The Dreams That Stuff is Made of

by Stephen Hawking

IX. SUMMARY OF RESULTS

  In this section the results of the preceding pages will be summarized, so far as they relate to the performance of practical calculations. In effect, this summary will consist of a set of rules for the application of the Feynman radiation theory to a certain class of problems.

  Suppose an electron to be moving in an external field with interaction energy given by (33). Then the interaction energy to be used in calculating the motion of the electron, including radiative corrections of all orders, is

  (70)

  with Hi given by (16), and the P notation as defined in (29).

  To find the effective n’th-order radiative correction to the potential acting on the electron, it is necessary to calculate the matrix elements of Jn for transitions from one one-electron state to another. These matrix elements can be written down most conveniently in the form of an operator Kn bilinear in and ψ, whose matrix elements for one-electron transitions are the same as those to be determined. In fact, the operator Kn itself is already the matrix element to be determined if the and ψ contained in it are regarded as one-electron wave functions.

  To write down Kn , the integrand Pn in Jn is first expressed in terms of its factors, ψ, and A, all suffixes being indicated explicitly, and the expression (37) used for jµ. All possible graphs G with (n + 1) vertices are now drawn as described in Section VII, omitting disconnected graphs, graphs with self-energy parts, and graphs with external vacuum polarization parts as defined in Section VIII. It will be found that in each graph there are at each vertex two electron lines and one photon line, with the exception of x0 at which there are two electron lines only; further, such graphs can exist only for even n. Kn is the sum of a contribution K(G) from each G.

  Given G, K(G) is obtained from Jn by the following transformations. First, for each photon line joining x and y in G, replace two factors Aµ(x) Aν(y) in P’n (regardless of their positions) by

  (71)

  with D’F given by (67) with R3 = 1, the function DF being defined by (42). Second, for each electron line joining x to y in G, replace two factors ψα(x)ψβ(y) in Pn (regardless of positions) by

  (72)

  with S’F given by (62) with R2 = 1, the function SF being defined by (44) and (45). Third, replace the remaining two factors P( in Pn by in this order. Fourth, replace by given by

  (73)

  or, more generally, by (69). Fifth, multiply the whole by (−1)l , where l is the number of closed loops in G as defined in Section VII.

  The above rules enable Kn to be written down very rapidly for small values of n. It should be observed that if Kn is being calculated, and if it is not desired to include effects of higher order than the n’th, then D’F , S’F , and in (71), (72), and (73) reduce to the simple functions DF , SF, and . Also, the integrand in Jn is a symmetrical function of x1, . . . , xn; therefore, graphs which differ only by a relabeling of the vertices x1, ..., xn give identical contributions to Kn and need not be considered separately.

  The extension of these rules to cover the calculation of matrix elements of (70) of a more general character than the one-electron transitions hitherto considered presents no essential difficulty. All that is necessary is to consider graphs with more than two “loose ends,” representing processes in which more than one particle is involved. This extension is not treated in the present paper, chiefly because it would lead to unpleasantly cumbersome formulas.

  X. EXAMPLE—SECOND-ORDER RADIATIVE CORRECTIONS

  As an illustration of the rules of procedure of the previous section, these rules will be used for writing down the terms giving second-order radiative corrections to the motion of an electron in an external field. Let the energy of the external field be

  (74)

  Then there will be one second-order correction term

  arising from the substitution (73) in the zero-order term (74). This is the well-known vacuum polarization or Uehling term.op

  The remaining second-order term arises from the second-order part J2 of (70). Written in expanded form, J2 is

  Next, all admissable graphs with the three vertices x0, x1, x2 are to be drawn. It is easy to see that there are only two such graphs, that G shown in Fig. 1, and the identical graph with x1 and x2 interchanged. The full lines are electron lines, the dotted line a photon line. The contribution K(G ) is obtained from J2 by substituting according to the rules of Section IX; in this case l = 0, and the primes can be omitted from (71), (72), (73) since only second-order terms are required. The integrand in K(G) can be reassembled into the form of a matrix product, suppressing the suffixes α, ..., ζ. Then, multiplying by a factor 2 to allow for the second graph, the complete second-order

  FIG. 1

  correction to (74) arising from J2 becomes

  This is the term which gives rise to the main part of the Lamb-Retherford line shift,oq the anomalous magnetic moment of the electron, or and the anomalous hyperfine splitting of the ground state of hydrogen.os

  The above expression L is formally simpler than the corresponding expression obtained by Schwinger, but the two are easily seen to be equivalent. In particular, the above expression does not lead to any great reduction in the labor involved in a numerical calculation of the Lamb shift. Its advantage lies rather in the ease with which it can be written down.

  In conclusion, the author would like to express his thanks to the Commonwealth Fund of New York for financial support, and to Professors. Schwinger and Feynman for the stimulating lectures in which they presented their respective theories.

