Many-worlds interpretation

The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics that rejects the non-deterministic and irreversible wavefunction collapse associated with measurement in the Copenhagen interpretation in favor of the conventional deterministic and CPT-reversible laws of quantum physics. The phenomena associated with measurement are explained by decoherence which occurs when a quantum mechanical system interacts with its environment. MWI reconciles how we perceive non-deterministic events (such as the random decay of a radioactive atom) with the deterministic equations of quantum physics. An implication of this reconcillation between determinism and non-determinism is that the universe is much larger than we would otherwise think, and that the world we see (including ourselves) is continuously branching into a greater and greater number of divergent copies. History, which prior to MWI had been viewed as a single "world-line", is rather a many-branched tree where every possible history is realised.

Many worlds is often referred as a theory, rather than just an interpretation, by those who propose that many worlds can make testable predictions. MWI is also regarded as a theory by those MWI proponents who argue that all other, non-MWI interpretations, are actually inconsistent, illogical or unscientific in their handling of measurements. Hugh Everett, MWI's originator, argued that his proposal was a metatheory, since it made statements about other interpretations of quantum theory.

Outline

Although several versions of MWI have been proposed since Hugh Everett's original work, they contain one key idea: the equations of physics that model the time evolution of systems without embedded observers are sufficient for modelling systems which do contain observers. The exact form of the quantum dynamics modelled, be it the non-relativistic Schrödinger equation, relativistic quantum field theory or some form of quantum gravity or string theory, does not alter the content of MWI since MWI is a metatheory applicable to all quantum theories and hence to all credible theories of physics. MWI's main conclusion is that the universe (or multiverse in this context) is composed of a quantum superposition of very many, possibly infinitely many, increasingly divergent, non-communicating parallel universes or quantum worlds.

The idea of MWI originated in Hugh Everett's Princeton Ph. D. thesis "The Theory of the Universal Wavefunction", developed under his thesis advisor John Wheeler, a shorter summary of which was published in 1957 entitled "Relative State Formulation of Quantum Mechanics" (Wheeler contributed the "relative state" title). The phrase "many worlds" is due to Bryce DeWitt, who was responsible for the wider popularisation of Everett's theory, which had been largely ignored for the first decade after publication. DeWitt's MWI terminology has become so much more popular than Everett's "Universal Wavefunction" or Everett-Wheeler's "Relative State Formutation" that many forget that this is only a difference of terminology; the content of all three papers is the same.

MWI is one of many multiverse hypotheses in physics and philosophy. It is currently considered a mainstream interpretation along with the Copenhagen and consistent histories interpretations.

Many worlds and the problem of interpretation

As with the other interpretations of quantum mechanics, the many-worlds interpretation is motivated by behavior that can be illustrated by the double-slit experiment. When particles of light (or anything else) are passed through the double slit, a calculation assuming wave-like behavior of light is needed to identify where the particles are likely to be observed. Yet when the particles are observed in this experiment, they appear as particles and not as non-localized waves. The Copenhagen interpretation of quantum mechanics proposed a process of "collapse" from wave behavior to particle-like behavior to explain this phenomenon of observation.

By the time John von Neumann wrote his famous treatise Mathematische Grundlagen der Quantenmechanik in 1932, the phenomenon of "wavefunction collapse" was accommodated into the mathematical formulation of quantum mechanics by postulating that there were two processes of wavefunction change:

1. The discontinuous nonunitary probabilistic change brought about by observation and measurement.
2. The deterministic unitary time evolution of an isolated system that obeys Schrödinger's equation.

The phenomenon of wavefunction collapse for (1) proposed by the Copenhagen interpretation was widely regarded as artificial and ad-hoc, and consequently an alternative interpretation in which the behavior of measurement could be understood from more fundamental physical principles was considered desirable.

Everett's Ph. D. work was intended to provide such an alternative interpretation. Everett proposed that for a composite system (for example that formed by a particle interacting with a measuring apparatus) the statement that a subsystem has a well-defined state is meaningless -- in modern parlance the subsystem states have become entangled. This led Everett to derive from the dynamics the notion of a relativity of states of one subsystem relative to another.

Everett's formalism for understanding the apparent process of wavefunction collapse as a result of observation is mathematically equivalent to a quantum superposition of wavefunctions; each element of the superpositon after an observation contains an observer whose relative state contains an associated "collapsed" object state. Since Everett stopped doing research in theoretical physics shortly after obtaining his Ph. D., much of the elaboration of his ideas was carried out by other researchers.

