Commentary on home: page 3: Quantum mechanics selects stable forms from primordial Hilbert space
Charles Darwin (1809 – 1882) explained that both variation and selection are necessary for creative evolution. The base vectors of the Hilbert space described in step 2 are random complex periodic functions like waves. Louis de Broglie saw that waves find stationary points when they add because the ups and downs can cancel. Charles Hermite (1822 – 1901) introduced the matrices that play an important role in these mappings of quantum vectors onto one another. Quantum operators select precise real stationary states from the random variations of Hilbert space, a process described by the Born rule. These abstract forms are converted to real particles using the energy created by the bifurcation of zero energy gravitation into the equal quantities of positive kinetic and and negative potential energy described in step 4. Charles Darwin - Wikipedia, Louis de Broglie - Wikipedia, Charles Hermite - Wikipedia, Born Rule - Wikipedia
Table Of Contents
1. Quantum Mechanics is invisible and occupies Hilbert Space
2. Quantization: Kirchhoff, Planck, Einstein And Von Neumann
3. Von Neumann’s axioms of abstract Hilbert space
5. Spectral Theory and Hermitian Operators
7. A simple statement of quantum theory
1. Quantum mechanics is invisible and lives in Hilbert space
Quantum mechanics is the fundamental scientific theory of the universe, but it presents a problem. The foundation of science is observation and measurement, but Hilbert space, the abstract mathematical foundation of quantum mechanics, cannot be directly observed. From an analogical point of view, it is like the mind of another person, something behind the behaviour which we can’t see. No matter how well we know cat, a dog, a horse or any other person, we can rarely predict exactly how they will respond to a particular event.
Richard Feynman captured this idea in his description of “quantum behaviour”, that is the behaviour of elementary particles driven by quantum mechanics. After describing the well known double-slit experiment in detail he comments:
One might still like to ask: “how does it work? What is the machinery behind the law?” No one has found any machinery behind the law. No one can “explain” any more than we have just “explained.” No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced. Double-slit experiment - Wikipedia, Richard Feynman (1965): FLP iii_1: Quantum Behaviour
seems a little pessimistic here. Quantum mechanics is quite an effective theory of the world based on linear algebra and explained in mathematical detail by John von Neumann using the work of David Hilbert. Unfortunately, however, our actions on the world are uncertain in that they may have a spectrum of results whose distribution is described by the Born rule. von Neumann (2018): Mathematical Foundations of Quantum Mechanics, Born rule - Wikipedia [link above]
Linear algebra is an effective and logical description of what is happening based in the Hilbert space described on page 2 which explains quantum behaviour. There we noted that Hilbert space is created by a random process. This is explains the uncertainty of quantum outcomes and opens the path for Darwinian evolution. Linear Algebra - Wikipedia
Hilbert space is a function space. In mathematics functions relate numbers to one other. We see this clearly when we plot functions like y = f(x) as a graph, with numbers along the x axis and the functions of these numbers on the y axis. Function space - Wikipedia
We may think of a function as a vector. The bases of this vector are plotted along the x axis and the y values corresponding to x are represented by y = f(x). In other words, we define a vector as a sequence of values corresponding to the elements of a basis. Each element of this basis corresponds to a dimension of the function space. In Euclidean space we see that the three dimensions are at right angles (orthogonal) to one another. In a function space we may have any number of independent dimensions.
This means that a continuous basis of real numbers require an infinite dimensional space, since there are infinity of real numbers in the real line. This is unnecessary in quantum mechanics. The basis states of Hilbert space in nature are discrete entities all orthogonal to one another.
2. Quantization: Kirchhoff, Planck, Einstein and von Neumann
the first hint of quantum mechanics came in 1860 when Gustav Kirchhoff formulated his law of thermal radiation:
For a body of any arbitrary material emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature. That universal function describes the perfect black-body emissive power. Kirchhoff's law of thermal radiation - Wikipedia
Kirchhoff’’s law started a search for the universal function. At the same time spectroscopists were extending the range of their measurements into the infra-red region of relatively low temperature. A number of candidates for the law were proposed that failed to fit the data at either high or low temperatures. The answer was finally found in 1900 by Max Planck. Planck's law - Wikipedia
Planck found that an explanation of the black body spectrum required the assumption that the radiation came in discrete quanta representing a spectrum of frequencies corresponding to different energies. The shape of the spectrum depends on the probabilities of emission of the different quanta. He found further that the energy of each quantum had fixed relationship to the frequency, the first equation of quantum mechanics, E = hf where h is a new universal constant, the quantum of action. h is very small and sets the minuscule scale of the fundamental particles and the primitive steps in any action in the world. Planck constant - Wikipedia
Einstein realised that Planck’s formula predicted the existence of the real physical particles that ultimately became known as photons. His work explained the photoelectric effect which has been observed by Heinrich Hertz in the 19th century. Albert Einstein (1905c): On a heuristic point of view concerning the production and transformation of light, Photoelectric effect - Wikipedia
The complex results of spectroscopy revealed that atoms and molecules, although invisibly small, have very complex interior structure. Two discoveries laid the foundations of the new science of quantum mechanics. The first, by Niels Bohr, gave the first hint of the electronic structure of the atom responsible for its spectrum. The second, by Louis de Broglie, suggested that quantum behaviour could be explained by associating waves with elementary particles. Bohr model - Wikipedia, Louis de Broglie (1929): Nobel lecture: The wave nature of the electron
3. von Neumann’s axioms of abstract Hilbert space
We have imagined quantum forms as complex vectors or functions with periodic structure, similar to snippets of sound or phonemes. The are formally distinct (like angels), mathematically orthogonal like the dimensions in Euclidean space. They are also one unit “long”, that is normalized. David Jones (2020): Angels: A History
All the sounds of the distinct instruments in an orchestra are orthogonal. Their outputs are summed or superposed when they are all playing together, each adding their distinctive input to the overall sound. Vocalists provide more complex inputs than individual instruments. Although the instruments are materially distinct their sounds are superposed in the air, analogous to the vector forms in Hilbert space. Like angels, sounds in air are distinguished only by their specific form, not by spatial separation. Aquinas, Summa I, 50, 4: Is every angel a different species?
