For elastic collisions, discrete wavefunctions can find particle-energy probability, and continuous wavefunctions can find particle-position probability {theory of double solution} {double-solution theory}. Continuous waves guide discrete particles, with discrete energies and momenta, to positions. Double-solution theory does not account for inelastic collisions.
Fluids have density and flux. Quantum mechanics is like hydrodynamical density and flow (with no rotation, no mutual interactions, and no radiation absorption), with particles in continuous fluid streamlines {fluid of wave motion} {wave motion fluid}. Density is like state probability and matter-wave amplitude. Flux is like particle speed, momentum, and energy and matter-wave frequency.
Hypercomplex algebras {Jordan algebra} can be non-associative over multiplication and describe particle entanglement.
Perhaps, wavefunctions are real and have latent positions and momenta, which measurement makes definite {latency theory}.
Projective geometry can be equivalent to a hierarchical network {lattice theory}|. Lattice theory is similar to fiber-bundle theory and similar to set theory. Hierarchical networks {lattice network} have highest node and lowest nodes. Two nodes can connect through intermediate-level nodes. Two nodes can have no connections. Projective geometry uses complex continuous functions. Lattice networks use real discrete values at lattice nodes, so calculations are simpler.
Lattice-network operations are commutative and associative, and can be distributive or not distributive.
quantum mechanics
Lattice theory is like quantum mechanics. Both are discontinuous, have intermediate states between states/nodes, and have different paths from one state/node to another state/node.
types
Node subsets can have least upper bounds and greatest lower bounds (complete lattice). Lattices can be graphs, polyhedra, or simplexes. Lattices can be quasi-ordered lattices, oriented graphs, or semilattices. Lattices can have independent branches (modular orthocomplemented lattice). Higher-dimension lattices can have vector-space factors (one-dimensional subspace), finite Abelian-group factors {cyclic component}, or combinatorial topologies.
Particles try all possible phase-space trajectories simultaneously {many-paths theory}|. States have different probabilities. Large and/or many objects have no observable deviation from average trajectories and states.
Perhaps, wavefunctions do not collapse. Universe evolves all possible wavefunction states and keeps them orthogonal and independent {many worlds} {many worlds theory} {relative state}. Observations/measurements split universe wavefunction and, after that, many independent universes continue. Wavefunctions do not collapse but have disjoint parts in new universes. Universe beginnings have definite measurements. Many universes and/or many minds exist and account for all possible wavefunction states. However, conventional probability and frequency ideas are lost. This idea does not connect independent states to show how probabilities arise. It does not show how states are orthogonal, only always entangle. It does not allow measurements in transformed coordinates, which are in fact possible.
Measurement is a process separate from unitary wavefunction evolution {measurement theory}|, because measurement causes state-vector reduction. Instruments, such as photodetectors or charge sensors, are not quantum mechanical. They detect momentum, energy, position, and time real positive values. Wavefunctions discontinuously precede and follow measurements. Measurements set initial conditions for wavefunctions.
Perhaps, particles and quanta are moving singularities in wave fields {neoclassical radiation theory}. Linear classical operators describe particles. Quadratic-interaction Hamiltonians describe fields. Both operator types couple particles to fields, allowing energy exchanges.
Zero-rest-mass-particle random motion follows Brownian-motion trajectories {path integral}. Particle wavefunction is sum of path integrals over Brownian-motion trajectories, because Brownian motion is a Schrödinger equation if time is large compared to relaxation time. Zero-rest-mass-particle systems have Gibbs-ensemble average values.
Perhaps, particles have no wavefunction, so there is no collapse. Particles really always have definite positions and momenta, but waves {pilot wave} direct them immediately throughout time and space (Bohm). However, particles have no hidden variables and so no pilot waves.
Perhaps, for measurements on two entangled spatially-separated particles, random effects always cause time delay long enough to allow information from first-particle measurement to travel to second particle before second measurement {retarded collapse} (Euan Squires) [1992]. However, retarded collapse makes measurements independent, and entangled-spin experiments show that measurements are dependent.
Finite-effect change requires finite-cause change, so particle probability-amplitude superposition is equivalent to unitary-transformation metric {unitary particle interpretation}. Unitary particle interpretation has no waves {Duane quantum rule}. Unitary particle interpretation is similar to corpuscular diffraction theory.
Assume particle probability-density function and transition-probability function. An integral equation {Fokker equation}, with random position coordinates, defines a Markov process. Fokker equation transforms to particle Schrödinger equation and indeterminacy relations.
Quantum-mechanics equations {Hamilton's equations} {Hamilton equations} relate particle positions and momenta. Potential-energy change plus kinetic-energy change equals zero, by conservation of energy. Energy conserves between kinetic-energy and potential-energy exchanges, so potential-energy change and kinetic-energy change are equal and opposite. Therefore, potential-energy change equals negative of kinetic-energy change. Potential energy depends on field position. Kinetic energy depends on momentum. Potential-energy change is energy gradient. Kinetic-energy change is momentum-change rate. Hamilton equation states that energy gradient, dH / dx, equals negative of momentum-change rate (force), dp / dt. Partial derivative of potential-energy function (Hamiltonian) with position is negative of derivative of momentum with time: DH / Dx = - dp / dt, where D is partial derivative, H is Hamiltonian potential energy, x is position, p is momentum, and t is time. Hamiltonians are wavefunctions that solve Hamilton equation.
Rearranging makes Hamiltonian potential-energy change dH equal negative of momentum change dp times position change dx divided by time change dt: dH = - dp * (dx / dt) = - m * dv * v = - m * v * dv, where v is velocity.
