Some thoughts on the nature of reality

Some other comment on an article on my other blog, inspired me to structure some thoughts that are spread over various blog posts. What follows below, is probably the first draft of an article or a paper I plan to write. Or, who knows, I might re-write my two introductory books on quantum physics and publish a new edition soon. 馃檪

Physical dimensions and Uncertainty

The physical dimension of the quantum of action (h or聽魔 = h/2蟺) is force (expressed in newton)聽times distance (expressed in meter)聽times time (expressed in seconds): N路m路s. Now, you may think this N路m路s dimension is kinda hard to imagine. We can imagine its individual components, right? Force, distance and time. We know what they are. But the product of all three? What is it, really?

It shouldn’t be all that hard to imagine what it might be, right? The N路m路s unit is also the unit in which angular momentum is expressed – and you can sort of imagine what that is, right? Think of a spinning top, or a gyroscope. We may also think of the following:

  1. [h] = N路m路s = (N路m)路s = [E]路[t]
  2. [h] = N路m路s = (N路s)路m = [p]路[x]

Hence, the physical dimension of action is that of energy (E) multiplied by time (t) or, alternatively, that of momentum (p) times distance (x). To be precise, the second dimensional equation should be written as [h] = [p]路[x], because both the momentum and the distance traveled will be associated with some direction. It’s a moot point for the discussion at the moment, though. Let’s think about the first equation first:聽[h] = [E]路[t]. What does it mean?

Energy… Hmm… In聽real life, we are usually not interested in the energy of a system as such, but by the energy it can deliver, or absorb, per second. This is referred to as the聽power聽of a system, and it’s expressed in J/s, or watt. Power is also defined as the (time) rate at which work is done. Hmm… But so here we’re multiplying energy and time. So what’s that? After Hiroshima and Nagasaki, we can sort of imagine the energy of an atomic bomb. We can also sort of imagine the power that’s being released by the Sun in light and other forms of radiation, which is about 385脳1024 joule per second. But energy times time? What’s that?

I am not sure. If we think of the Sun as a huge reservoir of energy, then the physical dimension of action is just like having that reservoir of energy guaranteed for some time, regardless of how fast or how slow we use it. So, in short, it’s just like the Sun – or the Earth, or the Moon, or whatever object – just being there, for some聽definite聽amount of time. So, yes: some聽definite amount of mass or energy (E) for some聽definite聽amount of time (t).

Let’s bring the mass-energy equivalence formula in here: E = mc2. Hence, the physical dimension of action can also be written as [h] = [E]路[t] = [mc]2路[t] = (kg路m2/s2)路s =聽kg路m2/s.聽What does that say? Not all that much – for the time being, at least. We can get this聽[h] = kg路m2/s through some other substitution as well. A force of one newton will give a mass of 1 kg an acceleration of 1 m/s per second. Therefore, 1 N = 1 kg路m/s2聽and, hence, the physical dimension of h, or the unit of angular momentum, may also be written as 1 N路m路s = 1 (kg路m/s2)路m路s = 1 kg路m2/s, i.e. the product of mass, velocity and distance.

Hmm… What can we do with that? Nothing much for the moment: our first reading of it is just that it reminds us of the definition of angular momentum – some mass with some velocity rotating around an axis. What about the distance? Oh… The distance here is just the distance from the axis, right? Right. But… Well… It’s like having some amount of linear momentum available over some distance – or in some space, right? That’s sufficiently significant as an interpretation for the moment, I’d think…

Fundamental units

This makes one think about what units would be fundamental – and what units we’d consider as being derived. Formally, the聽newton is a聽derived聽unit in the metric system, as opposed to the units of mass, length and time (kg, m, s). Nevertheless, I personally like to think of force as being fundamental:聽 a force is what causes an object to deviate from its straight trajectory in spacetime. Hence, we may want to think of the聽quantum of action as representing three fundamental physical dimensions: (1)聽force, (2)聽time and (3) distance – or space. We may then look at energy and (linear) momentum as physical quantities combining (1) force and distance and (2) force and time respectively.

Let me write this out:

  1. Force times length (think of a force that is聽acting on some object over some distance) is energy: 1 joule聽(J) =聽1 newtonmeter (N). Hence, we may think of the concept of energy as a projection聽of action in space only: we make abstraction of time. The physical dimension of the quantum of action should then be written as [h] = [E]路[t]. [Note the square brackets tell us we are looking at a聽dimensional聽equation only, so [t] is just the physical dimension of the time variable. It’s a bit confusing because I also use square brackets as parentheses.]
  2. Conversely, the magnitude of linear momentum (p = m路v) is expressed in newtonseconds: 1 kg路m/s = 1 (kg路m/s2)路s = 1 N路s. Hence, we may think of (linear) momentum as a projection of action in time only: we make abstraction of its spatial dimension. Think of a force that is acting on some object聽during some time.聽The physical dimension of the quantum of action should then be written as [h] = [p]路[x]

Of course, a force that is acting on some object during some time, will usually also act on the same object over some distance but… Well… Just try, for once, to make abstraction of one of the two dimensions here: time聽or聽distance.

It is a difficult thing to do because, when everything is said and done, we don’t live in space or in time alone, but in spacetime and, hence, such abstractions are not easy. [Of course, now you’ll say that it’s easy to think of something that moves in time only: an object that is standing still does just that – but then we know movement is relative, so there is no such thing as an object that is standing still in space聽in an absolute sense: Hence, objects never stand still in spacetime.] In any case, we should try such abstractions, if only because of the principle of least action聽is so essential and deep in physics:

  1. In classical physics, the path of some object in a force field will minimize聽the total action (which is usually written as S) along that path.
  2. In quantum mechanics, the same action integral will give us various values S – each corresponding to a particular path – and each path (and, therefore, each value of S, really) will be associated with a probability amplitude that will be proportional to some constant times e鈭抜路胃聽=聽ei路(S/魔). Because is so tiny, even a small change in S will give a completely different phase angle 胃. Therefore, most amplitudes will cancel each other out as we take the sum of the amplitudes over all possible paths: only the paths that nearly聽give the same phase matter. In practice, these are the paths that are associated with a variation in S of an order of magnitude that is equal to .

