Neil Turok is a professor of theoretical physics at the University of Edinburgh and former director of the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada. He returns to Perimeter for this lecture on 2023-10-25, presenting a model of the early universe based almost entirely on known particle physics, with minimal additions that are testable by experiment. Is it possible to dispense with most of the complexity that theorists have imagined to explain observational results?
A fascinating talk – albeit far above my pay grade. On the plus side, Prof. Turok can dispense with cosmic inflation following the Big Bang, which always sounded like a fudge. On the minus side, he has to invoke thermodynamics, which is its own minefield.
As so often, this kind of talk brings us back to the question of whether the universe can really be described by mathematics? And if someone wants to describe the universe mathematically, why does he have to start with the exponential number (e), the square root of minus one (i), and the ratio of a circle’s perimeter to its diameter (pi)? Why are those 3 particular numerical values so critical to the universe?
It seems to me those particular three values aren’t something we have to start with in order to describe the universe, but rather numbers we’ve found keep popping up when we observe phenomena in the universe and try to describe and/or predict their behaviour. For example, π appears all over the place in equations describing the propagation of electromagnetic waves, mechanics, gravitation, etc. But these are mostly equations which involve propagation of effects through space or, in other words, relating distance to area or volume. Because we live in a universe with three spatial dimensions which has no measurable intrinsic curvature, the ratio of radius to area or volume involves the constant π. If we lived in substantially curved spacetime, we would observe a relationship with a different constant depending upon the curvature. The ancients could use π in their calculations for surveying because the Earth’s surface was indistinguishable from flat for the distances and precision involved. Today, surveyors have to use spherical geometry to get the right answers.
Similarly, many processes in the universe, such as decay of a radioactive substance or flow of heat between bodies at different temperatures, are fundamentally random in nature, but depend upon the changing state of the system. Such an interaction is described accurately at a coarse-grained level by a logarithmic process, where the instantaneous rate depends upon the instantaneous state. Any such process will be found to behave based upon equations involving Euler’s constant e. We only observe such processes because we live in a universe which started with an extraordinarily low entropy which has been increasing ever since the big bang. If we lived in a universe at thermodynamic equilibrium, we would observe no such evolution (in the physical, not biological sense) and not discover the logarithmic description. (Of course, we would not exist, since life requires available energy which does not exist at equilibrium.)
Finally, propagation of waves in three dimensions (or in simpler cases where propagation is restricted to one or two dimensions) depends upon two items: amplitude and phase. In order to predict the observed behaviour of waves interacting with one another and fixed objects, you need to take into account both amplitude and phase at every point. You can’t do that with a single value (scalar)—it takes two numbers to encode amplitude and phase. As it happens, if you extend the real numbers by adding an “imaginary” component which extends orthogonally to the real number line, you get a system of numbers (the complex plane), in which the state of a wave (field) can be expressed at any point in space by a single complex number, and these complex numbers can be added, subtracted, and manipulated by other mathematical operations, all yielding the correct behaviour for waves. Complex numbers (and hence the square root of −1, which is multiplied by the “imaginary” [phase] component) are a compact and expressive way to deal with wave and field behaviour. Again, if we lived in a nonlinear medium in which waves interacted in different ways, we’d need a different and far more messy notation.
Eugene Wigner wrote a famous essay in 1960, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (full text [PDF, 9 pages]), which begins with the anecdote:
There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”
He goes on to describe how stunning it is that mathematics describes physical phenomena so well, and whether we should be surprised by this or whether it indicates a deep connection between mathematics and the structure of the universe. He concludes:
Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
Thank you! Wonderful!
To continue the puzzle, there is that famous relationship where an operation involving those two transcendental numbers and the imaginary number plus the basic unit number equals the zero which it took the human race so long to recognize.
It is hard to avoid the thought that something deeper lies behind this.
is often presented as more mysterious than it really is when you look at what’s going on.
The identity is a special case of Euler’s formula:
which was published in 1748 by Euler in a work based upon the infinite series expansions of the exponential and trigonometric functions. Once you have the series for the exponential, sine, and cosine in hand, the identity is immediately apparent. Why is this so? To a computer graphics programmer, it’s obvious because complex numbers are isomorphic to two-dimensional vectors, and multiplying two complex numbers is equivalent to rotation in the plane. Euler’s equation is simply what you get when you plug π into Euler’s formula, which in fact works for any complex number x.
This can be thought of as simply inter-converting Cartesian and polar co-ordinates for a point in the plane.
So, while Euler’s identity is beautiful and pulls together many seemingly unrelated mathematical constants, it is purely a mathematical transformation of one way of expressing rotations (complex numbers) into another (trigonometric functions), so I don’t think it reveals anything deep about the relationship between mathematics and physics.
For the antithesis of the “unreasonable effectiveness” argument, see Max Tegmark’s Our Mathematical Universe, in which he argues that mathematics describes the universe so well because the universe is actually built from mathematics and, in fact, every possible mathematical structure, even those so chaotic nothing built on them would be predictable or form complex structures, is realised somewhere in the multiverse. Since we are observers who are complex structures that require stability on the order of billions of years to evolve to the point where we begin to ask “Why?”, we shouldn’t be surprised to find ourselves inhabiting one of the branches in the multiverse where the mathematics permits such structures to exist and is simple and regular enough we can, slowly but surely, make sense of it.