page 23: Insight and belief

Plus ça change, plus c’est la même chose;
The more things change the more they remain the same.

A physical understanding is completely unmathematical, imprecise, an inexact thing but absolutely necessary to a physicist. Richard Feynman: Lectures on Physics II Chapter 2: Differential Calculus of Vector Fields

Table of contents

23.1: Apologia pro vita sua

23.2: Bernard Lonergan: Insight: Understanding and affirmation

23.3: Revelation: theology, physics and evolution

23.4: Perhaps quantum field theory not possible?

23.5: On coupling kinematic forms to dynamic energy

23.6: SLAC and deep inelastic scattering

Apologia pro vita sua

It seems appropriate to apologize for one's life if one has been given the best of everything and trashed it (at least in the eyes of the donors). John Henry Newman was a senior anglican cleric, marked for great things. During the reign of King Henry VIII the English Reformation Parliament (1529 - 1536) had passed laws abolishing Papal authority in England and declared Henry the head of the Church in England. As a result, Catholics had suffered numerous legal disabilities. These were not removed until the 1830s, leading to the Oxford Movement. The opinion that despite Henry Catholicism remains the true church led many Anglicans to convert to Catholicism, Newman among them. Oxford Movement - Wikipedia, Apologia Pro Vita Sua - Wikipedia, English Reformation - Wikipedia

I owe my apology to my mother. She lived and died as an extremely devout Catholic. Her influence, reinforced by my Catholic teachers, led me to join a Catholic religious order. My parents drove me to Melbourne, we said goodbye and I stepped into a modern day facsimile of medieval life, a discipline based on the Rule of St Augustine which was assumed by the Dominican Order when it was founded in 2016. The Order also has a book length Constitution which laid down the minutiae of monastic life and governance. Rule of Saint Augustine - Wikipedia, Order of Preachers: Book of Constitutions and Ordinations of the Friars of the Order of Preacher

Here I discovered a whole new world, the theology of Thomas Aquinas (1225-1275). His major work, the Summa Theologiae, fills a 3000 page volume on india paper, double column, small print, written in easy scholastic Latin. The Summa is an entrancing summary of the Christian faith seen through the eyes of a genius. It became my new bible. The most fascinating feature of the Summa for me was that it interpreted the faith using the best science then available, the work of Aristotle. More than 1000 years after his death, Aristotle's work had been introduced to the new universities of Christian Europe from the Eastern empire. Transmission of the Greek Classics - Wikipedia

Aquinas begins with a description of sacred doctrine, ie theology, itself. It is necessary, he says because our salvation depends entirely on God, and we need to know truths about our divine destiny which are beyond our human capacity. Bernard Lonergan, quoted below, calls them transcendental. Aquinas, Summa I, 1, 1: Is sacred doctrine necessary?

He then argues that sacred doctrine is a science. He understands science as a process of logical deduction from first principles rather than abductive reasoning from experience. He sees two classes of science, one derived from principles known by natural intelligence, like arithmetic and geometry, and the other derived from principles established by a higher science, the science of God and the blessed. Sacred doctrine falls into this second class, depending on principles revealed by God. The idea that science is based on observable evidence is not emphasized in his work. From Aquinas's point of view, divine revelation plays the role in sacred science that we would attribute to empirical evidence in any science. Aquinas, Summa, I, 1, 2: Is sacred doctrine is a science?, Abductive reasoning - Wikipedia

Having established a few other points about sacred doctrine, he turned to the principal question of theology: the existence of god, the invisible subject of sacred doctrine. Aquinas based his first proof for the existence of God almost verbatim on Aristotle's proof for the existence of the unmoved mover in the world. This entity, since it could move itself, must pure actuality, since Aristotle held that no potential can actualize itself. Pure actuality idea became Aquinas's definition of God. Unfortunately, although he used Aristotle's proof, he misapplied it and took it to mean that God is other than the world. A bit dishonest, I see now, but since everything I read in Aquinas was consistent with what I had been taught from childhood, this and the other elements of the famous five ways seemed reasonable enough. Aquinas, Summa: I, 2, 3: Does God exist?, Five Ways (Aquinas) - Wikipedia, Unmoved mover - Wikipedia Aristotle Metaphysics: God's happiness

I was still a believer. Aquinas's arguments were simple and straight forward. As he wrote in the preface to the Summa:

A teacher of Catholic doctrine must not only instruct advanced students, but beginners. Following the Apostle Paul in his first letter to the Corinthians chapter III who wrote "I fed you milk like babies in Christ, not solid food because you were not ready for it, " our intention in this work is to pass on the details of the Christian religion in a manner suitable for beginners. We have noticed that the various books available on this subject hinder novices in many ways, partly because of the multiplication of useless questions, articles and arguments; partly because things that are necessary for them to understand are not taught in a their natural order, instead following the books available or according to the opportunities they offer for disputation; and partly because frequent repetition generates boredom and confusion in the minds of the students. We will try to avoid these and similar problems, and with confidence in divine assistance, to lay out the content of sacred doctrine as briefly and clearly as the material allows. Fundación Tomás de Aquino: Corpus Thomisticum: Summa Theologiae

In this, he certainly succeeded for me. So far, so good. I received the habit, completed the Novitiate and was simply professed, incorporated into the Order for 3 years of obedience, poverty and chastity. My mother's dream was fulfilled. It caused her great pain to lose me although she bore ten more children. But there was a serpent in my Garden. Perhaps it was fitting that it should come dressed as a Jesuit.

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23.2: Bernard Lonergan: Insight: Understanding and affirmation

Our philosophical studies were not very demanding and they were supposed to be conducted in Latin. None of my class knew much Latin, so things stumbled along. Later, when I left the Order and requested some documentation of my short career I found that I got HDs for almost everything. We were not taught in the lucid Latin of Aquinas, but from scholastic manuals whose authors seemed to delight in tortuous imitation of classical latinity.

In my third year I discovered Bernard Lonergan's book, Insight. It was not recommended reading but there was a copy, possibly forgotten, in the library of the Dominican House in Wahroonga. Lonergan was a Canadian Jesuit who had made a close study of Aquinas's theory of knowledge and his use of the term Verbum (word), which is central to the doctrine of the Trinity (page 8: The theology of the Trinity). Bernard Lonergan (1992): Insight: A Study of Human Understanding , Lonergan (1997): Verbum: Word and Idea in Aquinas

Aquinas' proofs for the existence of God (= the non-divinity of the Universe) might be called physical, in that he starts each proof from a physical observation and uses his (mis) understanding of Aristotle's metaphysical theory of the world to show how this physical observation implies that the Universe cannot account for itself. Lonergan moves the question into the psychological realm:

The existence of God . . . is known as the conclusion to an argument, and while such arguments are many, all of them, I believe, are included in the following general form.

If the real is completely intelligible, God exists. But the real is completely intelligible. Therefore God exists.

The Universe, however, is not God, because it is not completely intelligible. This, Lonergan claims, is because there are positively given empirical data which lack intelligibility, the 'empirical residue' detected by 'inverse insight'. Insight, pp 43-56

. . . the five ways in which Aquinas proves the existence of God are so many particular cases of the general statement that the proportionate Universe is incompletely intelligible and that complete intelligibility is demanded. ibid., p 700

Lonergan has very little to say about quantum theory, and his statements are mainly made in the context of statistics which is far from the core of the theory. He says nothing about the problem solving intelligence of manifested by the theory. In Aristotelian terms, we may say that from a mathematical point of view, quantum mechanics pays the role of the agent intellect in solving the eigenvalue problem expressed by the equation Aψ = aψ where A is a linear operator and a is a real stationary eigenvalue corresponding to one of the eigenvectors of A. This, we might say, is the fundamental act of universal intelligence.

The ability of quantum mechanics to solve the eigenvalue problem is reminiscent of Misner, Thorne and Wheeler's statement on the progress of science:

Any forward step in human knowledge is truly creative in this sense: that theory, concept, law, and method of measurement—forever inseparable—are born into the world in union. Misner, Thorne & Wheeler (1973): Gravitation, page 71

One can imagine that quantum mechanics both devises a self-adjoint operator and solves for its eigenfunctions and eigenvalues in a coordinated move. Misner, Thorne & Wheeler (1973): Gravitation, page 71.

It is not clear how Lonergan, living in his incompletely intelligible Universe, could have learnt that the real must be completely intelligible. This "truth" is based on the ancient idea promoted by Aquinas that the key to intelligence is immateriality. Since God is maximally immaterial, they are also maximally intelligent and intelligible. It is interesting that in all his enormously detailed tome on intelligence and insight Lonergan never mentions the material brain and body that is the seat of our intelligence. He obviously believes we have an immaterial, immortal soul. He has no alternative if he is to hold the Christian belief in posthumous heaven and hell. Lonergan: ibid., page 695, Aquinas, Summa: I, 14, 1: Is there knowledge in God?

Lonergan's position is developed with subtlety and some insight into the results of modern science. It is based on a careful study of the psychology of Thomas Aquinas as applied to Trinity. Lonergan's interpretation of Aquinas was my starting point for this work.

Lonergan falls down, I believe, in his affirmation of the empirical residue which depends on turn on his division of reality into the proportionate Universe, open to human understanding, and the transcendental knowledge of God that, according to St Thomas Aquinas, consists in knowing that he is but not what he is (page 656). Here he relying on the opinion of Aquinas, common throughout Catholic doctrine, that God is beyond our understanding. There is, to the best of my knowledge, no evidence for this position (except that fact that we can be certain that there are things which we do not yet know). My lifetime of experience has convinced me that the foundation of the Roman Catholic Church and most other religions is a magnificent and sickening deception. This deception is founded on the claim that somehow things have been revealed to the theologians that are not open to ordinary human knowledge is. Mystery and secrecy are the keys to traditional theologies. Unfortunate people like myself, indoctrinated from birth, have died for and from these fictions.

