One-dimensional vibrating loops or line segments {string, physics} account for matter, force, and energy. Strings vibrate at specific frequencies, amplitudes, phases, and modes {superstring theory}| {string theory} to represent particles.
length and mass
String lengths are multiples of Planck length. Uncertainty principle requires strings to be longer than Planck length.
particles
Strings are fundamental particles and have no sub-structure or sub-particles. String vibration modes make elementary particles. String theory requires an infinite series of elementary particles with increasing masses. High-mass elementary particles are unstable. Particle interactions merge two strings into one string or split one string into two strings.
particles: dilaton
Force strengths depend on string 11th-space-time-dimension length (dilaton). Short dilatons represent weak nuclear forces. Long dilatons represent strong nuclear forces. Dilaton lengths represent electromagnetism, and dilaton length variations change electromagnetic fields.
Before universe origin, dilatons are long, and forces are strong. At universe origin, dilatons are short, and forces are weak. Observing intergalactic magnetic-field changes is a test for dilatons and so can indicate universe-origin conditions.
mass
String mass is proportional to string length. Shorter strings have higher vibration frequencies and so higher masses.
waves and resonance
Strings vibrate at light speed as high-frequency waves. Shorter strings have higher vibration frequencies. String waves dampen Planck-scale quantum fluctuations. Waves have Planck lengths and so are not observable. String-wave resonances make particles and forces.
space
Strings have dimension and vibrate, so strings occupy space. Strings require background space.
energy
Strings have high energy and mass, because they are stiff vibrators, with 10^39 tons tension {Planck tension}. However, quantum fluctuations decrease this energy.
In zero-rest-mass particles, string rest masses cancel, leaving low-energy vibrations. Zero-rest-mass strings have relativistic mass and so can have momentum and angular momentum.
In elementary particles with mass, almost all string rest masses cancel.
spatial dimensions
Uncertainty principle causes strings to have a smallest length, equal to Planck length. Strings occupy space because they vibrate.
String waves vibrate in three extended spatial dimensions, seven curled-up Planck-size spatial dimensions (Joyce manifolds), and one time dimension. Physical scalar and vector fields determine the number and properties of infinite and curled-up dimensions. String waves can vibrate in dimensions, wrap around dimensions, and travel around dimensions. Winding around large dimensions takes more energy, because string stretches. Moving around large dimensions takes less energy, because frequency is less. The effects balance each other.
spatial dimensions: winding
In a curled-up dimension, string vibrations occur along the dimension (dimensional vibration) and go around the dimension {winding mode}. They can go around more than once. Curled-up dimensions cannot be smaller and their vibrations have a minimum, so waves have quantum size and energy.
spatial dimensions: size
Curled-up dimensions have two radii, one across curl and one around dimension. Because vibrations increase as quantum fluctuations increase, if one dimension becomes smaller, other dimension becomes larger. Winding-mode vibrations depend directly on dimension radius. Dimensional vibrations depend inversely on radius. Because winding mode vibrations and dimensional vibrations are reciprocal, curled-up dimensions have same physics as dimensions with exchanged winding-mode and dimensional-vibration radius.
spatial dimensions: physical forces
Electromagnetic waves, matter waves, gluons, and W and Z particles travel only in three-dimensional space, not curled-up dimensions. Gravity waves can travel in all dimensions.
interactions
Strings can split into two strings, or two strings can merge into one string. Splitting and merging probabilities depend on a positive-number constant {string coupling constant}. Constant is less than one for weak coupling or greater than one for strong coupling.
finite
String theory has finite quantities and so removes field-theory infinity problems and quantum-mechanics renormalization problems. (However, eleven-dimensional quantum-field theory can have finite particles and quantities.)
strings
Strings are one-dimensional Planck-length-multiple line segments or circles, and M-theory p-branes are multi-dimensional Planck-length-multiple areas. Particles are Planck-length-multiple strings that harmonically vibrate in dimensions.
strings: general relativity
At greater-than-Planck-length distances and larger-than-energy-quantum energies, string theory and general relativity have same form.
strings: quantum mechanics
String-vibration wave equations have harmonic-frequency wavefunction solutions. Discrete wave frequencies represent energy quanta, which account for particle masses. String wavefunctions are essentially the same as particle quantum-mechanical wavefunctions.
strings: dimensions
Space has three infinite dimensions and some number of compactified dimensions. Compactified dimensions have relations.
strings: vibration components
As fundamental units, strings have no internal parts, structure, or forces. Because longitudinal vibrations require internal parts, structure, or forces, strings have no longitudinal vibrations.
