Many-worlds interpretation of quantum mechanics
|
|
The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics that averts the special role played by the measurement process in the Copenhagen interpretation by proposing several key ideas. The first of these is the existence of a state function for the entire universe which obeys Schrödinger's equation for all time and for which there is no wavefunction collapse due to measurement. The second idea is that the universal state is a quantum superposition of an infinite number of states of identical non-communicating "parallel universes". The ideas of MWI originated in Hugh Everett's Princeton Ph. D. thesis, but the phrase "many worlds" is due to Bryce DeWitt, who wrote more on the topic of Everett's original work. DeWitt's formulation has become so popular that many confuse it with Everett's original work.
| Contents |
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, 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:
- The discontinuous probabilistic change brought about by observation and measurement.
- The deterministic 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. This led Everett to suggest the notion of relativity of states of one subsystem relative to another.
Everett's formalism for understanding the process of wavefunction collapse as a result of observation is mathematically equivalent to a quantum superposition of wavefunctions. Everett left physics research shortly after obtaining his degree so 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.Many-worlds-bloch.png
Splittings-1.png
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. The novelty in DeWitt's viewpoint was that the various complete alternative histories could be superposed to form new quantum mechanical states.Many-worlds.png
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 can be thought of as causing the universal wavefunction to split 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 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.
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 an infinite number of 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.
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
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 HA; similarly particle B is described by a 2 dimensional Hilbert space HB. The composite system is described by the tensor product
- <math> H_{\mathrm{A}} \otimes H_{\mathrm{B}} <math>
which is 2 x 2 dimensional. If A and B are non-interacting, the set of pure tensors
- <math> |\phi \rangle \otimes | \psi \rangle <math>
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
- <math> \Phi = \sum_\ell | \phi_\ell \rangle \otimes | \psi_\ell \rangle <math>
To this state corresponds a linear operator HB → HA which maps pure states to pure states.
- <math> T_\Phi = \sum_\ell | \phi_\ell \rangle \otimes \langle \psi_\ell |.<math>
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 TΦ such that
- <math> T_\Phi = U S \quad <math>
and U is an isometric map defined on some subspace of HB. 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 HB with values in HA.
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
- <math> S = | \psi \rangle \langle \psi | <math>
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
- <math> S_1 = |E \psi \rangle \langle \psi E | + |F \psi \rangle \langle \psi F | <math>
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
- <math> \frac{1}{\|E \psi\|^2} | E \psi \rangle \quad \mbox{ or } \quad \frac{1}{\|F \psi\|^2} | F \psi \rangle. <math>
with respective probabilities
- <math> \|E \psi\|^2 \quad \mbox{ or } \quad \|F \psi\|^2. <math>
The mathematical way of presenting this mixed state is by taking the following convex combination of pure states:
- <math> \|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 |, <math>
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
- <math> \tilde{H} = H \otimes H_2 \cong H \oplus H <math>
where H2 is a two-dimensional Hilbert space with basis vectors <math> | 0 \rangle <math> and <math> | 1 \rangle <math>. 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
- <math> \phi = | E \psi \rangle \otimes | 0 \rangle + | F \psi \rangle \otimes | 1 \rangle \in \tilde{H} <math>
We can express this state in density matrix format as <math> | \phi \rangle \langle \phi | <math>. This multiplies out to:
- <math> \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) <math>
The partial trace of this mixed state is obtained by summing the operator coefficients of <math> | 0 \rangle \langle 0 | <math> and <math> | 1 \rangle \langle 1 | <math> 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. 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:
- <math> S_1 = E S E + F S F \quad <math>
The state S1 is the partial trace of
- <math> \bigg( E S E \, \otimes \, | 0 \rangle \langle 0 |\bigg) + \bigg( E S F \, \otimes \, | 0 \rangle \langle 1 |\bigg) + \bigg(F S E \, \otimes \, | 1 \rangle \langle 0 |\bigg) + \bigg(F S F \, \otimes \, | 1 \rangle \langle 1 |\bigg). <math>
Path-integral-mwi.png
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:
- <math> \gamma(S) = \sum_{i \in I} F_i S F_i^* <math>
where I is a finite or countably infinite index set. The operators Fi are called Kraus operators.
