機遇創生論 1.7.2

因果律 2.2

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這個大統一理論的成員,包括(但不止於):

精簡圖:

種子論
反白論
間書原理
完備知識論

自由決定論

它們可以大統一的成因,在於它們除了各個自成一國外,還可以合體理解和應用。

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「自由意志」問題方面,如果要討論的話,要先釐清「人有沒有自由意志」的意思,因為,它有超過一個常用的詮釋:

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1. 思:

人可不可以控制到,自己的「思想意志」?

如果可以的話,

2. 因:

人(的自由思想,)可不可以控制到,自己身體的行動?

如果可以的話,

3. 果:

人(的自由行動,)可不可以控制到,自己人生(或者世界歷史)的發展大方向?

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詳細一點的版本是:

1. 思:

意志有沒有自由?

究竟人的思想是有自由?還是其實,人的思想受制於教育等外在因素,所以都是不由自主的呢?

那就有如機械人的思想般,其實只是根據程式碼來運行。

2. 因:

即使假設意志有自由,人的身體是物體,它的運行仍然,受著物理力學定律所主宰,理應沒有自由。

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在「Laplace 因果律」的觀點下,兩者其實又沒有分別,因為「意志」都是,受物理定律的支配。「思想」是腦部的狀態,而腦部的生物化學作用,最底層也其實是,由物理定律控制。

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如果「Laplace 因果律」正確的話,人或者任何其他東西,也沒有自由。

至於「Laplace 因果律」正確與否,和人有自由與否,在我們以前的討論中,已得出了結論:

「自由論」和「決定論」,其實沒有實質上的分別。

所以,我把這個理論,稱為「自由決定論」。

如果對「因果律」和「自由意志」話題有興趣,想知道這結論的理據的話,請在本網誌搜尋「因果律」。

— Me@2020-12-29 05:46:05 PM

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2020.12.29 Tuesday (c) All rights reserved by ACHK

1994

This was my Art result. I had a great art teacher Mr Lo in that year. He taught us a lot of design concepts.

— Me@2020-12-29 10:52:04 AM

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2020.12.29 Tuesday (c) All rights reserved by ACHK

Ex 1.14 Lagrange equations for L’

