Love and Math
The Heart of Hidden Reality
2013 Edward Frenkel
Edward Frenkel (1968) is a Russian expatriate mathematician, currently teaching at Berkeley, working on the "Langlands Program", the unification of Number Theory, Curves over Finite Fields, and Reimann Surfaces. Wile's proof of Fermat's Last Theorem is discussed as an example of the combination of the first two.
This book is a combination of autobiography and descriptions of some complex mathematical thinking - it is more about the intellectual journey from young physics enthusiast to mathematical researcher, not a lucid explanation of the math. I suppose if you read it slowly, twice, you will have a basic understanding of some extremely challenging math. I did not do the hard work, but some of you might.
I enjoyed the book. I expect many but not all mathematicians will. Some might be frustrated by not fully understanding the subject matter. Others might find the autobiographical content too personal.
My lazy tendency is to throw stuff against the wall of my brain and see what sticks - a surprising amount does, even if I don't think I understand it at the time. Enough remains to combine with ideas from other areas, or to research in depth if needed. If you are more methodical than I am, you may not like the book. It will show you the promised land in the distance, but not take you there.
Tachyons
One casual remark on page 213, about low-dimensional string theories - Unfortunately, the resulting theory is plagued by some serious problems (in particular, it allows for the existence of tachyons, elementary particles moving faster than light, whose existence is prohibited by Einstein's relativity theory). This bothers me on many levels. Einstein's relativity theory may "contradict" some form of tachyons, but theories do not prohibit, any more than Newton's laws prohibit relativity. Einstein also "prohibited" non-locality, but non-locality is demonstrably true of quantum phenomena, and spent the rest of his life pursuing this error. Perhaps non-causality is permitted under some unusual circumstances, too.
The typical argument against tachyons is their potential use in causality-violating signalling systems, such as the one illustrated in Benford 1970. If tachyons cannot be used for practical signalling (for instance, because their typical interaction lengths with normal matter are on the scale of galaxies) that means that causality violation would be practically undetectable (as opposed to theoretically irksome). If tachyons had effects like attenuating photons over gigalightyear distances, or gravitationally concentrating near galaxies, then their effects on our universe might appear to be what we call a cosmological constant, or dark matter. Modern physics is full of mathematical epicycles, stuff thrown into the theories not to explain phenomena, but to make the math "beautiful". When "simple" quantum theory suggests cosmological constants 120 orders of magnitude larger than what seems to be the case, or predicts supersymmetric particles which experiments can't find, perhaps it is time to reform our mathematical aesthetics, or ditch assumptions that have no practical or experimental consequences.
The metric we inhabit is Minkowski spacetime, trajectories in spacetime described by 0 = x2 + y2 + z2 - c2t2. Einstein's energy relations can be derived from this. What about a spacetime described by 0 = x2 + y2 + z2 + c2t2? How would objects (perhaps not localized "particles" as we understand them) behave? How might that spacetime couple to our spacetime near the initial singularity, when most energy is light, and most matter is moving very close to the speed of light, and spacetime itself is a tiny Planck foam? How would our observed universe evolve from that initial linkage? I presume that is a stupid question, but among the scads of stupid questions I can imagine, there may be some smart ones, alternatives to the "just so" stories that dominate physics theories beyond the standard model (which explains everything we observe, except for very large scale cosmology, and the model itself).
But then, I am not a physicist. I appreciate what the condensed matter folk do. The particle theoreticians get the publicity, but they seem like "secular theologians" haunting modern science. I'm imagining a modern day Samuel Johnson, vexed by an unverifiable theory about intrinsically unobservable objects, swinging his foot through empty air and falling flat on his ass, saying "I refute it thus!"