Quantum Mechanics

"Hence coherence survives to the extent to which an experiment fails to distinguish between different eigenvalues of the observable being measured, and distinct final states of the object display interference if the final states of the apparatus overlap. […] When the apparatus ends up in one member of a set of orthogonal states, the experiment unambiguously determines whether or not the object is then in a particular eigenstate of the observable of interest, but the result becomes increasingly ambiguous with increasing overlap between the states of the apparatus; furthermore, states of the object with distinct eigenvalues interfere with a visibility that is a measure of the ambiguity with which the states assumed by the apparatus determine the states of the object."

"We see that the measuring process in quantum mechanics has a "two- faced" character: it plays different parts with respect to the past and future of the electron. With respect to the past, it "verifies" the probabilities of the various possible results predicted from the state brought about by the previous measurement. With respect to the future, it brings about a new state. Thus the very nature of the process of measurement involves a far-reaching principle of irreversibility."

"When we measure a real dynamical variable ξ, the disturbance involved in the act of measurement causes a jump in the state of the dynamical system. From physical continuity, if we make a second measurement of the same dynamical variable ξ immediately after the first, the result of the second measurement must be the same as that of the first. Thus after the first measurement has been made, there is no indeterminacy in the result of the second. Hence, after the first measurement has been made, the system is in an eigenstate of the dynamical variable ξ, the eigenvalue it belongs to being equal to the result of the first measurement. This conclusion must still hold if the second measurement is not actually made. In this way we see that a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement. We can infer that, with the dynamical system in any state, any result of a measurement of a real dynamical variable is one of its eigenvalues. Conversely, every eigenvalue is a possible result of a measurement of the dynamical variable for some state of the system, since it is certainly the result if the state is an eigenstate belonging to this eigenvalue. This gives us the physical significance of eigenvalues. The set of eigenvalues of a real dynamical variable are just the possible results of measurements of that dynamical variable and the calculation of eigenvalues is for this reason an important problem."

"What we have learnt from this chapter is that we cannot have a direct evidence of, i.e. directly measure, a quantum state of a single system. Our experience is only connected with the experimental values of observables, and any time we measure an observable we can only have a partial experience of a system under a certain perspective but we can never have a complete experience that would be represented by an observation of the state vector, which is – in a quantum-mechanical sense – a complete description of the system. In other words, the quantum state is not an observable in the classical sense. However, since this feature of the quantum state is not due to subjective ignorance but rather to an intrinsic characteristic of the microscopic world, there are no definitive reasons to deny the reality of a quantum state."

"By now the reader will have realized that measurement in quantum physics is fundamentally different from that in classical physics. In classical physics, a measurement reveals a pre-existing property of the physical system that is tested. If a car is driving at 180 km h−1 on the highway, the measurement of its speed by radar determines a property that exists prior to the measurement, which gives the police the legitimacy to give a ticket to the driver. On the contrary, the measurement of the Sx component of a spin-1/2 particle in the state |+〉 does not reveal a value of Sx existing before the measurement. The spread in the results of measuring Sx in this case is sometimes attributed to “uncontrollable perturbation of the spin due to the measurement process,” but the value of Sx does not exist before the measurement, and that which does not exist cannot be perturbed."

"Proton decays into a positron and neutral pion, p → e+π0, are a dominant decay mode in many GUT models. It also has a very clean experimental signature in a water Cherenkov detector with full reconstruction of the event. After decades of search, the sensitivity is still improving with advancement of detector technology and analysis technique. One of examples for such a technique is the background suppression with the neutron tagging. In the proton decay events, the probability of neutron emission is rather small, while in the atmospheric neutrino events, which is the dominant background of proton decay searches, often neutrons are produced. Thus, neutron tagging can provide an additional handle to suppress the background for the proton decay search and improve the sensitivity."

"Masashi Yokoyama and Proto-collaboration:"

"Besides non-zero neutrino masses, the other classic experimental implication of unification is proton instability, with a very long but perhaps not inaccessible lifetime. That prediction has not yet been verified, despite heroic efforts. The existing limits put significant pressure on the framework. Reading it optimistically: There is an excellent chance that further efforts along this line would be rewarded."

