Quantum Mechanics

"Although Heisenberg began thinking about an electron track in a , he came to realize that the problem of locating an electron is not one of instrumentation... [H]e found that it was never possible... to measure the precise path of an electron: for example, the droplet size is too large in a cloud chamber. Or if you try to "see" an electron with light (or x-rays), then the light photons strike the electron like billiard balls and randomly change the electron's position. ...It was ...a limit in principle on the accuracy ...a theoretical limit—an irreducible degree of uncertainty or Indeterminacy—in measuring the simultaneous position and motion of an electron. ...[T]he two measurements—of position and motion—counterbalance one another. The better you measure the position, the worse you will be able to measure the motion, and vice versa. If you could measure the position (or motion) perfectly, you would know nothing about the motion (position). The uncertainty principle thus seems to reconcile the particle-wave duality. The better you know the position... the more localized... the more it seems to act like a particle. Alternatively, if you know the motion or speed very well... [position] is diffuse, like a wave. ...[A] different point of view on duality but still ...a paradoxical fact of life."

"Instead of Newtonian certainty and determinism, quantum theory answers our questions with probability and statistics. Classical physics told us precisely where Mars was to be found. Quantum theory sends us to the gambling table to locate an electron in an atom. Then there's Heisenberg's uncertainty principle, which places an ultimate limit on our knowledge of the microworld and tells us that we can make no measurement without affecting the result."

"It must have been one evening after midnight when I suddenly remembered my conversation with Einstein and particularly his statement, "It is the theory which decides what we can observe." I was immediately convinced that the key to the gate that had been closed for so long must be sought right here. I decided to go on a nocturnal walk through Faelled Park and to think further about the matter. We had always said so glibly that the path of the electron in the cloud chamber could be observed. But perhaps what we really observed was something much less. Perhaps we merely saw a series of discrete and ill-defined spots through which the electron had passed. In fact, all we do see in the cloud chamber are individual water droplets which must certainly be much larger than the electron. The right question should therefore be: Can quantum mechanics represent the fact that an electron finds itself approximately in a given place and that it moves approximately with a given velocity, and can we make these approximations so close that they do not cause experimental difficulties?"

"Ultimately... one would hope to find a complete, consistent, unified theory that would include all... partial theories as approximations... "the unification of physics." Einstein spent most of his later years unsuccessfully searching... Einstein refused to believe in the reality of quantum mechanics, despite the important role he played in its development. Yet it seems that the uncertainty principle is a fundamental feature of the universe... A successful unified theory must... incorporate this principle."

"[T]he laws of physics need not break down at the origin of the universe. The state of the universe and its contents, like ourselves, are completely determined by the laws of physics, up to the limit set by the uncertainty principle. So much for free will."

"In September 1973, while I was visiting Moscow, I discussed s with two leading Soviet experts, Yakov Zeldovich and Alexander Starobinsky. They convince me that, according to the uncertainty principle, s should create and emit particles. I believed their arguments on physical grounds but did not like the mathematical way in which they calculated the emission. I therefore set about devising a better mathematical treatment... when I did the calculation I found to my surprise and annoyance, that even nonrotating black holes should create and emit particles at a steady rate. ...[What finally convinced me that the emission was real was that the spectrum of the emitted particles was exactly that which would be emitted by a hot body... at exactly... the correct rate to prevent violations of the second law."

"Une loi de Physique possède une certitude beaucoup moins immédiate et beaucoup plus difficile à apprécier qu'une loi de sens commun; mais elle surpasse cette dernière par la précision minutieuse et détaillée de ses prédictions. ...Cette minutie dans le détail, les lois de la Physique ne la peuvent acquérir qu'en sacrifiant quelque chose de la certitude fixe et absolue des lois de sens commun. Entre la précision et la certitude il y a une sorte de compensation; l'une ne peut croître qu'au détriment de l'autre. A law of physics possesses a certainty much less immediate and much more difficult to estimate than a law of common sense, but it surpasses the latter by the minute and detailed precision of its predictions. ...The laws of physics can acquire this minuteness of detail only by sacrificing something of the fixed and absolute certainty of common-sense laws. There is a sort of balance between precision and certainty; one cannot be increased except to the detriment of the other."

