This site documents the Divine Mind:
the Universe that enables us to create ourselves
page 3: Action: from the ancient unmoved mover to the modern quantum
page 3: Contents
3.1: A story from Ancient Greece3.2: Aristotle and the definition of God
3.3: From Newton to Lagrange
3.4: Max Planck and quantum mechanics
3.5: Action, logic and theology
3.1: A story from Ancient Greece
Action is what makes things happen, the transition from a possibility to a reality like the click of a switch or the sound of the ref's whistle that turn lights and games on and off. By defining them as pure action Aquinas recognizes that God is essentially an event, something happening like the world in motion, a tick in time.
When Plato proposed eternal forms to guide the structure of the world he was following a suggestion by Parmenides (5th - 6th century bce). We have fragments of a didactic poem from Parmenides describing his education by a Goddess. In this poem he proposed a foundation for science, based on the proposition that permanently true knowledge is only possible if the subject of knowledge is immutable or invariant. John Palmer: Parmenides
The Goddess contrasted the unshaken heart of well-rounded reality with the notions of mortals, in which there is no genuine trustworthiness. (Fr. 1.28b-32)
She went on to describe true reality, what is:
First, it is ungenerated and deathless. Further it it is whole and uniform, motionless . . .
. . . it is perfected from every side, like the bulk of a well-rounded globe, from the middle equal every way: for that it be neither any greater nor any smaller in this place or in that is necessary; for neither is there non-being, which would stop it reaching to its like, nor is What Is such that it might be more than What Is here and less there. Since it is all inviolate, for it is equal to itself from every side, it extends uniformly in limits.
This suite of attributes have remained central to the description of what is, also known as God, for 2500 years. Parmenides' student Zeno supported Parmenides' position with a series of subtle arguments to show that motion is impossible. Zeno's paradoxes - Wikipedia
This old idea is invalidated by the more modern Nyquist-Shannon sampling theorem: we can have true knowledge of changing situations if we update our knowledge fast enough. The key to catching a moving ball is to watch it closely and move accordingly. Perhaps Parmenides did not play ball games. We can capture flowing music or speech in static text if we write down the sequence of sounds as notes or words in a time based sequence. Nyquist-Shannon sampling theorem - Wikipedia
Plato guessed that Parmenides what is is a system of invisible, eternal forms which guide the structure of the world and our knowledge of it. Unlike Parmenides, Plato's student Aristotle considered motion to be an integral part of realty but he needed to preserve the stability of Plato's forms. He devised the theory of matter and form known as hylomorphism to explain changes in the world. Theory of Forms - Wikipedia, Hylomorphism - Wikipedia
We may see hylomorphism as a form of two factor authentication. In its abstract form, this theory explains that neither matter nor form can exist on their own, but together they make a real thing. Change is possible because in reality matters and forms could be swapped around to make different things. Thomas Ainsworth: Form vs. matter
Hylomorphism, in a more abstract form, is Aristotle's hypothesis of potency and act. Aristotle saw matter as potentially something and form as the element that made it actually something. The theory of potency and act is built around one axiom: no potential can actualize itself. This led him to postulate an unmoved mover as the source of all motion. Unmoved mover - Wikipedia
Aristotle saw motion as a transition from potential to actuality. It follows that nothing can move itself, everything is moved by something else. But there must be a source of motion, otherwise there would be no motion. This is the unmoved mover which is, by definition, pure actuality. In it and through it, all potentials are actualized.
