藍色情懷

玄之又玄話弦論：SA專訪弦論明星格林恩（Brian Greene） - 導讀：弦論是什麼？ 作者╱高涌泉 弦論是物理界的當紅炸子雞？還是讓人眼冒金星的玄學？現在，SA為你專訪弦論明星格林恩談談弦論未來可能的發展。而在這之前，我們也先一起來了解弦論是什麼？ 　　弦論（string theory）是一種量子重力論，也就是一種能兼顧微觀量子概念與重力現象的理論。由於欠缺實驗證據，目前還沒有任何量子重力論，是大家公認正確的理論， 不過很多人看好弦論最後會脫穎而出。在介紹弦論為什麼最被看好之前，我先解釋一下建構量子重力論的困難之處。 　　重力是自然界已知的四種力之一。 物理學家已經能夠非常成功地將重力之外的其他三種力──電磁力、弱力（更精確地講，電磁力已經與弱力統一成電弱力）、強力──納入一般量子場論架構中。例 如，量子電動力學（電弱理論的一部份）能精準地描述微觀電磁效應，其預測與實驗對比可以準確至10億分之一，描述強交互作用的量子色動力學則也相當準確。 所以四種基本力中只剩下重力尚未被量子規律馴化。 　　既然量子場論可以成功地處理重力以外的基本作用力，它也就自然成了追尋量子重力論的最佳參考 對象。無論是量子電動力學、量子色動力學或是任何其他場論，它們的基本理念大約都是：平坦的閔考斯基四維時空（即三維空間加上一維時間）是舞台，各式各樣 的點粒子（沒有大小的粒子）是演員；舞台是靜止的，而演員可以在舞台上四處遊走，也可以從舞台上消失或忽然現身；它們受到的唯一限制就是得遵循量子力學的 規範。以量子電動力學為例：舞台上的演員是帶電的各式粒子（如電子、質子等）以及不帶電的光子。帶電粒子會吸收或釋放光子，所以光子（精準點說，是虛光 子）可以在兩粒子之間傳遞；帶電粒子間的電磁交互作用就可以解釋成粒子在交換光子。同樣地，在量子色動力學裡，夸克帶有色荷（color charge，一種類似電荷的概念，代表強交互作用的強度），可以吸收或釋放膠子。夸克間的強交互作用力就來自夸克交換膠子。 　　如果以量子電動 力學為模範，重力現象應該可以解釋成有質量（即能量）的粒子在交換重力子：重力靠重力子傳遞，就好像電磁交互作用靠光子傳遞、強交互作用靠膠子傳遞、弱交 互作用靠W粒子傳遞。所以重力子和光子、膠子、W粒子一樣，都是傳遞子；但是重力子所帶的自旋角動量是其他傳遞子自旋角動量的兩倍，這項差別造成重力只會 是吸引力，而其他的力可以是吸力也可以是斥力。 　　建構量子重力論最簡單的方式，就是假設重力子遵循愛因斯坦廣義相對論方程式（就好像光子遵循馬 克士威方程式），然後將重力子放入基本粒子成員行列，再根據一般場論的規則（例如費曼圖），來計算微觀世界的重力效應。這麼直截了當的做法起碼已經有40 年以上的歷史了，但是這種進路卻無法通過一關嚴苛的考驗：在這樣的量子重力論中，會出現無法處理的無窮大，因此理論沒有能力計算出有意義的物理量。反之， 無窮大雖然也會出現在量子電動力學或量子色動力學裡，但它們的程度較輕微，可以被處理掉（隱藏起來），所以理論保留了預測的能力；我們稱這樣的理論為「可 重整理論」。量子重力理論無法對付無窮大，因此就是所謂的「不可重整理論」。在1980年代之前，量子重力論陣營的主要工作就是尋找「可重整化」的量子重 力論，但是一無所獲。 　　弦論是長久以來頭一個公認可以對付無窮大問題的量子重力論。一般相信，無窮大根本就不會出現在弦論裡，原因在於弦論有個非常奇特的假設：所有的基本粒子並不是沒有大小的點粒子，而是一段段、一圈圈的開弦或閉弦（參見右頁圖左）。這些弦非常小，大約只有10-33公 分，所以在一般尺度下看起來就像是一個點。弦有各式各樣的振動模式，不同的模式就對應到不同的基本粒子。兩段弦可以結合起來，一段弦也可以斷裂開來，這些 行為就對應到粒子間的交互作用（參見上圖右）。為什麼這樣的弦論是一種量子重力論？因為一但我們把量子法則施加在弦上頭，就會發現弦有一個特殊的振動態， 這個振動態沒有質量，它的自旋角動量，也剛好是光子自旋角動量的兩倍，所以就是重力子；包含了重力子的理論當然也就把重力作用（也就是愛因斯坦廣義相對 論）包括在內。除了重力子之外，弦論也包含了所有其他基本粒子。所以弦論不僅是一種量子重力論，而是一種涵蓋一切基本作用的統一理論。 　　從歷史 的角度看，弦論有很特殊的發展歷程：它在剛提出的時候其實是被當成一種描述強交互作用的理論，但是後來一方面因為量子色動力學出現，二來它包含了重力子， 所以人們就不再把弦論當成失敗的強交互作用理論，轉而將它看待成是最終理論。這種敗部復活的「蛻變」，的確史無前例。 　　弦論的數學結構很緊密， 所以有兩項很特別的推論：一、只有包含超對稱（這種對稱指的是物理定律在費米子與玻色子互換下仍維持不變）的弦論才沒有內在矛盾，否則會出現帶有負質量的 「迅子」（tachyon，速度超過光速的粒子）。所以一般談論到弦論就直稱「超弦」（超對稱弦）。二、超弦要求時空的維度必得是十維，也就是時空有九維 空間與一維時間。既然巨觀世界的時空是四維的，如果超弦是正確的，所餘下的六維空間就必然非常微小，以致平常是觀察不到的。由於弦論與超對稱的密切關係， 如果將來高能實驗找到超對稱存在的證據，雖然我們還是不能由此百分之百地肯定超弦的正確性，但是弦論的身價必然提升很多。 　　總體言之，單就目前 所知，弦論的優點首要在於它是沒有數學矛盾的量子重力論，這是非常難能可貴的，其他競爭的理論或多或少都有些缺點。其次，它還能把重力與其他作用力結合起 來，是真正的大一統理論，所以它能夠用一般的規範場論來理解重力；反之，它也可以讓我們用重力來解決困難的規範場論問題。另外，由於它擁有很豐富曲折的數 學架構，所以也促成很多數學上的發展。 　　至於弦論的缺點，首先當然是至目前止它還沒有任何實驗上的證據；此外，它除了超對稱與額外空間等概念 外，還提不出什麼精確而可以檢驗的預測，所以比較慎重的物理學家就認為，現在的弦論其實還談不上是個科學理論。就理論架構而言，弦論的缺點在於它還不是 「背景獨立」的理論，也就是說它和一般場論一樣，仍然先得把平坦的閔考斯基四維時空當成背景時空，才能開始發展理論。理想上，我們不應先預設任何特定的時 空，而讓理論來決定時空的幾何性質。所以弦論仍然有很長的路要走。 　　美國哥倫比亞大學物理系教授格林恩（Brian Greene）是弦論大將之一，他是英國牛津大學物理學博士，對於弦論中的數學問題有重要貢獻。1999年，他出版了《優雅的宇宙》（The Elegant Universe）一書，公認是目前最佳介紹弦論的科普作品，因而成為最著名的弦論代言人。今年11月，美國公共電視網推出了一個三小時介紹弦論的節目， 他當然成為主持人的不二人選。如果你想觀賞這個節目，可以到http://www.pbs.org/wgbh/nova/elegant/；而如果你想看看Scientific American對他的專訪，就請你再翻到下一頁。

