物理中的數學（一）：粒子運動學和微積分

受友人所托，要寫一些在物理中使用的數學。目標讀者是中學生。我想是有一點難度的，因為老實說，中學裡的物理課所用的數學是太簡單了。在大學的物理，最低限度也需要 Linear Algebra 和 Multi-variable calculus 才可以完成大學物理的本科課程。相比起中學那些代數，真正的物理所用的數學可以說是艱難多了。因此，如果沒有這些數學根底，也許很難會對我在這所談的物理或數學提起興趣吧！那就有點失去了寫這一系列文章的意義了。不過，無論如何，先寫寫吧。

在第一篇裡，我打算寫寫粒子運動學 (particle dynamics) 中所使用的微積分 (calculus)。當然，為了令（至少是會考的）中學生們對他們所學有所共鳴，我只會討論一下最簡單的一維粒子運動學，而所需的微積分也是中學生們會學到的。

不知道是否中學的課程愈來愈容易，我中學時所學到的粒子運動學只需要運用到代數，並不需要用到微積分的。最主要的原因是題目限制了當中的加速度 (acceleration) 是常數 (constant)。描速這種運動系統的方程就只需最基本的四條，如下：

• s=s_0+ut+\frac{1}{2}at^2
  
• v^2-u^2=2as
  
• v=u+at
  
• s=\left( \frac{u+v}{2} \right) t
  

($s$: displacement; $u$: initial velocity; $v$: final velocity; $a$: acceleration; $t$: time duration; $s_0$: initial displacement, assumed to be zero)

你可有留意總共有五個變數（$s_0$ 不算），但每一條中只有四個。因此，對於這種 constant acceleration 的系統，如果你知道了其中三個量，其餘的量也可以由以上的方程中找出來。簡單麼？

但是，現實宇宙中有多少東西只會作 constant acceleration 的運動呢？不是沒有，只是不太「現實」。自由落體 (Free falling) 在現實中會出現嗎？不是不會，不過在地球上的話就有 air friction 囉，要真正的 Free fall 嗎？大概要去月球吧！

好吧！要做「真實」一點的物理，我們一定不可以假設加速度是常數，而是一個隨時間改變的變數 $a(t)$。對應以上四條的式子就變成：

• v(t)=u+\int_{t_0}^{t} a(t) dt
• s(t)=s_0+\int_{t_0}^{t} v(t) dt
• v(t)=\frac{d}{dt}s(t)
• a(t)=\frac{d}{dt}v(t)

在這，我就不講解如何得到這些式子了，反正學過微積分的同學們都很容易理解。可是，你可能反而從這裡理解何謂微積分。移位 (displacement) 對時間的改變就是速度 (velocity)，速度對時間的改變就是加速度 (acceleration)。物理在這裡就提供了一個比較直觀的解釋，而不再是一般中學數學教師所講的，微分就是求 graph 的 slope，積分就是求 graph 「底下」的面積。這其實是很自然的事，因為當初微積分就是為了要解決物理問題而發明的。

在這討論了一維的事況，其實在三維都差不多，因為在牛頓力學中，x y z 三個方向是可分拆的 (decoupled)，即可以將其獨立出來，再運用以上的方法便可解決問題了。你可能會想，哪還不是一樣容易嗎？其實問題的難度不在維度，而是在粒子的數目。我們只討論了單粒子（或剛體）的運動學，如果是兩粒那還可以有 analytic solution，如果是三粒或以上，基本上是解不出來的，除非是一些特別的系統。這時候，物理學家就只好用其他的方法去解問題了。例如太陽系八大行星，以前就會用 perturbation 的方法來做，現在就用 numerical method 在電腦上做計算好了。

好吧，就談到這裡。下次應該會談些線性代數 (linear algebra) 的東西。

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IAS Distinguished Lecture Series: The Future of Physics

The speaker of the talk is Prof. David Gross, Nobel Laureate of Physics in 2004. Prof. Gross discuss about 25 questions that might guide Physics in the next 25 years. The questions range from cosmology to Biophysics.

The talk was held at LT-K, which is actually a small room for lecture. Due to presence of a distinguished speaker, LT-K was actually fully occupied and some audience needed to sit (on the ground) just in front of the screen, which of course is not a desired place to listen the talk. Later the talk was relocated to LT-D, again, fully occupied.

Well, the first question came: What is Physics? Of course, one could give the "official" answer yet it would be instructive to start the talk with something different. Physics is what physicists do. Here we have to define "physicists". They are (we are?) those people, while studying graduate school, use Jackson as EM textbook! To understand this joke, you ought to be within the Physics community. Why defining like this? Prof. Gross said that he wants to distinguish physicists from engineers. That's true. The math in Jackson's EM text is really too arduous for most of the engineers.

Below is the list of most of the questions mentioned in the talk, however, I think this list is incomplete. Anyway, for those who study Physics (including me), this list might be helpful.

