Introducing the divinity of the Universe
to pave the way for scientifically credible theology
Chapter 17: Gravitation + particles = Minkowski space
Synopsis
All the action in the Universe occurs in Hilbert space. The events we see in Minkowski space are observations of these processes. The creation of space enables a massive increase in the number of particles and the entropy of the Universe. All the fundamental particles are either bosons with integer spin (mostly massless travelling at the speed of light); or fermions with half even spin ½, etc., all massive and described by the Dirac equation. In standard quantum field theory the spin statistics theorem derives the difference between bosons and fermions from the special theory of relativity. Here we guess that the distinction between bosons and fermions is deeper in the structure of the Universe and attribute it to quantum mechanics, the universal agent. We see the properties of these particles as the source of the metric of Minkowski spacetime. Spin (physics) - Wikipedia, Spin-statistics theorem - Wikipedia
Contents
17.1: The birth of quantum field theory17.2: Zero-sum bifurcation
17.3: Bosons and fermions
17.4: The first particle: the photon
17.5: Dirac’s equation, fermions and the exclusion principle
17.6: The bifurcation into spacetime
17.1: The birth of quantum field theory
Quantum mechanics began as a study of atoms through their spectra. Its first big success, in 1913, was Bohr’s model of the atom. It worked quite well but only for one electron. The theorists struggled to deal with more electrons while the experimenters worked hard and found many new and puzzling results. Then in the late 20s Heisenberg and Schrödinger found a way ahead but it was not perfect. Elementary particles move at relativistic speeds so it became necessary to unite quantum mechanics with special relativity. This union led to quantum field theory. Steven Weinberg (1995): The Quantum Theory of Fields Volume I: Foundations, page 49
The new theory became plagued with infinities (Weinberg page 33). A solution was found in the process of renormalization. At least one of the masters, Richard Feynman, thought this was equivalent to sweeping dirt under the rug. Richard P. Feynman (1965): Nobel Lecture: The Development of the Space-Time View of Quantum Electrodynamics
In a modern review of the theory, philosopher Meinard Kuhlman sees many other problems:
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. Meinard Kuhlmann (Stanford Encyclopedia of Philosophy): Quantum Field Theory
Quantum field theory sees the world built on an entity called the vacuum which is the source of energy for the creation of fields. The postulated nature of the vacuum gives an energy density to the Universe which is about 10100 times greater than actually observed, the biggest computational error ever devised in physics. Vacuum state - Wikipedia
The field theoreticians Streater and Wightman point out that the classical notion of field originated in attempts to avoid the idea of action at a distance in the description of electromagnetic and gravitational phenomena. They define field: 1. It is observable; 2. it is defined by a set of functions on space-time. Streater & Wightman (2000): PCT, Spin, Statistics and All That
Here we differ, on the way to eventually abandoning quantum field theory as a suitable foundation for theology (see Chapter 26: An alternative to quantum field theory?). This may or not work but I am giving it a go. The way I see the world, spacetime is built on Hilbert space. Chapter 12: Is Hilbert space independent of Minkowski space? explains that spacetime did not exist before the origin of the first elementary particles.
To avoid all these problems I am inclined to avoid QFT. Instead I put the horse in front of the cart and make Hilbert space the foundation of the world. This frees quantum mechanics to work in its native Hilbert space without being distorted by relativity. But then how are we to explain the quantum origin of Minkowski space? The story begins with the creation of particles described in Chapter 16: Gravitation and the creation of dynamic particles.
17.2: Zero-sum bifurcation
The simplest measure of the creation of the Universe is the increase in its entropy. The entropy of the initial singularity is zero. the entropy of the current Universe is enormous. From a statistical point of view, the second law of thermodynamics guarantees the creativity of the Universe.
Here I propose that the Universe grows inside an initial singularity identified as naked gravitation. The initial singularity is also the initial symmetry of the Universe. The system grows more complex by repeatedly breaking this symmetry to create two new entities whose sum is zero. I first apply this idea for the creation of particles described in Chapter 16: Gravitation and the quantum creation of particles. There we imagine quantum mechanics extracting stationary eigenvectors out of the noisy chaos of the kinematic Hilbert space driven by the initial singularity.
