I’ve finished “On Space and Time” and my opinion of the book has only increased (see part 1 and part 2 for initial chapters). There were three chapters remaining, one by Alain Connes, the founder of Noncommutative geometry, Michael Heller, probably the coolest Roman Catholic priest I’ve read and a short epilogue by John Polkinghorne, an Anglican theologian.
Connes really tries to bring his ideas down to the level of us mortals but he continually fails. In fact as I’ll write later on, Heller’s chapter gives a much better description of Noncommutative geometry than the founder does. Connes begins with the combined Lagrangians for general relativity and the Standard model. He’s troubled by the fact that the Lagrangian for the Standard model is long, really long (there’s a nice picture of it in the chapter, but you can Google it and get the idea. I’d post something but I wouldn’t be able to tell if I found the correct Lagrangian or not). Though Connes is a pure mathematician he has a profound interest and respect for physics. Via Noncommutative geometry he’s found a way to form a simpler functional that is defined on a product space where M is spacetime but F is a discrete space. Lots of mathematics follows and I can only do it injustice at this time. It does yield lots of predictions however and it would be interesting to see if any of these have been verified or refuted since the time of the writing in 2008. I do want to include one quote however, Connes is discussing the simplicity of the result and the following sentence appears,
“Thus we imagine trying to explain this action principle to a Neanderthal man.”
It’s likely that you’ll take this out of context, but I thought it was funny because I imagine Connes would imagine me, the reader, along the lines of such a creature.
Michael Heller’s chapter is from the perspective of theology. For him, science informs theology and theology informs science. This might be my second favorite chapter in the text. Heller approaches a tender subject, God in science, masterfully. As an atheist I was impressed by the quality of thought and found no objections, in fact, I would greatly enjoy to read more of his writing.
Heller begins by discussing the notion of “the God of the gaps.” This is the tendency to see God in areas of human ignorance, such as the Big Bang. He cautions against this approach as it only erodes faith and increases the likelihood of it becoming increasingly irrelevant as science advances and fills in such gaps. He then spends some time reviewing some theology, in particular the notions of time and how time relates to God. St. Augustine and Aquinas are discussed and the notion of atemporality of God is introduced. The idea that God doesn’t sit in the time stream, that such notions of past or future may not make sense in a discussion of God. This discussion of time is one of the motivating factors for Heller’s interest in quantum gravity. What is the nature of time? General relativity had already altered our perspective of it, what will the future hold? How will this conception of time affect or inform the theology concerning the temporality of God? For Heller, science doesn’t compete with religion,
“But the correct theology is obliged to take into account what science has to say to us in this respect…Any theology that would choose to ignore this magnificent process is a blind way to nowhere.”
But he feels that science can benefit likewise and I can see his point,
“To see profit for science is perhaps less obvious, but we should take into account the fact that much of Western science, such as for instance Newton’s ideas, are imbued with things taken ultimately from theology, and it is better to be aware of this influence than not.”
Majid had similar sentiments in his chapter though of a more philosophical bent (without mention to theology) and this is certainly where I agree. I’m aware only of my own ignorance to the mutual impacts of science and philosophy. It affects our language, how we ask questions, which questions we ask and how we interpret the answers. It can be no coincidence that the brightest minds of mathematics and science take issues of philosophy very seriously and the those who disparage philosophy are often the crudest or more mundane minds.
Heller is no mere priest. He has a PhD in physics, specializing in cosmology and has published numerous scientific papers on general relativity, quantum mechanics and their unification. As such it turns out that he gives a rather nice overview of noncommutative geometry and the consequences if spacetime is noncommutative at some scale.
Manifolds are usually specified with charts and atlases, apparently there’s another method which involves the algebra of continuous functions from the manifold to the real numbers. I’m not aware of this method (yet). It ends up being coordinate free which is enticing and it ends being more amenable to generalization. If instead of functions to the reals you wre to look at the functions to a space of matrices then your function algebra would be noncommutative. At the most basic level this is the meaning of noncommutative in noncommutative geometry. This isn’t even the tip of the iceberg, this is akin to hearing a story about icebergs. When you abstract like this you lose, apparently, any notion of ‘local.’ You can distinguish points, there aren’t any points. This reminds me of Penrose tiles where you can’t tell which tiling you’re looking at from any finite section of it. In fact any picture of a Penrose tiling you’ve seen occurs infinitely often in every Penrose tiling. They are nonlocal and always described in introductions on noncommutative geometry. I’ll let Heller say a bit more,
“One of the most striking peculiarities of noncommutative spaces is their totally global character. In principle, no local concept can be given any meaning in the context of noncommutative geometry. Typical examples of local concepts are those of ‘point’ and neighborhood of a point. In general, noncommutative spaces do not consist of points. Let us look more closely at this peculiarity of noncommutative geometry. The basic idea of standard (commutative) geometry, in its map and atlas approach, is that every point is identifiable with the help of its coordinates. When we prefer the functional method, we can identify a point with all functions that vanish at this point. All such functions form what in algebra is called a maximal ideal of a given algebra of functions. In other words, points are identified with maximal ideals of a given functional algebra. And now a surprise! It can be shown that for strongly noncommutative algebras there are typically no maximal ideals. In such circumstances, there is nothing that could be identified with a point. Similar reasoning could be used with respect to other geometrical local concepts (like neighborhoods).”
This hit home for me as I’ve been struggling on and off to penetrate the mysteries of noncommutative geometry. Even the simple texts such as “Very Basic Noncommutative Geometry” by KhalKhali are dense and dispiriting. Heller echoes Majid again by suggesting that it doesn’t matter if noncommutative geometry ends up being part of quantum gravity, the benefit, conceptually, to both mathematics and physics, through the exchange of ideas and analysis of consequences is benefit enough. This is just too cool not to look into.
What are the consequences of nonlocality? Would the EPR paradox become trivial?
“The paradoxical behavior of two electrons from the EPR experiment is due to the fact that these two electrons, by their very quantum nature, explore the fundamental level in which there is no concept of distance in space”
Wild! He has a couple other examples I’ll leave to you to read about, but its provocative. Heller even introduces the notion (lightly) of noncommutative probability that I was unaware of until this article. The idea would follow from his above description of nonlocality. Typically for probability theory you consider events as elements (points) in a sample space and assign probabilities to those. I can’t imagine what noncommutative version would be but I can see how the lack of locality could cause problems.
It’s getting late and I’m going to wrap this up. If you’ve seen any of the Amazon reviews, many of them are dismissive of the last two chapters of the book. Polkinghorne’s was short and not very enlightening I thought, though relevant. Heller’s I thought was excellent. I’ll be buying this book so I can transfer all my pencil marks out of the libraries copy.