"It is commonly believed that grand unified theories (GUTs) predict proton decay. This is because the exchange of extra GUT gauge bosons gives rise to dimension 6 proton decay operators. We show that there exists a class of GUTs in which these operators are absent. Many string and supergravity models in the literature belong to this class."

"I expect that sooner or later we will be seeing another departure from the renormalizable Standard Model in the discovery of proton decay, or some other example of baryon nonconservation."

"Baryon-number-violating processes, including proton decay, and the existence of superpartners are dramatic, make-or-break predictions ... Either would open new worlds of phenomena to investigation. According to our best estimates, neither proton decay nor superpartners lie beyond the reach of a heroic search. They should be found, well within 100 years."

"Very high energy collisions occur naturally in cosmic ray interactions; they also occurred in the early moments of our universe according to big-bang cosmology. Both these sources provide useful information but they cannot compare with systematic experimentations in accelerator laboratories when this is possible."

"Stephen Hawking and I wrote an essay about future colliders that is relevant to both the CEPC and the FCC. We were encouraged by others to chime in because of discussions that arose in China about physics and economic and cultural issues surrounding building a future collider. The theoretical arguments for building a larger collider with several times the energy of the LHC are thus very strong, particularly in regard to solving the hierarchy problem. What would happen without data? Someone may get or even already have the solution, but no one will be convinced. With data pointing to the solution, we may be able to move on and obtain consensus about a comprehensive theory that incorporates the standard models of particle physics and cosmology and a quantum theory of general relativity, giving us a profound understanding of our universe."

"The discipline of collider physics involves going from the direct collider observables to the underlying lagrangian of the theory. One of the simplest questions one can ask is how to recognize the presence of new particles. In colliders the answer to this question is simple, one collects groups of particles (pairs, for example) and one plots the invariant mass. If a bump is seen in this distribution, one says that there is a new particle. One can also look at the angular distribution of the particles and read off the spin of the new particle."

"The hierarchy problem is hard to explain. ... Basically, the problem is that there are two main energy (or mass) scales in nature, but that situation shouldn’t be stable. One is the Planck scale, which is defined via fundamental constants: the speed of light, c; Planck’s quantum size, h; and Newton’s gravitational force strength, G. The associated energy scale is about 1019 GeV. The other is the electroweak scale, which is set by the masses of the Higgs and the W and Z bosons, at about 102 GeV. (Protons and atoms have smaller scales, but we understand how to derive those.) It is a conceptual problem, not a conflict with observations."

"Following 't Hooft, we can formulate a technical definition of naturalness: The smallness of a dimensionless parameter η would be considered natural only if a symmetry emerges in the limit η → 0. Thus, fermion masses could be naturally small, since, as you will recall from chapter II.1, a chiral symmetry emerges when a fermion mass is set equal to zero. On the other hand, no particular symmetry emerges when we set either the bare or renormalized mass of a scalar field equal to zero. This represents the essence of the hierarchy problem."

"There is virtually no chance that we will be able to do experiments involving processes at particle energies like 1016 GeV. With current technology the diameter of an accelerator is proportional to the energy given to the accelerated particles. To accelerate particles to an energy of 1016 GeV would require an accelerator a few light-years across."

"The hierarchy problem is somewhat unique. ... It's one of a trifecta of problems in the Standard Model that don't come from incontrovertible evidence ... Those three problems are ... the strong CP problem ... the cosmological constant problem ... and the hierarchy problem. ... These .. are ... problems where it's a conflict between our expectations for the size of parameters in quantum field theory and what we see."

"We know what kind of a dance to do experimentally to measure this number very accurately, but we don't what kind of a dance to do on a computer to make this number come out—without putting it in secretly!"

"... an understanding of the numerical value of the fine structure constant may emerge ... charge might be an emergent property generated by a simple interaction mechanism between point-like particles and the electromagnetic vacuum, similar to the process that generates the Lamb shift."

"The theoretical determination of the fine structure constant is certainly the most important of the unsolved problems of modern physics. To reach it, we shall, presumably, have to pay with further revolutionary changes of the fundamental concepts of physics with a still farther digression from the concepts of the classical theories."

