The wavefunction as an oscillation of spacetime

Post scriptum note added on 11 July 2016: This is one of the more speculative posts which led to my e-publication analyzing the wavefunction as an energy propagation. With the benefit of hindsight, I would recommend you to immediately the more recent exposé on the matter that is being presented here, which you can find by clicking on the provided link.

Original post:

You probably heard about the experimental confirmation of the existence of gravitational waves by Caltech’s LIGO Lab. Besides further confirming our understanding of the Universe, I also like to think it confirms that the elementary wavefunction represents a propagation mechanism that is common to all forces. However, the fundamental question remains: what is the wavefunction? What are those real and imaginary parts of those complex-valued wavefunctions describing particles and/or photons? [In case you think photons have no wavefunction, see my post on it: it’s fairly straightforward to re-formulate the usual description of an electromagnetic wave (i.e. the description in terms of the electric and magnetic field vectors) in terms of a complex-valued wavefunction. To be precise, in the mentioned post, I showed an electromagnetic wave can be represented as the  sum of two wavefunctions whose components reflect each other through a rotation by 90 degrees.]

So what? Well… I’ve started to think that the wavefunction may not only describe some oscillation in spacetime. I’ve started to think the wavefunction—any wavefunction, really (so I am not talking gravitational waves only)—is nothing but an oscillation of spacetime. What makes them different is the geometry of those wavefunctions, and the coefficient(s) representing their amplitude, which must be related to their relative strength—somehow, although I still have to figure out how exactly.

Huh? Yes. Maxwell, after jotting down his equations for the electric and magnetic field vectors, wrote the following back in 1862: “The velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments of MM. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

We now know there is no medium – no aether – but physicists still haven’t answered the most fundamental question: what is it that is oscillating? No one has gone beyond the abstract math. I dare to say now that it must be spacetime itself. In order to prove this, I’ll have to study Einstein’s general theory of relativity. But this post will already cover some basics.

The quantum of action and natural units

We can re-write the quantum of action in natural units, which we’ll call Planck units for the time being. They may or may not be the Planck units you’ve heard about, so just think of them as being fundamental, or natural—for the time being, that is. You’ll wonder: what’s natural? What’s fundamental? Well… That’s the question we’re trying to explore in this post, so… Well… Just be patient… 🙂 We’ll denote those natural units as FP, lP, and tP, i.e. the Planck force, Planck distance and Planck time unit respectively. Hence, we write:

ħ = FPlP∙tP

Note that FP, lP, and tP are expressed in our old-fashioned SI units here, i.e. in newton (N), meter (m) and seconds (s) respectively. So FP, lP, and tP have a numerical value as well as a dimension, just like ħ. They’re not just numbers. If we’d want to be very explicit, we could write: FP = FP [force], or FP = FP N, and you could do the same for land tP. However, it’s rather tedious to mention those dimensions all the time, so I’ll just assume you understand the symbols we’re using do not represent some dimensionless number. In fact, that’s what distinguishes physical constants from mathematical constants.

Dimensions are also distinguishes physics equations from purely mathematical ones: an equation in physics will always relate some physical quantities and, hence, when you’re dealing with physics equations, you always need to wonder about the dimensions. [Note that the term ‘dimension’ has many definitions… But… Well… I suppose you know what I am talking about here, and I need to move on. So let’s do that.] Let’s re-write that ħ = FPlP∙tP formula as follows: ħ/tP = FPlP.

FPlP is, obviously, a force times a distance, so that’s energy. Please do check the dimensions on the left-hand side as well: [ħ/tP] = [[ħ]/[tP] = (N·m·s)/s = N·m. In short, we can think of EP = FPlP = ħ/tP as being some ‘natural’ unit as well. But what would it correspond to—physically? What is its meaning? We may be tempted to call it the quantum of energy that’s associated with our quantum of action, but… Well… No. While it’s referred to as the Planck energy, it’s actually a rather large unit, and so… Well… No. We should not think of it as the quantum of energy. We have a quantum of action but no quantum of energy. Sorry. Let’s move on.

In the same vein, we can re-write the ħ = FPlP∙tP as ħ/lP = FP∙tP. Same thing with the dimensions—or ‘same-same but different’, as they say in Asia: [ħ/lP] = [FP∙tP] = N·m·s)/m = N·s. Force times time is momentum and, hence, we may now be tempted to think of pP = FP∙tP = ħ/lP as the quantum of momentum that’s associated with ħ, but… Well… No. There’s no such thing as a quantum of momentum. Not now in any case. Maybe later. 🙂 But, for now, we only have a quantum of action. So we’ll just call ħ/lP = FP∙tP the Planck momentum for the time being.

So now we have two ways of looking at the dimension of Planck’s constant:

  1. [Planck’s constant] = N∙m∙s = (N∙m)∙s = [energy]∙[time]
  2. [Planck’s constant] = N∙m∙s = (N∙s)∙m = [momentum]∙[distance]

In case you didn’t get this from what I wrote above: the brackets here, i.e. the [ and ] symbols, mean: ‘the dimension of what’s between the brackets’. OK. So far so good. It may all look like kids stuff – it actually is kids stuff so far – but the idea is quite fundamental: we’re thinking here of some amount of action (h or ħ, to be precise, i.e. the quantum of action) expressing itself in time or, alternatively, expressing itself in spaceIn the former case, some amount of energy is expended during some time. In the latter case, some momentum is expended over some distance.

Of course, ideally, we should try to think of action expressing itself in space and time simultaneously, so we should think of it as expressing itself in spacetime. In fact, that’s what the so-called Principle of Least Action in physics is all about—but I won’t dwell on that here, because… Well… It’s not an easy topic, and the diversion would lead us astray. 🙂 What we will do, however, is apply the idea above to the two de Broglie relations: E = ħω and p = ħk. I assume you know these relations by now. If not, just check one of my many posts on them. Let’s see what we can do with them.

The de Broglie relations

We can re-write the two de Broglie relations as ħ = E/ω and ħ = p/k. We can immediately derive an interesting property here:

ħ/ħ = 1 = (E/ω)/(p/k) ⇔ E/p = ω/k

So the ratio of the energy and the momentum is equal to the wave velocity. What wave velocity? The group of the phase velocity? We’re talking an elementary wave here, so both are the same: we have only one E and p, and, hence, only one ω and k. The E/p = ω/k identity underscores the following point: the de Broglie equations are a pair of equations here, and one of the key things to learn when trying to understand quantum mechanics is to think of them as an inseparable pair—like an inseparable twin really—as the quantum of action incorporates both a spatial as well as a temporal dimension. Just think of what Minkowski wrote back in 1907, shortly after he had re-formulated Einstein’s special relativity theory in terms of four-dimensional spacetime, and just two years before he died—unexpectely—from an appendicitis: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

So we should try to think of what that union might represent—and that surely includes looking at the de Broglie equations as a pair of matter-wave equations. Likewise, we should also think of the Uncertainty Principle as a pair of equations: ΔpΔx ≥ ħ/2 and ΔEΔt ≥ ħ/2—but I’ll come back to those later.

The ω in the E = ħω equation and the argument (θ = kx – ωt) of the wavefunction is a frequency in time (or temporal frequency). It’s a frequency expressed in radians per second. You get one radian by dividing one cycle by 2π. In other words, we have 2π radians in one cycle. So ω is related the frequency you’re used to, i.e. f—the frequency expressed in cycles per second (i.e. hertz): we multiply f by 2π to get ω. So we can write: E = ħω = ħ∙2π∙f = h∙f, with h = ħ∙2π (or ħ = h/2π).

Likewise, the k in the p = ħk equation and the argument (θ = kx – ωt) of the wavefunction is a frequency in space (or spatial frequency). Unsurprisingly, it’s expressed in radians per meter.

At this point, it’s good to properly define the radian as a unit in quantum mechanics. We often think of a radian as some distance measured along the circumference, because of the way the unit is constructed (see the illustration below) but that’s right and wrong at the same time. In fact, it’s more wrong than right: the radian is an angle that’s defined using the length of the radius of the unit circle but, when everything is said and done, it’s a unit used to measure some anglenot a distance. That should be obvious from the 2π rad = 360 degrees identity. The angle here is the argument of our wavefunction in quantum mechanics, and so that argument combines both time (t) as well as distance (x): θ = kx – ωt = k(x – c∙t). So our angle (the argument of the wavefunction) integrates both dimensions: space as well as time. If you’re not convinced, just do the dimensional analysis of the kx – ωt expression: both the kx and ωt yield a dimensionless number—or… Well… To be precise, I should say: the kx and ωt products both yield an angle expressed in radians. That angle connects the real and imaginary part of the argument of the wavefunction. Hence, it’s a dimensionless number—but that does not imply it is just some meaningless number. It’s not meaningless at all—obviously!

Circle_radians

Let me try to present what I wrote above in yet another way. The θ = kx – ωt = (p/ħ)·x − (E/ħ)·t equation suggests a fungibility: the wavefunction itself also expresses itself in time and/or in space, so to speak—just like the quantum of action. Let me be precise: the p·x factor in the (p/ħ)·x term represents momentum (whose dimension is N·s) being expended over a distance, while the E·t factor in the (E/ħ)·t term represents energy (expressed in N·m) being expended over some time. [As for the minus sign in front of the (E/ħ)·t term, that’s got to do with the fact that the arrow of time points in one direction only while, in space, we can go in either direction: forward or backwards.] Hence, the expression for the argument tells us that both are essentially fungible—which suggests they’re aspects of one and the same thing. So that‘s what Minkowski intuition is all about: spacetime is one, and the wavefunction just connects the physical properties of whatever it is that we are observing – an electron, or a photon, or whatever other elementary particle – to it.

Of course, the corollary to thinking of unified spacetime is thinking of the real and imaginary part of the wavefunction as one—which we’re supposed to do as a complex number is… Well… One complex number. But that’s easier said than actually done, of course. One way of thinking about the connection between the two spacetime ‘dimensions’ – i.e. t and x, with x actually incorporating three spatial dimensions in space in its own right (see how confusing the term ‘dimension’ is?) – and the two ‘dimensions’ of a complex number is going from Cartesian to polar coordinates, and vice versa. You now think of Euler’s formula, of course – if not, you should – but let me insert something more interesting here. 🙂 I took it from Wikipedia. It illustrates how a simple sinusoidal function transforms as we go from Cartesian to polar coordinates.

Cartesian_to_polar

Interesting, isn’t it? Think of the horizontal and vertical axis in the Cartesian space as representing time and… Well… Space indeed. 🙂 The function connects the space and time dimension and might, for example, represent the trajectory of some object in spacetime. Admittedly, it’s a rather weird trajectory, as the object goes back and forth in some box in space, and accelerates and decelerates all of the time, reversing its direction in the process… But… Well… Isn’t that how we think of a an electron moving in some orbital? 🙂 With that in mind, look at how the same movement in spacetime looks like in polar coordinates. It’s also some movement in a box—but both the ‘horizontal’ and ‘vertical’ axis (think of these axes as the real and imaginary part of a complex number) are now delineating our box. So, whereas our box is a one-dimensional box in spacetime only (our object is constrained in space, but time keeps ticking), it’s a two-dimensional box in our ‘complex’ space. Isn’t it just nice to think about stuff this way?

As far as I am concerned, it triggers the same profound thoughts as that E/p = ω/k relation. The  left-hand side is a ratio between energy and momentum. Now, one way to look at energy is that it’s action per time unit. Likewise, momentum is action per distance unit. Of course, ω is expressed as some quantity (expressed in radians, to be precise) per time unit, and k is some quantity (again, expressed in radians) per distance unit. Because this is a physical equation, the dimension of both sides of the equation has to be the same—and, of course, it is the same: the action dimension in the numerator and denominator of the ratio on the left-hand side of the E/p = ω/k equation cancel each other. But… What? Well… Wouldn’t it be nice to think of the dimension of the argument of our wavefunction as being the dimension of action, rather than thinking of it as just some mathematical thing, i.e. an angle. I like to think the argument of our wavefunction is more than just an angle. When everything is said and done, it has to be something physical—if onlyh because the wavefunction describes something physical. But… Well… I need to do some more thinking on this, so I’ll just move on here. Otherwise this post risks becoming a book in its own right. 🙂

Let’s get back to the topic we were discussing here. We were talking about natural units. More in particular, we were wondering: what’s natural? What does it mean?

Back to Planck units

Let’s start with time and distance. We may want to think of lP and tP as the smallest distance and time units possible—so small, in fact, that both distance and time become countable variables at that scale.

Huh? Yes. I am sure you’ve never thought of time and distance as countable variables but I’ll come back to this rather particular interpretation of the Planck length and time unit later. So don’t worry about it now: just make a mental note of it. The thing is: if tP and lP are the smallest time and distance units possible, then the smallest cycle we can possibly imagine will be associated with those two units: we write: ωP = 1/tP and kP = 1/lP. What’s the ‘smallest possible’ cycle? Well… Not sure. You should not think of some oscillation in spacetime as for now. Just think of a cycle. Whatever cycle. So, as for now, the smallest cycle is just the cycle you’d associate with the smallest time and distance units possible—so we cannot write ωP = 2/tP, for example, because that would imply we can imagine a time unit that’s smaller than tP, as we can associate two cycles with tP now.

OK. Next step. We can now define the Planck energy and the Planck momentum using the de Broglie relations:

EP = ħ∙ωP = ħ/tP  and pP = ħ∙kP = ħ/lP

You’ll say that I am just repeating myself here, as I’ve given you those two equations already. Well… Yes and no. At this point, you should raise the following question: why are we using the angular frequency (ωP = 2π·fP) and the reduced Planck constant (ħ = h/2π), rather than fP or h?

That’s a great question. In fact, it begs the question: what’s the physical constant really? We have two mathematical constants – ħ and h – but they must represent the same physical reality. So is one of the two constants more real than the other? The answer is unambiguously: yes! The Planck energy is defined as EP = ħ/tP =(h/2π)/tP, so we cannot write this as EP = h/tP. The difference is that 1/2π factor, and it’s quite fundamental, as it implies we’re actually not associating a full cycle with tP and lP but a radian of that cycle only.

Huh? Yes. It’s a rather strange observation, and I must admit I haven’t quite sorted out what this actually means. The fundamental idea remains the same, however: we have a quantum of action, ħ (not h!), that can express itself as energy over the smallest distance unit possible or, alternatively, that expresses itself as momentum over the smallest time unit possible. In the former case, we write it as EP = FPlP = ħ/tP. In the latter, we write it as pP = FP∙tP = ħ/lP. Both are aspects of the same reality, though, as our particle moves in space as well as in time, i.e. it moves in spacetime. Hence, one step in space, or in time, corresponds to one radian. Well… Sort of… Not sure how to further explain this. I probably shouldn’t try anyway. 🙂

The more fundamental question is: with what speed is is moving? That question brings us to the next point. The objective is to get some specific value for lP and tP, so how do we do that? How can we determine these two values? Well… That’s another great question. 🙂

The first step is to relate the natural time and distance units to the wave velocity. Now, we do not want to complicate the analysis and so we’re not going to throw in some rest mass or potential energy here. No. We’ll be talking a theoretical zero-mass particle. So we’re not going to consider some electron moving in spacetime, or some other elementary particle. No. We’re going to think about some zero-mass particle here, or a photon. [Note that a photon is not just a zero-mass particle. It’s similar but different: in one of my previous posts, I showed a photon packs more energy, as you get two wavefunctions for the price of one, so to speak. However, don’t worry about the difference here.]

Now, you know that the wave velocity for a zero-mass particle and/or a photon is equal to the speed of light. To be precise, the wave velocity of a photon is the speed of light and, hence, the speed of any zero-mass particle must be the same—as per the definition of mass in Einstein’s special relativity theory. So we write: lP/tP = c ⇔ lP = c∙tP and tP = lP/c. In fact, we also get this from dividing EP by pP, because we know that E/p = c, for any photon (and for any zero-mass particle, really). So we know that EP/pP must also equal c. We can easily double-check that by writing: EP/pP = (ħ/tP)/(ħ/lP) = lP/tP = c. Substitution in ħ = FPlP∙tP yields ħ = c∙FP∙tP2 or, alternatively, ħ = FPlP2/c. So we can now write FP as a function of lP and/or tP:

FP = ħ∙c/lP2 = ħ/(c∙tP2)

We can quickly double-check this by dividing FP = ħ∙c/lP2 by FP = ħ/(c∙tP2). We get: 1 = c2∙tP2/lP2 ⇔ lP2/tP2 = c2 ⇔ lP/tP = c.

Nice. However, this does not uniquely define FP, lP, and tP. The problem is that we’ve got only two equations (ħ = FPlP∙tP and lP/tP = c) for three unknowns (FP, lP, and tP). Can we throw in one or both of the de Broglie equations to get some final answer?

I wish that would help, but it doesn’t—because we get the same ħ = FPlP∙tP equation. Indeed, we’re just re-defining the Planck energy (and the Planck momentum) by that EP = ħ/tP (and pP = ħ/lP) equation here, and so that does not give us a value for EP (and pP). So we’re stuck. We need some other formula so we can calculate the third unknown, which is the Planck force unit (FP). What formula could we possibly choose?

Well… We got a relationship by imposing the condition that lP/tP = c, which implies that if we’d measure the velocity of a photon in Planck time and distance units, we’d find that its velocity is one, so c = 1. Can we think of some similar condition involving ħ? The answer is: we can and we can’t. It’s not so simple. Remember we were thinking of the smallest cycle possible? We said it was small because tP and lP were the smallest units we could imagine. But how do we define that? The idea is as follows: the smallest possible cycle will pack the smallest amount of action, i.e. h (or, expressed per radian rather than per cycle, ħ).

