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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