Brief overview

In Everett's formulation, a measuring apparatus M and an object system S form a composite system, each of which prior to measurement exists in well-defined (but time-dependent) states. Measurement is regarded as causing M and S to interact. After S interacts with M, it is no longer possible to describe either system by an independent state. According to Everett, the only meaningful descriptions of each system are relative states: for example the relative state of S given the state of M or the relative state of M given the state of S.

File:Many-worlds-bloch.png

Schematic representation of pair of "smallest possible" quantum mechanical systems prior to interaction : Measured system S and measurement apparatus M. Systems such as S are referred to as 1-qubit systems.

In DeWitt's formulation, the state of S after a sequence of measurements is given by a quantum superposition of states, each one corresponding to an alternative measurement history of S.

For example, consider the smallest possible truly quantum system S, as shown in the illustration. This describes for instance, the spin-state of an electron. Considering a specific axis (say the z-axis) the north pole represents spin "up" and the south pole, spin "down". The superposition states of the system are described by (the surface of) a sphere called the Bloch sphere. To perform a measurement on S, it is made to interact with another similar system M. After the interaction, the combined system is described by a state that ranges over a six-dimensional space (the reason for the number six is explained in the article on the Bloch sphere). This six-dimensional object can also be regarded as a quantum superposition of two "alternative histories" of the original system S, one in which "up" was observed and the other in which "down" was observed. Each subsequent binary measurement (that is interaction with a system M) causes a similar split in the history tree. Thus after three measurements, the system can be regarded as a quantum superposition of 8= 2 × 2 × 2 copies of the original system S.

The accepted terminology is somewhat misleading because it is incorrect to regard the universe as splitting at certain times; at any given instant there is one state in one universe.

Schematic illustration of splitting as a result of a repeated measurement.

Relative state

The goal of the relative-state formalism, as originally proposed by Everett in his 1957 doctoral dissertation, was to interpret the effect of external observation entirely within the mathematical framework developed by Dirac, von Neumann and others, discarding altogether the ad-hoc mechanism of wave function collapse. Since Everett's original work, there have appeared a number of similar formalisms in the literature. One such idea is discussed in the next section.

From the relative-state formalism, we can obtain a relative-state interpretation by two assumptions. The first is that the wavefunction is not simply a description of the object's state, but that it actually is entirely equivalent to the object, a claim it has in common with other interpretations. The second is that observation has no special role, unlike in the Copenhagen interpretation which considers the wavefunction collapse as a special kind of event which occurs as a result of observation.

The many-worlds interpretation is DeWitt's rendering of the relative state formalism (and interpretation). Everett referred to the system (such as an observer) as being split by an observation, each split corresponding to a possible outcome of an observation. These splits generate a possible tree as shown in the graphic below. Subsequently DeWitt introduced the term "world" to describe a complete measurement history of an observer, which corresponds roughly to a path starting at the root of that tree. Note that "splitting" in this sense, is hardly new or even quantum mechanical. The idea of a space of complete alternative histories had already been used in the theory of probability since the mid 1930s for instance to model Brownian motion.

File:Many-worlds.png

Partial trace as relative state. Light blue rectangle on upper left denotes system in pure state. Trellis shaded rectangle in upper right denotes a (possibly) mixed state. Mixed state from observation is partial trace of a linear superposition of states as shown in lower left-hand corner.

Under the many-worlds interpretation, the Schrödinger equation, or relativistic analog, holds all the time everywhere. An observation or measurement of an object by an observer is modeled by applying the Schrödinger wave equation to the entire system comprising the observer and the object. One consequence is that every observation can be thought of as causing the combined observer-object's wavefunction to change into a quantum superposition of two or more non-interacting branches, or "worlds". Since many observation-like events are constantly happening, there are an enormous number of simultaneously existing states.

If a system is composed of two or more subsystems, the system's state will be a superposition of products of the subsystems' states. Once the subsystems interact, their states are no longer independent. Each product of subsystem states in the overall superposition evolves over time independently of other products. The subsystems have become entangled and it is no longer possible to consider them independent of one another. Everett's term for this entanglement of subsystem states was a relative state, since each subsystem must now be considered relative to the other subsystems with which it has interacted.

File:Paths-many-worlds.png

Successive measurements with successive splittings

Comparative properties and experimental support

One of the salient properties of the many-worlds interpretation is that observation does not require an exceptional construct (such as wave function collapse) to explain it. Many physicists, however, dislike the implication that there are infinitely many non-observable alternate universes.

