Dynamical mechanics methods {quantum mechanics}| determine particle momentum through space {particle trajectory} or particle energy through time. In quantum mechanics, physical systems have states of objects, their properties, and events. Quantum-mechanics states are in phase space (coordinate space), rather than physical space. Phase space includes all particle positions and momenta and so includes physical space. Relativistic phase space includes physical space-time. Phase-space states represent discrete energy, momentum, angular momentum, length, time, and mass quanta. Adding energy, momentum, angular momentum, length, time, or mass increases energy, momentum, angular momentum, length, time, or mass by quantum leaps, not continuously.
particles and fields
Without quantum mechanics, in continuous space and time, particles have properties that cause forces, which make continuous force fields. Particles have continuous energies, times, positions, and momenta. Fields have potentials at all space locations and at all times. Particles have zero energy. By statistical mechanics, with or without quantum mechanics, energy tends to spread to all positions equally (energy equipartition). Because continuous fields have many more locations than particles, energy tends to go from particles into fields over time.
Without quantum mechanics, in continuous space and time, electrons orbiting atomic nuclei interact with other nuclei and electrons and their orbits decay, As they spiral into nucleus, they emit electromagnetic waves to maintain conservation of energy. As electrons become closer to nucleus, they emit higher-frequency electromagnetic waves because force is stronger. By energy equipartition, electrons eventually fall into nucleus, ending all atoms. However, the universe has stable atoms.
As they absorb electromagnetic waves from space and emit them to space, hot objects in cooler space tend to transfer heat to space, because absorption concentrates on one location but emission spreads to all other locations and so never returns to the location. By energy equipartition, all frequencies have equal probability. Without quantum mechanics, higher frequencies carry away most energy, and hot objects cool quickly. However, experiments show a Boltzmann distribution of frequencies, with higher frequencies having lower probabilities. Hot objects cool slower.
In quantum mechanics, particles have discrete states and discontinuous fields. In quantum mechanics, length, time, and mass have quanta, with non-zero minimum (Planck) length, time, and mass. Minimum length makes maximum momentum. Minimum time makes maximum energy. Minimum mass makes maximum frequency. For each property, particles have one quantum plus a number of quanta, up to a maximum number. Atom electrons must have minimum energy and so do decay into atom nuclei. Higher states have lower probabilities, because higher states are harder to reach. High-frequency electromagnetic waves are fewer. Quantum mechanics results in Boltzmann frequency distribution and observed slower cooling. Quantum mechanics matches atomic- and subatomic-particle behavior.
energy quanta
Particle-collision, light-absorption, and light-emission experiments show that particles absorb or emit energy in non-zero minimum amounts (quantum). Charge accelerations differ by a number of quanta. Minimum electromagnetic-wave energy varies directly with frequency. Higher energy means more electromagnetic waves. Particle and particle-system energy is one quantum plus a number of energy quanta. Energy levels differ by quanta and are discrete, not continuous.
particle wave
Simultaneously sending particles through two slits makes almost the same target patterns as sending electromagnetic waves, suggesting that particles behave like waves. Large particles at ordinary energies have very high-frequency waves with imperceptible wavelengths. Atomic- and subatomic-particles have low-frequency waves with observable wavelengths. Particles have waves in phase space, not in space-time. Particle waves have fundamental frequency, at lowest energy level, and higher harmonic frequencies at higher energy levels. Particle waves have discrete, not continuous, frequencies at overtones of the fundamental frequency. Wave energy varies directly with frequency, so particle energy levels are discrete.
Quantum-mechanics wave equations describe potential and kinetic energies in particle and force-field systems. Equation solutions are periodic wavefunctions that describe particle and field positions and momenta.
ground state
Particles cannot have zero energy, because they cannot have zero motion, because they have phase-space waves and waves propagate. Particles have a lowest energy state (ground state), which corresponds to the lowest-frequency (fundamental) particle wave. All other energy states are quantum amounts higher than ground state.
determinism
Previous particle positions and momenta determine future phase-space states. Quantum wave and particle mechanics is deterministic and follows normal causality.
observation
Observations interact with particle to put particle in one observed state. Observation immediately and discontinuously selects observed state from among possible states. From that moment, phase-space again follows determinism and causality. Old system wavefunction "collapses" to nothing, and new system wavefunction, about both system and measuring apparatus, begins.
Scientific experiments try to isolate observer from experimental system to prevent interactions. However, in experiments involving small things, observation has to cause disturbance and perturb observed system. Observer and observed become one new system. Observation causes measurement uncertainty about observed system.
Perception theory is about physical events that cause state perception. Quantum mechanics has no perception theory. Quantum theory does not describe observer, only observed system. Quantum mechanics leaves open the possibility of perceiving state superposition. However, people and instruments detect only states and never observe state superpositions. Quantum-mechanical waves are in phase space, not physical space.
particle systems
Without quantum mechanics, particle systems follow classical statistical Markov processes, such as diffusion and Brownian motion, in space-time. Diffusion and Brownian motion apply energy and momentum conservation to many particles. Many particles follow many paths (path distribution), completely determined by previous positions and momenta. Particles tend to have the highest-probability energy distribution.
Quantum-mechanics equations (Schrödinger equation) also derive from energy and momentum conservation and are similar to diffusion and Brownian-motion equations. Quantum-mechanics equations are about coordinate space or phase space, rather than physical space. Quantum-mechanics functions (wavefunction) are complex-number functions, which relate trigonometric and exponential functions. Because particles can be anywhere along infinite dimensions, wavefunctions are over infinite space and number of possible system phase-space states is infinite. In phase space, particles have possible trajectories (path distribution), each with different probability. Previous positions and momenta determine only probabilities of later paths. The sum of all probabilities equals 1 = 100% that the particle is somewhere. All possible states exist simultaneously and evolve independently. In quantum mechanics, particle measurement causes only one phase-space state/path to be observed. That state tends to have the highest probability.
particle systems: superposition
In quantum mechanics, particles have wavefunctions. Particle systems have wavefunctions that are particle-wavefunction linear combinations (superposition), just as electromagnetic waves superpose. Particle wavefunctions evolve independently, just as electromagnetic waves are independent. Waves do not have multiplicative or dependent effects on each other.
