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RSA公钥加密算法 | 此外金鑰 $d$ 必須足夠大，1990 年有人證明假如 $p$ 大於 $q$ 而小於 $2q$ （這是一個很常見的情況）而 $d<{\frac {1}{3}}\times N^{\frac {1}{4}}$ ，那麼從 $N$ 和 $e$ 可以很有效地推算出 $d$ 。此外 $e=2$ 永遠不應該被使用。
### 速度
比起 AES、3DES 和其它對稱演算法來說，RSA 要慢得多。實際的運用（如 TLS）一般結合了對稱加密（如 AES）和非對稱加密（如 RSA）兩者。
### 金鑰分配
和其它加密過程一樣，對 RSA 來說分配公鑰的過程是非常重要的。分配公鑰的過程必須能夠抵擋中間人攻擊。假設 Eve 交給 Bob 一個公鑰，並使 Bob 相信這是 Alice 的公鑰，並且她可以截下 Alice 和 Bob 之間的訊息遞移，那麼她可以將她自己的公鑰傳給 Bob，Bob 以為這是 Alice 的公鑰。Eve 可以將所有 Bob 遞移給 Alice 的訊息截下來，將這個訊息用她自己的金鑰解密，讀這個訊息，然後將這個訊息再用 Alice 的公鑰加密後傳給 Alice。理論上 Alice 和 Bob 都不會發現 Eve 在偷聽他們的訊息。今天人們一般用可靠的第三方機構簽發憑證來防止這樣的攻擊。
## 典型金鑰長度
NIST 建議的 RSA 密鑰長度為至少 2048 位元。
## 已公開的或已知的攻擊方法
### 大數因數分解
針對 RSA 最流行的攻擊一般是基於大數因數分解。1999 年，RSA-155（512 bits）被成功分解，花了五個月時間（約 8000 MIPS 年）和 224 CPU hours 在一台有 3.2G 中央記憶體的 Cray C916 電腦上完成。
RSA-155 表示如下：
```
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727842468219535093544305870490251995655335710209799226484977949442955603
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] |
RSA公钥加密算法 | 11658823406671259903148376558383270818131012258146392600439520994131344334162924536139
```
2009 年 12 月 12 日，編號為 RSA-768（768 bits, 232 digits）數也被成功分解。這一事件威脅了現通行的 1024-bit 金鑰的安全性，普遍認為使用者應儘快升級到 2048-bit 或以上。
RSA-768 表示如下：
```
123018668453011775513049495838496272077285356959533479219732245215172640050726
365751874520219978646938995647494277406384592519255732630345373154826850791702
6122142913461670429214311602221240479274737794080665351419597459856902143413
= 3347807169895689878604416984821269081770479498371376856891
2431388982883793878002287614711652531743087737814467999489×
3674604366679959042824463379962795263227915816434308764267
6032283815739666511279233373417143396810270092798736308917
```
### 時間攻擊
1995 年，丹・博內和大衛・布魯姆利提出了一種非常意想不到的攻擊方式：假如 Eve（竊密者）對 Alice 的硬體有充分的了解，而且知道它對一些特定的訊息加密時所需要的時間的話，那麼她可以很快地推導出 _d_。這種攻擊方式之所以會成立，主要是因為在進行加密時所進行的模指數運算是一個位元一個位元進行的，而位元為 1 所花的運算比位元為 0 的運算要多很多，因此若能得到多組訊息與其加密時間，就會有機會可以反推出私鑰的內容。
## 相關條目
* 公開金鑰加密
* 橢圓曲線密碼學
* 量子電腦
* 秀爾演算法
* 米勒 - 拉賓質數判定法
* 迪菲 - 赫爾曼密鑰交換
* 快速冪
* 擴充歐幾里得演算法
## 參考文獻
## 外部連結 | [
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RSA公钥加密算法 | * RSA, The Security Division of EMC（頁面存檔備份，存於網際網路檔案館）
* RSA 演算法詳解（頁面存檔備份，存於網際網路檔案館） | [
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] |
大統一論 | **大統一理論**（**Grand Unified Theory**，縮寫 **GUT**）是一種物理理論。物理學者希望能藉由單獨一種物理理論來合理解釋電磁交互作用、強交互作用和弱交互作用導致的物理現象。大一統理論算是物理界邁向萬有理論的踏腳石。請注意，大一統理論並不包括引力，試圖也將引力統一的理論稱為萬有理論。
## 歷史
愛因斯坦在提出相對論以後，從 20 年代開始就致力於尋找一種統一的理論來解釋當時已知的所有交互作用，也就是引力和電磁力，愛因斯坦晚年偏離物理界大方向自己研究大一統理論，直到他 1955 年逝世。在愛因斯坦的年代，強交互作用和弱交互作用的概念尚未形成，因此愛因斯坦的努力似乎過於超前。
電弱交互作用理論和強交互作用理論建立以後，由於他們都由相似的規範場論來描述，理論物理學家隨即試圖統一它們。哈沃德・喬吉和謝爾登・格拉肖於 1974 年提出了最早的 **SU (5) 大一統理論**。這是第一個統一了電弱交互作用和強交互作用的理論。值得指出的是，由於量子引力理論尚存在困難，目前一般稱為「大一統理論」的理論都與引力無關。
## 動機
在標準模型中，粒子的超荷對應於 U (1) 群的量子數。在數學理論中，U (1) 群可以有無限多個表示，與之相應，超荷可以是任意有理數。但是夸克和電子的電荷都是 1/3 的整數倍，因此它們的超荷都是 1/6 的整數倍。理論物理學家猜測這是因為標準模型的 $SU(3)\times SU(2)\times U(1)$ 其實是一個大的單群的子群。這樣，相關的數學理論就會把超荷限制在 1/6 的整數倍的數值上。
## 古典 SU (5) 理論
$SU(5)$ 理論是第一種大一統理論。它將標準模型的 $SU(3)\times SU(2)\times U(1)$ 嵌入在一個 $SU(5)$ 群中。該群兩個最小的不可約表示，**5** 表示和 **10** 表示剛好能夠將標準模型的一整代費米子納入其中， $SU(5)$ 群的超荷算符也剛好能給出所有粒子正確的超荷，因此理論物理學家一度認為，該模型極有可能是正確的。
值得注意的是， $SU(5)$ 模型是反常自動相消的。另外， $SU(5)$ 模型並沒有自動包括右手微中子，因此不能自動解釋微中子質量。
## 三種作用常數的統一與超對稱 | [
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大統一論 | 一些大一統理論通常要求標準模型三種耦合常數在某個能標上跑到同一個數值上去。簡單的 $SU(5)$ 模型無法做到這一點。但是在超對稱 $SU(5)$ 模型中，三種耦合常數在能標
: $\Lambda _{\text{GUT}}\approx 10^{16}\,{\text{GeV}}$ .
處比較精確地跑到了一點。理論物理學家相信，這不是巧合。這也是粒子物理學中引入超對稱的重要動機之一。
## 大一統理論的現狀
大一統理論通常會破壞重子數守恆，因此它們常常預言質子衰變。但是最簡單的 $SU(5)$ 模型已經被相關實驗排除。超對稱大一統理論預測了更長的質子壽命，因此相比簡單的 $SU(5)$ 模型而言稍微好一些。目前，沒有任何支持關於大一統理論的實驗證據。
## 參見
* 標準模型
* 萬有理論
* 規範理論
* 統一場論
* 希格斯場
## 外部連結
* Scholarpedia: Grand Unification（頁面存檔備份，存於網際網路檔案館） | [
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力場 | 在物理裡，**場**（英語：Field）是一個以時空為變數的物理量。空間中瀰漫著的基本交互作用被命名為「場」。場可以分為純量場、向量場和張量場等，依據場在時空中每一點的值是純量、向量還是張量而定。例如，古典重力場是一個向量場：標示引力場在時空中每一個的值需要三個量，此即為引力場在每一點的引力場向量分量。更進一步地，在每一範疇（純量、向量、張量）之中，場還可以分為「古典場」和「量子場」兩種，依據場的值是數字或量子算符而定。
場被認為是延伸至整個空間的，但實際上，每一個已知的場在夠遠的距離下，都會縮減至無法量測的程度。例如，在牛頓萬有引力定律裡，重力場的強度是和距離平方成反比的，因此地球的引力場會隨著距離很快地變得不可測得（在宇宙的尺度之下）。
定義場是一個「空間裡的數」，這不應該減損場在物理上所有的真實性（如定義「質量」為秤上的數字）。「場佔有空間。場含有能量、動量。場的存在排除了真正的真空。」真空中沒有物質，但並不是沒有場的。場形成了一個「空間的狀態」
當一個電荷移動時，另一個電荷並不會立刻感應到。第一個電荷會感應到一個反作用力，並獲得動量，但第二個電荷則沒有感應，直到第一個電荷移動的影響以光速傳遞到第二個電荷那裡，並給予其動量之後。場的存在解決了關於第二個電荷移動前，動量存在在哪裡的問題。因為依據動量守恆定律，動量必存在於某處。物理學家認為動量應該存在於場之中。如此的認定讓物理學家們相信電磁場是真實的存在，使得場的概念成為整個現代物理的範式。
## 歷史
艾薩克・牛頓在他的萬有引力定律簡單地把引力定義為由於質量引起的，施加於一對物體的力。當涉及到多個物體互相影響的系統，比如太陽系，原先的定義就不方便並且有侷限性了。在 18 世紀，一個新的概念提出來用來簡化這些一對對的引力。這就是引力場，在這個場的空間上的任何一單位質量都會受到那個點所有的引力。這個理論並不是對物理規律的顛覆，而是簡化了運算。你計算一個點上的總重力不管是考慮一對對重力，並把它們加起來還是首先把整個引力場計算出來，再算出哪一點上受到的力都是等價的。
## 古典場
古典場論仍然適用於非量子尺度的空間，依舊在科學研究中處於主流。材料的彈性、流體力學以及馬克士威方程式都是與此理論相關的例子。最簡單的場是力場。歷史上，場之一概念第一次被認可是法拉第用力線描述電場。隨後重力場的概念也用相似的方式提出。
### 牛頓重力場
用來描述兩個有質量的物體的吸引力的古典場論源於牛頓萬有引力定律。 | [
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力場 | ### 電磁場
麥可・法拉第在他研究磁學時最先意識到場作為一個物理量的重要性。他發現電場和磁場並非唯一能支配粒子運動的力場，但它們也具有獨立的物理實在性，因為它們可以攜帶能量。
這些想法最終使詹姆斯・克拉克・馬克士威通過電磁場的方程組建立了物理學中第一個統一的場論。這一方程組的現代形式被稱為馬克士威方程組。
#### 靜電場
一個帶電荷 q 的檢驗粒子在靜電場中會受到一個大小只與 q 有關的力 F 的作用。我們可以這樣描述電場 E：F=qE。通過這一關係和庫侖定律，我們可以得到，點電荷周圍的電場為：
: $\mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {q}{r^{2}}}{\hat {\mathbf {r} }}.$
靜電場是保守場，因此可以用一個純量勢函數 V (**r**) 來描述：
: $\mathbf {E} (\mathbf {r} )=-\nabla V(\mathbf {r} ).$
#### 靜磁場
#### 電動力學
### 廣義相對論所描述的引力場
## 量子場
## 場論
### 對稱性
#### 時空對稱
## 相關領域
* 古典場論
* 強場物理
* 場論（量子力學）
* 量子場論
* 統一場論（愛因斯坦）
* 電磁場
* 電場
* 磁場
* 能量場
* 引力場
## 參見
* 心理場論
* 統計場論
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动量定理 | 在古典力學裏，**動量**（momentum，**p**）被量化為物體的質量和速度的乘積（ $\mathbf {p} =m\mathbf {v} .$ ）。例如，一輛快速移動的重型卡車擁有很大的動量。若要使這重型卡車從零速度加速到移動速度，則需要使到很大的作用力；若要使重型卡車從移動速度減速到零，則也需要使到很大的作用力；若卡車輕一點或移動速度慢一點，則它的動量也會小一點。
動量在國際單位制中的單位為 kg・m/s。有關動量的更精確的量度的內容，請參見本頁的動量的現代定義部分。
一般而言，一個物體的動量指的是這個物體在它運動方向上保持運動的趨勢。動量實際上是牛頓第一定律的一個推論。動量是個向量，其方向與速度方向相同。動量同時也是一個守恆量，這表示為在一個封閉系統內動量的總和不可改變。在古典力學中，動量守恆暗含在牛頓定律中，但在狹義相對論中依然成立，（廣義）動量在電動力學、量子力學、量子場論、廣義相對論中也成立。
勒內・笛卡兒認為宇宙中總的「運動的量」是保持守恆的，這裡所說的「運動的量」被理解為「物體大小和速度的乘積」—— 但這不宜被解讀為現代動量定律的表達方式，因為笛卡爾並沒有把「質量」這個概念與物體「重量」和「大小」之間的關係區分開來，更重要的是他認為速率（純量）而不是速度（向量）是守恆的。因此對於笛卡兒來說：一個移動的物體從另一個表面彈回來的時候，該物體的方向發生了改變但速率沒有發生改變，運動的量應該沒有發生改變。
## 古典力學中的動量
物體在任何一個參考系中運動時，它都具有在_該參考系_中的動量。需要注意的是，動量是一個參考系決定量。也就是說，同一個物體在一個參考系中具有確定的動量，但在另一個參考系中卻有可能具有不同的動量。
物體動量的數值取決於兩個物理量的數值：運動物體在參考系中的質量與速度。在物理學中，動量以小寫的 $\mathbf {p} $ （黑體代表 **$\mathbf {p} $** 是一個向量）表示，動量的定義如下：
: $\mathbf {p} =m\mathbf {v} $
動量對時間的一階導數的定義如下： | [
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动量定理 | : ${\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} (m\mathbf {v} )}{\mathrm {d} t}}=m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}+v{\frac {\mathrm {d} m}{\mathrm {d} t}}$
其中 **p** 為動量，t 為時間，d 為微分算符。
當物體在運動中質量不變的情形下， ${\frac {\mathrm {d} m}{\mathrm {d} t}}=0$ ，此時，可以將動量對時間的一階導數簡寫作
: ${\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}$
一個物體的速度包括了該物體的速率與運動方向。因為動量由速度決定，所以動量也具有數量與方向，是一個空間向量。例如，要表示出 5 kg 的保齡球的動量的話，可以以它有以 2m/s 的速率向西運動的狀態來說明；但是，只認為該保齡球具有 10 kg・m/s 的動量的想法是不全面的，因為沒有表示出它的運動方向。
### 定理
**動量定理**指出：
$$
\sum \mathbf {I} =\Delta \mathbf {p}
$$
### 推導
設一個質量為 m 的物體，初速度為 v，那麼初動量為 p=mv, 在淨力 F 的作用下，經過一段時間 t 速度變為 $v^{'}$ ，末動量則變為 $p^{'}=mv^{'}$ 。物體的加速度為 $a={\frac {v^{'}-v}{t}}$ 。由牛頓第二定律 $F=ma={\frac {mv^{'}-mv}{t}}$ 可得 $Ft=mv^{'}-mv$ ，即 $Ft=p^{'}-p$ 。 | [
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动量定理 | 在動量定理的推導過程中，我們假定淨力 F 是恆定的，但是在實際生活當中要比這個複雜的多。如用球拍擊打球或是用腳踢踢球時作用力就不是恆定的。但可以證明，動量定理不但適用於定力，也可以隨時間而變化的變力，對於變力的情況，動量定理中的 F 應理解為在作用時間內的平均值。此時作用力 $\mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}$ 也稱作動量的變化率。
## 碰撞中的動量守恆
動量具有一個特殊屬性：只要是在一個封閉系統中，它總會保持恆定，即使是物體碰撞發生時。而對動能而言，非彈性碰撞的物體的動能將不會守恆。因此，當碰撞過後可利用動量守恆來計算未知速度。
在物理學上，這個特殊屬性被用來來解決兩個相碰物體的問題。因為動量始終保持恆定，碰撞前動量的總和一定與碰撞後動量的總和相等：
: : $m_{1}\mathbf {v} _{1{\text{i}}}+m_{2}\mathbf {v} _{2{\text{i}}}=m_{1}\mathbf {v} _{1{\text{f}}}+m_{2}\mathbf {v} _{2{\text{f}}}$
: 其中，i 表示碰撞前的初量，f 表示碰撞後的末量。要注意的是此時 $\mathbf {v} $ 為向量。
通常來說，我們只需知道碰撞前（或碰撞後）物體的速度便可計算出碰撞後（或碰撞前）物體的速度。碰撞有兩種類型，兩種類型中動量都守恆：
* 彈性碰撞中，動能保持恆定；
* 非彈性碰撞中，動能不保持恆定
## 彈性碰撞
彈性碰撞的一個較好的例子是兩個撞球之間的碰撞。當兩個球相碰時，除了動量保持恆定外，碰撞前後動能的總和也將保持不變：
: : ${\frac {1}{2}}m_{1}v_{1{\text{i}}}^{2}+{\frac {1}{2}}m_{2}v_{2{\text{i}}}^{2}={\frac {1}{2}}m_{1}v_{1{\text{f}}}^{2}+{\frac {1}{2}}m_{2}v_{2{\text{f}}}^{2}$
因為每個因式中都含有係數 ${\frac {1}{2}}$ ，所以亦可將該係數移除。
### 正向碰撞（一維）
正碰即對心碰撞（head on collision），兩物體沿著一條直線碰撞後仍沿原來直線運動，屬於彈性碰撞中的一種。
* 遵循動量守恆 | [
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动量定理 | : : $m_{1}v_{1{\text{i}}}+m_{2}v_{2{\text{i}}}=m_{1}v_{1{\text{f}}}+m_{2}v_{2{\text{f}}}$
* 遵循能量守恆
: : ${\frac {1}{2}}m_{1}v_{1{\text{i}}}^{2}+{\frac {1}{2}}m_{2}v_{2{\text{i}}}^{2}={\frac {1}{2}}m_{1}v_{1{\text{f}}}^{2}+{\frac {1}{2}}m_{2}v_{2{\text{f}}}^{2}$
聯立兩方程式可得出，兩物體最終速度
: : $v_{1{\text{f}}}={\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}v_{1{\text{i}}}+{\frac {2m_{2}}{m_{1}+m_{2}}}v_{2{\text{i}}}$
: : $v_{2{\text{f}}}={\frac {2m_{1}}{m_{1}+m_{2}}}v_{1{\text{i}}}+{\frac {m_{2}-m_{1}}{m_{1}+m_{2}}}v_{2{\text{i}}}$
### 斜向碰撞（二維）
可以分別以 $x$ 方向以及 $y$ 方向的動量守恆決定出碰撞前後的速度關係。
## 動量守恆定律
動量是守恆量。動量守恆定律表示為：一個系統不受外力或者所受外力之和為零，這個系統中所有物體的總動量保持不變。它的一個推論為：在沒有外力干預的情況下，任何系統的質心都將保持等速直線運動或靜止狀態不變。動量守恆定律可由力學能對空間平移對稱性推出。
在隔離系統（不存在外力）中總動量將是一個守恆量，這暗含在牛頓運動第一定律之中。
因為動量是向量，所以子彈從起先靜止的槍中射出後，儘管子彈和槍都在運動，但由於子彈的動量與槍的動量等值反向，它們相互抵消，使得子彈與槍形成的系統中動量的總和依然為零。
若有系統外合（淨）力為零，則系統內各質點交互作用力亦為零（可視為牛頓第三定律，作用力反作用力原理），故動量變化為零，所以動量守恆。動量守恆定律具有普適性，適用於宏觀、微觀系統，參考系。
## 動量的現代定義
### 相對論力學中的動量
在相對論力學中，動量被定義為：
: $\mathbf {p} =\gamma m\mathbf {u} $
其中：
* $m$ 表示運動物體的靜止質量； | [
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动量定理 | * $\gamma ={\frac {1}{\sqrt {1-u^{2}/c^{2}}}}$ ；
* _u_ 表示物體與觀察者之間的相對速度；
* _c_ 表示光速。
當物體在低速極限（u/c -> 0）下運動時，相對論力學的動量式可變化為牛頓力學的動量式： $m\mathbf {u} $ 。
阿爾伯特・愛因斯坦由勞倫茲變換下的四維矢量守恆發展提出了相對論的四維動量。其中四維矢量可從量子場論使用格林函數自然導出。四維動量被定義為：
: $\left({E \over c},p_{x},p_{y},p_{z}\right)$
其中， $p_{x}$ 表示_相對論_動量的 $x$ 分量， $E$ 表示系統的總能量：
: $E=\gamma mc^{2}\;$
令速度等於零，可得到一個物體的靜止質量和能量之間的關係 E=mc²。
矢量的「長度」保持恆定被定義為：
: $\mathbf {p} \cdot \mathbf {p} -E^{2}/c^{2}$
#### 無靜止質量物體的動量
無靜止質量物體，譬如光子亦有動量。計算的公式為：
: $p={\frac {h}{\lambda }}={\frac {E}{c}}$
: : 其中
: $h$ 表示普朗克常數；
: $\lambda $ 表示光子的波長；
: $E$ 表示光子的能量；
: $c$ 表示光速。
#### 動量的普適性
動量是平移守恆的諾特荷。因此，甚至連場也與其他物質一樣具有動量，而不止是粒子。但是，在彎曲時空（非閔考斯基式）中，動量根本沒有被定義。
### 量子力學中的動量
在量子力學中，動量被定義為波函數的一個算符。海森堡不確定性原理定義了單一觀測系統中一次測定動量和位置的精確極限。在量子力學中，動量與位置是一對共軛物理量。
對單個不帶電荷且沒有自旋的粒子來說，動量算符可被寫作：
: $\mathbf {p} ={\hbar \over i}\nabla =-i\hbar \nabla $
其中， $\nabla $ 表示梯度算符。這是動量算符的一個普通形式，而非最普遍的一個。
### 電磁學中的動量守恆
當電場和 / 或磁場移動時，它們帶有動量。電磁波（可見光、紫外線、無線電波等）也有動量，即使是沒有靜止質量的光子，也同樣帶有動量。這被應用在諸如太陽帆上。
## 參考文獻
## 參看條目
* 角動量 | [
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动量定理 | * 守恆定律
* 牛頓擺 | [
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角動量 | 在物理學中，**角動量**是與物體的位置向量和動量相關的物理量。對於某慣性參考系的原點 $\mathbf {O} $ ，物體的角動量是物體的位置向量和動量的叉積，通常寫做 $\mathbf {L} $ 。角動量是矢量，且是一贗矢量。
: $\mathbf {L} =\mathbf {r} \times \mathbf {p} $
其中， $\mathbf {r} $ 表示物體的位置向量， $\mathbf {L} $ 表示角動量。 $\mathbf {p} $ 表示動量。角動量 $\mathbf {L} $ 又可寫為：
: $\mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {r} \times (m\mathbf {v} )=\mathbf {r} \times ({\boldsymbol {\omega }}\times (m\mathbf {r} ))=mr^{2}{\boldsymbol {\omega }}=I{\boldsymbol {\omega }}$
其中， $I$ 表示質點的轉動慣量， ${\boldsymbol {\omega }}$ 是角速度向量。
假設作用於物體的外力矩和為零，則物體的角動量是守恆的。需要注意的是，由於成立的條件不同，角動量是否守恆與動量是否守恆沒有直接的聯繫。
