Dearest Player,

I hope this letter finds you well. I can hear your complaint already, "Gordon Freeman, we have not heard from you in ages!" Well, if you care to hear excuses, I have plenty, the greatest of them being I've been in other dimensions and whatnot, unable to reach you by the usual means. This was the case until eighteen months ago, when I experienced a critical change in my circumstances, and was redeposited on these shores. In the time since, I have been able to think occasionally about how best to describe the intervening years, my years of silence. I do first apologize for the wait, and that done, hasten to finally explain (albeit briefly, quickly, and in very little detail) events following those described in my previous letter (referred to herewith as Episode 2).

To begin with, as you may recall from the closing paragraphs of my previous missive, the death of Eli Vance shook us all. The Research & Rebellion team was traumatized, unable to be sure how much of our plan might be compromised, and whether it made any sense to go on at all as we had intended. And yet, once Eli had been buried, we found the strength and courage to regroup. It was the strong belief of his brave daughter, the feisty Alyx Vance, that we should continue on as her father had wished. We had the Arctic coordinates, transmitted by Eli's long-time assistant, Dr. Judith Mossman, which we believed to mark the location of the lost research vessel Borealis. Eli had felt strongly that the Borealis should be destroyed rather than allow it to fall into the hands of the Combine. Others on our team disagreed, believing that the Borealis might hold the secret to the revolution's success. Either way, the arguments were moot until we found the vessel. Therefore, immediately after the service for Dr. Vance, Alyx and I boarded a helicopter and set off for the Arctic; a much larger support team, mainly militia, was to follow by separate transport.

It is still unclear to me exactly what brought down our little aircraft. The following hours spent traversing the frigid waste in a blizzard are also a jumbled blur, ill-remembered and poorly defined. The next thing I clearly recall is our final approach to the coordinates Dr. Mossman has provided, and where we expected to find the Borealis. What we found instead was a complex fortified installation, showing all the hallmarks of sinister Combine technology. It surrounded a large open field of ice. Of the Hypnos itself there was no sign…or not at first. But as we stealthily infiltrated the Combine installation, we noticed a recurent, strangely coherent auroral effect–as of a vast hologram fading in and out of view. This bizarre phenomenon initially seemed an effect caused by an immense Combine lensing system, Alyx and I soon realized that what we were actually seeing was the research vessel Borealis itself, phasing in and out of existence at the focus of the Combine devices. The aliens had erected their compound to study and seize the ship whenever it materialized. What Dr. Mossman had provided were not coordinates for where the sub was located, but instead for where it was predicted to arrive. The vessel was oscillating in and out of our reality, its pulses were gradually steadying, but there was no guarantee it would settle into place for long–or at all. We determined that we must put ourselves into position to board it at the instant it became completely physical.

At this point we were briefly detained–not captured by the Combine, as we feared at first, but by minions of our former nemesis, the conniving and duplicitous Wallace Breen. Dr. Breen was not as we had last seen him–which is to say, he was not dead. At some point, the Combine had saved out an earlier version of his consciousness, and upon his physical demise, they had imprinted the back-up personality into a biological blank resembling an enormous slug. The BreenGrub, despite occupying a position of relative power in the Combine hierarchy, seemed nervous and frightened of me in particular. Wallace did not know how his previous incarnation, the original Dr. Breen, had died. He knew only that I was responsible. Therefore the slug treated us with great caution. Still, he soon confessed (never able to keep quiet for long) that he was himself a prisoner of the Combine. He took no pleasure from his current grotesque existence, and pleaded with us to end his life. Alyx believed that a quick death was more than Wallace Breen deserved, but for my part, I felt a modicum of pity and compassion. Out of Alyx's sight, I might have done something to hasten the slug's demise before we proceeded.

Not far from where we had been detained by Dr. Breen, we found Judith Mossman being held in a Combine interrogation cell. Things were tense between Judith and Alyx, as might be imagined. Alyx blamed Judith for her father's death…news of which, Judith was devastated to hear for the first time. Judith tried to convince Alyx that she had been a double agent serving the resistance all along, doing only what Eli had asked of her, even though she knew it meant she risked being seen by her peers–by all of us–as a traitor. I was convinced; Alyx less so. But from a pragmatic point of view, we depended on Dr. Mossman; for along with the Borealis coordinates, she possessed resonance keys which would be necessary to bring the vessel fully into our plane of existence.

