Quick Calculation 15.1

A First Course in String Theory

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Recall that a group is a set which is closed under an associative multiplication; it contains an identity element, and each element has a multiplicative inverse. Verify that \displaystyle{U(1)} and \displaystyle{U(N)}, as described above, are groups.

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What is \displaystyle{U(1)}?

— Me@2019-05-24 11:25:41 PM

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The set of all \displaystyle{1 \times 1} unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to \displaystyle{U(1)}, the first unitary group.

— Wikipedia on Circle group

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In mathematics, a complex square matrix \displaystyle{U} is unitary if its conjugate transpose \displaystyle{U^*} is also its inverse—that is, if

\displaystyle{U^{*}U=UU^{*}=I,}

where \displaystyle{I} is the identity matrix.

In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (\displaystyle{\dagger}) and the equation above becomes

\displaystyle{U^{\dagger }U=UU^{\dagger }=I.}

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

— Wikipedia on Unitary matrix

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2019.05.25 Saturday ACHK

Unitarity (physics)

Unitarity means that if a future state, F, of a system is unique, the corresponding past point, P,  is also unique, provided that there is no information lost on the transition from P to F.

— Me@2019-05-22 11:06:48 PM

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In quantum physics, unitarity means that the future point is unique, and the past point is unique. If no information gets lost on the transition from one configuration to another[,] it is unique. If a law exists on how to go forward, one can find a reverse law to it.[1] It is a restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1.

Since unitarity of a theory is necessary for its consistency (it is a very natural assumption, although recently questioned[2]), the term is sometimes also used as a synonym for consistency, and is sometimes used for other necessary conditions for consistency, especially the condition that the Hamiltonian is bounded from below. This means that there is a state of minimal energy (called the ground state or vacuum state). This is needed for the third law of thermodynamics to hold.

— Wikipedia on Unitarity (physics)

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2019.05.23 Thursday (c) All rights reserved by ACHK

PhD, 3.7.1

碩士 4.7.1 | On Keeping Your Soul, 2.2.7.1

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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。

(問:為什麼不講實際情況,而要講理想情況。講理想情況,即是脫離現實、執行不到。那不是浪費時間嗎?)

不是。

一來,「理想不能達到」,只是通常,並非必然。

二來,即使在「理想不能達到」的情況下,探討理想,都可以有益處;因為,那會令你知道,如何接近它。

(問:但是,現實可能同理想,相差「十萬八千里」。)

那樣,你仍然可以研究,如何把那「十萬八千里」的距離,縮減成「十萬七千九百九十九里」。

(問:那麼小的進步,又有什麼用呢?)

多麼小的進步,都可能帶來,鉅大的後果。例如,考試僅僅合格,和僅僅不合格,相差多少分?

還有,你並不是進步一點後,就原地踏步。

反而,進了一小步後,現實就立刻稍有不同,令你很有機會,可以繼而,進下一小步,甚至是下一大步。

簡言之,「見步行步,行步見步」,是一個不斷循環的過程。

(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 …

— Me@2019-05-18 03:14:02 PM

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2019.05.18 Saturday (c) All rights reserved by ACHK

Multiverse

A physics statement is meaningful only if it is with respect to an observer. So the many-world theory is meaningless.

— Me@2018-08-31 12:55:54 PM

— Me@2019-05-11 09:41:55 PM

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Answer me the following yes/no question:

In your multi-universe theory, is it possible, at least in principle, for an observer in one universe to interact with any of the other universes?

If no, then it is equivalent to say that those other universes do not exist.

If yes, then those other universes are not “other” universes at all, but actually just other parts of the same universe.