當物體的運動狀態（動量）發生變化，則表示物體受力作用，而作用力大小就等於動量 $\mathbf {P} $ 的時變率： $\mathbf {F} ={\frac {d\mathbf {P} }{dt}}$
當物體的轉動狀態發生改變時，表示物體受到力矩作用，而力矩就等於角動量的時變率： ${\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}}$
若物體（或系統）所受外力矩和為零，則物體（系統）的角動量守恆。例如靜電力或萬有引力均是徑向力，因此不會產生力矩。行星運動的交互作用力源自於萬有引力，故行星運動滿足角動量守恆，所對應的就是克卜勒定律中的第二定律。
需要特別說明的是，動量 $\mathbf {P} \equiv m\mathbf {v} $ ，也就是說，動量的方向和速度的方向一致。
伽利略・伽利萊首先引入角動量守恆的概念。
## 角動量量子化 | [
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角動量 | 在量子力學裡角動量是量子化的：系統的角動量不能任意地取某實數值而只能取以約化普朗克常數 $\hbar $ 為單位整數或半整數倍。粒子的運動軌道造成的角動量必須取 $\hbar $ 的整數倍。另外實驗證明大部分次原子粒子都擁有一種和運動無關的先天角動量叫自旋。自旋以 ${\frac {\hbar }{2}}$ 的倍數出現。
### 量子化角動量和不確定性原理
角動量是位移與動量的矢量積。而量子力學裡位移與同方向動量是非對易的因此各獨立方向的角動量分別非對易：
: $[L_{i},L_{j}]=i\hbar \epsilon _{ijk}L_{k}$
* $\epsilon _{ijk}$ 是列維 - 奇維塔符號。
* $[A,B]=AB-BA$ 是交換子。
根據海森堡不確定原理非對易的物理量不能同時測準。因此角動量矢量的各方向部可以各自但不能同時確定。雖然如此但是角動量矢量的長度是可和任意一部同時確定：
: $\left[L_{i},L^{2}\right]=0$
因此算符 $L^{2}$ 和 $L_{z}$ （任選一方向為 z）有共同的特徵波函數。 $L^{2}$ 在球座標系表現為：
: : $L^{2}=-{\frac {\hbar ^{2}}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)-{\frac {\hbar ^{2}}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}$
其中 $\theta $ 是位移與 $z$ 軸夾角， $\phi $ 是繞 $z$ 軸旋轉的角度。
它和 $L_{z}$ 的共同特徵函數
: : $L^{2}|l,m\rangle ={\hbar }^{2}l(l+1)|l,m\rangle $
: $L_{z}|l,m\rangle =\hbar m|l,m\rangle $
是球諧函數：
: : $\langle \theta ,\phi |l,m\rangle =Y_{l,m}(\theta ,\phi )$
$l$ 是某非負整數。 $-l\leq m\leq l$ 是絕對值不大於 $l$ 的整數。 | [
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角動量 | ### 能量均分與角動量量子化
古典力學內角動量是可以取任意連續值會導致熱力學上一些吊詭。角動量量子化給這些問題完美的答案，這也是角動量量子化有其必要性的證據之一。
在熱力學裡平均能量和系統自由度有關。例如忽略內部結構的單原子分子組成的理想氣體平均能量是 ${\frac {E}{N}}={\frac {3}{2}}k_{\mathrm {B} }T$ ：三維空間運動的分子的每個獨立運動方向分別給予平均能量 ${\frac {k_{\mathrm {B} }T}{2}}$ 。這是能量均分定理。
假設除了三維的平移運動，氣體的分子是由兩種原子組成。而原子可以相互環繞運動。為了簡化問題假設所有分子的原子對只能環繞 z 軸運動。它們旋轉的動能量是：
: $E={\frac {L_{z}^{2}}{2I}}$
$L_{z}$ 是分子旋轉的角動量， $I$ 是轉動慣量和原子的距離平方成正比。從運用統計力學的配分函數
: $Z=\int _{-\infty }^{\infty }dL_{z}e^{-\beta {\frac {L_{z}^{2}}{2I}}}={\sqrt {\frac {2\pi I}{\beta }}}$
（ $\beta ={\frac {1}{k_{\mathrm {B} }T}}$ 是溫度 $T$ 的倒數）可以得到古典旋轉運動對平均能量的貢獻：
: ${\frac {\langle E\rangle }{N}}=-{\frac {\partial \log Z}{\partial \beta }}={\frac {1}{2\beta }}={\frac {k_{\mathrm {B} }T}{2}}$
也就是新的旋轉自由度和每平移運動方向給與一樣的能量。
但是，旋轉的貢獻並不決定於分子的轉動慣量 $I$ 也就是和原子的距離無關。但這和我們期待原子距離或分子轉動慣量趨向 0 時回到無旋轉的結果相矛盾。這就是古典力學引起的弔詭：能量均分定理允許透過宏觀觀察得到所有微觀自由度的資訊：儘管由很多基本粒子組成的原子一般擁有遠高於宏觀觀察的自由度。
問題的解決來自角動量量子化。因為微觀角動量不能取任意的連續值因此以上用積分計算配分函數是不正確的。配分函數應該是一個和：
: $Z=\sum _{n=-\infty }^{\infty }e^{-\beta {\frac {n^{2}\hbar ^{2}}{2I}}}$ | [
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] |
角動量 | 在溫度很高（ $\beta \to 0$ ）或分子轉動慣量很大的情況下，每項間變化緩慢。用積分來進似近似以上和是可接受的。在這情況下選轉的確和一般自由度一樣。上段得到的結果是正確的。但在溫度很低或分子轉動慣量很小的情況下 $Z$ 主要貢獻來自 $|n|$ 小的前幾項：
: $Z\simeq 1+e^{-\beta {\frac {n^{2}\hbar ^{2}}{2I}}}+\cdots $
因此對平均溫度的貢獻是：
: ${\frac {\langle E\rangle }{N}}=-{\frac {\partial \log Z}{\partial \beta }}\simeq {\frac {n^{2}\hbar ^{2}}{2I}}$
而一個系統的量子旋轉特徵和古典旋轉特徵的交叉點出現在溫度可以給與幾個 $\hbar $ 角動量的能量：
: $T^{*}\approx {\frac {\hbar ^{2}}{2I}}$
## 參考文獻
## 參見
* 力矩
* 動量
* 守恆定律
* 角速度
* 角動量守恆定律
* 角動量算符
* 旋轉動能
* 轉動慣量
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自旋角动量 | 在量子力學中，**自旋**（英語：Spin）是粒子所具有的內稟性質，其運算規則類似於古典力學的角動量，並因此產生一個磁場。雖然有時會與古典力學中的自轉（例如行星公轉時同時進行的自轉）相類比，但實際上本質是迥異的。古典概念中的自轉，是物體對於其質心的旋轉，比如地球每日的自轉是順著一個通過地心的極軸所作的轉動。
首先對基本粒子提出自轉與相應角動量概念的是 1925 年由拉爾夫・克勒尼希、喬治・烏倫貝克與塞繆爾・古德斯米特三人所開創。他們在處理電子的磁場理論時，把電子想像為一個帶電的球體，自轉因而產生磁場。後來在量子力學中，透過理論以及實驗驗證發現基本粒子可視為是不可分割的點粒子，所以物體自轉無法直接套用到自旋角動量上來，因此僅能將自旋視為一種內稟性質，為粒子與生俱來帶有的一種角動量，並且其量值是量子化的，無法被改變（但自旋角動量的指向可以通過操作來改變）。
自旋對原子尺度的系統格外重要，諸如單一原子、質子、電子甚至是光子，都帶有正半奇數（1/2、3/2 等等）或含零正整數（0、1、2）的自旋；半整數自旋的粒子被稱為費米子（如電子），整數的則稱為玻色子（如光子）。複合粒子也帶有自旋，其由組成粒子（可能是基本粒子）之自旋透過加法所得；例如質子的自旋可以從夸克自旋得到。
## 概論
自旋角動量是系統的一個可觀測量，它在空間中的三個分量和軌道角動量一樣滿足相同的對易關係。每個粒子都具有特有的自旋。粒子自旋角動量遵從角動量的普遍規律，p=[J (J+1)]0.5h，此為自旋角動量量子數 ，J ＝ 0，1/2，1，3/2，……。自旋為半奇數的粒子稱為費米子，服從費米 - 狄拉克統計；自旋為 0 或整數的粒子稱為玻色子，服從玻色 - 愛因斯坦統計。複合粒子的自旋是其內部各組成部分之間相對軌道角動量和各組成部分自旋的矢量和，即按量子力學中角動量相加法則求和。已發現的粒子中，自旋為整數的，最大自旋為 4；自旋為半奇數的，最大自旋為 3/2。
自旋是微觀粒子的一種性質，沒有古典對應，是一種全新的內稟自由度。自旋為半奇數的物質粒子服從包立不相容原理。
## 發展史
自旋的發現，首先出現在鹼金屬元素的發射光譜課題中。於 1924 年，包立首先引入他稱為是「雙值量子自由度」（two-valued quantum degree of freedom），與最外殼層的電子有關。這使他可以形式化地表述包立不相容原理，即沒有兩個電子可以在同一時間共享相同的量子態。 | [
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自旋角动量 | 包立的「自由度」的物理解釋最初是未知的。拉爾夫・克勒尼希，朗德的一位助手，於 1925 年初提出它是由電子的自轉產生的。當包立聽到這個想法時，他予以嚴厲的批駁，他指出為了產生足夠的角動量，電子的假想表面必須以超過光速運動。這將違反相對論。很大程度上由於包立的批評，克勒尼希決定不發表他的想法。
當年秋天，兩個年輕的荷蘭物理學家產生同樣的想法，它們是烏倫貝克和塞繆爾・古德斯米特。在保羅・埃倫費斯特的建議下，他們以一個小篇幅發表了他們的結果。它得到了正面的反應，特別是在雷沃林・托馬斯消除了實驗結果與烏倫貝克和古德施密特的（以及克勒尼希未發表的）計算之間的兩個矛盾的係數之後。這個矛盾是由於電子指向的切向結構必須納入計算，附加到它的位置上；以數學語言來說，需要一個纖維叢描述。切向叢效應是相加性的和相對論性的（比如在 c 趨近於無限時它消失了）；在沒有考慮切向空間朝向時其值只有一半，而且符號相反。因此這個複合效應與後來的相差一個係數 2（參見：湯瑪斯進動）。
儘管他最初反對這個想法，包立還是在 1927 年形式化自旋理論，運用埃爾文・薛丁格和沃納・海森堡發現的現代量子力學理論。他開拓性地使用包立矩陣作為一個自旋算子的群表述，並且引入一個二元旋量波函數。
包立的自旋理論是非相對論性的。然而，在 1928 年，保羅・狄拉克發表狄拉克方程式，描述相對論性的電子。在狄拉克方程式中，一個四元旋量（所謂的「狄拉克旋量」）被用於電子波函數。在 1940 年，包立證明「自旋統計定理」，它表述費米子具有半整數自旋，玻色子具有整數自旋。
## 自旋量子數
### 基本粒子的自旋
對於像光子、電子、各種夸克這樣的基本粒子，理論和實驗研究都已經發現它們所具有的自旋無法解釋為它們所包含的更小單元圍繞質心的自轉（即使使用古典電子半徑，電子「赤道」處的速度也需要超光速才能解釋其自旋角動量）。由於這些不可再分的基本粒子可以認為是真正的點粒子，因此自旋與質量、電量一樣，是基本粒子的內稟性質。
在量子力學中，任何體系的角動量都是量子化的，其值只能為：
: $S=\hbar \,{\sqrt {s(s+1)}},$ | [
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自旋角动量 | 其中 $\hbar $ 是約化普朗克常數，而自旋量子數是整數或者半整數（0, 1/2, 1, 3/2, 2,……），自旋量子數可以取半整數的值，這是自旋量子數與軌道量子數的主要區別，後者的量子數取值只能為整數。自旋量子數的取值只依賴於粒子的種類，無法用現有的手段去改變其取值（不要與自旋的方向混淆，見下文）。
例如，所有電子具有 $s=1/2$ 的自旋，自旋為 1/2 的基本粒子還包括正電子、微中子和夸克，光子是自旋為 1 的粒子，理論假設的引力子是自旋為 2 的粒子，希格斯玻色子在基本粒子中比較特殊，它的自旋為 0。
### 次原子粒子的自旋
對於像質子、中子及原子核這樣的次原子粒子，自旋通常是指總的角動量，即次原子粒子的自旋角動量和軌道角動量的總和。次原子粒子的自旋與其它角動量都遵循同樣的量子化條件。
通常認為次原子粒子與基本粒子一樣具有確定的自旋，例如，質子是自旋為 1/2 的粒子，可以理解為這是該次原子粒子能量量低的自旋態，該自旋態由次原子粒子內部自旋角動量和軌道角動量的結構決定。
利用第一性原理推導出次原子粒子的自旋是比較困難的，例如，儘管我們知道質子是自旋為 1/2 的粒子，但是原子核自旋結構的問題仍然是一個活躍的研究領域。
### 原子和分子的自旋
原子的自旋是原子核自旋與核外電子自旋的疊加。分子的自旋是分子中未成對電子自旋之和，未成對電子的自旋導致原子和分子具有順磁性。
### 自旋與統計
粒子的自旋對於其在統計力學中的性質具有深刻的影響，具有半整數自旋的粒子遵循費米 - 狄拉克統計，稱為費米子，它們必須占據反對稱的量子態（參閱全同粒子），這種性質要求費米子不能占據相同的量子態，這被稱為包立不相容原理。另一方面，具有整數自旋的粒子遵循玻色 - 愛因斯坦統計，稱為玻色子，這些粒子可以占據對稱的量子態，因此可以占據相同的量子態。對此的證明稱為自旋統計定理，依據的是量子力學以及狹義相對論。事實上，自旋與統計的聯繫是狹義相對論的一個重要結論。
## 自旋的方向
### 自旋投影量子數與自旋多重態
在古典力學中，一個粒子的角動量不僅有大小（取決於粒子轉動的快慢），而且有方向（取決於粒子的旋轉軸）。量子力學中的自旋同樣有方向，但是是以一種更加微妙的形式出現的。
在量子力學中，對任意方向的角動量分量的測量只能取 $S_{i}=\hbar s_{i}$ ，其中 | [
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自旋角动量 | : $s_{i}\in \{-s,-(s-1),\dots ,s-1,s\},$
s 是之前章節討論過的自旋量子數。可以看出對於給定的 s， $s_{z}$ 可以取「2s+1」個不同的值，此即為自旋多重態。例如：對於自旋為 1/2 的粒子，"sz"只能取兩個不同的值，+1/2 或 - 1/2。相應的量子態為粒子自旋分別指向 + z 或 - z 方向，一般我們把這兩個量子態叫做"spin-up"和"spin-down"。
對於一個給定的量子態，可以給出一個自旋矢量 $\langle S\rangle $ ，它的各個分量是自旋沿著各坐標軸分量的數學期望值，即 $\langle S\rangle =[\langle s_{x}\rangle ,\langle s_{y}\rangle ,\langle s_{z}\rangle ]$ . 這個矢量描述自旋所指的「方向」，對應於古典物理下旋轉軸的概念。這個矢量在實際做量子力學計算時並不十分有用，因為它不能被直接精準測量：根據不確定性原理，_sx_、_sy_ 和 _sz_ 不能同時有確定值。但是對於被置於同一個量子態的大量粒子，例如使用斯特恩－革拉赫儀器得到的粒子，自旋矢量確實有良好定義的實驗意義。
### 自旋矢量
## 自旋與磁矩
具有自旋的粒子具有磁偶極矩，就如同古典電動力學中轉動的帶電物體。磁矩可以通過多種實驗手段觀察，例如，在斯特恩 - 革拉赫實驗中受到不均勻磁場的偏轉，或者測量粒子自身產生的磁場。
一個基本粒子，電量為 q，質量為 m，自旋為 S，則其內稟磁矩 $\mu $ 為
: $\mu =g\,{\frac {q}{2m}}\,S$
其中無因次量 g 稱為 g - 因數（g-factor），當僅有軌道角動量時，g=1。
電子是帶電荷的基本粒子，具有非零磁矩。量子電動力學理論成功以預測了電子的 g - 因數，其實驗測量值為−2.002 319 304 3622（15），括號中的兩位數字為測量的不確定度，來源於標準差，整數部分 2 來源於狄拉克方程式（狄拉克方程式是與將電子自旋與其電磁性質聯繫起來的基本方程式），小數部分（0.002 319 304…）來源於電子與周圍電磁場的交互作用，其中也包括電子自身的產生的電磁場。
## 量子力學中關於自旋的數學表示
### 自旋算符
與角動量算符類似，自旋滿足對易關係： | [
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自旋角动量 | : $[S_{i},S_{j}]=i\hbar \epsilon _{ijk}S_{k}$
其中 $\epsilon _{ijk}$ 為列維 - 奇維塔符號。 $S^{2}$ 與 $S_{z}$ 的本徵值（用狄拉克符號表示）為：
: $S^{2}|s,m\rangle =\hbar ^{2}s(s+1)|s,m\rangle $
: $S_{z}|s,m\rangle =\hbar m|s,m\rangle .$
自旋產生及湮沒算符作用於本徵矢量上可以得到：
: $S_{\pm }|s,m\rangle =\hbar {\sqrt {s(s+1)-m(m\pm 1)}}|s,m\pm 1\rangle ,$
其中 $S_{\pm }=S_{x}\pm iS_{y}$ 。
然而與軌道角動量所不同的是，自旋的本徵矢量不是球諧函數，它們不是 $\theta $ 和 $\phi $ 的函數，而且 $s$ 與 $m$ 不能取半整數值也只是一種約定，沒有具體的含義。
除了其它性質以外，量子力學描述的所有粒子具有內稟自旋（儘管可能出現量子數 $S=0$ 的情況）。自旋量子數的取值為約化普朗克常數 $\hbar $ 的整數倍或半整數倍，因此波函數可以寫為 $\psi =\psi (\mathbf {r} ,\sigma )\,,$ 而不是 $\psi =\psi (\mathbf {r} )$ ，其中 $\sigma $ 可以取值的集合為： $\sigma \in \{-S\cdot \hbar ,-(S-1)\cdot \hbar ,...,+(S-1)\cdot \hbar ,+S\cdot \hbar \}$ ，由此可以區分玻色子（S=0, 1 , 2 , ...）和費米子（S=1/2 , 3/2 , 5/2 , ...）。自旋角動量與軌道角動量之和為總角動量，在交互作用過程中總角動量守恆。
### 自旋與包立不相容原理
包立不相容原理指出，對於可分辨的 N 粒子體系，交換其中任意兩個粒子，則有： | [
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自旋角动量 | ： $\psi (\,...\,;\,\mathbf {r} _{i},\sigma _{i}\,;\,...\,;\mathbf {r} _{j},\sigma _{j}\,;\,...){\stackrel {!}{=}}(-1)^{2S}\cdot \psi (\,...\,;\,\mathbf {r} _{j},\sigma _{j}\,;\,...\,;\mathbf {r} _{i},\sigma _{i}\,;\,...)\,.$
因此，對於玻色子，前置因子 $(-1)^{2S}$ 可簡化為 + 1，而對於費米子為 - 1。在量子力學中，所有的粒子不是玻色子就是費米子，而在相對論量子場論中存在「超對稱」粒子，它們是玻色子成分和費米子成分的線性組合。對於二維體系，前置因子 $(-1)^{2S}$ 可以取為任何模為 1 的複數。
電子是自旋量子數 S=1/2 的費米子；光子是自旋量子數 S=1 的玻色子。這充分說明自旋這一特性無法完全用古典的內稟軌道角動量來解釋，也就是不能認為自旋是像陀螺一樣的自轉運動，因為軌道角動量只能導致 s 取整數值。電子一般情況下可以不考慮相對論效應，光子必須採用相對論來處理，而用來描述這些粒子的馬克士威方程組，也是滿足相對論關係的。
包立不相容原理非常重要，例如，化學家和生物學家常用的元素週期表就是遵循包立不相容原理制訂的。
### 自旋與旋轉
如上所述，量子力學指出角動量沿任意方向的分量只能取一系列離散值，量子力學中最普遍的描述粒子自旋的方法是，用一個歸一完備的複數集來表示內稟角動量在給定坐標軸方向投影出現的機率。例如，對於自旋 1/2 的粒子，用 $a_{\pm 1/2}$ 表示角動量投影出現的機率為 $\hbar /2$ 和 $-\hbar /2$ ，它們滿足：
: $|a_{1/2}|^{2}+|a_{-1/2}|^{2}\,=1.$
由於這些複數的取值依賴於坐標軸的選取，坐標軸轉動變換可以是非平凡的，因此要求採用線性的變換法則，以便將所有的轉動通過一個矩陣聯繫起來，這要求變換必須滿足乘法運算，而且必須保持內積不變，因此變換矩陣應當滿足：
: $\sum _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}(\sum _{n=-j}^{j}U_{nm}a_{n})^{*}(\sum _{k=-j}^{j}U_{km}b_{k})$ | [
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自旋角动量 | : $\sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.$
用數學語言表述，這些矩陣是 SO (3) 群的么正表示，每一個這樣的表示對應於 SU (2) 群的一個表示（SO (3) 群是 SU (2) 群的子群），SU (2) 群的每一個不可約表示對應一個維度。例如，自旋 1/2 的粒子在二維表示下作轉動變換，可以用包立矩陣表示為：
: ${\begin{pmatrix}a_{1/2}'\\a_{-1/2}'\end{pmatrix}}=\exp {(i\sigma _{z}\gamma /2)}\exp {(i\sigma _{y}\beta /2)}\exp {(i\sigma _{x}\alpha /2)}{\begin{pmatrix}a_{1/2}\\a_{-1/2}\end{pmatrix}}$
其中 $\alpha ,\beta ,\gamma $ 為歐拉角。
同樣地，可以用高維群表示描述粒子的高階自旋變換，參見包立矩陣相關章節。
### 自旋與勞侖茲變換
我們可以在勞侖茲變換下研究自旋的行為，但與 SO (3) 群不同，勞侖茲群 SO (3,1) 是非緊緻的，不存在有限維么正表示。
對於自旋 1/2 的粒子，有可能構造出保持內積不變的有限維表示。將每個粒子用一個四元狄拉克自旋量 $\psi $ 來表示，這些旋量在勞侖茲變換下遵守如下規則：
: $\psi '=\exp {\left({\frac {1}{8}}\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi $
其中 $\gamma _{\mu }$ 為伽瑪矩陣， $\omega _{\mu \nu }$ 是一個反對稱的 $4\times 4$ 矩陣，它將勞侖茲變換參數化。我們可以看到內積表示
: $\langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi $
保持不變。由於表示矩陣是非正定的，因此不是么正表示。
### 包立矩陣和自旋算符
量子力學中表示自旋這個可觀測量的算符為：
: $S_{x}={\hbar \over 2}\sigma _{x}$ | [
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自旋角动量 | : $S_{y}={\hbar \over 2}\sigma _{y}$
: $S_{z}={\hbar \over 2}\sigma _{z}$
對於自旋為 - 1/2 的情形， $\sigma _{x}$ , $\sigma _{y}$ 和 $\sigma _{z}$ 為三個包立矩陣，表示為
: $\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$
: $\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}$
: $\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$
### 沿 x, y 和 z 軸的自旋測量
每個包立矩陣的哈密頓量有兩個本徵值：+1 和 - 1。相應的歸一化本徵矢量為：
: $\psi _{x+}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}{1}\\{1}\end{pmatrix}},\psi _{x-}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}{1}\\{-1}\end{pmatrix}}$ ,
: $\psi _{y+}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}{1}\\{i}\end{pmatrix}},\psi _{y-}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}{1}\\{-i}\end{pmatrix}}$ ,
: $\psi _{z+}={\begin{pmatrix}1\\0\end{pmatrix}},\psi _{z-}={\begin{pmatrix}0\\1\end{pmatrix}}$ .
