All You Need to Know about Fundamental Quantum Mechanics

Quantum mechanics has got to be the hottest physics theory ever. Every successful theory in science is popular within that science - and quantum mechanics is the most accurate scientific theory of all time to date - but the popularity has decidedly left the Ivory Tower of academia. From quantum dishwashing to quantum real estate and practically everything in between, quantum mechanics seems to be the brainy version of adding sex appeal in marketing.

I can only imagine that advertising is the main reason for the flamboyant use of the word quantum, since it is essentially a fancy word for "smallest amount of some physical entity." For something to be quantised  means that it is made up of small indivisible quantities - for instance, spin of particles is quantised such that, for instance, electrons can only have some integer multiple of a half spin. So either these quantum products are "quantum" for their sexiness, or they are somehow quantised in such a way that is don't-ask-me-how relevant to the consumer. I suspect the former.

The other side of quantum mechanics (QM) in popular culture is not on the market but in the way people use QM to intellectualise their odd views about the world. For instance, there is esoteric views of "Quantum Healing" espoused by Deepak Chopra or wacky New Age perversions of the theory. Of course, to a physicist, all of that misuse of science amounts to a grand "quantum flapdoodle" (to use Murray Gell-Mann's term). In general, it is wisest to follow the advice of xkcd on this point when speaking to someone who has no background in actual physics:

But what if you do not want to be in the riff raff ignorant of quantum mechanics and rise to the lofty heights of someone who can honestly claim to not understand it? The fact of the matter is, quantum mechanics is weird because it operates on a different logic to what we are used to in "classical" physics. By all means, baffle your brains out by trying to picture interference patterns from double slit experiments with buckyballs (sixty carbon atoms arranged like a football) in terms of it acting like a wave and a particle. When you are suitably confused by that, perhaps you will appreciate that understanding the logic of QM gives far more insight, I think, into why QM seems so outlandish to us.

For those that are averse to mathematics, a career in quantum mechanics is not for you. Like most theories in physics, quantum mechanics has two parts:

1. Equations. 
2. Interpretations which explain how the symbols in the equations relate to real world phenomena.

But before you run away screaming that there are equations in physics, there is still some insight you can glean without actually looking at equations directly (something which some people avoid as much as looking at the sun for fear of burning their eyes). To begin with, the equations of QM are pretty well established. Unfortunately, the second element is not fundamentally agreed upon. Certainly, physicists can use the equations to make extraordinarily accurate predictions about the world from a "plug-and-chug" point of view, but there are wild divergences over what exactly is happening under the quantum mechanical bonnet. So a truly and absolutely conservative argument about quantum mechanics should probably only involve what the theory predicts - which unfortunately, has absolutely nothing to do with dish-washing, real estate, healing or, the darling of many quack worldviews, consciousness.

Still, the basic structure of the equations of quantum mechanics explains why we find it so un-intuitive: in QM, systems are described by states in Hilbert space. By contrast, classical systems are described by points in phase space. Even without really understanding what the Hilbert and phase spaces are, the underlying point is that the way we have to think about the inner workings of quantum compared to the more intuitive classical mechanics is fundamentally different. It would be, to use a crude analogy, like asking a car mechanic to work on a space shuttle: there are obviously certain similarities, but at the end of the day, car engines run off explosions which move pistons whereas rockets shoot fuel out their rear end to go forwards; they are incommensurate.

Let me finally state all the fundamental postulates of quantum mechanics:

• Physical systems are described by states in Hilbert space.
• Those systems evolve according to the Schrodinger equation (or equivalents, such as Heisenberg matrix mechanics and Feynman path integral formulation), given some Hamiltonian for the system (where the Hamiltonian is the functional of energy).

That is all. There is a rule for defining what something is and a rule for explaining how it changes in time. The state in Hilbert space is called the wavefunction and Schrodinger's equation is a wave equation, must like the one you would use in classical mechanics to describe a wave on a string, for instance.

Some physics-literate people may protest that I am missing an extra postulate. You see, part of the charm of quantum mechanics is that Hilbert space is mysterious and hidden. This means that you cannot actually measure wavefunctions - and this is quite the problem, because if you cannot measure wavefunctions and you hold that wavefunctions are what describe physical systems, then it would seem that physical systems cannot be measured. That has to be false, though, because we clearly measure things all the time. So they add in another postulate which explains measurements:

• The probability of measuring a system to be in some possible state is given by the Born rule, which "collapses" the wavefunction - in other words, measuring a system makes the wavefunction look like a very sharp spike at the value you measured.

