's Razor?</a><br />Q22 <a href=#conservation>Does many-worlds violate conservation of energy?</a><br />Q23 <a href=#probabilities>How do probabilities emerge within many-worlds?</a><br />Q24 <a href=#free-will>Does many-worlds allow free-will?</a><br />Q25 <a href=#i in this>Why am I in this world and not another?</a><br /><br /> <a href=#random>Why does the universe appear random?</a><br />Q26 <a href=#wavefunctions>Can wavefunctions collapse?</a><br />Q27 <a href=#linear>Is physics linear?</a><br /> <a href=#communicate>Could we ever communicate with the other worlds?</a><br /> <a href=#experience>Why do I only ever experience one world?</a><br /> <a href=#not aware>Why am I not aware of the world (and myself) splitting?</a><br /><br />Q28 <a href=#determine>Can we determine what other worlds there are?</a><br /> <a href=#knowable> Is the form of the Universal Wavefunction knowable?</a><br />Q29 <a href=#everett>Who was Everett?</a><br />Q30 <a href=#problems>What are the problems with quantum theory?</a><br />Q31 <a href=#copenhagen>What is the Copenhagen interpretation?</a><br />Q32 <a href=#epr>Does the EPR experiment prohibit locality?</a><br /><br /> <a href=#bell> What about Bell's Inequality?</a><br />Q33 <a href=#same>Is Everett's relative state formulation the same as many-worlds?</a><br />Q34 <a href=#relative>What is a relative state?</a><br />Q35 <a href=#splitter>Was Everett a "splitter"?</a><br />Q36 <a href=#unique>What unique predictions does many-worlds make?</a><br />Q37 <a href=#detect>Could we detect other Everett-worlds?</a><br /><br />Q38 <a href=#quantum gravity>Why <I>quantum</I> gravity?</a><br />Q39 <a href=#exact>Is linearity exact?</a><br />Q41 <a href=#boundary>Why can't the boundary conditions be updated to reflect my<br /> observations in this one world?</a><br /><hr /><br />A1 <a href=#references>References and further reading</a><br />A2 <a href=#dirac notation>Quantum mechanics and Dirac notation</a></pre><hr noshade /><br /><br /><blockquote><br /><br /><a name=faq><h3>Q0 Why this FAQ?</h3><br /><br />This FAQ shows how quantum paradoxes are resolved by the "many-worlds"<br />interpretation or metatheory of quantum mechanics. This FAQ does not<br />seek to <I>prove</I> that the many-worlds interpretation is the "correct"<br />quantum metatheory, merely to correct some of the common errors and<br />misinformation on the subject floating around.<p><br /><br />As a physics undergraduate I was struck by the misconceptions of my<br />tutors about many-worlds, despite that it seemed to resolve all the<br />paradoxes of quantum theory <b>[A]</b>. The objections raised to many-worlds<br />were either patently misguided <b>[B]</b> or beyond my ability to assess at the<br />time <b>[C]</b>, which made me suspect (confirmed during my graduate QFT<br />studies) that the more sophisticated rebuttals were also invalid. I<br />hope this FAQ will save other investigators from being lead astray by<br />authoritative statements from mentors.<p><br /><br />I have attempted, in the answers, to translate the precise mathematics<br />of quantum theory into woolly and ambiguous English - I would appreciate<br />any corrections. In one or two instances I couldn't avoid using some<br />mathematical (Dirac) notation, in particular in describing the Einstein-<br />Podolsky-Rosen (EPR) experiment and Bell's Inequality and in showing how<br />probabilities are derived, so I've included an appendix on the Dirac<br />notation.<p><br /><br /><b>[A]</b> See <a href=#epr>"Does the EPR experiment prohibit locality?"</a>, <a href=#bell>"What about Bell's<br />Inequality?"</a> and <a href=#splitsh>"When does Schrodinger's cat split?"</a> for how many-<br />worlds handles the most quoted paradoxes.<p><br /><br /><b>[B]</b> Sample objection: "Creation of parallel universes violates energy<br />conservation/Ockham's razor". (See <a href=#violate>"Does many-worlds violate<br />conservation of energy?"</a> and <a href=#ockham's>"Does many-worlds violate Ockham's Razor?"</a>)<p><br /><br /><b>[C]</b> eg "In quantum field theory the wavefunction becomes an operator". <br />Er, what does that mean? And is this relevant? (See <a href=#field>"What about<br />quantum field theory?")</a><p><hr noshade /><br /><br /><a name=believes in><h3>Q1 Who believes in many-worlds?</h3><br /><br /> <br />"Political scientist" L David Raub reports a poll of 72 of the "leading<br />cosmologists and other quantum field theorists" about the "Many-Worlds<br />Interpretation" and gives the following response breakdown [T].<p><br /><pre> <br />1) "Yes, I think MWI is true" 58%<br />2) "No, I don't accept MWI" 18%<br />3) "Maybe it's true but I'm not yet convinced" 13%<br />4) "I have no opinion one way or the other" 11%</pre><P><br /><br />Amongst the "Yes, I think MWI is true" crowd listed are Stephen Hawking<br />and Nobel Laureates Murray Gell-Mann and Richard Feynman. Gell-Mann and<br />Hawking recorded reservations with the name "many-worlds", but not with<br />the theory's content. Nobel Laureate Steven Weinberg is also mentioned<br />as a many-worlder, although the suggestion is not when the poll was<br />conducted, presumably before 1988 (when Feynman died). The only "No,<br />I don't accept MWI" named is Penrose.<p><br /><br />The findings of this poll are in accord with other polls, that many-<br />worlds is most popular amongst scientists who may rather loosely be<br />described as string theorists or quantum gravitists/cosmologists. It<br />is less popular amongst the wider scientific community who mostly remain<br />in ignorance of it.<p><br /><br />More detail on Weinberg's views can be found in _Dreams of a Final<br />Theory_ or _Life in the Universe_ Scientific American (October 1994),<br />the latter where Weinberg says about quantum theory:<br /><br /> "The final approach is to take the Schrodinger equation seriously<br /> [..description of the measurement process..] In this way, a<br /> measurement causes the history of the universe for practical<br /> purposes to diverge into different non-interfering tracks, one for<br /> each possible value of the measured quantity. [...] I prefer this<br /> last approach"<p><br /><br />In the <I>The Quark and the Jaguar</I> and <I>Quantum Mechanics in the Light<br />of Quantum Cosmology</I> [10] Gell-Mann describes himself as an adherent<br />to the (post-)Everett interpretation, although his exact meaning is<br />sometimes left ambiguous.<p><br /><br />Steven Hawking is well known as a many-worlds fan and says, in an<br />article on quantum gravity [H], that measurement of the gravitational<br />metric tells you which branch of the wavefunction you're in and<br />references Everett.<p><br /><br />Feynman, apart from the evidence of the Raub poll, directly favouring<br />the Everett interpretation, always emphasized to his lecture students<br />[F] that the "collapse" process could only be modelled by the<br />Schrodinger wave equation (Everett's approach).<p><br /><br /><b>[F]</b> Jagdish Mehra <I>The Beat of a Different Drum: The Life and Science<br /> Richard Feynman</I><br /><br /><br /><b>[H]</b> Stephen W Hawking <I>Black Holes and Thermodynamics</I> Physical Review<br /> D Vol 13 #2 191-197 (1976)<br /><br /><b>[T]</b> Frank J Tipler <I>The Physics of Immortality</I> 170-171<p><hr noshade /><br /><br /><a name=what is><h3>Q2 What is many-worlds?</h3><br /><br /> <br />AKA as the Everett, relative-state, many-histories or many-universes<br />interpretation or metatheory of quantum theory. Dr Hugh Everett, III,<br />its originator, called it the "relative-state metatheory" or the "theory<br />of the universal wavefunction" [1], but it is generally called "many-<br />worlds" nowadays, after DeWitt [4a],[5].<p><br /><br />Many-worlds comprises of two assumptions and some consequences. The<br />assumptions are quite modest:<br /><br /><b>1) The metaphysical assumption:</b> That the wavefunction does not merely<br /> encode the all the information about an object, but has an<br /> observer-independent objective existence and actually <I>is</I> the<br /> object. For a non-relativistic N-particle system the wavefunction<br /> is a complex-valued field in a 3-N dimensional space.<p><br /><br /><b>2) The physical assumption:</b> The wavefunction obeys the empirically<br /> derived standard linear deterministic wave equations at all times. <br /> The observer plays no special role in the theory and, consequently,<br /> there is no collapse of the wavefunction. For non-relativistic<br /> systems the Schrodinger wave equation is a good approximation to<br /> reality. (See <a href=#relativistic>"Is many-worlds a relativistic theory?"</a> for how the<br /> more general case is handled with quantum field theory or third quantisation.)<p><br /><br />The rest of the theory is just working out consequences of the above<br />assumptions. Measurements and observations by a subject on an object<br />are modelled by applying the wave equation to the joint subject-object<br />system. Some consequences are:<br /><br />1) That each measurement causes a decomposition or decoherence of the<br /> universal wavefunction into non-interacting and mostly non-<br /> interfering branches, histories or worlds. (See <a href=#decoherence>"What is<br /> decoherence?"</a>) The histories form a branching tree which<br /> encompasses all the possible outcomes of each interaction. (See<br /> <a href=#split>"Why do worlds split?"</a> and <a href=#do split>"When do worlds split?"</a>) Every<br /> historical what-if compatible with the initial conditions and<br /> physical law is realised.<p><br /><br />2) That the conventional statistical Born interpretation of the<br /> amplitudes in quantum theory is <I>derived</I> from within the theory<br /> rather than having to be <I>assumed</I> as an additional axiom. (See<br /> <a href=#probabilities>"How do probabilities emerge within many-worlds?"</a>)<p><br /><br />Many-worlds is a re-formulation of quantum theory [1], published in 1957<br />by Dr Hugh Everett III [2], which treats the process of observation or<br />measurement entirely within the wave-mechanics of quantum theory, rather<br />than an input as additional assumption, as in the Copenhagen<br />interpretation. Everett considered the wavefunction a real object. <br />Many-worlds is a return to the classical, pre-quantum view of the<br />universe in which all the mathematical entities of a physical theory are<br />real. For example the electromagnetic fields of James Clark Maxwell or<br />the atoms of Dalton were considered as real objects in classical<br />physics. Everett treats the wavefunction in a similar fashion. Everett<br />also assumed that the wavefunction obeyed the same wave equation during<br />observation or measurement as at all other times. This is the central<br />assumption of many-worlds: that the wave equation is obeyed universally<br />and at all times.<p><br /><br />Everett discovered that the new, simpler theory - which he named the<br />"relative state" formulation - predicts that interactions between two<br />(or more) macrosystems typically split the joint system into a<br />superposition of products of relative states. The states of the<br />macrosystems are, after the subsystems have jointly interacted,<br />henceforth correlated with, or dependent upon, each other. Each element<br />of the superposition - each a product of subsystem states - evolves<br />independently of the other elements in the superposition. The states<br />of the macrosystems are, by becoming correlated or entangled with each<br />other, impossible to understand in isolation from each other and must<br />be viewed as one composite system. It is no longer possible to speak<br />the state of one (sub)system in isolation from the other (sub)systems. <br />Instead we are forced to deal with the states of subsystems <I>relative</I><br />to each other. Specifying the state of one subsystem leads to a unique<br />specification of the state (the "relative state") of the other<br />subsystems. (See <a href=#relative>"What is a relative state?"</a>)<p><br /><br />If one of the systems is an observer and the interaction an observation<br />then the effect of the observation is to split the observer into a<br />number of copies, each copy observing just one of the possible results<br />of a measurement and unaware of the other results and all its observer-<br />copies. Interactions between systems and their environments, including<br />communication between different observers in the same world, transmits<br />the correlations that induce local splitting or decoherence into non-<br />interfering branches of the universal wavefunction. Thus the entire<br />world is split, quite rapidly, into a host of mutually unobservable but<br />equally real worlds.<p><br /><br />According to many-worlds all the possible outcomes of a quantum<br />interaction are realised. The wavefunction, instead of collapsing at<br />the moment of observation, carries on evolving in a deterministic<br />fashion, embracing all possibilities embedded within it. All outcomes<br />exist simultaneously but do not interfere further with each other, each<br />single prior world having split into mutually unobservable but equally<br />real worlds.<p><hr noshade /><br /><br /><a name=alternatives><h3>Q3 What are the alternatives to many-worlds?</h3><br /> <br />There is no other quantum theory, besides many-worlds, that is<br />scientific, in the sense of providing a reductionist model of reality,<br />and free of internal inconsistencies, that I am aware of. Briefly here<br />are the defects of the most popular alternatives:<p><br /><br />1) <b>Copenhagen Interpretation.</b> Postulates that the observer obeys<br /> different physical laws than the non-observer, which is a return<br /> to vitalism. The definition of an observer varies from one<br /> adherent to another, if present at all. The status of the<br /> wavefunction is also ambiguous. If the wavefunction is real the<br /> theory is non-local (not fatal, but unpleasant). If the<br /> wavefunction is not real then the theory supplies no model of<br /> reality. (See <a href=#problems>"What are the problems with quantum theory?"</a>)<p><br /><br />2) <b>Hidden Variables [B]. </b> Explicitly non-local. Bohm accepts that all<br /> the branches of the universal wavefunction exist. Like Everett<br /> Bohm held that the wavefunction is real complex-valued field which<br /> never collapses. In addition Bohm postulated that there were<br /> particles that move under the influence of a non-local "quantum-<br /> potential" derived from the wavefunction (in addition to the<br /> classical potentials which are already incorporated into the<br /> structure of the wavefunction). The action of the quantum-<br /> potential is such that the particles are affected by only one of<br /> the branches of the wavefunction. (Bohm derives what is<br /> essentially a decoherence argument to show this, see section 7,#I<br /> [B]).<p><br /><br /> The implicit, unstated assumption made by Bohm is that only the<br /> single branch of wavefunction associated with particles can contain<br /> self-aware observers, whereas Everett makes no such assumption. <br /> Most of Bohm's adherents do not seem to understand (or even be<br /> aware of) Everett's criticism, section VI [1], that the hidden-<br /> variable particles are not observable since the wavefunction alone<br /> is sufficient to account for all observations and hence a model of<br /> reality. The hidden variable particles can be discarded, along<br /> with the guiding quantum-potential, yielding a theory isomorphic<br /> to many-worlds, without affecting any experimental results.<p><br /><br /> [B] David J Bohm <I>A suggested interpretation of the quantum theory<br /> in terms of "hidden variables" I and II</I> Physical Review Vol<br /> 85 #2 166-193 (1952)<p><br /><br />3) <b> Quantum Logic.</b> Undoubtedly the most extreme of all attempts to<br /> solve the QM measurement problem. Apart from abandoning one or<br /> other of the classical tenets of logic these theories are all<br /> unfinished (presumably because of internal inconsistencies). Also<br /> it is unclear how and why different types of logic apply on<br /> different scales.