弦論明星，為你現身說「弦」 弦論往哪裡去？   - SA專訪弦論明星格林恩（Brian Greene） 撰文╱馬瑟（George Musser），SA編輯 翻譯／高湧泉，台大物理系教授

過 去一談到弦論，就讓人們腦筋打結，即使是弦論專家也會為它的複雜感到苦惱；至於其他物理學家，就在一旁嘲笑弦論拿不出任何實驗上的預測；而一般人則大致上 對弦論一無所知。科學家無法傳達究竟為什麼弦論那麼刺激：它為什麼可以實現愛因斯坦追求最終統一理論的夢想？對於「宇宙為什麼存在」這類深奧的問題，它又 是如何幫助我們理解的？但是到了1990年代中期，理論開始在觀念上統合在一起，而且一些可以檢驗、卻還不精確的預測也出現了。弦論開始引起外界的注意， 今年6月，就連伍迪艾倫也在《紐約客》週刊的專欄以弦論為諷刺題材──也許這是頭一回有人用「卡拉比–丘（成桐）空間」來談論辦公室戀情。

　　String theory used to get everyone all tied up in knots. Even its practitioners fretted about how complicated it was, while other physicists mocked its lack of experimental predictions. The rest of the world was largely oblivious. Scientists could scarcely communicate just why string theory was so exciting—why it could fulfill Albert Einstein's dream of the ultimate unified theory, how it could give insight into such deep questions as why the universe exists at all. But in the mid-1990s the theory started to click together conceptually. It made some testable, if qualified, predictions. The outside world began to pay attention. Woody Allen satirized the theory in a New Yorker column this past July—probably the first time anyone has used Calabi-Yau spaces to make a point about interoffice romance.

