vol 3: Development
chapter 3: Physics
page 10 Boson and fermion
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Boson and Fermion
Introduction
The fundamental element of quantum mechanical configuration space is the state, represented by a state vector in Hilbert space. From an abstract point of view all quantum states are identical, differentiated only by their state vectors. We might think of a state as an element of memory and the state vector as its address.
Quantum states represent possibilities which are realized when a state is 'excited' by energy. Such a realization of a possible state is a particle. Just as quantum states may have any measure of complexity, so particles may range from the fundamental particles like electrons and photons studied by physics to people, planets and beyond.
To model the universe, we have constructed an isolated quantum system with energy within which we are mathematically guaranteed at least one point that do not move, a fixed point. Energy, Fixed points
We now introduce communication by imagining our isolated quantum system, while remaining itself isolated, has split internally into a number of distinct partitions which communicate by exchanging messages with one another.
The partitions correspond to processes in a network and the messages to particles exchanged between the processes. Let us guess that the nodes of the network correspond to fermions and its connections to bosons. Fermion - Wikipedia, Boson - Wikipedia
Bosons
We imagine that bosonic behaviour dates from the epoch when the whole universe could be represented by a one dimensional line beginning at the initial singularity and parametrized by a number called proper time.
This universe has broken the symmetry of pure action into energy and time, which are related by the equation action = energy.time. This universe has time, energy and frequency, but no interior spatial extent and, of course, nothing outside.
This is consistent with the tendency of bosons to cluster in the same quantum mechanical state. In the current universe, there are two massless bosons, the photon and the graviton, which both move through space at the velocity of light, c.
At this epoch bosons are distinguished by their energy or frequency, forming a spectrum of frequencies running from zero to 'countable infinity. A quantum mechanical model approximating this spectrum is the quantum harmonic oscillator. Quantum harmonic oscillator - Wikipedia At this epoch also, since there is no transmission of messages through space, there are no observers and no possibility of error. This suggests that there is no need for quantization. Why is the observable universe universe quantized?
Fermions
Imagine that the next break in universal symmetry came with the advent of fermions in a two dimensional space-time, one dimension of time and one dimension of space. We guess that fermions, two dimensional spacetime and special relativity are all aspects of the fermion layer of the universal network which came into existence 'simultaneously'. While quantum mechanics alone is capable of describing the one dimensional space of bosons, we need quantum field theory to understand fermions. Quantum field theory is the union of special relativity, which models flat space-time, and quantum mechanics.
The formal quantum mechanical distinction between bosons and fermions lies in the behaviour of their wave functions. Feynman writes:
If a process involved two particles that are identical, reversing the one which arrives at a counter is an alternative which cannot be distinguished -- and like all cases of alternatives that cannot be distinguished -- interferes with the original, unexchanged case. The amplitude for the event is then the sum of the two interfering amplitudes, but interestingly enough, the interference is in some cases with the same phase, and in others with the opposite phase. Feynman page 4-1
Bosons interfere with the same phase, fermions with the opposite phase. It follows that many bosons tend can occupy the same state, but a given state can be occupied by one fermion only. This, the Pauli exclusion principle explains the stationary structures of the universe such as the electronic structure of atoms and everything built from atoms (ie everything we experience). Pauli exclusion principle - Wikipedia For electrons are fermions: two cannot occupy the same state.
One may see the Pauli exclusion principle as the creator of space. We may have many bosons in the same state, but if we are to have two fermions existing simultaneously, we need a space of two states, one of which is not the other.
Space, spin and statistics
Bosons and fermions not only have different statistics, but they also differ in a feature called spin. The 'spin-statistics' theorem tells us that bosons have integral spins like . . . -1, 0, +1, . . . , and fermions have half-integral spins ... -1/2, +1/2, . . . . Spin-statistics theorem - Wikipedia, John Baez Furthermore, assemblies of particles are bosons or fermions depending on the sum of the spins of their constituents.