The idea that things are composed of matter and form is at least as old as Aristotle (384-322 bc). The idea of form was introduced by his teacher Plato, whose theory of forms proposed that the nature of the world is determined by preexisting perfect forms. These forms are rather imperfectly embodied in the material components of the world. Some philosophers consider these forms themselves to be real, but Aristotle considered that real things comprise both matter and form. Substantial change is then possible because the same matter may appear in various forms. Aristotle's view is known as hylmorphism Greek for matter-formism. Theory of Forms - Wikipedia
The idea of form is very useful, in mathematics and in any task involving design. It is a lot cheaper and easier to sketch out designs and estimate their physical properties mathematically than to construct the real things and test them. Construction and testing are best saved until a promising design has been formulated. One can then learn with certainty whether the design works or not by building it.
This technique is embedded in reality in the relationship of abstract Hilbert space to physical particles and in the reproduction and evolution of living things whose physiological structures are represented by the genetic code in their genes. This code has quite close control of their growth and development. It serves to carry the specification of living things from generation to generation. It is often modified in various ways from generation to generation and the modifications tested by natural selection. Genetic code - Wikipedia, Natural selection - Wikipedia
Here we follow this formal approach to design and construction deeper into the foundation of the universe. The classical four dimensional space-time in which we live and move was first defined by Einstein in his special theory of relativity. He saw that every entity in non-accelerated (inertial) motion sees the same speed of light. Hermann Minkowski formulated Einstein’s idea in a succinct and easily applicable mathematical form so we now call this structure Minkowski spacetime. Einstein used this form to develop his general theory of relativity. Albert Einstein (1905): On the electrodynamics of moving bodies, Minkowski spacetime - Wikipedia
The magnificent observed success of Einstein’s general theory has established it as the standard explanation of the large scale geometrical structure of the universe. One of its notable consequences is the existence of black holes, massive structures which are black because their gravitational attraction is so great that even light cannot escape from them. Although we cannot see them directly, we can see the gravitational effects that they have on nearby stars. Black hole - Wikipedia, Hawking & Ellis (1975): The large scale structure of Space-time
Many have considered the classical Minkowski space in which we live and experience everything around us to be the fundamental structure of the world. This has led to a fundamental error in our understanding the world which has stood in the way of the theological recognition of the universe as divine by establishing an absolute distinction between matter and spirit.. This error arose when the physicists who discovered quantum mechanics believed that this theory overlies Minkowski space. Here I prefer the view that Minkowski space and all its contents are not the foundation of quantum mechanics but the result, analogous to the way visible life is the result of genetic coding.
Darwin saw that the foundation of biological evolution is variation in reproduction. Viable variations are captured and reproduced by natural selection. Here we place the variation founding the evolution of the universe in step 2, the random creation of Hilbert space in the initial symmetry. Here in step 3 we introduce quantum mechanics as the beginning of selection process.
4. Quantum Superposition
Paul Dirac described the superposition principle of quantum theory as follows:
The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, A say, and when made on the system in state B is certain to lead to some different result, B say. What will be the result of the observation when made on the system in the superposed state?return to top of pageThe answer is that the result will be sometimes A and sometimes B, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both A and B [i.e., either A or B]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states. P. A. M. Dirac (1983): The Principles of Quantum Mechanics (4th ed)
5. Spectral theory and hermitian operators
We may say that the fundamental variable in human life is time, and our fundamental problem is to predict the future. A powerful ancient view is embodied in the idea that invisible gods in the heavens decide our future. This idea probably led to the ideas of astrology. Astrologers claimed that their theory could predict the outcomes of certain events for their clients, like marriage or going to war. To do this they needed information about the motions of stars and planets to make their predictions.
This led in turn led to astronomy and physics, the careful study and explanation of celestial motion. Many newspapers still provide daily astrological predictions for people born under different star signs. My birth in January makes me a Capricorn.
Apart from such complex inputs into human affairs simple observations of the heavens also provide very useful information for navigation at sea and in featureless, natural landscapes, for the seasons, and for the behaviour of the plants and animals and other natural phenomena upon which human life depends.
Galileo’s telescopes and Newton’s mathematics laid the firm foundation of astronomy and physics which has brought us into the modern era of engineering and technology. This work depends heavily on differential and integral calculus. We compute the future of a function y = f(x) by integrating its differential df/dx with respect time on the assumption that time is a continuous variable based on an ancient assumption natura non facit saltus, nature does not make jumps.
John von Neumann points out, at the beginning of his book on the mathematical foundations of quantum mechanics, that reality is in fact discontinuous, comprising myriad tiny particles. The appearance of continuity arises from the fact that the quantum of action that measures these particles and their behaviour is exceedingly small. The huge number of steps involved in any perceptible event makes it appear continuous.
