PhD, 3.8.2

財政自由 1.3.2

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讀研究院的最大作用是,獲取獨家資料和人脈。

做到物理學家的其中一個,近乎先決條件是,認識一些一流的物理學家,從而可以跟他們對話。換句話說,讀研究院的主要目的是,爭取跟一些物理神人,對話的機會。
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(問:但是,你又提議最好在,已經有財政自由時,才讀研究院?那樣,豈不是起碼要三、四十歲時,才可以讀研究院?

那時,頭腦都已經,大大不如十多二十歲了。)

無錯。那是兩難。

有如人生目標一樣,不會只有一個。當兩個目標有衝突時,要麼取一捨一、要麼雙方妥協。有時靈機一觸的話,則可以協同互生,合而為一。

(問:那即是怎樣?)

每個人不同,沒有一定的答案。但是,你可以參考學術界中,成功或失敗人士的經驗,從而避開一些宏觀的錯誤。之於其他細節,只能隨機應變,見步行步,行步見步。

例如,有些人年輕時專心賺錢,暫時放棄讀書;打算老一點時,才重操學業。但是,老一點時,已經沒有心思了。

沒有「心」的原因是,年紀越大,機會成本越高。亦即是話,研究學術的時間,往往可以用於更偉大的地方。沒有「思」的原因是,年紀越大,一般而言,身體和智力也不如年輕時。

(問:你即是話,財政充裕前,讀研究院時,即使智力再高,往往沒有自由去,研究自己喜歡的課題。但是,要等到財政自由時,才研究學術的話,又未必仍然有足夠的智力。)

無錯。那是兩難。

有如人生目標一樣,不會只有一個。當兩個目標有衝突時,要麼取一捨一、要麼雙方妥協。有時靈機一觸的話,則可以協同互生,合而為一。

(問:協同互生?如何執行?可否舉一個例?)

— Me@2019-10-22 09:44:23 PM

— Me@2019-12-17 09:35:51 PM

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2019.12.18 Wednesday (c) All rights reserved by ACHK

University

d_2019_09_29__17_29_36_PM_

In this picture, he was at his 15.

Later on, he studied at another university.

— Me@2019-12-15 03:27:25 PM

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2019.12.15 Sunday (c) All rights reserved by ACHK

Varying the action, 2.2

\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L \circ \Gamma[q]) \delta_\eta \Gamma[q] \\ \end{aligned}}

\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}}

There are two kinds of tuples: up tuples and down tuples. We write tuples as ordered lists of their components; a tuple is delimited by parentheses if it is an up tuple and by square brackets if it is a down tuple.

— Structure and Interpretation of Classical Mechanics

So \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))} is really a dot product:

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\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}}

— Me@2019-12-14 06:11:22 PM

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2019.12.14 Saturday (c) All rights reserved by ACHK

Classical physics

Quantum Mechanics 6

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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:

  1. Classical coherent fields (e.g. light waves) appearing as a state of many particles (photons)

  2. 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

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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

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2019.12.14 Saturday (c) All rights reserved by ACHK

點石成金 8

The Metagame, 2

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無謂的事務,如果不可避免,你可試試加一個有謂的情境。

For a boring but unavoidable task, add an amazing context.

For an interesting but useless activity, add a meaningful context.

For example, I use video games to train my courage.

— Me@2011.08.24

— Me@2019-12-12

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2019.12.12 Thursday (c) All rights reserved by ACHK

Ken Chan 時光機 3.1

當年,他留下了傳呼機台的號碼。留言後,他會電話回覆,回答物理問題。

我在從事教學後才知道,用電話答物理概念問題,不會太花時間。但是,如果是答具體物理題目的話,其實十分費時,因為沒有紙筆的輔助。需要畫圖或者有繁複運算步驟時,大家也只能靠想像力。

同一題題目,假設當面解答,只需要 5 分鐘。但是,電話指點的話,就可以花上 15 分鐘至半小時不等。

但是,Ken Chan 仍然願意,以這個方式,為最多的學生,解答最多的問題,我對他十分感謝。

有一次,他在晚上十一時多回覆我的致電。他說先要關掉實驗機器。問了題目後,我順道問他讀書方法,訴苦說,來不及溫習應試。

他答了我很久,花了他大概半小時至一個小時,非常抱歉。

他講了一些東西。以下不依次序。

中五(會考年)每天溫習五、六小時,很正常的事。

你化學不好,為什麼不補化學?