根據量子力學基本假設，測量沿 x，y 或 z 軸的電子自旋的實驗只能得到相應坐標軸上自旋算符（ $S_{x}$ , $S_{y}$ , $S_{z}$ ）的本徵值： ${\hbar \over 2}$ 和 ${-\hbar \over 2}$ 粒子的量子態可以用一個具有兩個分量的自旋量來表示：
: $\psi ={\begin{pmatrix}{a+bi}\\{c+di}\end{pmatrix}}$ | [
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自旋角动量 | 當測量給定坐標軸方向（這裡取為 x 軸）的自旋時，測量到自旋為 ${\hbar \over 2}$ 的機率恰好為 $\mid \langle \psi \mid \psi _{x+}\rangle \mid ^{2}$ 。相應的測量到自旋為 ${-\hbar \over 2}$ 的機率恰好為 $\mid \langle \psi \mid \psi _{x-}\rangle \mid ^{2}$ 。經過測量，粒子的自旋將塌縮到相應的本徵態。結果導致，如果粒子在給定坐標軸方向的自旋已經被測量出確定的值，所有的測量將得到相同的本徵值（因為 $\mid \langle \psi _{x+}\mid \psi _{x+}\rangle \mid ^{2}=1$ ，依此類推），只要其它坐標軸方向的自旋還沒有被測量。
### 沿任意方向的自旋測量
沿任意方向的自旋算符很容易從包立矩陣導出，令 $u=(u_{x},u_{y},u_{z})$ 為任意單位矢量，則沿該方向的自旋算符為 $\sigma _{u}=\hbar (u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z})/2$ ，算符 $\sigma _{u}$ 具有本徵值 $\pm \hbar /2$ 。對於高自旋態，沿任意方向的自旋算符可以通過它與 x,y,z 軸三個方向的矢量的內積來確定。
對於自旋 - 1/2 的粒子，一個沿 $(u_{x},u_{y},u_{z})$ 方向的正交的自旋子為（除了導致 0/0 的自旋態）：
: ${\frac {1}{\sqrt {2+2u_{z}}}}{\begin{bmatrix}1+u_{z}\\u_{x}+iu_{y}\end{bmatrix}}.$
確定上述自旋子的一般方法：將矩陣 $\sigma _{u}$ 對角化，求取與本徵值相應的本徵矢量，這樣的本徵矢量就可以作為自旋子。
### 自旋測量的相容性
由於包立矩陣是不對易的，因此沿不同方向測量的自旋是不相容的，例如，在我們已知 x 軸方向的自旋的情況下，測量沿 y 軸方向的自旋，這樣會將我們先前在 x 軸方向的測量結果否定。這可以從包立矩陣的本徵矢量（本徵態）中看出來： | [
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自旋角动量 | : $\mid \langle \psi _{x+/-}\mid \psi _{y+/-}\rangle \mid ^{2}=\mid \langle \psi _{x+/-}\mid \psi _{z+/-}\rangle \mid ^{2}=\mid \langle \psi _{y+/-}\mid \psi _{z+/-}\rangle \mid ^{2}={\frac {1}{2}}$
因此，假如我們測量到沿 x 軸方向的自旋是 ${\hbar \over 2}$ ，這個粒子的自旋將塌縮為本徵態 $\mid \psi _{x+}\rangle $ ；當我們接著測量 y 軸方向的自旋時，自旋本徵態將塌縮到 $\mid \psi _{y+}\rangle $ 或者 $\mid \psi _{y-}\rangle $ ，塌縮到這兩個本徵態的機率都是 ${\frac {1}{2}}$ ，可以認為這是測量到了 ${-\hbar \over 2}$ 。當我們再次測量沿 x 軸的自旋，測量到 ${\hbar \over 2}$ 或者 ${-\hbar \over 2}$ 的機率各為 ${\frac {1}{2}}$ （ $\mid \langle \psi _{x+}\mid \psi _{y-}\rangle \mid ^{2}$ 和 $\mid \langle \psi _{x-}\mid \psi _{y-}\rangle \mid ^{2}$ ），這說明我們最初沿 x 軸方向的測量不再正確，因為此時沿 x 軸方向測量的自旋得到兩種本徵值的機率是相等的。
## 應用
自旋的直接的應用包括：核磁共振譜、電子順磁共振譜、質子密度的磁共振成像，以及巨磁電阻硬碟磁頭。自旋可能的應用有自旋場效應電晶體等。以電子自旋為研究對象，發展創新磁性材料和器件的學科分支稱為自旋電子學。
* 旋磁共振
* 核磁共振
* 電子順磁共振
## 相關條目
* 自旋 ½
* 包立矩陣
* 反常塞曼效應
* 自轉
* 進動
* 章動
* 自旋 - 軌道作用
## 參考資料
## 外部連結
* 聖地牙哥加州大學物理系視聽教學：自旋。 | [
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M理论 | **M 理論**（英語：M-theory）是物理學中將各種相容形式的超弦理論統一起來的理論。此理論最早由美國數學物理學家愛德華・維騰於 1995 年春季在南加州大學舉行的一次弦理論會議中提出。維騰的報告牽起了一股研究弦理論的熱潮，被稱為第二次超弦革命。
弦理論學者在維騰的報告之前已經識別出五種不同的超弦理論。儘管這些理論看上去似乎非常不一樣，但多位物理學家的研究指出這些理論有著微妙且有意義的關係。特別而言，物理學家發現這些看起來相異的理論其實可以透過兩種分別稱為 S 對偶和 T 對偶的數學變換所統合。維騰的猜想有一部份是基於這些對偶的存在，另有一部份則是基於弦理論與 11 維超重力場論的關係。
儘管尚未發現 M 理論的完整表述，這種理論應該能夠描述叫膜的二維及五維物體，而且也應該能描述低能量下的 11 維超引力。現今表述 M 理論的嘗試一般都是基於矩陣理論或 AdS/CFT 對偶。維騰表示根據個人喜好 M 應該代表 Magic（魔術理論）、Mystery（神秘理論）或 Membrane（膜理論），但應該要等到理論更基礎的表述出現後才能決定這個命名的真正意義。
有關 M 理論數學架構的研究已經在物理和數學領域產生了多個重要的理論成果。弦理論學界推測，M 理論有可能為研發統合所有自然基本力的統一理論提供理論框架。當嘗試把 M 理論與實驗聯繫起來時，弦理論學者一般會專注於使用額外維度緊緻化來建構人們所處的四維世界候選模型，但是到目前為止，物理學界還未能證實這些模型是否能產生出人們所能觀測到（例如在大型強子對撞機中）的物理現象。
## 背景
### 量子引力與弦
現代物理學中一個最深層問題就是量子引力。現在對引力的理解是來自阿爾伯特・愛因斯坦的廣義相對論，是古典物理學框架內的表述。然而，非引力的力則是由量子物理學的框架所描述，這是一套完全不同的表述，用於描述基於機率的物理現象。為了使廣義相對論與量子力學原則一致，因此需要一套引力的量子理論，但當使用量子理論的平常方式去描述引力時就出現了難題。 | [
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M理论 | 弦理論是一種嘗試使引力與量子力學一致的理論框架。粒子物理學的類點粒子在弦理論中被一種叫弦的一維物體所取代。弦理論描述弦是如何在空間中傳播及與其他弦之間的交互作用。在某一個形式的弦理論中只會有一種弦，看起來可能像普通線的線圈或線段，而且能夠以不同的方式振動。一根弦在比弦尺度大得多的距離下看起來就像是普通粒子，而其質量、電荷及其他性質則視乎弦的振動態而定。就這樣所有不同的基本粒子都可以被視為振動的弦。弦的其中一個振動態產生引力子，它是一種承載引力的量子力學粒子。
弦理論共有多個形式：I 型、IIA 型及 IIB 型，還有雜交弦理論的兩味（SO(32) 和 E8×E8）。這些不同的弦理論讓不同型的弦及低能量時所產生的粒子出現不同的對稱性。比方說，I 型理論包括開弦（有末端的段）和閉弦（形成密閉的圈），但 IIA 和 IIB 型則只包括閉弦。這五種理論中的每一種都是由 M 理論不同的特殊極限個案所產生的。這種理論與它的前身弦理論一樣，是引力的量子理論的一個例子。它用量子力學的規則描述了一種像人們所熟悉的引力那樣的力。
### 維度數
人們在日常生活中有熟悉的空間三維：長、闊、高。愛因斯坦的廣義相對論把時間視作於空間三維同等的維度；時間和空間在廣義相對論並不是分開的實體，而是統合成了四維時空。引力現象在這個框架下被視為時空幾何的後果。
儘管四維時空能很好地描述宇宙，但是物理學家還是有多個理由去研究其他維度。一些個案中的不同維度數時空模型在數學上比較容易處理，而且能夠更好地計算和更易瞭解整個模型。在凝聚體物理學中也有二維或三維時空能有用地描述現象的情況。最後還有實際上可以超過四維的情況，不過此時額外維度則需要避過觀測。
弦理論和 M 理論的一個顯著特徵就是需要額外的時空維度，以清除在數學上的矛盾。弦理論中的時空是 10 維的，而 M 理論的時空則為 11 維。為了使用這些理論來描述真實的物理現象，就必需想像這些額外維度不會被實驗觀測到的情況。
緊緻化是物理理論中一種修改維度數的方法。緊緻化假設一些額外維度被「捲曲」成圓環狀。在這些被捲起的時空趨向於非常小的極限時，就能得有效時空維度數較低的理論。緊緻化可以通過考慮多維物件來解釋，如橡膠水管。如果從足夠距離外看橡膠水管的話，它看起來就只有一維，就是長度。然而，向水管靠近的話，就會發現它的第二維圓周。因此在橡膠水管表面爬行的螞蟻能以二維方式移動。
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M理论 | 從 M 理論不同極限所得出的理論之間實際上有著非常有意義的關係。這些不同的物理理論間其中一個能夠存在的關係叫 S 對偶。一套理論中的一強交互作用粒子集，在某些個案下可被視為另一套完全不同理論中的一弱交互作用粒子集，這樣的一種關係就是 S 對偶。籠統地來說，強交互作用粒子有著頻密的結合和衰變，而弱交互作用粒子則並不經常。結果是 I 型弦理論在 S 對偶下與 SO(32) 雜交弦理論等效。而 IIB 型弦理論也類似地以 S 對偶與自身有意義地聯繫着。
而不同弦理論間的另一種關係就是 T 對偶。這裡要考慮的是在圓環狀額外維度中傳播的弦。T 對偶中在半徑為 R 的圓環中傳播的弦與在半徑為 1/R 中環狀中傳播的弦等效，即是說一種描述中的所有可觀測量在對偶描述中都有對應量。例如，弦在圓環中傳播時是有動量的，而它可以環繞圓環超過一次。弦環繞圓環的次數叫卷繞數。若一描述中弦的動量為 p，卷繞數為 n，則在對偶描述中它的動量為 n，卷繞數為 p。例如，IIA 型弦理論在 T 對偶下與 IIB 型等效，而雜交弦理論的兩個形式也是由 T 對偶所聯繫。
「對偶」這個詞一般來說指的是兩個看起來不一樣的的物理系統原來是有意義的等效系統。若兩種理論由對偶所聯繫的話，則是指一種理論可以通過某些方式變換成與另一種理論看起來一樣的理論。這樣就可以說兩種理論在變換下互為對偶。換句話來說，兩種理論是同一現象的數學上不同的描述而已。
### 超對稱
另一種對 M 理論起到作用的重要物理概念就是超對稱。這是一種存在於某些理論中的數學關係，是聯繫一種叫玻色子的粒子和另一種叫費米子的粒子的。籠統地來說，費米子是構成物質的，而玻色子則是傳遞交互作用的。在擁有超對稱性的理論中，每一種玻色子都有對應的費米子，反之亦然。將超對稱性作為局部對稱時，所得的量子力學理論會自動包括引力。這樣的理論叫超引力理論。
加入了超對稱概念的弦理論叫超弦理論。超弦理論有數種不同的形式，全部都歸入了 M 理論的框架。超弦理論在低能量時可用 10 維時空的超引力估算。類似地，M 理論在低能量時則可用 11 維時空的超引力估算。
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M理论 | 在弦理論及如超引力理論的相關理論中，膜是一種將點的概念推廣到更高維度的物體。比方說，點粒子可被視為零維上的膜，而弦則可被視為一維上的膜。同時，亦可考慮更高維的膜。它們在 p 維時就叫 p 膜。膜是動力學物體，因此它們能按照量子力學的規則在時空中傳播。它們能夠擁有質量及其他屬性，例如電荷。p 膜所掃出的 p+1 維體積叫「世界體積」。物理學家很多時候會研究與電磁場相似的場，而電磁場就是膜的世界體積內活動。「膜」一詞（brane）是源自於二維的膜（membrane）。
在弦理論中產生基本粒子的基礎物體為一維弦。儘管現時對 M 理論所描述的物理現象仍不是很瞭解，但是物理學家知道 M 理論所描述的是二維膜與五維膜。現時對 M 理論的絕大部份研究都是在嘗試能更好地理解這些膜的性質。
## 歷史與發展
### 卡魯扎 - 克萊因理論
包括阿爾伯特・愛因斯坦和赫爾曼・閔可夫斯基在內的物理學家和數學家在二十世紀初期開拓出使用四維幾何來描述物理世界的手段。這些研究的頂點就是愛因斯坦的廣義相對論表述，把引力與四維時空的幾何聯繫起來。
廣義相對論的成功使得研究人員著手研究用更高的維度幾何來解釋其他力。西奧多・卡魯扎於 1919 年的研究指出，在五維時空中電磁力與引力會統合成一種力。這項研究後來被奧斯卡・克萊因改進，並指出卡魯扎提出的額外維度能夠以半徑為 10−30 公分的環狀形式存在。
不論是卡魯扎 - 克萊因理論還是愛因斯坦後來的嘗試，統一場論的開發從沒有完全成功過。失敗原因的其中一部份是因為克魯扎 - 克萊因理論預測了一種從未見過的粒子，而另一部份則是它不能正確預測電子的質量電荷比。而且開發這些理論的時候，其他物理學家才開始發現量子力學，也就是最終能成功描述像電磁力的已知力的理論，還有到二十世紀中期才被發現的新核力。因此幾乎過了五十年，物理學界才重新認真對待額外維度。
### 超引力的早期研究
新的概念和數學工具為廣義相對論帶來新鮮的見解，亦為 1960 至 70 年代帶來了現在被稱為廣義相對論的黃金時代的光景。物理學家在 1970 年代中開始研究將廣義相對論與超對稱的高維度理論，也就是所謂的超引力理論。 | [
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M理论 | 廣義相對論並沒有為時空可行的維度數設置極限。雖然它一般是用四維空間表述，但是要以同樣的方程式寫下任何維度數的引力場也是可能的。由於超引力理論為維度數設置了上限，因此限制較多。維爾納・納姆於 1978 年的研究證明了具一致性的超對稱理論的時空維度上限為 11 維。巴黎高等師範學校的歐仁・克勒默、伯納德・朱利亞和喬爾・謝克於同年證明了超引力不但允許最高 11 維，而且實際上維數最多時理論才是最優雅的。
不少物理學家最初希望透過緊緻化 11 維超引力，來使得建構四維世界的擬真模型變得可行。他們希望這種模型能為自然的四種基本力提供統一的描述，這四種基本力是電磁力、強和弱核力，以及引力。物理學界對 11 維超引力的興趣很快就因為發現了各種問題而衰減。其中一個問題就是物理定律看起來像是會分辨順時針及逆時針方向，這個現象又叫手徵性。愛德華・維騰與其他研究者發現緊緻化 11 維並不能輕易地導出手徵性。
由於弦理論能夠統合粒子物理學與量子引力，因此不少物理學家在 1984 年的第一次超弦革命期間都轉而研究弦理論。弦理論與超引力理論不同的是，它能夠容納標準模型的手徵性，同時又能得出與量子效應一致的引力理論。另一個讓物理學家們在 1980 至 90 年代對弦理論趨之若鶩的原因就是弦理論具有的高度唯一性。普通的粒子理論要考慮任何的粒子集時，就只需使用能描述該粒子集古典行徑的任意拉格朗日量。而弦理論的可能性就窄得多了：截至 1990 年代為止，弦理論只有五種相容的形式。
### 弦理論間的關係
儘管相容的弦理論種類一隻手能數完，但是為甚麼不是只有一種相容表述仍然是一個謎。然而，當物理學家開始更仔細地檢驗這些理論時，他們發現這些理論的聯繫方式是既微妙又有意義的。
克勞斯・蒙托寧（Claus Montonen）與大衛・奧利夫於 1970 年代末為一些物理理論假設了一個屬性。這個假設的深度形式所考慮的是 N=4 超對稱楊米爾斯理論，它所描述的是夸克和膠子相近的粒子，而原子核就是由夸克和膠子所組成的。此理論中粒子間交互作用的強度由一個叫耦合常數的數所量度。蒙托寧與奧利夫的結果（也就是現在的蒙托寧 - 奧利夫對偶）說明了耦合常數為 g 的 N=4 超對稱楊米爾斯理論與耦合常數為 1/g 的同樣理論等效。也就是說，一個強交互作用的粒子系統（耦合常數大）在一個弱交互作用的粒子系統（耦合常數小）中有等效描述，反之亦然。 | [
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M理论 | 幾位物理學家在 1990 年代將蒙托寧 - 奧利夫對偶推廣成聯繫不同弦理論的 S 對偶。阿索克・森曾以四維雜交弦為背景來研究 S 對偶。克里斯・赫爾與保羅・唐森德成功證明了大耦合常數的 IIB 型弦理論與小耦合常數的同樣理論在 S 對偶下是等效的。理論物理學家們還發現了不同的弦理論是可以用 T 對偶來聯繫的。這種對偶意味著在不同時空幾何下傳播的弦可能在物理學上是等效的。
### 膜與五膜
弦理論延伸了普通的粒子物理學，它將零維的點粒子提升成一維的物體 —— 弦。因此理論物理學家在 1980 年代末期就很自然地研究起用二維的超膜或更高維度的膜來取代粒子。這些物體早在 1962 年就已被保羅・狄拉克研究過，到了 1980 年代再有一群為數不多的熱心物理學家重新研究它們。
超對稱嚴重地限制了膜維度的可能數。埃里克・伯格雪夫（Eric Bergshoeff）、埃爾金・塞茲金（Ergin Sezgin）及保羅・唐森德於 1987 年證明了十一維的超引力能容納二維的膜。這些物體在直覺上看起來就像在十一維時空中傳播的紙張或薄膜。邁克爾・達夫、保羅・賀維（Paul Howe）、稻見武夫及凱洛格・斯蒂爾（Kellogg Steele）在這項發現後不久就研究了十一維其中一維捲成環狀的緊致化情況。在這個設定中可以設想薄膜包住環狀維度。若環狀的半徑足夠小的話，則此膜看起來就跟十維時空的弦一樣。達夫與他的研究夥伴實際上亦證明了這種構造所得出的正是 IIA 型超弦理論的弦。
安德魯・施特羅明格於 1990 年發表了相近的結果，指出使用 5 維的弱交互作用膜來描述 10 維的強交互作用弦可能可行。物理學家在剛開始時並不能證明這項關係，原因有二。一方面蒙托寧 - 奧利夫對偶當時仍待證明，另一方面 5 維膜的量子特性當時在技術上仍然存在不少疑問。上述兩項難題的第一項由阿索克・森於 1993 年解決，他確立了某些物理理論需要同時帶電荷和磁荷的物體方能成立，正如蒙托寧和奧利夫在研究中預測的那樣。 | [
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M理论 | 儘管研究取得了這樣的進展，弦與 5 維膜的間的關係依然是一項假說，這是因為物理學家仍未成功將膜量子化。邁克爾・達夫、拉姆茲・胡里（Ramzi Khuri）、盧建新、魯本・米納西安（Ruben Minasian）和他們的研究團隊從 1991 年開始研究弦理論的一種特殊緊殊化，它將 10 維中的 4 維捲起來。如果只考慮包圍這些額外維度的 5 維膜的話，則那膜就正如 1 維弦一樣。於是這樣的話就能將之前弦與膜間的假說關係簡化成弦與弦之間的關係，而後者可被用於測試已經確立的理論技巧。
### 第二次超弦革命
普林斯頓高等研究院的愛德華・維騰在南加州大學 1995 年的弦理論研討會中講話時，提出了一套出人意表的理論，就是全部五種超弦理論實際上都是一種 11 維時空理論的不同極限個案。維騰的報告將之前關於 S 對偶、T 對偶和弦理論含有二維和五維膜的全部研究成果都綁在一起了。網際網路在維騰報告後的數月內出現了數以百計的新論文，確認了膜在新理論中是有著重要角色的。今日這股研究熱潮被稱為第二次超弦革命。
在維騰報告後其中一項重要的發展就是 1996 年維騰與弦理論家彼得・霍扎瓦合作的研究。維騰與霍扎瓦使用了兩個 10 維邊界的分量來研究特殊時空幾何上的 M 理論。他們的研究為 M 理論的數學架構提供了線索，同時亦建議了如何把 M 理論與真實世界的物理聯繫起來。
### 命名的由來
一些物理學家最初指出新理論是膜的基本理論，但是維騰對理論中膜的角色存疑。霍扎瓦與維騰在 1996 年的一篇論文中寫道：
> 由於那種 11 維理論是超膜理論，但是仍然有理由去懷疑那個詮釋，我們會不負責地叫它作 M 理論，把 M 與 membranes（膜）的關係留給未來解決。
>
>
在沒有理解 M 理論的真正意義和架構的情況下，維騰提議根據個人口味 M 應該代表 Magic（魔術理論）、Mystery（神秘理論）或 Membrane（膜理論），而這個命名的真正意義要等到理論更基礎的表述出現後才能下決定。
## 矩陣理論
### BFSS 矩陣理論
矩陣在數學上是數字或其他數據的矩形陣列。矩陣理論在物理學上就是將矩陣標記用在數學表述中重要地方的某種物理理論。矩陣模型描述了在量子力學框架下一組矩陣的性質。 | [
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M理论 | 矩陣理論的一個重要例子就是由湯姆・班克斯、威利・費席勒、史蒂芬・申克爾和李奧納特・色斯金於 1997 年所提出的 BFSS 矩陣理論。這套理論描述了一組 9 個大型矩陣的性質。他們在論文中除了展示其他證明，還成功證明了 11 維超重力描述了這套矩陣理論的低能量極限。這些計算引導了他們提出 BFSS 矩陣理論與 M 理論完全等價。因此可以使用 BFSS 矩陣理論作為正確 M 理論表述的原型，以及在相對簡單的設定下研究 M 理論特性的工具。
### 非交換幾何學
在幾何學中引入座標一般都是有用的。例如，為了研究歐幾里得平面上的幾何，可以把平面上的所有點與一對軸之間的距離定義為坐標 x 和 y。由於點的坐標在普通幾何中是一對數字，因此可以把它們相乘，而兩個座標的相乘結果並不取決於相乘的順序。那就是 xy=yx。乘法的這種性質叫交換律，而幾何學與坐標的交換代數這個關係是大部份現代幾何學的起點。
非交換幾何學是一門嘗試歸納這個情況的數學分科。這門學科改為考慮乘法不遵從交換律的數字相近物件（也就是說 xy 不一定等於 yx 的物件），例如矩陣，而不使用普通數字。這門學科的研究者想像這些非交換物件是一些更通用化「空間」概念中的坐標，然後利用與普通幾何學間的對比來證明這些通用空間中的定理。
阿蘭・科納、邁克爾・R・道格拉斯和阿爾伯特・施瓦茨在 1998 年發表的論文中指出矩陣理論的某些方面是由非交換量子場論所描述，那套特殊理論中的空間坐標是不符合交換性質的。它一方面確立了矩陣理論和 M 理論之間的聯繫，另一方面也確立了矩陣理論和非交換幾何學的聯繫。這很快就導致了非交換幾何學和其他各種物理理論間的重要聯繫相繼被發現。
## AdS/CFT 對偶
### 概述
一些物體具有時間空間範圍，如電磁場等，而應用在這些物體上的量子力學就是量子場論。量子場論是粒子物理學研究基本粒子的基礎理論，而其中這些粒子是由基本場的激發所描述。而凝聚態物理學也會使用量子場論來模擬類似於粒子的準粒子。
物理學家們可以透過反德西特 / 共形場論（AdS/CFT）對偶來表述 M 理論及研究其性質。AdS/CFT 對偶由胡安・馬爾達西那於 1997 年末提出，這項理論結果意味著在某些個案中 M 理論是等同於一種量子場論。AdS/CFT 對偶除了為數學家和物理學家就弦和 M 理論的數學結構提供了啟發之外，它還為解決在傳統計算技巧無效區間中的量子場論提供了多方面的線索。 | [
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M理论 | 在 AdS/CFT 對偶中，時空的幾何是由愛因斯坦場方程式中某些叫反德西特空間的真空解所描述。非常基本地來說，反德西特空間是一個數學模型，其中點與點間的距離概念（度規），與日常歐幾里德幾何中的距離概念不一樣。反德西特空間與雙曲空間有著密切的關係，而雙曲空間可用左圖的圓盤表示。左圖為由三角形和正方形所組成的密鋪。用某種方式可以為點間的距離下定義，使得所有三角形和正方形都是一樣大小的，並且圓周的外邊界與其內部任一點的距離為無限。
現在想像一疊雙曲圓盤，其中每一片圓盤代表某時間的宇宙態。而由此形成的幾何物體就是反德西特空間。它看起來像實心的圓柱體，其中每一片截面都是雙曲圓盤。右圖中時間以垂直方向行進。這圓柱體的表面在 AdS/CFT 對偶中有著重要的角色。反德西特空間跟雙曲圓盤一樣，它的彎曲方式使得內部任何一點與邊界面的距離為無限遠。
這樣的結構雖然描述了只有二維空間加一維時間的假想宇宙，但還是可推廣至適用於任何維數。雙曲空間實際上是可以超過二維的，把這些雙曲空間「疊起來」就能形成反德特空間的高維度模型。
反德西特空間的重要特點在於其邊界（三維德西特空間的邊界看起來像圓柱體）。這種邊界有一個特性，就是在任何點的局部範圍都和閔考斯基時空很像，而閔考斯基時空就是非重力物理所用的時空模型。因此可以構建一套「時空」由反德西特空間邊界提供的輔助理論。而這項觀察正是 AdS/CFT 對偶的起點，因為 AdS/CFT 對偶把反德西特空間的邊界視為共形場論的「時空」。對偶主張共形場論相等於反德西特空間主體的重力理論，也就是說兩者可以像有「字典」的那樣將計算互相翻譯。一套理論中的每一件實體在另一套理論中都有對應的實體。比方說，重力理論中的單一粒子可能對應邊界理論中的某粒子集。此外，兩套理論的量化預測也是一致的，例如說重力理論中兩個粒子碰撞的機率是 40%，那麼共形場論的對應粒子集碰撞機率也會是 40%。
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M理论 | AdS/CFT 對偶其中一項特別的實踐指出，積空間上 _AdS_7×_S_4 的 M 理論等同於 6 維邊界上所謂的 (2,0) 理論。這裏「(2,0)」指的是理論中出現的特定超對稱。在這個例子中，重力理論的時空實際上為 7 維（因此標記為 _AdS_7），以及額外 4 維的「緊緻」維度（由 _S_4 表示）。由於時空在現實世界中是 4 維的，最少在宏觀上是這樣的，因此對偶的這個形式並不能作為重力的現實模型。同樣道理，由於對偶另一邊的理論所描述的是 6 維世界，因此不能作為任何現實世界系統的可行模型。
然而，物理學家們已經證實 (2,0) 理論對研究量子場論的一般性質有重要作用。這套理論確實包括了不少數學上有趣的有效場論，而且引出了聯繫這些場論的新對偶。例如，路易斯・阿爾達伊（Luis Alday）、大衛・格約托和立川裕二（Yuji Tachikawa）證明了把曲面上理論緊緻化後，得出的是一種 4 維量子場論，而且還找到一種聯繫這套理論的物理和某些與曲面本身有關的物理概念的對偶，叫 AGT 對偶。理論學者們最近把這些概念開拓至研究以緊緻化降至 3 維的理論。
(2,0) 理論除了在量子場論的應用之外，它還在純粹數學中衍生出重要結果。例如，維騰使用 (2,0) 理論的存在性來為數學中一種叫幾何朗蘭茲綱領的假想關係作出「物理」解釋。維騰在後續的研究中證明了 (2,0) 理論可用於理解一種叫霍瓦諾夫同調的數學概念。霍瓦諾夫同調由米哈伊爾・霍瓦諾夫於 2000 年前後開發，是紐結理論的工具，紐結理論是研究各種不同紐結形狀並將之分類的數學分科。數學上 (2,0) 理論的另一種應用則是大衛・格約托、葛雷格・穆爾和安德魯・奈茨克（Andrew Neitzke）的研究，使用物理概念來推導出超凱勒幾何內的新成果。
### ABJM 超共形場論
AdS/CFT 對稱的另一項實踐，就是指出 _AdS_4×_S_7 等同於 3 維上一種叫 ABJM 理論的量子場論。在這個形式的對偶中，M 理論中其中 7 維捲了起來，餘下非緊緻化的 4 維。由於人們所處的宇宙的時空為 4 維，所以這個形式的對偶給出了某程度上較真實的重力描述。 | [
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] |
M理论 | 在這個形式的對偶中出現的 ABJM 理論還有其他各種有趣的理由。由奧弗爾・阿哈羅尼（Ofer Aharony）、奧朗・伯格曼（Oren Bergman）、丹尼爾・賈弗里斯（Daniel Jafferis）和胡安・馬爾達西那開拓的 ABJM 理論，與另一種量子場論
陳 - 西蒙斯理論有著密切的關係。後者因其與紐結理論而被維騰於 1980 年末推廣開來。除此之外，ABJM 理論還被用於解決凝聚態物理學難題用的半現實模型。
## 唯象學
### 概述
M 理論除了是一門吸引了巨大研究興趣的理論學科，還為真實世界物理模型的構築提供了結合廣義相對論和粒子物理學標準模型的框架。唯象學是物理學家利用較理論化的概念來構建真實自然模型的理論物理學分科。而弦唯象學則是弦理論中嘗試構建出基於弦和 M 理論的現實粒子物理模型的部份。
這些模型一般都是基於緊緻化。物理學家們以弦或 M 理論的 10 或 11 維空間為出發點，假定出額外維度的形狀。他們能透過所選取的大概形狀，構建出與粒子物理學標準模型粗略相近的模型，模型裏面還包括著未被發現的粒子，這些粒子一般都是與已知類似的粒子的超對稱伴子。其中一種從弦理論中推導出現實物理的常用方法就是以 10 維雜交理論為起點，並假設其 6 維額外維度的形狀類似於 6 維的卡拉比 - 丘流形。它是一種特殊的幾何形體，以數學家歐金尼奧・卡拉比和丘成桐命名。卡拉比 - 丘流形為從弦理論中提取現實物理提供了不少途徑。其他相近的方法可用於以 M 理論來構建出能某程度重現 4 維世界物理的模型。
一方面是因為理論和數學上的難點，而另一方面則是因為測試這些理論所需的實驗溫度極高（比可預見的將來中技術上可行的還要高），所以現時並沒有實驗證據能一致地指出這些模型中的任一種是自然基礎的正確描述。因此引起了社群中的一部份人批評這種統一手法，以及質疑繼續研究這種難題的價值。
### G2 流形的緊緻化
理論學者研究弦唯象學的另一手法，就是假設 M 理論 7 維額外維度的形狀與 G2 流形相似。這是一種由牛津大學數學家多米尼克・喬伊斯所構建的特殊 7 維形體。由於現時在數學上對 G2 流形的理解仍然很有限，因此物理學家要充份地開發這種弦唯象學的手法很有難度。 | [
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] |
M理论 | 比方說，數學家和物理學家常常對空間假設一項叫「光滑」的數學特性，但如果想從 G2 流形中得出 4 維世界的物理就不能假設這特性存在。另一個難題就是 G2 流形並不是複流形，因此物理學家在此不能使用數學分科複分析的工具。還有決定性的難題是 G2 流形在存在性、唯一性和性質上都有不少的未決問題，而且數學家還沒有一套對這種流形的系統性搜尋方式。
### 雜交 M 理論
由於 G2 流形有著不少的難點，因此嘗試以 M 理論來構建現實物理理論時，主要都是用 11 維時空緊緻化這種較直接的方式。維騰、霍扎瓦、伯特・奧夫拉特及其他研究者開發了一種叫雜 M 交理論的方法。這個方法把 M 理論 11 維中的其中 1 維想像成環狀。若環狀很小的話，則時空實際上變成了 10 維。然後就可以假設 10 維中的 6 維形成了卡拉比 - 丘流形。若這個卡拉比 - 丘流形也被視為微小的話，則所得的理論為 4 維。
雜交 M 理論已被用於構建膜宇宙學的模型，其中可觀測宇宙被視為存在於高維度環繞空間的一張膜上。它還衍生出不需要依賴宇宙暴漲理論的另類早期宇宙理論。
## 參考資料
### 註釋
### 資料來源
### 參考文獻
## 參閲
## 外部連結
* Superstringtheory.com （頁面存檔備份，存於網際網路檔案館） – The "Official String Theory Web Site", created by Patricia Schwarz. References on string theory and M-theory for the layperson and expert.