Rearranging makes negative of first derivative of Hamiltonian with momentum equal position derivative with time: - dH / dp = dx / dt = v. Velocity v = dx / dt equals negative of derivative of potential-energy change with momentum change dH / dp.
comparison
Hamilton's method substitutes two first-order differential equations for Lagrange's one second-order differential equation.
time
If particles are stationary, so positions do not depend on time, derivatives with time equal zero, and energy gradient equals zero, so energy is constant over all positions.
If particles move, so positions depend on time, use angle instead of position, and action instead of momentum, to find particle matter-wave frequencies and particle energies. Physical action is energy over time, so momentum is energy gradient over time. Angle indicates phase which indicates frequency, and angle varies directly with position, so position is angle gradient over time.
Quantum-mechanics equations {Lagrange equations} relate positions and velocities. Lagrange equations depend on energy conservation. Potential-energy change plus kinetic-energy change equals zero. In one space dimension, m * D((d^2x/dt^2) * dx) / Dx + m * dv / dt = 0. Because they use acceleration, Lagrange equations are second-order differential equations. Lagrange equations have same form for all three (equivalent) spatial coordinates. Lagrange equations have same form in all transformed coordinate systems, because kinetic energy plus potential energy is constant for both old and new coordinate systems.
In classical mechanics, particles have definite physical-space positions and momenta (velocities) through time. Particles have trajectories through physical space-time. For one-particle systems in physical three-dimensional space, classical configuration spaces have six continuous, infinite, and orthogonal dimensions: three for position and three for momentum. Classical configuration spaces have trajectories of successive states.
In quantum mechanics, particles do not have definite physical-space positions and momenta through time. Particle positions and momenta are functions of system energy, momenta, position, and time. Particles do not have trajectories through physical space-time but can be at any position and any momentum in physical space-time. For one-particle systems in physical three-dimensional space, quantum-mechanics configuration spaces have six continuous, infinite, and not necessarily orthogonal dimensions. Quantum-mechanics configuration-space points have scalar displacements that can vary over time. Frequency varies directly with particle energy. Adjacent-point scalar displacements vary over a wavelength. Wavelength varies inversely with particle momentum. Matter waves do not propagate or travel and so have no energy and are scalar waves. Maximum displacements (amplitudes) differ at different points, varying with system energy, momenta, position, and time. Matter waves have complex-number amplitudes because space-time has time coordinate of opposite sign from space coordinates, because of energy and momentum conservation, and because complex-number amplitudes result in constant-amplitude waves. Real-number waves travel outward and lose amplitude with distance. In the complex plane, multiplying by i rotates pi/2 radians (90 degrees). Complex-numbers represent rotation, frequency, phase, and magnitude.
Constants can be matrices.
Quantum-mechanics complex-number wave equations {Schrödinger equation}| relate energies and times. Schrödinger equations are similar to diffusion equations, but with a complex-number term, which makes them wave equations. Schrödinger equations require an imaginary term because they are about space-time and time has opposite sign to space components. Hermitian operators act on possible system-state Hilbert space to define observable quantities. Operator eigenvalues are possible physical-quantity measurements. Hamiltonian is total system-energy operator.
Isolated systems have constant total energy. By energy conservation, Schrödinger equations set constant total energy equal to potential energy plus kinetic energy. Potential energy varies with position. Kinetic energy varies with momentum. For waves, kinetic energy E varies directly with frequency f, and momentum p varies inversely with wavelength l: E = hf and p = h/l. Potential energy Wavefunction solutions represent system energy-level probabilities.
phase space
Physical systems have particles within boundaries. Particles have positions and momenta. Abstract phase space represents all particle positions and momenta. Particles deterministically follow trajectories through phase space. Particles have a succession of states (state vector) in phase space.
matter waves
Particles have matter waves. Matter waves resonate in phase space with harmonic wavelengths. Matter waves describe particle trajectories through phase space.
matter waves and particle energies
Matter-waves have frequencies, which determine particle energies. Waves must have frequency to be waves, so wave energy cannot be zero. Lowest-frequency resonating fundamental wave has lowest ground-state energy. Resonating waves also have fundamental-frequency overtones. Wave frequencies are not continuous but discrete. Particle energy levels are not continuous but discrete and separated by energy quanta. Energy-level differences decrease with higher frequency. Higher frequency waves have higher energy and have lower probability. Wave frequencies can increase indefinitely.
transitions
Schrödinger equations describe conservation of energy in particle systems and phase spaces and relate particle energies and times. Schrödinger equations have wavefunction solutions that define possible different-probability particle energy levels over time.
Schrödinger equations are about particle energy-level transitions. Particle can go from one energy level to another along infinitely many paths. For example, particle can go directly from one energy level to another {direct channel} or go to higher energy level and then drop down to lower energy level {cross channel}. Particles have matter waves, and each transition changes matter waves to a different frequency. For cross channels, net transition is superposition of matter-wave transitions.
transitions: probability
Going from one energy level to another has a probability that depends on energy difference and starting energy. Schrödinger-equation wavefunction solutions have transition complex-number amplitudes. For cross channels, total amplitude is complex-number sum of all transition amplitudes. Transition probabilities are absolute values of squared amplitudes. Squaring complex numbers makes real numbers. Absolute values make positive numbers. Therefore, transition probabilities are positive real numbers.
transitions: renormalization
Because number of paths is infinite, transition-probability sum seems infinite. However, higher frequencies have lower probabilities, so amplitude renormalization can make probability sum equal 1 = 100%.
energy
Potential energy PE is force F from field E times distance ds: PE = F * ds = E * H, where H is wavefunction. Kinetic energy KE depends on mass m and velocity v: KE = 0.5 * m * v^2 = 0.5 * (1/m) * p^2, where momentum p = m * v. Momentum squared is (h / (2 * pi))^2 times second derivative of wavefunction, because momentum squared depends on velocity squared: KE * H = 0.5 * (1/m) * (h /(2 * pi))^2 * (d^2)H / (dx)^2, where H is wavefunction, (d^2) is second derivative, h is Planck constant, p is momentum, and m is mass. Schrödinger equation sets sum of wavefunction potential-energy and kinetic-energy operators equal to wavefunction total energy operator {Hamiltonian operator}.
operators
Momentum over position, or energy over time, is physical action. Momentum and position operators, or energy and time operators, are commutative.
time
Wavefunctions can change over time (time-dependent Schrödinger equation).
frequency
Frequency is partial derivative of wavefunction with time.
spin
Schrödinger equation does not include particle spin, because waves cannot account for spin.
relativity
Schrödinger equation does not include relativistic effects, because waves cannot account for relativity.
