Many-worlds interpretation

The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics, based on Everett's relative-state formulation. The phrase "many worlds" is due to Bryce DeWitt, who wrote more on the topic of Hugh 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:

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).

Relative state

The relative-state formalism allows us to mathematically describe the wave function collapse within the existing mathematical framework developed by von Neumann and others. How this is actually done requires extensive use of linear algebra and is discussed separately in the next section. From the relative-state formalism, we obtain the 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 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 relative state formalism (and interpretation) is the same as the many-worlds formalism (and interpretation). Everett referred to the state of a system being split by measurement. These splits generate a tree as shown in the graphic below. Subsequently DeWitt introduced the term "world" to describe a complete path starting at the root of that tree.

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.

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. 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.

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 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.

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, called the relative state formalism, which by a process of iteration, leads to the many worlds formalism. It is then a short step from the many worlds formalism to many worlds interpretation.

For definiteness, let us assume that system is actually a particle such as an electron. The discussion of relative state and many worlds is no different in this case than if we considered the entire universe. 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

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

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

with respective probabilities

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

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

where H₂ is a two-dimensional Hilbert space with basis vectors   and  . 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

We can express this state in density matrix format as  . This multiplies out to:

The partial trace of this mixed state is obtained by summing the operator coefficients of   and   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. It is by definition the relative state of the subsystem.

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:

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:

with

Theorem. Let

Then

Moreover, the mapping V defined by

is such that

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.

Simple examples

Let's take a simple example. Consider two spin 1/2 particles, A and B, with their position information disregarded (but let's also assume both particles are at fixed spatially distinct positions). Then, we have a 2x2 dimensional Hilbert space. For our purposes, instead of focusing on the state vectors of this space, let's consider density matrices instead. If we have the entangled state which describes the state with a total angular momentum of zero, then the density matrix of this system is pure. However, it's also possible to consider the reduced density matrix describing particle A alone by taking the trace over the states of particle B. This reduced density matrix, unlike the original matrix actually describes a mixed state.

Now, let's take Schrodinger's cat as an example, where, for simplicity, we'll just assume there's a cat, which can either be dead or alive, a human observer (you, let's say) and "the rest of the universe". Initially, let's just assume the system is described by a pure state which is the product of the pure state describing the superposition of a dead and a live cat, a pure state describing you and a pure state describing the rest of the universe. Now, let's evolve this system forward by "opening the box the cat is in" (Actually, unless experimentalists, even thought experimentalists can come up with a box good enough to prevent ANY interaction whatsoever between what's inside and what's outside, there isn't even any need to open the box). Now, we have a pure state where you and the "rest of the universe" is affected by the state of the cat. But now, what happens if we choose to deal with the reduced system consisting of you only or you and the cat only? We have a mixed state! But really, if we consider the entire universe, we still have a pure state and it only "appears" to be mixed if we wish to localize our description to a restricted area. This is the essense of the relative state interpretation.

Also, since there are "simultaneously existing" states that occur through time, each universe is different. An example of this is a New York Post newspaper printed on July 6, 2004 that incorrectly stated that John Kerry picked Dick Gephardt as his runningmate in the 2004 election, when in fact Kerry had really picked John Edwards. One could say that the newspaper shows what reality would have been like in a universe in which Kerry picked Gephardt. Kerry's choice split the quantum "universe" in half: In one quantum branch John Edwards became his runningmate (the one we're living in) and in another quantum branch Gephardt became his runningmate. (See diagram above). There also exists an infinitesimal number of universes where you have become his running mate, by means of some very unlikely coincidences.

See also

• multiverse
• quantum decoherence
• quantum immortality

Links:

• Everett's Relative-State Formulation of Quantum Mechanics
• Michael Price's Everett FAQ

References:

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