Structure and Interpretation of Classical Mechanics

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Show by direct calculation that the Lagrange equations for \displaystyle{L'} are satisfied if the Lagrange equations for \displaystyle{L} are satisfied.

~~~

Equation (1.69):

\displaystyle{C \circ \Gamma[q'] = \Gamma[q]}

Equation (1.70):

\displaystyle{L' = L \circ C}

Equation (1.71):

\displaystyle{L' \circ \Gamma[q'] = L \circ C \circ \Gamma[q'] = L \circ \Gamma[q]}

The Lagrange equation:

\displaystyle{ \begin{aligned} D ( \partial_2 L \circ \Gamma[q]) - (\partial_1 L \circ \Gamma[q]) &= 0 \\ \end{aligned}}

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\displaystyle{ \begin{aligned}   &\partial_2 L \circ \Gamma[q] \\  &= \frac{\partial}{\partial v} L(t, x, v) \\   &= \frac{\partial}{\partial v} L'(t, x', v') \\   &= \frac{\partial}{\partial x'} L'(t, x', v') \frac{\partial x'}{\partial v} + \frac{\partial}{\partial v'} L'(t, x', v') \frac{\partial v'}{\partial v} \\   \end{aligned}}

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Since it is just a coordinate transformation \displaystyle{x = F(t, x')}, \displaystyle{x} has no explicitly dependent on \displaystyle{v'}. Similarly, if we consider the coordinate transformation \displaystyle{x' = G(t, x)}, \displaystyle{x'} has no explicitly dependent on \displaystyle{v}. So

\displaystyle{ \begin{aligned}   \frac{\partial x'}{\partial v} &= 0 \\   \end{aligned}}

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\displaystyle{ \begin{aligned}   &\partial_2 L \circ \Gamma[q] \\  &= \frac{\partial}{\partial v'} L'(t, x', v') \frac{\partial v'}{\partial v} \\   \end{aligned}}

— Me@2020-12-28 04:03:24 PM

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2020.12.28 Monday (c) All rights reserved by ACHK

The square root of the probability, 4

Eigenstates 3.4

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quantum ~ classical with the indistinguishability of cases

— Me@2020-12-23 06:19:00 PM

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In statistical mechanics, a semi-classical derivation of the entropy that does not take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive (is not proportional to the amount of substance in question). This leads to a paradox known as the Gibbs paradox, after Josiah Willard Gibbs who proposed this thought experiment in 1874‒1875. The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. A related paradox is the “mixing paradox”. If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, the paradox is averted.

— Wikipedia on Gibbs paradox

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2020.12.27 Sunday (c) All rights reserved by ACHK

尋覓 1.3

這段改編自 2010 年 10 月 14 日的對話。

.

但是,十年後的物理三十歲,一個的心理年齡已,發展到三十五歲,而另一個的心理,卻仍然停留在二十五歲。那樣的話,二人的感情,就未必再能維持。

除了相遇那年要夾(融洽)外,還要在之後的每一年,也是那樣夾。換句話說,即是兩人的變法要配合。所以,難度其實深了一層。

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然後,再深一層。其實呢,在看《潛行凶間》之前,我已有這個知識:

就算只考慮「現在的你」,或者「現在的他」,各人的心靈腦海之中,其實都有超過一個自己。只不過,眾多自我被困於,同一個頭顱之中。

最簡單的講法是,每人也起碼有兩個自己。

左腦的性格,和右腦的性格,其實是不同的。左腦喜歡計數邏輯思考,右腦鍾情音樂藝術創作。

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實情是,每一個人,也擁有超過兩個自己。簡單起見,假設你只有「甲、乙、丙」三個自己;而你的潛在情人,也只有「A、B、C」三個人格。那樣,你和他要完全夾的話,那就需要那九對組合也夾才可以。

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1. 甲A

2. 甲B

3. 甲C

4. 乙A

5. 乙B

6. 乙C

7. 丙A

8. 丙B

9. 丙C

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那為什麼拍拖那時,通常很開心?而結婚後,通常也不會太開心?

那是因為,拍拖時,通常只有「最好的你」和「最好的他」相處。結婚後,你會發現,原來在他的頭顱中,除了「最好的他」之外,還有很多隻怪獸。

友情方面,你可以選擇,只要對方的優點;
愛情方面,你不可以選擇,不要對方的缺點。

— Me@2010.06.01

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友情方面,你可以選擇,只要對方最好的優點;
愛情方面,你不可以選擇,不要對方最差的缺點。

–- Me@2010.06.01

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因為

朋友,只需接受部分;
情人,卻需接受全部。

— Me@2010.06.07

這就是我上次講,《潛行凶間》的劇情。《潛行凶間》的內容,有七成是真實的。

整體而言,今天的分手,對你來說,利遠大於弊,因為,他是你將來結婚對象的機會,實在太微;早分為妙。

— Me@2020-12-23 10:39:26 PM

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2020.12.25 Friday (c) All rights reserved by ACHK

Problem 2.4

A First Course in String Theory

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2.4 Lorentz transformations as matrices

A matrix L that satisfies (2.46) is a Lorentz transformation. Show the following.

(b) If \displaystyle{L} is a Lorentz transformation so is the inverse matrix \displaystyle{L^{-1}}.

(c) If \displaystyle{L} is a Lorentz transformation so is the transpose matrix \displaystyle{L^{T}}.

~~~

(b)

\displaystyle{   \begin{aligned}   (\mathbf{A}^\mathrm{T})^{-1} &= (\mathbf{A}^{-1})^\mathrm{T} \\  L^T \eta L &= \eta \\  \eta &= [L^T]^{-1} \eta L^{-1} \\  [L^T]^{-1} \eta L^{-1} &= \eta \\  [L^{-1}]^T \eta L^{-1} &= \eta \\  \end{aligned}}

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(c)

\displaystyle{   \begin{aligned}   L^T \eta L &= \eta \\  (L^T \eta L)^{-1} &= (\eta)^{-1} \\  L^{-1} \eta^{-1} (L^T)^{-1} &= \eta \\  L^{-1} \eta (L^T)^{-1} &= \eta \\  \eta &= L \eta L^T \\  L \eta L^T &= \eta \\  \end{aligned}}

— Me@2020-12-21 04:24:33 PM

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2020.12.21 Monday (c) All rights reserved by ACHK

Pointer state, 3

Eigenstates 3.3 | The square root of the probability, 3

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In calculation, if a quantum state is in a superposition, that superposition is a superposition of eigenstates.