Schrödinger-equation time evolution equals difference pattern between two phase-locked static waves {semiclassical}. If Schrödinger equation does not change with time, difference is zero. If Schrödinger equation changes with time, difference is a wave at beat frequency.
If Schrödinger equation does not change with time, space wavefunctions have finite single values in Hilbert space of complex-valued square-summable Lebesgue integrals {wave mechanics}.
Quantum mechanics can be deterministic if nature has hidden particles {hidden particle}|. Measurable particles and hidden particles superpose. Such particle ensembles have zero dispersion. Current sensitivities detect no hidden particles.
Related variables describe world. Perhaps, some variables {hidden variable}| are not measurable.
process
Inputs go to both hidden and observable variables. Hidden and observable variables make outputs, each with conditional probability. Bayesian statistics can estimate optimal variable probabilities {maximal a posteriori estimate} {MAP estimate}. Experiments show that hidden variables do not exist.
non-locality
By GHZ, hidden variables cannot be local. Local hidden variables cannot predict quantum-mechanical events correctly (Bell's theorem). If some variables are hidden, quantum physics must be non-local. Classical physics is local, so classical physics has no hidden variables.
If quantum object actions do not correlate before they interact, Bell's theorem requires that quantum physics must be non-local.
local
If quantum object actions correlate before they interact, they can correlate in past or future, and quantum mechanics can be local. Past correlation means they had common cause, but then all tiny events must have common cause, making complex metaphysics. Future correlation means future interaction itself supplies correlation. Both these cases are unlikely. Therefore, quantum object actions do not correlate before they interact, and quantum physics is non-local. Experiments (Alain Aspect) [1981] on photon spin show non-local quantum-mechanical statistical distribution. Therefore, local realist theories are incorrect.
Local hidden variables cannot predict quantum-mechanical events correctly {Bell's theorem} {Bell theorem}. If quantum object actions do not correlate before they interact, Bell's theorem requires that quantum physics must be non-local. Coupled particles have properties as predicted by quantum-mechanic entanglement, not properties predicted by independent random sums.
Fokker-Planck differential operators {density matrix} represent quantum-measurement processes. Discrete phase-space states (eigenstate) are independent and orthogonal and have real-number probabilities. States are phase-space vectors (state vector). State vectors have complex-number amplitudes, and probabilities are positive real-number absolute values of amplitude squares. State probability is tensor product of normalized state vector with complex conjugate, which eliminates phase. Tensor products are planes through complex Hilbert space. Renormalization can make sum of state probabilities equal one, and density-matrix-trace sum is one.
measurement
Measuring instruments are density-matrix projectors with one state vector, with real-number probability 1 = 100%. Product of physical-system density matrix and measuring-instrument density matrix makes density matrix with one trace value, the measurement.
transformations
Coordinate transformations do not change density matrices, because they are linear.
Quantum-mechanics theories {matrix quantum mechanics} {S-matrix theory} can use linear-equation systems, with indexed terms, to model electronic-transition energies.
transition matrix
Square matrices can represent linear-equation systems. Infinite square matrices can represent Hilbert spaces with infinitely many dimensions. Matrix rows and columns represent the same energy levels. Matrices are infinite, because particles can go to any energy level, and energy levels can go higher infinitely. Matrix cells represent possible particle-energy-level transitions and their probabilities. Matrix elements are time-dependent complex numbers in infinite Hilbert space. Squared-amplitude absolute values give probabilities of energy-level transitions.
Matrix cells include all direct and cross-channel electronic transitions. Cells (linear-equation terms) with both indices the same are for directly emitted or absorbed photons. Cells (linear-equation terms) with different indices are for cross channels.
Because transition-matrix amplitudes are renormalized, sum of all state probabilities is one. Transition matrices are mathematically equivalent to Schrödinger wave equations, because time-dependent complex numbers represent anharmonic oscillators.
quanta
Matrix cells represent discrete energy changes and so quanta. Matrices are not continuous.
deterministic
Particles move from energy state to energy state deterministically, with probabilities.
space
Transition matrices are not about space. There is no position or trajectory information.
space: no fields
Energy and momentum transfers are quanta. There are no fields.
space: uncertainty
Matrices use non-commutative symbol algebra, not wave-equation Hamiltonian-equation variables. The uncertainty principle depends on wave behavior. Non-commuting operators are certain, so matrix theory does not account for uncertainty.
time
Transition matrices can change over time.
tensor
Quantum-mechanical matrices are similar to general-relativity symmetric tensors. Hermitean-matrix principal-axis transformation is a unitary-Hilbert-space tensor. If transformation is independent of time, tensor is a diagonal matrix. However, quadratic distance form is invariant, so transformations are unitary, not orthogonal as in general relativity.
S-matrix theory additions {Regge calculus} can group hadron mesons and baryons. Hadron masses and angular momenta have groups {Regge hypothesis}. Hadron groups lie on a line {Regge trajectory} plotting angular momenta versus mass squared. Because mesons and baryons have same relation between mass and angular momentum, and both depend on quarks, their internal dynamics must be the same.
simplexes
Flat simplexes joined edge to edge, face to face, and vertex to vertex can approximate continuous space. For two-dimensional spaces, all curvature is at vertexes. For four-dimensional spaces, all curvature is at triangles. Curvature is where masses and particles are.
