From what I remember about quantum mechanics, energy can only take on certain values--integer multiples of a certain minimally-measurable quantity. Or, perhaps more specifically, that it can only beknown or measured(at the present time) that energy takes on integer multiples of this quantum value.

So what does this have to do with temperature? Well, from what I remember about thermodynamics, temperature is an average of the kinetic energy of molecules vibrating. If this energy can only take on integer multiples of some quantum valuec, then the arithmetic average can only take on rational values ofc, or if cleverer methods are used, roots of roots of roots (etc.) of rational values ofc.

What does this have to do with anything?

Well, when students learn about the Intermediate Value Theorem in real analysis, they are often introduced to the following problem:

Consider the continuous function over the globe mapping each point on Earth to the temperature at that point. Use the Intermediate Value Theorem to prove that there must exist two antipodal points that have the same temperature.

However, if temperature is quantized, then you could never for instance get the value of pi*cor e*c, since these numbers are transcendental and therefore are not the root of any polynomial with rational coefficients. Therefore no temperature function could be continuous over an interval [a,b] if f(a) is less than (and f(b) greater than) a transcendental multiple of c.

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