<p><br /><br />4) <b> Extended Probability</b> [M]. A bold theory in which the concept of<br /> probability is "extended" to include complex values [Y]. Whilst<br /> quite daring, I am not sure if this is logically permissable, being<br /> in conflict with the relative frequency notion of probability, in<br /> which case it suffers from the same criticism as quantum logic. <br /> Also it is unclear, to me anyway, how the resultant notion of<br /> "complex probability" differs from the quantum "probability<br /> amplitude" and thus why we are justified in collapsing the complex-<br /> valued probability as if it were a classical, real-valued<br /> probability.<p><br /><br /> [M] W Muckenheim <I>A review of extended probabilities</I> Physics<br /> Reports Vol 133 339- (1986)<br /> [Y] Saul Youssef <I>Quantum Mechanics as Complex Probability Theory</I><br /> hep-th 9307019<p><br /><br />5) <b>Transactional model [C]. </b> Explicitly non-local. An imaginative<br /> theory, based on the Feynman-Wheeler absorber-emitter model of EM,<br /> in which advanced and retarded probability amplitudes combine into<br /> an atemporal "transaction" to form the Born probability density. <br /> It requires that the input and output states, as defined by an<br /> observer, act as emitters and absorbers respectively, but not any<br /> internal states (inside the "black box"), and, consequently,<br /> suffers from the familiar measurement problem of the Copenhagen<br /> interpretation.<p><br /><br /> If the internal states <I>did</I> act as emitters/absorbers then the<br /> wavefunction would collapse, for example, around one of the double<br /> slits (an internal state) in the double slit experiment, destroying<br /> the observed interference fringes. In transaction terminology a<br /> transaction would form between the first single slit and one of the<br /> double slits and another transaction would form between the same<br /> double slit and the point on the screen where the photon lands. <br /> This never observed.<p><br /><br /> [C] John G Cramer <a href="http://mist.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html">The transactional interpretation of quantum<br /> mechanics</a> Reviews of Modern Physics Vol 58 #3 647-687 (1986)<p><br /><br /><b>6) Many-minds.</b> Despite its superficial similarities with many-worlds<br /> this is actually a very unphysical, non-operational theory. (See<br /> <a href=#minds>"What is many-minds?"</a>)<p><br /><br /><b>7) Non-linear theories in general.</b> So far no non-linear theory has<br /> any accepted experimental support, whereas many have failed<br /> experiment. (See <a href=#linear>"Is physics linear?"</a>) Many-worlds predicts that<br /> non-linear theories will always fail experiment. (See <a href=#exact>"Is<br /> linearity exact?"</a>)<p><hr noshade /><br /><br /><a name=is a><h3>Q4 What is a "world"?</h3><br /><br />Loosely speaking a "world" is a complex, causally connected, partially<br />or completely closed set of interacting sub-systems which don't<br />significantly interfere with other, more remote, elements in the<br />superposition. Any complex system and its coupled environment, with a<br />large number of internal degrees of freedom, qualifies as a world. An<br />observer, with internal irreversible processes, counts as a complex<br />system. In terms of the wavefunction, a world is a decohered branch of<br />the universal wavefunction, which represents a single macrostate. (See<br /><a href=#decoherence>"What is decoherence?"</a>) The worlds all exist simultaneously in a non-<br />interacting linear superposition.<p><br /><br />Sometimes "worlds" are called "universes", but more usually the latter<br />is reserved the totality of worlds implied by the universal<br />wavefunction. Sometimes the term "history" is used instead of "world". <br />(Gell-Mann/Hartle's phrase, see <a href=#many>"What is many-histories?"</a>).<p><hr noshade /><br /><br /><a name=measurement><h3>Q5 What is a measurement?</h3><br /> <br />A measurement is an interaction, usually irreversible, between<br />subsystems that correlates the value of a quantity in one subsystem with<br />the value of a quantity in the other subsystem. The interaction may<br />trigger an amplification process within one object or subsystem with<br />many internal degrees of freedom, leading to an irreversible high-level<br />change in the same object. If the course of the amplification is<br />sensitive to the initial interaction then we can designate the system<br />containing the amplified process as the "measuring apparatus", since the<br />trigger is sensitive to some (often microphysical) quantity or parameter<br />of the one of the other subsystems, which we designate the "object"<br />system. Eg the detection of a charged particle (the object) by a Geiger<br />counter (the measuring apparatus) leads to the generation of a "click"<br />(high-level change). The absence of a charged particle does not<br />generate a click. The interaction is with those elements of the charged<br />particle's wavefunction that passes <I>between</I> the charged detector<br />plates, triggering the amplification process (an irreversible electron<br />cascade or avalanche), which is ultimately converted to a click.<p><br /><br />A measurement, by this definition, does not require the presence of an<br />conscious observer, only of irreversible processes.<p><hr noshade /><br /><br /><a name=split><h3>Q6<br /> Why do worlds split?<br /><br /> <br /><a name=decoherence> <b>What is decoherence?</b></h3><br /><br />Worlds, or branches of the universal wavefunction, split when different<br />components of a quantum superposition "decohere" from each other [7a],<br />[7b], [10]. Decoherence refers to the loss of coherency or absence of<br />interference effects between the elements of the superposition. For two<br />branches or worlds to interfere with each other all the atoms, subatomic<br />particles, photons and other degrees of freedom in each world have to<br />be in the same state, which usually means they all must be in the same<br />place or significantly overlap in both worlds, simultaneously.<p><br /><br />For small microscopic systems it is quite possible for all their atomic<br />components to overlap at some future point. In the double slit<br />experiment, for instance, it only requires that the divergent paths of<br />the diffracted particle overlap again at some space-time point for an<br />interference pattern to form, because only the single particle has been<br />split.<p><br /><br />Such future coincidence of positions in all the components is virtually<br />impossible in more complex, macroscopic systems because all the<br />constituent particles have to overlap with their counterparts<br />simultaneously. Any system complex enough to be described by<br />thermodynamics and exhibit irreversible behaviour is a system complex<br />enough to exclude, for all practical purposes, any possibility of future<br />interference between its decoherent branches. An irreversible process<br />is one in, or linked to, a system with a large number of internal,<br />unconstrained degrees of freedom. Once the irreversible process has<br />started then alterations of the values of the many degrees of freedom<br />leaves an imprint which can't be removed. If we try to intervene to<br />restore the original status quo the intervention causes more disruption<br />elsewhere.<p><br /><br />In QM jargon we say that the components (or vectors in the underlying<br />Hilbert state space) have become permanently orthogonal due to the<br />complexity of the systems increasing the dimensionality of the vector<br />space, where each unconstrained degree of freedom contributes a<br />dimension to the state vector space. In a high dimension space almost<br />all vectors are orthogonal, without any significant degree of overlap. <br />Thus vectors for complex systems, with a large number of degrees of<br />freedom, naturally decompose into mutually orthogonal components which,<br />because they can never significantly interfere again, are unaware of<br />each other. The complex system, or world, has split into different,<br />mutually unobservable worlds.<p><br /><br />According to thermodynamics each activated degree of freedom acquires<br />kT energy. This works the other way around as well: the release of<br />approximately kT of energy increases the state-space dimensionality. <br />Even the quite small amounts of energy released by an irreversible<br />frictive process are quite large on this scale, increasing the size of<br />the associated Hilbert space.<p><br /><br />Contact between a system and a heat sink is equivalent to increasing the<br />dimensionality of the state space, because the description of the system<br />has to be extended to include all parts of the environment in causal<br />contact with it. Contact with the external environment is a very<br />effective destroyer of coherency. (See <a href=#environment>"What is the environment<br />basis?"</a>)<p><hr noshade /><br /><br /><a name=do split><h3>Q7 When do worlds split?</h3><br /><br /> <br />Worlds irrevocably "split" at the sites of measurement-like interactions<br />associated with thermodynamically irreversible processes. (See <a href=#measurement>"What<br />is a measurement?"</a>) An irreversible process will always produce<br />decoherence which splits worlds. (See <a href=#why>"Why do worlds split?"</a>, <a href=#decoherence>"What is<br />decoherence?"</a> and <a href=#splitsh>"When does Schrodinger's cat split?"</a> for a concrete<br />example.)<p><br /><br />In the example of a Geiger counter and a charged particle after the<br />particle has passed the counter one world contains the clicked counter<br />and that portion of the particle's wavefunction which passed though the<br />detector. The other world contains the unclicked counter with the<br />particle's wavefunction with a "shadow" cast by the counter taken out<br />of the particle's wavefunction.<p><br /><br />The Geiger counter splits when the amplification process became<br />irreversible, before the click is emitted. (See <a href=#measurement>"What is a<br />measurement?"</a>) The splitting is local (originally in the region of the<br />Geiger counter in our example) and is transmitted causally to more<br />distant systems. (See <a href=#local>"Is many-worlds a local theory?"</a> and <a href=#locality>"Does the<br />EPR experiment prohibit locality?"</a>) The precise moment/location of the<br />split is not sharply defined due to the subjective nature of<br />irreversibility, but can be considered complete when much more than kT<br />of energy has been released in an uncontrolled fashion into the<br />environment. At this stage the event has become irreversible.<p><br /><br />In the language of thermodynamics the amplification of the charged<br />particle's presence by the Geiger counter is an irreversible event. <br />These events have caused the decoherence of the different branches of<br />the wavefunction. (See<a href=#decoherence> "What is decoherence?" and <a href=#split>"Why do worlds<br />split?"</a>) Decoherence occurs when irreversible macro-level events take<br />place and the macrostate description of an object admits no single<br />description. (A macrostate, in brief, is the description of an object<br />in terms of accessible external characteristics.)<p><br /><br />The advantage of linking the definition of worlds and the splitting<br />process with thermodynamics is the splitting process becomes<br />irreversible and only permits forward-time-branching, following the<br />increase with entropy. (See <a href=#fuse>"Why don't worlds fuse, as well as split?"</a>) <br />Like all irreversible processes, though, there are exceptions even at<br />the coarse-grained level and worlds will occasionally fuse. A<br />necessary, although not sufficient, precondition for fusing is for all<br />records, memories etc. that discriminate between the pre-fused worlds or<br />histories be lost. This is not a common occurrence.<p><hr noshade /><br /><br /><a name=splitsh><h3>Q8 When does Schrodinger's cat split?</h3><br /> <br />Consider Schrodinger's cat. A cat is placed in a sealed box with a<br />device that releases a lethal does of cyanide if a certain radioactive<br />decay is detected. For simplicity we'll imagine that the box, whilst<br />closed, completely isolates the cat from its environment. After a while<br />an investigator opens the box to see if the cat is alive or dead. <br />According to the Copenhagen Interpretation the cat was neither alive nor<br />dead until the box was opened, whereupon the wavefunction of the cat<br />collapsed into one of the two alternatives (alive or dead cat). The<br />paradox, according to Schrodinger, is that the cat presumably knew if<br />it was alive *before* the box was opened. According to many-worlds the<br />device was split into two states (cyanide released or not) by the<br />radioactive decay, which is a thermodynamically irreversible process<br />(See <a href=#do split>"When do worlds split?"</a> and <a href=#split>"Why do worlds split?"</a>). As the<br />cyanide/no-cyanide interacts with the cat the cat is split into two<br />states (dead or alive). From the surviving cat's point of view it<br />occupies a different world from its deceased copy. The onlooker is<br />split into two copies only when the box is opened and they are altered<br />by the states of the cat.<P><br /><br />The cat splits when the device is triggered, irreversibly. The<br />investigator splits when they open the box. The alive cat has no idea<br />that investigator has split, any more than it is aware that there is a<br />dead cat in the neighbouring split-off world. The investigator can<br />deduce, after the event, by examining the cyanide mechanism, or the<br />cat's memory, that the cat split prior to opening the box.<p><hr noshade /><br /><br /><a name=sum><h3>Q9 What is sum-over-histories?</h3><br /> <br />The sum-over-histories or path-integral formalism of quantum mechanics<br />was developed by Richard Feynman in the 1940s [F] as a third<br />interpretation of quantum mechanics, alongside Schrodinger's wave<br />picture and Heisenberg's matrix mechanics, for calculating transition<br />amplitudes. All three approaches are mathematically equivalent, but the<br />path-integral formalism offers some interesting additional insights into<br />many-worlds.<p><br /><br />In the path-integral picture the wavefunction of a single particle at<br />(x',t') is built up of contributions of all possible paths from (x,t),<br />where each path's contribution is weighted by a (phase) factor of<br />exp(i*Action[path]/hbar) * wavefunction at (x,t), summed, in turn, over<br />all values of x. The Action[path] is the time-integral of the<br />lagrangian (roughly: the lagrangian equals kinetic minus the potential<br />energy) along the path from (x,t) to (x',t'). The final expression is<br />thus the sum or integral over all paths, irrespective of any classical<br />dynamical constraints. For N-particle systems the principle is the<br />same, except that the paths run through a 3-N space.<p><br /><br />In the path-integral approach every possible path through configuration<br />space makes a contribution to the transition amplitude. From this point<br />of view the particle explores every possible intermediate configuration<br />between the specified start and end states. For this reason the path-<br />integral technique is often referred to as "sum-over-histories". Since<br />we do not occupy a privileged moment in history it is natural to wonder<br />if alternative histories are contributing equally to transition<br />amplitudes in the future, and that each possible history has an equal<br />reality. Perhaps we shouldn't be surprised that Feynman is on record<br />as believing in many-worlds. (See <a href=#believes in>"Who believes in many-worlds?"</a>) What<br />is surprising is that Everett developed his many-worlds theory entirely<br />from the Schrodinger viewpoint without any detectable influence from<br />Feynman's work, despite Feynman and Everett sharing the same Princeton<br />thesis supervisor, John A Wheeler.<p><br /><br />Feynman developed his path-integral formalism further during his work<br />on quantum electrodynamics, QED, in parallel with Schwinger and Tomonoga<br />who had developed a less visualisable form of QED. Dyson showed that<br />these approaches were all equivalent. Feynman, Schwinger and Tomonoga<br />were awarded the 1965 Physics Nobel Prize for this work. Feynman's<br />approach was to show how any process, with defined in (initial) and out<br />(final) states, can be represented by a series of (Feynman) diagrams,<br />which allow for the creation, exchange and annihilation of particles. <br />Each Feynman diagram represents a different contribution to the complete<br />transition amplitude, provided that the external lines map onto the<br />required boundary initial and final conditions (the defined in and out<br />states). QED became the prototype for all the other, later, field<br />theories like electro-weak and quantum chromodynamics.