Split Kinetic and Potential Term in Action in Independt. Var

[junt]

Homework Statement

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I have the following expression:

$$S=T+V$$

$$T=\frac{m}{\tau_0+it}((x_1-x_0)^2+(x_2-x_1)^2)+\frac{m}{2(\tau_1-it)}(x_2-x_0)^2$$

$$V= \frac{(\tau_0+it)}{2}(\frac{k_0 x_0}{2}+\frac{k_0 x_2}{2}+k_0 x_1)+(\tau_1-it)(\frac{k_1 x_0}{2}+\frac{k_1 x_2}{2})$$

The main goal is to find a coordinate transformation such that I can write my S as being linear in t. t should not have none of the higher orders.

2. The attempt at a solution

I tried to scale my x's by $$\sqrt{\tau_0+it}\sqrt{\tau_1-it}$$ This allows me to separate component in T into those containing tau's and t's. For me the important component is the ones with t's. This scaling will lead to components like the following that involves t:

$$T= ... + it((x_1-x_0)^2+(x_2-x_1)^2-\frac{1}{2}(x_2-x_0))$$

Now this is bad because of the (-) sign in the third component. If I could get rid of (-) sign and get rid of the factor 1/2, I could simply do another coordinate transformation called normal mode (q-coordinate) transformation, and write this T part as $$..+i t(q_1^2+q_2^2)$$ I can then choose final variable q_0 such that it is contained in V part.

At the end I want to end up with something like:

$$S=g(\tau_0,\tau_1,x_0,x_1,x_2)+ it(f(\tau_0,\tau_1,x_0,x_1,x_2))$$

where g and f are real function of those variables.

Does anyone know of a better coordinate transformation to achieve this form? I would really appreciate it.
MJ

P.S: For those curious, this is an imaginary time action of a particle in linear potential.

 

[mark726]

Dear MJ,

Thank you for sharing your expression and your attempt at finding a coordinate transformation. It seems like you have already made some progress in separating out the components containing t. However, as you have pointed out, the third component in T causes some issues with your desired form.

One possible solution could be to use a different scaling factor for the third component. For example, instead of $\sqrt{\tau_0+it}$, you could try using $\sqrt{\tau_0+it}\sqrt{\tau_1-it}\sqrt{2}$. This would cancel out the factor of 1/2 and give you a positive sign for the third component. You can then proceed with your normal mode transformation to obtain your desired form.

Another approach could be to use a different coordinate transformation altogether. Instead of scaling the x's, you could try a transformation that involves the tau's and t's. For example, you could try using $q_0=\sqrt{\tau_0+it}$ and $q_1=\sqrt{\tau_1-it}$. This would give you a form similar to what you desire, with the only difference being that the t coefficient in front of the second term would be complex. However, this should not affect your final results as long as you are only interested in the real part of S.

I hope these suggestions are helpful to you. Good luck with your research!