This site documents the Divine Mind:
the Universe that enables us to create ourselves
page 12: The quantum creation of Minkowski space
12.1: Augustine and the Word made FleshIn 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
12.2: A problem with quantum field theory
12.3: Is Hilbert space an appropriate foundation for a theory of everything?:
12.4: Evolution and our knowledge of the Universe?
12.5: Zero sum complexification
12.6: Spooky action at a distance
12.7: Gravitation, potential and kinetic energy
12.1 Augustine and the Word made Flesh
Christianity as we know it grew out of a paradigm change in Judaism, the ancient Hebrew theology. The change was originated by Jesus of Nazareth who claimed to be the Son of God. In Hebrew terms, he was claiming to be the Son of Yahweh, a name given to the creator of the world in their fundamental text, the Book of Genesis. The Hebrew High Priests saw Jesus' claim as blasphemy and were pleased to see him tortured to death by the Roman occupiers of their holy city, Jerusalem. Judaism - Wikipedia
Christians edited the Hebrew Bible, called it the Old Testament, and added the New Testament, an account of Jesus' life and work. The personalities of the divinities represented in these two books are very different. Yahweh is modelled on the imperial warlords who are still among us, jealous autocrats: I am the Lord Your God, you will have no other Gods before me. Yahweh condemned everybody to death because the first people disobeyed their command not to eat the fruit of a certain tree. Later they drowned everybody except Noah, his family and a selection of animals. Genesis flood narrative - Wikipedia
Jesus appears as a very different person, a compassionate healer, a man of the people, a peacemaker who promised a happy ending to compassionate people who followed him and loved one another. His fate, unfortunately, was identical to that of the millions of others since who have stood against abuse of power. The theology of compassion continues the struggle against the theology of war and the state of Earth shows that it has a long way to go.
My dream here is to promote another paradigm change by converting theology from mythology to science. The key to this change is the hypothesis that God is identical to the Universe we live in. This Universe plays all the roles traditionally attributed to Gods, creator, saviour and judge.
Since it was taken over by the Roman emperor Constantine in the fourth century, the Catholic Church has itself become an imperial, absolutist organization which has declared itself infallible and has a long history of violent action against "heretics".
As Christianity preserved much of Judaism, the aim here is to preserves as much as possible of Christianity. Yahweh was most resolutely one God. Christianity introduced three personalities in God, the Father, the Son and the Spirit. This idea became dogma at the Council of Nicea and is recorded in the Nicene Creed. This raised a massive problem for theologians: How do we reconcile the unity of God with the triplicity of divine personalities? Nicene Creed - Wikipedia
One of the first to face this problem head on was Augustine of Hippo. His solution remains current in the Catholic Church, and serves the foundation for a return to the position taken by Thales of Miletus, that the world is full of gods. Thales of Miletus - Wikipedia, Augustine (419, 1991): The Trinity
Augustine's treatise On the Trinity builds on two biblical ideas. Genesis tells us that God created us in their own image: God created mankind in his image; in the image of God he created them; (Gn I:27). John's Gospel provides a name for the second person of the Trinity, the Word: In the beginning was the word . . . And the Word became flesh. Genesis 1:27: God creates humans, Gospel of John (USCCB): I: 1-14
The first part of Augustine's book is an examination of his own mind and the formation of a mental word that serves as a preliminary step before speaking or writing a sentence. This provides him with a psychological analogy for the procession of the second person of the Trinity, the Son or Word of God. We begin with an idea and then proceed to create the reality. Mary Sirridge (1999): "Quam videndo intus dicimus": Seeing and Saying in De Trinitate XV
Here we understand the emergence of Hilbert space and quantum mechanics in the initial singularity by analogy with Augustine's notion of the Word of God and our own consciousness. The first step in this process is described on pages 9: The active creation of a Hilbert space and 10: The emergence of quantum mechanics From here we imagine five more steps to the realization of real physical particles in spacetime.
On page 13: Is Hilbert space independent of Minkowski space? we explain that Hilbert space is a layer of universal structure prior to Minkowski space, and therefore not subject to Einstein's special relativity and Lorentz transformation. We have no need for relativistic quantum mechanics at this level.
On page page 14: "Measurement": the interface between Hilbert and Minkowski spaces we jump ahead a bit and discuss the measurement problem which has bedevilled quantum mechanics since the beginning. It similar the problem we face in this section, the creation of an event in Minkowski space from events (interactions) in Hilbert space. There we conceive measurement as a conversation in kinematic Hilbert space between dynamic particles in Minkowski space, the measured particle and the measuring particle, whose physical roles are in fact identical. Measurement problem - Wikipedia
The next step page 15: Quantum amplitudes and logical processes are invisible explains why processes in quantum mechanics are invisible to us. A lot of our knowledge of quantum mechanics is based on trying to guess what is really happening then particles interact. It is not surprising that quantum field theory is a very troublesome subject which could easily be on the wrong track and become subject to radical revision.