These vectors are formal or kinematic, something like the mathematical fields of QFT. Instead of particles being actualized by gaining energy from the vacuum, we imagine that naked gravitation bifurcates into potential and kinetic energy. The kinetic energy goes to converting formal eigenstates into real dynamic particles, fermions and bosons. The particles carry this energy and also the formal vector that defines them. This vector acts a bit like genes in the biological world determining the personality of individual creatures.
The potential energy resulting from this bifurcation becomes part of the gravitational potential well that stabilizes the Universe. It acts rather like a bank, supplying capital to realize an idea (eigenvector). A pendulum is a simple illustration of the interplay between real dynamic energy and gravitational potential energy.
17.3: Bosons and fermions
Chapter 16: Gravitation and the creation of dynamic particles describes the collaborative relationship between gravitation and quantum mechanics. Together they create real living particles selected from Hilbert space by quantum mechanics and endowed with energy from gravity. Modern physics identifies a family of 61 elementary particles which appear regularly in accelerator experiments. These particles are described by the QFT standard model. These particles fall into two groups, bosons and fermions. Elementary particle - Wikipedia
In Minkowski space the quantum of action has the dimensions of angular momentum. All the bosons except the Higgs particle have one quantum of angular momentum known as spin. One full turn of their polarization will bring them back to their original state. The spin of fermions is ½ so they need to make two full turns to reach their original state. This property of space is illustrated by the plate trick. Plate trick - Wikipedia
Darwin called his book The Origin of Species. His evolutionary idea explains the vast web of life that spans our planet. He knew nothing about genetics or the physiology of life but his idea was right. Fossils and our ability to read genes show us evolution in action. Billions of species of have diversified from our last universal common ancestor, the initial singularity of life. I want to take this idea right back to the beginning with a description of the origin of particles. A particle is any entity involved in communication, everything from elementary particles to people, planets and galaxies. Darwin’s theory leads the way with with living particles.
The differentiation of elementary particles into bosons and fermions is another example of zero-sum bifurcation. Bosons, like photons and gluons serve as messengers. Most bosons are massless and travel at the speed of light. Fermions, like electrons and protons serve as structural particles and obey the Pauli exclusion principle, no two identical fermions can occupy the same space. Boson - Wikipedia, Fermion - Wikipedia
17.4: The first particle: the photon
We read the history of life in material memories embodied in fossils and genes. We read the history of the Universe in photons which are in effect frozen fragments of Hilbert space. Special relativity tell us if we could see a photon it would have zero size and its clocks would be stopped. Photon - Wikipedia
Photons have two properties: energy, which determines their frequency and polarization, which is the direction of their spin. We imagine that the different frequencies of photons are represented by vectors selected by quantum mechanics from Hilbert space and endowed with energy by gravitation. They are the cosmic microwave background radiation which brings us information from the time when the Universe was only a few hundred thousand years old. Photon polarization - Wikipedia
17.5: Dirac’s equation, fermions and the exclusion principle
Paul Dirac led quantum mechanics to its first definitive step into special relativity. There space and time must be treated the same way. The space and time version of the Schrödinger equation does not do this. It treats time as linear, simply the symbol t. The momentum operator, on the other hand is quadratic, ∇2. Dirac’s idea was to linearize ∇2 by taking its square root. With a bit of mathematical magic he managed this and created the Dirac equation. Dirac equation - Wikipedia
There are four solutions to this equation which are new mathematical objects called spinors. Two of these could represent the well known negative electrons, fermions with spin-up and spin-down. The other two represent fermions with positive charge. Dirac did not know what these could be, but a solution arrived when Carl Anderson found positive electrons, positrons, in cosmic rays. Positron - Wikipedia
The peculiar property of fermions represented by the plate trick (noted in §17.3 above) is a property of Minkowski spacetime. It is not restricted to electrons (otherwise we could not do it with plates). Space might look simple, but as Atiyah says: the geometrical significance of spinors is still very mysterious. Dirac worked with the original space and time version of Schrödinger’s equation which has continued ever since to be very useful for low energy problems. Michael Atiyah in Peter Goddard (1998): Paul Dirac, The Man and His Work
The state of a particular particle is represented in Hilbert space by rays, Ψ which represent the same state even when they are rotated around the polar complex plane. A restricted version of the Schrödinger equation that deals with time and energy alone works perfectly in Hilbert space, describing these unitary kinematic rotations of rays in the space. The precise phase of a ray at a particular moment becomes relevant if this unitary evolution is interrupted by communication with another particle. This interruption is called measurement or observation, to be described in Chapter 20: Measurement: the interface between Hilbert and Minkowski spaces.
