As mentioned in my previous post, I thought it might be a good idea to release bits and pieces of my book on the go. It gives me the feeling I get something done, at least. 🙂
If you are reading this book, then you are like me. You want to understand. Something inside of you tells you that the idea that we will never be able to understand quantum mechanics “the way we would like to understand it” does not make any sense. The quote is from Richard Feynman – probably the most eminent of all post-World War II physicists – and, yes, this is an aggressive opening. But you are right: our mind is flexible. We can imagine weird shapes and hybrid models. Hence, we should be able to understand quantum physics in some kind of intuitive way.
But what is intuitive? A lot of the formulas in this book feel intuitive to me, but that is only because I have been working with them for quite a while now. They may not feel very intuitive to you. However, I have confidence you will also sort of understand what they represent – intuitively, that is – because all of the formulas I use represent something we can imagine in space and in time – and I mean 3D space here: our Universe. Not the Universe of strings and hidden dimensions. An intuitive understanding of things means an understanding in terms of their geometry and their physicality.
You bought the right book. No hocus-pocus here. All physicists – and popular writers on physics – will tell you it is not possible. You see, the wavefunction of a particle – say, an electron – has this weird 720-degree symmetry, which we cannot really imagine. Of course, we have these professors doing the Dirac belt trick on YouTube – and wonderful animations (Jason Hise – whom I’ve been in touch with – makes the best ones) but, still, these visualizations all assume some weird relation between the object and the subject. In short, we cannot really imagine an object with a 720-degree symmetry.
The good news is: you don’t have to. The early theorists made a small mistake: they did not fully exploit the power of Euler’s ubiquitous a·eiθ function. The mistake is illustrated below – but don’t worry if this looks like you won’t understand: we’ll come back to it. It is a very subtle thing. Quantum physicists will tell you they don’t really think of the elementary wavefunction as representing anything real but, in fact, they do. And they will tell you, rather reluctantly because they are not so sure about what is what, that it might represent some theoretical spin-zero particle. Now, we all know spin-zero particles do not exist. All real particles – electrons, photons, anything – have spin, and spin (a shorthand for angular momentum) is always in one direction or the other: it is just the magnitude of the spin that differs. So it’s completely odd that the plus (+) or the minus (-) sign of the imaginary unit (i) in the a·e±iθ function is not being used to include the spin direction in the mathematical description.
Figure 1: The meaning of +i and –i
Indeed, most introductory courses in quantum mechanics will show that both a·e–i·θ = a·e–i·(wt–kx) and a·e+i·θ = a·e+i·(wt–kx) are acceptable waveforms for a particle that is propagating in a given direction (as opposed to, say, some real-valued sinusoid). We would think physicists would then proceed to provide some argument showing why one would be better than the other, or some discussion on why they might be different, but that is not the case. The professors usually conclude that “the choice is a matter of convention” and, that “happily, most physicists use the same convention.” In case you wonder, this is a quote from the MIT’s edX course on quantum mechanics (8.01.1x).
That leads to the false argument that the wavefunction of spin-½ particles have a 720-degree symmetry. Again, you should not worry if you don’t get anything of what I write here – because I will come back to it – but the gist of the matter is the following: because they think the elementary wavefunction describes some theoretical zero-spin particle, physicists treat -1 as a common phase factor: they think we can just multiply a set of amplitudes – let’s say two amplitudes, to focus our mind (think of a beam splitter or alternative paths here) – with -1 and we’re going to get the same states. We find it rather obvious that that is not necessarily the case: -1 is not necessarily a common phase factor. We should think of -1 as a complex number itself: the phase factor may be +π or, alternatively, -π. To put it simply, when going from +1 to -1, it matters how you get there – and vice versa – as illustrated below.
Figure 2: e+iπ ¹ e–iπ
I know this sounds like a bad start for a book that promises to be easy – but I just thought it would be good to be upfront about why this book is very different than anything you’ve ever read about quantum physics. If we exploit the full descriptive power of Euler’s function, then all weird symmetries disappear – and we just talk standard 360-degree symmetries in space. Also, weird mathematical conditions – such as the Hermiticity of quantum-mechanical operators – can easily be explained as embodying some common-sense physical law. In this particular case (Hermitian operators), we are talking physical reversibility: when we see something happening at the elementary particle level, then we need to be able to play the movie backwards. Physicists refer to it as CPT-symmetry, but that’s what it is really: physical reversibility.
