William of Occam was a medieval philosopher and theologian. Just like his contemporary, Roger Bacon, he placed considerable emphasis on the study of nature through empirical methods. That’s why he was condemned as a heretic. Roger Bacon (not to be confused with Francis Bacon) had similar problems with the Church.
We know Occam because of the principle of parsimony in philosophical and scientific thought, aka Occam’s Razor: “When trying to explain something, it is vain to do with more what can be done with less.” No wonder he got denounced by the pope for “dangerous teachings”!
Why involve Occam here? Well… While physicists usually don’t like philosophers, some philosophy of science is not a bad thing, so I created this page for that: philosophical reflections. Let me offer my first one: Occam would have loved the math behind QM.
Indeed, looking back at the math and physics I just went through, I am deeply impressed by the beauty of the argument: the physics call for the math, and the math becomes the physics. We observe interference – not only of light but also of matter particles, notably electrons – and math gives us the tools we need to model that. In fact, math becomes the language in which we express the phenomenon.
Now, math surely is a language that respect Occam’s Razor: not one ‘word’ is wasted, and there’s extreme parsimony in the concepts used. Complex numbers are the key to understanding. In ordinary language, complex just means “consisting of many different and connected parts”, and oft-used synonyms are compound, composite, or multiplex. Complicated is not a synonym of complex: complex numbers are not complicated: it’s a shame our educational systems do not expose us to them at an earlier age, if at all.
One way of looking at the logical progression in mathematical thought could be something like this (and, yes, I know mathematicians will be appalled by the simplicity of the argument):
- Natural numbers count things: the legs of an insect, or the number of hairs on our head. The operations of addition, subtraction, multiplication and division come with the concept of counting. Division gives us fractional numbers (i.e. rational numbers).
- Positive real numbers measure magnitudes: our length, or the circumference of the Earth. The concept of a real number (including irrational real numbers, such as π and e) is easy to understand, indeed. [That’s why I am not mentioning set theory here.]
- Of course, positive numbers call for negative numbers, as we want to measure a difference. Positive numbers are associated with one direction of the x-axis and, therefore, we associate negative numbers with the opposite direction.
- Thinking about opposite directions calls for intermediate directions, or – what amounts to the same – we could say that one dimension (the real axis x) calls for another (the imaginary axis y). And so there we are: we have pairs of numbers, aka as vectors or complex numbers indeed – or, as Feynman calls them, arrows: things with a magnitude and a direction.
- Directions are defined by angles. If we change the direction of straight lines, they get curvature. We can define a circle as a ‘line’ (I should say a curve, of course) with a constant non-zero curvature.
- If we go around a circle, we get back to where we were: if we measure angles by the length of the corresponding arc of the (unit) circle, and if we then add or subtract the whole circumference of the (unit) circle, we’ll get the same angle again. Hence, if we use an angle as the argument of a function, we get a periodic function: an angle is a variable that goes around in a circle. [Again, I know I am using unacceptably informal language to describe things here but just go along with it as for now. As mentioned above, I want to describe some kind of logical progression of mathematical thought.]
- Waves are periodic functions. Waves interfere. Waves explain interference effects.
- Particle waves interfere in two different ways, which explains the fundamental dichotomy in Nature: force carriers (bosons, e.g. photons) versus particles (fermions, e.g. electrons). If we want to use the same mathematical shape to represent all particles (i.e. bosons and fermions), then the wave function will have to have two dimensions to allow for the two different types of interaction. Hence, the wave function is complex-valued: for each value of its argument, we get two values in return: the so-called real and imaginary part of the complex number respectively. [And remember that the terms ‘real’ and ‘imaginary’ do not say anything about the ‘reality’ of both parts: both are ‘real’, I’d say.]
It’s beautiful. One thing calls for another. That’s the way dialectical thought works: thesis, antithesis, synthesis.
But back to physics. Look at the symmetry of the two possible ways, (a) and (b), in which two identical particles interact. The functional form modeling both situations is, and obviously should be, the same function: f. It is only the argument that differs: in situation (a), the argument is θ, while in situation (b), it’s π–θ. That’s pure logic based on the symmetry involved: (b) is the same situation as (a), but we exchange the role of the detectors (or, which amounts to the same, the role of the particles). Detector 1 becomes detector 2, and vice versa. Or particle a becomes particle b, and vice versa.