Now, we usually think of packing energy, or momentum, instead of packing action, but that’s because… Well… Because we’re not good at thinking the way Minkowski wanted us to think: we’re not good at thinking of some kind of union of space and time. We tend to think of something moving in space, or, alternatively, of something moving in time—rather than something moving in spacetime. In short, we tend to separate dimensions. So that’s why we’d say the smallest possible cycle would pack an amount of energy that’s equal to EP = ħ∙ωP = ħ/tP, or an amount of momentum that’s equal to pP = ħ∙kP = ħ/lP. But both are part and parcel of the same reality, as evidenced by the E = m∙c2 = m∙cc = p∙c equality. [This equation only holds for a zero-mass particle (and a photon), of course. It’s a bit more complicated when we’d throw in some rest mass, but we can do that later. Also note I keep repeating my idea of the smallest cycle, but we’re talking radians of a cycle, really.]

So we have that mass-energy equivalence, which is also a mass-momentum equivalence according to that E = m∙c2 = m∙cc = p∙c formula. And so now the gravitational force comes into play: there’s a limit to the amount of energy we can pack into a tiny space. Or… Well… Perhaps there’s no limit—but if we pack an awful lot of energy into a really tiny speck of space, then we get a black hole.

However, we’re getting a bit ahead of ourselves here, so let’s first try something else. Let’s throw in the Uncertainty Principle.

The Uncertainty Principle

As mentioned above, we can think of some amount of action expressing itself over some time or, alternatively, over some distance. In the former case, some amount of energy is expended over some time. In the latter case, some momentum is expended over some distance. That’s why the energy and time variables, and the momentum and distance variables, are referred to as complementary. It’s hard to think of both things happening simultaneously (whatever that means in spacetime), but we should try! Let’s now look at the Uncertainty relations once again (I am writing uncertainty with a capital U out of respect—as it’s very fundamental, indeed!):

ΔpΔx ≥ ħ/2 and ΔEΔt ≥ ħ/2.

Note that the ħ/2 factor on the right-hand side quantifies the uncertainty, while the right-hand side of the two equations (ΔpΔx and ΔEΔt) are just an expression of that fundamental uncertainty. In other words, we have two equations (a pair), but there’s only one fundamental uncertainty, and it’s an uncertainty about a movement in spacetime. Hence, that uncertainty expresses itself in both time as well as in space.

Note the use of ħ rather than h, and the fact that the  1/2 factor makes it look like we’re splitting ħ over ΔpΔx and ΔEΔt respectively—which is actually a quite sensible explanation of what this pair of equations actually represent. Indeed, we can add both relations to get the following sum:

ΔpΔx + ΔEΔt ≥ ħ/2 + ħ/2 = ħ

Interesting, isn’t it? It explains that 1/2 factor which troubled us when playing with the de Broglie relations.

Let’s now think about our natural units again—about lP, and tP in particular. As mentioned above, we’ll want to think of them as the smallest distance and time units possible: so small, in fact, that both distance and time become countable variables, so we count x and t as 0, 1, 2, 3 etcetera. We may then imagine that the uncertainty in x and t is of the order of one unit only, so we write Δx = lP and Δt = tP. So we can now re-write the uncertainty relations as:

  • Δp·lP = ħ/2
  • ΔE·tP = ħ/2

Hey! Wait a minute! Do we have a solution for the value of lP and tP here? What if we equate the natural energy and momentum units to ΔE and Δp here? Well… Let’s try it. First note that we may think of the uncertainty in t, or in x, as being equal to plus or minus one unit, i.e. ±1. So the uncertainty is two units really. [Frankly, I just want to get rid of that 1/2 factor here.] Hence, we can re-write the ΔpΔx = ΔEΔt = ħ/2 equations as:

  • ΔpΔx = pPlP = FP∙tPlP = ħ
  • ΔEΔt = EP∙tP = FPlP∙tP = ħ

Hmm… What can we do with this? Nothing much, unfortunately. We’ve got the same problem: we need a value for FP (or for pP, or for EP) to get some specific value for lP and tP, so we’re stuck once again. We have three variables and two equations only, so we have no specific value for either of them. 😦

What to do? Well… I will give you the answer now—the answer you’ve been waiting for, really—but not the technicalities of it. There’s a thing called the Schwarzschild radius, aka as the gravitational radius. Let’s analyze it.

The Schwarzschild radius and the Planck length

The Schwarzschild radius is just the radius of a black hole. Its formal definition is the following: it is the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from the surface of the sphere would equal the speed of light (c). The formula for the Schwartzschild radius is the following:

RS = 2m·G/c2

G is the gravitational constant here: G ≈ 6.674×10−11 N⋅m2/kg2. [Note that Newton’s F = m·Law tells us that 1 kg = 1 N·s2/m, as we’ll need to substitute units later.]

But what is the mass (m) in that RS = 2m·G/c2 equation? Using equivalent time and distance units (so = 1), we wrote the following for a zero-mass particle and for a photon respectively:

  • E = m = p = ħ/2 (zero-mass particle)
  • E = m = p = ħ (photon)

How can a zero-mass particle, or a photon, have some mass? Well… Because it moves at the speed of light. I’ve talked about that before, so I’ll just refer you to my post on that. Of course, the dimension of the right-hand side of these equations (i.e. ħ/2 or ħ) symbol has to be the same as the dimension on the left-hand side, so the ‘ħ’ in the E = ħ equation (or E = ħ/2 equation) is a different ‘ħ’ in the p = ħ equation (or p = ħ/2 equation). So we must be careful here. Let’s write it all out, so as to remind ourselves of the dimensions involved:

  • E [N·m] = ħ [N·m·s/s] = EP = FPlP∙tP/tP
  • p [N·s] = ħ [N·m·s/m] = pP = FPlP∙tP/lP

Now, let’s check this by cheating. I’ll just give you the numerical values—even if we’re not supposed to know them at this point—so you can see I am not writing nonsense here:

  • EP = 1.0545718×10−34 N·m·s/(5.391×10−44 s) = (1.21×1044 N)·(1.6162×10−35 m) = 1.9561×10N·m
  • pP =1.0545718×10−34 N·m·s/(1.6162×10−35 m) = (1.21×1044 N)·(5.391×10−44 s) = 6.52485 N·s

You can google the Planck units, and you’ll see I am not taking you for a ride here. 🙂

The associated Planck mass is mP = EP/c2 = 1.9561×10N·m/(2.998×10m/s)2 = 2.17651×108 N·s2/m = 2.17651×108 kg. So let’s plug that value into RS = 2m·G/cequation. We get:

RS = 2m·G/c= [(2.17651×108 kg)·(6.674×10−11 N⋅m2/kg)/(8.988×1016 m2·s−2)

= 1.6162×1035 kg·N⋅m2·kg−2·m2·s−2 = 1.6162×1035 kg·N⋅m2·kg−2·m2·s−2 = 1.6162×1035 m = lP

Bingo! You can look it up: 1.6162×1035 m is the Planck length indeed, so the Schwarzschild radius is the Planck length. We can now easily calculate the other Planck units:

  • t= lP/c = 1.6162×1035 m/(2.998×10m/s) = 5.391×10−44 s
  • F= ħ/(tPlP)= (1.0545718×10−34 N·m·s)/[(1.6162×1035 m)·(5.391×10−44 s) = 1.21×10−44 N

Bingo again! 🙂

[…] But… Well… Look at this: we’ve been cheating all the way. First, we just gave you that formula for the Schwarzschild radius. It looks like an easy formula but its derivation involves a profound knowledge of general relativity theory. So we’d need to learn about tensors and what have you. The formula is, in effect, a solution to what is known as Einstein’s field equations, and that’s pretty complicated stuff.

However, my crime is much worse than that: I also gave you those numerical values for the Planck energy and momentum, rather than calculating them. I just couldn’t calculate them with the knowledge we have so far. When everything is said and done, we have more than three unknowns. We’ve got five in total, including the Planck charge (qP) and, hence, we need five equations. Again, I’ll just copy them from Wikipedia, because… Well… What we’re discussing here is way beyond the undergraduate physics stuff that we’ve been presenting so far. The equations are the following. Just have a look at them and move on. 🙂

Planck units

Finally, I should note one more thing: I did not use 2m but m in Schwarzschild’s formula. Why? Well… I have no good answer to that. I did it to ensure I got the result we wanted to get. It’s that 1/2 factor again. In fact, the E = m = p = ħ/2 is the correct formula to use, and all would come out alright if we did that and defined the magnitude of the uncertainty as one unit only, but so we used the E = m = p = ħ formula instead, i.e. the equation that’s associated with a photon. You can re-do the calculations as an exercise: you’ll see it comes out alright.

Just to make things somewhat more real, let me note that the Planck energy is very substantial: 1.9561×10N·m ≈ 2×10J is equivalent to the energy that you’d get out of burning 60 liters of gasoline—or the mileage you’d get out of 16 gallons of fuel! In short, it’s huge,  and so we’re packing that into a unimaginably small space. To understand how that works, you can think of the E = h∙f ⇔ h = E/f relation once more. The h = E/f ratio implies that energy and frequency are directly proportional to each other, with h the coefficient of proportionality. Shortening the wavelength, amounts to increasing the frequency and, hence, the energy. So, as you think of our cycle becoming smaller and smaller, until it becomes the smallest cycle possible, you should think of the frequency becoming unimaginably large. Indeed, as I explained in one of my other posts on physical constants, we’re talking the the 1043 Hz scale here. However, we can always choose our time unit such that we measure the frequency as one cycle per time unit. Because the energy per cycle remains the same, it means the quantum of action (ħ = FPlP∙tP) expresses itself over extremely short time spans, which means the EP = FPlP product becomes huge, as we’ve shown above. The rest of the story is the same: gravity comes into play, and so our little blob in spacetime becomes a tiny black hole. Again, we should think of both space and time: they are joined in ‘some kind of union’ here, indeed, as they’re intimately connected through the wavefunction, which travels at the speed of light.

The wavefunction as an oscillation in and of spacetime

OK. Now I am going to present the big idea I started with. Let me first ask you a question: when thinking about the Planck-Einstein relation (I am talking about the E = ħ∙ω relation for a photon here, rather than the equivalent de Broglie equation for a matter-particle), aren’t you struck by the fact that the energy of a photon depends on the frequency of the electromagnetic wave only? I mean… It does not depend on its amplitude. The amplitude is mentioned nowhere. The amplitude is fixed, somehow—or considered to be fixed.

Isn’t that strange? I mean… For any other physical wave, the energy would not only depend on the frequency but also on the amplitude of the wave. For a photon, however, it’s just the frequency that counts. Light of the same frequency but higher intensity (read: more energy) is not a photon with higher amplitude, but just more photons. So it’s the photons that add up somehow, and so that explains the amplitude of the electric and magnetic field vectors (i.e. E and B) and, hence, the intensity of the light. However, every photon considered separately has the same amplitude apparently. We can only increase its energy by increasing the frequency. In short, ω is the only variable here.

Let’s look at that angular frequency once more. As you know, it’s expressed in radians per second but, if you multiply ω by 2π, you get the frequency you’re probably more acquainted with: f = 2πω = f cycles per second. The Planck-Einstein relation is then written as E = h∙f. That’s easy enough. But what if we’d change the time unit here? For example, what if our time unit becomes the time that’s needed for a photon to travel one meter? Let’s examine it.

Let’s denote that time unit by tm, so we write: 1 tm = 1/c s ⇔ tm1 = c s1, with c ≈ 3×108. The frequency, as measured using our new time unit, changes, obviously: we have to divide its former value by c now. So, using our little subscript once more, we could write: fm = f/c. [Why? Just add the dimension to make things more explicit: f s1 = f/c tm1 = f/c tm1.] But the energy of the photon should not depend on our time unit, should it?

Don’t worry. It doesn’t: the numerical value of Planck’s constant (h) would also change, as we’d replace the second in its dimension (N∙m∙s) by c times our new time unit tm. However, Planck’s constant remains what it is: some physical constant. It does not depend on our measurement units: we can use the SI units, or the Planck units (FP, lP, and tP), or whatever unit you can think of. It doesn’t matter: h (or ħ = h/2π) is what is—it’s the quantum of action, and so that’s a physical constant (as opposed to a mathematical constant) that’s associated with one cycle.

Now, I said we do not associate the wavefunction of a photon with an amplitude, but we do associate it with a wavelength. We do so using the standard formula for the velocity of a wave: c = f∙λ ⇔ λ = c/f. We can also write this using the angular frequency and the wavenumber: c = ω/k, with k = 2π/λ. We can double-check this, because we know that, for a photon, the following relation holds: E/p = c. Hence, using the E = ħ∙ω and p = ħ∙k relations, we get: (ħ∙ω)/(ħ∙k) = ω/k = c. So we have options here: h can express itself over a really long wavelength, or it can do so over an extremely short wavelength. We re-write p = ħ∙k as p = E/c = ħ∙2π/λ = h/λ ⇔ E = h∙c/λ ⇔ h∙c = E∙λ. We know this relationship: the energy and the wavelength of a photon (or an electromagnetic wave) are inversely proportional to each other.

Once again, we may want to think of the shortest wavelength possible. As λ gets a zillion times smaller, E gets a zillion times bigger. Is there a limit? There is. As I mentioned above, the gravitational force comes into play here: there’s a limit to the amount of energy we can pack into a tiny space. If we pack an awful lot of energy into a really tiny speck of space, then we get a black hole. In practical terms, that implies our photon can’t travel, as it can’t escape from the black hole it creates. That’s what that calculation of the Schwarzschild radius was all about.

We can—in fact, we should—now apply the same reasoning to the matter-wave. Instead of a photon, we should try to think of a zero-mass matter-particle. You’ll say: that’s a contradiction. Matter-particles – as opposed to force-carrying particles, like photons (or bosons in general) – must have some rest mass, so they can’t be massless. Well… Yes. You’re right. But we can throw the rest mass in later. I first want to focus on the abstract principles, i.e. the propagation mechanism of the matter-wave.

Using natural units, we know our particle will move in spacetime with velocity Δx/Δt = 1/1 = 1. Of course, it has to have some energy to move, or some momentum. We also showed that, if it’s massless, and the elementary wavefunction is ei[(p/ħ)x – (E/ħ)t), then we know the energy, and the momentum, has to be equal to ħ/2. Where does it get that energy, or momentum? Not sure. I like to think it borrows it from spacetime, as it breaks some potential barrier between those two points, and then it gives it back. Or, if it’s at point x = t = 0, then perhaps it gets it from some other massless particle moving from x = t = −1. In both cases, we’d like to think our particle keeps moving. So if the first description (borrowing) is correct, it needs to keep borrowing and returning energy in some kind of interaction with spacetime itself. If it’s the second description, it’s more like spacetime bumping itself forward.

In both cases, however, we’re actually trying to visualize (or should I say: imagine?) some oscillation of spacetime itself, as opposed to an oscillation in spacetime.

Huh? Yes. The idea is the following here: we like to think of the wavefunction as the dependent variable: both its real as well as its imaginary part are a function of x and t, indeed. But what if we’d think of x and t as dependent variables? In that case, the real and imaginary part of the wavefunction would be the independent variables. It’s just a matter of perspective. We sort of mirror our function: we switch its domain for its range, and its range for its domain, as shown below. It all makes sense, doesn’t it? Space and time appear as separate dimensions to us, but they’re intimately connected through c, ħ and the other fundamental physical constants. Likewise, the real and imaginary part of the wavefunction appear as separate dimensions, but they’re intimately connected through π and Euler’s number, i.e. through mathematical constants. That cannot be a coincidence: the mathematical and physical ‘space’ reflect each other through the wavefunction, just like the domain and range of a function reflect each other through that function. So physics and math must meet in some kind of union—at least in our mind, they do!

dependent independent

So, yes, we can—and probably should—be looking at the wavefunction as an oscillation of spacetime, rather than as an oscillation in spacetime only. As mentioned in my introduction, I’ll need to study general relativity theory—and very much in depth—to convincingly prove that point, but I am sure it can be done.

You’ll probably think I am arrogant when saying that—and I probably am—but then I am very much emboldened by the fact some nuclear scientist told me a photon doesn’t have any wavefunction: it’s just those E and B vectors, he told me—and then I found out he was dead wrong, as I showed in my previous post! So I’d rather think more independently now. I’ll keep you guys posted on progress—but it will probably take a while to figure it all out. In the meanwhile, please do let me know your ideas on this. 🙂

Let me wrap up this little excursion with two small notes:

  • We have this E/c = p relation. The mass-energy equivalence relation implies momentum must also have an equivalent mass. If E = m∙c2, then p = m∙c ⇔ m = p/c. It’s obvious, but I just thought it would be useful to highlight this.
  • When we studied the ammonia molecule as a two-state system, our space was not a continuum: we allowed just two positions—two points in space, which we defined with respect to the system. So x was a discrete variable. We assumed time to be continuous, however, and so we got those nice sinusoids as a solution to our set of Hamiltonian equations. However, if we look at space as being discrete, or countable, we should probably think of time as being countable as well. So we should, perhaps, think of a particle being at point x = t = 0 first, and, then, being at point x = t = 1. Instead of the nice sinusoids, we get some boxcar function, as illustrated below, but probably varying between 0 and 1—or whatever other normalized values. You get the idea, I hope. 🙂

boxcar1

Post Scriptum on the Principle of Least Action: As noted above, the Principle of Least Action is not very intuitive, even if Feynman’s exposé of it is not as impregnable as it may look at first. To put it simply, the Principle of Least Action says that the average kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another. So we have a path or line integral here. In a gravitation field, that integral is the following:

Least action

The integral is not all that important. Just note its dimension is the dimension of action indeed, as we multiply energy (the integrand) with time (dt). We can use the Principle of Least Action to re-state Newton’s Law, or whatever other classical law. Among other things, we’ll find that, in the absence of any potential, the trajectory of a particle will just be some straight line.