The argument above involved geometry, and this brings me to a second mistake of the early quantum physicists: a total neglect of what I refer to as the form factor in physics. Why would an electron be some perfect sphere, or some perfect disk? In fact, we will argue it is not. It is a regular geometric shape – the Dirac-Kerr-Newman model suggests it’s an oblate spheroid – but so that’s not a perfect sphere. Once you acknowledge that, the so-called anomalous magnetic moment is not-so-anomalous anymore.
The mistake is actually more general than what I wrote above. We are thinking of the key constants in Nature as some number. Most notably, we think of Planck’s quantum of action (h ≈ 6.626×10−34 N·m·s) as some (scalar) number. Why would it be? It is – obviously – some vector quantity or – let me be precise – some matrix quantity: h is the product of a force (some vector in three-dimensional space), a distance (another three-dimensional concept) and time (one direction only). Somehow, those dimensions disappeared in the analysis. Vector equations became flat: vector quantities became magnitudes. Schrödinger’s equation should be rewritten as a vector or matrix equation. We do think of Planck’s quantum of action as some vector. We, therefore, think that the uncertainty – or the probabilistic nature of Nature, so to speak – is not in its magnitude: it’s in its direction. But we are getting ahead of ourselves here – as usual. We should go step by step. Let us first acknowledge where we came from.
Before doing so, I would like to make yet another remark – one that is actually not so relevant for what we are going to try to do this in this book – and that is to understand the QED sector of the Standard Model geometrically – or physically, I should say. The innate nature of man to generalize did not contribute to greater clarity – in my humble opinion, that is. Feynman’s weird Lecture (Volume III, Chapter 4) on the key difference between bosons and fermions does not have any practical value: it just confuses the picture.
Likewise, it makes perfect sense to me to think that each sector of the Standard Model requires its own mathematical approach. I will briefly summarize this idea in totally non-scientific language. We may say that mass comes in one ‘color’ only: it is just some scalar number. Hence, Einstein’s geometric approach to gravity makes total sense. In contrast, the electromagnetic force is based on the idea of an electric charge, which can come in two ‘colors’ (+ or -), so to speak. Maxwell’s equation seemed to cover it all until it was discovered the nature of Nature – sorry for the wordplay – might be discrete and probabilistic. However, that’s fine. We should be able to modify the classical theory to take that into account. There is no need to invent an entirely new mathematical framework (I am talking quantum field and gauge theories here). Now, the strong force comes in three colors, and the rules for mixing them, so to speak, are very particular. It is, therefore, only natural that its analysis requires a wholly different approach. Hence, I would think the new mathematical framework should be reserved for that sector. I don’t like the reference of Aitchison and Hey to gauge theories as ‘the electron-figure’. The electron figure is a pretty classical idea to me. Hence, I do hope one day some alien will show us that the application of the Dyson-Feynman-Schwinger-Tomonaga ‘electron-figure’ to what goes on inside of the nucleus of an atom was, perhaps, not all that useful. A simple exponential series should not be explained by calculating a zillion integrals over 891 Feynman diagrams. It should be explained by a simple set of equations. If I have not lost you by now, please follow me to the acknowledgments section.
 A fair amount of so-called thought experiments in quantum mechanics – and we are not (only) talking the more popular accounts on what quantum mechanics is supposed to be all about – do not model the uncertainty in Nature, but on our uncertainty on what might actually be going on. Einstein was not worried about the conclusion that Nature was probabilistic (he fully agreed we cannot know everything): a quick analysis of the full transcriptions of his oft-quoted remarks reveal that he just wanted to see a theory that explains the probabilities. A theory that just describes them didn’t satisfy him.
PS: The book might take a while – as I am probably going to co-publish with another author (it’s perhaps a bit too big for me alone). In the meanwhile, you can just read the articles that are going to make up its contents.