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By Root 2002 0

Everyone knew that you can’t trust quantum field theory on super-small distance scales. Jitters with wavelengths as small as the Planck scale, 10–33 centimeters, and smaller, have energy (and from m = E/c2, mass equivalent) so large that the gravitational force matters. To describe them properly requires a framework that melds quantum mechanics and general relativity. Conceptually, this shifts the discussion to string theory, or to any other proposed quantum theory that includes gravity. But the immediate and more pragmatic response among researchers was simply to declare that the calculations should disregard jitters on scales smaller than the Planck length. Failure to implement this exclusion would extend a quantum field theory calculation into a realm clearly beyond its range of validity. The expectation was that we will one day understand string theory or quantum gravity well enough to deal with the super-small jitters quantitatively, but the interim stopgap was to mathematically quarantine the most pernicious fluctuations. The import of the directive is clear: if you ignore jitters shorter than the Planck length, you’re left with only a finite number, so the total energy they contribute to a region of empty space is also finite.

Figure 6.3 There are infinitely many wave shapes in any volume and hence infinitely many distinct quantum jitters. This yields the problematic result of an infinite energy contribution.

That’s progress. Or, at the very least, it shifts the burden to future insights that would, fingers crossed, tame the super-small-wavelength quantum fluctuations. But even so, researchers found that the resulting answer for the energy jitters, while finite, was still gargantuan, about 1094 grams per cubic centimeter. This is far larger than what you’d get from compressing all the stars in all the known galaxies into a thimble. Focusing on an infinitesimal cube, one that measures a Planck length on each side, this stupendous density amounts to 10–5 grams per cubic Planck length, or 1 Planck mass per Planck volume (which is why these units, like kilos for potatoes and seconds for waiting, are the natural and sensible choice). A cosmological constant of this magnitude would drive such an enormously fast outward burst that everything from galaxies to atoms would be ripped apart. More quantitatively, astronomical observations had established a tight limit on how large a cosmological constant could be, if there were one at all, and the theoretical results exceeded the limit by a staggering factor of more than a hundred orders of magnitude. While a large finite number for the energy that suffuses space is better than an infinite one, physicists realized the dire need for dramatically reducing the result from their calculations.

Here’s where theoretical prejudice came to the fore. Assume for the moment that the cosmological constant is not just small. Assume it’s zero. Zero is a favorite number of theoreticians because there’s a tried and true way for it to emerge from calculations: symmetry. For example, imagine that Archie has enrolled in a continuing education course and for homework has to add together the sixty-third power of each of the first ten positive numbers, 163 + 263 + 363 + 463 + 563 + 663 + 763 + 863 + 963 + 1063, and then add the result to the sum of the sixty-third power of each of the first ten negative numbers, (–1)63 + (–2)63 + (–3)63 + (–4)63 + (–5)63 + (–6)63 + (–7)63 + (–8)63 + (–9)63 + (–10)63. What’s the final tally? As he laboriously calculates, getting ever-more frustrated, multiplying and then adding together numbers with more than five dozen digits, Edith chimes in: “Use symmetry, Archie.” “Huh?” What she means is that each term in the first collection has a symmetric balancing term in the second: 163 and (–1)63 sum to 0 (a negative raised to an odd power remains negative); 263 and (–2)63 sum to 0, and so on. The symmetry between the expressions results in a total cancellation, as if they were children of equal weight balancing on opposite sides of a seesaw. Needing no calculations