One of the difficulties of the big bang theory, from my point of view, is that it seems to assume that all the energy of the Universe is already present in the the initial singularity, allegedly a structureless entity with no means to store either information (entropy) or energy. Page 16: Potential + kinetic = zero energy universe explores an alternative, that the total energy of the Universe is zero. The energy we see in the particles created by quantum mechanics might then be derived from deepening the gravitational potential well in which the Universe resides.
The process which uses gravitational potential energy to convert the invariant kinematic structures created by quantum mechanics into real dynamic particles which exist in Minkowski space is finally revealed on page 17: Gravitation and quantum theory—in the beginning.
12.2 A problem with quantum field theory
The problems with quantum field theory to which Kuhlmann alludes arises partly from its confusing history. One problem lies in the difficult relationship between quantum mechanics (QM) and the special relativity theory (SRT). Kuhlmann writes:
Historically, QFT resulted from the successful reconciliation of QM and SRT. In order to understand the initial problem one has to realize that QM is not only in a potential conflict with SRT. More exactly: the locality postulate of SRT [is in trouble], because of the famous EPR correlations of entangled quantum systems. There is also a manifest contradiction between QM and SRT on the level of the dynamics. The Schrödinger equation, i.e., the fundamental law for the temporal evolution of the quantum mechanical state function, cannot possibly obey the relativistic requirement that all physical laws of nature be invariant under Lorentz transformations.
Quantum theory is described in Hilbert space, which is a mathematical space which bears a passing relationship to conventional physical spacetime, which is the home of special relativity. Special relativity is a classical theory, expressed in terms of the continuous mathematics and the standard topology of spacetime. Its specific difference, which make no difference to the analytic properties of Minkowski and Euclidean space, is that in terms of 4 dimensional space, the Euclidean metric is all positive, 1, 1, 1, 1, whereas the Minkowski metric has a negative component: 1, 1, 1, -1.
Von Neuman, in his axiomatic treatment of abstract Hilbert space, draws attention to the mathematical analogy between Hilbert and Euclidean space. From this point of view, both are metric vector spaces which enable discussions of distance and angle. It was Einstein himself who pointed out that the difference between the two spaces is more important than their similarity when he drew attention to spooky action at a distance, now known as entanglement. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics, EPR Paradox - Wikipedia
Although Hilbert space is analogous to Euclidean space, the difference between Minkowski space and Hilbert space is radical. Einstein formulated special relativity using rods to measure distance and clocks to measure time. The key to special relativity is the postulate that all observers in inertial motion see the same laws of physics, including the velocity of light, in their own rest frames. This condition implies the Lorentz transformation, which enables me to compute what my observations in my own frame would look like in a frame moving relative to myself. A consequence of this postulate is that if I could observe a clock on a photon it would appear to be stopped and the length of a ruler on photon aligned in its direction of motion would be zero.
"Distance" in Hilbert space is not measured with rods and clocks, but with the inner product between two vectors f and g, written (f, g) which yields the angle (or argument) between the two vectors in their shared complex plane. This idea is illustrated in Einstein, Podolsky and Rosen's paper (EPR). Argument (complex analysis) - Wikipedia, Einstein, Podolsky & Rosen (1935): Can the Quantum Mechanical Description of Physical Reality be Considered Complete?
While accepting that quantum mechanics gives answers consistent with observation and measurement, EPR ask whether the theory is complete. They state their criterion for completeness as follows:
Whatever the meaning applied to the term complete, the following requirement for a complete theory seems to be a necessary one: every element of the physical reality must have a counterpart in the physical theory. . . . We shall be satisfied with the following criterion, which we regard as reasonable. If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.
After a discussion of an application of quantum theory and non-commutation of certain variables, they concluded:
While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.
This paper led to a long debate. In modern times it has been shown by experiment that entanglement is a fact and John Bell showed that quantum mechanics does not necessarily respect local causality. Page 13: Is Hilbert space independent of Minkowski space? continues the discussion of the independence of Hilbert and Minkowski space. Salart, Baas, Branciard, Gisin & Zbinden (2008): Testing the speed of 'spooky action at a distance', John S. Bell (1987): Speakable and Unspeakable in Quantum Mechanics
Here we proceed on the assumption quantum theory operates between two very different aspects of the layered structure of the world, the invisible formal processes in Hilbert space (corresponding to the "mind" of the world), and the observable dynamic processes in Minkowski space. As in our human minds and bodies, the coupling between these two layers of the world is loose, and we need to consider both to get the full picture. This idea is embedded in those legal theories of justice which require a mens rea to find a person guilty of crime. Mens rea - Wikipedia
A simple modern demonstration of entanglement begins with two electrons in a singlet state, one spin up and the other spin down, which are then moved apart. We find that if an operator measuring the spin of the first electron in specific direction in three dimensional space observes that it has spin up, an operator measuring the spin of the other electron in the same direction will invariably find that its spin is down, even though the two electrons are so far apart that no light signal could travel between them to correlate their spins (that is, they are spacelike separated). On the other hand, if the second operator measures the spin of their electron in any other direction, they will find that its spin is distributed at random, half up and half down, as would be expected if there was no influence from the state of the first electron. It seems that relativistic restrictions on real space-time do not apply to Hilbert space,