* Not Even Wrong （頁面存檔備份，存於網際網路檔案館） – Peter Woit's blog on physics in general, and string theory in particular.
### 通俗文化
* bbc-horizon: parallel-uni （頁面存檔備份，存於網際網路檔案館） – 2002 feature documentary by BBC Horizon, episode Parallel Universes focus on history and emergence of M-theory, and scientists involved. | [
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M理论 | * pbs.org-nova: elegant-uni （頁面存檔備份，存於網際網路檔案館） – 2003 Emmy Award-winning, three-hour miniseries by Nova with Brian Greene, adapted from his The Elegant Universe (original PBS broadcast dates: October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003). | [
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加速运动 | : 在這篇文章內，向量與純量分別用粗體與斜體顯示。例如，位置向量通常用 $\mathbf {r} \,\!$ 表示；而其大小則用 $r\,\!$ 來表示。
**加速度（英語：Acceleration）** 是物理學中的一個物理量，是一個矢量，主要應用於古典物理當中，一般用字母 $\mathbf {a} $ 表示，在國際單位制中的單位為米每二次方秒（ $\mathrm {m/s^{2}} $ ）。加速度是速度矢量對於時間的變化率，描述速度的方向和大小變化的快慢。
在古典力學中，牛頓第二定律說明了力和加速度成正比，這定律又稱為「加速度定律」。假設施加於物體的淨外力為零，則加速度為零，速度為常數，由於動量是質量與速度的乘積，所以動量守恆。在電動力學裏，呈加速度運動的帶電粒子會發射電磁輻射。
## 簡述
簡單地說，速度描述了位置是如何變化的，而加速度描述了速度是如何變化的。比如，水平地向前扔出一個物體，起初它的速度朝向正前，然而由於重力它開始在向前的同時向下墜落，即其速度改變了。這裡改變物體速度的主要是地球的重力引起的重力加速度。
加速度具有向量性質，即需要用大小和方向同時描述一個加速度。在光滑水平面上向前運動的物體，如果向左或向右施以力，即給予了不同的加速度，則其速度會發生變化（包含了速率及方向），然而向左的加速度和向右的加速度顯然引起了不同的效果。同樣，施力的大小不同，引起的加速度不同，最終的結果也不一樣，亦可以從向量的加成性來看。作為一個矢量，加速度的疊加和分解分別遵循平行四邊形法則和三角形法則。
具體而言，加速度描述的是速度隨時間的變化率。需要注意的是，由於速度也是矢量，因此加速度不為零的物體速度的大小（稱之為速率）也不一定會發生變化，實際上，如果加速度保持與速度垂直，速度大小就一直不會改變，同時方向一直改變。這種情況在生活中最常見的是圓周運動，比如在被拴在一端固定的線的另一端的一個小物體在線保持繃直時做的運動，又比如帶電粒子在僅受靜磁場的勞侖茲力 $\mathbf {F} =q\mathbf {v} \times \mathbf {B} $ 時做的運動。
### 直線運動中的平均加速度和瞬時加速度
設質點 A 呈一維運動， $t$ 時刻位於 $x(t)$ 處，經過 $\Delta t$ 時間後位於 $x(t+\Delta t)$ 處，則定義質點 A 在 $t$ 時刻的瞬時速度（簡稱速度）為 | [
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加速运动 | : $v(t)=\lim _{\Delta t\to 0}{\frac {x(t+\Delta t)-x(t)}{\Delta t}}={\frac {\mathrm {d} x}{\mathrm {d} t}}$
其中， ${\frac {\mathrm {d} x}{\mathrm {d} t}}$ 表示位移對時間的一階導數，在時間 - 位移圖上表現為求斜率。
首先，定義 $t$ 時刻到 $t+\Delta t$ 時刻之間的**平均加速度**為
: ${\bar {a}}={\frac {v(t+\Delta t)-v(t)}{\Delta t}}$
平均加速度粗略地表示了在該段時間內物體速度的變化情況。如果 $\Delta t$ 越小，該段時間內速度的波動就越小，描述的速度變化情況也就越精細，從而定義質點 A 在 $t$ 時刻的**瞬時加速度**為
: $a(t)=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} x}{\mathrm {d} t^{2}}}$
瞬時加速度，簡稱加速度。進而有
: $v(t_{1})=\int _{t_{0}}^{t_{1}}a(t)\mathrm {d} t+v(t_{0})$
在直線運動時，矢量約化為帶符號的純量，其絕對值表示該物理量的大小。速度為正表示向右，速度為負表示向左（二維空間座標中）。加速度與速度方向相同（即符號相同）時表示物體不斷加速，不同則表示物體不斷減速。
右圖畫出了三個質點在 $t=0$ 從坐標原點以相同的速度 $v_{0}$ 出發，由於分別擁有正、零、負的加速度而導致其位置 $x$ 關於時間的曲線。可以將其想像為在光滑桌面上，三個木塊以相同初速度，沿斜面向下、沿水平面、沿斜面向上滑行。
在位移 - 時間圖上，加速度由曲線的凹凸性表示，加速度為正的部分表現為凸函數，反之為凹函數，亦可以從微分的角度來分析。
### 曲線運動中的加速度
設質點 A 在空間中運動，原點 O 指向 A 的矢量 $\mathbf {r} $ 為其矢徑，則可類似定義其速度矢量和加速度矢量為 | [
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加速运动 | : $\mathbf {v} (t)={\frac {\mathrm {d} \mathbf {r} (t)}{\mathrm {d} t}}$
: $\mathbf {a} (t)={\frac {\mathrm {d} \mathbf {v} (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \mathbf {r} (t)}{\mathrm {d} t^{2}}}$
右圖表現的是一個質點沿一曲線運動的軌跡，表示出了兩次微分的過程，為了清晰表示，這裡使用差分（ $\Delta t$ 並不趨於 0）近似代替了微分，因此表現的是平均速度和平均加速度。可以看出，加速度與速度都具有方向和大小，並且即使在同一時刻兩者方向也不一定相同。加速度與速度方向平行的分量表示速度大小的變化率（相同則加速，相反則減速），而與速度垂直的分量表示速度方向的變化率（速度矢量轉動的角速度）。
在 $\Delta t$ 足夠小時，可以將那一小段曲線運動（稱作元弧）近似看作直線運動或圓周運動。
### 伽利略變換
在古典物理下，即速度遠小於光速、研究宏觀物體時，可以使用伽利略變換來研究不同參考系間的加速度的聯繫，簡單來說就是座標間的轉換，但仍有保持一定的不變量：
: $\mathbf {a} =\mathbf {a} '+\mathbf {a} _{\text{rel}}$
其中， $\mathbf {a} $ 為物體在參考系 $S$ 下的加速度， $\mathbf {a} '$ 為物體在參考系 $S'$ 下的加速度， $\mathbf {a} _{\text{rel}}$ 為參考系 $S'$ 在參考系 $S$ 下的加速度。
考慮站在地面看火車上的人拋出一個小球，這個公式表達：小球相對於地面的加速度 $\mathbf {a} $ ，等於小球相對於火車的加速度 $\mathbf {a} '$ 加上火車相對於地面的加速度 $\mathbf {a} _{\text{rel}}$ 。這個式子是矢量表達式，即三個加速度矢量的方向不在同一條直線上時，要使用矢量加法計算。
### 牛頓第二定律
加速度最主要的應用之一是牛頓第二定律。簡單地說，牛頓第二定律表明，感受到淨外力的作用，物體的加速度與淨外力成正比，與質量成反比，加速度方向沿淨力方向，在國際單位制中表示為
: $\mathbf {F} =m\mathbf {a} $ | [
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加速运动 | 其中 $\mathbf {F} $ 表示物體所受淨外力， $m$ 為物體質量， $\mathbf {a} $ 為物體的加速度。
在古典物理下，牛頓第二定律廣泛適用。此外，牛頓第二定律要求所處參考係為慣性參考系。由於古典物理的研究幾乎都會或多或少地涉及到物體在力的作用下的運動，又由於牛頓第二定律和伽利略變換極具簡潔性，所以，牛頓第二定律是古典動力學裏的重要基礎定律，質點運動亦然。
### 慣性力
當帶質量物體加速時，慣性是物體維持原有運動狀態的傾向，慣性力是對於這傾向的衡量。因為慣性力實際上並不存在，只有原本將該物體加速的作用力實際存在，因此慣性力又稱為假想力。更具體而言，根據牛頓第二定律，
: $\sum _{i}\mathbf {F} _{i}=m\mathbf {a} $
其中， $\mathbf {F} _{i}$ 是第 $i$ 個作用力， $\mathbf {a} $ 是物體的加速度。
重新編排，可以得到
: $\sum _{i}\mathbf {F} _{i}-m\mathbf {a} =0$
設定慣性向量 $\mathbf {I} =-m\mathbf {a} $ ，這物體的動力系統滿足方程式
: $\sum _{i}\mathbf {F} +\mathbf {I} =0$
想像這慣性向量為由於加速度運動而產生的一種力，稱為慣性力。因為慣性力與所有作用於這物體的外力的向量總和為零，這動力系統可以視為處於動力平衡狀態。藉著這機制，可以將動力系統約化為靜力系統，用靜力學發展出的方法來解析動力問題。
慣性力在平日生活中其實很常見，例如，停止不動的火車突然向前方加速，則所有站立乘客都會向後方傾移，這便是慣性力效應，從另外一個角度而言也是為了提供乘客們有足夠的摩擦力來進行移動。
### 加加速度
將位移對於時間進行一階求導得到了速度，二階求導得到了加速度。可能會想到，可以通過進行三階求導來得到一個諸如加加速度的物理量。 | [
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加速运动 | 在工程學中經常需要用到加加速度，特別是在交通工具設計以及材料等問題。交通工具在加速時將使乘客產生不適感，這種不適感不僅來自於**加速度**，也與加加速度有關。在這種情況中，加速度反應人體器官在加速度運動時所感受到的力（見牛頓第二定律），加加速度則反應這作用力的變化快慢。較大的加加速度將會使人體產生相當的不適感，例如在電梯升降，汽車、火車等加速和轉彎的過程中（在這些情況中加速度和加加速度的效應一般會同時存在）。人體需要時間適應加速度的變化，假若加加速度超過安全標準，則可能會發生像車禍造成的頸部扭傷（whiplash）一類的人體傷害。因而在設計交通工具時加加速度是必須考慮的因素。
在物理學裏，加加速度現在主要應用在混沌理論領域。
### 角加速度
角加速度涉及繞著固定軸轉動的物體，例如，想像一個圓盤和一個垂直固定於其中心的木棍，兩隻手合拍住木棍並前後磨擦，造成木棍與圓盤共同轉動（例如，在地上高速旋轉的陀螺，繞著固定點轉動）。在圓盤上做一個標記（如一條半徑），則繞著固定軸轉動的物體可以簡單地用純量角弧（即該標記轉動的角弧）來做定量描述。
旋轉運動可以與直線運動相類比：位移、速度、加速度，分別對應於角弧、角速度、角加速度。直線運動中已有的定律和方法可以直接帶入，用於旋轉運動，例如，使用已有的勻加速直線運動理論來研究勻角加速度固定軸轉動。
在國際單位制中，角加速度的單位為弧度每二次方秒（ $\mathrm {rad/s^{2}} $ ）。其定義式為
: $\alpha (t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \theta (t)}{\mathrm {d} t^{2}}}$
其中， $\alpha $ 為物體角加速度， $\omega $ 為物體角速度， $\theta $ 為物體轉過的角弧。
## 加速度的分解
處理關於空間加速度矢量的問題，除了直接計算矢量之外，更多的時候可以將加速度按照適當坐標軸分解，即將矢量形式的加速度表示為相互獨立的不同方向上的純量的形式。因為純量的計算要容易很多，因此這是解決問題常用的方法。
### 按坐標系分解
#### 平面直角坐標系
在平面直角坐標系中，
: $\mathbf {a} (t)=a_{x}(t)\mathbf {i} +a_{y}(t)\mathbf {j} $ | [
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加速运动 | 其中 $\mathbf {i} $ 和 $\mathbf {j} $ 分別為 _x_、_y_ 坐標軸上的單位矢量，皆為常矢量。
這種分解方式的優點在於，形式簡便，思維簡單；因為單位矢量不會變化，故質點在二個方向上的投影等價於直線運動，並將其疊加，使得問題完全化為代數問題，並且可以直接使用直線運動的已有結論。
#### 極坐標系
在二維空間裡，極坐標系用半徑坐標 $r$ 、角坐標 $\theta $ 來表示質點的位置。半徑坐標是極點與質點的直線距離；角坐標是極點與質點的連線對於極軸的角弧。在任意點的兩個單位矢量分別為沿半徑向外的 $\mathbf {e} _{r}$ 和垂直於半徑指向角坐標正方向的 $\mathbf {e} _{\theta }$ 。不論是直角坐標或是極坐標都可以互相來駔變換，坐標和坐標之間有一定的轉換量，使用以方便所在的坐標系為主。
從極坐標 $r$ 和 $\theta $ 可以計算出直角坐標：
: $x=r\cos \theta $
: $y=r\sin \theta $
在極坐標系中，位置 $\mathbf {r} $ 、速度 $\mathbf {v} $ 、加速度 $\mathbf {a} $ 分別為
: $\mathbf {r} =r\mathbf {e} _{r}$
: $\mathbf {v} =r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {dr}{\mathrm {d} t}}\mathbf {e} _{r}$
: $\mathbf {a} =\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }$
##### 極坐標系分解的推導 | [
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加速运动 | 兩個單位矢量 $\mathbf {e} _{r}$ 、 $\mathbf {e} _{\theta }$ 會隨質點所處位置不同而變化，並可通過分析得出結論，在質點運動的時候
: $\mathrm {d} \mathbf {e} _{r}=\mathrm {d} \theta \mathbf {e} _{\theta }$
: $\mathrm {d} \mathbf {e} _{\theta }=-\mathrm {d} \theta \mathbf {e} _{r}$
其中，經過微小時間 $\mathrm {d} t$ 後，單位矢量 $\mathbf {e} _{r}$ 、 $\mathbf {e} _{\theta }$ 的微小變化分別為 $\mathrm {d} \mathbf {e} _{r}$ 、 $\mathrm {d} \mathbf {e} _{\theta }$ 。
應用微分的萊布尼茲法則
: ${\frac {\mathrm {d} }{\mathrm {d} t}}(a\ b)=a\ {\frac {\mathrm {d} b}{\mathrm {d} t}}+b\ {\frac {\mathrm {d} a}{\mathrm {d} t}}$
在極坐標系下，速度 $\mathbf {v} $ 、加速度 $\mathbf {a} $ 分別為
: ${\begin{aligned}\mathbf {v} &={\frac {\mathrm {d} }{\mathrm {d} t}}(r\mathbf {e} _{r})\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\\\end{aligned}}$ | [
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加速运动 |
: ${\begin{aligned}\mathbf {a} &={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\&={\frac {\mathrm {d} }{\mathrm {d} t}}\left(r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\right)+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\right)\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{\theta }}{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{r}}{\mathrm {d} t}}\\&=-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\mathbf {e} _{r}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\\&=\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }\\\end{aligned}}$ | [
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加速运动 | #### 按自然坐標系分解
假設一個質點移動於二維平面。在質點軌道的任意位置，二維自然坐標系的一個坐標軸方向（切向）保持與軌道切線方向平行，另一個坐標軸方向（法向）則與軌道法線平行。分解按右圖。向量可以無限地做拆解，所以只需要選擇對於分析最有利的為主！通常以切線方向和法線方向來分解。
簡單地說，加速度的切向分量 $a_{\text{t}}$ 表示速度大小的變化，加速度的法向分量 $a_{\text{n}}$ 表示速度方向的變化，即
: $a_{\text{t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}$
: $a_{\text{n}}={\frac {v^{2}}{\rho }}$
其中， $v$ 為此時刻的速度大小， $\rho $ 為此時刻的曲率半徑。
### 按功能分解
#### 等速率圓周運動 向心加速度
若質點以不變的速率（速度大小）沿著圓周繞著圓心運動，則質點呈等速率圓周運動，質點具有向心加速度 $\mathbf {a} _{n}$ ，其方向保持沿半徑方向向裏（因此不斷變化），大小為
: $a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}$
其中， $\omega $ 是角速度。
這公式也可以從極坐標系分解中，代入與等速率圓周運動相關的特殊值得到。更一般的情況下（非等速率圓周運動），以矢量來表示，
: $\mathbf {a} _{\text{n}}=\mathbf {\omega } \times (\mathbf {\omega } \times \mathbf {r} )$
在矢量式中，令沿半徑向外為正。在平面的情況下，該矢量式約化為上述純量式，這時會得到一個負號，通常以圓座標來表示最為合適。
假設，在一根繩子的一端系上一個小物體（比如石頭），另一端握在手中，大致保持手不動而水平旋轉，則手會明確地感受到繩子的拉力，該拉力的反作用力在繩子的另一端表現為向心力，提供小物體的向心加速度。當轉得越快，向心力會越大，可以定性地驗證上述向量式。從這個實驗，可以看出，向心加速度總是使物體趨向於朝著圓心做運動；如果沒有繩子施予向心力，物體一定會飛奔出去。 | [
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加速运动 | 再舉一個例子，在遊樂場的巨大旋轉圓盤上，大部分遊客都會站立不穩，總是會向外摔倒，這是因為缺乏向心力施予於遊客。在旋轉圓盤的非慣性系中，遊客會感受到慣性力，但由於缺乏向心力，無法達成平衡狀態，因此被向外「甩」出去，這慣性力又稱為離心力，人們以這個原理製成了離心機。
上述公式不但對於從等速率圓周運動成立，也可以應用於各種圓周運動、甚至任意曲線運動，只是上述的 $v$ 和 $\omega $ 應理解為該時刻的瞬時物理量， $r$ 應以曲率半徑 $\rho $ 替代，表示的是物體的加速度在垂直於路徑方向的分量。
##### 向心加速度的推導
如等速率圓周運動圖所示，某一時刻質點速度為 $\mathbf {v} _{1}$ ，極短時間 $dt$ 過後，質點沿圓周前行，速度變成 $\mathbf {v} _{2}$ 。因為是等速率圓周運動，速度大小保持為 $v$ ，但方向會保持與圓相切（垂直於半徑 $r$ ），不斷改變。關注圖的左上部分，當 $\beta $ 使用弧度值時，速度大小的改變 $dv$ 在 $d\beta $ 極小時近似於 $vd\beta $ ，方向大約沿半徑向裏，則該段時間內的平均加速度大小為
: $a_{\text{n}}=v{\frac {\mathrm {d} \beta }{\mathrm {d} t}}=v\omega $
其中，角速度 $\omega ={\frac {\mathrm {d} \beta }{\mathrm {d} t}}$ 。
以上「近似」在 $dt\to 0$ 時精確成立。又因為
: $v=\omega r$
所以，
: $a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}$
#### 科里奧利效應
給定固定參考系 _S_ 與旋轉參考系 _S'_，從固定參考系 _S_ 觀察，旋轉參考系 _S'_ 以勻角速度轉動。移動於旋轉參考系 _S'_ 的質點因為運動速度而產生的偏轉效應，稱為「科里奧利效應」，這是為紀念法國科學家賈斯帕 - 古斯塔夫・科里奧利而命名。 | [
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加速运动 | 舉例而言，設想一個巨大的圓盤在地上繞著定點勻角速度轉動（定點運動），而這定點為圓盤的圓心，在圓盤上沿半徑方向有一個直導軌，一個物體被限制在導軌上運動，從圓心勻速向外移動。從地上（固定參考系）觀察，物體的軌跡不是一條直線，而是一條弧形或者螺旋形路線，物體也會感受到導軌的約束力，其方向垂直於導軌，並且指向圓盤旋轉方向（不是角速度向量的方向），這約束力促使物體朝著圓盤旋轉方向加速，使物體的軌跡呈弧形或螺旋形。從圓盤（旋轉坐標系）觀察，物體所感受到的科里奧利力會與導軌施予的約束力相抵消，因此，物體只會呈直線運動。假若，導軌不存在，則物體會逆著圓盤旋轉方向以科里奧利加速度運動。
在科里奧利效應裏，參考系 S' 的物體的柯里奧利加速度與感受到的科里奧利力分別為
: $\mathbf {a} _{\text{cor}}=-2{\boldsymbol {\Omega }}\times \mathbf {v} $
: $\mathbf {F} _{\text{cor}}=-2m{\boldsymbol {\Omega }}\times \mathbf {v} $
其中 $\mathbf {a} _{\text{cor}}$ 為科里奧利加速度， ${\boldsymbol {\Omega }}$ 為參考系 S' 在參考系 S 中的角速度矢量， $\mathbf {v} $ 為物體在參考系 _S'_ 中的速度矢量。
在氣象學裏，科里奧利力使得熱帶氣旋在沒有強引導氣流影響下移向兩極。 熱帶氣旋靠近兩極部分含有東風，科里奧利力會將東風拉向兩極；靠近赤道部分含有西風，科里奧利力會將西風拉向赤道。在地球上越接近赤道科里奧利力會越弱，所以科里奧利力影響熱帶氣旋靠近兩極部分會較靠近赤道部分為多。因此，在沒有其他引導氣流抵消科里奧利力的情況下，北半球的熱帶氣旋一般會向北移動，而南半球的熱帶氣旋則會向南移動。
科里奧利力也會開啟氣旋系統的旋轉，但驅動高速度旋轉的主要因素，不是科里奧利力，而是凝結熱。
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加速运动 | 給定固定參考系 _S_ 與旋轉參考系 _S'_，從固定參考系 _S_ 觀察，旋轉參考系 _S'_ 以非勻角速度 ${\boldsymbol {\Omega }}$ 轉動。從旋轉參考系 _S'_ 觀察，物體因這非勻角速度而感受到的虛設力（fictitious force）稱為「歐拉力」，產生的加速度稱為「歐拉加速度」。歐拉力 $\mathbf {F} _{\text{Euler}}$ 與歐拉加速度 $\mathbf {a} _{\text{Euler}}$ 之間的關係式為
: $\mathbf {F} _{\text{Euler}}=m\mathbf {a} _{\text{Euler}}$
設想一個巨大的圓盤在地上繞著定點轉動，而這定點為圓盤的圓心。在圓盤的非圓心位置固定一個物體。圓盤呈非勻角速度運動，則從旋轉參考系 S' 觀察，延著物體的圓形軌跡切向（不是角速度向量的方向），此物體的受力是歐拉力。
歐拉加速度的一般公式為
: $\mathbf {a} _{\text{Euler}}=-{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} $
其中， $\mathbf {r} $ 是物體在旋轉參考系 S' 的位置。
歐拉力為
: $\mathbf {F} _{\text{Euler}}=-m{\frac {\mathrm {d} \mathbf {\Omega } }{\mathrm {d} t}}\times \mathbf {r} $
## 幾種特殊的運動
以下為幾種特殊的運動，因為在不同的模型下質點常被不同地近似處理，並且可以得出的結論較之上面的積分式常能極大地簡省計算量，故有研究的價值。最常運用的就是拋體運動，以及自由落體。
### 勻速直線運動
若某質點保持加速度 $a=0\,\!$ ，則其速度 $\mathbf {v} \,\!$ 的大小和方向不會變化，質點將保持在同一直線上以同一速率（速度大小）運動，這種運動被稱作勻速直線運動。特殊地，若速度 $v=0\,\!$ ，則質點靜止。 | [
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加速运动 | 勻速直線運動主要出現在牛頓第一定律中，該定律表示：「不受任何力或受淨力為零的物體作勻速直線運動。」由於自然界中大部分力的隨距離增大而減小，故離所有其它物體足夠遠的某一物體的運動能夠在足夠的精度下被近似為勻速直線運動。這種近似常被用於尋找慣性參考系和粒子物理學的運算當中。
