《時間簡史08｜宇宙的起源和命運1（中英文）》
In an attempt to find a model of the universe in which many different initial configurations could have evolved tosomething like the present universe, a scientist at the Massachusetts Institute of Technology, Alan Guth, suggestedthat the early universe might have gone through a period of very rapid expansion. This expansion is said to be“inflationary,” meaning that the universe at one time expanded at an increasing rate rather than the decreasing ratethat it does today. According to Guth, the radius of the universe increased by a million million million million million (1with thirty zeros after it) times in only a tiny fraction of a second.
為了試圖尋找一個能從許多不同的初始結構演化到象現在這樣的宇宙的宇宙模型， 麻省理工學院的科學家阿倫·固斯提出， 早期宇宙可能存在過一個非常快速膨脹的時期。 這種膨脹叫做“暴漲”， 意指宇宙在一段時間裡， 不像現在這樣以減少的、而是以增加的速率膨脹。 按照固斯理論， 在遠遠小於1秒的時間裡， 宇宙的半徑增大了100萬億億億（1後面跟30個0）倍。
Guth suggested that the universe started out from the big bang in a very hot, but rather chaotic, state. These hightemperatures would have meant that the particles in the universe would be moving very fast and would have highenergies. As we discussed earlier, one would expect that at such high temperatures the strong and weak nuclearforces and the electromagnetic force would all be unified into a single force. As the universe expanded, it would cool,and particle energies would go down. Eventually there would be what is called a phase transition and the symmetrybetween the forces would be broken: the strong force would become different from the weak and electromagneticforces. One common example of a phase transition is the freezing of water when you cool it down. Liquid water issymmetrical, the same at every point and in every direction. However, when ice crystals form, they will have definitepositions and will be lined up in some direction. This breaks water’s symmetry.
固斯提出， 宇宙是以一個非常熱而且相當紊亂的狀態從大爆炸開始的。 這些高溫表明宇宙中的粒子運動得非常快並具有高能量。
In the case of water, if one is careful, one can “supercool” it: that is, one can reduce the temperature below thefreezing point (OºC) without ice forming. Guth suggested that the universe might behave in a similar way: thetemperature might drop below the critical value without the symmetry between the forces being broken. If thishappened, the universe would be in an unstable state, with more energy than if the symmetry had been broken. Thisspecial extra energy can be shown to have an antigravitational effect: it would have acted just like the cosmologicalconstant that Einstein introduced into general relativity when he was trying to construct a static model of theuniverse. Since the universe would already be expanding just as in the hot big bang model, the repulsive effect ofthis cosmological constant would therefore have made the universe expand at an ever-increasing rate. Even inregions where there were more matter particles than average, the gravitational attraction of the matter would havebeen outweighed by the repulsion of the effective cosmological constant. Thus these regions would also expand inan accelerating inflationary manner. As they expanded and the matter particles got farther apart, one would be leftwith an expanding universe that contained hardly any particles and was still in the supercooled state. Anyirregularities in the universe would simply have been smoothed out by the expansion, as the wrinkles in a balloon aresmoothed away when you blow it up. Thus the present smooth and uniform state of the universe could have evolvedfrom many different non-uniform initial states.
處理水的時候， 只要你足夠小心， 就能使之“過冷”， 也就是可以將溫度降低到冰點（0℃）以下而不結冰。 固斯認為， 宇宙的行為也很相似：宇宙溫度可以低到臨界值以下，
In such a universe, in which the expansion was accelerated by a cosmological constant rather than slowed down bythe gravitational attraction of matter, there would be enough time for light to travel from one region to another in theearly universe. This could provide a solution to the problem, raised earlier, of why different regions in the earlyuniverse have the same properties. Moreover, the rate of expansion of the universe would automatically becomevery close to the critical rate determined by the energy density of the universe. This could then explain why the rateof expansion is still so close to the critical rate, without having to assume that the initial rate of expansion of theuniverse was very carefully chosen.
在這樣一個其膨脹由宇宙常數加速、而不由物質的引力吸引使之減慢的宇宙中， 早期宇宙中的光線就有足夠的時間從一個地方傳到另一個地方。 這就解答了早先提出的， 為何在早期宇宙中的不同區域具有同樣性質的問題。 不但如此， 宇宙的膨脹率也自動變得非常接近於由宇宙的能量密度決定的臨界值。 這樣， 不必去假設宇宙初始膨脹率曾被非常仔細地選擇過， 就能解釋為何現在的膨脹率仍然是如此地接近於臨界值。
The idea of inflation could also explain why there is so much matter in the universe. There are something like tenmillion million million million million million million million million million million million million million (1 with eightyzeros after it) particles in the region of the universe that we can observe. Where did they all come from? The answeris that, in quantum theory, particles can be created out of energy in the form of particle/antiparticle pairs. But that justraises the question of where the energy came from. The answer is that the total energy of the universe is exactlyzero. The matter in the universe is made out of positive energy. However, the matter is all attracting itself by gravity.Two pieces of matter that are close to each other have less energy than the same two pieces a long way apart,because you have to expend energy to separate them against the gravitational force that is pulling them together.Thus, in a sense, the gravitational field has negative energy. In the case of a universe that is approximately uniformin space, one can show that this negative gravitational energy exactly cancels the positive energy represented by thematter. So the total energy of the universe is zero.
暴漲的思想還能解釋為何宇宙存在這麼多物質。 在我們能觀察到的宇宙裡大體有1億億億億億億億億億億（1後面跟80個0）個粒子。 它們從何而來？答案是， 在量子理論中， 粒子可以從粒子/反粒子對的形式由能量中創生出來。 但這只不過引起了能量從何而來的問題。 答案是， 宇宙的總能量剛好是零。 宇宙的物質是由正能量構成的；然而， 所有物質都由引力互相吸引。 兩塊互相靠近的物質比兩塊分得很開的物質具有更少的能量， 因為你必須消耗能量去克服把它們拉在一起的引力而將其分開。 這樣， 在一定意義上， 引力場具有負能量。 在空間上大體一致的宇宙的情形中， 人們可以證明， 這個負的引力能剛好抵消了物質所代表的正能量，
Now twice zero is also zero. Thus the universe can double the amount of positive matter energy and also double thenegative gravitational energy without violation of the conservation of energy. This does not happen in the normalexpansion of the universe in which the matter energy density goes down as the universe gets bigger. It does happen,however, in the inflationary expansion because the energy density of the supercooled state remains constant whilethe universe expands: when the universe doubles in size, the positive matter energy and the negative gravitationalenergy both double, so the total energy remains zero. During the inflationary phase, the universe increases its sizeby a very large amount. Thus the total amount of energy available to make particles becomes very large. As Guthhas remarked, “It is said that there’s no such thing as a free lunch. But the universe is the ultimate free lunch.”
零的兩倍仍為零。 這樣宇宙可以同時將其正的物質能和負的引力能加倍， 而不破壞其能量的守恆。 在宇宙的正常膨脹時， 這並沒有發生。 這時當宇宙變大時， 物質能量密度下降。 然而， 這種情形確實發生於暴漲時期。 因為宇宙膨脹時， 過冷態的能量密度保持不變：當宇宙體積加倍時， 正物質能和負引力能都加倍， 總能量保持為零。 在暴漲相， 宇宙的尺度增大了一個非常大的倍數。 這樣， 可用以製造粒子的總能量變得非常大。 正如固斯所說的：“都說沒有免費午餐這件事， 但是宇宙是最徹底的免費午餐。 ”
The universe is not expanding in an inflationary way today. Thus there has to be some mechanism that wouldeliminate the very large effective cosmological constant and so change the rate of expansion from an acceleratedone to one that is slowed down by gravity, as we have today. In the inflationary expansion one might expect thateventually the symmetry between the forces would be broken, just as super-cooled water always freezes in the end.The extra energy of the unbroken symmetry state would then be released and would reheat the universe to atemperature just below the critical temperature for symmetry between the forces. The universe would then go on toexpand and cool just like the hot big bang model, but there would now be an explanation of why the universe wasexpanding at exactly the critical rate and why different regions had the same temperature.
今天宇宙不是以暴漲的方式膨脹。 這樣， 必須有一種機制， 它可以消去這一非常大的有效宇宙常數， 從而使膨脹率從加速的狀態，改變為正如同今天這樣由引力減慢下的樣子。人們可以預料，在宇宙暴漲時不同力之間的對稱最終會被破壞，正如過冷的水最終會凝固一樣。這樣，未破缺的對稱態的額外能量就會釋放，並將宇宙重新加熱到剛好低於使不同力對稱的臨界溫度。以後，宇宙就以標準的大爆炸模式繼續膨脹並變冷。但是，現在找到了何以宇宙剛好以臨界速率膨脹，並在不同的區域具有相同溫度的解釋。
In Guth’s original proposal the phase transition was supposed to occur suddenly, rather like the appearance of icecrystals in very cold water. The idea was that “bubbles” of the new phase of broken symmetry would have formed inthe old phase, like bubbles of steam surrounded by boiling water. The bubbles were supposed to expand and meetup with each other until the whole universe was in the new phase. The trouble was, as I and several other peoplepointed out, that the universe was expanding so fast that even if the bubbles grew at the speed of light, they wouldbe moving away from each other and so could not join up. The universe would be left in a very non-uniform state,with some regions still having symmetry between the different forces. Such a model of the universe would notcorrespond to what we see.