"The title of my talk may seem a bit ambitious, but please note the plural “constants”. To calculate the fine structure constant, 1/137, we would need a realistic model of just about everything, and this we do not have. In this talk I want to return to the old question of what it is that determines gauge couplings in general, and try to prepare the ground for a future realistic calculation."

"From quantum theory there follows the existence of so called zero-point oscillations; for example each oscillator in its lowest is not completely at rest but always is moving about its equilibrium position. Therefore electromagnetic oscillations also can never cease completely. Thus the quantum nature of the electromagnetic field has as its consequence zero point oscillations of the field strength in the lowest energy state, in which there are no light quanta in space... The zero point oscillations act on an electron in the same way as ordinary electrical oscillations do. They can change the eigenstate of the electron, but only in a transition to a state with the lowest energy, since empty space can only take away energy, and not give it up. In this way spontaneous radiation arises as a consequence of the existence of these unique field strengths corresponding to zero point oscillations. Thus spontaneous radiation is induced radiation of light quanta produced by zero point oscillations of empty space."

"We here face a fundamental problem of outstanding importance. Its solution may still require a radical change in our theories beyond our present imagining."

"In his Theorie der Wärmestrahlung, Planck emphasized that the existence of a zero-point energy was completely foreign to classical physics. However, it seemed to be a ghost-like entity which it was difficult to connect to experiments."

"I fear that your hatred of the zero-point energy extends to the electrodynamic emission hypothesis that I introduced and that leads to it. But what’s to be done? For my part, I hate discontinuity of energy even more than discontinuity of emission."

"Zero-point energy is now dead as a doornail."

"The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one is emitted it can be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are an infinite number of light quanta in the zero state..."

"One hopes will soon demonstrate the incorrectness of the hypothesis of zero-point energy, the theoretical untenability of which became glaringly obvious to me soon after the publication of the paper I coauthored with Mr. Stern."

"It from bit. Otherwise put, every it—every particle, every field of force, even the spacetime continuum itself—derives its function, its meaning, its very existence entirely—even if in some contexts indirectly—from the apparatus-elicited answers to yes or no questions, binary choices, bits. It from bit symbolizes the idea that every item of the physical world has at bottom—at a very deep bottom, in most instances—an immaterial source and explanation; that what we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and this is a participatory universe."

"In my view the most fundamental statement of quantum mechanics is that the wavefunction, or more generally the density matrix, represents our knowledge of the system we are trying to describe. I shall return later to the question "whose knowledge?". It is well known that we have to use a wavefunction if we have a "pure state" i.e. if our knowledge of the system is complete, in the sense that any further knowledge is barred by the uncertainty principle. Failing such complete knowledge we must use a density matrix, which therefore contains both quantum and classical ignorance. The wavefunction is a special case of a density matrix, and I shall here talk about "density matrix" when I mean "wavefunction or density matrix"."

"Information? Whose information? Information about what?"

"I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles."

"So, what is quantum mechanics? Even though it was discovered by physicists, it’s not a physical theory in the same sense as electromagnetism or general relativity. In the usual “hierarchy of sciences” – with biology at the top, then chemistry, then physics, then math – quantum mechanics sits at a level between math and physics that I don’t know a good name for. Basically, quantum mechanics is the operating system that other physical theories run on as application software (with the exception of general relativity, which hasn’t yet been successfully ported to this particular OS). There’s even a word for taking a physical theory and porting it to this OS: “to quantize.” But if quantum mechanics isn’t physics in the usual sense – if it’s not about matter, or energy, or waves, or particles – then what is it about? From my perspective, it’s about information and probabilities and observables, and how they relate to each other."

"One serious mystery of decoherence is where the Born probabilities come from, or even what they are probabilities of. What does the integral over the squared modulus of the amplitude density have to do with anything? … So what could it mean, to associate a "subjective probability" with a component of one factor of a combined amplitude distribution that happens to factorize? … But what does the integral over squared moduli have to do with anything? On a straight reading of the data, you would always find yourself in both blobs, every time. How can you find yourself in one blob with greater probability? What are the Born probabilities, probabilities of? Here's the map—where's the territory? I don't know. It's an open problem. This problem is even worse than it looks, because the squared-modulus business is the only non-linear rule in all of quantum mechanics."

