User:John R. Brews/Sample
In physics and chemistry, charge is fundamentally related to fields and forces, and is a property of pieces of matter that leads to forces between spatially separate pieces of matter that likewise manifest that particular property. There are a wide variety of such charges. In the Standard Model, there are three types of charge: color, weak isospin and weak hypercharge. These include the electric charge underlying electric current that enters Maxwell's equations for the electromagnetic field. In addition, there is mass that enters gravitation.
These charges are conserved quantities and are related to currents describing their flux or motion. The conservation law relating the charge to its current is of the form:
where div is the vector divergence operator, J is the vector current density, and ρ is the charge density. For a volume enclosed by a surface, this equation can be expressed by the statement that any change in the charge contained inside the closed surface is due to a current of said charge either entering or exiting through that surface.
Such conservation laws are examples of Noether's theorem, which states that every symmetry of a physical theory is related to a conservation law of this kind. This theorem is closely related to Curie's principle:
- The symmetry of an isolated system cannot decrease as the system evolves with time.
The best known of these conservation laws are the conservation of momentum (the current is momentum density, the charge is mass density), related to translational symmetry of the laws of mechanics, conservation of angular momentum, related to the rotational symmetry of the laws of mechanics, and conservation of energy, related to the independence of the laws of mechanics from time translations. Such symmetries are intuitive for point particle mechanics, but for the physics of general fields some symmetries are quite non-intuitive.
A formal description of Noether's theorem as related to charge is that a current ja = (ja0, ja) satisfying:
which implies the conservation of the charge Q defined by:
is a natural consequence of the ja being generators of a Lie group that is a symmetry group of the physical system.
Charge and exchange forces
Forces between particles are mediated by exchange of shared properties. For example, two nucleons in the same state of motion can exchange electric charge, producing an exchange force. The Yukawa theory of nuclear force posited that nucleons (protons p and neutrons n) could exchange electric charge by trading pions according to the reactions:
and forces between like particles could be introduced by exchange of zero-charge pions:
These reactions do not conserve mass or energy, they are virtual reactions. One common (although not universally accepted) "explanation" why violation is permissible is that such reactions occur very rapidly, and for very short times the energy uncertainty relation allows violation of these conservation rules.
Besides electric charge, other properties can be exchanged, such as spin (Bartlett exchange), or position (Majorana exchange).
The swapping of shared properties is a symmetry operation, the exchange of identical particles, and as such is related to conserved quantities via Noether's theorem. For example, the nucleon can be thought of as a two-state particle with an isospin that is +1/2 for a neutron and −1/2 for a proton, so the change of one to the other is an isospin exchange, and symmetry of a theory under isospin exchange indicates the theory conserves isospin. In a quantized version of such a theory, isospin exchange could be moderated by the pion reactions above.
Only if isospin symmetry in the theory can be produced by a continuous transformation (one depending upon some continuously variable parameter), does it lead to an isospin current conservation law.
In electrodynamics, two types of charge are known, magnetic and electric. The distinguishing property of electric charge is that electric charges can be isolated, while while an isolated magnetic charge or magnetic monopole never has been observed. Electric charges interact with magnetic charges only when in relative motion one to the other.
The conservation of electric charge follows directly from Maxwell's equations. It also can be derived from Noether's theorem as a result of a gauge invariance of Maxwell's theory when that theory is expressed in terms of a vector potential. Although this approach has continuity with much of modern field theory, it is somewhat unintuitive, as the "symmetry" of the recast Maxwell equations is simply due to introduction of a mathematical device that adds an unnecessary degree of freedom into the formulation thereby introducing this symmetry artificially. Below is a digression on this topic.
- The basic electric field E and magnetic field B of Maxwell's equations can be replaced by introduction of a scalar potential φ and a vector potential A using the relations:
- Although the potentials uniquely determine the fields, the reverse is not true. Different potentials produce the same fields; in particular the potentials denoted by primes below produce the same fields:
- Here Γ = Γ(r, t) is any continuous function of the space-time coordinates r, t. Consequently, a theory based upon potentials instead of fields has the additional symmetry that it is unchanged by substitution of primed potentials instead of the original potentials. This change of potentials from unprimed to primed is called a gauge transformation and this new symmetry leads directly to the continuity equation for electric charge:
- This equation is a direct consequence of the Maxwell equations defining charge and current densities (in Heaviside-Lorentz units):
- However, using the potential formulation, the continuity equation is required if the theory is to be gauge invariant, and this requirement is consistent with Noether's theorem.
In a quantized theory based upon the potential formulation of Maxwell's equations, the electrical force between charged particles is an exchange force mediated by trading charge-neutral photons. The electromagnetic potentials exist as vibrations with certain allowed amplitudes determined by the number of photons employed, and field amplitudes are increased or decreased by adding or subtracting photons. Thus, the force exerted upon a charged particle as determined by the field it experiences, depends upon the number of photons in the corresponding potentials.
Weak forces are mediated by the electric charged W+ and W− particles and the electric charge neutral Z0 particle, all with spin 1. The weak interaction is of short range, being effective over a distance of approximately 10−3 fm. Analysis of the weak force parallels that of the electromagnetic force, apart from the huge mass of the exchanged particles compared to the photon. The "weak force" charge introduced that couples to this force is called flavor. It is customaryu to refer to lepton flavor, rather than lepton charge, and individual lepton flavors are attributed to each family: electron flavor Le for electrons; muon flavor Lμ for muons; tau flavor Lτ for taus. p. 38 However, the terminology is somewhat confused. Some authors do refer to both "weak charge" and to "lepton flavor". The issue may be that quarks and leptons behave differently under the weak force?
