On the Interpretation of Quantum Mechanics

Andrew Steane, March 1998

The essential problem of the interpretation of quantum mechanics is that exhibited by the Schrodinger cat experiment. We need an interpretive approach which is both in agreement with experiment and also clear and elegant. What is happening at the moment is that we make use of a variety of interpretations, all of which agree with experiment, but which suffer from varying degrees of vagueness or awkwardness.

The most obvious way to think about Schrodinger's cat is to adopt one of the following hypotheses:

(i) completely unitary dynamics leading to highly complicated quantum states
(ii) non-unitary change of the quantum state (`collapse') through new non-linear dynamics
(iii) additions to, and different interpretation of, the mathematical apparatus of quantum mechanics

The first is the hypothesis in which Schrodinger's equation is perfectly ok and describes all physical interactions all the time, so the final state of nucleus, poison, cat, other observers, old uncle Tom Cobbley and all is a quantum superposition state. The second is the hypothesis that some as yet not fully understand non-linear dynamics occurs in physical systems of sufficient mass or complexity, leading to a final state in which the cat is either alive or dead, and certainly not in a superpussition. The third is a framework such as the de Broglie Bohm pilot wave theory, or the decoherent histories approach.

It seems to me that Occam's razor (``don't accumulate unneccesary hypotheses'') would shave away (ii) and (iii) if we could be convinced that (i) is a clear and elegant description of the world around us. I do not find (iii) appealing because those approaches seem to me highly inelegant, especially when the attempt is made to render them Lorentz invarient.

The problem with (i) is that it seems to be at odds with what goes on in the real world. The way to avoid this appearance is to argue that a state such as

|phi> = |unemitted>|alive>|warm> + |emitted>|dead>|cold> (1)

corresponds directly and simply to the experimentally observed situation that the cat is either alive or dead. It is this point of view which I adopt. The three-system ket describes the state of the radioactive nucleus, the cat, and the rest of the universe (which is hotter when the cat is alive since the live cat continuously converts work into heat). I will abreviate this ket to |phi> = |u a w> + |e d c>.

My point of view implies the following apparent contradiction: a quantum system in the state |u a w> + |d d c> is also either in the state |u a w> or the state |e d c>. This is only an apparent contradiction, which turns on the use of the phrase ``in the state''. The correct statement, which is clear and non-contradictory, is:

a quantum system described by the state |u a w> + |d d c> is also described either by the state |u a w> or the state |d d c>

In order for this statement to be useful, we need to solve the preferred basis problem. For example, consider the mathematical identity

|u a w> + |e d c> = |n+ s+ f+> + |n+ s- f-> + |n- s+ f-> + |n- s- f+> (2)

where

|n+> = |u> + |e>, |n-> = |u> - |e>
|s+> = |a> + |d>, |s-> = |a> - |d>,
|f+> = |w> + |c>, |f-> = |w> - |c>.

Unless we introduce a further piece of interpretive apparatus, we are in danger of supposing that the system described by |phi> is also described by |n+ s+ f+> or each of the other components in (2), which would mean we have not solved the original problem. However, this is easily solved, as follows:

Postulate 1. A quantum system and environment described by the state |phi> is also described by one of the states of the basis in which the reduced density matrix of |phi> is diagonal after the environment has been traced over.

The `collapse' from |phi> to one of |u a w> and |e d c> is now no different from the abrupt change in a classical probability distribution when more information becomes available. The definition of the `environment', which plays a crucial role in Postulate 1, is

Definition 1. The environment is defined to be any system (or collection of systems) whose identification as environment does not make postulate 1 inconsistent during the evolution to be described by |phi>.

In practice this means that the definition of `environment' can depend on how far into the future we intend the quantum state to describe the quantum system. There is no problem with that, since the quantum state merely provides information about the quantum system, the system is not really ``in the state x'', though sometimes that is a useful shorthand.

The usual choice for environment is any system which does not thereafter couple to the other systems described by |phi> in such a way as to disentangle itself (i.e. end up in a product with the non-environment). In Schrodinger's cat experiment, almost all the rest of the cat can act as `environment' to any small part of the cat.

This framework may seem to leave the `real world' rather unconcrete, its state depending on what the environment might do in the future. Actually the situation is that the real world is as real and concrete as those words can mean, but quantum mechanics offers us not the direct mathematical machinery underlying the real world, but rather a description of the information which we can gain about the real world. The world is always, in the nature of things, going to be one step removed from our description of it, but that does not make it any the less real. Rather it is our insight which is contingent on the information we possess.

In my point of view, quantum states such as |u a w> + |e d c> are perfectly legitimate. An apparent difficulty is that we might imagine we don't encounter states like |u a w> + |e d c> in our conscious experience. Once we recall that the ket is merely a description, not the underlying reality, this problem vanishes. The reality is that we experience cats either alive or dead, and the state |u a w> + |e d c> is the description of our experience, furnished by our physical theory. We can tell when this description is a quantum superposition and when a mixture by appealing to Postulate 1. The only way this can lead to problems is if your conscious experience does not have an environment, but this never happens and indeed is probably a self-contradictory circumstance.

Having provided our interpretive formalism for quantum theory, there remains the following insight into the universe we live in. This is not an axiom of quantum mechanics, but an empirical observation about the universe. The dynamics of the quantum systems in the universe is such that most of the universe acts as `environment', i.e. can fulfil the role required for the `environement' in quantum theory, permanently, or at least for a timespan of the order of the age of the universe, for a large set of systems, whose rich behaviour can be captured mathematically by principles of stationary phase. The large and rich set of systems and behaviour is classical physics, and the truth of this observation is linked to the second law of thermodynamics and to Feynman's path integral formalism.

This interpretive approach to quantum mechanics combines features of the Copenhagen interpretation, the Everett `relative state' interpretation, the environment-induced decoherence, and consistent histories, but is not identical to any of them.