However, real superposition does not just include eigenstates that make macroscopic senses.

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That is the major mistake of the many-worlds interpretation of quantum mechanics.

— Me@2017-12-30 10:24 AM

— Me@2018-07-03 07:24 PM

— Me@2020-12-18 06:12 PM

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Mathematically, a quantum superposition is a superposition of eigenstates. An eigenstate is a quantum state that is corresponding to a macroscopic state. A superposition state is a quantum state that has no classical correspondence.

The macroscopic states are the only observable states. An observable state is one that can be measured directly or indirectly. For an unobservable state, we write it as a superposition of eigenstates. We always write a superposition state as a superposition of observable states; so in this sense, before measurement, we can almost say that the system is in a superposition of different (possible) classical macroscopic universes.

However, conceptually, especially when thinking in terms of Feynman’s summing over histories picture, a quantum state is more than a superposition of classical states. In other words, a system can have a quantum state which is a superposition of not only normal classical states, but also bizarre classical states and eigen-but-classically-impossible states.

A bizarre classical state is a state that follows classical physical laws, but is highly improbable that, in daily life language, we label such a state “impossible”, such as a human with five arms.

An eigen-but-classically-impossible state is a state that violates classical physical laws, such as a castle floating in the sky.

For a superposition, if we allow only normal classical states as the component eigenstates, a lot of the quantum phenomena, such as quantum tunnelling, cannot be explained.

If you want multiple universes, you have to include not only normal universes, but also the bizarre ones.

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Actually, even for the double-slit experiment, “superposition of classical states” is not able to explain the existence of the interference patterns.

The superposition of the electron-go-left universe and the electron-go-right universe does not form this universe, where the interference patterns exist.

— Me@2020-12-16 05:18:03 PM

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One of the reasons is that a quantum superposition is not a superposition of different possibilities/probabilities/worlds/universes, but a superposition of quantum eigenstates, which, in a sense, are square roots of probabilities.

— Me@2020-12-18 06:07:22 PM

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2020.12.18 Friday (c) All rights reserved by ACHK

機遇創生論 1.7

因果律 2.1

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這個大統一理論的成員,包括(但不止於):

精簡圖:

種子論
反白論
間書原理
完備知識論

自由決定論

它們可以大統一的成因,在於它們除了各個自成一國外,還可以合體理解和應用。

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(問:「種子論」,其實就即是「自由決定論」?)

可以那樣說。但是重點不同。

「種子論」重於討論,人生如何逹到成功。(留意,這裡的「成功」,是在你自己定義下的成功,而不是在世俗標準下。)

「自由決定論」則重於研究,宇宙既然依物理定律運行,那就代表,人的一舉一動,甚至思想意志,在宇宙創生那刻,就已經決定了?

(問:「自由決定論」即是問,世間上,有沒有「自由意志」?)

不太是。

「自由決定論」的重點跟人(或者其他意識體)沒有直接的關係。

「自由決定論」的重點在於研究,「Laplace 因果律」是否正確。

「Laplace 因果律」就是:

我們只要掌握某一個時刻,宇宙狀態的所有資料,我們就可以推斷到,宇宙在任何其他時刻的狀態。

這個話題中的細節,我們以前已經詳細討論過,所以這裡不再跟進。

「如果『因果律』是正確,人就沒有自由」只是「因果律」的一個例子。而這個例子因為直接和人相關,所以,人們特別重視。但是,即使那樣,那仍不是「因果律」的重點。

亦即是話,「自由意志」問題,只是「因果律」問題的支節。

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(另一話題:)

「自由意志」問題方面,如果要討論的話,要先釐清「人有沒有自由意志」的意思,因為,它有超過一個常用的詮釋:

.

1. 思:

人可不可以控制到,自己的「思想意志」?

如果可以的話,

2. 因:

人(的自由思想,)可不可以控制到,自己身體的行動?

如果可以的話,

3. 果:

人(的自由行動,)可不可以控制到,自己人生(或者世界歷史)的發展大方向?