A byproduct of these fictions, as Constantine realized, is the production of a homogenized and cooperative human community, a necessary prerequisite for a stable empire. One can see a direct link between theological fiction and religious war. It is not surprising, therefore, that imperial organizations like the Roman Catholic Church have commonly used violence to encourage belief in their stories. The theologically motivated violent suppression of women is widespread around the world. European wars of religion - Wikipedia, Albigensian Crusade - Wikipedia

The Catholic Church recently reinforced the view that there is world beyond human understanding. In the Dogmatic Constitution on Divine Revelation the Fathers wrote:

6. Through divine revelation, God chose to show forth and communicate Himself and the eternal decisions of His will regarding the salvation of men. That is to say, He chose to share with them those divine treasures which totally transcend the understanding of the human mind. Second Vatican Council: Dogmatic Constitution on Divine Revelation "Dei Verbum"

This project relies on two arguments to deny the Church's claim of transcendental knowledge. The first is derived from the work of Aquinas himself, repeating and confirming the age old belief that God is pure spirit, structureless and absolutely simple. From this point of view, there is nothing to be known about God but that it exists. This is the property of the initial singularity which enables us on this site to identify it with the traditional God. Neither this god nor the singularity has the structure necessary to represent information. Aquinas, Summa, I, 3, 7: Is God altogether simple?

The second, discussed on page 21: Matter and spirit relies on our modern understanding that information is a physical reality which requires material entities, like the letters in this text, for representation. The ancient view that pure spirit is the foundation of knowledge and understanding appears to be a mistake possibly derived from Aristotle's belief that our faculty of intelligence must be matter free in order for us to be able to understand all material things by analogy with the idea that our eyes must be colourless to be receptive of all colours. Is there knowlege in God?, Christopher Shields (Stanford Encyclopedia of Philosophy b): The Active Mind of De Anima III 5

The fact that the classical God of Aquinas has no means to embody information means that they could not have created the world to a predetermined plan as Plato proposed. The only way that the structure of the Universe could have emerged from an omnipotent singularity is by the process of variation and selection integral to evolution. This history suggests that Lonergan's idea of empirical residue has no foundation, since biologists have discovered that every atom in every molecule in living creatures has been selected to occupy a particular role in the processes of life. Although this is relatively clear in biology, we can extend the idea to the evolution of all other material objects in the Universe, each having properties selected to provide it with a role in the overall structure of the world. We assume that the initial set of fundamental particles that constitute the standard model have evolved to fulfil this role. As with ancient manuscripts written on fragile media, the survival of a structure requires that it be reproducible. Aquinas, Summa I, 15, 1: Are there ideas in God?, Standard model - Wikipedia

Lonergan relies on insight or understanding, but he sees understanding as only half of knowledge. The other half is judgement that the understanding is true. He discusses various human ("proportionate") means to establish truth, but as a faithful Christian and a member of a religious order, he simply has to take the dogmata of the Church on faith, bolstered by the knowledge that the First Vatican Council has declared the Pope infallible when he expounds matter of faith and morals.

My conclusion from the study or Lonergan's work was that the notion of a God separate from the Universe is meaningless. I began to talk about this, and so came into conflict with the authoritarian delusions of the Catholic Church and was rightfully, by their standards, ejected from the Order.

This was a painful process, particularly for my mother, who could not understand why the Church had rejected her gift of a child. Like John Henry Newman, however, I did what I had to do and tried to explain to my mother, in the 50 years between my expulsion from the Dominican Order and her death, that what happened to me was for the best. Despite its manifest faults, my respect for my mother's faith made it impossible for me to share with her my detailed opinions of the Church.

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23.3: Revelation: theology, physics and evolution

Bernard Lonergan was a Roman Catholic theologian working to defend the Catholic view that God, the creator, exists outside the Universe and reveals themself to us through ancient prophetic literature. The primary foundation of the Catholic faith is the Bible. This document is surrounded by an enormous secondary literature, the Deposit of Faith. A recent addition to this literature is the Catechism of the Catholic Church. Deposit of faith - Wikipedia, John Paul II (1992): Apostolic Constitution Fidei Depositum on the publication of the Catechism of the Catholic Church prepared following the Second Vatican Ecumenical Council

This site is based on the hypothesis that the Universe itself is divine. From this point of view the analogue of the deposit of faith is all of our well established science in addition to all human experience since our species emerged, a fraction of which is recorded in archaeology, art and literature. This revelation is updated from moment to moment by our individual experiences. The internet now allows continuous instantaneous connection enabling billions of people to share their experience in near real time. In our divine world, all of this is divine revelation, information may serve as input to theology, very much greater than the traditional deposit of faith.

This flow of information is beyond comprehension, but we can narrow the field by assuming that in our divine Universe physicists and theologians are talking about the same thing, whether they appreciate this or not. In order to narrow the field still further I have been working on a plausible story of the creation of the world based on the physical science we have gathered since the time of Galileo. His conflict with the Church about the nature of the solar system became the foundation of our current cosmology which aspires to a theory of everything, beginning from an eternal structureless initial singularity. Now, with space based instrumentation, we are continually receiving awe inspiring images of the gigantic Universe that surrounds us. We are not alien creatures placed here by a remote creator. This is the system that created everything that surrounds, us from fundamental particles to our planet.

Theology and physics are among the oldest sciences, well attested in ancient Greek literature. At this period it seems to have been generally believed that the world was eternal and that its overall structure was unchanging. Although people in close contact with plants and animals may have been aware that they underwent small changes during a lifetime of domestication, historical memories were not long enough to record the emergence and extinction of new species, rise and fall of mountains or the drift of continents. Sea level rise and fall, however, appears to be recorded in the memory of Australian indigenous people. Nick Reid, Patrick Nunn & M Sharpe (2014): Indigenous Australian stories and sea-level change

Lonergan tells us on page 658 that the immanent source of transcendence in man is his detached, disinterested, unrestricted desire to know. The authors of Genesis knew this when they wrote So God created man in his own image ( I:27). I interpret this statement in terms of quantum theory. While working toward this position I noticed some problems in quantum field theory which still represents the mainstream of development in modern physics. Although I criticize Lonergan for flogging a dead horse, he opened my eyes to a the need for logical and psychological explanation of the physical world. Galileo was right to see mathematics as language of physics, but we must go beyond arithmetic and geometry to get as firm grip on reality. We need to take into account the rapid development in logic and mathematics which occurred around the beginning of the twentieth century.

This possibility first occurred to me about 20 years ago when I saw that if the Universe is divine, physics and theology are talking about the same subject. At that point it was about 40 years since I had last studied Lonergan's Insight but I remembered that he had tried to use a psychological approach, criticized above, to prove the existence of the God of Aquinas, and consequently (in his terms) the existence of transcendental knowledge. I occurred to me that any reasonable theory of everything, to have a hope of comprehending the Universe, must be logical and psychological rather than geometric, algebraical and arithmetic. This was about the time when Feynman's idea that quantum mechanics is not so much a theory of physics as a theory of communication and computation was emerging and some were dreaming that quantum computation would be more powerful than Turing computation. I bought my first edition copy of Nielsen and Chuang in 2002. Richard P. Feynman (1985): Quantum mechanical computers, Nielsen (2000): Quantum Computation and Quantum Information

This, we might say, is the fundamental act of universal intelligence. A primary insight in theory of linear operators lies at the foundation of quantum theory. It is faced by both nature and science, the study of instances of and solutions to the equation

α|P> = a|P>

where α is a linear operator and a is a number. The number a and the ket |P> are unknowns which we have to be chosen either by nature (in the process of creation) or by physicists (in the search for understanding). P A M Dirac (1983): The Principles of Quantum Mechanics (4th ed) pp 29 sqq.

My intuitive feeling about this equation arises is the following scenario:

On introspection I begin with a blank mind, no sharp imagery and then, in a Cartesian moment which is analogous to the construction and solution of the equation above, suddenly an operator, an eigenfunction and a spectrum of eigenvalues associated with the eigenfunction appear flowing out of the tip of my pen as a real time observation of an idea that occurred about three minutes ago. Manley, D. B., & Taylor, C. S. (1996): Descartes Meditations - Trilingual Edition

This set up and solution of the eigenvalue problem is the act of quantum mechanical insight, formulating the problem and the answer simultaneously. I have noted on page 7: Networks, brains and consciousness that our brains also work by superposition rather like quantum mechanics. In quantum mechanics, the superposition is strictly linear; in our brains the superposition evolves non-linearly in time by the adjustment of synaptic weights. Eigenvalue algorithm - Wikipedia, Synaptic weight - Wikipedia, John C. Eccles (1958): Innovation in Science: The Physiology of Imagination

How does this happen? I feel that it is an act of evolution by natural selection. The idea is explained in more detail on pages 9: The active creation of Hilbert space and 10: The emergence of quantum mechanics. Erwin Schrödinger (1926): Quantization is an eigenvalue problem

The history of quantum mechanics is often recounted. Planck discovered the quantum of action; Einstein realized that Planck's quantum of energy is a real photon; Bohr used the fact that in Minkowski space the quantum has dimensions of angular momentum so the superposition of waves can lead to stationary state to produce a first version of atomic structure; Heisenberg, Born, Jordan and Schrödinger discovered quantum mechanics as we know it and Dirac and von Neumann saw that the heart of the matter lies in the self-adjoint Hermitian operators with real diagonals acting in a Hilbert space to solve the eigenvalue problem.

John von Neumann spent a number of years studying quantum mechanics and put it on a sound mathematical footing. As van Hove notes, von Neumann's major contribution to quantum mechanics came in 1929 with the spectral theorem for hyper-maximal symmetric operators in Hilbert space.' Leon van Hove (1957): von Neumann's Contributions to Quantum Theory

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23.4 Perhaps quantum field theory is not possible?

