A Proof of God using Quantum Mechanics [Not an article for beginners]

The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction and denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternative histories and futures are real, each representing an actual “world” (or “universe”). It is also referred to as MWI, the relative state formulation, the Everett interpretation, the theory of the universal wavefunctionmany-universes interpretation, or just many-worlds.

The original relative state formulation is due to Hugh Everett in 1957.[2][3] Later, this formulation was popularized and renamed many-worlds by Bryce Seligman DeWitt in the 1960s and 1970s.[1][4][5][6] The decoherence approaches to interpreting quantum theory have been further explored and developed,[7][8][9] becoming quite popular. MWI is one of many multiverse hypotheses in physics and philosophy. It is currently considered a mainstream interpretation along with the other decoherence interpretations, theCopenhagen interpretation,[10] and deterministic interpretations such as the Bohmian mechanics.

Before many-worlds, reality had always been viewed as a single unfolding history. Many-worlds, however, views reality as a many-branched tree, wherein every possible quantum outcome is realised.[11] Many-worlds claims to reconcile the observation of non-deterministic events, such as the random radioactive decay, with the fully deterministicequations of quantum physics.

In many-worlds, the subjective appearance of wavefunction collapse is explained by the mechanism of quantum decoherence, which resolves all of thecorrelation paradoxes of quantum theory, such as the EPR paradox[12][13] and Schrödinger’s cat,[1] since every possible outcome of every event defines or exists in its own “history” or “world”.

In lay terms, the hypothesis states there is a very large–perhaps infinite[14]–number of universes, and everything that could possibly have happened in our past, but did not, has occurred in the past of some other universe or universes.

In order to preserve the consistency of quantum mechanics, Everett concluded that the standard collapse formulation could not be used to describe systems that contain observers; that is, it could only be used to describe a system where all observers are external to the described system. And, for Everett, this restriction on the applicability of quantum mechanics was unacceptable. Everett wanted a formulation of quantum mechanics that could be applied to any physical system whatsoever, one that described observers and their measuring devices the same way that it described every other physical system.

3. Everett’s Proposal

In order to solve the quantum measurement problem Everett proposed dropping the collapse dynamics (Rule 4b) from the standard collapse theory and proposed taking the resulting physical theory as providing a complete and accurate description of all physical systems whatsoever. Everett then intended to deduce the standard statistical predictions of quantum mechanics (the predictions that depend on Rule 4b in the standard collapse formulation of quantum mechanics) as the subjective experiences of observers who are themselves treated as ordinary physical systems within the new theory.

Everett says:

We shall be able to introduce into [the relative-state theory] systems which represent observers. Such systems can be conceived as automatically functioning machines (servomechanisms) possessing recording devices (memory) and which are capable of responding to their environment. The behavior of these observers shall always be treated within the framework of wave mechanics. Furthermore, we shall deduce the probabilistic assertions of Process 1 [rule 4b] as subjective appearances to such observers, thus placing the theory in correspondence with experience. We are then led to the novel situation in which the formal theory is objectively continuous and causal, while subjectively discontinuous and probabilistic. While this point of view thus shall ultimately justify our use of the statistical assertions of the orthodox view, it enables us to do so in a logically consistent manner, allowing for the existence of other observers (1973, p. 9).

Everett’s goal then was to show that the memory records of an observer as described by quantum mechanics without the collapse dynamics would somehow agree with those predicted by the standard formulation with the collapse dynamics. The main problem in understanding what Everett had in mind is in figuring out how this correspondence between the predictions of the two theories was supposed to work.

In order to see what happens, let us try Everett’s no-collapse proposal for a simple measurement interaction. One can measure the x-spin of a physical system. More specifically, a spin-1/2 system will be found to be either “x-spin up” or “x-spin down” when its x-spin is measured. So suppose that J is a good observer who measures the x-spin of a spin-1/2 system S. For Everett, being a good x-spin observer means that J has the following two dispositions (the arrows below represent the time-evolution described by the deterministic dynamics of Rule 4a):

equation 1

equation 2

If J measures a system that is determinately x-spin up, then J will determinately record “x-spin up”; and if J measures a system that is determinately x-spin down, then J will determinately record “x-spin down” (and we assume, for simplicity, that the spin of the object system S is undisturbed by the interaction).

Now consider what happens when J observes the x-spin of a system that begins in a superposition of x-spin eigenstates:

equation 3

The initial state of the composite system then is:

equation 4

Here J is determinately ready to make an x-spin measurement, but the object system S, according to Rule 3, has no determinate x-spin. Given J‘s two dispositions and the fact that the deterministic dynamics is linear, the state of the composite system after J‘s x-spin measurement will be:

equation 5

On the standard collapse formulation of quantum mechanics, somehow during the measurement interaction the state would collapse to either the first term of this expression (with probability equal to a squared) or to the second term of this expression (with probability equal to b squared). In the former case, J ends up with the determinate measurement record “spin up”, and in the later case J ends up with the determinate measurement record “spin down”. But on Everett’s proposal no collapse occurs. Rather, the post-measurement state is simply this entangled superposition of J recording the result “spin up” and S being x-spin upand J recording “spin down” and S being x-spin down. Call this state E for Everett. On the standard eigenvalue-eigenstate link (Rule 3) E is not a state where determinately records “spin up”, neither is it a state where Jdeterminately records “spin down”. So the puzzle for an interpretation of Everett is to explain the sense in which J‘s entangled superposition of mutually incompatible records is supposed to agree with the empirical prediction made by the standard collapse formulation of quantum mechanics. The standard collapse theory, again, predicts that J either ends up with the fully determinate measurement record “spin up” or the fully determinate record “spin down”, with probabilities equal to a-squared and b-squared respectively.