In quantum mechanics, however, we have uncertainty, as expressed in the ΔpΔx ≥ ħ/2 and ΔEΔt ≥ ħ/2 relations. Now, that uncertainty may express itself in time, or in distance, or in both. That’s where things become tricky. 🙂 I’ve written on this before, but let me copy Feynman himself here, with a more exact explanation of what’s happening (just click on the text to enlarge):

Feynman

The ‘student’ he speaks of above, is himself, of course. 🙂

Too complicated? Well… Never mind. I’ll come back to it later. 🙂

Re-visiting the speed of light, Planck’s constant, and the fine-structure constant

Note: I have published a paper that is very coherent and fully explains what the fine-structure constant actually is. There is nothing magical about it. It’s not some God-given number. It’s a scaling constant – and then some more. But not God-given. Check it out: The Meaning of the Fine-Structure Constant. No ambiguity. No hocus-pocus.

Jean Louis Van Belle, 23 December 2018

Original post:

A brother of mine sent me a link to an article he liked. Now, because we share some interest in physics and math and other stuff, I looked at it and…

Well… I was disappointed. Despite the impressive credentials of its author – a retired physics professor – it was very poorly written. It made me realize how much badly written stuff is around, and I am glad I am no longer wasting my time on it. However, I do owe my brother some explanation of (a) why I think it was bad, and of (b) what, in my humble opinion, he should be wasting his time on. 🙂 So what it is all about?

The article talks about physicists deriving the speed of light from “the electromagnetic properties of the quantum vacuum.” Now, it’s the term ‘quantum‘, in ‘quantum vacuum’, that made me read the article.

Indeed, deriving the theoretical speed of light in empty space from the properties of the classical vacuum – aka empty space – is a piece of cake: it was done by Maxwell himself as he was figuring out his equations back in the 1850s (see my post on Maxwell’s equations and the speed of light). And then he compared it to the measured value, and he saw it was right on the mark. Therefore, saying that the speed of light is a property of the vacuum, or of empty space, is like a tautology: we may just as well put it the other way around, and say that it’s the speed of light that defines the (properties of the) vacuum!

Indeed, as I’ll explain in a moment: the speed of light determines both the electric as well as the magnetic constants μand ε0, which are the (magnetic) permeability and the (electric) permittivity of the vacuum respectively. Both constants depend on the units we are working with (i.e. the units for electric charge, for distance, for time and for force – or for inertia, if you want, because force is defined in terms of overcoming inertia), but so they are just proportionality coefficients in Maxwell’s equations. So once we decide what units to use in Maxwell’s equations, then μand ε0 are just proportionality coefficients which we get from c. So they are not separate constants really – I mean, they are not separate from c – and all of the ‘properties’ of the vacuum, including these constants, are in Maxwell’s equations.

In fact, when Maxwell compared the theoretical value of c with its presumed actual value, he didn’t compare c‘s theoretical value with the speed of light as measured by astronomers (like that 17th century Ole Roemer, to which our professor refers: he had a first go at it by suggesting some specific value for it based on his observations of the timing of the eclipses of one of Jupiter’s moons), but with c‘s value as calculated from the experimental values of μand ε0! So he knew very well what he was looking at. In fact, to drive home the point, it may also be useful to note that the Michelson-Morley experiment – which accurately measured the speed of light – was done some thirty years later. So Maxwell had already left this world by then—very much in peace, because he had solved the mystery all 19th century physicists wanted to solve through his great unification: his set of equations covers it all, indeed: electricity, magnetism, light, and even relativity!

I think the article my brother liked so much does a very lousy job in pointing all of that out, but that’s not why I wouldn’t recommend it. It got my attention because I wondered why one would try to derive the speed of light from the properties of the quantum vacuum. In fact, to be precise, I hoped the article would tell me what the quantum vacuum actually is. Indeed, as far as I know, there’s only one vacuum—one ’empty space’: empty is empty, isn’t it? 🙂 So I wondered: do we have a ‘quantum’ vacuum? And, if so, what is it, really?

Now, that is where the article is really disappointing, I think. The professor drops a few names (like the Max Planck Institute, the University of Paris-Sud, etcetera), and then, promisingly, mentions ‘fleeting excitations of the quantum vacuum’ and ‘virtual pairs of particles’, but then he basically stops talking about quantum physics. Instead, he wanders off to share some philosophical thoughts on the fundamental physical constants. What makes it all worse is that even those thoughts on the ‘essential’ constants are quite off the mark.

So… This post is just a ‘quick and dirty’ thing for my brother which, I hope, will be somewhat more thought-provoking than that article. More importantly, I hope that my thoughts will encourage him to try to grind through better stuff.

On Maxwell’s equations and the properties of empty space

Let me first say something about the speed of light indeed. Maxwell’s four equations may look fairly simple, but that’s only until one starts unpacking all those differential vector equations, and it’s only when going through all of their consequences that one starts appreciating their deep mathematical structure. Let me quickly copy how another blogger jotted them down: 🙂

god-said-maxwell-equation

As I showed in my above-mentioned post, the speed of light (i.e. the speed with which an electromagnetic pulse or wave travels through space) is just one of the many consequences of the mathematical structure of Maxwell’s set of equations. As such, the speed of light is a direct consequence of the ‘condition’, or the properties, of the vacuum indeed, as Maxwell suggested when he wrote that “we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena”.

Of course, while Maxwell still suggests light needs some ‘medium’ here – so that’s a reference to the infamous aether theory – we now know that’s because he was a 19th century scientist, and so we’ve done away with the aether concept (because it’s a redundant hypothesis), and so now we also know there’s absolutely no reason whatsoever to try to “avoid the inference.” 🙂 It’s all OK, indeed: light is some kind of “transverse undulation” of… Well… Of what?

We analyze light as traveling fields, represented by two vectors, E and B, whose direction and magnitude varies both in space as well as in time. E and B are field vectors, and represent the electric and magnetic field respectively. An equivalent formulation – more or less, that is (see my post on the Liénard-Wiechert potentials) – for Maxwell’s equations when only one (moving) charge is involved is:

E

B

This re-formulation, which is Feynman’s preferred formula for electromagnetic radiation, is interesting in a number of ways. It clearly shows that, while we analyze the electric and magnetic field as separate mathematical entities, they’re one and the same phenomenon really, as evidenced by the B = –er×E/c equation, which tells us the magnetic field from a single moving charge is always normal (i.e. perpendicular) to the electric field vector, and also that B‘s magnitude is 1/times the magnitude of E, so |B| = B = |E|/c = E/c. In short, B is fully determined by E, or vice versa: if we have one of the two fields, we have the other, so they’re ‘one and the same thing’ really—not in a mathematical sense, but in a real sense.

Also note that E and B‘s magnitude is just the same if we’re using natural units, so if we equate c with 1. Finally, as I pointed out in my post on the relativity of electromagnetic fields, if we would switch from one reference frame to another, we’ll have a different mix of E and B, but that different mix obviously describes the same physical reality. More in particular, if we’d be moving with the charges, the magnetic field sort of disappears to re-appear as an electric field. So the Lorentz force F = Felectric + Fmagnetic = qE + qv×B is one force really, and its ‘electric’ and ‘magnetic’ component appear the way they appear in our reference frame only. In some other reference frame, we’d have the same force, but its components would look different, even if they, obviously, would and should add up to the same. [Well… Yes and no… You know there’s relativistic corrections to be made to the forces to, but that’s a minor point, really. The force surely doesn’t disappear!]

All of this reinforces what you know already: electricity and magnetism are part and parcel of one and the same phenomenon, the electromagnetic force field, and Maxwell’s equations are the most elegant way of ‘cutting it up’. Why elegant? Well… Click the Occam tab. 🙂

Now, after having praised Maxwell once more, I must say that Feynman’s equations above have another advantage. In Maxwell’s equations, we see two constants, the electric and magnetic constant (denoted by μand ε0 respectively), and Maxwell’s equations imply that the product of the electric and magnetic constant is the reciprocal of c2: μ0·ε= 1/c2. So here we see εand only, so no μ0, so that makes it even more obvious that the magnetic and electric constant are related one to another through c.

[…] Let me digress briefly: why do we have c2 in μ0·ε= 1/c2, instead of just c? That’s related to the relativistic nature of the magnetic force: think about that B = E/c relation. Or, better still, think about the Lorentz equation F = Felectric + Fmagnetic = qE + qv×B = q[E + (v/c)×(E×er)]: the 1/c factor is there because the magnetic force involves some velocity, and any velocity is always relative—and here I don’t mean relative to the frame of reference but relative to the (absolute) speed of light! Indeed, it’s the v/c ratio (usually denoted by β = v/c) that enters all relativistic formulas. So the left-hand side of the μ0·ε= 1/c2 equation is best written as (1/c)·(1/c), with one of the two 1/c factors accounting for the fact that the ‘magnetic’ force is a relativistic effect of the ‘electric’ force, really, and the other 1/c factor giving us the proper relationship between the magnetic and the electric constant. To drive home the point, I invite you to think about the following:

  • μ0 is expressed in (V·s)/(A·m), while εis expressed in (A·s)/(V·m), so the dimension in which the μ0·εproduct is expressed is [(V·s)/(A·m)]·[(A·s)/(V·m)] = s2/m2, so that’s the dimension of 1/c2.
  • Now, this dimensional analysis makes it clear that we can sort of distribute 1/c2 over the two constants. All it takes is re-defining the fundamental units we use to calculate stuff, i.e. the units for electric charge, for distance, for time and for force – or for inertia, as explained above. But so we could, if we wanted, equate both μ0 as well as εwith 1/c.
  • Now, if we would then equate c with 1, we’d have μ0 = ε= c = 1. We’d have to define our units for electric charge, for distance, for time and for force accordingly, but it could be done, and then we could re-write Maxwell’s set of equations using these ‘natural’ units.

In any case, the nitty-gritty here is less important: the point is that μand εare also related through the speed of light and, hence, they are ‘properties’ of the vacuum as well. [I may add that this is quite obvious if you look at their definition, but we’re approaching the matter from another angle here.]

In any case, we’re done with this. On to the next!

On quantum oscillations, Planck’s constant, and Planck units 

The second thought I want to develop is about the mentioned quantum oscillation. What is it? Or what could it be? An electromagnetic wave is caused by a moving electric charge. What kind of movement? Whatever: the charge could move up or down, or it could just spin around some axis—whatever, really. For example, if it spins around some axis, it will have a magnetic moment and, hence, the field is essentially magnetic, but then, again, E and B are related and so it doesn’t really matter if the first cause is magnetic or electric: that’s just our way of looking at the world: in another reference frame, one that’s moving with the charges, the field would essential be electric. So the motion can be anything: linear, rotational, or non-linear in some irregular way. It doesn’t matter: any motion can always be analyzed as the sum of a number of ‘ideal’ motions. So let’s assume we have some elementary charge in space, and it moves and so it emits some electromagnetic radiation.

So now we need to think about that oscillation. The key question is: how small can it be? Indeed, in one of my previous posts, I tried to explain some of the thinking behind the idea of the ‘Great Desert’, as physicists call it. The whole idea is based on our thinking about the limit: what is the smallest wavelength that still makes sense? So let’s pick up that conversation once again.

The Great Desert lies between the 1032 and 1043 Hz scale. 1032 Hz corresponds to a photon energy of Eγ = h·f = (4×10−15 eV·s)·(1032 Hz) = 4×1017 eV = 400,000 tera-electronvolt (1 TeV = 1012 eV). I use the γ (gamma) subscript in my Eγ symbol for two reasons: (1) to make it clear that I am not talking the electric field E here but energy, and (2) to make it clear we are talking ultra-high-energy gamma-rays here.

In fact, γ-rays of this frequency and energy are theoretical only. Ultra-high-energy gamma-rays are defined as rays with photon energies higher than 100 TeV, which is the upper limit for very-high-energy gamma-rays, which have been observed as part of the radiation emitted by so-called gamma-ray bursts (GRBs): flashes associated with extremely energetic explosions in distant galaxies. Wikipedia refers to them as the ‘brightest’ electromagnetic events know to occur in the Universe. These rays are not to be confused with cosmic rays, which consist of high-energy protons and atomic nuclei stripped of their electron shells. Cosmic rays aren’t rays really and, because they consist of particles with a considerable rest mass, their energy is even higher. The so-called Oh-My-God particle, for example, which is the most energetic particle ever detected, had an energy of 3×1020 eV, i.e. 300 million TeV. But it’s not a photon: its energy is largely kinetic energy, with the rest mass m0 counting for a lot in the m in the E = m·c2 formula. To be precise: the mentioned particle was thought to be an iron nucleus, and it packed the equivalent energy of a baseball traveling at 100 km/h! 

But let me refer you to another source for a good discussion on these high-energy particles, so I can get get back to the energy of electromagnetic radiation. When I talked about the Great Desert in that post, I did so using the Planck-Einstein relation (E = h·f), which embodies the idea of the photon being valid always and everywhere and, importantly, at every scale. I also discussed the Great Desert using real-life light being emitted by real-life atomic oscillators. Hence, I may have given the (wrong) impression that the idea of a photon as a ‘wave train’ is inextricably linked with these real-life atomic oscillators, i.e. to electrons going from one energy level to the next in some atom. Let’s explore these assumptions somewhat more.

Let’s start with the second point. Electromagnetic radiation is emitted by any accelerating electric charge, so the atomic oscillator model is an assumption that should not be essential. And it isn’t. For example, whatever is left of the nucleus after alpha or beta decay (i.e. a nuclear decay process resulting in the emission of an α- or β-particle) it likely to be in an excited state, and likely to emit a gamma-ray for about 10−12 seconds, so that’s a burst that’s about 10,000 times shorter than the 10–8 seconds it takes for the energy of a radiating atom to die out. [As for the calculation of that 10–8 sec decay time – so that’s like 10 nanoseconds – I’ve talked about this before but it’s probably better to refer you to the source, i.e. one of Feynman’s Lectures.]

However, what we’re interested in is not the energy of the photon, but the energy of one cycle. In other words, we’re not thinking of the photon as some wave train here, but what we’re thinking about is the energy that’s packed into a space corresponding to one wavelength. What can we say about that?

As you know, that energy will depend both on the amplitude of the electromagnetic wave as well as its frequency. To be precise, the energy is (1) proportional to the square of the amplitude, and (2) proportional to the frequency. Let’s look at the first proportionality relation. It can be written in a number of ways, but one way of doing it is stating the following: if we know the electric field, then the amount of energy that passes per square meter per second through a surface that is normal to the direction in which the radiation is going (which we’ll denote by S – the s from surface – in the formula below), must be proportional to the average of the square of the field. So we write S ∝ 〈E2〉, and so we should think about the constant of proportionality now. Now, let’s not get into the nitty-gritty, and so I’ll just refer to Feynman for the derivation of the formula below:

S = ε0c·〈E2

So the constant of proportionality is ε0c. [Note that, in light of what we wrote above, we can also write this as S = (1/μ0·c)·〈(c·B)2〉 = (c0)·〈B2〉, so that underlines once again that we’re talking one electromagnetic phenomenon only really.] So that’s a nice and rather intuitive result in light of all of the other formulas we’ve been jotting down. However, it is a ‘wave’ perspective. The ‘photon’ perspective assumes that, somehow, the amplitude is given and, therefore, the Planck-Einstein relation only captures the frequency variable: Eγ = h·f.

Indeed, ‘more energy’ in the ‘wave’ perspective basically means ‘more photons’, but photons are photons: they have a definite frequency and a definite energy, and both are given by that Planck-Einstein relation. So let’s look at that relation by doing a bit of dimensional analysis:

  • Energy is measured in electronvolt or, using SI units, joule: 1 eV ≈ 1.6×10−19 J. Energy is force times distance: 1 joule = 1 newton·meter, which means that a larger force over a shorter distance yields the same energy as a smaller force over a longer distance. The oscillations we’re talking about here involve very tiny distances obviously. But the principle is the same: we’re talking some moving charge q, and the power – which is the time rate of change of the energy – that goes in or out at any point of time is equal to dW/dt = F·v, with W the work that’s being done by the charge as it emits radiation.
  • I would also like to add that, as you know, forces are related to the inertia of things. Newton’s Law basically defines a force as that what causes a mass to accelerate: F = m·a = m·(dv/dt) = d(m·v)/dt = dp/dt, with p the momentum of the object that’s involved. When charges are involved, we’ve got the same thing: a potential difference will cause some current to change, and one of the equivalents of Newton’s Law F = m·a = m·(dv/dt) in electromagnetism is V = L·(dI/dt). [I am just saying this so you get a better ‘feel’ for what’s going on.]
  • Planck’s constant is measured in electronvolt·seconds (eV·s) or in, using SI units, in joule·seconds (J·s), so its dimension is that of (physical) action, which is energy times time: [energy]·[time]. Again, a lot of energy during a short time yields the same energy as less energy over a longer time. [Again, I am just saying this so you get a better ‘feel’ for these dimensions.]
  • The frequency f is the number of cycles per time unit, so that’s expressed per second, i.e. in herz (Hz) = 1/second = s−1.

So… Well… It all makes sense: [x joule] = [6.626×10−34 joule]·[1 second]×[f cycles]/[1 second]. But let’s try to deepen our understanding even more: what’s the Planck-Einstein relation really about?