This section is written with Sunny Auyang's book How is Quantum Field Theory Possible? in mind. This project began when the Roman Catholic Dominican Order expelled me in 1967 for entertaining a heretical view on the nature of God. As a young monk inspired by Pope John XXIII's movement for Aggiornomento I proposed that theology could become a real science and become much more credible if the Universe were divine. Sunny Auyang (1995): How is Quantum Field Theory Possible?

An immediate consequence of this hypothesis is that physics and theology have the same subject. On previous pages, starting with the Catholic doctrine of the Trinity, I have explained in a series of hypothetical steps how I imagine an omnipotent initial singularity, identical to both the Christian God and the initial singularity proposed by the big bang theory, could evolve into our current Universe. page 8: The theology of the Trinity. I feel that much Catholic doctrine is false but I wish to retain as many of the old ideas as possible while I build a new picture of creation. It was only when I began to follow the idea that theology and physics are talking about the same Universe that I began to have difficulty with quantum field theory. Now, having rejected much of Catholic theology, I find good reasons to reject much of quantum field theory too. Naturally I accept all the data that physicists have collected over the last few centuries. It is the theory that worries me. Quantum field theory builds on a number of mathematical ideals that cannot be realized physically, like infinity and particles of zero size. The last section of this page describes the Stanford Linear Accelerator and the collection of some of the data that established the Standard Model. I use this work and the problems of its interpretation to propose an alternative to field theory on page 24: The end of field theory?

Auyang begins with Faraday's electromagnetic field, which she describes as a new form of matter, filling the whole of spacetime and explaining the observed world.

Here we immediately come across a situation reminiscent of the clash of ideas apparent in theology that God is both absolutely simple and structureless, yet nevertheless omniscient. This cannot be consistent with the modern idea of information since an absolutely simple God, like the initial singularity, has no way to carry a symbolic representation of all available information (See page 21: Matter and spirit).

Auyang writes:

"Field" has at least two senses in the physics literature. A field is a continuous dynamical system, or a system with infinite degrees of freedom. A field is also a dynamic variable characterizing such a system or an aspect of the system. Fields are continuous, but not amorphous; a field comprises discrete and concrete point entities, each indivisible, but each having intrinsic characteristics. The description of a field and its properties is local, concentrating on a point entity and its infinitesimal displacement (page 47). . . .

Symbolically: A field φ(t, x) is a dynamical variable for a continuous system whose points are indexed by t and x, time and space (page 48).

The notion that fields or anything else can be both continuous and discrete seems to me to be a problem in quantum field theory. This problem often manifests in attempts to describe a logically quantized Universe with continuous mathematics.

In his book QED: The Strange Story of Light and Matter Richard Feynman provides and entertaining and semi-technical description of quantum electrodynamics. Much of it is built around Feynman diagrams, and his path intergal approach to quantum theory, two of his significant contributions to the theory. Richard Feynman (1988): QED: The Strange Story of Light and Matter, Path integral formulation - Wikipedia

In his final lecture he emphasizes first, that the (1988) framework of amplitudes then had a very long history and is very well established:

So this framework of amplitudes has no experimental doubt about it; you can have all the philosophical worries you want about it as to what the amplitudes mean (if, indeed, they mean anything at all), but because physics is an experimental science and the framework agrees with experiment, it is good enough for us so far.

He then raises a couple of problems related to zero sized particles and infinite degrees of freedom:

There is, however a problem that arises when we try to apply the framework of amplitudes to very small distances, approaching zero. Things go wrong.

One should be able to go down to zero distance in order to be mathematically consistent but that's where there is no n [mass of the electron] or j [coupling constant between the electron and the photon] that makes any sense; that's where the trouble is (page 127).

So it appears that the only things that depend on the small distance between coupling points are the values of n and j— theoretical numbers that are not directly observable anyway; everything else, which can be observed, seems not to be affected.

The other problem relates to the infinite degrees of freedom. Every degree of freedom comes with some energy, so infinite degrees of freedom would seem to imply infinite energy. The way out of this problem is renormalization.

The shell game that we play to find n and j is technically called "renormalization." But no matter how clever the word, it is what I would call a dippy process! Having to resort to hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self consistent. It's surprising that the theory had not been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate (page 128). Renormalization - Wikipedia, Quantum electrodynamics - Wikipedia

The problem with the consistency of quantum field theory remains. In the introduction to their book on axiomatic quantum field theory, Streater and Wightman write:

In fact the Main Problem of quantum field theory turned out to be to kill it or cure it: either to show that the idealizations involved in the fundamental notions of the theory (relativistic invariance, quantum mechanics, local fields, etc.) are incompatible in some physical sense, or to recast the theory in such a form that it provides a practical language for the description of elementary particle dynamics. Streater & Wightman (2000): PCT, Spin, Statistics and All That

The Clay Mathematics Institute and the American Mathematical Society are trying buy a cure by offering a million dollar prize for anyone who can improve the situation. They note that one does not have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so!. It may be that like Lonergan above, they are flogging a dead horse. Time will tell. Carlson, Jaffe & Wiles (2006): The Millennium Prize Problems

Since it seems very unlikely that reality is composed of continuous fields made of discrete particles, particles of zero size and states of infinite energy, the fact that the theory must be cured of these evils by devious practices suggests that although the theory yields some good numerical results is not a particularly good fit to reality.

Most of the trouble in quantum field theory seems to have arisen from the attempt to unite quantum theory with the classical special theory of relativity. By the end of the 1920s non-relativistic quantum mechanics was more or less complete, but the biggest challenge lay ahead: to make it consistent with special relativity.

Although the resulting theory worked well at low energies great difficulties arose in trying to construct a relativistic theory. Quantum field theory remains our best hope to date. At the beginning of his treatise on QFT Steven Weinberg wrote:

The point of view of this book is that quantum field theory is the way it is because . . . it is the only way to reconcile the principles of quantum mechanics with those of special relativity (vol 1, page xxi. Steven Weinberg (1995): The Quantum Theory of Fields Volume I: Foundations

I wanted to avoid the problems of quantum field theory. This led me to a tale of two spaces, suggested by my theological background. Traditionally we are a compound of mind and body, or maybe body and soul. The Catholic Church has told us that as a consequence of original sin the body no longer obeys the mind, or as Paul of Tarsus puts it, the flesh wars against the spirit. In more modern terminology we might say that the mind adds a new degree of freedom to the mind/body composite. We consider ourselves very much more versatile than less mental forms of life.

A few year ago I began to see that we can apply this paradigm to the whole Universe, proposing quantum theory operating in Hilbert space as the mind of the Universe and Einstein's dynamism expressed in Minkowski space as the body of the Universe. Two spaces, two degrees of freedom. I then began to think in Trinitarian and evolutionary terms and proposed to myself that Hilbert space and quantum mechanics are in effect the realm of variation, the realm of possibility, the mind of the Universe; and Minkowski space is the realm of selection, picking the ideas that work by being able to reproduce themselves. This paradigm appeared to be universal. Mobile phones, for instance, are a product of imaginations like Steve Jobs, and they have now been able, through their irresistible attraction to humans and the development of the technology necessary to reproduce them, to capture about half the population of the world in a little over a decade. Paul the Apostle: Galatians, chapter 5, USCCB

The theological discussion of the multiplication of Christian divine personalities provided me with a starting point for a comprehensive picture of an omnipotent divine world able to create itself. This devlopment of this idea begins on page 8: The theology of the Trinity and continues on page 9: The active creation of Hilbert space with the creation of Hilbert space within the initial singularity. This singularity is a structureless entity which we can represent mathematically by the empty set, ∅. As the dimension the Hilbert space grows to 1, 2, 3 . . . , we assume that this space provides a fertile milieu for the evolution of the Universe.

Next comes quantum mechanics, which by a process of variation and selection in Hilbert space begins to create operators which begin to select eigenvectors of stationary phase, the precursors of of fundamental particles, bosons and fermions. The properties of these particles once they become energized and dynamic will develop space by the exclusion principle. This space will not be continuous in the classical sense but pixellated by the quantum of action.

An important feature of this structure at this point is that it is purely a quantum mechanical creation in Hilbert space. Hilbert space itself is independent of Minkowski space, existing prior to the emergence of space and time as described on page 13: Is Hilbert space independent of Minkowski space?. Because Hilbert space exists prior to and outside Minkowski space, we do not need relativistic quantum mechanics. After some further development gravitation is introduced as the divinity at the root of the Universe. on Page 17: Gravitation and quantum theory—in the beginning

I see a number advantages in this scenario:

1: In the first incarnation of quantum theory it was thought that Minkowski space was the domain of Hilbert space. I seems to me that this destroyed the independence of Hilbert space and quantum theory which had brought the Universe a new degree of freedom. By locking Hilbert space to Minkowski space, we have in effect denied the Universe its mind and this has had the effect of making quantum field theory as we currently have it self contradictory and impossible.

2: I have quoted Kuhlmann's critique of Quantum Field theory a number of times on this site, and I hope that the the distinction between Hilbert and Minkowski spaces will solve some of the problems of representation he sees:

In conclusion one has to recall that one reason why the ontological interpretation of [Quantum Field Theory] is so difficult is the fact that it is exceptionally unclear which parts of the formalism should be taken to represent anything physical in the first place. And it looks as if that problem will persist for quite some time. Meinard Kuhlmann (Stanford Encyclopedia of Philosophy): Quantum Field Theory

One of my ways of expressing the difference between Hilbert space and Minkowski space is to think of Hilbert space (and mathematics in general) as kinematic, unable to move itself but, like the manually generated imagery in animated cartoons, able to do almost anything, not constrained by laws like conservation of energy or the impossibility of linear superposition. A clear distinction between the representation of kinematic and dynamic objects may clear up some of the problems in field theory.

