vol III: Development

chapter 4: Physics

page 11: Quantum field theory

From quantum mechanics to quantum field theory

People have long believed that invisible influences control the visible world. For some these may be gods or demons, but for physicists they are fields. We are all familiar with the gravitational field, the source of the force that pulls us toward the Earth. We feel this force wherever we go. Everything wants to fall down, that is toward the centre of the earth.

Gravitation was studied in detail by Isaac Newton. He studied the motions of the planets using data and models assembled by people such as Ptolemy, Copernicus and Kepler. Newton concerned himself only with the motions of the planets without speculating about the influence that kept them in their orbits. Whatever its nature, he found that the gravitational attraction between any two bodies was proportional to the product of their masses divided by the square of the distance between them. As fas as we know, every form of energy in the universe, including gravitation itself, is feels gravitation. Ptolemy - Wikipedia, Nicolaus Copernicus - Wikipedia, , Isaac Newton - Wikipedia

The other main field of interest to early physicists was electromagnetism. Michael Faraday was the first to use the field concept to describe the phenomena he observed. He used iron filings to visualize the field around a magnet in 1851. Later James Clerk Maxwell applied calculus to the dynamics of the electromagnetic field to show that light and radiation in general are electromagnetic phenomena. Classical electromagnetism - Wikipedia, Michael Faraday - Wikipedia, Michael Faraday, James Clerk Maxwell - Wikipedia

Quantum field theories may be seen as modern versions of the theories of Plato and Aristotle. Plato imagined an invisible space of forms which determined the structure of the the visible world. Particle physicists imagine set of fields filling all of space and time, each corresponding to a species of observed particle. Aristotle understood motion and change to be the embodiment of different forms in the same matter. He called this generation and corruption. Physicists replace matter with energy, form with field and imagine particles as fields embodied with energy. They speak of the creation and annihilation of particles when the relevant fields become energized and de-energized. Theory of Forms - Wikipedia, Hylomorphism - Wikipedia

The difference between Aristotle and modern physicists, of course, is that 2500 years of scientific development have put ancient ideas in a new context. We now use mathematical techniques to model the nature of the fields and the particles they create and the frequencies of creation and annihilation. Quantum field theory is an attempt to produce a marriage quantum mechanics and relativity which is both formally consistent and consistent with observation.

From classical mechanics to quantum mechanics

Quantum mechanics, like Newtonian mechanics, provides a general paradigm for the study of motion. Newtonian mechanics struck trouble when it came to deal with electromagnetism. This problem ultimately led to the development of quantum theory. Quantum theory, in its turn, revealed itself to be very difficult to harmonize with Einstein's special theory of relativity. Quantum field theory has evolved to deal with this difficulty and is the foundation with a fairly comprehensive picture of how the Universe might work at a fundamental level, the Standard Model. Standard model - Wikipedia

Classical mechanics and classical electrodynamics began to meet in 1860 when Gustav Kirchoff proposed his law of thermal radiation: that there should be a universal function relating the spectrum of the radiation emitted by a hot body to its temperature. Early efforts based on classical theory has led to equations that worked at low temperatures but failed at high temperatures and vice versa. It was not until 1900 that Max Planck found a formula that fitted the data across the observed spectrum. In his search for this relationship, he was forced, as 'an act of despair' to assume that the energy and frequency of radiation have a fixed relationship expressed by the equation E = hf. h has since become known as Planck's constant. This was the first step toward quantum mechanics. Gustav Kirchoff - Wikipedia, , Ultraviolet catastrophe - Wikipedia, Max Planck - Wikipedia

Initially quantum mechanics, like classical mechanics, dealt with the dynamics of durable particles or mass points. Initial studies were concerned with the electronic structure of atoms based on spectroscopic information. Using the energy of the radiation emitted and absorbed by atoms, physicists sought to understand the electronic structure of atoms. Once is became clear that radiation comprised discrete particles (photons), interest turned to studying the frequencies of emission (creation) and absorption (annihilation) of photons as well as the frequencies of the photons themselves. Classical mechanics - Wikipedia

This was a change of emphasis that led to the difficult problems that quantum field theory has helped to solve. How are material particles created and annihilated? The answer lies in Einstein's discovery that mass and energy are equivalent, the constant of proportionality being the velocity of light, c in the equation E = mc². This made it possible for material particles to be created by the energy in a field, and for the energy of annihilated particles to return to the field. Since fields also carry action and momentum, they provide an invisible reservoir to maintain the conservation laws during particle creation and annihilation.