Everett confesses that a post-measurement state like E is puzzling:

As a result of the interaction the state of the measuring apparatus is no longer capable of independent definition. It can be defined only relative to the state of the object system. In other words, there exists only a correlation between the states of the two systems. It seems as if nothing can ever be settled by such a measurement (1957b, p. 318).

And he describes the problem he consequently faces:

This indefinite behavior seems to be quite at variance with our observations, since physical objects always appear to us to have definite positions. Can we reconcile this feature of wave mechanical theory built purely on [Rule 4a] with experience, or must the theory be abandoned as untenable? In order to answer this question we consider the problem of observation itself within the framework of the theory (1957b, p. 318).

Then he describes his solution to this determinate-record (determinate-experience) problem:

Let one regard an observer as a subsystem of the composite system: observer + object-system. It is then an inescapable consequence that after the interaction has taken place there will not, generally, exist a single observer state. There will, however, be a superposition of the composite system states, each element of which contains a definite observer state and a definite relative object-system state. Furthermore, as we shall see, each of these relative object system states will be, approximately, the eigenstates of the observation corresponding to the value obtained by the observer which is described by the same element of the superposition. Thus, each element of the resulting superposition describes an observer who perceived a definite and generally different result, and to whom it appears that the object-system state has been transformed into the corresponding eigenstate. In this sense the usual assertions of [the collapse dynamics (Rule 4b)] appear to hold on a subjective level to each observer described by an element of the superposition. We shall also see that correlation plays an important role in preserving consistency when several observers are present and allowed to interact with one another (to ‘consult’ one another) as well as with other object-systems (1973, p. 10).

To this end Everett presents a principle that he calls the fundamental relativity of quantum mechanical states. On this principle, one can say that in state EJrecorded “x-spin up” relative to S being in the x-spin up state and that J recorded “x-spin down” relative to S being in the x-spin down state. But this principle cannot by itself provide Everett with the determinate measurement records (or the determinate measurement experiences) predicted by the standard collapse formulation of quantum mechanics. The standard formulation predicts that on measurement the quantum-mechanical state of the composite system will collapse to precisely one of the following two states:

equation 6

and that there is thus a single, simple matter of fact about which measurement result J recorded. On Everett’s account it is unclear whether J ends up recording one result or the other or perhaps somehow both.

The problem is that there is a gap in Everett’s exposition between what he set out to explain and what he ultimately ends up saying. He set out to explain why observers get precisely the same sort of measurement records in his no-collapse formulation of quantum mechanics as predicted by the standard collapse formulation of quantum mechanics, but he ends up describing a post-measurement observer who apparently does not have any particular measurement record. And since it is unclear exactly how Everett intends to explain determinate measurement records, it is also unclear how he intends to explain why one should expect one’s determinate measurement records to exhibit the standard quantum statistics. It is this gap in Everett’s exposition that has encouraged subsequent reconstructions of his theory.

A satisfactory reconstruction of Everett’s relative-state interpretation would address each of the following three problems and ideally do so in a way that somehow takes quantum mechanics without the collapse dynamics as descriptively complete:

First, it would explain the sense in which an observer has a determinate measurement record when the post-measurement quantum-mechanical state E is not an eigenstate of there being a single determinate record. Alternatively, it might somehow explain why there appears to be a determinate physical record when they is in fact no such record.

Second, it would account for the standard probabilistic predictions of quantum mechanics. In order to do this in a straightforward way, one must have already solved the determinate record problem so that there is something to which quantum probabilities might apply. Further, assigning standard quantum probabilities to possible determinate measurement outcomes presupposes that only one of the determinate records is in fact realized for an observer. Finally, assigning standard quantum probabilities to my future measurement records requires an understanding of how to identify which future Everett branch represents me. Alternatively, if all Everett branches describe measurement records that in fact obtain and hence that occur with classical probability one, one might seek to develop a new, nonstandard notion of probability that will give the right quantum probabilities even when one knows that every physically possible measurement outcome is fully realized (see Saunders 1998 for an example of a nonstandard notion of probability that applies to relative facts). The challenge then would be to explain why one should care about the new notion of probability when, at least on the face of it, the nonstandard probability of an event cannot matter for the purposes of rational action since every physically possible event in fact occurs. A closely related approach is to seek to develop a new, nonstandard notion of rational choice that makes sense even when one knows that every physically possible outcome will occur (see Deutsch 1999, Wallace 2003 and 2007, and Greaves 2006 for variations on this strategy). But then the question is why one should want a nonstandard notion of rational choice in the first place. One might argue that if one is committed to every Everett branch in fact being realized, one must have either a nonstandard account of probability or rational choice, or both. But then why should a classically rational agent ever opt for such a commitment? Hence the third problem.

So third, and finally, a satisfactory reconstruction of Everett would allow one to explain how one might have empirical justification for accepting the relative-state formulation of quantum mechanics if the physical world were in fact faithfully described by the theory. That is, the theory is only satisfactory if it is empirically coherent (See Barrett 1996 for a discussion of the condition of empirical coherence). A theory that is not empirically coherent might be true, it is just that one would never have good empirical grounds for accepting it as true. One might, of course, seek to develop new, nonstandard criteria for rational acceptance in order to make one’s favorite reading of Everett rationally acceptable, but then one might likewise make any physical theory rationally acceptable (see Wallace 2006 for an example of an argument that Everettian quantum mechanics is rationally acceptable on empirical grounds).

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