在固斯的原先設想中，有點像在非常冷的水中出現冰晶體，相變是突然發生的。其想法是，正如同沸騰的水圍繞著蒸汽泡，新的對稱破缺相的“泡泡”在原有的對稱相中形成。泡泡膨脹並互相碰撞，直到整個宇宙變成新相。麻煩在於，正如同我和其他幾個人所指出的，宇宙膨脹得如此之快，甚至即使泡泡以光速漲大，它們也要互相分離，並因此不能合併在一起。結果宇宙變成一種非常不一致的狀態，有些區域仍具有不同力之間的對稱。這樣的模型跟我們所觀察到的宇宙並不吻合。
the inflationary model and its problems at the Sternberg Astronomical Institute. Before this, I had got someone elseto give my lectures for me, because most people could not understand my voice. But there was not time to preparethis seminar, so I gave it myself, with one of my graduate students repeating my words. It worked well, and gave memuch more contact with my audience. In the audience was a young Russian, Andrei Linde, from the LebedevInstitute in Moscow. He said that the difficulty with the bubbles not joining up could be avoided if the bubbles were sobig that our region of the universe is all contained inside a single bubble. In order for this to work, the change fromsymmetry to broken symmetry must have taken place very slowly inside the bubble, but this is quite possibleaccording to grand unified theories. Linde’s idea of a slow breaking of symmetry was very good, but I later realizedthat his bubbles would have to have been bigger than the size of the universe at the time! I showed that instead thesymmetry would have broken everywhere at the same time, rather than just inside bubbles. This would lead to auniform universe, as we observe. I was very excited by this idea and discussed it with one of my students, Ian Moss.As a friend of Linde’s, I was rather embarrassed, however, when I was later sent his paper by a scientific journal andasked whether it was suitable for publication. I replied that there was this flaw about the bubbles being bigger thanthe universe, but that the basic idea of a slow breaking of symmetry was very good. I recommended that the paper ¿published as it was because it would take Linde several months to correct it, since anything he sent to the Westwould have to be passed by Soviet censorship, which was neither very skillful nor very quick with scientific papers.Instead, I wrote a short paper with Ian Moss in the same journal in which we pointed out this problem with the bubbleand showed how it could be resolved.
1981年10月，我去莫斯科參加量子引力的會議。會後，我在斯特堡天文研究所做了一個有關暴漲模型和它的問題的講演。聽眾席中有一年輕的蘇聯人——莫斯科列別提夫研究所的安德雷·林德——他講，如果泡泡是如此之大，以至於我們宇宙的區域被整個地包含在一個單獨的泡泡之中，則可以避免泡泡不能合併在一起的困難。為了使這個行得通，從對稱相向對稱破缺相的改變必須在泡泡中進行得非常慢，而按照大統一理論這是相當可能的。林德的緩慢對稱破缺思想是非常好的，但過後我意識到，他的泡泡在那一時刻必須比宇宙的尺度還要大！我指出，那時對稱不僅僅在泡泡裡，而且在所有的地方同時被破壞。這會導致一個正如我們所觀察到的一致的宇宙。我被這個思想弄得非常激動，並和我的一個學生因·莫斯討論。然而，當我後來收到一個科學雜誌社寄來的林德的論文，徵求是否可以發表時，作為他的朋友，我感到相當難為情。我回答說，這裡有一個關於泡泡比宇宙還大的瑕疵，但是裡面關於緩慢對稱破缺的基本思想是非常好的。我建議將此論文照原樣發表。因為林德要花幾個月時間去改正它，並且他寄到西方的任何東西都要通過蘇聯的審查，這種對於科學論文的審查既無技巧可言又很緩慢。我和因·莫斯便越俎代庖，為同一雜誌寫了一篇短文。我們在該文中指出這泡泡的問題，並提出如何將其解決。
The day after I got back from Moscow I set out for Philadelphia, where I was due to receive a medal from theFranklin Institute. My secretary, Judy Fella, had used her not inconsiderable charm to persuade British Airways togive herself and me free seats on a Concorde as a publicity venture. However, I .was held up on my way to theairport by heavy rain and I missed the plane. Nevertheless, I got to Philadelphia in the end and received my medal. Iwas then asked to give a seminar on the inflationary universe at Drexel University in Philadelphia. I gave the sameseminar about the problems of the inflationary universe, just as in Moscow.
我從莫斯科返回的第二天，即去費城接受佛蘭克林研究所的獎章。我的秘書裘蒂·費拉以其不差的魅力說服了英國航空公司向她和我免費提供協和式飛機的宣傳旅行座席。然而，在去機場的路上被大雨耽擱，我沒趕上航班。儘管如此，我最終還是到了費城並得到獎章。之後，應邀作了關於暴漲宇宙的講演。正如在莫斯科那樣，我用大部分時間講授關於暴漲模型的問題。
A very similar idea to Linde’s was put forth independently a few months later by Paul Steinhardt and AndreasAlbrecht of the University of Pennsylvania. They are now given joint credit with Linde for what is called “the newinflationary model,” based on the idea of a slow breaking of symmetry. (The old inflationary model was Guth’soriginal suggestion of fast symmetry breaking with the formation of bubbles.)
幾個月之後，賓州大學的保羅·斯特恩哈特和安德魯斯·阿爾伯勒希特獨立地提出和林德非常相似的思想。現在他們和林德分享以緩慢對稱破缺的思想為基礎的所謂“新暴脹模型” 的榮譽。（舊的暴脹模型是指固斯關於形成泡泡後快速對稱破缺的原始設想。）
The new inflationary model was a good attempt to explain why the universe is the way it is. However, I and severalother people showed that, at least in its original form, it predicted much greater variations in the temperature of themicrowave background radiation than are observed. Later work has also cast doubt on whether there could be aphase transition in the very early universe of the kind required. In my personal opinion, the new inflationary model isnow dead as a scientific theory, although a lot of people do not seem to have heard of its demise and are still writingpapers as if it were viable. A better model, called the chaotic inflationary model, was put forward by Linde in 1983. Inthis there is no phase transition or supercooling. Instead, there is a spin 0 field, which, because of quantumfluctuations, would have large values in some regions of the early universe. The energy of the field in those regionswould behave like a cosmological constant. It would have a repulsive gravitational effect, and thus make thoseregions expand in an inflationary manner. As they expanded, the energy of the field in them would slowly decreaseuntil the inflationary expansion changed to an expansion like that in the hot big bang model. One of these regionswould become what we now see as the observable universe. This model has all the advantages of the earlierinflationary models, but it does not depend on a dubious phase transition, and it can moreover give a reasonable sizefor the fluctuations in the temperature of the microwave background that agrees with observation.
新暴漲模型是一個好的嘗試，它能解釋宇宙為何是這種樣子。然而我和其他幾個人指出，至少在它原先的形式，它預言的微波背景輻射的溫度起伏比所觀察到的情形要大得多。後來的工作還對極早期宇宙中是否存在這類所需要的相變提出懷疑。我個人的意見是，現在新暴漲模型作為一個科學理論是氣數已盡。雖然有很多人似乎沒有聽進它的死訊，還繼續寫文章，好像那理論還有生命力。林德在1983年提出了一個更好的所謂紊亂暴漲模型。這裡沒有相變和過冷，而代之以存在一個自旋為0的場，由於它的量子漲落，在早期宇宙的某些區域有大的場量。在那些區域中，場的能量起到宇宙常數的作用，它具有排斥的引力效應，因此使得這些區域以暴漲的形式膨脹。當它們膨脹時，它們中的場的能量慢慢地減小，直到暴漲改變到猶如熱大爆炸模型中的膨脹時為止。這些區域之一就成為我們看到的宇宙。這個模型具有早先暴漲模型的所有優點，但它不是取決於使人生疑的相變，並且還能給出微波背景輻射的溫度起伏，其幅度與觀測相符合。
This work on inflationary models showed that the present state of the universe could have arisen from quite a largenumber of different initial configurations. This is important, because it shows that the initial state of the part of theuniverse that we inhabit did not have to be chosen with great care. So we may, if we wish, use the weak anthropicprinciple to explain why the universe looks the way it does now. It cannot be the case, however, that every initialconfiguration would have led to a universe like the one we observe. One can show this by considering a verydifferent state for the universe at the present time, say, a very lumpy and irregular one. One could use the laws ofscience to evolve the universe back in time to determine its configuration at earlier times. According to the singularitytheorems of classical general relativity, there would still have been a big bang singularity. If you evolve such auniverse forward in time according to the laws of science, you will end up with the lumpy and irregular state youstarted with. Thus there must have been initial configurations that would not have given rise to a universe like theone we see today. So even the inflationary model does not tell us why the initial configuration was not such as toproduce something very different from what we observe. Must we turn to the anthropic principle for an explanation?Was it all just a lucky chance? That would seem a counsel of despair, a negation of all our hopes of understandingthe underlying order of the universe.
暴漲模型的研究指出：宇宙現在的狀態可以從相當大量的不同初始結構引起的。這是重要的，因為它表明不必非常細心地選取我們居住的那部份宇宙區域的初始狀態。所以，如果願意的話，我們可以利用弱人擇原理解釋宇宙為何是這個樣子。然而，絕不是任何一種初始結構都會產生像我們所觀察到的宇宙。這一點很容易說明，考慮現在宇宙處於一個非常不同的態，例如一個非常成團的、非常無規則的態，人們可以利用科學定律，在時間上將其演化回去，以確定宇宙在更早時刻的結構。按照經典廣義相對論的奇點定理，仍然存在一個大爆炸奇點。如果你在時間前進方向上按照科學定律演化這樣的宇宙，你就會得到你一開始給定的那個成團的無規則的態。這樣，必定存在不會產生我們今天所觀察到的宇宙的初始結構。所以，就連暴漲模型也沒有告訴我們，為何初始結構不是那種產生和我們觀測到的非常不同的宇宙的某種態。我們是否應該轉去應用人擇原理以求解釋呢？難道所有這一切僅僅是因為好運氣？看來，這只是無望的遁詞，是對我們理解宇宙內在秩序的所有希望的否定。
In order to predict how the universe should have started off, one needs laws that hold at the beginning of time. If theclassical theory of general relativity was correct, the singularity theorems that Roger Penrose and I proved show thatthe beginning of time would have been a point of infinite density and infinite curvature of space-time. All the knownlaws of science would break down at such a point. One might suppose that there were new laws that held atsingularities, but it would be very difficult even to formulate such laws at such badly behaved points, and we wouldhave no guide from observations as to what those laws might be. However, what the singularity theorems reallyindicate is that the gravitational field becomes so strong that quantum gravitational effects become important:classical theory is no longer a good description of the universe. So one has to use a quantum theory of gravity todiscuss the very early stages of the universe. As we shall see, it is possible in the quantum theory for the ordinarylaws of science to hold everywhere, including at the beginning of time: it is not necessary to postulate new laws forsingularities, because there need not be any singularities in the quantum theory.