"Suppose... motion of an electron in the absence of a field of force, is to be investigated... by testing the validity of [no force implies zero acceleration]...\frac{d^2q}{dt^2} = 0, \quad ...18(3)...q ...the position of the particle at time t. The... procedure is to measure the position and momentum of the electron at... time t = t_0... to obtain two "initial conditions" which can be inserted in the solution of 18(3)... then calculate the position and momentum at some later time... and see if the calculation agrees with... observation... Suppose we observe... with light of wavelength \lambda. ...[D]iffraction of the wave sets the limit to the accuracy of a position measurement...\vartriangle q \sim \frac{\lambda}{2sin\theta}, \quad ...18(4)where \vartriangle q is the probable error in... q, and \theta is the semi-angle of the cone of rays accepted by the microscope... [and] \sim means "at least of the order of magnitude of". The experiment of Compton... shows that the interaction... involves an exchange of momentum. We may assume that the momenta... were exactly known before their interaction, but... [those] after the interaction depends on the accuracy [of the] momentum exchanged during the interaction. [T]he photon enters the microscope, and... we know its direction... within an angle 2\theta. Any attempt [to reduce] the effective aperture... increases \vartriangle q. Thus... the momentum of the photon in the plane [in which q is measured] perpendicular to the axis of the microscope... is uncertain by an amount\vartriangle p \sim \frac{2h\nu}{c}sin\theta \quad. ...18(5)The momentum of the particle after the interaction is uncertain by \vartriangle p. Combining... we have\vartriangle p \vartriangle q \sim \frac{\lambda}{2sin\theta} \frac{2h\nu}{c} sin\theta,i.e.,\vartriangle p \vartriangle q \sim h \quad. ...18(6)"

"Relativity principles require us to associate mass with the energy of radiation, and it is reasonable to suppose... an exchange of ... [T]he exchange of momentum between free electrons and radiation is very similar to the exchange... when two particles collide. ...[A] beam of light should be considered as an assembly of "units", each or which [using light (\nu), (h), speed of light (c)] possesses energy (W), momentum (p), and mass (m), given byW = h\nu; \; p = \frac{h\nu}{c}; \; m = \frac{h\nu}{c^2}. \quad 17(14)...This general picture was first suggested by Einstein... The units are now called photons... [T]he spreading of light by diffraction cannot be permanently concentrated in a small volume like the energy of a material particle. ...The pressure p, exerted by a parallel beam incident normally on a body which completely absorbs it, is...p = \rho_p, \quad ...17(15) where \rho_p is the energy per unit volume of the incident radiation. ...[C]onsider the radiation pressure of a parallel beam of light, incident on an absorbing body... the light is of frequency \nu and... there are N quanta per unit volume. Then...\rho_p = Nh\nu. \quad ...17(18)[A]ll the quanta in a cylinder of volume c [speed of light multiplied by unit area] cubic centimetres are incident upon unit area of the surface in one second, the pressure...p = NcP, \quad ...17(19)where P is the momentum of one photon. Combining...P = \frac{h\nu}{c} = \frac{h}{\lambda}.[Experimental] results... for isotropic radiation are in agreement..."

"The uncertainty has deep implications. For example, it means that a quantum particle does not move along a well-defined path through space. ...The smearing of position and momentum leads to an inherent indeterminism in the behaviour of quantum systems. ...[T]he experimenter may fire an electron at a target and find that it scatters to the left, then, on repeating... the next electron scatters to the right. Quantum mechanics still enables the relative probabilities of the alternatives to be specified precisely. ...It can make definite predictions about ensembles of identical systems, but it can generally tell us nothing definite about an individual system."

"At the heart of the quantum revolution is Heisenberg's uncertainty principle... roughly... all physical quantities... are subject to unpredictable fluctuations, so that their values are not precisely defined. ...[e.g., we are] free to measure [position x and the momentum p of a quantum particle] to arbitrary precision, but they cannot possess precise values simultaneously. The spread, or uncertainty, in their values, denoted by Ax and Ap... are such that [their] product... cannot be less than... [after Max Planck], numerically very small... so that quantum effects are generally only important in the atomic domain... not... in daily life. ...[T]his uncertainty is inherent in nature and not merely the result of technological limitations in measurement."