The paragraph above summarizes, in essence, Feynman’s path integral formulation of quantum mechanics. We may, therefore, think of the quantum of action聽expressing聽itself (1) in time only, (2) in space only, or – much more likely – (3) expressing itself in both dimensions at the same time. Hence, if the quantum of action gives us the order of magnitude聽of the uncertainty – think of writing something like S 卤 , we may re-write our dimensional [] = [E]路[t] and [] = [p]路[x] equations as the uncertainty equations:

  • 螖E路螖t =
  • 螖p路螖x =

You should note here that it is best to think of the uncertainty relations as a聽pair聽of equations, if only because you should also think of the concept of energy and momentum as representing different aspects聽of the same reality, as evidenced by the (relativistic) energy-momentum relation (E2聽= p2c2聽鈥 m02c4). Also, as illustrated below, the actual path – or, to be more precise, what we might associate with the concept of the actual path – is likely to be some mix of 螖x and 螖t. If 螖t is very small, then 螖x will be very large. In order to move over such distance, our particle will require a larger energy, so 螖E will be large. Likewise, if 螖t is very large, then 螖x will be very small and, therefore, 螖E will be very small. You can also reason in terms of 螖x, and talk about momentum rather than energy. You will arrive at the same conclusions: the 螖E路螖t = h and 螖p路螖x = h聽relations represent two aspects of the same reality – or, at the very least, what we might聽think聽of as reality.

Uncertainty

Also think of the following: if聽螖E路螖t =聽h聽and 螖p路螖x =聽h, then聽螖E路螖t =聽螖p路螖x and, therefore,聽螖E/螖p must be equal to 螖x/螖t. Hence, the聽ratio聽of the uncertainty about x (the distance) and the uncertainty about t (the time) equals the聽ratio聽of the uncertainty about E (the energy) and the uncertainty about p (the momentum).

Of course, you will note that the actual uncertainty relations have a factor 1/2 in them. This may be explained by thinking of both negative as well as positive variations in space and in time.

We will obviously want to do some more thinking about those physical dimensions. The idea of a force implies the idea of some object – of some mass on which the force is acting. Hence, let’s think about the concept of mass now. But… Well… Mass and energy are supposed to be equivalent, right? So let’s look at the concept of energy聽too.

Action, energy and mass

What is聽energy, really? In聽real life, we are usually not interested in the energy of a system as such, but by the energy it can deliver, or absorb, per second. This is referred to as the聽power聽of a system, and it’s expressed in J/s. However, in physics, we always talk energy – not power – so… Well… What is the energy of a system?

According to the de Broglie聽and Einstein – and so many other eminent physicists, of course – we should not only think of the kinetic energy of its parts, but also of their potential energy, and their rest聽energy, and – for an atomic system – we may add some internal energy, which may be binding energy, or excitation energy (think of a hydrogen atom in an excited state, for example). A lot of stuff. 馃檪 But, obviously, Einstein’s mass-equivalence formula comes to mind here, and summarizes it all:

E = m路c2

The m in this formula refers to mass – not to meter, obviously. Stupid remark, of course… But… Well… What is energy, really? What is mass,聽really? What’s that聽equivalence聽between mass and energy,聽really?

I don’t have the definite answer to that question (otherwise I’d be famous), but… Well… I do think physicists and mathematicians should invest more in exploring some basic intuitions here. As I explained in several posts, it is very tempting to think of energy as some kind of two-dimensional oscillation of mass. A force over some distance will cause a mass to accelerate. This is reflected in the聽dimensional analysis:

[E] = [m]路[c2] = 1 kg路m2/s2聽= 1 kg路m/s2路m = 1 N路m

The kg and m/s2聽factors make this abundantly clear: m/s2聽is the physical dimension of acceleration: (the change in) velocity per time unit.

Other formulas now come to mind, such as the Planck-Einstein relation: E = h路f = 蠅路魔. We could also write: E = h/T. Needless to say, T = 1/f聽is the聽period聽of the oscillation. So we could say, for example, that the energy of some particle times the period of the oscillation gives us Planck’s constant again. What does that mean? Perhaps it’s easier to think of it the other way around: E/f = h = 6.626070040(81)脳10鈭34聽J路s. Now, f聽is the number of oscillations聽per second. Let’s write it as聽f聽= n/s, so we get:

E/f聽= E/(n/s) = E路s/n聽= 6.626070040(81)脳10鈭34聽J路s 鈬 E/n聽= 6.626070040(81)脳10鈭34聽J

What an amazing result! Our wavicle – be it a photon or a matter-particle – will always聽pack聽6.626070040(81)脳10鈭34joule聽in聽one聽oscillation, so that’s the numerical聽value of Planck’s constant which, of course, depends on our fundamental聽units (i.e. kg, meter, second, etcetera in the SI system).

Of course, the obvious question is: what’s one聽oscillation? If it’s a wave packet, the oscillations may not have the same amplitude, and we may also not be able to define an exact period. In fact, we should expect the amplitude and duration of each oscillation to be slightly different, shouldn’t we? And then…

Well… What’s an oscillation? We’re used to聽counting聽them:聽n聽oscillations per second, so that’s聽per time unit. How many do we have in total? We wrote about that in our posts on the shape and size of a photon. We know photons are emitted by atomic oscillators – or, to put it simply, just atoms going from one energy level to another. Feynman calculated the Q of these atomic oscillators: it鈥檚 of the order of 108聽(see his聽Lectures,聽I-33-3: it鈥檚 a wonderfully simple exercise, and one that really shows his greatness as a physics teacher), so… Well… This wave train will last about 10鈥8聽seconds (that鈥檚 the time it takes for the radiation to die out by a factor 1/e). To give a somewhat more precise example,聽for sodium light, which has a frequency of 500 THz (500脳1012聽oscillations per second) and a wavelength of 600 nm (600脳10鈥9聽meter), the radiation will lasts about 3.2脳10鈥8聽seconds. [In fact, that鈥檚 the time it takes for the radiation鈥檚 energy to die out by a factor 1/e, so(i.e. the so-called decay time 蟿), so the wavetrain will actually last聽longer, but so the amplitude becomes quite small after that time.]聽So… Well… That鈥檚 a very short time but… Still, taking into account the rather spectacular frequency (500 THz) of sodium light, that makes for some 16 million oscillations and, taking into the account the rather spectacular speed of light (3脳108聽m/s), that makes for a wave train with a length of, roughly,聽9.6 meter. Huh? 9.6 meter!? But a photon is supposed to be pointlike, isn’it it? It has no length, does it?