談 起掀開弦論神秘面紗的功勞，很少人能比得上格林恩；他是哥倫比亞大學物理教授，同時也是弦論大將之一。格林恩在1999年出版的《優雅的宇宙》，在《紐約 時報》暢銷書排行榜曾位居第四，而且也是普立茲獎入圍作品。他是美國公共電視網Nova系列專輯的主持人，也剛完成一本關於時間與空間本質的書。 Scientific American編輯馬瑟最近和格林恩邊吃細弦般的義大利麵邊聊弦論。以下是這次「餐訪」的節錄版：

　　Few people can take more credit for demystifying string theory than Brian Greene, a Columbia University physics professor and a major contributor to the theory. His 1999 book The Elegant Universe reached number four on the New York Times best-seller list and was a finalist for the Pulitzer Prize. Greene is now host of a three-part Nova series on PBS and has just completed a book on the nature of space and time. Scientific American staff editor George Musser recently spoke with him over a plate of stringy spaghetti. Here is an abridged, edited version of that conversation.

SA: 有時候當我們的讀者一聽到「弦論」或是「宇宙論」時，他們會雙手一攤，說：「我永遠也搞不懂這些東西。」

　　SCIENTIFIC AMERICAN: Sometimes when our readers hear the words “string theory” or “cosmology,” they throw up their hands and say, “I'll never understand it.”

格林恩: 我的確知道人們在一開始接觸弦論或宇宙論時會感受到一些壓力。我和很多人談過，但我發現他們對於這些概念的基本興趣是那麼廣泛與深刻，因此，比起其他更容易的題材，人們在這上面會願意多用一點心。

　　BRIAN GREENE: I've definitely encountered a certain amount of intimidation at the outset when it comes to ideas like string theory or cosmology. But what I have found is that the basic interest is so widespread and so deep in most people that I've spoken with, that there is a willingness to go a little bit further than you might with other subjects that are more easily taken in.

SA: 在《優雅的宇宙》中，我注意到你在許多地方會先扼要地描述物理觀念，然後才談論細節。

　　SA: I noticed that at several points in The Elegant Universe, you first gave a rough idea of the physics concepts and then the detailed version.

格林恩: 我發現那是很有用的方法，特別是在比較困難的部份。如此一來，讀者就可以選擇：如果你只需要簡要的說明，那這樣就夠了，你可以跳過底下比較難的部份；如果 你還不滿意，就讀下去。我喜歡用一種以上的方式來說明事情，因為我認為，當你碰到抽像的觀念時，你需要更多方式來瞭解它們。從科學觀點來看，如果你只追隨 一種方法，我想你在研究上突破的能力便會受到影響。我認為那正是突破的要點。大家都用一種方法看問題，而你則從後面看過來。不同的方法會揭露出其他進路所 看不到的東西。

　　BG: I found that to be a useful way of going about it, especially in the harder parts. It gives the reader permission: If the rough idea is the level at which you want to take it in, that's great; feel free to skip this next stuff. If not, go for it. I like to say things more than one way. I just think that when it comes to abstract ideas, you need many roads into them. From the scientific point of view, if you stick with one road, I think you really compromise your ability to make breakthroughs. I think that's really what breakthroughs are about. Everybody's looking at a problem one way, and you come at it from the back. That different way of getting there somehow reveals things that the other approach didn't.

SA: 能不能給我們一些「走後門」的例子？

　　SA: What are some examples of that back-door approach?

格林恩: 嗯，最重要的例子也許是維頓（Edward Witten）的突破。維頓只是走上山往下看，看到了別人看不到的關聯，因而統一了五種先前被認為是完全不同的弦論。所有的東西都已經有了，他只是採用了不同的觀點，然後「砰」，把一切都兜在一起，那就是天才。

　　BG: Well, probably the biggest ones are Ed Witten's breakthroughs. Ed [of the Institute for Advanced Study in Princeton, N.J.] just walked up the mountain and looked down and saw the connections that nobody else saw and in that way united the five string theories that previously were thought to be completely distinct. It was all out there; he just took a different perspective, and bang, it all came together. And that's genius.

