“The Strong Law of Small Numbers” is a humorous paper by mathematician Richard K. Guy and also the so-called law that it proclaims: “There aren’t enough small numbers to meet the many demands made of them.” In other words, any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few. Guy’s 1988 paper gives 35 examples in support of this thesis. This can lead inexperienced mathematicians to conclude that these concepts are related, which in fact they are not.

— Wikipedia on Strong Law of Small Numbers

2012.05.17 Thursday ACHK

You Are More Likely to Be Remembered by Your Expository Work

Let us look at two examples, beginning with Hilbert. When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert’s name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry, and for his text on integral equations. The term “Hilbert space” was introduced by Stone and von Neumann in recognition of Hilbert’s textbook on integral equations, in which the word “spectrum” was first defined at least twenty years before the discovery of quantum mechanics. Hilbert’s textbook on integral equations is in large part expository, leaning on the work of Hellinger and several other mathematicians whose names are now forgotten.

Similarly, Hilbert’s Foundations of Geometry, the book that made Hilbert’s name a household word among mathematicians, contains little original work and reaps the harvest of the work of several geometers, such as Kohn, Schur (not the Schur you have heard of), Wiener (another Wiener), Pasch, Pieri, and several other Italians.

Again, Hilbert’s Zahlbericht, a fundamental contribution that revolutionized the field of number theory, was originally a survey that Hilbert was commissioned to write for publication in the Bulletin of the German Mathematical Society.

— Ten Lessons I Wish I Had Been Taught

— Gian-Carlo Rota

2012.05.11 Friday ACHK

There exist heavier particle species which are relevant for shorter distance scales. Most of the matter around us is composed of electrons, protons, and neutrons, or – using the more elementary description – electrons, up-quarks, and down-quarks (which are attracted by forces mediated by photons and gluons). However, there exist many other particle species similar to electrons – the so-called leptons – and many other quarks. Many of those particles are unstable, and therefore unimportant in the composition of stable materials.

But even if heavier particles are stable, they are less important than the light ones because it is hard to create them and because their potential existence only affects the phenomena at ever shorter distances. Elementary particles heavier than the Planck mass or so – \(10^{-8}\) kilograms or so – also exist and there are many of them. However, they may be interpreted as black hole microstates and their description in terms of Einstein’s general theory of relativity becomes more natural than their description in terms of quantum field theory.

String/M-theory provides us with many detailed interpolations between the regular light particle species and the black holes – e.g. Kaluza-Klein modes i.e. particles moving in extra dimensions; excited string states and branes, and others.

— Ten new things that science has learned about matter

— Lubos Motl

2012.05.09 Tuesday ACHK

After a minute spent with a different colloquium at the following week (whose title was in Latin), Andy Strominger quickly jumps to his main point.

In the early 21st century, black holes seem to play the same universal exploratory role in physics that the quantum harmonic oscillator used to play almost 100 years earlier. The reason of their universal importance is that they’re the simplest and the most complex objects in the Universe at the same moment.

It sounds almost like poetry but it’s actually true although one needs to appreciate many aspects of black holes that general relativity, its semiclassical quantization, and later string theory discovered to appreciate the depth of the summary above.

— Black holes: harmonic oscillators of the 21st century

— Lubos Motl

2012.05.08 Tuesday ACHK

Particle physics phenomenology is the part of theoretical particle physics that deals with the application of theory to high-energy particle physics experiments. Within the Standard Model, phenomenology is the calculating of detailed predictions for experiments, usually at high precision (e.g., including radiative corrections). Beyond the Standard Model, phenomenology addresses the experimental consequences of new models: how their new particles could be searched for, how the model parameters could be measured, and how the model could be distinguished from other, competing models. Phenomenology may in some sense be regarded as forming a bridge between the rarefied, highly mathematical world of theoretical physics proper (such as quantum field theories and theories of the structure of space-time) and experimental particle physics.

— Wikipedia on Phenomenology (particle physics)

2012.05.06 Sunday ACHK

In quantum mechanics and quantum field theory, the vacuum is defined as the state (that is, the solution to the equations of the theory) with the lowest possible energy (the ground state of the Hilbert space). In quantum electrodynamics this vacuum is referred to as ‘QED vacuum’ to separate it from the vacuum of quantum chromodynamics, denoted as QCD vacuum. QED vacuum is a state with no matter particles (hence the name), and also no photons, no gravitons, etc.

Theoretically, in QCD vacuum multiple vacuum states can coexist. The starting and ending of cosmological inflation is thought to have arisen from transitions between different vacuum states. For theories obtained by quantization of a classical theory, each stationary point of the energy in the configuration space gives rise to a single vacuum. String theory is believed to have a huge number of vacua – the so-called string theory landscape.

— Wikipedia on Vacuum

2012.05.04 Friday ACHK

In quantum gravity, the metric tensor is a quantum observable. Also, the metric tensor determines the structure of the light cones and the rules for causality, and therefore the causal structure becomes uncertain and confusing. At any rate, string theory is smarter than we are and it is able to avoid these conceptual problems.

— Causality and entanglement

— Lubos Motl

2012.04.30 Monday ACHK

Naturalness means that the dimensionless parameters in physics are likely to be comparable to one – and very unlikely to be much greater or much smaller than one – unless an explanation why they’re very different from one exists.

— Lubos Motl

2012.04.29 Sunday ACHK

At that time, Feynman already realized that the electromagnetic field is an infinite-dimensional harmonic oscillator. Well, the Dirac field is simply its Grassmann-valued counterpart and every other field is a sort of an infinite-dimensional oscillator, too.

— Feynman’s thesis: arrival of path integrals

— Lubos Motl

2012.04.28 Saturday ACHK

The only special thing about the Standard Model is that it’s the “minimal” theory (when it comes to counting of the fields etc.) that is compatible with data we had known before the LHC.

— Theory vs phenomenology in the days of experimental reckoning

— Lubos Motl

2012.04.24 Tuesday ACHK

The black hole information paradox results from the combination of quantum mechanics and general relativity. It suggests that physical information could permanently disappear in a black hole, allowing many physical states to evolve into the same state. This is controversial because it violates a commonly assumed tenet of science — that in principle complete information about a physical system at one point in time should determine its state at any other time. A fundamental postulate of quantum mechanics is that complete information about a system is encoded in its wave function. The evolution of the wave function is determined by a unitary operator, and unitarity implies that information is conserved in the quantum sense.

— Wikipedia on Black hole information paradox

2012.04.23 Monday ACHK