2007, 2

Posted on January 30, 2021 by rpflee

— Me@2021-01-30 05:00:21 PM

Fundamental polygon

Posted on January 30, 2021 by rpflee

A First Course in String Theory

2.5 Constructing simple orbifolds

(b) Consider a torus $\displaystyle{T^2}$ , presented as the $\displaystyle{(x,y)}$ plane with the identifications $\displaystyle{x \sim x + 2}$ and $\displaystyle{y \sim y+2}$ . Choose $\displaystyle{-1 < x, y, \le 1}$ as the fundamental domain. The orbifold $\displaystyle{T^2/\mathbb{Z}_2}$ is defined by imposing the $\displaystyle{\mathbb{Z}_2}$ identification $\displaystyle{(x,y) \sim (-x,-y)}$ .

Prove that there are four points on the torus that are left fixed by the $\displaystyle{\mathbb{Z}_2}$ transformation. Show that the orbifold $\displaystyle{T^2/\mathbb{Z}_2}$ is topologically a two-dimensional sphere, naturally presented as a square pillowcase with seamed edges.

~~~

To find the fixed points, we consider the cases when $\displaystyle{-x = x + 2m}$ and $\displaystyle{-y = y + 2n}$ , where $\displaystyle{m,n \in \mathbb{Z}}$ . Since the length of the interval is only 2, we can consider only the cases when $\displaystyle{m,n = 0, 1}$ . Then the only solutions are

$\displaystyle{(0,0)}$
$\displaystyle{(0,1)}$
$\displaystyle{(1,0)}$
$\displaystyle{(1,1)}$

— Me@2021-01-17 04:14:44 PM

— Wikipedia on Surface (topology)

The formula for this topology is $\displaystyle{ABB^{-1}A^{-1}ABB^{-1}A^{-1} = ABB^{-1}A^{-1} }$ , which is a sphere.

— Me@2021-01-29 06:10:46 PM

Cosmic computer

Posted on January 29, 2021 by rpflee

Consistent histories, 9

There is a cosmic computer there

which is responsible to make sure that

quantum mechanics (laws) will always give consistent measurement results,

such as the ones of the EPR entangled pairs.

NO. That is wrong.

Instead, quantum mechanics itself is THAT cosmic computer that renders all the measurement results consistent.

— Me@2021-01-27 3:54 PM

Feeling

Posted on January 27, 2021 by rpflee

feeling ~ receiving data non-sequentially

— Me@2017-06-03 02:53:16 PM

feeling

~ receiving data in parallel

~ receiving data at once

— Me@2021-01-27 08:19:33 PM

For example, you would not feel that it is actually that the air molecules keep hitting you. You would not feel the speed and force of each individual particle. Instead, you have overall feelings of “pressure” and “temperature”.

Actually, you do not feel the temperature. Instead, you feel “cold”, “cool”, “warm”, or “hot”, etc.

— Me@2021-01-27 08:19:33 PM

feeling

~ turning data into information

~ statistics in real time

— Me@2021-01-27 08:47:39 PM

機遇創生論, 2.2

Posted on January 27, 2021 by rpflee

這段改編自 2010 年 4 月 18 日的對話。

這個大統一理論的成員，包括（但不止於）：

精簡圖：

種子論
反白論
間書原理
完備知識論

自由決定論

它們可以大統一的成因，在於它們除了各個自成一國外，還可以合體理解和應用。

大種子論？

（安：我剛才都是想到「大種子論」。）

但是，大的是理論，而不是種子。

種子大論？

（安：但「種子大論」比較奇怪。）

都是「大種子論」比較好一點。還有，遲一些有更大的理論時，可以叫做，巨大種子論、超巨大種子論、特超巨大種子論等等。

中文名就有了定案。那樣，英文名呢？你有沒有提議？

我暫時想不到，好的英文名字。所以暫時擱置。我們還是先討論其他問題。

這個「大種子論」，和我們之前的「心靈作業系統」理論，如何合體？

亦即之話，如何將「心靈作業系統」裝嵌於「大種子論」之中？

其實兩個理論，似乎沒有什麼關係。我很難想像，如何可以統一。

（安：這個比較難，因為，一個就講人生的某些方面，而另一個則講人的某些性質。換句話說，一個是人生理論，另一個則是人心理論。我暫時看不出，它們如何可以，自然地合體。）

可不可以這樣：「心靈作業系統」健全的人，才可以運行到「大種子論」。「大種子論」是一個健全「心靈作業系統」的必要構成部分。

（安：等等。你剛才那兩句，互相矛盾。你第一句的意思是，一個是「心靈作業系統」，另一個即是「心靈應用程式」。「大種子論」歸類成「應用程式」。但是，在第二句，你又把「大種子論」歸類成「系統程式」（的一部分）。）

有了「大種子論」的話，「心靈作業系統」的運行，就會暢順很多。其原因是在「大種子論」中，有「完備知識論」，講述如何得到「非全部，但完備」的知識體系。心中沒有完備知識，「心靈作業系統」的運作，自然事倍功半。

但是，這樣合體法，有沒有大意義呢？

（安：我們原本的「心靈作業系統」，似乎不是講這些東西。

我們以前討論的是，人既有的心理結構，而不是研究「如何編寫，一個良好的『心靈作業系統』」。如果根據前者，即是「原著」的話，「大種子論」並不是，「心靈作業系統」的一部分。

反而，「心靈作業系統」應該是「大種子論」一部分。了解普遍人性，才有機會駕馭人生。）

— Me@2021-01-23 09:17:24 PM

1990s, 6

