Wave function of the Universe. Offprint from: Physical Review D, Vol. 28, No. 12, 15 December 1983.

[Brookhaven, NY: The American Physical Society], 1983.

First edition, very rare offprint, and Hawking’s own file copy, of this famous paper describing the 'no-boundary proposal’ for the origin of the universe. “Stephen sought to understand the whole universe in scientific terms. As he said famously, ‘My goal is simple. It is a complete understanding of the universe’ … From 1982 onwards, Stephen concentrated his efforts on the deeper puzzle of the boundary conditions required to bring about inflation and the probability of them coming about … The singularity theorems proved by Stephen, Penrose and others showed conclusively that the classical Einstein equation implied that the universe began in a hot Big Bang. But the singularity theorems also showed that the beginning could not be described by a classical space-time geometry obeying the Einstein equation with three space directions and one time direction at each point. Rather, they showed something more sweeping: the classical Einstein equation breaks down at the Big Bang and, along with that, the notion that it could be described by a classical space-time … [it was therefore necessary to try] to describe the quantum birth of the universe. Stephen first put forward a proposal along these lines at a conference in the Vatican in 1981, where he suggested that the universe began with a regular Euclidean geometry having four space dimensions which made a quantum transition to a Lorentzian geometry with three space dimensions and one time dimension that we have today. To put this idea on a solid footing required a quantum state—a wave function of the universe” (Carr et al.). “‘Murray Gell-Mann used to ask me,’ Hartle said, referring to the late Nobel Prize-winning physicist, ‘if you know the wave function of the universe, why aren’t you rich?’” (Wachover). “Because one is using Euclidean space-times, in which the time direction is on the same footing as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time would be like the surface of the earth, only with two more dimensions … the quantum theory of gravity has opened up a new possibility, in which there would be no boundary to space-time and so there would be no need to specify the behavior at the boundary. There would be no singularities at which the laws of science broke down, and no edge of space-time at which one would have to appeal to God or some new law to set the boundary conditions for space-time. One could say: ‘The boundary condition of the universe is that it has no boundary.’ The universe would be completely self-contained and not affected by anything outside itself. It would neither be created nor destroyed, it would just BE” (Hawking). “The scientific importance of the no-boundary proposal is not just as a successful theory of the origin of the basic structure of the universe—it has also had a significant effect on how we think about the universe and our place in it” (Carr et al.). We are not aware of any other copy of this offprint having appeared on the market.

Provenance: the estate of Professor Stephen Hawking (1942-2018).

“In 1981, many of the world’s leading cosmologists gathered at the Pontifical Academy of Sciences, a vestige of the coupled lineages of science and theology located in an elegant villa in the gardens of the Vatican. Stephen Hawking chose the august setting to present what he would later regard as his most important idea: a proposal about how the universe could have arisen from nothing” (Wachover).

At that time the standard theory of the origin of the universe was the ‘Big Bang’ model. This had apparently been confirmed by the discovery of the cosmic microwave background by Arno Penzias and Robert Wilson in 1965. But the Big Bang theory had some problems. Not only was there the question of what happened before the Big Bang, but the theory predicted a universe that was not nearly as smooth and uniform as ours. In 1980, the cosmologist Alan Guth realized that the Big Bang’s problems could be fixed with an add-on: an initial, exponentially fast expansion known as cosmic inflation lasting some 10-33 seconds, which would have rendered the universe smooth and flat before expansion resumed at a slower rate allowing gravity to create the relatively few inhomogeneities we see in the current universe. Inflation quickly became the leading theory of our cosmic origins. Yet problems remained: inflation requires extremely specific initial conditions, and the question becomes, why were the initial conditions just those required to make the expansion theory work?

In his Vatican lecture, “Hawking proposed a solution to the problem of initial conditions: there is no beginning at all. According to the record of the Vatican conference, the Cambridge physicist, then 39 and still able to speak with his own voice, told the crowd, “There ought to be something very special about the boundary conditions of the universe, and what can be more special than the condition that there is no boundary?”

