20 | February | 2018 | Physics Town

Quick Calculation 14.4b | A First Course in String Theory

Massive level in the open superstring

Consider the first and second excited levels of the open superstring ( $\alpha' M^2 = 1$ and $\alpha' M^2 = 2$ ). List the states in NS sector and the states in the R sector. Confirm that you get the same number of states.

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When $\alpha' M^2 = 2$ , by Equation (14.54), the possible states are

$\{ \alpha_{-2}^I, \alpha_{-1}^I \alpha_{-1}^J, d_{-1}^I d_{-1}^J \} | R_a \rangle, || ...$

$\{ \alpha_{-1}^I d_{-1}^J, d_{-2}^I \} | R_{\bar a} \rangle, || ...$

For $\alpha_{-2}^I | R_a \rangle$ , the number of states is 8.

For $\alpha_{-1}^I \alpha_{-1}^J | R_a \rangle$ , the number of states is $\frac{8 \times 7}{2} + 8 = 36$ .

For $d_{-1}^I d_{-1}^J | R_a \rangle$ , the number of states is $\frac{8 \times 7}{2} = 28$ .

For $\alpha_{-1}^I d_{-1}^J | R_{\bar a} \rangle$ , the number of states is $8 \times 8 = 64$ .

For $d_{-2}^I | R_{\bar a} \rangle$ , the number of states is 8.

However, since each of $a$ and ${\bar a}$ has 8 possible values, there is an additional multiple of 8.

The total number of states is $8 \left[ 8 + 36 + 28 + 64 + 8 \right]$ .

You can check this answer against Equation (14.67):

$f_{NS} (x) = \frac{1}{\sqrt{x}} + 8 + 36 \sqrt{x} + 128 x + 402 x \sqrt{x} + 1152 x^2 + ...$

— Me@2018.02.20 10:57 AM

Energy conservation, 6.2 | Energy 5.2

time ~ change

energy ~ the ability of _keeping_ changing

constant velocity ~ the amount of an object’s change of position, measured with respect to its observer’s unit of change, is constant

$s = \Delta x$

$v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}$

kinetic energy ~ the amount of the ability of keeping changing an object’s position

$\frac{1}{2} m v^2$ ~ the square of (the amount of change of position, relative to the observer’s unit of change)

Energy difference is _not_ exactly a measurement of the amount of change, time interval is.

— Me@2018-02-20 09:39:30 AM

S	M	T	W	T	F	S
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28

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Daily Archives: February 20, 2018

Problem 14.3b6

The language of Change 1.2