"If quantum mechanics is right, there is no way to get around the uncertainty principle. The reason that the electron’s probability wave spread so much after we confined it, Heisenberg would argue, is that its momentum became almost completely indeterminate. In a manner of speaking, it headed off in all directions."

"All the indistinguishable photons illuminate the array of N slits, or grating, simultaneously. If only one photon propagates, at any given time, then that individual photon illuminates the whole array of N slits simultaneously."

"My favorite deep, elegant and beautiful explanation is Albert Einstein's 1905 proposal that light consists of energy quanta, today called photons. Actually, it is little known, even among physicists, but extremely interesting how Einstein came to this position. It is often said that Einstein invented the concept to explain the photoelectric effect. Certainly, that is part of Einstein's 1905 publication, but only towards its end. The idea itself is much deeper, more elegant and, yes, more beautiful."

"Light is something like raindrops—each little lump of light is called a photon—and if the light is all one color, all the "raindrops" are the same."

"The theory of the light quantum, which Einstein initiated in 1905 and which is with us still, is certainly the most revolutionary development in the history of physics and arguably in the history of science. The creators of the theory, men such as Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli, and Paul Dirac, were often struck by the apparent "absurdity"—the utterly noncommonsensical aspects—of the world depicted by the quantum theory."

"We know something about Einstein's genius we didn't know before."

"While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible."

"One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical realities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously... can be predicted, they are not simultaneously real. ...No reasonable definition of reality could be expected to permit this."

"Starting then with the assumption that the wave function does give a complete description of reality, we arrive to the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."

"As a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. ...Thus, it is possible to assign two different wave functions... to the same reality."

"In quantum mechanics it is usually assumed that the wave function does contain a complete description of the physical reality of the system in the state to which it corresponds. ...We shall show, however, that this assumption, together with the criterion of reality given above, leads to a contradiction."

"The usual conclusion... in quantum mechanics is that when the momentum of a particle is known, its coordinate has no physical reality. More generally, it is shown in quantum mechanics that, if the operators corresponding to two physical quantities... do not commute... then the precise knowledge of one of them precludes such knowledge of the other. Furthermore, any attempt to determine the latter experimentally will alter the state of the system in such a way as to destroy the knowledge of the first. From this follows that either (1) the quantum mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to the two physical quantities do not commute the two quantities cannot have simultaneous reality."

"The elements of the physical reality cannot be determined by a priori philosophical considerations, but must be found by an appeal to results of experiments and measurements. ...We shall be satisfied with the following criterion... If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, there exists an element of physical reality corresponding to this physical quantity. It seems to us that this criterion, while far from exhausting all possible ways of recognizing a physical reality, at least provides us with one such way..."

"In a complete theory there is an element corresponding to each element in reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by the wave function is not complete."

"In conclusion, although Einstein, Podolsky, and Rosen presented a picture of the world that was disproven experimentally, I would still regard them as having won a moral victory: The then-common interpretation of quantum mechanics did indeed have a one person measuring at A, seeing a single outcome, and then making a certain prediction about a unique outcome at B; and this is indeed incompatible with relativity, and wrong. Though people are still arguing about that."

"Aage Bohr expressed the same point to me the last time I talked to him. He just couldn’t understand why there were all these conferences—conference after conference—on EPR, when it is just the way it works. It is just exactly the wrong thing to be asking about. There is a conference coming up in Finland in August with some people I’d love to talk to, but I’ve written them to say that I’m not going. If you keep trying to pull apples off the apple tree, after a while it doesn’t do. I hope that I am not being too propagandistic in speaking of the idea that when we see it all it will be so simple we’ll all say, ‘How stupid we’ve been all this time!’ We’ve got to look for the right word, the right image. So you try one word for a day, for a week, for a month, or for a year, and then you give it up and try another one."

"Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate and therefore virtually capable of being "best possibly known," i.e., of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known—at least not in the way that it could possibly be known more completely—it is due to the interaction itself. Attention has recently been called * to the obvious but very disconcerting fact that even though we restrict the disentangling measurements to one system, the representative obtained for the other system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are entirely arbitrary. It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter's mercy in spite of his having no access to it."