For hadrons, exchange-transition scattering-amplitude sum equals direct-channel-transition scattering-amplitude sum {dual resonance theory}. Hadrons are zero-rest-mass-string quantum states. String ends move at light speed. Strings can break, rejoin, rotate, and oscillate. String tension is potential energy. Quarks are at string ends, so strings are one-dimensional gauge fields. Dual-resonance theory requires hadrons {pomeron} with no quarks. Dual-resonance theory predicts infinite hadrons, with heavier masses {Regge recurrences}. Dual-resonance theory predicts that maximum temperature is 10^12 K.
Perhaps, rather than calculus of continuous variables, discrete algebra {algebraic physics} can describe physical laws using groups or matrices.
Perhaps, rather than calculus of continuous variables, spins or other quanta can be space, time, energy, and/or mass units, making discrete-number physics {combinatorial physics}.
Abstract Euclidean or non-Euclidean space {configuration space} {phase space, quantum mechanics} can have any number of dimensions and discrete or continuous points, with vectors from origin to points.
physical space and classical configuration space
Particles have center-of-gravity positions and momenta. In three-dimensional physical space, particle positions have three coordinates. Positions are real numbers, over an infinite range. In three-dimensional physical space, particle momenta have three coordinates. Momenta are real numbers, over an infinite range. Classical configuration space has six dimensions for each particle. In three-dimensional physical space, one particle has six-dimension configuration space: three dimensions for space coordinates and three dimensions for momentum coordinates. Two particles have twelve-dimension configuration space. For an N-particle system, classical configuration space has 6*N dimensions. Systems must have a finite number of particles, because universe is not infinitely big. Classical configuration space has Euclidean topology.
Phase space represents particle positions and momenta. For one particle, particle physical-space position coordinates can be the same as particle configuration-space position coordinates. For more than one particle, particle physical-space position coordinates are put on different configuration-space dimensions. For one particle, particle physical-space momentum coordinates are the same as measured in physical space at that position. For more than one particle, particle physical-space momentum coordinates are put on different configuration-space dimensions. In general, configuration space includes physical space for only one particle.
Particle positions and momenta are independent dimensions, because particles are independent. In classical physical space, a particle has a real-number density function, and particles have independent real-number density functions that add to make system density function.
To simplify, assume one particle and that the y-axis and z-axis positions and momenta are zero, so configuration space has x-axis perpendicular to x-momentum-axis. Assume that one particle moves in the positive direction along the x-axis. For no external forces and so constant momentum, configuration space has a straight-line trajectory parallel to the x-axis. For constant external force in the positive direction along the x-axis and so increasing momentum, configuration space has a straight-line trajectory with positive slope to the x-axis. For two particles under the same conditions, configuration space has four independent dimensions and two independent straight-line trajectories.
To account for rotations and angular momenta, configuration space can have three more dimensions for each particle.
quantum mechanics
In quantum mechanics, particle positions and momenta have three complex-number coordinates. Configuration space has six dimensions for each particle, but each dimension has two dependent components: real and imaginary. If particles interact, particle dimensions are not independent. For example, when processes create two photons, photon spins entangle.
In quantum-mechanics configuration space, the system density function is not the sum of particle complex-number wave functions. Quantum-mechanical configuration space has non-Euclidean topology.
states
Configuration-space points represent all possible physical-system states. Assume one particle and that y-axis and z-axis positions and momenta are zero, so configuration space has x-axis perpendicular to x-momentum-axis. Assume that one particle moves in the positive direction along x-axis. For no external forces and so constant momentum, quantum-mechanical configuration space has evenly-spaced points along a straight-line trajectory parallel to x-axis. For constant external force in the positive direction along x-axis and so increasing momentum, quantum-mechanical configuration space has unevenly-spaced points along a straight-line trajectory with positive slope to x-axis. Assume that particle is inside a box, and particle has elastic collisions with box walls, then particle has higher probability of being in the box than outside.
Number of possible states is infinite, because matter waves are infinitely long, because configuration-space dimensions are infinite. Particle positions are anywhere along dimension, because matter waves are infinitely long. Particle momenta are anywhere along dimension, because mass can increase indefinitely.
states: lattice
In continuous physical space, number of positions is infinite. Using a lattice of points, separated by a fixed distance, makes number of positions over an interval finite, for computer calculation.
time
Over time, system coordinates stay orthogonal, and states that are orthogonal stay orthogonal. Scalar products stay constant {unitary evolution, spaces}. Relations between vectors do not change.
time: steps
Over continuous time, number of times is infinite. Using time steps, separated by a fixed interval, can make number of times over an interval finite.
momentum or energy levels
Over continuous momentum or energy, number of levels is infinite. Using quanta, separated by a fixed interval, can make number of levels over an interval finite.
spin angular momentum levels
Spin angular momenta can be 0, +1/2, -1/2, 1, -1, +3/2, -3/2, and so on. For particle systems, total spin angular-momentum levels can be 0 (0, +1/2, -1/2, 1, -1, +3/2, or -3/2, and so on), 2 (+1/2 or -1/2), 3 (+1, 0, or -1), 4 (+3/2, +1/2, -1/2, or -3/2), and so on.
waves
Classical configuration space has no matter waves, because it has only real numbers and so no real-number/imaginary-number interactions. Quantum-mechanical configuration space has complex numbers and resonating matter waves. Complex-number wavefunctions represent all possible particle positions and momenta, or energies and times, and their probabilities. Matter waves cause space, time, energy, and momentum quanta and the uncertainty principle. Possible configuration-space points are possible particle states (state vector), because they are wavefunction solutions. Matter waves only relate to electromagnetic waves for a system with one photon. Matter waves are not in physical space, do not travel, and have no energy.