<p><br /><br />[F] Richard P Feynman <I>Space-time approach to non-relativistic quantum<br /> mechanics</I> Reviews of Modern Physics, Vol 20: 267-287 (1948)<hr noshade /><br /><br /><a name=many><h3>Q10<br /> What is many-histories?</h3><br /> <br /> <a name=environment> <b>What is the environment basis?</b><br /> <br />There is considerable linkage between thermodynamics and many-worlds,<br />explored in the "decoherence" views of Zurek [7a], [7b] and Gell-Mann<br />and Hartle [10], Everett [1], [2] and others [4b]. (See <a href=#decoherence>"What is<br />decoherence?"</a>)<p> <br /><br /><br />Gell-Mann and Hartle, in particular, have extended the role of<br />decoherence in defining the Everett worlds, or "histories" in their<br />nomenclature. They call their approach the "many-histories" approach,<br />where each "coarse-grained or classical history" is associated with a<br />unique time-ordered sequence of sets of irreversible events, including<br />measurements, records, observations and the like. (See <a href=#measurement>"What is a<br />measurement?"</a>) Fine-grained histories effectively relax the<br />irreversible criterion. Mathematically the many-histories approach is<br />isomorphic to Everett's many-worlds.<p><br /><br />The worlds split or "decohere" from each other when irreversible events<br />occur. (See <a href=#split>"Why do worlds split?"</a> and <a href=#do split>"When do worlds split?"</a>) <br />Correspondingly many-histories defines a multiply-connected hierarchy<br />of classical histories where each classical history is a "child" of any<br />parent history which has only a subset of the child defining<br />irreversible events and a parent of any history which has a superset of<br />such events. Climbing up the tree from child to parent moves to<br />progressively coarser grained consistent histories until eventually the<br />top is reached where the history has <I>no</I> defining events (and thus<br />consistent with everything!). This is Everett's universal wavefunction. <br />The bottom of the coarse-grained tree terminates with the maximally<br />refined set of decohering histories. The classical histories each have<br />a probability assigned to them and probabilities are additive in the<br />sense that the sum of the probabilities associated a set classical<br />histories is equal to the probability associated with the unique parent<br />history defined by the set. (Below the maximally refined classical<br />histories are the fine grained or quantum histories, where probabilities<br />are no longer additive and different histories significantly interfere<br />with each other. The bottom level consists of complete microstates,<br />which fully specified states.)<p><br /><br />The decoherence approach is useful in considering the effect of the<br />environment on a system. In many ways the environment, acting as a heat<br />sink, can be regarded as performing a succession of measurement-like<br />interactions upon any system, inducing associated system splits. All<br />the environment basis is a basis chosen so as to minimise the cross-<br />basis interference terms. It makes any real-worlds calculation easy,<br />since the cross terms are so small, but it does not <I>uniquely</I> select<br />a basis, just eliminates a large number.<p><hr noshade /><br /><br /><a name=how many><h3>Q11 How many worlds are there?</h3><br /> <br />The thermodynamic Planck-Boltzmann relationship, S = k*log(W), counts<br />the branches of the wavefunction at each splitting, at the lowest,<br />maximally refined level of Gell-Mann's many-histories tree. (See <a href=#many>"What<br />is many-histories?"</a>) The bottom or maximally divided level consists of<br />microstates which can be counted by the formula W = exp (S/k), where S<br />= entropy, k = Boltzmann's constant (approx 10^-23 Joules/Kelvin) and<br />W = number of worlds or macrostates. The number of coarser grained<br />worlds is lower, but still increasing with entropy by the same ratio,<br />i.e. the number of worlds a single world splits into at the site of an<br />irreversible event, entropy dS, is exp(dS/k). Because k is very small<br />a great many worlds split off at each macroscopic event.<p><hr noshade /><br /><br /><a name=local><h3>Q12 Is many-worlds a local theory?</h3><br /> <br />The simplest way to see that the many-worlds metatheory is a local<br />theory is to note that it requires that the wavefunction obey some<br />relativistic wave equation, the exact form of which is currently<br />unknown, but which is presumed to be locally Lorentz invariant at all<br />times and everywhere. This is equivalent to imposing the requirement<br />that locality is enforced at all times and everywhere. Ergo many-worlds<br />is a local theory.<p><br /><br />Another way of seeing this is examine how macrostates evolve. <br />Macrostates descriptions of objects evolve in a local fashion. Worlds<br />split as the macrostate description divides inside the light cone of the<br />triggering event. Thus the splitting is a local process, transmitted<br />causally at light or sub-light speeds. (See <a href=#locality>"Does the EPR experiment<br />prohibit locality?"</a> and <a href=#do split>"When do worlds split?"</a>)<p><hr noshade /><br /><br /><a name=deterministic><h3>Q13 Is many-worlds a deterministic theory?</h3><br /> <br />Yes, many-worlds is a deterministic theory, since the wavefunction obeys<br />a deterministic wave equation at all times. All possible outcomes of<br />a measurement or interaction (See<a href=#measurement> "What is a measurement?"</a>) are embedded<br />within the universal wavefunction although each observer, split by each<br />observation, is only aware of single outcomes due to the linearity of<br />the wave equation. The world appears indeterministic, with the usual<br />probabilistic collapse of the wavefunction, but at the objective level,<br />which includes all outcomes, determinism is restored.<p><br /><br />Some people are under the impression that the only motivation for many-<br />worlds is a desire to return to a deterministic theory of physics. This<br />is not true. As Everett pointed out, the objection with the standard<br />Copenhagen interpretation is not the indeterminism per se, but that<br />indeterminism occurs only with the intervention of an observer, when the<br />wavefunction collapses. (See <a href=#copenhagen>"What is the Copenhagen interpretation?"</a>)<p><hr noshade /><br /><br /><a name=relativistic><h3>Q14<br /> Is many-worlds a relativistic theory?<br /><br /><br /><a name=field> What about quantum field theory?<br /><br /> <br /><a name=quantum gravity> What about quantum gravity?<br /></h3><br /> <br /><br />It is trivial to relativise many-worlds, at least to the level of<br />special relativity. All relativistic theories of physics are quantum<br />theories with linear wave equations. There are three or more stages to<br />developing a fully relativised quantum field theory:<p><br /><br />First quantisation: the wavefunction of an N particle system is a<br />complex field which evolves in 3N dimensions as the solution to either<br />the many-particle Schrodinger, Dirac or Klein-Gordon or some other wave<br />equation. External forces applied to the particles are represented or<br />modelled via a potential, which appears in the wave equation as a<br />classical, background field.<p><br /><br />Second quantisation: AKA (relativistic) quantum field theory (QFT)<br />handles the creation and destruction of particles by quantising the<br />classical fields and potentials as well as the particles. Each particle<br />corresponds to a field, in QFT, and becomes an operator. E.g. the<br />electromagnetic field's particle is the photon. The wavefunction of a<br />collection of particles/fields exists in a Fock space, where the number<br />of dimensions varies from component to component, corresponding to the<br />indeterminacy in the particle number. Many-worlds has no problems<br />incorporating QFT, since a theory (QFT) is not altered by a metatheory<br />(many-worlds), which makes statements <I>about</I> the theory.<p><br /><br />Third quantisation: AKA quantum gravity. The gravitational metric is<br />quantised, along with (perhaps) the topology of the space-time manifold. <br />The role of time plays a less central role, as might be expected, but<br />the first and second quantisation models are as applicable as ever for<br />modelling low-energy events. The physics of this is incomplete,<br />including some thorny, unresolved conceptual issues, with a number of<br />proposals (strings, supersymmetry, supergravity...) for ways forward,<br />but the extension required by many-worlds is quite trivial since the<br />mathematics would be unchanged.<p><br /><br />One of the original motivations of Everett's scheme was to provide a<br />system for quantising the gravitational field to yield a quantum<br />cosmology, permitting a complete, self-contained description of the<br />universe. Indeed many-words actually <I>requires</I> that gravity be<br />quantised, in contrast to other interpretations which are silent about<br />the role of gravity. (See "Why <a href=#quantum gravity><I>quantum</I> gravity?"</a>)<p><hr noshade /><br /><br /><a name=where are><h3>Q15 Where are the other worlds?</h3><br /> <br /><br />Non-relativistic quantum mechanics and quantum field theory are quite<br />unambiguous: the other Everett-worlds occupy the same space and time as<br />we do.<p><br /><br />The implicit question is really, why aren't we aware of these other<br />worlds, unless they exist "somewhere" else? To see why we aren't aware<br />of the other worlds, despite occupying the same space-time, see "Why do<br />I only ever experience one world?" Some popular accounts describe the<br />other worlds as splitting off into other, orthogonal, dimensions. These<br />dimensions are the dimensions of Hilbert space, not the more familiar<br />space-time dimensions.</p><br /><br />The situation is more complicated, as we might expect, in theories of<br />quantum gravity (See <a href=#quantum gravity>"What about quantum gravity?"</a>), because gravity can<br />be viewed as perturbations in the space-time metric. If we take a<br />geometric interpretation of gravity then we can regard differently<br />curved space-times, each with their own distinct thermodynamic history,<br />as non-coeval. In that sense we only share the same space-time manifold<br />with other worlds with a (macroscopically) similar mass distribution. <br />Whenever the amplification of a quantum-scale interaction effects the<br />mass distribution and hence space-time curvature the resultant<br />decoherence can be regarded as splitting the local space-time manifold<br />into discrete sheets.<p><hr noshade /><br /><br /><a name=interpretation><h3>Q16 Is many-worlds (just) an interpretation?</h3><br /> <br />No, for four reasons:<p><br /><br />First, many-worlds makes predictions that differ from the other so-<br />called interpretations of quantum theory. Interpretations do not make<br />predictions that differ. (See <a href=#unique>"What unique predictions does many-worlds<br />make?"</a>) In addition many-worlds retrodicts a lot of data that has no<br />other easy interpretation. (See <a href=#retrodictions>"What retrodictions does many-worlds<br />make?"</a>)<p><br /><br />Second, the mathematical structure of many-worlds is not isomorphic to<br />other formulations of quantum mechanics like the Copenhagen<br />interpretation or Bohm's hidden variables. The Copenhagen<br />interpretation does not contain those elements of the wavefunction that<br />correspond to the other worlds. Bohm's hidden variables contain<br />particles, in addition to the wavefunction. Neither theory is<br />isomorphic to each other or many-worlds and are not, therefore, merely<br />rival "interpretations".<p><br /><br />Third, there is no scientific, reductionistic alternative to many-<br />worlds. All the other theories fail for logical reasons. (See <a href=#alternatives>"Is<br />there any alternative theory?"</a>)<p><br /><br />Fourth, the interpretative side of many-worlds, like the subjective<br />probabilistic elements, are derived from within the theory, rather than<br />added to it by assumption, as in the conventional approach. (See <a href=#probabilities>"How<br />do probabilities emerge within many-worlds?"</a>)<p><br /><br />Many-worlds should really be described as a theory or, more precisely,<br />a metatheory, since it makes statements that are applicable about a<br />range of theories. Many-worlds is the unavoidable implication of any<br />quantum theory which obeys some type of linear wave equation. (See <a href=#linear>"Is<br />physics linear?"</a>)<p><hr noshade /><br /><br /><a name=fuse><h3> Q17<br /> Why don't worlds fuse, as well as split?<br /><br /><a name=irreversible>Do splitting worlds imply irreversible physics?<br /></h3><br />This is really a question about why thermodynamics works and what is the<br />origin of the "arrow of time", rather than about many-worlds.<p><br /><br />First, worlds almost never fuse, in the forward time direction, but<br />often divide, because of the way we have defined them. (See <a href=#decoherence>"What is<br />decoherence?"</a>, <a href=#split>"Why do worlds split?"</a> and <a href=#do split>"When do worlds split?"</a>) The<br />Planck-Boltzmann formula for the number of worlds (See <a href=#how many>"How many worlds<br />are there?"</a>) implies that where worlds to fuse together then entropy<br />would decrease, violating the second law of thermodynamics.<p><br /><br />Second, this does not imply that irreversible thermodynamics is<br />incompatible with reversible (or nearly so) microphysics. The laws of<br />physics are reversible (or CPT invariant, more precisely) and fully<br />compatible with the irreversibility of thermodynamics, which is solely<br />due to the boundary conditions (the state of universe at some chosen<br />moment) imposed by the Big Bang or whatever we chose to regard as the<br />initial conditions. (See <a href=#boundary>"Why can't the boundary conditions be updated<br />to reflect my observations in this one world?"</a>)<p><hr noshade /><br /><br /><a name=retrodictions><h3>Q18 What retrodictions does many-worlds make?</h3><br /> <br />A retrodiction occurs when already gathered data is accounted for by a<br />later theoretical advance in a more convincing fashion. The advantage<br />of a retrodiction over a prediction is that the already gathered data<br />is more likely to be free of experimenter bias. An example of a<br />retrodiction is the perihelion shift of Mercury which Newtonian<br />mechanics plus gravity was unable, totally, to account for whilst<br />Einstein's general relativity made short work of it.<p><br /><br />Many-worlds retrodicts all the peculiar properties of the (apparent)<br />wavefunction collapse in terms of decoherence. (See <a href=#decoherence>"What is<br />decoherence?"</a>, <a href=#collapse>"Can wavefunctions collapse?"</a>, <a href=#do split>"When do worlds split?"</a>and <a href=#split>"Why do worlds split?"</a>) No other quantum theory has yet accounted for<br />this behaviour scientifically. (See <a href=#alternatives>"What are the alternatives to many-<br />worlds?"</a>)<p><hr noshade /><br /><br /><a name=differentiate><h3>Q19 Do worlds differentiate or split?</h3><br /> <br />Can we regard the separate worlds that result from a measurement-like<br />interaction (See <a href=#measurement>"What is a measurement?"</a>) as having previous existed<br />distinctly and merely differentiated, rather than the interaction as<br />having split one world into many? This is definitely not permissible<br />in many-worlds or any theory of quantum theory consistent with<br />experiment. Worlds do not exist in a quantum superposition<br />independently of each other before they decohere or split. The<br />splitting is a physical process, grounded in the dynamical evolution of<br />the wave vector, not a matter of philosophical, linguistic or mental<br />convenience (see <a href=#split>"Why do worlds split?"