為了預言宇宙應該是如何開始的，人們需要在時間開端處有效的定律。羅傑·彭羅斯和我證明的奇點定理指出，如果廣義相對論的經典理論是正確的，則時間的開端是具有無限密度和無限空間——時間曲率的一點，在這一點上所有已知的科學定律都失效。人們可以設想存在在奇點處成立的新定律，但是在如此不守規矩的點處，甚至連表述這樣的定律都是非常困難的，而且從觀察中我們沒有得到關於這些定律應是什麼樣子的任何提示。然而，奇點定理真正表明的是，該處引力場變得如此之強，以至於量子引力效應變得重要：經典理論不再能很好地描述宇宙。所以，人們必須用量子引力論去討論宇宙的極早期階段。我們將會看到，在量子力學中，通常的科學定律有可能在任何地方都有效，包括時間開端這一點在內：不必針對奇點提出新的定律，因為在量子理論中不須有任何奇點。
We don’t yet have a complete and consistent theory that combines quantum mechanics and gravity. However, weare fairly certain of some features that such a unified theory should have. One is that it should incorporateFeynman’s proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle doesnot have just a single history, as it would in a classical theory. Instead, it is supposed to follow every possible path inspace-time, and with each of these histories there are associated a couple of numbers, one represent-ing the size ofa wave and the other representing its position in the cycle (its phase). The probability that the particle, say, passesthrough some particular point is found by adding up the waves associated with every possible history that passesthrough that point. When one actually tries to perform these sums, however, one runs into severe technicalproblems. The only way around these is the following peculiar prescription: one must add up the waves for particlehistories that are not in the “real” time that you and I experience but take place in what is called imaginary time.Imaginary time may sound like science fiction but it is in fact a well-defined mathematical concept. If we take anyordinary (or “real”) number and multiply it by itself, the result is a positive number. (For example, 2 times 2 is 4, butso is – 2 times – 2.) There are, however, special numbers (called imaginary numbers) that give negative numberswhen multiplied by themselves. (The one called i, when multiplied by itself, gives – 1, 2i multiplied by itself gives – 4,and so on.)
我們仍然沒有一套完整而協調的理論，它將量子力學和引力結合在一起。然而，我們相當清楚這樣一套統一理論所應該具有的某些特徵。其中一個就是它必須和費因曼提出的按照對歷史求和的量子力學表述相一致。在這種方法裡，一個粒子不像在經典理論中那樣，不僅只有一個歷史。相反的，它被認為是通過空間——時間裡的每一可能的路徑，每一條途徑有一對相關的數，一個代表波的幅度，另一個代表它的相位。粒子通過一指定點的概率是將通過此點的所有可能途徑的波迭加而求得。然而，當人們實際去進行這些求和時，就遇到了嚴重的技術問題。回避這個問題的唯一獨特的方法是：你必須不是對發生在你我經驗的“實”的時間內的，而是對發生在所謂“虛”的時間內的粒子的途徑的波進行求和。虛時間可能聽起來像科學幻想，但事實上，它是定義得很好的數學概念。如果你取任何平常的（或“實的”）數和它自己相乘，結果是一個正數。（例如2乘2是4，但-2乘-2也是這麼多）。然而，有一種特別的數（叫虛數），當它們自乘時得到負數。（在這兒的虛數單位叫做i，它自乘時得-1，2i自乘得-4，等等。）
One can picture real and imaginary numbers in the following way: The real numbers can be represented by a linegoing from left to right, with zero in the middle, negative numbers like – 1, – 2, etc. on the left, and positive numbers,1, 2, etc. on the right. Then imaginary numbers are represented by a line going up and down the page, with i, 2i, etc.above the middle, and – i, – 2i, etc. below. Thus imaginary numbers are in a sense numbers at right angles toordinary real numbers.
人們可以用下面的辦法來圖解實數和虛數：實數可以用一根從左至右的線來代表，中間是零點，像-1，-2等負數在左面，而像1，2等正數在右面。而虛數由書頁上一根上下的線來代表，i，Zi在中點以上，而-i，-2i在中點以下。這樣，在某種意義上可以說，虛數和實數夾一直角。
To avoid the technical difficulties with Feynman’s sum over histories, one must use imaginary time. That is to say, forthe purposes of the calculation one must measure time using imaginary numbers, rather than real ones. This has aninteresting effect on space-time: the distinction between time and space disappears completely. A space-time inwhich events have imaginary values of the time coordinate is said to be Euclidean, after the ancient Greek Euclid,who founded the study of the geometry of two-dimensional surfaces. What we now call Euclidean space-time is verysimilar except that it has four dimensions instead of two. In Euclidean space-time there is no difference between thetime direction and directions in space. On the other hand, in real space-time, in which events are labeled by ordinary,real values of the time coordinate, it is easy to tell the difference – the time direction at all points lies within the lightcone, and space directions lie outside. In any case, as far as everyday quantum mechanics is concerned, we mayregard our use of imaginary time and Euclidean space-time as merely a mathematical device (or trick) to calculateanswers about real space-time.
人們必須利用虛時間，以避免在進行費因曼對歷史求和的技術上的困難。也就是為了計算的目的人們必須用虛數而不是用實數來測量時間。這對時空有一有趣的效應：時間和空間的區別完全消失。事件具有虛值時間座標的時空被稱為歐幾裡德型的，它是採用建立了二維面幾何的希臘人歐幾裡德的名字命名的。我們現在稱之為歐幾裡德時空的東西除了是四維而不是二維以外，其餘的和它非常相似。在歐幾裡德時空中，時間方向和空間方向沒有不同之處。另一方面，在通常用實的時間座標來標記事件的實的時空裡，人們很容易區別這兩種方向——在光錐中的任何點是時間方向，之外為空間方向。就日常的量子力學而言，在任何情況下，我們利用虛的時間和歐幾裡德時空可以認為僅僅是一個計算即時空的答案的數學手段（或技巧）。
A second feature that we believe must be part of any ultimate theory is Einstein’s idea that the gravitational field isrepresented by curved space-time: particles try to follow the nearest thing to a straight path in a curved space, butbecause space-time is not flat their paths appear to be bent, as if by a gravitational field. When we apply Feynman’ssum over histories to Einstein’s view of gravity, the analogue of the history of a particle is now a complete curvedspace-time that represents the history of the whole universe. To avoid the technical difficulties in actually performingthe sum over histories, these curved space-times must be taken to be Euclidean. That is, time is imaginary and isindistinguishable from directions in space. To calculate the probability of finding a real space-time with some certainproperty, such as looking the same at every point and in every direction, one adds up the waves associated with allthe histories that have that property.
我們相信，作為任何終極理論的一部分而不可或缺的第二個特徵是愛因斯坦的思想，即引力場是由彎曲的時空來代表：粒子在彎曲空間中試圖沿著最接近於直線的某種途徑走，但因為時空不是平坦的。它們的途徑看起來似乎被引力場折彎了。當我們用費因曼的路徑求和方法去處理愛因斯坦的引力觀點時，和粒子的歷史相類似的東西則是代表整個宇宙歷史的完整的彎曲的時空。為了避免實際進行歷史求和的技術困難，這些彎曲的時空必須採用歐幾裡德型的。也就是，時間是虛的並和空間的方向不可區分。為了計算找到具有一定性質，例如在每一點和每一方向上看起來都一樣的實的時空的概率，人們將和所有具有這性質的歷史相關聯的波迭加起來即可。
In the classical theory of general relativity, there are many different possible curved space-times, each correspondingto a different initial state of the universe. If we knew the initial state of our universe, we would know its entire history.Similarly, in the quantum theory of gravity, there are many different possible quantum states for the universe. Again,if we knew how the Euclidean curved space-times in the sum over histories behaved at early times, we would knowthe quantum state of the universe.
在廣義相對論的經典理論中，有許多不同的可能彎曲的時空，每一個對應於宇宙的不同的初始態。如果我們知道宇宙的初始態，我們就會知道它的整個歷史。類似地，在量子引力論中，存在許多不同的可能的宇宙量子態。如果我們知道在歷史求和中的歐幾裡德彎曲時空在早先時刻的行為，我們就會知道宇宙的量子態。
In the classical theory of gravity, which is based on real space-time, there are only two possible ways the universecan behave: either it has existed for an infinite time, or else it had a beginning at a singularity at some finite time inthe past. In the quantum theory of gravity, on the other hand, a third possibility arises. Because one is usingEuclidean space-times, in which the time direction is on the same footing as directions in space, it is possible forspace-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time wouldbe like the surface of the earth, only with two more dimensions. The surface of the earth is finite in extent but itdoesn’t have a boundary or edge: if you sail off into the sunset, you don’t fall off the edge or run into a singularity. (Iknow, because I have been round the world!)
在以實的時空為基礎的經典引力論中，宇宙可能的行為只有兩種方式：或者它已存在了無限長時間，或者它在有限的過去的某一時刻的奇點上有一個開端。而在量子引力論中，還存在第三種可能性。因為人們是用歐幾裡德時空，在這兒時間方向和空間方向是同等的，所以時空只有有限的尺度，卻沒有奇點作為它的邊界或邊緣是可能的。時空就像是地球的表面，只不過多了兩維。地球的表面積是有限的，但它沒有邊界或邊緣：如果你朝著落日的方向駕船，你不會掉到邊緣外面或陷入奇點中去。（因為我曾經環球旅行過，所以知道！）
If Euclidean space-time stretches back to infinite imaginary time, or else starts at a singularity in imaginary time, wehave the same problem as in the classical theory of specifying the initial state of the universe: God may know howthe universe began, but we cannot give any particular reason for thinking it began one way rather than another. Onthe other hand, the quantum theory of gravity has opened up a new possibility, in which there would be no boundaryto space-time and so there would be no need to specify the behavior at the boundary. There would be nosingularities at which the laws of science broke down, and no edge of space-time at which one would have to appealto God or some new law to set the boundary conditions for space-time. One could say: “The boundary condition ofthe universe is that it has no boundary.” The universe would be completely self-contained and not affected byanything outside itself. It would neither be created nor destroyed, It would just BE.
如果歐幾裡德時空延伸到無限的虛時間，或者在一個虛時間奇點處開始，我們就有了和在經典理論中指定宇宙初態的同樣問題，即上帝可以知道宇宙如何開始，但是我們提不出任何特別原因，認為它應以這種而不是那種方式開始。另一方面，量子引力論開闢了另一種新的可能性，在這兒時空沒有邊界，所以沒有必要指定邊界上的行為。這兒就沒有使科學定律失效的奇點，也就是不存在在該處必須祈求上帝或某些新的定律給空間一時間設定邊界條件的時空邊緣。人們可以說：“宇宙的邊界條件是它沒有邊界。”宇宙是完全自足的，而不被任何外在於它的東西所影響。它既不被創生，也不被消滅。它就是存在。
It was at the conference in the Vatican mentioned earlier that I first put forward the suggestion that maybe time andspace together formed a surface that was finite in size but did not have any boundary or edge. My paper was rathermathematical, however, so its implications for the role of God in the creation of the universe were not generallyrecognized at the time (just as well for me). At the time of the Vatican conference, I did not know how to use the “noboundary” idea to make predictions about the universe. However, I spent the following sum-mer at the University ofCalifornia, Santa Barbara. There a friend and colleague of mine, Jim Hartle, worked out with me what conditions theuniverse must satisfy if space-time had no boundary. When I returned to Cambridge, I continued this work with two ofmy research students, Julian Luttrel and Jonathan Halliwell.