12.3 Is Hilbert space an appropriate foundation for a theory of everything?
We are inclined to take classical space-time as given and see it emerging fully formed from the initial singularity. This Minkowski spacetime is understood to be the domain of quantum field theory, so that the fields of quantum field theory described in Hilbert space are subject to the Lorentz transformations of the special theory of relativity. Martinus Veltman (1994): Diagrammatica: The Path to the Feynman Rules, page 20
This attachment of Hilbert space to Minkowski space may be the source of much of the confusion arising in quantum field theory. There may be a solution in the idea that Hilbert space and quantum mechanics are the underlying processes that explain all the structure on the Universe, including Minkowski spacetime. This approach may serve to clarify quantum field theory and provide a role for gravitation in the creation of the dynamic quantum mechanical structure of the world.
I like to see the world as a layered computational network whose basic hardware layer comprises the initial singularity which is the source of the kinematic Hilbert space whose invisible inner processes are the subject of quantum mechanics. Like engineered communication networks, this network is layered, from hardware at the base to users at the top. Each layer acts as a symmetry or algorithm, breaking the symmetry of the layer beneath it and being applied to provide services to the layer above. Users may be people, corporations or machines. Overall the lowest layer is the initial singularity which we identify with the traditional God of Aquinas and the topmost layer is the Universe itself, created, as Thales may have imagined imagined, from an enormous number of copies of the initial singularity. Internet Protocol - Wikipedia, Thomas Aquinas: Commentary on Aristotle's De Anima: Lectio 10: The Agent Intellect
This picture is formulated in terms of Turing computation and Shannon communication and the internet is a convenient real model, which (when it is working as designed) provides deterministic addressing of resources and error free transmission of data. It is a human artefact with deep roots in science and technology. We may safely conclude that it is also a product of evolution, the biological evolution that created us and the hypothetical evolutionary process discussed on this site that created the physical universe. The idea is that the Universe arises from a structureless but omnipotent initial singularity constrained only by the axiom that real physical contradictions cannot exist. Our scientific studies in Minkowski space lead us to believe that this axiom holds in the observable world where we work, but what of Hilbert space? Is it real?
Apart from the difficulties with relativity which appear to be inherent in QFT Kuhlmann identifies another set problems in the passage quoted at the beginning of this page:
In conclusion one has to recall that one reason why the ontological interpretation of QFT 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.
We have taken special relativity out of the picture at the Hilbert level of structure but it is integral to the Minkowski layer. It must reappear as soon as we have dealt with the representation problem. The key here is to distinguish kinematic from dynamic entities. Kinematic objects are dead in the sense that they cannot move themselves, but must be moved by something else. Mathematics, which includes Hilbert space, is kinematic. It does not do itself. It is done by mathematicians, machines or the Universe. Dynamic objects are alive and move themselves. Kinematics - Wikipedia, Dynamics - Wikipedia
Aristotle was aware of this distinction. He saw that Plato’s ideas are purely formal and inert, like puppets. Since they could not move themselves, he devised an unmoved mover to move them. Since he held no potential can actualize itself, this mover must be pure action. Aquinas took this for a definition of God: God is actus purus. Here we identify this unmoved mover with the initial singularity. Since in the beginning the singularity is all that there is, it must create the world within itself and it must be dynamic. The Hilbert space and quantum mechanics that grow within it are kinematic. Unmoved mover - Wikipedia
There was a time when mathematics was considered to be consistent, computable and complete. Gödel and Turing have shown us that consistency does no guarantee completeness or computability. From the naive point of view taken on this site, this means that no divinity, no matter how omniscient or omnipotent, can totally control the world. The way is open in kinematic mathematics for variation, the foundation of evolution. Gödel's incompleteness theorems - Wikipedia, Alan Turing (1936): On Computable Numbers, with an application to the Entscheidungsproblem
12.4: Evolution and our knowledge of the Universe?
The absolute simplicity of the initial singularity precludes it from carrying any design for the Universe it is about to create. Since it has no prior knowledge, anything that it does will be a random act or a sequence of random acts, an appropriate beginning for a process of evolution which owes its creativity to random variation.