### 勻變速直線運動
若某作質點作直線運動並保持加速度 $a\,\!$ 恆定，則質點作勻變速直線運動。在這種情況下，若 $t=0\,\!$ 時刻速度為 $v_{0}\,\!$ ， $t\,\!$ 時刻速度為 $v(t)\,\!$ ，位移為 $s(x)\,\!$ ，則可由上面積分式得出
: $v(t)=v_{0}+at\,.$
: ${\begin{aligned}s(t)&=v_{0}t+{\frac {1}{2}}at^{2}\\&={\frac {v(t)+v_{0}}{2}}t\\\end{aligned}}$
以及得出
: $a={\frac {v(t)^{2}-v_{0}^{2}}{2s(t)}}$
#### 自由落體運動 重力加速度
自由落體運動是指初速度為 0，加速度恆為豎直向下
的重力加速度 g 的運動，在地球上大約有 $g=9.8\operatorname {m\cdot s^{-2}} $ 。自由落體運動是勻變速直線運動的一種特殊情況。自由落體運動是將地球上的物體下落的狀況進行理想化的抽象模型，當物體在地面附近，且所受空氣阻力遠小於其重力時，在一定精度內可被視作自由落體運動。
### 加速度恆定的運動
加速度是一個矢量，因此「加速度恆定」暗示加速度的大小和方向都不隨時間變化。
當加速度 $\mathbf {a} \,$ 與速度 $\mathbf {v} \,$ 不在同一條直線上時，選取適當的坐標系，可以將其按照平面直角坐標系分解，使質點的運動在其中一個坐標軸上的投影為勻速直線運動，另一個方向上為勻變速直線運動。根據獨立作用原理，兩者的合運動（即質點的軌跡）為一條拋物線的一部分。
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加速运动 | 拋體運動具體包括平拋運動和斜（上、下）拋運動，和自由落體運動類似，它是在地球上向不同方向拋出的物體在忽略空氣阻力的情況下的運動狀況進行理想化的抽象模型。物體擁有一個非豎直方向的不為零初速度 $\mathbf {v_{0}} \,\!$ ，和豎直向下、大小恆定為重力加速度 g 的加速度，落地前的軌跡為一條拋物線的一部分。這也正是拋物線名字的由來。
### 簡諧運動
再一個例子是簡諧運動，即質點在正弦或餘弦函數形式下的一維運動，一般形式為
: $x=A\cos(\omega t+\phi _{0})\,.$
其中， $A\,$ 為振幅， $\omega \,$ 為角頻率， $\phi _{0}\,$ 為初相位。將其對時間求導後可得出
: $v=-A\omega \sin(\omega t+\phi _{0})\,$
: $a=-A\omega ^{2}\cos(\omega t+\phi _{0})$
由此也可以得出一些有趣的結論，如在任一時刻，
: $a=-x\omega ^{2}\,$
: $A^{2}=\left({\frac {a}{\omega ^{2}}}\right)^{2}+\left({\frac {v}{\omega }}\right)^{2}\,$
## 加速度的應用
### 電磁輻射
加速度的另一個重要應用之處是帶電粒子的電磁輻射（即手機和收音機使用所需要的信號來源）。通過對馬克士威方程組的研究，可以將帶電粒子產生電磁輻射的規律概括性地定性總結為：帶電粒子的加速度產生電磁輻射，並且電磁輻射的強度和加速度大小正相關。電磁輻射常見於用帶電粒子的碰撞實驗中。這類實驗的一個早期著名例子是拉塞福用電子碰撞金箔的實驗，這個實驗導致了對原子結構的深入探索。而這類實驗至今廣泛見於在各種大型粒子對撞機中，帶電粒子以很高的速度運動，經受撞擊後變慢、靜止甚至反彈回來，這個過程中顯然速度發生劇烈改變，一定經受了加速度不為零的過程，也一定會放出輻射。這樣產生的輻射被稱為軔致輻射。 | [
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加速运动 | 加速度產生電磁輻射的另一個很典型的例子是迴旋加速器（迴旋輻射）。帶電粒子在迴旋加速器中作圓周運動，每半圈加速一次，同時運動半徑增大從而形成螺旋軌道，最後以很高的速度射出。圓周運動需要向心加速度來維持，當速度相當高時，加速度太大以至於因為電磁輻射損失的能量過多，導致迴旋加速器實際對粒子的加速作用有上限。光速可比擬時，這時因為相對論效應而需使用同步加速器。這樣產生的光能量高、偏振高，並且集中在一個很小的錐角里（相對論效應導致的前燈效應），因此是很好的大型物理用同步輻射光源。
### 狹義相對論
狹義相對論用於速度可以和光速相比擬時的運動，並且要求參考系是慣性系。在狹義相對論中，加速度的定義沒有改變。然而，由於在狹義相對論中，不同的參考系有不同的時間和空間度量標準，即當前參考系中的加速度為**當前**參考系中的位移對**當前**參考系中的時間的二階導數，因此在參考系變換（勞侖茲變換）時變得複雜很多。
設有兩個參考系 $S$ 、 $S'$ ，在空間直角坐標系中，三個坐標軸相對應平行，在 $t=t'=0$ 時刻兩坐標系原點對齊，在 $S$ 中 $S'$ 以速率 $v$ 沿 _x_ 正方向運動。
同一事件在兩個參考系中的時空坐標 $(x,y,z,t)$ 、 $(x',y',z',t')$ 變換如下：
: ${\begin{cases}x=\gamma (x'+vt)\\y=y'\\z=z'\\t=\gamma \left(t'+{\cfrac {v}{c^{2}}}x\right)\\\end{cases}}$
其中，單撇符號 $'$ 標示該物理量是在 $S'$ 下的測量， $\gamma ={\tfrac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ 是勞侖茲因子，用 $\left(u_{x},u_{y},u_{z}\right)$ 表示一質點的速度， $\left(a_{x},a_{y},a_{z}\right)$ 表示其加速度。表示定義式如下
: ${\begin{cases}{\cfrac {\mathrm {d} x}{\mathrm {d} t}}=u_{x}\\{\cfrac {\mathrm {d} u_{x}}{\mathrm {d} t}}=a_{x}\end{cases}}$
y、z 方向的定義式與之類似。綜合該定義式，利用坐標轉換的 t 部分，將坐標轉換的 x、y、z 連續兩次進行一階求導。 | [
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加速运动 | 通過展開，可以得到
: ${\begin{cases}a_{x}={\cfrac {a'_{x}}{\gamma ^{3}\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\\a_{y}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{y}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{y}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\a_{z}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{z}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{z}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\\end{cases}}$
其中， $\gamma =1/{\sqrt {1-v^{2}/c^{2}}}$ 是勞侖茲因子。
可以看出，在狹義相對論中，加速度的變換公式冗長而複雜，各分量的公式也極不相似。再加上如果要考慮到力，雖然 $\mathbf {F} =m\mathbf {a} $ 仍然成立，但質量也會變得隨參考系的不同而不同。以上原因導致加速度在牛頓力學中那種因為簡潔而具有的優越性，在狹義相對論中不復存在，必須使用更有功能的數學工具，即張量分析。
### 廣義相對論
在廣義相對論中和在量子力學中，大都是從能量、動量等的角度出發（類似於分析力學），而很少會像牛頓第二定律一樣涉及到作用力；實際上，即使在需要表示出「位移的二階導數」這一個量的時候，會趨向於直接使用 ${\ddot {x}}$ ，等價於 ${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}$ ，來求解微分方程式。因此，加速度在進階理論中較少被用到。 | [
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加速运动 | 運用到加速度的其中一個例子是等效原理，簡單地說，它敘述了觀測者不能在局部的區域內分辨出由加速度所產生的慣性力或由物體所產生的引力。比如，觀測者站在地球上靜止的電梯廂中向前方拋球，球會向下墜落，是因為地球的引力；而在遠離任何星體的宇宙中的一個電梯廂，在以重力加速度 $g$ 向上（定義觀測者踩到的地面為下）加速運動時，觀測者拋出一個球，仍然會向「下」墜落，是因為慣性力。作為在封閉電梯廂中的觀測者無法分辨這兩種情況，愛因斯坦據此提出，引力與慣性力等價。等效原理是廣義相對論中的支柱性原理之一。
## 參見
* 慣性力
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靜電力 | : 在這篇文章內，向量與純量分別用粗體與斜體顯示。例如，位置向量通常用 $\mathbf {r} \,\!$ 表示；而其大小則用 $r\,\!$ 來表示。檢驗變數或場變數的標記的後面沒有單撇號「 $'\,\!$ 」；源變數的標記的後面有單撇號「 $'\,\!$ 」。
**庫侖定律**（Coulomb's law）為法國物理學家查爾斯・庫侖於 1785 年發現的物理學定律；庫侖證明兩帶電體間有交互作用力，且其定量關係可以方程式表示。庫侖定律是電學發展史上的第一個定量規律，電學的研究從此由定性進入定量階段，是電學史上重要里程碑。
庫侖定律表明，在真空中兩個靜止點電荷之間的交互作用力，與兩電荷間距離的平方成反比，且與兩電荷電量的乘積成正比，作用力方向在它們的連線上，同號電荷相斥，異號電荷相吸。庫侖定律的標量形式可以表示為
: $F=k_{e}{qq' \over r^{2}}$ ；
其中， $F$ 是作用力， $k_{e}$ 是庫侖常數， $q$ 與 $q'$ 為兩個帶有正負號的電荷， $r$ 是兩個電荷彼此之間的距離。
在真空中，庫倫定律可以表達為
: $F={qq' \over 4\pi \varepsilon _{0}r^{2}}$ ；
其中， $\varepsilon _{0}$ 為真空的電容率。
## 發現過程及地位
早在 1760 年，丹尼爾・白努利就曾懷疑靜電的吸引行為遵循平方反比定律。
1766 年，英格蘭化學家約瑟夫・普利斯特里收到好友班傑明・富蘭克林來信告知他的一項新發現：將軟木塞球置入帶電金屬杯內部後，軟木塞球不會出現任何異樣行為。富蘭克林希望普利斯特里重複做這實驗以檢試這事實是否正確。因此，普利斯特里設計出並完成了一個實驗，該實驗顯示，帶電空心金屬容器的內部表面並未帶有任何電荷，測量不出任何靜電力。他於是在隔年發布推論，電荷之間的交互作用力具有類似於萬有引力的平方反比形式，這是因為，假若地球的形狀是一個空心球殼，則在其內部的物體不會感受到一邊的吸引力強過於另一邊地吸引力。
蘇格蘭物理學家約翰・羅比遜於 1769 年首次通過實驗直接觀測到，兩個帶電球體彼此之間作用於對方的物理行為，他發現，兩個帶電球體之間的作用力與它們之間距離的 2.06 次方成反比。很可惜的是，羅比遜並未察覺這發現的重要性。 | [
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靜電力 | 1770 年代早期，著名英國物理學家亨利・卡文迪什通過巧妙的實驗，得出了帶電體之間的作用力依賴於帶電量與距離，並得出靜電力與距離的 $2\pm {\frac {1}{50}}$ 次方成反比，只是卡文迪什沒有公布這個結果。
後來，馬克士威利用與卡文迪什類似的方法，得出靜電力與距離的 $2\pm {\frac {1}{21600}}$ 次方成反比的結果。
庫侖定律是電學的基本定律，其中平方反比關係是否精確成立尤其重要，而根據現代量子場論，靜電力的平方反比關係是與光子的靜質量是否精確為零相關的，所以，對靜電力的平方反比關係的精確驗證，關係著現代物理學基本理論的基礎。當前對庫侖定律平方反比關係的驗證越來越精確，如 1971 年進行的一次實驗，給出庫侖定律與平方反比關係的偏差小於 $2.7\times 10^{-16}$ 。
## 純量形式
庫侖定律的純量形式只描述兩個點電荷彼此交互作用的靜電力的大小。一個電量為 $q'$ 的點電荷作用於另一個電量為 $q$ 的點電荷，其靜電力 $F$ 的大小，可以用方程式表達為：
: $F=k_{\mathrm {e} }{\frac {qq'}{r^{2}}}$ ，
其中， $r$ 是兩個點電荷之間的距離， $k_{\mathrm {e} }$ 是**庫侖常數**。
庫侖常數與真空電容率的關係方程式為
: ${\begin{aligned}k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}={\frac {c_{0}^{2}\mu _{0}}{4\pi }}&=c_{0}^{2}\times 10^{-7}\ \mathrm {H\ m} ^{-1}\\&=8.987\ 551\ 787\ 368\ 176\ 4\times 10^{9}\ \mathrm {N\ m^{2}\ C} ^{-2}.\end{aligned}}$
正值的 $F$ 表示排斥力；而負值則表示牽引力。
採用國際單位制，真空電容率 $\epsilon _{0}$ 的值是 $8.854\ 187\ 817\times 10^{-12}$ F·m−1。採用厘米 - 克 - 秒制，**單位電荷**（esu），又稱為靜庫侖（statcoulomb），定義為使庫侖常數 $k_{\mathrm {e} }$ 為 1 的數值。 | [
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靜電力 | 庫侖定律的純量公式表明，力量的大小直接地與兩個點電荷的電量成正比，又與兩個點電荷之間距離的平方成反比。根據實驗數據，距離的指數，與 $-2$ 的偏差，低於十億分之一。
## 向量形式
給予兩個電量分別為 $q$ 、 $q'$ ，位置分別為 $\mathbf {r} $ 、 $\mathbf {r} '$ 的點電荷。為了要得到點電荷 $q'$ 作用於點電荷 $q$ 的力量 $\mathbf {F} $ 的大小與方向，必須使用庫侖定律的向量形式：
: $\mathbf {F} ={\cfrac {1}{4\pi \epsilon _{0}}}{\cfrac {qq'\ (\mathbf {r} -\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{3}}}$ 。
假若兩個點電荷同性（電荷的正負號相同），則其電量的乘積 $qq'$ 是正值，兩個點電荷互相排斥。反之，假若兩個點電荷異性（電荷的正負號相反），則其電量的乘積 $qq'$ 是負值，兩個點電荷互相吸引。
### 電場
根據勞侖茲力定律，
: $\mathbf {F} =q[\mathbf {E} +\mathbf {v} \times \mathbf {B} ]$ 。
其中， $\mathbf {F} $ 是勞侖茲力， $\mathbf {E} $ 是電場， $\mathbf {v} $ 是電荷的運動速度， $\mathbf {B} $ 是磁場。
假設，電荷靜止不動：
: $\mathbf {v} =0$ ，
則 $\mathbf {F} =q\mathbf {E} $ 。
所以，一個電量為 $q'$ ，位置為 $\mathbf {r} '$ 的點電荷，所產生的電場 $\mathbf {E} $ 在位置 $\mathbf {r} $ 為
: $\mathbf {E} ={1 \over 4\pi \epsilon _{0}}{\frac {q'\ (\mathbf {r} -\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{3}}}$ 。
假若電荷是正值，電場的方向是從點電荷以徑向朝外指出；假若是負值，則電場的方向是反方向。電場的單位是 V/m 或 N/C。
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靜電力 | 由 $N$ 個點電荷所組成的一個系統，其作用於一個電量為 $q$ ，位置為 $\mathbf {r} $ 的檢驗電荷的靜電力，可以用疊加原理來計算：
: $\mathbf {F} ={\cfrac {q}{4\pi \epsilon _{0}}}\sum _{i=1}^{N}{\cfrac {q_{i}'\ (\mathbf {r} -\mathbf {r} _{i}')}{|\mathbf {r} -\mathbf {r} _{i}'|^{3}}}$ ；
其中， $q_{i}'$ 和 $\mathbf {r} _{i}'$ 分別是第 $i$ 個點電荷的電量和位置。
### 連續電荷分佈
對於一個連續電荷分佈，我們可以將每一個無窮小的空間元素視為一個電量為 $dq$ 的點電荷，做無限求和。這程序等價於連續電荷分佈的區域積分。
線電荷分佈（例如，一根帶電的直線）的電量為
: $dq'=\lambda (\mathbf {r^{\prime }} )dl^{\prime }$ ；
其中， $\lambda (\mathbf {r^{\prime }} )$ 是位於 $\mathbf {r^{\prime }} $ 的線電荷密度（每單位長度所帶的電量）， $dl^{\prime }$ 是一個無窮小線元素。
表面電荷分佈（例如，兩平行金屬板電容器的一片帶電的金屬板）的電量為
: $dq'=\sigma (\mathbf {r^{\prime }} )da^{\prime }$ ；
其中， $\sigma (\mathbf {r^{\prime }} )$ 是位於 $\mathbf {r^{\prime }} $ 的面電荷密度（每單位面積所帶的電量）， $da^{\prime }$ 是一個無窮小面積元素。
體積電荷分佈（例如，一個帶電的圓球）的電量為
: $dq'=\rho (\mathbf {r^{\prime }} )d\tau ^{\prime }$ ；
其中， $\rho (\mathbf {r^{\prime }} )$ 是位於 $\mathbf {r^{\prime }} $ 的體電荷密度（每單位體積所帶的電量）， $d\tau ^{\prime }$ 是一個無窮小體積元素。
作用於一個電量為 $q$ 的檢驗電荷的靜電力 $\mathbf {F} $ ，可以表達為 | [
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0.8894335031509399,
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0.5009872317314148,
-0.07747571915388107,
-0.4256415069103241
] |
靜電力 | : $\mathbf {F} (\mathbf {r} )=q\int dq'\ {\frac {\mathbf {r} -\mathbf {r} ^{\prime }}{|\mathbf {r} -\mathbf {r} ^{\prime }|^{3}}}$ 。
其中， $\mathbf {r} $ 是檢驗電荷的位置， $dq'$ 是位於 $\mathbf {r} ^{\prime }$ 的無窮小電荷元素。
## 靜電近似
在上述兩種表述裏，只有當點電荷是處於固定狀態的時候，庫侖定律才是完全正確的；假若點電荷處於緩慢的運動狀態，則只能說庫侖定律是大概正確。這條件稱為靜電近似。當幾個點電荷處於相對運動狀態的時候，根據愛因斯坦的相對論，會有磁場產生，這連帶地改變了作用於點電荷的力量。
## 物理量表格
## 參閱
* 鏡像法
* 庫侖
* 電磁學
* 電場
* 物理學定律列表
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酵素 | **酶**（英語：Enzyme），又稱**酵素**，是一類大分子生物催化劑。酶能加快化學反應的速度（即具有催化作用）。由酶催化的反應中，反應物稱爲受質，生成的物質稱爲產物。幾乎所有細胞內的代謝過程都離不開酶。酶能大大加快這些過程中各化學反應進行的速率，使代謝產生的物質和能量能滿足生物體的需求。細胞中酶的類型對可在該細胞中發生的代謝途徑的類型起決定作用。對酶進行研究的學科稱爲酶學（enzymology）。
目前已知酶可以催化超過 5000 種生化反應。大部分酶是蛋白質，有少部分酶是具有催化活性的 RNA 分子，這些酶被稱為核酶。酶的特異性是由其獨特的三級結構決定的。
和所有的催化劑一樣，酶通過降低反應活化能加快化學反應的速率。一些酶可以將受質轉化爲產物的速率提高數百萬倍。一個比較極端的例子是乳清苷 - 5'- 磷酸脫羧酶。該酶可以使在無催化劑條件下需要進行數百萬年的化學反應在幾毫秒內完成。從化學原理上講，酶和其它所有催化劑一樣，反應不會使其物質量發生變化。酶亦不能改變化學平衡，這一點和其它催化劑也是一樣的。酶和其它催化劑的不同之處在於，它們的專一性要強得多。一些分子可以影響酶的活性。如酶抑制劑能降低酶的活性，酶活化劑能提高酶的活性。許多藥物及毒物是酶的抑制劑。當超出或小於適宜的溫度和 pH 值後，酶的活性會顯著下降。
酶在工業和人們的日常生活中的應用也非常廣泛。例如，藥廠用特定的合成酶來合成抗生素；洗衣粉中添加酶能加速附著在衣物上的蛋白質、澱粉或脂肪漬的分解；嫩肉粉中加入木瓜蛋白酶能將蛋白質分解爲稍小的分子，使肉的口感更嫩滑。
## 發現及研究史
酶的發現來源於人們對發酵機理的逐漸了解。早在 18 世紀末和 19 世紀初，人們就認識到食物在胃中被消化，用植物的提取液可以將澱粉轉化為糖，但對於其對應的機理則並不了解。
十八世紀義大利的生物學家斯巴蘭贊尼在火雞和鵝的砂囊中發現可以分解碎肉和麥粒的液體，後來法國的兩位化學家沛因與 & 貝索茲則由大麥和水磨碎的混合物中提煉出能分解澱粉的物質。
1878 年，德國生理學家威廉・屈內首次提出了酶的概念。隨後，酶被用於專指胃蛋白酶等一類非活體物質，而酵素（ferment）則被用於指由活體細胞產生的催化活性。 | [
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酵素 | 這種對酶的錯誤認識很快得到糾正。1897 年，德國科學家愛德華・比希納開始對不含細胞的酵母提取液進行發酵研究，通過在柏林洪堡大學所做的一系列實驗最終證明發酵過程並不需要完整的活細胞存在。他將其中能夠發揮發酵作用的酶命名為發酵酶。這一貢獻打開了通向現代酶學與現代生物化學的大門，其本人也因「發現無細胞發酵及相應的生化研究」而獲得了 1907 年的諾貝爾化學獎。在此之後，酶和酵素兩個概念合二為一，並依據比希納的命名方法，酶的發現者們根據其所催化的反應將它們命名。通常酶的英文名稱是在催化受質或者反應類型的名字最後加上 - ase 的後綴，而對應中文命名也採用類似方法，即在名字最後加上「酶」。例如，乳糖酶是能夠剪切乳糖的酶；DNA 聚合酶能夠催化 DNA 聚合反應。
人們在認識到酶是一類不依賴於活體細胞的物質後，下一步工作就是鑑定其生化組成成分。許多早期研究者指出，一些蛋白質與酶的催化活性相關；但包括諾貝爾獎得主里夏德・維爾施泰特在內的部分科學家認為酶不是蛋白質，他們辯稱那些蛋白質只是酶分子的攜帶者，蛋白質本身並不具有催化活性。1926 年，美國生物化學家詹姆斯・薩姆納完成了一個決定性的實驗。他首次從刀豆得到尿素酶結晶，並證明了尿素酶的蛋白質本質。其後，薩姆納在 1931 年在過氧化氫酶的研究中再次證實了酶為蛋白質。約翰・霍華德・諾思羅普和溫德爾・梅雷迪思・斯坦利通過對胃蛋白酶、胰蛋白酶和胰凝乳蛋白酶等消化性蛋白酶的研究，最終確認蛋白質可以是酶。以後陸續發現的兩千餘種酶均證明酶的化學本質是蛋白質。以上三位科學家因此獲得 1946 年度諾貝爾化學獎。
由於蛋白質可以結晶，通過 X 射線晶體學就可以對酶的三維結構進行研究。第一個獲得結構解析的酶分子是溶菌酶，一種在眼淚、唾液和蛋清中含量豐富的酶，其功能是溶解細菌外殼。溶菌酶結構由大衛・菲利浦所領導的研究組解析，並於 1965 年發表。這一成果的發表標誌著結構生物學研究的開始，高解析度的酶三維結構使得對於酶在分子水平上的工作機制的了解成為可能。
1980 年代，托馬斯・切赫和雪梨・奧爾特曼分別從四膜蟲的 rRNA 前體的加工研究和細菌的核糖核酸酶 P 複合物的研究中都發現 RNA 本身具有自我催化作用，並提出了核酶的概念。這是第一次發現蛋白質以外的具有催化活性的生物分子。
1989 年，其二人也因此獲得諾貝爾化學獎。
## 命名規則 | [
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酵素 | 酶的命名是衍生自其受質或是要催化的化學反應，在字尾會加上_-ase_。例如乳糖酶、醇去氫酶及 DNA 聚合酶。但有些化學反應可以由幾種不同的酶催化，這些酶稱為同工酶，而上述的命名法無法處理同工酶的情形。
國際生物化學與分子生物學聯盟提出了酶的命名法，也就是 EC 編號。每一個酶用一個四位數的數字表示，前面再加上 "EC"。第一位數字是酶依酶促反應的機制來分類。
依照第一位數字，可以分為以下六類：
* EC 1 氧化還原酶：催化氧化還原反應的酶類，例如乳酸去氫酶、琥珀酸去氫酶、細胞色素氧化酶、過氧化氫酶、過氧化物酶等。
* EC 2 轉移酶：轉移官能基（例如甲基或是磷酸基團）的酶類，例如甲基轉移酶、胺基轉移酶、己醣激酶、磷酸化酶等。
* EC 3 水解酶：催化受質發生水解反應的酶類，例如澱粉酶、蛋白酶、脂肪酶、磷酸酶等。
* EC 4 解離酶：用氧化及水解反應以外的方式移去基團的酶類，例如碳酸酐酶、醛縮酶、檸檬酸合酶等。
* EC 5 異構酶：催化分子同分異構反應的酶類，例如磷酸丙醣異構酶、消旋酶等。
* EC 6 連接酶：用共價鍵結合二個分子的酶類，例如麩胺醯胺合成酶、丙酮酸羧化酶等。。
上述分類會再依受質、生成物及化學結構來分類。用四位數字可以完整的描述一個酶。例如，己醣激酶（EC 2.7.1.1）是轉移酶（EC 2），會將磷酸基團（EC 7）加到六碳醣中，是一個含有醇基的分子（EC 2.7.1）。
## 結構
酶大都是球狀蛋白，以單體或聚成複合物對反應進行催化。和其他的蛋白質一樣，酶的三維結構是通過多肽鏈摺疊形成的。胺基酸的序列（一級結構）能決定蛋白質的三維結構，進而影響酶的催化活性。儘管結構決定功能是一條具普適性的規則，一種新的酶的活性不能僅僅通過其結構預測。加熱時或與化學變性劑接觸時，酶結構會發生去摺疊（即變性），原有的結構被打亂，活性也往往隨之喪失。在溫度超過正常水平時，酶就會變性。因此，不難推斷生活在火山環境（比如熱泉）中的細菌的酶具有很強的耐熱性。這些酶使高溫條件下酶促反應的發生成爲可能，在工業上具有很高的利用價值。 | [