我正是在早先提到的那次梵帝岡會議上第一次提出，時間和空間可能會共同形成一個在尺度上有限而沒有任何邊界或邊緣的面。然而我的論文數學氣息太濃，所以文章中包含的上帝在創造宇宙的作用的含義在當時沒有被普遍看出來（對我也正是如此）。在梵蒂岡會議期間，我不知道如何用“無邊界”思想去預言宇宙。然而，第二年夏天我在加州大學的聖他巴巴拉分校渡過。我的一位朋友兼合作者詹姆·哈特爾在那裡，他和我共同得出了如果時空沒有邊界時宇宙應滿足的條件。回到劍橋後，我和我的兩個研究生朱麗安·拉卻爾和約納遜·哈裡威爾繼續從事這項工作。
I’d like to emphasize that this idea that time and space should be finite “without boundary” is just a proposal: it cannotbe deduced from some other principle. Like any other scientific theory, it may initially be put forward for aesthetic ormetaphysical reasons, but the real test is whether it makes predictions that agree with observation. This, how-ever, isdifficult to determine in the case of quantum gravity, for two reasons. First, as will be explained in Chapter 11, we arenot yet sure exactly which theory successfully combines general relativity and quantum mechanics, though we knowquite a lot about the form such a theory must have. Second, any model that described the whole universe in detailwould be much too complicated mathematically for us to be able to calculate exact predictions. One therefore has tomake simplifying assumptions and approximations – and even then, the problem of extracting predictions remains a formidable one.
我要著重說明，時空是有限而無界的思想僅僅只是一個設想，它不能從其他原理匯出。正如任何其他的科學理論，它原先可以是出於美學或形而上學的原因而被提出，但是對它的真正檢驗在於它所給出的預言是否與觀測相一致。然而，在量子引力的情況下，由於以下兩個原因這很難確定。首先，正如將在十一章所要解釋的，雖然我們對能將廣義相對論和量子力學結合在一起的理論所應具有的特徵，已經知道得相當多，但我們還不能準確地認定這樣一個理論。其次，任何詳盡描述整個宇宙的模型在數學上都過於複雜，以至於我們不能通過計算做出準確的預言。所以，人們不得不做簡化的假設和近似——並且甚至這樣，要從中引出預言仍是令人生畏的問題。
Each history in the sum over histories will describe not only the space-time but everything in it as well, including anycomplicated organisms like human beings who can observe the history of the universe. This may provide anotherjustification for the anthropic principle, for if all the histories are possible, then so long as we exist in one of thehistories, we may use the anthropic principle to explain why the universe is found to be the way it is. Exactly whatmeaning can be attached to the other histories, in which we do not exist, is not clear. This view of a quantum theoryof gravity would be much more satisfactory, however, if one could show that, using the sum over histories, ouruniverse is not just one of the possible histories but one of the most probable ones. To do this, we must perform thesum over histories for all possible Euclidean space-times that have no boundary.
在對歷史求和中的每一個歷史不只描述時空，而且描述在其中的任何東西——包括像能觀察宇宙歷史的人類那樣複雜的生物。這可對人擇原理提供另一個支援，因為如果任何歷史都是可能的，就可以用人擇原理去解釋為何我們發現宇宙是現今這樣子。儘管我們對自己並不生存於其中的其他歷史究竟有什麼意義還不清楚。然而，如果利用對歷史求和可以顯示，我們的宇宙不只是一個可能的，而且是最有可能的歷史，則這個量子引力論的觀點就會令人滿意得多。為此，我們必須對所有可能的沒有邊界的歐幾裡德時空進行歷史求和。
Under the “no boundary” proposal one learns that the chance of the universe being found to be following most of thepossible histories is negligible, but there is a particular family of histories that are much more probable than theothers. These histories may be pictured as being like the surface of the earth, with the distance from the North Polerepresenting imaginary time and the size of a circle of constant distance from the North Pole representing the spatialsize of the universe. The universe starts at the North Pole as a single point. As one moves south, the circles oflatitude at constant distance from the North Pole get bigger, corresponding to the universe expanding with imaginarytime Figure 8:1. The universe would reach a maximum size at the equator and would contract with increasingimaginary time to a single point at the South Pole. Ever though the universe would have zero size at the North andSouth Poles, these points would not be singularities, any more than the North aid South Poles on the earth aresingular. The laws of science will hold at them, just as they do at the North and South Poles on the earth.
人們從“無邊界”假定得知，宇宙沿著大多數歷史的機會是可以忽略不計的，但是有一族特別的歷史比其他的歷史有更多機會。這些歷史可以描繪得像是地球的表面。在那兒與北極的距離代表虛的時間，並且離北極等距離的圓周長代表宇宙的空間尺度。宇宙是從作為單獨一點的北極開始的。當你一直往南走去，離開北極等距離的緯度圈變大，這是和宇宙隨虛時間的膨脹相對應（圖8.1）。宇宙在赤道處達到最大的尺度，並且隨著虛時間的繼續增加而收縮，最後在南極收縮成一點。儘管宇宙在北南二極的尺度為零，這些點不是奇點，並不比地球上的北南二極更奇異。科學定律在這兒有效，正如同它仍在地球上的北南二極有效一樣。
The history of the universe in real time, however, would look very different. At about ten or twenty thousand millionyears ago, it would have a minimum size, which was equal to the maximum radius of the history in imaginary time. Atlater real times, the universe would expand like the chaotic inflationary model proposed by Linde (but one would notnow have to assume that the universe was created somehow in the right sort of state). The universe would expand toa very large size Figure 8:1 and eventually it would collapse again into what looks like a singularity in real time. Thus,in a sense, we are still all doomed, even if we keep away from black holes. Only if we could picture the universe interms of imaginary time would there be no singularities.
然而，在實的時間裡宇宙的歷史顯得非常不一樣。大約在100或200億年以前，它有一個最小的尺度，這相當於在虛時間裡的最大的半徑。在後來的即時間裡，宇宙就像由林德設想的紊亂暴漲模型那樣地膨脹（但是現在人們不必假定宇宙是從某一類正確的狀態產生出來）。宇宙會膨脹到一個非常大的尺度，並最終重新坍縮成為在即時間裡看起來像是奇點的一個東西。這樣，在某種意義上說，即使我們躲開黑洞，仍然是註定要毀滅的。只有當我們按照虛時間來描繪宇宙時才不會有奇點。
If the universe really is in such a quantum state, there would be no singularities in the history of the universe inimaginary time. It might seem therefore that my more recent work had completely undone the results of my earlierwork on singularities. But, as indicated above, the real importance of the singularity theorems was that they showedthat the gravitational field must become so strong that quantum gravitational effects could not be ignored. This in turnled to the idea that the universe could be finite in imaginary time but without boundaries or singularities. When onegoes back to the real time in which we live, however, there will still appear to be singularities. The poor astronautwho falls into a black hole will still come to a sticky end; only if he lived in imaginary time would he encounter nosingularities.
如果宇宙確實處在這樣的一個量子態裡，在虛時間裡宇宙就沒有奇點。所以，我近期的工作似乎完全使我早期研究奇點的工作成果付之東流。但是正如上面所指出的，奇點定理的真正重要性在於，它們指出引力場必然會強到不能無視量子引力效應的程度。這接著導致也許在虛時間裡宇宙的尺度有限但沒有邊界或奇點的觀念。然而，當人們回到我們生活於其中的即時間，那兒仍會出現奇點。陷進黑洞那位可憐的太空人的結局仍然是極可悲的；只有當他在虛時間裡生活，才不會遭遇到奇點。
This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just afigment of our imaginations. In real time, the universe has a beginning and an end at singularities that form aboundary to space-time and at which the laws of science break down. But in imaginary time, there are nosingularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is justan idea that we invent to help us describe what we think the universe is like. But according to the approach Idescribed in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: itexists only in our minds. So it is meaningless to ask: which is real, “real” or “imaginary” time? It is simply a matter ofwhich is the more useful description.
上述這些也許暗示所謂的虛時間是真正的即時間，而我們叫做即時間的東西恰恰是子虛烏有的空想的產物。在即時間中，宇宙的開端和終結都是奇點。這奇點構成了科學定律在那兒不成立的時空邊界。但是，在虛時間裡不存在奇點或邊界。所以，很可能我們稱之為虛時間的才真正是更基本的觀念，而我們稱作即時間的反而是我們臆造的，它有助於我們描述宇宙的模樣。但是，按照我在第一章 所描述的方法，科學理論僅僅是我們用以描述自己所觀察的數學模型，它只存在於我們的頭腦中。所以去問諸如這樣的問題是毫無意義的：“實”的或“虛”的時間，哪一個是實在的？這僅僅是哪一個描述更為有用的問題。
One can also use the sum over histories, along with the no boundary proposal, to find which properties of theuniverse are likely to occur together. For example, one can calculate the probability that the universe is expanding atnearly the same rate in all different directions at a time when the density of the universe has its present value. In thesimplified models that have been examined so far, this probability turns out to be high; that is, the proposed noboundary condition leads to the prediction that it is extremely probable that the present rate of expansion of theuniverse is almost the same in each direction. This is consistent with the observations of the microwave backgroundradiation, which show that it has almost exactly the same intensity in any direction. If the universe were expandingfaster in some directions than in others, the intensity of the radiation in those directions would be reduced by anadditional red shift.