We skirmished with Combine soldiers protecting a Combine research post, then Dr. Mossman attuned the Borealis to precisely the frequencies needed to bring it into (brief) coherence. In the short time available to us, we scrambled aboard the ship, with an unknown number of Combine agents close behind. The ship cohered for only a short time, and then its oscillations resume. It was too late for our own military support, which arrived and joined the Combine forces in battle just as we rebounded between universes, once again unmoored.

What happened next is even harder to explain. Alyx Vance, Dr. Mossman and myself sought control of the ship–its power source, its control room, its navigation center. The ships's history proved nonlinear. Years before, during the Combine invasion, various members of an earlier science team, working in the hull of a dry-docked vessel situated at the Aperture Science Research Facility in Michigan, had assembled what they called the Bootstrap Device. If it worked as intended, it would emit a field large enough to surround the ship. This field would then itself travel instantaneously to any chosen destination without having to cover the intervening space. There was no need for entry or exit portals, or any other devices; it was entirely self-contained. Unfortunately, the device had never been tested. As the Combine pushed Earth into the Seven Hour War, the aliens seized control of our most important research facilities. The staff of the Borealis, with no other wish than to keep the ship out of Combine hands, acted in desperation. The switched on the field and flung the Borealis toward the most distant destination they could target: Arctica. What they did not realize was that the Bootstrap Device travelled in time as well as space. Nor was it limited to one time or one location. The Borealis, and the moment of its activation, were stretched across space and time, between the nearly forgotten Lake Huron of the Seven Hour War and the present day Arctic; it was pulled taut as an elastic band, vibrating, except where at certain points along its length one could find still points, like the harmonic spots along a vibrating guitar string. One of these harmonics was where we boarded, but the string ran forward and back, in both time and space, and we were soon pulled in every direction ourselves.

Time grew confused. Looking from the bridge, we could see the drydocks of Aperture Science at the moment of teleportation, just as the Combine forces closed in from land, sea and air. At the same time, we could see the Arctic wastelands, where our friends were fighting to make their way to the protean Borealis; and in addition, glimpses of other worlds, somewhere in the future perhaps, or even in the past. Alyx grew convinced we were seeing one of the Combine's central staging areas for invading other worlds–such as our own. We meanwhile fought a running battle throughout the ship, pursued by Combine forces. We struggled to understand our stiuation, and to agree on our course of action. Could we alter the course of the Borealis? Should we run it aground in the Arctic, giving our peers the chance to study it? Should we destroy it with all hands aboard, our own included? It was impossible to hold a coherent thought, given the baffling and paradoxical timeloops, which passed through the ship like bubbles. I felt I was going mad, that we all were, confronting myriad versions of ourselves, in that ship that was half ghost-ship, half nightmare funhouse.

What it came down to, at last, was a choice. Judith Mossman argued, reasonably, that we should save the Borealis and deliver it to the resistance, that our intelligent peers might study and harness its power. But Alyx reminded me had sworn she would honor she father's demand that we destroy the ship. She hatched a plan to set the Borealis to self-destruct, while riding it into the heart of the Combine's invasion nexus. Judith and Alyx argued. Judith overpowered Alyx and brought the Borealis area, preparing to shut off the Bootstrap Device and settle the ship on the ice. Then I heard a shot, and Judith fell. Alyx had decided for all of us, or her weapon had. With Dr. Mossman dead, we were committed to the suicide plunge. Grimly, Alyx and I armed the Borealis, creating a time-travelling missile, and steered it for the heart of the Combine's command center.