Quantum mechanics and special relativity

The development of quantum mechanics took quite some time from Planck's first discovery, but by the late 1920s a clear and consistent mathematical formalism had been developed, although there was much debate about its interpretation. This formalism led to many difficulties when confronted with the finite velocity of light and special relativity. Quantum mechanics - Wikipedia

The Dirac equation

An important expression of quantum dynamics is the Schrödinger equation which is a generalizes Planck-Einstein equation E = ℏω to include many states of energy and momentum. It is a partial differential equation that serves as a fixed point in quantum mechanics analogous to Newton's second law in classical mechanics. The Schrödinger equation describes the deterministic evolution of the quantum wave function in a given situation, but this function is not observable. Max Born found that the absolute square of the quantum amplitude was the probability of observing the state described by the wave function. This is a universal feature of quantum theory and the foundation of Einstein's feeling that quantum mechanics is incomplete, since it does not make deterministic predictions, as classical theories often do. Schrödinger equation - Wikipedia, Einstein, Podolsky and Rosen

The Schrödinger equation is not invariant under Lorentz transformations, that is it does not take special relativity into account. A significant step toward harmonizing quantum mechanics and special relativity was taken by Paul Dirac. The Dirac equation does not change under relativistic (Lorentz) transformations and predicts the existence of antimatter. Dirac equation - Wikipedia, Antimatter - Wikipedia, Lorentz transformation - Wikipedia

The special theory of relativity requires that space and time be treated on the same basis. Schrödinger's differential equation, when applied to space and time, contains a first order time derivative and a second order space derivative so it cannot fulfill this requirement. Dirac sought a way to avoid this difference. Special relativity - Wikipedia

Dirac started with the relativistic equation which combines energy and momentum to give the four momentum E²/c² - p² = m²c². Klein and Gordon had converted this into a relativistic wave equation by making the usual quantum mechanical substitutions E = i∂/∂t and p = −i∇. The result is a second order equation in both space and time. Unfortunately it admits solutions with negative probability, which appear unrealistic. It is used now to describe particles with spin 0. The particles of interest to quantum electrodynamics, the first quantum field theory, are the photon with spin 1 and the electron with spin ½. Klein-Gordon equation - Wikipedia

Dirac overcame this problem by producing differential equation first order in space and time which was in effect the square root of the Klein Gordon equation. The result was a wave function ψwith four components which he wrote

(βmc² + (∑₁³α_np_n))ψ(x, t) = iℏ∂ψ/∂t

The trick to successfully extracting the square root lies in the matrices (now known as the gamma matrices) α and β (see the Wikipedia page on the Dirac equation cited above for details).

The four components of the wave function were found to represent the two spin states of an electron and a positron. Although Dirac was unaware of it at the time he used them, the gamma matrices are a representation of a Clifford algebra which couples Minkowski space-time to the Dirac spinors which represent the electron and the positron. This establishes a close relationship between spin ½ particles and the structure of space-time. Gamma matrices - Wikipedia, Minkowski space - Wikipedia, Clifford algebra - Wikipedia

From relativistic quantum mechanics to quantum field theory

The Dirac equation was a giant step forward, but not perfect. It gave very good results for the energy of the hydrogen electron, but improved measurement technology revealed the 'Lamb shift' which the equation did not account for. Lamb shift - Wikipedia

The Lamb shift revealed that the properties of a particle are somewhat dependent on its environment. The properties of an electron bound in an atom are found to differ slightly from the properties of a free electron. The explanation for this has come to be known as 'vacuum energy'. The bound electron was understood to be coupling to virtual particles arising from the vacuum which changed its properties.

Bosons and fermions

Bosons and fermions are distinguished by a property called spin, which has the dimensions of action or angular momentum. Bosons have integral spin in units of Planck's constant and fermions have half integral spin.

Here we think of each field as a communication network defined by a certain communication protocol. The overall structure of the Universe may be conceived as a network of fermions communicating by exchanging bosons with one another. Fermions define the structure of the universe through the exclusion principle, which says no two fermions can occupy one quantum state. They must spread out, like the electrons in an atom. Bosons, on the other hand, tend to attract one another into the same state. In the universal network, fermions serve as sources, bosons as messages. Fermion - Wikipedia, Boson - Wikipedia, Spin-statistics theorem - Wikipedia, John Baez: Spin, statistics

The quantum harmonic oscillator

The development of black body theory after Planck's initial discovery centered around a quantum mechanical structure known as the quantum harmonic oscillator. The quantum harmonic oscillator explains the spectrum of black body radiation using the properties of the photon, which is a boson.