Posted on January 25, 2021 by rpflee

— Photomyne colorization

— Me@2021-01-25 03:49:32 PM

Superposition always exists, 2.2.1

Posted on January 25, 2021 by rpflee

Decoherence and the Collapse, 2.1 | Quantum decoherence 7.2.1

But wait! Doesn’t this mean that the “consciousness causes collapse” theory is wrong? The spin bit was apparently able to cause collapse all by itself, so assuming that it isn’t a conscious system, it looks like consciousness isn’t necessary for collapse! Theory disproved!

No. As you might be expecting, things are not this simple. For one thing, notice that this ALSO would prove as false any other theory of wave function collapse that doesn’t allow single bits to cause collapse (including anything about complex systems or macroscopic systems or complex information processing). We should be suspicious of any simple argument that claims to conclusively prove a significant proportion of experts wrong.

To see what’s going on here, let’s look at what happens if we don’t assume that the spin bit causes the wave function to collapse. Instead, we’ll just model it as becoming fully entangled with the path of the particle, so that the state evolution over time looks like the following:

$\displaystyle{|O, \uparrow \rangle \to \frac{1}{\sqrt{2}} |A, \downarrow \rangle + \frac{1}{\sqrt{2}} |B, \uparrow \rangle \to \frac{1}{\sqrt{2}}\sum_i \left( \alpha_i | i, \downarrow \rangle + \beta_i |i, \uparrow \rangle \right) = | \Psi \rangle}$

…

The interference has vanished, even though we never assumed that the wave function collapsed!

…

And all that’s necessary for that is environmental decoherence, which is exactly what we had with the single spin bit!

…

A particle can be in a superposition of multiple states but still act as if it has collapsed!

…

You might be tempted to say at this point: “Well, then all the different theories of wave function collapse are empirically equivalent! At least, the set of theories that say ‘wave function collapse = total decoherence + other necessary conditions possibly’. Since total decoherence removes all interference effects, the results of all experiments will be indistinguishable from the results predicted by saying that the wave function collapsed at some point!”

But hold on! This is forgetting a crucial fact: decoherence is reversible, while wave function collapse is not!!!

…

Now the two branches of the wave function have “recohered,” meaning that what we’ll observe is back to the interference pattern!

— Decoherence is not wave function collapse

— MARCH 17, 2019

— SQUARISHBRACKET

— Rising Entropy

This is the original link:

Decoherence is not wave function collapse

In case the original link does not work, use the Internet Archive version:

https://web.archive.org/web/20210124095054/https://risingentropy.com/decoherence-is-not-wave-function-collapse/

— Me@2021-01-24 07:14:50 PM

A particle can be in a superposition of …

Note that it is not that the particle is in a superposition. Instead, it is that the system is in a superposition.

— Me@2021-01-24 07:16:49 PM

2021.01.25 Monday ACHK

Superposition always exists, 2.2.2

Posted on January 24, 2021 by rpflee

Decoherence and the Collapse, 2.2 | Quantum decoherence 7.2.2

superposition ~ indistinguishability

superposition state ~ logically indistinguishable states (forming one SINGLE quantum state)

logically indistinguishable ~ indistinguishable by definition ~ indistinguishable due to “the experiment setup is without detector” part of the definition

By the Leibniz’s Law (Identity of indiscernibles), logically indistinguishable cases are actually the same one SINGLE case, represented by one SINGLE quantum state.