我答:補得太多的話,我沒有足夠溫習。

他說:咁又係。(那又是。)

他問我附加數學有哪些課題。

如果讀了那幾個課題就不會有大問題。

— Me@2019-12-08 10:36:20 AM

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2019.12.11 Wednesday (c) All rights reserved by ACHK

Problem 13.5b

A First Course in String Theory

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13.6 Unoriented closed strings

This problem is the closed string version of Problem 12.12. The closed string \displaystyle{X^{\mu} (\tau, \sigma)} with \displaystyle{\sigma \in [0, 2 \pi]} and fixed \displaystyle{\tau} is a parameterized closed curve in spacetime. The orientation of a string is the direction of the increasing \displaystyle{\sigma} on this curve.

Introduce an orientation reversing twist operator \displaystyle{\Omega} such that

\displaystyle{\Omega X^I(\tau, \sigma) \Omega^{-1}} = X^I (\tau, 2 \pi - \sigma)

Moreover, declare that

\displaystyle{\Omega x_0^- \Omega^{-1} = x_0^-}

\displaystyle{\Omega p^+ \Omega^{-1} = p^+}

(b) Used the closed string oscillator expansion (13.24) to calculate

\displaystyle{\Omega x_0^I \Omega^{-1}}

\displaystyle{\Omega \alpha_0^I \Omega^{-1}}

\displaystyle{\Omega \alpha_n^I \Omega^{-1}}

\displaystyle{\Omega \bar \alpha_n^I \Omega^{-1}}

~~~

Equation (13.24):

\displaystyle{X^{\mu} (\tau, \sigma) = x_0^\mu + \sqrt{2 \alpha'} \alpha_0^\mu \tau + i \sqrt{\frac{\alpha'}{2}} \sum_{n \ne 0} \frac{e^{-in\tau}}{n} (\alpha_n^\mu e^{i n \sigma} + \bar \alpha_n^\mu e^{-in \sigma})}

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\displaystyle{\begin{aligned}   X^{\mu} (\tau, \sigma) &= x_0^\mu + \sqrt{2 \alpha'} \alpha_0^\mu \tau + i \sqrt{\frac{\alpha'}{2}} \sum_{n \ne 0} \frac{e^{-in\tau}}{n} (\alpha_n^\mu e^{i n \sigma} + \bar \alpha_n^\mu e^{-in \sigma}) \\   X^I (\tau, 2 \pi - \sigma)  &= x_0^I + \sqrt{2 \alpha'} \alpha_0^I \tau + i \sqrt{\frac{\alpha'}{2}} \sum_{n \ne 0} \frac{e^{-in\tau}}{n} \left( \alpha_n^I e^{- in\sigma} + \bar \alpha_n^I e^{i n \sigma)} \right) \\   \end{aligned}}

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\displaystyle{\Omega X^I(\tau, \sigma) \Omega^{-1}} = X^I (\tau, 2 \pi - \sigma)

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By comparing \displaystyle{\Omega X^I(\tau, \sigma) \Omega^{-1}} with \displaystyle{X^I (\tau, 2 \pi - \sigma)}, we have:

\displaystyle{\begin{aligned}   \Omega x_0^I \Omega^{-1} &= x_0^I \\  \Omega \alpha_0^I \Omega^{-1} &= \alpha_0^I \\  \Omega \alpha_n^I \Omega^{-1} &= \bar \alpha_n^I \\  \Omega \bar \alpha_n^I \Omega^{-1} &= \alpha_n^I \\   \end{aligned}}

— Me@2019-11-24 04:33:23 PM

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2019.11.24 Sunday (c) All rights reserved by ACHK

PhD, 3.8.1

財政自由 1.3.1

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The secret to creativity is knowing how to hide your sources.

— Not Einstein

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In 1924, while working as a Reader (Professor without a chair) at the Physics Department of the University of Dhaka, Bose wrote a paper deriving Planck’s quantum radiation law without any reference to classical physics by using a novel way of counting states with identical particles. This paper was seminal in creating the very important field of quantum statistics. Though not accepted at once for publication, he sent the article directly to Albert Einstein in Germany. Einstein, recognising the importance of the paper, translated it into German himself and submitted it on Bose’s behalf to the prestigious Zeitschrift für Physik. As a result of this recognition, Bose was able to work for two years in European X-ray and crystallography laboratories, during which he worked with Louis de Broglie, Marie Curie, and Einstein.

