Problem 13.5b

A First Course in String Theory


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})}


\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}}


\displaystyle{\Omega X^I(\tau, \sigma) \Omega^{-1}} = X^I (\tau, 2 \pi - \sigma)


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



2019.11.24 Sunday (c) All rights reserved by ACHK

PhD, 3.8.1

財政自由 1.3.1


The secret to creativity is knowing how to hide your sources.

— Not Einstein


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






















— Me@2019-10-29 10:20:33 PM


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



2019.10.29 Tuesday (c) All rights reserved by ACHK


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.





— Christopher Francis Patten



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]}


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]}


— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics


\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}}


— Me@2019-10-12 03:42:01 PM



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



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。



  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?












— Me@2019-09-30 01:09:36 PM



2019.09.30 Monday (c) All rights reserved by ACHK

Problem 13.6b

A First Course in String Theory | Topology, 2 | Manifold, 2


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.


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


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


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



2019.09.26 Thursday ACHK

Pointer state, 2

Eigenstates 3.2


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


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


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


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


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


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


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



2019.09.22 Sunday (c) All rights reserved by ACHK

財政自由 1.2





















— Me@2019-09-10 08:33:38 PM



2019.09.14 Saturday (c) All rights reserved by ACHK


Event Realism | 事件實在論 6.1


exist = can be found


— Me@2013.09.25


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

— Me@2019.09.05


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



實在 ~ 堅實地存在



— Me@2019-09-05 09:08:41 PM



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



2019.09.04 Wednesday ACHK

Quantum observer 2

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


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


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


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



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.


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.


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




I’m glad.


I’m glad he’s gone.

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


This was my home.


His home.

— The Time Machine (2002 film)



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。



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。


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。




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。



— Me@2019-08-30 09:31:52 PM



2019.09.01 Sunday (c) All rights reserved by ACHK