Side note: I will not go into why I think the Born rule is an unnecessary addition to the theory other than to say that I think the Everettian Quantum Mechanics is correct.

You can write that all in terms of the mathematical formalism, which is of course a necessary step, but if you leave this blogpost understanding nothing more than that fundamental difference between quantum and classical mechanics (ie, Hilbert vs. phase spaces), you will understand more than practically anyone outside of science. But why leave maths out of it when you can put it in for good measure? Here are the postulates in their mathematical glory:

Physical systems are described by states in Hilbert space which are written in Dirac notation with "bra"s (1) and "ket"s (2) (which together make bra-kets, or brackets):
$$\begin{equation}\label{eq:bra}\langle\phi\rvert \end{equation}$$ $$\begin{equation}\label{eq:ket}\lvert\psi\rangle \end{equation}$$

The bras are just the Hermitian conjugates of the kets - they correspond to the same vectors in the opposite sides of dual space.

Observables are the things you measure, and in quantum mechanics, they are described by Hermitian ("self-adjoint") operators such that: $$ \hat A = \hat A^\dagger$$

The possible results of measuring some observable are the eigenvalues of that operator. In other words, if you take the momentum operator: $$\mathbf {\hat p} = -i\hbar\mathbf{\nabla}$$

These systems evolve (change over time) according to the Schrodinger equation:

$$\displaystyle i\hbar\frac{\partial \psi}{\partial t} = \frac{-\hbar^2}{2m}  \nabla^2 \psi + V \psi$$

Or compactly and in terms of bras and kets:

$$i\hbar\frac{\partial\lvert\psi\rangle}{\partial t} = \hat H \lvert\psi\rangle$$ 

Finally, the Born rule can be written as (where x is just standing in for any observable, not necessarily an x coordinate):

$$P(x=a) = |\langle\phi(a)\rvert\psi(a)\rangle|^2$$

There you have it; that is all there is to fundamental quantum mechanics. Use your knowledge for good.

Implicit Bias and Scepticism

I listened to a fascinating lecture given by Jennifer Saul at the Royal Institute of Philosophy in the UK which brought up an excellent link between the psychological study of implicit bias (for which she gave numerous examples) and its implications for epistemology, in particular, scepticism. I recommend giving the talk a listen (YouTube link here) because it goes into much greater detail than I will here. I am going to focus on the fascinating link Saul made between the fact that we know we have these ingrained biases with our inability to overcome them by willpower and its implications for knowledge and rational thought.

Saul tentatively defines the term implicit bias as the collection of largely unconscious associations which people are prone to which affect how we perceive and interact with the world. The biases are not quite beliefs, they are not conscious, and yet they affect our thought processes. They are not mitigated by stated beliefs: for instance, the famous African-American civil rights activist Jesse Jackson said:

“There is nothing more painful to me at this stage in my life than to walk down the street and hear footsteps and start thinking about robbery. Then look around and see somebody White and feel relieved.” (citation)

That is implicit bias in action, and the painful experience of Jackson is not only common, it is practically universal. Unlike in Jackson's case, it is almost always sub-conscious. This presents an unfortunate sceptical problem: we have good reason to believe that our faculties are deficient when it comes to decisions that we believe are made rationally.

Jackson's example is one of a feeling produced, but others may bring out the force of the problem more clearly if they highlight how it is our conscious decision making that is affected by these ingrained biases against people of particular races, genders, sexual orientations and so forth. But the research is clear that there are pervasive and insidious cases of discrimination among self-labelled egalitarians who, try as they may to make calm, reflective decisions when choosing between biased options, still tend to make the biased on - whether it be gender, racial or even height discrimination, among countless other features to discriminate upon.

The social problem is clear in that these implicit biases produce stagnant structures of discrimination. However, there is a more philosophical problem that Saul highlights:

"I will be arguing that what we know about implicit biases shows us that we have very good reason to believe that we cannot properly trust our knowledge-seeking faculties. This does not mean that we might be mistaken about everything, or even everything in the external world (so it is weaker than traditional scepticism). But it does mean that we have good reason to believe that we are mistaken about a great deal (so it is stronger than traditional forms of scepticism). A further way in which bias-related doubt is stronger than traditional scepticism: this is doubt that demands action. With traditional scepticism, we feel perfectly fine about setting aside the doubts we have felt when we leave the philosophy seminar room. But with bias-related doubt, we don’t feel fine about this at all. We feel a need to do something to improve our epistemic situation." (citation)

I recommend giving a long and hard think to the paper she delivered (cited just above) because unfortunately, this is not a problem that exists external to us as individuals: the problem is me, the problem is you.