Strings vibrate transversely across all dimensions, so vibrations have many components. For curled-up dimensions, vibration frequency varies directly with radius inverse, so shorter strings have wave higher frequencies. For curled-up dimensions, strings can also vibrate transversely around the dimension (wind), and winding-vibration frequency varies directly with radius, so longer strings have higher winding-wave frequencies. Therefore, winding vibrations are duals to the other transverse vibrations.
In each dimension, strings vibrate at resonant frequencies, determined by string length and tension. Vibration components have complex-number frequencies, which determine real and imaginary wave-energy components, which determine real and imaginary particle-mass components. Because complex-number operations can result in positive or negative values, superpositions of wave components can make positive or negative wave energies and positive or negative particle masses. Quantum energy fluctuations can also be positive or negative, so virtual particles can have positive or negative masses.
Strings cannot have zero length, because then they are not strings, which must have tension between endpoints. Strings cannot have no vibrations, because they must have tension and endpoints and so fundamental frequency.
strings: particles
String theory accounts for all Standard-Model particles.
Electric charge is at open-string endpoints or spread around closed string. Zero-rest-mass oriented open strings are U(n) bosons. Photons are open strings, are vectors with spin 1, and have wave amplitude zero. For local interactions, photons have clockwise or counterclockwise transverse-wave amplitude-vector rotation around oriented-string long axis to account for integer spin. Photons have no mass, so strings have zero-point lowest-energy state. Photons have photons as antiparticles. For global interactions, electric charge can be positive or negative and that corresponds to orientation, requiring otherwise-same opposite oriented open strings or otherwise-same clockwise and counterclockwise motions around closed strings. Electric charge does not have anti-charge, only exactly opposite positive and negative charge. Electrons and positrons have clockwise or counterclockwise transverse-wave amplitude-vector rotation around long axis to account for half-integer spin.
Strangeness is at open-string endpoints or spread around closed string. Vector bosons are open strings and are vectors with spin 1. Strangeness can be zero or one, corresponding to absence or presence, requiring non-opposite oriented open strings or non-opposite clockwise and counterclockwise motions around closed strings (violating parity). Strangeness does not have anti-strangeness. For local interactions, Z intermediate vector bosons have clockwise or counterclockwise transverse-wave amplitude-vector rotation around oriented-string long axis to account for integer spin. W intermediate vector bosons have both clockwise and counterclockwise transverse-wave amplitude-vector rotation around oriented-string long axis to account for zero spin. Intermediate vector bosons have mass, so strings have intermediate-energy state. Intermediate vector bosons have intermediate vector bosons as antiparticles. For global interactions, pions have clockwise or counterclockwise transverse-wave amplitude-vector rotation around oriented-string long axis to account for half-integer spin.
Color charge is at open-string endpoints or spread around closed string. Gluons are open strings and are vectors with spin 1. Color has three vectors that add to zero, corresponding to an equiangular triangle, requiring complex-number oriented open strings or complex-number clockwise and counterclockwise motions around closed strings. Colors have anti-colors. For local interactions, gluons have clockwise or counterclockwise transverse-wave amplitude-vector rotation around long axis to account for integer spin. Gluons have mass, so strings have high-energy state. Gluons have gluons as antiparticles. For global interactions, quarks have clockwise or counterclockwise transverse-wave amplitude-vector rotation around long axis to account for half-integer spin.
Gravitons are closed strings, are symmetric tensors with spin 2, and have wave amplitude zero. Mass-energy is positive and scalar. Mass has negative scalar anti-mass. For local interactions, gravitons have two clockwise or counterclockwise transverse-wave amplitude-vector spins, one around each tensor axis. Gravitons have no mass, so strings have zero-point energy. Gravitons have gravitons as antiparticles. Fermion and boson waves together make tensor that has one symmetry, which makes tensor gauge and so closed string with spin 2.
Dilatons are closed strings and are scalars with spin 0. Axions are closed strings and are antisymmetric tensors with spin 0. Zero-rest-mass unoriented open strings are SO(n) or Sp(n) bosons.
strings: virtual particles
At small distances, string-theory quantum mechanics allows virtual particles. Strings always change to strings, never to no strings, because, by uncertainty principle, zero-length strings have infinite energy. The no-string (vacuum) state cannot exist.
One zero-point-energy string can become two virtual-particle strings. Two virtual-particle strings can become one zero-point-energy string.
Strings preserve all symmetries and conservation laws.
Real-particle strings have longer lengths, lower energies, and longer lifetimes. Virtual-particle strings have short lengths, high energies, and short lifetimes.
strings: particle properties
String (Planck-multiple) lengths and (high) tensions determine transverse vibration modes and account for particle energies, masses, rotations, and other properties.
