Quantum Mechanics 6
.
As Gene and Sidney Coleman have pointed out, the term “interpretation of quantum mechanics” is a misnomer encouraging its users to generate logical fallacies. Why? It’s because we should always use a theory, or a more accurate, complete, and universal theory, to interpret its special cases, to interpret its approximations, to interpret the limits, and to interpret the phenomena it explains.
However, there’s no language “deeper than quantum mechanics” that could be used to interpret quantum mechanics. Unfortunately, what the “interpretation of quantum mechanics” ends up with is an attempt to find a hypothetical “deeper classical description” underneath the basic wheels and gears of quantum mechanics. But there’s demonstrably none. Instead, what makes sense is an “interpretation of classical physics” in terms of quantum mechanics. And that’s exactly what I am going to focus in this text.
Plan of this blog entry
After a very short summary of the rules of quantum mechanics, I present the widely taught “mathematical limit” based on the smallness of Planck’s constant. However, that doesn’t really fully explain why the world seems classical to us. I will discuss two somewhat different situations which however cover almost every example of a classical logic emerging from the quantum starting point:
- Classical coherent fields (e.g. light waves) appearing as a state of many particles (photons)
- Decoherence which makes us interpret absorbed particles as point-like objects and which makes generic superpositions of macroscopic objects unfit for well-defined questions about classical facts
— How classical fields, particles emerge from quantum theory
— Lubos Motl
.
There is no interpretation problem for quantum mechanics. Instead, if there is a problem, it should be the interpretation of classical mechanics problem.
— Lubos Motl
— paraphrased
— Me@2011.07.28
.
.
2019.12.14 Saturday (c) All rights reserved by ACHK
![\displaystyle{\left[ \bar L_m^\perp, x_0^I \right] = - i \sqrt{\frac{\alpha'}{2}} \bar \alpha^I_m}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cleft%5B+%5Cbar+L_m%5E%5Cperp%2C+x_0%5EI+%5Cright%5D+%3D+-+i+%5Csqrt%7B%5Cfrac%7B%5Calpha%27%7D%7B2%7D%7D+%5Cbar+%5Calpha%5EI_m%7D&bg=ffffff&fg=333333&s=0&c=20201002)
,![\displaystyle{\left[ L_m^\perp, x_0^I \right] = - i \sqrt{\frac{\alpha'}{2}} \alpha^I_m}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cleft%5B+L_m%5E%5Cperp%2C+x_0%5EI+%5Cright%5D+%3D+-+i+%5Csqrt%7B%5Cfrac%7B%5Calpha%27%7D%7B2%7D%7D+%5Calpha%5EI_m%7D&bg=ffffff&fg=333333&s=0&c=20201002)
.
,
.![\displaystyle{ \begin{aligned} \left[ \bar L_m^\perp, x_0^I \right] &= \frac{1}{2} \sum_{p \in \mathbb{Z}} \left[ \bar \alpha^J_p \bar \alpha^J_{m-p}, x_0^I \right] \\ &= \frac{1}{2} \sum_{p \in \mathbb{Z}} \bar \alpha^I_p \left[ \bar \alpha^J_{m-p}, x_0^I \right] + \frac{1}{2} \sum_{p \in \mathbb{Z}} \left[ \bar \alpha^J_p, x_0^I \right] \bar \alpha^I_{m-p} \\ &= \frac{1}{2} \bar \alpha^J_m \left[ \bar \alpha^J_{0}, x_0^I \right] + \frac{1}{2} \left[ \bar \alpha^J_0, x_0^I \right] \bar \alpha^J_{m} \\ \end{aligned} }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D+%5Cleft%5B+%5Cbar+L_m%5E%5Cperp%2C+x_0%5EI+%5Cright%5D+%26%3D+%5Cfrac%7B1%7D%7B2%7D+%5Csum_%7Bp+%5Cin+%5Cmathbb%7BZ%7D%7D+%5Cleft%5B+%5Cbar+%5Calpha%5EJ_p+%5Cbar+%5Calpha%5EJ_%7Bm-p%7D%2C+x_0%5EI+%5Cright%5D+%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+%5Csum_%7Bp+%5Cin+%5Cmathbb%7BZ%7D%7D+%5Cbar+%5Calpha%5EI_p+%5Cleft%5B+%5Cbar+%5Calpha%5EJ_%7Bm-p%7D%2C+x_0%5EI+%5Cright%5D+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Csum_%7Bp+%5Cin+%5Cmathbb%7BZ%7D%7D+%5Cleft%5B+%5Cbar+%5Calpha%5EJ_p%2C+x_0%5EI+%5Cright%5D+%5Cbar+%5Calpha%5EI_%7Bm-p%7D+%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+%5Cbar+%5Calpha%5EJ_m+%5Cleft%5B+%5Cbar+%5Calpha%5EJ_%7B0%7D%2C+x_0%5EI+%5Cright%5D+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Cleft%5B+%5Cbar+%5Calpha%5EJ_0%2C+x_0%5EI+%5Cright%5D+%5Cbar+%5Calpha%5EJ_%7Bm%7D+%5C%5C++%5Cend%7Baligned%7D+%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![\displaystyle{ \begin{aligned} \left[ \bar L_m^\perp, x_0^I \right] &= - \frac{1}{2} \bar \alpha^J_m \left[ i \sqrt{\frac{\alpha'}{2}} \eta^{IJ} \right] - \frac{1}{2} \left[ i \sqrt{\frac{\alpha'}{2}} \eta^{IJ} \right] \bar \alpha^J_{m} \\ &= - i \sqrt{\frac{\alpha'}{2}} \eta^{IJ} \bar \alpha^I_m \\ \end{aligned} }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D++%5Cleft%5B+%5Cbar+L_m%5E%5Cperp%2C+x_0%5EI+%5Cright%5D++%26%3D+-+%5Cfrac%7B1%7D%7B2%7D+%5Cbar+%5Calpha%5EJ_m+%5Cleft%5B+i+%5Csqrt%7B%5Cfrac%7B%5Calpha%27%7D%7B2%7D%7D+%5Ceta%5E%7BIJ%7D+%5Cright%5D+-+%5Cfrac%7B1%7D%7B2%7D+%5Cleft%5B+i+%5Csqrt%7B%5Cfrac%7B%5Calpha%27%7D%7B2%7D%7D+%5Ceta%5E%7BIJ%7D+%5Cright%5D+%5Cbar+%5Calpha%5EJ_%7Bm%7D+%5C%5C++%26%3D+-+i+%5Csqrt%7B%5Cfrac%7B%5Calpha%27%7D%7B2%7D%7D+%5Ceta%5E%7BIJ%7D+%5Cbar+%5Calpha%5EI_m+%5C%5C++%5Cend%7Baligned%7D+%7D&bg=ffffff&fg=333333&s=0&c=20201002)
and 
are transverse coordinate indices, neither of them can be zero.![\displaystyle{ \begin{aligned} \left[ \bar L_m^\perp, x_0^I \right] &= - i \sqrt{\frac{\alpha'}{2}} \bar \alpha^I_m \\ \end{aligned} }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D++%5Cleft%5B+%5Cbar+L_m%5E%5Cperp%2C+x_0%5EI+%5Cright%5D++%26%3D++-+i+%5Csqrt%7B%5Cfrac%7B%5Calpha%27%7D%7B2%7D%7D+%5Cbar+%5Calpha%5EI_m+%5C%5C++%5Cend%7Baligned%7D+%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L \circ \Gamma[q]) \delta_\eta \Gamma[q] \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%28D+L+%5Ccirc+%5CGamma%5Bq%5D%29+%5Cdelta_%5Ceta+%5CGamma%5Bq%5D+%5C%5C+%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} [\partial_0 L (t, q, v), \partial_1 L (t, q, v), \partial_2 L (t, q, v)] (0, \eta(t), D\eta(t)) \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%5B%5Cpartial_0+L+%28t%2C+q%2C+v%29%2C+%5Cpartial_1+L+%28t%2C+q%2C+v%29%2C+%5Cpartial_2+L+%28t%2C+q%2C+v%29%5D+%280%2C+%5Ceta%28t%29%2C+D%5Ceta%28t%29%29+%5C%5C++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![\displaystyle{\left[\partial_0 L (t, q, v), \partial_1 L (t, q, v), \partial_2 L (t, q, v)\right] (0, \eta(t), D\eta(t))}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cleft%5B%5Cpartial_0+L+%28t%2C+q%2C+v%29%2C+%5Cpartial_1+L+%28t%2C+q%2C+v%29%2C+%5Cpartial_2+L+%28t%2C+q%2C+v%29%5Cright%5D+%280%2C+%5Ceta%28t%29%2C+D%5Ceta%28t%29%29%7D&bg=ffffff&fg=333333&s=0&c=20201002)
is really a dot product:![\displaystyle{ \begin{aligned} & \int_{t_1}^{t_2} (D L \circ \Gamma[q]) \delta_\eta \Gamma[q] \\ &= \int_{t_1}^{t_2} [\partial_0 L (t, q, v), \partial_1 L (t, q, v), \partial_2 L (t, q, v)] (0, \eta(t), D\eta(t)) \\ &= \int_{t_1}^{t_2} [\partial_1 L (t, q, v) \eta(t) + \partial_2 L (t, q, v) D\eta(t)] \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D+%26+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%28D+L+%5Ccirc+%5CGamma%5Bq%5D%29+%5Cdelta_%5Ceta+%5CGamma%5Bq%5D+%5C%5C+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%5B%5Cpartial_0+L+%28t%2C+q%2C+v%29%2C+%5Cpartial_1+L+%28t%2C+q%2C+v%29%2C+%5Cpartial_2+L+%28t%2C+q%2C+v%29%5D+%280%2C+%5Ceta%28t%29%2C+D%5Ceta%28t%29%29+%5C%5C+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%5B%5Cpartial_1+L+%28t%2C+q%2C+v%29+%5Ceta%28t%29+%2B+%5Cpartial_2+L+%28t%2C+q%2C+v%29+D%5Ceta%28t%29%5D+%5C%5C+%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0&c=20201002)
Physics Town 
You must be logged in to post a comment.