人們還可以利用對歷史求和以及無邊界假設去發現宇宙的哪些性質可能發生。例如，人們可以計算，當宇宙具有現在密度的某一時刻，在所有方向上以幾乎同等速率膨脹的概率。在迄今已被考察的簡化的模型中，發現這個概率是高的；也就是，無邊界假設導致一個預言，即宇宙現在在每一方向的膨脹率幾乎相同是極其可能的。這與微波背景輻射的觀測相一致，它指出在任何方向上具有幾乎完全同樣的強度。如果宇宙在某些方向比其他方向膨脹得更快，在那些方向輻射的強度就會被一個附加的紅移所減小。
Further predictions of the no boundary condition are currently being worked out. A particularly interesting problem isthe size of the small departures from uniform density in the early universe that caused the formation first of thegalaxies, then of stars, and finally of us. The uncertainty principle implies that the early universe cannot have beencompletely uniform because there must have been some uncertainties or fluctuations in the positions and velocitiesof the particles. Using the no boundary condition, we find that the universe must in fact have started off with just theminimum possible non-uniformity allowed by the uncertainty principle. The universe would have then undergone aperiod of rapid expansion, as in the inflationary models. During this period, the initial non-uniformities would havebeen amplified until they were big enough to explain the origin of the structures we observe around us. In 1992 theCosmic Background Explorer satellite (COBE) first detected very slight variations in the intensity of the microwavebackground with direction. The way these non-uniformities depend on direction seems to agree with the predictionsof the inflationary model and the no boundary proposal. Thus the no boundary proposal is a good scientific theory inthe sense of Karl Popper: it could have been falsified by observations but instead its predictions have beenconfirmed. In an expanding universe in which the density of matter varied slightly from place to place, gravity wouldhave caused the denser regions to slow down their expansion and start contracting. This would lead to the formationof galaxies, stars, and eventually even insignificant creatures like ourselves. Thus all the complicated structures thatwe see in the universe might be explained by the no boundary condition for the universe together with the uncertaintyprinciple of quantum mechanics.
人們正在研究無邊界條件的進一步預言。一個特別有趣的問題是，早期字宙中物質密度對其平均值的小幅度偏離，這些偏離首先引起星系，然後是恒星，最後是我們自身的形成。不確定性原理意味著，早期宇宙不可能是完全均勻的，因為粒子的位置和速度必定有一些不確定性或起伏。利用無邊界條件，我們發現，宇宙事實上必須是從僅僅由不確定性原理允許的最小的可能的非均勻性開始的。然後，正如在暴脹模型中預言的一樣，宇宙經歷了一段快速膨脹時期。在這個期間，初始的非均勻性被放大到足以解釋在我們周圍觀察到的結構的起源。1992年宇宙背景探險者衛星（COBE）首次檢測到微波背景隨方向的非常微小的變化。這種非均勻性隨方向的變化方式似乎和暴脹模型以及無邊界設想的預言相符合。這樣，在卡爾·波普的意義上，無邊界設想是一種好的科學理論：它的預言可以被觀測證偽，但是卻被證實了。在一個各處物質密度稍有變化的膨脹字宙中，引力使得較緊密區域的膨脹減慢，並使之開始收縮。這就導致星系、恒星和最終甚至像我們自己這樣微不足道的生物的形成。因而，我們在宇宙中看到的所有複雜的結構，可由宇宙無邊界條件和量子力學中的不確定性原理給予解釋。
The idea that space and time may form a closed surface without boundary also has profound implications for the roleof God in the affairs of the universe. With the success of scientific theories in describing events, most people havecome to believe that God allows the universe to evolve according to a set of laws and does n ot intervene in theuniverse to break these laws. However, the laws do not tell us what the universe should have looked like when itstarted – it would still be up to God to wind up the clockwork and choose how to start it off. So long as the universehad a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having noboundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?
空間和時間可以形成一個沒有邊界的閉曲面的思想，對於上帝在宇宙事務中的作用還有一個深遠的含義。隨著科學理論在描述事件的成功，大部分人進而相信上帝允許宇宙按照一套定律來演化，而不介入其間促使宇宙觸犯這些定律。然而，定律並沒有告訴我們，字宙的太初應該像什麼樣子——它依然要靠上帝去卷緊發條，並選擇如何去啟動它。只要宇宙有一個開端，我們就可以設想存在一個造物主。但是，如果宇宙確實是完全自足的，沒有邊界或邊緣，它就既沒有開端也沒有終結——它就是存在。那麼，還會有造物主存身之處嗎？
從而使膨脹率從加速的狀態，改變為正如同今天這樣由引力減慢下的樣子。人們可以預料，在宇宙暴漲時不同力之間的對稱最終會被破壞，正如過冷的水最終會凝固一樣。這樣，未破缺的對稱態的額外能量就會釋放，並將宇宙重新加熱到剛好低於使不同力對稱的臨界溫度。以後，宇宙就以標準的大爆炸模式繼續膨脹並變冷。但是，現在找到了何以宇宙剛好以臨界速率膨脹，並在不同的區域具有相同溫度的解釋。In Guth’s original proposal the phase transition was supposed to occur suddenly, rather like the appearance of icecrystals in very cold water. The idea was that “bubbles” of the new phase of broken symmetry would have formed inthe old phase, like bubbles of steam surrounded by boiling water. The bubbles were supposed to expand and meetup with each other until the whole universe was in the new phase. The trouble was, as I and several other peoplepointed out, that the universe was expanding so fast that even if the bubbles grew at the speed of light, they wouldbe moving away from each other and so could not join up. The universe would be left in a very non-uniform state,with some regions still having symmetry between the different forces. Such a model of the universe would notcorrespond to what we see.
在固斯的原先設想中，有點像在非常冷的水中出現冰晶體，相變是突然發生的。其想法是，正如同沸騰的水圍繞著蒸汽泡，新的對稱破缺相的“泡泡”在原有的對稱相中形成。泡泡膨脹並互相碰撞，直到整個宇宙變成新相。麻煩在於，正如同我和其他幾個人所指出的，宇宙膨脹得如此之快，甚至即使泡泡以光速漲大，它們也要互相分離，並因此不能合併在一起。結果宇宙變成一種非常不一致的狀態，有些區域仍具有不同力之間的對稱。這樣的模型跟我們所觀察到的宇宙並不吻合。
the inflationary model and its problems at the Sternberg Astronomical Institute. Before this, I had got someone elseto give my lectures for me, because most people could not understand my voice. But there was not time to preparethis seminar, so I gave it myself, with one of my graduate students repeating my words. It worked well, and gave memuch more contact with my audience. In the audience was a young Russian, Andrei Linde, from the LebedevInstitute in Moscow. He said that the difficulty with the bubbles not joining up could be avoided if the bubbles were sobig that our region of the universe is all contained inside a single bubble. In order for this to work, the change fromsymmetry to broken symmetry must have taken place very slowly inside the bubble, but this is quite possibleaccording to grand unified theories. Linde’s idea of a slow breaking of symmetry was very good, but I later realizedthat his bubbles would have to have been bigger than the size of the universe at the time! I showed that instead thesymmetry would have broken everywhere at the same time, rather than just inside bubbles. This would lead to auniform universe, as we observe. I was very excited by this idea and discussed it with one of my students, Ian Moss.As a friend of Linde’s, I was rather embarrassed, however, when I was later sent his paper by a scientific journal andasked whether it was suitable for publication. I replied that there was this flaw about the bubbles being bigger thanthe universe, but that the basic idea of a slow breaking of symmetry was very good. I recommended that the paper ¿published as it was because it would take Linde several months to correct it, since anything he sent to the Westwould have to be passed by Soviet censorship, which was neither very skillful nor very quick with scientific papers.Instead, I wrote a short paper with Ian Moss in the same journal in which we pointed out this problem with the bubbleand showed how it could be resolved.
1981年10月，我去莫斯科參加量子引力的會議。會後，我在斯特堡天文研究所做了一個有關暴漲模型和它的問題的講演。聽眾席中有一年輕的蘇聯人——莫斯科列別提夫研究所的安德雷·林德——他講，如果泡泡是如此之大，以至於我們宇宙的區域被整個地包含在一個單獨的泡泡之中，則可以避免泡泡不能合併在一起的困難。為了使這個行得通，從對稱相向對稱破缺相的改變必須在泡泡中進行得非常慢，而按照大統一理論這是相當可能的。林德的緩慢對稱破缺思想是非常好的，但過後我意識到，他的泡泡在那一時刻必須比宇宙的尺度還要大！我指出，那時對稱不僅僅在泡泡裡，而且在所有的地方同時被破壞。這會導致一個正如我們所觀察到的一致的宇宙。我被這個思想弄得非常激動，並和我的一個學生因·莫斯討論。然而，當我後來收到一個科學雜誌社寄來的林德的論文，徵求是否可以發表時，作為他的朋友，我感到相當難為情。我回答說，這裡有一個關於泡泡比宇宙還大的瑕疵，但是裡面關於緩慢對稱破缺的基本思想是非常好的。我建議將此論文照原樣發表。因為林德要花幾個月時間去改正它，並且他寄到西方的任何東西都要通過蘇聯的審查，這種對於科學論文的審查既無技巧可言又很緩慢。我和因·莫斯便越俎代庖，為同一雜誌寫了一篇短文。我們在該文中指出這泡泡的問題，並提出如何將其解決。
The day after I got back from Moscow I set out for Philadelphia, where I was due to receive a medal from theFranklin Institute. My secretary, Judy Fella, had used her not inconsiderable charm to persuade British Airways togive herself and me free seats on a Concorde as a publicity venture. However, I .was held up on my way to theairport by heavy rain and I missed the plane. Nevertheless, I got to Philadelphia in the end and received my medal. Iwas then asked to give a seminar on the inflationary universe at Drexel University in Philadelphia. I gave the sameseminar about the problems of the inflationary universe, just as in Moscow.
我從莫斯科返回的第二天，即去費城接受佛蘭克林研究所的獎章。我的秘書裘蒂·費拉以其不差的魅力說服了英國航空公司向她和我免費提供協和式飛機的宣傳旅行座席。然而，在去機場的路上被大雨耽擱，我沒趕上航班。儘管如此，我最終還是到了費城並得到獎章。之後，應邀作了關於暴漲宇宙的講演。正如在莫斯科那樣，我用大部分時間講授關於暴漲模型的問題。
A very similar idea to Linde’s was put forth independently a few months later by Paul Steinhardt and AndreasAlbrecht of the University of Pennsylvania. They are now given joint credit with Linde for what is called “the newinflationary model,” based on the idea of a slow breaking of symmetry. (The old inflationary model was Guth’soriginal suggestion of fast symmetry breaking with the formation of bubbles.)