Because wavefunctions are complex-number functions, wavefunctions can add in two ways, constructive interference, A + B, and destructive interference, A - B = B - A. It is like positive and negative momentum, as in a reverberating system. Both superpositions are possible phase-space states, with probabilities.
particle systems: normalization
For linear equations, dividing all terms by any number results in equivalent equations, with same equation solutions. Dividing by any number only changes term coefficients/weights. Therefore, only coefficient/weight ratios determine equation meaning. Making sum of squared coefficients/weights equal one makes total probability 1 = 100% (normalization). Normalizing weights reflects the physical meaning of quantum-mechanics linear equations, that all state probabilities add to 100%.
particle systems: action at a distance
Physical processes can create two particles simultaneously, making two-particle systems. Both particles share one system wavefunction and have related energy levels. Measurement on one particle immediately affects system wavefunction and, by conservation laws, determines states of both particles, even if other particle is far away. State determination happens faster than light speed (action at a distance), appearing to send information faster than light and so violate relativity. However, observer knowledge of newly determined state happens only after information travels at light speed back to observer, so relativity is intact.
particle systems: energy partitioning
As in classical mechanics, for a specific total energy, quantum-mechanics phase-space wave interactions transfer total energy among wave frequencies so that energy distribution (Boltzmann distribution) is the wave-frequency distribution with maximum number of system states. Only waves with frequencies that make their energy less than half total energy can be in the distribution.
wave-particle duality
Quantum mechanics combines ideas about particles and waves (wave-particle duality). Particles have energies. Waves have wavelengths and positions. To calculate energies, quantum mechanics uses particle properties. To calculate positions, quantum mechanics uses wave properties.
waves
Quantum-mechanical particle phase-space waves extend infinitely in space and time. Wave equations have no initial conditions or boundaries.
waves: wave packets
Superposing many similar-frequency waves cancels amplitudes in most places but increases amplitude in a small space-time interval (wave packet). Wave packets are particles. Particles have many similar-frequency waves. Wave-packet frequency varies directly with particle energy. Wave-packet amplitude varies directly with particle-wave phase range, because narrower phase range makes higher amplitude.
waves: particle energies
Waves have frequency and wavelength. Quantum-mechanics wave-equation solutions are particle wavefunctions. Periodic solutions are true at lowest frequency and at all integer multiples (harmonics/overtones) of that frequency, because those waves have the same phase. Other frequencies have different phases and are not solutions. Wavefunction frequency varies directly with particle energy. Waves with higher frequency have higher energy, because field change is more when wavelength is less, so momentum change is more. Therefore, wavefunctions represent a series of possible particle energies. Energy levels differ by energy quanta. Particle-wave frequencies and energies are not continuous but discrete.
waves: position
Waves have frequency and wavelength and occupy space and time intervals. Waves cannot be at points. Particles and fields are waves and so have no definite position.
waves: shapes
Reverberations cause resonance and standing waves. Standing waves have different shapes depending on space boundaries and spin and orbit rotations. String waves have nodes at ends. For overtones of fundamental frequency, string waves have nodes at regular intervals. Molecule electrons have spherical s orbits and p, d, and f orbits with nodes. Wave shapes reflect average field density and probabilities that particle is in those space regions. In dynamic systems, wave shapes can vary over time.
waves: uncertainty
If wave has higher wavelength, position interval is wider and is less certain, but momentum and energy change, measured as less steep wave slope, is slower and so more certain. If wave has lower wavelength, position interval is narrower and is more certain, but momentum and energy change, measured as steeper wave slope, is faster and so less certain. Therefore, both position and momentum cannot be specific, and one or the other, or both, are uncertain.
waves: complex-number exponential functions
Wavefunctions are sine or cosine (trigonometric) functions. For frequency f, amplitude A, and position x, field = A * sin(2 * pi * f / x). For period t, field = A * sin(2 * pi * x / t). For wavelength l, field = A * sin(2 * pi * x / l).
Because e^(i*a) = cos(a) + i * sin(a), where a is angle in radians and is real, wavefunctions are complex-number exponential functions. sin(a) = (e^i * a - e^-i * a) / (2*i). cos(a) = (e^i * a + e^-i * a) / 2. Phase-space particle wavefunctions can superpose constructively and destructively, because they are complex-number functions.
quantum mechanics and projective geometry
Quantum mechanics has elements of projective geometry, which account for its non-local properties because projective geometry has no distance or between-ness. (General relativity is about metric geometry. Quantum mechanics is relativistic, so it also has metric properties.)
Particles have finite discrete physical-property values {quantum, quantity} {quanta, quantity}. Physical-property values do not vary over a continuous range but have definite values. For example, particle energies have discrete levels and do not have intermediate energies. Discrete physical-property values differ by an amount.
particle properties
Masses are aggregations of particles and so have quanta. As they change speed, masses add or subtract relativistic mass by quanta. Charges are aggregations of electrons and protons and so have quanta. As they change speed, charges add or subtract relativistic charge by quanta. Colors are aggregations of quark colors and so have quanta. As they change speed, colors add or subtract relativistic color by quanta. Strangenesses are aggregations of strangeness and so have quanta. As they change speed, strangenesses add or subtract relativistic strangeness by quanta.
Photons, gluons, bosons, and gravitons (exchange particles) have discrete energies, momenta, and angular momenta (such as spin). Light does not change frequency or wavelength as it travels in vacuum, or as it encounters other electric charges or magnetic fields. Gluons, bosons, and gravitons do not change as they travel or encounter fields. Therefore, forces, energies, and momenta have quanta.
maximum value
Relativity limits values to below maximum, because only infinite energy can make massive particles reach light speed.
Doppler effect
Because light speed is always constant, light sources moving toward or away change light frequency and wavelength. The change occurs at the source, so light does not change frequency and wavelength. as it travels or as it encounters other electric charges or magnetic fields.
Light travels at constant speed. If wavelength decreases, frequency increases. If wavelength increases, frequency decreases. If object is moving away, Doppler effect makes wavelength increase and frequency decrease. If object is moving closer, Doppler effect makes wavelength decrease and frequency increase. Faster motions make greater Doppler effects.
Time dilation is not about Doppler effect, because light is not clock, and light travels at light speed, not lower speed.
ground-state energy
Particle energies have a minimum value (ground state) above zero, because particles have phase-space waves, and waves propagate and so have minimum motion. Particles must move so they cannot have zero energy. Propagating waves have frequency and wavelength. Waves cannot have zero frequency, so waves have a lowest frequency (fundamental frequency) and so lowest possible energy. Electromagnetic-wave energy is frequency times Planck constant.
Because waves have wavelengths, they have uncertain position. Because waves have frequencies, they have uncertain momentum, and uncertain momentum requires minimum energy (uncertainty principle).
energy levels
Waves with the same phase satisfy the Schrödinger wave equation. Therefore, particles can have phase-space waves with harmonic frequencies. Fundamental-frequency harmonics determine allowed energy levels. See Figure 1. Higher frequencies have more energy.