"One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns a (complex) number, called the “amplitude,” to each possible measurement outcome. The Born Rule is then very simple: it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. (The wave function is just the set of all the amplitudes.) … The Born Rule is certainly correct, as far as all of our experimental efforts have been able to discern. But why? … The status of the Born Rule depends greatly on one’s preferred formulation of quantum mechanics. … Everett, … is claiming that all the weird stuff about “measurement” and “wave function collapse” in the conventional way of thinking about quantum mechanics isn’t something we need to add on; it comes out automatically from the formalism. The trickiest thing to extract from the formalism is the Born Rule."

"... the Born rule just comes out of the blue without being derived from the time-dependent Schrödinger equation — which is supposed to account for everything."

"If one translates this result into terms of particles, only one interpretation is possible: \Phi_{n,m} (\alpha, \beta, \gamma)[* addition in proof: More careful consideration shows that the probability is proportional to the square of the quantity \Phi_{n,m}] gives the probability for the electron, arriving from the z-direction, …"

"And Po'mi has just asked: "Why should the subjective probability of finding ourselves in a side of the split world, be exactly proportional to the square of the thickness of that side?"When the initial hubbub quiets down, the respected Nharglane of Ebbore asks: "Po'mi, what is it exactly that you found?" "Using instruments of the type we are all familiar with," Po'mi explains, "I determined when a splitting of the world was about to take place, and in what proportions the world would split. I found that I could not predict exactly which world I would find myself in—" "Of course not," interrupts De'da, "you found yourself in both worlds, every time -" "—but I could predict probabilistically which world I would find myself in. Out of all the times the world was about to split 2:1, into a side of two-thirds width and a side of one-third width, I found myself on the thicker side around 4 times out of 5, and on the thinner side around 1 time out of 5. When the world was about to split 3:1, I found myself on the thicker side 9 times out of 10, and on the thinner side 1 time out of 10.""

"A few years after writing the preface of that book, Popper fell into an opposite, and equally serious error about an "EPR situation," On this occasion, contrary to the preceding one, there is an over- rather than an underevaluation of the EPR analysis. On p. 27 of the same book, Popper proposes an experiment that constitutes a variant of the EPR argument, asserting that if the the Copenhagen interpretation is correct, the experiment just analyzed would allow for sending signals faster than the speed of light. This work is one of a lengthy series we will discuss later, in which it is maintained that quantum formalism would permit us to use the reduction of the wave packet to violate one of the postulates at the basis of relativity (i.e., that the speed of light cannot be exceeded). Now, despite the peculiarity of the situation addressed by EPR, this conclusion is fundamentally erroneous and arise from an incorrect use of quantum formalism. I recall a spirited discussion I once had with Popper at the International Center for Theoretical Physics at Miramare in 1983. Professor Abdus Salam informed me that on the occasion of Popper's visit (for delivering a lecture on the foundations of quantum mechanics), he would be very pleased if the Center would have on hand some competent person in the field, and asked me to take part in the discussion. I knew Popper's work well and told Professor Salam that my intervention could be critical. Salam's reply was simple: "I have full confidence in you, and if you think you are right, you should explain your position without any fear." Popper presented his thought experiment (a variant of the EPR argument), which, according to him, left us with only two alternatives: either the orthodox interpretation was correct, and it would then be possible to send signals faster than the speed of light, or there would not be any action at a distance and the experiment would constitute a falsification of quantum theory. At the end of the conference I explained to him in simple, but mathematically precise terms, the reasons why his point of departure was erroneous: he had not correctly applied the rules of the theory and in fact, the impossibility of sending superluminal signals would confirm the theory rather than falsify it—the exact opposite of what he maintained. At the end of my intervention he only said that he could not answer my objection since he did not have a mastery of the mathematics of the formalism, but was still convinced that the theory implied the possibility of superluminal signals. This strange, and, as we shall see, fundamentally erroneous idea has been supported by various researchers in various scientific works, and published in prestigious journals."

"Above all, the ominous clouds of those phenomena that we are with varying success seeking to explain by means of the quantum of action, are throwing their shadows over the sphere of physical knowledge, threatening no one knows what new revolution."