That’s where relativity helps us out: as I wrote in one of my posts, relativistic length contraction may explain the apparent paradox. Using the reference frame of the photon聽– so if we’d be traveling at speed c,鈥 riding鈥 with the photon, so to say, as it鈥檚 being emitted – then we’d 鈥榮ee鈥 the electromagnetic transient as it鈥檚 being radiated into space.

However, while we can associate some mass聽with the energy of the photon, none of what I wrote above explains what the (rest) mass of a matter-particle could possibly be.There is no real answer to that, I guess. You’ll think of the Higgs field now but… Then… Well. The Higgs field is a scalar field. Very simple: some number that’s associated with some position in spacetime. That doesn’t explain very much, does it? 馃槮 When everything is said and done, the scientists who, in 2013 only, got the Nobel Price for their theory on the Higgs mechanism, simply tell us mass is some number. That’s something we knew already, right? 馃檪

The reality of the wavefunction

The wavefunction is, obviously, a mathematical construct: a聽description聽of reality using a very specific language. What language? Mathematics, of course! Math may not be universal (aliens might not be able to decipher our mathematical models) but it’s pretty good as a global聽tool of communication, at least.

The real聽question is: is the description聽accurate? Does it match reality and, if it does, how聽good聽is the match? For example, the wavefunction for an electron in a hydrogen atom looks as follows:

蠄(r, t) = ei路(E/魔)路tf(r)

As I explained in previous posts (see, for example, my recent post聽on reality and perception), the聽f(r) function basically provides some envelope for the two-dimensional ei路胃聽=聽ei路(E/魔)路t聽= cos胃 + isin胃聽oscillation, with r= (x, y, z),聽胃 = (E/魔)路t聽= 蠅路t聽and 蠅 = E/魔. So it presumes the聽duration of each oscillation is some constant. Why? Well… Look at the formula: this thing has a constant frequency in time. It’s only the amplitude that is varying as a function of the r= (x, y, z) coordinates. 馃檪 So… Well… If each oscillation is to always聽pack聽6.626070040(81)脳10鈭34joule, but the amplitude of the oscillation varies from point to point, then… Well… We’ve got a problem. The wavefunction above is likely to be an approximation of reality only. 馃檪 The associated energy is the same, but… Well… Reality is probably聽not聽the nice geometrical shape we associate with those wavefunctions.

In addition, we should think of the聽Uncertainty Principle: there聽must聽be some uncertainty in the energy of the photons when our hydrogen atom makes a transition from one energy level to another. But then… Well… If our photon packs something like 16 million oscillations, and the order of magnitude of the uncertainty is only of the order of聽h聽(or 魔 = h/2蟺) which, as mentioned above, is the (average) energy of one聽oscillation only, then we don’t have much of a problem here, do we? 馃檪

Post scriptum: In previous posts, we offered some analogies – or metaphors – to a two-dimensional oscillation (remember the V-2 engine?). Perhaps it’s all relatively simple. If we have some tiny little ball of mass – and its center of mass has to stay where it is – then any rotation – around any axis – will be some combination of a rotation around our聽x- and z-axis – as shown below. Two axes only. So we may want to think of a two-dimensional聽oscillation as an oscillation of the polar and azimuthal angle. 馃檪

oscillation of a ball

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Thinking again…

One of the comments on my other blog made me think I should, perhaps, write something on waves again. The animation below shows the聽elementary聽wavefunction聽蠄 =聽aei胃聽= 蠄 =聽aei路胃聽聽= aei(蠅路t鈭択路x)聽= ae(i/魔)路(E路t鈭抪路x)聽.AnimationWe know this elementary wavefunction cannotrepresent a real-life聽particle. Indeed, the aei路胃聽function implies the probability of finding the particle – an electron, a photon, or whatever – would be equal to P(x, t) = |蠄(x, t)|2聽= |ae(i/魔)路(E路t鈭抪路x)|2聽= |a|2路|e(i/魔)路(E路t鈭抪路x)|2聽= |a|2路12= a2everywhere. Hence, the particle would be everywhere – and, therefore, nowhere really. We need to localize the wave – or build a wave packet. We can do so by introducing uncertainty: we then add聽a potentially infinite number of these elementary wavefunctions with slightly different values for E and p, and various amplitudes a. Each of these amplitudes will then reflect the聽contribution聽to the composite wave, which – in three-dimensional space – we can write as:

蠄(r, t) = ei路(E/魔)路tf(r)

As I explained in previous posts (see, for example, my recent post聽on reality and perception), the聽f(r) function basically provides some envelope for the two-dimensional ei路胃聽=聽ei路(E/魔)路t聽= cos胃 + isin胃聽oscillation, with r= (x, y, z),聽胃 = (E/魔)路t聽= 蠅路t聽and 蠅 = E/魔.

Note that it looks like the wave propagates聽from left to right – in the聽positive direction of an axis which we may refer to as the x-axis. Also note this perception results from the fact that, naturally, we’d associate time with the聽rotation聽of that arrow at the center – i.e. with the motion in the illustration,聽while the spatial dimensions are just what they are: linear spatial dimensions. [This point is, perhaps, somewhat less self-evident than you may think at first.]