Classically, there are no such logically indistinguishable cases because classically, all particles are distinguishable. So the probability distribution in the newly invented non-classical state should be completely different from any probability distributions provided by classical physics. Such cases of a new kind are called quantum states.

A quantum state’s probability distribution can be calculated from its wave function.

“Why that single quantum state is represented by a superposition of eigenstates and why its wave function is governed by the Schrödinger equation” is ANOTHER set of questions, whose correct answers may or may not be found in the Wikipedia article Theoretical and experimental justification for the Schrödinger equation.

Superpositions always exist. Logically indistinguishable cases are always there. You just trade some logically indistinguishable cases with some other logically indistinguishable cases.

The “superpositions” are superpositions in definition, in language, in logic, in calculation, and in mathematics, but not in physical reality, not in physical spacetime.

— Me@2021-01-24 09:29:13 PM

… is in a superposition

Posted on January 24, 2021 by rpflee

Quantum decoherence 5.3.2 | Wheeler’s delayed choice experiment 1.2

For example, in the double-slit experiment, if no detector is installed, the system is in a quantum superposition state.

It is not that each individual photon is in a superposition. Instead, it is that the system of the whole experimental setup is in a superposition.

— Me@2021-01-23 12:57 AM

[guess]

However, “what ‘the whole experimental setup‘ is” is not 100% objective. In other words, it is a little bit subjective.

“The whole experimental setup”, although largely objective, is partially defined with respect to an observer.

— Me@2021-01-23 12:58 AM

So quantum probability/indistinguishability effect is partly observer-dependent, although the subjectivity is just tiny compared with that of the classical probability in a mixed state.

— Me@2021-01-23 12:59 AM

[guess]

Consistent histories, origin

Posted on January 23, 2021 by rpflee

The square root of the probability, 6

There is no wave function collapse.

For example, in the double-slit experiment, with-detector and without-detector are actually two different physics systems. Different experimental setups provide different probability distributions, encoded in the wave functions. So different experimental setups result in different wave functions.

That is the key to understanding strange quantum phenomena such as EPR. A classical system has consistent results is no magic.

You create either a system with a detector or a system without a detector. With a detector, it will have only distinguishable-at-least-in-definition states, aka classical states. A system with only classical states is a classical system. Then, why so shocked when a classical system has consistent results?

Quantum mechanics is “strange”, but not “that strange”. It is not so strange that it is unexplainable.

— Me@2021-01-20 07:11 PM

— Me@2021-01-22 08:48 AM

2004, 2

Posted on January 22, 2021 by rpflee

— Me@2021-01-22 07:24:57 PM

Ex 1.15 Central force motion

Posted on January 22, 2021 by rpflee

Structure and Interpretation of Classical Mechanics

Find Lagrangians for central force motion in three dimensions in rectangular coordinates and in spherical coordinates. First, find the Lagrangians analytically, then check the results with the computer by generalizing the programs that we have presented.