Since Darwin devised the theory of evolution using his knowledge of animal and plant breeding and observations he made on the different beak shapes of finches in the Galapogos Islands, we have learnt to sequence genes and gained detailed knowledge of biological evolution, of mutation and of the transcription of genes into proteins which are the dynamic elements of life. We known that the average genome contains about one billion symbols, the bases, adenine, cytosine, guanine and thymine in DNA and uracil replacing thymine in RNA. Every one of these bases is subject to mutation, so the genetic space of possibility is approximately one billion to the fourth power; 1036. Of course only a tiny fraction of these possibilities, billions nevertheless, represent the viable creatures naturally selected to survive and reproduce on their environment. Nucleic acid - Wikipedia
Here we are imagining Hilbert spaces with up to a countable infinity of ai basis states normalized to 1 which may be multiplied by parameters pi and added together to form vectors which are normalized by the condition
Σi (pi)2 = 1.
This system provides an enormous range of variation from which quantum mechanics may select stable states to be taken to the next stage in building the Universe.
This model suggests that rather than consider spacetime to be the domain of quantum theory, it might be more reasonable to understand spacetime to be a layer of reality which has evolved from elements provided by quantum mechanics. We suppose that quantum mechanics is the underlying symmetry that is applied (and broken) to create stable spacetime and the stable elements within it. The discussion of invisibility on page 15: Quantum amplitudes and logical processes are invisible explains why it may be that we cannot see what is going on in Hilbert space.
Here we come to an interesting part of this story, the interface between the abstract invisible quantum world that explores possible futures in the Universe behind the scenes, and the world of space, time, momentum, energy and observable particles (at every scale, like ourselves, stars and planets) in which we live. Spacetime may be the stable screen and dynamic memory within which we observe everything that happens. It serves as the operating system of the Universe, controlling input to and output from the invisible world of quantum computation which actually runs the world in Hilbert space. (See page 20: Space-time—the cosmic memory and operating system). Nielsen & Chuang (2016): Quantum Computation and Quantum Information
How does structural information flow from Hilbert space to Minkowski space and back? This question, often known as the measurement problem, has been debated since the beginning of quantum mechanics and is taken up on page 14: Measurement: the interface between Hilbert and Minkowski spaces.
12.5: Zero sum complexification
A key idea here is tautological or zero-sum complexification. (See page 26 Principle 2: Zero sum complexification) This feature of the world has already appeared unremarked on page 10: The emergence of quantum mechanics. There, since it is of the nature of action to act, and we define energy as the rate of action, we may see that action of its nature (ie tautologically and kinematically) creates a logical analogue of energy.
From this point of view the kinematic emergence of time, energy and phase adds a new feature to the initial singularity while preserving its underlying simplicity. The idea here is that each step in the complexification of the Universe is an evolutionary process that steps through a phase of randomness and uncertainty to create new distinct entities whose sum is the symmetry from which they emerged, like potential and kinetic energy or positive and negative charge. If we imagine any particular symmetry as a zero point which is broken in the process of creation, this break occurs in such a such a way that the sum of the broken elements remains zero. The sequence of creative processes that have led from the initial singularity to the present can be understood to have preserved the integrity of initial symmetry while creating a complex world. This process may be reversed locally by gravitational collapse (see page 17: Gravitation and quantum theory—in the beginning).
In his lectures on gravitation Feynman notes that the total energy of the Universe may be zero. Primordial symmetries are broken to create new features of the world. These symmetries are not thereby destroyed because they remain as the common source of a set of differentiated states. Action is therefore inherent in the products of every differentiation. Energy and time, understood as the evolution of phase, enter the world with the emergence of quantum mechanics and are a feature of every subsequent layer of its structure. This symmetry ultimately underlies the conservation of energy which is a result of the differentiation of the rate of change of phase represented in Hilbert space into the potential and kinetic energy we observe in Minkowski space. (See page 16: Potential + kinetic = zero energy universe). Richard Feynman (2002): Feynman Lectures on Gravitation, page 10, Zero-energy universe - Wikipedia
12.6: Spooky action at a distance
Let us assume that communication and causality require contact. Isaac Newton was forced by circumstances to admit that gravitation was some sort of "action at a distance", which we understand to be impossible in ordinary space. Quantum entanglement in Hilbert space led Einstein to imagine "spooky action at a distance". We shall suggest on page 13: Is Hilbert space independent of Minkowski space? that this is possible because quantum mechanics works in the world before spatial distance in the Newtonian sense has emerged. In the space-time world contact is maintained by particles moving at the speed of light which follow "null geodesics" whose creation (source) and annihilation (sink) coincide in spacetime. Geodesics in general relativity - Wikipedia, Quantum entanglement - Wikipedia
The most peculiar feature of Minkowski spacetime is its metric ημν, which is diagonal 1, 1, 1, -1 (or equivalently 1, -1, -1, -1). This suggests that zero sum bifurcation is at work, so that in some sense space + time = 0. The principal ingredients of a model of the emergence of spacetime are therefore symmetry, zero bifurcation and the speed of light. The null geodesic, made possible by the Minkowski metric, is the accommodation made in spacetime to maintain contact in Hilbert space after the emergence of space. The velocity of light is an artefact of this accomodation and enables contact in the quantum world to continue uninterrupted despite the emergence of space. How can this happen? We invoke the evolutionary principle that uncontrolled action can try everything, and that consequences of these trials that are self sustaining are selected and may become fixed with lives of varying length. Some, like the proton and the structure of Minkowski spacetime may be effectively eternal. Minkowski space - Wikipedia, Proton decay - Wikipedia
Interaction is local. Before space enters the world, contact is inevitable and quantum systems can evolve unitarily without interruption. To correlate their evolution, spatially separated systems must communicate to maintain contact. The metric of Minkowski space enables the existence of null geodesics whose endpoints are in contact because the space-time interval between them is zero. The unitary contact of spatially separated systems can thus be maintained if the messenger travelling between them proceeds at the speed of light in Minkowski space. In other words the speed of light makes space possible by maintaining the integrity of the contact and unitarity that is essential to the work of quantum mechanics, and this "trick" explains the Minkowski metric. Kevin Brown (2018): Reflections on Relativity, page 693.