Now, the axis which points upwards is usually referred to as the z-axis, and the third and final axis – which points towards聽us –聽would then be the y-axis, obviously.聽Unfortunately, this definition would violate the so-called right-hand rule for defining a proper reference frame: the figures below shows the two possibilities – a left-handed and a right-handed reference frame – and it’s the right-handed reference (i.e. the illustration on the right) which we have to use in order to correctly define all directions, including the direction of聽rotation聽of the argument of the wavefunction.400px-Cartesian_coordinate_system_handednessHence, if we don’t change the direction of the y– and z-axes – so we keep defining the z-axis as the axis pointing upwards, and the y-axis as the axis pointing towards聽us – then the positive direction of the x-axis would actually be the direction from right to left, and we should say that the elementary wavefunction in the animation above seems to propagate in the negativex-direction. [Note that this left- or right-hand rule is quite astonishing: simply swapping the direction of聽one聽axis of a left-handed frame makes it right-handed, and vice versa.]

Note my language when I talk about the direction of propagation of our wave. I wrote: it looks like, or it seems to聽go in this or that direction. And I mean that: there is no real travelinghere. At this point, you may want to review a post I wrote for my son, which explains the basic math behind waves, and in which I also explained the animation below.

wave_opposite-group-phase-velocity

Note how the peaks and troughs of this pulse seem to move leftwards, but the wave packet (or the聽group聽or the聽envelope聽of the wave鈥攚hatever you want to call it) moves to the right. The point is: the pulse itself doesn’t聽travel left or right. Think of the horizontal axis in the illustration above as an oscillating guitar string: each point on the string just moves up and down. Likewise, if our repeated pulse would represent a physical wave in water, for example, then the water just stays where it is: it just moves up and down. Likewise, if we shake up some rope, the rope is not going anywhere: we just started some motion聽that is traveling down the rope.聽In other words, the phase velocity is just a mathematical concept. The peaks and troughs that seem to be traveling are just mathematical points that are 鈥榯raveling鈥 left or right. That鈥檚 why there鈥檚 no limit on the phase velocity: it can聽– and, according to quantum mechanics, actually will聽–聽exceed the speed of light. In contrast, the group聽velocity – which is the actual speed of the particle that is being represented by the wavefunction – may approach聽– or, in the case of a massless photon, will actually equal聽–聽the speed of light, but will never exceed聽it, and its聽direction聽will, obviously, have a聽physical聽significance as it is, effectively, the direction of travel of our particle – be it an electron, a photon (electromagnetic radiation), or whatever.

Hence, you should not think the聽spin聽of a particle – integer or half-integer – is somehow related to the direction of rotation of the argument of the elementary wavefunction. It isn’t: Nature doesn’t give a damn about our mathematical conventions, and that’s what the direction of rotation of the argument of that wavefunction is: just some mathematical convention. That’s why we write aei(蠅路t鈭択路x)聽rather than聽aei(蠅路t+k路x)聽or聽aei(蠅路t鈭択路x): it’s just because of the right-hand rule for coordinate frames, and also because Euler defined the counter-clockwise direction as the聽positive direction of an angle. There’s nothing more to it.

OK. That’s obvious. Let me now return to my interpretation of Einstein’s E = m路c2聽formula (see my previous posts on this). I noted that, in the reference frame of the particle itself (see my basics page), the elementary wavefunction ae(i/魔)路(E路t鈭抪路x)聽reduces to ae(i/魔)路(E’路t’): the origin of the reference frame then coincides with (the center of) our particle itself, and the wavefunction only varies with the time in the inertial reference frame (i.e. the proper聽time t’), with the rest energy of the object (E’) as the time scale factor. How should we interpret this?

Well… Energy is force times distance, and force is defined as that what causes some mass聽to聽accelerate. To be precise, the聽newton聽– as the unit of force – is defined as the聽magnitude of a force which would cause a mass of one kg to accelerate with one meter per second聽per second. Per second per second. This is not a typo: 1 N corresponds to 1 kg times 1 m/s聽per second, i.e. 1 kg路m/s2. So… Because energy is force times distance, the unit of energy聽may be expressed in units of kg路m/s2路m, or kg路m2/s2, i.e. the unit of mass times the unit of聽velocity squared. To sum it all up:

1 J = 1 N路m = 1 kg路(m/s)2

This reflects the physical dimensions聽on both sides of the聽E = m路c2聽formula again but… Well… How should we聽interpret聽this? Look at the animation below once more, and imagine the green dot is some tiny聽mass聽moving around the origin, in an equally tiny circle. We’ve got聽two聽oscillations here: each packing聽half聽of the total energy of… Well… Whatever it is that our elementary wavefunction might represent in reality聽– which we don’t know, of course.

circle_cos_sin

Now, the blue and the red dot – i.e. the horizontal and vertical projection聽of the green dot –聽accelerate up and down. If we look carefully, we see these dots accelerate聽towards聽the zero point and, once they’ve crossed it, they聽decelerate, so as to allow for a reversal of direction: the blue dot goes up, and then down. Likewise, the red dot does the same. The interplay between the two oscillations, because of the 90掳 phase difference, is interesting: if the blue dot is at maximum speed (near or at the origin), the red dot reverses speed (its speed is, therefore, (almost) nil), and vice versa. The metaphor of our frictionless V-2 engine, our perpetuum mobile,聽comes to mind once more.

The question is: what’s going on, really?

My answer is: I don’t know. I do think that, somehow, energy should be thought of as some two-dimensional oscillation of something – something which we refer to as聽mass, but we didn’t define mass very clearly either. It also, somehow, combines linear and rotational motion. Each of the two dimensions packs half of the energy of the particle that is being represented by our wavefunction. It is, therefore, only logical that the physical unit聽of both is to be expressed as a force over some distance – which is, effectively, the physical dimension of energy – or the rotational equivalent of them: torque聽over some angle.聽Indeed, the analogy between linear and angular movement is obvious: the聽kinetic聽energy of a rotating object is equal to K.E. = (1/2)路I路蠅2. In this formula, I is the rotational inertia聽– i.e. the rotational equivalent of mass – and 蠅 is the angular velocity – i.e. the rotational equivalent of linear聽velocity. Noting that the (average) kinetic energy in any system must be equal to the (average) potential energy in the system, we can add both, so we get a formula which is structurally聽similar to the聽E = m路c2聽formula. But is聽it the same? Is the effective mass of some object the sum of an almost infinite number of quanta聽that incorporate some kind of聽rotational聽motion? And – if we use the right units – is the angular velocity of these infinitesimally small rotations effectively equal to the speed of light?