“The ‘no-boundary proposal,’ which Hawking and his frequent collaborator, James Hartle, fully formulated in a 1983 paper [offered here], envisions the cosmos having the shape of a shuttlecock. Just as a shuttlecock has a diameter of zero at its bottommost point and gradually widens on the way up, the universe, according to the no-boundary proposal, smoothly expanded from a point of zero size. Hartle and Hawking derived a formula describing the whole shuttlecock – the so-called ‘wave function of the universe’ that encompasses the entire past, present and future at once – making moot all contemplation of seeds of creation, a creator, or any transition from a time before. ‘Asking what came before the Big Bang is meaningless, according to the no-boundary proposal, because there is no notion of time available to refer to,’ Hawking said in another lecture at the Pontifical Academy in 2016, a year and a half before his death. ‘It would be like asking what lies south of the South Pole’ …

“Hawking and Hartle … defined the no-boundary wave function describing such a universe using an approach invented by Hawking’s hero, the physicist Richard Feynman. In the 1940s, Feynman devised a scheme for calculating the most likely outcomes of quantum mechanical events. To predict, say, the likeliest outcomes of a particle collision, Feynman found that you could sum up all possible paths that the colliding particles could take, weighting straightforward paths more than convoluted ones in the sum. Calculating this ‘path integral’ gives you the wave function: a probability distribution indicating the different possible states of the particles after the collision.

“Likewise, Hartle and Hawking expressed the wave function of the universe – which describes its likely states – as the sum of all possible ways that it might have smoothly expanded from a point. The hope was that the sum of all possible ‘expansion histories’ … would yield a wave function that gives a high probability to a huge, smooth, flat universe like ours … The problem is that the path integral over all possible expansion histories is far too complicated to calculate exactly. Countless different shapes and sizes of universes are possible … to actually solve for the wave function using Feynman’s method, Hartle and Hawking had to drastically simplify the situation, ignoring even the specific particles that populate our world. They considered the path integral over all possible toy universes in ‘minisuperspace,’ defined as the set of all universes with a single energy field coursing through them: the energy that powered cosmic inflation” (Wachover).

A further difficulty is that, in calculating the Feynman path integrals that represent the sum over expansion histories, “one runs into severe technical problems. The only way around these is the following peculiar prescription: one must add up the waves for particle histories that are not in the ‘real’ time that you and I experience but take place in what is called ‘imaginary’ time … one must measure time using imaginary numbers, rather than real ones. This has an interesting effect on space-time: the distinction between time and space disappears completely. A space-time in which events have imaginary values of the time coordinate is said to be Euclidean … To avoid the technical difficulties in actually performing the sum over [expansion] histories, these curved space-times must be taken to be Euclidean …

“In the classical theory of gravity, which is based on real space-time, there are only two possible ways the universe can behave: either it has existed for an infinite time, or else it had a beginning at a singularity at some finite time in the past. In the quantum theory of gravity, on the other hand, a third possibility arises. Because one is using Euclidean space-times, in which the time direction is on the same footing as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge … If the universe really is in such a quantum state, there would be no singularities in the history of the universe in imaginary time … When one goes back to the real time in which we live, however, there will still appear to be singularities.

“This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just a figment of our imaginations. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is just an idea that we invent to help us describe what we think the universe is like” (Hawking).

In the shuttlecock picture, in which two of the space dimensions of our universe have been suppressed, the circular cross-sections perpendicular to the axis of the shuttlecock represent space at a particular time, while time is the perpendicular direction up or down the shuttlecock. At the bottommost point of the shuttlecock, however, there is no upward direction, all directions tangential to the shuttlecock are parallel to the spatial cross-sections (just as one can travel south at every point of the earth’s surface, except at the south pole). Thus, at the bottommost point of the shuttlecock, which represents the origin of the universe, there is only space and no time – because time has become imaginary. If time remained real right back to the origin point, the appropriate picture would be not a shuttlecock but a cone, with a sharp point – a singularity – at the bottommost point. In the shuttlecock picture, real time comes into existence as soon as we leave the origin point. But a shuttlecock is a classical object (!), and when quantum theory is taken into account it turns out that the universe makes a ‘quantum transition’ from imaginary time to real time soon after the universe begins to expand from the origin point. (In quantum theory, no smaller interval of time than the ‘Planck time’ (approximately 10-43 seconds) has any meaning.)

The sum over histories, together with the no-boundary proposal, can be tested by observations. “For example, one can calculate the probability that the universe is expanding at nearly the same rate in all different directions at a time when the density of the universe has its present value. In the simplified models that have been examined so far, this probability turns out to be high; that is, the proposed no-boundary condition leads to the prediction that it is extremely probable that the present rate of expansion of the universe is almost the same in each direction. This is consistent with the observations of the microwave background radiation, which show that it has almost exactly the same intensity in any direction. If the universe were expanding faster in some directions than in others, the intensity of the radiation in those directions would be reduced by an additional red shift.