As of 2002, there were no practical experiments that would distinguish between many-worlds and Copenhagen, and in the absence of observational data, the choice is one of personal taste. However, one area of research is devising experiments which could distinguish between various interpretations of quantum mechanics, although there is some skepticism whether it is even meaningful to ask such a question. Indeed, it can be argued that there is a mathematical equivalence between Copenhagen (as expressed for instance in a set of algorithms for manipulating density states) and many-worlds (which gives the same answers as Copenhagen using a more elaborate mathematical picture) which would seem to make such an endeavor impossible. However, this algorithmic equivalence may not be true on a cosmological scale. It has been proposed that in a world with infinite alternate universes, the universes which collapse would exist for a shorter time than universes which expand, and that would cause detectable probability differences between many-worlds and the Copenhagen interpretation.

In the Copenhagen interpretation, the mathematics of quantum mechanics allows one to predict probabilities for the occurrence of various events. In the many-worlds interpretation, all these events occur simultaneously. What meaning should be given to these probability calculations? And why do we observe, in our history, that the events with a higher computed probability seem to have occurred more often? One answer to these questions is to say that there is a probability measure on the space of all possible universes, where a possible universe is a complete path in the tree of branching universes. This is indeed what the calculations give. Then we should expect to find ourselves in a universe with a relatively high probability rather than a relatively low probability: even though all outcomes of an experiment occur, they do not occur in an equal way.

As an interpretation which (like other interpretations) is consistent with the equations, it is hard to find testable predictions of MWI. There is a rather more dramatic test than the one outlined above for people prepared to put their lives on the line: use a machine which kills them if a random quantum decay happens. If MWI is true, they will still be alive in the world where the decay didn't happen and would feel no interruption in their stream of consciousness. By repeating this process a number of times, their continued consciousness would be arbitrarily unlikely unless MWI was true, when they would be alive in all the worlds where the random decay was on their side. From their viewpoint they would be immune to this death process. Clearly, if MWI does not hold, they would be dead in the one world. Other people would generally just see them die and would not be able to benefit from the result of this experiment.

The many-worlds interpretation should not be confused with the many-minds interpretation which postulates that it is only the observers' minds that split instead of the whole world.

Everett's many-worlds interpretation and axiomatics

The existence of many worlds in superposition is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the collapse of the wave packet: All the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical (not just formally mathematical, as in other interpretations) quantum superposition. (Such a superposition of consistent state combinations of different systems is called an entangled state.)

Hartle (1968) showed that in Everett's relative-state theory, Born's probability law

The probability of an observable $A$ to have the value $a$ in a normalized state $|\psi \rangle$ is the absolute square of the eigenvalue component of the state corresponding to the eigenvalue a: $P(a)=|\langle a|\psi \rangle |^{2}$

no longer has to be considered an axiom or postulate. It can rather be derived from the other axioms of quantum mechanics. All that has to be assumed is that if the state $|\psi \rangle$ is an eigenstate $|a,i\rangle$ of the observable $A$, then the result $a$ of the measurement is certain. This means that a second axiom of quantum mechanics can be removed. Hartle's derivation only works in a theory (like Everett's) that does not cut away ("collapse") any superposition components of the wave function. In other interpretations it is not comprehensible why the absolute square is used and not some other arbitrary, more complicated expression of the eigenvalue component say, the square root or some polynomial of its norm.

The consequence is that Everett's concept is more than just an interpretation, it's rather an alternative formulation of quantum theory requiring fewer axioms.

One might argue that postulating the existence of many worlds is some kind of axiomatic assumption, but the concept of quantum superpositions is a common indispensable part of all interpretations of quantum theory, as is most clearly illustrated in the path integral formulation of quantum mechanics. Everett's theory just considers it a real phenomenon in nature and applies it to macroscopic systems in the same way as to microscopic systems.

MWI describes measurements as a formation of an entangled state which is a perfectly linear process (in terms of quantum superpositions) without any collapse of the wave function. For illustration, consider a Stern-Gerlach experiment and an electron or a silver atom passing this apparatus with a spin polarization in the x direction and thus a superposition of a spin up and a spin down state in z-direction. As a measuring apparatus, take a bubble or tracking chamber or another nonabsorbing particle detector; let the electron pass the apparatus and reach the same site in the end on either way so that except for the z-spin polarization the state of the electron is finally the same regardless of the path taken (see The Feynman Lectures on Physics for a detailed discussion of such a setup). Before the measurement, the state of the electron and the measuring apparatus is:

$|\psi _{1}\rangle$	$=|e^{-}{\mbox{: x-spin}}\uparrow \rangle \otimes |{\mbox{m: initialized}}\rangle$
	$={\frac {1}{\sqrt {2}}}(|e^{-}{\mbox{: z-spin}}\uparrow \rangle +|e^{-}{\mbox{: z-spin}}\downarrow \rangle )\otimes |{\mbox{m: init.}}\rangle$
	$={\frac {1}{\sqrt {2}}}(|e^{-}{\mbox{: z-spin}}\uparrow \rangle \otimes |{\mbox{m: init.}}\rangle +|e^{-}{\mbox{: z-spin}}\downarrow \rangle \otimes |{\mbox{m: init.}}\rangle )$