</a> and <a href=#do split>"When do worlds split?"</a>) <br />If you try to treat the worlds as pre-existing and separate then the<br />maths and probabilistic behaviour all comes out wrong. Also the<br />differentiation theory isn't deterministic, in contradiction to the wave<br />equations which are deterministic, since many-minds says that:<p><pre><br /><br /> AAAAAAAAAAAAAAABBBBBBBBBBBBBBB --------------> time<br /> (Worlds differentiate)<br /> AAAAAAAAAAAAAAACCCCCCCCCCCCCCC<br /><br />occurs, rather than:<br /> BBBBBBBBBBBBBBB<br /> B<br /> AAAAAAAAAAAAAA (Worlds split)<br /> C<br /> CCCCCCCCCCCCCCC<br /><br />according to many-worlds.</pre><p><br /><br />This false differentiation model, at the mental level, seems favoured<br />by adherents of many-minds. (See <a href=#minds>"What is many-minds?"</a>)<p><hr noshade /><br /><br /><a name=minds><h3>Q20 What is many-minds?</h3><br /> <br />Many-minds proposes, as an extra fundamental axiom, that an infinity of<br />separate minds or mental states be associated with each single brain<br />state. When the single physical brain state is split into a quantum<br />superposition by a measurement (See <a href=#measurement>"What is a measurement?"</a>) the<br />associated infinity of minds are thought of as differentiating rather<br />than splitting. The motivation for this brain-mind dichotomy seems<br />purely to avoid talk of minds splitting and talk instead about the<br />differentiation of pre-existing separate mental states. There is no<br />physical basis for this interpretation, which is incapable of an<br />operational definition. Indeed the differentiation model for physical<br />systems is specifically not permitted in many-worlds. Many-minds seems<br />to be proposing that minds follow different rules than matter. (See <a href=#differentiate>"Do<br />worlds differentiate or split?"</a>)<p><br /><br />In many-minds the role of the conscious observer is accorded special<br />status, with its fundamental axiom about infinities of pre-existing<br />minds, and as such is philosophically opposed to many-worlds, which<br />seeks to remove the observer from any privileged role in physics. <br />(Many-minds was co-invented by David Albert, who has, apparently, since<br />abandoned it. See Scientific American July 1992 page 80 and contrast<br />with Albert's April '94 Scientific American article.)<p><br /><br />The two theories must not be confused. <p><hr noshade /> <br /><br /><a name=ockham's><h3>Q21 Does many-worlds violate Ockham's Razor?</h3><br /> <br />William of Ockham, 1285-1349(?) English philosopher and one of the<br />founders of logic, proposed a maxim for judging theories which says that<br />hypotheses should not be multiplied beyond necessity. This is known as<br />Ockham's razor and is interpreted, today, as meaning that to account for<br />any set of facts the simplest theories are to be preferred over more<br />complex ones. Many-worlds is viewed as unnecessarily complex, by some,<br />by requiring the existence of a multiplicity of worlds to explain what<br />we see, at any time, in just one world.<p><br /><br />This is to mistake what is meant by "complex". Here's an example. <br />Analysis of starlight reveals that starlight is very similar to faint<br />sunlight, both with spectroscopic absorption and emission lines. <br />Assuming the universality of physical law we are led to conclude that<br />other stars and worlds are scattered, in great numbers, across the<br />cosmos. The theory that "the stars are distant suns" is the simplest<br />theory and so to be preferred by Ockham's Razor to other geocentric<br />theories.<p><br /><br />Similarly many-worlds is the simplest and most economical quantum theory<br />because it proposes that same laws of physics apply to animate observers<br />as has been observed for inanimate objects. The multiplicity of worlds<br />predicted by the theory is not a weakness of many-worlds, any more than<br />the multiplicity of stars are for astronomers, since the non-interacting<br />worlds emerge from a simpler theory.<p><br /><br />(As an historical aside it is worth noting that Ockham's razor was also<br />falsely used to argue in favour of the older heliocentric theories<br /><I>against</I> Galileo's notion of the vastness of the cosmos. The notion<br />of vast empty interstellar spaces was too uneconomical to be believable<br />to the Medieval mind. Again they were confusing the notion of vastness<br />with complexity [15].)<p><hr noshade /><br /><br /><a name=violate><h3>Q22 Does many-worlds violate conservation of energy?</h3><br /> <br />First, the law conservation of energy is based on observations within<br />each world. All observations within each world are consistent with<br />conservation of energy, therefore energy is conserved.<p><br /><br />Second, and more precisely, conservation of energy, in QM, is formulated<br />in terms of weighted averages or expectation values. Conservation of<br />energy is expressed by saying that the time derivative of the expected<br />energy of a closed system vanishes. This statement can be scaled up to<br />include the whole universe. Each world has an approximate energy, but<br />the energy of the total wavefunction, or any subset of, involves summing<br />over each world, weighted with its probability measure. This weighted<br />sum is a constant. So energy is conserved within each world and also<br />across the totality of worlds.<P><br /><br />One way of viewing this result - that observed conserved quantities are<br />conserved across the totality of worlds - is to note that new worlds are<br />not created by the action of the wave equation, rather existing worlds<br />are split into successively "thinner" and "thinner" slices, if we view<br />the probability densities as "thickness".<p><hr noshade /><br /><br /><a name=probabilities><h3>Q23 How do probabilities emerge within many-worlds?</h3><br /> <br />Everett demonstrated [1], [2] that observations in each world obey all<br />the usual conventional statistical laws predicted by the probabilistic<br />Born interpretation, by showing that the Hilbert space's inner product<br />or norm has a special property which allows us to makes statements about<br />the worlds where quantum statistics break down. The norm of the vector<br />of the set of worlds where experiments contradict the Born<br />interpretation ("non-random" or "maverick" worlds) vanishes in the limit<br />as the number of probabilistic trials goes to infinity, as is required<br />by the frequentist definition of probability. Hilbert space vectors<br />with zero norm don't exist (see below), thus we, as observers, only<br />observe the familiar, probabilistic predictions of quantum theory. <br />Everett-worlds where probability breaks down are never realised.<p><br /><br />Strictly speaking Everett did not prove that the usual statistical laws<br />of the Born interpretation would hold true for all observers in all<br />worlds. He merely showed that no other statistical laws could hold true<br />and asserted the vanishing of the Hilbert space "volume" or norm of the<br />set of "maverick" worlds. DeWitt later published a longer <I>derivation</I><br />of Everett's assertion [4a], [4b], closely based on an earlier,<br />independent demonstration by Hartle [H]. What Everett asserted, and<br />DeWitt/Hartle derived, is that the collective norm of all the maverick<br />worlds, as the number of trials goes to infinity, vanishes. Since the<br />only vector in a Hilbert space with vanishing norm is the null vector<br />(a defining axiom of Hilbert spaces) this is equivalent to saying that<br />non-randomness is never realised. All the worlds obey the usual Born<br />predictions of quantum theory. That's why we never observe the<br />consistent violation of the usual quantum statistics, with, say, heat<br />flowing from a colder to a hotter macroscopic object. Zero-probability<br />events never happen.<p><br /><br />Of course we have to assume that the wavefunction is a Hilbert space<br />vector in the first place but, since this assumption is also made in the<br />standard formulation, this is not a weakness of many-worlds since we are<br />not trying to justify all the axioms of the conventional formulation of<br />QM, merely those that relate to probabilities and collapse of the<br />wavefunction.<p><br /><br />In more detail the steps are:<p><br /><br /><B>1)</B> Construct the tensor product of N identical systems in state |psi>,<br /> according to the usual rules for Hilbert space composition<br /> (repeated indices summed):<pre><br /><br /><br /> |PSI_N> = |psi_1>*|psi_2>*...... |psi_N> where<br /> |psi_j> = jth system prepared in state |psi><br /> = |i_j><i_j|psi> (ie the amplitude of the ith eigenstate<br /> is independent of which system it is in)<br /> so that <br /> |PSI_N> = |i_1>|i_2>...|i_N><i_1|psi><i_2|psi>...<i_N|psi><p><br /><br /><b>2)</B> Quantify the deviation from the "expected" Born-mean for each<br /> component of |PSI_N> with respect to the above |i_1>|i_2>...|i_N><br /> basis by counting the number of occurrences of the ith<br /> eigenstate/N. Call this number RF(i). Define the Born-deviation<br /> as D = sum(i)( (RF(i) - |<i|psi>|^2)^2 ). Thus D, loosely<br /> speaking, for each N length sequence, quantifies by how much the<br /> particular sequence differs from the Born-expectation.<p><br /><br /><B>3)</B> Sort out terms in the expansion of |PSI_N> according to whether D<br /> is less/equal to (.LE.) or greater than (.GT.) E, where E is a<br /> real, positive constant. Collecting terms together we get:<br /><br /><br /> |PSI_N> = |N,"D.GT.E"> + |N,"D.LE.E"><br /> worlds worlds<br /> for which for which<br /> D > E D <= E<p><br /><br /><B>4)</B> What DeWitt showed was that:<br /><br /> <N,"D.GT.E"|N,"D.GT.E"> < 1/(NE) (proof in appendix of [4b])<br /> Thus as N goes to infinity the right-hand side vanishes for all<br /> positive values of E. (This mirrors the classical "frequentist"<br /> position on probability which states that if event i occurs with<br /> probability p(i) then the proportion of N trials with outcome i<br /> approaches p(i)/N as N goes to infinity [H]. This has the<br /> immediate benefit that sum(i) p(i) = 1.) The norm of |N,"D.LE.E">,<br /> by contrast, approaches 1 as N goes to infinity.<p><br /><br /> Note: this property of D is not shared by other definitions, which<br /> is why we haven't investigated them. If, say, we had defined, in<br /> step 2), A = sum(i)( (RF(i) - |<i|psi>|)^2 ), so that A measures<br /> the deviation from |psi|, rather than |psi|^2, then we find that<br /> <A> does not have the desired property of vanishing as N goes to<br /> infinity.<p><br /><br /><br /><B>5)</B> The norm of the collection of non-random worlds vanishes and<br /> therefore must be identified with some complex multiple of the null<br /> vector.<p><br /><br /><B>6)</B> Since (by assumption) the state vector faithfully models reality<br /> then the null vector cannot represent any element of reality, since<br /> it can be added to (or subtracted from) any other state vector<br /> without altering the other state vector.<p><br /><br /><B>7)</B> Ergo the non-random worlds are not realised, without making any<br /> additional physical assumptions, such the imposition of a measure.<p><br /><br /> <b> Note:</b> no finite sequence of outcomes is excluded from happening,<br /> since the concept of probability and randomness only becomes<br /> precise only as N goes to infinity [H]. Thus, heat <I>could</I> be<br /> observed to flow from a cold to hotter object, but we might have<br /> to wait a very long time before observing it. What <I>is</I> excluded<br /> is the possibility of this process going on forever.<p></pre></I></I><br /><br /><br />The emergence of Born-style probabilities as a consequence of the<br />mathematical formalism of the theory, without any extra interpretative<br />assumptions, is another reason why the Everett metatheory should not be<br />regarded as just an interpretation. (See <a href=#interpretation>"Is many-worlds (just) an<br />interpretation?"</a>) The interpretative elements are forced by the<br />mathematical structure of the axioms of Hilbert space.<p><br /><br />[H] JB Hartle <I>Quantum Mechanics of Individual Systems</I> American<br /> Journal of Physics Vol 36 #8 704-712 (1968) Hartle has<br /> investigated the N goes to infinity limit in more detail and more<br /> generally. He shows that the relative frequency operator, RF,<br /> obeys RF(i) |psi_1>|psi_2>.... = |<i|psi>|^2 |psi_1>|psi_2>....,<br /> for a normed state. Hartle regarded his derivation as essentially<br /> the same as Everett's, despite being derived independently.<p></pre><hr noshade /><br /><br /><a name=free-will><h3>Q24 Does many-worlds allow free-will?</h3><br /> <br />Many-Worlds, whilst deterministic on the objective universal level, is<br />indeterministic on the subjective level so the situation is certainly<br />no better or worse for free-will than in the Copenhagen view. <br />Traditional Copenhagen indeterministic quantum mechanics only slightly<br />weakens the case for free-will. In quantum terms each neuron is an<br />essentially classical object. Consequently quantum noise in the brain<br />is at such a low level that it probably doesn't often alter, except very<br />rarely, the critical mechanistic behaviour of sufficient neurons to<br />cause a decision to be different than we might otherwise expect. The<br />consensus view amongst experts is that free-will is the consequence of<br />the mechanistic operation of our brains, the firing of neurons,<br />discharging across synapses etc. and fully compatible with the<br />determinism of classical physics. Free-will is the inability of an<br />intelligent, self-aware mechanism to predict its own future actions due<br />to the logical impossibility of any mechanism containing a complete<br />internal model of itself rather than any inherent indeterminism in the<br />mechanism's operation.<p><br /><br />Nevertheless, some people find that with all possible decisions being<br />realised in different worlds that the prima face situation for free-<br />will looks quite difficult. Does this multiplicity of outcomes destroy<br />free-will? If both sides of a choice are selected in different worlds<br />why bother to spend time weighing the evidence before selecting? The<br />answer is that whilst all decisions are realised, some are realised more<br />often than others - or to put to more precisely each branch of a<br />decision has its own weighting or measure which enforces the usual laws<br />of quantum statistics.<p><br /><br />This measure is supplied by the mathematical structure of the Hilbert<br />spaces. Every Hilbert space has a norm, constructed from the inner<br />product, - which we can think of as analogous to a volume - which<br />weights each world or collection of worlds. A world of zero volume is<br />never realised. Worlds in which the conventional statistical<br />predictions consistently break down have zero volume and so are never<br />realised. (See <a href=#probabilities>"How do probabilities emerge within many-worlds?"</a>)<p> <br /><br />Thus our actions, as expressions of our will, correlate with the weights<br />associated with worlds. This, of course, matches our subjective<br />experience of being able to exercise our will, form moral judgements and<br />be held responsible for our actions.<p><hr noshade /><br /><br /><a name=#I in this><h3>Q25<br /> Why am I in this world and not another?<br /><br /> <br /><a name=#random> Why does the universe appear random?</h3><br /> <br />These are really the same questions. Consider, for a moment, this<br />analogy:<p><br /><br />Suppose Fred has his brain divided in two and transplanted into two<br />different cloned bodies (this is a gedanken operation! <B>[*]</B>). Let's<br />further suppose that each half-brain regenerates to full functionality<br />and call the resultant individuals Fred-Left and Fred-Right. Fred-Left<br />can ask, why did I end up as Fred-Left? Similarly Fred-Right can ask,<br />why did I end up as Fred-Right? The only answer possible is that there<br />was <I>no</I> reason. From Fred's point of view it is a subjectively<br /><I>random</I> choice which individual "Fred" ends up as. To the surgeon the<br />whole process is deterministic. To both the Freds it seems random.