— Me@2020-12-11 06:43:58 PM

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2020.12.11 Friday (c) All rights reserved by ACHK

How to Find Lagrangians

Lagrange’s equations are a system of second-order differential equations. In order to use them to compute the evolution of a mechanical system, we must find a suitable Lagrangian for the system. There is no general way to construct a Lagrangian for every system, but there is an important class of systems for which we can identify Lagrangians in a straightforward way in terms of kinetic and potential energy. The key idea is to construct a Lagrangian L such that Lagrange’s equations are Newton’s equations \displaystyle{\vec F = m \vec a}.

— 1.6 How to Find Lagrangians

— Structure and Interpretation of Classical Mechanics

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2020.12.06 Sunday ACHK

Logical arrow of time, 6.4.2

Logical arrow of time, 6.1.2

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The source of the macroscopic time asymmetry, aka the second law of thermodynamics, is the difference between prediction and retrodiction.

In a prediction, the deduction direction is the same as the physical/observer time direction.

In a retrodiction, the deduction direction is opposite to the physical/observer time direction.

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— guess —

If a retrodiction is done by a time-opposite observer, he will see the entropy increasing. For him, he is really making a prediction.

— guess —

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— Me@2013-10-25 3:33 AM

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A difference between deduction and observation is that in observation, the probability is updated in real time.

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each update time interval ~ infinitesimal

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In other words, when you observe a system, you get new information about that system in real time.

Since you gain new knowledge of the system in real time, the probability assigned to that system is also updated in real time.

— Me@2020-10-13 11:27:59 AM

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2020.12.04 Friday (c) All rights reserved by ACHK

尋覓 1.2

這段改編自 2010 年 10 月 14 日的對話。

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但是,要找到你,心目中的那個外星人,不易做到。

一來,地球上,地球人很多,外星人很少。

二來,外星人中,都可以有好有壞。即使同樣是外星人,也可以是來自不同的星球。

三來,你二十歲未到,還未有足夠的人生閱歷,去辨認哪些人是,有誠信的外星人。你等年紀大一點,例如大學時代,才拍拖,可能會好一點。所以,宏觀而言,你現在的分手,對你的人生是好事。

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記住,第一個要點是,不要找地球人,而要找外星人;不只要找外星人,而要找來自,高級星球的外星人。

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第二個要點是,在時間上,所謂的「一個人」,並不是真的是,同一個人。

什麼意思呢?

例如,如果你把我十九歲時寫的文章,和我三十歲時寫的文章,比較一下的話,你會覺得,那兩篇文章是,來者不同二人的手筆。那兩文有著不同的風格,不同的看法,如果事前不告訴你,它們其實來自同一作者的話,你不會那樣估計。

亦即是話,即使假設你和現在的男朋友,互相為對方的理想對象,二人其實仍然會,各自隨時間變化。你不能保證,你們各自變了十年後,你們各自的新版本,仍然會是相愛。

那已經不是「你們」的故事,而是「他們」的旅程;而「他們」,亦可能已經,不能再是在一起了。
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心理成熟程度,簡稱「心理年齡」。

心理年齡相差越大,相處的難度越高。

方便起見,假設你們相遇時,物理年齡一樣,都是二十歲;而心理年齡亦相同,都是二十歲。

但是,十年後的物理三十歲,一個的心理年齡已,發展到三十五歲,而另一個的心理,卻仍然停留在二十五歲。那樣的話,二人的感情,就未必再能維持。

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除了相遇那年要夾(融洽)外,還要在之後的每一年,也是那樣夾。換句話說,即是兩人的變法要配合。所以,難度其實深了一層。

— Me@2020-11-29 10:36:46 PM

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2020.12.01 Tuesday (c) All rights reserved by ACHK

Problem 2.3b5

A First Course in String Theory

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2.3 Lorentz transformations, derivatives, and quantum operators.