Aristotle's works gradually entered the newly formed Christian universities of Europe when contact was established between the two halves of the old Roman empire. Albert the Great and Thomas Aquinas used Aristotle's work to set up a new philosophical foundation for Christian theology. Aquinas developed a new Catholic model of God (which has since become standard) from Aristotle's theological treatment of the unmoved mover:
But if there is something which is capable of moving things or acting on them, but is not actually doing so, there will not necessarily be movement; for that which has a potency need not exercise it. Nothing, then, is gained even if we suppose eternal substances, as the believers in the Forms do [ie Plato], unless there is to be in them some principle which can cause change; nay, even this is not enough, nor is another substance besides the Forms enough; for if it is not to act, there will be no movement. Further even if it acts, this will not be enough, if its essence is potency; for there will not be eternal movement, since that which is potentially may possibly not be. There must, then, be such a principle, whose very essence is actuality. Further, then, these substances must be without matter; for they must be eternal, if anything is eternal. Therefore they must be actuality. Recovery of Aristotle - Wikipedia, Aristotle Metaphysics XII, vi, 2
3.2: Aristotle and the definition of God
Aristotle was among the first to study action. He coined two words for it, energeia (ενεργεια) and entelecheia (εντελεχεια). Energeia may be translated as being-at-work and entelecheia as completeness, the end of work. Both these terms are contrasted to potentiality dunamis (δυναμις), which can mean either active power or passivity. In Aristotle's theory the potential is passive. In modern physics, it is active, identical but opposite to kinetic energy, the two forms adding up to zero, like energy and anti-energy. Aristotle's axiom does not hold. The action of a pendulum transforms actual (kinetic) energy into potential and back again as it swings up and down. It may be that the total energy of the Universe is zero, the actual energy of matter being exactly balanced by the potential energy of gravitation. Zero-energy universe - Wikipedia
Aristotle thought that the unmoved mover was an integral part of the Cosmos and that the world is eternal so it had no need for a creator. Aquinas used Aristotle's argument to establish the existence of God but, faithful to his religion, placed this creator outside the Universe. The doctrine of the Summa has never been supeseded in the Church. It remains officially endorsed in Canon Law. Aquinas' definition of God as actus purus is, by papal fiat, standard. Aquinas's philosophy has been codified, reduced to 24 tutae normae directivae (safe directive norms), and so sterilized, by the Church, but there is plenty of inspiration in his enormous body of writing. Aristotle, Metaphysics 1072b3 sqq., Aquinas Summa I, 2, 3: Does God exist?, Holy See: Code of Canon Law: Canon 252 § 3, Pedro Lumbreras: The Twenty-Four Fundamental Theses of Official Catholic Philosophy [Sacred Congregtion of Studies, 24 July 1914]
Seven hundred years later, the Catholic theologian Bernard Lonergan set out to reconceive Aquinas' arguments for the existence of God. Lonergan's idea was to recast Aquinas's ideas in epistemological or cognitive rather than physical terms. Lonergan's story follows a time honoured path. We all agree that the world exists, but we can see (some say) that it cannot account for its own existence. There must therefore be a Creator to explain the existence of the world which many of us agree to call God. Lonergan seems to me to beg the question right at the beginning by assuming the existence of two classes of intelligible being. Proportionate being is proportional to our capacity to think so that we can understand it. Transcendental being, the realm of Gods and Angels, is beyond us. Lonergan (1992): Insight, A Study of Human Understanding Chapter 19.
Lonergan set out to argue that God is other than the Universe by following the epistemological path pioneered by Parmenides, using the act of human understanding, insight, as his starting point. God, he assumed, must be perfectly intelligible. But the world is not perfectly intelligible. It contains meaningless data, empirical residue, so it cannot be divine.
For me the weak spot in this argument is the idea that the world contains meaningless data. The theory of evolution suggests that there is a reason for every detail. Every tiny feature of living bodies, from eyebrows to genetic code right down to the choice and positioning of individual atoms has been explored, tested, accepted or rejected by the evolutionary variation and selection over billions of years and countless generations. The world is dense with meaning. Jeffrey Nicholls (1967): How universal is the Universe?, Jeffrey Nicholls (2019): A prolegomenon to scientific theology
Although I feel that Lonergan fails to prove that God is not the Universe, his work led me to think of the Universe in cognitive terms. The story presented here is intended to use an epistemological approach consolidate the view, in opposition to Lonergan and the dogma of the Roman Catholic Church, that the Universe plays all the roles attributed to traditional Gods. It is at once simple, omniscient, omnipotent and cognitive, knowing itself by acting upon itself. Although we live in spacetime, the idea here is that spacetime itself is a product of the underlying cognitive processes described by quantum theory.
Implicit in our ancient views are the dichotomies of God and the World, Heaven and Earth, Matter and Spirit. The leading idea is that matter is dead and inert. It cannot move itself. It cannot be the seat of understanding. It cannot be creative. Since the advent of modern physics, founded on relativity and quantum theory, these ideas are exactly wrong. Physics based on quantum theory describes a Universe in perpetual motion as a gigantic network of communication (which is a form of computation) equivalent to a mind.