"When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or ψ-functions) have become entangled."

"This onslaught came down upon us as a bolt from the blue. Its effect on Bohr was remarkable... as soon as Bohr had heard my report of Einstein’s argument, everything else was abandoned: we have to clear up such a misunderstanding at once. We should reply by taking up the same example and showing the right way to speak about it. In great excitement, Bohr immediately started dictating to me the outline of such a reply. Very soon, however, he became hesitant. ‘No, this won’t do, we must try all over again... we must make it quite clear...’ So it went on for a while, with growing wonder at the unexpected subtlety of the argument."

"Finally, in that period the 'EPR paper' appeared, a collaboration with Nathan Rosen and Boris Podolsky which deals with the foundations of quantum mechanics. There are a number of physicists who consider this a fundamental contribution to that subject. I am not one of those."

"The only part of this article that will ultimately survive, I believe, is this last phrase, which so poignantly summarizes Einstein's views on quantum mechanics in his later years. The content of this paper has been referred to on occasion as the Einstein-Podolsky-Rosen paradox. It should be stressed that this paper contains neither a paradox nor any flaw of logic. It simply concludes that objective reality is incompatible with the assumption that quantum mechanics is complete. This conclusion has not affected subsequent developments in physics, and it is doubtful that it ever will. [...] Experimentalists have actively participated, as well. A number of experimental tests of quantum mechanics in general and also of the predictions of specific alternative schemes have been made. This has not led to any surprises."

"Einstein had drawn attention to nonlocality in 1935 in an effort to show that quantum mechanics must be flawed. ...Einstein proposed a thought experiment—now called the EPR experiment—involving two particles that spring from a common source and fly in opposite directions. According to the standard model of quantum mechanics, neither particle has a definite position or momentum before it is measured; but by measuring the momentum of one particle, the physicist instantaneously forces the other particle to assume a fixed position... Deriding this effect as "spooky action at a distance," Einstein argued that it violated both common sense and his own theory of special relativity, which prohibits the propagation of effects faster than the speed of light; quantum mechanics must therefore be an incomplete theory. In 1980, however, a group of French physicists carried out a version of the EPR experiment and showed that it did indeed give rise to spooky action. (The reason that the experiment does not violate special relativity is that one cannot exploit nonlocality to transmit information.)"

"The intent of the Einstein-Podolsky-Rosen paper was to show that quantum mechanics... could not be the final word regarding the physics of the microcosmos. ...they wanted to show that every particle does possess a definite position and a definite velocity at any given instant of time, and thus they wanted to conclude that the uncertainty principle reveals a fundamental limitation of the quantum mechanical approach. ...quantum mechanics provides only a partial description of the universe. ...an incomplete theory of physical reality and, perhaps, merely a stepping-stone toward a deeper framework waiting to be discovered."

"The false report that measuring one of the photons immediately affects the other leads to all sorts of unfortunate conclusions. ...the alleged effect... would violate the requirement of relativity theory that no signal... can travel faster than the speed of light. If it were to do so, it would appear to observers in some states of motion that the signal were traveling backward in time."

"The principal distortion disseminated ... is the implication, or even the explicit claim, that measuring the polarization, circular or plane, of one of the photons somehow affects the other photon. In fact, the measurement does not cause any physical effect to propagate from one photon to the other. ...If on one branch of history, the plane polarization of one photon is measured and thereby specified with certainty, then on the same branch of history the circular polarization of the other photon is also specified with certainty. On a different branch of history the circular polarization of one of the photons may be measured, in which case the circular polarization of both photons is specified with certainty. On each branch, the situation is like that of Bertlmann's socks, described by John Bell... Bertlmann... always wears one pink and one green sock. If you see just one... you know immediately the other... Yet no signal is propogated... Likewise no signal passes from one photon to the other in the experiment that confirms quantum mechanics. No action at a distance takes place."

"Many years ago David Bohm proposed replacing the hypothetical "completeness" experiment of Einstein, Podolsky, and Rosen...by a modified and more practical version. Bohm's experiment (called the EPRB...) involves the decay of a particle into two photons... the photons travel in opposite directions, have equal energy, and have identical polarizations. ...John Bell revealed that the EPRB... could be used to distinguish quantum mechanics from hypothetical hidden variable theories. Bell's theorem (also called Bell's inequalities) concerns a particular quantity that specifies the correlation between the polarizations of the two photons."