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酵素 | 酶通常比受質大得多。酶的肽鏈長度從 62 個胺基酸殘基的 4 - 草醯巴豆酸異構酶的單體到長度超過 2,500 個胺基酸殘基的動物脂肪酸合酶。酶的結構只有一小部分（大約 2-4 個胺基酸）是直接與催化相關的。這部分稱爲催化位點。催化位點通常與一個或多個與受質結合的結合位點相連。催化位點與結合位點共同組成了酶的活性位點。酶的其餘部分起維持活性位點準確的方向以及動力學特性的作用。
在一些酶中，催化與任何一個胺基酸都沒有關係。這類酶另有與催化輔助因子結合的位點。一些酶亦可能包含異位位點。小分子與異位位點的結合可使酶發生構象改變，進而使酶的活性降低或升高。
一些具有生物催化活性的 RNA 分子稱爲核酶。這類分子可能單獨發揮催化作用，也可能在與蛋白質結合成複合物的條件下發揮催化作用。最常見的核酶應是核糖體。核糖體是蛋白質以及具有催化活性的 RNA 的複合物。核糖體的活性位點完全由 RNA 組成，而蛋白質僅起支架的作用。
## 機制
### 受質結合
酶在催化化學反應前必須要與受質結合。酶具有很強的專一性，通常僅能與寥寥數種受質結合，催化一種或幾種反應。專一性通過結合區的形狀、電荷、疏水 / 親水性與受質互補實現。因此，酶可以用來區分化學選擇性上、區域選擇性上、立體專一性上有所不同的結構相似的分子。
一些與基因組複製與表達相關的酶具有很高的專一性和準確性，表現在一些酶所具有的校對機制。以 DNA 聚合酶爲例，這類酶先催化 DNA 鏈的合成，再檢查新加上的鹼基是否正確。校對機制確保了酶的極高準確性，哺乳動物的高保真 DNA 聚合酶在每一億次反應中才會出一次錯誤。RNA 聚合酶、胺醯 - tRNA 合成酶、核糖體也有與 DNA 聚合酶類似的校對機制。
相對應地，另外一些酶則表現出多受質特異性（的現象。這類酶專一性弱，能與一系列生理上相關的受質反應。很多此類酶偶然還會出現一定程度的副反應活性，而這可能是進化出新功能酶的起點。
#### 「鑰匙和鎖」模型
爲了解釋觀察到的酶的特異性，1894 年，赫爾曼・埃米爾・費歇爾提出，酶和受質靠著互補的幾何形狀精準地結合在一起。這一理論即通常所說的「鑰匙和鎖」模型。這一早期的理論解釋了酶的專一性，但卻沒能解釋酶的過渡態爲何能穩定存在。
#### 誘導契合模型 | [
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酵素 | 1958 年，丹尼爾・科甚蘭提出了一個對鑰匙與鎖模型進行修正的理論：酶的結構相對靈活，受質與酶（活性位點）作用時，活性位點會不斷改變結構。受質不是簡單地與一個剛性的活性位點結合。組成活性位點的胺基酸側鏈的準確有序排布保證酶能執行催化功能。醣苷酶等酶，當受質分子與活性位點結合時，受質分子亦會發生輕微的形狀改變。直到受質與酶發生完全結合，分子形狀和電荷排佈都最終確定，活性位點都會不斷發生結構變化誘導契合可以通過結構校對機制在噪音和競爭物存在的條件下增強分子識別的保真度。
### 催化機理
酶可以通過多種方式加快化學反應的進行速度，基本機理都是降低反應的活化能（ΔG‡，吉布斯自由能）。
1. 穩定過度態：
* 產生一個與過度態所帶電荷互補的環境，以降低其能量。
2. 提供不同的反應途徑：
* 先和受質發生初步反應，與受質形成共價鍵連接，產生一個能量較低的中間態。
3. 降低受質基態的穩定性：
* 結合受質後引起受質扭曲，從而降低受質基態與過渡態之間的能差。
* 通過改變受質的排列方式，減少反應的熵變，此機制對催化的貢獻較小。。
酶可以同時使用以上多種催化機制來催化反應。比如，胰蛋白酶先通過一個催化三聯體進行共價催化產生中間態，再藉助氧負離子洞穩定過渡態的電荷排佈，水解過程的完成則依賴有序排列的水分子受質。
### 動態結構
酶的結構並不是剛性的、恆定不變的。酶的內部結構會發生複雜的動力學變化。酶的結構可能在反應過程中發生變化，單個胺基酸殘基、一個轉角、一個二級結構單位，乃至整個結構域的位置都可能發生改變。酶內部結構的變化能使差異度不大、且能發生互變的多種構象體在熱力學平衡狀態下共存，從而形成所謂構象系綜。在系綜狀態下，酶的每一結構狀態或者構象體都可能與酶的功能的某一部分有關。舉例來說，二氫葉酸還原酶的不同構象就分別和受質結合、催化、輔助因子釋放、產物釋放相關。
### 異位調節
別構中心是酶結構上的結合口袋，遠離活性中心，可以與細胞環境中的一些分子結合。和別構中心結合的分子往往可以改變酶的構象或是酶的動態結構，從而影響酶活性中心的反應速率。因此，別構反應可以使酶被活化或被抑制。別構調節對代謝通路非常重要，代謝酶往往可以和通路上游或是下游的代謝物發生別構效應，而產生反饋調控，即根據通路其他部分上的代謝通量來調整酶的活性。
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酵素 | 一些酶並不需要額外的組分就能就能正常發揮作用，另外一些酶則要在和輔助因子結合後才能顯示出活性。輔助因子可以是無機物（如金屬離子、鐵硫簇），也可以是有機物（比如黃素和血基質）。有機輔助因子如果在反應中會與酶分離則爲輔酶，如果與酶緊密結合則爲輔基。有機的輔基可能與酶發生共價結合（丙酮酸羧化酶與生物素之間即發生共價結合）。
碳酸酐酶即是一類含有輔助因子的酶。理察森圖顯示，碳酸酐酶的肽鏈環繞在一個鋅離子（值得注意的是，鋅離子亦是活性位點的一部分）周圍，並與這個鋅離子結合在一起這些與酶緊密結合的離子或分子通常位於活性位點之中，並且能夠參與催化反應。例如，黃素以及血紅素即會參與氧化還原反應。
需要輔助因子才可以發揮作用的酶，處於未與輔助因子狀態時稱爲「脫輔酶」（_apoenzymes_ 或 _apoproteins_），當其與輔助因子結合後則稱爲「全酶」（_holoenzyme_）。不過，值得注意的是，名詞全酶亦可以指含有多個亞基的酶，如 DNA 聚合酶。不過，在本文中，全酶是指含有所有發揮活性所需的亞基的酶。
### 輔酶
輔酶是一類與酶結合的小分子有機物，輔酶與酶結合的強度因輔酶和酶的種類而異，或強或弱。輔酶能夠將化學基團從一個酶轉移到另一個酶上。NADH、NADPH 爲兩種常見的輔酶。核黃素、硫胺、葉酸等維生素類輔酶人體不能合成，需要通過膳食補充。輔酶能攜帶化學基團，如 NAD 或 NADP 能攜帶負氫離子、ATP 能攜帶磷酸基團、輔酶 A 能攜帶乙醯基團、葉酸基團能攜帶甲醯基、次甲基，或甲基、S - 腺苷甲硫胺酸能攜帶甲基。
輔酶在酶促反應發生後化學結構會發生改變。因此，可以將看作一類普遍存在的特殊受質。許多酶都有與之匹配的輔酶，2015 年已知大約 1000 種酶使用 NADH 作爲輔酶。
輔酶通常能不斷再生，濃度能始終維持在一個恆定不變的水平上。舉例來說，NADPH 能通過磷酸戊醣途徑再生，S - 腺苷甲硫胺酸能通過甲硫胺酸腺苷轉移酶催化的反應生成。持續不斷的再生意味著總量不多的輔酶也能以很快的速度消耗。舉例來說，人類每天會更新重量與自身體重相等的 ATP。
## 熱力學 | [
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酵素 | 與其他催化劑一樣，酶並不改變反應的平衡常數，而是通過降低反應的活化能來加快反應速率（見右圖）。通常情況下，反應在酶存在或不存在的兩種條件下，其反應方向是相同的，只是前者的反應速度更快一些。但必須指出的是，在酶不存在的情況下，受質可以通過其他不受催化的「自由」反應生成不同的產物，原因是這些不同產物的形成速度更快。
酶可以連接兩個或多個反應，因此可以用一個熱力學上更容易發生的反應去「驅動」另一個熱力學上不容易發生的反應。例如，細胞常常通過 ATP 被酶水解所產生的能量來驅動其他化學反應。
酶可以同等地催化正向反應和逆向反應，而並不改變反應自身的化學平衡。例如，碳酸酐酶可以催化如下兩個可逆反應，催化哪一種反應則是依賴於反應物濃度。
: $\mathrm {CO_{2}+H_{2}O{}^{\mathrm {\quad *} }\!\!\!\!\!\!\!\!{\overrightarrow {\qquad }}H_{2}CO_{3}} $ （組織內；高 CO2 濃度下）
: $\mathrm {H_{2}CO_{3}{}^{\mathrm {\quad *} }\!\!\!\!\!\!\!\!{\overrightarrow {\qquad }}CO_{2}+H_{2}O} $ （肺中；低 CO2 濃度下）
反應式中「\*」表示「碳酸酐酶」
當然，如果反應平衡極大地趨向於某一方向，比如釋放高能量的反應，而逆反應不可能有效的發生，則此時酶實際上只催化熱力學上允許的方向，而不催化其逆反應。
## 動力學
酶動力學是研究酶結合受質能力和催化反應速率的科學。研究者通過酶反應分析法來獲得用於酶動力學分析的反應速率資料。
1902 年，維克多・亨利提出了酶動力學的定量理論；隨後該理論得到他人證實並擴展為米氏方程。亨利最大貢獻在於其首次提出酶催化反應由兩步組成：首先，受質可逆地結合到酶上，形成酶 - 受質複合物；然後，酶完成對對應化學反應的催化，並釋放生成的產物（見左圖）。 | [
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酵素 | 酶可以在一秒鐘內催化數百萬個反應。例如，乳清酸核苷 5'- 磷酸脫羧酶所催化的反應在無酶情況下，需要七千八百萬年才能將一半的受質轉化為產物；而同樣的反應過程，如果加入這種脫羧酶，則需要的時間只有 25 毫秒。酶催化速率依賴於反應條件和受質濃度。如果反應條件中存在能夠將蛋白解鏈的因素，如高溫、極端的 pH 和高的鹽濃度，都會破壞酶的活性；而提高反應體系中的受質濃度則會增加酶的活性。在酶濃度固定的情況下，隨著受質濃度的不斷升高，酶催化的反應速率也不斷加快並趨向於最大反應速率（_V_max，見右圖的飽和曲線）。出現這種現象的原因是，當反應體系中受質的濃度升高，越來越多自由狀態下的酶分子結合受質形成酶 - 受質複合物；當所有酶分子的活性位點都被受質飽和結合，即所有酶分子形成酶 - 受質複合物時，催化的反應速率達到最大。當然，_V_max 並不是酶唯一的動力學常數，要達到一定反應速率所需的受質濃度也是一個重要的動力學指標。這一動力學指標即米氏常數（_K_m），指的是達到 _V_max 值一半的反應速率所需的受質濃度（見右圖）。對於特定的受質，每一種酶都有其特徵 _K_m 值，表示受質與酶之間的結合強度（_K_m 值越低，結合越牢固，親和力越高）。另一個重要的動力學指標是 _k_cat，定義為一個酶活性位點在一秒鐘內催化受質的數量，用於表示酶催化特定受質的能力。
酶的催化效率可以用 _k_cat/_K_m 來衡量。這一表示式又被稱為特異性常數，其包含了催化反應中所有步驟的反應常數。由於特異性常數同時反映了酶對受質的親和力和催化能力，因此可以用於比較不同酶對於特定受質的
催化效率或同一種酶對於不同受質的催化效率。特異性常數的理論最大值，又稱為擴散極限，約為 108 至 109 M−1s−1；此時，酶與受質的每一次碰撞都會導致受質被催化，因此產物的生成速率不再為反應速率所主導，而分子的擴散速率起到了決定性作用。酶的這種特性被稱為「催化完美性」或「動力學完美性」。相關的酶的例子有磷酸丙醣異構酶、碳酸酐酶、乙醯膽鹼酯酶、過氧化氫酶、延胡索酸酶、β- 內醯胺酶和超氧化物歧化酶。
米氏方程是基於質量作用定律而確立的，而該定律則基於自由擴散和熱動力學驅動的碰撞這些假定。然而，由於酶 / 受質 / 產物的高濃度和相分離或者一維 / 二維分子運動，許多生化或細胞進程明顯偏離質量作用定律的假定。在這些情況下，可以應用分形米氏方程。 | [
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酵素 | 存在一些酶，它們的催化產物動力學速率甚至高於分子擴散速率，這種現象無法用目前公認的理論來解釋。有多種理論模型被提出來解釋這類現象。其中，部分情況可以用酶對受質的附加效應來解釋，即一些酶被認為可以通過雙偶極電場來捕捉受質以及將受質以正確方位擺放到催化活性位點。另一種理論模型引入了基於量子理論的穿隧效應，即質子或電子可以穿過活化能壘（就如同穿過隧道一般），但關於穿隧效應還有較多爭議。有報導發現色胺中質子存在量子穿隧效應。因此，有研究者相信在酶催化中也存在著穿隧效應，可以直接穿過反應能壘，而不是像傳統理論模型的方式通過降低能壘達到催化效果。有相關的實驗報導提出在一種醇去氫酶的催化反應中存在穿隧效應，但穿隧效應是否在酶催化反應中普遍存在並未有定論。
## 抑制作用
酶的催化活性可以被多種抑制劑所降低。
### 可逆抑制作用
可逆抑制作用的類型有多種，它們的共同特點在於抑制劑對酶活性的抑制反應具有可逆性。
#### 競爭性抑制作用
抑制劑與受質競爭結合酶的活性位點（抑制劑和受質不能同時結合到活性位點），也就意味著它們不能同時結合到酶上。對於競爭性抑制作用，催化反應的最大反應速率值沒有變，但是需要更高的受質濃度，反映在表觀 _K_m 值的增加。
#### 非競爭性抑制作用
非競爭性抑制抑制劑可以與受質同時結合到酶上，即抑制劑不結合到活性位點。酶 - 抑制劑複合物（EI）或酶 - 抑制劑 - 受質複合物（EIS）都沒有催化活性。與競爭性抑制作用相比，非競爭性抑制作用不能通過提高受質濃度來達到所需反應速度，即表觀最大反應速率 _V_max 的值變小；而同時，由於抑制劑不影響受質與酶的結合，因此 _K_m 值保持不變。
#### 反競爭性抑制作用
反競爭性抑制作用比較少見：抑制劑不能與處於自由狀態下的酶結合，而只能和酶 - 受質複合物（ES）結合，在酶反應動力學上表現為 _V_max 和 _K_m 值都變小。這種抑制作用可能發生在多亞基酶中。
#### 複合抑制作用 | [
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酵素 | 這種抑制作用與非競爭性抑制作用比較相似，區別在於 EIS 複合物殘留有部分酶的活性。在許多生物體中，這類抑制劑可以作為負反饋機制的組成部分。若一個酶體系生產了過多的產物，那麼產物就會抑制合成該產物的酶體系中第一個酶的活性，這就可以保證一旦合成足夠多的產物後，該產物的合成速率會下降或停止。受這種抑制作用調控的酶通常為多亞基酶，並具有與調控產物結合的異位結合位點。這種抑制作用的反應速率與受質濃度的關係圖不再是雙曲線形而是 S 形。
### 不可逆抑制作用
不可逆抑制劑可以與酶結合形成共價連接，而其他抑制作用中酶與抑制劑之間都是非共價結合。這種抑制作用是不可逆的，酶一旦被抑制後就無法再恢復活性狀態。這類抑制劑包括二氟甲基鳥胺酸（一種可用於治療寄生蟲導致的昏睡症的藥物）、苯甲基磺醯氟（PMSF）、青黴素和阿司匹林。這些藥物都是與酶活性位點結合併被活化，然後與活性位點處的一個或多個胺基酸殘基發生不可逆的反應形成共價連接。
### 抑制劑的用途
酶抑制劑常被用作藥物，同樣也可以被作為毒藥使用。而藥物和毒藥之間的差別通常非常小，大多數的藥物都有一定程度的毒性，正如帕拉塞爾蘇斯所言：「所有東西都有毒，沒有什麼是無毒的」（「In all things there is a poison, and there is nothing without a poison」）。相同的，抗生素和其他抗感染藥物只是特異性地對病原體而不是對宿主有毒性。
一個獲得廣泛應用的抑制劑藥物是阿司匹林，它可以抑制環加氧酶的活性，而環加氧酶可以生產炎症反應信使前列腺素，因此，阿司匹林可以起到抑制疼痛與炎症的作用。而劇毒毒藥氰化物可以通過結合細胞色素氧化酶位點處的銅和鐵原子不可逆地抑制酶活性，從而抑制細胞的呼吸作用。
## 生物學功能
在生物體內，酶發揮著非常廣泛的功能。訊息傳遞和細胞活動的調控都離不開酶，特別是激酶和磷酸酶的參與。酶也能產生運動，通過催化肌球蛋白上 ATP 的水解產生肌肉收縮，並且能夠作為細胞骨架的一部分參與運送胞內物質。一些位於細胞膜上的 ATP 酶作為離子泵參與主動運輸。一些生物體中比較奇特的功能也有酶的參與，例如螢光素酶可以為螢火蟲發光。病毒中也含有酶，或參與侵染細胞（如 HIV 整合酶和反轉錄酶），或參與病毒顆粒從宿主細胞的釋放（如流感病毒的神經胺酸酶）。 | [
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酵素 | 酶的一個非常重要的功能是參與在動物消化系統的工作。以澱粉酶和蛋白酶為代表的一些酶可以將進入消化道的大分子（澱粉和蛋白質）降解為小分子，以便於腸道吸收。澱粉不能被腸道直接吸收，而酶可以將澱粉水解為麥芽糖或更進一步水解為葡萄糖等腸道可以吸收的小分子。不同的酶分解不同的食物受質。在草食性反芻動物的消化系統中存在一些可以產生纖維素酶的細菌，纖維素酶可以分解植物細胞壁中的纖維素，從而提供可被吸收的養料。
### 代謝
多個酶以某一特定的順序發揮功能，共同構成了代謝途徑。在代謝途徑中，前一個酶的產物是後一個酶的受質；每個酶催化反應後，產物被傳遞到另一個酶。有些情況下，不同的酶可以平行地催化同一個反應，從而允許進行更為複雜的調控：比如一個酶可以以較低的活性持續地催化該反應，而另一個酶在被誘導後可以較高的活性進行催化。
酶的存在確定了整個代謝按正確的途徑進行；而一旦沒有酶的存在，代謝既不能按所需步驟進行，也無法以足夠的速度完成合成以滿足細胞的需要。實際上如果沒有酶，代謝途徑，如糖解，無法獨立進行。例如，葡萄糖可以直接與 ATP 反應使得其一個或多個碳原子被磷酸化；在沒有酶的催化時，這個反應進行得非常緩慢以致可以忽略；而一旦加入己醣激酶，在 6 位上的碳原子的磷酸化反應獲得極大加速，雖然其他碳原子的磷酸化反應也在緩慢進行，但在一段時間後檢測可以發現，絕大多數產物為葡萄糖 - 6 - 磷酸。於是每個細胞就可以通過這樣一套功能性酶來完成代謝途徑的整個反應網絡。
### 活性控制
細胞內有五種控制酶催化活性的機制：
1. 根據外界環境的變化，細胞可以增強或減弱酶的生產（即酶相關基因的轉錄和轉譯）。這屬於一種基因調控，被稱為酶的誘導和抑制。例如，當環境中出現如青黴素這樣的抗生素時，部分細菌可以對抗生素產生抗性，其原因就在於細菌體內的 β- 半乳醣苷酶被誘導而大量生產，這種酶可以水解青黴素分子上關鍵的 β- 乳胺環。另一個例子是在人體肝臟中存在一類酶對於藥物代謝非常重要的酶，細胞色素 P450；對這一類酶的誘導或抑制，會導致藥物相互作用。
2. 通過將特定的酶分隔在特定的細胞組分中，細胞可以完成不同的代謝途徑。例如，脂肪酸的合成是由細胞溶質、內質網和高基氏體中的一系列酶所完成，而脂肪酸的降解（以提供能量）是在粒線體中由另一系列酶通過 β- 氧化來完成。 | [
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酵素 | 3. 酶可以被抑制劑與活化劑所調控。例如，一個代謝途徑中的終產物常常是這一途徑中第一個酶的抑制劑，從而調控這一代謝途徑的產物量。這種調控機制被稱為負反饋機制，因為終產物的合成量是受其自身濃度調控。負反饋機制可以根據細胞的需要，有效地調節中間代謝物的合成速率，從而使細胞的能量和物質的分配更為高效，並防止多餘產物的合成。控制酶的作用，可以在生物體內維持一個穩定的內部環境（即體內平衡）。
4. 轉譯後修飾也可以調控酶的活性。這些修飾包括磷酸化、肉豆蔻酸化和醣基化。例如，細胞接受胰島素信號後，對包括肝糖合酶在內的多個酶進行磷酸化，幫助控制肝糖的合成或降解，使得細胞可以對血糖的變化產生反應。另一個轉譯後修飾的例子是多肽鏈的剪切。胰凝乳蛋白酶，一種消化性蛋白酶，是產生於胰臟中的無活性的胰凝乳蛋白酶原，這一蛋白通過運輸到達胃後才被活化。這種方式有效地防止了胰凝乳蛋白酶在進入腸之前消化胰臟或其他組織。這種無活性的酶的前體被命名為酶原。
5. 還有一些酶可以通過定位到不同環境後而被活化，比如從還原態的環境（細胞質）到氧化態環境（細胞周質空間），從高 pH 環境到低 pH 環境等。流感病毒的紅血球凝集素蛋白就是一個例子：當它接觸到宿主細胞囊泡的酸性環境時，它的構象立刻發生變化，導致其獲得活化。
### 相關疾病
酶的活性必須嚴格控制以維持體內平衡，對於能夠影響一個關鍵酶的功能的任何基因缺陷（如突變導致活性變化，過量表現、過低表現或刪除突變）都可能導致遺傳性疾病發生。許多事實顯示，一種致命疾病的病因可以只是由於人體中的數千種酶中的一種發生功能故障。
* 苯丙酮尿症：此種病症是典型的酶相關病例之一。病因是苯丙胺酸羥化酶（其功能是催化苯丙胺酸降解過程中的第一步）上一個胺基酸位點發生了突變，導致體內苯丙胺酸和相關產物的水平過高，如果沒有得到合適的治療，會進一步導致智能障礙。
* 紫質病：該病是由於血基質生物合成途徑中特定酶的酶活性過低（基因突變或其他原因導致），使得中間產物紫質的產生和排泄異常，在一定誘因（如陽光照射）下，可導致皮膚或其他組織器官發生病變。
* 當生殖細胞中編碼 DNA 修復相關酶的基因發生突變，其結果會導致遺傳性癌症綜合病徵，如著色性干皮症。DNA 修復酶的缺陷導致人體喪失修復突變基因的能力。發生的突變不斷積累，最終使得患者有多種癌症發生。 | [
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酵素 | 酶的口服給藥，可用於治療多種疾病（如胰腺功能不全和乳糖不耐受症）。由於酶作為蛋白質可能在消化道環境中失活或被降解，因此一種非侵入性的成像方法被開發用於監測作為藥物的酶在消化道中的活性變化。
## 工業應用
酶因爲能高效催化特定反應，已在化工等行業得到廣泛應用。總的來說，酶的應用因爲它們能催化的反應數目少、在有機溶劑中以及高溫環境下不穩定而受到限制。因此，酶工程這一熱門學科應運而生。酶工程旨在藉助合理的設計或體外進化的方法研發具有新特性的酶。目前，酶工程學已取得了一定成果，研究人員甚至已「從頭」（即不以任何自然界中的酶爲模板）設計出了一些能催化在自然界中不能發生的反應的酶。
## 醫藥應用
由於六大類酵素只有水解酶在人體細胞外作用，其他均在細胞內發揮功能，所以大多數口服酵素都沒有證據有生理功效或病理療效。這是因為絕大部分的酵素都是蛋白質，而易受腸胃道酸鹼值及人體本身消化道的消化酵素破壞。目前有醫療上較可靠證據的，只有屬於水解酶的鳳梨酶，且必須藉由特殊劑型設計以腸溶衣保護，才能可能發揮療效。由於經口攝入的胜肽或蛋白質，只有二肽及三肽能完整地被腸胃道吸收進入血循，所以醫藥專家仍對此存疑。
## 參見
* 酶列表
* 酶資料庫
* BRENDA
* ExPASy
* IntEnz
* KEGG
* MetaCyc
## 參考文獻
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耗散结构理论 | **耗散系統**（英語：dissipative system）是指遠離熱力學平衡狀態的開放系統，此系統和外環境交換能量、物質和熵而繼續維持平衡，對這種結構的研究，解釋了自然界許多以前無法解釋的現象。
耗散結構一詞由比利時物理學家、化學家伊里亞・普里高津發明。普里高津創立了耗散結構理論，研究一個系統從混沌無序向有序轉化的機理、條件和規律的科學，他為此曾獲 1977 年諾貝爾化學獎。
常見的耗散結構包括對流、氣旋、熱帶氣旋及生物體。像鐳射、瑞利–貝納爾對流及 B-Z 反應也是耗散結構的例子。
## 簡介
耗散結構的特點是自發對稱性破缺（各向異性）以及複雜，甚至混沌的結構。普里高津考慮的耗散結構有其動態的機制，因此可以視為熱力學上的穩態，有時也可以用適當的非平衡熱力學中的極值定理來描述。
以前的物理理論認為，只有能量最低時，系統最穩定，否則系統將消耗能量，產生熵，而使系統不穩定。耗散結構理論認為在高能量的情況下，開放系統也可以維持穩定。例如生物體，以前按照熱力學定律，是一種極不穩定的結構，不斷地產生熵而應自行解體，但實際是反而能不斷自我完善。其實生物體是一種開放結構，不斷從環境中吸收能量和物質，而向環境放出熵，因而能以破壞環境的方式保持自身系統的穩定。城市也是一種耗散結構。
牛頓的萬有引力描述的是無始無終按規律運行的美好世界，而熱力學第二定律描述的是一切終將走向滅亡的熱寂，相較之下，耗散結構描述在遠離平衡態的開放系統中「生」的機制，但其先決假定條件是存在提供能量、物質，並且可以交換熵的外環境。
## 熱力學描述
一開放系統的熵變化可以表示如下：
: $\,dS=dS_{i}+dS_{e}$
熵變化可以分解為系統內（ $\,dS_{i}$ ）及系統外的（ $\,dS_{e}$ ，和環境交換的熵）。
在封閉系統中系統無法和環境交換熵，因此（ $dS=dS_{i}$ ），根據熱力學第二定律 $dS_{i}\geq 0$ （等號成立時表示平衡），因此 $dS\geq 0$ 。
不過在開放系統中，系統可以和環境交換熵，因此可以形成一個穩態的結構，假設總熵不變 $dS=0$ ，根據熱力學第二定律 $dS_{i}\geq 0$ ，因此可得
: $\,dS_{e}=-dS_{i}<0$ （負熵流）
## 控制理論中的耗散系統 | [
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耗散结构理论 | 在系統及控制理論中，耗散系統是滿足「耗散不等式」的動力系統，假設其狀態、輸入及輸出分別為 $x(t)$ 、 $u(t)$ 及 $y(t)$ 。
假設一個函數 $w=u\cdot y$ ，其針對任何輸入 $u$ 及初始狀態 $x(0)$ ，在任意有限時間內的積分都為有限值，將此函數稱為供應率函數，則一個系統為耗散系統的條件是存在一個連續的非負函數 $V(x)$ （稱為儲存函數），使得針對任意輸入 $u$ 及初始狀態 $x(0)$ ，以下的不等式（耗散不等式）都成立：
: $V(x(t))-V(x(0))\leq \int _{0}^{t}u(\tau )\cdot y(\tau )d\tau $ ,
耗散系統的耗散不等式也可以表示為以下的形式：
: ${\frac {dV(x(t))}{dt}}\leq u(t)\cdot y(t)$
物理的解釋可將 $V(x)$ 視為是系統的能量，而 $u\cdot y$ 是單位時間輸入系統的能量。
此表示方式和李雅普諾夫穩定性有很強的關係，在系統有特定可控制性及可觀察性的條件時，儲存函數可以作為李雅普諾夫函數。
簡單來說，耗散理論可以用來設計線性及非線性系統的回授控制。耗散系統理論是由 V.M. Popov、J.C. Willems、D.J. Hill 及 P. Moylan 等學者提出。對於線性非時變系統，耗散系統即為正實轉移函數，而且 Kalman–Yakubovich–Popov 引理可以聯繫正實系統的相空間及頻域相關特性。由於耗散理論在應用上的重要性．其仍為系統及控制研究的熱門領域之一。
## 量子力學中的耗散系統
量子力學及其他以哈密頓力學為基礎的經典動態系統，具有時間可逆轉性，其本質無法描述耗散系統。理論上可以將系統進行弱耦合，以諧振子為例，可以將許多處於熱平衡，但頻率各自不同的諧振子視為一個整體，整體有很寬的頻譜，記錄整體平均的情形。會得到一個主方程，是林德布拉德方程 (Lindblad equation) 的特例，而林德布拉德方程可視為劉維爾定理的量子力學版本。
## 相關條目
## 參考資料
* Davies, Paul _The Cosmic Blueprint_ Simon & Schuster, New York 1989 (abridged— 1500 words) (abstract— 170 words) — self-organized structures. | [
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] |
耗散结构理论 | * B. Brogliato, R. Lozano, B. Maschke, O. Egeland, Dissipative Systems Analysis and Control. Theory and Applications. Springer Verlag, London, 2nd Ed., 2007.