To answer that question, let’s think some more about the wave function. As you know, it’s customary to express the frequency as an angular frequency ω, as used in the wave function A(x, t) = A0·sin(kx − ωt). The angular frequency is the frequency expressed in radians per second. That’s because we need an angle in our wave function, and so we need to relate x and t to some angle. The way to think about this is as follows: one cycle takes a time T (i.e. the period of the wave) which is equal to T = 1/f. Yes: one second divided by the number of cycles per second gives you the time that’s needed for one cycle. One cycle is also equivalent to our argument ωt going around the full circle (i.e. 2π), so we write:  ω·T = 2π and, therefore:

ω = 2π/T = 2π·f

Now we’re ready to play with the Planck-Einstein relation. We know it gives us the energy of one photon really, but what if we re-write our equation Eγ = h·f as Eγ/f = h? The dimensions in this equation are:

[x joule]·[1 second]/[cyles] = [6.626×10−34 joule]·[1 second]

⇔ = 6.626×10−34 joule per cycle

So that means that the energy per cycle is equal to 6.626×10−34 joule, i.e. the value of Planck’s constant.

Let me rephrase truly amazing result, so you appreciate it—perhaps: regardless of the frequency of the light (or our electromagnetic wave, in general) involved, the energy per cycle, i.e. per wavelength or per period, is always equal to 6.626×10−34 joule or, using the electronvolt as the unit, 4.135667662×10−15 eV. So, in case you wondered, that is the true meaning of Planck’s constant!

Now, if we have the frequency f, we also have the wavelength λ, because the velocity of the wave is the frequency times the wavelength: = λ·f and, therefore, λ = c/f. So if we increase the frequency, the wavelength becomes smaller and smaller, and so we’re packing the same amount of energy – admittedly, 4.135667662×10−15 eV is a very tiny amount of energy – into a space that becomes smaller and smaller. Well… What’s tiny, and what’s small? All is relative, of course. 🙂 So that’s where the Planck scale comes in. If we pack that amount of energy into some tiny little space of the Planck dimension, i.e. a ‘length’ of 1.6162×10−35 m, then it becomes a tiny black hole, and it’s hard to think about how that would work.

[…] Let me make a small digression here. I said it’s hard to think about black holes but, of course, it’s not because it’s ‘hard’ that we shouldn’t try it. So let me just mention a few basic facts. For starters, black holes do emit radiation! So they swallow stuff, but they also spit stuff out. More in particular, there is the so-called Hawking radiation, as Roger Penrose and Stephen Hawking discovered.

Let me quickly make a few remarks on that: Hawking radiation is basically a form of blackbody radiation, so all frequencies are there, as shown below: the distribution of the various frequencies depends on the temperature of the black body, i.e. the black hole in this case. [The black curve is the curve that Lord Rayleigh and Sir James Jeans derived in the late 19th century, using classical theory only, so that’s the one that does not correspond to experimental fact, and which led Max Planck to become the ‘reluctant’ father of quantum mechanics. In any case, that’s history and so I shouldn’t dwell on this.]

600px-Black_body

The interesting thing about blackbody radiation, including Hawking radiation, is that it reduces energy and, hence, the equivalent mass of our blackbody. So Hawking radiation reduces the mass and energy of black holes and is therefore also known as black hole evaporation. So black holes that lose more mass than they gain through other means are expected to shrink and ultimately vanish. Therefore, there’s all kind of theories that say why micro black holes, like that Planck scale black hole we’re thinking of right now, should be much larger net emitters of radiation than large black holes and, hence, whey they should shrink and dissipate faster.

Hmm… Interesting… What do we do with all of this information? Well… Let’s think about it as we continue our trek on this long journey to reality over the next year or, more probably, years (plural). 🙂

The key lesson here is that space and time are intimately related because of the idea of movement, i.e. the idea of something having some velocity, and that it’s not so easy to separate the dimensions of time and distance in any hard and fast way. As energy scales become larger and, therefore, our natural time and distance units become smaller and smaller, it’s the energy concept that comes to the fore. It sort of ‘swallows’ all other dimensions, and it does lead to limiting situations which are hard to imagine. Of course, that just underscores the underlying unity of Nature, and the mysteries involved.

So… To relate all of this back to the story that our professor is trying to tell, it’s a simple story really. He’s talking about two fundamental constants basically, c and h, pointing out that c is a property of empty space, and h is related to something doing something. Well… OK. That’s really nothing new, and surely not ground-breaking research. 🙂

Now, let me finish my thoughts on all of the above by making one more remark. If you’ve read a thing or two about this – which you surely have – you’ll probably say: this is not how people usually explain it. That’s true, they don’t. Anything I’ve seen about this just associates the 1043 Hz scale with the 1028 eV energy scale, using the same Planck-Einstein relation. For example, the Wikipedia article on micro black holes writes that “the minimum energy of a microscopic black hole is 1019 GeV [i.e. 1028 eV], which would have to be condensed into a region on the order of the Planck length.” So that’s wrong. I want to emphasize this point because I’ve been led astray by it for years. It’s not the total photon energy, but the energy per cycle that counts. Having said that, it is correct, however, and easy to verify, that the 1043 Hz scale corresponds to a wavelength of the Planck scale: λ = c/= (3×10m/s)/(1043 s−1) = 3×10−35 m. The confusion between the photon energy and the energy per wavelength arises because of the idea of a photon: it travels at the speed of light and, hence, because of the relativistic length contraction effect, it is said to be point-like, to have no dimension whatsoever. So that’s why we think of packing all of its energy in some infinitesimally small place. But you shouldn’t think like that. The photon is dimensionless in our reference frame: in its own ‘world’, it is spread out, so it is a wave train. And it’s in its ‘own world’ that the contradictions start… 🙂

OK. Done!

My third and final point is about what our professor writes on the fundamental physical constants, and more in particular on what he writes on the fine-structure constant. In fact, I could just refer you to my own post on it, but that’s probably a bit too easy for me and a bit difficult for you 🙂 so let me summarize that post and tell you what you need to know about it.

The fine-structure constant

The fine-structure constant α is a dimensionless constant which also illustrates the underlying unity of Nature, but in a way that’s much more fascinating than the two or three things the professor mentions. Indeed, it’s quite incredible how this number (α = 0.00729735…, but you’ll usually see it written as its reciprocal, which is a number that’s close to 137.036…) links charge with the relative speeds, radii, and the mass of fundamental particles and, therefore, how this number also these concepts with each other. And, yes, the fact that it is, effectively, dimensionless, unlike h or c, makes it even more special. Let me quickly sum up what the very same number α all stands for:

(1) α is the square of the electron charge expressed in Planck units: α = eP2.

(2) α is the square root of the ratio of (a) the classical electron radius and (b) the Bohr radius: α = √(re /r). You’ll see this more often written as re = α2r. Also note that this is an equation that does not depend on the units, in contrast to equation 1 (above), and 4 and 5 (below), which require you to switch to Planck units. It’s the square of a ratio and, hence, the units don’t matter. They fall away.

(3) α is the (relative) speed of an electron: α = v/c. [The relative speed is the speed as measured against the speed of light. Note that the ‘natural’ unit of speed in the Planck system of units is equal to c. Indeed, if you divide one Planck length by one Planck time unit, you get (1.616×10−35 m)/(5.391×10−44 s) = m/s. However, this is another equation, just like (2), that does not depend on the units: we can express v and c in whatever unit we want, as long we’re consistent and express both in the same units.]

(4) α is also equal to the product of (a) the electron mass (which I’ll simply write as me here) and (b) the classical electron radius re (if both are expressed in Planck units): α = me·re. Now think that’s, perhaps, the most amazing of all of the expressions for α. [If you don’t think that’s amazing, I’d really suggest you stop trying to study physics. :-)]

Also note that, from (2) and (4), we find that:

(5) The electron mass (in Planck units) is equal me = α/r= α/α2r = 1/αr. So that gives us an expression, using α once again, for the electron mass as a function of the Bohr radius r expressed in Planck units.

Finally, we can also substitute (1) in (5) to get:

(6) The electron mass (in Planck units) is equal to me = α/r = eP2/re. Using the Bohr radius, we get me = 1/αr = 1/eP2r.

So… As you can see, this fine-structure constant really links all of the fundamental properties of the electron: its charge, its radius, its distance to the nucleus (i.e. the Bohr radius), its velocity, its mass (and, hence, its energy),…

So… Why is what it is?

Well… We all marvel at this, but what can we say about it, really? I struggle how to interpret this, just as much – or probably much more 🙂 – as the professor who wrote the article I don’t like (because it’s so imprecise, and that’s what made me write all what I am writing here).

Having said that, it’s obvious that it points to a unity beyond these numbers and constants that I am only beginning to appreciate for what it is: deep, mysterious, and very beautiful. But so I don’t think that professor does a good job at showing how deep, mysterious and beautiful it all is. But then that’s up to you, my brother and you, my imaginary reader, to judge, of course. 🙂

[…] I forgot to mention what I mean with ‘Planck units’. Well… Once again, I should refer you to one of my other posts. But, yes, that’s too easy for me and a bit difficult for you. 🙂 So let me just note we get those Planck units by equating not less than five fundamental physical constants to 1, notably (1) the speed of light, (2) Planck’s (reduced) constant, (3) Boltzmann’s constant, (4) Coulomb’s constant and (5) Newton’s constant (i.e. the gravitational constant). Hence, we have a set of five equations here (ħ = kB = ke = G = 1), and so we can solve that to get the five Planck units, i.e. the Planck length unit, the Planck time unit, the Planck mass unit, the Planck energy unit, the Planck charge unit and, finally (oft forgotten), the Planck temperature unit. Of course, you should note that all mass and energy units are directly related because of the mass-energy equivalence relation E = mc2, which simplifies to E = m if c is equated to 1. [I could also say something about the relation between temperature and (kinetic) energy, but I won’t, as it would only further confuse you.]

OK. Done! 🙂

Addendum: How to think about space and time?

If you read the argument on the Planck scale and constant carefully, then you’ll note that it does not depend on the idea of an indivisible photon. However, it does depend on that Planck-Einstein relation being valid always and everywhere. Now, the Planck-Einstein relation is, in its essence, a fairly basic result from classical electromagnetic theory: it incorporates quantum theory – remember: it’s the equation that allowed Planck to solve the black-body radiation problem, and so it’s why they call Planck the (reluctant) ‘Father of Quantum Theory’ – but it’s not quantum theory.

So the obvious question is: can we make this reflection somewhat more general, so we can think of the electromagnetic force as an example only. In other words: can we apply the thoughts above to any force and any movement really?

The truth is: I haven’t advanced enough in my little study to give the equations for the other forces. Of course, we could think of gravity, and I developed some thoughts on how gravity waves might look like, but nothing specific really. And then we have the shorter-range nuclear forces, of course: the strong force, and the weak force. The laws involved are very different. The strong force involves color charges, and the way distances work is entirely different. So it would surely be some different analysis. However, the results should be the same. Let me offer some thoughts though:

  • We know that the relative strength of the nuclear force is much larger, because it pulls like charges (protons) together, despite the strong electromagnetic force that wants to push them apart! So the mentioned problem of trying to ‘pack’ some oscillation in some tiny little space should be worse with the strong force. And the strong force is there, obviously, at tiny little distances!
  • Even gravity should become important, because if we’ve got a lot of energy packed into some tiny space, its equivalent mass will ensure the gravitational forces also become important. In fact, that’s what the whole argument was all about!
  • There’s also all this talk about the fundamental forces becoming one at the Planck scale. I must, again, admit my knowledge is not advanced enough to explain how that would be possible, but I must assume that, if physicists are making such statements, the argument must be fairly robust.

So… Whatever charge or whatever force we are talking about, we’ll be thinking of waves or oscillations—or simply movement, but it’s always a movement in a force field, and so there’s power and energy involved (energy is force times distance, and power is the time rate of change of energy). So, yes, we should expect the same issues in regard to scale. And so that’s what’s captured by h.

As we’re talking the smallest things possible, I should also mention that there are also other inconsistencies in the electromagnetic theory, which should (also) have their parallel for other forces. For example, the idea of a point charge is mathematically inconsistent, as I show in my post on fields and charges. Charge, any charge really, must occupy some space. It cannot all be squeezed into one dimensionless point. So the reasoning behind the Planck time and distance scale is surely valid.

In short, the whole argument about the Planck scale and those limits is very valid. However, does it imply our thinking about the Planck scale is actually relevant? I mean: it’s not because we can imagine how things might look like  – they may look like those tiny little black holes, for example – that these things actually exist. GUT or string theorists obviously think they are thinking about something real. But, frankly, Feynman had a point when he said what he said about string theory, shortly before his untimely death in 1988: “I don’t like that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation—a fix-up to say, ‘Well, it still might be true.'”

It’s true that the so-called Standard Model does not look very nice. It’s not like Maxwell’s equations. It’s complicated. It’s got various ‘sectors’: the electroweak sector, the QCD sector, the Higgs sector,… So ‘it looks like it’s got too much going on’, as a friend of mine said when he looked at a new design for mountainbike suspension. 🙂 But, unlike mountainbike designs, there’s no real alternative for the Standard Model. So perhaps we should just accept it is what it is and, hence, in a way, accept Nature as we can see it. So perhaps we should just continue to focus on what’s here, before we reach the Great Desert, rather than wasting time on trying to figure out how things might look like on the other side, especially because we’ll never be able to test our theories about ‘the other side.’

On the other hand, we can see where the Great Desert sort of starts (somewhere near the 1032 Hz scale), and so it’s only natural to think it should also stop somewhere. In fact, we know where it stops: it stops at the 1043 Hz scale, because everything beyond that doesn’t make sense. The question is: is there actually there? Like fundamental strings or whatever you want to call it. Perhaps we should just stop where the Great Desert begins. And what’s the Great Desert anyway? Perhaps it’s a desert indeed, and so then there is absolutely nothing there. 🙂

Hmm… There’s not all that much one can say about it. However, when looking at the history of physics, there’s one thing that’s really striking. Most of what physicists can think of, in the sense that it made physical sense, turned out to exist. Think of anti-matter, for instance. Paul Dirac thought it might exist, that it made sense to exist, and so everyone started looking for it, and Carl Anderson found in a few years later (in 1932). In fact, it had been observed before, but people just didn’t pay attention, so they didn’t want to see it, in a way. […] OK. I am exaggerating a bit, but you know what I mean. The 1930s are full of examples like that. There was a burst of scientific creativity, as the formalism of quantum physics was being developed, and the experimental confirmations of the theory just followed suit.

In the field of astronomy, or astrophysics I should say, it was the same with black holes. No one could really imagine the existence of black holes until the 1960s or so: they were thought of a mathematical curiosity only, a logical possibility. However, the circumstantial evidence now is quite large and so… Well… It seems a lot of what we can think of actually has some existence somewhere. 🙂

So… Who knows? […] I surely don’t. And so I need to get back to the grind and work my way through the rest of Feynman’s Lectures and the related math. However, this was a nice digression, and so I am grateful to my brother he initiated it. 🙂

The Strange Theory of Light and Matter (III)

This is my third and final comments on Feynman’s popular little booklet: The Strange Theory of Light and Matter, also known as Feynman’s Lectures on Quantum Electrodynamics (QED).

The origin of this short lecture series is quite moving: the death of Alix G. Mautner, a good friend of Feynman’s. She was always curious about physics but her career was in English literature and so she did not manage the math. Hence, Feynman introduces this 1985 publication by writing: “Here are the lectures I really prepared for Alix, but unfortunately I can’t tell them to her directly, now.”

Alix Mautner died from a brain tumor, and it is her husband, Leonard Mautner, who sponsored the QED lectures series at the UCLA, which Ralph Leigton transcribed and published as the booklet that we’re talking about here. Feynman himself died a few years later, at the relatively young age of 69. Tragic coincidence: he died of cancer too. Despite all this weirdness, Feynman’s QED never quite got the same iconic status of, let’s say, Stephen Hawking’s Brief History of Time. I wonder why, but the answer to that question is probably in the realm of chaos theory. 🙂 I actually just saw the movie on Stephen Hawking’s life (The Theory of Everything), and I noted another strange coincidence: Jane Wilde, Hawking’s first wife, also has a PhD in literature. It strikes me that, while the movie documents that Jane Wilde gave Hawking three children, after which he divorced her to marry his nurse, Elaine, the movie does not mention that he separated from Elaine too, and that he has some kind of ‘working relationship’ with Jane again.

Hmm… What to say? I should get back to quantum mechanics here or, to be precise, to quantum electrodynamics.

One reason why Feynman’s Strange Theory of Light and Matter did not sell like Hawking’s Brief History of Time, might well be that, in some places, the text is not entirely accurate. Why? Who knows? It would make for an interesting PhD thesis in History of Science. Unfortunately, I have no time for such PhD thesis. Hence, I must assume that Richard Feynman simply didn’t have much time or energy left to correct some of the writing of Ralph Leighton, who transcribed and edited these four short lectures a few years before Feynman’s death. Indeed, when everything is said and done, Ralph Leighton is not a physicist and, hence, I think he did compromise – just a little bit – on accuracy for the sake of readability. Ralph Leighton’s father, Robert Leighton, an eminent physicist who worked with Feynman, would probably have done a much better job.

I feel that one should not compromise on accuracy, even when trying to write something reader-friendly. That’s why I am writing this blog, and why I am writing three posts specifically on this little booklet. Indeed, while I’d warmly recommend that little book on QED as an excellent non-mathematical introduction to the weird world of quantum mechanics, I’d also say that, while Ralph Leighton’s story is great, it’s also, in some places, not entirely accurate indeed.

So… Well… I want to do better than Ralph Leighton here. Nothing more. Nothing less. 🙂 Let’s go for it.

