LHC news: CMS released two papers with 20/fb of the 8 TeV 2012 data. Remarkably enough, both the dijet paper (Figure 5) and the dilepton paper (Figure 5) show 2-sigma excesses for a possible new object of mass 1,750 GeV or so. They're much weaker signals than what is expected from the typical theories for the objects searched in these papers but the overlap of these excesses is still a bit interesting.After a break, I answered a package of questions at the Physics Stack Exchange. It's sometimes fun and some of the questions are even interesting but there are also some omnipresent sources of frustration.
Let me mention some of them.
Pretty much every day, there is a question or two that tries to announce the discovery that quantum mechanics has been overthrown and/or may be employed to send faster-than-light signals, and so on. See e.g. user1247 yesterday. Or another question by the same user that tries to overthrow the postulates of quantum mechanics within quantum field theory "only". Or joshphysics who is convinced that observables can't be observable. And so on.
This is a topic that will probably never go away. A physics discussion forum that is open to the laymen must probably inevitably become a place for street rallies where protesters protest against what they hate about modern physics – and its very foundations, principles of quantum mechanics, are of course among the most hated aspects of modern physics.
Very rarely, one feels that they're starting to get the basic points which are really simple if you're impartial. All physically meaningful questions are questions about observables. All observables are represented by linear operators on the Hilbert space. The a priori allowed values of their measurements are given by the eigenvalues. Only the probabilities of each possible outcome may be predicted through a simple and universal formula, the Born rule – in one form or another. This is the framework for modern physics.
If a question can't be formulated in this way, as a question about probabilities of observables (linear operators), it doesn't belong to science. If an answer to a question uses different rules than the quantum-evolution-based and Born-rule-backed calculations of probabilities of propositions about observables (or some legitimately derived approximate theories that ultimately boil down to quantum mechanics), it's wrong. These claims have no exceptions, loopholes, and they don't allow any excuses. Everything else are "technical details" – what the observables are, how they commute or not commute with each other, how they act on various states and how they evolve – and one needs lots of experience. But the conceptual foundations may be summarized in the few principles fully captured by the few sentences above.
The foundations of quantum mechanics are almost never taught correctly, especially outside top universities. The teacher usually never understands modern physics himself or herself so even if the material is presented in some way, it's sold along with a completely wrong focus and "comments surrounding the material". It leads the students to reinforce their misconceptions. It leads them to believe that the foundations of quantum mechanics are a controversial optional luxury that may be replaced by something else at any point. It leads them to believe that \(|\psi(x)|^2\) doesn't have to be interpreted as the probability density.
Perhaps it's "ugly" and it may also be the number of sex partners that a woman has had, a voltage in a vibrator, or anything else. They think that everything goes and anything goes. They don't care that the probabilistic interpretation of the objects is a directly observable experimental fact and any modification of it would instantly invalidate the whole theory. Virtually every question about quantum mechanics starts with some deeply emotional words proving the writer's discomfort with some very basic principles of quantum mechanics.
Sometimes I tell myself not to answer because it's completely unproductive – it's just beyond the people's abilities; the emotions sometimes look so deep that I decide that the question was asked by a psychiatrically ill person who should better not be told the truth because it may lead to dangerous situations. Sometimes I react differently when I tell myself: It's just not possible that the folks are this breathtakingly stupid and can't understand these simple points. In the latter case, I am usually destined to a journey towards a pedagogical failure. The people simply are breathtakingly stupid, prejudiced without any limitations, closed-minded, thoroughly incompatible with modern physics.
But that doesn't mean that everything is pleasing about the questions by those who shut up, who kind of correctly use the formalism of quantum mechanics, and who ask different questions. As the title of this blog entry indicates, this blog entry is about the practicing, exercises, particular examples, applications of abstract physical knowledge. Needless to say, this is ultimately what physics and science is good for. It's the bulk through which physics manifests itself in the human society.
Physicists have the working knowledge of all the things in the Universe – OK, I mean all the important things in the Universe – but they only have it if they can actually think. If they know not only the laws and principles of physics but if they also operationally mean what the laws and principles mean and if they have mastered a sufficient chunk of maths that is needed to connect the statements about physical phenomena with each other and with the mathematical expressions, structures, and propositions.