Laws of Nature as Patterns

An old definition of science could be that science is the study of cause and effect in nature. The idea that causality is the central idea in knowledge goes back at least to Aristotle's four cause analysis of the world and in many ways this is what people believe science does today. Instead of directly citing cause and effect, it seems to me that the intermediate notion of the "laws of nature" is interjected; science studies the laws of nature, which are about cause and effect in nature. Simple, right?

I want to propose an alternate idea based on how I view science and in particular, based on how I see modern science handling data. There's a caveat before I begin: as usual, I think much more about the canonical status of physics in science than I do of other equally legitimate branches of natural science. So what I say relates most to physics and (perhaps) the applicability slowly decreases as we move towards more qualitative sciences. Or it might not - my point is that I am making a stronger case for science as most exemplified in the hard, mathematical sciences.

Scientists do experiments and get data. This data is put often into tables or graphs and analysed, whereby models are produced which explain the correlations in the data by causal mechanisms. The model, if it is a good one, will make other predictions which can be tested to see if it is correct, and the more it passes those tests, the more credence it is given. That is the scientific method of observation, hypothesis, testing, conclusion. There are two very similar problems I see with this process: an error in data analysis and a deeper philosophical issue that Hume would have noted.

It is drilled into every data analysis/scientific statistics student that a correlation does not mean a causation and yet this explication of the scientific method clearly makes that jump. The justification is simple: eliminate as many variables as possible and the remaining correlations must be causation. That is simply mistaken and the history of science is rife with examples of deeper explanations being found of natural phenomena which destroyed the previously perceived causal mechanism.

Hume would have heartily agreed with this objection and would defiantly disagree with anyone who tried to maintain that, whilst correlation does not equal causation, a lot of correlation does, in fact, equal causation. His problem was two-fold: the assumption that the correlations of the past will hold in the future is only based on the observation that the correlations of the past have so far held true in the future. But that is circular, since in effect it says that the future is causally equivalent to the past because the past is causally equivalent to the past. But again, Hume had a deeper problem that simply the problem of induction. His biggest reason for scepticism is that causality was not a superficial relation between objects, in fact, we never really perceive causality at all, we perceive effects and infer causation. He called this "customary conjunction", or basically, correlation. For all this scepticism (and Hume unlike others branded with this title really was a sceptic), Hume did still believe in cause and effect, he simply thought it was beyond our knowledge.

I am not sure if I believe in cause and effect, but I propose right now the weaker claim that science does not study it. Science studies data to produce laws of nature which are expressed mathematically because, at bottom, the laws of nature are patterns in observable variables or parameters. In other words, the laws of nature are patterns of numbers that describe nature. Let me give an illustrating idea to stir the intuition of this proposal and consider how this relates to epistemology and the metaphysics of causality (if causality exists at all).

Think of the positive integers: 1, 2, 3, 4, ... They form a nice set whose properties can be analysed to yield fascinating mathematics. Relations can be defined on this set giving it order, taking a pair of numbers to another in the set by multiplication or addition, and so on. The tools of mathematics are about seeing mathematical structure in the integers, seeing what symmetries it might have, trying to see what patterns it has. A simple pattern in the integers is that the numbers are alternating odd, even, odd, even.

What if we tried to apply the tools of old fashioned science to the integers? It would yield fictitious language about causality to something which exists independent of causes: the fact that 2 is even is not caused by the fact that 1 is odd, even though we could conceivably speak of it that way. Many of the properties of the positive integers can be spoken of in terms of causality, but it is a fiction of our language, not a fact about the set itself.

So, I claim, it is in science. Equations like Newton's laws are mathematical statements which should not be thought of in terms of causality but in terms of patterns. It is still possible to make if-then statements: If a force is applied, then an acceleration will occur. That is a statement about what the second law predicts and codifies. It is also important to note that I am not simply saying "science produces equations that have no necessary connection to what really exists." This view is not scientific anti-realism, it is congruent with a critical realism about science.