I am not sure. Not at all, really. But, so far, I can’t think of any explanation of the wavefunction that would make more sense than this one. I just need to keep trying to find better ways to聽articulate聽or聽imagine聽what might be going on. 馃檪 In this regard, I’d like to add a point – which may or may not be relevant. When I talked about that guitar string, or the water wave, and wrote that each point on the string – or each water drop – just moves up and down, we should think of the physicality of the situation: when the string oscillates, its聽length聽increases. So it’s only because our string is flexible that it can vibrate between the fixed points at its ends. For a rope that’s聽not聽flexible, the end points would need to move in and out with the oscillation. Look at the illustration below, for example: the two kids who are holding rope must come closer to each other, so as to provide the necessary space inside of the oscillation for the other kid. 馃檪kid in a ropeThe next illustration – of how water waves actually propagate – is, perhaps, more relevant. Just think of a two-dimensional equivalent – and of the two oscillations as being transverse聽waves, as opposed to longitudinal.聽See how string theory starts making sense? 馃檪

rayleighwaveThe most fundamental question remains the same: what is it,聽exactly, that is oscillating here? What is the聽field? It’s always some force on some charge – but what charge, exactly? Mass? What is it? Well… I don’t have the answer to that. It’s the same as asking: what is聽electric聽charge,聽really? So the question is: what’s the聽reality聽of mass, of electric charge, or whatever other charge that causes a force to聽act聽on it?

If you聽know, please let聽me聽know. 馃檪

Post scriptum: The fact that we’re talking some聽two-dimensional oscillation here – think of a surface now – explains the probability formula: we need to聽square聽the absolute value of the amplitude to get it. And normalize, of course. Also note that, when normalizing, we’d expect to get some factor involving聽蟺 somewhere, because we’re talking some聽circular聽surface – as opposed to a rectangular one. But I’ll let聽you聽figure that out. 馃檪

An introduction to virtual particles (2)

When reading quantum mechanics, it often feels like the more you know, the less you understand. My reading of the Yukawa theory of force, as an exchange of virtual particles (see my previous post), must have left you with many questions. Questions I can’t answer because… Well… I feel as much as a fool as you do when thinking about it all. Yukawa first talks about some potential – which we usually think of as being some scalar聽function – and then聽suddenly this potential becomes a wavefunction. Does that make sense? And think of the mass of that ‘virtual’ particle: the rest mass of a neutral pion is about 135 MeV. That’s an awful lot – at the (sub-)atomic scale that is: it’s equivalent to the rest mass of some 265 electrons!

But… Well… Think of it: the use of a static potential when solving Schr枚dinger’s equation for the electron orbitals around a hydrogen nucleus (a proton, basically) also raises lots of questions: if we think of our electron as a point-like particle being first here and then there, then that’s also not very consistent with a static (scalar) potential either!

One of the weirdest aspects of the Yukawa theory is that these emissions and absorptions of virtual particles violate the energy conservation principle. Look at the animation once again (below): it sort of assumes a rather heavy particle – consisting of a d- or u-quark and its antiparticle – is emitted聽– out of nothing, it seems – to then vanish as the antiparticle is destroyed when absorbed. What about the energy balance here: are we talking six quarks (the proton and the neutron), or six plus two?Nuclear_Force_anim_smallerNow that we’re talking mass, note a neutral pion (蟺0) may either be a u奴 or a d膽combination, and that the mass of a u-quark and a d-quark is only 2.4 and 4.8 MeV – so the聽binding聽energy of the constituent parts of this 蟺0聽particle is enormous: it accounts for most of its mass.

The thing is… While we’ve presented the 蟺0聽particle as a virtual聽particle here, you should also note we find 蟺0聽particles in cosmic rays. Cosmic rays are particle rays, really: beams of highly energetic particles. Quite a bunch of them are just protons that are being ejected by our Sun. [The Sun also ejects electrons – as you might imagine – but let’s think about the protons here first.] When these protons hit an atom or a molecule in our atmosphere, they usually break up in various particles, including our聽蟺0聽particle, as shown below.聽850px-Atmospheric_Collision

 

So… Well… How can we relate these things? What is聽going on, really, inside of that nucleus?

Well… I am not sure. Aitchison and Hey聽do their utmost to try to explain the pion – as a聽virtual聽particle, that is – in聽terms of聽energy fluctuations聽that obey the Uncertainty Principle for energy and time:聽螖E路螖t聽鈮ヂ/2. Now, I find such explanations difficult to follow. Such explanations usually assume any measurement instrument – measuring energy, time, momentum of distance – measures those variables on some discrete scale, which implies some uncertainty indeed. But that uncertainty is more like an imprecision, in my view. Not something fundamental. Let me quote Aitchison and Hey:

“Suppose a device is set up capable of checking to see whether energy is, in fact, conserved while the pion crosses over.. The crossing time 螖t must be at least r/c, where r is the distance apart of the nucleons. Hence, the device must be capable of operating on a time scale smaller than 螖t to be able to detect the pion, but it need not be very much less than this. Thus the energy uncertainty in the reading by the device will be of the order 螖E聽鈭悸/螖t) = 魔路(c/r).”

As said, I find such explanations really difficult, although I can sort of sense some of the implicit assumptions. As I mentioned a couple of times already, the E = m路c2聽equation tells us energy is mass in motion, somehow: some weird two-dimensional oscillation in spacetime. So, yes, we can appreciate we need some聽time unit聽to聽count聽the oscillations – or, equally important, to measure their聽amplitude.