At the same moment, I would agree that too much exercising becomes useless, too. Physics – and research-level maths – is ultimately not about following some procedures and protocols infinitely many times. We want to find new ones. When we have mastered some old classes of problems, we understandably lose our interest in them. This is how physics differs from chess or recreational mathematics. We just don't want to play chess millions of times because it's ultimately almost the same thing. We don't want to solve billions of Sudokus. A person who has the curiosity of a physicist simply wants to learn new things that are qualitatively different from the things he has already learned.
So I want to say that it is indeed natural if a physicist doesn't want to spend too much time with practicing the same thing. Engineers or athletes spend much more time by doing the same things all the time – which may ultimately be a good idea for practical or financial reasons. A physicist wants to get as far as he can in his mastery of the Universe or its chosen part. On the other hand, and this is what the title is about, a certain amount of practicing is simply necessary even for the most exercise-hating physicists because it's needed to guarantee that the knowledge is genuine and usable.
When we study a course or a book, it doesn't mean that we must understand every letter of it. An analogous claim holds for research, too. We shouldn't get completely stuck and terminate all learning or further thinking when we encounter first equation or question that seems confusing to us. After some attempts to lift the fog, we must live with it, remember that the confusion was there, and try to continue, probably using the assumption that the answer obtained differently (or the claim by the author of a textbook or the instructor in a course) is right.
There are also gaps that may seem technically demanding and we may sometimes uncritically accept claims by the instructor, textbook, or fellow researchers that "if you calculate this and that, you obtain this". Because of assorted sociological criteria, such a claim may look plausible and we sometimes need to save the time. I would say that a good scientist should ultimately rely on almost no examples of "blind faith" of this kind. He or she should verify and/or "rediscover" everything that he or she wants to use as a starting point for further research or calculations.
However, in some cases, e.g. when you're learning a body of knowledge someone has packaged for you and you were not necessarily interested in every product in the package, it's natural to skip certain aspects, especially if the other material doesn't seem to depend on them (much). But the gaps in our knowledge that is created in this way should stay "under control". You must "almost know" that if you will need to learn some method or verify the proof of a claim or something of the sort, you will return "here or there" and spend roughly XY hours of time and you will be done.
I would argue that these gaps must be "mostly exceptions", perhaps like the holes in a Swiss cheese that still holds together. I don't say that the critical percentage is 50% but there is some rough percentage and if the number of holes is just too large, the Swiss cheese of your knowledge decays into hundreds of Swiss cheese balls that don't hold together.
When the amount of ignorance and the number of holes grow too large, you not only fail to know many particular things but you also lose the idea about how many things you're actually ignorant about and how to ask questions that would give you a chance to fill the holes, and so on. Your physics knowledge becomes unusable.
I had this feeling when user6818 asked four questions about Polchinski's textbook on string theory. How to derive the Green's functions on a sphere, disk, projective sphere, why this is canceled, and so on? The questions are based on Chapter 6, Volume I.
Superficially, they're legitimate, even high-brow questions. What's problematic about them is that every person – even person who knows no physics, not to mention string theory – could ask such questions. You pick an equation in a book and ask "Why is it true?" But in physics, statements such as equations for Green's functions don't have simple, generally understandable (by everyone), and self-explanatory one-sentence answers. They're derivations that may be compressed or need to be inflated depending on one's knowledge or experience or the lack of it, derivations that always depend on lots of other background.
When one asks why is [relatively straightforward] eqn. (6.2.17) right – without specifying any details about his or her confusion etc. – it suggests that he or she has made no attempt to derive the equation. And it seems that he or she probably can't solve or prove even similar, simpler equations. And although I don't have any rigorous proof, I would bet that these worries are justified by the facts – in this case and many others.
To meaningfully answer such questions and to have an idea how many details the answer should discuss, one needs to know how much the person who asks something actually understands. Does he understand why the logarithm solves the Poisson equation in two dimensions? Can he use substitutions while solving differential equations similar to the known ones? Does he know how to calculate Gaussian integrals? Does he understand why the holomorphic functions have a vanishing Laplacian? Does he know that the sphere is conformally equivalent to the plane, that the disk is conformally equivalent to the half-plane, that the projective sphere may be obtained by identifying opposite points on the sphere as well? Does he know what the boundary conditions at infinity must be for the plane to conformally represent the sphere? Does he really want to explain the blind mechanical calculation proving some independence on the conformal factor or the conceptual reasons? Does he understand the general concept of "solving differential equations", especially the fact that there exists no mechanical procedure that would lead to the solution of any equation?