17.6: The bifurcation into spacetime
Standard quantum field theory assumes that Hilbert space is a field laid over the classical spacetime described by special relativity. This Minkowski space is first described in Chapter 4: A new beginning. Minkowski space - Wikipedia
We now guess how quantum mechanics creates Minkowski space. The trick is another application of the zero sum bifurcation described in Chapter 16: Gravitation and the creation of dynamic particles. This time we see the quantum mechanical parameter phase or time bifurcated into real time and space.
There no real space and time in Hilbert space. It is an abstract kinematic space driven in the first instance by the initial singularity. All new particles are children of the initial singularity, each of them associated with the particular Hilbert space from which they were created. All these new particles can reproduce themselves like the the initial singularity, creating a chain reaction like the big bang.
The structure of Minkowski space is determined by its metric. A spacetime interval ds in this space is measured by the equation:
ds2 = dr2 – c2dt2
where r is a vector in three dimensional space. This equation shows that ds = 0 for a particle travelling at the speed of light so that r = ct. This null geodesic is the path followed by massless bosoms, and appears to be outside space and time. This suggests that the null geodesic is in effect a finger of Hilbert space in Minkowski space. Particles travelling at c carry quantum states unchanged from their point of creation to their point of annihilation. These points are generally massive fermions.
Only massless bosons can travel at the speed of light. Massive identical fermions are constrained by the exclusion principle to stay some distance apart, so contributing to the geometric spatial extension of Minkowski space. The appearance of the quantum of action in spacetime pixellates spacetime according to the well-know uncertainty relations ΔE.Δt ≅ Δx.Δp ≅ ℏ. Pauli exclusion principle - Wikipedia
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Notes and references
Further reading
Books
Goddard (1998), Peter , and Stephen Hawking, Abraham Pais, Maurice Jacob, David Olive, and Michael Atiyah, Paul Dirac, The Man and His Work, Cambridge University Press 1998 Jacket: Paul Adrien Maurice Dirac was one of the founders of quantum theory and the aithor of many of its most important subsequent developments. He is numbered alongside Newton, Maxwell, Einstein and Rutherford as one of the greatest physicists of all time.
This volume contains four lectures celebrating Dirac's life and work and the text of an address given by Stephen Hawking, which were given on 13 November 1995 on the occasion of the dedication of a plaque to him in Westminster Abbey. In the first lecture, Abraham Pais describes from personal knowledge Dirac's character and his approach to his work. In the second lecture, Maurice Jacob explains not only how and why Dirac was led to introduce the concept of antimatter, but also its central role in modern particle physics and cosmology. In the third lecture, David Olive gives an account of Dirac's work on magnetic monopoles and shows how it has had a profound influence in the development of fundamental physics down to the present day. In the fourth lecture, Sir Michael Atiyah explains the widespread significance of the Dirac equation in mathematics, its roots in algebra and its implications for geometry and topology.'