Quantum field theory, following Einstein’s general theory of relativity, assumes that the Minkowski space in which we live (described by special relativity) is continuous. It therefore frequently uses calculus for its computations. Here we assume that the Hilbert space described in Commentary on Home Page Step 2: The generation of random abstract Hilbert space is not continuous. We cannot apply calculus. Instead we apply difference equations which are appropriate for stepwise processes. This enables to avoid the problem that Einstein faced at the end of his life, th difficulty of describing the particulate world using a continuous mathematical model.
In 1933 Einstein gave a lecture at Oxford University on the method of theoretical physics. He expressed the view that:
Our experience up to date justifies us in feeling sure that in nature is actualized the ideal of mathematical simplicity. It is my conviction that pure mathematical construction enables us to discover the concepts and the laws connecting them which give us the key to the understanding of the phenomena of nature. [. . .] To justify this confidence of mine, I must necessarily avail myself of mathematical concepts. The physical world is represented as a four-dimensional continuum [my emphasis]. Albert Einstein (1933): Herbert Spencer lecture 1933: On the Method of Theoretical Physics
he continues:
The most difficult point for such a field-theory at present is how to include the atomic structure of matter and energy. For the theory in its basic principles is not an atomic one in so far as it operates exclusively with continuous functions of space, in contrast to classical mechanics whose most important feature, the material point, squares with the atomistic structure of matter.
He then turns to the Born probabilistic interpretation of quantum theory and explains his dissatisfaction with the theory:
I cannot help confessing that I myself accord to this interpretation no more than a transitory significance. I still believe in the possibility of giving a model of reality, a theory, that is to say, which shall represent events themselves and not merely the probability of their occurrence. Born rule - Wikipedialink above
Although he looked for a mathematically continuous unified theory until his death in april 1955, he had no success.
He wrote some of his final words on the unified theory at the end of an article summarizing his latest work. It appeared in the scientific american in april 1950:
Is the manifold of solutions for the system EE3 as extensive as must be required for a physical theory? This purely mathematical problem is as yet unsolved.The skeptic will say: “it may well be true that this system of equations is reasonable from a logical standpoint. but this does not prove that it corresponds to nature.” You are right, dear skeptic. Experience alone can decide on truth. Yet we have achieved something if we have succeeded in formulating a meaningful and precise question. Albert Einstein (1950_04): On the Generalized Theory of Gravitation
This ability of mathematics to answer difficult physical problems remains a very common view among physicists, but we cannot really apply mathematics until we have some sort of model of what is actually happening. An accountant cannot write meaningful accounts for a business until they know how it works. Where is the money coming from? Where is it going? Who pays who for what? Until they know how the business works their calculations can make no sense.
In his 1933 article Einstein misses this point. He is starting with an assumption about the mathematics and trying to bend the physics to fit. In fact, as very detailed laboratory work in quantum mechanics had demonstrated, this approach is not safe. First we need the data and a model, then we can apply the mathematics.
Unfortunately physicists working on quantum field theory and the standard model seem fixed on the idea that Minkowski space and the quantum field presumed to occupy this space are continuous. This has caused serious mathematical problems with infinity which are presumed to have been solved with renormalization. In his Nobel prize lecture Richard Feynman suggests that this may be sweeping a real problem under the carpet.
I am deeply impressed by the depth and complexity of the mathematical ideas that are applied in physics. I do not understand much of it and I wonder if it is necessary. We are not trying to account for a multinational multi billion dollar business that employs thousands of people. We are dealing with the very simplest level of the universe, starting from an initial singularity which, if we are to believe Aquinas and Einstein, has no structure at all. Richard P. Feynman (1965): Nobel Lecture: The Development of the Space-Time View of Quantum Electrodynamics
Feynman’s fellow laureate, Sin Itiro Tomonaga also studied the problem of mathematical failure in electrodynamics and an application renormalization.
He wrote:
As there are an infinite number of space points in field theory in contrast to the finite number of particles in particle theory, the number of time variables appearing in the probability amplitude became infinite. [. . .] This discouraging state of affairs generated in many people a strong distrust of quantum field theory. There were even those with the extreme view that the concept of field reaction itself had nothing to do at all with the true law of nature.
However, further work with his students revealed the “very pleasant thing that no divergence is involved in the theory except for the two infinities of the electronic mass and charge.” This revealed a way forward [. . .]:
The theory does not of course yield a resolution of the infinities. That is, since those parts of the modified mass and charge due to field reactions contain divergence, it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are, as I mentioned above, after all the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically. When a theory is incompetent in part, it is a common procedure to rely on experiment for that part. This procedure is called the renormalization of mass and charge, and our method has brought the possibility that the theory will lead to finite results by the renormalization even if it contains defects.return to top of page
6. A new beginning
The apparent success of of Faraday’s electromagnetic field and Einstein’s gravitation has fixed the notions of continuous fields and calculus at the centre of physics. Calculus depends on the magical process of taking limits in order to make sense of dividing zero by zero. We are also lacking an explanation for the existence of spacetime. Given the hint from quantum mechanics that the world is not continuous but proceeds by discrete steps of jumps, it is worth exploring the alternative presented here: we begin with a structureless formal entity and rely on fixed point theory to introduce discrete orthogonal fixed points in a formally structureless initial symmetry.