"Quantum mechanics, created early this century in response to certain experimental facts which were inexplicable according to previously held ideas (...'classical physics'), caused three great revolutions. In the first place it opened up a completely new set range of phenomena to which the methods of physics could be applied. ...The second revolution was the apparent breakdown of determinism, which had always been an unquestioned ingredient and an inescapable prediction of classical physics. ...The outcome of any particular experiment is not, even in principle, predictable, but is chosen at random from a set of possibilities; all that can be predicted is the probability of particular results when the experiment is repeated many times. ...even if we had complete knowledge of the initial state... The third revolution ...challenged the basic belief, implicit in all science and indeed in almost the whole of human thinking, that there exists an objective reality ...that does not depend for its existence on its being observed."

"In the brief period between 1900 and 1935 there occurred one of the most astonishing outbursts of scientific creativity in all of history. ...no other historical era has crammed so much scientific creativity, so many discoveries of new ideas and techniques, into so few years. Although a few outstanding individuals dominate... they were assisted in their work by an army of talented scientists and technicians."

"In the years 1925 and 1926 modern quantum theory came into being fully fledged. These anni mirabiles remain an episode of great significance in the folk memory of the theoretical physics community... Werner Heisenberg had been struggling to understand the details of atomic spectra. ...Heisenberg's discovery came to be known as matrix mechanics. ...In 1925 matrices were... mathematically exotic to the average theoretical physicist... Prince Louis de Broglie... made the bold suggestion that if undulating light also showed particle-like properties... one should expect particles such as electrons to manifest wavelike properties... by generalizing the Planck formula. The latter had made the particlelike property of energy proportional to the wavelike property of frequency. De Broglie suggested that... particlelike... momentum... should analogously be related to... wavelength, with Planck's universal constant again... These equivalences provided... for translating from particles to waves, and vice versa. In 1924, de Broglie laid out these ideas in his doctoral thesis. ...To attain a full dynamical theory, a further generalization... allowed the incorporation of interactions... This is the problem that Schrödinger succeeded in solving. Early in 1926 he published the famous equation... led to its discovery by exploiting an analogy drawn from optics."

"Quantum mechanics is revolutionary because it overturned scientific concepts that seemed to be so obvious and so well confirmed by experience that they were beyond reasonable question, but it is an incomplete because we still do not know precisely where quantum mechanics will lead us—nor even why it must be true!"

"The development of the quantum ideas themselves occurred in discrete jumps, quantum leaps of the creative insights of a few people."

"The start of the revolution that produced the old quantum theory is moved from the end of 1900 to 1906... The preceding crisis...resulted from the difficulties in reconciling Planck's derivation with the tenets of classical physics. Planck's change in vocabulary—from "resonator" to "oscillator" and from "element" to "quantum"—is the central symptom of incommensurability. It signals the changed meaning of the quantity hv from a mental subdivision of the energy continuum to a physically separable atom of energy. That my critics continue to apply the term "energy quantum" to pre-1906 papers and lectures in which Planck consistently used "energy element" reveals something of the difficulty of reversing the gestalt switch that took place during that year and those which followed. ...Boltzmann's probabilistic derivation of the entropy of a gas ...illustrates the problem to which the concept of paradigm was a response. The derivation was not reduced to rules but instead served as a model to be applied by means of analogy. As a result, when its application was transferred from gases to radiation, Planck and Lorentz could invent different analogies with which to effect change."

"Intuitively, Planck knew that his work was as important as Newton's in paving a new physics, as he privately confided to his son in 1901. Because of his own conservative beliefs, Planck felt stymied by the revolutionary nature of his own ideas... In truth, Planck was only underestimating the importance of his work, if anything. In retrospect... it started a revolution in science—the quantum revolution—compared to which the Copernican revolution pales."

"The quantum revolution describes our deepest insight, so far, into the physical structure of nature. It is comparable only with the Copernican revolution, switching from a finally oriented anthropomorphic description of physical phenomena to one using general laws with initial or boundary conditions, connected with the names Kepler, Galileo, and Newton, or with the change from tangible mass points as basic structures to Faraday's and Maxwell's field concepts and, shortly before quantum theory, with the relativization of space and time by the lonely genius Einstein."

"The realization of the importance of entanglement and the clarification of the quantum description of single objects have been at the root of a second quantum revolution, and the John Bell was its prophet."