3: As described in §3 above, it is becoming clear that quantum mechanics is more than a simple physical theory. It is a theory of communication and computation which makes it well suited to be considered the mind of the Universe.

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23.5: On coupling kinematic forms to dynamic energy

We have excluded the dynamic constraints on Minkowski space expressed by the special theory of relativity from the domain of quantum mechanics since it exists formally outside space and time. We now turn to the relationship of the two spaces, Hilbert and Minkowski, and introduce the special theory of relativity and its companion general relativity which are both classical theories. This discussion extends page 14: "Measurement": the interface between Hilbert and Minkowski spaces

The term measurement is in inverted commas because it is rather ancient term dating from the days when quantum theory was a laboratory curiosity rather than playing a central role in modern technology and everything that happens anywhere in the universe. Here we understand every meeting between between observable particles where information is exchanged as a quantum measurement or observation. An infinitesimal fraction of these events occur under laboratory conditions.

We noted on page 15: Quantum amplitudes and logical processes are invisible. We do not see Hilbert space, we only see real particles in Minkowski space. In high energy physics we shoot particles at one another and watch the particles that come out. In the early days of quantum mechanics we stimulated materials with heat and recorded the spectra of the photons coming out. This spectral information was all we had to go on to try to work out what was going on.

People made millions of observations and collected large volumes of data, but this, by itself, was not enough to understand what was happening. As Lonergan points out, collecting data is not enough. Insights, like Archimedes', is what we need to understand what is happening. Gustav Kirchoff found the path to quantum mechanics in the 1860s and Planck took the first step in 1900. Thomas Kuhn (1987): Black-Body Theory and the Quantum Discontinuity 1894-1912

The development of non-relativistic quantum mechanics was effectively complete by the end of the 1920s and the mathematical method for computing the outcome of quantum events refined to finding real solutions of the eigenvalue problem described in §23.3 above. The key elements in the quantum theory are the linear operators, also understood as dynamical variables α in the equation quoted above.

If we take the view proposed by Nielsen and Huang that quantum theory is a theory of computation and communication we may understand the linear operators to be quantum software acting as codecs in the quantum communications between the particles involved in an interaction. If we think in terms of classical communication described by Shannon, the linear operators of quantum theory are performing a task analogous to the work of the classical computers we use to apply Shannon's methods to encode and decode messages in a manner which makes them resistant to error. The association of quantum observables with real eigenvalues interfaces the invisible quantum process described by quantum mechanics with the real world observable in Minkowski space. Nielsen & Chuang (2016): Quantum Computation and Quantum Information

The idea that quantum mechanics is a theory of communication and computation leads us to think of the fundamental interactions as codec's represented by hermitian operators which have been discovered through an evolutionary process (see page 13: Is Hilbert space independent of Minkowski space?). Some proponents of unified field theory feel that at sufficiently high energy all the three forces described by quantum field theory will become one. The energy scale is predicted to be enormously greater than the energy that will ever be available to us in our laboratories, but may have been the case in the first moments of the big bang when temperatures correspond to the [Planck time] (10³⁴ K): Page 18.2: Fixed Points—Motion and stillness: Parmenides vs Heraclitus . This unification seems unlikely if the interactions are in fact quantum codecs. We know that the operation of classical codecs is constrained by software snd so the outcomes are independent of the speed of the machine on which it is executed. The same may be true with quantum codecs imagined as linear operators (fixed algorithms) whose rate of execution is a function of the energy of interaction so that the effect of the codec will be independent of the temperature of interaction which is a proxy for the energy of interaction.

Laboratory quantum observations are made over a wide range of energies ranging from those made on particles confined in low temperature electromagnetic traps to the highest energies we can manage with accelerators and the higher energies available in cosmic rays and observable cosmic events. E. K. Anderson et al (2023_09_27): Observation of the effect of gravity on the motion of antimatter

Experiments conducted with the Large Hadron Collider in the vicinity of 10 TeV produce large showers of particles with varying probabilities which may be understood to represent the spectrum of the dynamic variable operating in the collision. The purpose of using high energies is to observe the occurrence of particles with high mass / low probability which test the boundaries of theory. Whether the high relativistic masses of the colliding particles have any effect on the precise nature of the quantum linear operators involved may not be discernible if the only effect of high energies to reveal more improbable elements of the spectra of these operators.

this page 23: toc

23.6: SLAC and deep inelastic scattering

Electrons and photons go hand in hand and it seems clear that photons are the subtlest and most interesting particles in the world. In the 1940s, as high energy electrons began to become available from accelerators, and it was clear that in order to resolve the structure of the proton high energies would be necessary. A byproduct of high energy electrons is high energy photon, and a feature of high energy photons is short wavelength and high resolving power, as we have learnt from the electron microscope. Another advantage of electrons and photons is that in the 1940s quantum electrodynamics became the first and most successful quantum field theory. We know a lot about them. They explain almost everything we see, including the details of how our bodies work. Precision tests of QED - Wikipedia, Proton - Wikipedia

At 1 MeV it became apparent that the proton is not an elementary particle but had internal structure. This motivated the construction of the 50 GeV Stanford linear accelerator with the intention of resolving the internal structure of the proton. SLAC National Accelerator Laboratory - Wikipedia

The construction of the accelerator and the elaborate system of spectroscopes necessary to measure the energy and deflection of electrons from the hydrogen target was completed and began doing physics in 1967. The first inelastic scattering experiments began in August 1967. Richard E. Taylor (1990): Nobel Lecture: I. Deep Inelastic Scattering: The Early Years

The beam energy of 21 GeV then available from the new accelerator made it possible to study the proton at much higher resolution. The surprising results suggested the presence, contrary to expectation, of pointlike constituents within the proton. Henry W. Kendall (1990): Nobel lecture: II: Deep Inelastic Scattering: Experiments on the Proton and the Observation of Scaling

High energy experiments had produced a surprisingly large number of different hadrons (the "particle zoo" era). Murray Gell-Mann and George Zweig attempted to classify these using the SU(3) group and the "eightfold way" which suggested that the different hadrons were different combinations of theoretical constituents which came to be called quarks. The quark theory began simply as means of classification: the inelastic scattering experiments gradually endowed the quarks with reality. Particle zoo - Wikipedia, Eightfold way (physics) - Wikipedia

Initially the reality of quarks was disputed on a number of grounds. First, because high energy collisions failed to yield free quarks; second, because they could not be freed seemed to imply that they were very massive, which did not fit the data; third, since there were three of them in a hadron, they violated the exclusion principle for spin ½ fermions; and finally, if there were three of them, they needed to have fractional electric charges, which seemed very unlikely. Pauli exclusion principle - Wikipedia

The first project on the SLAC accelerator studied the elastic scattering of electrons from protons. This produced the expected result, that the proton is a relatively homogeneous body whose diameter is about 1.6 x 10^-12 millimetres.

The inelastic scattering experiment produced two surprises. First, there appeared to be freely moving point particles within the proton (quarks?). The rate of inelastic scattering (which deposited energy within the proton) was one to two orders of magnitude greater than elastic scattering where the electron simply bounced off the proton. The second surprise was a feature known as scaling. The forces between the "quarks" seemed to decrease as they became closer together, the opposite of what we normally expect for charged particles. They were asymptotically free. Jerome I. Friedman (1990): Nobel Lecture III: Deep Inelastic Scattering: Comparisons with the quark model, Asymptotic freedom - Wikipedia

These unexpected results led to the production of a number of models intended to explain the data without accepting the reality of quarks. None of these succeeded. A model developed by Feynman while studying proton proton collisions which introduced the notion of partons within the proton motivated renewed interest in the interaction between the constituents of protons. His model provided a key insight into the results of the deep inelastic electron scattering experiments.

Feynman's idea relied on relativistic time dilation to provide time for the electron to interact with a partons which are momentarily free from interaction with one another.

By the 1970s the notion that quarks were constituents of hadrons and mesons was gaining ground. A new line of evidence was presented in 1972 drawn from neurtrino scattering experiments conducted at CERN with the 24 GEV synchrotron and the large heavy liquid bubble chamber Gargamelle. As the 1970's progressed, belief in the quark model slowly solidified. Eventually the data became completely convincing. It could not be explained away, it had to be explained. Gargamelle - Wikipedia, CDHS Experiment - Wikipedia

The final chapter was written by Gross and Wilzcek in the early 70s. Their work provides an explanation of the confusing experimental data and brought us to the closure of quantum field theory, quantum chromodynamics. Quantum chromodynamics - Wikipedia, David J. Gross (2004): Nobel lecture: The Discovery of Asymptotic Freedom and the Emergence of QCD, Frank Wilczek (2004): Nobel lecture: Asymptotic Freedom: from Paradox to Paradigm

Following this achievement, Wilczek wrote a book which led for me to the final demolition of field theory. I take this story up on the next page and then complete my story with my own version of Aristotle's magnificent journey from physics to theology which provided Christianity with its first contact with science. Frank Wilczek (2008): The Lightness of Being: Mass, Ether, and the Unification of Forces

(revised Thursday 21 March 2024)