The state is factorizable into a tensor factor for the electron and another factor for the measurement apparatus. After the measurement, the state is:

$|\psi _{2}\rangle ={\frac {1}{\sqrt {2}}}($	$|e^{-}{\mbox{: z-spin}}\uparrow \rangle \otimes |{\mbox{m: bubbles along the upper path}}\rangle \ +$
	$|e^{-}{\mbox{: z-spin}}\downarrow \rangle \otimes |{\mbox{m: bubbles along the lower path}}\rangle )$

The state is no longer factorizable -- regardless of the vector basis chosen. As an illustration, understand that the following state is factorizable:

$|\phi \rangle =|e^{-}{\mbox{: a}}\rangle \otimes |{\mbox{m: c}}\rangle +|e^{-}{\mbox{: a}}\rangle \otimes |{\mbox{m: d}}\rangle +|e^{-}{\mbox{: b}}\rangle \otimes |{\mbox{m: c}}\rangle +|e^{-}{\mbox{: b}}\rangle \otimes |{\mbox{m: d}}\rangle$

since it can be written as

$|\phi \rangle =(|e^{-}{\mbox{: a}}\rangle +|e^{-}{\mbox{: b}}\rangle )\otimes (|{\mbox{m: c}}\rangle +|{\mbox{m: d}}\rangle )$

(which might be not so obvious if another vector basis is chosen for the states).

The state of the above experiment is decomposed into a sum of two so-called entangled states ("worlds") both of which will have their individivual history without any interaction between the two due to the physical linearity of quantum mechanics (the superposition principle): All processes in nature are linear and correspond to linear operators acting on each superposition component individually without any notice of the other components being present.

This would also be true for two non-entangled superposed states, but the latter can be detected by interference which is not possible for different entangled states (without reversing the entanglement first): Different entangled states cannot interfere; interactions with others systems will only result in a further entanglement of them as well. In the example above, the state of a Schrödinger cat watching the scene will be factorizable in the beginning (before watching)

$|\psi _{1}'\rangle ={\frac {1}{\sqrt {2}}}($	$|e^{-}{\mbox{: z-spin}}\uparrow \rangle \otimes |{\mbox{m: upper bubbels}}\rangle \ +$
	$|e^{-}{\mbox{: z-spin}}\downarrow \rangle \otimes |{\mbox{m: lower bubbles}}\rangle )\otimes |{\mbox{cat: ignorant of the bubbles}}\rangle$

but not in the end:

$|\psi _{2}'\rangle ={\frac {1}{\sqrt {2}}}($	$|e^{-}{\mbox{: z-spin}}\uparrow \rangle \otimes |{\mbox{m: upper bubbels}}\rangle \otimes |{\mbox{cat: remembers upper bubbles}}\rangle \ +$
	$|e^{-}{\mbox{: z-spin}}\downarrow \rangle \otimes |{\mbox{m: lower bubbles}}\rangle \otimes |{\mbox{cat: remembers lower bubbles}}\rangle )$

This example also shows that it's not the whole world that is split up into "many worlds", but only the part of the world that is entangled with the considered quantum event. This splitting tends to extend by interactions and can be visualised by a zipper or a DNA molecule which are in a similar way not completely opened instantaneously but gradually, element by element.

Imaginative readers will even see the zipper structure and the extending splitting in the formula:

$|\psi _{2}'\rangle =($	$\alpha \ |e^{-}{\mbox{:}}e_{1}\rangle \otimes |{\mbox{m:}}m_{1}\rangle \otimes |{\mbox{c:}}c_{1}\rangle \otimes |{\mbox{a:}}a_{1}\rangle \ +\quad \Longrightarrow$
	$\beta \ |e^{-}{\mbox{:}}e_{2}\rangle \otimes \,|{\mbox{m:}}m_{2}\rangle \,\otimes \,|{\mbox{c:}}c_{2}\rangle \,\otimes |{\mbox{a:}}a_{2}\rangle \ )\ \otimes |{\mbox{b:}}b_{0}\rangle \otimes |{\mbox{d:}}d_{0}\rangle \otimes |{\mbox{f:}}f_{0}\rangle \otimes |{\mbox{g:}}g_{0}\rangle \otimes |{\mbox{h:}}h_{0}\rangle$

If a system state is entangled with many other degrees of freedom (such as those in amplifiers, photographs, heat, sound, computer memory circuits, neurons, paper documents) in an experiment, this amounts to a thermodynamically irreversible process which is constituted of many small individually reversible processes at the atomic or subatomic level as is generally the case for thermodynamic irreversibility in classical or quantum statistical mechanics. Thus there is -- for thermodynamic reasons -- no way for an observer to completely reverse the entanglement and thus observe the other worlds by doing interference experiments on them. On the other hand, for small systems with few degrees of freedom this is feasible, as long as the investigated aspect of the system remains unentangled with the rest of the world.