From a formal point of view, we may imagine the points introduced into the symmetry by it reflecting upon itself as described by Brouwer’s theorem, as being quanta of action, the normalized orthogonal basis states of a Hilbert space. Brouwer fixed-point theorem - Wikipedia
We can draw insight here from two ancient theological ideas. First is the explanation of the Trinity of discrete divine persons in the unitary divinity of Christianity. In his book on the Trinity Augustine of Hippo developed the idea that the persons of the Trinity are reflections of the divinity upon itself analogous to his own experience of reflecting upon his own personality. Augustine (419, 1991) & Edmund Hill (translator): The Trinity
The second idea is the notion of omnipotence often attributed to divinities. The Christian theologian Thomas Aquinas interprets the notion of omnipotence as the ability to execute any action that does not inherently involve contradiction. The bases of Hilbert space are rendered independent by their orthogonality. Their interaction is described by hermitian operators of countable dimension. We expect that this mechanism can perform any consistent formal operation, which Hilbert understands to be the domain of mathematics. Aquinas, Summa I, 25, 3: Is God omnipotent?, Formalism (philosophy of mathematics) - Wikipedia, Nielsen & Chuang (2016): Quantum Computation and Quantum Information
To these we can add the quantum of action and the observed pixellation of Minkowski space at th scale of the quantum of action, h: ΔE . Δt ≈ Δp . Δx ≈ h which gives us a measure of the microscopic size of the quantum structure underlying the observable structure of the universe. This explains its enormous computational power at the macroscopic scale. The processing rate in a grain of sand seems to be equivalent to all the computers on Earth. Jeffrey Nicholls (2017): Computing power of a grain of sand: a calculation
return to top of page7. A simple statement of quantum theory
Quantum mechanics entered physics in 1900 when Max Planck (1858 – 1947) found that black body radiation is emitted discrete packets, it is quantized. This was a radical break from classical physics, which is based on the assumption that all the operations of nature are continuous, obeying the dictum Natura non facit saltus: Nature does not make jumps.
As de Broglie noted in the very early days of quantum mechanics, the fixed points that we can observe in the world, like the frequencies of the radiations emitted and absorbed by atoms, are measured in real numbers. These numbers occur when the wave motions represented by complex numbers cancel one another out to give real quantities represented by real numbers. This occurs because the self-adjoint hermitian operators used in quantum mechanics to control the superposition of complex vectors give precise values of the real numbers occurring when these superpositions lead to the suppression of the waves and reveal fixed points. Louis de Broglie - Wikipedia [link above], Louis de Broglie (1929): Nobel lecture: The Wave Nature of the Electron [link above], Matter Wave - Wikipedia
An important feature of quantum mechanics is emphasized by the so called measurement problem. This “problem” appeared early in the interpretation of quantum mechanics and distinguishes it sharply from classical mechanics. The problem is that unlike measurements made in classical physics that give just one answer (more or less accurate depending on the measuring instrument used) quantum mechanics can give many different answers [and accurate] to the same question, but it gives only one answer a time. Measurement problem - Wikipedia
From the quantum mechanical point of view, the world is a bit like a witness who gives different answers to the same question when it is asked over and over again. Nevertheless long experience shows that all the answers are correct but that some are more probable than others. The answer to this problem takes us to the heart of Einstein’s misunderstanding of quantum mechanics. The fact that the world only gives one answer at a time has been called a quantum jump or the collapse of the wave function. Wojciech Hubert Zurek (2008): Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical
Zurek explains that that the alleged collapse of the wave function is a necessary consequence of the transmission of information between two quantum systems.
First we note that every measurement in the classical world is a case of the world (of which we are part) measuring itself. The laboratory distinction between observer and observed is fictitious, in the sense that every quantum process is simply the operation of a communication channel in Hilbert space between two observable sources (particles)in Minkowski space.
Claude Shannon’s mathematical theory of communication treats the space of all possible classical communications between two sources, but its results apply to each particular communication. His work is based on the unitarity of language which tells us that the sum of the probabilities of a functioning set of linguistic symbols is 1. The mathematical foundation of quantum mechanics also depends on unitarity. The sum of the probabilities of the outcomes of a quantum event is also 1. This fact was used by von Neumann to establish the equivalence of the wave and matrix versions of quantum theory developed in the 1920s and to introduce David Hilbert’s linear algebra as the natural mathematical foundation of the theory. Claude Shannon (1949): Communication in the Presence of Noise, John von Neumann (2018), Mathematical Foundations of Quantum Mechanics [link above]
We often think of a quantum measurement as an interaction between a classical and a quantum system, but in reality it is the interaction in Hilbert space of the quantum systems associated with two classical particles observed in Minkowski space. One classical system is the source of a state which we call the measurement operator. This system interacts with an unknown state attached to another classical system, yielding a classically observable result, the particle(s) created by this interaction.
Invisible isolated quantum processes represented by a wave function are believed to be in the continual motion described by the Schrödinger quation. Schrödinger equation - Wikipedia
A measurement is understood to interrupt an isolated system by injecting another process, represented by a measurement operator, into the isolated system. This is analogous to one person interrupting another by starting a conversation.
Zurek begins with a concise definition of standard quantum mechanics in six propositions. The first three describe its mathematical mechanism:
(1) The quantum state of a system is represented by a vector in its Hilbert space;
(2) A complex system is represented by a vector in the tensor product of the Hilbert spaces of the constituent systems;
(3) The evolution of isolated quantum systems is unitary, governed by the Schrödinger equation:iℏ ∂|ψ⟩ / ∂t = H |ψ⟩
where H is the energy (or Hamiltonian) operator.
