Daily Archives: April 25, 2012

Black hole information paradox, 2

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It shouldn’t be so surprising that unitarity survives completely while causality doesn’t. After all, the basic postulates of quantum mechanics, including unitarity, the probabilistic interpretation of the amplitudes, and the linearity of the operators representing observables, seem to be universally necessary to describe physics of any system that agrees with the basic insights of the quantum revolution.

On the other hand, geometry has been downgraded into an effective, approximate, emergent aspect of reality. The metric tensor is just one among many fields in our effective field theories including gravity.

— Black hole information puzzle

— Lubos Motl

2012.04.25 Wednesday ACHK

微積分驗算 1.2

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這段改編自 2010 年 6 月 2 日的對話。

又例如，求 d/dx (1/x) 時，你既可以直接計算，又可以用「消滅分母法」：

y = 1/x

x y = 1

左右兩邊同時對 x 求導：

（Differentiate both sides with respect to x:）

(dx/dx) y + x dy/dx = d(1)/dx

y + x dy/dx = 0

dy/dx = -y/x

dy/dx = (-1/x)/x

dy/dx = -1/x^2

凡是有分母的 function（函數）做 differentiation（求導），你都可以用這個「消滅分母法」去驗算。

再例如，求 d/dx (sqrt{x}) 時，你既可以直接計算，又可以用「消滅開方法」：

y = sqrt{x}

y^2 = x

左右兩邊同時對 x 求導：

（Differentiate both sides with respect to x:）

d/dx (y^2) = dx/dx

2y dy/dx = 1

dy/dx = 1/2y

dy/dx = 1/(2 sqrt{x})

凡是有開方的 function（函數）做 differentiation（求導），你都可以用這個「消滅開方法」去驗算。

— Me@2012.04.25

2012.04.25 Wednesday (c) All rights reserved by ACHK