It is important to dispel the objection that I am merely playing with words and the if-then statements are exactly equivalent to causality statements. But I refer back to the case of mathematics: it seems clear that "if you add one to an odd number you get and even number" is not equivalent to "adding one to an odd number causes an even number." The first is true, and the second uses perverted language to try and express, it seems, the first statement.

The pattern view has several advantages: for starters, it is epistemically conservative and codifies what the science actually shows rather than trying to make it jump over the correlation-causation barrier in the data. It is robust to quibbles over the meaning and nature of causality, in particular, it allows for a less stringent requirement on the necessary conjunction between a state of affairs and its antecedents; the pattern view may allow, once specified, the derivation of if-then statements, but it does not have to. Patterns can in principle be random or ordered.

It also avoids infinite causal regress problems. It seems intuitive to some (though not all) that causal chains must have a beginning (whether the causal series be per se or per accidens as Thomists would distinguish). But it does not seem to be obvious that patterns need to have a beginning: sure, the pattern of odd, even, odd, even in the positive integers has a beginning because it has a first member. But the integers have no first member because they stretch from negative infinity to infinity - and yet they still have the pattern of odd, even, odd, even. What is not definable is whether the first element was odd or even, because there is no first element.

But even if we take the positive integers, it still does not need for there to be something before the first member for the pattern to continue ad infinitum. To ask "what caused the first member of the positive integers" is a senseless question. Similarly, it may very well be the case that nature had a beginning and it is self-contained. The objection that "but it had to have had a cause, science demonstrates that!" is simply not true. The universe could be just like a set which starts at time zero and continues to infinity without  quibbling over whether there was anything at t = -1.

Periodization in Swimming

I was swimming at UQ for a month or so before I noticed a pattern: Wednesdays were a lot tougher on me than Tuesdays. I quickly then figured out that the regular squad training sessions followed a week-long plan of sprints on Monday, technique focus on Tuesday, heart-rate set on Wednesday and aerobic set on Thursday (see here for a bit on different types of sets). That was my introduction to the idea of periodization: a one week repeating cycle hitting the major systems in turn.

Since then my training periodization has developed a lot more depth and the one week repeating cycle no longer suffices. Periodization is about creating a plan for the training season with variations of focus in training. Before discussing periodization it is useful to define a few of the key words.

Macrocycle: the longest type of cycle, for many swimmers the macrocycle is about a year, or encompassing the whole season (of whatever length that may be).

Mesocycle: the mid-length cycle, mesocycles are around four to eight weeks and are based around the idea that it takes about six weeks (give or take) to produce significant adaptations to a training regime.

Microcycle: the shortest cycle, microcycles are usually a week long. In my experience, the biggest reason for incorporating the idea of a microcycle is to ensure that swimmers have sufficient rest - by the time I get to Saturday morning training, for instance, I am giving the last of what is in my training-tank for the week, so I really need the weekend rest to recover for the next week of training starting Monday morning.

All coaches use some kind of periodization unless the training sessions are literally random. Even the simple week progression I started with at UQ's adult squad sessions were a (basic) form of periodization. But periodization has benefits far beyond making it easier for the coach to pick what kind of set the swimmers will endure! As I see it, there are two main features of periodization:

1. Training is about overloading a system so that it adapts for the better, and a single training session cannot achieve this. So a benefit of periodization is to decide that the first mesocycle of the season can be dedicated to, for instance, endurance work. This provides enough of a stimulus for the swimmer to positively adapt and improve.

2. Periodization allows the coach to plan when the swimmers will be at their peak. Often there will be some kind of end-of-season meet where the swimmer will want to achieve their maximal performance, and intelligent planning allows the training schedule to produce a well-timed peak.

Having noted the importance of good periodization, what is a coach to do? A plethora of periodization plans exist, but they tend to fall into one of two camps:

- Linear: This is your most intuitive kind of periodization plan, where you take each mesocycle and dedicate it to a particular system. A coach might start with a mesocycle dedicated to endurance, then another to V02 max training, then heart-rate, then speed work and finish off with a week long taper before peaking at the end of the season. The major benefit to linear periodization is that it incorporates an ideal length of overload and allows the swimmer to peak at the right time of the season. This comes with drawbacks, however: the idea behind linear periodization is to peak at the end, so in-season racing is going to be decidedly subpar. Additionally, there is the possibility of atrophy of systems as time progresses and it has been ever-longer since that system was worked - in the example, endurance might be flailing by the time speed work is begun. Finally, what if injury or personal circumstance leads to missing a few weeks of practice? Suddenly an athlete may have missed training an entire system crucial to their race performance.