<p><br /><br />Same with many-worlds. There was no reason "why" you ended up in this<br />world, rather than another - you end up in all the quantum worlds. It<br />is a subjectively random choice, an artefact of your brain and<br />consciousness being split, along with the rest of the world, that makes<br />our experiences seem random. The universe is, in effect, performing<br />umpteen split-brain operations on us all the time. The randomness<br />apparent in nature is a consequence of the continual splitting into<br />mutually unobservable worlds.<p><br /><br />(See <a href=#probabilities>"How do probabilities emerge within many-worlds?"</a> for how the<br />subjective randomness is moderated by the usual probabilistic laws of<br />QM.)<p><br /><br /><B>[*]</B> Split brain experiments <I>were</I> performed on epileptic patients<br />(severing the corpus callosum, one of the pathways connecting the<br />cerebral hemispheres, moderated epileptic attacks). Complete<br />hemispherical separation was discontinued when testing of the patients<br />revealed the presence of two distinct consciousnesses in the same skull. <br />So this analogy is only partly imaginary.<p><hr noshade /><br /><br /><a name=wavefunctions><h3> Q26 Can wavefunctions collapse?</h3><br /> <br />Many-worlds predicts/retrodicts that wavefunctions appear to collapse<br />(See <a href=#epr>"Does the EPR experiment prohibit locality?"</a>), when measurement-<br />like interactions (See <a href=#measurement>"What is a measurement?"</a>) and processes occur via<br />a process called decoherence (See <a href=#decoherence>"What is decoherence?"</a>), but claims<br />that the wavefunction does not <I>actually</I> collapse but continues to<br />evolve according to the usual wave-equation. If a <I>mechanism</I> for<br />collapse could be found then there would be no need for many-worlds. <br />The reason why we doubt that collapse takes place is because no one has<br />ever been able to devise a physical mechanism that could trigger it.<p><br /><br />The Copenhagen interpretation posits that observers collapse<br />wavefunctions, but is unable to define "observer". (See <a href=#copenhagen>"What is the<br />Copenhagen interpretation?"</a> and <a href=#alternatives>"Is there any alternative theory?"</a>) <br />Without a definition of observer there can be no mechanism triggered by<br />their presence.<p><br /><br />Another popular view is that irreversible processes trigger collapse. <br />Certainly wavefunctions <I>appear</I> to collapse whenever irreversible<br />processes are involved. And most macroscopic, day-to-day events are<br />irreversible. The problem is, as with positing observers as a cause of<br />collapse, that any irreversible process is composed of a large number<br />of sub-processes that are each individually reversible. To invoke<br />irreversibility as a <I>mechanism</I> for collapse we would have to show that<br />new <I>fundamental</I> physics comes into play for complex systems, which is<br />quite absent at the reversible atom/molecular level. Atoms and<br />molecules are empirically observed to obey some type of wave equation. <br />We have no evidence for an extra mechanism operating on more complex<br />systems. As far as we can determine complex systems are described by<br />the quantum-operation of their simpler components interacting together. <br />(Note: chaos, complexity theory, etc., do not introduce new fundamental<br />physics. They still operate within the reductionistic paradigm -<br />despite what many popularisers say.)<p><br /><br />Other people have attempted to construct non-linear theories so that<br />microscopic systems are approximately linear and obey the wave equation,<br />whilst macroscopic systems are grossly non-linear and generates<br />collapse. Unfortunately all these efforts have made additional<br />predictions which, when tested, have failed. (See <a href=#linear>"Is physics linear?"</a>)<p><br /><br />(Another reason for doubting that any collapse actually takes place is<br />that the collapse would have to propagate instantaneously, or in some<br />space-like fashion, otherwise the same particle could be observed more<br />than once at different locations. Not fatal, but unpleasant and<br />difficult to reconcile with special relativity and some conservation<br />laws.)<p><br /><br />The simplest conclusion, which is to be preferred by Ockham's razor, is<br />that wavefunctions just <I>don't</I> collapse and that all branches of the<br />wavefunction exist.<p><hr noshade /><br /><br /><a name=linear><h3> Q27<br /> Is physics linear?<br /><br /><br /> <br /> <a name=communicate>Could we ever communicate with the other worlds?<br /><br /> <br /> <a name=experience>Why do I only ever experience one world?<br /><br /> <br /> <a name=not aware>Why am I not aware of the world (and myself) splitting?</h3><br /> <br />According to our present knowledge of physics whilst it is possible to<br />detect the presence of other nearby worlds, through the existence of<br />interference effects, it is impossible travel to or communicate with<br />them. Mathematically this corresponds to an empirically verified<br />property of all quantum theories called linearity. Linearity implies<br />that the worlds can interfere with each other with respect to a<br />external, unsplit, observer or system but the interfering worlds can't<br />influence each other in the sense that an experimenter in one of the<br />worlds can arrange to communicate with their own, already split-off,<br />quantum copies in other worlds.<p><br /><br />Specifically, the wave equation is linear, with respect to the<br />wavefunction or state vector, which means that given any two solutions<br />of the wavefunction, with identical boundary conditions, then any linear<br />combination of the solutions is another solution. Since each component<br />of a linear solution evolves with complete indifference as to the<br />presence or absence of the other terms/solutions then we can conclude<br />that no experiment in one world can have any effect on another<br />experiment in another world. Hence no communication is possible between<br />quantum worlds. (This type of linearity mustn't be confused with the<br />evident non-linearity of the equations with respect to the <I>fields</I>.)<p><br /><br />Non communication between the splitting Everett-worlds also explains why<br />we are not aware of any splitting process, since such awareness needs<br />communication between worlds. To be aware of the world splitting you<br />would have to be receiving sensory information from, and thereby effect<br />by the reverse process, more than one world. This would enable<br />communication between worlds, which is forbidden by linearity. Ergo,<br />we are not aware of any splitting precisely because we are split into<br />non-interfering copies along with the rest of the world.<p><br /><br />See also <a href=#exact>"Is linearity exact?"</a><p><hr noshade /><br /><br /><a name=determine><h3>Q28<br /> Can we determine what other worlds there are?<br /><br /> <br /><a name=knowable> Is the form of the Universal Wavefunction knowable?</h3><br /> <br /><br />To calculate the form of the universal wavefunction requires not only<br />a knowledge of its dynamics (which we have a good approximation to, at<br />the moment) but also of the boundary conditions. To actually calculate<br />the form of the universal wavefunction, and hence make inferences about<br /><I>all</I> the embedded worlds, we would need to know the boundary conditions<br />as well. We are presently restricted to making inferences about those<br />worlds with which have shared a common history up to some point, which<br />have left traces (records, fossils, etc.) still discernible today. This<br />restricts us to a subset of the extant worlds which have shared the same<br />boundary conditions with us. The further we probe back in time the less<br />we know of the boundary conditions and the less we can know of the<br />universal wavefunction.<P><br /><br />This limits us to drawing conclusions about a restricted subset of the<br />worlds - all the worlds which are consistent with our known history up<br />to a some common moment, before we diverged. The flow of historical<br />events is, according to chaos/complexity theory/thermodynamics, very<br />sensitive to amplification of quantum-scale uncertainty and this<br />sensitivity is a future-directed one-way process. We can make very<br />reliable deductions about the past from the knowledge future/present but<br />we can't predict the future from knowledge the past/present. <br />Thermodynamics implies that the future is harder to predict than the<br />past is to retrodict. Books get written about this "arrow of time"<br />problem but, for the purposes of this discussion, we'll accept the<br />thermodynamic origin of time's arrow is as given. The fossil and<br />historical records say that dinosaurs and Adolf Hitler once existed but<br />have less to say about the future.<P><br /><br />Consider the effects of that most quantum of activities, Brownian<br />motion, on the conception of individuals and the knock-on effects on the<br />course of history. Mutation itself, one of the sources of evolutionary<br />diversity, is a quantum event. For an example of the<br />biological/evolutionary implications see Stephen Jay Gould's book<br /><I>Wonderful Life</I> for an popular exploration of the thesis that the path<br />of evolution is driven by chance. According to Gould evolutionary<br />history forms an enormously diverse tree of possible histories - all<br />very improbable - with our path being selected by chance. According to<br />many-worlds all these other possibilities are realised. Thus there are<br />worlds in which Hitler won WW-II and other worlds in which the dinosaurs<br />never died out. We can be as certain of this as we are that Hitler and<br />the dinosaurs once existed in our own past.<P><br /><br />Whether or not we can ever determine the totality of the universal<br />wavefunction is an open question. If Steven Hawking's work on the no-<br />boundary-condition condition is ultimately successful, or it emerges<br />from some theory of everything, and many think it will, then the actual<br />form of the <I>total</I> wavefunction could, in principle, we determined from<br />a complete knowledge of physical law itself.<P><hr noshade /><br /><br /><a name=everett><h3>Q29 Who was Everett?</h3><br /> <br />Hugh Everett III (1930-1982) did his undergraduate study in chemical<br />engineering at the Catholic University of America. Studying von<br />Neumann's and Bohm's textbooks as part of his graduate studies, under<br />Wheeler, in mathematical physics at Princeton University in the 1950s<br />he became dissatisfied (like many others before and since) with the<br />collapse of the wavefunction. He developed, during discussions with<br />Charles Misner and Aage Peterson (Bohr' assistant, then visiting<br />Princeton), his "relative state" formulation. Wheeler encouraged his<br />work and preprints were circulated in January 1956 to a number of<br />physicists. A condensed version of his thesis was published as a paper<br />to <I>The Role of Gravity in Physics</I> conference held at the University<br />of North Carolina, Chapel Hill, in January 1957.<p><br /><br />Everett was discouraged by the lack of response from others,<br />particularly Bohr, whom he flew to Copenhagen to meet but got the<br />complete brush-off from. Leaving physics after completing his Ph.D.,<br />Everett worked as a defense analyst at the Weapons Systems Evaluation<br />Group, Pentagon and later became a private contractor, apparently quite<br />successfully for he became a multimillionaire. In 1968 Everett worked<br />for the Lambda Corp. His published papers during this period cover<br />things like optimising resource allocation and, in particular,<br />maximising kill rates during nuclear-weapon campaigns.<p><br /><br />From 1968 onwards Bryce S DeWitt, one of the 1957 Chapel Hill conference<br />organisers, but better known as one of the founders of quantum gravity,<br />successfully popularised Everett's relative state formulation as the<br />"many-worlds interpretation" in a series of articles [4a],[4b],[5].<p><br /><br />Sometime in 1976-9 Everett visited Austin, Texas, at Wheeler or DeWitt's<br />invitation, to give some lectures on QM. The strict no-smoking rule in<br />the auditorium was relaxed for Everett (a chain smoker); the only<br />exception ever. Everett, apparently, had a very intense manner,<br />speaking acutely and anticipating questions after a few words. Oh yes,<br />a bit of trivia, he drove a Cadillac with horns.<p><br /><br />With the steady growth of interest in many-worlds in the late 1970s<br />Everett planned returning to physics to do more work on measurement in<br />quantum theory, but died of a heart attack in 1982. Survived by his<br />wife.<p><hr noshade /><br /><br /><a name=problems><h3>Q30 What are the problems with quantum theory?</h3><br /> <br />Quantum theory is the most successful description of microscopic systems<br />like atoms and molecules ever, yet often it is not applied to larger,<br />classical systems, like observers or the entire universe. Many<br />scientists and philosophers are unhappy with the theory because it seems<br />to require a fundamental quantum-classical divide. Einstein, for<br />example, despite his early contributions to the subject, was never<br />reconciled with assigning to the act of observation a physical<br />significance, which most interpretations of QM require. This<br />contradicts the reductionist ethos that, amongst other things,<br />observations should emerge only as a consequence of an underlying<br />physical theory and not be present at the axiomatic level, as they are<br />in the Copenhagen interpretation. Yet the Copenhagen interpretation<br />remains the most popular interpretation of quantum mechanics amongst the<br />broad scientific community. (See <a href=#copenhagen>"What is the Copenhagen<br />interpretation?"</a>)<p><hr noshade /><br /><br /><a name=copenhagen><h3>Q31 What is the Copenhagen interpretation?</h3><br /> <br />An unobserved system, according to the Copenhagen interpretation of<br />quantum theory, evolves in a deterministic way determined by a wave<br />equation. An observed system changes in a random fashion, at the moment<br />of observation, instantaneously, with the probability of any particular<br />outcome given by the Born formula. This is known as the "collapse" or<br />"reduction" of the wavefunction. The problems with this approach are:<br /><br /><br />(1) The collapse is an instantaneous process across an extended<br /> region ("non-local") which is non-relativistic.<br /><br />(2) The idea of an observer having an effect on microphysics is<br /> repugnant to reductionism and smacks of a return to pre-scientific<br /> notions of vitalism. Copenhagenism is a return to the old vitalist<br /> notions that life is somehow different from other matter, operating<br /> by different laws from inanimate matter. The collapse is triggered<br /> by an observer, yet no definition of what an "observer" is<br /> available, in terms of an atomic scale description, even in<br /> principle.<p><br /><br />For these reasons the view has generally been adopted that the<br />wavefunction associated with an object is not a real "thing", but merely<br />represents our <I>knowledge</I> of the object. This approach was developed<br />by Bohr and others, mainly at Copenhagen in the late 1920s. When we<br />perform an measurement or observation of an object we acquire new<br />information and so adjust the wavefunction as we would boundary<br />conditions in classical physics to reflect this new information. This<br />stance means that we can't answer questions about what's actually<br />happening, all we can answer is what will be the probability of a<br />particular result if we perform a measurement. This makes a lot of<br />people very unhappy since it provides no model for the object.<p><br /><br />It should be added that there are other, less popular, interpretations<br />of quantum theory, but they all have their own drawbacks, which are<br />widely reckoned more severe. Generally speaking they try to find a<br />mechanism that describes the collapse process or add extra physical<br />objects to the theory, in addition to the wavefunction. In this sense<br />they are more complex. (See <a href=#alternatives>"Is there any alternative theory?"