Bosons and Fermions in the network
From a network points of view, bosons serve as messages and the fermions as algorithmic processors, moving from state to state as they send and receive messages. Here we interpret the identity of particles as the identity of processes, so that we can interpret identical particles as different instances of the same process.
These two different classes of particle seem to be the first step toward structure of the cosmic network. The fermion - boson dichotomy thus corresponds to the analysis of networks into processing elements (wires, computer chips, fermions) and transport elements (messages, bosons).
There are many different fermions and bosons in the standard model. Standard model - Wikipedia Here we mention only one of each, the photon and the electron.
The photon and the electron
The photon is the most obvious boson in our lives, bringing light and energy from the sun, forming the physical carrier of much of our everyday communication (eg sight, wireless). Communication through photons binds atoms, molecules and large structures like our bodies and the earth into a coherent whole. Photon - Wikipedia
Photons couple (communicate) with electrically charged particles, such as the proton and the electron. Proton - Wikipedia, Electron - Wikipedia Photons and electrons may collide with one another in free space or they may be bound together with protons and neutrons to form atoms.
In an atom, negatively charged electrons move in a potential space created by positively charged protons. Every electronic transition is accompanied by the absorption or emission of a photon, revealing, through photon frequencies (= energies) the intricate interior structure of the atom. Even the simplest atom can exist in a countable infinity of states and emit and absorb a corresponding spectrum of photons.
New states
From the quantum mechanical point of view, the interaction (communication) of two particles is represented by state vectors in a Hilbert space which is the tensor product of the Hilbert spaces of the individual particles.
In the layered network model, we imagine higher layers being growing out of lower layers through the control of communication in the lower layers by the emergent higher layers. On this picture, we might consider bosons as the layer below fermions, used by fermions to execute their processes. The emergence of fermions may be seen as the emergency of a two dimensional spacetime from a one dimensional state characterized by energy and time.
We can construct the tensor product space by replacing every point of one space with the whole of the other space, thus creating something much bigger than the sum of the two states.
Communication modelled as the interaction of fermion and boson states is thus associated with the complexification of the universe.
(revised 23 November 2008)
Further reading
Books
| Feynman, Richard P, and Robert B Leighton, Matthew Sands, The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Addison Wesley 1970 Foreword: 'This set of lectures tries to elucidate from the beginning those features of quantum mechanics which are the most basic and the most general. ... In each instance the ideas are introduced together with a detailed discussion of some specific examples - to try to make the physical ideas as real as possible.' Matthew Sands Amazon back |
| Schwinger, Julian, and (editor), Selected Papers on Quantum Electrodynamics, Dover 1958 Jacket: In this volume the history of quantum electrodynamics is dramatically unfolded through the original words of its creators. It ranges from the initial successes, to the first signs of crisis, and then, with the stimulus of experimental discovery, the new triumphs leading to an unparalleled quantitative accord between theory and experiment. In terminates with the present position in quantum electrodynamics as part of the larger subject of theory of elementary particles, faced with fundamental problems and future prospect of even more revolutionary discoveries.' Amazon back |
| Tomonaga, Sin-itiro, The Story of Spin, University of Chicago Press 1997 Jacket: 'The Story of Spin, as told by Sin-itiro Tomonaga and lovingly translated by Takeshi Oka, is a brilliant and witty account of the development of modern quantum theory, which takes electron spin as a pivotal concept. Reading these twelve lectures on the fundamental aspects of physics is a joyful experience that is rare indeed.' Laurie Brown, Northwestern University. Amazon back |
Papers
| Pauli, W, "The Connection between Spin and Sratistics", Physical Review, 58, , 1940, page 716. 'In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a spacelike distance are commutable, the necessity of Bose-Einstein statitics with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classes which have commutable multiplication like +1, -1, +e, -e, with e2 = 1.. back |
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 |
| Classical mechanics - Wikipedia Classical mechanics - Wikipedia, the free encyclopedia 'Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains, and is one of the oldest and largest subjects in science and technology.' back |