The interference pattern which one will observe – so this is a reference to reality 🙂 – will be something like either the blue or the red graph below (the only difference between the four situations below is the frequency of the wave function).
If the particles are bosons, we get the blue probability function: we only add the real parts of the (complex-valued) wave functions f(θ) and f(π–θ) before taking the absolute square. The imaginary parts cancel out. Conversely, if the particles are fermions, we get the red probability function: in that case, the real parts cancel out, and so we only add the imaginary parts of f(θ) and f(π–θ), before taking the absolute square to calculate the combined probability function P(θ).
If the frequency of the wave function becomes a large number, then the interference pattern will be hard to detect, as shown in the last of the four diagrams. In that case, we have a classical interaction, like between two billiard balls. Now, it is, in fact, likely that the frequency is a (very) large number, because the 1/ħ factor in the de Broglie relation ω = E/ħ equals 15×1014 if E is expressed in electronvolts. Hence, the angular frequency of a 1 eV photon is 15×1014 radians per second, which corresponds to a light frequency of 2π×15×1014 ≈ 100×1014 Hz. That’s light with fairly low energy: infrared light, or thermal radiation. In other words, this frequency is not very high. Indeed, matter particles (e.g. electrons) will have frequencies that are much higher, because, unlike photons, they have mass and, hence, a lot more energy (that’s just the E = mc2 equivalence relation).
That’s why interference between electrons was only observed (and predicted) in the 1920s (Davisson-Germer experiment), i.e. very recently on the timeline of human evolution. 🙂 These high frequencies also explain why we don’t see interference in the day-to-day world that we humans inhabit.
What is that I want to say? Nothing. As mentioned above, I just wanted to note the parsimony in the quantum-mechanical explanation of reality and, hence, the beauty of the theory: all is needed, nothing is superfluous, and, therefore, it’s perfect logic. I think that’s great.
Let me make another point that’s interesting from a philosophical point of view: it’s about all of the symmetries involved in physics.
I’ll once again use unacceptably informal language to describe things here. Let me first make a link with the previous section and mention Euler’s formula. With Euler’s formula, we go from ‘linear’ (rectangular) to ‘circular’ (polar) coordinates, and vice versa. Think of it as going from a ‘linear’ to a ‘circular’ world. From space to curved space, I’d say (although that’s, once again, a terribly inaccurate expression). Indeed, a complex number (z) can be written in two ways:
z = x + iy = reiθ = r(cosθ + isinθ), with r = |z| and θ = arg(z)
The eiθ = cosθ + sinθ shows that the eiθ function is ‘two-dimensional’ (it consists of a sine and a cosine) but that it has one ‘degree of freedom’ only: θ, a variable that goes around and around and around in a circle (in other words, it’s an angle). That’s why mathematicians will not say complex numbers are ‘two-dimensional’. What’s a dimension? That’s not a term that’s defined unambiguously in math. I’d say: a complex number has two dimensions but it’s one number only. 🙂
In any case, if two-dimensional numbers (complex numbers) are ideally suited to describe what’s usually referred to as quantum electrodynamics (QED) – so that’s interactions involving photons and electrons – what about quantum chromodynamics (QCD) – so that’s interactions involving the strong force (quarks and gluons), instead of the electromagnetic force?
Instead of just one boson, we’ve got three in QCD, referred to as gluons, with a specific ‘color’. To be precise, we actually have 8 independent types of gluons: one of three ‘colors’ (r, g and b) can mix with one of three ‘anti-colors’. While that makes for 9 types, theoretically, only 8 of them are linearly independent. In addition, we know that we have quarks that come in various flavors (usually u or d, but we also have s, c, b and t) and combine in basically two ways: protons and neutrons (baryons) consist of three quarks, while mesons consist of two quarks only, and we should note that mesons behave like bosons too. So, yes, we have a lot of symmetries here too, and there is a special type of four-dimensional numbers – quaternions, which we can look at as pairs of complex numbers – that simplify the analysis. However, it’s obvious that QCD is infinitely more complex than QED and, hence, I’d need to have a look at it myself before I write anything more about it.