The other three show how the mathematical formalism in Hilbert space couples to the observed world:
(4) Immediate repetition of a measurement yields the same outcome;
(5) Measurement outcomes are restricted to an orthonormal set { |sk⟩} of k eigenstates of the measured observable [ie the measurement operator associated with the classical measuring system];
(6) The probability of finding a given outcome is given by the born rulepk = |⟨ sk |ψ⟩|2
where |ψ⟩ is the preexisting state of the [measured] system. Born rule, op.cit.
Let us ignore zurek’s detailed calculation and jump to his information theoretical conclusion.
He writes:
The aim of this paper is to point out that already the (symmetric and uncontroversial) postulates (1) - (3) necessarily imply selection of some preferred set of orthogonal states – that they impose the broken symmetry that is at the heart of the collapse postulate (4).Selection of an orthonormal basis induced by information transfer – the need for spontaneous symmetry breaking that arises from the unitary axioms of quantum mechanics (1, 3) is a general and intriguing result.
This means that if information is to be transferred in an interaction all the basis states of the interacting particles cannot talk at once. One basis must be selected by breaking the symmetry of the unitary transformation believed to be occurring in the undisturbed evolution of the quantum minds of the particles.
Nature, it seems, works like the chair at a well organized meeting. She permits just one person to speak at a time.
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notes and references
Further reading
Books
Augustine (419, 1991), and Edmond Hill (Introduction, translation and notes), and John E Rotelle (editor), The Trinity, New City Press 399-419, 1991 Written 399 - 419: De Trinitate is a radical restatement, defence and development of the Christian doctrine of the Trinity. Augustine's book has served as a foundation for most subsequent work, particularly that of Thomas Aquinas.
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Hawking (1975), Steven W, and G F R Ellis, The Large Scale Structure of Space-Time, Cambridge UP 1975 Preface: Einstein's General Theory of Relativity . . . leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.'
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Jones (2020), David Albert, Angels: A History, Oxford University Press 2010 What are angels? Where were they first encountered? Can we distinguish angels from gods, faeries, ghosts, and aliens? And why do they remain so popular? In this introduction to the history of angels, David Albert Jones outlines some of the more prominent stories and speculations about angels in Judaism, Islam, Christianity and post-Christian spiritualities. He reflects on the way angels are portrayed in art, whether as young men in the Hebrew Scriptures, androgynous winged creatures of the pre-Raphaelites, or the masculine statue of the Angel of the North. He also considers angels in films such as Wim Wenders' Wings of Desire, and Frank Capra's It's a Wonderful Life, as well as angels in literature. From the idea of the angel as a messenger, through to the image of angels sent to protect and help those in need, this is an examination of the implications of angels. It asks why people find the idea of them so attractive, helpful or consoling, and why they remain so powerful in modern culture. In this thought-provoking introduction, Jones considers the view that reflecting on angels can teach us something about human existence. Whether or not we believe that they exist in their own right, angels can still illuminate our thoughts.
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Nielsen (2016), Michael A., and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2016 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002.
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von Neumann (2018), John, and Nicholas A. Wheeler (editor), Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 2018 ' Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics--a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. In this new edition of this classic work, mathematical physicist Nicholas Wheeler has completely reset the book in TeX, making the text and equations far easier to read. He has also corrected a handful of typographic errors, revised some sentences for clarity and readability, provided an index for the first time, and added prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson. The result brings new life to an essential work in theoretical physics and mathematics.'
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Links
Albert Einstein (1905), On the Electrodynamics of Moving Bodies, An english translation of the paper that founded Special relativity. 'Examples of this sort, [in the contemporary application of Maxwell's electrodynamics to moving bodies] together with the unsuccessful attempts to discover any motion of the earth relatively to the ``light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.' back |
Albert Einstein (1905c), On a heuristic point of view concerning the production and transformation of light, ' The wave theory of light, which operates with continuous spatial functions, has proved itself splendidly in describing purely optical phenomena and will probably never be replaced by another theory. One should keep in mind, however, that optical observations apply to time averages and not to momentary values, and it is conceivable that despite the complete confirmation of the theories of diffraction, reflection, refraction, dispersion, etc., by experiment, the theory of light, which operates with continuous spatial functions, may lead to contradictions with experience when it is applied to the phenomena of production and transformation of light. Indeed, it seems to me that the observations regarding "black-body" light, and other groups of phenomena associated with the production or conversion of light can be understood better if one assumes that the energy of light is discontinuously distributed in space.' back |
Albert Einstein (1933), Herbert Spencer Lecture 1933: On the Method of Theoretical Physics , ' It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience. back |
Albert Einstein (1950_04), On the Generalized Theory of Gravitation, ' This article was published with the title “On the Generalized Theory of Gravitation” in Scientific American Magazine Vol. 182 No. 4 (April 1950), p. 13 |
Aquinas, Summa I, 50, 4, Is every angel a different species?, ' . . . such things as agree in species but differ in number, agree in form, but are distinguished materially. If, therefore, the angels be not composed of matter and form, as was said above (Article 2), it follows that it is impossible for two angels to be of one species; just as it would be impossible for there to be several whitenesses apart, or several humanities, since whitenesses are not several, except in so far as they are in several substances.' back |