The classical harmonic oscillator, like a pendulum, is a mass m oscillating around an equilibrium point, x = 0. The mass feels a force toward equilibrium that varies with its distance from equilibrium, so we write F = −kx. This is a simple example of linear feedback, which leads, in the absence of friction, to a continued oscillation with fixed amplitude and frequency once the oscillator is set in motion. The classical harmonic oscillator may also rest in its equilibrium position, like a pendulum hanging straight down. Harmonic Oscillator - Wikipedia

The quantum harmonic oscillator developed by analogy to the classical oscillator. The Schrödinger equation for the quantum oscillator can be solved exactly and reveals interesting properties. While the classical oscillator may have a continuous range of energies, the quantum oscillator has a fixed 'ladder' of energies, one for each value of n: (n +½)ℏω. Consequently it does not have an equilibrium point with zero energy. The minimum energy, called zero point energy, is ½ℏω when n = 0. The 'restoring force' in a classical oscillator is gravitation, in the case of the pendulum, or the elasticity of a spring or other elastic medium. In the quantum oscillator, it is the tendency for bosons to be attracted to the same state, so that the black body spectrum has a maximum at a certain frequency and tends to zero at high and low temperatures. Quantum harmonic oscillator - Wikipedia, Zero-point energy - Wikipedia

Vacuum energy

Quantized fields may be imagined as infinite superpositions of quantum oscillators. Excited fields (particles) may have more than one quantum of energy, but the fact that there is a both minimum energy for a quantum oscillators and an infinity of them raises that 'cosmological constant problem': the energy of the vacuum would seem to be exceedingly large. Observation does not support this view, and many authors have sought a way to avoid the problem, but it probably remains an open question. Stephen Weinberg, Vacuum energy - Wikipedia

The path integral in quantum theory

Richard Feynman made many contributions to the development of quantum field theory. With Sin-Itiro Tomonaga and Julian Schwinger, he won the 1965 physics Nobel Prize for his contribution to quantum electrodynamics. Feynman's approach is built around two techniques, the path integral method and feynman diagrams. Feynman sees the path integral as a third way of doing quantum mechanics, different from the methods of Schrödinger and Heisenberg, but equivalent to them. Richard P. Feynman: Nobel lecture

The Feynman path integral approach to quantization of classical fields is followed here because it implements Hamilton's principle which is in turn an implementation of the principle of natural selection. Since we are working on the assumption that the Universe is everything, we can admit no outside constraints: all the constraints that determine the structure of the universe must arise within it, all derived ultimately from the requirement that the universe be consistent. Path integral formulation - Wikipedia

The Lagrangian approach to mechanics satisfies the feeling that the universe has been constructed by the counsel and dominion of an intelligent and powerful being, as Newton felt. Here we work on the assumption that this being is the universe itself and that a process of evolutionary selection explains the structure of a self made universe. The path integral method is an algorithm to select the actual path taken from one point to another from the infinity of possible paths available in a continuous space-time. Isaac Newton: General Scholium, Yourgrau & Mandelstam: Variational Principles

The path integral formulation is based on two features of quantum mechanics. The first is superposition. Probability amplitudes belonging to indistinguishable events add. The second couples sequential events. The probability amplitude associated with a sequence of events is obtained by multiplying the amplitudes for the individual events. These features are discussed in detail by Feynman in his explorationof the two slit experiment. Feynman, Leighton & Sands FLP III:01

The Planck-Einstein relationship E = ℏω implies that each cycle of a quantum amplitude is equivalent to one quantum of action. The path integral approach, as Zee suggests, imagines an infinity of screens with so many slits that the screens disappear. The action along every possible path is is computed by multiplying the actions of infinitesimal steps action along every possible path from the initial to the final state and adding them to obtain the amplitude of the transition from the initial to the final state. The computation of the action involves the relevant wavefunction (Schrödinger, Dirac) in the computation of each path. This approach, as in all other approaches to quantum theory, assumes that the spacetime manifold is smooth and differentiable. Zee

Feynman diagrams

The path integral enables one to compute the probability amplitude for a certain event. An event, however, may involve many events, which may be represented graphically by diagrams invented by Feynman. At high energies, many particles may be created and annihilated in the course of a single event of interest. The assumption that the space-time domain of the quantum fields is continuous may introduce infinities into the situation, so that determining the final probability amplitude for an event may involve much computation. Feynman diagram - Wikipedia, Flip Taneda: Lets draw Feynman diagrams