Mathematical spaces {complex vector space} {Hilbert space, quantum mechanics} can have complex-number vectors that originate at origin.
dimensions
Mathematical spaces can have from zero to infinite number of dimensions (coordinates), all of same type. Mathematical-space points have values for all coordinates.
vectors
Complex vectors are not lines, like real vectors, but are planes because they have two components, real and imaginary. Complex vectors can vary over time and so are waves with phase and amplitude. Phase goes from 0 to 2 * pi. Vector length is wave amplitude.
Hilbert-space vectors represent same state no matter what length, because only space direction is a physical property.
vectors: normalization
Because only direction matters, normalized vectors can all have amplitude one (unit vector), making square equal one.
vectors: scalar product
Vectors have scalar products with themselves {Hermitean scalar product}, to make squared length. Scalar products commute, so relations are symmetrical. If two coordinate vectors have scalar product zero, they are orthogonal and independent. Two vectors typically are not orthogonal, but spin states of spin-1/2 particles are orthogonal, as are integer multiples of spin 1/2.
transformations
If coordinate relations are linear, coordinate systems can transform, using translation, rotation, and reflection.
Hilbert-space states can have different coordinates {transformation theory}.
Relativity is important at high speed or gravity. Quantum mechanics is important at small distances and energies. Theories {quantum relativity} try to unite relativity and quantum mechanics.
Space-time time and quantum-mechanics time are not compatible. By uncertainty principle and complementarity principle, relativistic space-time and quantum-mechanics space-time are not compatible. In quantum mechanics, space-time is one history in superspace, with all possible histories inside, which all interact to give actual space-time. Space-time geometry has probability and phase and cannot be at any location. In relativity, physics is local, and space-time is relativistic.
fluid
From far away, fluids and crystals are continuous as in relativity, but from nearby they are discrete as in quantum mechanics. Fluids can model space-time curvature. Sound propagating in turbulently flowing fluid has similarities to light propagating in curved space-time.
fluid: black hole
Black-hole-radiation Hawking effect occurs at continuous event horizon at vacuum ground-state energy. Sound waves must have wavelength longer than distance between molecules. Hawking-effect photons start with wavelength less than black-hole diameter. Gravity pulls emitted photons, so wavelength becomes longer.
fluid: low temperature
At near-zero temperature, sounds can have phonon quanta. Flow changes are slow compared to molecular changes, so phonons have ground-state energy. In non-accelerating fluid, wavelength, frequency, and speed stay constant. In accelerating fluid, wavelength and speed increase. As wavelength approaches distance between molecules, molecular interactions cause speed in different fluids to differ. If speed stays constant {Type I behavior}, quantum effects do not matter. If speed decreases {Type II behavior}, phonons just outside event horizon can go below horizon speed and first fall in but then go out. If speed increases {Type III behavior}, phonons just inside event horizon can exceed horizon speed and escape.
fluid: surface waves
Surface waves on deeper and shallower flowing water can model event-horizon behavior.
fluid: inertia
Fluids have inertia, which affects motions. Electromagnetism has self-energy. Perhaps, inertia and self-energy relate.
unification by harmonic oscillators
In quantum mechanics, continuous fields are virtual-particle streams. Fields can carry waves. Infinite-length virtual-particle streams can be harmonic oscillators. Perhaps, quantum-mechanical waves are virtual-particle harmonic oscillators.
General relativity uses tensors to represent continuous fields. Tensors can represent harmonic oscillators. Perhaps, general-relativity tensors are harmonic oscillators.
Perhaps, harmonic oscillators unify general relativity and quantum mechanics by combining waves and quanta.
Crystals are lattices. Quantized space-times can be lattices {crystals, general relativity} {general relativity, crystals}.
Crystal defects are disinclinations or dislocations. Dislocations are disinclination and anti-disinclination pairs. Disinclinations are dislocation series.
Zero-curvature space-time lattices have no crystal defects. Curved space-time lattices have disinclinations. Space-time lattice torsions have dislocations (line defects such as edge and screw dislocations).
In crystals, dislocations are disinclination and anti-disinclination pairs, and disinclinations are dislocation series, so crystal curvature and torsion are interchangeable. Perhaps, force fields are series of units, and units have disinclinations. However, general relativity does not allow torsion, only curvature.
Perhaps, gravity is weak because it involves shorter unit distances than electromagnetism.
At 10^16 GeV, all forces except gravitation are equal in strength. At 10^18 GeV, all forces are equal in strength. Why is this unifying energy so high {hierarchy problem}?
Nozzles {Laval nozzle}, such as rocket nozzles, can have narrowing, in which fluid exceeds sound speed but makes no shock wave. Narrowing pushes sound going upstream back. Original sound wavelength is distance between molecules. Above boundary, pushing back increases wavelength. Below boundary, pushing back makes sound faster than it can travel. At boundary, at near-zero temperature, sound emits thermal-phonon pairs. One pair member can go up flow, and one down flow. At near-zero temperature, narrow region acts like black-hole event horizon.
At Planck length, space-time is energetic and discontinuous and has nodes, loops (quantum loop), kinks, knots, intersections, and links {quantum foam}, depending on spins.
Weak force and electromagnetic force can unite with special relativity {quantum electroweak theory}. Field has photons and has Z and W particles, not force lines. Field can change from photons and Z and W particles to particles and back. Weak force has symmetry.
Perhaps, space-time averages all possible 4-simplex matter-wave superpositions {Euclidean quantum gravity}. If space and time are equivalent dimensions, time has no direction, and physics has no causality. If space and time are not equivalent dimensions, time has direction, and physics has causality, so simplexes connect {causal dynamical triangulations}.