[…] But… Well… This falls short of a more聽fundamental聽explanation of what’s going on. I like to think of Uncertainty in terms of Planck’s constant itself:聽魔 or聽h聽or – as you’ll usually see it – as half聽of that value: /2. [The Stern-Gerlach experiment implies it’s /2, rather than h/2 or 魔 or聽h聽itself.] The physical dimension of Planck’s constant is action: newton times distance times time. I also like to think action can express itself in two ways: as (1) some amount of energy (螖E: some force of some distance) over some time (螖t) or, else, as (2) some momentum (螖p: some force during some time) over some distance (螖s). Now, if we equate 螖E with the energy of the pion (135 MeV), then we may calculate the order of magnitude聽of聽螖t from 螖E路螖t 鈮 /2 as follows:

聽螖t = (/2)/(135 MeV) 鈮 (3.291脳10鈭16聽eV路s)/(134.977脳106聽eV)聽鈮 0.02438脳10鈭22聽s

Now, that’s an聽unimaginably聽small time unit – but much and much聽larger than the Planck time (the Planck time unit is about 5.39 脳 10鈭44 s). The corresponding distance聽r聽is equal to r聽= 螖t路c聽= (0.02438脳10鈭22聽s)路(2.998脳108聽m/s) 鈮 0.0731脳10鈭14聽m = 0.731 fm. So… Well… Yes. We got the answer we wanted… So… Well… We should be happy about that but…

Well… I am not. I don’t like this indeterminacy. This randomness in the approach. For starters, I am very puzzled by the fact that the lifetime of the actual0聽particle we see in the debris聽of proton collisions with other particles as cosmic rays enter the atmosphere is like聽8.4脳10鈭17 seconds, so that’s like 35聽million聽times longer than the 螖t =聽0.02438脳10鈭22聽s we calculated above.

Something doesn’t feel right. I just can’t see the logic here.聽Sorry. I’ll be back. :-/

An introduction to virtual particles

We are going to聽venture beyond quantum mechanics as it is usually understood – covering electromagnetic interactions only. Indeed, all of my posts so far – a bit less than 200, I think 馃檪 – were all centered around electromagnetic interactions – with the model of the hydrogen atom as our most precious gem, so to speak.

In this post, we’ll be talking the strong force – perhaps not for the first time but surely for the first time at this level of detail. It’s an entirely different world – as I mentioned in one of my very first posts in this blog. Let me quote what I wrote there:

“The math describing the ‘reality’ of electrons and photons (i.e. quantum mechanics and quantum electrodynamics), as complicated as it is, becomes even more complicated 鈥 and, important to note, also much less accurate聽鈥 when it is used to try to describe the behavior of 聽quarks. Quantum chromodynamics (QCD) is a different world. […]聽Of course, that should not surprise us, because we’re talking very different order of magnitudes here: femtometers (10鈥15 m), in the case of electrons, as opposed to attometers (10鈥18 m)聽or even zeptometers (10鈥21聽m) when we’re talking quarks.”

In fact, the femtometer scale is used to measure the radius聽of both protons as well as electrons and, hence, is much smaller than the atomic scale, which is measured in nanometer (1 nm = 10鈭9聽m). The so-called Bohr radius for example, which is a measure for the size of an atom, is measured in nanometer indeed, so that’s a scale that is a聽million聽times larger than the femtometer scale. This聽gap聽in the scale effectively separates entirely different worlds. In fact, the gap is probably as large a gap as the gap between our macroscopic world and the strange reality of quantum mechanics. What happens at the femtometer scale,聽really?

The honest answer is: we don’t know, but we do have models聽to describe what happens. Moreover, for want of better models, physicists sort of believe these models are credible. To be precise, we assume there’s a force down there which we refer to as the聽strong聽force. In addition, there’s also a weak force. Now, you probably know these forces are modeled as聽interactions聽involving an聽exchange聽of聽virtual聽particles. This may be related to what Aitchison and Hey refer to as the physicist’s “distaste for action-at-a-distance.” To put it simply: if one particle – through some force – influences some other particle, then something must be going on between the two of them.

Of course, now you’ll say that something is聽effectively going on: there’s the electromagnetic field, right? Yes. But what’s the field? You’ll say: waves. But then you know electromagnetic waves also have a particle aspect. So we’re stuck with this weird theoretical framework: the conceptual distinction between particles and forces, or between particle and field, are not so clear. So that’s what the more advanced theories we’ll be looking at – like quantum field theory – try to bring together.

Note that we’ve been using a lot of confusing and/or ambiguous terms here: according to at least one leading physicist, for example, virtual particles should not be thought of as particles! But we’re putting the cart before the horse here. Let’s go step by step. To better understand the ‘mechanics’ of how the strong and weak interactions are being modeled in physics, most textbooks – including Aitchison and Hey, which we’ll follow here – start by explaining the original ideas as developed by the Japanese physicist Hideki Yukawa, who received a Nobel Prize for his work in 1949.

So what is it all about? As said, the ideas聽– or the聽model聽as such, so to speak – are more important than Yukawa’s original application, which was to model the force between a proton and a neutron. Indeed, we now explain such force as a force between quarks, and the force carrier is the gluon, which carries the so-called聽color聽charge. To be precise, the force between protons and neutrons – i.e. the so-called nuclear聽force – is聽now聽considered to be a rather minor聽residual force: it’s just what’s left of the actual聽strong force that binds quarks together. The Wikipedia article on this聽has some聽good text and聽a really nice animation on this. But… Well… Again, note that we are only interested in the聽model right now. So how does that look like?