</a>)<p><hr noshade /><br /><br /><a name=epr><h3>Q32 <br /> Does the EPR experiment prohibit locality?<br /><br /><br /> <br /><a name=bell> What about Bell's Inequality?</h3><br /> <br />The EPR experiment is widely regarded as the definitive gedanken<br />experiment for demonstrating that quantum mechanics is non-local<br />(requires faster-than-light communication) or incomplete. We shall see<br />that it implies neither.<p><br /><br />The EPR experiment was devised, in 1935, by Einstein, Podolsky and Rosen<br />to demonstrate that quantum mechanics was incomplete [E]. Bell, in<br />1964, demonstrated that any hidden variables theory, to replicate the<br />predictions of QM, must be non-local [B]. QM predicts strong<br />correlations between separated systems, stronger than any local hidden<br />variables theory can offer. Bell encoded this statistical prediction<br />in the form of some famous inequalities that apply to any type of EPR<br />experiment. Eberhard, in the late 1970s, extended Bell's inequalities<br />to cover any local theory, with or without hidden variables. Thus the<br />EPR experiment plays a central role in sorting and testing variants of<br />QM. All the experiments attempting to test EPR/Bell's inequality to<br />date (including Aspect's in the 1980s [As]) are in line with the<br />predictions of standard QM - hidden variables are ruled out. Here is<br />the paradox of the EPR experiment. It seems to imply that any physical<br />theory must involve faster-than-light "things" going on to maintain<br />these "spooky" action-at-a-distance correlations and yet still be<br />compatible with relativity, which seems to forbid FTL.<p><br /><br />Let's examine the EPR experiment in more detail.<p><br /><br />So what did EPR propose? The original proposal was formulated in terms<br />of correlations between the positions and momenta of two once-coupled<br />particles. Here I shall describe it in terms of the spin (a type of<br />angular momentum intrinsic to the particle) of two electrons. [In this<br />treatment I shall ignore the fact that electrons always form<br />antisymmetric combinations. This does not alter the results but does<br />simplify the maths.] Two initially coupled electrons, with opposed<br />spins that sum to zero, move apart from each other across a distance of<br />perhaps many light years, before being separately detected, say, by me<br />on Earth and you on Alpha Centauri with our respective measuring<br />apparatuses. The EPR paradox results from noting that if we choose the<br />same (parallel) spin axes to measure along then we will observe the two<br />electrons' spins to be anti-parallel (i.e. when we communicate we find<br />that the spin on our electrons are correlated and opposed). However if<br />we choose measurement spin axes that are perpendicular to each other<br />then there is no correlation between electron spins. Last minute<br />alterations in a detector's alignment can create or destroy correlations<br />across great distances. This implies, according to some theorists, that<br />faster-than-light influences maintain correlations between separated<br />systems in some circumstances and not others.<p><br /><br />Now let's see how many-worlds escapes from this dilemma.<p><br /><br />The initial state of the wavefunction of you, me and the electrons and<br />the rest of the universe may be written:<p><pre><br /><br /> |psi> = |me> |electrons> |you> |rest of universe><br /> on in on<br /> Earth deep Alpha<br /> space Centauri<br />or more compactly, ignoring the rest of the universe, as:<br /> |psi> = |me, electrons, you> <br />And<br /> |me> represents me on Earth with my detection apparatus.<br /> |electrons> = (|+,-> - |-,+>)/sqrt(2) <br /> represents a pair electrons, with the first electron travelling<br /> towards Earth and the second electron travelling towards Alpha<br /> Centauri.<br /><br /> |+> represents an electron with spin in the +z direction<br /> |-> represents an electron with spin in the -z direction<p></pre><br /><br />It is an empirically established fact, which we just have to accept,<br />that we can relate spin states in one direction to spin states in other<br />directions like so (where "i" is the sqrt(-1)):<br /><pre><br /> |left> = (|+> - |->)/sqrt(2) (electron with spin in -x direction)<br /> |right> = (|+> + |->)/sqrt(2) (electron with spin in +x direction)<br /> |up> = (|+> + |->i)/sqrt(2) (electron with spin in +y direction)<br /> |down> = (|+> - |->i)/sqrt(2) (electron with spin in -y direction)<br />and inverting:<br /> |+> = (|right> + |left>)/sqrt(2) = (|up> + |down>)/sqrt(2)<br /> |-> = (|right> - |left>)/sqrt(2) = (|down> - |up>)i/sqrt(2)</pre><p><br /><br />(In fancy jargon we say that the spin operators in different directions<br />form non-commuting observables. I shall eschew such obfuscations.)<p><br /><br />Working through the algebra we find that for pairs of electrons:<p><pre><br /><br /> |+,-> - |-,+> = |left,right> - |right,left><br /> = |up,down>i - |down,up></pre><p><br /><br />I shall assume that we are capable of either measuring spin in the x or<br />y direction, which are both perpendicular the line of flight of the<br />electrons. After having measured the state of the electron my state is<br />described as one of either:<br /><pre><br /> |me[l]> represents me + apparatus + records having measured <br /> and recorded the x-axis spin as "left"<br /> |me[r]> ditto with the x-axis spin as "right"<br /> |me[u]> ditto with the y-axis spin as "up"<br /> |me[d]> ditto with the y-axis spin as "down"</pre><br /><br />Similarly for |you> on Alpha Centauri. Notice that it is irrelevant<br /><I>how</I> we have measured the electron's spin. The details of the<br />measurement process are irrelevant. (See <a href=#measurement>"What is a measurement?"</a> if<br />you're not convinced.) To model the process it is sufficient to assume<br />that there is a way, which we have further assumed does not disturb the<br />electron. (The latter assumption may be relaxed without altering the<br />results.)<p><br /><br />To establish familiarity with the notation let's take the state of the<br />initial wavefunction as:<p><pre><br /><br /> |psi>_1 = |me,left,up,you><br /> / \<br /> / \<br /> first electron in left second electron in up state<br /> state heading towards heading towards you on<br /> me on Earth Alpha Centauri</pre><p><br /> <br />After the electrons arrive at their detectors, I measure the spin<br />along the x-axis and you along the y-axis. The wavefunction evolves<br />into |psi>_2:<p><pre><br /><br /> local <br /> |psi>_1 ============> |psi>_2 = |me[l],left,up,you[u]> <br /> observation<br /><br /></pre><p>which represents me having recorded my electron on Earth with spin left<br />and you having recorded your electron on Alpha Centauri with spin up. <br />The index in []s indicates the value of the record. This may be held<br />in the observer's memory, notebooks or elsewhere in the local<br />environment (not necessarily in a readable form). If we communicate our<br />readings to each other the wavefunctions evolves into |psi>_3:<p><pre><br /><br /> remote <br /> |psi>_2 ============> |psi>_3 = |me[l,u],left,up,you[u,l]> <br /> communication<br /><br /><br /></pre><p>where the second index in []s represents the remote reading communicated<br />to the other observer and being recorded locally. Notice that the<br />results both agree with each other, in the sense that my record of your<br />result agrees with your record of your result. And vice versa. Our<br />records are consistent.<p><br /><br />That's the notation established. Now let's see what happens in the more<br />general case where, again,:<p><pre><br /><br /> |electrons> = (|+,-> - |-,+>)/sqrt(2).<br /><br /></pre><p>First we'll consider the case where you and I have previously arranged<br />to measure the our respective electron spins along the same x-axis.<p><br /><br />Initially the wavefunction of the system of electrons and two<br />experimenters is:<p><pre><br /><br /> |psi>_1 <br /> = |me,electrons,you><br /> = |me>(|left,right> - |right,left>)|you> /sqrt(2)<br /> = |me,left,right,you> /sqrt(2)<br /> - |me,right,left,you> /sqrt(2)</pre><p><br /><br />Neither you or I are yet unambiguously split.<p><br /><br />Suppose I perform my measurement first (in some time frame). We get<p><pre><br /><br /> |psi>_2<br /> = (|me[l],left,right> - |me[r],right,left>)|you> /sqrt(2)<br /> = |me[l],left,right,you> /sqrt(2)<br /> - |me[r],right,left,you> /sqrt(2)</pre><P><br /><br />My measurement has split me, although you, having made no measurement,<br />remain unsplit. In the full expansion the terms that correspond to you<br />are identical.<p><br /><br />After the we each have performed our measurements we get:<p><pre><br /><br /> |psi>_3<br /> = |me[l],left,right,you[r]> /sqrt(2)<br /> - |me[r],right,left,you[l]> /sqrt(2)</pre><p><br /><br />The observers (you and me) have been split (on Earth and Alpha Centauri)<br />into relative states (or local worlds) which correlate with the state<br />of the electron. If we now communicate over interstellar modem (this<br />will take a few years since you and I are separated by light years, but<br />no matter). We get:<p><pre><br /><br /> |psi>_4<br /> = |me[l,r],left,right,you[r,l]> /sqrt(2)<br /> - |me[r,l],right,left,you[l,r]> /sqrt(2)</pre><p><br /><br />The world corresponding to the 2nd term in the above expansion, for<br />example, contains me having seen my electron with spin right and knowing<br />that you have seen your electron with spin left. So we jointly agree,<br />in both worlds, that spin has been conserved.<p><br /><br />Now suppose that we had prearranged to measure the spins along different<br />axes. Suppose I measure the x-direction spin and you the y-direction<br />spin. Things get a bit more complex. To analyse what happens we need<br />to decompose the two electrons along their respective spin axes.<p><pre><br /><br /> |psi>_1 =<br /> |me,electrons,you><br /> = |me>(|+,-> - |-,+>)|you>/sqrt(2) <br /> = |me> (<br /> (|right>+|left>)i(|down>-|up>)<br /> - (|right>-|left>)(|down>+|up>)<br /> ) |you> /2*sqrt(2) <br /> = |me> (<br /> |right>(|down>-|up>)i<br /> + |left> (|down>-|up>)i<br /> - |right>(|down>+|up>)<br /> + |left> (|down>+|up>)<br /> ) |you> /2*sqrt(2) <br /> = |me> (<br /> |right,down> (i-1) - |right,up> (1+i)<br /> + |left,up> (1-i) + |left,down> (1+i) <br /> ) |you> /2*sqrt(2) <br /> = (<br /> + |me,right,down,you> (i-1)<br /> - |me,right,up,you> (i+1)<br /> + |me,left,up,you> (1-i)<br /> + |me,left,down,you> (1+i) <br /> ) /2*sqrt(2) </pre><P><br /><br />So after you and I make our local observations we get:<p><pre><br /><br /> |psi>_2 =<br /> (<br /> + |me[r],right,down,you[d]> (i-1) <br /> - |me[r],right,up,you[u]> (i+1) <br /> + |me[l],left,up,you[u]> (1-i) <br /> + |me[l],left,down,you[d]> (1+i)<br /> ) /2*sqrt(2)</pre><p><br /><br />Each term realises a possible outcome of the joint measurements. The<br />interesting thing is that whilst we can decompose it into four terms<br />there are only two states for each observer. Looking at myself, for<br />instance, we can rewrite this in terms of states relative to *my*<br />records/memories.<p><pre><br /><br /> |psi>_2 = <br /> ( <br /> |me[r],right> ( |down,you[d]> (i-1) - |up,you[u]> (i+1) )<br /> + |me[l],left> ( |up,you[u]> (1-i) + |down,you[d]> (1+i) )<br /> ) /2*sqrt(2)</pre><p><br /><br />And we see that there are only two copies of <I>me</I>. Equally we can<br />rewrite the expression in terms of states relative to <I>your</I><br />records/memory.<p><pre><br /><br /> |psi>_2 =<br /> ( <br /> ( |me[l],left> (1-i) - |me[r],right> (i+1) ) |up,you[u]> <br /> + ( |me[r],right> (i-1) + |me[l],left> (1+i) ) |down,you[d]><br /> ) /2*sqrt(2)<p></pre><br /><br />And see that there are only two copies of <I>you</I>. We have each been<br />split into two copies, each perceiving a different outcome for our<br />electron's spin, but we have not been split by the measurement of the<br />remote electron's spin. <p><br /><br /><I>After</I> you and I communicate our readings to each other, more than four<br />years later, we get:<p><pre><br /><br /> |psi>_3 =<br /> (<br /> + |me[r,d],right,down,you[d,r]> (i-1) <br /> - |me[r,u],right,up,you[u,r]> (i+1) <br /> + |me[l,u],left,up,you[u,l]> (1-i) <br /> + |me[l,d],left,down,you[d,l]> (1+i)<br /> ) /2*sqrt(2)</pre><p><br /><br />The decomposition into four worlds is forced and unambiguous after<br />communication with the remote system. Until the two observers<br />communicated their results to each other they were each unsplit by each<br />others' measurements, although their own local measurements had split<br />themselves. The splitting is a local process that is causally<br />transmitted from system to system at light or sub-light speeds. (This<br />is a point that Everett stressed about Einstein's remark about the<br />observations of a mouse, in the Copenhagen interpretation, collapsing<br />the wavefunction of the universe. Everett observed that it is the mouse<br />that's split by its observation of the rest of the universe. The rest<br />of the universe is unaffected and unsplit.)<p><br /><br />When all communication is complete the worlds have finally decomposed<br />or decohered from each other. Each world contains a consistent set of<br />observers, records and electrons, in perfect agreement with the<br />predictions of standard QM. Further observations of the electrons will<br />agree with the earlier ones and so each observer, in each world, can<br />henceforth regard the electron's wavefunction as having collapsed to<br />match the historically recorded, locally observed values. This<br />justifies our operational adoption of the collapse of the wavefunction<br />upon measurement, without having to strain our credibility by believing<br />that it actually happens.<p><br /><br />To recap. Many-worlds is local and deterministic. Local measurements<br />split local systems (including observers) in a subjectively random<br />fashion; distant systems are only split when the causally transmitted<br />effects of the local interactions reach them. We have not assumed any<br />non-local FTL effects, yet we have reproduced the standard predictions<br />of QM.<p><br /><br />So where did Bell and Eberhard go wrong? They thought that all theories<br />that reproduced the standard predictions must be non-local. It has been<br />pointed out by both Albert [A] and Cramer [C] (who both support<br />different interpretations of QM) that Bell and Eberhard had implicity<br />assumed that every possible measurement - even if not performed - would<br />have yielded a <I>single</I> definite result. This assumption is called<br />contra-factual definiteness or CFD [S]. What Bell and Eberhard really<br />proved was that every quantum theory must either violate locality <I>or</I><br />CFD. Many-worlds with its multiplicity of results in different worlds<br />violates CFD, of course, and thus can be local.<p><br /><br />Thus many-worlds is the only local quantum theory in accord with the<br />standard predictions of QM and, so far, with experiment.<p><br /><br />[A] David Z Albert, <I>Bohm's Alternative to Quantum Mechanics</I><br /> Scientific American (May 1994)<br /><br /><br />[As] Alain Aspect, J Dalibard, G Roger <I>Experimental test of Bell's<br /> inequalities using time-varying analyzers</I> Physical Review Letters<br /> Vol 49 #25 1804 (1982).<br /><br />[C] John G Cramer <I>The transactional interpretation of quantum<br /> mechanics</I> Reviews of Modern Physics Vol 58 #3 647-687 (1986)<br /><br />[B] John S Bell: <I>On the Einstein Podolsky Rosen paradox</I> Physics 1<br /> #3 195-200 (1964).<br /><br />[E] Albert Einstein, Boris Podolsky, Nathan Rosen: <I>Can<br /> quantum-mechanical description of physical reality be considered<br /> complete?</I> Physical Review Vol 41 777-780 (15 May 1935).<br /><br /><br />[S] Henry P Stapp <I>S-matrix interpretation of quantum-theory</I> Physical<br /> Review D Vol 3 #6 1303 (1971)<br /><br /><br /><a name=same><h3>Q33 Is Everett's relative state formulation the same as many-worlds?</h3><br /> <br />Yes, Everett's formulation of the relative state metatheory is the same<br />as many-worlds, but the language has evolved a lot from Everett's<br />original article [2] and some of his work has been extended, especially<br />in the area of decoherence. (See <a href=#decoherence>"What is decoherence?"