A point that I master somewhat better and that may also arouse your curiosity somewhat more are the ‘symmetries’ related to time and space, or the topic of symmetries in physics in general. [Again, I’ll use the term ‘symmetry’ very loosely.]
A number of remarks can be made here.
The first thing to note is that the argument in a wave function f will usually have both time and space as variables: f = f(x, t). From a mathematical point of view, they are fully interchangeable through the speed of the wave which, as you know, is determined through the wavelength (λ) and frequency (ν): c = λν. Indeed, a wave function f = f(x, t) can always be written as:
f(x, t) = f(x – ct).
Note that c can be any velocity. We should use v but that may cause confusion because of the widespread use of the ν symbol to denote the frequency. As shown below, if a wave travels a distance Δx in space, in a time equal to Δt, then its amplitude will be the same as it was cΔt time ago. That’s why we have the minus sign. If you don’t immediately ‘see’ this, think about the obvious identity below:
f(x – ct) = f[x+Δx – c(t+Δt)]
Hence, we can ‘freeze’ a wave in time or, alternatively, in space, and we won’t be able to tell in what direction the wave is traveling. The math of the function basically treats both variables the same and shows that, to some extent, time and space are indeed interchangeable as dimensions, although I should add that it’s only special relativity theory that shows how intimately they are related (see my post(s) on special relativity theory).
So that’s one type of ‘symmetry’ related to space and time. I should note, however, that, when physicists talk symmetries, they usually mean something totally different. They’ll talk about charge conjugation (C), parity symmetry (P), and time reversal (T). I’ve written a number of posts on that already (see, for example, time reversal and CPT symmetry) and, hence, I won’t repeat myself here. In essence, it means the laws of physics should still make sense if:
- We reverse all charges (not only electric but also color charges and whatever other ‘charges’ are there),
- We mirror the world (i.e. we reflect what’s going on in space), and
- We change the arrow of time (i.e. we put a minus sign in front of the time variable).
In my posts on this topic, I noted that the weak force is the weird force because, unlike the other forces, it does not respect any of the individual symmetries, nor does it respect the combined CPT symmetry. Hence, the arrow of time is not only something that comes out of the so-called ‘Entropy Law‘ (i.e. the 2nd law of thermodynamics), with which you’re surely familiar. No. There’s also something else there, buried deeply in what is often referred to as Quantum Flavordynamics (QFD), but what’s better known as the theory of the weak interaction.
The weak interaction is the force explaining atomic decay processes. The topic is truly complicated and, hence, I’ll just refer you to my posts on it. I’ll just quote Feynman’s words on the absence of the usual symmetries in the analysis when it comes to the weak force:
“Why is nature so nearly symmetrical? No one has any idea why. The only thing we might suggest is something like this: There is a gate in Japan, a gate in Neiko, which is sometimes called by the Japanese the most beautiful gate in all Japan; it was built in a time when there was great influence from Chinese art. This gate is very elaborate, with lots of gables and beautiful carving and lots of columns and dragon heads and princes carved into the pillars, and so on. But when one looks closely he sees that in the elaborate and complex design along one of the pillars, one of the small design elements is carved upside down; otherwise the thing is completely symmetrical. If one asks why this is, the story is that it was carved upside down so that the gods will not be jealous of the perfection of man. So they purposely put an error in there, so that the gods would not be jealous and get angry with human beings. We might like to turn the idea around and think that the true explanation of the near symmetry of nature is this: that God made the laws only nearly symmetrical so that we should not be jealous of His perfection!”
I’ve discussed this statement before, and I hinted that I didn’t like the reference to God here. That being said, it is a very deep point, which I discussed at length in my post on broken symmetries. From a philosophical point of view, it rules out the possibility of ‘different worlds’. There’s only one: ours. And it’s beautiful: nearly symmetric indeed, in many respects, but not completely symmetric. We like perfect symmetries but, frankly, nearly perfect symmetries are also in line with our sense of aesthetics, aren’t they? 🙂