Quantum mechanics can unify with general relativity {quantum gravity}|. Quantum gravity is unitary.
gravity
Gravity curves space-time, and space-time determines mass motions {Wheeler-DeWitt equation}. Gravitons and interactions among gravitons determine curvature, but interactions are small if curvature is much larger than Planck length. Interactions take all possible paths, because no information is available about interaction.
gravity: metrics
For cosmology, measurements must be from within and so local. Metrics can have no singularities. Euclidean metrics can be local and can have two types, connected and disconnected.
Connected metrics are broad bounded space-time regions, with a local measurement region. Connected metrics have a boundary, and so boundary conditions. Connected metrics have few paths.
Disconnected metrics are compact unbounded space-time regions, with all local measurements. Disconnected metrics have no boundary, and so no boundary conditions. Disconnected metrics have almost all paths.
wavefunction
Universe wavefunction determines particle positions and depends on three spatial-dimension metrics and on particle. It does not depend on time, because compact metric has no preferred time. It does not depend on coordinate choice, becasue coordinates are equivalent.
Observers can see only part of space, so universe has mixed quantum state, which implies decoherence and classical physics. Superpositions do not happen, because gravitational effects cancel superpositions.
Fermions have Fermi-Dirac statistics, and bosons have Bose-Einstein statistics, and there are no other particle types {spin statistics theorem}, because quantum field theory functionals either commute or anti-commute.
To relate fermions to bosons, theories {supergravity}| can use three spatial dimensions, one time dimension, and seven more spatial dimensions to form high-curvature and high-energy-density seven-spheres. Supergravity is supersymmetry using curved spatial dimensions, seven curled-up dimensions, and gravity.
To describe phenomena that involve massive objects at short distances, such as black holes and Big Bang, theories {theory of everything}| {final theory} must unite general relativity and quantum mechanics. String theory derives from quantum mechanics.
Quantum mechanics can combine with general relativity to make quantum field theory {relativistic quantum mechanics}| {quantum field theory}. Relativistic quantum mechanics accounts for all force types, allows particle creation and destruction, is invariant under Lorentz transformations, requires negative energy levels, and predicts antiparticles. Quantum-field theories modify relativity with quantum mechanics and include quantum electrodynamics, quantum chromodynamics, and grand unified theories.
Non-relativistic quantum mechanics does not require particle spin and does not require Hilbert space. By relativity, observed values cannot affect each other faster than light. Relativistic quantum mechanics requires Hilbert space. In (relativistic) quantum field theory, functionals of quantum fields either commute or anti-commute, because otherwise they would interact faster than light. Relativistic quantum mechanics requires particle spin, to allow commutation and anti-commutation. Fermions anti-commute, and bosons commute. In (relativistic) quantum field theory, these are the only allowed particle types. Other non-commutative relations allow faster than light affects, because of their other components. Relativistic quantum-mechanics operator commutation properties determine Pauli exclusion principle. (Non-relativistic quantum-mechanics operator commutation properties determine Heisenberg uncertainty principle.)
Electromagnetic waves are vector waves, but non-relativistic quantum-mechanics wavefunctions are scalar waves. Scalar waves have no polarization, so non-relativistic quantum-mechanics wavefunctions cannot represent spin. Relativistic quantum-mechanics wavefunctions are scalar waves with spinors and so are vector waves. Vector waves have polarization and can be plane-polarized or circularly polarized, and spin applies to circular polarization. Relativistic quantum-mechanics wavefunctions can represent particle spin. Circular-polarization rate represents particle spin.
Theories {unified field theory}| try to unite all forces and particles. Strong, weak, and electromagnetic forces unify at 10^28 K at distances of 10^-31 meters, when universe was 10^-39 second old, if supersymmetry is true and superpartners exist. Weak and electromagnetic forces unify at 10^15 K.
Theories {grand unified theories}| {Grand Unification} (GUTS) use a new gauge boson that affects both quarks and leptons and so unifies strong and electromagnetic forces.
requirements
Complete unified theory must have perfect symmetry at high temperature, high energy, and short distances and have different and lower symmetry for current universe. Theory must relate three quark and lepton generations {horizontal symmetry}. Maintaining symmetry to preserve conservation laws requires forces.
First symmetry loss creates the twelve hyperweak-force bosons. Next symmetry loss creates the eight strong-force gluons. Next symmetry loss creates the three weak-force intermediate vector bosons. These symmetry losses give bosons their masses.
unity
Particles can have inner electric field surrounded by region with particle creations and annihilations that decrease field. Inner electric field is stronger than electromagnetism and decreases by less than radius squared.
Particles can have inner strong or weak force field surrounded by region with particle creations and annihilations that increase field. Inner field is weaker than strong or weak force and decreases by more than radius.
Decrease of strong nuclear forces and increases of electric and weak forces can meet to unify all forces.
weak and strong forces
Rotation between weak and strong forces became constant when symmetry broke at an angle {Cabibbo angle}.
weak force and electromagnetism
Weinberg-angle coupling constant for isospin and electroweak hypercharge has value close to that predicted by grand unified theory.