The formation of an entangled state is a linear operation in terms of quantum superpositions. Consider for example the vector basis $|e^{-}{\mbox{: a}}\rangle \otimes |{\mbox{m: c}}\rangle ,|e^{-}{\mbox{: a}}\rangle \otimes |{\mbox{m: d}}\rangle ,|e^{-}{\mbox{: b}}\rangle \otimes |{\mbox{m: c}}\rangle ,|e^{-}{\mbox{: b}}\rangle \otimes |{\mbox{m: d}}\rangle$

and the non-entangled initial state $|\psi _{1}\rangle =|e^{-}{\mbox{: a}}\rangle \otimes |{\mbox{m: c}}\rangle$

The linear (and unitary and thus reversible) operation (in terms of quantum superpositions) corresponding to the matrix

${\begin{bmatrix}0&0&1&0\\1/{\sqrt {2}}&1/{\sqrt {2}}&0&0\\-1/{\sqrt {2}}&1/{\sqrt {2}}&0&0\\0&0&0&1\end{bmatrix}}$

(in the above vector basis) will result in the entangled state $|\psi _{2}\rangle ={\frac {1}{\sqrt {2}}}(|e^{-}{\mbox{: a}}\rangle \otimes |{\mbox{m: d}}\rangle -|e^{-}{\mbox{: b}}\rangle \otimes |{\mbox{m: c}}\rangle )$

A simple example

We consider formally the example presented in the introduction. Consider a pair of spin 1/2 particles, A and B, in which we only consider the spin observable (in particular with their position information disregarded). As an isolated system, particle A is described by a 2 dimensional Hilbert space H_A; similarly particle B is described by a 2 dimensional Hilbert space H_B. The composite system is described by the tensor product

$H_{\mathrm {A} }\otimes H_{\mathrm {B} }$

which is 2 x 2 dimensional. If A and B are non-interacting, the set of pure tensors

$|\phi \rangle \otimes |\psi \rangle$

is invariant under time evolution; in fact, since we only consider the spin observables which for isolated particles are invariant, time has no effect prior to interaction. However, after interaction, the state of the composite system is a possibly entangled state, that is one which is no longer a pure tensor.

The most general entangled state is a sum

$\Phi =\sum _{\ell }|\phi _{\ell }\rangle \otimes |\psi _{\ell }\rangle$

To this state corresponds a linear operator H_B → H_A which maps pure states to pure states.

$T_{\Phi }=\sum _{\ell }|\phi _{\ell }\rangle \otimes \langle \psi _{\ell }|.$

This mapping (essentially modulo normalization of states) is the relative state mapping defined by Everett, which associates a pure state of B the corresponding relative (pure) state of A. More precisely, there is a unique polar decomposition of T_Φ such that

$T_{\Phi }=US\quad$

and U is an isometric map defined on some subspace of H_B. U is actually the relative state mapping. See also Schmidt decomposition.

Note that the density matrix of the composite system is pure. However, it is also possible to consider the reduced density matrix describing particle A alone by taking the partial trace over the states of particle B. This reduced density matrix, unlike the original matrix actually describes a mixed state. This particular example is the basis for the EPR paradox.

The previous example easily generalizes to arbitrary systems A, B without any restriction on the dimension of the corresponding Hilbert spaces. In general, the relative state is an isometric linear mapping defined on a subspace of H_B with values in H_A.

Partial trace and relative state

The state transformation of a quantum system resulting from measurement, such as the double slit experiment discussed above, can be easily described mathematically in a way that is consistent with most mathematical formalisms. We will present one such description, also called reduced state, based on the partial trace concept, which by a process of iteration, leads to a kind of branching many worlds formalism. It is then a short step from this many worlds formalism to a many worlds interpretation.

For definiteness, let us assume that system is actually a particle such as an electron. The discussion of reduced state and many worlds is no different in this case than if we considered any other physical system, including an "observer system". In what follows, we need to consider not only pure states for the system, but more generally mixed states; these are certain linear operators on the Hilbert space H describing the quantum system. Indeed, as the various measurement scenarios point out, the set of pure states is not closed under measurement. Mathematically, density matrices are statistical mixtures of pure states. Operationally a mixed state can be identified to a statistical ensemble resulting from a specific lab preparation process.