  Notes added in proof (To Section II). The argument of Section II is an over-simplification of the method of Tomonaga,1 and is unsound. There is an error in the derivation of (3); derivatives occurring in H(r) give rise to non-commutativity between H(r) and field quantities at r’ when r is a point on σ infinitesimally distant from r ’. The argument should be amended as follows. Φ is defined only for flat surfaces t(r) = t, and for such surfaces (3) and (6) are correct. ψ is defined for general surfaces by (12) and (10), and is verified to satisfy (9). For a flat surface, Φ and ψ are then shown to be related by (7). Finally, since H1 does not involve the derivatives in H, the argument leading to (3) can be correctly applied to prove that for general σ the state-vector ψ(σ) will completely describe results of observations of the system on σ.

  (To Section III). A covariant perturbation theory similar to that of Section III has previously been developed by E. C G. Stueckelberg, Ann. d. Phys. 21, 367 (1934); Nature, 153, 143 (1944).

  (To Section V). Schwinger’s “affective potential” is not Hr given by (25), but is . Here Q is a “square-root” of S(∞) obtained by expanding by the binomial theorem. The physical meaning of this is that Schwinger specifies states neither by Ω nor by Ω’, but by an intermediate state-vector Ω" = QΩ = Q−1Ω’, whose definition is symmetrical between past and future. is also symmetrical between past and future. For one-particle states, HT and are identical.

  Equation (32) can most simply be obtained directly from the product expansion of S(∞).

  (To Section VII). Equation (62) is incorrect. The function S’F is well-behaved, but its fourier transform has a logarithmic dependence on frequency, which makes an expansion precisely of the form (62) impossible.

  (To Section X). The term L still contains two divergent parts. One is an “infra-red catastrophe” removable by standard methods. The other is an “ultraviolet” divergence, and has to be interpreted as an additional charge-renormalization, or, better, cancelled by part of the charge-renormalization calculated in Section VIII.

  Chapter Nine

  In this section, we present two lecture series and a historical account by some of the founders of quantum theory. In 1925–26 Max Born presented a series of lectures at MIT entitled “Problems of Atomic Dynamics.” This was immediately after Heisenberg had, with Born’s aid, worked out the first quantum theory of the atom, and it was during this time that Schrodinger published his wave theory of quantum mechanics. These lectures present t
he atomic theory right as it was being worked out. Born starts by introducing the classical theory of mechanics and then shows how the Bohr model of the atom was able to explain the hydrogen atom. He explains why Bohr’s model was unsuccessful for other elements including helium. Finally, he presents the matrix method for the new quantum theory, and shows how to derive fundamental commutation relationships upon which the uncertainty principle is founded. During the course of the lecture series insights by other researchers were being discovered. For example, Pauli’s introduction of the spin quantum number and Schrodinger’s and Dirac’s formulations of quantum theory all happened over the course of this series, and it is interesting to see the evolution of Born’s ideas as the new quantum theory took shape.

  The second work in this chapter is a historical account by the physicist George Gamow of his experiences working with the founders of quantum theory. In the late 1920s Gamow received a fellowship to study at Niels Bohr’s institute in Copenhagen. There he worked with many of quantum theory’s greatest pioneers, including, of course, Bohr himself. In “Thirty Years that Shook Physics,” he gives an insider’s perspective on the historical development of quantum theory. He shares numerous anecdotes about his experiences with these great thinkers, providing a refreshingly human face to this esoteric but fascinating field.

  The last work in this section is a series of four lectures delivered by Paul Dirac at Yeshiva University outlining developments in quantum theory. In the first lecture, Dirac starts with the Hamiltonian formulation of classical mechanics and shows how to develop quantum mechanics by implementing a quantization principle. In the second lecture, Dirac generalizes the method in the first lecture by showing how to obtain a quantum field theory from a classical field theory. Dirac is ultimately concerned with obtaining a quantum theory that can incorporate Einstein’s theory of general relativity. From general relativity we know that gravity curves space, and thus Dirac wanted to know if it is possible to derive a relativistic quantum theory on curved surfaces. In the third lecture, Dirac examines this question and decides that in general it is not possible to obtain relativistic quantum theory on a curved surface. Finally, he shows that it is possible to develop a relativistic quantum theory on a flat surface. Thus, we can obtain a quantum theory which is consistent with special but not general relativity. The search for a quantum theory that is consistent with general relativity or a quantum gravity theory is still an unsolved problem in physics. Finding this theory is, perhaps, the primary goal of theoretical physics. Currently the most favored type of quantum gravity theory is known as string theory, but it still remains to be shown if string theory is an accurate description of reality. It will be fascinating to see in the coming years which quantum gravity theory best describes our universe, because once this theory is found, we will for the first time have a fundamental understanding of all known physical laws.

  PROBLlEMS OF ATOMIC DYNAMICS

  BY

  MAX BORN

  PREFACE

  The lectures which constitute this book are given just as they were presented at the Massachusetts Institute of Technology from November 14, 1925, to January 22, 1926, without any amplification. They do not purport to be a text-book—for of these we have enough—but rather an exposition of the present status of research in those regions of physics in which I myself have made investigations, and of which I therefore believe that I can take a comprehensive view. In the short time that was at my disposal, I could neither seek for completeness nor consider minutiæ. It was my purpose to present methods, objects of investigation, and the most important results. I have avoided references and have only occasionally named individual authors. I take this occasion to ask the pardon of all those colleagues whose names I have omitted to mention.