“Further predictions of the no-boundary condition are currently being worked out. A particularly interesting problem is the size of the small departures from uniform density in the early universe that caused the formation first of the galaxies, then of stars, and finally of us. The [Heisenberg] uncertainty principle implies that the early universe cannot have been completely uniform because there must have been some uncertainties or fluctuations in the positions and velocities of the particles. Using the no-boundary condition, we find that the universe must in fact have started off with just the minimum possible non-uniformity allowed by the uncertainty principle. The universe would have then undergone a period of rapid expansion, as in the inflationary models. During this period, the initial non-uniformities would have been amplified until they were big enough to explain the origin of the structures we observe around us. In 1992 the Cosmic Background Explorer satellite (COBE) first detected very slight variations in the intensity of the microwave background with direction. The way these non-uniformities depend on direction seems to agree with the predictions of the inflationary model and the no-boundary proposal … In an expanding universe in which the density of matter varied slightly from place to place, gravity would have caused the denser regions to slow down their expansion and start contracting. This would lead to the formation of galaxies, stars, and eventually even insignificant creatures like ourselves. Thus all the complicated structures that we see in the universe might be explained by the no-boundary condition for the universe together with the uncertainty principle of quantum mechanics.

“The idea that space and time may form a closed surface without boundary also has profound implications for the role of God in the affairs of the universe. With the success of scientific theories in describing events, most people have come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the universe to break these laws. However, the laws do not tell us what the universe should have looked like when it started – it would still be up to God to wind up the clockwork and choose how to start it off. So long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply BE. What place, then, for a creator?” (Hawking). 

“Hartle and Hawking’s paper became one of the most widely cited papers in the field, ranking second on a list of citations on the Stanford Linear Accelerator Center’s online archive (behind Hawking’s follow-up to his initial black hole radiation paper). (Of the 88 papers listed on the all-time list, 11 were written or co-written by Hawking)” (Larsen, p. 67).

Hawking continued to work on the no-boundary proposal. “In a series of papers over many years he and his collaborators [showed] that the no-boundary proposal predicts an early period of inflation. Specifically, Stephen and Jonathan Halliwell showed that the no-boundary proposal describes an ensemble of universes in which inflation triggers the emergence of a classical Lorentzian space-time from the quantum fuzz at the beginning, along with fluctuations that are initially predicted to be in their ground state. Thus, the no-boundary proposal provides a foundation for inflationary cosmology. [Moreover, the] no-boundary proposal does not posit classical backgrounds; it predicts them quantum mechanically, providing a unified origin for both classical backgrounds and quantum fluctuations—a remarkable, simple and beautiful achievement” (Carr et al.).

The no-boundary proposal has also led to a deeper understanding of quantum mechanics itself. “[Hawking] thought we understood quantum mechanics well enough and, indeed, he was successful in applying it without worrying about any foundational questions. But the no-boundary proposal motivated new formulations of quantum mechanics that were adequate for cosmology … The usual textbook (Copenhagen) formulations of quantum mechanics are inadequate for cosmology, not least because they predict probabilities of measurements made by observers. But in the very early universe no measurements were being made and there were no observers around to make them. A formulation of quantum mechanics general enough for cosmology was started by Hugh Everett [the ‘many universes’ interpretation] and developed by many. That effort led to the ‘decoherent (or consistent) histories’ approach to quantum theory and is adequate for quantum cosmology. It implies, however, that the Copenhagen interpretation of quantum theory is an approximation for measurement situations” (ibid.).

Carr, Ellis, Gibbons, Hartle, Hertog, Penrose, Perry, & Thorne, ‘Stephen William Hawking CH CBE. 8 January 1942 – 14 March 2018,’ Bibliographical Memoirs of Fellows of the Royal Society, Vol. 66, June 2019 (royalsocietypublishing.org/doi/10.1098/rsbm.2019.0001). Hawking, A Brief History of Time, 1988. Larsen, Stephen Hawking: A Biography, 2005. Wachover, ‘Physicists debate Hawking’s idea that the universe had no beginning,’ Quantamagazine, June 6, 2019 (quantamagazine.org/physicists-debate-hawkings-idea-that-the-universe-had-no-beginning-20190606/).

Large 8vo (285 x 210mm), pp. 2960-2975. Stapled self-wrappers (punch holes in inner margin, not affecting text).

Item #5075

Price: $8,500.00