幾個月之後，賓州大學的保羅·斯特恩哈特和安德魯斯·阿爾伯勒希特獨立地提出和林德非常相似的思想。現在他們和林德分享以緩慢對稱破缺的思想為基礎的所謂“新暴脹模型” 的榮譽。（舊的暴脹模型是指固斯關於形成泡泡後快速對稱破缺的原始設想。）
The new inflationary model was a good attempt to explain why the universe is the way it is. However, I and severalother people showed that, at least in its original form, it predicted much greater variations in the temperature of themicrowave background radiation than are observed. Later work has also cast doubt on whether there could be aphase transition in the very early universe of the kind required. In my personal opinion, the new inflationary model isnow dead as a scientific theory, although a lot of people do not seem to have heard of its demise and are still writingpapers as if it were viable. A better model, called the chaotic inflationary model, was put forward by Linde in 1983. Inthis there is no phase transition or supercooling. Instead, there is a spin 0 field, which, because of quantumfluctuations, would have large values in some regions of the early universe. The energy of the field in those regionswould behave like a cosmological constant. It would have a repulsive gravitational effect, and thus make thoseregions expand in an inflationary manner. As they expanded, the energy of the field in them would slowly decreaseuntil the inflationary expansion changed to an expansion like that in the hot big bang model. One of these regionswould become what we now see as the observable universe. This model has all the advantages of the earlierinflationary models, but it does not depend on a dubious phase transition, and it can moreover give a reasonable sizefor the fluctuations in the temperature of the microwave background that agrees with observation.
新暴漲模型是一個好的嘗試，它能解釋宇宙為何是這種樣子。然而我和其他幾個人指出，至少在它原先的形式，它預言的微波背景輻射的溫度起伏比所觀察到的情形要大得多。後來的工作還對極早期宇宙中是否存在這類所需要的相變提出懷疑。我個人的意見是，現在新暴漲模型作為一個科學理論是氣數已盡。雖然有很多人似乎沒有聽進它的死訊，還繼續寫文章，好像那理論還有生命力。林德在1983年提出了一個更好的所謂紊亂暴漲模型。這裡沒有相變和過冷，而代之以存在一個自旋為0的場，由於它的量子漲落，在早期宇宙的某些區域有大的場量。在那些區域中，場的能量起到宇宙常數的作用，它具有排斥的引力效應，因此使得這些區域以暴漲的形式膨脹。當它們膨脹時，它們中的場的能量慢慢地減小，直到暴漲改變到猶如熱大爆炸模型中的膨脹時為止。這些區域之一就成為我們看到的宇宙。這個模型具有早先暴漲模型的所有優點，但它不是取決於使人生疑的相變，並且還能給出微波背景輻射的溫度起伏，其幅度與觀測相符合。
This work on inflationary models showed that the present state of the universe could have arisen from quite a largenumber of different initial configurations. This is important, because it shows that the initial state of the part of theuniverse that we inhabit did not have to be chosen with great care. So we may, if we wish, use the weak anthropicprinciple to explain why the universe looks the way it does now. It cannot be the case, however, that every initialconfiguration would have led to a universe like the one we observe. One can show this by considering a verydifferent state for the universe at the present time, say, a very lumpy and irregular one. One could use the laws ofscience to evolve the universe back in time to determine its configuration at earlier times. According to the singularitytheorems of classical general relativity, there would still have been a big bang singularity. If you evolve such auniverse forward in time according to the laws of science, you will end up with the lumpy and irregular state youstarted with. Thus there must have been initial configurations that would not have given rise to a universe like theone we see today. So even the inflationary model does not tell us why the initial configuration was not such as toproduce something very different from what we observe. Must we turn to the anthropic principle for an explanation?Was it all just a lucky chance? That would seem a counsel of despair, a negation of all our hopes of understandingthe underlying order of the universe.
暴漲模型的研究指出：宇宙現在的狀態可以從相當大量的不同初始結構引起的。這是重要的，因為它表明不必非常細心地選取我們居住的那部份宇宙區域的初始狀態。所以，如果願意的話，我們可以利用弱人擇原理解釋宇宙為何是這個樣子。然而，絕不是任何一種初始結構都會產生像我們所觀察到的宇宙。這一點很容易說明，考慮現在宇宙處於一個非常不同的態，例如一個非常成團的、非常無規則的態，人們可以利用科學定律，在時間上將其演化回去，以確定宇宙在更早時刻的結構。按照經典廣義相對論的奇點定理，仍然存在一個大爆炸奇點。如果你在時間前進方向上按照科學定律演化這樣的宇宙，你就會得到你一開始給定的那個成團的無規則的態。這樣，必定存在不會產生我們今天所觀察到的宇宙的初始結構。所以，就連暴漲模型也沒有告訴我們，為何初始結構不是那種產生和我們觀測到的非常不同的宇宙的某種態。我們是否應該轉去應用人擇原理以求解釋呢？難道所有這一切僅僅是因為好運氣？看來，這只是無望的遁詞，是對我們理解宇宙內在秩序的所有希望的否定。
In order to predict how the universe should have started off, one needs laws that hold at the beginning of time. If theclassical theory of general relativity was correct, the singularity theorems that Roger Penrose and I proved show thatthe beginning of time would have been a point of infinite density and infinite curvature of space-time. All the knownlaws of science would break down at such a point. One might suppose that there were new laws that held atsingularities, but it would be very difficult even to formulate such laws at such badly behaved points, and we wouldhave no guide from observations as to what those laws might be. However, what the singularity theorems reallyindicate is that the gravitational field becomes so strong that quantum gravitational effects become important:classical theory is no longer a good description of the universe. So one has to use a quantum theory of gravity todiscuss the very early stages of the universe. As we shall see, it is possible in the quantum theory for the ordinarylaws of science to hold everywhere, including at the beginning of time: it is not necessary to postulate new laws forsingularities, because there need not be any singularities in the quantum theory.
為了預言宇宙應該是如何開始的，人們需要在時間開端處有效的定律。羅傑·彭羅斯和我證明的奇點定理指出，如果廣義相對論的經典理論是正確的，則時間的開端是具有無限密度和無限空間——時間曲率的一點，在這一點上所有已知的科學定律都失效。人們可以設想存在在奇點處成立的新定律，但是在如此不守規矩的點處，甚至連表述這樣的定律都是非常困難的，而且從觀察中我們沒有得到關於這些定律應是什麼樣子的任何提示。然而，奇點定理真正表明的是，該處引力場變得如此之強，以至於量子引力效應變得重要：經典理論不再能很好地描述宇宙。所以，人們必須用量子引力論去討論宇宙的極早期階段。我們將會看到，在量子力學中，通常的科學定律有可能在任何地方都有效，包括時間開端這一點在內：不必針對奇點提出新的定律，因為在量子理論中不須有任何奇點。
We don’t yet have a complete and consistent theory that combines quantum mechanics and gravity. However, weare fairly certain of some features that such a unified theory should have. One is that it should incorporateFeynman’s proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle doesnot have just a single history, as it would in a classical theory. Instead, it is supposed to follow every possible path inspace-time, and with each of these histories there are associated a couple of numbers, one represent-ing the size ofa wave and the other representing its position in the cycle (its phase). The probability that the particle, say, passesthrough some particular point is found by adding up the waves associated with every possible history that passesthrough that point. When one actually tries to perform these sums, however, one runs into severe technicalproblems. The only way around these is the following peculiar prescription: one must add up the waves for particlehistories that are not in the “real” time that you and I experience but take place in what is called imaginary time.Imaginary time may sound like science fiction but it is in fact a well-defined mathematical concept. If we take anyordinary (or “real”) number and multiply it by itself, the result is a positive number. (For example, 2 times 2 is 4, butso is – 2 times – 2.) There are, however, special numbers (called imaginary numbers) that give negative numberswhen multiplied by themselves. (The one called i, when multiplied by itself, gives – 1, 2i multiplied by itself gives – 4,and so on.)
我們仍然沒有一套完整而協調的理論，它將量子力學和引力結合在一起。然而，我們相當清楚這樣一套統一理論所應該具有的某些特徵。其中一個就是它必須和費因曼提出的按照對歷史求和的量子力學表述相一致。在這種方法裡，一個粒子不像在經典理論中那樣，不僅只有一個歷史。相反的，它被認為是通過空間——時間裡的每一可能的路徑，每一條途徑有一對相關的數，一個代表波的幅度，另一個代表它的相位。粒子通過一指定點的概率是將通過此點的所有可能途徑的波迭加而求得。然而，當人們實際去進行這些求和時，就遇到了嚴重的技術問題。回避這個問題的唯一獨特的方法是：你必須不是對發生在你我經驗的“實”的時間內的，而是對發生在所謂“虛”的時間內的粒子的途徑的波進行求和。虛時間可能聽起來像科學幻想，但事實上，它是定義得很好的數學概念。如果你取任何平常的（或“實的”）數和它自己相乘，結果是一個正數。（例如2乘2是4，但-2乘-2也是這麼多）。然而，有一種特別的數（叫虛數），當它們自乘時得到負數。（在這兒的虛數單位叫做i，它自乘時得-1，2i自乘得-4，等等。）
One can picture real and imaginary numbers in the following way: The real numbers can be represented by a linegoing from left to right, with zero in the middle, negative numbers like – 1, – 2, etc. on the left, and positive numbers,1, 2, etc. on the right. Then imaginary numbers are represented by a line going up and down the page, with i, 2i, etc.above the middle, and – i, – 2i, etc. below. Thus imaginary numbers are in a sense numbers at right angles toordinary real numbers.
人們可以用下面的辦法來圖解實數和虛數：實數可以用一根從左至右的線來代表，中間是零點，像-1，-2等負數在左面，而像1，2等正數在右面。而虛數由書頁上一根上下的線來代表，i，Zi在中點以上，而-i，-2i在中點以下。這樣，在某種意義上可以說，虛數和實數夾一直角。
To avoid the technical difficulties with Feynman’s sum over histories, one must use imaginary time. That is to say, forthe purposes of the calculation one must measure time using imaginary numbers, rather than real ones. This has aninteresting effect on space-time: the distinction between time and space disappears completely. A space-time inwhich events have imaginary values of the time coordinate is said to be Euclidean, after the ancient Greek Euclid,who founded the study of the geometry of two-dimensional surfaces. What we now call Euclidean space-time is verysimilar except that it has four dimensions instead of two. In Euclidean space-time there is no difference between thetime direction and directions in space. On the other hand, in real space-time, in which events are labeled by ordinary,real values of the time coordinate, it is easy to tell the difference – the time direction at all points lies within the lightcone, and space directions lie outside. In any case, as far as everyday quantum mechanics is concerned, we mayregard our use of imaginary time and Euclidean space-time as merely a mathematical device (or trick) to calculateanswers about real space-time.
