How Come Quantum Theory?

"A central aim of physics is to understand the world around us. It is not quite clear exactly what this means. However, physicists often take the pragmatic viewpoint that understanding the world means to be able to predict its behavior from a small set of well motivated principles," writes Lucien Hardy.

But is this really a valid equivalent? For example, haven't we all lived with parents and partners who could predict our moves but had nounderstanding of them? Is this simply a problem of subjectivity? Or, is this a universal distinction between how and why questions? Moreover, does John Wheeler's use of the "How come?" question somehow allow a weasel-like slide between why and how? Consider, for example, the theme of today's essay: How come the quantum?

Thus, Hardy continues, "Newton's laws of motion are few in number and seem to be intuitively reasonable. Similarly, Einstein's special theory of relativity is founded on two sensible axioms (that the laws of physics should be the same in all inertial frames and that the speed of light is the same when measured in different inertial frames)."

Now here's a question of a different sort: Is it a rational assumption to assume that the universe would behave rationally? Does that not imply, along the lines of the old philosophers, that rationality is somehow inherent in the universe? And, if it is, what does this mean? Furthermore, how do we know that this reasonableness is not simply an artifact of our intended perception or, worse yet, merely a by-product of our ignorance or limited perception? For, as we all know from experience, the more complex a phenomenon, the harder it is to see.

However, Hardy goes on to say that the "reasonableness of these two theories contrasts sharply with the situation in quantum theory. While the usual formulation of quantum theory is based on only a few axioms, these axioms are not intuitively reasonable. They refer to mathematical entities like Hilbert spaces, and operators on Hilbert spaces. Such abstract objects have no obvious physical origin. This raises the question of whether it is possible to posit a different set of axioms that are more physically reasonable. If we can then we might feel that we have answered John Wheeler's question: 'Why the quantum?'"

Continue reading to find out how Hardy goes about answering that question. Hardy's paper is presented as part of a special series in anticipation of The Science & Ultimate Reality Symposium in Princeton, a symposium in honor of the 90th year of John Archibald Wheeler.

—Editor

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The vast majority of scientists take quantum mechanics for granted. Students will often ask why there is indeterminism in the atomic domain, or where Schrodinger's equation comes from. The standard answer is simply, the world just happens to be that way. But some researchers aren't satisfied with this reply. John Wheeler asks: How come the quantum? What he seeks is a reason for the strange rules of quantum mechanics.

So is there anything special about the logical and mathematical structure of quantum mechanics that singles it out as a natural way for the universe to be put together? Could we imagine making small changes to quantum mechanics without wrecking the form of the world as we know it? Is there a deeper principle at work that translates into quantum mechanics at the level of familiar physics?

Lucien Hardy of the Centre for Quantum Computation at the University of Oxford has attempted to construct quantum mechanics from a set of formal axioms. Boiled down to its essentials, quantum mechanics is one possible set of probabilistic rules with some added properties. The question then arises as to whether it has specially significant consequences. Could it be that quantum mechanics is the simplest mechanics consistent with the existence of conscious beings? Or might quantum mechanics be the structure that optimizes the information processing power of the universe? Or is it just an arbitrary set of properties that this world possesses reasonlessly?

—Paul Davies

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Summary:

A set of five physically reasonable principles (or axioms) for quantum theory are put forward. Four of the axioms are obviously consistent withboth classical probability theory and quantum theory. The remaining axiom requires that there exists a continuous reversible transformation between any pair of pure states. It is the quite natural requirement of continuity that rules out classical probability theory and gives us quantum theory. It is suggested that this approach provides a possible answer to Wheeler's question: Why the quantum?

A central aim of physics is to understand the world around us. It is not quite clear exactly what this means. However, physicists often take the pragmatic viewpoint that understanding the world means to be able to predict its behavior from a small set of well motivated principles. Thus, Newton's laws of motion are few in number and seem to be intuitively reasonable. Similarly, Einstein's special theory of relativity is founded on two sensible axioms (that the laws of physics should be the same in all inertial frames and that the speed of light is the same when measured indifferent inertial frames). The reasonableness of these two theories contrasts sharply with the situation in quantum theory. While the usual formulation of quantum theory is based on only a few axioms, these axioms are not intuitively reasonable. They refer to mathematical entities like Hilbert spaces, and operators on Hilbert spaces. Such abstract objects have no obvious physical origin. This raises the question of whether it is possible to posit a different set of axioms that are more physically reasonable. If we can then we might feel that we have answered John Wheeler's question: Why the quantum?

Once stripped of all its incidental structure, quantum theory is simply a new type of probability theory. Its predecessor, classical probability theory, can be based on far more reasonable principles and is readily understood even by the layman. The purpose of this paper is to describe some recent work in which it has been shown that quantum theory can actually be based on a small number of quite reasonable principles (or axioms). We consider systems with a countable number of distinguishable states (in quantum theory this number corresponds to the dimension of the associated Hilbert space) and we consider only kinematics (we do not obtain a specific Hamiltonian since that would pertain to a particular application of quantum theory). Of the axioms, four are obviously consistent with both classical probability theory and quantum theory. The remaining axiom requires that there exists a continuous reversible transformation between any pair of pure states. It is the requirement of continuity here that rules out classical probability theory and gives us quantum theory.

An example illustrates this. Consider a classical bit. This is a system with two distinguishable states that can be labelled 0 and 1. To go from the 0 state to the 1 state it is necessary to jump. There is no continuous transformation. Now consider a qubit (a quantum system with two distinguishable states). There can exist superpositions of the |0> and |1>state with arbitrary weightings. Thus, it is possible to go from the |0>state to the |1> state in a continuous way (by going along a trajectory through these superpositions). When viewed from this point of view we see that, in fact, quantum theory is actually more reasonable than classical probability theory because it has a very natural continuity property. What is even more remarkable is that imposing this continuity property in addition to a set of quite reasonable axioms which are consistent with classical probability theory actually leads naturally to quantum theory. Thus, an answer to Wheeler's question (Why the quantum?) is that we require quantum theory because we want to get rid of those dammed classical jumps!

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