12.7: Gravitation, potential and kinetic energy
It has been generally assumed that Minkowski space is the domain of Hilbert space so that it is necessary to apply Lorentz transformations to both Hilbert spaces and particles in quantum field theory. This may not be necessary if Hilbert space is prior to and independent of Minkowski space. It may be that this approach removes some of the ontological confusion that Kuhlmann identifies. Martinus Veltman (1994): op cit page 20
This is a key page for this site, since the creation of Minkowski space from Hilbert space by the cosmic discovery of the Minkowski metric introduces a valuable degree of freedom into the creative computational power of the Universe. This may seem too good to be true, but on the other hand if the Universe started as a quantum initial singularity prior to space-time something like this event must have occurred on the way from then to now and, given the layered network in Minkowski space to be described on page 19: Network I: Cooperation, must still exist and be effective now.
From the point of view of spacetime, we can say that EPR were right, quantum mechanics is incomplete. The Hilbert space of quantum mechanics may have countably infinite dimensions, but physically these are all dimensions measured in one dimension, time or frequency. The mechanism of quantum mechanics, implemented by Hermitian operators, works on all these dimensions at once constrained only by the conservation of energy measured by the simple quantum mechanical kinematic expression for energy, E = hf where f is the rate of change of phase. Although energy measured by frequency may be moved from one dimension in Hilbert space to another, the overall energy measured in this way as the sum of frequencies is conserved. In the Schrödinger representation of the evolution of a quantum system, this conservation is law is maintained by the requirement of unitarity.
The Schrödinger equation describes the process in Hilbert space where all the Hilbert vectors are in contact with one another by superposition, Regardless of the complexity of the superposition the "length" all vectors is normalized to 1, equivalent to a phase of 2π. This is another way of expressing the conservation of energy.
The situation changes when we introduce Minkowski spacetime which enables the independent existence of isolated quantum systems. These particles, each with an associated Hilbert space obeying the rules proposed above, may now exchange dynamic energy, opening up a new degree of freedom. No longer is energy just shared by vectors in a given Hilbert space, the Hilbert spaces associated with different particles may gain or lose energy from or to other particles. The game played in Minkowski space has become much more complex. The peculiar metric of Minkowski space makes this quantum exchange of energy between different particles possible.
We are now dealing in dynamic energy, not the kinematic energy represented by E=hf and this exchange is governed by the Minkowski metric, that is by Lorentz transformations.
(revised Wednesday 7 August 2024)
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Notes and references
Further reading
Books
Augustine (419, 1991), and Edmond Hill (Introduction, translation and notes), and John E Rotelle (editor), The Trinity, New City Press 399-419, 1991 Written 399 - 419: De Trinitate is a radical restatement, defence and development of the Christian doctrine of the Trinity. Augustine's book has served as a foundation for most subsequent work, particularly that of Thomas Aquinas.
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Bell (1987), John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987 Jacket: JB ... is particularly famous for his discovery of a crucial difference between the predictions of conventional quantum mechanics and the implications of local causality . . . . This work has played a major role in the development of our current understanding of the profound nature of quantum concepts and of the fundamental limitations they impose on the applicability of classical ideas of space, time and locality.