— Me@2019-05-11 09:43:40 PM

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2019.05.11 Saturday (c) All rights reserved by ACHK

追憶逝水年華, 3

In Search of Lost Time, 3 | (反對)開夜車 3.1 | 止蝕 4

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其實,我中五那年,試過企圖「開夜車」。詳細過程怎樣,已不太記得。但是,總體的感受,仍然是深刻。

那時,我有很多天真的想法,例如:

1. 只要每晚睡少四小時,就每晚可以多四小時溫習。

2. 只要每晚可以多四小時溫習,就可以追回之前,落後了的進度。

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實情是,「貪心」帶來「貧窮」:

很多時,我也是覺得自己,溫習的進度落後了,來不及準備那年的公開試。所以,想透過凌晨(例如)三時起牀,來追回之前,落後了的進度。但是,我卻幾乎每天凌晨,也起不到牀,導致溫習大計失敗,遺撼非常。繼而,因為失落了,更多的時間,我就覺得,更加需要在下一天,凌晨起牀。

惡性循環,持續了很久,浪費了我大量的時間。其實,只要我睡眠適量,作息定時,反而有不少機會,有上佳的成績。

現在回想起來,仍然驚嘆著,為何年輕時,可以那麼愚蠢。

只可以說,人在惶恐時,多麼糊塗的事,都可以發生。

— Me@2019-05-09 10:04:55 PM

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2019.05.10 Friday (c) All rights reserved by ACHK

Ex 1.8 Implementation of $\delta$

\displaystyle{ \begin{aligned} \delta_\eta f[q] &= \lim_{\epsilon \to 0} \left( \frac{f[q+\epsilon \eta]-f[q]}{\epsilon} \right) \\ \end{aligned}}

The variation may be represented in terms of a derivative.

— Structure and Interpretation of Classical Mechanics

\displaystyle{ \begin{aligned} g( \epsilon ) &= f[q + \epsilon \eta] \\ \delta_\eta f[q] &= \lim_{\epsilon \to 0} \left( \frac{g(\epsilon) - g(0)}{\epsilon} \right) \\ &= D g(0) \\ \end{aligned}}

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A lambda expression evaluates to a procedure. The environment in effect when the lambda expression is evaluated is remembered as part of the procedure; it is called the closing environment.

— Structure and Interpretation of Classical Mechanics

(define (((delta eta) f) q)
  (let ((g (lambda (epsilon) (f (+ q (* epsilon eta))))))
    ((D g) 0))) 

— Me@2019-05-05 10:47:46 PM

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2019.05.05 Sunday (c) All rights reserved by ACHK

Classical probability, 7

Classical probability is macroscopic superposition.

— Me@2012.04.23

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That is not correct, except in some special senses.

— Me@2019-05-02

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That is not correct, if the “superposition” means quantum superposition.

— Me@2019-05-03 08:44:11 PM

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The difference of the classical probability and quantum probability is the difference of a mixed state and a pure superposition state.

In classical probability, the relationship between mutually exclusive possible measurement results, before measurement, is OR.

In quantum probability, if the quantum system is in quantum superposition, the relationship between mutually exclusive possible measurement results, before measurement, is neither OR nor AND.

— Me@2019-05-03 06:04:27 PM

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2019.05.03 Friday (c) All rights reserved by ACHK

PhD, 3.6

碩士 4.6 | On Keeping Your Soul, 2.2.6

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有生以來,你遇過的教師有多少個?

而值得尊重的,又有多少個?

— Me@2019-04-30 11:22:05 PM

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2019.04.30 Tuesday (c) All rights reserved by ACHK

Varying a path

Suppose that we have a function \displaystyle{f[q]} that depends on a path \displaystyle{q}. How does the function vary as the path is varied? Let \displaystyle{q} be a coordinate path and \displaystyle{q + \epsilon \eta} be a varied path, where the function \displaystyle{\eta} is a path-like function that can be added to the path \displaystyle{q}, and the factor \displaystyle{\epsilon} is a scale factor. We define the variation \displaystyle{ \delta_\eta f[q]} of the function \displaystyle{f} on the path \displaystyle{q} by

\displaystyle{\delta_\eta f [q] = \lim_{\epsilon \to 0} \left( \frac{f[q + \epsilon \eta] - f[q]}{\epsilon} \right)}

The variation of \displaystyle{f} is a linear approximation to the change in the function \displaystyle{f} for small variations in the path. The variation of \displaystyle{f} depends on \displaystyle{\eta}.

— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics

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Exercise 1.7. Properties of \displaystyle{\delta}

The meaning of \displaystyle{\delta_\eta (fg)[q]} is

\displaystyle{\delta_\eta (f[q]g[q])}

— Me@2019-04-27 07:02:38 PM

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2019.04.27 Saturday ACHK

Mixed states, 4

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How is quantum superposition different from mixed state?

The state

\displaystyle{|\Psi \rangle = \frac{1}{\sqrt{2}}\left(|\psi_1\rangle +|\psi_2\rangle \right)}

is a pure state. Meaning, there’s not a 50% chance the system is in the state \displaystyle{|\psi_1 \rangle } and a 50% it is in the state \displaystyle{|\psi_2 \rangle}. There is a 0% chance that the system is in either of those states, and a 100% chance the system is in the state \displaystyle{|\Psi \rangle}.

The point is that these statements are all made before I make any measurements.

— edited Jan 20 ’15 at 9:54

— Mehrdad

— answered Oct 12 ’13 at 1:42

— Andrew

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Given a state, mixed or pure, you can compute the probability distribution \displaystyle{P(\lambda_n)} for measuring eigenvalues \displaystyle{\lambda_n}, for any observable you want. The difference is the way you combine probabilities, in a quantum superposition you have complex numbers that can interfere. In a classical probability distribution things only add positively.

— Andrew Oct 12 ’13 at 14:41

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— How is quantum superposition different from mixed state?

— Physics StackExchange

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2019.04.23 Tuesday ACHK

(反對)開夜車 2.5

本文章並(!)不(!)可作為醫學建議。如需醫學意見,請諮詢專業人士。

(問:「只要飛蚊不惡化」?你怎能保證?)

不能。那正正就是,我更大的苦惱。

一日不能知道,我當年開始患飛蚊症的真正原因,我也會不安,擔心症狀加劇。

「飛蚊症」只是某個或某些疾病的一個「症」,而不是「疾病」本身。一日不能知道,我飛蚊症的病因,我也不能有力預防,症狀的加劇。

(問:你不是說,長期夜睡少睡,導致你的飛蚊症嗎?)

那只是估計,不是肯定。

那一段時間,「出事」前的一兩個星期,我剛好時常睡不著,導致睡了很少。所以,長期夜睡少睡,很可能是主要的原因。

當然,如果平日的睡眠充足,並不會因為單一事件,某一晚的夜睡少睡,就立刻有飛蚊。如果人體是那麼沒有彈性,人類這種生物,可能一早就已經絕種了。

但是,如果是長期夜睡少睡,身體自然會越來越虛弱。到達某個臨界點時,意外事件就可以隨時發生。從相反的角度來講,如果平日睡眠適量,就等於刪除了,飛蚊症的其中一個主要誘因,從而大大減低了,其發生的機會率。

本文章並(!)不(!)可作為醫學建議。如需醫學意見,請諮詢專業人士。

— Me@2019-04-13 03:34:33 PM

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2019.04.15 Monday (c) All rights reserved by ACHK

Physical laws are low-energy approximations to reality, 1.3.1

d_2019_04_07__22_39_01_PM_

Symmetry breaking is important.

When there is symmetry-breaking, the system goes to a low-energy state.

Each possible low-energy state can be regarded as a new “physical world”.

One “physical world” cannot jump to another, unless through quantum tunnelling. But the probability of quantum tunnelling happening is low.

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Low-energy physics theories, such as harmonic oscillator, are often simple and beautiful.

— Professor Renbao Liu

— Me@2019-04-08 10:46:32 PM

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2019.04.09 Tuesday (c) All rights reserved by ACHK