Medieval scholars, engineers, farmers and parents gradually learnt that ancient texts and pure reason could not fully explain the world. This attitude was strongly supported by astronomy, the first science based on instrumental observation. Galileo's telescope led to radical developments in astronomy, and some conflict with ancient religious beliefs. Galileo's opinion that mathematics is the language of the Universe reached a high point in Isaac Newton's description of gravitation. Newton showed that the Moon in the heavens and apples on Earth are guided by the same invisible structure. Wisely, he declined to speculate about what this structure might be and left it in the hands of the divinity, the traditional answer to all insoluble problems. Galileo affair - Wikipedia, Isaac Newton (1729, 1972): Philosophiae Naturalis Principia Mathematica, Isaac Newton (1726): General Scholium to Principia
3.3: From Newton to Lagrange
Newton's work sparked huge developments in mathematics and physics which still continue. On a more philosophical level, Maupertuis (1698-1759) speculated that a wise creator would have made the world as efficient as possible, and started a train of thought that lead to Hamilton's principle, the amalgamation of the calculus of variations with Langrangian mechanics. We might also imagine that the effectiveness of Hamilton's principle is the result of natural evolution selecting the most efficient mechanical processes to dominate the physical world.
Joseph-Louis Lagrange sought to express classical Newtonian mechanics in a form that would make it easier to study many body problems like the solar system. His work, Mecanique Analytique, placed mechanics on an algebraic rather than a geometric foundation. Mécanique analytique Volume 1
In the Lagrangian approach the action S associated with an event x that takes place between times t1 and t2 is expressed by the action functional
S(x) = ∫ L dt
The Lagrangian L = (T(t) − V(t)), where T and V are functions of the kinetic and potential energy of the system. Lagrangian mechanics postulates Hamilton's principle that the actual trajectory taken by a particle whose motion is constrained by T and V coincides with a stationary value of S (a fixed point in the action) which may be found using Euler's calculus of variations. Lagrangian mechanics - Wikipedia, Hamilton's principle - Wikipedia, Calculus of variations - Wikipedia
Lagrangian mechanics is very versatile and serves as a bridge between classical and quantum mechanics, quantum field theory and physical problems in general. On this basis, we might understand mechanics in spacetime as the study of action in the relationship between kinetic and potential energy.
3.4: Max Planck and quantum mechanics
Quantum mechanics began with Planck's discovery, in 1900, that action is quantized and that the quantum of action h is the constant of proportionality between the energy of radiation and its frequency. This is now the fundamental equation of quantum theory, E = ℏω where ℏ is the reduced Planck constant, h / 2π and the frequency is expressed in radians per second, ω, understood as the rate of change of the phase of a quantum state ∂ | φ> / ∂ t. Max Planck: On the Law of Distribution of Energy in the Normal Spectrum
The quantum of action is very small and has the same dimensions as angular momentum in classical physics: ML2T-1, since energy has the dimension ML2T-2 and frequency the dimension of inverse time T-1. In classical mechanics the Lagrangian action S (x) is a continuous variable, whereas in quantum theory it is discrete, so that every event is associated with an integral number of quanta of action nh.
The quantum of action is a now precisely fixed natural constant which is used as a foundation for the natural set of units. We might look upon it as an invariant solution to a Lagrangian variation problem established at the very root of universal structure. NIST: Kilogram, Mass and Planck's Constant
For Aristotle and Aquinas action is a metaphysical term, but here we see that it has a physical realization, providing a bridge between physics and metaphysics in a way analogous to its role in coupling classical and quantum mechanics. Dirac found that this role goes deeper, and Feynman used it to create a new expression of quantum mechanics, the path integral formulation. Feynman & Hibbs (1965): Quantum Mechanics and Path Integrals, Path integral formulation - Wikipedia
Quantum mechanics came of age in the 1920's in two versions known as wave mechanics (the Schrödinger equation) and matrix mechanics. These were shown to be equivalent by Schrödinger, given a clear abstract symbolic expression by Dirac and a sound mathematical foundation by von Neumann using linear operators in abstract Hilbert space. Dirac notes that the major features of quantum mechanics are linearity and superposition. Schrödinger equation - Wikipedia, Matrix mechanics - Wikipedia, Paul Dirac (1983): The Principles of Quantum Mechanics, chapter 1, John von Neumann (2014): Mathematical Foundations of Quantum Mechanics
Feynman introduced a third approach to quantum mechanics which has since found favour because it provides a more direct route to quantum field theory and string theory. His path integral formulation seeks a stationary superposition of the contributions of all possible space-time paths between an initial and a final state. In principle, this set of paths spans the whole classical Universe so the formulation depends implicitly on the idea, discussed on page 13: Is Hilbert space independent of Minkowski space?, that the quantum world is prior to and independent of Minkowski space.