對我而言，這意味著一個基本的發現。從某個觀點，宇宙帶領我們前往真理，因為正是那些真理在控制我們 所能看到的東西。如果我們全部受控於我們所見，就會被引導至同一個方向。所以，究竟能不能達成突破，關鍵經常只在於一點點洞察力（無論是真的洞察力或數學 洞察力），看是否能夠將東西以不同的方式結合起來。

　　To me that suggests what a fundamental discovery is. The universe in a sense guides us toward truths, because those truths are the things that govern what we see. If we're all being governed by what we see, we're all being steered in the same direction. Therefore, the difference between making a breakthrough and not often can be just a small element of perception, either true perception or mathematical perception, that puts things together in a different way.

SA: 如果沒有天才介入，你認為還會有這些發現嗎？

　　SA: Do you think that these discoveries would have been made without the intervention of genius?

格林恩: 嗯，這很難講。以弦論來說，我認為是會的。因為裡頭的謎正一點一滴越變越清楚。也許會晚五年或十年，但是我認為這些結果還是會出現。不過如果談到廣義相對 論，我就不知道了。廣義相對論這一跳實在太遠，是對於空間、時間與重力重新省思的里程碑，所以如果沒有愛因斯坦，我不確定廣義相對論會在什麼時候以什麼方 式出現。

　　BG: Well, it's tough to say. In the case of string theory, I think so, because the pieces of the puzzle were really becoming clearer and clearer. It may have been five or 10 years later, but I suspect it would have happened. But with general relativity, I don't know. General relativity is such a leap, such a monumental rethinking of space, time and gravity, that it's not obvious to me how and when that would have happened without Einstein.

SA: 你認為弦論中有沒有類似的大躍進？

　　SA: Are there examples in string theory that you think are analogous to that huge leap?

格林恩: 我想我們還在等待那種大躍進。弦論是奠基在眾多的小點子上面，這些小點子來自很多人，它們正慢慢連接成非常可觀的理論結構。但是坐在結構頂端的究竟是什麼 樣的概念，我們實在還不清楚。一但我們搞清楚了，它應該會像個閃亮的燈塔，照亮整個結構，而且我相信還會解答目前仍舊無解的關鍵問題。

　　BG: I think we're still waiting for a leap of that magnitude. String theory has been built up out of a lot of smaller ideas that a lot of people have contributed and been slowly stitching together into an ever more impressive theoretical edifice. But what idea sits at the top of that edifice, we still don't really know. When we do have that idea, I believe that it will be like a beacon shining down; it will illuminate the edifice, and it will also, I believe, give answers to critical questions that remain unresolved.

　　SA：在廣義相對論裡，你有「等效原理」與「廣義協變性」扮演燈塔的角色；在標準模型中，則是「規範不變性」。你在《優雅的宇宙》中提議，對於弦論而言，那個原理就是「全像原理」（參見《科學人》2003年9月號〈資訊˙黑洞˙全像宇宙〉）。你現在怎麼想？

　　SA: In the case of relativity, you had the equivalence principle and general covariance in that beacon role. In the Standard Model, it's gauge invariance. In The Elegant Universe you suggested the holographic principle could be that principle for string theory [see also “Information in the Holographic Universe,” by Jacob D. Bekenstein; Scientific American, August]. What's your thinking on that now?

　　格林恩：嗯，全像原理在過去幾年只有變得更為重要、更可信。在1990年代中期，就在全像的想法提出後 不久，支持這個想法的證據還相當抽象與含糊，它們多半是基於黑洞的特性：黑洞熵存在於黑洞表面；因此，自由度或許也存在於表面；因此，一切有「視界」的區 域或許也是如此；也許對於宇宙視界也同樣成立；也許我們所居住的宇宙這一角落，其真正的自由度是在很遠的地方。很棒的奇怪想法，可是證據很有限。

　　BG: Well, the past few years have only seen the holographic principle rise to a yet greater prominence and believability. Back in the mid-'90s, shortly after the holographic ideas were suggested, the supporting ideas were rather abstract and vague, all based upon features of black holes: Black hole entropy resides on the surface; therefore, maybe the degrees of freedom reside on the surface; therefore, maybe that's true of all regions that have a horizon; maybe it's true of cosmological horizons; maybe we're living within a cosmological region which has its true degrees of freedom far away. Wonderfully strange ideas, but the supporting evidence was meager.

　　但是馬多西納（Juan Maldacena，普林斯頓高等研究院的物理學者）改變了一切。他在弦論中發現了一個明確的例子，其中立體空間（即我們認為是真實的場域）內的物理會完 整地顯現在環繞的邊界上。不論是空間內或邊界上，兩種方式都能夠真實地描述所發生的事件，儘管兩種描述的細節差異很大。其中之一是五維的描述，另一則是四 維的描述。所以甚至像維度這種概念，似乎都不是靠得住的東西，因為對於你所觀察到的物理，還可以從另一種維度給予精確地描述。

　　But that changed with the work of Juan Maldacena [of the Institute for Advanced Study in Princeton, N.J.], in which he found an explicit example within string theory, where physics in the bulk—that is, in the arena that we consider to be real—would be exactly mirrored by physics taking place on a bounding surface. There'd be no difference in terms of the ability of either description to truly describe what's going on, yet in detail the descriptions would be vastly different. One would be in five dimensions, the other in four. So even the number of dimensions seems not to be something which you can count on, because there can be alternative descriptions that would accurately reflect the physics you're observing.