Aquinas, Summa: I, 9, 1, Is god is completely immutable?, 'I answer that, from what precedes, it is shown that God is altogether immutable. First, because it was shown above that there is some first being, whom we call God; and that this first being must be pure act, without the admixture of any potentiality, for the reason that, absolutely, potentiality is posterior to act. Now everything which is in any way changed, is in some way in potentiality. Hence it is evident that it is impossible for God to be in any way changeable. . . . ' back |
Black hole - Wikipedia, Black hole - Wikipedia, the free encyclopedia, ' A black hole is a region of spacetime where gravity is so strong that nothing, including light and other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of no escape is called the event horizon. Although it has a great effect on the fate and circumstances of an object crossing it, it has no locally detectable features according to general relativity. In many ways, a black hole acts like an ideal black body, as it reflects no light. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is of the order of billionths of a kelvin for stellar black holes, making it essentially impossible to observe directly. ' back |
Bohr model - Wikipedia, Bohr model - Wikipedia, the free encyclopedia, 'In atomic physics, the Rutherford–Bohr model or Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with attraction provided by electrostatic forces rather than gravity.' back |
Born rule - Wikipedia, Born rule - Wikipedia, the free encyclopedia, ' The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see bra-ket notation), then the measured result will be one of the eigenvalues λ of A, and the probability of measuring a given eigenvalue λi will equal <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'.' back |
Brouwer fixed-point theorem - Wikipedia, Brouwer fixed point theorem - Wikipedia, the free encyclopedia, 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compact convex set to itself, there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset K of Euclidean space to itself.' back |
Charles Darwin - Wikipedia, Charles Darwin - Wikipedia, the free encyclopedia, ' Charles Robert Darwin FRS FRGS FLS FZS JP (12 February 1809 – 19 April 1882) was an English naturalist, geologist, and biologist, widely known for his contributions to evolutionary biology. His proposition that all species of life have descended from a common ancestor is now generally accepted and considered a fundamental concept in science.' back |
Charles Hermite - Wikipedia, Charles Hermite - Wikipedia, the free encyclopedia, 'Hermite introduced the notion of an orthogonal matrix, which is equal to the inverse of its transpose, in 1854, though the modern formal definition was first stated by 1878 by Ferdinand Georg Frobenius In 1855, Hermite proved that the eigenvalues of matrices equal to their own complex conjugate transposes, Hermitian matrices, are always real, thereby generalizing the 1829 result of Cauchy for n × n real symmetric matrices. The notion of Hermitian matrices was later extended for infinitely variables and became an important topic in the study of differential and integral equations thanks to the vision of David Hilbert. Such linear transformations, dubbed Hermitian operators, were used in the rigorous mathematical formulation of quantum mechanics first by Norbert Wiener and Max Born, and subsequently by Hilbert, Lothar Nordheim, and John von Neumann.' back |
Claude E Shannon (1948), A Mathematical Theory of Communication, ' The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.' back |
Claude Shannon (1949), Communication in the presence of noise, 'A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two “function spaces,” and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of “ideal” systems which transmit at this maximum rate are discussed. The equivalent number of binary digits per second for certain information sources is calculated.' [C. E. Shannon , “Communication in the presence of noise,” Proc. IRE, vol. 37, pp. 10–21, Jan. 1949.] back |
Complex number - Wikipedia, Complex number - Wikipedia, the free encyclopedia, 'A complex number is a number that can be expressed in the form a + bi, where a. and b are real numbers and |
Double-slit experiment - Wikipedia, Double-slit experiment - Wikipedia, the free encyclopedia, ' In the double-slit experiment, light is shone at a solid thin plate that has two slits cut into it. A photographic plate is set up to record what comes through those slits. One or the other slit may be open, or both may be open. . . . The most baffling part of this experiment comes when only one photon at a time is fired at the barrier with both slits open. The pattern of interference remains the same as can be seen if many photons are emitted one at a time and recorded on the same sheet of photographic film. The clear implication is that something with a wavelike nature passes simultaneously through both slits and interferes with itself — even though there is only one photon present. (The experiment works with electrons, atoms, and even some molecules too.) ' back |
Formalism (philosophy of mathematics) - Wikipedia, Formalism (philosophy of mathematics) - Wikipedia, the free encyclopedia, 'In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." |
Function space - Wikipedia, Function space - Wikipedia, the free encyclopedia, 'In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both' back |
Genetic code - Wikipedia, Genetic code - Wikipedia, the free encyclopedia, ' Genetic code is a set of rules used by living cells to translate information encoded within genetic material (DNA or RNA sequences of nucleotide triplets or codons) into proteins. Translation is accomplished by the ribosome, which links proteinogenic amino acids in an order specified by messenger RNA (mRNA), using transfer RNA (tRNA) molecules to carry amino acids and to read the mRNA three nucleotides at a time. The genetic code is highly similar among all organisms and can be expressed in a simple table with 64 entries. |
Hylomorphism - Wikipedia, Hylomorphism - Wikipedia, the free encyclopedia, 'Hylomorphism (Greek ὑλο- hylo-, "wood, matter" + -morphism < Greek μορφή, morphē, "form") is a philosophical theory developed by Aristotle, which analyzes substance into matter and form. Substances are conceived of as compounds of form and matter.' back |