"Quantum key distribution as described above was the product of my graduate infatuation with quantum theory. It all happened back in Oxford in the late 1980s and early 1990s. I do not remember exactly what prompted me to visit the Clarendon Laboratory library one rainy day, browse the dusty shelves and pick up the original Einstein, Podolsky and Rosen paper for casual reading. However, I do remember this one sentence in the paper that drew my attention: "...If, without in any way disturbing a system, we can predict with certainty ... the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." This is a definition of perfect eavesdropping! I guess I was lucky to read it in this particular way. The rest was just about rephrasing the subject in cryptographic terms."

"If Monod and Weinberg are truly speaking for the twentieth century, then I prefer the eighteenth. But in fact Monod and Weinberg, both of them first-rate scientists and leaders of research in their specialties, are expressing a point of view which does not take into account the subtleties and ambiguities of twentieth-century physics. The roots of their philosophical attitudes lie in the nineteenth century, not in the twentieth. The taboo against mixing knowledge with values arose during the nineteenth century out of the great battle between the evolutionary biologists led by Thomas Huxley and the churchmen led by Bishop Wilberforce. Huxley won the battle, but a hundred years later Monod and Weinberg were still fighting the ghost of Bishop Wilberforce.… For the biologists, every step down in size was a step toward increasingly simple and mechanical behavior. A bacterium is more mechanical than a frog, and a DNA molecule is more mechanical than a bacterium. But twentieth-century physics has shown that further reductions in size have an opposite effect. If we divide a DNA molecule into its component atoms, the atoms behave less mechanically than the molecule... ... If we divide an atom into nucleus and electrons, the electrons are less mechanical than the atom. There is a famous experiment, originally suggested by Einstein, Podolsky and Rosen in 1935 as a thought experiment to illustrate the difficulties of quantum theory, which demonstrates that the notion of an electron existing in an objective state independent of the experimenter is untenable. The experiment has been done in various ways with various kinds of particles, and the results show clearly that the state of a particle has a meaning only when a precise procedure for observing the state is prescribed. Among physicists there are many different philosophical viewpoints, and many different ways of interpreting the role of the observer in the description of subatomic processes. But all physicists agree with the experimental facts which make it hopeless to look for a description independent of the mode of observation. When we are dealing with things as small as atoms and electrons, the observer or experimenter cannot be excluded from the description of nature. In this domain, Monod's dogma, "The cornerstone of the scientific method is the postulate that nature is objective," turns out to be untrue. … We are saying only that if as physicists we try to observe in the finest detail the behavior of a single molecule, the meaning of the words "chance" and "mechanical" will depend upon the way we make our observations. The laws of subatomic physics cannot even be formulated without some reference to the observer. "Chance" cannot be defined except as a measure of the observer's ignorance of the future. The laws leave a place for mind in the description of every molecule."

"The two photons are entangled and according to local realism, their polarization planes should become independent... a typical EPR situation. Already in 1948, observations... agreed with quantum mechanics, not with local realism."

"Due to the lucidity and apparently incontestable character of the argument, the paper of Einstein, Podolsky and Rosen created a stir among physicists and has played a large role in general philosophical discussion. Certainly the issue is of a very subtle character and suited to emphasize how far, in quantum theory, we are beyond the reach of pictorial visualization."

"It deals with an imaginary experiment, but like so many other works of the imagination this one has surpassed the intentions of its creators. In the first place, it is unlikely that they thought that the experiment they were proposing, or any facsimile, would ever be carried out. This, thanks largely to the work inspired by Bell, has now happened. Indeed, our physics journals are now resplendent with new and ever more ingenious versions of the Einstein-Podolsky-Rosen experiment, along with increasingly accurate experimental results. In the second place, and this is also an aftermath of Bell’s work, the Einstein-Podolsky-Rosen experiment has made its way into much of the popular folklore about the quantum theory. (It is usually referred to in the literature, familiarly, as the EPR experiment.)"

