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The continuum



 

I consider it quite possible that physics cannot be based on the field concept  , i.e., on continuous structures  . In that case nothing remains of my entire castle in the air gravitation theory  included, [and of] the rest of modern physics(467)[28].

Continuity is among the oldest of mathematical problems. From a technical standpoint most issues were resolved long ago by defining continuous structures as limiting cases of discrete structures as the elements in the discrete structure increase without limit. However this technical solution does not address let alone resolve the philosophical issues raised by the paradoxes of Zeno  . Those question how it is possible to construct a continuous interval from an infinite number of infinitesimal intervals. There is no mathematics to do that. Mathematics based on taking the limit allows us to construct a discrete algebra  that is the   of an unbounded number of discrete systems. It is a discrete algebra because it operates on discrete abstractions like lines, planes, points and circles and not on a completed infinite collection of infinitesimal objects. Phrases like ``taking the limit  '' or ``passing over to the limit  '' are misleading. We never reach the limit  . All we do is show that certain properties can be made to hold with arbitrarily high accuracy if we can construct a sufficiently large discrete system.

The question of whether time, space and everything definable in space-time is discrete is an open one. It is not clear what it would mean to say that time and space are continuous. The things that obey the algebra of continuity  are not continuous structures  but discrete abstractions. Yet there is nothing abstract about the way physical space-time  is built from smaller regions.

Quantum mechanics suggest that the information content of any finite space-time region with finite energy is finite. If true we can fully model physical reality with a discrete model. If this is so it would again be difficult to know what it means to say that space is continuous  .

Such philosophical musing may be interesting but the proof is in the pudding. Only if discrete mathematics leads to testable theories at variance with continuous theories will such questions move into the realm of physics. Einstein thought this may happen.

The field concept is central to Einstein's approach to physics. It suggests we can model physics by looking at what happens at an arbitrarily small region of space. It assumes that there are some simple rules that define how the state evolves  in such a region. If we know these rules and the initial conditions we can predict the future with certainty (given sufficient computing resources). The field concept and the locality  it implies are fundamental to relativity  . What happens at a point in space is completely determined by the immediate region of that point. Distant objects can only have an effect through the fields they generate and that propagate through space.

Relativity implies continuity. No matter how small a region we select the same rules apply. Quantum mechanics establishes a scale to the universe  that defies our ordinary ideas about continuity. As we move to smaller time and distance scales the complexity of what can be observed according to increases without limit. This alone suggest that at a sufficiently small scale will fail. There seems to be no way to reconcile these two theories within the existing framework of either. Perhaps it is such considerations that led Einstein to suspect that ultimately physics cannot be based on continuous structures.

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