* J.C. Willems. Dissipative dynamical systems, part I: General theory; part II: Linear systems with quadratic supply rates. Archive for Rationale mechanics Analysis, vol.45, pp. 321–393, 1972.
## 外部連結
* The dissipative systems model （頁面存檔備份，存於網際網路檔案館） The Australian National University | [
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弦理论 | **弦理論**(英語：String theory)，又稱**弦論**，是發展中理論物理學的起始，是一在量子力學及相對論、微積分等相對發展完善後，試圖透過單一解釋的系統統一物質和基本作用力的萬有理論。
弦理論雛形奠基於二十世紀中葉後半由 _Gabriele Veneziano_ 主張並提出，原始目的是找到詮釋強交互作用力之數學函數，但據此數學函數南部楊一部博士發現可將不具空間之零維點粒子視為細小的弦，進而提出強子弦模型。弦論在一段時間不備受關注，除其複雜的高維度詮釋外，量子色動力學等場論早已能完美解釋強交互作用力。而後弦論被少數科學家發現其解釋若套用到重力則可以完美解釋關於重力無法納入大統一理論中的窘境，史稱第一次弦論革命，而後第二次弦論革命解決對偶性問題，正式與 Standard Model（2012 年 7 月 4 日，_CERN LHC CMS&ATLAS_ 向量玻色子散射實驗雙盲共同發表成果發掘出的標準模型希格斯玻色子使其成為下述三大萬物理論最具權威性的理論) 及環圈量子重力場論併肩成為大統一理論備受矚目的可能性選項，其嚴謹幻妙的數學式、不需重整化的構思及對稱性讓許多物理學家徜徉於其中。
弦理論用一段段「能量弦線」作最基本單位以說明宇宙里所有微觀粒子如電子、夸克、微中子都由這一維的「能量線」所組成；換而言之，弦論主張「弦」以不同的振動模式對應到自然界的各種基本粒子。
較早時期所建立的粒子學說則是認為所有物質是由零維的點粒子所組成，也是目前廣為接受的物理模型，也很成功的解釋和預測相當多的物理現象和問題，但是此理論所根據的粒子模型卻遇到一些無法解釋的問題。比較起來，弦理論的基礎是波動模型，因此能夠避開前一種理論所遇到的問題。更深的弦理論學說不只是描述弦狀物體，還包含了點狀、薄膜狀物體，更高維度的空間，甚至平行宇宙。弦理論目前尚未能做出可以實驗驗證的準確預測。
## 發展歷史
弦理論的雛形是在 1968 年由加布里埃萊・韋內齊亞諾提出。有說法稱，他原本是要找能描述原子核內的強作用力的數學函數，然後在一本老舊的數學書裡找到了有 200 年歷史的歐拉貝他函數，這函數能夠描述他所要求解的強作用力。但根據維內奇諾本人的說法，這個函數是他多年努力的結果，而那些「偶然發現」以及「從數學書中發現」的傳言令他本人很不高興。 | [
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弦理论 | 不久後李奧納特・色斯金發現，這函數可理解為一小段類似橡皮筋那樣可扭曲抖動的有彈性的「線段」，這在日後則發展成「弦理論」。目前弦論學家普遍認為強子散射振幅公式是弦論的開端，此一公式即來自於 Γ 函數與 B 函數，描述兩個強子一開始是兩條弦，然後融合成一條，再分裂出兩條。在這些弦掃過的區域稱為世界面，可以用量子力學算這整個過程的機率振幅。
另外，同在 CERN 工作的鈴木真彥（Mahiko Suzuki）幾乎同時而又獨立地查閱相關資料，並且也發現了貝他函數，當他將該消息告訴 CERN 的一位資深物理學家後，得到的回應卻是：「另一個年輕物理學家（即韋內齊亞諾）已經在幾個星期前發現了相同的函數。」並勸鈴木不要發表他的結果。
弦論除了可以解釋強作用力，也能消除點粒子的無窮大問題。由於粒子的交互作用可以用費曼圖描述，然而粒子的交互作用點卻等同於奇異點，換句話說，它會引起無窮大的問題。雖然量子場論中的重整化理論可以解決無窮大，然而在量子的微觀尺度，卻是充滿隨機的量子漲落，結構層次的改變將使得重整化無法適用。這是因為廣義相對論中傳遞重力的介質可以視為整體時空，當時空背景為量子尺度時，結構將會不穩定，且若將量子力學的計算方式強行套用在廣義相對論則會產生限制。因此，若用弦來描述粒子交互作用的費曼圖，基本上不會產生奇異點，這是由於弦的運動軌跡是世界面。故弦論為量子重力的候選者，有望完成物理界所追求的萬有理論。
雖然弦理論最開始是要解出強作用力的作用模式，但是後來的研究則發現了所有的粒子（含反粒子），如正反夸克，正反電子（電子、正電子），正反微中子等等，以及四種基本作用力粒子（膠子、中間玻色子、光子、引力子），都能用類似方法表示成一小段的不停振動的能量弦線，而各種粒子彼此之間的差異只是這弦線的長度、振動參數和形狀的不同而已。
## 發展
### 玻色弦理論
最早期的弦論叫做玻色弦理論，南部陽一郎給予最早的作用量，但是該作用量在場論的框架內難以量子化。此後亞歷山大・泊里雅科夫給予一個等效的作用量，其幾何含義是把時空坐標視為一個世界面的純量場，並且在世界面上滿足廣義相對論的一般坐標變換規則。除此之外，如果要求這個作用量同時滿足在外爾變換下不變，那麼自然的會要求這個世界面是一個二維的曲面。 | [
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弦理论 | 玻色弦理論是最簡單的一個弦論的模型，它最重要的物理圖像是認為物理粒子不是單純的點粒子，而是由於弦的振動產生的激發態。顯然它有很大的缺點，其一是它只簡單描述純量玻色子，沒有將費米子引入框架內；其二沒有包含一般量子場論中的規範對稱性；其三是當研究它的質量譜時候發現，它的真空態是一組質量平方小於零的不穩定迅子。所有這些問題在推廣到超弦理論後得到很好的解釋。
### 開弦
1995 年，加利福尼亞大學聖塔芭芭拉分校的約瑟夫・波爾欽斯基發現弦理論一個相當晦澀的特點。他發現開放的弦的端點（開弦）在陷在某些特別的時空區域時無法完全自由地移動。波爾欽斯基隨後猜測這些特殊的空間正是被 P 膜所占據。這些「黏性」的膜就叫做狄利克雷 - P - 膜，或者 D-P - 膜。他的計算表明 D-P - 膜正是對弦端點施加的力的來源，目的是將其限制於其所存在的 P 維空間內。
### 閉弦
但不是所有的弦都屬於 P - 膜。閉弦類似於引力子，可以隨意在膜間移動。在四種力（強交互作用，弱交互作用，電磁交互作用和引力交互作用）的粒子中，引力子因此很特別。研究人員推測這或許就是為什麼對其他三種力的研究都沒有辦法找到高維空間的存在。這三種力的媒介粒子就是將它們自己限制在 P 膜裡的開弦。現階段所需要做的就是對引力子進行更詳實的研究來證明其他維度的空間的存在。
### 超弦理論
另外，「弦理論」這一用詞所指的原本包含 26 維的玻色弦理論，和加入超對稱性的超弦理論。在近日的物理界，「弦理論」 一般是專指「超弦理論」，為了方便區分，較早的「玻色弦理論」則以全名稱呼。
1990 年代，受對偶性的啓發，愛德華・維頓造了個 11 維的 M 理論，把 5 種版本的 10 維超弦理論與 11 維超重力論推演成 M 理論的 6 個形。這些發現帶動了第二次超弦革命有數百篇論文出現。
### 弦理論與大統一理論
弦理論會吸引這麼多注意，大部分的原因是因為它很有可能會成為大統一理論。弦理論也可能是量子引力的解決方案之一，含有量很大。除了引力之外，它很自然的成功描述各式作用力，包含電磁力和自然界存在的其他各種作用力。超弦理論還包含組成物質的基本粒子之一的費米子。至於弦理論能不能成功的解釋基於目前物理界已知的所有作用力和物質所組成的宇宙，這還是未知數。至今研究員仍未能找到一個弦論模型，其低能極限為標準模型。
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弦理论 | 額外維是相對於「四維時空」而提出的一個概念，一般泛指的是理論在四維時空基礎上擴展出來的其它維度。
愛因斯坦提出宇宙是空間加時間組成的「四維時空」。1926 年，德國數學物理學家西奧多・卡魯扎在四維時空上再添加一個空間維，也就是添加一個第五維，把愛因斯坦的相對論方程式加以改寫，改寫後的方程式可以把當時已知的兩種基本力即「電磁力」和「引力」很自然地統一在同一個方程式中。至此，理論中存在額外添加的維度統稱為「額外維」。超弦理論中是一維時間十維空間或十一維空間。
### D - 膜
由於超弦理論的時空維數為 10 維，所以很自然的可以認為有 6 個額外的維度需要被緊化。當對閉弦緊化時，可以發現所謂的 T 對偶；而對開弦緊化則可以發現開弦的端點是停留在這些超曲面上的，並且滿足狄利克雷邊界條件。所以這些超曲面一般被稱為「D 膜」。
研究員稱 D 膜的動力學為「矩陣理論」（M 理論），是為「M」字來源之一。
## 物理或是哲學
在未獲實驗證實之前，不能完全算是物理學。無法獲得實驗證明的原因之一是目前尚沒有人對弦理論有足夠的了解而做出正確的預測，另一個則是目前的高速粒子加速器還不夠強大。
科學家們使用目前的和正在籌備中的新一代的高速粒子加速器試圖尋找超弦理論裡主要的超對稱性學說所預測的超對稱粒子。但是就算是超粒子真的找到了，這仍不能算是可以證實弦理論的強力證據，因為那也只是找到一個本來就存在於這個宇宙的粒子而已，不過這至少表示研究方向還不是錯誤的。
## 問題與爭論
雖然歷史上，弦理論是物理學的分支之一，但仍有一些人主張，弦理論目前不可實驗的情況，意味著它應該（嚴格地說）被更多地歸為一個數學框架而非科學。一個有效的理論，必須通過實驗與觀察，並被經驗地證明。不少物理學家們主張要通過一些實驗途徑去證實弦理論。一些科學家希望藉助歐洲核子研究組織的大型強子對撞機，以獲得相應的實驗數據 —— 儘管許多人相信，任何關於量子引力的理論都需要更高數量級的能量 (弦理論預測約 1034J 的能量可以打開所有維度) 來直接探查。此外，弦理論雖然被部分物理學家認同，但它擁有非常多的等可能性的解決方案。因此，一些科學家主張弦理論或許不是可證偽的，並且沒有預言的力量。 | [
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弦理论 | 由於任何弦理論所作出的那些與其他理論都不同的預測都未經實驗證實的，該理論的正確與否尚待驗證。為了看清微粒中弦的本性所需要的能量級，要比目前實驗可達到的高出許多。弦理論具有很多在數學上很有意思的特徵（features of mathematical interest）並自然地包含了標準模型的大多數特性，比如非阿貝爾群與手徵性費米子（chiral fermions）。因為弦理論在可預知的未來可能難以被實驗證明，一些科學家 問，弦理論甚至是否應該被叫做一個科學理論。它現在還不能在波普爾的哲學含義中被證偽。但這也暗示了弦理論更多地被看做建設模型的框架。在同樣的形式中，量子場論是一個框架。
弦理論的思想為物理學帶來了一個建議上超越標準模型的巨大影響。例如，雖然超對稱性是組成弦理論的重要一部分，但是那些與弦理論沒有明顯聯繫的超對稱模型，科學家們也有研究。因此，如果超對稱性在大型強子對撞機中被偵測到，它不會被看做弦理論的一個直接證明。然而，如果超對稱性未被偵測出，由於弦理論中存在只有以更加更加高的能量才能看出超對稱性的真空，所以它的缺乏不會證明弦理論是錯誤的。相反，如果日食期間觀測到太陽的引力未使光按預測的角度偏轉，那麼愛因斯坦的廣義相對論將被證明是錯誤的。實驗證明了廣義相對論的正確性。
在更數學的層次上，另一個問題是，如同很多量子場論，弦理論的很大一部分仍然是微擾地（perturbatively）用公式表達的（即為對連續的逼近，而非一個精確的解）。雖然非微擾技術有相當大的進步 —— 包括猜測時空中滿足某些漸進性的完整定義 —— 仍然缺乏一個非微擾且充分的理論定義。
物理學中，弦理論有關應用的一個中心問題是，弦理論最好的理解背景保存著大部分從「時不變的時空」得出的的超對稱性潛在理論：目前，弦理論無法處理好時間依賴與宇宙論背景的問題。
前面提到的兩點涉及一個更深奧的問題：在弦理論目前的構想中，由於弦理論對背景的依賴 —— 它描述的是關於固定時空背景的微擾膨脹，它可能不是真正基礎的。一些人把背景獨立（background independence）看做對於一個量子重力理論的基礎要求；自從廣義相對論已經是背景獨立的以來，尤其如此。
## 參見
* 超弦理論
* 對偶性 (弦論)
* 弦場論
* M 理論
## 資料來源
## 參考出處
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弦理论 | * The Elegant Universe Online （頁面存檔備份，存於網際網路檔案館） (含 3 小時的影片)
* 弦論景觀 （1,2,3/4） | [
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白洞 | 在廣義相對論中，**白洞**（英語：White hole），是一種理論推測出來的時空區域，物質與光線無法進入這個區域中，但是可以從這個區域中向外放射。白洞的性質與黑洞相反，光與物質可以進入黑洞中，但是無法從黑洞中離開。這個白洞存在的理論最早是由伊戈爾・德米特里耶維奇・諾維科夫在 1964 年根據對史瓦西解的計算，而提出這個假設，並由尼古拉・卡爾達肖夫發展理論。目前已經有許多證據顯示黑洞存在，到現在還沒有任何證據表明白洞存在，因此白洞仍然只是一種由理論推導而出的假想星體。白洞的存在也違反熱力學定理，因為熱力學認為熵不是保持不變就是增加，但白洞會使熵減少。於 2012 年發表的一篇論文認為宇宙形成最初的大爆炸是短暫噴發的白洞。在論文中，作者還認為白洞理論可以解釋 2006 年發現的伽瑪射線暴 ——GRB 060614。假設黑洞和白洞彼此連接，在其中連接的通道叫蟲洞，這成為許多科幻小說的主題。
## 概述
## 起源
## 大霹靂／超大質量白洞
## 參見
* 時間箭頭
* 大反彈
* 黑洞
* 黑洞宇宙觀
* 共形循環宇宙學
* 暗物質
* 奇異物質
* 裸奇異點
* 負質量
* 普朗剋星
* 量子力學
* 時空
* 恆星
* 蟲洞
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奇異星 | **夸克星**（英語：Quark star）由奇異物質組成，是一種理論假設可能存在的引力緻密星體，需要更多的觀測數據及關鍵遺失環結理論推導來佐證其真實性。
實驗驗證方面，關鍵的奇異物質理論至今還是假說，至 2013 年五月為止，沒有任何可能的夸克星類型被證實或理論可以完全自洽，基礎成分「H 雙重子」亦未被尋獲，最後一組對「H 雙重子」進行搜尋實驗的是日本 KEK（高能加速器研究機構）與日本原子能研究開發機構（JAEA）的合作項目 J-PARC，目前尚未有結論。
2013 年 6 月 17 日，北京質譜儀 BES III 與日本 KEK 的 Belle 團隊在研究疑似粲夸克偶素（Charmonium）的 Y（4260）時，分別獨立發現 Zc（3900），實驗報告於美國物理通訊上發表，Zc（3900）的夸克態可能是 ccud 或是介子分子混雜態（hadron molecule），是目前跡象最明確有可能被正式認定的第一個四夸克態粒子（雙夸克反雙夸克態）。Zc（3900）如果確認成立，其意義十分重大，將正式確立多夸克態物理的成立，確認一整門新物理學的出現，多夸克態一旦成立，則夸克水平的星體均可能成立，但不見得是奇異夸克星，也有可能是混雜態夸克星或是孤子星產生機率更高，這對近代天體物理發展而言是一項很大的突破，一整個族系的多夸克態星體均有可能被列入天體物理的研究範圍內。 | [
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奇異星 | 對夸克星模型產生矛盾的現有物理實驗當中，在 2013 年 1 月，質子大小再度被確認為 0.84087 飛米，以 μ- 氫原子（Hydrogen muon）作為測量基準，置信度為 7σ，遠比使用氫原子精確許多，推翻百年以來推算的大小 0.8768 飛米，完成驗證程序，正式為物理學界承認（2010 年，德國馬克斯・普朗克量子光學研究所（MPQ）首度測量 μ- 氫原子所得數據大約為 0.8418 飛米，其後被物理學界稱為質子大小謎團）。該數值導致量子電動力學當中的一些物理常量可能必須修改，例如「里德伯常量」。質子的夸克態為 uud，質子大小修正幅度達 4%，這意味過去推導的「H 雙重子」uuddss 物態方程，在數值計算上幾乎是全面錯誤的，短距力的效應在夸克星模型當中被低估許多。由此可以確信的是現有的夸克星模型全部都是需要修正的，這包含了夸克星半徑的推算、引力緻密程度及內部能階所能產生各類衰變粒子所造成的星體穩定性問題，2013 年以前推導的夸克星模型沒有任何一個是正確的，引用新數值重新計算的工作還在進行中，尚未有相關的新論文出現。
理論發展方面，2013 年 3 月中，CERN 宣布了希格斯玻色子的能階大約在 125.3-126.0GeV 之間，如果 CERN 以外的第三方對照組實驗的數據同樣驗證此一數值（現代科學程序上要求 CERN 以外的機構重覆檢驗正確性，至少要有 CERN 以外的一個單位或多個單位進行重覆證實，CERN 的發現並非最終結論），則此一能階則表示夸克星核心將會頻繁地形成希格斯玻色子及比較強烈的真空極化效應，甚至會形成穩定的希格斯玻色子物質團，夸克星的組成將不再是單純的奇異物質團，模型還必須考慮到與希格斯玻色子的交互作用，舊有推導的夸克星模型則幾乎全面都存在錯誤。考慮到夸克星是最可能進一步坍縮成更高密度的引力緻密星體，核心當中含有高密度的希格斯玻色子應當是一個正確的物理推論結果，提供了完美解釋了進一步坍縮的成因，過往的夸克星模型通常避開此一量子效應，在希格斯玻色子能階確認以後，夸克星模型無可避免地需要進行全面修正。 | [
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奇異星 | 在質量生成貢獻度方面，希格斯玻色子一般只貢獻大約 10% 以下，90% 以上是由夸克與膠子之間的力所賦予，質子質量當中，夸克僅佔 5%，膠子不具質量，其餘質量貢獻為夸克與膠子之間的交互作用所貢獻，由於 H 雙重子尚未尋獲，無法得知其實際質量，在夸克星的密度及強引力參數下，夸克與膠子之間的交互作用對質量的貢獻比例是否會發生重大改變，成為夸克星模型當中的關鍵要素，對於其是否進一步坍縮或是維持長期結構穩定，以及星體總質量的生成因素，有關鍵性的影響，同時也全面影響夸克星的演化結構，舊有的理論物態方程均未考慮到此一因素，明顯需要進行大幅度修正。
希格斯玻色子的發現，將會使得夸克星研究成為新物理學及「巨觀宇宙結構研究」的關鍵性角色，夸克星引力及質量生成機制涉及使用廣義相對論的部份必須幾乎全面修改，物態轉換過程的進一步研究，對於證明廣義相對論是一個錯誤的物理理論有很大的幫助，目前夸克星機制的矛盾，大多數都來自於使用廣義相對論假設，假定廣義相對論存在錯誤的假設，並且採用新的量子引力延展理論，例如霍拉瓦重力或是純量不變量（Scalar invariant）系列約十餘種延展理論，在高能階區域進行修正，對於尋找正確的夸克星模型及證明「經典黑洞理論」是錯誤的天體物理理論會有很大的幫助，而正確的夸克星模型則對暗物質、巨引源、超級星系長城及巨觀宇宙結構有決定性的影響。
## 夸克星模型
以夸克水平為基礎的星體在理論模型上至少有三種，「奇異夸克星」、「孤子星」及「玻色星」。
### 奇異夸克星
「奇異夸克星」是科普文章通稱的「夸克星」，成分以奇異物質為主，主要建立在威滕假說上，專業學者文章多以「奇異星」來區分其差異，強調出其為奇異物質所組成的夸克星，由於「奇異星」有時會跟「奇特星」（Exotic star）發生混淆，而「孤子星」及「玻色星」本身都有專有名詞，故一般稱「夸克星」係指「奇異夸克星」，而專業研究者之間因為有共通語言，因此學術論文中則大多以「奇異星」來避免泛指所有類型的夸克星。
「奇異夸克星」在「希格斯玻色子」的能階確認以後，是否能夠稱為「夸克星」已經開始形成一個重要的疑問，希格斯玻色子的能階明顯導致「奇異夸克星」不是單純地由「奇異物質團」所構造成的，原有的理論需要大幅度進行修正。其次，質子大小的實驗數據，導致原有夸克星的短距力計算需要全部重新推導，新的物態方程尚未有任何研究報告發表。
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奇異星 | 「孤子星」（Soliton Star）以諾貝爾獎得主李政道所推出的「非拓樸性孤子」（Non-topological soliton, NTS）為理論基礎（拓樸性孤子的模型目前有 Skyrmion），主要是以純粹「費米子」具有孤子波性質的「孤子」來組成夸克星，被認為是「暗物質」的最佳候選者。由於宇宙間有 95% 以上的物質屬於暗物質（26.8%）或暗能量（68.3%），孤子星為暗物質的最佳候選者，「孤子星模型」則在天體物理學當中形成一大門派，在宇宙學上是非常重要的一個分支，解釋了宇宙間觀測到的質量遺失問題。
### 玻色星
「玻色星」則為以純粹玻色子來組成夸克水平的星體（複合玻色子），由於普通的星體一般是以費米子為主的重子所組成，星爆不能供應足夠的玻色子，玻色星被認為不能由星爆產生，而是由大爆炸時期所遺留下來的暗物質，或是存在於星系核當中作為「巨質量玻色星」。因為希格斯玻色子的加入，「巨質量玻色星」應該是最常見的形式，星系核在這一理論當中被認為是玻色星而非黑洞所組成的，此即為「銀河中心星系核是由暗物質所組成」的說法來源，此一說法比「銀核是由黑洞所組成」更加合理，矛盾較少，同時作為星系核的玻色星無法任意被製造出來，也是觀測當中沒有見過黑洞吸聚物質因而產生嬰兒銀河的合理解釋。玻色星的性質相當奇怪，活動模式也非常多樣化，許多人關注的黑洞、孤子星、夸克星及重力真空星的活動與玻色星相較之下可說堪稱無聊至極，由此可見玻色星具有很高的研究價值。
另外還有一些理論尚不成熟的部分類型夸克星模型推出，例如：「裸奇異星」、「混雜態夸克星」與「夸克行星」。
### 黑星
有些觀點認為，作為黑洞替代方案最佳選擇之一的黑星（Black star）：「重力真空星」，其真空極化外殼組成成分因為是透過玻色–愛因斯坦凝聚態所產生的，由於大部分天體都是由重子所組成的，而重子的成份是由夸克所組成的，因此天體坍縮後形成的重力真空星也應該是由夸克所組成的，所以「重力真空星」應該也是屬於夸克星的一種類型，不過「重力真空星」並未推導出其內部實際組成物質。 | [
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奇異星 | 此外「重力真空星」雖無奇點，但是卻有一個類似「事件視界」的「擬事界」，星體活動近似於黑洞，使得外部觀測者沒有任何手段來區分重力真空星與黑洞的差別，要透過觀測來證明其組成物質為夸克，存在巨大的技術難度，難以提供確切證據說明理論的正確性，此外理論中隱含使用了「時間量子」（chronon）的維度緊化，用以解釋緻密星的時間停滯現象與坍縮空間壁的產生過程，而「時間量子」在物理實驗中尚未被發現，因此要說服大部分天體物理學家做此歸類，恐怕還需要更多的理論推導與實驗觀測。
### 分類與實驗觀測
孤子星及玻色星經常被歸類為暗物質星，重力真空星則傾向於被歸類於暗能量星，由於人類離真正意義的宇宙航行能力相距甚遠，無法實際近距離觀測暗物質星，短期內的未來，「孤子星」、「玻色星」及「黑星」無法驗證推論是否正確。
關於「奇異夸克星」，並沒有任何報告指出科學家找到自然界中的奇異物質，目前以 LHC 擁有的 1.4×1013 電子伏特的能階而言，尚無法製造出夸克星的基本組成物質 H 雙重子（陽春版奇異物質）。目前 LHC ALICE 偵測器及日本春天八號（SPring-8）等各大重離子加速器均有此搜尋計畫，而天文觀測的數據只說明夸克星候選星可能不是中子星，而沒有指出其與夸克星性質有吻合之處。要確定一顆星體是否為夸克星，還需要非常多的努力。
在相對論性重離子對撞機、LHC、阿爾法磁譜儀及分子科學研究所等單位對奇異物質的實驗報告出爐前，輕言斷定有夸克星的存在並非恰當的舉動。
## 奇異物質假說
### 假說
正常含有的奇夸克的物質是不穩定的，奇夸克較上夸克及下夸克重，例如 Λ0 粒子 會透過弱作用力衰變成只含有上夸克及下夸克。Bodmer 及愛德華・維騰提出的奇異物質假說認為大批的夸克聚集在一起，未考慮重力作用的條件下，這些聚集的夸克由於「包立不相容原理」進入穩定態，不受此侷限，最低能階的狀態是擁有三者相同數目夸克的均能階，這個能階被稱為「奇異物質」[4]（頁面存檔備份，存於網際網路檔案館）[5] 。