　　所以對我而言，馬多西納的工作使得抽象的想法更為具體。它讓你相信抽象的概念，即使弦論的細節改變了， 我和其他很多人（雖然不是所有的人）認為，全像的想法還是會成立，並且繼續引導我們。這是否就是我們追求的那一個原理？我不知道，我認為不是。但是我想它 是追尋關鍵概念路途上很重要的踏腳石。這和理論的細節無關，它只是在描述，這是兼具量子力學和重力的世界中，一個十分普遍的特性。

　　So to my mind, that makes the abstract ideas now concrete; it makes you believe the abstract ideas. And even if the details of string theory change, I think, as many others do—not everyone, though—that the holographic idea will persist and will guide us. Whether it truly is the idea, I don't know. I don't think so. But I think that it could well be one of the key stepping-stones towards finding the essential ideas of the theory. It steps outside the details of the theory and just says, Here's a very general feature of a world that has quantum mechanics and gravity.

SA: 讓我們談一下環圈量子重力論（loop quantum gravity）與其他理論。你一向說弦論是唯一的量子重力論。你還這麼想嗎？

　　SA: Let's talk a bit about loop quantum gravity and some of the other approaches. You've always described string theory as the only game in town when it comes to quantum gravity. Do you still feel that way?

格林恩: 嗯，我想弦論是目前最好玩有趣的理論！持平而論，環圈量子重力論陣營已經獲得很大的進展。不過，我認為還有很多非常基本的問題沒有令人滿意的答案。但是它 的確是個可能成功的理論，而且很多極端聰明的人都在從事環圈量子重力的研究，這是很好的事。我希望，終究我們是在發展同一套理論，只是所採用的角度不一樣 而已，而這也是斯莫林（加拿大滑鐵盧圓周研究院的物理學者）所鼓吹的。或許在往量子重力論的路上，我們走我們的，他們走他們的路，而兩條路終會在某個地方 相會。因為他們的很多長處恰是我們的短處，而我們的很多長處也恰是他們的短處。

　　BG: Well, I think it's the most fun game in town! But to be fair, the loop-quantum-gravity community has made tremendous progress. There are still many very basic questions that I don't feel have been answered, not to my satisfaction. But it's a viable approach, and it's great there are such large numbers of extremely talented people working on it. My hope—and it has been one that Lee Smolin [of the Perimeter Institute in Waterloo, Canada] has championed—is that ultimately we're developing the same theory from different angles. It's far from impossible that we're going down our route to quantum gravity, they're going down their route to quantum gravity, and we're going to meet someplace. Because it turns out that many of their strengths are our weaknesses. Many of our strengths are their weaknesses.

弦論的缺點之一是所謂的「背景相依」（background-dependent）。我們必須假設一 個存在的時空，弦就在這時空中運動。然而你會希望，在一個真正的量子重力論中，時空會從基本方程式中出現。他們（環圈量子重力的研究者）的理論中的確有一 個「背景獨立」的數學架構，其中的時空的確是以更基本的方式從理論本身出現。從另一方面講，我們（弦論）可以在大尺度的結構下，和愛因斯坦方程式直接連上 關係。我們可以從方程式看到這一點，而他們的理論不容易和平常的重力連接起來。所以，或許我們可以結合兩邊的長處。

　　One weakness of string theory is that it's so-called background-dependent. We need to assume an existing spacetime within which the strings move. You'd hope, though, that a true quantum theory of gravity would have spacetime emerge from its fundamental equations. They [the loop-quantum gravity researchers], however, do have a background-independent formulation in their approach, where spacetime does emerge more fundamentally from the theory itself. On the other hand, we are able to make very direct contact with Einstein's general relativity on large scales. We see it in our equations. They have some difficulty making contact with ordinary gravity. So naturally, you'd think maybe one could put together the strengths of each.

SA: 有沒有人在這麼做？

　　SA: Has that effort been made?