Strong nuclear force can unite with special relativity {quantum chromodynamics}| (QCD).
color
Long-range color force causes short-range strong nuclear force. Like electric charge, color conserves.
electric charge
Particles with integral electric charge have no color, because their colors add to white or black. Particles with fractional electric charge have color, because their colors do not add to white or black. For example, pions have up quark and down antiquark, so charge is -1 (-2/3 + -1/3), and color and complementary color add to white. Protons have two up quarks and one down quark, so charge adds to +1 (+2/3 + +2/3 + -1/3), and colors red, green, and blue add to white. In particles, two up quarks must have different colors, because same colors repel.
strength
Close quarks interact weakly, because net color is zero. Farther quarks interact more strongly, because net color is more.
free quarks
Fractional-charge colorful particles cannot exist by themselves, because they cannot break free of strong force. For high energy and temperature, distances are short, and quarks and gluons do not strongly interact {asymptotic freedom}.
vectors
Quantum chromodynamics uses three complex gauge-field vectors, for red, green, and blue, and so is non-Abelian. Cyan, magenta, and yellow are vectors in opposite directions. Colors add by vector addition, so vectors make a color wheel in complex plane.
gauge
Quantum chromodynamics is a hadron gauge theory and uses the SU(3) symmetry group. Strong force has symmetry, because quark color does not matter, only net color.
strong-force exchange particle
Strong-force field has gluons, not force lines, and can change from gluons to particles and back.
lattice
Three-dimensional lattices can approximate continuous space as discontinuous nodes. Nodes represent possible quark locations. Paths between nodes represent quark interactions, and lattice lines are forces connecting quarks. Because strong force is constant with distance after short distance, number of lines between two quarks is constant.
string theory
Strings in five-dimensional dynamic space, and particles in four-dimensional boundary of QCD-force space, have equivalent mathematics. When QCD forces are strong, strings interact weakly. In string theory, QCD viscosity is like black-hole gravity-wave absorption.
Electromagnetism can unite with special relativity {quantum electrodynamics}| (QED) {relativistic quantum field theory}. From electron charge and mass, quantum electrodynamics can predict all charged-particle interactions. Quantum electrodynamics describes electromagnetic photon-electron/proton/ion interactions using quantum mechanics. Possible paths have amplitudes and probabilities. Path number is infinite, but some cancel and some end (sum over histories). Feynman diagrams illustrate paths.
field
Electric field has photons, not force lines. Electromagnetic force has symmetry.
photons
Photons are electric-field excitations. Sources emit photons, and sinks absorb photons. Field can change from photons to particles and back.
quasiparticle
Electrons {quasiparticle, electron} move through material with higher or lower mass than rest mass, because they interact more or less with material electric fields. Electrons moving at relativistic speed tunnel through barriers {Klein paradox}. Electrons {Dirac quasiparticle} moving at relativistic speeds have low effective mass, because they have accompanying virtual antiparticles, which subtract mass, that materialize from vacuum. In vacuum, time is short, so frequency and energy are high enough to make particle-antiparticle pairs. Antiparticles attract to fields that repel particles, so Dirac quasiparticles tunnel.
string theory
String theory derives from quantum-electrodynamics approximation methods {perturbation theory}.
special relativity
Quantum mechanics can combine with special relativity, for use in flat space-time or in time-independent space-time. Time can include imaginary time, which rotates time axis {Wick rotation} and transforms Minkowski into Euclidean space. Gravitons have features that are not gravitational-field excitations.
At energy levels that are low compared to interacting-particle mass, forces are negligible {effective field theory}. Gravitation has negligible force.
Quantum electrodynamics, quantum chromodynamics, and quantum electroweak theory form unified theory {particle physics standard model} {standard model of particle physics} {standard theory}.
particles
Quarks, leptons, and intermediate vector bosons are wave bundles in fields. Top quark has 175 GeV. Proton has 1 GeV.
Why are there three particle generations, rather than just one? The first generation makes consistent theory with need for higher-mass particles.
Particle masses, charges, and spins relate by the Yang-Mills gauge group in the particle Standard Model. That gauge group is the direct product of the Special Unitary group for three gluons, Special Unitary group for two intermediate vector bosons, and Unitary group for one photon: SU(3) x SU(2) x U(1). Therefore, the Yang-Mills gauge group has SU(3), SU(2), and U(1) as subgroups. SU(3) is for strong-force quark and gluon color, is non-Abelian, and has no invariant subgroups, so its matrix is traceless. SU(2) is for weak-force pion and W-and-Z boson strangeness, is non-Abelian, and has no invariant subgroups, so its matrix is traceless. U(1) is for electromagnetic electron and positron electric charge and is Abelian and normal. Unitary groups have unitary square matrices, as generators. Special groups have square-matrix determinants = 1.
field
Standard theory is renormalizable quantum-field theory. Quantum-field theory is for energies that are high compared to particle mass, so it is not about gravitation.
gauge symmetry
Only quantum differences are important, not absolute values.
gauge symmetry: renormalization
Redefining 18 physical constants {renormalizable} can remove infinite quantities.
other forces: mass
Gravitation is about mass. Standard Model does not predict quark and lepton masses, unless it adds a scalar field. Scalar field probably has quanta and so Higgs particles, with masses of 100 to 300 GeV.
other forces: supersymmetry
Perhaps, a new force allows protons to be unstable with half-life 10^31 to 10^34 years. Perhaps, new force gives mass 10^-11 GeV to neutrinos.
In quantum-field theories, matter positive frequencies can go forward in time, and antimatter negative frequencies can go backward in time {twistor, quantum mechanics}| (Penrose). In Minkowski space, twistors are spinors and complex-conjugate spinors.
Riemann sphere
Complex numbers graph to planes. Plane can be at Riemann sphere equator. Pole point can be at infinity. Line from pole through plane can intersect Riemann sphere. Real numbers are on equator. Positive frequencies are in upper hemisphere. Riemann sphere is twistor space. Twistor space has two plane dimensions and three space-time-point dimensions. Adding spin makes six real dimensions {projective twistor space}.
space-time and quantum mechanics
Perhaps, general relativity and quantum mechanics unify using twistors. Space-time relates to quantum-mechanics complex amplitudes through Riemann spheres. Riemann-sphere space-time points have light-ray sets. Space-time events are Riemann-sphere directions, showing which past events can affect future event. In twistor space, light rays are points, so twistor space is not local. Photons have right or left circular polarization {helicity}. Half-spin particles have up and down spin superpositions, as observer sees Riemann sphere. Riemann spheres can have inscribed icosahedrons, which define 20 sphere points. Points join three edges, which can be like three space dimensions. Points combine two independent entangled fermion spins, with spin +1/2 or -1/2. Riemann tensor has 20 components in flat space-time. Perhaps, complex numbers can relate general relativistic space-time to spin quantum mechanics [Penrose, 2004]. At different velocities, transformation groups {Möbius transformation, twistor} can find curvature.
