a list compiled by Alex Kasman (College of Charleston)
|A time machine story based on a combination of Hilbert's Hotel analogy and the "Fermi Sea". We read of the travels of the main character to the ancient past, to the San Francisco earthquake and to the 1960's (where he spends a lot of time living as a hippie) before learning of his tragic fate.
As the story explains, mathematician Paul Dirac famously predicted the existence of anti-matter when he combined special relativity with quantum mechanics. (Notes: The story calls him a physicist, but his training and employment were in mathematics. So, I think of him as a mathematician who made important contributions to physics. Also, some may be confused by the story's claim that Dirac combined relativity with quantum physics since this is what many people are trying to do today...but note that it is general relativity which is the subject of this research now.) Although it at first may have appeared to be nothing but a mathematical trick, anti-matter has been found experimentally and in complete agreement with Dirac's predictions. The "Fermi Sea" is part of the theoretical framework that Dirac used. The idea is that there is an equivalence between a space filled with a certain particle except at a few points and a space empty except having the corresponding anti-particles at those same points. (The name "Fermi" is here because this idea only works for the type of particles known as "fermions" for which only one particle can be in a given state at a given time.) In other words, a box full of electrons except for one little gap is the same as an empty box with a positron in the position of the gap.
So, when working with Dirac's "Fermi Sea", you imagine that the vacuum -- instead of just being empty space -- is filled up with anti-matter of every possible negative energy state. Since it is filled up to height "0", this is like the "sea level". Then, what you can do is pick a handful of those negative energy particles and replace them with holes and add a corresponding number of positive energy states (e.g. electrons) and you've created some particles on the vacuum. By a strange coincidence, I'm writing this now from a mathematical physics workshop at Stony Brook at which I've been doing computations with the Fermi sea and so have been doing precisely this sort of thing.
In fact, using the creation/annihilation operators to make some holes at negative energy levels and adding some positive ones does seem a bit like the idea of "Hilbert's Hotel" where one makes room for more people in the filled hotel...but the bizarre properties of infinity are not really needed here. I mean, I suppose you could think of the creation of a "hole" in the sea as moving all of the negative states below it down one step to make room for it, but this would require infinite energy. Instead, Dirac's picture is that a filled positive energy state and a filled negative energy state can combine to make a hole (matter-antimatter reaction). And that seems to be what this story is talking about.
Similarly, the "time travel" aspect of the story makes sense if you know a little mathematical physics but not too much. There is a correspondence between time reversal and matter-antimatter duality. In particular, antimatter does behave like matter running backwards in time...but if it is indeed a symmetry (see CPT Symmetry) as we now suspect, you could not go back in time and do things...it's just an equivalent way to interpret the same physical situation.
Anyway, I'm taking this all far too seriously. It's a fine, thought provoking time travel yarn that includes discussion of a bit of math and theoretical physics.
Published in Isaac Asimov's Science Fiction Magazine, 1988 and reprinted in Time Machines.
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Exciting News: The 1,600th entry was recently added to this database of mathematical fiction! Also, for those of you interested in non-fictional math books
let me (shamelessly) plug the recent release of the second edition of my soliton theory textbook.
(Maintained by Alex Kasman,
College of Charleston)
(Maintained by Alex Kasman, College of Charleston)