Adjacent wave frequencies differ by fundamental frequency. The energy quantum varies directly with a function of particle phase-space wave fundamental frequency. As frequencies increase, energy differences decrease.
frequency
Wavefunctions with harmonic frequencies solve wave equation. Waves that solve the wave equation resonate in the system, like standing waves that constructively superpose to have net amplitude. Non-standing waves have zero amplitude. Possible standing waves have harmonic frequencies.
quanta
Particle energies, momenta, orbital and spin angular momenta, masses, forces, fields, velocities, accelerations, orbital radii, orbital periods, orbital frequencies, and properties have discrete levels separated by quanta. See Figure 2.
amplitude
Quantum-mechanical waves have amplitude. For any frequency, amplitude relates to probability that particles currently have that wave frequency.
system size
High-energy systems follow quantum mechanics, but phase-space wave wavelengths are too small to detect, so such systems do not appear to have quanta. Physical systems with very small energy or momentum differences, such as subatomic particles, atoms, and molecules, have measurable phase-space wave wavelengths, and such systems require quanta to describe their behavior correctly. See Figure 3. Some quantum-mechanical systems have large space and time differences.
In quantum mechanics, fields {quantized field} have quanta. Particles are like field singularities, vortexes, or discontinuities.
In quantum mechanics, particle and field quanta are at the lowest reductionist level. There is no subquantum world {subquanta}. Subquanta are smaller than Planck time, distance, charge, and mass. Subquantum interactions occur within Planck time, distance, charge, and mass. At subquantum sizes, space, time, forces, and energies do not exist or are indistinguishable. There is no gravity, electromagnetism, stromg or weak nuclear force, distance, time, or mass.
Particle energy E and particle phase-space wave frequency f (in cycles per second) are directly proportional by constant h {Planck constant}| {Planck's constant}: E = h * f, so h = E / f. Planck constant unit is energy times time, and action in physics is energy times time, so Planck constant is quantum of action {quantum of action} {action quantum}. h = 6.626 * 10^-34 Joule-seconds or 4.136 * 10^-15 eV-s.
Particle momentum p and particle phase-space wave wavelength w are inversely proportional by Planck constant: h = p * w. For light, E = h * f = h * c / w, so h = p * w. Momentum times distance is action in physics.
For angular frequency, radians per second, Planck constant divides by 2 * pi {reduced Planck constant} {Dirac constant} {h-bar}: h-bar = h / (2 * pi). h-bar is the quantum of angular momentum.
In quantum mechanics, phase space includes particle positions and momenta and so includes physical space. Particle systems have phase-space waves that determine probabilities of particle positions and momenta at times. In bounded space regions, such as atoms, molecules, and boxes, particles have resonating phase-space waves, with stationary points at boundaries, whose frequencies are harmonics. For example, a particle in a box has phase-space waves, with stationary points at box walls, which have fundamental frequency, twice fundamental frequency, thrice fundamental frequency, and so on. Phase-space wave frequencies determine energies, so system energies are discrete and in series: E0, E1, E2, and so on. Energy-level differences are quanta that are functions of fundamental frequency.
Because energy has quanta, momentum and angular momentum (including spin) have quanta. Electron experiments have determined the angular-momentum quantum unit to be h-bar / (2)^0.5. Momentum has quantum: h / (phase-space wave wavelength). Energy has quantum: h * (phase-space wave frequency). Electron experiments have determined that action has quanta, so energy times time, and momentum times distance, have quanta.
Because a continuous quantity times a discontinuous quantity would make a continuous quantity, for action to have quanta, time and length must have quanta. The quantum-mechanical uncertainty principle depends on particle-wave properties, relates indeterminacies in particle energy and time (or momentum and position), and so relates energy uncertainty to time uncertainty: dE * dt >= h. In space-time, maximum particle energy is where particle gravity has quantum effects and makes space-time discontinuous: 1.22 * 10^19 GeV. By the uncertainty principle, minimum time is then 10^-43 seconds (and minimum length is 10^-35 meters).
Maximum particle energy, 1.22 * 10^19 GeV, is where gravity has quantum effects and makes space and time discontinuous. Field theory no longer applies. Space is foam-like and loops and distorts, due to spin, and has no dimensionality.
Particles have phase-space waves. Particle momentum varies directly with particle-wave wavelength. Wavelength varies directly with time. Because momentum uncertainty times length uncertainty must be less than Planck constant, by the uncertainty principle, at maximum particle energy, quantum length unit {Planck length} is 1.6 * 10^-33 centimeters (1.6 x 10^-35 meters). Because space-time is no longer continuous, phase-space waves cannot have frequency greater than 10^43 Hz and wavelength less than 10^-35 meters.
Planck length depends on gravity strength and so gravitational constant g, electromagnetism strength and so light speed c, and action quantum Planck constant h: (h-bar * g / c^3)^0.5, where h-bar is Planck constant h divided by (2 times pi). h is the quantum of action, and h-bar is the quantum of angular momentum, so Planck length is the quantum of length. Planck length is distance light travels in Planck time.
Planck area quantum is 10^-66 cm^2. Planck volume quantum is 10^-99 cm^3.
Planck-length-diameter black-hole mass {Planck mass} is 10^-5 gram. Particle gravity has quantum effects and makes space-time discontinuous. Because particles are waves, if position uncertainty equals Planck length, gravity uncertainty is highest. Field theory no longer applies. Space is foam-like, due to spin, and has no dimensionality.
At universe origin or soon after, universe had Planck-length diameter. Space-time was discontinuous. Field theory no longer applies. Space is foam-like, due to spin, and has no dimensionality. When universe grew larger than Planck-length diameter, space became continuous, and temperature {Planck temperature} was 10^32 K.
Maximum particle energy, 1.22 * 10^19 GeV, is where gravity has quantum effects and makes space and time discontinuous. Field theory no longer applies. Time loops and distorts, due to spin, and has no dimensionality.
Particles have phase-space waves. Particle energy varies directly with particle-wave frequency. Frequency varies inversely with time. Because energy uncertainty times time uncertainty must be less than Planck constant, by the uncertainty principle, at maximum particle energy, minimum time unit {Planck time} is 5.391 * 10^-44 seconds. Because space-time is no longer continuous, phase-space waves cannot have frequency greater than 10^43 Hz and wavelength less than 10^-35 meters.
Planck time depends on gravity strength and so gravitational constant g, electromagnetism strength and so light speed c, and action quantum Planck constant h: (h-bar * g / c^5)^0.5, where h-bar is Planck constant h divided by (2 times pi). h is the quantum of action, and h-bar is the quantum of angular momentum, so Planck time is the quantum of time. Planck time is time light travels Planck length.
Time {chronon}| {time quantum} for light to travel (classical) electron radius is 10^-24 seconds.
Event-quantum time intervals {instanton}| are non-linear waves, lasting for one electronic transition or one quantum tunneling.
Past events determine, or at least constrain, future events. Because other events are too far away in space-time, only a subset of past events affects an event. In phase spaces, past states determine, or at least constrain, future states {consistent histories}. Because other states are too far away in phase space, only a subset of past states affects a state. In most systems, most states do not affect a future state. In quantum mechanics, systems have an infinite number of different consistent histories, each with a probability. Without quantum mechanics, completely determined systems have one consistent history.