</a>) This has<br />confused some people into thinking that Everett's "relative state<br />metatheory" and DeWitt's "many-worlds interpretation" are different<br />theories.<p><br /><br />Everett [2] talked about the observer's memory sequences splitting to<br />form a "branching tree" structure or the state of the observer being<br />split by a measurement. (See <a href=#measurement>"What is a measurement?"</a>) DeWitt<br />introduced the term "world" for describing the split states of an<br />observer, so that we now speak of the observer's world splitting during<br />the measuring process. The maths is the same, but the terminology is<br />different. (See <a href=#is a>"What is a world?"</a>)<p><br /><br />Everett tended to speak in terms of the measuring apparatus being split<br />by the measurement, into non-interfering states, without presenting a<br />detailed analysis of *why* a measuring apparatus was so effective at<br />destroying interference effects after a measurement, although the topics<br />of orthogonality, amplification and irreversibility were covered. (See<br /><a href=#measurement>"What is a measurement?"</a>, <a href=#split>"Why do worlds split?"</a> and <a href=#when>"When do worlds<br />split?"</a>) DeWitt [4b], Gell-Mann and Hartle [10], Zurek [7a] and others<br />have introduced the terminology of "decoherence" (See <a href=#decoherence>"What is<br />decoherence?"</a>) to describe the role of amplification and irreversibility<br />within the framework of thermodynamics.<p><hr noshade /><br /><br /><a name=relative><h3>Q34 What is a relative state?</h3><br /> <br />The relative state of something is the state that something is in,<br /><br /><I>conditional</I> upon, or relative to, the state of something else. What<br />the heck does that mean? It means, amongst other things, that states<br />in the same Everett-world are all states relative to each other. (See<br /><a href=#dirac>"Quantum mechanics and Dirac notation"</a> for more precise details.)<p><br /><br />Let's take the example of Schrodinger's cat and ask what is the relative<br />state of the observer, after looking inside the box? The relative state<br />of the observer (either "saw cat dead" or "saw cat alive") is<br />conditional upon the state of the cat (either "dead" or "alive").<p><br /><br />Another example: the relative state of the last name of the President<br />of the Unites States, in 1995, is "Clinton". Relative to what? <br />Relative to you and me, in this world. In some other worlds it will be<br />"Bush", "Smith", etc. ....... Each possibility is realised in some world<br />and it is the relative state of the President's name, relative to the<br />occupants of that world.<p><br /><br />According to Everett almost all states are relative states. Only the<br />state of the universal wavefunction is not relative but absolute.<p><hr noshade /><br /><br /><a name=splitter><h3>Q35 Was Everett a "splitter"?</h3><br /> <br />Some people believe that Everett eschewed all talk all splitting or<br />branching observers in his original relative state formulation [2]. <br />This is contradicted by the following quote from [2]:<br /><br /> [...] Thus with each succeeding observation (or interaction),<br /> the observer state "branches" into a number of different<br /> states. Each branch represents a different outcome of the<br /> measurement and the <I>corresponding</I> eigenstate for the object-<br /> system state. All branches exist simultaneously in the<br /> superposition after any given sequence of observations.<B>[#]</B><br /> The "trajectory" of the memory configuration of an observer<br /> performing a sequence of measurements is thus not a linear<br /> sequence of memory configurations, but a branching tree, with<br /> all possible outcomes existing simultaneously in a final<br /> superposition with various coefficients in the mathematical<br /> model. [...]<p><br /><br /> <B>[#]</B> Note added in proof-- In reply to a preprint of this<br /> article some correspondents have raised the question of the<br /> "transition from possible to actual," arguing that in<br /> "reality" there is-as our experience testifies-no such<br /> splitting of observers states, so that only one branch can<br /> ever actually exist. Since this point may occur to other<br /> readers the following is offered in explanation.<br /><br /><br /> The whole issue of the transition from "possible" to<br /> "actual" is taken care of in the theory in a very simple way-<br /> there is no such transition, nor is such a transition<br /> necessary for the theory to be in accord with our experience.<br /> From the viewpoint of the theory <I>all</I> elements of a<br /> superposition (all "branches") are "actual," none are any more<br /> "real" than the rest. It is unnecessary to suppose that all<br /> but one are somehow destroyed, since all separate elements of<br /> a superposition individually obey the wave equation with<br /> complete indifference to the presence or absence ("actuality"<br /> or not) of any other elements. This total lack of effect of<br /> one branch on another also implies that no observer will ever<br /> be aware of any "splitting" process.<br /><br /> Arguments that the world picture presented by this theory<br /> is contradicted by experience, because we are unaware of any<br /> branching process, are like the criticism of the Copernican<br /> theory that the mobility of the earth as a real physical fact<br /> is incompatible with the common sense interpretation of nature<br /> because we feel no such motion. In both case the arguments<br /> fails when it is shown that the theory itself predicts that<br /> our experience will be what it in fact is. (In the Copernican<br /> case the addition of Newtonian physics was required to be able<br /> to show that the earth's inhabitants would be unaware of any<br /> motion of the earth.)<p><hr noshade /><br /><br /><a name=unique><h3>Q36 What unique predictions does many-worlds make?</h3><br /> <br />A prediction occurs when a theory suggests new phenomena. Many-worlds<br />makes at least three predictions, two of them unique: about linearity,<br />(See <a href=#exact>"Is linearity exact?"</a>), quantum gravity (See <a href=#quantum gravity>"Why <I>quantum</I><br /><br />gravity?"</a>) and reversible quantum computers (See <a href=#detect>"Could we detect other<br />Everett-worlds?"</a>).<p><hr noshade /><br /><br /><a name=detect><h3>Q37 Could we detect other Everett-worlds?</h3><br /> <br />Many-Worlds predicts that the Everett-worlds do not interact with each<br />other because of the presumed linearity of the wave equation. However<br />worlds <I>do</I> interfere with each other, and this enables the theory to<br />be tested. (Interfere and interact mean different things in quantum<br />mechanics. Pictorially: Interactions occur at the vertices within<br />Feynman diagrams. Interference occurs when you add together different<br />Feynman diagrams with the same external lines.)<p><br /><br />According to many-worlds model worlds split with the operation of every<br />thermodynamically irreversible process. The operation of our minds are<br />irreversible, carried along for the ride, so to speak, and divide with<br />the division of worlds. Normally this splitting is undetectable to us. <br />To detect the splitting we need to set an up experiment where a mind is<br />split but the world <I>isn't</I>. We need a reversible mind.<p><br /><br />The general consensus in the literature [11], [16] is that the<br />experiment to detect other worlds, with reversible minds, will be doable<br />by, perhaps, about mid-21st century. That date is predicted from two<br />trendlines, both of which are widely accepted in their own respective<br />fields. To detect the other worlds you need a reversible machine<br />intelligence. This requires two things: reversible nanotechnology and<br />AI.<p><br /><br /><b>1) Reversible nanoelectronics.</b> This is an straight-line extrapolation<br />based upon the log(energy) / logic operation figures, which are<br />projected to drop below kT in about 2020. This trend has held good for<br />50 years. An operation that thermally dissipates much less than kT of<br />energy is reversible. (This implies that frictive or dissipative forces<br />are insignificant by comparison with other processes.) If more than kT<br />of energy is released then, ultimately, new degrees of freedom are<br />activated in the environment and the change becomes irreversible.<p><br /><br /><b>2) AI.</b> Complexity of human brain = approx 10^17 bits/sec, based on the<br />number of neurons (approx 10^10) per human brain, average number of<br />synapses per neuron (approx 10^4) and the average firing rate (approx<br />10^3 Hz). Straight line projection of log(cost) / logic operation says<br />that human level, self-aware machine intelligences will be commercially<br />available by about 2030-2040. Uncertainty due to present human-level<br />complexity, but the trend has held good for 40 years.<p><br /><br />Assuming that we have a reversible machine intelligence to hand then the<br />experiment consists of the machine making three reversible measurements<br />of the spin of an electron (or polarisation of a photon). (1) First it<br />measures the spin along the z-axis. It records either spin "up" or spin<br />"down" and notes this in its memory. This measurement acts just to<br />prepare the electron in a definite state. (2) Second it measures the<br />spin along the x-axis and records either spin "left" or spin "right" and<br />notes <I>this</I> in its memory. The machine now reverses the entire x-axis<br />measurement - which must be possible, since physics is effectively<br />reversible, if we can describe the measuring process physically -<br />including reversibly erasing its memory of the second measurement. (3)<br />Third the machine takes a spin measurement along the z-axis. Again the<br />machine makes a note of the result.<p> <br /><br /><br />According to the Copenhagen interpretation the original (1) and final<br />(3) z-axis spin measurements have only a 50% chance of agreeing because<br />the intervention of the x-axis measurement by the conscious observer<br />(the machine) caused the collapse of the electron's wavefunction. <br />According to many-worlds the first and third measurements will<I>always</I><br />agree, because there was no intermediate wavefunction collapse. The<br />machine was split into two states or different worlds, by the second<br />measurement; one where it observed the electron with spin "left"; one<br />where it observed the electron with spin "right". Hence when the<br />machine reversed the second measurement these two worlds merged back<br />together, restoring the original state of the electron 100% of the time.<p><br /><br />Only by accepting the existence of the other Everett-worlds is this 100%<br />restoration explicable.<P><hr noshade /><br /><br /><a name=quantum gravity><h3>Q38 Why <I>quantum</I> gravity?</h3><br /> <br />Many-worlds makes a very definite prediction - gravity must be<br />quantised, rather than exist as the purely classical background field<br />of general relativity. Indeed, no one has conclusively directly<br />detected (classical) gravity waves (as of 1994), although their<br />existence has been indirectly observed in the slowing of the rotation<br />of pulsars and binary systems. Some claims have been made for the<br />detection of gravity waves from supernova explosions in our galaxy, but<br />these are not generally accepted. Neither has anyone has directly<br />observed gravitons, which are predicted by quantum gravity, presumably<br />because of the weakness of the gravitational interaction. Their<br />existence has been, and is, the subject of much speculation. Should,<br />in the absence of any empirical evidence, gravity be quantised at all? <br />Why not treat gravity as a classical force, so that quantum physics in<br />the vicinity of a mass becomes quantum physics on a curved Riemannian<br />background? According to many-worlds there <I>is</I> empirical evidence for<br />quantum gravity.<p><br /><br />To see why many-worlds predicts that gravity must be quantised, let's<br />suppose that gravity is not quantised, but remains a classical force. <br />If all the other worlds that many-worlds predicts exist then their<br />gravitational presence should be detectable -- we would all share the<br />same background gravitational metric with our co-existing quantum<br />worlds. Some of these effects might be undetectable. For instance if<br />all the parallel Earths shared the same gravitational field small<br />perturbations in one Earth's orbit from the averaged background orbit<br />across all the Everett-worlds would damp down, eventually, and remain<br />undetectable.<p><br /><br />However theories of galactic evolution would need considerable<br />revisiting if many-worlds was true and gravity was not quantised, since,<br />according to the latest cosmological models, the original density<br />fluctuations derive from quantum fluctuations in the early universe,<br />during the inflationary era. These quantum fluctuations lead to the<br />formation of clusters and super-clusters of galaxies, along with<br />variations in the cosmic microwave background (detected by Smoots et al)<br />which vary in location from Everett-cosmos to cosmos. Such fluctuations<br />could not grow to match the observed pattern if all the density<br />perturbations across all the parallel Everett-cosmoses were<br />gravitationally interacting. Stars would bind not only to the observed<br />galaxies, but also to the host of unobserved galaxies.<p><br /><br />A theory of classical gravity also breaks down at the scale of objects<br />that are not bound together gravitationally. Henry Cavendish, in 1798,<br />measured the torque produced by the gravitational force on two separated<br />lead spheres suspended from a torsion fibre in his laboratory to<br />determine the value of Newton's gravitational constant. Cavendish<br />varied the positions of other, more massive lead spheres and noted how<br />the torsion in the suspending fibre varied. Had the suspended lead<br />spheres been gravitationally influenced by their neighbours, placed in<br />different positions by parallel Henry Cavendishs in the parallel<br />Everett-worlds, then the torsion would have been the averaged sum of all<br />these contributions, which was not observed. In retrospect Cavendish<br />established that the Everett-worlds are not detectable gravitationally. <br />More recent experiments where the location of attracting masses were<br />varied by a quantum random (radioactive) source have confirmed these<br />findings. [W]<p><br /><br />A shared gravitational field would also screw up geo-gravimetric<br />surveys, which have successfully detected the presence of mountains,<br />ores and other density fluctuations at the Earth's surface. Such<br />surveys are not sensitive to the presence of the parallel Everett-Earths<br />with different geological structures. Ergo the other worlds are not<br />detectable gravitationally. That gravity must be quantised emerges as<br />a unique prediction of many-worlds.<p><br /><br />[W] Louis Witten <I>Gravitation: an introduction to current research</I> <br /> New York, Wiley (1962).<br /><br /><br /> <I>Essays in honor of Louis Witten on his retirement. Topics on<br /> quantum gravity and beyond</I>: University of Cincinnati, USA, 3-4<br /> April 1992 / editors, Freydoon Mansouri & Joseph J. Scanio. <br /> Singapore ; River Edge, NJ : World Scientific, c1993 ISBN 981021290<p><hr noshade /><br /><br /><a name=exact><h3>Q39 Is linearity exact?</h3><br /> <br />Linearity (of the wavefunction) has been verified to hold true to better<br />than 1 part in 10^27 [W]. If slight non-linear effects were ever<br />discovered then the possibility of communication with, or travel to, the<br />other worlds would be opened up. The existence of parallel Everett-<br />worlds can be used to argue that physics must be <I>exactly</I> linear, that<br />non-linear effects will never be detected. (See <a href=#linear>"Is physics linear"</a> for<br />more about linearity.)