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Notes and references

Auyang (1995), Sunny Y., How is Quantum Field Theory Possible?, Oxford University Press 1995 Jacket: 'Quantum field theory (QFT) combines quantum mechanics with Einstein's special theory of relativity and underlies elementary particle physics. This book presents a philosophical analysis of QFT. It is the first treatise in which the philosophies of space-time, quantum phenomena and particle interactions are encompassed in a unified framework.'
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Carlson (2006), James, and Arthur Jaffe & Andrew Wiles, The Millennium Prize Problems, Clay Mathematics Institute and American Mathematical Society 2006
1: The Birch and Swinnerton-Dyer Conjecture: Andrew Wiles
2: The Hodge Conjecture: Pierre Deligne
3: The Existence and Smoothness of the Navier-Stokes Equation: Charles L Fefferman
4: The Poincare Conjecture: John Milnor
5: The P versus NP Problem: Stephen Cook
6: The Riemann Hypothesis: Enrico Bombieri
7: Quantum Yang-Mills Theory: Arthur Jaffe and Edward Whitten
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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 (1988), Richard, QED: The Strange Story of Light and Matter, Princeton UP 1988 Jacket: 'Quantum electrodynamics - or QED for short - is the 'strange theory' that explains how light and electrons interact. Thanks to Richard Feynmann and his colleagues, it is also one of the rare parts of physics that is known for sure, a theory that has stood the test of time. . . . In this beautifully lucid set of lectures he provides a definitive introduction to QED.'
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Feynman (2007), Richard, Feynman Lectures on Computation, Perseus Publishing 2007 Amazon Editorial Reviews Book Description 'The famous physicist's timeless lectures on the promise and limitations of computers When, in 1984-86, Richard P. Feynman gave his famous course on computation at the California Institute of Technology, he asked Tony Hey to adapt his lecture notes into a book. Although led by Feynman, the course also featured, as occasional guest speakers, some of the most brilliant men in science at that time, including Marvin Minsky, Charles Bennett, and John Hopfield. Although the lectures are now thirteen years old, most of the material is timeless and presents a "Feynmanesque" overview of many standard and some not-so-standard topics in computer science such as reversible logic gates and quantum computers.'
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Kuhn (1987), Thomas S, Black-Body Theory and the Quantum Discontinuity 1894-1912, University of Chicago Press 1987 Jacket: '[This book] traces the emergence of discontinuous physics during the early years of this century. Breaking with historiographic tradition, Kuhn maintains that, though clearly due to Max Planck, the concept of discontinuous energy change does not originate in his work. Instead it was introduced by physicists trying to understand the success of his brilliant new theory of black-body radiation.'
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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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Lonergan (1997), Bernard J F, and Robert M. Doran, Frederick E. Crowe (eds), Verbum: Word and Idea in Aquinas (Collected Works of Bernard Lonergan volume 2), University of Toronto Press 1997 Jacket: 'Verbum is a product of Lonergan's eleven years of study of the thought of Thomas Aquinas. The work is considered by many to be a breakthrough in the history of Lonergan's theology . . .. Here he interprets aspects in the writing of Aquinas relevant to trinitarian theory and, as in most of Lonergan's work, one of the principal aims is to assist the reader in the search to understand the workings of the human mind.'
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Misner (1973), Charles W, and Kip S Thorne, John Archibald Wheeler, Gravitation, Freeman 1973 Jacket: 'Einstein's description of gravitation as curvature of spacetime led directly to that greatest of all predictions of his theory, that the universe itself is dynamic. Physics still has far to go to come to terms with this amazing fact and what it means for man and his relation to the universe. John Archibald Wheeler. . . . this is a book on Einstein's theory of gravity. . . . '
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Nielsen (2000), Michael A, and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2000 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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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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Streater (2000), Raymond F, and Arthur S Wightman, PCT, Spin, Statistics and All That, Princeton University Press 2000 Amazon product description: 'PCT, Spin and Statistics, and All That is the classic summary of and introduction to the achievements of Axiomatic Quantum Field Theory. This theory gives precise mathematical responses to questions like: What is a quantized field? What are the physically indispensable attributes of a quantized field? Furthermore, Axiomatic Field Theory shows that a number of physically important predictions of quantum field theory are mathematical consequences of the axioms. Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.'
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Weinberg (1995), Steven, The Quantum Theory of Fields Volume I: Foundations, Cambridge University Press 1995 Jacket: 'After a brief historical outline, the book begins anew with the principles about which we are most certain, relativity and quantum mechanics, and then the properties of particles that follow from these principles. Quantum field theory then emerges from this as a natural consequence. The classic calculations of quantum electrodynamics are presented in a thoroughly modern way, showing the use of path integrals and dimensional regularization. The account of renormalization theory reflects the changes in our view of quantum field theory since the advent of effective field theories. The book's scope extends beyond quantum elelctrodynamics to elementary partricle physics and nuclear physics. It contains much original material, and is peppered with examples and insights drawn from the author's experience as a leader of elementary particle research. Problems are included at the end of each chapter. '
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Wilczek (2008), Frank, The Lightness of Being: Mass, Ether, and the Unification of Forces, Basic Books 2008 ' In this excursion to the outer limits of particle physics, Wilczek explores what quarks and gluons, which compose protons and neutrons, reveal about the manifestation of mass and gravity. A corecipient of the 2004 Nobel Prize in Physics, Wilczek knows what he’s writing about; the question is, will general science readers? Happily, they know what the strong interaction is (the forces that bind the nucleus), and in Wilczek, they have a jovial guide who adheres to trade publishing’s belief that a successful physics title will not include too many equations. Despite this injunction (against which he lightly protests), Wilczek delivers an approachable verbal picture of what quarks and gluons are doing inside a proton that gives rise to mass and, hence, gravity. Casting the light-speed lives of quarks against “the Grid,” Wilczek’s term for the vacuum that theoretically seethes with quantum activity, Wilczek exudes a contagious excitement for discovery. A near-obligatory acquisition for circulating physics collections.' --Gilbert Taylor
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Links

Abductive reasoning - Wikipedia, Abductive reasoning - Wikipedia, the free encyclopedia, ' Abductive reasoning (also called abduction, abductive inference, or retroduction) is a form of logical inference that seeks the simplest and most likely conclusion from a set of observations. It was formulated and advanced by American philosopher Charles Sanders Peirce beginning in the latter half of the 19th century. Abductive reasoning, unlike deductive reasoning, yields a plausible conclusion but does not definitively verify it. Abductive conclusions do not eliminate uncertainty or doubt, which is expressed in retreat terms such as "best available" or "most likely". While inductive reasoning draws general conclusions that apply to many situations, abductive conclusions are confined to the particular observations in question.' back

Albigensian Crusade - Wikipedia, Albigensian Crusade - Wikipedia, the free encyclopedia, 'When Innocent III's diplomatic attempts to roll back Catharism met with little success and after the papal legate Pierre de Castelnau was murdered (allegedly by an agent serving the Cathar count of Toulouse), Innocent III declared a crusade against Languedoc, offering the lands of the schismatics to any French nobleman willing to take up arms. The violence led to France's acquisition of lands with closer cultural and linguistic ties to Catalonia (see Occitan). An estimated 200,000 to 1,000,000 people were massacred during the crusade.' back

Apologia Pro Vita Sua - Wikipedia, Apologia Pro Vita Sua - Wikipedia, the free encyclopedia, ' Apologia Pro Vita Sua (Latin: A defence of one's own life) is John Henry Newman's defence of his religious opinions, published in 1864 in response to Charles Kingsley of the Church of England after Newman quit his position as the Anglican vicar of St. Mary's, Oxford.' back

Aquinas Summa I, 2, 3, Does God exist?, 'I answer that, The existence of God can be proved in five ways.
The first and more manifest way is the argument from motion. It is certain, and evident to our senses, that in the world some things are in motion. Now whatever is in motion is put in motion by another . . . For motion is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. . . . Therefore, whatever is in motion must be put in motion by another. If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; . . .Therefore it is necessary to arrive at a first mover, put in motion by no other; and this everyone understands to be God.' back

Aquinas, Summa I, 1, 1, Is sacred doctrine necessary?, 'I answer that, It was necessary for man's salvation that there should be a knowledge revealed by God besides philosophical science built up by human reason. Firstly, indeed, because man is directed to God, as to an end that surpasses the grasp of his reason: "The eye hath not seen, O God, besides Thee, what things Thou hast prepared for them that wait for Thee" (Isaiah 64:4). But the end must first be known by men who are to direct their thoughts and actions to the end. Hence it was necessary for the salvation of man that certain truths which exceed human reason should be made known to him by divine revelation.' back

Aquinas, Summa I, 25, 3, Is God omnipotent?, '. . . God is called omnipotent because He can do all things that are possible absolutely; which is the second way of saying a thing is possible. For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey.' back

Aquinas, Summa, I, 1, 2, Is sacred doctrine is a science?, 'I answer that, Sacred doctrine is a science. We must bear in mind that there are two kinds of sciences. There are some which proceed from a principle known by the natural light of intelligence, such as arithmetic and geometry and the like. There are some which proceed from principles known by the light of a higher science: thus the science of perspective proceeds from principles established by geometry, and music from principles established by arithmetic. So it is that sacred doctrine is a science because it proceeds from principles established by the light of a higher science, namely, the science of God and the blessed.' back

Aquinas, Summa, I, 3, 7, Is God altogether simple?, 'I answer that, The absolute simplicity of God may be shown in many ways. First, from the previous articles of this question. For there is neither composition of quantitative parts in God, since He is not a body; nor composition of matter and form; nor does His nature differ from His "suppositum"; nor His essence from His existence; neither is there in Him composition of genus and difference, nor of subject and accident. Therefore, it is clear that God is nowise composite, but is altogether simple. . . . ' back

Aquinas, Summa: I, 14, 1, Is there knowledge in God?, ' I answer that, In God there exists the most perfect knowledge. . . . it is clear that the immateriality of a thing is the reason why it is cognitive; and according to the mode of immateriality is the mode of knowledge. Hence it is said in De Anima ii that plants do not know, because they are wholly material. But sense is cognitive because it can receive images free from matter, and the intellect is still further cognitive, because it is more separated from matter and unmixed, as said in De Anima iii. Since therefore God is in the highest degree of immateriality as stated above (Question 7, Article 1), it follows that He occupies the highest place in knowledge.' back