Feynman diagrams plot the courses of both real observable particles and the unobservable 'virtual particles' which are believed to mediate the interactions of the real particles. The ephemeral existence of virtual particles is made possible by the quantum uncertainty principle, which allows momentary violations of the conservation laws of action, energy and momentum as long as the laws are observed in the longer run. Virtual particle - Wikipedia

Gauge symmetries and the Standard Model

Science attempts to find the invariants or fixed points in the dynamics of the world which can be written down. In the past such fixed points have been conceived as laws of nature, but the modern view sees them as symmetries. The 'Standard Model' which embraces almost all particle phenomena is founded on a comprehensive set of symmetries known a gauge symmetries. It has been found that the requirement that field theory be invariant under gauge transformations can be used to completely determine the dynamics of quantum gauge fields.

In physics a symmetry is said to exist exist when a formal and invisible change to a system produces no observable change. This situation points to redundancy in the formalism.

Quantum field theory takes the space-time domain described by special relativity as given, and constructs its fields upon this domain. Space-time provides three foundational symmetries which the particles arising from these fields are observed to honour: Conservation of action / spin: ie symmetry with respect to rotation; conservation of energy: symmetry with respect to the flow of time; and conservation of momentum: symmetry with respect to spatial position. Action, energy and momentum are observable, but many features of fields are not and must be guessed by modellers to see if they yield results that correspond to observation. The principal 'unobservables' in quantum mechanics are probability amplitudes and virtual particles. Probability amplitude - Wikipedia

The absolute phase of the wave function cannot be observed. All we can observe are the effects of phase differences between two systems observing each other. The overall phases in a system can arise from any arbitrary starting point without changing the probabilities of events predicted by the quantum theory. This is called a global gauge invariance or global symmetry. Gauge theory - Wikipedia

In contrast to the global gauge symmetry of quantum mechanics, quantum electrodynamics is built on the notion of local gauge symmetry, which allows phase locally. To maintain causality, however, local changes are attributed to the emission of absorption of a gauge particle, which in electrodynamics is the photon. A similar approach has proven successful in the theories of the weak and strong forces.

The four fields and gauge theory

There are four fields, called gravitation, electromagnetic, strong and weak. Gravitation is responsible for the overall structure of the Universe described by the general theory of relativity. The electromagnetic field, carried by the photon, and its interactions with electrically charged matter, is described by quantum electrodynamics. Quantum electrodynamics explains most of our everyday experience other than the effects of gravitation. Both gravitation and electromagnetic fields have unlimited range, and so can be sensed at the macroscopic level. We are all familiar with the pull of gravity which keeps us on the earth, and have played with magnets, so feeling an electromagnetic field. Peskin & Schroeder, Electromagnetic force - Wikipedia, Gravitation - Wikipedia

The other two fields, strong and weak, have extremely short range, but are nevertheless essential to the functioning of the Universe. Quantum field theory shows how the particles of the Universe form a complete set, spanning all possible communications of all possible information. Strong interaction - Wikipedia, Weak interaction - Wikipedia

(revised 16 June 2016)

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Feynman, Richard, QED: The Strange Story of Light and Matter, Princeton UP 1988 Jacket: 'Quantum electrodynamics - or QED for short - is the 'strange theory' that explains how light and electrons interact. Thanks to Richard Feynmann and his colleagues, it is also one of the rare parts of physics that is known for sure, a theory that has stood the test of time. . . . In this beautifully lucid set of lectures he provides a definitive introduction to QED.'  Amazon   back

Peskin, Michael E, and Dan V Schroeder, An Introduction to Quantum Field Theory, Westview Press 1995 Amazon Product Description 'This book is a clear and comprehensive introduction to quantum field theory, one that develops the subject systematically from its beginnings. The book builds on calculation techniques toward an explanation of the physics of renormalization.'   Amazon   back

Yourgrau, Wolfgang, and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory, Dover 1979 Variational principles serve as filters for parititioning the set of dynamic possibilities of a system into a high probability and a low probability set. The method derives from De Maupertuis (1698-1759) who formulated the principle of least action, which states that physical laws include a rule of economy, the principle of least action. This principle states that in a mathematically described dynamic system will move so as to minimise action. Yourgrau and andelstam explains the application of this principle to a variety of physical systems.   Amazon   back