格林恩: 很緩慢。真正精通兩邊理論的人很少。兩個體系都太龐大，你可以單在你的理論花上一輩子時間，竭盡你工作的每一分每一秒，卻還無法知道這個體系的所有進展。但是很多人正朝著這個方向走，開始思考這方面的問題，相互間的討論也已經開始。

　　BG: Slowly. There are very few people who are really well versed in both theories. These are both two huge subjects, and you can spend your whole life, every moment of your working day, just in your own subject, and you still won't know everything that's going on. But many people are heading down that path and starting to think along those lines, and there have been some joint meetings.

SA: 如果你有「背景相依」，那麼要如何才能真正深刻地瞭解什麼是空間與時間？

　　SA: If you have this background dependence, what hope is there to really understand, in a deep sense, what space and time are?

格 林恩: 唔，你可以一步一步來。例如，即使有「背景相依」，我們也學到了像「鏡對稱」這樣的東西，也就是說兩種時空可以有相同的物理。我們也學到時空的拓樸可以改 變：空間以傳統上不可置信的方式演化。我們學到了主宰微觀世界的是「不可交換幾何」，其中的座標不是實數，因此相乘的結果和相乘的順序有關。所以（關於時 空）你會得到一些暗示。你會隱約在這裡看見一點，那裡看見一點，還有它們底下到底是怎麼一回事。但是我認為，如果沒有「背景獨立」的數學架構，將很難把這 些點點滴滴湊在一起。

　　BG: Well, you can chip away at the problem. For instance, even with background dependence, we've learned things like mirror symmetry—there can be two spacetimes, one physics. We've learned topology change—that space can evolve in ways that we wouldn't have thought possible before. We've learned that the microworld might be governed by noncommutative geometry, where the coordinates, unlike real numbers, depend upon the order in which you multiply them. So you can get hints. You can get isolated glimpses of what's truly going on down there. But I think without the background-independent formalism, it's going to be hard to put the pieces together on their own.

　　SA：「鏡對稱」非常深奧，因為它讓物理與時空幾何分離開來。把這兩者聯繫起來一向是愛因斯坦的想法。

　　SA: The mirror symmetry is incredibly profound, because it divorces spacetime geometry from physics. The connection between the two was always the Einsteinian program.

　　格林恩：沒錯，不過它並沒有把物理與時空幾何分離得很徹底。它只是在說，事情你只知道了一半。幾何和物 理的關係很密切，但這是二對一的對應。我們不應說物理與幾何，而是物理與「幾何–幾何」，你可以選擇要用哪一個幾何。有時候，某一個幾何會比另一個讓你看 得更清楚。這又是觀看同一種物理的不同方式：兩種不同的幾何，一種物理。人們發現，只用一種幾何，是無法回答關於某些物理與幾何系統的數學問題。但是如果 引進以前沒注意到的「鏡對稱」，忽然間，非常困難的問題，在另一種幾何裡會變成不可思議的簡單。

　　BG: That's right. Now, it doesn't divorce them completely. It simply says that you're missing half of the story. Geometry is tightly tied to physics, but it's a two-to-one map. It's not physics and geometry. It's physics and geometry-geometry, and which geometry you want to pick is up to you. Sometimes using one geometry gives you more insight than the other. Again, different ways of looking at one and the same physical system: two different geometries and one physics. And people have found there are mathematical questions about certain physical and geometrical systems that people couldn't answer using the one geometry. Bring in the mirror geometry that had previously gone unrealized, and, all of a sudden, profoundly difficult questions, when translated, were mind-bogglingly simple.

　　SA：你能不能描述一下「不可交換幾何」？

　　SA: Can you describe noncommutative geometry?

　　格林恩：自從笛卡兒以來，我們就知道用座標來標定一個點是很有用的做法，例如用經緯度來標定地球表面的 位置，或是用在高中所學的三個笛卡兒座標x, y, z來標定三維空間中的點。我們總是想像這些數字就像一般數字，當你把它們相乘（這是物理上常見的運算），其相乘的順序不會影響到出來的結果：3×5等於5 ×3。我們似乎發現，當你在很小的尺度之下設定空間座標的時候，牽涉到的不像3或5這一類的數字（乘在一起時，答案與相乘順序無關）。但有一類新的數字， 相乘的答案與順序是有關的。

　　BG: Since the time of Descartes, we've found it very powerful to label points by their coordinates, either on Earth by their latitude and longitude or in three-space by the three Cartesian coordinates, x, y and z, that you learn in high school. And we've always imagined that those numbers are like ordinary numbers, which have the property that, when you multiply them together—which is often an operation you need to do in physics—the answer doesn't depend on the order of operation: 3 times 5 is 5 times 3. What we seem to be finding is that when you coordinatize space on very small scales, the numbers involved are not like 3's and 5's, which don't depend upon the order in which they're multiplied. There's a new class of numbers that do depend on the order of multiplication.