Aquinas, Summa: I, 2, 3, Does God exist?, 'I answer that, The existence of God can be proved in five ways. The first and more manifest way is the argument from motion. . . . ' back

Aristotle Metaphysics, God's happiness, ' If, then, the happiness which God always enjoys is as great as that which we enjoy sometimes, it is marvellous ; and if it is greater, this is still more marvellous. Nevertheless it is so. Moreover, life belongs to God. For the actuality of thought is life, and God is that actuality ; and the essential actuality of God is life most good and eternal. We hold, then, that God is a living being, eternal, most good ; and therefore life and a continuous eternal existence belong to God; for that is what God is.' (1072b25-31) back

Asymptotic freedom - Wikipedia, Asymptotic freedom - Wikipedia, the free encyclopedia, 'In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i.e. length scales that asymptotically converge to zero (or, equivalently, energy scales that become arbitrarily large). Asymptotic freedom implies that in high-energy scattering the quarks move within nucleons, such as the neutron and proton, mostly as free non-interacting particles. It allows physicists to calculate the cross sections of various events in particle physics reliably using parton techniques.' back

Binary number - Wikipedia, Binary number - Wikipedia, the free encyclopedia, ' A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one). The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.' 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

CDHS Experiment - Wikipedia, CDHS Experiment - Wikipedia, the free encyclopedia, ' CDHS was a neutrino experiment at CERN taking data from 1976 until 1984. The experiment was officially referred to as WA1. CDHS was a collaboration of groups from CERN, Dortmund, Heidelberg, Saclay and later Warsaw. The collaboration was led by Jack Steinberger. The experiment was designed to study deep inelastic neutrino interactions in iron. back

Christopher Shields (Stanford Encyclopedia of Philosophy b), The Active Mind of De Anima III 5 , ' After characterizingnous the mind (nous) and its activities in De Animaiii 4, Aristotle takes a surprising turn. In De Anima iii 5, he introduces an obscure and hotly disputed subject: the active mind or active intellect (nous poiêtikos). Controversy surrounds almost every aspect of De Anima iii 5, not least because in it Aristotle characterizes the active mind—a topic mentioned nowhere else in his entire corpus—as ‘separate and unaffected and unmixed, being in its essence actuality’ (chôristos kai apathês kai amigês, tê ousia energeia; DA iii 5, 430a17–18) and then also as ‘deathless and everlasting’ (athanaton kai aidion; DA iii 5, 430a23). This comes as no small surprise to readers of De Anima, because Aristotle had earlier in the same work treated the mind (nous) as but one faculty (dunamis) of the soul (psuchê), and he had contended that the soul as a whole is not separable from the body (DA ii 1, 413a3–5). 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 is the imaginary unit, which satisfies the equation i² = −1. In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane (also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part.' back

Computer - Wikipedia, Computer - Wikipedia, the free encyclopedia, ' A computer is a machine that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming. Modern computers have the ability to follow generalized sets of operations, called programs. These programs enable computers to perform an extremely wide range of tasks. A "complete" computer including the hardware, the operating system (main software), and peripheral equipment required and used for "full" operation can be referred to as a computer system.' back

David J. Gross (2004), Nobel lecture: The Discovery of Asymptotic Freedom and the Emergence of QCD, ' The emergence of QCD is a wonderful example of the evolution from farce to triumph. During a very short period, a transition occurred from experimental discovery and theoretical confusion to theoretical triumph and experimental confirmation. In this Nobel lecture I shall describe the turn of events that led to the discovery of asymptotic freedom, which in turn led to the formulation of QCD, the final element of the remarkably comprehensive theory of elementary particle physics – the Standard Model. I shall then briefly describe the experimental tests of the theory and the implications of asymptotic freedom.' back

Deposit of faith - Wikipedia, Deposit of faith - Wikipedia, the free encyclopedia, ' The deposit of faith . . . is the body of revealed truth in the scriptures and sacred tradition proposed by the Roman Catholic Church for the belief of the faithful. . . . Catholic usage: The "sacred deposit" of the faith (depositum fidei) refers to the teachings of the Catholic Church that are believed to be handed down since the time of the Apostles – namely scripture and sacred tradition. . . . According to Dei Verbum, "Sacred tradition and Sacred Scripture form one sacred deposit of the word of God, committed to the Church [...] both of them, flowing from the same divine wellspring, in a certain way merge into a unity and tend toward the same end.".' back

E. K. Anderson et al (2023_09_27), Observation of the effect of gravity on the motion of antimatter, ' Abstract: Einstein’s general theory of relativity from 1915 remains the most successful description of gravitation. From the 1919 solar eclipse to the observation of gravitational waves, the theory has passed many crucial experimental tests. However, the evolving concepts of dark matter and dark energy illustrate that there is much to be learned about the gravitating content of the universe. Singularities in the general theory of relativity and the lack of a quantum theory of gravity suggest that our picture is incomplete. It is thus prudent to explore gravity in exotic physical systems. Antimatter was unknown to Einstein in 1915. Dirac’s theory appeared in 1928; the positron was observed in 1932. There has since been much speculation about gravity and antimatter. The theoretical consensus is that any laboratory mass must be attracted by the Earth, although some authors have considered the cosmological consequences if antimatter should be repelled by matter. In the general theory of relativity, the weak equivalence principle (WEP) requires that all masses react identically to gravity, independent of their internal structure. Here we show that antihydrogen atoms, released from magnetic confinement in the ALPHA-g apparatus, behave in a way consistent with gravitational attraction to the Earth. Repulsive ‘antigravity’ is ruled out in this case. This experiment paves the way for precision studies of the magnitude of the gravitational acceleration between anti-atoms and the Earth to test the WEP. back

Eigenvalue algorithm - Wikipedia, Eigenvalue algorithm - Wikipedia, the free encyclopedia, ' In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.' back

Eightfold way (physics) - Wikipedia, Eightfold way (physics) - Wikipedia, the free encyclopedia, ' n physics, the eightfold way is an organizational scheme for a class of subatomic particles known as hadrons that led to the development of the quark model. Working alone, both the American physicist Murray Gell-Mann and the Israeli physicist Yuval Ne'eman proposed the idea in 1961. The name comes from Gell-Mann's (1961) paper and is an allusion to the Noble Eightfold Path of Buddhism. By 1947, physicists believed that they had a good understanding of what the smallest bits of matter were. There were electrons, protons, neutrons, and photons (the components that make up the vast part of everyday experience such as atoms and light) along with a handful of unstable (i.e., they undergo radioactive decay) exotic particles needed to explain cosmic rays observations such as pions, muons and hypothesized neutrino. . . . But the discovery of the (neutral) kaon in late 1947 and the subsequent discovery of a positively charged kaon in 1949 extended the meson family in an unexpected way and in 1950 the lambda particle did the same thing for the baryon family. . . . This was the beginning of a chaotic period in particle physics that has become known as the "particle zoo" era. The eightfold way represented a step out of this confusion and towards the quark model, which proved to be the solution.' back

Empiricism - Wikipedia, Empiricism - Wikipedia, the free encyclopedia, ' In philosophy, empiricism is an epistemological view which holds that true knowledge or justification comes only or primarily from sensory experience. It is one of several competing views within epistemology, along with rationalism and skepticism. Empiricism emphasizes the central role of empirical evidence in the formation of ideas, rather than innate ideas or traditions.Empiricists may argue that traditions (or customs) arise due to relations of previous sensory experiences.' back

English Reformation - Wikipedia, English Reformation - Wikipediam the free encyclopedia, ' The English Reformation took place in 16th-century England when the Church of England was forced by its monarchs and elites to break away from the authority of the pope and the Catholic Church. These events were part of the wider European Reformation, a religious and political movement that affected the practice of Christianity in Western and Central Europe. . . .. The English Reformation began as more of a political affair than a theological dispute.[note 1] In 1527, Henry VIII requested an annulment of his marriage, but Pope Clement VII refused. In response, the Reformation Parliament (1529–1536) passed laws abolishing papal authority in England and declared Henry to be head of the Church of England. Final authority in doctrinal disputes now rested with the monarch. Though a religious traditionalist himself, Henry relied on Protestants to support and implement his religious agenda.' back

Erwin Schrödinger (1926), Quantization is an eigenvalue problem, '§1. In this communication I would like first to show, in the simplest case of the (non-relativistic and unperturbed) hydrogen atom, that the usual prescription for quantisation can be substituted by another requirement in which no word about “integer numbers” occurs anymore. Rather, the integerness emerges in the same natural way as, for example, the integerness of the number of knots of a vibrating string. The new interpretation is generalisable and touches, as I believe, very deeply the true essence of the quantisation prescription.' back

European wars of religion - Wikipedia, European war of religion - Wikipedia, the free encyclopedia, ' The conflicts began with the minor Knights' Revolt (1522), followed by the larger German Peasants' War (1524–1525) in the Holy Roman Empire. Warfare intensified after the Catholic Church began the Counter-Reformation in 1545 against the growth of Protestantism. The conflicts culminated in the Thirty Years' War, which devastated Germany and killed one-third of its population, a mortality rate twice that of World War I. The Peace of Westphalia broadly resolved the conflicts by recognising three separate Christian traditions in the Holy Roman Empire: Roman Catholicism, Lutheranism, and Calvinism.' back

Feynman, Leighton and Sands FLP II_02, Chapter 2: Differential Calculus of Vector Fields, ' Ideas such as the field lines, capacitance, resistance, and inductance are, for such purposes, very useful. So we will spend much of our time analyzing them. In this way we will get a feel as to what should happen in different electromagnetic situations. On the other hand, none of the heuristic models, such as field lines, is really adequate and accurate for all situations. There is only one precise way of presenting the laws, and that is by means of differential equations. They have the advantage of being fundamental and, so far as we know, precise. If you have learned the differential equations you can always go back to them. There is nothing to unlearn.' back