~~~

The Lagrangians in rectangular coordinates:

$\displaystyle{ \begin{aligned} &L (t; x, y, x; v_x, v_y, v_z) \\ &= \frac{1}{2} m \left( v_x^2 + v_y^2 + v_z^2 \right) - U \left( \sqrt{x^2 + y^2 + z^2} \right) \\ \end{aligned}}$

The Lagrangians in spherical coordinates:

$\displaystyle{ \begin{aligned} &L (t; r, \theta, \phi; \dot r, \dot \theta, \dot \phi) \\ &= \frac{1}{2} m r^2 \dot \phi^2 (\sin \theta)^2 + \frac{1}{2} m \dot \theta^2 r^2 + \frac{1}{2} m \dot r^2 - U (r) \\ &= \frac{1}{2} m \left( \dot r^2 + r^2 \dot \theta^2 + r^2 (\sin^2 \theta) \dot \phi^2 \right) - U (r) \\ \end{aligned}}$


(define ((F->C F) local)
  (->local (time local)
           (F local)
           (+ (((partial 0) F) local)
              (* (((partial 1) F) local) 
                 (velocity local)))))

(define (p->r local)
  (let ((polar-tuple (coordinate local)))
    (let ((r (ref polar-tuple 0)) 
          (theta (ref polar-tuple 1))
	  (phi (ref polar-tuple 2)))
      (let ((x (* r (sin theta) (cos phi))) 
            (y (* r (sin theta) (sin phi)))
	    (z (* r (cos theta))))
        (up x y z)))))

(define ((L-central-rectangular m U) local)
  (let ((q (coordinate local))
        (v (velocity local)))
    (- (* 1/2 m (square v))
       (U (sqrt (square q))))))

(define (L-central-polar m U)
  (compose (L-central-rectangular m U) (F->C p->r)))

(show-expression
  ((L-central-polar 'm (literal-function 'U))
   (->local 't 
             (up 'r 'theta 'phi) 
             (up 'rdot 'thetadot 'phidot))))

— Me@2021-01-22 02:59:11 PM

Quantum Computing, 2.2

Posted on January 21, 2021 by rpflee

2021.01.21 Thursday ACHK

Fallacy of infinity thinking, prequel

Posted on January 20, 2021 by rpflee

Why should you NOT murder one innocent person in order to save millions of people?

(Note: This question is NOT the same as “Could we give up one innocent person’s life in order to save millions of people’s?”)

Could we murder one innocent person in order to save millions of people?

— Me@2017-06-20 01:04:56 PM

— Me@2021-01-19 06:15:54 PM

NO.

If we can murder one innocent person in order to save all other people, then no one is safe after all, because anyone could be THAT innocent person, being sacrificed at any time.

(In the situation that we cannot save all the people at the same time, which person or which group has higher or lower priorities depends on context. There is no universal answer.)

Instead, if we protect each person’s life, then all the people’s lives are protected.

— Me@2021-01-19 06:01:24 PM

If you start with protecting each one, then every one person and thus the whole society will be protected.

If you start with protecting the whole society at all costs, then no one will be safe, because anyone could be that cost; any innocent person could be sacrificed at any time.

— Me@2021-01-20 6:48 AM

鐵達尼極限 2.2

Posted on January 19, 2021 by rpflee

尋覓 1.5

這段改編自 2010 年 10 月 14 日的對話。

但是，如果有一個人，心中有一個太陽，自己會發熱發亮的話，他就不怕人情冷暖。亦即是話，如果你一個人時，生活已經十分精采的話，你就不需要愛情。

留意，「不需要」不代表「不應有」。一個人不需要愛情，而仍然選擇愛情的話，可以是因為技術上的問題，例如，剛巧對象與他相愛，而他又想有自己的兒女。

Enjoy everything, need nothing, …

.

… especially for human relationships.

– Conversations with God

然後，可以再追問，為何有些人會，想有自己的兒女？

在這之前，你反而應該先追問，「愛情的感覺」從何而來？

我現在是講普通人，而不是講外星人。

大部分情況下，愛情的感覺，其實是大自然對人的戲弄。這就是傳說中的「愛情陷阱」。

某些原因，大自然想人類（或其他生物）繁殖。

（那某些原因，其實就是「基因」。「基因」只顧自己生命的延續，企圖不斷複製自己，即使犧牲「基因載體」的利益、幸福，甚至生命，也在所不惜。「基因載體」者，人或其他生物也。）

所以，大自然令到，年輕男子和年輕女子，有愛情的感覺，從而互相吸引。不幸的是，那愛情感覺，大自然不會提供一生，只會提供暫時。你今天的分手，就是其中一個例子，大自然中止愛情感覺。