String endpoints rotate around center, or closed strings rotate, so strings account for particle spin. String orientations that differ only in direction can represent clockwise and counterclockwise spin. String orientations that differ in direction and other properties can represent parity or no parity. Closed strings can account for zero-rest-mass spin-2 particles (graviton).
matrix theory
Perhaps, space is intertwined strings or zero-branes {matrix theory} and so is not background-independent.
quantum loop
Perhaps, quantum loops are the background for strings. Larger loops can be strings. Perhaps, strings are waves in spin networks.
comparison to points
Strings have one dimension, are Planck length or higher, and have waves. Points have zero dimension, are smaller than Planck scale, and have no waves.
Point particles {point particle} are zero-dimensional points with quanta. Point particles have quantum-mechanical waves. Point particles preserve symmetries and conservation laws.
virtual particles
Point-particle quantum mechanics allows virtual particles. Two virtual point particles can appear (particle creation) from space-vacuum energy (negative-energy-particle field) fluctuations. Two virtual point particles can become space-vacuum photon energy (particle annihilation).
energy
Because point particles have only one point, and space has no compactified dimensions, point particles have only positive real-number energies and masses. Point particles have no size and do not change size, so they cannot have infinite energy.
particle properties
Point particles have no mechanism for particle spin, orientation, or other properties and no mechanism to make spin-2 bosons (gravitons).
Point particles can have any rest mass and so can have zero rest mass.
space dimensions
Point particles have one point of zero dimension and so do not require unobserved dimensions. Point particles have no mechanism to specify number of space-time dimensions.
Strings can be closed loops {closed string} or have ends that freely move {open string}. Open-string ends have boundary conditions {Dirichlet boundary conditions}, typically different for different dimensions. For example, electron strings move in the three infinite dimensions but do not move in the seven curled-up dimensions.
High-speed strings appear to have discrete line segments {string bit}. String bits have quantized length, energy, momentum, and angular momentum. String-bit minimum length is Planck length. Planck length string bit has minimum momentum. Longer string bits have multiples of Planck length and multiples of minimum momentum.
Expanded string quantum-field theory {M-theory}| adds vibrating disks, blobs, toruses, and higher-dimension branes, to explain elementary particles, forces, and energies. M-theory branes create space-time (background independence). M-theory has ten space dimensions (three infinite and seven curled up) and one time dimension. People do not yet know M-theory physical principles.
Spaces can be lattices {Wilson's loops} {Wilson loops}. Particles are at nodes. Gravity, electric, and strong-force field lines are on lattice lines between nodes. For electric fields, field lines diverge, decrease with distance, and approximate continuity. For color-charge fields, field lines stay apart and have constant force.
String theory assumes space-time {background dependence}, in which strings move. M-theory branes create space-time, so M-theory has background independence.
Space-time points have zero dimension and do not vibrate, so real-number coordinates represent them. Strings and quantum loops have one dimension and vibrate. Strings and quantum loops have uncertain positions and motions, by the uncertainty principle. Complex-number matrices can represent string (and quantum-loop) positions and motions. Complex-number matrix operations do not commute {non-commutative geometry}. At large scales, diagonal matrices can approximate complex-number matrices, and diagonal matrices have commutative operations (commutative geometry).
Force strengths depend on no-spin closed-string 11th-space-time-dimension scalar length {dilaton}. Short dilatons represent weak nuclear forces. Long dilatons represent strong nuclear forces. Dilaton lengths represent electromagnetism, and dilaton length variations change electromagnetic fields.
Before universe origin, dilatons are long, and forces are strong. At universe origin, dilatons are short, and forces are weak. Observing intergalactic magnetic-field changes is a test for dilatons and so can indicate universe-origin conditions.
Magnetic-field photons can make dilaton-related antisymmetric-tensor no-spin closed-strings {axion} that have less than one millionth electron mass, no charge, and zero average quantum field. Magnetic-field axions can make photons. Therefore, axions allow strong nuclear forces to maintain charge-parity (CP) symmetry between antiparticles and particles.
Cosmic-microwave-background temperature fluctuations are small, have Gaussian distribution, and have same amplitude for large space regions. Cosmic-microwave-background temperature fluctuations arise mostly from density differences and partly from gravity waves. However, string theories without axions allow no density differences. Axions determine large-scale universe temperature fluctuations [Adams, 2002].
Strings, disks, blobs, toruses, and higher-dimension objects {brane} can vibrate.
Rather than strings, fundamental elements can be membranes {supermembrane} {2-brane} {3-brane} {p-brane}|, of dimension 2, 3, or any natural number p. Minimum length is Planck length. Branes vibrate and have high-energy waves. Branes require ten space dimensions (three infinite and seven curled up) and one time dimension.
Open-string endpoints can stay fixed at one point (Dirichlet boundary condition) (zero-dimension brane), which maintains vacuum gauge invariance. Endpoints can move along straight or curved lines (Neumann boundary condition) (one-dimension brane), which fixes particles. Endpoints can move around surfaces (two-dimension brane) or higher branes.