Feynman began with Dirac's search for a feature of quantum theory corresponding to classical Lagrangian mechanics. Dirac found that the classical action could be used in a complex exponential to describe the evolution of a quantum state. Feynman imagined, by analogy with the two slit model, that the actual path taken by a particle is a stationary superposition of all possible paths where the contribution from a particular path is postulated to be an exponential whose (imaginary) phase is the classical action, in units of h. P. A. M. Dirac (1933): The Lagrangian in Quantum Mechanics, Richard P. Feynman (1948): Space-Time Approach to Non-Relativistic Quantum Mechanics
The path integral relies on the three general principles of quantum mechanics formulated by Feynman:
Feynman lectures on physics III Chapter 3: Probability Amplitudes1. The probability that a particle will arrive at x, when let out at the source s, can be represented quantitatively by the absolute square of a complex number called a probability amplitude — in this case, the “amplitude that a particle from s will arrive at x.”
2. When a particle can reach a given state by two possible routes, the total amplitude for the process is the sum of the amplitudes for the two routes considered separately. There is interference.
3. When a particle goes by some particular route the amplitude for that route can be written as the product of the amplitude to go part way with the amplitude to go the rest of the way.
The path integral computes the probability amplitude for a particle to go from s to x by dividing every possible path into infinitesimal segments and multiplying the amplitudes for the particle to cross each segment according to principle 3 to get the amplitude for the whole path, adding these amplitudes as required by principle 2 and computing the probability represented by the resulting amplitude by principle 1. The process works and contributes to computations in quantum field theory which precisely match observation. But, we might ask, does nature really work this way? If we are to consider the quantum of action as an atomic event, can we trust the mathematical fiction that a quantum path can be sliced into an infinity of infinitesimal events?
3.5: Action, logic and theology
On page 2: Introduction I wrote:
When modern physicists talk about creating a theory of everything, what they have in mind is a unified theory of the four fundamental forces, gravitation, electromagnetism, the weak force and the strong force. From the theological point of view this is a very limited ambition. A real theory of everything would be a theology, able to describe every detail of creation. Theology itself paints the broad picture, relying on all the other sciences and arts to fill in the details. At present we have a good picture of the development of the Universe since it was about 300 000 years old. Eventually, perhaps, a theological picture will take us back to the beginning.
All the discussion above has assumed the existence of spacetime and is built on our spacetime experiences, but action is the definition of god and the traditional God predates spacetime.
Quantum mechanics in Minkowski spacetime interprets the time rate of action as energy (E = hω) and the spatial rate of action as momentum (p = h / λ for a photon wavelength λ). In the Hilbert space that underlies Minkowski space, the quantum of action behaves as a logical operator. This is represented in the quantum theory of communication and computation as a linear operator which can be written out mathematically as a matrix. Nielsen & Chuang (2016): Quantum Computation and Quantum Information
Each quantum event, like the interaction of an electron and a photon, involves the transfer of one quantum of action. When an atom emits a photon with spin 1, corresponding to one quantum of action, we understand that the source of the photon is an atomic electron which changes to an orbital with one Plank unit unit less angular momentum and a loss of energy which is reflected in the frequency of the photon emitted.
Although this event looks quite simple, the quantum field theory calculation of the energy involved, and therefore the frequency of the emitted photon can be quite complex, and this complexity is represented by the mathematical model of the event. For high precision this may involve the hundreds or thousands of terms in the relevant Feynman diagrams, each representing a possible path of interaction. Nevertheless the exchange of action involved is precisely one quantum and it is required, for the conservation of probability, that the process is normalized so that length of the complex vectors created by the superposition of all the diagrams is is precisely 1.