Zee, Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2003 Amazon book description: 'An esteemed researcher and acclaimed popular author takes up the challenge of providing a clear, relatively brief, and fully up-to-date introduction to one of the most vital but notoriously difficult subjects in theoretical physics. A quantum field theory text for the twenty-first century, this book makes the essential tool of modern theoretical physics available to any student who has completed a course on quantum mechanics and is eager to go on. Quantum field theory was invented to deal simultaneously with special relativity and quantum mechanics, the two greatest discoveries of early twentieth-century physics, but it has become increasingly important to many areas of physics. These days, physicists turn to quantum field theory to describe a multitude of phenomena. Stressing critical ideas and insights, Zee uses numerous examples to lead students to a true conceptual understanding of quantum field theory--what it means and what it can do. He covers an unusually diverse range of topics, including various contemporary developments,while guiding readers through thoughtfully designed problems. In contrast to previous texts, Zee incorporates gravity from the outset and discusses the innovative use of quantum field theory in modern condensed matter theory. Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used.   Amazon   back

Links

Antimatter - Wikipedia, Antimatter - Wikipedia, the free encyclopedia, 'In particle physics, antimatter is material composed of antiparticles, which have the same mass as particles of ordinary matter but have opposite charge and other particle properties such as lepton and baryon number, quantum spin, etc. Encounters between particles and antiparticles lead to the annihilation of both, giving rise to varying proportions of high-energy photons (gamma rays), neutrinos, and lower-mass particle–antiparticle pairs. . . . There is considerable speculation as to why the observable universe is apparently composed almost entirely of ordinary matter, as opposed to a more symmetric combination of matter and antimatter. This asymmetry of matter and antimatter in the visible universe is one of the greatest unsolved problems in physics.[2] The process by which this asymmetry between particles and antiparticles developed is called baryogenesis.' back

Born rule - Wikipedia, Born rule - Wikipedia, the free encyclopedia, 'The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see bra-ket notation), then the measured result will be one of the eigenvalues λ of A, and the probability of measuring a given eigenvalue λi will equal <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'. back

Circle group - Wikipedia, Circle group - Wikipedia, the free encyclopedia, 'In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane or simply the unit complex numbers.' back

Classical electromagnetism - Wikipedia, Classical electromagnetism - Wikipedia, the free encyclopedia, 'Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides an excellent description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible.' 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

Clifford algebra - Wikipedia, Clifford algebra - Wikipedia, the free encyclopedia, 'In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English geometer William Kingdon Clifford. 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

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

Eigenvalues and eigenvectors - Wikipedia, Eigenvalues and eigenvectors - Wikipedia, the free encyclopedia, 'An eigenvector of a square matrix A is a non-zero vector vthat, when the matrix multiplies yields a constant multiple of v, the latter multiplier being commonly denoted by λ. That is: Av = λv' back

Einstein, Podolsky and Rosen, 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

Electromagnetic force - Wikipedia, Electromagnetic force - Wikipedia, the free encyclopedia, 'In physics, the electromagnetic force is the force that the electromagnetic field exerts on electrically charged particles. It is the electromagnetic force that holds electrons and protons together in atoms, and which hold atoms together to make molecules. The electromagnetic force operates via the exchange of messenger particles called photons and virtual photons. The exchange of messenger particles between bodies acts to create the perceptual force whereby instead of just pushing or pulling particles apart, the exchange changes the character of the particles that swap them.' 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

Feynman diagram - Wikipedia, Feynman diagram - Wikipedia, the free encyclopedia, 'In quantum field theory a Feynman diagram is an intuitive graphical representation of a contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory' back

Feynman, Leighton & Sands FLP III:01, Quantum Behaviour, 'The gradual accumulation of information about atomic and small-scale behavior during the first quarter of the 20th century, which gave some indications about how small things do behave, produced an increasing confusion which was finally resolved in 1926 and 1927 by Schrödinger, Heisenberg, and Born. They finally obtained a consistent description of the behavior of matter on a small scale. We take up the main features of that description in this chapter.' back

Flip Taneda, Let's draw Feynman diagrams, 1. Let’s draw Feynman diagrams! (this post) 2. More Feynman diagrams. 3. Introducing the muon. 4. The Z boson and resonances. 5. Neutrinos. 6. The W boson, mixing things up. 7. Meet the quarks. 8. World of glue. 9. QCD and confinement. 10. Known knowns of the Standard Model. (summary) 11. When Feynman Diagrams Fail. 12. An idiosyncratic introduction to the Higgs. 13. A diagrammatic hint of masses from the Higgs 14. Higgs and the vacuum: Viva la “vev” 14. Helicity, Chirality, Mass, and the Higgs 16. The Birds and the Bs 17. The spin of gauge bosons 18. Who ate the Higgs? 19. Unitarization of vector boson scattering 20. Private lives of Standard Model particles (summary) back

