© Fernando Caracena, 2019
Dreams of a Final Theory
This is the title of a book written by the Nobel laureate Steven Weinberg [ 1992, 334 pp. Pantheon Books, NY, NY. ]. This book is a modern manifestation of the dream that goes back to the ancient philosophers who sought to understand the nature of the universe. This dream has been progressively shaped in definition as a quest for the Theory of Everything (TOE). Such a theory in physics should be mathematically based, so that it would make quantitative predictions, which would be verified by experiments and observations. As physics progresses, however, serious obstacles appear to finding a final theory. One area where these obstacles appear is in the idea of being able to make accurate predictions. Below, the difficulties of prediction are discussed.
Once I took great interest in this subject. It is highly technical. Unless you are a mathematician or a serious student of the subject, you probably do not want to wade into the mathematical details of the theory. If you are curious to see what the beast looks like, take a look at the Wikipedia article about it. The most important part of it that I want to emphasize here is a basic feature of the theory.
Catastrophe Theory concerns the solution space of a set of complicated equations in mathematics that may constitute a model of some physical aspect of reality. Call a run of that theory, say on a computer, as a model simulation. Such a theory includes a set of parameters that may be chosen and fixed as constants during the mathematical simulation. The result of many choices of the model parameters result in some kind of abstract solution space, which you can think of as a curved hyper-surface, or membrane. In this case, I am using 'membrane' in a specialized way, which means a hyper-surface of a model embedded in a space of all possible solutions of that type of model.
The important property of catastrophe theory that I want to mention here is the sudden bifurcation that can occur in the solution space of a model as its parameters are varied. Up to this critical, bifurcation point the solution space is a continuous surface; but after the critical point is breached, the solutions of the model wander off into two independent surfaces that no longer connect with each other continuously. The idea of a book that I read sometime ago on catastrophe theory was more restrictive than perhaps the more general theory by that name, which merely specifies a topological change in the solution space at the critical point.
Space-Time Solution Space
The mathematics of catastrophe theory is rather abstruse, but it contains ideas vital to the future of human beings. That is the task of science, to look ahead and advise us on the best course of action in confronting the future. Physical theories contain parameters. These are called fundamental physical constants. A smaller set of dimensionless parameters, the fundamental constants, determine the solution space of physical theories.
Since the whole universe is governed by physics, one might inquire about what would have happened in the universe had the fundamental constants been different at that time of the Big Bang. The big question asked here is if there is a Cosmological Goldilocks relationship among these constants that allows for the possibility of life in the universe? A Goldilocks hypothesis is used by astronomers in searching for exoplanets that could contain life; but the cosmological Goldilocks hypothesis pointed to here concerns the type of universe that has any possibility of sustaining life as we know it. We know one solution of this problem because our own universe supports life, so that the set of our fundamental constants constitute a possible solution of a cosmological Goldilocks problem.
A the present time, we have the mathematical tools to formulate some of the big questions and see how they involve each other. However, the sheer complexity of the problems raised, makes their solution intractable. Fortunately , we do have a good tool in the digital computer to begin to address the big questions, but in a limited way. One approach developed so far, is complexity theory, which involves computer simulations. The previously designated hyperlinked video explaining this theory is an excellent introduction, but be aware that it contains a conceptual flaw in its characterization of Newtonian physics. Newton's theories are not fundamentally linear. For example, the partial differential equations that constitute weather prediction models are nonlinear, but are written within a Newtonian framework.
Newtonian physics constitutes a whole paradigm, which allows complexity of solutions. Modern physics consisting of Relativity and the quantum theory constitute another paradigm that also allows complexity. Modern physics contains an unresolved issue in that it has not been able to combine its two theories into a single theory. One of these components, Quantum theory, contains a feature which Einstein perceived as a flaw that rendered it incomplete, probability, which is expressed by Albert Einstein's famous quote is, "God does not play dice with the universe."
So far, all experiments involving quanta result in events that are totally random, viz. unpredictable. Yet there is still some order here, which the quantum theory is able to handle and still salvage some predictability. Unfortunately, the predictions of the theory are about the evolution of probability distributions, not the actual events. Sidney Harris's cartoon about "Then a miracle occurs" well applies here. This is what really got to Einstein who in a personal letter wrote the following:
I do not believe in a personal God and I have never denied this but have expressed it clearly. If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it.
Perhaps, Einstein's thinking in this, was influenced by Baruch Sinoza's philosopy. The feeling here is that a true physical theory should be strictly deterministic and not probabilistic, since probability is useful in making predictions based on partial ignorance of the causes of an event.
Another troubling aspect of the Quantum theory was addressed by Hugh Everett in his Many-world' s interpretation of quantum theory. The classical physics paradigm supported Newtonian physics allows a deterministic description of the entire universe. This type of physics is consistent with Spinoza' s philosophy. However, In classical physics's being superseded by modern physics, the scope of application was reduced to subsystems within the universe. Each quantum event introduced a random direction to the state of the universe thus making its evolution of the universe. With the many-world's interpretation Everett attempted to reinstate a universal predictability, albeit probabilistically.
The interesting feature of the many-world's interpretation is that it makes every quantum event a critical point in catastrophe theory, which preserves the causality of probability amplitudes, but at the expense of creating a geometrically expanding, huge number of universal outcomes, or parallel universes.
God may not play Dice, but He may make the world Superficially simpler for Us.