In short: spin-1/2 particles are quaternions!

Thus the meaning of j is time reversal.

— John Baez

2010.05.29 Saturday ACHK

# 0-dimensional singularity: magnetic monopole
# 1-dimensional singularity: cosmic string
# 2-dimensional singularity: domain wall

— Wikipedia on Gravitational singularity

2010.05.28 Friday ACHK

Similar to the statement that

Nothing in biology makes sense except in the light of evolution

the nPOV asserts that

Nothing in mathematics makes sense except in the light of higher category theory.

— nLab

2010.05.26 Wednesday ACHK

Some interactions cannot be simply described by a potential energy V(x). The magnetic force is the most obvious example.

— Kenneth Young

2010.05.17 Monday ACHK

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Sometimes physicists write the definition in terms of a limit and the Dirac delta function, :

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Relationship between the mathematical and physical definitions

The mathematicians’ definition and the physicists’ definition of the functional derivative differ only in the physical interpretation. Since the mathematical definition is based on a relationship that holds for all test functions f, it should also hold when f is chosen to be a specific function. The only handwaving difficulty is that specific function was chosen to be a delta function — which is not a valid test function.

In the mathematical definition, the functional derivative describes how the entire functional, , changes as a result of a small change in the function . Observe that the particular form of the change in is not specified. The physics definition, by contrast, employs a particular form of the perturbation — namely, the delta function — and the ‘meaning’ is that we are varying only about some neighborhood of y. Outside of this neighborhood, there is no variation in .

— Wikipedia on Functional derivative

2010.05.08 Saturday ACHK

— Wikipedia on First variation

2010.05.07 Friday ACHK

Last step

This last step is best explained first in a certain limit. In order to describe our world, strings must be extremely tiny objects. So when one studies string theory at low energies, it becomes difficult to see that strings are extended objects — they become effectively zero-dimensional (pointlike). Consequently, the quantum theory describing the low energy limit is a theory that describes the dynamics of these points moving in spacetime, rather than strings. Such theories are called quantum field theories.

However, since string theory also describes gravitational interactions, one expects the low-energy theory to describe particles moving in gravitational backgrounds. Finally, since superstring string theories are supersymmetric, one expects to see supersymmetry appearing in the low-energy approximation. These three facts imply that the low-energy approximation to a superstring theory is a supergravity theory.

— Wikipedia on M-theory

2010.05.06 Thursday ACHK

M-theory is not yet complete; however it can be applied in many situations (usually by exploiting string theoretic dualities [clarification needed]). The theory of electromagnetism was also in such a state in the mid-19th century; there were separate theories for electricity and magnetism and, although they were known to be related, the exact relationship was not clear until James Clerk Maxwell published his equations, in his 1864 paper A Dynamical Theory of the Electromagnetic Field. Witten has suggested that a general formulation of M-theory will probably require the development of new mathematical language.

— Wikipedia on M-theory

2010.05.05 Wednesday ACHK

In physics, the AdS/CFT correspondence (anti-de-Sitter space/conformal field theory correspondence), sometimes called the Maldacena duality, is the conjectured equivalence between a string theory defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more.

— Wikipedia on AdS/CFT correspondence

2010.05.04 Tuesday ACHK

Physicists learned rather unexpectedly, beginning in the early 1970s, that the problem of quantum gravity could be overcome by introducing a new sort of fuzziness. One replaces “point particles” by “strings”. Of course, the point particles and strings must both be treated quantum mechanically. Quantum effects are proportional to Planck’s constant , and stringy effects are proportional to a new constant (equal to approximately ) that determines the size of strings.

— Edward Witten

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2010.05.03 Monday ACHK

The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.

— John Baez

2010.05.02 Sunday ACHK

Golden ratio

The number 5 is quirky and intriguing, thanks in large part to its relation with the golden ratio, the “most irrational” of irrational numbers.

— John Baez

2010.05.01 Saturday ACHK

Nature of the fractional derivative

An important point is that the fractional derivative at a point x is a local property only when a is an integer; in non-integer cases we cannot say that the fractional derivative at x of a function f depends only on the graph of f very near x, in the way that integer-power derivatives certainly do. Therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out. To use a metaphor, the fractional derivative requires some peripheral vision.

— Wikipedia on Fractional calculus

2010.04.30 Friday ACHK

The second superstring revolution was the intense wave of breakthroughs in string theory that took place approximately between 1994 and 1997.

The different versions of superstring theory were unified, as long hoped, by new equivalences. These are known as S-duality, T-duality, U-duality, mirror symmetry, and conifold transitions. The different theories of strings were also connected to a new 11-dimensional theory called M-theory.

New objects called branes were discovered as inevitable ingredients of string theory. Their analysis – especially the analysis of a special type of branes called D-branes – led to the AdS/CFT correspondence, the microscopic understanding of the thermodynamic properties of black holes, and many other developments.

— Wikipedia on Second superstring revolution

2010.04.28 Wednesday ACHK