First, we’ve got the equivalent of the electric charge: the nucleon is supposed to have some ‘strong’ charge, which we’ll write as gs. Now you know the formulas for the聽potential聽energy – because of the gravitational force – between two masses, or the聽potential聽energy between two charges – because of the electrostatic force. Let me jot them down once again:

  1. U(r) =聽鈥揋路M路m/r
  2. U(r) = (1/4蟺蔚0)路q1路q2/r

The two formulas are exactly the same. They both assume U = 0 for聽r聽鈫 鈭. Therefore, U(r) is always negative. [Just think of q1聽and q2聽as opposite charges, so the minus sign is not explicit – but it is also there!] We know that聽U(r)聽curve will look like the one below: some work (force times distance) is needed to move the two charges some distance聽away from each other – from point 1 to point 2, for example. [The distance r is x here – but you got that, right?]potential energy

Now, physics textbooks – or other articles you might find, like on Wikipedia – will sometimes mention that the strong force is non-linear, but that’s very confusing because… Well… The electromagnetic force – or the gravitational force – aren’t linear either: their strength is inversely proportional to the square聽of the distance and – as you can see from the formulas for the potential energy – that 1/r factor isn’t lineareither. So that isn’t very helpful. In order to further the discussion, I should now write down Yukawa’s聽hypothetical聽formula for the potential energy between a neutron and a proton, which we’ll refer to, logically, as the n-p potential:n-p potentialThe 鈭抔s2聽factor is, obviously, the equivalent of the q1路q2聽product: think of the proton and the neutron having equal but opposite ‘strong’ charges. The 1/4蟺 factor reminds us of the Coulomb constant:聽ke聽= 1/4蟺蔚0. Note this constant ensures the physical dimensions of both sides of the equation make sense: the dimension of 蔚0聽is N路m2/C2, so U(r) is – as we’d expect – expressed in newton路meter, or聽joule. We’ll leave the question of the units for gs聽open – for the time being, that is. [As for the 1/4蟺 factor, I am not sure why Yukawa put it there. My best guess is that he wanted to remind us some constant should be there to ensure the units come out alright.]

So, when everything is said and done, the big new thing is the er/a/r聽factor, which replaces the usual 1/r dependency on distance. Needless to say, e is Euler’s number here –聽not聽the electric charge. The two green curves below show what the er/a聽factor does to the classical 1/r function for a聽= 1 and聽a聽= 0.1 respectively: smaller values for聽a聽ensure the curve approaches zero more rapidly. In fact, for聽a聽= 1,聽er/a/r聽is equal to 0.368 for聽r聽= 1, and remains significant for values r聽that are greater than 1 too.聽In contrast, for聽a聽= 0.1, er/a/r聽is equal to 0.004579 (more or less, that is) for r聽= 4 and rapidly goes to zero for all values greater than that.

graph 1graph 2Aitchison and Hey call聽a, therefore, a聽range parameter: it effectively defines the聽range聽in which the n-p potential has a significant value: outside of the range, its value is, for all practical purposes, (close to) zero. Experimentally, this range was established as being more or less equal to r聽鈮 2 fm.聽Needless to say, while this range factor may do its job, it’s obvious Yukawa’s formula for the n-p potential comes across as being somewhat random: what’s the theory behind? There’s none, really. It makes one think of the logistic function: the logistic function fits many statistical patterns, but it is (usually) not obvious why.

Next in Yukawa’s argument is the establishment of an equivalent, for the nuclear force, of the Poisson equation in electrostatics: using the聽E = 鈥桅 formula, we can re-write Maxwell’s 鈭団E= 蟻/蔚0聽equation (aka Gauss’ Law) as聽鈭団 =聽鈥撯垏鈥⑩垏= 鈥2鈬斅2=聽鈥撓/蔚0聽indeed. The divergence聽operator聽the聽鈥 operator gives us the聽volume聽density of the flux of E out of an infinitesimal volume around a given point. [You may want to check one of my post on this. The formula becomes somewhat more obvious if we re-write it as 鈭団EdV = 鈥(蟻路dV)/蔚0: 鈭団EdV is then, quite simply, the flux of E out of the infinitesimally small volume dV, and the right-hand side of the equation says this is given by the product of the charge inside (蟻路dV) and 1/蔚0, which accounts for the permittivity of the medium (which is the vacuum in this case).] Of course, you will also remember the 桅 notation: is just the gradient (or vector derivative) of the (scalar) potential 桅, i.e. the electric (or electrostatic) potential in a space around that infinitesimally small volume with charge density 蟻. So… Well… The Poisson equation is probably not聽so聽obvious as it seems at first (again, check聽my post on it聽on it for more detail) and, yes, that 鈥 operator – the divergence聽operator – is a pretty impressive mathematical beast. However, I must assume you master this topic and move on. So… Well… I must now give you the equivalent of Poisson’s equation for the nuclear force. It’s written like this:Poisson nuclearWhat the heck? Relax. To derive this equation, we’d need to take a pretty complicated d茅tour, which we won’t do. [See Appendix G of Aitchison and Grey if you’d want the details.] Let me just point out the basics:

1. The Laplace operator (鈭2) is replaced by one that’s nearly the same: 鈭2聽鈭 1/a2. And it operates on the same concept: a potential, which is a (scalar) function of the position r. Hence, U(r) is just the equivalent of聽桅.

2. The right-hand side of the equation involves Dirac’s delta function. Now that’s a weird mathematical beast. Its definition seems to defy what I refer to as the ‘continuum assumption’ in math. 聽I wrote a few things about it in one of my posts on Schr枚dinger’s equation聽– and I could give you its formula – but that won’t help you very much. It’s just a weird thing. As Aitchison and Grey聽write, you should just think of the whole expression as a finite range analogue聽of Poisson’s equation in electrostatics. So it’s only for extremely small聽r聽that the whole equation makes sense. Outside of the range defined by our range parameter聽a, the whole equation just reduces to 0 = 0 – for all practical purposes, at least.