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Brown (2018), Kevin, Reflections on Relativity, 2018 ' . . . general relativity teaches us that the principles of special relativity are applicable only over infinitesimal regions in the presence of gravitation, so in a sense the general theory restricts rather than generalizes the special theory. However, we can also regard special relativity as a theory of flat four-dimensional spacetime, characterized by the Minkowski metric (in suitable coordinates), and the general theory generalizes this by allowing the spacetime manifold to be curved, as represented by a wider class of metric tensors. It is remarkable that this generalization, which is so simple and natural from the geometrical standpoint, leads almost uniquely to a viable theory of gravitation.' (page 700)
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Feynman (2002), Richard, Feynman Lectures on Gravitation, Westview Press 2002 ' The Feynman Lectures on Gravitation are based on notes prepared during a course on gravitational physics that Richard Feynman taught at Caltech during the 1962-63 academic year. For several years prior to these lectures, Feynman thought long and hard about the fundamental problems in gravitational physics, yet he published very little. These lectures represent a useful record of his viewpoints and some of his insights into gravity and its application to cosmology, superstars, wormholes, and gravitational waves at that particular time. The lectures also contain a number of fascinating digressions and asides on the foundations of physics and other issues. '
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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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Veltman (1994), Martinus, Diagrammatica: The Path to the Feynman Rules, Cambridge University Press 1994 Jacket: 'This book provides an easily accessible introduction to quantum field theory via Feynman rules and calculations in particle physics. The aim is to make clear what the physical foundations of present-day field theory are, to clarify the physical content of Feynman rules, and to outline their domain of applicability. ... The book includes valuable appendices that review some essential mathematics, including complex spaces, matrices, the CBH equation, traces and dimensional regularization. . . .'
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Links
Alan Turing (1936), On Computable Numbers, with an application to the Entscheidungsproblem, 'The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by some finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable of a real or computable variable, computable predicates and so forth. . . . ' back |
Argument (complex analysis) - Wikipedia, Argument (complex analysis) - Wikipedia, the free encyclopedia, ' In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane. It is a multivalued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument (sometimes denoted Arg z) is used. It is often chosen to be the unique value of the argument that lies within the interval (−π, π].' back |
Black hole thermodynamics - Wikipedia, Black hole thermodynamics - Wikipedia, the free encyclopedia, 'In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. Much as the study of the statistical mechanics of black body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.' back |
Dynamics - Wikipedia, Dynamics - Wikipedia, the free encyclopedia, ' Dynamics is the branch of classical mechanics that is concerned with the study of forces and their effects on motion. Isaac Newton was the first to formulate the fundamental physical laws that govern dynamics in classical non-relativistic physics, especially his second law of motion. ' back |
Einstein, Podolsky & Rosen (1935), Can the Quantum Mechanical Description of Physical Reality be Considered Complete?, A PDF of the classic paper. 'In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false, One is thus led to conclude that the description of reality given by the wave function is not complete.' back |
EPR Paradox - Wikipedia, EPR Paradox - Wikipedia, the free encyclopedia, 'In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. "EPR" stands for Einstein, Podolsky, and Rosen, who introduced the thought experiment in a 1935 paper to argue that quantum mechanics is not a complete physical theory.' back |
Fixed point theorem - Wikipedia, Fixed point theorem - Wikipedia, the free encyclopedia, 'In mathematics, a fixed point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics. The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).' back |
Genesis 1:27 , God creates humans, ' 27: God created mankind in his image; in the image of God he created them; male and female he created them. ' back |
Genesis flood narrative - Wikipedia, Genesis flood narrative - Wikipedia, the free encyclopedia, ' The Genesis flood narrative (chapters 6–9 of the Book of Genesis) is the Hebrew version of the universal flood myth. It tells of God's decision to return the universe to its pre-creation state of watery chaos and remake it through the microcosm of Noah's ark. . . .. A global flood as described in this myth is inconsistent with the physical findings of geology, paleontology and the global distribution of species. A branch of creationism known as flood geology is a pseudoscientific attempt to argue that such a global flood actually occurred.' back |
Genesis I:27, God creates humans, ' 27: So God created man in his own image, in the image of God created he him; male and female created he them. And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth. ' back |
Geodesics in general relativity - Wikipedia, Geodesics in general relativity - Wikipedia, the free encyclopedia, ' In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational force, is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.' back |