  The lectures on the theory of lattices are essentially an abstract of certain sections of my book, Atomtheorie des festen Zustandes, and of subsequent works on this topic. In the same manner, the earlier lectures on the structure of the atom are closely related to my book Atommechanik, but I soon made the transition to a different point of view. At the time I began this course of lectures, Heisenberg’s first paper on the new quantum theory had just appeared. Here his masterly treatment gave the quantum theory an entirely new turn. The paper of Jordan and myself, in which we recognized the matrix calculus as the proper formulation for Heisenberg’s ideas, was in press, and theThis work was originally published in 1926 by the Massachusetts Institute of Technology.

  manuscript of a third paper by the three of us was almost completed. Though the results contained in this third paper left no doubt in my mind as to the superiority of the new methods to the old, I could not bring myself to plunge directly into the new quantum mechanics. To do this would not only be to deny to Bohr’s great achievement its due need of credit, but even more to deprive the reader of the natural and marvelous development of an idea. I have consequently begun by presenting the Bohr theory as an application of classical mechanics, but have emphasized more than is usual its weaknesses and conceptual difficulties. It is perhaps superfluous to state that this is only done to establish the necessity of a new conception, and is not intended as a hostile criticism of Bohr’s immortal work. As the course proceeded, further achievements of the new method came to my notice. I was able to introduce some of these into the lectures. Pauli’s theory of the hydrogen atom is a case in point. Of others, such as the treatment of the theory of aperiodic processes in terms of a general calculus of operators, developed by N. Wiener and myself, I was able to give a sketch. These sections are not so much a report on scientific results as an enumeration of the problems which seem of most interest to us theoretical physicists.

  I wrote the original text in German. It was then translated into English by Dr. W. P. Allis and Mr. Hans Müller, and read through by me. Mr. F. W. Sears revised the second part; finally Dr. M. S. Vallarta went carefully through the complete text in order to verify the formulas and make the English idiomatic. I hereby express my sincere thanks to all these gentlemen, who have spent much labor on this work of revision, and have sacrificed much valuable time, as well as to Assistant Dean H. E. Lobdell, who has taken great pains in the supervision of the work of publication.

  I feel it as a great honor that this book appears as a publication of the Massachusetts Institute of Technology. For this I wish to express my thanks to President S. W. Stratton and Professor C. L. Norton, the Head of the Department of Physics. To Professor Paul Heymans, who has not merely extended to my wife and myself the hospitality of his house, but also shared his office with me during the three months of my stay at the Institute, I wish to express my gratitude in visible form by the dedication of this little book.

  MAX BORN

  MASSACHUSETTS INSTITUTE OF TECHNOLlOGY

  January, 1926

  SERIES I THE STRUCTURE OF THE ATOM

  LECTURE 1

  Comparison between the classical continuum theory and the quantum theory—Chief experimental results on the structure of the atom—General principles of the quantum theory—Examples.

  Physics today is everywhere based on the theory of atoms. Through experimental and theoretical researches we have reached the conviction that matter is not infinitely divisible, but that there exist ultimate units of matter which cannot be further divided. However, it is not the atoms of the chemists that we feel authorized in calling “indivisible”; on the contrary they are very complicated structures composed of smaller elements. These are, from the point of view of recent investigations, the atoms of electricity, the (negative) electrons and the (positive) protons. It is conceivable that at a later epoch science will change its point of view and penetrate to still smaller elements; in this case the philosophical significance of atomistics could no longer be valued as highly. The last units would not be anything absolute, but only a measure of the present status of science. But I do not think that is so; I believe that we can hope that we have not to do with an endless chain of divisions, but that we are near the end of a fin
ite chain, perhaps we have even attained it. The reasons that can be given for this optimism lie less in the experimental evidence for the reality of atoms, protons and electrons, which the new physics has furnished, than in the special character of the laws which govern the interactions of elementary electric particles. These laws have indeed properties which permit us to conclude that we are near their final formulation.

  Such an assertion may seem too bold, because all philosophies of all ages have taught that human knowledge is incomplete, that each goal of knowledge is attained only at the cost of new puzzles. Up to the present, in physics as in other sciences, every result that our age has proclaimed as absolute has had to fall after a few years, decades or centuries, because new investigations have brought new knowledge and we have become used to consider the true laws of nature as unattainable ideals to which the so-called laws of physics are only successive approximations. Now, when I say that certain formulations of the laws of the atomistics of today have a character which is in a certain sense final, this does not fit in with our scheme of successive approximations and it becomes necessary that I offer an explanation. This special character that the atom possesses is the appearance of whole numbers. We pretend not only that in any body, for instance a piece of metal, there exist a certain finite number of atoms or electrons, but further that the properties of a single atom and the processes which occur during the interaction of several atoms are capable of being described by whole numbers. This is the substance of the quantum theory, the fundamental significance of which is based not only in its practical application but above all in its philosophical consequences considered here. To illustrate this idea we consider a small body free to move in a straight line. According to the usual ideas it can be at any time at any point. To fix this point we give the coördinate x measured from a point 0.