In quantum mechanics, particle-system phase-space states have probabilities. States that do not happen have as much information as states that do happen. In quantum mechanics, because they had probability to happen, states that did not happen {counterfactual, quantum mechanics} can cause physical events/states on the same or other particles, because they collapse the wavefunction without interacting with the particle property/event/state. Measuring for a particle state that does not happen {null measurement} {interaction-free measurement} can gain information about another system particle or state without affecting that particle or state.
Particles simultaneously try all possible phase-space trajectories. Trajectories go directly (direct-channel) or indirectly (cross-channel) from one system energy level to another. The probability that the system reaches an energy level is the sum {sum over paths} {sum over histories} of renormalized path probabilities for direct-channel and cross-channel paths to that energy level.
Because bosons have integer spins, when previously independent identical-state bosons interchange, their wavefunctions stay the same as the other. Bosons have Bose-Einstein statistics. Therefore, interactions can bring two now-interdependent bosons to the same state. In a system, two bosons can be in the same state.
Because fermions have half-unit spins, when previously independent identical-state fermions interchange, their wavefunctions become the negative of the other. Fermions have Fermi-Dirac statistics. Therefore, no interaction can bring two now-interdependent fermions to the same state. In a system, no two fermions can be in the same state {exclusion principle} (Pauli exclusion principle).
To show directly that physics is non-local, measure entangled-electron spins {Bell experiment}. Electrons are indistinguishable. Around any measuring axis, electron spins have only two, clockwise or counterclockwise, angular-momentum states. For systems with zero total angular momentum, one electron has spin +1/2 and the other has spin -1/2. Experimenters can only measure one electron's spin, after wavefunction collapse, so system wavefunction before collapse had both electrons having both spins in superposition. Electrons 1 and 2 have spins along axes x, y, and z. If axes are indistinguishable and electrons combine randomly, states are 1x+2x-, 1x-2x+; 1x+2y-, 1x-2y+; 1x+2z-, 1x-2z+; 1y+2y-, 1y-2y+; 1y+2z-, 1y-2z+; 1z+2z-, 1z-2z+, so 6/12 of states involve x-axis, and 6/12 do not. If axes are distinguishable and electrons combine randomly, states are 1x+2x-, 1x-2x+; 1x+2y-, 1x-2y+; 1x+2z-, 1x-2z+; 1y+2x-, 1y-2x+; 1y+2y-, 1y-2y+; 1y+2z-, 1y-2z+; 1z+2x-, 1z-2x+; 1z+2y-, 1z-2y+; 1z+2z-, 1z-2z+, so 10/18 of states involve x-axis, and 8/18 do not. However, system has equal probability to start with 1x+2x- and 1x-2x+, so 2 x-axis states must be left out, making 9/18 of states involve x-axis, and 9/18 do not. If axes are indistinguishable and electrons entangle, states are xx, xy, xz, yy, yz, zz, so 3/6 of states involve x-axis, and 3/6 do not. If axes are distinguishable and electrons entangle, states are xx, xy, yx, xz, zx, yy, yz, zy, zz, so 5/9 of states involve x-axis, and 4/9 do not. Bell experiment result confirms the last conditions, so, if there are no hidden variables, electrons entangle, and physics is non-local.
In hypothetical experiments with closed boxes, particle decay triggers processes that kill cats {Schrödinger's cat} {Schrödinger cat}. If decay probability is one-half, is cat half-alive and half-dead inside box until observed? Is cat dead when particle decays, observed or not? In which state is cat after event and before observation? Less and less classical objects and events can replace cats, until replacements are quantum events and particle decay triggers a quantum state, so when does quantum-event wavefunction collapse?
In classical physical space, particles have definite positions and momenta, not probabilities of positions and momenta. If physical space has no external forces, positions and momenta are independent. If physical space has force fields, position change changes momentum in only one way, according to energy conservation. Because particles have definite positions and momenta, and classical configuration space has only real numbers, classical configuration space has no real-number/imaginary-number interactions and so no waves. The Hamiltonian function represents energy as a function of momentum (kinetic energy) and space (potential energy) coordinates.
In quantum-mechanical physical space, particles have probabilities of positions and momenta. Quantum-mechanical physical space has energy conservation, but positions and momenta are not independent, so energy-conservation equation (Schrödinger equation) and S-matrix theory, which relate kinetic-energy change and momentum to potential-energy change and position, have complex numbers. Exponentials with complex-number exponents represent cosine and sine waves. (Maxwell's equations relate kinetic-energy change and momentum to potential-energy change and position, and solutions are electromagnetic waves.) Frequency is time derivative. Wave number is spatial derivative. The time derivative introduces an imaginary number to multiply the time derivative to give a real number.
Quantum-mechanical configuration space (phase space) has complex-number particle position and momentum coordinates. Along each configuration-space dimension, real and imaginary numbers interact to make helical scalar waves {matter wave}| {de Broglie wave} {probability wave}.
scalar
Electromagnetic waves are vector waves, because electric and magnetic forces and fields have direction, electromagnetic waves propagate in a direction, and energy travels in that direction. Matter waves are scalar, because they are not about forces or fields, have no energy, and do not propagate and so do not travel and are standing waves. Scalar waves have amplitude but no direction.
phase space
Matter waves are not in physical space.
wavelength
Wavelength determines possible particle positions and momenta, at maximum-displacement positions. Frequency and phase affect amplitude.
amplitude and probability
Amplitude determines probability that particle is at that position or momentum.
frequency and kinetic energy
Particle kinetic energy E determines matter-wave frequency f: E = h * f, where h is Planck constant. For higher energies, matter waves have higher frequencies and lower wavelengths. Particle momentum p determines matter-wave wavelength w: h = p * w. Theoretical matter-wave velocity v increases with particle kinetic energy: v = f * w = (E/h) * (h/p) = E/p.
transverse wave
Real and imaginary number interactions make transverse waves around each phase-space dimension.
length
Matter waves are in configuration space, which has infinitely-long dimensions, so matter waves are infinitely long. By uncertainty principle, matter waves extend through all space, but with low amplitude outside physical system.
no propagation and no energy
Because they are infinitely long, matter waves do not propagate, are standing waves, and have no travel, no velocity, no energy, and no leading or trailing edge. Matter waves resonate in phase space.
positions, points, and intervals
Waves require one wavelength to be a wave, so there is no definite position. For waves, positions cannot be points but are one-wavelength or half-wavelength intervals.
solidity
Matter waves have width of at least one wavelength, so they cause matter to spread over space, not be at points. Matter waves make matter have area, and matter appears solid.
momentum and position
In quantum mechanics, unlike classical mechanics, momentum and position are not independent, because amplitude relates to position, frequency relates to momentum, wave amplitude-change rate relates to wave frequency-change rate, wavelength relates to position uncertainty, and amplitude-change rate relates to momentum uncertainty.
particle sizes
Large objects have high matter-wave frequencies. At high frequencies, matter-wave properties are undetectable, because wavelengths are too small, so classical mechanics applies. Small objects have low matter-wave frequencies, so atomic particles have detectable quantum properties.
waves and quanta
Resonating waves have fundamental frequency and harmonic overtones. Particles have matter waves with harmonic frequencies. Harmonic frequencies correspond to a series of positions or energy/momentum levels, separated by equal amounts (quantum).
