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The real line and configuration space

At the start of this chapter we observed that the continuous real line that we see is a creation of our brain and nervous system. Everything we see and touch is made up of fundamental particles. Although objects in space-time are discrete, space-time itself remains continuous in special and general relativity. Those theories are so dependent on the classical continuum that Einstein recognized any fully discrete theory would imply relativity was only approximately true and would make false predictions at the scale of space-time discreteness (see Section 7.1).

Space-time is very strange in quantum mechanics. It remains continuous but it has a peculiar connectivity because of quantum entanglement. In classical physics and relativity space is separable. You can fully describe what happens in any localized region over a brief time interval without taking into account distant events. This is not possible in quantum mechanics.

The nonrelativistic version of quantum mechanics exists, not in physical space, but in an abstract higher dimensional structure known as configuration space where there is a single time dimension and a separate set of spatial dimensions for every particle. See Figure 7.5. The connection between configuration space and physical space is through a probability distribution which gives the probability that a given configuration of particles will be observed.

In physical space we do not have anything like the classical real line. What exactly we do have is not clear since the actualization of probabilities in configuration space to events in physical space is not part of any existing scientific theory.

A mathematical model from a scientific theory may have little to do with how nature is structured. Obviously it must provide an accurate approximation in its experimental predictions. Classical mechanics is very accurate for a wide range of experiments but quantum mechanics has shown that the structure of physical reality must differ radically, Some physicists argue that it is naive realism to expect a correspondence between nature and our mathematical models. While one must admit that anything is possible, and there are many aspects of existing theory that make it seem difficult to construct such a correspondence, I suspect that those are problems in the existing theories and our understanding of nature. Mathematics can model what nature does to extraordinarily accuracy and this leads me to suspect that nature has a mathematical structure.

Quantum entanglement is at the core of the strangeness in contemporary physics. The evidence that distant events influence each other, in ways that can never be explained by a local mechanism, is dramatic and compelling but not totally conclusive (see Section 8.6). Experiments, as always, will decide the issue but what we make of experiments and how decisive they need to be is a matter of judgment. One of the factors that goes into such judgments is our sense of what alternative possibilities exist. Fully discrete models would be radically different than anything previously investigated. They hold the possibility of the more complete theory Einstein sought. There may be an experimental path that leads to such a theory. All of these issues fall under the problem of integrating relativity and quantum mechanics which is the subject of the next chapter.

Figure 7.5: Configuration space
\begin{figure} % latex2html id marker 1700 \sf Two particles $p_1$\ and $p_2$\ i... ...cBellIneq}. \begin{center}\index{configuration space} \end{center}\end{figure}

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