Richard Feynman, one of the inventors of quantum electrodynamics, had this to say about the current form of precise calculations in his Nobel Prize Lecture:
I don’t think we have a completely satisfactory relativistic quantum-mechanical model, even one that doesn’t agree with nature, but, at least, agrees with the logic that the sum of probability of all alternatives has to be 100%. Therefore, I think that the renormalization theory is simply a way to sweep the difficulties of the divergences of electrodynamics under the rug. I am, of course, not sure of that. Richard P. Feynman (1965): Nobel Lecture: The Development of the Space-Time View of Quantum Electrodynamics
The problem facing physicists is that they are trying to explain phenomena in spacetime which are being worked out behind the scenes in a structure which antedates and underlies spacetime. From a theological point of view we may see that quantum theory is not so much an arithmetic as a logical theory, and is therefore closely related to mental function and opens the way for us to consider the physical Universe as the physical mind of God, analogous to our human brains that implement our minds. The authors of Genesis I: 27 may have been hinting at this when they wrote: So God created man in his own image, in the image of God created he him; male and female created he them.
(Revised Monday 22 April 2024)
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Notes and references
Further reading
Books
Dirac (1983), P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)
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Feynman (1965), Richard P, and Albert P Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill 1965 Preface: 'The fundamental physical and mathematical concepts which underlie the path integral approach were first developed by R P Feynman in the course of his graduate studies at Princeton, ... . These early inquiries were involved with the problem of the infinite self-energy of the electron. In working on that problem, a "least action" principle was discovered [which] could deal successfully with the infinity arising in the application of classical electrodynamics.' As described in this book. Feynman, inspired by Dirac, went on the develop this insight into a fruitful source of solutions to many quantum mechanical problems.
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Lonergan (1992), Bernard J F, Insight: A Study of Human Understanding (Collected Works of Bernard Lonergan : Volume 3), University of Toronto Press 1992 '. . . Bernard Lonergan's masterwork. Its aim is nothing less than insight into insight itself, an understanding of understanding'
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Newton (1729, 1972), Isaac, Philosophiae Naturalis Principia Mathematica, Harvard University Press 1729, 1972 One of the most important contributions to human knowledge. First translated from the Latin by Andrew Motte in 1729,
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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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Links
Aquinas Summa I, 2, 3, Does God exist?, 'I answer that, The existence of God can be proved in five ways. |
Aristotle 1071b12, Metaphysics book XII, vi, 2: The role of the unmoved mover, ' But even if we are to suppose that there is something which is kinetic and productive, although it does not actually move or produce, there will not necessarily be motion; for that which has potentiality may not actualise it. Thus it will not help matters if we posit eternal substances, as do the exponents of the Forms, unless there is in them some principle which can cause change. . . . therefore there must be a principle of this kind whose essence is actuality.' back |
Aristotle, Metaphysics, Metaphysics, Book XII, vii, 'But since there is something which moves while itself unmoved, existing actually, this can in no way be otherwise than as it is. For motion in space is the first of the kinds of change, and motion in a circle the first kind of spatial motion; and this the first mover produces. The first mover, then, exists of necessity; and in so far as it exists by necessity, its mode of being is good, and it is in this sense a first principle.' 1072b6 sqq back |
Calculus of variations - Wikipedia, Calculus of variations - Wikipedia, the free encylopedia, ' The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.' back |
Conservation of energy - Wikipedia, Conservation of energy - Wikipedia, the free encyclopedia, 'In physics, the law of conservation of energy states that the total energy of an isolated system cannot change—it is said to be conserved over time. Energy can be neither created nor destroyed, but can change form, for instance chemical energy can be converted to kinetic energy in the explosion of a stick of dynamite. back |
Feynman, Leighton & Sands FLP III:03, Chapter 3: Probability Amplitudes, 'We will begin in this chapter by dealing with some general quantum mechanical ideas. Some of the statements will be quite precise, others only partially precise. It will be hard to tell you as we go along which is which, but by the time you have finished the rest of the book, you will understand in looking back which parts hold up and which parts were only explained roughly. The chapters which follow this one will not be so imprecise. In fact, one of the reasons we have tried carefully to be precise in the succeeding chapters is so that we can show you one of the most beautiful things about quantum mechanics—how much can be deduced from so little.' back |