Five Ways (Aquinas) - Wikipedia, Five Ways (Aquinas) - Wikipedia, the free encyclopedia, ' The Quinque viæ (Latin for "Five Ways") (sometimes called "five proofs") are five logical arguments for the existence of God summarized by the 13th-century Catholic philosopher and theologian St. Thomas Aquinas in his book Summa Theologica. . . . Because most of his ways can be traced to arguments presented by Jewish philosophers (especially Maimonides, whose work Aquinas was known to be intimately familiar with), Aquinas is recognized as having popularized aspects of Jewish theology within Christianity.' back

Fundación Tomás de Aquino, Corpus Thomisticum: Summa Theologiae, Prooemium: Quia Catholicae veritatis doctor non solum provectos debet instruere, sed ad eum pertinet etiam incipientes erudire, secundum illud apostoli I ad Corinth. III, tanquam parvulis in Christo, lac vobis potum dedi, non escam; propositum nostrae intentionis in hoc opere est, ea quae ad Christianam religionem pertinent, eo modo tradere, secundum quod congruit ad eruditionem incipientium. Consideravimus namque huius doctrinae novitios, in his quae a diversis conscripta sunt, plurimum impediri, partim quidem propter multiplicationem inutilium quaestionum, articulorum et argumentorum; partim etiam quia ea quae sunt necessaria talibus ad sciendum, non traduntur secundum ordinem disciplinae, sed secundum quod requirebat librorum expositio, vel secundum quod se praebebat occasio disputandi; partim quidem quia eorundem frequens repetitio et fastidium et confusionem generabat in animis auditorum. Haec igitur et alia huiusmodi evitare studentes, tentabimus, cum confidentia divini auxilii, ea quae ad sacram doctrinam pertinent, breviter ac dilucide prosequi, secundum quod materia patietur. back

Gargamelle - Wikipedia, Gargamelle - Wikipedia, the free encyclopedia, ' Gargamelle was a heavy liquid bubble chamber detector in operation at CERN between 1970 and 1979. It was designed to detect neutrinos and antineutrinos, which were produced with a beam from the Proton Synchrotron (PS) between 1970 and 1976, before the detector was moved to the Super Proton Synchrotron (SPS). . . .. Gargamelle is famous for being the experiment where neutral currents were discovered. Found in July 1973, neutral currents were the first experimental indication of the existence of the Z0 boson, and consequently a major step towards the verification of the electroweak theory. . . .. Initially the first priority of the Gargamelle had been to measure the neutrino and antineutrino cross-sections and structure functions. The reason for this was to test the quark model of the nucleon. Firstly the neutrino and antineutrino cross-sections were shown to be linear with energy, which is what one expects for the scattering of point-like constituents in the nucleon. Combining the neutrino and antineutrino structure functions allowed the net number of quarks in the nucleon to be determined, and this was in good agreement with 3. In addition comparing the neutrino results with results from Stanford Linear Accelerator Center (SLAC) in the US, using an electron beam, one found that quarks had fractional charges, and experimentally proved the values of these charges: +2⁄3 e, −1⁄3 e. The results were published in 1975, providing crucial evidence for the existence of quarks.' back

Hadron - Wikipedia, Hadron - Wikipedia, the free encyclopedia, In particle physics, a hadron is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the electric force. Most of the mass of ordinary matter comes from two hadrons: the proton and the neutron, while most of the mass of the protons and neutrons is in turn due to the binding energy of their constituent quarks, due to the strong force.' 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

Henry W. Kendall (1990), Nobel lecture: II: Deep Inelastic Scattering: Experiments on the Proton and the Observation of Scaling, ' There are three lectures that, taken together, describe the MIT-SLAC experiments. The first, written by R.E.Taylor sets out the early history of the construction of the two mile accelerator, the proposals made for the construction of the electron scattering facility, the antecedent physics experiments at other laboratories, and the first of our scattering experiments which determined the elastic proton structure form factors. This paper describes the knowledge and beliefs about the nucleon’s internal structure in 1968, including the conflicting views on the validity of the quark model and the “bootstrap” models of the nucleon. . . . The last lecture,by J. I. Friedman is concerned with the later measurements of inelastic electron-neutron and electron-proton measurements and the details of the physical theory - the constituent quark model - which the experimental scattering results stimulated and subsequently, in conjunction with neutrino studies, confirmed.' back

Jerome I. Friedman (1990), Nobel Lecture III: Deep Inelastic Scattering: Comparisons with the quark model, ' The first suggestion that deep inelastic scattering might provide evidence of elementary constituents was made by Bjorken in his 1967 Varenna lectures. Studying the sum rule predictions derived from current algebra, he stated, ". . . We find these relations so perspicuous that, by an appeal to history, an interpretation in terms of elementary constituents is suggested." . . . The constituent model which opened the way for a simple dynamical interpretation of the deep inelastic results was the parton model of Feynman. He developed this model to describe hadon hadron interactions in which the constituents of one hadron interact with those of the other.' back

John C. Eccles (1958), Innovation in Science: The Physiology of Imagination, ' Our task here is to see how far our present ideas on the working of the brain can be related to the experiences of mind. The way to the imagination, the highest level of mental experience, lies through the lower levels of sensory experience, imagery, hallucination and memory, and that is the path we shall traverse. All that we shall learn must itself, of course, be the product of perceiving, reasoning and imagining by our brains! back

John Paul II (1992), Apostolic Constitution Fidei Depositum on the publication of the Catechism of the Catholic Church prepared following the Second Vatican Ecumenical Council, 'IV. The Doctrinal Value of the Text The Catechism of the Catholic Church, which I approved 25 June last and the publication of which I today order by virtue of my Apostolic Authority, is a statement of the Church's faith and of catholic doctrine, attested to or illumined by Sacred Scripture, the Apostolic Tradition and the Church's Magisterium. I declare it to be a sure norm for teaching the faith and thus a valid and legitimate instrument for ecclesial communion and a sure norm for teaching the faith.' back

Leon van Hove (1957), von Neumann's Contributions to Quantum Theory, ' Von Neumann's major contributions to quantum mechanics are his development of the mathematical framework of the theory and his formal study of quantum statistics, quantum measuring processes and their interrelations. Whereas the latter study was essentially complete in 1927 (except for the quantum ergodic theorem of 1929) the work on the mathematical foundations of quantum mechanics came to its culmination in 1929 with the spectral theorem for hyper- maximal symmetric operators in Hilbert space.' back

Manley, D. B., & Taylor, C. S. (1996), Descartes Meditations - Trilingual Edition, ' The publication of this English-Latin-French edition of Descartes' Meditations on First Philosophy is quite simply an experiment in electronic scholarship. We decided to make this edition available and to encourage its free distribution for scholarly purposes. The idea behind the experiment is to see how others involved in electronic scholarship might put these texts to use. We have no predetermined ideas of what such use may be when transformed from this origin. The texts have no hypertext annotations except for those used for navigation. We invite others to download this edition and to create their own hypertext annotated editions and then to publish those additions on their own Web servers for everyone to use.' back

Meinard Kuhlmann (Stanford Encyclopedia of Philosophy), Quantum Field Theory, ' Quantum Field Theory (QFT) is the mathematical and conceptual framework for contemporary elementary particle physics. In a rather informal sense QFT is the extension of quantum mechanics (QM), dealing with particles, over to fields, i.e. systems with an infinite number of degrees of freedom. (See the entry on quantum mechanics.) In the last few years QFT has become a more widely discussed topic in philosophy of science, with questions ranging from methodology and semantics to ontology. QFT taken seriously in its metaphysical implications seems to give a picture of the world which is at variance with central classical conceptions of particles and fields, and even with some features of QM.' back

Nick Reid, Patrick Nunn & M Sharpe (2014), Indigenous Australian stories and sea-level change, ' Oral traditions, especially contrasted with written history, are typically portrayed as inaccurate. Commenting on native title claims in the US, Simic (2000) made the specific claim: “As a general rule, unwritten legends that refer to events more than 1,000 years in the past contain little, if any, historical truth”. So can preliterate Indigenous languages tell us anything factual about the distant past, or does the transmission of historical facts become inevitably corrupted? Changes in sea levels around the Australian coast are now well established. Marine geographers can now point to specific parts of the Australian coast and know with some confidence what the sea levels were at a particular time before the present. This paper reports on a substantial body of Australian Aboriginal stories that appear to represent genuine and unique observations of post-glacial increases in sea level, at time depths that range from about 13,400–7,500 years BP. This paper makes the case that endangered Indigenous languages can be repositories for factual knowledge across time depths far greater than previously imagined, forcing a rethink of the ways in which such traditions have been dismissed.' back

Order of Preachers, Book of Constitutions and Ordinations of the Friars of the Order of Preachers, ' IV. We also undertake as sharers of the apostolic mission the life of the Apostles in the form conceived by St. Dominic, living with one mind the common life, faithful in the profession of the evangelical counsels, fervent in the common celebration of the liturgy, especially of the Eucharist and the divine office as well as other prayer, assiduous in study, and persevering in regular observance. All these practices contribute not only to the glory of God and our sanctification, but serve directly the salvation of mankind, since they prepare harmoniously for preaching, furnish its incentive, form its character, and in turn are influenced by it. These elements are closely interconnected and carefully balanced, mutually enriching one another, so that in their synthesis the proper life of the Order is established: a life in the fullest sense apostolic, in which preaching and teaching must proceed from an abundance of contemplation.' back