(b) Show that the objects \displaystyle{\frac{\partial}{\partial x^\mu}} transform under Lorentz transformations in the same way as the \displaystyle{a_\mu} considered in (a) do. Thus, partial derivatives with respect to conventional upper-index coordinates \displaystyle{x^\mu} behave as a four-vector with lower indices – as reflected by writing it as \displaystyle{\partial_\mu}.

~~~

Denoting \displaystyle{ \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma}} as \displaystyle{L^{~\nu}_{\mu}} is misleading, because that presupposes that \displaystyle{ \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma}} is directly related to the matrix \displaystyle{L}.

To avoid this bug, instead, we denote \displaystyle{ \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma}} as \displaystyle{M ^\nu_{~\mu}}. So

\displaystyle{ \begin{aligned} (x')^\mu &= L^\mu_{~\nu} x^\nu \\ (x')^\mu (x')_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\beta \sigma} x_\beta \right) \\ (x')^\mu (x')_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( M^{\beta}_{~\mu} x_\beta \right) \\ x^\mu x_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( M^{\beta}_{~\mu} x_\beta \right) \\ \end{aligned}}

\displaystyle{ \begin{aligned} \nu \neq \mu:&~~~~~~\sum_{\mu = 0}^4 L^\mu_{~\nu} M^{\beta}_{~\mu} &= 0 \\ \nu = \mu:&~~~~~~\sum_{\mu = 0}^4 L^\mu_{~\nu} M^{\beta}_{~\mu} &= 1 \\ \end{aligned}}

Using the Kronecker Delta and Einstein summation notation, we have

\displaystyle{ \begin{aligned} L^\mu_{~\nu} M^{\beta}_{~\mu} &= M^{\beta}_{~\mu} L^\mu_{~\nu} \\ &= \delta^{\beta}_{~\nu} \\ \end{aligned}}

So

\displaystyle{ \begin{aligned} \sum_{\mu=0}^{4} L^\mu_{~\nu} M^{\beta}_{~\mu} &= \delta^{\beta}_{~\nu} \\ \end{aligned}}

\displaystyle{ \begin{aligned}   M^{\beta}_{~\mu} &= [L^{-1}]^{\beta}_{~\mu} \\   \end{aligned}}

In other words,

\displaystyle{ \begin{aligned}    \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma} &= [L^{-1}]^{\beta}_{~\mu} \\   \end{aligned}}

— Me@2020-11-23 04:27:13 PM

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One defines (as a matter of notation),

{\displaystyle {\Lambda _{\nu }}^{\mu }\equiv {\left(\Lambda ^{-1}\right)^{\mu }}_{\nu },}

and may in this notation write

{\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }.}

Now for a subtlety. The implied summation on the right hand side of

{\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }A_{\mu }}

is running over a row index of the matrix representing \displaystyle{\Lambda^{-1}}. Thus, in terms of matrices, this transformation should be thought of as the inverse transpose of \displaystyle{\Lambda} acting on the column vector \displaystyle{A_\mu}. That is, in pure matrix notation,

{\displaystyle A'=\left(\Lambda ^{-1}\right)^{\mathrm {T} }A.}

— Wikipedia on Lorentz transformation

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So

\displaystyle{ \begin{aligned}   M^{\beta}_{~\mu} &= [L^{-1}]^{\beta}_{~\mu} \\   \end{aligned}}

In other words,

\displaystyle{ \begin{aligned}    \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\beta \sigma} &= [L^{-1}]^{\beta}_{~\mu} \\   \end{aligned}}

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Denote \displaystyle{[L^{-1}]^{\beta}_{~\mu}} as

\displaystyle{ \begin{aligned}   N^{~\beta}_{\mu} \\   \end{aligned}}

In other words,

\displaystyle{ \begin{aligned}   N^{~\beta}_{\mu} &= M^{\beta}_{~\mu} \\   [N^T] &= [M] \\   \end{aligned}}

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The Lorentz transformation:

\displaystyle{ \begin{aligned}   (x')^\mu &= L^\mu_{~\nu} x^\nu \\   (x')_\mu &= \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\beta \sigma} x_\beta \\   \end{aligned}}

.

\displaystyle{ \begin{aligned}   (x')^\mu &= L^\mu_{~\nu} x^\nu \\   (x')_\mu &= N^{~\nu}_{\mu} x_\nu \\   \end{aligned}}