I. Probability amplitudes: what are they?

The greatest achievement of that little QED publication is that it manages to avoid any reference to wave functions and other complicated mathematical constructs: all of the complexity of quantum mechanics is reduced to three basic events or actions and, hence, three basic amplitudes which are represented as ‘arrows’—literally.

Now… Well… You may or may not know that a (probability) amplitude is actually a complex number, but it’s not so easy to intuitively understand the concept of a complex number. In contrast, everyone easily ‘gets’ the concept of an ‘arrow’. Hence, from a pedagogical point of view, representing complex numbers by some ‘arrow’ is truly a stroke of genius.

Whatever we call it, a complex number or an ‘arrow’, a probability amplitude is something with (a) a magnitude and (b) a phase. As such, it resembles a vector, but it’s not quite the same, if only because we’ll impose some restrictions on the magnitude. But I shouldn’t get ahead of myself. Let’s start with the basics.

A magnitude is some real positive number, like a length, but you should not associate it with some spatial dimension in physical space: it’s just a number. As for the phase, we could associate that concept with some direction but, again, you should just think of it as a direction in a mathematical space, not in the real (physical) space.

Let me insert a parenthesis here. If I say the ‘real’ or ‘physical’ space, I mean the space in which the electrons and photons and all other real-life objects that we’re looking at exist and move. That’s a non-mathematical definition. In fact, in math, the real space is defined as a coordinate space, with sets of real numbers (vectors) as coordinates, so… Well… That’s a mathematical space only, not the ‘real’ (physical) space. So the real (vector) space is not real. 🙂 The mathematical real space may, or may not, accurately describe the real (physical) space. Indeed, you may have heard that physical space is curved because of the presence of massive objects, which means that the real coordinate space will actually not describe it very accurately. I know that’s a bit confusing but I hope you understand what I mean: if mathematicians talk about the real space, they do not mean the real space. They refer to a vector space, i.e. a mathematical construct. To avoid confusion, I’ll use the term ‘physical space’ rather than ‘real’ space in the future. So I’ll let the mathematicians get away with using the term ‘real space’ for something that isn’t real actually. 🙂

End of digression. Let’s discuss these two mathematical concepts – magnitude and phase – somewhat more in detail.

A. The magnitude

Let’s start with the magnitude or ‘length’ of our arrow. We know that we have to square these lengths to find some probability, i.e. some real number between 0 and 1. Hence, the length of our arrows cannot be larger than one. That’s the restriction I mentioned already, and this ‘normalization’ condition reinforces the point that these ‘arrows’ do not have any spatial dimension (not in any real space anyway): they represent a function. To be specific, they represent a wavefunction.

If we’d be talking complex numbers instead of ‘arrows’, we’d say the absolute value of the complex number cannot be larger than one. We’d also say that, to find the probability, we should take the absolute square of the complex number, so that’s the square of the magnitude or absolute value of the complex number indeed. We cannot just square the complex number: it has to be the square of the absolute value.

Why? Well… Just write it out. [You can skip this section if you’re not interested in complex numbers, but I would recommend you try to understand. It’s not that difficult. Indeed, if you’re reading this, you’re most likely to understand something of complex numbers and, hence, you should be able to work your way through it. Just remember that a complex number is like a two-dimensional number, which is why it’s sometimes written using bold-face (z), rather than regular font (z). However, I should immediately add this convention is usually not followed. I like the boldface though, and so I’ll try to use it in this post.] The square of a complex number z = a + bi is equal to z= a+ 2abi – b2, while the square of its absolute value (i.e. the absolute square) is |z|= [√(a+ b2)]2 = a+ b2. So you can immediately see that the square and the absolute square of a complex numbers are two very different things indeed: it’s not only the 2abi term, but there’s also the minus sign in the first expression, because of the i= –1 factor. In case of doubt, always remember that the square of a complex number may actually yield a negative number, as evidenced by the definition of the imaginary unit itself: i= –1.

End of digression. Feynman and Leighton manage to avoid any reference to complex numbers in that short series of four lectures and, hence, all they need to do is explain how one squares a length. Kids learn how to do that when making a square out of rectangular paper: they’ll fold one corner of the paper until it meets the opposite edge, forming a triangle first. They’ll then cut or tear off the extra paper, and then unfold. Done. [I could note that the folding is a 90 degree rotation of the original length (or width, I should say) which, in mathematical terms, is equivalent to multiplying that length with the imaginary unit (i). But I am sure the kids involved would think I am crazy if I’d say this. 🙂 So let me get back to Feynman’s arrows.

B. The phase

Feynman and Leighton’s second pedagogical stroke of genius is the metaphor of the ‘stopwatch’ and the ‘stopwatch hand’ for the variable phase. Indeed, although I think it’s worth explaining why z = a + bi = rcosφ + irsinφ in the illustration below can be written as z = reiφ = |z|eiφ, understanding Euler’s representation of complex number as a complex exponential requires swallowing a very substantial piece of math and, if you’d want to do that, I’ll refer you to one of my posts on complex numbers).

Complex_number_illustration

The metaphor of the stopwatch represents a periodic function. To be precise, it represents a sinusoid, i.e. a smooth repetitive oscillation. Now, the stopwatch hand represents the phase of that function, i.e. the φ angle in the illustration above. That angle is a function of time: the speed with which the stopwatch turns is related to some frequency, i.e. the number of oscillations per unit of time (i.e. per second).

You should now wonder: what frequency? What oscillations are we talking about here? Well… As we’re talking photons and electrons here, we should distinguish the two:

  1. For photons, the frequency is given by Planck’s energy-frequency relation, which relates the energy (E) of a photon (1.5 to 3.5 eV for visible light) to its frequency (ν). It’s a simple proportional relation, with Planck’s constant (h) as the proportionality constant: E = hν, or ν = E/h.
  2. For electrons, we have the de Broglie relation, which looks similar to the Planck relation (E = hf, or f = E/h) but, as you know, it’s something different. Indeed, these so-called matter waves are not so easy to interpret because there actually is no precise frequency f. In fact, the matter wave representing some particle in space will consist of a potentially infinite number of waves, all superimposed one over another, as illustrated below.

Sequential_superposition_of_plane_waves

For the sake of accuracy, I should mention that the animation above has its limitations: the wavetrain is complex-valued and, hence, has a real as well as an imaginary part, so it’s something like the blob underneath. Two functions in one, so to speak: the imaginary part follows the real part with a phase difference of 90 degrees (or π/2 radians). Indeed, if the wavefunction is a regular complex exponential reiθ, then rsin(φ–π/2) = rcos(φ), which proves the point: we have two functions in one here. 🙂 I am actually just repeating what I said before already: the probability amplitude, or the wavefunction, is a complex number. You’ll usually see it written as Ψ (psi) or Φ (phi). Here also, using boldface (Ψ or Φ instead of Ψ or Φ) would usefully remind the reader that we’re talking something ‘two-dimensional’ (in mathematical space, that is), but this convention is usually not followed.

Photon wave

In any case… Back to frequencies. The point to note is that, when it comes to analyzing electrons (or any other matter-particle), we’re dealing with a range of frequencies f really (or, what amounts to the same, a range of wavelengths λ) and, hence, we should write Δf = ΔE/h, which is just one of the many expressions of the Uncertainty Principle in quantum mechanics.

Now, that’s just one of the complications. Another difficulty is that matter-particles, such as electrons, have some rest mass, and so that enters the energy equation as well (literally). Last but not least, one should distinguish between the group velocity and the phase velocity of matter waves. As you can imagine, that makes for a very complicated relationship between ‘the’ wavelength and ‘the’ frequency. In fact, what I write above should make it abundantly clear that there’s no such thing as the wavelength, or the frequency: it’s a range really, related to the fundamental uncertainty in quantum physics. I’ll come back to that, and so you shouldn’t worry about it here. Just note that the stopwatch metaphor doesn’t work very well for an electron!

In his postmortem lectures for Alix Mautner, Feynman avoids all these complications. Frankly, I think that’s a missed opportunity because I do not think it’s all that incomprehensible. In fact, I write all that follows because I do want you to understand the basics of waves. It’s not difficult. High-school math is enough here. Let’s go for it.

One turn of the stopwatch corresponds to one cycle. One cycle, or 1 Hz (i.e. one oscillation per second) covers 360 degrees or, to use a more natural unit, 2π radians. [Why is radian a more natural unit? Because it measures an angle in terms of the distance unit itself, rather than in arbitrary 1/360 cuts of a full circle. Indeed, remember that the circumference of the unit circle is 2π.] So our frequency ν (expressed in cycles per second) corresponds to a so-called angular frequency ω = 2πν. From this formula, it should be obvious that ω is measured in radians per second.

We can also link this formula to the period of the oscillation, T, i.e. the duration of one cycle. T = 1/ν and, hence, ω = 2π/T. It’s all nicely illustrated below. [And, yes, it’s an animation from Wikipedia: nice and simple.]

AngularFrequency

The easy math above now allows us to formally write the phase of a wavefunction – let’s denote the wavefunction as φ (phi), and the phase as θ (theta) – as a function of time (t) using the angular frequency ω. So we can write: θ = ωt = 2π·ν·t. Now, the wave travels through space, and the two illustrations above (i.e. the one with the super-imposed waves, and the one with the complex wave train) would usually represent a wave shape at some fixed point in time. Hence, the horizontal axis is not t but x. Hence, we can and should write the phase not only as a function of time but also of space. So how do we do that? Well… If the hypothesis is that the wave travels through space at some fixed speed c, then its frequency ν will also determine its wavelength λ. It’s a simple relationship: c = λν (the number of oscillations per second times the length of one wavelength should give you the distance traveled per second, so that’s, effectively, the wave’s speed).

Now that we’ve expressed the frequency in radians per second, we can also express the wavelength in radians per unit distance too. That’s what the wavenumber does: think of it as the spatial frequency of the wave. We denote the wavenumber by k, and write: k = 2π/λ. [Just do a numerical example when you have difficulty following. For example, if you’d assume the wavelength is 5 units distance (i.e. 5 meter) – that’s a typical VHF radio frequency: ν = (3×10m/s)/(5 m) = 0.6×108 Hz = 60 MHz – then that would correspond to (2π radians)/(5 m) ≈ 1.2566 radians per meter. Of course, we can also express the wave number in oscillations per unit distance. In that case, we’d have to divide k by 2π, because one cycle corresponds to 2π radians. So we get the reciprocal of the wavelength: 1/λ. In our example, 1/λ is, of course, 1/5 = 0.2, so that’s a fifth of a full cycle. You can also think of it as the number of waves (or wavelengths) per meter: if the wavelength is λ, then one can fit 1/λ waves in a meter.

waveform-showing-wavelength

Now, from the ω = 2πν, c = λν and k = 2π/λ relations, it’s obvious that k = 2π/λ = 2π/(c/ν) = (2πν)/c = ω/c. To sum it all up, frequencies and wavelengths, in time and in space, are all related through the speed of propagation of the wave c. More specifically, they’re related as follows:

c = λν = ω/k

From that, it’s easy to see that k = ω/c, which we’ll use in a moment. Now, it’s obvious that the periodicity of the wave implies that we can find the same phase by going one oscillation (or a multiple number of oscillations back or forward in time, or in space. In fact, we can also find the same phase by letting both time and space vary. However, if we want to do that, it should be obvious that we should either (a) go forward in space and back in time or, alternatively, (b) go back in space and forward in time. In other words, if we want to get the same phase, then time and space sort of substitute for each other. Let me quote Feynman on this: “This is easily seen by considering the mathematical behavior of a(tr/c). Evidently, if we add a little time Δt, we get the same value for a(tr/c) as we would have if we had subtracted a little distance: ΔcΔt.” The variable a stands for the acceleration of an electric charge here, causing an electromagnetic wave, but the same logic is valid for the phase, with a minor twist though: we’re talking a nice periodic function here, and so we need to put the angular frequency in front. Hence, the rate of change of the phase in respect to time is measured by the angular frequency ω. In short, we write:

θ = ω(t–x/c) = ωt–kx

Hence, we can re-write the wavefunction, in terms of its phase, as follows:

φ(θ) = φ[θ(x, t)] = φ[ωt–kx]

Note that, if the wave would be traveling in the ‘other’ direction (i.e. in the negative x-direction), we’d write φ(θ) = φ[kx+ωt]. Time travels in one direction only, of course, but so one minus sign has to be there because of the logic involved in adding time and subtracting distance. You can work out an example (with a sine or cosine wave, for example) for yourself.

So what, you’ll say? Well… Nothing. I just hope you agree that all of this isn’t rocket science: it’s just high-school math. But so it shows you what that stopwatch really is and, hence, – but who am I? – would have put at least one or two footnotes on this in a text like Feynman’s QED.

Now, let me make a much longer and more serious digression:

Digression 1: on relativity and spacetime

As you can see from the argument (or phase) of that wave function φ(θ) = φ[θ(x, t)] = φ[ωt–kx] = φ[–k(x–ct)], any wave equation establishes a deep relation between the wave itself (i.e. the ‘thing’ we’re describing) and space and time. In fact, that’s what the whole wave equation is all about! So let me say a few things more about that.

Because you know a thing or two about physics, you may ask: when we’re talking time, whose time are we talking about? Indeed, if we’re talking photons going from A to B, these photons will be traveling at or near the speed of light and, hence, their clock, as seen from our (inertial) frame of reference, doesn’t move. Likewise, according to the photon, our clock seems to be standing still.

Let me put the issue to bed immediately: we’re looking at things from our point of view. Hence, we’re obviously using our clock, not theirs. Having said that, the analysis is actually fully consistent with relativity theory. Why? Well… What do you expect? If it wasn’t, the analysis would obviously not be valid. 🙂 To illustrate that it’s consistent with relativity theory, I can mention, for example, that the (probability) amplitude for a photon to travel from point A to B depends on the spacetime interval, which is invariant. Hence, A and B are four-dimensional points in spacetime, involving both spatial as well as time coordinates: A = (xA, yA, zA, tA) and B = (xB, yB, zB, tB). And so the ‘distance’ – as measured through the spacetime interval – is invariant.

Now, having said that, we should draw some attention to the intimate relationship between space and time which, let me remind you, results from the absoluteness of the speed of light. Indeed, one will always measure the speed of light c as being equal to 299,792,458 m/s, always and everywhere. It does not depend on your reference frame (inertial or moving). That’s why the constant c anchors all laws in physics, and why we can write what we write above, i.e. include both distance (x) as well as time (t) in the wave function φ = φ(x, t) = φ[ωt–kx] = φ[–k(x–ct)]. The k and ω are related through the ω/k = c relationship: the speed of light links the frequency in time (ν = ω/2π = 1/T) with the frequency in space (i.e. the wavenumber or spatial frequency k). There is only degree of freedom here: the frequency—in space or in time, it doesn’t matter: ν and ω are not independent.  [As noted above, the relationship between the frequency in time and in space is not so obvious for electrons, or for matter waves in general: for those matter-waves, we need to distinguish group and phase velocity, and so we don’t have a unique frequency.]

Let me make another small digression within the digression here. Thinking about travel at the speed of light invariably leads to paradoxes. In previous posts, I explained the mechanism of light emission: a photon is emitted – one photon only – when an electron jumps back to its ground state after being excited. Hence, we may imagine a photon as a transient electromagnetic wave–something like what’s pictured below. Now, the decay time of this transient oscillation (τ) is measured in nanoseconds, i.e. billionths of a second (1 ns = 1×10–9 s): the decay time for sodium light, for example, is some 30 ns only.

decay time

However, because of the tremendous speed of light, that still makes for a wavetrain that’s like ten meter long, at least (30×10–9 s times 3×10m/s is nine meter, but you should note that the decay time measures the time for the oscillation to die out by a factor 1/e, so the oscillation itself lasts longer than that). Those nine or ten meters cover like 16 to 17 million oscillations (the wavelength of sodium light is about 600 nm and, hence, 10 meter fits almost 17 million oscillations indeed). Now, how can we reconcile the image of a photon as a ten-meter long wavetrain with the image of a photon as a point particle?

The answer to that question is paradoxical: from our perspective, anything traveling at the speed of light – including this nine or ten meter ‘long’ photon – will have zero length because of the relativistic length contraction effect. Length contraction? Yes. I’ll let you look it up, because… Well… It’s not easy to grasp. Indeed, from the three measurable effects on objects moving at relativistic speeds – i.e. (1) an increase of the mass (the energy needed to further accelerate particles in particle accelerators increases dramatically at speeds nearer to c), (2) time dilation, i.e. a slowing down of the (internal) clock (because of their relativistic speeds when entering the Earth’s atmosphere, the measured half-life of muons is five times that when at rest), and (3) length contraction – length contraction is probably the most paradoxical of all.

Let me end this digression with yet another short note. I said that one will always measure the speed of light c as being equal to 299,792,458 m/s, always and everywhere and, hence, that it does not depend on your reference frame (inertial or moving). Well… That’s true and not true at the same time. I actually need to nuance that statement a bit in light of what follows: an individual photon does have an amplitude to travel faster or slower than c, and when discussing matter waves (such as the wavefunction that’s associated with an electron), we can have phase velocities that are faster than light! However, when calculating those amplitudes, is a constant.

That doesn’t make sense, you’ll say. Well… What can I say? That’s how it is unfortunately. I need to move on and, hence, I’ll end this digression and get back to the main story line. Part I explained what probability amplitudes are—or at least tried to do so. Now it’s time for part II: the building blocks of all of quantum electrodynamics (QED).