人們必須利用虛時間，以避免在進行費因曼對歷史求和的技術上的困難。也就是為了計算的目的人們必須用虛數而不是用實數來測量時間。這對時空有一有趣的效應：時間和空間的區別完全消失。事件具有虛值時間座標的時空被稱為歐幾裡德型的，它是採用建立了二維面幾何的希臘人歐幾裡德的名字命名的。我們現在稱之為歐幾裡德時空的東西除了是四維而不是二維以外，其餘的和它非常相似。在歐幾裡德時空中，時間方向和空間方向沒有不同之處。另一方面，在通常用實的時間座標來標記事件的實的時空裡，人們很容易區別這兩種方向——在光錐中的任何點是時間方向，之外為空間方向。就日常的量子力學而言，在任何情況下，我們利用虛的時間和歐幾裡德時空可以認為僅僅是一個計算即時空的答案的數學手段（或技巧）。
A second feature that we believe must be part of any ultimate theory is Einstein’s idea that the gravitational field isrepresented by curved space-time: particles try to follow the nearest thing to a straight path in a curved space, butbecause space-time is not flat their paths appear to be bent, as if by a gravitational field. When we apply Feynman’ssum over histories to Einstein’s view of gravity, the analogue of the history of a particle is now a complete curvedspace-time that represents the history of the whole universe. To avoid the technical difficulties in actually performingthe sum over histories, these curved space-times must be taken to be Euclidean. That is, time is imaginary and isindistinguishable from directions in space. To calculate the probability of finding a real space-time with some certainproperty, such as looking the same at every point and in every direction, one adds up the waves associated with allthe histories that have that property.
我們相信，作為任何終極理論的一部分而不可或缺的第二個特徵是愛因斯坦的思想，即引力場是由彎曲的時空來代表：粒子在彎曲空間中試圖沿著最接近於直線的某種途徑走，但因為時空不是平坦的。它們的途徑看起來似乎被引力場折彎了。當我們用費因曼的路徑求和方法去處理愛因斯坦的引力觀點時，和粒子的歷史相類似的東西則是代表整個宇宙歷史的完整的彎曲的時空。為了避免實際進行歷史求和的技術困難，這些彎曲的時空必須採用歐幾裡德型的。也就是，時間是虛的並和空間的方向不可區分。為了計算找到具有一定性質，例如在每一點和每一方向上看起來都一樣的實的時空的概率，人們將和所有具有這性質的歷史相關聯的波迭加起來即可。
In the classical theory of general relativity, there are many different possible curved space-times, each correspondingto a different initial state of the universe. If we knew the initial state of our universe, we would know its entire history.Similarly, in the quantum theory of gravity, there are many different possible quantum states for the universe. Again,if we knew how the Euclidean curved space-times in the sum over histories behaved at early times, we would knowthe quantum state of the universe.
在廣義相對論的經典理論中，有許多不同的可能彎曲的時空，每一個對應於宇宙的不同的初始態。如果我們知道宇宙的初始態，我們就會知道它的整個歷史。類似地，在量子引力論中，存在許多不同的可能的宇宙量子態。如果我們知道在歷史求和中的歐幾裡德彎曲時空在早先時刻的行為，我們就會知道宇宙的量子態。
In the classical theory of gravity, which is based on real space-time, there are only two possible ways the universecan behave: either it has existed for an infinite time, or else it had a beginning at a singularity at some finite time inthe past. In the quantum theory of gravity, on the other hand, a third possibility arises. Because one is usingEuclidean space-times, in which the time direction is on the same footing as directions in space, it is possible forspace-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time wouldbe like the surface of the earth, only with two more dimensions. The surface of the earth is finite in extent but itdoesn’t have a boundary or edge: if you sail off into the sunset, you don’t fall off the edge or run into a singularity. (Iknow, because I have been round the world!)
在以實的時空為基礎的經典引力論中，宇宙可能的行為只有兩種方式：或者它已存在了無限長時間，或者它在有限的過去的某一時刻的奇點上有一個開端。而在量子引力論中，還存在第三種可能性。因為人們是用歐幾裡德時空，在這兒時間方向和空間方向是同等的，所以時空只有有限的尺度，卻沒有奇點作為它的邊界或邊緣是可能的。時空就像是地球的表面，只不過多了兩維。地球的表面積是有限的，但它沒有邊界或邊緣：如果你朝著落日的方向駕船，你不會掉到邊緣外面或陷入奇點中去。（因為我曾經環球旅行過，所以知道！）
If Euclidean space-time stretches back to infinite imaginary time, or else starts at a singularity in imaginary time, wehave the same problem as in the classical theory of specifying the initial state of the universe: God may know howthe universe began, but we cannot give any particular reason for thinking it began one way rather than another. Onthe other hand, the quantum theory of gravity has opened up a new possibility, in which there would be no boundaryto space-time and so there would be no need to specify the behavior at the boundary. There would be nosingularities at which the laws of science broke down, and no edge of space-time at which one would have to appealto God or some new law to set the boundary conditions for space-time. One could say: “The boundary condition ofthe universe is that it has no boundary.” The universe would be completely self-contained and not affected byanything outside itself. It would neither be created nor destroyed, It would just BE.
如果歐幾裡德時空延伸到無限的虛時間，或者在一個虛時間奇點處開始，我們就有了和在經典理論中指定宇宙初態的同樣問題，即上帝可以知道宇宙如何開始，但是我們提不出任何特別原因，認為它應以這種而不是那種方式開始。另一方面，量子引力論開闢了另一種新的可能性，在這兒時空沒有邊界，所以沒有必要指定邊界上的行為。這兒就沒有使科學定律失效的奇點，也就是不存在在該處必須祈求上帝或某些新的定律給空間一時間設定邊界條件的時空邊緣。人們可以說：“宇宙的邊界條件是它沒有邊界。”宇宙是完全自足的，而不被任何外在於它的東西所影響。它既不被創生，也不被消滅。它就是存在。
It was at the conference in the Vatican mentioned earlier that I first put forward the suggestion that maybe time andspace together formed a surface that was finite in size but did not have any boundary or edge. My paper was rathermathematical, however, so its implications for the role of God in the creation of the universe were not generallyrecognized at the time (just as well for me). At the time of the Vatican conference, I did not know how to use the “noboundary” idea to make predictions about the universe. However, I spent the following sum-mer at the University ofCalifornia, Santa Barbara. There a friend and colleague of mine, Jim Hartle, worked out with me what conditions theuniverse must satisfy if space-time had no boundary. When I returned to Cambridge, I continued this work with two ofmy research students, Julian Luttrel and Jonathan Halliwell.
我正是在早先提到的那次梵帝岡會議上第一次提出，時間和空間可能會共同形成一個在尺度上有限而沒有任何邊界或邊緣的面。然而我的論文數學氣息太濃，所以文章中包含的上帝在創造宇宙的作用的含義在當時沒有被普遍看出來（對我也正是如此）。在梵蒂岡會議期間，我不知道如何用“無邊界”思想去預言宇宙。然而，第二年夏天我在加州大學的聖他巴巴拉分校渡過。我的一位朋友兼合作者詹姆·哈特爾在那裡，他和我共同得出了如果時空沒有邊界時宇宙應滿足的條件。回到劍橋後，我和我的兩個研究生朱麗安·拉卻爾和約納遜·哈裡威爾繼續從事這項工作。
I’d like to emphasize that this idea that time and space should be finite “without boundary” is just a proposal: it cannotbe deduced from some other principle. Like any other scientific theory, it may initially be put forward for aesthetic ormetaphysical reasons, but the real test is whether it makes predictions that agree with observation. This, how-ever, isdifficult to determine in the case of quantum gravity, for two reasons. First, as will be explained in Chapter 11, we arenot yet sure exactly which theory successfully combines general relativity and quantum mechanics, though we knowquite a lot about the form such a theory must have. Second, any model that described the whole universe in detailwould be much too complicated mathematically for us to be able to calculate exact predictions. One therefore has tomake simplifying assumptions and approximations – and even then, the problem of extracting predictions remains a formidable one.
我要著重說明，時空是有限而無界的思想僅僅只是一個設想，它不能從其他原理匯出。正如任何其他的科學理論，它原先可以是出於美學或形而上學的原因而被提出，但是對它的真正檢驗在於它所給出的預言是否與觀測相一致。然而，在量子引力的情況下，由於以下兩個原因這很難確定。首先，正如將在十一章所要解釋的，雖然我們對能將廣義相對論和量子力學結合在一起的理論所應具有的特徵，已經知道得相當多，但我們還不能準確地認定這樣一個理論。其次，任何詳盡描述整個宇宙的模型在數學上都過於複雜，以至於我們不能通過計算做出準確的預言。所以，人們不得不做簡化的假設和近似——並且甚至這樣，要從中引出預言仍是令人生畏的問題。
Each history in the sum over histories will describe not only the space-time but everything in it as well, including anycomplicated organisms like human beings who can observe the history of the universe. This may provide anotherjustification for the anthropic principle, for if all the histories are possible, then so long as we exist in one of thehistories, we may use the anthropic principle to explain why the universe is found to be the way it is. Exactly whatmeaning can be attached to the other histories, in which we do not exist, is not clear. This view of a quantum theoryof gravity would be much more satisfactory, however, if one could show that, using the sum over histories, ouruniverse is not just one of the possible histories but one of the most probable ones. To do this, we must perform thesum over histories for all possible Euclidean space-times that have no boundary.