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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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Links
Boson - Wikipedia, Boson - Wikipedia, the free encyclopedia, 'In particle physics, bosons are particles with an integer spin, as opposed to fermions which have half-integer spin. From a behaviour point of view, fermions are particles that obey the Fermi-Dirac statistics while bosons are particles that obey the Bose-Einstein statistics. They may be either elementary, like the photon, or composite, as mesons. All force carrier particles are bosons. They are named after Satyendra Nath Bose. In contrast to fermions, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the same place in space.' back |
Dirac equation - Wikipedia, Dirac equation - Wikipedia, the free encyclopedia, 'In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles such as electrons and quarks, for which parity is a symmetry, and is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It accounted for the fine details of the hydrogen spectrum in a completely rigorous way.' back |
Elementary particle - Wikipedia, Elementary particle - Wikipedia, the free encyclopedia, ' In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include the fundamental fermions (quarks, leptons, antiquarks, and antileptons), which generally are "matter particles" and "antimatter particles", as well as the fundamental bosons (gauge bosons and the Higgs boson), which generally are "force particles" that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle.' back |
Fermion - Wikipedia, Fermion - Wikipedia, the free encyclopedia, 'In particle physics, fermions are particles with a half-integer spin, such as protons and electrons. They obey the Fermi-Dirac statistics and are named after Enrico Fermi. In the Standard Model there are two types of elementary fermions: quarks and leptons. . . . In contrast to bosons, only one fermion can occupy a quantum state at a given time (they obey the Pauli Exclusion Principle). Thus, if more than one fermion occupies the same place in space, the properties of each fermion (e.g. its spin) must be different from the rest. Therefore fermions are usually related with matter while bosons are related with radiation, though the separation between the two is not clear in quantum physics. 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 |
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 |
Photon - Wikipedia, Photon - Wikipedia, the free encyclopedia, ' A photon (from Ancient Greek φῶς, φωτός (phôs, phōtós) 'light') is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always move at the speed of light in vacuum, 299792458 m/s . . .. The photon belongs to the class of bosons.' back |
Photon polarization - Wikipedia, Photon polarization - Wikipedia, the free encyclopdia, 'Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photon eigenstates have either right or left circular polarization. A photon that is in a superposition of eigenstates can have linear, circular, or elliptical polarization. . . . The connection with quantum mechanics is made through the identification of a minimum packet size, called a photon, for energy in the electromagnetic field. The identification is based on the theories of Planck and the interpretation of those theories by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.' back |
Plate trick - Wikipedia, Plate trick - Wikipedia, the free encyclopedia, ' In mathematics and physics, the plate trick, also known as Dirac's string trick (after Paul Dirac, who introduced and popularized it), the belt trick, or the Balinese cup trick, is any of several demonstrations of the idea that rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while a second rotation of 360 degrees, a total rotation of 720 degrees, does.Mathematically, it is a demonstration of the theorem that SU(2) (which double-covers SO(3)) is simply connected. To say that SU(2) double-covers SO(3) essentially means that the unit quaternions represent the group of rotations twice over. A detailed, intuitive, yet semi-formal articulation can be found in the article on tangloids. back |
Positron - Wikipedia, Positron - Wikipedia, the free encyclopedia, ' The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 e, a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides with an electron, annihilation occurs. If this collision occurs at low energies, it results in the production of two or more photons. Positrons can be created by positron emission radioactive decay (through weak interactions), or by pair production from a sufficiently energetic photon which is interacting with an atom in a material.' back |
Richard P. Feynman (1965), Nobel Lecture: The Development of the Space-Time View of Quantum Electrodynamics, Nobel Lecture, December 11, 1965: We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn’t any place to publish, in a dignified manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing. Since winning the prize is a personal thing, I thought I could be excused in this particular situation, if I were to talk personally about my relationship to quantum electrodynamics, rather than to discuss the subject itself in a refined and finished fashion. Furthermore, since there are three people who have won the prize in physics, if they are all going to be talking about quantum electrodynamics itself, one might become bored with the subject. So, what I would like to tell you about today are the sequence of events, really the sequence of ideas, which occurred, and by which I finally came out the other end with an unsolved problem for which I ultimately received a prize.' back |
Spin (physics) - Wikipedia, Spin (physics) - Wikipedia, the free encyclopedia, 'In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. |
Spin-statistics theorem - Wikipedia, Spin-statistics theorem - Wikipedia, the free encyclopedia, 'In quantum mechanics, the spin–statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum (that is, the contribution to the total angular momentum that is not due to the orbital motion of the particle). All particles have either integer spin or half-integer spin (in units of the reduced Planck constant ħ). The theorem states that: The wave function of a system of identical integer-spin particles has the same value when the positions of any two particles are swapped. Particles with wave functions symmetric under exchange are called bosons. The wave function of a system of identical half-integer spin particles changes sign when two particles are swapped. Particles with wave functions antisymmetric under exchange are called fermions.' back |
Vacuum state - Wikipedia, Vacuum state - Wikipedia, the free encyclopedia, 'In quantum field theory, the vacuum state (also called the vacuum) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. Zero-point field is sometimes used as a synonym for the vacuum state of an individual quantized field.' back |