Waves change frequency without passing through intermediate frequencies. No intermediate frequencies means no intermediate positions or energies/momenta. Matter waves explain why particles have discrete energy levels, separated by quanta, and why, during energy-level transitions, particles never have in-between energy levels. Particles also have discrete locations, separated by quantum distances.
physical systems
In free space, particle matter waves have a small range of frequencies and superpose to make a wave packet. Particle systems superpose particle matter waves to make system matter waves. Non-interacting particles have dependent matter waves that add non-linearly (entangle). In atoms and molecules, electrons, neutrons, and protons have phase-space matter waves that represent transitions among atomic orbits.
Electrons cannot be near nucleus, because then electron matter-wave interacts with proton matter-wave, and atom collapses.
philosophy
Perhaps, matter waves are particles, only associate with particles, are mathematical descriptions, or are all that observers can know.
Matter-wave wavelength equals Planck constant divided by momentum {de Broglie relation}|.
In quantum mechanics, matter-wave amplitude determines probability that particle is at that position. Matter waves are infinite and so have positive amplitude at all space points. Therefore, unlike classical mechanics, particles have a probability of being outside potential-energy barriers {tunneling}|.
At barriers, particle waves reflect back or refract through. Particles with higher matter-wave frequency and more energy have more refraction. As difference between barrier potential energy and particle energy increases, reflection {anti-tunneling} increases.
Matter waves are infinitely long. Because particle matter waves have fundamental frequency and its harmonics, particles have an infinite number of different-frequency matter waves. Because particles interact with other universe masses and charges, particles have matter waves differing in wavelength by infinitesimally small amounts. Superposition of an infinite number of infinitely long waves, differing in wavelength by infinitesimally small amounts, makes significant amplitude {wave packet}| in one region and insignificant amplitudes in all other regions. Particles are matter-wave packets.
time
Over time, as superposition makes different results, wave packets can disappear and reappear. Wave superposition can narrow or broaden wave-packet duration. and broadening frequency range.
size
Wave packets have three to ten oscillations, with maximum amplitude in center and no amplitude at edges. Longest wavelengths are in middle and smallest wavelengths are at edges. If wavelength range is small, packet is wide. If wavelength range is large, packet is narrow.
speed
Wave packets travel at particle speed, but wave-packet component waves travel at slower and faster speeds.
frequencies
Matter-wave-packet frequency varies directly with particle energy. Wave superposition can narrow or broaden wave-packet frequency range. If wave packet has many frequencies, volume is small, but energy is big. If wave packet has few frequencies, volume is large, and energy is small.
dispersion
Due to dispersion, wave packets spread out lengthwise and transversely.
In classical mechanics, positions and momenta (and energies and times) are independent variables, but in quantum mechanics, they are dependent variables and interact in wavefunctions. In classical mechanics, when two or more particles interact, system properties sum particle properties. In quantum mechanics, when two or more particles interact, system properties multiply and sum particle properties, and particle wavefunctions combine constructively and/or destructively to make a system wavefunction {entanglement}|. If two (indistinguishable) particles entangle, they both travel together on all possible state paths available to them, and they interfere with each other's independent-particle wavefunctions along each path. For example, two particles created simultaneously form one system with one wavefunction.
Entanglement does not put particles into unchanging states (that observers measure later). Neither do particle states continually change state as they move through space-time (not like independent neutrinos, which change properties as they travel). Therefore, observation method, time, and space position and orientation do not determine observed particle state. In quantum mechanics, particles have probabilities, depending on particles and system, of taking all possible space-time and particle-interaction paths, and measurement finds that the particle has randomly gone into one of the possible particle states.
system wavefunction
When two particle wavefunctions add, system-wavefunction frequency is the beat frequency of the two particle-wavefunction frequencies, and is lower than those frequencies. System wave packet has smaller spatial extension than particle wave packets, and has higher amplitude (more energy) at beat-frequency wavelengths. Quantum-mechanical particle and system wavefunctions have non-zero fundamental frequency and its harmonic frequencies and have non-zero amplitudes over all space and time. Systems spread out over space and time.
system wavefunction decoherence
After entanglement, system wavefunction lasts until outside disturbances, such as measurement, particle collision or absorption, and electromagnetic, gravitational, or nuclear force field, interact with one or more particles. At that definite time and position, system wavefunction separates into independent particle wavefunctions (decoherence). Whole system wavefunction ends simultaneously over whole extent.
measurement
By uncertainty principle, experimenters can precisely measure either particle energy or particle time (or momentum or position) but not both. After two entangled particles separate, separate instruments can measure each particle's energy (or momentum) precisely and simultaneously and then communicate to determine the exact difference.
measurement: speculation
Perhaps, unobserved particles and systems are two-dimensional (but still in three-dimensional space). Observation then puts particles and systems into three dimensions. People observe only three-dimensional space. For example, observers see that gloves are right-handed or left-handed. Perhaps, unobserved quantum-mechanical-size gloves actually have no thickness and so have only two dimensions, so unobserved right-handed and left-handed gloves are the same, because they can rotate in three-dimensional space to superimpose and be congruent. Perhaps, unobserved clockwise and counter-clockwise particle spins are two-dimensional and so are equivalent. (Note that a two-dimensional glove appears right-handed or left-handed depending on whether the observation point is above or below the glove.)
Perhaps, unobserved particles and systems randomly, continually, and instantaneously turn inside out (and outside in), in three-dimensional space. Observation stops the process. For example, turning a right-handed glove inside out makes a left-handed glove, and vice versa. Perhaps, unobserved quantum-mechanical-size gloves continually and instantaneously turn inside out in three-dimensional space and so are equally right-handed and left-handed. Perhaps, unobserved clockwise and counter-clockwise particle spins continually interchange. (Note that a glove appears right-handed or left-handed depending on when the process stops.)
Perhaps, unobserved quantum-mechanical-size particle and system states are indeterminate and follow quantum-mechanical rules because space-time is not conventional four-dimensional space-time. Observation requires conventional three-dimensional space, and randomly makes definite three-dimensional particle and system states, with probabilities. Perhaps, time is not real-number time, but complex-number or hypercomplex-number time. Real-number times are separate, but imaginary-number times are not. Perhaps, space is not real-number space, but complex-number or hypercomplex-number space. Real-number distances are separate, but imaginary-number distances are not.