In an attempt to mitigate these factors, the other camp suggests:

- Non-linear: Instead of linearly progressing through the systems, the non-linear plan says to focus on multiple at a time, whilst still varying over the course of the season. Sometimes referred to as an undulating periodization, this type of plan rotates through more frequently over the course of a season, and so allows for multiple peaks (whilst some kind of maximal peak can still be achieved by taper at the end of the season). It also allows for more flexibility when swimmers miss blocks of training. It seems, however, to have mitigated one of the major benefits of the linear plan: there may not be enough training stimulus for highly trained athletes to improve. Furthermore, some types of systems' training are hard (if at all possible) to combine: sprint sets and endurance/aerobic sets are fundamentally different, so producing training adaptations in both seems difficult.

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Added to these considerations, the coach needs to address the particular needs of the individual swimmer. I, for instance, would probably benefit from an increased focus on endurance work. This would allow me to have a greater capacity for training, so provide a platform for training the other systems. I furthermore breakdown in breaststroke over too long a distance, so it would give me the ability to push through fatigue with proper form (particularly turns).

Outside the pool, there is also much to be said for the periodization of dryland activities. That in itself is a universe of controversy - some coaches even reject the need for dryland!

How to Elegantly Refute Astrology with Quantum Mechanics

Most people reject astrology out of hand these days, saying that the alignment of the planets does not affect our lives whatsoever. If pressed, they would defend themselves perhaps by claiming that astrology is unscientific or stubbornly say that it is just wrong.

The idea that the alignment of the planets has no effect on us is, in fact, wrong. It is very much a scientific point of view that these sorts of celestial bodies and their positions can influence us from afar and we have known that this is the case for quite some time. A point for the astrologers. Unfortunately for astrology, the known effect of those celestial bodies is from their gravitational fields and the influence that their gravity has on us is vanishingly small. Sure, it's not zero, but it is dwarfed by the gravity from the table closest to you, and does not act at the moment of birth in particular. No astrologer could therefore point to gravity as a means of affecting your life by the alignment of planets.

But, the astrologer might come back with a certain scepticism of the science. Sure, they might say, science knows of no forces that can act at that sort of range outside of electromagnetism and gravity. Sure, there is no contribution really from electromagnetism or gravity. But that does not mean that science will not one day discover a new force which shows that the planets affect our lives in meaningful ways. So, the astrologer says, we should remain agnostic of astrology.

Sean Carroll has supplied to me the insight to finally put to bed that claim. He points out that, with the discovery of the Higgs Boson, we now know all the fundamental physics relevant to our lives. Sure, there's dark energy and dark matter - but that does not affect our everyday lives. Sure, there could be forces operating below what we can detect - but if we cannot detect them, they cannot influence us. If they could exert some kind of cause on us then we would be able to detect it. Is it really possible to make that claim?

It relies on a very simple idea: the world behaves as described by quantum mechanics and we know that if there were some other force we would have found it. In fact, the physics relevant to our lives has been condensed beautifully to a few particles and forces. We are made of the kind of stuff (neutrons, electrons, protons) that are affected by certain forces (gravitational, electromagnetic and nuclear). Could there be other forces? Definitely. But they would not interact enough with our ordinary matter to be of causal significance. Are there other particles? We know for a fact there are. But we are not made of those other particles. Nor is any of the stuff we commonly use.

Carroll explains by means of a quote from Donald Rumsfield that there are three levels of knowledge: the known knowns, the unknown knowns and the unknown unknowns. These are the things we know that we know, the things we know that we don't know and the things that we don't know we don't know. The beauty of this refutation of astrology is not in pointing out that astrology is not a "known", because we already knew that. The beauty is that we now know how far our knowledge extends and the range of our knowledge excludes astrology.

The claims of astrology are like claims of an elephant in my kitchen. I know that the microwave is there, I can see it  - a known known. I do not know whether there is any salt in the pantry (a known unknown). But despite all the things I do not know about my kitchen, I know there is not an elephant. There is no astrology in the universe. Astrology is wrong.

