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By Root 656 0

in its quantum form, with photons as the elementary building blocks, was essentially understood in the 1930s. All other fundamental forces except gravity followed with a coherent mathematical description soon after they were discovered in particle interactions. Gravity had been known for centuries and should duly receive a quantum treatment, too. Luminaries such as Paul Dirac, Richard Feynman, and others tried their best but failed. In the 1960s, John Wheeler and Bryce DeWitt started their own dedicated program, but here also there was no clear road toward its rules for temporal change in sight, despite renewed efforts in the 1980s by Stephen Hawking and others. What was missing in all these attempts was a proper understanding of the fundamental nature, down to its atomic scales, of space and time. This is what loop quantum gravity has provided.

Formulating suitable equations in a mathematically useful form, as it was realized in the mid-1990s by loop quantum gravity, surely represented important progress. When Thiemann introduced his equations, the mood was indeed extremely optimistic, almost euphoric. In principle, these equations could be used to determine the structure of space-time in its finest details. Analyzing the past of an atomic space-time, one could calculate backward in time from any given arrangement of loops to find where it would have come from. One could try to understand the universe at very early times, perhaps in a state without loops such as might have existed at the big bang singularity. Or maybe the equations of loop quantum gravity could show deviations from those of general relativity at such extreme energy densities of the early universe, possibly preventing the state of vanishing volume and thus the singularity altogether. With sufficient control over the equations, there would be no limit to the number of intriguing questions one could address. Shortly after Thiemann’s feat, Lewandowski organized a conference at the Banach Institute in Warsaw, Poland, where, according to lore, the prevailing opinion was that one could now solve all the fundamental questions of the cosmos, for its ruling laws were known. (However, I cannot report this from direct experience: I was still studying at that time and did not attend the conference.) Alas, the equations’ complexity quickly struck down all hopes, and to date not a single exact, or just approximate, solution able to describe an entire universe is known.

Such difficulties should not come as a surprise if one considers the immensity of the task. Posing and then solving atomic equations for the dynamic changes of the loops of quantum gravity is an enormously complex procedure, at a mathematical as well as conceptual level. Among other things, it illustrates the special and confusing role of time in fundamental modeling. There is no space-time to start with; even in the classical theory of general relativity, space-time and its structure is a derived concept that follows after solving Einstein’s equation. In quantum gravity, the task is to find the analogous equation, successfully taking into account the tiniest atomic details. When solving quantum gravity, what we classically understand as space-time arises as a collapsed version of infinitely many spaces with different sizes and geometries, all in a quantum superposition. According to the rules of quantum theory, they all exist “at once,” but time is yet to emerge. One can think of this superposition as a dissection of space-time, like a deck of cards, into its spatial configurations at constant times, yet unordered owing to the absence of time.

Once such a mathematical solution has been achieved, one can analyze it further. At this stage, all the spatial configurations may be ordered, for instance by arranging them by their volumes. When this is done, evolution comes in by computing other quantities for the ordered configurations, such as the matter density. One could, for instance, derive a functional relationship between the density and the volume of space quite similar to the solutions one obtains