Oxford Movement - Wikipedia, Oxford Movement - Wikipedia, the free encyclopedia, 'The Oxford Movement was a movement of high church members of the Church of England which began in the 1830s and eventually developed into Anglo-Catholicism. The movement, whose original devotees were mostly associated with the University of Oxford, argued for the reinstatement of some older Christian traditions of faith and their inclusion into Anglican liturgy and theology. They thought of Anglicanism as one of three branches of the "one, holy, catholic, and apostolic" Christian church. Many key participants subsequently converted to Roman Catholicism. ' back

P. A. M. Dirac, The Lagrangian in Quantum Mechanics, 'Quantum mechanics was built upon a foundation of analogy with the Hamiltonian theory of classical mechanics. . . . Now there is an alternative formulation for classical dynamics, provided by the Lagrangian. This requires one to work in terms of coordinates and velocities instead of coordinates and momenta. The two formulations are, of course, closely related, but there are reasons for believing that the Lagrangian one is more fundamental.' . . .. A little consideration shows, however, that one cannot expect to be able to take over the classical Lagrangian equations in any very direct way. These equations involve partial derivatives of the Lagrangian with respect to coordinates and velocities and no meaning can be given to such derivatives in quantum mechanics.' back

Particle zoo - Wikipedia, Particle zoo - Wikipedia, ther free encyclopedia, ' In particle physics, the term particle zoo is used colloquially to describe the relatively extensive list of known subatomic particles by comparison to the variety of species in a zoo. In the history of particle physics, the topic of particles was considered to be particularly confusing in the late 1960s. Before the discovery of quarks, hundreds of strongly interacting particles (hadrons) were known and believed to be distinct elementary particles. It was later discovered that they were not elementary particles, but rather composites of quarks. The set of particles believed today to be elementary is known as the Standard Model and includes quarks, bosons and leptons. The term "subnuclear zoo" was coined or popularized by Robert Oppenheimer in 1956 at the VI Rochester International Conference on High Energy Physics.' 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

Paul the Apostle, Galatians, chapter 5, USCCB, ' 16 I say, then: live by the Spirit and you will certainly not gratify the desire of the flesh. For the flesh has desires against the Spirit, and the Spirit against the flesh; these are opposed to each other, so that you may not do what you want. But if you are guided by the Spirit, you are not under the law. Now the works of the flesh are obvious: immorality, impurity, licentiousness, idolatry, sorcery, hatreds, rivalry, jealousy, outbursts of fury, acts of selfishness, dissensions, factions, occasions of envy, drinking bouts, orgies, and the like. I warn you, as I warned you before, that those who do such things will not inherit the kingdom of God. ' back

Pauli exclusion principle - Wikipedia, Pauli exclusion principle - Wikipedia, the free encyclopedia, 'The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions (particles with half-integer spin) may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles. The principle was formulated by Austrian physicist Wolfgang Pauli in 1925.' back

Precision tests of QED - Wikipedia, Precision tests of QED - Wikipedia, the free encyclopedia, 'Quantum electrodynamics (QED), a relativistic quantum field theory of electrodynamics, is among the most stringently tested theories in physics. . . . Tests of a theory are normally carried out by comparing experimental results to theoretical predictions. . . . The agreement found this way is to within ten parts in a billion . . . . This makes QED one of the most accurate physical theories constructed thus far.' back

Proton - Wikipedia, Proton - Wikipedia, the free encyclopedia, 'In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit . . . , a diameter of about 1.65 x 10^-15 m, and a mass of 938.27231(28) MeV/c2 (1.6726 X 10^{- 27} kg), 1.007 276 466 88(13) u or about 1836 times the mass of an electron. Protons are spin 1/2 fermions and are composed of three quarks, making them baryons. The two up quarks and one down quark of the proton are held together by the strong force, mediated by gluons' back

Quantum chromodynamics - Wikipedia, Quantum chromodynamics - Wikipedia, the free encyclopedia, ' In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called color. Gluons are the force carrier of the theory, like photons are for the electromagnetic force in quantum electrodynamics.' back

Quantum electrodynamics - Wikipedia, Quantum electrodynamics - Wikipedia, the free encyclopedia, 'In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.' back

Quantum field theory - Wikipedia, Quantum field theory - Wikipedia, the free encyclopedia, 'Quantum field theory (QFT) provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or (especially in a condensed matter context) of many-body systems. . . . In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta - necessarily chunked ripples in a field that 'look like' particles. Fermions, like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta). back

Renormalization - Wikipedia, Renormalization - Wikipedia, the free encyclopedia, ' Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of quantities to compensate for effects of their self-interactions. But even if it were the case that no infinities arose in loop diagrams in quantum field theory, it could be shown that renormalization of mass and fields appearing in the original Lagrangian is necessary.' back

Richard E. Taylor (1990), Nobel Lecture: I. Deep Inelastic Scattering: The Early Years, ' Soon after the 1990 Nobel Prize in Physics was announced Henry Kendall, Jerry Friedman and I agreed that we would each describe a part of the deep inelastic experiments in our Nobel lectures. The division we agreed upon was roughly chronological. I would cover the early times, describing some of the work that led to the establishment of the Stanford Linear Accelerator Center where the experiments were performed, followed by a brief account of the construction of the experimental apparatus used in the experiments and the commissioning of the spectrometer facility in early elastic scattering experiments at the Center.' back

Richard P. Feynman (1985), Quantum mechanical computers, ' We are here considering ideal machines; the effects of imperfections will be considered later. This study is one of principle; our aim is to exhibit some Hamiltonian for a system which could serve as a computer. We are not concerned with whether we have the most efficient system, nor how we could best implement it.' back

Second Vatican Council, Dogmatic Constitution on Divine Revelation 'Dei Verbum', Solemnly promulgated by His Holiness Pope Paul VI on November 18, 1965. ' Preface: 1. Hearing the word of God with reverence and proclaiming it with faith, the sacred synod takes its direction from these words of St. John: "We announce to you the eternal life which dwelt with the Father and was made visible to us. What we have seen and heard we announce to you, so that you may have fellowship with us and our common fellowship be with the Father and His Son Jesus Christ" (1 John 1:2-3). Therefore, following in the footsteps of the Council of Trent and of the First Vatican Council, this present council wishes to set forth authentic doctrine on divine revelation and how it is handed on, so that by hearing the message of salvation the whole world may believe, by believing it may hope, and by hoping it may love.' back

Semiconductor - Wikipedia, Semiconductor - Wikipedia, the free encyclopedia, ' A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. Its conducting properties may be altered in useful ways by introducing impurities ("doping") into the crystal structure. When two differently doped regions exist in the same crystal, a semiconductor junction is created. The behavior of charge carriers, which include electrons, ions, and electron holes, at these junctions is the basis of diodes, transistors, and most modern electronics.' back

SLAC National Accelerator Laboratory - Wikipedia, SLAC National Accelerator Laboratory - Wikipedia, the free encyclopedia, ' Founded in 1962 as the Stanford Linear Accelerator Center, the facility is located on 172 ha (426 acres) of Stanford University-owned land on Sand Hill Road in Menlo Park, California—just west of the university's main campus. The main accelerator is 3.2 km (2 mi) long—the longest linear accelerator in the world—and has been operational since 1966.
Research at SLAC has produced three Nobel Prizes in Physics:
1976: The charm quark—see J/ψ meson
1990: Quark structure inside protons and neutrons
1995: The tau lepton . ' back

Standard model - Wikipedia, Standard model - Wikipedia, the free encyclopedia, 'The Standard Model of particle physics is a theory that describes three of the four known fundamental interactions between the elementary particles that make up all matter. It is a quantum field theory developed between 1970 and 1973 which is consistent with both quantum mechanics and special relativity. To date, almost all experimental tests of the three forces described by the Standard Model have agreed with its predictions. However, the Standard Model falls short of being a complete theory of fundamental interactions, primarily because of its lack of inclusion of gravity, the fourth known fundamental interaction, but also because of the large number of numerical parameters (such as masses and coupling constants) that must be put "by hand" into the theory (rather than being derived from first principles) . . . ' back

Synaptic weight - Wikipedia, Synaptic weight - Wikipedia, the free encyclopedia, ' In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research.' back

Transmission of the Greek Classics - Wikipedia, Transmission of the Greek Classics - Wikipedia, the free encyclopedia, ' Classical Greek philosophy consisted of various original works ranging from those from Ancient Greece (e.g. Aristotle) to those Greco-Roman scholars in the classical Roman Empire (e.g. Ptolemy). Though these works were originally written in Greek, for centuries the language of scholarship in the Mediterranean region, many were translated into Syriac, Arabic, and Persian during the Middle Ages and the original Greek versions were often unknown to the West. With increasing Western presence in the East due to the Crusades, and the gradual collapse of the Byzantine Empire during the Late Middle Ages, many Byzantine Greek scholars fled to Western Europe, bringing with them many original Greek manuscripts, and providing impetus for Greek-language education in the West and further translation efforts of Greek scholarship into Latin.' 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

Whitehead and Russell (1910), Principia Mathematica, Jacket: 'Principia Mathematica was first published in 1910-1913; this is the fifth impression of the second edition of 1925-7. The Principia has long been recognized as one of the intellectual landmarks of the century. It was the first book to show clearly the close relationship between mathematics and formal logic. Starting with a minimal number of axioms, Whitehead and Russell display the structure of both kinds of thought. No other book has had such an influence on the subsequent history of mathematical philosophy .' back

page 23: Insight and belief

Apologia pro vita sua

23.2: Bernard Lonergan: Insight: Understanding and affirmation

23.3: Revelation: theology, physics and evolution

23.4 Perhaps quantum field theory is not possible?

23.5: On coupling kinematic forms to dynamic energy

23.6: SLAC and deep inelastic scattering

Notes and references

Further reading

Books

Links