| Doppler effect - Wikipedia Doppler effect - Wikipedia 'The Doppler effect, named after Christian Doppler, is the change in frequency and wavelength of a wave as perceived by an observer moving relative to the source of the waves. For waves that propagate in a wave medium, such as sound waves, the velocity of the observer and of the source are relative to the medium in which the waves are transmitted. The total Doppler effect may therefore result from motion of the source, motion of the observer, or motion of the medium. Each of these effects is analysed separately. For waves which do not require a medium, such as light or gravity in special relativity, only the relative difference in velocity between the observer and the source needs to be considered.' back |
| Electron - Wikipedia Electron - Wikipedia, the free encyclopedia 'The electron is a fundamental subatomic particle that carries a negative electric charge. It is a spin ? lepton that participates in electromagnetic interactions, and its mass is approximately 1 / 1836 of that of the proton. Together with atomic nuclei, which consist of protons and neutrons, electrons make up atoms. Their interaction with adjacent nuclei is the main cause of chemical bonding.' 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 |
| John Baez Spin, Statistics, CPT and All That Jazz 'This is a little grab-bag of proofs of the spin-statistics theorem. Quantum mechanics says that if you turn a particle around 360°, its wavefunction changes by a phase of either +1 (that is, not at all) or -1. It also says that if you interchange two particles of the same type, their joint wavefunction changes by a phase of +1 or -1. The spin-statistics theorem says that these are not independent choices: you get the same phase in both cases! The phase you get by rotating a particle is related to its spin, while the phase you get by switching two goes by the funny name of "statistics". The spin-statistics theorem says how these are related. The theorem lays out two possibilities. Some particles change phase by +1 when you rotate one by 360° or switch two of them. These are called bosons. They include photons, the W and Z boson, and gluons. Others change phase by -1 when you rotate one by 360° or switch two of them. These are called fermions. They include protons, neutrons, electrons and neutrinos..' back |
| Maxwell's Equations - GSU Maxwell's Equations - Hyperphysics 'Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory treatment of the subject, except perhaps as summary relationships.' back |
| Maxwell's equations - Wikipedia Maxwell's equations - Wikipedia, the free encyclopedia 'In classical electromagnetism, Maxwell's equations are a set of four equations that describe the properties of the electric and magnetic fields and relate them to their sources, charge density and current density. Maxwell used the equations to show that light is an electromagnetic wave.' back |
| Photon - Wikipedia Photon - Wikipedia, the free encyclopedia ;In physics, the photon is an elementary particle, the quantum of the electromagnetic field and thus the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force.' 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 [1], and a mass of 938.27231(28) MeV/c2 (1.6726 ? 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 harmonic oscillator - Wikipedia Quantum harmonic oscillator - Wikipedia, the free encyclopedia 'The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it is one of the few quantum mechanical systems for which a simple exact solution is known.' back |
| Quantum mechanics - Wikipedia Quantum mechanics - Wikipedia, the free encyclopedia 'Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics. The effects of quantum mechanics are typically not observable on macroscopic scales, but become evident at the atomic and subatomic level. Quantum theory generalizes all classical theories, including mechanics and electromagnetism, and provides accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits.' back |
| Special relativity - Wikipedia Special relativity - Wikipedia, the free encyclopedia 'This theory has a wide range of consequences which have been experimentally verified. Special relativity overthrows Newtonian notions of absolute space and time by stating that time and space are perceived differently by observers in different states of motion. It yields the equivalence of matter and energy, as expressed in the mass-energy equivalence formula E = mc2, where c is the speed of light in a vacuum. The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared to the speed of light.' back |
| Spin-statistics theorem - Wikipedia Spin-statistics theorem - Wikipedia, the free encyclopedia 'The theorem states that: o The wave functions of a system of identical integer-spin particles, spin 0, 1, 2, 3, has the same value when the positions of any two particles are exchanged. Particles with wavefunctions symmetric under exchange are called bosons. o The wave functions of a system of identical half-integer-spin s = 1/2, 3/2, 5/2, are anti-symmetric under exchange, meaning that the wavefunction changes sign when the positions of any pair of particles are swapped. Particles whose wavefunction changes sign are called fermions.' 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 |
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