Quick Calculation 15.1

Posted on May 26, 2019 by rpflee

A First Course in String Theory

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.

~~~

What is $\displaystyle{U(1)}$ ?

— Me@2019-05-24 11:25:41 PM

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

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

2019.05.25 Saturday ACHK

Unitarity (physics)

Posted on May 23, 2019 by rpflee

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

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)

Free

Posted on May 18, 2019 by rpflee

So long as a man can look into the eyes of his oppressor, he is free.

— Me and the Big Guy

2019.05.18 Saturday ACHK

PhD, 3.7.1

Posted on May 18, 2019 by rpflee

碩士 4.7.1 | On Keeping Your Soul, 2.2.7.1

（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

1996-1997

Posted on May 13, 2019 by rpflee

— 1996-1997

The Principle of Stationary Action

Posted on May 13, 2019 by rpflee

Varying the action

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

The Principle of Stationary Action

$\displaystyle{\delta_\eta S[q] (t_1, t_2) = 0}$

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

— Structure and Interpretation of Classical Mechanics

2019.05.13 Monday ACHK

Multiverse

Posted on May 11, 2019 by rpflee

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

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

In Search of Lost Time, 4

Posted on May 11, 2019 by rpflee

回到過去 4 | 止蝕 5 | Regret, 2

You can’t change the beginning, but you can change the ending.

—

You can’t go back and change the beginning, but you can start where you are and change the ending.

—

2019.05.11 Saturday ACHK

追憶逝水年華, 3

Posted on May 10, 2019 by rpflee

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

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。~~

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

Physical laws are low-energy approximations to reality, 1.3.2

Posted on May 6, 2019 by rpflee

d_2019_01_31__23_13_13_PM_

QCD, Maxwell, Dirac equation, spin wave excitation, superconductivity, …

~ low energy physics

symmetry breaking

$\displaystyle{\downarrow}$

local minimum

$\displaystyle{\downarrow}$

simple physics

— Me@2019-05-06 11:12:02 PM

Ex 1.8 Implementation of $\delta$

Posted on May 5, 2019 by rpflee

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

The variation may be represented in terms of a derivative.

— Structure and Interpretation of Classical Mechanics

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

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

Classical probability, 7

Posted on May 3, 2019 by rpflee

Classical probability is macroscopic superposition.

— Me@2012.04.23

That is not correct, except in some special senses.

— Me@2019-05-02

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

— Me@2019-05-03 08:44:11 PM

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

Forget oneself

Posted on May 2, 2019 by rpflee

神來之筆 3

忘我

~ the state of releasing yourself

~ to let god take control of you

— Me@2009.07.25

PhD, 3.6

Posted on May 1, 2019 by rpflee

碩士 4.6 | On Keeping Your Soul, 2.2.6

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

Physics Town

continues

Monthly Archives: May 2019