Many-Worlds Interpretation FAQ

Against ``Against Many-Worlds Interpretations''

By T.Sakaguchi

A.Kent has submitted the paper[1] entitled ``Against Many-Worlds Interpretations'', in which we can see frequently (almost endlessly) raised claims on Everett's many-worlds interpretation (MWI). However, these claims apparently came from misconceptions about Everett's original MWI and I think we should correct them. The paper neither is intended for a debate on the different versions of MWIs, nor criticize recent post-Everettians view, such as consistent histories approach. I just show what the Everett MWI is.

In the paper[1] (Subsection ``A. Everett'' in Section II. ``THE CASE AGAINST''), the author tries to illustrate problems on the Everett MWI[2] by a simple example (typographic errors are corrected, all quotes appear in color):

Suppose that a previously polarized spin-1/2 particle has just had its spin measured by a macroscopic Stern-Gerlach device, on an axis chosen so that the probability of measuring spin +1/2 is 2/3. The result can be idealized by the wavefunction

phi = a phi_0 Phi_0 + b phi_1 Phi_1                         (2)

where phi_0 is the spin +1/2 state of the particle, Phi_0 the state of the device having measured spin +1/2; phi_1, Phi_1 likewise correspond to spin -1/2; |a|^2 = 2/3, |b|^2 = 1/3. Now in trying to interpret this result we encounter the following problems: ...

First of all, the above wavefunction does not describe any quantum-mechanical measurement process which is distinguished from the classical one by the occurrence of the collapse-of-wavefunction (COW) phenomenon. It simply describes a state after the interaction between the spin and the Stern-Gerlach device occurred. Quantum mechanics cannot contradict with the above description, even the Copenhagen interpretation agrees with until an observation by an observer takes place.

In Everett's MWI, it is essential to consider an OBSERVER state, in which all physical informations which were observed by the observer are embedded. Therefore, the above equation should be replaced by

phi = a phi_0 Phi_0 Psi_0 + b phi_1 Phi_1 Psi_1             (2a)

where, Psi_0 and Psi_1 denote the observer states who observed the spin +1/2 and -1/2 respectively. This is a direct consequence of (A) the linearity of the time evolution operator U(t) which includes the interaction between the object and observer systems and of (B) the relation which should be satisfied if the object system is prepared in the EIGENSTATES phi_i and the observer system is prepared to observe the observable (the spin component):

U(t) phi_i Phi_i Psi = phi_i Phi_i Psi_i.                   (2b)

The author continues:

Firstly, no choice of basis has been specified; we could expand phi in the 1-dimensional basis phi or any of the orthogonal 2-dimensional bases

{ cos(theta) phi_0 Phi_0  + sin(theta) phi_1 Phi_1,
  sin(theta) phi_0 Phi_0  - cos(theta) phi_1 Phi_1 }        (3)

or indeed in multi-dimensional or unorthogonal bases. Of course, the information is in the wavefunction is basis-independent, and one is free to choose any particular basis to work with. But if one intends to make a physical interpretation only in one particular basis, using quantities (such as $|a|^2$ and $|b|^2$) which are defined by that basis, one needs to define this process (and, in particular, the preferred basis) by an axiom. This Everett fails to do.

The author apparently fails in understanding what the Everett MWI is. As stated above, Eq.(2) by itself tells nothing about what was observed by an observer unless we specify the observer.

In the state (2a), we can see that there are two observer states Psi_0 and Psi_1, each of which is evolved from the same observer state Psi. These two observer states correspond to the two observers seeing different outcomes, i.e., spin+1/2 and spin-1/2. There is no OBJECTIVE COW, but it APPEARS to the observer that the state of the OBJECT system has collapsed into one of the eigenstates of the observable, while the whole system which includes the observer system is still in a superposition. This is the Everett MWI.

We do not have to require any loss of coherence between the branched states: The process described by Eq.(2b), in which there is no room for indicating any kind of decoherence, defines the observer states unambiguously. This is why there is no need for a preferred basis. (Given a Hamiltonian, eigenstates of the object system, initial state of the measurement device, and initial state of the observer system, we can check if Eq.(2b) holds true.)

The following claim comes from author's misconception.

Secondly, suppose that the basis (phi_0 Phi_0 , phi_1 Phi_1 ) is somehow selected. Then one can perhaps INTUITIVELY view the corresponding components of phi as describing a pair of independent worlds. But this intuitive interpretation goes beyond what the axioms justify; the axioms say nothing about the existence of multiple physical worlds corresponding to wavefunction components.

As explained above, there is no OBJECTIVE multiple physical worlds corresponding to wavefunction components, the world is subjective to each observer. Let us go on to the next claim:

Thirdly, in any case, no physical meaning has been attached to the constants |a|^2 and |b|^2. They are not to be interpreted as the probabilities that their respective branches are realized; this is the whole point of Everett's proposal. It can not be said that a proportion |a|^2 of the total number of worlds is in state phi_0 Phi_0; there is nothing in the axioms to justify this claim. (Note that if the two worlds picture were justified, then each state would correspond to one world, and it must be explained why each measurement does not have probability 1/2.) Nor can one argue that the probability of a particular observer finding herself in the world with state phi_0 Phi_0 is |a|^2; this conclusion again is unsupported by the axioms.

The probability is NOT a branching rate into many-worlds, but the relative frequency which is counted by the observer in EACH world[4]. It is easy to show that the observer who is described in EACH element of the SUPERPOSITION agrees that the relative frequency equals to |a|^2 : |b|^2 in the limit that the number of trials goes to infinity. (Quantum mechanics tells nothing about the outcome of each trial: No one can see any interference pattern in a double slit experiment with only a few of dots on the screen.)

Finally, the author considers the MEASURE and claims:

... mu(Phi^{epsilon}=0) cannot imply that Phi^{epsilon} is physically irrelevant, when no hypothesis has been made about the physical meaning of mu.

where, mu(Phi^{epsilon}) is introduced by the author as a measure. However, it is not the Everett measure, but is a norm which is equal to (Everett measure)^{1/2} (see author's claim below Axiom 1E). The claim seems to arise from a confusion of two totally different concepts; `norm of a state vector' and `measure of a subset in a superposition'.

In order to derive QUANTITATIVE results in a theory, we need to have a measure. Even in a classical theory, we have a measure, though it is trivial one[2]. The physical meaning, or more precisely, the physical DEFINITION, of the Everett measure is that if a state appears in a superposition with measure zero then the state is irrelevant, i.e., we can ignore it. If we have no measure in a theory there is no hope of finding any quantitative result in it (think of measure in path integral, for example). What Everett showed in his paper is that we can construct a consistent quantum theory of a closed system IF we take the Everett measure.


References:
[1] A.Kent, gr-qp/9703089,
    Int.J.Mod.Phys. A5, 1745 (1990).
[2] H.Everett III, Rev.Mod.Phys. 29, 454 (1957).
[3] J.von Neumann, Mathematische Grundlagen der Quanten-mechanik,
    (Berlin, Julius Springer 1932).
[4] T.Sakaguchi, quant-ph/9506042

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