# Hyperspace

In 2016 Oxford University Press started a *Landmark Science* series with cheap reprints of classic books, the 'must-reads', about topics that have shaped current science. Here I review two of the more mathematical/physical ones from the first release if seven books in April 2016: this book and *The Emperor's New Mind* (R. Penrose).

Like Penrose, Kaku gives in this book also an overview of how physics and our understanding of the laws that govern nature and the cosmos. Where Penrose's main objective was to understand the human mind and brain and uses a lot of technical material, Kaku's story is much more accessible and sticks more to the quest of a unifying theory that captures all the laws of physics. As physicists of the 20th century constructed more complex theories that picked up more of the laws that describe phenomena, considered of different nature before, they needed to extend the dimension of the space in which they had to be described. This need for higher dimensions provides the thread that runs through the book and that also was the inspiration for the title.

Kaku divides his book in four parts. In the first part he prepares the reader for the roller coaster he is about to enter. The four forces that need to be unified are Maxwell electromagnetic force, strong and weak nuclear force, and gravitational force. We are introduced to the queer higher dimensional world and a world that is not flat. We get a sneak preview of where this may lead us: black holes, wormholes, and time travel. He more or less assumes we are familiar with the classical theory of Newtonian mechanics and presents a first unification by introducing the Riemannian tensor for the Kaluza-Klein theory which consists of the 4-dimensional Einstein block, bordered by a row and column to include Maxwell's theory.

In part two, we get the more recent history in search for the unification. It starts with the observations that have led to the development of quantum theory that, when followed consequently, has strange and unexplainable consequences. The Yang-Mills field theory replaced the classical Maxwell theory. Here we also meet with Einstein's reluctance towards quantum theory's 'spooky action at a distance'. Till then the theory was carved in marble he said, quantum theory was made from wood, and he wanted to build a theory that was just marble, no wood. This metaphore of opposing viewpoints is taken along by Kaku as his book develops. Einstein's relativity theory where gravity is explained by space-time curvature is of a different nature that does not mix well with quantum field theory. That is why the Standard Model could explain and unify the electromagnetic and the strong and weak nuclear forces but leaves out gravitation which is too weak a force to fit. Gradually more wood is turned into marble, i.e. using geometric arguments, by bordering the Riemann tensor with the Yang-Mills field. The many subatomic quarks and leptons were rather messy and symmetry was lost but that problem was solved by supergravity (and supersymmetry), lifting the tensor to an even higher *N*-dimensional space, so that more symmetry could be assumed.

Enter string theory, exit supergravity in the effort to include gravity. This is where Kaku himself has contributed. Here the general *N* is `naturally' forced to be 10 for mathematical reasons. According to Kaku, this is something that could be explained by studying modular functions in Ramanujan's lost notebook. Recall that Kaku wrote his book in 1994, and the theory was still young. Since experimental verification is not possible with the energy we have available here on Earth, we have to look outside to cosmic phenomena and the origin of the universe. Kaku even takes the time to go through several historical proofs for the existence of God as a creator of it all.

The two remaining parts will certainly be appreciated by lovers of Science Fiction. Kaku explores in part 3 what the theory so far would make possible in principle. So we learn the possibilities created by black holes, wormholes and time travel. The possible interpretation of quantum theory dealing with many universes. The last part speculates on the `what if' aspect of the future. What if our civilization ever reaches a type II level where it becomes possible to control the amounts of energy of our sun, or even type III where it becomes possible to work with the energy of an entire galaxy. Do such civilizations exist and if so, why didn't they contact us? Perhaps they never reach it because of disastrous events happened as they did on our planet according to geological observations. And what will eventually happen with the whole universe?

Although the book is about physics, the battlefield is on the mathematical structure that is used. Topology is what should bind the Riemannian geometry of Einstein and the Lie groups of quantum physics. Mathematics and physics were in close harmony for many centuries. They started to develop in different directions late 19th and early 20th century. It turned out that at the dawn of the 21st century they need each other more than ever, which raises the philosophical question of all times: why this `unreasonable effectiveness of mathematics in the natural sciences' as E. Wigner formulated it. If you are looking for the mathematics, that you will not find in this book, but it will convince you, if that is still needed, that mathematics is the key to understand nature.

Of course, theoretical physics and mathematics did not stop evolving after this book was written, gravitational waves were observed in 2016, the Higgs boson was identified in 2012, the tau neutrino in 2000, etc. Neither did Kaku stop form 'spreading the news'. He is present in the media and wrote several other popular books since, just a couple of examples: *Parallel Worlds: A Journey through Creation, Higher Dimensions, and the Future of the Cosmos* (2004), *Physics of the Impossible: A Scientific Exploration into the World of Phasers, Force Fields, Teleportation, and Time Travel* (2008), and a recent one that relates to the book of Penrose mentioned in the beginning: *The Future of the Mind: The Scientific Quest to Understand, Enhance, and Empower the Mind* (2014). However, the current *Hyperspace* book was his biggest success so far. So one should be aware that as Kaku brings the reader of this book to the level of superstrings, this is in the context of the euphoria of the early 1990s. Nevertheless, it is still mostly entertaining to read this book about the evolution of theoretical physics up to that point. It is the best documentation for the layman of this history that I know of. He does avoid the technical stuff that most of those readers will probably skip anyway, and yet it gives so much details that takes you along on *A Scientific Odyssey Trough Parallel Universes, Time Warps, and The Tenth Dimension* which is rightfully the subtitle of the book.

**Submitted by Adhemar Bultheel |

**2 / May / 2016