　　事實上，它們並不真的那麼新鮮，因為長久以來，我們已知道一種稱為矩陣的東西；如果將兩個矩陣相乘，它 的結果會和相乘的順序有關。也就是說，如果A和B是矩陣，則A×B不（必然）等於B×A。弦論似乎在說，用單一個實數可以描述的點，得由另一種幾何物體取 代，而描述這種幾何物體得用上矩陣。當尺度越來越大，矩陣就越像是對角矩陣，而對角矩陣的性質之一，正是它們的乘積與相乘順序無關：如果A和B是對角矩 陣，則A×B等於B×A。但是當你越來越深入微觀世界時，矩陣的非對角元素會越來越大，也越來越重要。「不可交換幾何」是個全新幾何領域，由一些人花了好 幾年的時間在發展，不過，這些人並沒有必然把物理的應用放在心上。法國數學家鞏訥（Alain Connes）就寫了一本厚厚的書《不可交換幾何》。歐幾里得、高斯與黎曼和其他了不起的幾何學家，都是在可交換幾何的框架下做研究；現在，鞏訥與其他人 開始發展這個更新的不可交換幾何結構。

　　They're actually not that new, because for a long time we have known of an entity called the matrix. Sure as shooting, matrix multiplication depends upon the order of multiplication. A times B does not equal B times A if A and B are matrices. String theory seems to indicate that points described by single numbers are replaced by geometrical objects described by matrices. On big scales, it turns out that these matrices become more and more diagonal, and diagonal matrices do have the property that they commute when you multiply. It doesn't matter how you multiply A times B if they're diagonal matrices. But then if you venture into the microworld, the off-diagonal entries in the matrices get bigger and bigger and bigger until way down in the depths, they are playing a significant part. Noncommutative geometry is a whole new field of geometry that some people have been developing for years without necessarily an application of physics in mind. The French mathematician Alain Connes has this big thick book called Noncommutative Geometry. Euclid and Gauss and Riemann and all those wonderful geometers were working in the context of commutative geometry, and now Connes and others are taking off and developing the newer structure of noncommutative geometry.

　　SA：我聽不太懂，也許它本來就應該令人迷惑，你必須用一個矩陣或某些非純數來標定時空中的點？那是什麼意思？

　　SA: It is baffling to me—maybe it should be baffling—that you would have to label points with a matrix or some nonpure number. What does that mean?

　　格林恩：我們應該這麼看：並沒有「點」這回事，點是一種近似的概念。如果有一個點，你的確應該用一個數 字來標定它。但是新的看法是，在足夠小的尺度下，點這個語言（概念）變成很差的近似，所以沒有什麼用處。當我們談到幾何中的點時，我們其實談的是東西如何 能夠通過這些點。終究重要的是物體的運動，這些運動其實可以比滑來滑去更為複雜。矩陣可以用來描述所有這種運動。所以我們標定物體的方式不是藉由它所通過 的點，而是利用這個矩陣的自由度來標定它的運動。

　　BG: The way to think about it is: There is no notion of a point. A point is an approximation. If there is a point, you should label it by a number. But the claim is that, on sufficiently small scales, that language of points becomes such a poor approximation that it just isn't relevant. When we talk about points in geometry, we really talk about how something can move through points. It's the motion of objects that ultimately is what's relevant. Their motion, it turns out, can be more complicated than just sliding back and forth. All those motions are captured by a matrix. So rather than labeling an object by what point it's passing through, you need to label its motion by this matrix of degrees of freedom.

SA: 你現在對於「人本原理」以及多重宇宙的看法是什麼？在《優雅的宇宙》中，你在討論弦論的解說能力是否受到某種限制時，曾提到這些問題。

　　SA: What is your current thinking on anthropic and multiverse-type ideas? You talked about it in The Elegant Universe in the context of whether there is some limit to the explanatory power of string theory.

格林恩: 我和其他很多人從來就不太喜歡任何這種「人本」的想法，主要是因為我覺得在科學史上，你在任何時刻都可以說：「好，就到此為止了，不可能再往前進了。每一 個未解問題的最終答案都是『事情是這個樣子的原因是：如果不是這樣子，我們就不會在這裡問這個問題』。」所以這好像在躲避問題。也許這樣講不太好，這不必 然在逃避問題，但我覺得這樣有點危險，因為你只須再多努力五年就可以回答那些還未解的問題，而不必只是記下「它們就是這樣了」。所以我的顧慮是：人們因為 有這個退路而不再努力。

　　BG: I and many others have never been too happy with any of these anthropic ideas, largely because it seems to me that at any point in the history of science, you can say, “Okay, we're done, we can't go any further, and the final answer to every currently unsolved question is: 'Things are the way they are because had they not been this way, we wouldn't have been here to ask the question.' ” So it sort of feels like a cop-out. Maybe that's the wrong word. Not necessarily like a cop-out; it feels a little dangerous to me, because maybe you just needed five more years of hard work and you would have answered those unresolved questions, rather than just chalking them up to, “That's just how it is.” So that's my concern: that one doesn't stop looking by virtue of having this fallback position.

