Daily Archives: October 6, 2015

Exercise 6a

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You Could Have Invented Monads! (And Maybe You Already Have.)

Show that f * unit = unit * f

——————————

(f * g) (x, xs)
= ((bind f) . (bind g)) (x, xs)

bind f x = concat (map f x)

(f * unit) (x:xs)
= bind f (bind unit (x:xs))
= bind f (concat (map unit (x:xs)))
= bind f (concat (map unit [x1, x2, x3, ...]))
= bind f (concat ([[x1], [x2], [x3], ...]))
= bind f [x1, x2, x3, ...]
= concat (map f [x1, x2, x3, ...])
= concat [f x1, f x2, f x3, ...]
= [f x1, f x2, f x3, ...]

(unit * f) (x:xs)
= ((bind unit) . (bind f)) (x:xs)
= bind unit (bind f (x:xs))
= bind unit (concat (map f (x:xs)))
= bind unit (concat (map f [x1, x2, ...]))
= bind unit (concat [f x1, f x2, ...])
= bind unit [f x1, f x2, ...]
= concat (map unit [f x1, f x2, ...])
= concat [[f x1], [f x2], ...]
= [f x1, f x2, ...]

— Me@2015.07.20 09:00 PM

2015.10.06 Tuesday (c) All rights reserved by ACHK

Problem 14.3b2

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A First Course in String Theory
 
 
14.3 Massive level in the open superstring

~~~

How come R sector has a factor 16 while NS sector has not?

Equation (14.66):

f_{NS}(x) = \frac{1}{\sqrt{x}} \prod_{n=1}^\infty \left( \frac{1+x^{n - \frac{1}{2}}}{1-x^n} \right)^8

Equation (14.68):

f_R(x) = 16 \prod_{n=1}^\infty \left( \frac{1+x^n}{1-x^n} \right)^8

p.319 “The overall multiplicative factor appears because each combination of oscillators gives rise to 16 states by acting on each of the available ground states.”

p.319 “We note that the R coefficients are actually double the corresponding NS coefficients. This is not a coincidence, as we will see in the following section.”

p.320 “We have seen that the Ramond sector has world-sheet supersymmetry: there are equal numbers of fermionic and bosonic states at each mass level.”
 
 
With the factor 16, how come the R coefficients are only double, but not 16 times as big as the corresponding NS coefficients?
 
It is caused by the difference of x^n and x^{n-\frac{1}{2}}.

— Me@2015.10.06 08:23 AM
 
 
 
2015.10.06 Tuesday (c) All rights reserved by ACHK