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

財政自由 1.2

(問:那就即是話,要在還年青時,就賺到一生夠用的金錢?

為什麼要選擇,那麼難的目標?)

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(問:但是,財政自由,又可以如何實現呢?)

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首先,要有一個概念,就是:

長遠而言,為生不可靠「售賣時間」,而要靠「創造價值」。

例如,你是一位音樂演奏家。如果你的收入,主要來自去現場演出的話,那就為之,以「售賣時間」為生。

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假設,作為音樂演奏家的你,今次的主要收入,再不是來自去現場演出,而是靠灌錄唱片來售賣的話,你就是以「創造價值」為生。那就再不只是「一分耕耘,一分收穫」了。你的收入,再不取決於,你花了多少時間演奏;而是取決於,你演奏的質素。那樣,你的收入,再不只是,靠主動去賺取,還同時有另一部分,可以被動地接收。

前者可以稱為「主動收入」;而後者則可稱為「自動收入」。

(問:依靠「售賣時間」而來的收入,就為之「主動收入」?

倚賴「創造價值」而來的收入,就即是「自動收入」?)

無錯。

(問:有自動生成收入的話,就當然十分高興。但是,那些「自動收入」,又從何而來呢?

你講的例子,我很難執行。

我又不是音樂家。即使是音樂家,大賣唱片,或者開到大型演奏會的,萬中無一。)

那並不是非黑即白。我並不是叫你,將你所有的靠「售賣時間」的收入來源,立刻刪除,百分百改為「創造價值」。

第一步應是,正職以外先加多,起碼一個收入來源;不論那一個新的賺錢方法,在開始時,賺到的有多微薄。

重點是,那方法的方向正確,長遠而言,可以令你收入中的自動部分,比重越來越多。

— Me@2019-09-10 08:33:38 PM

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2019.09.14 Saturday (c) All rights reserved by ACHK

事件實在論,更正

Event Realism | 事件實在論 6.1

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exist = can be found

無後果,就不再存在。

— Me@2013.09.25

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If the consequences of an event cannot be found anymore, that event no longer exists.

— Me@2019.09.05

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The surprising implications of the original delayed-choice experiment led Wheeler to the conclusion that “no phenomenon is a phenomenon until it is an observed phenomenon”, which is a very radical position. Wheeler famously said that the “past has no existence except as recorded in the present“, and that the Universe does not “exist, out there independent of all acts of observation”.

— Wikipedia on Wheeler’s delayed choice experiment

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「事件」並不完全「實在」。

實在 ~ 堅實地存在

仍然有後果的事件,才為之「仍然存在」。

永久地有後果的事件,才為之「實在」。

— Me@2019-09-05 09:08:41 PM

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2019.09.05 Thursday (c) All rights reserved by ACHK

A Tale of Two L’s

Lagrange’s equations are traditionally written in the form

\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}} \right ) = \frac {\partial L}{\partial q}}

or, if we write a separate equation for each component of \displaystyle{q}, as

\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q^i}} \right ) = \frac {\partial L}{\partial q^i}}

In this way of writing Lagrange’s equations the notation does not distinguish between \displaystyle{L}, which is a real-valued function of three variables \displaystyle{(t, q, \dot q)}, and \displaystyle{L \circ \Gamma[q]}, which is a real-valued function of one real variable \displaystyle{t}.

— Structure and Interpretation of Classical Mechanics

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2019.09.04 Wednesday ACHK

Quantum observer 2

Consistent histories, 6.2 | Relational quantum mechanics, 2 | Eigenstates 2.3.2.2

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Would an observer see itself being in a superposition?

In a sense, tautologically, an observer is not a superposition of itself, because “an observer” can be defined as “a consistent history”.

an observer ~ a consistent history

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Because “state” is expressed in RQM as the correlation between two systems, there can be no meaning to “self-measurement”.

— Wikipedia on Relational quantum mechanics

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Would an observer see itself being in a superposition?

When we say that “before observation, observable B is in a superposition of some eigenstates”, you have to specify

  1. it is a superposition of what?

  2. it is a superposition with respect to what apparatuses or experimental setups?

— Me@2018-02-05 12:45 AM

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2019.09.03 Tuesday (c) All rights reserved by ACHK

The Time Machine

This is the place.

So it is.

_But there’s nothing here.

Well, it was different then. My laboratory was all around here. The kitchen was up there where that tree is. Not that Mrs. Watchit ever let me go in there.

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I don’t know what to tell you, sir. He’s been gone this whole week.

_And you’ve no idea where he went?

No, sir.

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_[Alexander] This would be my greenhouse. There was a garden outside.

Gren’tormar’tas?

Yes.

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I’m glad.

_Sir?

I’m glad he’s gone.

Maybe he’s finally found some place where he can be happy.

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This was my home.

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His home.

— The Time Machine (2002 film)

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2019.09.01 Sunday ACHK

Ken Chan 時光機 2.1

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。

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1.1 除了是物理補習天王外,

2.1 他宣稱有在大學教書。是教授、講師,還是其他,我就不知道。我忘記了,他有沒有講過。

2.2 他在大學做研究。據我理解,他當時研究的是有關激光的實驗物理。

有一次,我打電話給他問題目。他說他正在做實驗,需要先行關掉,用來做實驗的機器。

2.3 他有時要往大陸作學術演講。

3.1 然後,他亦是某某什麼工程學會的主席。

名銜太多,不能盡錄。

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雖然,那是九十年代,但是,他十年後的廣告,大概可作參考。

畢業於香港大學工程學系,並擁有多個學位、專業資格及榮銜,包括: 香港大學太古學者、英國皇家物理學會、香港工程師學會、電機電子工程師學會、中國機械工程師學會……等等。

廣告中的那些學會,不知是哪個意思。例如,他是電機電子工程師學會的什麼?

只是會員?還是管理層?

— Me@2019-08-30 09:31:52 PM

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