— Wikipedia on Satyendra Nath Bose

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(問:根據你的講法,好像大部分情況下,都不應該讀研究院似的。)

在理想的情況下,你可能應該讀研究院。

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(問:那樣,你心目中的理想情況是什麼?)

假設你已經有財政自由,你就有可能,適合讀研究院;…

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讀研究院的最大作用是,獲取獨家資料和人脈。

做到物理學家的其中一個,近乎先決條件是,認識一些一流的物理學家,從而可以跟他們對話。換句話說,讀研究院的主要目的是,爭取跟一些物理神人,對話的機會。

(問:那金錢和時間成本奇高。間中約他們暢談可以嗎?)

那比較困難。

如果你不是他們原本的朋友、同事或學生的話,大概不會有足夠時間,分配給你。

(問:自己一個做研究,一定不可以嗎?)

那十分困難。

即使是獨行俠愛因思坦,他的獨行俠形象,也是假的。大概而言,那只是他的大眾形象、公關技巧。

實情是,他在學術上,有一個開放謙卑的態度,十分願意吸收他人的思想,無論對方當時的名氣是怎麼樣。

例如,他有一段時期會,參加維也納學團的學術討論聚會。

又例如,有來自印度一所大學的,一位尚未世界知名的物理學家,企圖發表一篇文章,但給學術期刊拒絕了。於是,他把那篇文章,寄給了愛因思坦。

雖然素未謀面(?),愛因思坦仍然用心閱讀,發現該文有料到,十分有意思。不單如此,他更親自把文章由英文翻譯成德文。在他的引薦下,德國一著名物理期刊出版了該文。

那位印度物理學家,就是後來舉世聞名的玻色。

愛因思坦一生幾大曠世鉅著之一,玻色-愛因斯坦統計規律,就是緣起於他和玻色的這次合作。

— Me@2019-10-29 10:20:33 PM

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Bose adapted this lecture into a short article called Planck’s Law and the Hypothesis of Light Quanta and submitted it to the Philosophical Magazine. However, the referee’s report was negative, and the paper was rejected. Undaunted, he sent the manuscript to Albert Einstein requesting publication in the Zeitschrift für Physik. Einstein immediately agreed, personally translated the article from English into German (Bose had earlier translated Einstein’s article on the theory of General Relativity from German to English), and saw to it that it was published. Bose’s theory achieved respect when Einstein sent his own paper in support of Bose’s to Zeitschrift für Physik, asking that they be published together. The paper came out in 1924.

— Wikipedia on Bose–Einstein statistics

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2019.10.29 Tuesday (c) All rights reserved by ACHK

寧化飛灰,不作浮塵

And what of that truth which more than anything else gives me confidence in Hong Kong? The truth is this. The qualities, the beliefs, the ideals that have made Hong Kong’s present will still be here to shape Hong Kong’s future.

Hong Kong, it seems to me, has always lived by the author, Jack London’s credo:

“I would rather be ashes than dust,
I would rather my spark should burn out in a brilliant blaze,
Than it should be stifled in dry rot.
I would rather be a superb meteor,
With every atom of me in magnificent glow,
Than a sleepy and permanent planet.”

Whatever the challenges ahead, nothing should bring this meteor crashing to earth, nothing should snuff out its glow. I hope that Hong Kong will take tomorrow by storm. And when it does, History will stand and cheer.

最能使我對香港信心十足的事實又是甚麼呢?那便是港人的優良特質、信念和理想,不僅為-香港奠下了今天的基業,而且必會繼續為香港開創美好明天。

在我看來,香港一直在生活中實踐作家傑克˙倫敦的信條:

「寧化飛灰,不作浮塵。
寧投熊熊烈火,光盡而滅;
不伴寂寂朽木,默然同腐。
寧為耀目流星,迸發萬丈光芒;
不羨永恒星體,悠悠沉睡終古。」

前路不管有何挑戰,都不會,我重複,都不會使這顆流星飛墜,光華從此消逝。我深願香港-能奮然而起,征服未來,那時候,歷史也必為之動容,起立喝采。

— Christopher Francis Patten

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2019.10.13 Sunday ACHK

Varying the action, 2.1

Equation (1.28):

\displaystyle{S[q](t_1, t_2) = \int_{t_1}^{t_2} L \circ \Gamma[q]}

Equation (1.30):

\displaystyle{h[q] = L \circ \Gamma[q]}

