Physical laws are low-energy approximations to reality, 1.6



too many particles

when you cool down the system, you see something that your equations cannot predict

only by experiment, you discover that you can go to there

the system state goes from one local minimum to another


Since each theory is valid only when near a particular energy local minimum, we cannot use it to predict other local minima (other physical realities). That’s why we need to keep doing experiments that are designed for stress-testing existing theories. For example, one experiment can put a system in extreme high pressure. Another experiment can put a system in extreme high temperature.

Once a theory breaks down in one of those experiments, we see phenomena that we have never seen before. From there, we construct new theories to explain those phenomena.


Once a theory breaks down in one of those experiments, we see phenomena that we have never seen before. In this sense, experimental physics is much richer.

Computational physics goes further in only one direction. Only experiments let you try randomly.

— Me@2019-08-18 07:51:05 PM



2019.08.18 Sunday (c) All rights reserved by ACHK

Literal numbers

All primitive mathematical procedures are extended to be generic over
symbolic arguments. When given symbolic arguments, these procedures
construct a symbolic representation of the required answer. There are
primitive literal numbers. We can make a literal number that is
represented as an expression by the symbol “a” as follows:

(literal-number 'a)        ==>  (*number* (expression a))

The literal number is an object that has the type of a number, but its
representation as an expression is the symbol “a”.

(type (literal-number 'a))          ==>  *number*

(expression (literal-number 'a))    ==>  a

— SCMUTILS Reference Manual



2019.08.17 Saturday ACHK

Multiple dimensions of time

What would be the implications of multiple dimensions of time?

That means the (past) history itself can change, as commonly seen in time travel stories.

But wouldn’t that be the case with one dimension also?

In reality, there is only one dimension of time, meaning that the state of a system keeps changing, forming the timeline. But the timeline itself cannot be changed once formed. In other words, (past) history cannot be changed.

— Me@2019-08-11 04:07:48 PM



2019.08.11 Sunday (c) All rights reserved by ACHK

(反對)開夜車 4.2







— Me@2019-07-30 11:11:42 PM



2019.08.03 Saturday (c) All rights reserved by ACHK

Quick Calculation 15.1.2

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.



A group is a set, G, together with an operation \displaystyle{\bullet} (called the group law of G) that combines any two elements a and b to form another element, denoted \displaystyle{a \bullet b} or \displaystyle{ab}. To qualify as a group, the set and operation, \displaystyle{(G, \bullet)}, must satisfy four requirements known as the group axioms:


For all a, b in G, the result of the operation, \displaystyle{a \bullet b}, is also in G.


For all a, b and c in G, \displaystyle{(a \bullet b) \bullet c = a \bullet (b \bullet c)}.

Identity element

There exists an element e in G such that, for every element a in G, the equation \displaystyle{e \bullet a = a \bullet e = a} holds. Such an element is unique, and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G, commonly denoted \displaystyle{a^{-1}} (or \displaystyle{-a}, if the operation is denoted “+”), such that \displaystyle{a \bullet b = b \bullet a = e}, where e is the identity element.

— Wikipedia on Group (mathematics)


The axioms for a group are short and natural… Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.

— Richard Borcherds in Mathematicians: An Outer View of the Inner World



2019.07.28 Sunday ACHK

Alfred Tarski, 3

The undefinability theorem shows that this encoding cannot be done for semantic concepts such as truth. It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage capable of expressing the semantics of some object language must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language.

— Wikipedia on Tarski’s undefinability theorem


Tarski’s 1969 “Truth and proof” considered both Gödel’s incompleteness theorems and Tarski’s undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.

— Wikipedia on Alfred Tarski



2019.07.20 Saturday ACHK

PhD, 3.7.2

碩士 4.7.2 | On Keeping Your Soul,






  1. 如果你是自資,就即是不拿學校的資助。那樣,你就不是僱員。研究以外的工作,例如做助教等,可以一概不理。

  2. 同理,你的博士導師再不會是,你工作上的上司。反而,你是消費者,他是你的僱員。













— Me@2019-07-06 10:57:22 PM



2019.07.10 Wednesday (c) All rights reserved by ACHK

Ex 1.7 Properties of $\delta$

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)}

— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics


Prove that

\displaystyle{\delta_\eta F \circ g[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]}


\displaystyle{RHS = \lim_{\Delta t \to 0} \left( \frac{F \circ g[q](t+\Delta t) - F \circ g[q](t)}{\Delta t} \right) \lim_{\epsilon \to 0} \left( \frac{g[q+\epsilon \eta]-g[q]}{\epsilon} \right)}

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

Since \displaystyle{F} is path-independent,

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

Let \displaystyle{ g[q+\epsilon \eta] = g + \Delta g}.

\displaystyle{ \begin{aligned} LHS   &= \lim_{\epsilon \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\epsilon} \right) \\   &= \lim_{\epsilon \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\   \end{aligned}}

When \displaystyle{ \epsilon \to 0}, \displaystyle{ \Delta g \to 0 }.

\displaystyle{ \begin{aligned} LHS   &= \lim_{\substack{\epsilon \to 0 \\ \Delta g \to 0}} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\   &= \lim_{\Delta g \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]} \lim_{\epsilon \to 0} \frac{g[q + \epsilon \eta] - g[q]}{\epsilon} \right) \\   &= DF \left( g[q] \right) \delta_\eta g[q] \\   &= RHS \\  \end{aligned}}

— Me@2019-06-24 10:55:28 PM



2019.06.25 Tuesday (c) All rights reserved by ACHK

From classical to quantum

From this viewpoint, the move from a classical to a quantum mechanical system is not a move from a comutative to a non-commutative algebra \displaystyle{\mathcal{A}} of a real-valued observables, but, instead, a move from a commutative algebra to a partial commutative algebra of observables.

Of course, every non-commutative algebra determines an underlying partial commutative algebra and also its diagram of commutative subalgebras.

That fact that assuming the structure of a non-commutative algebra is the wrong assumption has already been observed in the literature (see, for example, [19]),

but it is often replaced by another wrong assumption, namely that of assuming the structure of a Jordan algebra.

These differing assumptions on the structure of \displaystyle{\mathcal A} affect the size of its automorphisum group and, hence, of the allowable symmetries of the system (the weaker the assumed structure on \displaystyle{\mathcal A}, the larger is its automorphism group).

— The Mathematical Foundations of Quantum Mechanics

— David A. Edwards



2019.06.18 Tuesday ACHK

(反對)開夜車 4.1









  • 只可以間中,不可以經常。

  • 日間中途要有小睡。

  • 平均而言,你仍必須要有,充足的睡眠。亦即是話,某一天睡少了,必須於在當個星期,還回「睡債」。

    • 例如,如果你的充足睡眠是,每天七小時,而你在某一天只睡了六小時的話,你就有義務,在當個星期的另一天,睡多一小時。




— Me@2019-06-06 08:23:56 PM



2019.06.08 Saturday (c) All rights reserved by ACHK

Quick Calculation 15.1

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


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)

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)



2019.05.23 Thursday (c) All rights reserved by ACHK

PhD, 3.7.1

碩士 4.7.1 | On Keeping Your Soul,
















假設你已經有財政自由 …

— Me@2019-05-18 03:14:02 PM



2019.05.18 Saturday (c) All rights reserved by ACHK