Jeffrey Nicholls (2017), Computing power of a grain of sand: the calculation, Quantum theory has taught us that there is more computing power in a grain of sand than all the computers on the planet. Here is the calculation. We let our grain of sand weigh a microgram. Einstein's formula E = mc2 tells us that this is [equivalent to] (10-9 kilograms) × (3E8 metres per second)2 = 9 ×107 Joules, say 100 million. The Planck-Einstein formula f = E/h relates frequency to energy. Filling in the numbers, f = 108 Joule / (6.63×10-34 Joule-seconds) = ≈ 1041 cycles per second, that is computations per second. If every one of us, about 10 billion, or 1010 has a teraflop computer capable of performing 1012 operations per second the total power is 1022 computations per second, ie about 1019 times slower than the grain of sand. back |
Kirchhoff's law of thermal radiation - Wikipedia, Kirchhoff's law of thermal radiation - Wikipedia, the free encyclopedia, 'Kirchhoff's law states that: For a body of any arbitrary material, emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature, the perfect black-body emissive power. back |
Laplace's demon - Wikipedia, Laplace's demon - Wikipedia, the free encyclopedia, ' In the history of science, Laplace's demon was a notable published articulation of causal determinism on a scientific basis by Pierre-Simon Laplace in 1814. According to determinism, if someone (the demon) knows the precise location and momentum of every particle in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of classical mechanics. |
Linear Algebra - Wikipedia, Linear algebra - Wikipedia, the free encyclopedia, 'Linear algebra is the branch of mathematics concerning linear equations such as a1 x1 + ⋯ + anxn = b. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to function spaces.' back |
Louis de Broglie - Wikipedia, Louis de Broglie - Wikipedia, the free encyclopedia, ' Louis-Victor-Pierre-Raymond, 7th duc de Broglie . . . 15 August 1892 – 19 March 1987) was a French physicist who made groundbreaking contributions to quantum theory. In his 1924 PhD thesis he postulated the wave nature of electrons and suggested that all matter has wave properties. This concept is known as the de Broglie hypothesis, an example of wave-particle duality, and forms a central part of the theory of quantum mechanics.' back |
Louis de Broglie (1929), Nobel Lecture: The Wave Nature of the Electron, ' The necessity of assuming for light two contradictory theories-that of waves and that of corpuscles - and the inability to understand why, among the infinity of motions which an electron ought to be able to have in the atom according to classical concepts, only certain ones were possible: such were the enigmas confronting physicists at the time I resumed my studies of theoretical physics. Now a purely corpuscular theory does not contain any element permitting the definition of frequency. This also renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity. On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too. . . . Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time. The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted. It has thus been possible to predicta whole group of phenomena without their actually having been discovered. And it is on this concept of the duality of waves and corpuscles in Nature, expressed in a more or less abstract form, that the whole recent development of theoretical physics has been founded and that all future development of this science will apparently have to be founded.' back |
Matter wave - Wikipedia, Matter wave - Wikipedia, the fre ncyclopedia, 'When I conceived the first basic ideas of wave mechanics in 1923–1924, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that Einstein had introduced for photons in his theory of light quanta in 1905. |
Measurement problem - Wikipedia, Measurement problem - Wikipedia, the free encyclopedia, ' In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum systems have superpositions but quantum measurements only give one definite result. |
Minkowski spacetime - Wikipedia, Minkowski spacetime - Wikipedia, the free encyclopedia, ' By1908 Minkowski recalized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval x2 + y2 + z2 − c2 t2.' back |
Natura non facit saltus - Wikipedia, Natura non facit saltus - Wikipedia, the free encyclopedia, ' Natura non facit saltus (Latin for "nature does not make jumps") has been an important principle of natural philosophy. It appears as an axiom in the works of Gottfried Leibniz (New Essays, IV, 16: "la nature ne fait jamais des sauts", "nature never makes jumps"), one of the inventors of the infinitesimal calculus (see Law of Continuity). It is also an essential element of Charles Darwin's treatment of natural selection in his Origin of Species. The Latin translation comes from Linnaeus' Philosophia Botanica. |
Natural selection - Wikipedia, Natural selection - Wikipedia, the free encyclopedia, 'Natural selection is the differential survival and reproduction of individuals due to differences in phenotype; it is a key mechanism of evolution. The term "natural selection" was popularised by Charles Darwin, who intended it to be compared with artificial selection, now more commonly referred to as selective breeding. . . . Natural selection is one of the cornerstones of modern biology. The concept was published by Darwin and Alfred Russel Wallace in a joint presentation of papers in 1858, and set out in Darwin's influential 1859 book On the Origin of Species,[3] in which natural selection was described as analogous to artificial selection, a process by which animals and plants with traits considered desirable by human breeders are systematically favoured for reproduction.' back |
Photoelectric effect - Wikipedia, Photoelectric effect - Wikipedia, the free encyclopedia, 'The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. . . . The experimental results disagree with classical electromagnetism, which predicts that continuous light waves transfer energy to electrons, which would then be emitted when they accumulate enough energy. An alteration in the intensity of light would theoretically change the kinetic energy of the emitted electrons, with sufficiently dim light resulting in a delayed emission. The experimental results instead show that electrons are dislodged only when the light exceeds a certain frequency—regardless of the light's intensity or duration of exposure.' back |
Planck constant - Wikipedia, Planck constant - Wikipedia, the free encyclopedia, ' Since energy and mass are equivalent, the Planck constant also relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to replacing the metal cylinder, called the International Prototype of the Kilogram (IPK), that had defined the kilogram since 1889. . . . For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.626 070 150 × 10-34 J⋅s exactly. ' back |