Galileo affair - Wikipedia, Galileo affair - Wikipedia, the free encyclopedia, ' The Galileo affair (Italian: il processo a Galileo Galilei) began around 1610 and culminated with the trial and condemnation of Galileo Galilei by the Roman Catholic Inquisition in 1633. Galileo was prosecuted for his support of heliocentrism, the astronomical model in which the Earth and planets revolve around the Sun at the centre of the Solar System. ' back |
Hamilton's principle - Wikipedia, Hamilton's principle - Wikipedia, the free encyclopedia, 'In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action . . . It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it.' back |
Holy See, Code of Canon Law: Canon 252 §3, ' There are to be classes in dogmatic theology, always grounded in the written word of God together with sacred tradition; through these, students are to learn to penetrate more intimately the mysteries of salvation, especially with St. Thomas as a teacher. There are also to be classes in moral and pastoral theology, canon law, liturgy, ecclesiastical history, and other auxiliary and special disciplines, according to the norm of the prescripts of the program of priestly formation.' back |
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 |
Interpretation of quantum mechanics - Wikipedia, Interpretation of quantum mechanics - Wikipedia, the free encyclopedia, 'An interpretation of quantum mechanics is a statement which attempts to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has received thorough experimental testing, many of these experiments are open to different interpretations. There exist a number of contending schools of thought, differing over whether quantum mechanics can be understood to be deterministic, which elements of quantum mechanics can be considered "real", and other matters.' back |
Isaac Newton (1726), General Scholium, 'Published for the first time as an appendix to the 2nd (1713) edition of the Principia, the General Scholium reappeared in the 3rd (1726) edition with some amendments and additions. As well as countering the natural philosophy of Leibniz and the Cartesians, the General Scholium contains an excursion into natural theology and theology proper. In this short text, Newton articulates the design argument (which he fervently believed was furthered by the contents of his Principia), but also includes an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity, which Newton saw as a post-biblical corruption. The English translation here is that of Andrew Motte (1729). Italics and orthography as in original. back |
Jeffrey Nicholls (1967), How universal is the universe?, ' 61 The future is beyond our comprehension, but we can get an idea of it and speed its coming by studying what we already have. Contemplating the size and wonder of the universe as it stands in the light of its openness to the future must surely be a powerful incentive to men to love God. We have come a long way since the little world of St Thomas. Ours is open to all things, even participating in god. This is what I mean by universal. ' back |
Jeffrey Nicholls (2019), A prolegomenon to scientific theology, ' This thesis is an attempt to carry speculative theology beyond the apogee it reached in the medieval work of Thomas Aquinas into the world of empirical science. Since the time of Aquinas, our understanding of the Universe has increased enormously. The ancient theologians not only conceived a perfect God, but they also saw the world as a very imperfect place. Their reaction was to place God outside the world. I argue (on classical scientific grounds) that we live in a Universe which approaches infinity in size and complexity, is as perfect as can be, and fulfils all the roles traditionally attributed to God, creator, lawmaker and judge.' back |
John Palmer (Stanford Encyclopedia of Philosophy), Parmenides, ' Immediately after welcoming Parmenides to her abode, the goddess describes as follows the content of the revelation he is about to receive: |
John von Neumann (2014), Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford. Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC back |
Joseph-Louis Lagrange (1811), Mécanique analytique Volume 1, ' On a déjà plusieurs Traités de Mécanique , mais le plan de celui - ci est entièrement neuf . Je me suis proposé de réduire la théorie de cette Science , et l'art de résoudre les problèmes qui s'y rapportent , à des formules générales, dont le simple développement donne toutes les équations nécessaires pour la solution de chaque problème.' back |
Lagrangian mechanics - Wikipedia, Lagrangian mechanics - Wikipedia, the free encyclopedia, ' Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle. Given a system of point masses and a pair, t1 and t2 Lagrangian mechanics postulates that the system's trajectory (describing evolution of the system over time) . . . must be a stationary point of the action functional S = ∫ L dt. By convention, L = T − V, where T and V are the kinetic and potential energy of the system, respectively.' back |