\displaystyle{\delta_\eta S[q](t_1, t_2) = \int_{t_1}^{t_2} \delta_\eta h[q]}

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Let \displaystyle{F} be a path-independent function and \displaystyle{g} be a path-dependent function; then

\displaystyle{\delta_\eta h[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]~~~~~\text{with}~~~~~h[q] = F \circ g[q].~~~~~(1.26)}

\displaystyle{\delta_\eta F \circ g[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]}

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— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics

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\displaystyle{ \begin{aligned} &\delta_\eta S[q] (t_1, t_2) \\ &= \int_{t_1}^{t_2} \delta_\eta \left( L \circ \Gamma[q] \right) \\  \end{aligned}}

Assume that \displaystyle{L} is a path-independent function, so that we can use Eq. 1.26:

\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L \circ \Gamma[q]) \delta_\eta \Gamma[q] \\  \end{aligned}}

\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L \circ \Gamma[q]) (0, \eta(t), D\eta(t)) \\  &= \int_{t_1}^{t_2} (D L \left[ \Gamma[q] \right]) (0, \eta(t), D\eta(t)) \\  \end{aligned}}

Assume that \displaystyle{L} is a path-independent function, so that any value of \displaystyle{L} depends on the value of \displaystyle{\Gamma} at that moment only, instead of depending on the whole path \displaystyle{\Gamma}:

\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L (\Gamma[q])) (0, \eta(t), D\eta(t)) \\  &= \int_{t_1}^{t_2} (D L (t, q, v)) (0, \eta(t), D\eta(t)) \\  &= \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}}

What kind of product is it here? Is it just a dot product? Probably not.

\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} [\partial_1 L (t, q, v) \eta(t) + \partial_2 L (t, q, v) D\eta(t)] \\  \end{aligned}}

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— Me@2019-10-12 03:42:01 PM

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2019.10.13 Sunday (c) All rights reserved by ACHK

Introduction to Differential Equations

llamaz 1 hour ago [-]

I think the calculus of variations might be a better approach to introducing ODEs in first year.

You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of \displaystyle{\frac{dy}{dx} = 0}, you end up with an ODE (viz. the Euler-Lagrange equation).

This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system.

Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you’ve motivated ODEs and given the students a reason to solve ODEs with constant coefficients.

— Hacker News

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2019.10.02 Wednesday ACHK

Ken Chan 時光機 2.2

那時,Ken Chan 有一項特點,令我覺得奇怪。

他有極多的職位。當時,我不明白,他哪有那麼多的時間。

長大後,我發現,其實,有很大機會,那只是語言技倆。例如:

  1. 當時他眾多職位之中,全部是真的嗎?

  2. 即使全部是真的,有多少是實職?又有多少,只是名銜而已?

  3. 即使全部是實職,有多少需要親力親為?又有多少,只是出主意、提意見而已?

長大後,我發現,一句說話,即使根據字面意思,不是直接的假話,也可以十分誤導。

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例如,

愛迪生一生中,發明了千多樣科技産品。

沒有誤導的版本是,

愛迪生一生中,透過他的公司,發明了千多樣科技産品。

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又例如,

達文西一生中,在多門學問,也有鉅大的成就。

如實反映的版本是,

達文西一生中,在他工作室團隊的附助下,在多門學問,也有鉅大的成就。

— Me@2019-09-30 01:09:36 PM

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2019.09.30 Monday (c) All rights reserved by ACHK

Problem 13.6b

A First Course in String Theory | Topology, 2 | Manifold, 2

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13.6 Orientifold Op-planes