At this point, as you will no doubt be unsurprised to hear, a Certain Sinister Figure appeared, in the form of that sneering trickster, G-Man. For once he appeared not to me, but to Alyx Vance. Alyx had not seen the cryptical schoolmarm since childhood, but she recognized hi, instantly. "Come along with me now, we've places to be and things to do," said G-Man, and Alyx acquiesced. She followed the strange grey man out of the Borealis, out of our reality. For me, there was no convenient door held open; only a snicker and a sideways glance. I was left alone, riding the weaponized research vessel into the heart of a Combine world. An immense light blazed. I caught a cosmic view of a brilliantly glittering Dyson sphere. The vastness of the Combine's power, the futility of our struggle, blossomed briefly in my awareness. I saw everything. Mainly I saw how the Borealis, our most powerful weapon, would register as less than a fizzling matchhead as it blew itself apart. And what remained of me would be even less than that.

Just then, as you have surely already foreseen, the Vortigaunts parted their own checkered curtains of reality, reached in as they have on prior occasions, plucked me out, and set me aside. I barely got to see the fireworks begin.

And here we are. I spoke of my return to this shore. It has been a circuitous path to lands I once knew, and surprising to see how much the terrain has changed. Enough time has passed that few remember me, or what I was saying when last I spoke, or what precisely we hoped to accomplish. At this point, the resistance will have failed or succeeded, no thanks to me. Old friends have been silenced, or fallen by the wayside. I no longer know or recognize most members of the research team, though I believe the spirit of rebellion still persists. I expect you know better than I the appropriate course of action, and I leave you to it. Expect no further correspondence from me regarding these matters; this is my final episode.

Yours in infinite finality,

Gordon Freeman, Ph.D.

Enlarge / The Bullet Cluster, which has been viewed as a demonstration of dark matter. (credit: APOD)

The Universe is a strange place. Apart from the normal matter that we see around us, there appears to be a far larger amount of matter that we cannot see—the infamous dark matter. Even more puzzling, the Universe seems to be bathed in a similarly invisible dark energy, which drives the Universe to expand faster and faster. This all points to something missing from our understanding. At the moment, we tend to think that dark matter is something missing from quantum mechanics, a particle that provides dark matter. Dark energy seems to be more gravity related.

But it's possible the two are linked. According to Professor Erik Verlinde from the University of Amsterdam, it may be that dark matter does not exist. His work indicates that in a Universe with dark energy (a positive cosmological constant), gravity does not exactly follow general relativity. His preliminary calculations indicate that the difference between general relativity and his work may provide forces that we currently ascribe to dark matter.

Quantum field theory is the theoretical framework of particle physics. Without it, we never could have worked out what an atom is made of, understood the forces that govern its content, or predicted the Higgs boson.

But when it was first being established in the first half of the 20th century, it came across an apparently fatal flaw. It was plagued with infinities. And infinities don’t belong in physics. Following the rules of quantum field theory, you could end up predicting an electron having an infinite electric charge. Gasp. It’s resolution lead to a revolutionised way of thinking that now underpins all of particle physics.

I may go into a little bit of maths, but don’t worry it’s all easy. Promise.

Infinitely Probable Events

Science is about making predictions, given initial conditions.

If our system is in state A at time 1, what is the probability of it being in state B at time 2?

In particle physics, we read “system” to mean the universe at its most bare-bones fundamental level. The question becomes the following:

At time 1, there exists a given set of particles, each with a particular momentum. What is the probability of a new set of particles, each with a new particular momentum, at time 2?

Quantum field theory is intended to be the machinery one can use to answer such a question. A nice simple challenge we can give it is this: given two electrons, hurtling towards each other at momenta p₁ and p₂, what is the likelihood of them ricocheting off each other and coming out of the collision with the new momenta q₁ and q₂?

Fig 1: Feynmann Diagram of two electrons exchanging a photon

The easiest (most likely) way for this to happen is shown in fig. 1, this thing is called a Feynman diagram. Electron 1 emits a photon (the particle responsible for the electromagnetic force). This flies over to electron 2 with momentum k, and gets absorbed. We can use the principle of conservation of momentum to uniquely determine k. The principle states that total momentum must be the same at the beginning and end of all events. Applying this to electron 1 emitting the photon, initial momentum = final momentum implies p₁ = q₁ + k. Then, rearranging gets us to k = p₁ – q₁. Since we’re given p₁ and q₁, we can use this equation to work out exactly what k will be.