Matrix mechanics - Wikipedia, Matrix mechanics - Wikipedia, the free encyclopedia, 'Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps occur. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, and is the basis of Dirac's bra-ket notation for the wave function. back |
Max Planck, On the Law of Distribution of Energy in the Normal Spectrum, Annalen der Physik, vol. 4, p. 553 ff (1901) 'The recent spectral measurements made by O. Lummer and E. Pringsheim and even more notable those by H. Rubens and F. Kurlbaum which together confirmed an earlier result obtained by H. Beckmann show that the law of energy distribution in the normal spectrum, first derived by W. Wien from molecular-kinetic considerations and later by me from the theory of electromagnetic radiation, is not valid generally.' back |
Minkowski space - Wikipedia, Minkowski space - Wikipedia, the free encyclopedia, ' By 1908 Minkowski realized 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 |
NIST, Kilogram, Mass and Planck's Constant, ' For many observers, the connection between mass on the scale of a liter of water and a constant deriving from the very earliest days of quantum mechanics may not be immediately obvious. The scientific context for that connection is suggested by a deep underlying relationship between two of the most celebrated formulations in physics. One is Einstein's famous E =mc2, where E is energy, m is mass and c is the speed of light. The other expression, less well known to the general public but fundamental to modern science, is E = hν, the first "quantum" expression in history, stated by Max Planck in 1900. Here, E is energy, ν is frequency (the ν is not a “v” but instead the lowercase Greek letter nu), and h is what is now known as the Planck constant.' back |
Nyquist-Shannon sampling theorem - Wikipedia, Nyquist-Shannon sampling theorem - Wikipedia, the free encyclopedia, ' In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals (often called "analog signals") and discrete-time signals (often called "digital signals"). It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.' back |
P. A. M. Dirac (1933), The Lagrangian in Quantum Mechanics, ' . . . there is an alternative formulation [to the Hamiltonian] in classical dynamics, provided by the Lagrangian. This requires one to work in terms of coordinates and velocities instead of coordinates and momenta. The two formulation are closely related but there are reasons for believing that the Lagrangian one is more fundamental. . . . Secondly the lagrangian method can easily be expressed relativistically, on account of the action function being a relativistic invariant; . . .. ' [This article was first published in Physikalische Zeitschrift der Sowjetunion, Band 3, Heft 1 (1933), pp. 64–72.] back |
P. Lumbreras, The Twenty-Four Fundamental Theses of Official Catholic Philosophy [Sacred Congregtion of Studies, 24 July 1914], ' In our preceding paper we proved by documents of recent Popes that the Church, in exercising her right, has adopted the scholastic philosophy as her official philosophical teaching, that by scholastic philosophy the Church understands not only chiefly but exclusively the philosophy of St. Thomas, and that St. Thomas' philosophy stands for at least the twenty-four theses approved and published by the Sacred Congregation of Studies.' back |
Path integral formulation - Wikipedia, Path integral formulation - Wikipedia, the free encyclopedia, 'The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude. . . . This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970s which unified quantum field theory with statistical mechanics. . . . ' back |
Potential energy - Wikipedia, Potential energy - Wikipedia, the free encyclopedia, 'In physics, potential energy is the energy of an object or a system due to the position of the body or the arrangement of the particles of the system. The SI unit for measuring work and energy is the joule (symbol J). The term potential energy was coined by the 19th century Scottish engineer and physicist William Rankine although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with a set of forces that act on a body in a way that depends only on the body's position in space.' back |
Recovery of Aristotle - Wikipedia, Recovery of Aristotle - Wikipedia, the free encclopedia , ' The "Recovery of Aristotle" (or Rediscovery) refers to the copying or re-translating of most of Aristotle's books (of ancient Greece), from Greek or Arabic text into Latin, during the Middle Ages, of the Latin West. The Recovery of Aristotle spanned about 100 years, from the middle 12th century into the 13th century, and copied or translated over 42 books (see: Corpus Aristotelicum), including Arabic texts from Arabic authors, where the previous Latin versions had only two books in general circulation: Categories and On Interpretation (De Interpretatione).' back |