~~~

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for “orbit-manifold”) is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure.

The underlying space locally looks like the quotient space of a Euclidean space under the linear action of a finite group.

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In string theory, the word “orbifold” has a slightly new meaning. For mathematicians, an orbifold is a generalization of the notion of manifold that allows the presence of the points whose neighborhood is diffeomorphic to a quotient of \displaystyle{\mathbf{R}^n} by a finite group, i.e. \displaystyle{\mathbf{R}^n/\Gamma}. In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit space \displaystyle{M/G} where \displaystyle{M} is a manifold (or a theory), and \displaystyle{G} is a group of its isometries (or symmetries) — not necessarily all of them. In string theory, these symmetries do not have to have a geometric interpretation.

— Wikipedia on Orbifold

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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an \displaystyle{n}-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension \displaystyle{n}.

— Wikipedia on Manifold

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In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints.

— Wikipedia on Topological space

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2019.09.26 Thursday ACHK

Pointer state, 2

Eigenstates 3.2

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Microscopically, a state can be definite or indefinite. Even if it is indefinite, the overlapping of superpositions of states of a lot of particles, or the superposition of a lot of system-microstates gives a definite macrostate.

If a state is definite, it is corresponding to one single system-macrostate directly.

I am referring to the physical definition, not the mathematical definition.

— Me@2012-12-31 09:28:08 AM

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If a microstate is definite, it is called an “eigenstate”. It is corresponding to one single system-macrostate directly.

However, the microstate is NOT the macrostate. The microstate is just corresponding to that macrostate.

— Me@2019-09-20 07:02:10 AM

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In quantum Darwinism and similar theories, pointer states are quantum states, sometimes of a measuring apparatus, if present, that are less perturbed by decoherence than other states, and are the quantum equivalents of the classical states of the system after decoherence has occurred through interaction with the environment. ‘Pointer’ refers to the reading of a recording or measuring device, which in old analog versions would often have a gauge or pointer display.

— Wikipedia on Pointer state

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In quantum mechanics, einselections, short for environment-induced superselection, is a name coined by Wojciech H. Zurek for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descriptions of reality from quantum descriptions.

In this approach, classicality is described as an emergent property induced in open quantum systems by their environments. Due to the interaction with the environment, the vast majority of states in the Hilbert space of a quantum open system become highly unstable due to entangling interaction with the environment, which in effect monitors selected observables of the system.

After a decoherence time, which for macroscopic objects is typically many orders of magnitude shorter than any other dynamical timescale, a generic quantum state decays into an uncertain [in the sense of classical probability] state which can be decomposed into a mixture of simple pointer states. In this way the environment induces effective superselection rules. Thus, einselection precludes stable existence of pure superpositions of pointer states. These ‘pointer states’ are stable despite environmental interaction. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement and superposition.

Advocates of this approach argue that since only quasi-local, essentially classical states survive the decoherence process, einselection can in many ways explain the emergence of a (seemingly) classical reality in a fundamentally quantum universe (at least to local observers). However, the basic program has been criticized as relying on a circular argument (e.g. R. E. Kastner). So the question of whether the ‘einselection’ account can really explain the phenomenon of wave function collapse remains unsettled.

— Wikipedia on Einselection

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Here I simply review the basic approach to ‘deriving’ einselection via decoherence, and point to a key step in the derivation that makes it a circular one.

— Ruth E. Kastner

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We should not derive einselection via decoherence. Instead, they should be regarded as different parts or different presentations of the same theory.

In other words, “einselection” and “decoherence” are synonyms.

— Me@2019-09-21 05:53:53 PM

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There has been significant work on correctly identifying the pointer states in the case of a massive particle decohered by collisions with a fluid environment, often known as collisional decoherence. In particular, Busse and Hornberger have identified certain solitonic wavepackets as being unusually stable in the presence of such decoherence.

— Wikipedia on Einselection

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2019.09.22 Sunday (c) All rights reserved by ACHK