In 1935 Yukawa invented the meson theory for explaining the forces holding atomic nucleii together, an assemblage of neutrons and protons. The theory led to the experimental observation of the pion or π-meson and the muon or μ-meson. The behavior of nuclear forces was explained as an exchange of mesons. Today, mesons are considered to be quark-antiquark pairs, and a more refined theory of nuclear interactions is based upon quantum chromodynamics. Nuclear forces are not considered fundamental today, but are a consequence of the underlying strong forces between quarks, also called chromodynamic forces or color forces. On that basis, nuclear forces are an exchange force fundamentally based upon color, and only approximated by the Yukawa theory.
In the Standard Model of particle physics, quantum chromodynamics describes the strong force, also called the color force or chromo force, and relates it to the color charge as a property of quarks and gluons. Similar to magnetic charge, color is not seen directly, as all observable particles have no overall color. As with electric and magnetic charge, color charge can be multiple valued, conventionally called red, green or blue. Color charge is not assigned a numerical value; however, a superposition in equal amounts of all three colors leads to a "neutral" color charge, a somewhat stretched analogy with the superposition of red, green and blue light to produce white light. Thus, protons and neutrons, which consist of three quarks with all three colors are color-charge neutral. Quark combinations are held together by exchange of combinations of eight different gluons that also are color charged.
The charges above are related to fields and forces and to a local (coordinate dependent) Noether's theorem. Other charges are known, however, that are connected to global or discrete symmetries (no continuous parametric dependence, such as a coordinate dependence) and so to a global Noether's theorem, and have no relation to forces or fields.
One such charge in elementary particle theory is the baryonic charge, B, also referred to as a number, with value +1 for all baryons (notably, neutrons and protons, but also others like the Λ and Σ particles) and −1 for all antibaryons and zero for non-baryons. Quarks are an exception, and have a baryon number of 1/3. Unlike electric charge, which serves as a source for the electromagnetic field, baryon charge is not related to an associated "baryonic" field.
Finally, we mention the leptonic charge (also called lepton number) carried by leptons: electrons, muons, taus, and their associated neutrinos. Lepton charge depends upon the flavor of the lepton Le, Lμ, Lτ with values +1 for the electron, muon and tau meson, and −1 for their antiparticles. The total lepton number L of a complex is:
Non-leptons have a total lepton number L of zero. Within the Standard Model, lepton number is conserved for strong and electromagnetic interactions; however, it is not necessarily conserved in weak particle reactions.
- John F. Donoghue, Eugene Golowich, Barry R. Holstein (1994). Dynamics of the standard model. Cambridge University Press, p. 24. ISBN 0521476526.
- Mark Burgess (2004). “Chapter 12: Charge and current”, Classical covariant fields. Cambridge University Press, pp. 325 ff. ISBN 0521813638.
- Some care is needed in looking at this principle because of the phenomenon of spontaneous symmetry breaking. For example, as a cubic ferroelectric material like BaTiO3 is cooled below its Curie point, its cubic symmetry is replaced by a tetragonal ferroelectric symmetry as the frequency corresponding to a tetragonal elastic distortion tends to zero (Goldstone's theorem). The overall cubic symmetry of the crystal is retained because the crystal breaks into finite domains, each with a differently oriented tetragonal axis, so that statistically the symmetry of an infinite crystal still is cubic. For a general discussion, see F. Strocchi (2008). Symmetry breaking, 2nd ed. Springer. ISBN 3540735925.
- Nina Byers (1994). The Life and Times of Emmy Noether: Contributions of Emmy Noether to Particle Physics. UCLA/94/TEP/42; hep-th/9411110. Presented at the International Conference on THE HISTORY OF ORIGINAL IDEAS AND BASIC DISCOVERIES IN PARTICLE PHYSICS, Erice, Italy, 29 July - 4 August 1994.
- Hari Jeevan Arnikar (1995). Essentials of nuclear chemistry, 4th ed. New Age International, p. 16 ff. ISBN 8122407129.
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- WN Cottingham, DA Greenwood (2007). “Gauge transformations”, An introduction to the Standard Model of particle physics, 2nd ed. Cambridge University Press, p. 41. ISBN 978-0-521-85249-4.
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- Yorikiyo Nagashima, Yoichiro Nambu (2010). Elementary Particle Physics: Volume 1: Quantum Field Theory and Particles, Volume 1. Wiley-VCH, p. 554. ISBN 3527409629.
- Stephen Webb (2004). Out of this world: colliding universes, branes, strings, and other wild ideas of modern physics. Springer, p. 190. ISBN 0387029303.
- Andrew Watson (2004). The quantum quark. Cambridge University Press, pp. 170 ff. ISBN 0521829070.
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- Noether's theorem
- essay on charge
- gauge theory
- symmetry breaking
- gauge theory
- derivation of Noether from change in lagrangian
- Lagrangian is a function of the symmetry current
- Lagrangian of QCD and Noether's theorem
- exchange forces
- derive Coulomb's law as an exchange
- flavor not conserved
- three coupling constants: α, αs, αW
- weak charge also called flavor charge
- lepton and baryon number
- Griffiths: gauge theories
- the phase is just the gauge function
- Gauge field interactions
- generators of group induced by a current are charges
- generator current&f=false does a conserved current imply a symmetry? p. 129
- Noether current; basics
- basics again
- conserved quantities and forces
- conservation of current
- Noether current
- formal math statement
- conserved vector current
- coupling factors e, gI, gY
- Donoghue on charge
- Dodd on charge
- Griffiths on charge
- lepton flavor, not lepton charge