Quantum mechanics can combine with special relativity {gauge theory}.
boson
Forces have force fields and exchange bosons. Bosons are quanta. Field quanta are bosons. Gauge transformations are boson exchanges. Boson exchange carries energy and momentum quanta between fermions. Field is for relativity, and quanta are for quantum mechanics.
Higgs particles are bosons that generate masses for particles. Hadrons are bosons in multiplets for charge and isotopic spin.
groups
Conservation laws determine symmetries and gauge transformations, which form mathematical groups. Quantum electrodynamics is lepton gauge theory and uses symmetry group U(1). Quantum chromodynamics is hadron gauge theory and uses symmetry group SU(3). Electroweak theory [1973] is gauge theory for weak interactions and electromagnetism and uses symmetry group SU(2) x U(1).
Symmetry {gauge symmetry}| requires that only quantum differences are important, not absolute values.
Continuous point sets are manifolds {base space}. Manifold points can have internal spaces {fiber space}, with internal dimensions {fiber, mathematics}. Fiber spaces are manifolds. Fibers do not intersect. Fibers project to points {canonical projection}.
fiber bundles
Combined base and fiber space {fiber bundle}| {bundle} has dimension number equal to sum of fiber-space and base-space dimensions. Base space can be curve. Curve points have line tangents to curve. Tangents are fiber spaces.
Curved-surface points have planes tangent to surface. Tangent planes are fiber spaces.
vector bundle
Fiber spaces can be vector spaces {vector bundle}.
twisting
If fiber spaces are the same for all base-space points, base space and fiber space can make product space {untwisted bundle}. If fiber spaces are not all the same, base space and fiber space can make a symmetrical locally untwisted product space {twisted bundle} with a mathematical group. For example, particle spins can be fiber bundles. Base-space spins go to fiber-space phase relations.
curvature
Curvature can be connections between fibers in fiber bundles, with rule {path-lifting rule} for getting to fiber-space point from base-space point.
gauge fields
Gauge fields can be connections between fiber-bundle fibers. Bundles can have locally constant values {bundle connection}, which are like gauge connections. Connections represent field phase shifts {path lift}.
tangent bundle
Base spaces can have tangent vectors as fiber spaces {tangent bundle} or covectors as fiber spaces {cotangent bundle}. Base spaces can be two-dimensional spheres. Fiber spaces can be circles. Bundles {Hopf fibration} {Clifford bundle} can be three-dimensional spheres.
Quantum mechanics can combine with general relativity by gauge-theory extension {relativistic gauge theory}. Base field or space represents physical space-time events. Total field or space represents quantum wavefunctions or symmetry transformations. Base-space points project to total-space points to make fibers.
Perhaps, fermions and bosons can interchange using a new force {technicolor theory}| {supersymmetry} {Supersymmetric Standard Model} (SSM). Fermions and bosons have quarks, which are fermions. Supersymmetry unites half-integer-spin fermions and integer-spin bosons.
fermion
Particles with odd number of quarks are fermions, which have half-integer spins. Fermions have negative ground-state energy.
boson
Particles with even number of quarks are bosons, which have integer spins. Bosons have positive ground-state energy.
stability
Fermion-boson interaction can cancel ground-state energies, leaving small stable energies.
force
Fermions and bosons can have a new force. The new exchange particles have 1000-GeV energies, with range from 10^2 GeV to 10^16 GeV. Because force strength depends on particle energy, the new force is the strongest force.
spin: superpartner
Particles pair with massive superpartners with spin 1/2 more or less than particle spin. Fermions have boson superpartners, such as squark, sneutrino, and selectron. Bosons have fermion superpartners, such as gravitino, higgsino, photino, gluino, wino, and zino.
spin: change and symmetry
Perhaps, besides space, time, and orientation symmetries, angular-momentum components {spin symmetry} can unite all forces and particles.
spin: space dimensions
Supersymmetry spin change requires extra spatial dimensions {Grassmann dimension, spin}.
spin: symmetry
Supersymmetry uses graded Lie algebra {superalgebra}.
detection
Instruments have not yet detected superpartners or fermion decay to bosons. Perhaps, universe origin had supersymmetry but universe now has broken symmetry.
hierarchy problem
At 10^16 GeV, all forces except gravitation are equal in strength. At 10^18 GeV, all forces are equal in strength. Why is this unifying energy so high (hierarchy problem)? Supersymmetry uses high energies and can resolve this problem.
supergravity
Supersymmetry applies to flat space-time Yang-Mills-field strong and weak nuclear forces and to electromagnetic fields, but can extend to gravity.
Standard Model
Supersymmetry can add to Standard Model. Standard-Model particles have superpartners {Minimal Supersymmetric Standard Model}.
In a supersymmetry model {interacting boson model}| [Arima and Iachello, 1975], atomic nuclei can have nucleon pairs. Even numbers of protons and neutrons, as in platinum, can have three dynamical-symmetry classes. Even numbers of protons and odd numbers of neutrons, and vice versa, and odd numbers of protons and numbers, relate to even-even case. Interacting bosons make nuclei behavior independent of particles and of special relativity, except for mass. If boson and fermion numbers are constant, supersymmetry can predict odd-odd case for heavy atoms, such as gold 196 with 79 p and 117 n, which has doublet ground state.
Particles have massive paired particles {superpartner}, with spin 1/2 more or less than particle spin.
Supersymmetry requires extra dimensions {Grassmann dimension, supersymmetry}.
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Date Modified: 2022.0225