Theorem. Let
- <math> \Phi(S) = \sum_{i,j} F_i S F_j^* \, \otimes \, | i \rangle \langle j | <math>
Then
- <math> \gamma(S) = \operatorname{Tr}_H(\Phi(S)). <math>
Moreover, the mapping V defined by
- <math> V | \psi \rangle = \sum_\ell | F_\ell \psi \rangle \, \otimes \, | \ell \rangle <math>
is such that
- <math> \Phi(S) = V S V^* \quad <math>
If γ is a trace-preserving quantum operation, then V is an isometric linear map
- <math> V : H \rightarrow H \otimes \ell^2(I) \cong H \oplus H \oplus \cdots \oplus H <math>
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
- <math> \gamma(S) = \operatorname{Tr}_H V S V^* \quad <math>
and
- <math> \delta(S) = \operatorname{Tr}_H W S W^*, \quad <math>
where
- <math> V | \psi \rangle = \sum_{\ell \in I}| F_\ell \psi \rangle \, \otimes \, | \ell \rangle <math>
and
- <math> W | \phi \rangle = \sum_{i \in J}| G_i \phi \rangle \, \otimes \, | i \rangle <math>
then an easy calculation shows
- <math> \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\}. <math>
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 world" interpretations. It is often noted (see the Barrett reference) that Everett himself was not entirely clear as to what he meant. Moreover, popularizers have often used many-worlds to justifiy claims about the relationship between consciousness and the material world. Apart from these new-agey interpretations, "many world"-like interpretations are now considered fairly mainstream.
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 highly unscientific polls have been taken at other conferences: see for instance Michael Nielsen's blog [1] (http://www.qinfo.org/people/nielsen/blog/archive/000060.html) report on one such poll. Nielsen remarks that it appeared most of the conference attendees "thought the poll was a waste of time".
However, the physicist Asher Peres in his 1993 textbook expresses a great deal of skepticism towards MWI which is shared by many physicists. In fact, he questions whether many worlds is really an "interpretation" at all (particularly in a section with the title Everett's interpretation and other bizarre theories). 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 worse than the problem it is supposed to solve.
As such, because the interpretation results from an equivalence between two mathematical formalisms, and is considered unfalsifiable (because the multiple parallel universes are non-communicating), critics consider the many-worlds interpretation metaphysical rather than a testable scientific theory. Moreover, notwithstanding the personal opinions or speculation of individual physicists (or indeed even a statistical majority of physicists), subjective polls of "acceptance" such as the above cannot be interpreted as evidence of the correctness or incorrectness of a particular theory; as such, for example, the mere fact that any particular person or percentage of people "accept" the many-worlds interpretation should not be considered evidence of its accuracy.
Many worlds in literature and science fiction
Main article: Many-worlds and possible worlds in literature and art
The many-worlds interpretation (and the unrelated 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 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.
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
The following provide more speculative interpretations:
External links
- Against Many-Worlds Interpretations (http://arxiv.org/abs/gr-qc/9703089)
- Everett's Relative-State Formulation of Quantum Mechanics (http://plato.stanford.edu/entries/qm-everett/)
- Michael Price's Everett FAQ (http://www.hedweb.com/manworld.htm)
- Max Tegmark's web page (http://www.hep.upenn.edu/~max/everett.html)
- Many Worlds & Parallel Universes (http://timetravelportal.com/viewtopic.php?t=288)
- Many Worlds is a "lost cause" (http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XII) according to R. F. Streater
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 S. DeWitt, R. Neill Graham, eds, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973)
- 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
- David Deutsch, The Fabric of Reality: The Science of Parallel Universes And Its Implications, Penguin Books (August 1, 1998), ISBN 014027541X.