粒子核基本上是由三個夸克所組成的，根據這個理論，「奇異物質」比正常的粒子核更加穩定，並且所有的粒子核都有衰變成「奇異物質」傾向，只不過這個過程有可能比宇宙年齡還久。奇異物質的穩定程度依據大小決定，太大的奇異物質會因其表面張力，傾向於變成小號的奇異物質，如果超過一定的臨界值，則奇異物質便轉變成越大越穩定的狀態。這就是「奇異物質假說」，也就是夸克星的立論基礎。 | [
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奇異星 | 這個假說並未獲得證實，導致夸克星實際上也只是一個假說。
事實上，「奇異物質」的最小號版本「H 雙重子」（有時也稱為 ΛΛ 雙重子態，S=-2,I=0,B=2,JF=0+，夸克態 udsuds 或 uuddss），是由 Robert L. Jaffe 在 1977 年開啟的系列工作所提出的，其後的研究者又提出了 D\*、N-ω、ω-ω 雙重子態及其他的更低能階多夸克穩定態。
「奇異物質」名稱的來源是羅伯特・賈菲（Robert L. Jaffe）在 1984 年的論文 "Strange Matter"，「奇異滴」（Strangelet）是由愛德華・維騰所命名，實際上兩者在專業的含意上有少許的差異，「奇異滴」代表的大批的夸克聚集在一起具有奇異性的多夸克物質。「奇異物質」與「奇異滴」是兩個不同的詞彙，不可混為一談，主要是奇異性的物理性質差異，最小版本的奇異物質「H 雙重子」，低密度及低引力的條件下，並不具備奇異性，同時壽命也很短（2 × 10-9 s）。
至今為止全球實驗室尋找超過三十年，沒有任何一個得到實驗的證實，一個「H 雙重子」都沒有找到，包含 NASA 的月球土壤樣本也沒有找到「奇異物質」，其他的雙重子態亦不曾被驗證過。目前對「奇異物質」的任何實驗並不足以產生任何危害公眾的災難性結果。
理論上來說，LHC 及 RHIC 各大重離子加速器尋找「H 雙重子」的活動沒有任何危險性，奇異性的證明需要有技術手段來凝聚大批高密度 H 雙重子，使其產生奇異物質理論預測的反應，這種實驗無法在重離子加速器上面進行，因為重離子加速器使粒子速度太高、而密度太低，重離子加速器形同對粒子加熱，而 H 雙重子的形成過程卻是需要冷凍，現有的技術能力無法在夸克禁閉突破以後，在 10-8 秒的瞬間將夸克冷凍凝結成雙重子，因而重離子加速器要製造 H 雙重子的機率非常的低，地球上所有的實驗室目前都還不具備使「奇異物質」產生奇異性的實驗條件，此外低密度 H 雙重子壽命短，保存 H 雙重子的技術手段尚未有任何發展可能，無法實際進行奇異物質奇異性的實驗證明。而且奇異性實驗的費用極度高昂，任何國家無法單一承擔所有實驗費用，在各國財政受金融因素導致極度困難的當前，短期內要證實奇異物質理論的正確性，恐怕難度甚大。
### 中子星內轉化奇異物質的反應
奇異物質一般相信是在中子星的過程當中所產生的，並且扮演主要的催化坍縮成因，幾個主體反應如下： | [
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奇異星 | * n → u + d + d（夸克禁閉突破，中子被瓦解成夸克簡並態。）
* u (1) + d → u (2) + s（非輕子弱作用過程，夸克膠子電漿中產生奇夸克相變，能階大約在 170MeV，一般稱之為「火球」，夸克星模型經常使用「火球模型」進行分析。）
* u + d + s + u + d + s + g → H0（直接六夸克反應，同位旋 I3=+1⁄2-1⁄2+0+1⁄2-1⁄2+0=0，同位旋為整數，因此是玻色子，遵守玻色愛因斯坦統計。）
或者
* u + d + s → Λ0（Λ0 重子，奇異物質均能階態的基本形式。）
* u + d + s + (g) → R baryon（超對稱 R 重子，超膠子（g）是膠子的超對稱伴子，奇異物質均能階態的基本形式，記號為 S0，僅在超對稱理論成立下才會發生，屬於 SIMP 粒子。）
* Λ0 + Λ0 → H0
此即為命名 H 雙重子的名稱來源，由兩個 Λ0 重子所組成的粒子，兩個重子所組成的一個多夸克態粒子，即雙重子態，也是當前夸克星理論是否成立的最嚴重阻礙。
此後，H0 與 Λ0 再繼續進行其他反應，進入其他更好的穩定態，一直到發生奇異性為止，便可以形成奇異物質。不過，這個過程實際上還大有細節上的各種疑問存在，各項機制不曾被仔細研究過，實驗數據無法取得，目前沒有人有能力提出真正的機制解釋，也是主要的爭議焦點之一。
低密度、低引力及聲速下的稀薄 H0 粒子並非奇異物質，並且容易發生衰變：
* H0 + γ → Λ0 + Λ0
* H0 + γ → Ξ0 + n
* H0 + γ → Ξ0 + Δ0
* H0 + γ → Ξ- + p
* H0 + γ → Ξ- + Δ+
* H0 + γ → Σ- + p
或是衰變成介子。
不穩定性使得儲存成為技術上極大的困難，低密度下形成奇異物質的機率微乎極微，聚集大批高密度 H0 粒子是一項技術上及財力上的極大困難，導致奇異物質實驗成為短期內不可能的任務。
### 實驗費用極度高昂
這個形成奇異物質反應（威滕假說）的實驗費用極度高昂，具有高度危險性，克服避免產生具有毀滅力的帶負電荷奇異物質的技術難度高，任何單一國家無法獨立進行，以其危險性而言，可能全球所有國家政府都不會批准這樣的實驗，出於安全性的考慮，最後可能只能在離地球軌道極遠的太空實驗室當中進行，除非發展出可行的廉價安全實驗方法，否則難以驗證正確性。 | [
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奇異星 | H0 粒子的實驗則單一實驗室可以進行，沒有危險性。
### 高度爭議性
天體物理學家相信這個中子星內的奇異物質發生過程是由於 H0 粒子開始進入更穩定的多夸克態，多餘的能量向外釋出會導致中子星的外殼受到內部的能量流衝擊，外部結構因為壓力嚴重失衡而瓦解，將中子星殼層全數向外拋出，產生極超新星爆發。
這個過程是許多非主流學派論述的主要爭議焦點，過程中的質量遺失問題發生過許多次明來暗去的學術戰爭，各學派幾乎都使用尚未經過驗證的理論推導，非主流的量子蟲洞學派認為產生量子蟲洞及質量傳送效應，因醜聞而不被信賴的扭旋場論則堅持發生了超光速現象導致質量遺失，而古典理論則無法妥善解釋這個問題，只是用敷衍的方式搪塞質量在爆發時全數拋出而無法觀測到，詳細而確定的發生細節，由於實驗數據的難以取得，至今尚未有定論，具有強烈的爭議性。
### 地球上的實驗方法
地球上的實驗方式並不需要直接使用上述反應式，只要可以滿足構成：
u + d + s + u + d + s + (g) → H0（實驗室目前無法製造的原因之一是目前對膠子沒有技術能力控制，因此看似簡單的反應，實驗數十年卻無法達成。）
即可。
任何可以供應這樣反應的連鎖反應都可以進行這樣的實驗，沒有一定非要用中子來進行實驗。
重離子加速器實驗數十年無法找到的原因，也源自於達成這個反應所需要的實驗條件難度很高，三十六年來使用各種可以想像的手段，大型實驗搜尋活動超過三十多次，實驗報告超過七千次，均沒有辦法克服困難成功達成製造 H0 粒子，主要因素是目前對於膠子實驗控制的技術能力還在起步階段，沒有辦法順利將膠子與夸克順利連接成 H0 粒子，同時重離子加速器因為必須將粒子加速，粒子高速運動下，使得突破夸克禁閉後的瞬時密度無法提高到創造六個夸克瞬間集結成一個粒子的密度條件。
Λ0 + Λ0 → H0（目前的主要實驗方法）
大部分實驗使用將 Λ0 重子加速來進行實驗，目前理論預測的主要穩定島集中在 H (2220)，h (2250)，實驗數據已經排除 2.202GeV 以下的可能性。
另一個主要的實驗構思是使用零號元素（Neutrium），或者是稱為四中子（Tetraneutrons）的物質，或是更進一步使用多中子物質（Polyneutron）。H0 粒子無法儲存，因而不可能對奇異物質進行實驗，但多中子物質卻還有機會及技術能力來達成，透過瞬間高密度高能雷射加壓產生局部的中子星內環境，達成下述反應： | [
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奇異星 | * 4n (Neutrium) → 4u + 8d
* 4u(1) + 4d → 4u(2) + 4s
* 4u + 4d + 4s → 4Λ0
* 2Λ0 + 2Λ0 → 2H0
* nH0 → S2n（Strangelet，奇異滴反應）
使得一個零號元素變成兩個 H0 粒子，然後再創造高密度加壓環境使 H0 粒子進入更穩定的多夸克態直到轉變成奇異物質。非理論主流封閉而不對外發表論文的量子蟲洞學派曾經進行過類似的實驗，以低溫玻色愛因斯坦凝聚態進行高密度高能雷射加壓，試圖產生量子蟲洞，透過非正式管道流出的非公開實驗結果說明這種方法可能因為需要突破夸克禁閉，而導致場勢的能階提昇而無法進入穩定態（該實驗因資金不足無法達成精度及指向性而最終宣告探測失敗）。
nH0 → S2n 奇異滴反應如果是連鎖反應，則是個極端危險的實驗，學術研究如果確定其發生可能性後，應當禁止此項實驗於地球上進行。
第三系列主要的方法是使用 B 介子或 K 介子進行合成實驗。
實驗室一般都不採用直接偵測 H0 粒子的方法，而是採取偵測 H0 粒子衰變後的粒子散射來進行偵測，例如：
H0 → Σ- + p → n + π- + p
或是
H0 → Λ + π- + p
連鎖反應，由於鮑立不相容原理，ΣN 的反應傾向大於 Λn。
另一種搜尋方法曾經被使用過，197Au + 197Au 對撞，這個對撞理論上產生三個可能的連鎖反應序列：
* Λ + Λ → H0
* Σ + Σ → H0
* Ξ + N → H0
### 系統複雜度變因多
上述反應式目前來說，還存在許多爭議性，例如：
* n → u + d + d
* u(1) + d → u(2) + s
沒有任何物理學家會相信這個反應在數量上是完美均衡的，也就是說這個過程實際上不可能只產生 H0 粒子，還會產生「孤子態」、其他「非奇異性雙重子」及「介子」，純粹以奇異物質物態方程構築的奇異夸克星是不可能存在的，包含了其他物質狀態方程，並且考慮了量子重力修正的夸克星才是正確的夸克星模型。
不均衡導致的結果之一如下：
4n → 4u + 8d
3u(1) + 3d → 3u(2) + 3s
3u + 3d + 3s → 3Λ0
Λ0 + Λ0 → H0 | [
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奇異星 | 最後形成：Neutrium → H0 + Λ0 + n，實際上這是最有可能發生的狀況之一，中子星演化成孤子星的可能性也是存在的，中子星物理學當中稱此為「混和氣態中子星」。
這導致「血統純正的奇異夸克星」實際上實驗天體物理學家是不可能相信它可以真實存在，該反應的系統複雜度大幅度增加，星體穩定度的計算難以評估，理論天體物理學家因為推導上的便利性，閱讀者喜歡看優雅的物理公式，完全一廂情願地相信有可能發生，並且盡量避免討論「血統不純正的奇異夸克星」，而傾向於讓推導方程看似完美無暇，而實驗天體物理學家堅持不可能，推導的物態方程繁複而難懂，模式繁多而難以理解，機理錯綜複雜，關連的反應式參數超過上千個，連專業的專家同行都不見得看的懂，處處顧忌、處處疑問、處處懷疑，而理論物理學家通常喜歡只使用低於二十個關連的反應式參數，經常將實驗上生成率低於 10-8 的參數設定為 100%，讓學術論文容易在具權威性期刊上發表，引來實驗天體物理學家的猛烈抨擊，實驗天體物理學家的論文通常極度難懂而篇幅極長，審稿者沒有能力斷定正確性，因而在權威性期刊不常出現，這導致兩方意見完全相左，爭吵長年不斷，也就使得夸克星的真實存在性再添一筆可長期爭吵的項目。
### 目前並未發現禁止雙重子態成立的物理機制
雖然一直有物理學家懷疑存在不明因素禁止雙重子態發生，導致無法發現 H0 粒子（不可能發現一個不會存在的粒子），目前並未發現禁止雙重子態成立的物理機制，但是卻也一直無法成功製造出任何的雙重子態，目前在高能物理上，依然還是個懸案。
高能物理學家一直保持樂觀認為雙重子態成立，並且持續搜尋多夸克態粒子的存在，主要除了理論允許以外，未發現任何物理機制禁止雙重子態成立也是一個重要因素。
### 中子星演化原則上支持奇異物質假說成立
中子星演化的五種可能性，理論原則上都支持奇異物質假說成立。
中子星→夸克星
中子星→重力真空星
中子星→先子球 （假設先子理論成立）
中子星→模糊球
中子星→黑洞
中子星→先子星→黑洞
上述五種中子星的可能演化途徑均支持奇異物質假說成立。 | [
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奇異星 | 中子星如果可以演化成夸克星，則奇異物質假說必定成立。中子星如果可以演化成重力真空星，則真空極化的現象的出現，密度的條件要求，亦需要奇異物質假說必定成立。中子星演化成先子星所需要的密度，除了奇異物質理論及非拓樸性孤子理論以外，目前沒有任何其他的可行理論選擇。中子星演化成模糊球，及中子星演化成黑洞，或是先透過演化成先子星再演變成黑洞，其中所需要的真空極化產生空間壁，進而導致引力坍縮，都需要透過先產生奇異物質所擁有的密度來建立所需要的場。
### 具有可見的科學及經濟上的高度價值
對中子星機理更加深入地研究，對夸克星機制的了解有很大的幫助，許多在夸克星研究上遭遇重大阻礙的學者，多數再度轉回中子星做更深入的多夸克態機制研究，這樣的研究路徑對現行物理層次有極大的提升，目前也吸引了許多優秀科學家投入，而 H0 粒子本身理論上也在材料科學及高效能引擎上具有巨大潛力。
六夸克態總計有大約三百萬種（2,985,984）可能的粒子束縛態，去除掉組合狀態，依然有高達數十萬種可能的雙重子束縛態，對新物質材料及新科技而言是個接近無盡的可發揮空間，實驗設計延伸的應用技術，在經濟潛力上可能遠勝於對希格氏玻色子的投入，而材料本身則在經濟上高度可能具有重大價值，也是當前各國為何投入研究多夸克態物理的重要原因之一，研究深度是各國在科學及技術上的重要指標，是一個可以看得到未來的重要研究課題，也是目前許多國家支持的重點研究方向。
### H 雙重子實驗直接影響夸克星理論是否成立
H 雙重子（H dibaryon）除了對多夸克態物理十分重要以外，實驗數據也直接影響各種夸克星理論是否成立，這包含物態方程會直接影響夸克星半徑上限的計算，各種效應在哪個能階出現等等。
H 雙重子及六夸克態的實驗只要有實驗設計能力就可以進行，許多小型實驗室都可以獨立進行實驗，大部分並沒有危險性，目前的實驗數據都表明六夸克態高度可能成立。
奇異物質實驗，除了有高危險性以外，目前則所有大型實驗室都不具備條件進行實驗，實驗的困難性，包含了製造、儲存並聚集雙重子，製造高密度環境產生奇異性，分離帶電負電荷奇異物質，隔離並且將之拋離地球的設備，這些條件都遠遠脫離現行技術能力之外，短期內進行這樣的實驗，可能性很低。理論物理學界可能要長期忍耐奇異物質物性實驗報告數據缺乏下，對相關理論進行猜測並推論，並且忍受實驗數據缺乏下的所造成的見解爭議。 | [
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奇異星 | 現行對夸克星及奇異物質的所有說法，在 H 雙重子實驗數據出現之前，可信度必須保持一定程度的懷疑，如果成功尋獲後，實驗數據會導致幾乎所有的相關理論都必須依照數據進行修正。因此，對目前所有相關研究宣稱，均可採取質疑或不相信態度。
## 形成機制
### 前身星
假定這項奇異物質物質的理論是正確的，那麼夸克星的生成來源就可以直接由恆星二次星爆生成或是由中子星演化而成。具備這種條件的星體，基本上以能夠發展成沃爾夫 - 拉葉星為主，某些特定條件下，赫羅圖上其他種類的星體也能發展成為**夸克星**，並不一定需要經歷中子星的過程。是否形成夸克星的決定關鍵，在於星爆時的恆星動力學條件，巨質量的恆星亦可能形成夸克星。
### 由中子星發展成夸克星
由中子星活動對內核所造成的瞬間壓力增大，例如伽瑪射線爆或星震，使得內核部位開始產生奇異物質，並且中心密度開始增大，活動時間到達一定臨界時，使得內核到達再度坍縮的臨界，此時中子星會開始發生坍縮的活動。
### 由沃爾夫 - 拉葉星發展成夸克星
沃爾夫 - 拉葉星是一種快速演化、壽命很短的大質量恆星，質量都超過太陽質量的 20 倍，會因為星爆而產生超新星或極超新星爆發，爆發的結果會完全粉碎、形成引力緻密星或是星雲。
### 由高光度藍變星直接發展成夸克星
**夸克星**也有可能從巨質量高光度藍變星發生連續超新星爆炸時，由於中子星核形成時，超過臨界點，但沒有形成黑洞，而在形成中子星的瞬間，數十分鐘內，再次發展成更大的極超新星爆發，從而直接從恆星演化成**夸克星**。高光度藍變星亦可能爆發力量太強，形成完全毀滅的狀況，甚至連黑洞都沒有留下，只留下星雲。
### 由紅超巨星直接發展成夸克星
質量稍低的紅超巨星會變成 II 型超新星，而質量稍大的紅超巨星則會變成沃爾夫 - 拉葉星，兩者均有機會再發展成夸克星。
### II 型超新星爆發展成夸克星 | [
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奇異星 | 核塌縮超新星的質量，質量至少是太陽質量的 9 倍，SN 1987A 正是 II 型超新星，因此它在 1989-1990 年間，曾經以高速旋轉中子星被誤認為正式的夸克星（Kristian 等人），並在「自然」雜誌上面發表，後來發表該論文的同一組科學家承認錯誤，並不存在高速旋轉中子星，並將之修正為未知星體（1991 年提出修正，撤回原有宣稱，該訊號是儀器的寄生信號，但發生錯誤的兩年當中卻意外地大幅度推動夸克星的研究成果，一個美麗的錯誤），這一錯誤幾乎延續兩年被認為是正確的。亞洲方面，因誤傳認為是 2009 年由亞洲學者所首先提出，實際上該名學者已經於隔年自行提出修正，相同的錯誤在二十年前就發生過，而在 2000 年，Kristian 等人再次提出 SN 1987A 高速旋轉中子星的候選訊號（2.14 毫秒），也就是說 SN 1987A 是夸克星並非不可能，只是技術能力還不足而已。SN 1987A 在 1989 年就曾經發生過天文學家咖啡喝太多興奮過度所導致的集體失眠誤判，而該星體的集體失誤，由於天體物理學家的期待與幻想，並不是只有被誤認為夸克星一項。
### 由藍超巨星直接發展成夸克星
藍超巨星一般質量是 10-50 個太陽，表面溫度為 20,000－50,000°C。SN 1987A 的前身星是一顆藍超巨星，當 SN 1987A 爆發時，傳統上認為只有紅超巨星才會發生超新星爆發的觀點改變，其後修正了許多的恆星模型來說明 SN 1987A 的現象。由於，沒有發現預期的星體，目前普遍認為它發展成為夸克星或黑洞，一般認為前者的可能性比較高，不過也並未排除形成孤子星的可能性。假設它發展成夸克星，則其發生機制應該是發生了連續兩次的超新星爆發，第一次坍縮時，形成近似中子星的天體，假設奇異物質假說是正確的，由於形成超過臨界點的巨型奇異物質球類物體，在產生第一次坍縮的瞬間，不斷吸收夸克，在巨型奇異夸克周圍則吸引了奇異物質團，當巨型奇異夸克到達引力發生二度坍縮的臨界，發生奇異物質團的拋出星爆。
## 物理機制
**夸克星**並非巨無霸版本的中子，事實上它比較像是一顆巨碩的強子，傳統理論基本上同步受「量子色動力學」及「引力」的作用，也就是「量子重力理論」，然而目前物理學的進展，並未實際有能力探索到這個等級的物理，量子重力效應是否真實存在依然是個物理學上的疑問，因此所有關於夸克星的說法可信度極有爭議性。是否為巨無霸粒子天體，也是個還有爭議的主要課題。
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奇異星 | 蘭斯 - 蒂林效應圈（Ergosphere，又稱 Frame Dragging 或是 Lense Thirring Effect），轉動狀態的質量會對其周圍的時空產生拖拽的現象，這種現象被稱作參考系拖拽，延展的理論為「引力磁性」。
夸克星是緻密天體，而極超新星爆發後，殘餘的角動量，將使得夸克星是一個轉動的緻密天體，一般而言是高速旋轉。夸克星如果形成，周圍必然會形成蘭斯 - 蒂林效應圈，也就是南北極與赤道在時空效應上有所不同，由於夸克星密度高於中子星，這會產生一些奇妙的效應來讓我們有機會斷定其實實在在是一顆夸克星的特徵。要斷定中子星、夸克星及黑洞的分別，正常的夸克星多數是高速旋轉的，檢定蘭斯 - 蒂林效應圈是一個重要的手段，這三者的參考系拖拽效應有一定程度的區別。
觀測者可以利用光圈效應及蘭斯 - 蒂林效應圈，觀測進入或脫離夸克星的光子的運動，透過間接的手段，例如粒子含量的分佈及潘洛斯過程（旋轉黑洞的能量拉出過程，這個效應在夸克星當中是適用的），來間接了解其重力的分佈，透過重力的分佈重新建立出其蘭斯 - 蒂林效應圈，夸克星的轉動速度透過其他方法可以取得，透過這種方法，可以區分出該星體到底應該是中子星、夸克星或黑洞。只有雙星以上的系統才能夠進行這樣的觀測。
### 袋模型的推論
MIT 袋模型是目前用以推導夸克星的最主要基礎流行理論。無袋模型修正下，夸克星的模型顯示出與觀測數據差距甚大，模型與觀測的吻合程度並不一致。
對夸克星有效的分析方法目前除去「袋模型」，還有半古典修正。
#### 星體半徑的計算
透過托爾曼－奧本海默－沃爾科夫方程式（Tolman-Oppenheimer-Volkoff limit）來計算，中子星的半徑與質量的關係是三次方成反比，考慮以上夸克及下夸克為主的中子所造成的各類反應作用力所造成的結果。夸克星則相反，半徑與質量的關係是三次方成正比，考慮以上夸克、下夸克及奇夸克的超核系統為主所造成的各類反應及相關作用力的結果，這個推論並未考慮量子引力效應，因此只能被視為是理論推導的結果，目前還無法透過觀測來驗證。
這個推論目前已經開始發生爭議，特別是狀態方程的適用性問題，幾種新的說法開始出現，包含夸克星的半徑上限比中子星半徑上限大，不過都沒有實驗與觀測數據來證明，可信度都不高，特別是奇異物質物性根本並沒有任何檢驗報告，也就導致說法大都屬於理論猜測。 | [
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奇異星 | 夸克星的直徑估算上限是 10,000 公尺以內，一般應當是大約 4,000－7,000 公尺，加上大氣的因素，觀測上大約小於 15,000 公尺的中子星就可以考慮是否其為夸克星候選星。考慮到大小及距離，實驗與觀測數據難以取得。
2013 年 1 月，使用 μ- 氫原子對質子大小測定的實驗，導致 2013 年以前推導的夸克星模型物態方程必須全部重新計算，上述數值係由舊理論所推導而得，正確性是必須懷疑的。由於質子大小對量子電動力學的修正尚未完成，理論物理學界還無法找到正確的推導，新的夸克星半徑估算亦未被提出。
#### 希格斯玻色子對夸克星的穩定性效應
希格斯玻色子的出現對於夸克星的穩定結構有重要的影響。
H 雙重子的能階大約在 2.220GeV 到 2.250GeV 左右，而希格斯玻色子的能階大約在 125.3GeV 到 126GeV 左右。當 H 雙重子邁向更低能階的奇異物質團演化時，更多的能量被釋放出來，夸克星內部某些密度更高的臨界區域則能持續到達 125.3GeV 的穩定能量流釋放（126 / 2.25 = 56.0，量級僅相差二，既意味發生機率極高，簡單的估計方式，夸克星內部至少有 1.785% 的區域能階持續處於 126GeV 左右或以上，忽略希格斯玻色子量子效應的夸克星模型不可能是正確的夸克星模型），當這種情形發生的時候，大批的希格斯玻色子則被撞擊而成為粒子型態，這造成至少三種主要量子效應。
第一個量子效應是由於能量因為希格斯玻色子的出現而被吸收掉，使得夸克星不會因為區域能量流過高而立即被瓦解掉，因此可以預期夸克星的壽命比原有推算的模型更長，而能量流被吸收時，則應當會釋放伽瑪射線爆。
第二個量子效應是由於希格斯玻色子的出現，原有的奇異物質團的結構造成破壞，原有的理論是整個夸克星都是奇異物質團，但是由於希格斯玻色子的出現，這個論點目前則不成立，可能性較高的是夸克星是一團一團的奇異物質團所組成的，而非單一奇異物質團所組成，此即為奇異物質團穩定性的破壞。以此效應而言，基本上可以否決夸克星是個巨型粒子天體的說法，同時也可以否決掉夸克星密度均勻說。 | [
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