There are tons of questions – the list above isn't complete. If the answer to some of the question(s) above is "No", the thing may be given a meaningful answer. But if someone just asks why is (6.2.17) and three more formulae right, it looks like the answers to all the questions above are "No". What I mean is that the person must be taught complex numbers, calculus, integrals, conformal transformations, symmetries in physics, two-dimensional Riemann manifolds and their relationships, and many other things pretty much from scratch, and in a more detailed way than Polchinski's book. (Of course, the equation (6.2.17) isn't the first equation where the ignorance about any of these topics should show up – but that is just another detail that makes the question about a seemingly random equation in the middle of the book slightly more surprising.) That's a big task, however, because Joe Polchinski spent a decade by writing his perfectionist textbook on string theory so you may need 50+ years to write the more detailed one.
But would it make sense? I don't think so. If someone really needed to explain all the things above – and I don't claim it's the case of user6818 – he or she simply shouldn't study Chapter 6 of Polchinski's book because he doesn't have the background for that. It's useless to learn some material if you can't use it. And if you can't understand how the material is derived – and the derivation is just an application of a "simpler" material you should have known before – then it shows that you probably can't apply this material (because you can't apply even simpler one) so it's useless to learn it.
The number of questions on Physics Stack Exchange where the answer could very well be 100 times longer than the question itself is significant. (A particular user named Anirbit has asked 60+ questions that mostly fall into this category: "Explain everything in a paper to me".) I am afraid that it is a waste of time to be answering these questions – questions of the type "explain every line of a paper or chapter to me" even though the paper or book have already been written with the assumption that they speak for themselves. A meaningful communication and explanation only occurs when the two sides are at least a little bit on the same frequency and/or if the "teacher" has some idea about the things that the "student" knows. Without such a context, it is impossible to teach physics. And if you teach physics to someone who will ask you "Why it is so?" almost after every claim he sees (and sometimes again – many times – after you already answer it), it's like training armless boys to become construction workers. A teacher may build a house out of bricks but it won't make any impact on the skills of the student simply because he can't do it. So if the "pedagogical house" is useless, you may better avoid this futile pedagogic exercise.
In this respect, physics differs from literature or many other subjects – that often include natural sciences – where the structure of the background isn't hierarchical or is much less hierarchical than in physics. In other words, you don't need much and you may immediately learn some isolated insight from advanced research. You may have never read a book but you may memorize two sentences from a play by Shakespeare and some naive people will instantly think that you're a cultural human being. And so on.
But in physics, this doesn't work. String theory is arguably the tip of a pyramid of knowledge that has almost as many floors as the Empire State Building, if I count if in a fine-grained way. Memorization of an isolated insight or rule is almost worthless in physics because the meaning and power only emerges when many prerequisites are understood.
Needless to say, this is why some people hate physics – and maths – at school. If I don't count gyms and similar things, almost all other subjects at school are about memorization, a universal method that requires lots of RAM (or hard disk space) and almost no CPU or GPU, if I compare the students to computers. At most, some subjects require that the students learn how to follow a relatively mechanical procedure or two to "derive something".
Paradoxically enough, it's the same people who don't like maths and physics for their "requirements of creativity and practicing" who most frequently complain that maths and physics are mechanical, dull, narrow-minded, isolated from practice, and that they reduce people to mindless mechanical machines. When you look rationally at the situation, you notice that the truth is exactly the opposite. These critics of maths and physics are the mindless unthinking machines that only do mechanical things and they hate maths and physics exactly because they can't be mastered in this way! ;-)
But I was thinking about folks who would never open Chapter 6 of Polchinski's textbook, of course. ;-)
Let me return back and say that to learn a subject such as quantum field theory or string theory, one needs to practice, rediscover, verify his own predictions (against well-known insights if not experiments), and think about the implications a lot. Ignorance about particular things is permissible if not inevitable in science. But even ignorance has to be tamed and brought "under some control". We must have an idea how many things we probably misunderstand, how many things about them may be known to other people, how or where to find the answers if necessary, how much time it may take to find the answers or verify some results, and – perhaps most importantly – we must roughly know to what extent the things we're ignorant about may affect the things we think we know (where are the boundaries of our ignorance).
The holes in the Swiss cheese should never break our knowledge into a large collection of disconnected marbles. If that happens, physics ceases to be physics. It ceases to be the lively mechanism to find and incorporate all important insights about everything in the Universe.
And that's the memo.