更常的例子是，在你有了子女後，大自然覺得，你已經失去了利用價値。所以，祂不再需要，提供「愛情感覺」給你。你和當時另一半，再不會有愛情的感覺，暫時。

— Me@2021-01-03 04:18:55 PM

1985

Posted on January 18, 2021 by rpflee

— Photomyne colorization

— Me@2021-01-18 10:14:30 PM

Problem 2.5b1

Posted on January 18, 2021 by rpflee

A First Course in String Theory

2.5 Constructing simple orbifolds

~~~

The wikipedia page “Fundamental polygon”, specifically the subsection entitled “group generators”, has a serious mathematical error. You cannot derive a presentation for the fundamental group from the fundamental polygon using the side labels in the manner described on that page (and which you have copied), unless all of the vertices of the polygon are identified to the same point. In the picture you provided and which can be seen on that page, one opposite pair of vertices of the square is identified to one point on the sphere, the other opposite pair of vertices is identified to a different point on the sphere.

There is still a way to derive a presentation for the fundamental group from a fundamental polygon, but it is not the way described on the wikipedia page. In the sphere example of your question, you have to ignore one of the two letters $\displaystyle{A}$ , $\displaystyle{B}$ , keeping only the other letter. For example, ignoring $\displaystyle{A}$ and keeping $\displaystyle{B}$ , you get a presentation $\displaystyle{ \langle B \mid B B^{-1} = 1 \rangle }$ , which is a presentation of the trivial group. The way you tell which to ignore and which to keep is by taking the quotient of the boundary of the polygon which is a graph with vertices and edges, choosing a maximal tree in that graph, ignoring all edge labels in the maximal tree, and keeping all edge labels not in the maximal tree.

On that wikipedia page, the Klein bottle and the torus examples are correct and you do not have to ignore any edge labels: all vertices are identified to a single point and the maximal tree is just a point. The sphere and the projective plane examples are incorrect: the four vertices are identified to two separate points, the maximal tree has one edge, and you have to ignore one edge label. The example of a hexagon fundamental domain for the torus is also incorrect: the six vertices are identified to two separate points, the maximal tree has one edge, and you have to ignore one edge label.

edited Jul 23 ’14 at 17:17

answered Jul 23 ’14 at 17:11

Lee Mosher

yes, i thought that the fundamental polygon is this quotient space. – user159356 Jul 23 ’14 at 17:28

That’s backward: in your example, the sphere is the quotient space of the fundamental polygon, not the other way around. – Lee Mosher Jul 23 ’14 at 17:30

— Mathematics Stack Exchange

2021.01.18 Monday ACHK

Summing over histories, 2

Posted on January 17, 2021 by rpflee

The square root of the probability, 5

If there is more than one way to achieve the present state, present == sum over all possible pasts, with weightings.

— Me@2011.06.26

This is false for a physical state. This is only true for wave functions, which are NOT probabilities.

Wave functions are used for calculating probabilities; but they are not themselves probabilities.

Wave functions are quantum states, but not physical states.

Wave functions are logical and mathematical, but not physical.

A physical state is something observable, something can be measured, at least in principle.

A physical state is something that exists in spacetime, a wave function is not.

— Me@2021-01-16 06:12:08 PM

Zooming out, 2

Posted on January 16, 2021 by rpflee

In this morning’s dream, I could hold my lucid dream without feeling a headache. I felt comfortable as awake.

I was in a city and listening to an Inception-style music.

I tried to fold the city as in Inception so that I could have walked on a vertical land.

However, it did not work. The city got folded together, not as an L shape, [?]but as an L with a horizontal line head shape.[?]

Then I got myself waking up by flying-falling downwards.

— Me@2011.08.03

機遇創生論, 2.1

Posted on January 15, 2021 by rpflee

這段改編自 2010 年 4 月 18 日的對話。

不行。雖然尚算準確，但是不夠精采。

還有，「人生攻略理論」令人聯想到很多東西，而大部分也不是，我們那個合體大理論的內容。

「緣份攻略」都不行，因為感覺有點怪。

（安：那就不如叫做「緣份理論」。）

「理論」很空泛。不應把「理論」，視為名字的一部分。

（安：不如叫做「超級種子理論」，或者「廣義種子理論」？）

種子論最終定版？

種子論傳 …

種子論 X 傳 …

字母 X 可代表，前中後左中右上下東南西北，例如，種子論前傳、種子論左傳等等。原本的種子論，就為之正傳，種子正傳。即是好像 Microsoft DirectX 可以代表 Direct3D、DirectDraw、DirectMusic、DirectPlay、DirectSound 等等。.

但是「種子 X 傳」相當「拗口」，讀法不暢順。

（安：無錯，那相當「拗口」。)