Relations as Sets

I recently came across a powerful idea in maths which could have far-reaching consequences for how I conteptualise things even in every day life (insofar as I think about things logically in every day life!) The idea is really simple: relations between two things form a set of ordered pairs. These can be numerical, conceptual, whatever you want! Let's get started on some examples to illustrate what I mean:

Simple maths case: Fractions

Fractions are relations between two numbers, the numerator and denominator. Take the simple example of a half, \(\frac{1}{2}\). This is the ordered pair $(1,2)$. In this set (currently just containing $(1,2)$) the first number indicates the numerator and the second number is the denominator. In fact, you can easily create an infinite set full of halfs by defining the equivalence class as all those ordered pairs which give \(\frac{1}{2}\): \(\frac{2}{4}\), \(\frac{4}{8}\), \(\frac{-8}{-16}\), etc. This would form the set of ordered pairs: $(1,2), (2,4), (4,8), (-8,-16),$ ... So you can see that something simple like \(\frac{1}{2}\) creates a whole set based not on numerical equivalence but on the relation between the first and second number - the denominator is double the numerator. Relations make sets.

But it's more than just mathematical relations which produce sets. Any relation does.

Real life cases: Friendships

You could create a friendship set with all the different pairs of friends. Say Alice and Bob are friends - then there would be an element in that set: $(Alice, Bob)$. Maybe you think all friendships are reciprocal - so if Alice is friends with Bob, that implies that Bob is friends with Alice. Then if $(Alice, Bob)$ is in the set, so is $(Bob, Alice)$. That's true for Facebook friends, at least - there are no one-way friends on Facebook. That's not the case for followers on Twitter, though: you can have the follower set $(Follower, Followed)$ where $(Alice, Bob)$ but not $(Bob, Alice)$.

I find this kind of interesting because it means friendship groups and networks can be thought of in terms of sets and analysed accordingly. So far though, I have only described ordered pairs. It is pretty simple to generalise this to sets with relations between any number of things. This would increase the dimension (ie, the number of elements within the brackets of each object). Let me list an example of sets of different dimensions to do with friendship:

Dimension 1: The set of all my friends is not a set of pairs but of individuals. So my friendship set might look like ${Me, Pillow, Brother, ...}$
Dimension 2: As before, the set of friendships has two elements (even if they are the same element - I am one of my best friends, so $(Me, Me)$ is an element of the friendship set).
Dimension 3: An example of a three dimensional friendship set could be friendship triplets, but it would probably be more interesting to create a set which included some different relation about three people. Three people in a love triangle might make a three-element per object set. It could be $(Lover \#1, Lover \#2, Beloved)$. So the set of all love triangles is a three dimensional set. Or another might be an introduction set, where Alice introducing Bob to Catherine would be the element $(Alice, Bob, Catherine)$.

And so on. No matter how complicated the relation, I suspect it could somehow be represented as a set. This is just a curious way of looking at relations when it comes to everyday life, but it holds particular significance for computer science. Here's an example:

Functions and their Plots

A function can be thought of as the relation between an input and an output. Viewed that way, functions create sets of ordered numbers - say you have $y = f(x)$, then over all the domain of $x$, there is some output y which creates all the points $(x,y)$. If the function is something simple, like $f(x) = x$, then the set would contain elements like $(1,1), (2,2), ( \pi, \pi)$, ...

Just like before, you can get higher dimensional sets from functions by just having more input variables. If $z=f(x,y,t)$ then the element would be $(x,y,t,z)$. This is exactly how a mathematics software like MATLAB/Octave graph their functions. You define all your $x$ and $y$ and you enter the command $plot(x,y)$ to give you a plot with the mathematical relation of the ordered pairs $(x,y)$. You could ask Octave to just give you $(x,y)$ as a matrix specifying all the elements of the set (discretely, of course - computers do not deal with continuous functions when doing numerical calculations).

Why Cares?

Most people will not care, and will just be happy to know that the mathematicians, scientists, engineers and computer scientists will use this idea when necessary. However, I think it does offer a cool way of looking at relations between objects and it gives a formalised representation of the relationships, which means it is easier to analyse formally and mathematically. It is a lot harder to be logically sloppy when dealing with set theory directly!

