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Many-worlds interpretation

The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics, based on Hugh Everett's relative-state formulation. The phrase "many worlds" is due to Bryce DeWitt , who wrote more on the topic of Everett's original work, and this particular version has become so popular that many confuse it with Everett's own work.

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 that the light is behaving as a wave is needed to identify where the particles are likely to be observed. Yet when the particles are observed, they appear as particles and not as non-localized waves. In the Copenhagen interpretation of quantum mechanics, when the position of a particle is measured it appears to "collapse" from wave behavior to particle-like behavior.

It should be noted that, despite the name, which conjures up less the image of science than of science fiction, many worlds is a very widely accepted interpretation. According to 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, MWI is widely accepted:

Yes, I think MWI is true	58%
No, I don't accept MWI	18%
Maybe it's true but I'm not yet convinced	13%
I have no opinion one way or the other	11%

Among the supporters of MWI are Stephen Hawking and Murray Gell-Mann. Among the skeptics are Roger Penrose. Richard Feynman is also said to have accepted MWI (although not in this poll, since he died in 1988).

Also from a 1997 paper by Max Tegmark (see reference to his web page below)

A (highly unscientific) poll taken at the 1997 UMBC (University of Maryland Baltimore County, Ed.) quantum mechanics workshop gave the once all-dominant Copenhagen interpretation less than half of the votes. The Many Worlds interpretation (MWI) scored second, comfortably ahead of the Consistent Histories and Bohm interpretations.

Contents

1 Relative state

2 Acceptance and experimental support

3 A simple example

4 Partial trace and relative state

4.1 Decohered states as relative states
4.2 Multiple branching
4.3 General quantum operations

5 Many worlds and science fiction

6 See also

7 External Links

8 References

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 formalisms in the literature, which use similar terminology and have similar objectives, but which are hard to reconcile with each other. We point out that one way to do away with the language of wave function collapse, based on the idea of partial trace is discussed separately in the next section.

From any of the relative-state formalisms, 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. In Everett's formluation, any measuring apparatus M that interacts with an object system S forms a composite system; after the interaction, it is no longer possible to describe the state of S or that of M by an independent wave-function. The only meaningful descriptions of each 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.

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 observation tree as shown in the graphic below. Subsequently DeWitt introduced the term "world" to describe a complete measurement record 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 path space interpretation of stochastic processes is in many ways the same idea. The novelty in DeWitt's viewpoint was the various worlds could be superposed to form quantum mechanical states.

Theory that non-interacting branches (or "quantum branches") lead to simultaneously existing states when choices are made in time. Each decision branches off into 2 separate alternate realities.

Under the many-worlds interpretation, the Schrödinger equation holds all the time everywhere. An observation or measurement of an object by an observer is modelled by applying the Schrödinger wave equation to the entire system comprising the observer and the object. One consequence is that every observation causes the universal wavefunction to decohere into 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 typically 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.

Acceptance and experimental support

Mathematically and physically, the many-worlds interpretation is simpler than the Copenhagen interpretation. The act of observation or measurement is not magical, and the interpretation of probabilities as the squared amplitude of the wave function is a direct consequence of the theory rather than a necessary axiom. However, many physicists dislike the implication that there are an infinite number of non-observable alternate universes. Some physicists have noted that there appears to be an increase in support for the many-worlds interpretation largely because many-worlds seems to allow for predictions on the process of quantum decoherence in a natural way rather than adding it in an ad-hoc manner.

Nevertheless, 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 active area of research is devising experiments which could distinguish between various interpretations of quantum mechanics, although there is also skepticism whether it is possible to devise such experiments. 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 illustrated in the graphic. 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. There are still many unsolved philosophical problems with this interpretation.

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.

A simple example

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 on H_B 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, 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

$T_\Phi = U S \quad$

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 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 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:

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 branching worlds 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.

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:

$S_1 = E S E + F S F \quad$

The state S₁ is the partial trace of

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

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:

$\gamma(S) = \sum_i F_i S F_i^*$

with

$\sum_i F_i S F_i^* \leq 1$

Theorem. Let

$\Phi(S) = \sum_{i,j} F_i S F_j^* \, \otimes \, | i \rangle \langle j |$

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

$\Phi(S) = V S V^* \quad$

This theorem 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.

Many worlds and science fiction

Fanciful illustrations of the many worlds interpretation are given by science fiction stories in which individuals are capable of viewing events on alternate paths of the universe or even travelling between alternate worlds or parallel universes; see, for instance, "Sliders". Aside from violating fundamental principles of casuality 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, considers alternate outcomes of historical events. For instance, a New York Post newspaper printed on July 6, 2004 incorrectly stated that the presumptive Democratic nominee for president, U.S. Sen. John F. Kerry had chosen U.S. Rep. Richard A. "Dick" Gephardt as his running mate in the 2004 election, when in fact Kerry had chosen U.S. Sen. John R. Edwards. One could imagine that the newspaper shows what a universe in which Kerry picked Gephardt would have been like. Kerry's choice split the "universe" in half: In one branch John Edwards became his running mate (the one we're living in) and in another branch Gephardt became his running mate. There also exists a relatively infinitesimally small number of universes where the reader has become his running mate, by means of some very unlikely coincidences. From the point of view of quantum mechanics, these examples 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 but which is generally agreed to be completely extraneous to quantum theory, namely the question of the nature of individual choice.

It has been controversially claimed that an interesting but dangerous experiment which would also clearly distinguish between the Many Worlds interpretation and all other interpretations involves a quantum suicide machine and a physicist who cares enough about the issue to risk his own life. At best, this would only decide the issue for the brave physicist; bystanders would learn nothing.

External Links

Everett's Relative-State Formulation of Quantum Mechanics

Michael Price's Everett FAQ

Max Tegmark's web page

Many Worlds & Parallel Universes

References

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)

The Many-Worlds Interpretation of Quantum Mechanics by Bryce S. DeWitt, R. Neill Graham, eds, Princeton Series in Physics, Princeton University Press (1973)

John Wheeler, Assessment of Everett's "Relative State Formulation of Quantum Theory" Reviews of Modern Physics vol 29, (1957) pp 463-465

Categories: Quantum mechanics

Last updated: 10-24-2004 05:10:45

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