.

\displaystyle{ \begin{aligned}   x^\mu &= [L^{-1}]^\mu_{~\nu} (x')^\nu \\   (x')_\mu &= M^{\nu}_{~\mu} x_\nu \\   \end{aligned}}

.

\displaystyle{ \begin{aligned}   x^\mu &= [L^{-1}]^\mu_{~\nu} (x')^\nu \\   (x')_\mu &= [L^{-1}]^{\nu}_{~\mu} x_\nu \\   \end{aligned}}

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\displaystyle{ \begin{aligned}   \frac{\partial}{\partial (x')^\mu} &= \frac{\partial x^\nu}{\partial (x')^\mu} \frac{\partial}{\partial x^\nu} \\   &= \frac{\partial x^0}{\partial (x')^\mu} \frac{\partial}{\partial x^0} + \frac{\partial x^1}{\partial (x')^\mu} \frac{\partial}{\partial x^1} + \frac{\partial x^2}{\partial (x')^\mu} \frac{\partial}{\partial x^2} + \frac{\partial x^3}{\partial (x')^\mu} \frac{\partial}{\partial x^3} \\   \end{aligned}}

Now we consider \displaystyle{f} as a function of \displaystyle{x^{\mu}}‘s:

\displaystyle{f(x^0, x^1, x^2, x^3)}

Since \displaystyle{x^{\mu}}‘s and \displaystyle{(x')^{\mu}}‘s are related by Lorentz transform, \displaystyle{f} is also a function of \displaystyle{(x')^{\mu}}‘s, although indirectly.

\displaystyle{f(x^0((x')^0, (x')^1, (x')^2, (x')^3), x^1((x')^0, ...), x^2((x')^0, ...), x^3((x')^0, ...))}

For notational simplicity, we write \displaystyle{f} as

\displaystyle{f(x^\alpha((x')^\beta))}

Since \displaystyle{f} is a function of \displaystyle{(x')^{\mu}}‘s, we can differentiate it with respect to \displaystyle{(x')^{\mu}}‘s.

\displaystyle{ \begin{aligned}   \frac{\partial}{\partial (x')^\mu} f(x^\alpha((x')^\beta))) &= \sum_{\nu = 0}^4 \frac{\partial x^\nu}{\partial (x')^\mu} \frac{\partial}{\partial x^\nu}  f(x^\alpha) \\   \end{aligned}}

Since

\displaystyle{ \begin{aligned}   x^\nu &= [L^{-1}]^\nu_{~\beta} (x')^\beta \\   \end{aligned}},

\displaystyle{ \begin{aligned}   \frac{\partial f}{\partial (x')^\mu}   &= \sum_{\nu = 0}^4 \frac{\partial}{\partial (x')^\mu} \left[  \sum_{\beta = 0}^4 [L^{-1}]^\nu_{~\beta} (x')^\beta \right] \frac{\partial f}{\partial x^\nu} \\   &= \sum_{\nu = 0}^4 \sum_{\beta = 0}^4 [L^{-1}]^\nu_{~\beta} \frac{\partial (x')^\beta}{\partial (x')^\mu} \frac{\partial f}{\partial x^\nu} \\   &= \sum_{\nu = 0}^4 \sum_{\beta = 0}^4 [L^{-1}]^\nu_{~\beta} \delta^\beta_\mu \frac{\partial f}{\partial x^\nu} \\   &= \sum_{\nu = 0}^4 [L^{-1}]^\nu_{~\mu} \frac{\partial f}{\partial x^\nu} \\   &= [L^{-1}]^\nu_{~\mu} \frac{\partial f}{\partial x^\nu} \\   \end{aligned}}

Therefore,

\displaystyle{ \begin{aligned}   \frac{\partial}{\partial (x')^\mu} &= [L^{-1}]^\nu_{~\mu} \frac{\partial}{\partial x^\nu} \\   \end{aligned}}

It is the same as the Lorentz transform for covariant vectors:

\displaystyle{ \begin{aligned}   (x')_\mu &= [L^{-1}]^{\nu}_{~\mu} x_\nu \\   \end{aligned}}

— Me@2020-11-23 04:27:13 PM

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2020.11.24 Tuesday (c) All rights reserved by ACHK

Global symmetry, 2

In physics, a global symmetry is a symmetry that holds at all points in the spacetime under consideration, as opposed to a local symmetry which varies from point to point.

Global symmetries require conservation laws, but not forces, in physics.

— Wikipedia on Global symmetry

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2020.11.22 Sunday ACHK

Light, 3

無額外論 7

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The one in the mirror is your Light.

— Me@2011.06.24

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Thou shalt have no other gods before Me.

— one of the Ten Commandments

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God teach you through your mind; help you through your actions.

— Me@the Last Century

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2020.11.21 Saturday (c) All rights reserved by ACHK