Quick Calculation 3.8

A First Course in String Theory

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Show that this condition fixes uniquely \displaystyle{\alpha = \gamma = 1/2}, and \displaystyle{\beta = - 3/2}, thus reproducing the result in (3.90).

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Eq. (3.93):

\displaystyle{l_P = (G)^\alpha (c)^\beta (\hbar)^\gamma}

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\displaystyle{l_P = \left( \frac{l_p^3}{m_p t_P^2} \right)^\alpha \left( \frac{l_P}{t_P} \right)^\beta \left( \frac{m_P l_P^2}{t_P} \right)^\gamma}

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\displaystyle{\begin{aligned}   3 \alpha + \beta + 2\gamma &= 1 \\   -\alpha + \gamma &= 0 \\   - 2 \alpha - \beta - \gamma &= 0 \\   \end{aligned}}

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var('a b c')

solve([3*a+b+2*c==1, -a+c==0, -2*a-b-c==0], a, b, c)

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\displaystyle{\begin{aligned}   \alpha &= \frac{1}{2} \\ \\  \beta &= \frac{-3}{2} \\ \\  \gamma &= \frac{1}{2} \\ \\  \end{aligned}}

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Eq. (3.90):

\displaystyle{l_P = \sqrt{\frac{G \hbar}{c^3}}}

— Me@2022-06-23 10:46:22 AM

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