"While Einstein's belief in an objective reality is similar to that of Weinberg and Sokal, his arguments for his conception of reality are not. In fact, Einstein was no "naive realist," despite such caricaturing of his stand by the Copenhagen orthodoxy. He ridiculed the "correspondence" view of reality that many scientists accept uncritically. Einstein fully realized that the world is not presented to us twice-first as it is, and second, as it is theoretically described-so we can compare our theoretical "copy" with the "real thing." The world is given to us only once - through our best scientific theories. So Einstein deemed it necessary to ground his concept of objective reality in the invariant characteristics of our best scientific theories."

"We physicists are always checking to see if there is something the matter with the theory. That’s the game, because if there is something the matter, it’s interesting! But so far, we have found nothing wrong with the theory of quantum electrodynamics. It is, therefore, I would say, the jewel of physics—our proudest possession. The theory of quantum electrodynamics is also the prototype for new theories that attempt to explain nuclear phenomena, the things that go on inside the nuclei of atoms. If one were to think of the physical world as a stage, then the actors would be not only electrons, which are outside the nucleus in atoms, but also quarks and gluons and so forth—dozens of kinds of particles—inside the nucleus. And though these “actors” appear quite different from one another, they all act in a certain style—a strange and peculiar style—the “quantum” style. At the end, I’ll tell you a little bit about the nuclear particles. In the meantime, I’m only going to tell you about photons—particles of light—and electrons, to keep it simple. Because it’s the way they act that is important, and the way they act is very interesting."

"Once we got past the obscurities produced by spontaneous symmetry breaking in the weak interactions and color trapping in the strong interactions, the Standard Model was revealed to us as a theory that was really not very different from quantum electrodynamics. We had more gauge fields, not just the electromagnetic field but gluon and W and Z fields. There were more fermions, not just the electron but a whole host of charged leptons and neutrinos and quarks. But the Standard Model seemed to be quantum electrodynamics writ large."

"The history of QED in the period from 1946 to 1949 has many similarities with the history of the developments of quantum mechanics from 1925 to 1927, when Schrödinger and Heisenberg had propounded two different approaches to quantum mechanics. The correspondence can be taken to be: Feynman is to Heisenberg what Tomonaga-Schwinger is to Schrödinger, with Dyson initially playing the role of Pauli, Schrödinger, and Eckart in proving the equivalence of the two approaches."

"When after some weeks I had a chance to talk to Oppenheimer, I was astonished to discover that his reasons for being uninterested in my work were quite the opposite of what I had imagined. I had expected that he would disparage my program as merely unoriginal, a minor adumbration of Schwinger and Feynman. On the contrary, On the contrary, he considered it to be fundamentally on the wrong track. He thought adumbrating Schwinger and Feynman to be a wasted effort, because he did not believe that the ideas of Schwinger and Feynman had much to do with reality. I had known that he had never appreciated Feynman, but it came as a shock to hear him now violently opposing Schwinger, his own student, whose work he had acclaimed so enthusiastically six months earlier. He had somehow become convinced during his stay in Europe that physics was in need of radically new ideas, that this quantum electrodynamics of Schwinger and Feynman was just another misguided attempt to patch up old ideas with fancy mathematics."

"With the help of W Furry, Weisskopf ... showed that the inclusion of positrons in the electron self-energy calculation reduced the degree of divergence to ln(1/r0), where r0 again represents a minimum cut-off distance. Thus positrons were a partial, but not sufficient, help. Another infinite quantity occurring in quantum electrodynamics arises as a result of the production of virtual electron-positron pairs by a photon. Like the electron self-energy, this vacuum polarization divergence depends on the logarithm of a cut-off parameter. Despite the infinite nature of vacuum polarization, it was possible to calculate its effect on the Coulomb interaction, for instance in a hydrogen atom, by comparing the interaction at large and shorter distances."

"The theory of quantum electrodynamics describes nature as absurd from the point of view of common sense. And it fully agrees with experiment. So I hope you can accept nature as She is—absurd."

"The sum-over-paths formulation is particularly convenient for integrating out one set of coordinates to concentrate on the remaining set. Thus the photon propagator in quantum electrodynamics is obtained ... by "integrating out" the photon variables, leaving electrons and positrons, both real and virtual, to interact by means of the covariant function \delta (x^2) + (\pi i x^2)^-1."