Gamma matrices - Wikipedia, Gamma matrices - Wikipedia, the free encyclopedia, 'In mathematical physics, the gamma matrices, {γ0, γ1, γ2,γ3}, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(R). When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of space time acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate space-time computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-1/2particles.' back

Gauge boson - Wikipedia, Gauge boson - Wikipedia, the free encyclopedia, 'In particle physics, a gauge boson is a force carrier, a bosonic particle that carries any of the fundamental interactions of nature, commonly called forces.[1][2] Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge bosons—usually as virtual particles' back

Gauge theory - Wikipedia, Gauge theory - Wikipedia, the free encyclopedia, 'In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations. . . . The term gauge refers to redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field.' back

General relativity - Wikipedia, General relativity - Wikipedia, the free encyclopedia, 'General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916.[1] It is the current description of gravitation in modern physics. General relativity generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.' back

Gravitation - Wikipedia, Gravitation - Wikipedia, the free encyclopedia, 'Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped. Gravitation causes dispersed matter to coalesce, and coalesced matter to remain intact, thus accounting for the existence of the Earth, the Sun, and most of the macroscopic objects in the universe.' back

Gustav Kirchoff - Wikipedia, Gustav Kirchoff - Wikipedia, the free encyclopedia, 'Gustav Robert Kirchhoff (12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He coined the term "black body" radiation in 1862, and two different sets of concepts (one in circuit theory, and one in spectroscopy) are named "Kirchhoff's laws" after him; there is also a Kirchhoff's Law in thermochemistry. The Bunsen–Kirchhoff Award for spectroscopy is named after him and his colleague, Robert Bunsen.' back

Hamilton's principle - Wikipedia, Hamilton's principle - Wikipedia, the free encyclopedia, 'IIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action (see that article for historical formulations). It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and has even been extended to quantum mechanics, quantum field theory and criticality theories.' back

Harmonic Oscillator - Wikipedia, Harmonic Oscillator - Wikipedia, the free encyclopedia, 'In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: F = kx, where k is a positive constant. If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).' back

Hylomorphism - Wikipedia, Hylomorphism - Wikipedia, the free encyclopedia, 'Hylomorphism (Greek ὑλο- hylo-, "wood, matter" + -morphism < Greek μορφή, morphē, "form") is a philosophical theory developed by Aristotle, which analyzes substance into matter and form. Substances are conceived of as compounds of form and matter.' back

Isaac Newton, The General Scholium to Isaac Newton's Principia mathematica, 'Published for the first time as an appendix to the 2nd (1713) edition of the Principia, the General Scholium reappeared in the 3rd (1726) edition with some amendments and additions. As well as countering the natural philosophy of Leibniz and the Cartesians, the General Scholium contains an excursion into natural theology and theology proper. In this short text, Newton articulates the design argument (which he fervently believed was furthered by the contents of his Principia), but also includes an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity, which Newton saw as a post-biblical corruption. The English translation here is that of Andrew Motte (1729). Italics and orthography as in original.' back

Isaac Newton - Wikipedia, Isaac Newton - Wikipedia, the free encyclopedia, 'Sir Isaac Newton PRS (25 December 1642 – 20 March 1726/27 was an English physicist and mathematician (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations for classical mechanics.' back

James Clerk Maxwell - Wikipedia, James Clerk Maxwell - Wikipedia, the free encyclopedia, 'James Clerk Maxwell FRS FRSE (13 June 1831 – 5 November 1879) was a Scottish scientist in the field of mathematical physics. His most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as manifestations of the same phenomenon. Maxwell's equations for electromagnetism have been called the "second great unification in physics" after the first one realised by Isaac Newton.' back

Johannes Kepler - Wikipedia, Johannes Kepler - Wikipedia, the free encyclopedia, 'Johannes Kepler (. . . December 27, 1571 – November 15, 1630) was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation.' 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

Julian Schwinger, Nobel Lecture: Relativistic Quantum Field Theory, 'The distinctive features of relativistic quantum mechanics flow from the dea that each small element of three-dimensional space at a given time is physically independent of all other such volume elements.' back

Kirchoff's law of thermal radiation - Wikipedia, Kirchoff's law of thermal radiation - Wikipedia, 'Kirchhoff's law states that: For a body of any arbitrary material, emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature, the perfect black-body emissive power. back