<p><br /><br />The argument for exactness uses a version of the weak anthropic<br />principle and proceeds thus: the exploitation of slight non-linear<br />quantum effects could permit communication with and travel to the other<br />Everett-worlds. A sufficiently advanced "early" civilisation [F] might<br />colonise uninhabited other worlds, presumably in an exponentially<br />spreading fashion. Since the course of evolution is dictated by random<br />quantum events (mutations, genetic recombination) and environmental<br />effects (asteroidal induced mass extinctions, etc.) it seems inevitable<br />that in a minority, although still a great many, of these parallel<br />worlds life on Earth has already evolved sapient-level intelligence and<br />developed an advanced technology millions or even billions of years ago. <br />Such early arrivals, under the usual Darwinian pressure to expand, would<br />spread across the parallel time tracks, if they had the ability,<br />displacing their less-evolved quantum neighbours.<p><br /><br />The fossil record indicates that evolution, in our ancestral lineage,<br />has proceeded at varying rates at different times. Periods of rapid<br />development in complexity (e.g. the Cambrian explosion of 530 millions<br />years ago or the quadrupling of brain size during the recent Ice Ages)<br />are interspersed with long periods of much slower development. This<br />indicates that we are not in the fast lane of evolution, where all the<br />lucky breaks turned out just right for the early development of<br />intelligence and technology. Ergo none of the more advanced<br />civilisations that exist in other worlds have ever been able to cross<br />from one quantum world to another and interrupt our long, slow<br />biological evolution.<p><br /><br />The simplest explanation is that physics is sufficiently linear to<br />prevent travel between Everett worlds. If technology is only bounded<br />by physical law (the Feinberg principle [F]) then linearity would have<br />to be exact.<p><br /><br />[F] Gerald Feinberg. <I>Physics and Life Prolongation</I> Physics Today Vol<br /> 19 #11 45 (1966). "A good approximation for such [technological]<br /> predictions is to assume that everything will be accomplished that<br /> does not violate known fundamental laws of science as well as many<br /> things that do violate these laws."<p><br /><br />[W] Steven Weinberg <I>Testing Quantum Mechanics</I> Annals of Physics Vol<br /> 194 #2 336-386 (1989) and <I>Dreams of a Final Theory</I> (1992)<p><hr noshade /><br /><br /><a name=boundary><h3>Q40 Why can't the boundary conditions be updated to reflect my <br />observations in this one world?</h3><br /> <br />What is lost by this approach is a unique past assigned to each future. <br />If you time-evolve the world-we-now-see backwards in time you get a<br />superposition of earlier starting worlds. Similarly if you time evolve<br />a single (initial) world forward you get a superposition of later<br />(final) worlds.<p><br /><br />For example consider a photon that hits a half-silvered mirror and turns<br />into a superposition of a transmitted and a reflected photon. If we<br />time-evolve one of these later states backwards we get not the original<br />photon, but the original photon plus a "mirror image" of the original<br />photon. (Try the calculation and see.) Only if we retain both the<br />reflected and transmitted photons, with the correct relative phase, do<br />we recover the single incoming photon when we time-reverse everything. <br />(The mirror image contributions from both the final states have opposite<br />signs and cancel out, when they are evolved backwards in time to before<br />the reflection event.)<p><br /><br />All the starting states have to have their relative phases co-ordinated<br />or correlated just right (i.e. coherently) or else it doesn't work out. <br />Needless to say the chances that the initial states should be arranged<br />coherently just so that they yield the one final observed state are<br />infinitesimal and in violation of observed thermodynamics, which states,<br />in one form, that correlations only increase with time.<p><hr noshade size=5 /><br /><br /><a name=references><h3>A1 References and further reading</h3><pre><br /> <br />[1] Hugh Everett III <I>The Theory of the Universal Wavefunction,<br /> Princeton thesis</I> (1956?)<br /> The original and most comprehensive paper on many-worlds. <br /> Investigates and recasts the foundations of quantum theory in<br /> information theoretic terms, before moving on to consider the<br /> nature of interactions, observation, entropy, irreversible<br /> processes, classical objects etc. 138 pages. Only published in<br /> [5].<br />[2] Hugh Everett III <I>"Relative State" Formulation of Quantum<br /> Mechanics</I> Reviews of Modern Physics Vol 29 #3 454-462, (July<br /> 1957) A condensation of [1] focusing on observation.<br />[3] John A Wheeler <I>Assessment of Everett's "Relative State"<br /> Formulation of Quantum Theory</I>, Reviews of Modern Physics Vol<br /> 29 #3 463-465 (July 1957) Wheeler was Everett's PhD<br /> supervisor.<br />[4a] Bryce S DeWitt <I>Quantum Mechanics and Reality</I> Physics Today,<br /> Vol 23 #9 30-40 (September 1970) An early and accurate<br /> popularisations of Everett's work. The April 1971 issue has<br /> reader feedback and DeWitt's responses.<br />[4b] Bryce S DeWitt <I>The Many-Universes Interpretation of Quantum<br /> Mechanics</I> in <I>Proceedings of the International School of Physics<br /> "Enrico Fermi" Course IL: Foundations of Quantum Mechanics</I><br /><br /> Academic Press (1972)<br />[5] Bryce S DeWitt, R Neill Graham eds <I>The many-worlds<br /> Interpretation of Quantum Mechanics</I>_. Contains<br /> [1],[2],[3],[4a],[4b] plus other material. Princeton Series<br /> in Physics, Princeton University Press (1973) ISBN 0-691-<br /> 08126-3 (hard cover), 0-691-88131-X (paper back) The<br /> definitive guide to many-worlds, if you can get hold of a<br /> copy, but now (1994) only available xeroxed from microfilm<br /> (ISBN 0-7837-1942-6) from Books On Demand, 300 N Zeeb Road,<br /> Ann Arbor, MI 48106-1346, USA. Tel: +01-313 761 4700 or 800<br /> 521 0600.<br />[15] Frank J Tipler <I>The many-worlds interpretation of quantum mechanics<br /> in quantum cosmology</I> in <I>Quantum Concepts of Space and Time</I> eds<br /> Roger Penrose and Chris Isham, Oxford University Press (1986). Has<br /> a discussion of Ockham's razor.<hr /><br /><b>On quantum theory, measurement and decoherence generally:</b><br />[6] John A Wheeler, Wojciech H Zurek eds <I>Quantum Theory and<br /> Measurement</I> Princeton Series in Physics, Princeton University<br /> Press (1983) ISBN 0-691-08316-9. Contains 49 classic<br /> articles, including [2], covering the history and development<br /> of interpretations of quantum theory. <br />[7a] Wojciech H Zurek <I>Decoherence and the Transition from the<br /> Quantum to the Classical</I>, Physics Today, 36-44 (October<br /> 1991). The role of thermodynamics and the properties of large<br /> ergodic systems (like the environment) are related to the<br /> decoherence or loss of interference effects between superposed<br /> macrostates.<br />[7b] Wojciech H Zurek <I>Preferred States, Predictability, Classicality,<br /> and the Environment-Induced Decoherence</I> Progress of Theoretical<br /> Physics, Vol 89 #2 281-312 (1993) A fuller expansion of [7a]<br />[8] Max Jammer <I>The Philosophy of Quantum Mechanics</I> Wiley, New<br /> York (1974) Almost every interpretation of quantum mechanics<br /> is covered and contrasted. Section 11.6 contains a lucid<br /> review of many-worlds theories.<br />[9] Bethold-Georg Englert, Marlan O Scully, Herbert Walther <I>Quantum<br /> optical tests of complementarity</I> Nature, Vol 351, 111-116 (9 May<br /> 1991). Demonstrates that quantum interference effects are destroyed<br /> by irreversible object-apparatus correlations ("measurement"), not<br /> by Heisenberg's uncertainty principle itself. See also <I>The<br /> Duality in Matter and Light</I> Scientific American, (December 1994)<br />[10] Murray Gell-Mann, James B Hartle <I>Quantum Mechanics in the Light<br /> of Quantum Cosmology</I> Proceedings of the 3rd International<br /> Symposium on the Foundations of Quantum Mechanics (1989) 321-343. <br /> They accept the Everett's decoherence analysis, and have extended<br /> it further.<hr /><br /><br /><b>Tests of the Everett metatheory:</b><br />[11] David Deutsch <I>Quantum theory as a universal physical theory</I><br /> International Journal of Theoretical Physics, Vol 24 #1<br /> (1985). Describes an experiment which tests for the existence<br /> of superpositions of *consciousness (in an AI).<br />[16] David Deutsch <I>Three connections between Everett's interpretation<br /> and experiment</I> Quantum Concepts of Space and Time, eds Roger<br /> Penrose and Chris Isham, Oxford University Press (1986). Discusses<br /> a testable split observer experiment and quantum computing.<hr /><br /><b>On quantum computers:</b><br />[12] David Deutsch <I>Quantum theory, the Church-Turing principle and the<br /> universal quantum computer</I> Proceedings of the Royal Society of<br /> London, Vol. A400, 96-117 (1985).<br />[13] David Deutsch <I>Quantum computational networks</I> Proceedings of<br /> the Royal Society of London, Vol. A425, 73-90 (1989).<br />[14] David Deutsch and R. Jozsa _<I>Rapid solution of problems by<br /> quantum computation</I> Proceedings of the Royal Society of<br /> London, Vol. A439, 553-558 (1992).<br />[17] Julian Brown <I>A Quantum Revolution for Computing</I> New Scientist,<br /> pages 21-24, 24-September-1994</pre><p><hr noshade /><br /><br /><a name=dirac><h3>A2 Quantum mechanics and Dirac notation </h3><br /> <br />Note: this is a very inadequate guide. Read a more comprehensive text<br />ASAP. For a more technical exposition of QM the reader is referred to<br />the standard textbooks. Here are 3 I recommend:<p><br /><br />Richard P Feynman <I>QED: the strange story of light and matter</I> ISBN 0-<br />14-012505-1. (Requires almost no maths and is universally regarded as<br />outstanding, despite being about quantum electrodynamics.)<p><br /><br />Richard P Feynman <I>The Feynman Lectures in Physics</I> Volume III Addison-<br />Wesley (1965) ISBN 0-201-02118-8-P. The other volumes are worth reading<br />too!<p><br /><br />Daniel T Gillespie <I>A Quantum Mechanics Primer: An Elementary<br />Introduction to the Formal Theory of Non-relativistic Quantum Mechanics</I> <br />(Takes an axiomatic, geometric approach and teaches all the Hilbert<br />space stuff entirely by analogy with Euclidean vector spaces. Not sure<br />if it is still in print.)<p><br /><br />Quantum theory is the most successful theory of physics and chemistry<br />ever. It accounts for a wide range of phenomena from black body<br />radiation, atomic structure and chemistry, which were very puzzling<br />before quantum mechanics was first developed (c1926) in its modern form. <br />All theories of physics are quantum physics, with whole new fields, like<br />the semiconductor and microchip technology, based upon the quantum<br />effects. This FAQ assumes familiarity with the basics of quantum theory<br />and with the associated "paradoxes" of wave-particle duality. It will<br />not explain the uncertainty principle or delve into the significance of<br />non-commuting matrix operators. Only those elements of quantum theory<br />necessary for an understanding of many-worlds are covered here.<p><br /><br />Quantum theory contains, as a central object, an abstract mathematical<br />entity called the "wavefunction" or "state vector". Determining the<br />equations that describe its form and evolution with time is an<br />unfinished part of fundamental theoretical physics. Presently we only<br />have approximations to some "correct" set of equations, often referred<br />to whimsically as the Theory of Everything.<p><br /><br />The wavefunction, in bracket or Dirac notation, is written as |symbol>,<br />where "symbol" labels the object. A dog, for example, might be<br />represented as |dog>.<p><br /><br />A general object, labelled "psi" by convention, is represented as |psi><br />and called a "ket". Objects called "bra"s, written <psi|, may be formed<br />from kets. An arbitrary bra <psi'| and ket |psi> may be combined<br />together to form the bracket, <psi'|psi>, or inner product, which is<br />just a fancy way of constructing a complex number. Amongst the<br />properties of the inner product is:<p><pre><br /><br /> <psi'|(|psi1>*a_1 + |psi2>*a_2) = <psi'|psi1>*a_1 + <psi'|psi2>*a_2<p></pre><br /><br />where the a_i are arbitrary complex numbers. This is what is meant by<br />saying that the inner product is linear on the right or ket side. It<br />is made linear on the left-hand or bra side by defining <p><pre><br /><br /> <psi|psi'> = complex conjugate of <psi'|psi></pre><p><br /><br />Any ket may be expanded as:<p><pre><br /><br /> |psi> = sum |i>*<i|psi> <br /> i<br /> = |1>*<1|psi> + |2>*<2|psi> + ...</pre><br /><br />where the states |i> form an orthonormal basis, with <i|j> = 1 for i =<br />j and = 0 otherwise, and where i labels some parameter of the object<br />(like position or momentum).<p><br /><br />The probability amplitudes, <i|psi>, are complex numbers. It is<br />empirically observed, first noted by Max Born and afterwards called the<br />Born interpretation, that their magnitudes squared represent the<br />probability that, upon observation, that the value of the parameter,<br />labelled by i, will be observed if the system is the state represented<br />by |psi>. It is also empirically observed that after observing the<br />system in state |i> that we can henceforth replace the old value of the<br />wavefunction, |psi>, with the observed value, |i>. This replacement is<br />known as the collapse of the wavefunction and is the source of much<br />philosophical controversy. Somehow the act of measurement has selected<br />out one of the components. This is known as the measurement problem and<br />it was this phenomenon that Everett addressed.<p><br /><br />When a bra, <psi|, is formed from a ket, |psi>, and both are inner<br />productted together the result, <psi|psi>, is a non-negative real<br />number, called the norm of the vector. The norm of a vector provides<br />a basis-independent way of measuring the "volume" of the vector.<p><br /><br />The wavefunction for a joint system is built out of products of the<br />components from the individual subsystems.<p> <br /><br />For example if the two systems composing the joint system are a cat and<br />a dog, each of which may be in two states, alive or dead, and the state<br />of the cat and the dog were <I>independent</I> of each other then we could<br />write the total wavefunction as a product of terms. If<br /><pre><br /> |cat> = |cat alive> * c_a + |cat dead> * c_d<br />and <br /> |dog> = |dog alive> * d_a + |dog dead> * d_d<br />then<br /> |dog+cat> = |cat>x|dog> where x = tensor product<br /> = (|cat alive> * c_a + |cat dead> * c_d)<br /> x (|dog alive> * d_a + |dog dead> * d_d)<br /> = |cat alive> x |dog alive> * c_a * d_a <br /> + |cat alive> x |dog dead> * c_a * d_d<br /> + |cat dead> x |dog alive> * c_d * d_a<br /> + |cat dead> x |dog dead> * c_d * d_d<br /> = |cat alive, dog alive> * c_a * d_a <br /> + |cat alive, dog dead> * c_a * d_d<br /> + |cat dead, dog alive> * c_d * d_a<br /> + |cat dead, dog dead> * c_d * d_d<p></pre><br /><br />More generally, though, we states of subsystems are not independent of<br />each other we have to use a more general formula:<p><pre><br /><br /> |dog+cat> = |cat alive, dog alive> * a_1<br /> + |cat alive, dog dead> * a_2<br /> + |cat dead, dog alive> * a_3<br /> + |cat dead, dog dead> * a_4<p></pre><br /><br />This is sometimes described by saying that the states of the cat and dog<br />have become entangled. It is fairly trivial to define the state of the<br />cat and the dog with respect to each other. For instance we could re-<br />express the above expansion with respect to the cat's two states as:<p><pre><br /><br /> |dog+cat> = <br /> |cat alive>x(|dog alive> * a_1 + |dog dead> * a_2)<br /> + |cat dead>x(|dog alive> * a_3 + |dog dead> * a_4)<p></pre><br /><br />We term the state of the dog the <I>relative state</I> (Everett invented this<br />terminology) with respect to the cat, specifying which cat state (alive<br />or dead) we are interested in. This thus the dog's relative state with<br />respect to the cat alive state is:<p><pre><br /><br /> (|dog alive> * a_1 + |dog dead> * a_2)/sqrt(|a_1|^2 + |a_2|^2)<p></pre><br /><br />where the sqrt term has been added to normalise the relative state.<p>
Michael Clive Price
February 1995
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