The Types of Sets in Swimming

Training at an elite level, or to achieve an elite level, is not easy. It is incredibly taxing physically, mentally and socially. In terms of time, swimming training is particularly intensive - for me, it involves two swimming sessions a day from Monday-Friday, around half an hour of dryland (every form of exercise not done in the pool is referred to as "dryland") each of those days before the evening training session, plus an hour in the gym working on strength on Tuesdays and Thursdays. Then on the weekend, there's a slightly longer morning training on Saturday, half an hour of core-specific work right afterwards then and I'm done for the day. On Sunday, I go to another couple of dryland sessions, one focusing on mobility and flexibility in the morning, the other a faster-paced strength session in the afternoon.

So even if you include the daily half-hour dryland into the evening session, that's sixteen distinct training sessions a week. Which is a lot. But swimming training is not just about hard work; it's also about smart work. I do not think coaches are given enough credit for the time and expertise that goes into structuring and implementing a training program: in the past, when the science was ambiguous, coaches trained their athletes in the dark - quite the challenge - and now there is so much science that a great coach is a very intelligent person having invested enormous time resources resulting in significant expertise in all sorts of sports science ranging from bio-mechanics to psychology.

Swimming sets are probably a little less complicated than metaphysics.

Not being such an intelligent coach (or a coach at all, for that matter), I cannot claim to know that much about the complicated thought-process that for some coaches has become second-nature in creating a training schedule. There's an aerobic season and a speed season - but you're always training aerobically and speed within both seasons. There's the complicated taper time. There's all sorts of types of training and with each, a range of opinion among the swimming community as to what works best - is USRPT really all that it's cracked up to be? Should garbage yardage be the backbone of a swimming regime? How much time should be spent on dryland? I might have opinions but I do not have answers.

I want to start a series of posts explaining the logic behind the training program that my coach(s) implement with me. To start off with, it's important to understand the kind of sets that make up training in the pool. In the pool, I would say there are really four distinct categories of set and a fifth miscellaneous set which combines elements from the others. These are:

- Drill sets: Here I work on the technical aspects of the stroke and of the race, focusing on making sure the technique is conducive to the fastest times by reducing drag, ensuring that the stroke is effective, and so on. Whilst usually less physically demanding, they are important to set the stage for optimal swimming. Water is an unforgiving medium with much more drag than air to slow me down, and an intense training schedule means poor form repeated over hundreds of kilometers is bound to result in injury. So perfecting technique is crucial to swimming.

Perfecting the breathing yawn face is a lifelong process.
Alas, no medals are given for how bored snapshots of swimmers' faces look.
(Photo: Swimming World)

- Aerobic sets: These are longer sets where, as I understand it, the aim is to be at 70-80% of my maximum heart rate and keep it there for a prolonged period of time. This improves both my aerobic and cardiovascular capacity, or in other words, improves my lungs and my heart. Even though I am a breaststroke specialist, I do not have special organs for breaststroke as opposed to every other stroke, so I will almost always do aerobic sets with freestyle. This allows me to rest the breaststroke specific muscles, develop a more rounded body, and still train my ability to use oxygen and my heart effectively. But for the heart there is also...

- Heart rate sets: These are some of the toughest sets, in my opinion, as they are mid-to-high intensity in terms of speed with minimal rest in between. Heart rate sets are about getting the heart rate up and keeping it there, so as to maximise the ability of the heart to work effectively under physical duress. They may not be as long as aerobic sets, but they are significantly faster.

- Sprint sets: My personal favourites, sprint sets are about going fast for a short amount of time and distance. Part of the reason I like sprint sets so much is that I can vary elements of my race and get instant feedback on how it impacts my performance - as the coach says, the clock is your best friend, who is always there and never lies to you. More than just testing fields for subtle technique changes, sprint sets are about putting together the rest of the training to make sure that it all comes together in an effective and fast end-product. Drill sets may be important, but two-kicks-one-pull breaststroke will never be an Olympic event!

- Combinations: Whilst I think the above four categories cover the main types of sets, it is important to understand that they can be combined. Heart rate sets, for instance, can be made longer (more like an aerobic set) or shorter (more like sprints). Sprint sets need not be full-stroke, they could be sprint kick sets, for instance - and kicking is basically a drill. Similarly there could be aerobic drill sets.

A typical session in the pool will, for me, usually involve some sort of drill set and usually one other element from these categories (sometimes two). How do you arrange the focus of a training session over the course of a season and its off-season? That's another question.