II. The building blocks: P(A to B), E(A to B) and j

The three basic ‘events’ (and, hence, amplitudes) in QED are the following:

1. P(A to B)

P(A to B) is the (probability) amplitude for a photon to travel from point A to B. However, I should immediately note that A and B are points in spacetime. Therefore, we associate them not only with some specific (x, y, z) position in space, but also with a some specific time t. Now, quantum-mechanical theory gives us an easy formula for P(A to B): it depends on the so-called (spacetime) interval between the two points A and B, i.e. I = Δr– Δt= (x2–x1)2+(y2–y1)2+(z2–z1)– (t2–t1)2. The point to note is that the spacetime interval takes both the distance in space as well as the ‘distance’ in time into account. As I mentioned already, this spacetime interval does not depend on our reference frame and, hence, it’s invariant (as long as we’re talking reference frames that move with constant speed relative to each other). Also note that we should measure time and distance in equivalent units when using that Δr– Δtformula for I. So we either measure distance in light-seconds or, else, we measure time in units that correspond to the time that’s needed for light to travel one meter. If no equivalent units are adopted, the formula is I = Δrc·Δt2.

Now, in quantum theory, anything is possible and, hence, not only do we allow for crooked paths, but we also allow for the difference in time to differ from  the time you’d expect a photon to need to travel along some curve (whose length we’ll denote by l), i.e. l/c. Hence, our photon may actually travel slower or faster than the speed of light c! There is one lucky break, however, that makes all come out alright: it’s easy to show that the amplitudes associated with the odd paths and strange timings generally cancel each other out. [That’s what the QED booklet shows.] Hence, what remains, are the paths that are equal or, importantly, those that very near to the so-called ‘light-like’ intervals in spacetime only. The net result is that light – even one single photon – effectively uses a (very) small core of space as it travels, as evidenced by the fact that even one single photon interferes with itself when traveling through a slit or a small hole!

[If you now wonder what it means for a photon to interfere for itself, let me just give you the easy explanation: it may change its path. We assume it was traveling in a straight line – if only because it left the source at some point in time and then arrived at the slit obviously – but so it no longer travels in a straight line after going through the slit. So that’s what we mean here.]

2. E(A to B)

E(A to B) is the (probability) amplitude for an electron to travel from point A to B. The formula for E(A to B) is much more complicated, and it’s the one I want to discuss somewhat more in detail in this post. It depends on some complex number j (see the next remark) and some real number n.

3. j

Finally, an electron could emit or absorb a photon, and the amplitude associated with this event is denoted by j, for junction number. It’s the same number j as the one mentioned when discussing E(A to B) above.

Now, this junction number is often referred to as the coupling constant or the fine-structure constant. However, the truth is, as I pointed out in my previous post, that these numbers are related, but they are not quite the same: α is the square of j, so we have α = j2. There is also one more, related, number: the gauge parameter, which is denoted by g (despite the g notation, it has nothing to do with gravitation). The value of g is the square root of 4πε0α, so g= 4πε0α. I’ll come back to this. Let me first make an awfully long digression on the fine-structure constant. It will be awfully long. So long that it’s actually part of the ‘core’ of this post actually.

Digression 2: on the fine-structure constant, Planck units and the Bohr radius

The value for j is approximately –0.08542454.

How do we know that?

The easy answer to that question is: physicists measured it. In fact, they usually publish the measured value as the square root of the (absolute value) of j, which is that fine-structure constant α. Its value is published (and updated) by the US National Institute on Standards and Technology. To be precise, the currently accepted value of α is 7.29735257×10−3. In case you doubt, just check that square root:

j = –0.08542454 ≈ –√0.00729735257 = –√α

As noted in Feynman’s (or Leighton’s) QED, older and/or more popular books will usually mention 1/α as the ‘magical’ number, so the ‘special’ number you may have seen is the inverse fine-structure constant, which is about 137, but not quite:

1/α = 137.035999074 ± 0.000000044

I am adding the standard uncertainty just to give you an idea of how precise these measurements are. 🙂 About 0.32 parts per billion (just divide the 137.035999074 number by the uncertainty). So that‘s the number that excites popular writers, including Leighton. Indeed, as Leighton puts it:

“Where does this number come from? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly!”

Is it Leighton, or did Feynman really say this? Not sure. While the fine-structure constant is a very special number, it’s not the only ‘special’ number. In fact, we derive it from other ‘magical’ numbers. To be specific, I’ll show you how we derive it from the fundamental properties – as measured, of course – of the electron. So, in fact, I should say that we do know how to make this number come out, which makes me doubt whether Feynman really said what Leighton said he said. 🙂

So we can derive α from some other numbers. That brings me to the more complicated answer to the question as to what the value of j really is: j‘s value is the electron charge expressed in Planck units, which I’ll denote by –eP:

j = –eP

[You may want to reflect on this, and quickly verify on the Web. The Planck unit of electric charge, expressed in Coulomb, is about 1.87555×10–18 C. If you multiply that j = –eP, so with –0.08542454, you get the right answer: the electron charge is about –0.160217×10–18 C.]

Now that is strange.

Why? Well… For starters, when doing all those quantum-mechanical calculations, we like to think of j as a dimensionless number: a coupling constant. But so here we do have a dimension: electric charge.

Let’s look at the basics. If is –√α, and it’s also equal to –eP, then the fine-structure constant must also be equal to the square of the electron charge eP, so we can write:

α = eP2

You’ll say: yes, so what? Well… I am pretty sure that, if you’ve ever seen a formula for α, it’s surely not this simple j = –eP or α = eP2 formula. What you’ve seen, most likely, is one or more of the following expressions below :

Fine-structure constant formula

That’s a pretty impressive collection of physical constants, isn’t it? 🙂 They’re all different but, somehow, when we combine them in one or the other ratio (we have not less than five different expressions here (each identity is a separate expression), and I could give you a few more!), we get the very same number: α. Now that is what I call strange. Truly strange. Incomprehensibly weird!

You’ll say… Well… Those constants must all be related… Of course! That’s exactly the point I am making here. They are, but look how different they are: mmeasures mass, rmeasures distance, e is a charge, and so these are all very different numbers with very different dimensions. Yet, somehow, they are all related through this α number. Frankly, I do not know of any other expression that better illustrates some kind of underlying unity in Nature than the one with those five identities above.

Let’s have a closer look at those constants. You know most of them already. The only constants you may not have seen before are μ0Rand, perhaps, ras well as m. However, these can easily be defined as some easy function of the constants that you did see before, so let me quickly do that:

  1. The μ0 constant is the so-called magnetic constant. It’s something similar as ε0 and it’s referred to as the magnetic permeability of the vacuum. So it’s just like the (electric) permittivity of the vacuum (i.e. the electric constant ε0) and the only reason why this blog hasn’t mentioned this constant before is because I haven’t really discussed magnetic fields so far. I only talked about the electric field vector. In any case, you know that the electric and magnetic force are part and parcel of the same phenomenon (i.e. the electromagnetic interaction between charged particles) and, hence, they are closely related. To be precise, μ0ε0 = 1/c= c–2. So that shows the first and second expression for α are, effectively, fully equivalent. [Just in case you’d doubt that μ0ε0 = 1/c2, let me give you the values: μ0 = 4π·10–7 N/A2, and ε0 = (1/4π·c2)·10C2/N·m2. Just plug them in, and you’ll see it’s bang on. Moreover, note that the ampere (A) unit is equal to the coulomb per second unit (C/s), so even the units come out alright. 🙂 Of course they do!]
  2. The ke constant is the Coulomb constant and, from its definition ke = 1/4πε0, it’s easy to see how those two expressions are, in turn, equivalent with the third expression for α.
  3. The Rconstant is the so-called von Klitzing constant. Huh? Yes. I know. I am pretty sure you’ve never ever heard of that one before. Don’t worry about it. It’s, quite simply, equal to Rh/e2. Hence, substituting (and don’t forget that h = 2πħ) will demonstrate the equivalence of the fourth expression for α.
  4. Finally, the re factor is the classical electron radius, which is usually written as a function of me, i.e. the electron mass: re = e2/4πε0mec2. Also note that this also implies that reme = e2/4πε0c2. In words: the product of the electron mass and the electron radius is equal to some constant involving the electron (e), the electric constant (ε0), and c (the speed of light).

I am sure you’re under some kind of ‘formula shock’ now. But you should just take a deep breath and read on. The point to note is that all these very different things are all related through α.

So, again, what is that α really? Well… A strange number indeed. It’s dimensionless (so we don’t measure in kg, m/s, eV·s or whatever) and it pops up everywhere. [Of course, you’ll say: “What’s everywhere? This is the first time I‘ve heard of it!” :-)]

Well… Let me start by explaining the term itself. The fine structure in the name refers to the splitting of the spectral lines of atoms. That’s a very fine structure indeed. 🙂 We also have a so-called hyperfine structure. Both are illustrated below for the hydrogen atom. The numbers n, JI, and are quantum numbers used in the quantum-mechanical explanation of the emission spectrum, which is  also depicted below, but note that the illustration gives you the so-called Balmer series only, i.e. the colors in the visible light spectrum (there are many more ‘colors’ in the high-energy ultraviolet and the low-energy infrared range).

Fine_hyperfine_levels

Prism_5902760665342950662

To be precise: (1) n is the principal quantum number: here it takes the values 1 or 2, and we could say these are the principal shells; (2) the S, P, D,… orbitals (which are usually written in lower case: s, p, d, f, g, h and i) correspond to the (orbital) angular momentum quantum number l = 0, 1, 2,…, so we could say it’s the subshell; (3) the J values correspond to the so-called magnetic quantum number m, which goes from –l to +l; (4) the fourth quantum number is the spin angular momentum s. I’ve copied another diagram below so you see how it works, more or less, that is.

hydrogen spectrum

Now, our fine-structure constant is related to these quantum numbers. How exactly is a bit of a long story, and so I’ll just copy Wikipedia’s summary on this: ” The gross structure of line spectra is the line spectra predicted by the quantum mechanics of non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy levels only depend on the principal quantum number n. However, a more accurate model takes into account relativistic and spin effects, which break the degeneracy of the the energy levels and split the spectral lines. The scale of the fine structure splitting relative to the gross structure energies is on the order of ()2, where Z is the atomic number and α is the fine-structure constant.” There you go. You’ll say: so what? Well… Nothing. If you aren’t amazed by that, you should stop reading this.

It is an ‘amazing’ number, indeed, and, hence, it does quality for being “one of the greatest damn mysteries of physics”, as Feynman and/or Leighton put it. Having said that, I would not go as far as to write that it’s “a magic number that comes to us with no understanding by man.” In fact, I think Feynman/Leighton could have done a much better job when explaining what it’s all about. So, yes, I hope to do better than Leighton here and, as he’s still alive, I actually hope he reads this. 🙂

The point is: α is not the only weird number. What’s particular about it, as a physical constant, is that it’s dimensionless, because it relates a number of other physical constants in such a way that the units fall away. Having said that, the Planck or Boltzmann constant are at least as weird.

So… What is this all about? Well… You’ve probably heard about the so-called fine-tuning problem in physics and, if you’re like me, your first reaction will be to associate fine-tuning with fine-structure. However, the two terms have nothing in common, except for four letters. 🙂 OK. Well… I am exaggerating here. The two terms are actually related, to some extent at least, but let me explain how.

The term fine-tuning refers to the fact that all the parameters or constants in the so-called Standard Model of physics are, indeed, all related to each other in the way they are. We can’t sort of just turn the knob of one and change it, because everything falls apart then. So, in essence, the fine-tuning problem in physics is more like a philosophical question: why is the value of all these physical constants and parameters exactly what it is? So it’s like asking: could we change some of the ‘constants’ and still end up with the world we’re living in? Or, if it would be some different world, how would it look like? What if was some other number? What if ke or ε0 was some other number? In short, and in light of those expressions for α, we may rephrase the question as: why is α what is is?

Of course, that’s a question one shouldn’t try to answer before answering some other, more fundamental, question: how many degrees of freedom are there really? Indeed, we just saw that ke and εare intimately related through some equation, and other constants and parameters are related too. So the question is like: what are the ‘dependent’ and the ‘independent’ variables in this so-called Standard Model?

There is no easy answer to that question. In fact, one of the reasons why I find physics so fascinating is that one cannot easily answer such questions. There are the obvious relationships, of course. For example, the ke = 1/4πεrelationship, and the context in which they are used (Coulomb’s Law) does, indeed, strongly suggest that both constants are actually part and parcel of the same thing. Identical, I’d say. Likewise, the μ0ε0 = 1/crelation also suggests there’s only one degree of freedom here, just like there’s only one degree of freedom in that ω/k = relationship (if we set a value for ω, we have k, and vice versa). But… Well… I am not quite sure how to phrase this, but… What physical constants could be ‘variables’ indeed?

It’s pretty obvious that the various formulas for α cannot answer that question: you could stare at them for days and weeks and months and years really, but I’d suggest you use your time to read more of Feynman’s real Lectures instead. 🙂 One point that may help to come to terms with this question – to some extent, at least – is what I casually mentioned above already: the fine-structure constant is equal to the square of the electron charge expressed in Planck units: α = eP2.

Now, that’s very remarkable because Planck units are some kind of ‘natural units’ indeed (for the detail, see my previous post: among other things, it explains what these Planck units really are) and, therefore, it is quite tempting to think that we’ve actually got only one degree of freedom here: α itself. All the rest should follow from it.

[…]

It should… But… Does it?

The answer is: yes and no. To be frank, it’s more no than yes because, as I noted a couple of times already, the fine-structure constant relates a lot of stuff but it’s surely not the only significant number in the Universe. For starters, I said that our E(A to B) formula has two ‘variables’:

  1. We have that complex number j, which, as mentioned, is equal to the electron charge expressed in Planck units. [In case you wonder why –eP ≈ –0.08542455 is said to be an amplitude, i.e. a complex number or an ‘arrow’… Well… Complex numbers include the real numbers and, hence, –0.08542455 is both real and complex. When combining ‘arrows’ or, to be precise, when multiplying some complex number with –0.08542455, we will (a) shrink the original arrow to about 8.5% of its original value (8.542455% to be precise) and (b) rotate it over an angle of plus or minus 180 degrees. In other words, we’ll reverse its direction. Hence, using Euler’s notation for complex numbers, we can write: –1 = eiπ eiπ and, hence, –0.085 = 0.085·eiπ = 0.085·eiπ. So, in short, yes, j is a complex number, or an ‘arrow’, if you prefer that term.]
  2. We also have some some real number n in the E(A to B) formula. So what’s the n? Well… Believe it or not, it’s the electron mass! Isn’t that amazing?

You’ll say: “Well… Hmm… I suppose so.” But then you may – and actually should – also wonder: the electron mass? In what units? Planck units again? And are we talking relativistic mass (i.e. its total mass, including the equivalent mass of its kinetic energy) or its rest mass only? And we were talking α here, so can we relate it to α too, just like the electron charge?

These are all very good questions. Let’s start with the second one. We’re talking rather slow-moving electrons here, so the relativistic mass (m) and its rest mass (m0) is more or less the same. Indeed, the Lorentz factor γ in the m = γm0 equation is very close to 1 for electrons moving at their typical speed. So… Well… That question doesn’t matter very much. Really? Yes. OK. Because you’re doubting, I’ll quickly show it to you. What is their ‘typical’ speed?

We know we shouldn’t attach too much importance to the concept of an electron in orbit around some nucleus (we know it’s not like some planet orbiting around some star) and, hence, to the concept of speed or velocity (velocity is speed with direction) when discussing an electron in an atom. The concept of momentum (i.e. velocity combined with mass or energy) is much more relevant. There’s a very easy mathematical relationship that gives us some clue here: the Uncertainty Principle. In fact, we’ll use the Uncertainty Principle to relate the momentum of an electron (p) to the so-called Bohr radius r (think of it as the size of a hydrogen atom) as follows: p ≈ ħ/r. [I’ll come back on this in a moment, and show you why this makes sense.]

Now we also know its kinetic energy (K.E.) is mv2/2, which we can write as p2/2m. Substituting our p ≈ ħ/r conjecture, we get K.E. = mv2/2 = ħ2/2mr2. This is equivalent to m2v2 = ħ2/r(just multiply both sides with m). From that, we get v = ħ/mr. Now, one of the many relations we can derive from the formulas for the fine-structure constant is re = α2r. [I haven’t showed you that yet, but I will shortly. It’s a really amazing expression. However, as for now, just accept it as a simple formula for interim use in this digression.] Hence, r = re2. The rfactor in this expression is the so-called classical electron radius. So we can now write v = ħα2/mre. Let’s now throw c in: v/c = α2ħ/mcre. However, from that fifth expression for α, we know that ħ/mcre = α, so we get v/c = α. We have another amazing result here: the v/c ratio for an electron (i.e. its speed expressed as a fraction of the speed of light) is equal to that fine-structure constant α. So that’s about 1/137, so that’s less than 1% of the speed of light. Now… I’ll leave it to you to calculate the Lorentz factor γ but… Well… It’s obvious that it will be very close to 1. 🙂 Hence, the electron’s speed – however we want to visualize that – doesn’t matter much indeed, so we should not worry about relativistic corrections in the formulas.

Let’s now look at the question in regard to the Planck units. If you know nothing at all about them, I would advise you to read what I wrote about them in my previous post. Let me just note we get those Planck units by equating not less than five fundamental physical constants to 1, notably (1) the speed of light, (2) Planck’s (reduced) constant, (3) Boltzmann’s constant, (4) Coulomb’s constant and (5) Newton’s constant (i.e. the gravitational constant). Hence, we have a set of five equations here (ħ = kB = ke = G = 1), and so we can solve that to get the five Planck units, i.e. the Planck length unit, the Planck time unit, the Planck mass unit, the Planck energy unit, the Planck charge unit and, finally (oft forgotten), the Planck temperature unit. Of course, you should note that all mass and energy units are directly related because of the mass-energy equivalence relation E = mc2, which simplifies to E = m if c is equated to 1. [I could also say something about the relation between temperature and (kinetic) energy, but I won’t, as it would only further confuse you.]