在對歷史求和中的每一個歷史不只描述時空，而且描述在其中的任何東西——包括像能觀察宇宙歷史的人類那樣複雜的生物。這可對人擇原理提供另一個支援，因為如果任何歷史都是可能的，就可以用人擇原理去解釋為何我們發現宇宙是現今這樣子。儘管我們對自己並不生存於其中的其他歷史究竟有什麼意義還不清楚。然而，如果利用對歷史求和可以顯示，我們的宇宙不只是一個可能的，而且是最有可能的歷史，則這個量子引力論的觀點就會令人滿意得多。為此，我們必須對所有可能的沒有邊界的歐幾裡德時空進行歷史求和。
Under the “no boundary” proposal one learns that the chance of the universe being found to be following most of thepossible histories is negligible, but there is a particular family of histories that are much more probable than theothers. These histories may be pictured as being like the surface of the earth, with the distance from the North Polerepresenting imaginary time and the size of a circle of constant distance from the North Pole representing the spatialsize of the universe. The universe starts at the North Pole as a single point. As one moves south, the circles oflatitude at constant distance from the North Pole get bigger, corresponding to the universe expanding with imaginarytime Figure 8:1. The universe would reach a maximum size at the equator and would contract with increasingimaginary time to a single point at the South Pole. Ever though the universe would have zero size at the North andSouth Poles, these points would not be singularities, any more than the North aid South Poles on the earth aresingular. The laws of science will hold at them, just as they do at the North and South Poles on the earth.
人們從“無邊界”假定得知，宇宙沿著大多數歷史的機會是可以忽略不計的，但是有一族特別的歷史比其他的歷史有更多機會。這些歷史可以描繪得像是地球的表面。在那兒與北極的距離代表虛的時間，並且離北極等距離的圓周長代表宇宙的空間尺度。宇宙是從作為單獨一點的北極開始的。當你一直往南走去，離開北極等距離的緯度圈變大，這是和宇宙隨虛時間的膨脹相對應（圖8.1）。宇宙在赤道處達到最大的尺度，並且隨著虛時間的繼續增加而收縮，最後在南極收縮成一點。儘管宇宙在北南二極的尺度為零，這些點不是奇點，並不比地球上的北南二極更奇異。科學定律在這兒有效，正如同它仍在地球上的北南二極有效一樣。
The history of the universe in real time, however, would look very different. At about ten or twenty thousand millionyears ago, it would have a minimum size, which was equal to the maximum radius of the history in imaginary time. Atlater real times, the universe would expand like the chaotic inflationary model proposed by Linde (but one would notnow have to assume that the universe was created somehow in the right sort of state). The universe would expand toa very large size Figure 8:1 and eventually it would collapse again into what looks like a singularity in real time. Thus,in a sense, we are still all doomed, even if we keep away from black holes. Only if we could picture the universe interms of imaginary time would there be no singularities.
然而，在實的時間裡宇宙的歷史顯得非常不一樣。大約在100或200億年以前，它有一個最小的尺度，這相當於在虛時間裡的最大的半徑。在後來的即時間裡，宇宙就像由林德設想的紊亂暴漲模型那樣地膨脹（但是現在人們不必假定宇宙是從某一類正確的狀態產生出來）。宇宙會膨脹到一個非常大的尺度，並最終重新坍縮成為在即時間裡看起來像是奇點的一個東西。這樣，在某種意義上說，即使我們躲開黑洞，仍然是註定要毀滅的。只有當我們按照虛時間來描繪宇宙時才不會有奇點。
If the universe really is in such a quantum state, there would be no singularities in the history of the universe inimaginary time. It might seem therefore that my more recent work had completely undone the results of my earlierwork on singularities. But, as indicated above, the real importance of the singularity theorems was that they showedthat the gravitational field must become so strong that quantum gravitational effects could not be ignored. This in turnled to the idea that the universe could be finite in imaginary time but without boundaries or singularities. When onegoes back to the real time in which we live, however, there will still appear to be singularities. The poor astronautwho falls into a black hole will still come to a sticky end; only if he lived in imaginary time would he encounter nosingularities.
如果宇宙確實處在這樣的一個量子態裡，在虛時間裡宇宙就沒有奇點。所以，我近期的工作似乎完全使我早期研究奇點的工作成果付之東流。但是正如上面所指出的，奇點定理的真正重要性在於，它們指出引力場必然會強到不能無視量子引力效應的程度。這接著導致也許在虛時間裡宇宙的尺度有限但沒有邊界或奇點的觀念。然而，當人們回到我們生活於其中的即時間，那兒仍會出現奇點。陷進黑洞那位可憐的太空人的結局仍然是極可悲的；只有當他在虛時間裡生活，才不會遭遇到奇點。
This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just afigment of our imaginations. In real time, the universe has a beginning and an end at singularities that form aboundary to space-time and at which the laws of science break down. But in imaginary time, there are nosingularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is justan idea that we invent to help us describe what we think the universe is like. But according to the approach Idescribed in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: itexists only in our minds. So it is meaningless to ask: which is real, “real” or “imaginary” time? It is simply a matter ofwhich is the more useful description.
上述這些也許暗示所謂的虛時間是真正的即時間，而我們叫做即時間的東西恰恰是子虛烏有的空想的產物。在即時間中，宇宙的開端和終結都是奇點。這奇點構成了科學定律在那兒不成立的時空邊界。但是，在虛時間裡不存在奇點或邊界。所以，很可能我們稱之為虛時間的才真正是更基本的觀念，而我們稱作即時間的反而是我們臆造的，它有助於我們描述宇宙的模樣。但是，按照我在第一章 所描述的方法，科學理論僅僅是我們用以描述自己所觀察的數學模型，它只存在於我們的頭腦中。所以去問諸如這樣的問題是毫無意義的：“實”的或“虛”的時間，哪一個是實在的？這僅僅是哪一個描述更為有用的問題。
One can also use the sum over histories, along with the no boundary proposal, to find which properties of theuniverse are likely to occur together. For example, one can calculate the probability that the universe is expanding atnearly the same rate in all different directions at a time when the density of the universe has its present value. In thesimplified models that have been examined so far, this probability turns out to be high; that is, the proposed noboundary condition leads to the prediction that it is extremely probable that the present rate of expansion of theuniverse is almost the same in each direction. This is consistent with the observations of the microwave backgroundradiation, which show that it has almost exactly the same intensity in any direction. If the universe were expandingfaster in some directions than in others, the intensity of the radiation in those directions would be reduced by anadditional red shift.
人們還可以利用對歷史求和以及無邊界假設去發現宇宙的哪些性質可能發生。例如，人們可以計算，當宇宙具有現在密度的某一時刻，在所有方向上以幾乎同等速率膨脹的概率。在迄今已被考察的簡化的模型中，發現這個概率是高的；也就是，無邊界假設導致一個預言，即宇宙現在在每一方向的膨脹率幾乎相同是極其可能的。這與微波背景輻射的觀測相一致，它指出在任何方向上具有幾乎完全同樣的強度。如果宇宙在某些方向比其他方向膨脹得更快，在那些方向輻射的強度就會被一個附加的紅移所減小。
Further predictions of the no boundary condition are currently being worked out. A particularly interesting problem isthe size of the small departures from uniform density in the early universe that caused the formation first of thegalaxies, then of stars, and finally of us. The uncertainty principle implies that the early universe cannot have beencompletely uniform because there must have been some uncertainties or fluctuations in the positions and velocitiesof the particles. Using the no boundary condition, we find that the universe must in fact have started off with just theminimum possible non-uniformity allowed by the uncertainty principle. The universe would have then undergone aperiod of rapid expansion, as in the inflationary models. During this period, the initial non-uniformities would havebeen amplified until they were big enough to explain the origin of the structures we observe around us. In 1992 theCosmic Background Explorer satellite (COBE) first detected very slight variations in the intensity of the microwavebackground with direction. The way these non-uniformities depend on direction seems to agree with the predictionsof the inflationary model and the no boundary proposal. Thus the no boundary proposal is a good scientific theory inthe sense of Karl Popper: it could have been falsified by observations but instead its predictions have beenconfirmed. In an expanding universe in which the density of matter varied slightly from place to place, gravity wouldhave caused the denser regions to slow down their expansion and start contracting. This would lead to the formationof galaxies, stars, and eventually even insignificant creatures like ourselves. Thus all the complicated structures thatwe see in the universe might be explained by the no boundary condition for the universe together with the uncertaintyprinciple of quantum mechanics.
人們正在研究無邊界條件的進一步預言。一個特別有趣的問題是，早期字宙中物質密度對其平均值的小幅度偏離，這些偏離首先引起星系，然後是恒星，最後是我們自身的形成。不確定性原理意味著，早期宇宙不可能是完全均勻的，因為粒子的位置和速度必定有一些不確定性或起伏。利用無邊界條件，我們發現，宇宙事實上必須是從僅僅由不確定性原理允許的最小的可能的非均勻性開始的。然後，正如在暴脹模型中預言的一樣，宇宙經歷了一段快速膨脹時期。在這個期間，初始的非均勻性被放大到足以解釋在我們周圍觀察到的結構的起源。1992年宇宙背景探險者衛星（COBE）首次檢測到微波背景隨方向的非常微小的變化。這種非均勻性隨方向的變化方式似乎和暴脹模型以及無邊界設想的預言相符合。這樣，在卡爾·波普的意義上，無邊界設想是一種好的科學理論：它的預言可以被觀測證偽，但是卻被證實了。在一個各處物質密度稍有變化的膨脹字宙中，引力使得較緊密區域的膨脹減慢，並使之開始收縮。這就導致星系、恒星和最終甚至像我們自己這樣微不足道的生物的形成。因而，我們在宇宙中看到的所有複雜的結構，可由宇宙無邊界條件和量子力學中的不確定性原理給予解釋。
The idea that space and time may form a closed surface without boundary also has profound implications for the roleof God in the affairs of the universe. With the success of scientific theories in describing events, most people havecome to believe that God allows the universe to evolve according to a set of laws and does n ot intervene in theuniverse to break these laws. However, the laws do not tell us what the universe should have looked like when itstarted – it would still be up to God to wind up the clockwork and choose how to start it off. So long as the universehad a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having noboundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?
空間和時間可以形成一個沒有邊界的閉曲面的思想，對於上帝在宇宙事務中的作用還有一個深遠的含義。隨著科學理論在描述事件的成功，大部分人進而相信上帝允許宇宙按照一套定律來演化，而不介入其間促使宇宙觸犯這些定律。然而，定律並沒有告訴我們，字宙的太初應該像什麼樣子——它依然要靠上帝去卷緊發條，並選擇如何去啟動它。只要宇宙有一個開端，我們就可以設想存在一個造物主。但是，如果宇宙確實是完全自足的，沒有邊界或邊緣，它就既沒有開端也沒有終結——它就是存在。那麼，還會有造物主存身之處嗎？