The Imaginary Collapse of the Wavefunction

Posted on 22 March 2009

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The so-called “collapse” of the wave function in quantum theory is often illustrated by the wave/particle duality. When a photon propagates through a double-slit apparatus, it behaves like a wave. Yet, if it is observed, the non-local wave is collapsed into a single localized particle. However, both theory and experiment show that this is not a clear-cut either/or  distinction, as it is misleadingly presented in traditional discussions of the double slit experiment. The interference pattern is not simply there or not, but is gradually deteriorated as more information about which slit the particle went through can be extracted from the photon measurement. This suggests that, in general, there is never any discontinuous or sudden collapse of the wavefunction. All that is ever happening is that we’re pushing information around with measurement interactions in a completely continuous (unitary) way.

Not only is collapse of the wave function totally unverifiable and nonphysical, but another big problem with collapse is that it is in blatant violation of the Schrödinger equation! Any other scientific hypothesis that both violates known laws of physics and is not verifiable would normally be immediately rejected as pseudo-science. Why, then, has the notion of collapse stuck? Perhaps because one consequence of rejecting collapse would seem to be that it would lead us inevitably to the many worlds interpretation. Strange as the many worlds interpretation may be, however, it does have the virtue of being consistent with the laws of physics, at least as we know them so far.

The many worlds interpretation is often rejected as outrageous because it seems to imply that all the separate “worlds” have some actual existence, just like ours. But, it’s more like none of the “worlds” have actual existence, including ours. To make an analogy with the theory of relativity, it’s not like there are many actual velocities of the earth in space, each existing as its own separate actualized “world.” Rather, it’s that the earth has no actual objectively existing velocity at all. Velocity only has meaning relative to a reference frame, and reality does not have any privileged reference frame. We happen to observe things in the reference frame of the Earth where that velocity is zero. If we were on the Moon, things would be different. Is there really some mystery here? How is this so different from quantum theory? The original “relative state” formulation of quantum theory seems to be in line with this view, and calling it a “many worlds” theory is just as misleading as calling relativity theory a “many worlds” theory. It’s just “many reference frames” and one world. One might complain that the “one world” is a strange one, but that’s no less true in relativity theory where nothing has any objective mass, length, time, etc. The only objective realities are the four-dimensional invariants. These are almost as weird as coherent superpositions.

It is good to remember that physical theories in general are abstractions, describing a reality that is beyond our direct experience. We experience our immediate sensations of sight, sound, etc., and never directly experience the abstractions of “atoms” or “fields” which are only indirectly inferred from experience. (The same is actually true of a “chair” or “rock” as well.) These may be useful abstractions, but we never actually experience them directly, and can never know if they really exist the way we think. In fact, we don’t really know that they exist at all. We could be a brain in a vat or having a lucid dream right now. Science tries to balance the belief in some objective reality with the fact that we can never know the thing in itself. As Heisenberg wrote,

We have to remember that what we observe is not nature in itself but nature exposed to our method of questioning.

It is actually more radical than Heisenberg suggests. Consider again the double-slit experiment. A simple photon which “measures” which slit the particle went through does not actually collapse the wave function to be localized in just one region of space. It merely entangles itself with the system.  Provided no decoherence has taken place so that the coherence of the original system is not washed out in many degrees of freedom of the measurement system, then there is no sense in which an irreversible measurement interaction has taken place. So one is still free to decide what will ultimately be measured. Because there has not been any interaction with a particular well-defined measurement apparatus (by which I mean a device that involves decoherence) the attributes of the system are likewise still undefined.

The above situation with regard to a quantum system is analogous to not having defined any particular well-defined reference frame in relativity. If I do not specify a reference frame for an observation of a monolith floating in space, then it has no definite well-defined value for various properties such as velocity, mass and length. Once the reference frame is specified, however, then one can meaningfully talk about definite values for these quantities. Similarly, once one specifies a particular measurement apparatus (that involves decoherence), then one can say there is a well-defined meaning to talking about certain properties. The coherence is lost and there is no practical possibility to erase that measurement choice after the interaction with the measurement apparatus and choose instead to measure a complementary observable. And all observers will agree on what is measured.

In connection with this, Pauli has this interesting statement:

Just as in the theory of relativity a group of mathematical transformations connects all possible coordinate systems, so in quantum mechanics a group of mathematical transformations connects the possible experimental arrangements.

And Bohr writes:

In neither case [of quantum theory or relativity theory] does the appropriate widening of our conceptual framework imply any appeal to the observing subject, which would hinder unambiguous communication of experience. In relativistic argumentation, such objectivity is secured by due regard to the dependence of the phenomena on the reference frame of the observer, while in complementary description all subjectivity is avoided by proper attention to the circumstances required for the well-defined use of elementary physical concepts.

Admittedly, the analogy with relativity only goes so far. In the case of relativity, the choice of reference frame is sufficient to provide a unique and definite value for physical attributes. In quantum systems, on the other hand, although the interaction with a particular decohering measurement apparatus gives a particular observable well-defined meaning, it still does not result in a definite value (i.e., the wavefunction is not collapsed). The analogy with relativity, it seems, is a similarity between the choice of reference frame and the choice of a particular decohering measurement apparatus. These choices are sufficient to give well-defined meaning to certain physical quantities. The difference seems to be that in quantum theory, even though the quantities may have well-defined meaning, they still have not been actualized. For example, once the atom has interacted with the Geiger counter and poison bottle, it makes sense to say that Schrödiner’s cat is either alive or dead (there is no longer any coherence that would allow one to perform a measurement of a complementary observable to the alive/dead observable).

The actualization of a particular value could be described in terms of the many worlds interpretation as the choice of which world “you” get identified with. In relativity, though, one can actually imagine something analogous, but we don’t regard it as a mystery for some reason: The description of the world according to relativity does not specify which moment in spacetime we should be experiencing as “here and now”. So, what determines which point in Minkowski space is “actualized” in our experience as here and now? Why should we experience this here and now rather than some other? This question seems quite similar to the question of why we experience ourselves in one of the many worlds as opposed to some other. What “collapses” us into a particular here and now? Clearly, there is no such collapse, just as there is no collapse in quantum theory. The theory is an abstraction from the here and now. If we get confused and think that we really live in the abstraction, then we become perplexed at how the specific here and now is mysteriously “collapsed” from all the possibilities in the general, abstract world we’ve dreamed up.

There is also an interesting similarity between the role of decoherence, which effectively cuts us off from ever detecting any of the worlds that have decohered from ours, and space-like separation in relativity. There are spacelike separated regions of spacetime that can not have any interaction or communication with us. So, what justification is there for saying that they exist at all? They can never be observed or verified to exist. Is this really any different than the other branches of the universal wave function that we can no longer detect because of decoherence?

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Posted in: Philosophy, Science