不過「人本」的想法的確是比過去更進步了，現在已有一些具體的點子，裡頭牽涉到多重宇宙，其中每個宇 宙都可能有不同的性質。我們之所以在這個宇宙之中，很可能是因為這個宇宙的性質適合我們存在，而我們之所以不在別的宇宙裡，是因為我們無法生存在那樣的宇 宙裡。這樣的講法比較不那麼唯心。

　　But you know, it's definitely the case that the anthropic ideas have become more developed. They're now real proposals whereby you would have many universes, and those many universes could all have different properties, and it very well could be that we're simply in this one because the properties are right for us to be here, and we're not in those others because we couldn't survive there. It's less of just a mental exercise.

SA: 弦論以及一般的現代物理，似乎逼近一個非如此不可的邏輯結構；理論是這個樣子，因為沒有別的可能形式。一方面，這和「人本」的方向相反，但是另一方面，理論還是有彈性可以引導你到「人本」的方向。

　　SA: String theory, and modern physics generally, seem to be approaching a single logical structure that had to be the way it is; the theory is the way it is because there's no other way it could be. On the one hand, that would argue against an anthropic direction. But on the other hand, there's a flexibility in the theory that leads you to an anthropic direction.

格林恩: 這個彈性也許在那裡，也許不在。它可能只是我們欠缺全面理解所造成的假像。不過以我們今天所瞭解的去推斷，弦論的確可以導出很多不同的世界。我們的世界可 能只是其中之一，而且還不是非常特別的一個。所以，是的，這和追求一個絕對、沒有商量餘地的目標是互相矛盾的。

　　BG: The flexibility may or may not truly be there. That really could be an artifact of our lack of full understanding. But were I to go by what we understand today, the theory seems to be able to give rise to many different worlds, of which ours seems to be potentially one, but not even necessarily a very special one. So yes, there is a tension with the goal of absolute, rigid inflexibility.

SA: 如果有研究生還在摸索，你怎麼指引他們方向？

　　SA: If you had other grad students waiting in the wings, what would you steer them to?

格林恩: 嗯，我想大的問題前面我們已經談過了。例如，我們能不能瞭解空間和時間來自何處？我們能不能弄懂弦論或「M理論」的基本概念？我們能不能證明這個基本想法 可以導致一個獨特的理論，這個獨特理論有獨特解，也就是我們所知的這個世界？有沒有可能藉由天文觀測或是加速器實驗來檢驗這些想法？

　　BG: Well, the big questions are, I think, the ones that we've discussed. Can we understand where space and time come from? Can we figure out the fundamental ideas of string theory or M-theory? Can we show that this fundamental idea yields a unique theory with the unique solution, which happens to be the world as we know it? Is it possible to test these ideas through astronomical observations or through accelerator-based experiment?

甚至，我們能不能回頭瞭解，為什麼量子力學必得是我們所知世界不可或缺的一部份？任何可能成功的理論在其深層都得依賴一些東西，例如：空間、時間、量子力學，其中有哪些是真正關鍵的？有哪些可以省略掉而仍可能得到和我們類似的世界？

　　Can we even take a step further back and understand why quantum mechanics had to be part and parcel of the world as we know it? How many of the things that we rely on at a very deep level in any physical theory that has a chance of being right—such as space, time, quantum mechanics—are truly essential, and how many of them can be relaxed and potentially still yield the world that appears close to ours?

物理有沒有可能走上另外一條路─雖然面貌完全不同，但是仍能成功解釋所有的實驗？我不知道，但是我想 這是個真正有趣的問題。從數據與數學邏輯出發，有多少我們認為真正基本的東西是唯一可能的結論？又有多少是可以有其他可能性，而我們只是剛好走上其中一條 恰好被我們發現的路而已？另一個行星上的生物會不會有完全不同於我們的定律，而那些定律又和我們的定律一樣成功？

　　Could physics have taken a different path that would have been experimentally as successful but completely different? I don't know. But I think it's a real interesting question to ask. How much of what we believe is truly fundamentally driven in a unique way by data and mathematical consistency, and how much of it could have gone one way or another, and we just happened to go down one path because that's what we happened to discover? Could beings on another planet have completely different sets of laws that somehow work just as well as ours?