Membranes {Dirichlet-brane} {D-brane} can contain string ends. Black holes have many D-branes, and black-hole temperature is number of possible D-brane arrangements.
Curled-up dimensions have curved cylindrical handles {handle, string}, which have branes wrapped around them.
Curled-up dimensions have spikes {throat, string}, which have branes at tips.
Perhaps, three-dimensional space {braneworld} is a brane.
Triangles {2-simplex} have two dimensions and three lines/sides. Tetrahedrons {3-simplex} have three dimensions and four triangles/faces. Simplexes {4-simplex} can have four dimensions and five tetrahedrons/faces. 4-simplex connections dynamically determine number of space-time dimensions.
Space can be tiny causally-connected (over time) 4-simplexes {causal dynamical triangulation} (CDT). CDT allows only causally possibly configurations. CDT results in three large-scale spatial dimensions and one Planck-scale spatial dimension, making four space dimensions. CDT can account for all forces and particles.
If dynamical triangulation can be causal or non-causal, space has infinite dimensions or two dimensions.
Six curled-up dimensions can combine in different ways to make thousands of different spaces {Calabi-Yau spaces} or shapes {Calabi-Yau shapes}. Calabi-Yau shapes have different numbers of holes, different numbers of even-dimension holes, and different numbers of odd-dimension holes. For Calabi-Yau shapes with same total hole number, interchanging number of even-dimension holes and odd-dimension holes results in same physics {mirror manifold}. String vibration sizes and frequencies depend on the difference between odd-dimensional hole number and even-dimensional hole number.
One curled-up spatial dimension is a circle (one-dimensional torus), with one hole.
Two curled-up spatial dimensions are a sphere (two-dimensional torus), with one hole, or three-dimensional torus, with two holes. Two-or-more-dimensional toruses can have one complex dimension. Calabi-Yau manifolds with one complex dimension have at least one hole. Compact and simply connected Calabi-Yau manifolds with one complex dimension are elliptic curves.
In three-dimensional space, three curled-up spatial dimensions are solid sphere, with zero holes, or solid torus, with one hole. In four-dimensional space, three curled-up spatial dimensions are hollow-sphere-cross-section hollow sphere, with two holes; hollow-sphere-cross-section hollow torus, with three holes; or hollow-torus-cross-section hollow torus, with four holes. Three curled-up real dimensions make a volume. See Figure 1.
Four-dimensional space has six regular and convex structures {4-polytope} {polychoron}, which have one of the five Platonic solids on their three-dimensional boundaries [Ludwig Schläfli, 1850]: pentachoron, tesseract, hexadecachoron, icositetrachoron, hecatonicosachoron, and hexaicosichoron. Compact simply connected Calabi-Yau four-dimensional manifolds {K3 surface} have two complex dimensions and at least two holes.
Calabi-Yau six-dimensional manifolds have three complex dimensions and at least three holes and have thousands of variations: for example, all zeros of a homogeneous quintic polynomial.
Calabi-Yau shapes can tear {flop-transition} {topology-changing transition} to make topologically distinct Calabi-Yau shapes. Particle properties then change slowly and non-catastrophically.
Seven curled-up spatial dimensions can combine in different ways to make thousands of manifolds {Joyce manifolds}, with four holes to thousands of holes. String-theory one-dimensional strings, and M-theory multi-dimensional branes, vibrate in three infinite and seven curled-up spatial dimensions.
Infinite dimensions can change, because their scales can change {scale factor}. Distances between objects can increase or decrease, as space relativistically expands or contracts.
Scalar fields {moduli, string} {modulus, string} cause curled-up-dimension shapes and sizes.
Modulus parameters determine Calabi-Yau spaces, but modulus can go to zero {moduli problem}.
Gravity can be in all space dimensions, including non-infinite high-curvature dimensions {Randall-Sundrum model}, making gravity weak compared to other forces, which are not in curled-up dimensions.
String theory for particles is equivalent to quantum chromodynamics for fields {duality hypothesis}.
String theories can use different geometries to describe the same situations {geometrical duality}. A large finite dimension can behave equivalently to a small one, because a wrapped-around-dimension string can exchange with an unwrapped-in-dimension string, or because Calubi-Yau shapes can exchange number of odd-dimensional holes with number of even-dimensional holes.
Because strings have two motion modes, which can exchange, in each dimension, which can exchange, strings with small and large dimensions have same physical effects {T-duality}. Because strings have two motion modes, which can exchange, in each dimension, which can exchange, closed strings and open strings also have T-duality.
Because strings have two motion modes, which can exchange, in each dimension, which can exchange, physical systems can have either or both of two space-time geometries {mirror symmetry}.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225