Quantum field theory can be used to work out the probability of each individual part of the Feynman diagram. Electron 1 emitting the photon, the photon travelling from electron 1 to 2 with momentum k, and electron 2 absorbing it. This produces the so-called Feynman rules, a translation between parts of the diagram and probabilities of each part taking place. The probability of the entire event can be found by just multiplying probabilities of each component event. The probability of the photon emission, multiplied by the probability of it’s travel to electron 2, multiplied by the probability of it’s absorption, gets you the overall probability. Nobel prizes all ’round.

But wait. This is not the only way you can put in two electrons of momenta p₁ and p₂ and get two electrons out with momenta q₁ and q₂. There are a number of different ways the two electrons could interact, in order to produce the same outcome. For example, this:

Fig.2: Feynmann Diagram of two electrons exchanging a photon which splits into an electron positron pair on the way.

The photon splits into two new particles, which recombine to return the photon. Similarly to before we know exactly what the photon momentum k is, using k = p₁ – q₁, and the values for p₁ and q₁  which we are given in the problem. But now, there is no guiding principle to decide what the momenta of the electron and the positron in the middle will have. We know that k₁ + k₂= k from conservation of momentum, but this is one equation containing two unknowns. Compare it to how we worked out k in the first diagram, in which case there was only one unknown, so we could use all the other known values (p₁ and q₁) to get the unknown one (k). If we fix k₂ by saying k₂ = k – k₁, we have one unfixed degree of freedom left, k₁, which could take on any value. k₁ could even have negative values, these represent the electron moving in the opposite direction to all the other particles.

k1 is not uniquely determined by the given initial and final momenta of the electrons. This becomes significant when working out the overall probability of fig.2 occurring.

To work out the overall probability, one needs to use the Feynmann rules to translate each part of the diagram into a probability, then combine them. The probability of electron 1 emitting photon, multiplied by the probability of photon moving to where it splits up, multiplied by the probability of photon splitting into the electron & positron, etc.  But this time, since the middle electron could have any momentum, one needs to add up the probability of that part for all values of k₁. There is an infinite spectrum of possible k₁ values so there are an infinite number of ways fig.2 could occur.

Let’s step back for a moment. In general, if there are lots of different events (call them E₁, E₂, ….) that could cause the same overall outcome O to occur, then the probability of O, prob(O), is

prob(O) = prob(E₁) + prob(E₂) + …

If there are an infinite number of ways O could occur, then it becomes and infinite sum of probabilities, and as long as each of the probabilities are not zero, then prob(O) becomes infinite.

This is what happens with our particles. Since there is an infinite number of momentum values the middle electron could have, there is an infinite number of probabilities that must be added up to get the probability of fig.2 occurring, so the probability of fig.2 is infinite.

What could that even mean? A probability should be a number between 0 (definitely won’t happen) and 1 (definitely will happen). Such predictions of infinite probabilities renders a theory useless, quantum field theory is doomed. The Higgs boson is a conspiracy invented by the Chinese.

Renormalization or How to ignore all your problems

This wasn’t the end of quantum field theory- since there is a way of resolving this problem. Kind of. The solution, or rather the family of solutions, are referred to as renormalization. It comes in many different manifestations, but it all boils down to something along the lines of the following. We pretend that k1, our unconstrained electron momentum, can only have a value below some maximum allowed size we’ll call Λ. Then, we don’t need to add up probabilities from situations where k1 goes arbitrarily high. We’re left with a finite number of possibilites, therefore a finite probability for the whole event. More generally, we can solve all problems like this by making Λ a universal maximum momentum for all particles involved in an interaction. Λ is called a momentum cutoff.

This solves the issue, we end up with sensible predictions for all processes. And as long as we make Λ suitably larger than the momentum of the initial and final electrons, the answer matches results of experiments to high precision. But I’ll understand if you feel a little unsatisfied by this. How come we can just ignore the possibility of electrons having momentum higher than Λ? To win you over, I’ll tell you a bit about what Λ physically means.

In quantum mechanics, an electron is both a particle and a wave. One of the first realisations in quantum mechanics was that the wavelength of an electron wave is inversely proportional to it’s momentum; wavelength = 1/momentum. A high momentum corresponds to a small wavelength, and vice versa. Ignoring particles with momentum higher than Λ, is the same as ignoring waves with wavelength smaller than 1/Λ. Since all particles can also be seen as waves, the universe is made completely  out of waves. If you ignore all waves of wavelength smaller than 1/Λ, you’re effectively ignoring “all physics” at lengths smaller than 1/Λ.

