# 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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# Free

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

— Me and the Big Guy

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2019.05.18 Saturday ACHK

# PhD, 3.7.1

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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。）

（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？）

（問：但是，現實可能同理想，相差「十萬八千里」。）

（問：那麼小的進步，又有什麼用呢？）

（問：那樣，你心目中的理想情況是什麼？）

— Me@2019-05-18 03:14:02 PM

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# The Principle of Stationary Action

Varying the action

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

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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

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2019.05.13 Monday 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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# In Search of Lost Time, 4

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

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

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2019.05.11 Saturday 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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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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— Me@2019-05-09 10:04:55 PM

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# Physical laws are low-energy approximations to reality, 1.3.2

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

~ low energy physics

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symmetry breaking

$\displaystyle{\downarrow}$

local minimum

$\displaystyle{\downarrow}$

simple physics

— Me@2019-05-06 11:12:02 PM

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# 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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# 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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# Forget oneself

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~ the state of releasing yourself

~ to let god take control of you

— Me@2009.07.25

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# PhD, 3.6

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— Me@2019-04-30 11:22:05 PM

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# 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

— 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

# The language of Change, 2

time ~ change (noun)

action ~ change (verb) or counter-change

— Me@2018-02-19 10:24 AM

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# （反對）開夜車 2.5

（問：「只要飛蚊不惡化」？你怎能保證？）

「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？）

— Me@2019-04-13 03:34:33 PM

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# Physical laws are low-energy approximations to reality, 1.3.1

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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