Now, you may or may not remember that the Planck time and length units are unimaginably small, but that the Planck mass unit is actually quite sizable—at the atomic scale, that is. Indeed, the Planck mass is something huge, like the mass of an eyebrow hair, or a flea egg. Is that huge? Yes. Because if you’d want to pack it in a Planck-sized particle, it would make for a tiny black hole. 🙂 No kidding. That’s the physical significance of the Planck mass and the Planck length and, yes, it’s weird. 🙂

Let me give you some values. First, the Planck mass itself: it’s about 2.1765×10−8 kg. Again, if you think that’s tiny, think again. From the E = mc2 equivalence relationship, we get that this is equivalent to 2 giga-joule, approximately. Just to give an idea, that’s like the monthly electricity consumption of an average American family. So that’s huge indeed! 🙂 [Many people think that nuclear energy involves the conversion of mass into energy, but the story is actually more complicated than that. In any case… I need to move on.]

Let me now give you the electron mass expressed in the Planck mass unit:

  1. Measured in our old-fashioned super-sized SI kilogram unit, the electron mass is me = 9.1×10–31 kg.
  2. The Planck mass is mP = 2.1765×10−8 kg.
  3. Hence, the electron mass expressed in Planck units is meP = me/mP = (9.1×10–31 kg)/(2.1765×10−8 kg) = 4.181×10−23.

We can, once again, write that as some function of the fine-structure constant. More specifically, we can write:

meP = α/reP = α/α2rP  = 1/αrP

So… Well… Yes: yet another amazing formula involving α.

In this formula, we have reP and rP, which are the (classical) electron radius and the Bohr radius expressed in Planck (length) units respectively. So you can see what’s going on here: we have all kinds of numbers here expressed in Planck units: a charge, a radius, a mass,… And we can relate all of them to the fine-structure constant

Why? Who knows? I don’t. As Leighton puts it: that’s just the way “God pushed His pencil.” 🙂

Note that the beauty of natural units ensures that we get the same number for the (equivalent) energy of an electron. Indeed, from the E = mc2 relation, we know the mass of an electron can also be written as 0.511 MeV/c2. Hence, the equivalent energy is 0.511 MeV (so that’s, quite simply, the same number but without the 1/cfactor). Now, the Planck energy EP (in eV) is 1.22×1028 eV, so we get EeP = Ee/EP = (0.511×10eV)/(1.22×1028 eV) = 4.181×10−23. So it’s exactly the same as the electron mass expressed in Planck units. Isn’t that nice? 🙂

Now, are all these numbers dimensionless, just like α? The answer to that question is complicated. Yes, and… Well… No:

  1. Yes. They’re dimensionless because they measure something in natural units, i.e. Planck units, and, hence, that’s some kind of relative measure indeed so… Well… Yes, dimensionless.
  2. No. They’re not dimensionless because they do measure something, like a charge, a length, or a mass, and when you chose some kind of relative measure, you still need to define some gauge, i.e. some kind of standard measure. So there’s some ‘dimension’ involved there.

So what’s the final answer? Well… The Planck units are not dimensionless. All we can say is that they are closely related, physically. I should also add that we’ll use the electron charge and mass (expressed in Planck units) in our amplitude calculations as a simple (dimensionless) number between zero and one. So the correct answer to the question as to whether these numbers have any dimension is: expressing some quantities in Planck units sort of normalizes them, so we can use them directly in dimensionless calculations, like when we multiply and add amplitudes.

Hmm… Well… I can imagine you’re not very happy with this answer but it’s the best I can do. Sorry. I’ll let you further ponder that question. I need to move on.  

Note that that 4.181×10−23 is still a very small number (23 zeroes after the decimal point!), even if it’s like 46 million times larger than the electron mass measured in our conventional SI unit (i.e. 9.1×10–31 kg). Does such small number make any sense? The answer is: yes, it does. When we’ll finally start discussing that E(A to B) formula (I’ll give it to you in a moment), you’ll see that a very small number for n makes a lot of sense.

Before diving into it all, let’s first see if that formula for that alpha, that fine-structure constant, still makes sense with me expressed in Planck units. Just to make sure. 🙂 To do that, we need to use the fifth (last) expression for a, i.e. the one with re in it. Now, in my previous post, I also gave some formula for re: re = e2/4πε0mec2, which we can re-write as reme = e2/4πε0c2. If we substitute that expression for reme  in the formula for α, we can calculate α from the electron charge, which indicates both the electron radius and its mass are not some random God-given variable, or “some magic number that comes to us with no understanding by man“, as Feynman – well… Leighton, I guess – puts it. No. They are magic numbers alright, one related to another through the equally ‘magic’ number α, but so I do feel we actually can create some understanding here.

At this point, I’ll digress once again, and insert some quick back-of-the-envelope argument from Feynman’s very serious Caltech Lectures on Physics, in which, as part of the introduction to quantum mechanics, he calculates the so-called Bohr radius from Planck’s constant h. Let me quickly explain: the Bohr radius is, roughly speaking, the size of the simplest atom, i.e. an atom with one electron (so that’s hydrogen really). So it’s not the classical electron radius re. However, both are also related to that ‘magical number’ α. To be precise, if we write the Bohr radius as r, then re = α2r ≈ 0.000053… times r, which we can re-write as:

α = √(re /r) = (re /r)1/2

So that’s yet another amazing formula involving the fine-structure constant. In fact, it’s the formula I used as an ‘interim’ expression to calculate the relative speed of electrons. I just used it without any explanation there, but I am coming back to it here. Alpha again…

Just think about it for a while. In case you’d still doubt the magic of that number, let me write what we’ve discovered so far:

(1) α is the square of the electron charge expressed in Planck units: α = eP2.

(2) α is the square root of the ratio of (a) the classical electron radius and (b) the Bohr radius: α = √(re /r). You’ll see this more often written as re = α2r. Also note that this is an equation that does not depend on the units, in contrast to equation 1 (above), and 4 and 5 (below), which require you to switch to Planck units. It’s the square of a ratio and, hence, the units don’t matter. They fall away.

(3) α is the (relative) speed of an electron: α = v/c. [The relative speed is the speed as measured against the speed of light. Note that the ‘natural’ unit of speed in the Planck system of units is equal to c. Indeed, if you divide one Planck length by one Planck time unit, you get (1.616×10−35 m)/(5.391×10−44 s) = m/s. However, this is another equation, just like (2), that does not depend on the units: we can express v and c in whatever unit we want, as long we’re consistent and express both in the same units.]

(4) Finally – I’ll show you in a moment – α is also equal to the product of (a) the electron mass (which I’ll simply write as me here) and (b) the classical electron radius re (if both are expressed in Planck units): α = me·re. Now think that’s, perhaps, the most amazing of all of the expressions for α. If you don’t think that’s amazing, I’d really suggest you stop trying to study physics. 🙂

Note that, from (2) and (4), we find that:

(5) The electron mass (in Planck units) is equal me = α/r= α/α2r = 1/αr. So that gives us an expression, using α once again, for the electron mass as a function of the Bohr radius r expressed in Planck units.

Finally, we can also substitute (1) in (5) to get:

(6) The electron mass (in Planck units) is equal to me = α/r = eP2/re. Using the Bohr radius, we get me = 1/αr = 1/eP2r.

So… As you can see, this fine-structure constant really links ALL of the fundamental properties of the electron: its charge, its radius, its distance to the nucleus (i.e. the Bohr radius), its velocity, its mass (and, hence, its energy),… In short,

IT IS ALL IN ALPHA!

Now that should answer the question in regard to the degrees of freedom we have here, doesn’t it? It looks like we’ve got only one degree of freedom here. Indeed, if we’ve got some value for α, then we’ve have the electron charge, and from the electron charge, we can calculate the Bohr radius r (as I will show below), and if we have r, we have mand re. And then we can also calculate v, which gives us its momentum (mv) and its kinetic energy (mv2/2). In short,

ALPHA GIVES US EVERYTHING!

Isn’t that amazing? Hmm… You should reserve your judgment as for now, and carefully go over all of the formulas above and verify my statement. If you do that, you’ll probably struggle to find the Bohr radius from the charge (i.e. from α). So let me show you how you do that, because it will also show you why you should, indeed, reserve your judgment. In other words, I’ll show you why alpha does NOT give us everything! The argument below will, finally, prove some of the formulas that I didn’t prove above. Let’s go for it:

1. If we assume that (a) an electron takes some space – which I’ll denote by r 🙂 – and (b) that it has some momentum p because of its mass m and its velocity v, then the ΔxΔp = ħ relation (i.e. the Uncertainty Principle in its roughest form) suggests that the order of magnitude of r and p should be related in the very same way. Hence, let’s just boldly write r ≈ ħ/p and see what we can do with that. So we equate Δx with r and Δp with p. As Feynman notes, this is really more like a ‘dimensional analysis’ (he obviously means something very ‘rough’ with that) and so we don’t care about factors like 2 or 1/2. [Indeed, note that the more precise formulation of the Uncertainty Principle is σxσ≥ ħ/2.] In fact, we didn’t even bother to define r very rigorously. We just don’t care about precise statements at this point. We’re only concerned about orders of magnitude. [If you’re appalled by the rather rude approach, I am sorry for that, but just try to go along with it.]

2. From our discussions on energy, we know that the kinetic energy is mv2/2, which we can write as p2/2m so we get rid of the velocity factor. [Why? Because we can’t really imagine what it is anyway. As I said a couple of times already, we shouldn’t think of electrons as planets orbiting around some star. That model doesn’t work.] So… What’s next? Well… Substituting our p ≈ ħ/r conjecture, we get K.E. = ħ2/2mr2. So that’s a formula for the kinetic energy. Next is potential.

3. Unfortunately, the discussion on potential energy is a bit more complicated. You’ll probably remember that we had an easy and very comprehensible formula for the energy that’s needed (i.e. the work that needs to be done) to bring two charges together from a large distance (i.e. infinity). Indeed, we derived that formula directly from Coulomb’s Law (and Newton’s law of force) and it’s U = q1q2/4πε0r12. [If you think I am going too fast, sorry, please check for yourself by reading my other posts.] Now, we’re actually talking about the size of an atom here in my previous post, so one charge is the proton (+e) and the other is the electron (–e), so the potential energy is U = P.E. = –e2/4πε0r, with r the ‘distance’ between the proton and the electron—so that’s the Bohr radius we’re looking for!

[In case you’re struggling a bit with those minus signs when talking potential energy  – I am not ashamed to admit I did! – let me quickly help you here. It has to do with our reference point: the reference point for measuring potential energy is at infinity, and it’s zero there (that’s just our convention). Now, to separate the proton and the electron, we’d have to do quite a lot of work. To use an analogy: imagine we’re somewhere deep down in a cave, and we have to climb back to the zero level. You’ll agree that’s likely to involve some sweat, don’t you? Hence, the potential energy associated with us being down in the cave is negative. Likewise, if we write the potential energy between the proton and the electron as U(r), and the potential energy at the reference point as U(∞) = 0, then the work to be done to separate the charges, i.e. the potential difference U(∞) – U(r), will be positive. So U(∞) – U(r) = 0 – U(r) > 0 and, hence, U(r) < 0. If you still don’t ‘get’ this, think of the electron being in some (potential) well, i.e. below the zero level, and so it’s potential energy is less than zero. Huh? Sorry. I have to move on. :-)]

4. We can now write the total energy (which I’ll denote by E, but don’t confuse it with the electric field vector!) as

E = K.E. + P.E. =  ħ2/2mr– e2/4πε0r

Now, the electron (whatever it is) is, obviously, in some kind of equilibrium state. Why is that obvious? Well… Otherwise our hydrogen atom wouldn’t or couldn’t exist. 🙂 Hence, it’s in some kind of energy ‘well’ indeed, at the bottom. Such equilibrium point ‘at the bottom’ is characterized by its derivative (in respect to whatever variable) being equal to zero. Now, the only ‘variable’ here is r (all the other symbols are physical constants), so we have to solve for dE/dr = 0. Writing it all out yields:

dE/dr = –ħ2/mr+ e2/4πε0r= 0 ⇔ r = 4πε0ħ2/me2

You’ll say: so what? Well… We’ve got a nice formula for the Bohr radius here, and we got it in no time! 🙂 But the analysis was rough, so let’s check if it’s any good by putting the values in:

r = 4πε0h2/me2

= [(1/(9×109) C2/N·m2)·(1.055×10–34 J·s)2]/[(9.1×10–31 kg)·(1.6×10–19 C)2]

= 53×10–12 m = 53 pico-meter (pm)

So what? Well… Double-check it on the Internet: the Bohr radius is, effectively, about 53 trillionths of a meter indeed! So we’re right on the spot! 

[In case you wonder about the units, note that mass is a measure of inertia: one kg is the mass of an object which, subject to a force of 1 newton, will accelerate at the rate of 1 m/s per second. Hence, we write F = m·a, which is equivalent to m = F/a. Hence, the kg, as a unit, is equivalent to 1 N/(m/s2). If you make this substitution, we get r in the unit we want to see: [(C2/N·m2)·(N2·m2·s2)/[(N·s2/m)·C2] = m.]

Moreover, if we take that value for r and put it in the (total) energy formula above, we’d find that the energy of the electron is –13.6 eV. [Don’t forget to convert from joule to electronvolt when doing the calculation!] Now you can check that on the Internet too: 13.6 eV is exactly the amount of energy that’s needed to ionize a hydrogen atom (i.e. the energy that’s needed to kick the electron out of that energy well)!

Waw ! Isn’t it great that such simple calculations yield such great results? 🙂 [Of course, you’ll note that the omission of the 1/2 factor in the Uncertainty Principle was quite strategic. :-)] Using the r = 4πε0ħ2/meformula for the Bohr radius, you can now easily check the re = α2r formula. You should find what we jotted down already: the classical electron radius is equal to re = e2/4πε0mec2. To be precise, re = (53×10–6)·(53×10–12m) = 2.8×10–15 m. Now that’s again something you should check on the Internet. Guess what? […] It’s right on the spot again. 🙂

We can now also check that α = m·re formula: α = m·r= 4.181×10−23 times… Hey! Wait! We have to express re in Planck units as well, of course! Now, (2.81794×10–15 m)/(1.616×10–35 m) ≈ 1.7438 ×1020. So now we get 4.181×10−23 times 1.7438×1020 = 7.29×10–3 = 0.00729 ≈ 1/137. Bingo! We got the magic number once again. 🙂

So… Well… Doesn’t that confirm we actually do have it all with α?

Well… Yes and no… First, you should note that I had to use h in that calculation of the Bohr radius. Moreover, the other physical constants (most notably c and the Coulomb constant) were actually there as well, ‘in the background’ so to speak, because one needs them to derive the formulas we used above. And then we have the equations themselves, of course, most notably that Uncertainty Principle… So… Well…

It’s not like God gave us one number only (α) and that all the rest flows out of it. We have a whole bunch of ‘fundamental’ relations and ‘fundamental’ constants here.

Having said that, it’s true that statement still does not diminish the magic of alpha.

Hmm… Now you’ll wonder: how many? How many constants do we need in all of physics?

Well… I’d say, you should not only ask about the constants: you should also ask about the equations: how many equations do we need in all of physics? [Just for the record, I had to smile when the Hawking of the movie says that he’s actually looking for one formula that sums up all of physics. Frankly, that’s a nonsensical statement. Hence, I think the real Hawking never said anything like that. Or, if he did, that it was one of those statements one needs to interpret very carefully.]

But let’s look at a few constants indeed. For example, if we have c, h and α, then we can calculate the electric charge e and, hence, the electric constant ε= e2/2αhc. From that, we get Coulomb’s constant ke, because ke is defined as 1/4πε0… But…

Hey! Wait a minute! How do we know that ke = 1/4πε0? Well… From experiment. But… Yes? That means 1/4π is some fundamental proportionality coefficient too, isn’t it?

Wow! You’re smart. That’s a good and valid remark. In fact, we use the so-called reduced Planck constant ħ in a number of calculations, and so that involves a 2π factor too (ħ = h/2π). Hence… Well… Yes, perhaps we should consider 2π as some fundamental constant too! And, then, well… Now that I think of it, there’s a few other mathematical constants out there, like Euler’s number e, for example, which we use in complex exponentials.

?!?

I am joking, right? I am not saying that 2π and Euler’s number are fundamental ‘physical’ constants, am I? [Note that it’s a bit of a nuisance we’re also using the symbol for Euler’s number, but so we’re not talking the electron charge here: we’re talking that 2.71828…etc number that’s used in so-called ‘natural’ exponentials and logarithms.]

Well… Yes and no. They’re mathematical constants indeed, rather than physical, but… Well… I hope you get my point. What I want to show here, is that it’s quite hard to say what’s fundamental and what isn’t. We can actually pick and choose a bit among all those constants and all those equations. As one physicist puts its: it depends on how we slice it. The one thing we know for sure is that a great many things are related, in a physical way (α connects all of the fundamental properties of the electron, for example) and/or in a mathematical way (2π connects not only the circumference of the unit circle with the radius but quite a few other constants as well!), but… Well… What to say? It’s a tough discussion and I am not smart enough to give you an unambiguous answer. From what I gather on the Internet, when looking at the whole Standard Model (including the strong force, the weak force and the Higgs field), we’ve got a few dozen physical ‘fundamental’ constants, and then a few mathematical ones as well.