Klein-Gordon equation - Wikipedia, Klein-Gordon equation - Wikipedia, the free encyclopedia, 'It is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a Schrödinger equation for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density. Still, with the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wavefunction, but the particle propagates both forwards and backwards in time. Any solution to the Dirac equation is automatically a solution to the Klein–Gordon equation, but the converse is not true.' back

Lamb shift - Wikipedia, Lamb shift - Wikipedia, the free encyclopedia, 'In physics, the Lamb shift, named after Willis Lamb (1913–2008), is a difference in energy between two energy levels 2S1/2 and 2P1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy. Interaction between vacuum energy fluctuations and the hydrogen electron in these different orbitals is the cause of the Lamb Shift, as was shown subsequent to its discovery.' back

Lorentz transformation - Wikipedia, Lorentz transformation - Wikipedia, the free encyclopedia, 'In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik Lorentz. It reflects the surprising fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events.' back

Max Planck - Wikipedia, Max Planck - Wikipedia, the free encyclopedia, 'Max Karl Ernst Ludwig Planck, FRS (23 April 1858 – 4 October 1947) was a German theoretical physicist whose work on quantum theory won him the Nobel Prize in Physics in 1918. Planck made many contributions to theoretical physics, but his fame as a physicist rests primarily on his role as an originator of quantum theory, which revolutionized human understanding of atomic and subatomic processes.' 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, 'Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862. back

Michael Faraday, Michael Faraday's iron filings, 'You cannot see the magnetic force around a magnet, but you can see the effects of its presence when an iron nail sticks to a magnet. In 1851, Faraday experimented to prove the existence of lines of force. He demonstrated their existence by coating sheets of paper with a thin layer of melted wax. He then placed these sheets of paper on top of bar magnets and gently poured tiny powder like pieces of iron (iron filings) over the top. The iron filings are were attracted to the magnetic forces around each magnet, revealing their existence and showing that the strength of the magnetic forces was concentrated around and between the poles at the end of the magnets. Faraday completed his experiment by gently heating the waxed paper to set the iron filings on the page enabling him to keep his results for further study.' back

Michael Faraday - Wikipedia, Michael Faraday - Wikipedia, the free encyclopedia, 'Michael Faraday . . . FRS (22 September 1791 – 25 August 1867) was an English scientist who contributed to the fields of electromagnetism and electrochemistry. His main discoveries include those of electromagnetic induction, diamagnetism and electrolysis.' back

Minkowski space - Wikipedia, Minkowski space - Wikipedia, the free encyclopedia, 'In mathematical physics, Minkowski space or Minkowski spacetime is a combination of Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.' back

Nicolaus Copernicus - Wikipedia, Nicolaus Copernicus - Wikipedia, the free encyclopedia, 'Nicolaus Copernicus (19 February 1473 – 24 May 1543) was a Renaissance mathematician and astronomer who formulated a model of the universe that placed the Sun rather than the Earth at the center of the universe.' back

Path integral formulation - Wikipedia, Path integral formulation - Wikipedia, the free encyclopedia, 'The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude. . . . This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970s which unified quantum field theory with statistical mechanics. . . . ' back

Photon - Wikipedia, Photon - Wikipedia, the free encyclopedia, 'A photon is an elementary particle, the quantum of all forms of electromagnetic radiation including light. It is the force carrier for electromagnetic force, even when static via virtual photons. The photon has zero rest mass and as a result, the interactions of this force with matter at long distance are observable at the microscopic and macroscopic levels.' back

Probability amplitude - Wikipedia, Probability amplitude - Wikipedia, the free encyclopedia, 'In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.' 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

Ptolemy - Wikipedia, Ptolemy - Wikipedia, the free encyclopedia, 'Claudius Ptolemy . . . was a Greco-Egyptian writer, known as a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology' back

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

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

Quantum harmonic oscillator - Wikipedia, Quantum harmonic oscillator - Wikipedia, the free encyclopedia, 'The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.' back

Renormalization - Wikipedia, Renormalization - Wikipedia, the free encyclopedia, 'In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities.' back

Richard P. Feynman, 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

Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, 'IIn quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back

Special relativity - Wikipedia, Special relativity - Wikipedia, the free encyclopedia, 'Special relativity . . . is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré and others) in the paper "On the Electrodynamics of Moving Bodies". It generalizes Galileo's principle of relativity—that all uniform motion is relative, and that there is no absolute and well-defined state of rest (no privileged reference frames)—from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be. Special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source.' back