Decohered states as relative states

Suppose we have an ensemble of particles, prepared in such a way that its state S is pure. This means that there is a unit vector ψ in H (unique up to phase) such that S is the operator given in bra-ket notation by

$S=|\psi \rangle \langle \psi |$

Now consider an experimental setup to determine whether the particle has a particular property: For example the property could be that the location of the particle is in some region A of space. The experimental setup can be regarded either as a measurement of an observable or as a filter. As a measurement, it measures the observable Q which takes the value 1 if the particle is found in A and 0 otherwise. As a filter, it filters in those particles in the ensemble which have the stated property of being in A and filtering out the others.

Mathematically, a property is given by a self-adjoint projection E on the Hilbert space H: Applying the filter to an ensemble of particles, some of the particles of the ensemble are filtered in, and others are filtered out. Now it can be shown that the operation of the filter "collapses" the pure state in the following sense: it prepares a new mixed state given by the density operator

$S_{1}=|E\psi \rangle \langle \psi E|+|F\psi \rangle \langle \psi F|$

where F = 1 - E.

To see this, note that as a result of the measurement, the state of the particle immediately after the measurement is in an eigenvector of Q, that is one of the two pure states

${\frac {1}{\|E\psi \|^{2}}}|E\psi \rangle \quad {\mbox{ or }}\quad {\frac {1}{\|F\psi \|^{2}}}|F\psi \rangle .$

with respective probabilities

$\|E\psi \|^{2}\quad {\mbox{ or }}\quad \|F\psi \|^{2}.$

The mathematical way of presenting this mixed state is by taking the following convex combination of pure states:

$\|E\psi \|^{2}\times {\frac {1}{\|E\psi \|^{2}}}|E\psi \rangle \langle E\psi |+\|F\psi \|^{2}\times {\frac {1}{\|F\psi \|^{2}}}|F\psi \rangle \langle F\psi |,$

which is the operator S₁ above.

Remark. The use of the word collapse in this context is somewhat different that its use in explanations of the Copenhagen interpretation. In this discussion we are not referring to collapse or transformation of a wave into something else, but rather the transformation of a pure state into a mixed one.

The considerations so far, are completely standard in most formalisms of quantum mechanics. Now consider a "branched" system whose underlying Hilbert space is

${\tilde {H}}=H\otimes H_{2}\cong H\oplus H$

where H₂ is a two-dimensional Hilbert space with basis vectors $|0\rangle$ and $|1\rangle$. The branched space can be regarded as a composite system consisting of the original system (which is now a subsystem) together with a non-interacting ancillary single qubit system. In the branched system, consider the entangled state

$\phi =|E\psi \rangle \otimes |0\rangle +|F\psi \rangle \otimes |1\rangle \in {\tilde {H}}$

We can express this state in density matrix format as $|\phi \rangle \langle \phi |$. This multiplies out to:

${\bigg (}|E\psi \rangle \langle E\psi |\ \otimes \ |0\rangle \langle 0|{\bigg )}\,+\,{\bigg (}|E\psi \rangle \langle F\psi |\ \otimes \ |0\rangle \langle 1|{\bigg )}\,+\,{\bigg (}|F\psi \rangle \langle E\psi |\ \otimes \ |1\rangle \langle 0|{\bigg )}\,+\,{\bigg (}|F\psi \rangle \langle F\psi |\ \otimes \ |1\rangle \langle 1|{\bigg )}$

The partial trace of this mixed state is obtained by summing the operator coefficients of $|0\rangle \langle 0|$ and $|1\rangle \langle 1|$ in the above expression. This results in a mixed state on H. In fact, this mixed state is identical to the "post filtering" mixed state S₁ above.

To summarize, we have mathematically described the effect of the filter for a particle in a pure state ψ in the following way:

• The original state is augmented with the ancillary qubit system.

• The pure state of the original system is replaced with a pure entangled state of the augmented system and

• The post-filter state of the system is the partial trace of the entangled state of the augmented system.

Multiple branching

In the course of a system's lifetime we expect many such filtering events to occur. At each such event, a branching occurs. In order for this to be consistent with the branching structure as depicted in the illustration above, we must show that if a filtering event occurs in one path from the root node of the tree, then we may assume it occurs in all branches. This shows that the tree is highly symmetric, that is for each node n of the tree, the shape of the tree does not change by interchanging the subtrees immediately below that node n.