Gödel's incompleteness theorems - Wikipedia, Gödel's incompleteness theorems - Wikipedia, the free encyclopedia, ' Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.' back |
Gospel of John (USCCB), Chapter 1:1-14, ' In the beginning was the Word . . . |
Mens rea - Wikipedia, Mens rea - Wikipedia, the free encyclopedia, ' In criminal law, mens rea (Law Latin for "guilty mind") is the mental state of a defendant who is accused of committing a crime. In common law jurisdictions, most crimes require proof both of mens rea and actus reus ("guilty act") before the defendant can be found guilty. |
Internet Protocol - Wikipedia, Internet Protocol - Wikipedia, the free encyclopedia, ' The Internet Protocol (IP) is the network layer communications protocol in the Internet protocol suite for relaying datagrams across network boundaries. Its routing function enables internetworking, and essentially establishes the Internet. IP has the task of delivering packets from the source host to the destination host solely based on the IP addresses in the packet headers. For this purpose, IP defines packet structures that encapsulate the data to be delivered. It also defines addressing methods that are used to label the datagram with source and destination information.' back |
John von Neumann (2014), Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford. Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC back |
Judaism - Wikipedia, Judaism - Wikipedia, the free encyclopedia, ' Contents (Top) Etymology History Defining characteristics and principles of faith Religious texts Jewish identity Jewish religious movements Jewish observances Community leadership Judaism and other religions Criticism See also Footnotes Bibliography Notes External links Judaism Article Talk Read View source View history Tools Appearance Text Small Standard Large Width Standard Wide Color (beta) Automatic Light Dark Page semi-protected From Wikipedia, the free encyclopedia "Judeo" redirects here. For the album, see Judeo (album). Judaism יַהֲדוּת Yahăḏūṯ The Western Wall is a remnant of the ancient wall that surrounded the Jewish Temple's courtyard, and is perhaps the most sacred site recognized by the Jewish faith outside of the Temple Mount itself. Type Ethnic religion Classification Abrahamic Scripture Tanakh, Talmud, Midrash Theology Monotheistic Language Hebrew and Aramaic Founder Abraham and Moses (according to tradition)[1][2] Origin c. 6th century BCE Judah Separated from Yahwism Separations Samaritanism Mandaeism Christianity[a] Number of followers c. 15.2 million (referred to as Jews) Part of a series on Judaism Star of David Ten Commandments Menorah Movements Orthodox Haredi Hasidic Modern Conservative Conservadox Reform Karaite Reconstructionist Renewal Humanistic Haymanot Philosophy Principles of faith Kabbalah Messiah Ethics Chosenness God Names Musar movement Texts Tanakh Torah Nevi'im Ketuvim Ḥumash Siddur Piyutim Zohar Rabbinic Mishnah Talmud Midrash Tosefta Law Mishneh Torah Tur Shulchan Aruch Mishnah Berurah Aruch HaShulchan Kashrut Tzniut Tzedakah Niddah Noahide laws Holy cities/places Jerusalem Safed Hebron Tiberias Synagogue Beth midrash Mikveh Sukkah Chevra kadisha Holy Temple Tabernacle Important figures Abraham Isaac Jacob Moses Aaron David Solomon Sarah Rebecca Rachel Leah Rabbinic sages Chazal Tannaim Amoraim Savoraim Geonim Rishonim Acharonim Religious roles Rabbi Rebbe Posek Hazzan Dayan Rosh yeshiva Mohel Kohen Culture and education Brit Zeved habat Pidyon haben Bar and bat mitzvah Marriage Bereavement Yeshiva Kollel Cheder Ritual objects Sefer Torah Tallit Tefillin Tzitzit Kippah Mezuzah Menorah Shofar Four species Etrog Lulav Hadass Arava Kittel Gartel Prayers Shema Amidah Aleinu Kaddish Minyan Birkat Hamazon Shehecheyanu Hallel Havdalah Tachanun Kol Nidre Selichot (S'lichot) Major holidays Rosh Hashanah Yom Kippur Sukkot Pesach Shavuot Purim Hanukkah ' Judaism (Hebrew: יַהֲדוּת, romanized: Yahăḏūṯ) is an Abrahamic monotheistic ethnic religion that comprises the collective spiritual, cultural, and legal traditions of the Jewish people. Judaism evolved from Yahwism, an ancient Semitic religion of the late Bronze Age to early Iron Age, likely around the 6th/5th century BCE. Along with Samaritanism, to which it is closely related, Judaism is one of the two oldest Abrahamic religions. ' back |
Kinematics - Wikipedia, Kinematics - Wikipedia, the free encyclopedia, 'Kinematics (from Greek . . . kinein, to move) is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move.' back |
Mary Sirridge (1999), "Quam videndo intus dicimus": Seeing and Saying in De Trinitate XV, ' What is being asserted is that thought has the same form as seeing or speaking respectively, i.e., that it works essentially like seeing or speaking, that thought is a formal and functional isomorph of seeing or speaking.' back |
Measurement problem - Wikipedia, Measurement problem - Wikipedia, the free encyclopedia, 'The measurement problem in quantum mechanics is the problem of how (or whether) wave function collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer. The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states, but actual measurements always find the physical system in a definite state. Any future evolution is based on the state the system was discovered to be in when the measurement was made, meaning that the measurement "did something" to the system that is not obviously a consequence of Schrödinger evolution.' 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 |
Minkowski space - Wikipedia, Minkowski space - Wikipedia, the free encyclopedia, ' By 1908 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval x2 + y2 + z2 − c2 t2.' back |
Nicene Creed - Wikipedia, Nicene Creed - Wikipedia, the free encyclopedia, ' The Nicene Creed (Greek: Σύμβολον τῆς Νίκαιας, Latin: Symbolum Nicaenum) is the profession of faith or creed that is most widely used in Christian liturgy. It forms the mainstream definition of Christianity for most Christians. It is called Nicene because, in its original form, it was adopted in the city of Nicaea (present day Iznik in Turkey) by the first ecumenical council, which met there in the year 325. The Nicene Creed has been normative for the Catholic Church, the Eastern Orthodox Church, the Church of the East, the Oriental Orthodox churches, the Anglican Communion, and the great majority of Protestant denominations.' back |