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

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

Stephen Weinberg, The Cosmological Constant Problems, 'Abstract. The old cosmological constant problem is to understand why the vacuum energy is so small; the new problem is to understand why it is comparable to the present mass density. Several approaches to these problems are reviewed. Quintessence does not help with either; anthropic considerations offer a possibility of solving both. In theories with a scalar field that takes random initial values, the anthropic principle may apply to the cosmological constant, but probably to nothing else.' back

Steven Weinberg, The Search for Unity: Notes for a History of Quantum Field Theory, 'In its essentials this point of view has survived to the present day, and forms the central dogma of quantum field theory. The esential reality is a set of fields, subject to the rules of special relativity and quantum mechanics; all else is derived as a consequence of the quantum dynamics of these fields' back

Strong interaction - Wikipedia, Strong interaction - Wikipedia, the free encyclopedia, 'In particle physics, the strong interaction (also called the strong force, strong nuclear force, or color force) is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force. At atomic scale, it is about 100 times stronger than electromagnetism, which in turn is orders of magnitude stronger than the weak force interaction and gravitation. ' back

Theory of Forms - Wikipedia, Theory of Forms - Wikipedia, the free encyclopedia, 'Plato's theory of Forms or theory of Ideas asserts that non-material abstract (but substantial) forms (or ideas), and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form or idea is often capitalized. Plato speaks of these entities only through the characters (primarily Socrates) of his dialogues who sometimes suggest that these Forms are the only true objects of study that can provide us with genuine knowledge; thus even apart from the very controversial status of the theory, Plato's own views are much in doubt. Plato spoke of Forms in formulating a possible solution to the problem of universals.' back

Tim Gershon, PX430: Gauge Theories for Particle Physics: Handout 1, 'Handout 1: Revision, Notation and The Gauge Principle Relativistic quantum mechanics is a prerequisite for this module, and therefore we will assume knowledge of the material covered in PX408, available from http://www2.warwick.ac.uk/fac/sci/physics/teach/module home/px408/ back

Ultraviolet catastrophe - Wikipedia, Ultraviolet catastrophe - Wikipedia, the free encyclopedia, The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, but the concept originated with the 1900 derivation of the Rayleigh–Jeans law. The phrase refers to the fact that the Rayleigh-Jeans law accurately predicts experimental results at radiative frequencies below 105 GHz, but begins to diverge with empirical observations as these frequencies reach the ultraviolet region of the electromagnetic spectrum.[1] Since the first appearance of the term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence.' back

Vacuum energy - Wikipedia, Vacuum energy - Wikipedia, the free encyclopedia, 'The effects of vacuum energy can be experimentally observed in various phenomena such as spontaneous emission, the Casimir effect and the Lamb shift, and are thought to influence the behavior of the Universe on cosmological scales. Using the upper limit of the cosmological constant, the vacuum energy of free space has been estimated to be 10−9 joules . . . per cubic meter. However, in both quantum electrodynamics (QED) and stochastic electrodynamics (SED), consistency with the principle of Lorentz covariance and with the magnitude of the Planck constant requires it to have a much larger value of 10113 joules per cubic meter. This huge discrepancy is known as the vacuum catastrophe.' back

Virtual particle - Wikipedia, Virtual particle - Wikipedia, the free encyclopedia, 'In physics, a virtual particle is a transient fluctuation that exhibits many of the characteristics of an ordinary particle, but that exists for a limited time. The concept of virtual particles arises in perturbation theory of quantum field theory where interactions between ordinary particles are described in terms of exchanges of virtual particles. Any process involving virtual particles admits a schematic representation known as a Feynman diagram, in which virtual particles are represented by internal lines.' back

Weak interaction - Wikipedia, Weak interaction - Wikipedia, the free encyclopedia, 'Weak interaction (often called the weak force or sometimes the weak nuclear force), is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars. Weak interactions affect all known fermions; that is, particles whose spin (a property of all particles) is a half-integer.' back

Zero-point energy - Wikipedia, Zero-point energy - Wikipedia, the free encyclopedia, 'In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system. The concept was first proposed by Albert Einstein and Otto Stern in 1913. The term "zero-point energy" is a translation of the German Nullpunktsenergie. All quantum mechanical systems have a zero point energy. The term arises commonly in reference to the ground state of the quantum harmonic oscillator and its null oscillations. In quantum field theory, it is a synonym for the vacuum energy, an amount of energy associated with the vacuum of empty space. In cosmology, the vacuum energy is taken to be the origin of the cosmological constant. Experimentally, the zero-point energy of the vacuum leads directly to the Casimir effect, and is directly observable in nanoscale devices.' back