Richard P. Feynman (1948), Space-Time Approach to Non-Relativistic Quantum Mechanics, ' Abstract Non-relativistic quantum mechanics is formulated here in a different way. It is, however, mathematically equivalent to the familiar formulation. In quantum mechanics the probability of an event which can happen in several different ways is the absolute square of a sum of complex contributions, one from each alternative way. The probability that a particle will be found to have a path x(t) lying somewhere within a region of space time is the square of a sum of contributions, one from each path in the region. The contribution from a single path is postulated to be an exponential whose (imaginary) phase is the classical action (in units of ℏ) for the path in question. The total contribution from all paths reaching x, t from the past is the wave function ψ(x, t). This is shown to satisfy Schroedinger's equation. The relation to matrix and operator algebra is discussed. Applications are indicated, in particular to eliminate the coordinates of the field oscillators from the equations of quantum electrodynamics.' back |
Richard P. Feynman (1965), Nobel Lecture: The Development of the Space-Time View of Quantum Electrodynamics, Nobel Lecture, December 11, 1965: 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 |
Richard P. Feynman (1965), Nobel Lecture: The Development of the Space-Time View of Quantum Electrodynamics, Nobel Lecture, December 11, 1965: I did gather from my readings, however, that two things were the source of the difficulties with the quantum electrodynamical theories. The first was an infinite energy of interaction of the electron with itself. And this difficulty existed even in the classical theory. The other difficulty came from some infinites which had to do with the infinite numbers of degrees of freedom in the field. As I understood it at the time (as nearly as I can remember) this was simply the difficulty that if you quantized the harmonic oscillators of the field (say in a box) each oscillator has a ground state energy of (½hω) and there is an infinite number of modes in a box of every increasing frequency ω, and therefore there is an infinite energy in the box. I now realize that that wasn’t a completely correct statement of the central problem; it can be removed simply by changing the zero from which energy is measured. At any rate, I believed that the difficulty arose somehow from a combination of the electron acting on itself and the infinite number of degrees of freedom of the field.' 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 |
Theory of Forms - Wikipedia, Theory of Forms - Wikipedia, the free encyclopedia, 'Plato's theory of Forms or theory of Ideas asserts that non-material abstract (but substantial) forms (or ideas), and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form or idea is often capitalized. Plato speaks of these entities only through the characters (primarily Socrates) of his dialogues who sometimes suggest that these Forms are the only true objects of study that can provide us with genuine knowledge; thus even apart from the very controversial status of the theory, Plato's own views are much in doubt. Plato spoke of Forms in formulating a possible solution to the problem of universals.' back |
Thomas Ainsworth (Stanford Encyclopedia of Philosophy), Form vs. Matter, 'Aristotle famously contends that every physical object is a compound of matter and form. This doctrine has been dubbed “hylomorphism”, a portmanteau of the Greek words for matter (hulê) and form (eidos or morphê). Highly influential in the development of Medieval philosophy, Aristotle’s hylomorphism has also enjoyed something of a renaissance in contemporary metaphysics.' back |
Unmoved mover - Wikipedia, Unmoved mover - Wikipedia, the free encyclopedia, ' The unmoved mover (Ancient Greek: ὃ οὐ κινούμενον κινεῖ, lit. 'that which moves without being moved' or prime mover (Latin: primum movens) is a concept advanced by Aristotle as a primary cause (or first uncaused cause) or "mover" of all the motion in the universe. As is implicit in the name, the unmoved mover moves other things, but is not itself moved by any prior action. In Book 12 (Greek: Λ) of his Metaphysics, Aristotle describes the unmoved mover as being perfectly beautiful, indivisible, and contemplating only the perfect contemplation: self-contemplation. He equates this concept also with the active intellect. This Aristotelian concept had its roots in cosmological speculations of the earliest Greek pre-Socratic philosophers and became highly influential and widely drawn upon in medieval philosophy and theology. St. Thomas Aquinas, for example, elaborated on the unmoved mover in the Quinque viae. ' back |
Zeno's paradoxes - Wikipedia, Zeno's paradoxes - Wikipedia, the free encyclopedia, 'Zeno's paradoxes are a set of problems generally thought to have been devised by Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion.' back |
Zero-energy universe - Wikipedia, Zero-energy universe - Wikipedia, the free encyclopedia, 'The zero-energy universe hypothesis proposes that the total amount of energy in the universe is exactly zero: its amount of positive energy in the form of matter is exactly cancelled out by its negative energy in the form of gravity. . . . The zero-energy universe theory originated in 1973, when Edward Tryon proposed in the journal Nature that the universe emerged from a large-scale quantum fluctuation of vacuum energy, resulting in its positive mass-energy being exactly balanced by its negative gravitational potential energy.' back |