Now, of course, you know that the neutron and the proton are not supposed to just sit there. They’re also in these sort of intricate dance which – for the electron case – is described by some wavefunction, which we derive as a solution from Schr枚dinger’s equation. So U(r) is going to vary not only in space but also in time and we should, therefore, write it as U(r, t). Now, we will, of course, assume it’s going to vary in space and time as some聽wave聽and we may, therefore, suggest some wave聽equation聽for it. To appreciate this point, you should review some of the posts I did on waves. More in particular, you may want to review the post I did on traveling fields, in which I showed you the following:聽if we see an equation like:f8then the function(x, t) must have the following general functional form:solutionAny聽function 蠄聽like that will work – so it will be a solution to the differential equation – and we’ll refer to it as a wavefunction. Now, the equation (and the function) is for a wave traveling in聽one dimension only (x) but the same post shows we can easily generalize to waves traveling in three dimensions. In addition, we may generalize the analyse to include聽complex-valued聽functions as well. Now, you will still be shocked by Yukawa’s field equation for U(r, t) but, hopefully, somewhat less so after the above reminder on how wave equations generally look like:Yukawa wave equationAs said, you can look up the nitty-gritty in Aitchison and Grey聽(or in its appendices) but, up to this point, you should be able to sort of appreciate what’s going on without getting lost in it all. Yukawa’s next step – and all that follows – is much more baffling. We’d think U, the nuclear potential, is just some scalar-valued wave, right? It varies in space and in time, but… Well… That’s what classical waves, like water or sound waves, for example do too. So far, so good. However, Yukawa’s next step is to associate a聽de Broglie-type wavefunction with it. Hence, Yukawa imposes聽solutions of the type:potential as particleWhat?Yes. It’s a big thing to swallow, and it doesn’t help most physicists refer to U as a聽force field. A force and the potential that results from it are two different things. To put it simply: the聽force聽on an object is聽not聽the same as the聽work聽you need to move it from here to there. Force and potential are聽related聽but聽different聽concepts. Having said that, it sort of make sense now, doesn’t it? If potential is energy, and if it behaves like some wave, then we must be able to associate it with a聽de Broglie-type particle. This U-quantum, as it is referred to, comes in two varieties, which are associated with the ongoing聽absorption-emission process that is supposed to take place inside of the nucleus (depicted below):

p + U聽鈫 n and聽n + U+聽鈫 p

absorption emission

It’s easy to see that the聽U聽and聽U+聽particles are just each other’s anti-particle. When thinking about this, I can’t help remembering Feynman, when he enigmatically wrote – somewhere in his Strange Theory of Light and Matter聽– that聽an anti-particle might just be the same particle traveling back in time.聽In fact, the聽exchange聽here is supposed to happen within a聽time window聽that is so short it allows for the brief聽violation聽of the energy conservation principle.

Let’s be more precise and try to find the properties of that mysterious U-quantum. You’ll need to refresh what you know about operators to understand how substituting Yukawa’s聽de Broglie聽wavefunction in the complicated-looking differential equation (the wave聽equation) gives us the following relation between the energy and the momentum of our new particle:mass 1Now, it doesn’t take too many gimmicks to compare this against the relativistically correct energy-momentum relation:energy-momentum relationCombining both gives us the associated (rest) mass of the U-quantum:rest massFor聽a聽鈮 2 fm,聽mU聽is about 100 MeV. Of course, it’s always to check the dimensions and calculate stuff yourself. Note the physical dimension of聽/(ac) is N路s2/m = kg (just think of the F = m路a formula). Also note that N路s2/m = kg = (N路m)路s2/m2聽= J/(m2/s2), so that’s the [E]/[c2] dimension.聽The calculation – and interpretation – is somewhat tricky though: if you do it, you’ll find that:

/(ac) 鈮 (1.0545718脳10鈭34聽N路m路s)/[(2脳10鈭15聽m)路(2.997924583脳108聽m/s)] 鈮 0.176脳10鈭27聽kg

Now, most physics handbooks continue that terrible habit of writing particle weights in eV, rather than using the correct eV/c2聽unit. So when they write: mU聽is about 100 MeV, they actually mean to say that it’s 100 MeV/c2. In addition, the eV is not聽an SI unit. Hence, to get that number, we should first write 0.176脳10鈭27kg as some value expressed in J/c2, and then convert the joule聽(J) into electronvolt (eV). Let’s do that. First, note that c2聽鈮 9脳1016聽m2/s2, so 0.176脳10鈭27聽kg聽鈮埪1.584脳10鈭11聽J/c2. Now we do the conversion from joule聽to electronvolt. We聽get: (1.584脳10鈭11聽J/c2)路(6.24215脳1018聽eV/J)聽鈮 9.9脳107聽eV/c2聽= 99 MeV/c2.聽Bingo!聽So that was Yukawa’s prediction for the聽nuclear force quantum.

Of course, Yukawa was wrong but, as mentioned above, his ideas are now generally accepted. First note the mass of the U-quantum is quite considerable:聽100 MeV/c2聽is a bit more than 10% of the individual proton or neutron mass (about 938-939 MeV/c2). While the聽binding energy聽causes the mass of an atom to be less than the mass of their constituent parts (protons, neutrons and electrons), it’s quite remarkably that the deuterium atom – a hydrogen atom with an extra neutron – has an excess mass of about 13.1 MeV/c2, and a binding energy with an equivalent mass of only 2.2 MeV/c2. So… Well… There’s something there.

As said, this post only wanted to introduce some basic ideas. The current model of nuclear physics is represented by the animation below, which I took from the Wikipedia article on it. The U-quantum appears as the pion here – and it does聽not聽really turn the proton into a neutron and vice versa. Those particles are assumed to be stable. In contrast, it is the聽quarks聽that change聽color聽by exchanging gluons between each other. And we know look at the exchange particle – which we refer to as the pion聽–聽between the proton and the neutron as consisting of two quarks in its own right: a quark and a anti-quark. So… Yes… All weird. QCD is just a different world. We’ll explore it more in the coming days and/or weeks. 馃檪Nuclear_Force_anim_smallerAn alternative – and simpler – way of representing this exchange of a virtual particle (a neutral聽pion聽in this case) is obtained by drawing a so-called Feynman diagram:Pn_scatter_pi0OK. That’s it for today. More tomorrow. 馃檪