Renormalization is a “coarse graining” or “pixelation” of our description of space, the calculation has swept details smaller than 1/Λ under the rug.

Making exceptions like this have in fact been a feature of all models of nature throughout history. When you’re in physics class doing experiments with pendulums, you know that the gravitational pull of Jupiter isn’t going to effect the outcome of your experiment, so broadly speaking, long-range interactions aren’t relevant. You also know that the exact nature of the bonds between atoms in the weight of your pendulum isn’t worth thinking about, so short-range interactions also aren’t relevant. The swing of the pendulum can be modelled accurately by considering only physics at the same scale as it, stuff happening on the much larger and much smaller scale can be ignored. In essence you are also using renormalization.

Renormalization is just a mathematically explicit formulation of this principle.

The Gradual Probing of Scales

Renormalization teaches us how to think about the discovery of new laws of physics.

The fact that experiments on the pendulum aren’t effected by small scales means we cannot use the pendulum to test small scale theories like quantum mechanics. In order to find out what’s happening at small scales, you need to study small things.

Since particles became a thing, physicists have been building more and more powerful particle accelerators, which accelerate particles to high momenta and watch them interact. As momenta increase, the wavelength of the particles get smaller, and the results of the experiments are probing smaller and smaller length scales. Each time a bigger accelerator is required in order to accelerate particles to higher speeds, and each jump is a huge engineering challenge. This race to the small scales has culminated in the gargantuan 27km ring buried under Geneva called the Large Hadron Collider (LHC). This has achieved particle momenta high enough to probe distances of around 10 zeptometers (0.000000000000000000001 meters), the current world record.

Galileo didn’t know anything about quantum mechanics when he did his pioneering pendulum experiments. But it didn’t stop him from understanding those pendulums damn well. In the present day, we still don’t know how physics works at distances under 10 zeptometers, but we can still make calculations about electrons interacting.

From this point of view, it seems like we absolutely should impose a maximum momentum/minimum distance when working out the probabilities of Feynmann diagrams. We don’t know what’s going on at distances smaller than 1/Λ. We need to remain humble and always have in mind that any theory of nature we build is only right within its regime of validity. If we didn’t involve this momentum cutoff, we would be claiming that our theory still works at those smaller scales, which we don’t know to be true. Making such a mistake causes infinite probabilities, which suggests that there is indeed something lurking in those small scales that is beyond what we know now…

The road to the Planck scale

There are currently a bunch of theories about what is going on at the small untested length scales. We can make educated guesses about what scales these prospective new features of nature should become detectable at.

Fig. 3: Length scales

There has been a fashionable theory floating around for a while called supersymmetry, which says, broadly, that matter and the forces between bits of matter are in a sense interchangeable. It’s some well sick theoretical physics that I won’t go into here. The effects of this theory is believed to become visible at scales only slightly smaller than the ones we’ve already tested. It may even be detected at the LHC!

There’s a family of theories pertaining to even smaller sizes, called grand unified theories. These claim that if we can see processes at some way smaller scale, many of the fundemental forces will be revealed to be manifestations of a single unified force. The expected scale where this happens is about a billion billion times smaller than what we’ve currently tested, so will take a billion billion times more energy to probe. Don’t hold your breath on that one.

Finally, there’s reason to believe that there exists a smallest possible scale. This is known as the Planck length. If any object is smaller than the planck length, it would collapse into a quantum black hole, then immediately evaporate, removing any evidence of its existence. This is the scale where the quantum nature of gravity becomes important, and if you want to test that, you’ll need a particle collider 100 billion billion times more powerful than the LHC.

If we want to learn about these mysterious smaller scales, we’re going to need some mighty big colliders. Perhaps impossibly big. Maybe we need some new innovation which makes the probing of scales easier. Maybe the challenge for the next generation of particle physicists will be a rethink of how we test particle physics all together.

More on drawing Feynmann Diagrams

More on renormalisation

Supersymmetry

Grand unified theories

Planck length