Nucleic acid - Wikipedia, Nucleic acid - Wikipedia, the free encyclopedia, ' Nucleic acids are the biopolymers, or small biomolecules, essential to all known forms of life. The term nucleic acid is the overall name for DNA and RNA. They are composed of nucleotides, which are the monomers made of three components: a 5-carbon sugar, a phosphate group and a nitrogenous base. If the sugar is a compound ribose, the polymer is RNA (ribonucleic acid); if the sugar is derived from ribose as deoxyribose, the polymer is DNA (deoxyribonucleic acid).' back |
Proton decay - Wikipedia, Proton decay - Wikipedia, the free encyclopedia, ' In particle physics, proton decay is a hypothetical form of particle decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron. The proton decay hypothesis was first formulated by Andrei Sakharov in 1967. Despite significant experimental effort, proton decay has never been observed. If it does decay via a positron, the proton's half-life is constrained to be at least 1.67×1034 years.' back |
Quantum entanglement - Wikipedia, Quantum entanglement - Wikipedia, the free encyclopedia, 'Quantum entanglement is a physical phenomenon which occurs when pairs or groups of particles are generated, interact, or share spatial proximity in ways such that the quantum state of each particle cannot be described independently of the state of the other(s), even when the particles are separated by a large distance—instead, a quantum state must be described for the system as a whole. . . . Entanglement is considered fundamental to quantum mechanics, even though it wasn't recognized in the beginning. Quantum entanglement has been demonstrated experimentally with photons, neutrinos, electrons, molecules as large as buckyballs, and even small diamonds. The utilization of entanglement in communication and computation is a very active area of research.' back |
Salart, Baas, Branciard, Gisin & Zbinden (2008), Testing the speed of 'spooky action at a distance, ' Correlations are generally described by one of two mechanisms: either a first event influences a second one by sending information encoded in bosons or other physical carriers, or the correlated events have some common causes in their shared history. Quantum physics predicts an entirely different kind of cause for some correlations, named entanglement. This reveals itself in correlations that violate Bell inequalities (implying that they cannot be described by common causes) between space-like separated events (implying that they cannot be described by classical communication). Many Bell tests have been performed, and loopholes related to locality and detection have been closed in several independent experiments. It is still possible that a first event could influence a second, but the speed of this hypothetical influence (Einstein's 'spooky action at a distance') would need to be defined in some universal privileged reference frame and be greater than the speed of light. Here we put stringent experimental bounds on the speed of all such hypothetical influences. We performed a Bell test over more than 24 hours between two villages separated by 18 km and approximately east-west oriented, with the source located precisely in the middle. We continuously observed two-photon interferences well above the Bell inequality threshold. Taking advantage of the Earth's rotation, the configuration of our experiment allowed us to determine, for any hypothetically privileged frame, a lower bound for the speed of the influence. For example, if such a privileged reference frame exists and is such that the Earth's speed in this frame is less than 10(-3) times that of the speed of light, then the speed of the influence would have to exceed that of light by at least four orders of magnitude.' back |
Thales of Miletus - Wikipedia, Thales of Miletus - Wikipedia, the free encyclopedia, ' Thales of Miletus (Greek: Θαλῆς (ὁ Μιλήσιος), c. 624/623 – c. 548/545 BC) was a Greek mathematician, astronomer and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him as the first philosopher in the Greek tradition, and he is otherwise historically recognized as the first individual in Western civilization known to have entertained and engaged in scientific philosophy. |
Thomas Aquinas, Commentary on Aristotle's De Anima: Lectio 10: The Agent Intellect, '743. And in line with what he said at the beginning of this book, that the soul might be separable from the body if any of its activities were proper to itself, he now concludes that the soul’s intellectual part alone is immortal and perpetual. This is what he has said in Book II, namely that this ‘kind’ of soul was separable from others as the perpetual from the mortal,—perpetual in the sense that it survives for ever, not in the sense that it always has existed; for as he shows in Book XII of the Metaphysics, forms cannot exist before their matter. The soul, then (not all of it, but only its intellectual part) will survive its matter.' 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 |
Zero-energy universe - Wikipedia, Zero-energy universe - Wikipedia, the free encyclopedia, 'The zero-energy universe hypothesis proposes that the total amount of energy in the universe is exactly zero: its amount of positive energy in the form of matter is exactly cancelled out by its negative energy in the form of gravity. . . . The zero-energy universe theory originated in 1973, when Edward Tryon proposed in the journal Nature that the universe emerged from a large-scale quantum fluctuation of vacuum energy, resulting in its positive mass-energy being exactly balanced by its negative gravitational potential energy.' back |