That’s a lot, you’ll say. Yes. At the same time, it’s not an awful lot. Whatever number it is, it does raise a very fundamental question: why are they what they are? That brings us back to that ‘fine-tuning’ problem. Now, I can’t make this post too long (it’s way too long already), so let me just conclude this discussion by copying Wikipedia on that question, because what it has on this topic is not so bad:

“Some physicists have explored the notion that if the physical constants had sufficiently different values, our Universe would be so radically different that intelligent life would probably not have emerged, and that our Universe therefore seems to be fine-tuned for intelligent life. The anthropic principle states a logical truism: the fact of our existence as intelligent beings who can measure physical constants requires those constants to be such that beings like us can exist.

I like this. But the article then adds the following, which I do not like so much, because I think it’s a bit too ‘frivolous’:

“There are a variety of interpretations of the constants’ values, including that of a divine creator (the apparent fine-tuning is actual and intentional), or that ours is one universe of many in a multiverse (e.g. the many-worlds interpretation of quantum mechanics), or even that, if information is an innate property of the universe and logically inseparable from consciousness, a universe without the capacity for conscious beings cannot exist.”

Hmm… As said, I am quite happy with the logical truism: we are there because alpha (and a whole range of other stuff) is what it is, and we can measure alpha (and a whole range of other stuff) as what it is, because… Well… Because we’re here. Full stop. As for the ‘interpretations’, I’ll let you think about that for yourself. 🙂

I need to get back to the lesson. Indeed, this was just a ‘digression’. My post was about the three fundamental events or actions in quantum electrodynamics, and so I was talking about that E(A to B) formula. However, I had to do that digression on alpha to ensure you understand what I want to write about that. So let me now get back to it. End of digression. 🙂

The E(A to B) formula

Indeed, I must assume that, with all these digressions, you are truly despairing now. Don’t. We’re there! We’re finally ready for the E(A to B) formula! Let’s go for it.

We’ve now got those two numbers measuring the electron charge and the electron mass in Planck units respectively. They’re fundamental indeed and so let’s loosen up on notation and just write them as e and m respectively. Let me recap:

1. The value of e is approximately –0.08542455, and it corresponds to the so-called junction number j, which is the amplitude for an electron-photon coupling. When multiplying it with another amplitude (to find the amplitude for an event consisting of two sub-events, for example), it corresponds to a ‘shrink’ to less than one-tenth (something like 8.5% indeed, corresponding to the magnitude of e) and a ‘rotation’ (or a ‘turn’) over 180 degrees, as mentioned above.

Please note what’s going on here: we have a physical quantity, the electron charge (expressed in Planck units), and we use it in a quantum-mechanical calculation as a dimensionless (complex) number, i.e. as an amplitude. So… Well… That’s what physicists mean when they say that the charge of some particle (usually the electric charge but, in quantum chromodynamics, it will be the ‘color’ charge of a quark) is a ‘coupling constant’.

2. We also have m, the electron mass, and we’ll use in the same way, i.e. as some dimensionless amplitude. As compared to j, it’s is a very tiny number: approximately 4.181×10−23. So if you look at it as an amplitude, indeed, then it corresponds to an enormous ‘shrink’ (but no turn) of the amplitude(s) that we’ll be combining it with.

So… Well… How do we do it?

Well… At this point, Leighton goes a bit off-track. Just a little bit. 🙂 From what he writes, it’s obvious that he assumes the frequency (or, what amounts to the same, the de Broglie wavelength) of an electron is just like the frequency of a photon. Frankly, I just can’t imagine why and how Feynman let this happen. It’s wrong. Plain wrong. As I mentioned in my introduction already, an electron traveling through space is not like a photon traveling through space.

For starters, an electron is much slower (because it’s a matter-particle: hence, it’s got mass). Secondly, the de Broglie wavelength and/or frequency of an electron is not like that of a photon. For example, if we take an electron and a photon having the same energy, let’s say 1 eV (that corresponds to infrared light), then the de Broglie wavelength of the electron will be 1.23 nano-meter (i.e. 1.23 billionths of a meter). Now that’s about one thousand times smaller than the wavelength of our 1 eV photon, which is about 1240 nm. You’ll say: how is that possible? If they have the same energy, then the f = E/h and ν = E/h should give the same frequency and, hence, the same wavelength, no?

Well… No! Not at all! Because an electron, unlike the photon, has a rest mass indeed – measured as not less than 0.511 MeV/c2, to be precise (note the rather particular MeV/c2 unit: it’s from the E = mc2 formula) – one should use a different energy value! Indeed, we should include the rest mass energy, which is 0.511 MeV. So, almost all of the energy here is rest mass energy! There’s also another complication. For the photon, there is an easy relationship between the wavelength and the frequency: it has no mass and, hence, all its energy is kinetic, or movement so to say, and so we can use that ν = E/h relationship to calculate its frequency ν: it’s equal to ν = E/h = (1 eV)/(4.13567×10–15 eV·s) ≈ 0.242×1015 Hz = 242 tera-hertz (1 THz = 1012 oscillations per second). Now, knowing that light travels at the speed of light, we can check the result by calculating the wavelength using the λ = c/ν relation. Let’s do it: (2.998×10m/s)/(242×1012 Hz) ≈ 1240 nm. So… Yes, done!

But so we’re talking photons here. For the electron, the story is much more complicated. That wavelength I mentioned was calculated using the other of the two de Broglie relations: λ = h/p. So that uses the momentum of the electron which, as you know, is the product of its mass (m) and its velocity (v): p = mv. You can amuse yourself and check if you find the same wavelength (1.23 nm): you should! From the other de Broglie relation, f = E/h, you can also calculate its frequency: for an electron moving at non-relativistic speeds, it’s about 0.123×1021 Hz, so that’s like 500,000 times the frequency of the photon we we’re looking at! When multiplying the frequency and the wavelength, we should get its speed. However, that’s where we get in trouble. Here’s the problem with matter waves: they have a so-called group velocity and a so-called phase velocity. The idea is illustrated below: the green dot travels with the wave packet – and, hence, its velocity corresponds to the group velocity – while the red dot travels with the oscillation itself, and so that’s the phase velocity. [You should also remember, of course, that the matter wave is some complex-valued wavefunction, so we have both a real as well as an imaginary part oscillating and traveling through space.]

Wave_group (1)

To be precise, the phase velocity will be superluminal. Indeed, using the usual relativistic formula, we can write that p = γm0v and E = γm0c2, with v the (classical) velocity of the electron and what it always is, i.e. the speed of light. Hence, λ = h/γm0v and = γm0c2/h, and so λf = c2/v. Because v is (much) smaller than c, we get a superluminal velocity. However, that’s the phase velocity indeed, not the group velocity, which corresponds to v. OK… I need to end this digression.

So what? Well, to make a long story short, the ‘amplitude framework’ for electrons is differerent. Hence, the story that I’ll be telling here is different from what you’ll read in Feynman’s QED. I will use his drawings, though, and his concepts. Indeed, despite my misgivings above, the conceptual framework is sound, and so the corrections to be made are relatively minor.

So… We’re looking at E(A to B), i.e. the amplitude for an electron to go from point A to B in spacetime, and I said the conceptual framework is exactly the same as that for a photon. Hence, the electron can follow any path really. It may go in a straight line and travel at a speed that’s consistent with what we know of its momentum (p), but it may also follow other paths. So, just like the photon, we’ll have some so-called propagator function, which gives you amplitudes based on the distance in space as well as in the distance in ‘time’ between two points. Now, Ralph Leighton identifies that propagator function with the propagator function for the photon, i.e. P(A to B), but that’s wrong: it’s not the same.

The propagator function for an electron depends on its mass and its velocity, and/or on the combination of both (like it momentum p = mv and/or its kinetic energy: K.E. = mv2 = p2/2m). So we have a different propagator function here. However, I’ll use the same symbol for it: P(A to B).

So, the bottom line is that, because of the electron’s mass (which, remember, is a measure for inertia), momentum and/or kinetic energy (which, remember, are conserved in physics), the straight line is definitely the most likely path, but (big but!), just like the photon, the electron may follow some other path as well.

So how do we formalize that? Let’s first associate an amplitude P(A to B) with an electron traveling from point A to B in a straight line and in a time that’s consistent with its velocity. Now, as mentioned above, the P here stands for propagator function, not for photon, so we’re talking a different P(A to B) here than that P(A to B) function we used for the photon. Sorry for the confusion. 🙂 The left-hand diagram below then shows what we’re talking about: it’s the so-called ‘one-hop flight’, and so that’s what the P(A to B) amplitude is associated with.

Diagram 1Now, the electron can follow other paths. For photons, we said the amplitude depended on the spacetime interval I: when negative or positive (i.e. paths that are not associated with the photon traveling in a straight line and/or at the speed of light), the contribution of those paths to the final amplitudes (or ‘final arrow’, as it was called) was smaller.

For an electron, we have something similar, but it’s modeled differently. We say the electron could take a ‘two-hop flight’ (via point C or C’), or a ‘three-hop flight’ (via D and E) from point A to B. Now, it makes sense that these paths should be associated with amplitudes that are much smaller. Now that’s where that n-factor comes in. We just put some real number n in the formula for the amplitude for an electron to go from A to B via C, which we write as:

P(A to C)∗n2∗P(C to B)

Note what’s going on here. We multiply two amplitudes, P(A to C) and P(C to B), which is OK, because that’s what the rules of quantum mechanics tell us: if an ‘event’ consists of two sub-events, we need to multiply the amplitudes (not the probabilities) in order to get the amplitude that’s associated with both sub-events happening. However, we add an extra factor: n2. Note that it must be some very small number because we have lots of alternative paths and, hence, they should not be very likely! So what’s the n? And why n2 instead of just n?

Well… Frankly, I don’t know. Ralph Leighton boldly equates n to the mass of the electron. Now, because he obviously means the mass expressed in Planck units, that’s the same as saying n is the electron’s energy (again, expressed in Planck’s ‘natural’ units), so n should be that number m = meP = EeP = 4.181×10−23. However, I couldn’t find any confirmation on the Internet, or elsewhere, of the suggested n = m identity, so I’ll assume n = m indeed, but… Well… Please check for yourself. It seems the answer is to be found in a mathematical theory that helps physicists to actually calculate j and n from experiment. It’s referred to as perturbation theory, and it’s the next thing on my study list. As for now, however, I can’t help you much. I can only note that the equation makes sense.

Of course, it does: inserting a tiny little number n, close to zero, ensures that those other amplitudes don’t contribute too much to the final ‘arrow’. And it also makes a lot of sense to associate it with the electron’s mass: if mass is a measure of inertia, then it should be some factor reducing the amplitude that’s associated with the electron following such crooked path. So let’s go along with it, and see what comes out of it.

A three-hop flight is even weirder and uses that n2 factor two times:

P(A to E)∗n2∗P(E to D)∗n2∗P(D to B)

So we have an (n2)= nfactor here, which is good, because two hops should be much less likely than one hop. So what do we get? Well… (4.181×10−23)≈ 305×10−92. Pretty tiny, huh? 🙂 Of course, any point in space is a potential hop for the electron’s flight from point A to B and, hence, there’s a lot of paths and a lot of amplitudes (or ‘arrows’ if you want), which, again, is consistent with a very tiny value for n indeed.

So, to make a long story short, E(A to B) will be a giant sum (i.e. some kind of integral indeed) of a lot of different ways an electron can go from point A to B. It will be a series of terms P(A to E) + P(A to C)∗n2∗P(C to B) + P(A to E)∗n2∗P(E to D)∗n2∗P(D to B) + … for all possible intermediate points C, D, E, and so on.

What about the j? The junction number of coupling constant. How does that show up in the E(A to B) formula? Well… Those alternative paths with hops here and there are actually the easiest bit of the whole calculation. Apart from taking some strange path, electrons can also emit and/or absorb photons during the trip. In fact, they’re doing that constantly actually. Indeed, the image of an electron ‘in orbit’ around the nucleus is that of an electron exchanging so-called ‘virtual’ photons constantly, as illustrated below. So our image of an electron absorbing and then emitting a photon (see the diagram on the right-hand side) is really like the tiny tip of a giant iceberg: most of what’s going on is underneath! So that’s where our junction number j comes in, i.e. the charge (e) of the electron.

So, when you hear that a coupling constant is actually equal to the charge, then this is what it means: you should just note it’s the charge expressed in Planck units. But it’s a deep connection, isn’t? When everything is said and done, a charge is something physical, but so here, in these amplitude calculations, it just shows up as some dimensionless negative number, used in multiplications and additions of amplitudes. Isn’t that remarkable?

d2 d3

The situation becomes even more complicated when more than one electron is involved. For example, two electrons can go in a straight line from point 1 and 2 to point 3 and 4 respectively, but there’s two ways in which this can happen, and they might exchange photons along the way, as shown below. If there’s two alternative ways in which one event can happen, you know we have to add amplitudes, rather than multiply them. Hence, the formula for E(A to B) becomes even more complicated.

D5d4

Moreover, a single electron may first emit and then absorb a photon itself, so there’s no need for other particles to be there to have lots of j factors in our calculation. In addition, that photon may briefly disintegrate into an electron and a positron, which then annihilate each other to again produce a photon: in case you wondered, that’s what those little loops in those diagrams depicting the exchange of virtual photons is supposed to represent. So, every single junction (i.e. every emission and/or absorption of a photon) involves a multiplication with that junction number j, so if there are two couplings involved, we have a j2 factor, and so that’s 0.085424552 = α ≈ 0.0073. Four couplings implies a factor of 0.085424554 ≈ 0.000053.

Just as an example, I copy two diagrams involving four, five or six couplings indeed. They all have some ‘incoming’ photon, because Feynman uses them to explain something else (the so-called magnetic moment of a photon), but it doesn’t matter: the same illustrations can serve multiple purposes.

d6 d7

Now, it’s obvious that the contributions of the alternatives with many couplings add almost nothing to the final amplitude – just like the ‘many-hop’ flights add almost nothing – but… Well… As tiny as these contributions are, they are all there, and so they all have to be accounted for. So… Yes. You can easily appreciate how messy it all gets, especially in light of the fact that there are so many points that can serve as a ‘hop’ or a ‘coupling’ point!

So… Well… Nothing. That’s it! I am done! I realize this has been another long and difficult story, but I hope you appreciated and that it shed some light on what’s really behind those simplified stories of what quantum mechanics is all about. It’s all weird and, admittedly, not so easy to understand, but I wouldn’t say an understanding is really beyond the reach of us, common mortals. 🙂

Post scriptum: When you’ve reached here, you may wonder: so where’s the final formula then for E(A to B)? Well… I have no easy formula for you. From what I wrote above, it should be obvious that we’re talking some really awful-looking integral and, because it’s so awful, I’ll let you find it yourself. 🙂

I should also note another reason why I am reluctant to identify n with m. The formulas in Feynman’s QED are definitely not the standard ones. The more standard formulations will use the gauge coupling parameter about which I talked already. I sort of discussed it, indirectly, in my first comments on Feynman’s QED, when I criticized some other part of the book, notably its explanation of the phenomenon of diffraction of light, which basically boiled down to: “When you try to squeeze light too much [by forcing it to go through a small hole], it refuses to cooperate and begins to spread out”, because “there are not enough arrows representing alternative paths.”

Now that raises a lot of questions, and very sensible ones, because that simplification is nonsensical. Not enough arrows? That statement doesn’t make sense. We can subdivide space in as many paths as we want, and probability amplitudes don’t take up any physical space. We can cut up space in smaller and smaller pieces (so we analyze more paths within the same space). The consequence – in terms of arrows – is that directions of our arrows won’t change but their length will be much and much smaller as we’re analyzing many more paths. That’s because of the normalization constraint. However, when adding them all up – a lot of very tiny ones, or a smaller bunch of bigger ones – we’ll still get the same ‘final’ arrow. That’s because the direction of those arrows depends on the length of the path, and the length of the path doesn’t change simply because we suddenly decide to use some other ‘gauge’.

Indeed, the real question is: what’s a ‘small’ hole? What’s ‘small’ and what’s ‘large’ in quantum electrodynamics? Now, I gave an intuitive answer to that question in that post of mine, but it’s much more accurate than Feynman’s, or Leighton’s. The answer to that question is: there’s some kind of natural ‘gauge’, and it’s related to the wavelength. So the wavelength of a photon, or an electron, in this case, comes with some kind of scale indeed. That’s why the fine-structure constant is often written in yet another form:

α = 2πree = rek

λe and kare the Compton wavelength and wavenumber of the electron (so kis not the Coulomb constant here). The Compton wavelength is the de Broglie wavelength of the electron. [You’ll find that Wikipedia defines it as “the wavelength that’s equivalent to the wavelength of a photon whose energy is the same as the rest-mass energy of the electron”, but that’s a very confusing definition, I think.]

The point to note is that the spatial dimension in both the analysis of photons as well as of matter waves, especially in regard to studying diffraction and/or interference phenomena, is related to the frequencies, wavelengths and/or wavenumbers of the wavefunctions involved. There’s a certain ‘gauge’ involved indeed, i.e. some measure that is relative, like the gauge pressure illustrated below. So that’s where that gauge parameter g comes in. And the fact that it’s yet another number that’s closely related to that fine-structure constant is… Well… Again… That alpha number is a very magic number indeed… 🙂

abs-gauge-press

Post scriptum (5 October 2015):

Much stuff is physics is quite ‘magical’, but it’s never ‘too magical’. I mean: there’s always an explanation. So there is a very logical explanation for the above-mentioned deep connection between the charge of an electron, its energy and/or mass, its various radii (or physical dimensions) and the coupling constant too. I wrote a piece about that, much later than when I wrote the piece above. I would recommend you read that piece too. It’s a piece in which I do take the magic out of ‘God’s number’. Understanding it involves a deep understanding of electromagnetism, however, and that requires some effort. It’s surely worth the effort, though.