In order to show this branching uniformity property, note that the same calculation carries through even if original state S is mixed. Indeed, the post filtered state will be the density operator:

$S_{1}=ESE+FSF\quad$

The state S₁ is the partial trace of

${\bigg (}ESE\,\otimes \,|0\rangle \langle 0|{\bigg )}+{\bigg (}ESF\,\otimes \,|0\rangle \langle 1|{\bigg )}+{\bigg (}FSE\,\otimes \,|1\rangle \langle 0|{\bigg )}+{\bigg (}FSF\,\otimes \,|1\rangle \langle 1|{\bigg )}.$

This means that to each subsequent measurement (or branching) along one of the paths from the root of the tree to a leaf node corresponds to a homologous branching along every path. This guarantees the symmetry of the many-worlds tree relative to flipping child nodes of each node.

File:Path-integral-mwi.png

Superposition over paths through observation tree

General quantum operations

In the previous two sections, we have represented measurement operations on quantum systems in terms of relative states. In fact there is a wider class of operations which should be considered: these are called quantum operations. Considered as operations on density operators on the system Hilbert space H, these have the following form:

$\gamma (S)=\sum _{i\in I}F_{i}SF_{i}^{*}$

where I is a finite or countably infinite index set. The operators F_i are called Kraus operators.

Theorem. Let

$\Phi (S)=\sum _{i,j}F_{i}SF_{j}^{*}\,\otimes \,|i\rangle \langle j|$

Then

$\gamma (S)=\operatorname {Tr} _{H}(\Phi (S)).$

Moreover, the mapping V defined by

$V|\psi \rangle =\sum _{\ell }|F_{\ell }\psi \rangle \,\otimes \,|\ell \rangle$

is such that

$\Phi (S)=VSV^{*}\quad$

If γ is a trace-preserving quantum operation, then V is an isometric linear map

$V:H\rightarrow H\otimes \ell ^{2}(I)\cong H\oplus H\oplus \cdots \oplus H$

where the Hilbert direct sum is taken over copies of H indexed by elements of I. We can consider such maps Φ as imbeddings. In particular:

Corollary. Any trace-preserving quantum operation is the composition of an isometric imbedding and a partial trace.

This suggests that the many worlds formalism can account for this very general class of transformations in exactly the same way that it does for simple measurements.

Branching

In general we can show the uniform branching property of the tree as follows: If

$\gamma (S)=\operatorname {Tr} _{H}VSV^{*}\quad$

and

$\delta (S)=\operatorname {Tr} _{H}WSW^{*},\quad$

where

$V|\psi \rangle =\sum _{\ell \in I}|F_{\ell }\psi \rangle \,\otimes \,|\ell \rangle$

and

$W|\phi \rangle =\sum _{i\in J}|G_{i}\phi \rangle \,\otimes \,|i\rangle$

then an easy calculation shows

$\delta \circ \gamma (S)=\operatorname {Tr} _{H}{\bigg \{}{\bigg (}W\otimes \operatorname {id} _{\ell ^{2}(I)}\,\circ \,V{\bigg )}S{\bigg (}W\otimes \operatorname {id} _{\ell ^{2}(I)}\,\circ \,V{\bigg )}^{*}{\bigg \}}.$

This also shows that in between the measurements given by proper (that is, non-unitary) quantum operations, one can interpolate arbitrary unitary evolution.

Acceptance of the many-worlds interpretation

There is a wide range of claims that are considered "many worlds" interpretations. It is often noted (see the Barrett reference) that Everett himself was not entirely clear as to what he meant. Moreover, several books that could be considered scientific popularizations of quantum mechanics have often used many-worlds to justify claims about the relationship between consciousness and the material world. Apart from these new-agey interpretations, "many worlds"-like interpretations are now considered fairly mainstream within the quantum physics community.

For example, a poll of 72 leading physicists conducted by the American researcher David Raub in 1995 and published in the French periodical Sciences et Avenir in January 1998 recorded that nearly 60% thought many worlds interpretation was "true". Max Tegmark (see reference to his web page below) also reports the result of a poll taken at a 1997 quantum mechanics workshop. According to Tegmark, "The many worlds interpretation (MWI) scored second, comfortably ahead of the consistent histories and Bohm interpretations." Other such unscientific polls have been taken at other conferences: see for instance Michael Nielsen's blog [1] report on one such poll. Nielsen remarks that it appeared most of the conference attendees "thought the poll was a waste of time".

One of MWI's strongest advocates is David Deutsch. According to Deutsch the single photon interference pattern observed in the double slit experiment, can be explained by interference of photons in multiple universes. Viewed in this way, the single photon interference experiment is indistinguishable from the multiple photon interference experiment. In a more practical vein, in one of the earliest papers on quantum computing (Deutsch 1985), he suggested that parallelism that results from the validity of MWI could lead to "a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it".