• How did the universe begin?
• How far can we probe?
• Can string theory determine initial conditions?
• Was there a time before the big bang?
• Is time itself an emergent concept?
• How does time begin?
• What is the nature of dark matters (~21%)?
• How do we detect dark matter?
• Can we produce them in the laboratory?
• What is the nature of dark energy (~75%)?
• Is it constant or varying in time?
• Or is it just the Einstein's cosmological constant, Λ?
• How do stars form?
• How do we understand spectrum of masses, frequency of binaries and clusters?
• How do planets form?
• What is the frequency of habitable planets?
• Is General Relativity valid in all scales?
• Would it breakdown at small distance and/or strong field?
• Is gravitational waves detectable?
• Can we determine Kerr Metric of a blackhole experimentally?
• Is quantum mechanics the ultimate description of nature or will it fail somewhere?
• Fail at very short distances? (Resolved by string theory)
• Fail in large complex systems? (the Schrodinger's cat)
• Problems with conscious systems (problems of measurements)
• Problems with Standard Model
• Is there any pattern for the masses and interaction coefficients of the elementary particles?
• Does quantum fluctuation give rise to an energy (= cosmology constant)?
• Λ_obs ~ 10^-4eV⁴ ≠ 0
• Λ_theory ~ 10⁶⁰eV⁴
• Why is there such a large inconsistency between the observed and theoretical values?
  
• Are diamonds forever? (Is nucleus stable?) Why?
• Proton's lifetime > 10³² years. Why is it so stable?
• Can we significantly improve this measurement?
• What is the origin of the enormous hierarchy of energy scales?
• The scale (length, energy & time) of our living universe differs a lot from the natural scale (Planck scale)
• Is there low energy supersymmetry?
• Due to supersymmetry, there is a super-partner (super-particle) for each observed particle, e.g. quark → squark; electron → slectron
• These super-particles are supposed to have the same masses as the partners. But we actually can't observe them, why is there a broken symmetry?
• Can we solve quantum chromodynamics at long distance?
• Possibly by dual string theory
• What is string theory?
• Actually string theory cannot be solved in approximative way. Yet now we cannot solve it in an exact manner.
• What is space-time?
• Is space-time an emergent concept?
• Are there generic non-Fermi-liquid?
• For quantum computers, how can we deal with the background noise?
• In an quiet way - isolated from the noise
• In an deaf way - "topological" quantum bits that are delocalized and that are totally decoupled from the noise
• Can we construct an applicable quantum computer? (we need >10,000 QuBits)
• Is there room temperature (or even higher temperature) superconductor?
• Is there room temperature ferromagnet made by electronic (semiconductors) materials?
• Theoretical Biology Problem
• How to we think of analyze and exhibit dynamics over many wide-ranging time scales?
• Genomics
• Can theory of evolution be quantitative or predictive?
  For example, there may be a genomics question like this in the year of 2107:
  What is the outlook of the creature that have the DNA sequence of AAAGTCTGAC...... > Answer <
• What is consciousness and/or memory?
• Can one measure the onset of consciousness in infant? For instance, consciousness increases continuously with age? Or discontinuously?
• Computational Physics
• Will computer replace analytic techniques?
• When will we have creative theoretical physicists of computers?
• How do we train them? (With Jackson?)
• In high energy Physics, will the experimental apparatus become unaffordable or too difficult to be built?
• Can we find any alternative ways to prove/disprove the theories?
  
• Will Physics be a useful subject in the next 25 years? (Yes, for sure)
  

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Prof. Gross is specialized in high energy Physics, so inevitably the content of the talk had to have some bias towards those area, such as string theory and standard model etc.

In my view, most of the questions related to high energy Physics can't be resolved in a short time. Physicists still have to work really hard.

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Field Theory

連續幾天都在看 Schweber, 也看了其他的作參考. 在這簡短的記下一些重點和進度吧, 好等以後讓自己容易翻查.

首先是場 (Field) 的概念. 以前學過的 de Broglie hypothesis 提道粒子 (Particle) 其實有著波 (Wave) 的特性, 如果光 (Light, electromagnetic waves) 可以以光子 (Photons), 即粒子的形式存在, 那如果宇宙有著一個粒子和波的對稱性, 那麼一些以前當作粒子的東西, 應該也會有著波的特質. 因此 de Broglie 就在他的博士論文寫下了

\lambda=h/p

這著名的方程. 這就是波粒二象性 (Particle-wave duality).

場就是波的概念的推廣. 波一般來說只是在時空 (Spacetime) 中每一點指定一個純量 (Scalar), 而場就不一定只是純量, 而可以是張量 (tensor). 我們可以有純量場, 向量場, 張量場. 場的概念是由古典電磁學中提出的. 作用力 (Interaction) 不是瞬間傳播的, 而是由場來傳播的, 所以便會因為場的特性 (被運動方程 equation of motion 支配), 使傳播速度不為無限. 其中兩個熟識的例子就是電磁場 (向量) 和愛因斯坦場 (張量) :

G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}.

場是連續的 (Continuous). 波可以根據運動方程, 存在於連續的媒介 (Media) -- 即場. 有不同的場和不同的運動方程, 所得出的波亦不同.

運動方程可由不同的方式得出, 可以是牛頓第二定律, 或者是由作用量原理 (Action Principle) 得出.

以上所提到的, 全都是古典力學中對場的描述 (Classical Field Theory). 我們要看到波粒二象性, 就要包含量子力學, 我們便要對場作出量子化 (Quantization), 這就是下次的題目.

Latex Equation Editor

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