Observations measure real-number part of complex-number variables. Perhaps, wavefunction imaginary-number part continues after observation.
Perhaps, Necker cubes illustrate the effects of observation. Observer angle to Necker cube determines whether observer sees right-facing or left-facing Necker cube. Effects may be linear with angle or depend on cosine of angle.
interacting electrons and spin
If a process creates two electrons, momentum sum is the same before and after creation, by momentum conservation, and electrons move away from each other at same velocity along a straight line. Angular-momentum sum is the same before and after creation, by angular-momentum conservation. (If two separate electrons entangle, momentum sum and angular-momentum sum are the same before and after interaction.)
By quantum mechanics, measured spin is always +1/2 or -1/2. Because the electrons are in a system, one cannot know which has +1/2 spin and which -1/2 spin. Both electrons share a system wavefunction that superposes the state (wavefunction) in which first electron has spin +1/2 unit and second has spin -1/2 unit and the state (wavefunction) in which first electron has spin -1/2 unit and second has spin +1/2 unit, with zero total angular momentum in any direction. Two wavefunctions can superpose constructively (add) or destructively (subtract). Because two electrons are distinguishable, the two wavefunctions add, so system wavefunction is anti-commutative.
One possibility is that one particle has positive 1/2 unit spin along z-axis (motion line), and other particle has negative 1/2 unit spin along z-axis. See Figure 1.
After two particles interact and move apart, separate spin detectors can measure around any axis for first particle and around any axis for second particle, simultaneously or in succession. For example, the axes can be z-axis (motion line), x-axis, and y-axis. See Figure 2. Measuring spin around an axis fixes one electron's spin at +1/2 (or -1/2) and fixes the other electron's spin around an axis at -1/2 (or +1/2), to conserve angular momentum.
spin: possible axis and spin combinations
By quantum mechanics, left electron has spin +1/2 half the time and spin -1/2 half the time, around any axis, say z-axis. Around same z-axis, right electron always has opposite spin: left=z+ right=z- or left=z- right=z+. Around x-axis, right electron has opposite spin (while y-axis has same spin), same spin (while y-axis has opposite spin), opposite spin (while y-axis has opposite spin), or same spin (while y-axis has same spin): x-y+z-, x+y-z-, x-y-z-, x+y+z-; x-y+z+, x+y-z+, x-y-z+, x+y+z+. For right-electron z-axis compared to left-electron z-axis, spins are opposite all of the time: z+z-, z+z-, z+z-. z+z-. For right-electron x-axis or y-axis compared to left-electron z-axis, spins are same 1/2 of time and opposite 1/2 of time: z+x-, z+y+; z+x+, z+y-; z+x-, z+y-; z+x+, z+y+. See Figure 3. Because quantum mechanics has random probabilities, left and right electrons have same spin half the time and opposite spin half the time.
However, quantum mechanics with non-randomness (due to local real hidden factors) makes a different prediction. Non-random hidden factors correlate right and left spins, to conserve angular momentum. If left=x+y+z+, right=x-y-z-. If left=x-y+z+, right=x+y-z-. If left=x+y-z+, right=x-y+z-. If left=x-y-z+, right=x+y+z-. If left=x-y-z-, right=x+y+z+. If left=x+y-z-, right=x-y+z+. If left=x-y+z-, right=x+y-z+. If left=x+y+z-, right=x-y-z+. See Figure 4. For right-electron z-axis compared to left-electron z-axis, spins are opposite all the time. For right-electron x-axis or y-axis compared to left-electron z-axis, spins are same 4/9 of time and opposite 5/9 of time, higher than the 1/2 level for quantum mechanics. Local hidden variable theories correlate events through hidden variable(s), making probabilities non-random. Quantum mechanics has no more-fundamental factors and introduces uncertainties, and so is random. Therefore, correlated outcomes in classical theories have different probabilities than in quantum mechanics. Experiments show that outcomes are random, so there are no local hidden factors and/or no real hidden factors.
if infinite light speed
Perhaps, entanglement over large distances and times has no non-locality problems if light speed is infinite, as in Newton's gravitational theory. Assume that relativity is true but with light speed infinite. Time is zero for light, and speed is always infinite for all observers, so all objects are always in contact. However, light speed is finite.
action at distance
Wavefunctions do not represent physical forces or energy exchanges, so space and time do not matter. If system wavefunction does not decohere, system particles and fields remain connected, even over long duration and far distances. Experiments that measure energy and time differences, or momentum and position differences, show that particles remained entangled over far distances and long times, and that wavefunction collapse immediately affects all system particles and fields, no matter how distant (action at a distance). Seemingly, new information about one particle travels instantly to second particle. See Particle Interference, Scientific American 269(August): 52-60 [1993].
However, information about collapse only travels at light speed, preserving special relativity theory that physical effects faster than light speed are not possible. Observers must wait for light to travel to them before they become aware of information changes. All physical laws require local interaction through field-carrying particle exchanges, which result in space curvatures. All physical communication happens when particles are in contact and interact, so there is no actual action at a distance.
teleportation
After particle entanglement, particle wavefunctions have specific relations. By manipulating particle properties at interaction and at wavefunction collapse, experimenters can transfer particle properties from one particle to another particle, even far away, though the particles have no physical connection at collapse time.
Bombs can have photon or light pressure triggers. Bombs explode if trigger does not jam, but jamming happens often. How can testers find at least one working bomb without exploding it {Elitzur-Vaidman bomb-testing problem} [1993] (Avshalom C. Elitzur and Lev Vaidman)? Using photon entanglement can find good bomb without triggering it.
Particles can seemingly move from one place to another without ever being between the two places {teleportation}|. Teleportation requires that both locations share a particle pair {EPR pair}. Particles are identical, with entangled properties. For example, if one photon splits into two photons, new photons can be same-state superpositions. If instrument observes one particle's state later, it then knows other particle's state. If EPR pair exists, putting one pair member into one state can result in property disappearance at one location and other-pair-member property appearance at another location.