Asher Peres was an outspoken critic of MWI, for example in a section in his 1993 textbook with the title Everett's interpretation and other bizarre theories. In fact, Peres questioned whether MWI is really an "interpretation" or even if interpretations of quantum mechanics are needed at all. Indeed, the many-worlds interpretation can be regarded as a purely formal transformation, which adds nothing to the instrumentalist (i.e. statistical) rules of the quantum mechanics. Perhaps more significantly, Peres seems to suggest that positing the existence of an infinite number of non-communicating parallel universes is highly suspect as it violates Occam's Razor. Proponents of MWI argue precisely the opposite, by applying Occam's Razor to the set of assumptions rather than multiplicity of universes. In Max Tegmark's formulation, the alternative to many worlds is many words (notably von Neumann's collapse postulate).

MWI is considered by some to be unfalsifiable, because the multiple parallel universes are non-communicating in the sense that no information can be passed between them. Moreover, it has also been noted (for instance by Peres himself) that polls regarding the acceptance of a particular interpretation within the scientific community, such as those mentioned above, cannot be used as evidence supporting a specific interpretation's validity. However others note that science is a group activity (for instance, peer review) and that polls are a systematic way of revealing the thinking of the scientific community.

Many worlds in literature and science fiction

Main article: Parallel universe (fiction)

The many-worlds interpretation (and the somewhat related concept of possible worlds) have been associated to numerous themes in literature, art and science fiction.

Aside from violating fundamental principles of causality and relativity, these stories are extremely misleading since the information-theoretic structure of the path space of multiple universes (that is information flow between different paths) is very likely extraordinarily complex. Also see Michael Price's FAQ referenced in the external links section below where these issues (and other similar ones) are dealt with more decisively.

Another kind of popular illustration of many worlds splittings, which does not involve information flow between paths, or information flow backwards in time considers alternate outcomes of historical events. From the point of view of quantum mechanics, these stories however are deficient for at least two reasons:

• There is nothing inherently quantum mechanical about branching descriptions of historical events. In fact, this kind of case-based analysis is a common planning technique and it can be analysed quantitatively by classical probability.
• The use of historical events complicates matters by introduction of an issue which is generally believed to be completely extraneous to quantum theory, namely the question of the nature of individual choice.

Speculative implications of many worlds

It has been claimed that there is an experiment that would clearly differentiate between the many-worlds interpretation and other interpretations of quantum mechanics. It involves a quantum suicide machine and an experimenter willing to risk death. However, at best, this would only decide the issue for the experimenter; bystanders would learn nothing.

The many-worlds interpretation has some similarity to modal realism in philosophy, which is the view that the possible worlds used to interpret modal claims actually exist.

See also

• Afshar experiment
• multiverse
• quantum decoherence
• multiple histories
• many-minds interpretation

The following provide more speculative interpretations:

• quantum immortality
• holomovement

External links

• Michael Price's Everett FAQ -- a very clear presentation of the theory with some additional insights
• Against Many-Worlds Interpretations
• Everett's Relative-State Formulation of Quantum Mechanics
• Many-Worlds Interpretation of Quantum Mechanics
• Max Tegmark's web page
• Many Worlds & Parallel Universes
• Many Worlds is a "lost cause" according to R. F. Streater
• The many worlds of quantum mechanics

References

• Jeffrey A. Barrett, The Quantum Mechanics of Minds and Worlds, Oxford University Press, 1999.
• Hugh Everett, Relative State Formulation of Quantum Mechanics, Reviews of Modern Physics vol 29, (1957) pp 454-462.
• Christopher Fuchs, Quantum Mechanics as Quantum Information (and only a little more), arXiv:quant-ph/0205039 v1, (2002)
• Bryce DeWitt, R. Neill Graham, eds, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN 069108131X
• Asher Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993.
• John Archibald Wheeler, Assessment of Everett's "Relative State Formulation of Quantum Theory", Reviews of Modern Physics, vol 29, (1957) pp 463-465
• John Archibald Wheeler, Geons, Black Holes & Quantum Foam, ISBN 0393319911. pp 268-270
• David Deutsch, The Fabric of Reality: The Science of Parallel Universes And Its Implications, Penguin Books (August 1, 1998), ISBN 014027541X.
• David Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proceedings of the Royal Society of London A 400, (1985) , pp. 97–117
• James Hartle, Quantum Mechanics of Individual Systems, Amer. Jour. Phys., vol 36 (1968), # 8