Planck's Law - Wikipedia, Planck's Law - Wikipedia, the free encyclopedia, ' In physics, Planck's law describes the spectral radiance of electromagnetic radiation at all wavelengths from a black body at temperature T. As a function of frequency ν. back |
Recurrence relation - Wikipedia, Recurrence relation - Wikipedia, the free encyclopedia, 'In mathematics and computer science, a recurrence relation is an equation according to which the n {\displaystyle n}th term of a sequence of numbers is equal to some combination of the previous terms. Often, only k {\displaystyle k} previous terms of the sequence appear in the equation, for a parameter k {\displaystyle k} that is independent of n {\displaystyle n}; this number k {\displaystyle k} is called the order of the relation. If the values of the first k {\displaystyle k} numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In linear recurrences, the nth term is equated to a linear function of the k {\displaystyle k} previous terms. A famous example is the recurrence for the Fibonacci numbers, Fn = F(n − 1) + F(n − 2).' back |
Richard Feynman (1965), FLP III_1: Quantum Behaviour, ' Summary:
1. The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number ϕ which is called the probability amplitude: |
Richard P. Feynman (1965), Nobel Lecture: The Development of the Space-Time View of Quantum Electrodynamics, ' We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn’t any place to publish, in a dignified manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing. Since winning the prize is a personal thing, I thought I could be excused in this particular situation, if I were to talk personally about my relationship to quantum electrodynamics, rather than to discuss the subject itself in a refined and finished fashion. Furthermore, since there are three people who have won the prize in physics, if they are all going to be talking about quantum electrodynamics itself, one might become bored with the subject. So, what I would like to tell you about today are the sequence of events, really the sequence of ideas, which occurred, and by which I finally came out the other end with an unsolved problem for which I ultimately received a prize.' back |
Rolf Landauer (1991), Information is a Physical Entity, Abstract Information is inevitably tied to a physical representation and therefore to restrictions and possibilities related to the laws of physics and the parts available in the universe. Quantum mechanical superpositions of information bearing states can be used, and the real utility of that needs to be understood. Quantum parallelism in computation is one possibility and will be assessed pessimistically. The energy dissipation requirements of computation, of measurement and of the communications link are discussed. The insights gained from the analysis of computation has caused a reappraisal of the perceived wisdom in the other two fields. A concluding section speculates about the nature of the laws of physics, which are algorithms for the handling of information, and must be executable in our real physical universe.' back |
Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, ' In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. . . . In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back |
Self-adjoint operator - Wikipedia, Self-adjoint operator - Wikipedia, the free encyclopedia, ' In mathematics, a self-adjoint operator on a complex vector space V with inner product ⟨ ⋅ , ⋅ ⟩ is a linear map A (from V to itself) that is its own adjoint. That is, ⟨ A x , y ⟩ = ⟨ x , A y ⟩ = for all x , y ∊ V. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. . . . Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space.' back |
Self-adjoint operator - Wikipedia, Self-adjoint operator - Wikipedia, the free encyclopedia, ' In mathematics, a self-adjoint operator on a complex vector space V with inner product ⟨ ⋅ , ⋅ ⟩ is a linear map A (from V to itself) that is its own adjoint. That is, ⟨ A x , y ⟩ = ⟨ x , A y ⟩ for all x , y ∈ V If V {\displaystyle V} is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A ∗ By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. |
Sergei Treil (2026_01_21), Linear Algebra Done Wrong, 'Besides being a first course in linear algebra it is also supposed to be
a first course introducing a student to rigorous proof, formal definitions—in short, to the style of modern theoretical (abstract) mathematics. The target audience explains the very specific blend of elementary ideas and concrete examples, which are usually presented in introductory linear algebra texts with more abstract definitions and constructions typical for advanced books. |
Sin-Itiro Tomonaga (1965), Nobel Lecture: Development of Quantum Electrodynamics: Personal recollections, ' (1) In 1932, when I started my research career as an assistant to Nishina, Dirac published a paper in the Proceedings of the Royal Society, London1. In this paper, he discussed the formulation of relativistic quantum mechanics, especially that of electrons interacting with the electromagnetic field. At that time a comprehensive theory of this interaction had been formally completed by Heisenberg and Pauli, but Dirac was not satisfied with this theory and tried to construct a new theory from a different point of view. Heisenberg and Pauli regarded the (electromagnetic) field itself as a dynamical system amenable to the Hamiltonian treatment; its interaction with particles could be described by an interaction energy, so that the usual method of Hamiltonian quantum mechanics could be applied. On the other hand, Dirac thought that the field and the particles should play essentially different roles. That is to say, according to him, “the role of the field is to provide a means for making observations of a system of particles” and therefore “we cannot suppose the field to be a dynamical system on the same footing as the particles and thus be something to be observed in the same way as the particles” [. . .]' back |
Theory of Forms - Wikipedia, Theory of Forms - Wikipedia, the free encyclopedia, ' The Theory of Forms or Theory of Ideas, also known as Platonic idealism or Platonic realism, is a philosophical theory credited to the Classical Greek philosopher Plato. |
Wojciech Hubert Zurek (2008), Quantum origin of quantum jumps: Breaking of unitary symmetry induced by information transfer and the transition from quantum to classical, 'Submitted on 17 Mar 2007 (v1), last revised 18 Mar 2008 (this version, v3)) Measurements transfer information about a system to the apparatus, and then further on – to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide framework for the “wavepacket collapse”, designating terminal points of quantum jumps, and defining the measured observable by specifying its eigenstates.' back |