Instruments can measure momentum, position, energy, and time by absorbing energy and using clocks and rulers. However, instruments cannot simultaneously or precisely measure both particle momentum and position {uncertainty principle}| {Heisenberg uncertainty principle} {indeterminacy principle}, because measuring one alters information about the other. Instruments cannot simultaneously or precisely measure both particle energy and time, because they relate to momentum and position.
situation
The uncertainty principle is about measurement precision on one particle at one time and place. The uncertainty principle does not apply to different measurements on same particle over time. The uncertainty principle does not apply to simultaneous momentum and position, or energy and time, measurements on different particles.
wave packet
Particles have wavefunctions, so measurements are about wave packets. As particle moves through time and space, total uncertainty increases, because wave packet spreads out.
wave properties
Uncertainty follows from wave properties, because wave position and momentum, or time and energy, inversely relate. Energy and momentum depend on wave frequency. Position and time depend on wave amplitude. Measuring wave frequency or wavelength precisely prevents measuring wave amplitude precisely. Measuring wave amplitude precisely prevents measuring wave frequency or wavelength precisely. If momentum or position is specific, position or momentum must be uncertain. If energy or time is specific, time or energy must be uncertain.
At space points, wavefunctions that have high amplitude have precise position and timing. However, wavefunction slope is steep, so amplitude change between nearby points is large, so velocity change, momentum change, and energy change are large and so uncertain at that position. See Figure 1.
Wavefunctions with wide wave packets have large uncertainty. Wavefunction slopes are not steep, and amplitude change at nearby points is small, so velocity change, momentum change, and energy change are small in that region. Momentum is precise, while position is imprecise. Alternatively, energy is precise, while timing is imprecise. See Figure 2.
Waves that have just one frequency and wavelength have one momentum and energy. Only one wave can have no superposition and no cancellation anywhere in space or time, making wave equally present throughout all space and time, and so completely uncertain in position and time. See Figure 3.
Wavefunctions that have almost all frequencies and wavelengths have precise position and time, because waves cancel everywhere, except one space or time point. Wavefunctiond that have almost all frequencies and wavelengths have almost all momentum and energy levels, making wave momentum and energy very uncertain. See Figure 4.
Waves that have some frequencies and wavelengths have moderate uncertainty in momentum and energy and moderate uncertainty in position and time, because waves cancel, except at moderate-size wave packet.
Waves with two or three frequencies and wavelengths have beat frequencies where waves superpose. Beat frequency makes precise momentum and energy, but time and position are uncertain. See Figure 5.
Waves with harmonic frequencies and wavelengths have beat frequencies where waves superpose. Beat frequencies make precise momentum and energy, but time and position are uncertain.
measurement processes
Besides wavefunction effects, physical processes limit precision. To find precise frequency for energy and momentum takes time and space, so position and time information are uncertain. To find precise position and time takes high amplitude, so position and time information are uncertain. Uncertainty's physical cause is discontinuity, whereas uncertainty's quantum-mechanical cause is wave-particle duality, because particles are about momentum and energy and waves are about position and time, as shown above.
mathematics
Quantum of action is h, and energy over time is action. Therefore, energy uncertainty dE times time uncertainty dt equals at least Planck constant divided by 4 * pi: dE * dt >= h / (4 * pi).
dE = F * dx = (dp / (4 * pi * dt)) * dx, so dE * dt * (4 * pi) = dp * dx. Position uncertainty dx times momentum uncertainty dp equals at least Planck constant: dx * dp >= h.
dx = 4 * pi * dF, and dp = dN / 2. Phase uncertainty dF times phonon number uncertainty dN equals Planck constant divided by 2 * pi: dF * dN = h / (2 * pi).
energy levels
Electrons in lower atomic orbitals have higher frequency, kinetic energy, and angular momentum and lower time period and orbital diameter. Electrons in higher atomic orbitals have lower frequency, kinetic energy, and angular momentum and higher time period and orbital diameter. Therefore, higher orbitals have higher position uncertainty and lower momentum uncertainty.
For low-orbital and high-orbital electrons, photon absorption can cause electronic transition to adjacent higher energy level, increasing position uncertainty and decreasing momentum uncertainty. For low-orbital and high-orbital electrons, photon emission can cause electronic transition to adjacent lower energy level, decreasing position uncertainty and increasing momentum uncertainty.
For low-orbital and high-orbital electrons, photon absorption can cause electronic transition to non-adjacent higher energy levels, increasing position uncertainty and decreasing momentum uncertainty. For low-orbital and high-orbital electrons, photon emission can cause electronic transition to non-adjacent lower energy levels, decreasing position uncertainty and increasing momentum uncertainty.
Besides fundamental Heisenberg uncertainty, electron, proton, and neutron configuration changes affect measured amounts. Electronic transitions conserve energy, momentum, and angular momentum, so absorption and emission do not necessarily have the same photon frequency. Electrons cannot transition to same orbital.
two particles
Though instruments cannot measure either's time or energy, instruments can measure two particles' energy difference and time difference precisely and simultaneously. Such measurement can define one-ness and two-ness.
confinement
By uncertainty principle, particles confined to smaller regions or times have greater momentum and energy. In confined regions, even in vacuum, energy is high, allowing particle creation and annihilation.
matrices
In quantum mechanics, particle position and momentum are quantized and so are matrices (not scalars or vectors), with complex-number elements. Because particles have probabilities of being anywhere in space, matrix rows and columns have infinite numbers of elements, and matrices are square matrices. In quantum mechanics, position and momentum are not necessarily independent, but depend on the whole particle system.
Matrices represent electronic transitions between energy levels. Matrix rows are one energy level, and matrix columns are the other energy level. Matrix elements represent the probability of that electronic transition. Matrix elements are periodic to represent the possible quanta. The diagonal represents transitions between the same energy level and so has value zero. Near the diagonal represents transitions between adjacent energy levels and so has higher values. Far from the diagonal represents transitions between non-adjacent energy levels and so has lower values. Energy levels have ground state and no upper limit, so the matrices have infinite numbers of elements. There is no zero energy level.
For non-infinite-dimension square matrices with real elements, PQ = QP (commutative). For infinite-dimension and/or non-square and/or complex-number-element matrices, PQ <> QP (non-commutative). Matrix multiplication is typically non-commutative.
In quantum mechanics, particle action is the product of the momentum P and position Q matrices: action = PQ. For infinite-dimension square matrices with complex-number elements, PQ - QP = -i*h*I, where I is identity matrix and h is Planck constant, because action has Planck-constant units and complex number multiplication rotates the axes by pi/2 radians.
