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This is the content of section 1.4 "Instantons and Large N" by Marcos Mariño.

The (inverted) potential is following form. $$ V(q)=-\frac{1}{2}q^2+\frac{1}{4}q^4 $$ I want to show the period of motion $$ \beta=2\int_{q_-}^{q_+}\frac{\mathrm{d}q}{\sqrt{2(E-V(q))}} $$ is equal to $$ \beta=2\sqrt{2}\left(\frac{2-m}{2}\right)^{1/2}\mathrm{K}(k). $$

$K(k)$ is the complete elliptic integral of first kind. In addition, $k^2=m=1-\frac{q_-^2}{q_+^2}$ and $q_\pm=\sqrt{1\pm\sqrt{1+4E}}$.

The substitutions ($q=s+q_-,s=(q_+-q_-)t,t=\sin\theta$) don't work well.

The EOM solution is $q(t)=q_+\mathrm{dn}(u;k)$ where $u=\frac{q_+}{\sqrt{2}}(t-t_0),k^2=1-\frac{q_-^2}{q_+^2}$.

The textbook says $E=-\frac{1-k^2}{(2-k^2)^2}$.

How do I transform that integral?

Anzu Ariake
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1 Answers1

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Use following transformations: $$ q^2=q_+^2-(q_+^2-q_-^2)s^2,\;s=-t. $$ The first one's integration range must be from $s=-1$ to $s=0$.

Anzu Ariake
  • 37