Though electrons and protons have strong electrical attraction, and outside electrical attractions and repulsions can disturb atom orbitals, electrons do not spiral into protons and collapse atoms. Because particles have matter waves, by the uncertainty principle, orbiting electrons cannot spiral into atomic nucleus {atom, stability}. See Figure 1.
waves
Particles have matter waves, whose harmonic frequencies relate to particle energy levels.
uncertainty
Waves by definition must be at least one wavelength long. Therefore, particle waves have location uncertainty of at least one wavelength. Particle waves have time uncertainty of at least one period, which is one wavelength divided by light speed. Particle waves have momentum uncertainty of at least Planck constant divided by wavelength. Particle waves have energy uncertainty of at least Planck constant divided by period. Particle waves make the uncertainty principle.
energy
By uncertainty principle, particles must move, and so they cannot have zero energy. Particles cannot have zero energy because they cannot have zero motion, because that violates conservation of both energy and momentum. Lowest particle energy is first-quantum-level ground-state energy.
orbits
Electron orbits have quantum distances from nucleus and take quantum durations to orbit nucleus. In lowest orbital, electron position uncertainty has same diameter as orbital. Electron can be anywhere in that region around nucleus. In lowest orbital, electron time uncertainty is same period as orbital rotation. Electron can be anywhere in that interval. In lowest orbital, electron is already at closest possible distance and smallest possible time.
transitions
From lowest orbital, electrons cannot go to lower orbits, because there are no lower energy levels. They cannot lose more energy, because if energy decreases then time increases, by uncertainty principle, making orbital go higher. They cannot lose more distance because if distance decreases then energy must increase, by uncertainty principle, making orbital go higher. Therefore, lowest orbital has lowest energy, smallest distance, and shortest time. Lowest orbital already includes nucleus region, so it cannot be smaller.
kinetic and potential energy
In quantum mechanics and classical mechanics, electric-field positions relate to potential energies. In quantum mechanics, unlike classical mechanics, kinetic energy cannot completely convert to potential energy, and vice versa. Kinetic energy and potential energy have minimum energy level and cannot be zero.
energy quantum
First energy quantum is difference between ground-state energy and next-highest-orbital energy. Second energy quantum is difference between next-highest-orbital energy and third-orbital energy. Energy quanta are not equal. Energy quanta decrease at higher orbitals. Energy quanta relate to wave harmonic frequencies. Higher adjacent wave frequencies have smaller energy differences.
atom nucleus
Atomic nucleus occupies only 10^-5 volume inside lowest-electron-orbital volume. Nucleus protons and neutrons have energy, momentum, position, and time uncertainty and so have ground-state energies. Nucleus protons and neutrons have quantum energy levels.
Lowest-orbital electrons and highest-orbital neutrons and protons never collide, because electrons have lower orbiting energies, and higher orbital radii, than neutrons and protons.
electron-proton collision
At high-enough energy and beam collimation, electrons can collide with atomic nuclei, because increased energy can narrow position, by uncertainty principle. Such electrons are not orbiting, so this situation is not about atom stability.
Particle in enclosed space {particle in box} must have velocity, because particle has fixed position, so uncertainty is in momentum. If enclosed space is smaller, velocity must be more.
Electric field and magnetic field cannot be at rest {quantum fluctuation}, because then they have precise position and precise zero momentum and so violate uncertainty principle. All fields have random motion, even in vacuum where net energy is zero.
At quantum level, empty-space field fluctuation {vacuum polarization}| is infinite.
Two parallel uncharged metal plates attract each other by reducing vacuum-energy fluctuations and number of wavelengths between them {Casimir effect} {Casimir force}: energy density = c / d^4, where c is constant and d is plate distance. Energy at plate is zero. Interior energy density decreases, so exterior energy density increases and pushes plates together. Fewer particle histories with closed time-like loops are between plates.
Particles cannot be at rest {zero point motion}|, because then they have precise position and precise zero momentum and so violate uncertainty principle. All particles have random motion, even in vacuum where net energy is zero.
For energy transfers, particles act like particles. For determining locations, particles act like waves {wave-particle duality}|.
Matter waves have spatial/momentum effects and time/energy effects, which instruments cannot detect simultaneously {complementarity, quantum mechanics}. Particles have energy, and waves have positions. Instruments cannot determine particle properties and wave properties simultaneously. Experiments can be only complementary, because particles always have both wave and particle properties.
Wave, photon, or particle sources can send collimated beams through one or two slits, to a measuring surface {slit experiment, quantum mechanics} {two-slit experiment}. For one slit, beam makes medium intensity line across from slit. For two slits, beam makes line with four times medium intensity across from slit. It makes alternating intense and clear lines on both sides. First intense line to side has two times medium intensity. Second intense line has medium intensity. Third intense line has lower intensity, and so on. Beam waves constructively and destructively interfere.
quanta
Particles sent through two consecutive pinholes create concentric rings on screen, as waves do. Particles sent through two adjacent pinholes make stripes perpendicular to line between pinholes on far screen, as waves do. If one slit closes, ring pattern appears. If slits alternate between closed and open, two ring patterns appear. If detector is at one slit, ring pattern appears. If detectors are poor, feeble stripe pattern appears. If half-silvered mirror is after one slit in particle-stream path, and both paths reflect from mirrors, stripe pattern appears.
wave
Particle motions are not single trajectories but diffract, as waves do. Wave theory accounts for all results. Matrix theory can account for results if slits act together to make periodicity.
Paths entangle, so electrons that pass through beam splitter and go past solenoid coil have quantum interference {Aharonov-Bohm effect}, though no electromagnetic field is outside solenoid coil.
Detectors can be after location at which particles must choose which path to take and can turn on after particles pass decision point {delayed-choice experiment} (Wheeler) [1980].
In two-slit experiments (Scully and Drühl) [1982], tagger {quantum eraser} can be in front of each slit to make spin clockwise or counterclockwise along axis. Screen can detect particle location and spin. There is no interference. Waves are present, but they cancel. Before screen, place spin tagger that always results in same spin. There is interference. Waves do not cancel.
down conversion
A photon can become two photons, each with half the energy {down-converter}. In beam-splitter experiments (Scully and Drühl) [1982], a down-converter can be on each path, to make one photon that continues on that path {signal photon} and one photon {idler photon} that is detected {delayed-choice quantum eraser}. Waves do not interfere.
When information about idler photon is random, because idler photon splits and goes on ambiguous paths, waves interfere. Instruments can receive the information before or after signal photons hit, by any amount of time or space. Waves are always present, but they can cancel.
detector
In two-slit experiments, particles make interference pattern when observed. If detector capable of knowing if particle went through left, right, none, or both slits is after slits, and it indicates that each particle goes only through either left or right slit, never both or none, there is no interference pattern.
If detector can operate without affecting particle in any way, and observer observes it, there is still no interference pattern.
If observer does not observe detector, there is interference pattern, even if detector puts the information in memory awhile and then deletes memory. This suggests that just gaining information is enough to end interference [Seager, 1999].
The quantum mechanics wave equation, which relates kinetic and potential energy to total energy, has complex-number, single-valued, continuous, and finite solutions {wavefunction}|. The wave equation, and its wavefunction solutions, are about abstract phase space, which includes space-time and describes system momenta and position or energy and time states. Wavefunctions represent possible physical-system energy levels and positions, and their probabilities. Wavefunctions correlate particle energies and times or particle momenta and positions. Wavefunctions typically depend on position, because energy includes potential energy. Wavefunctions typically depend on time, because energy includes kinetic energy. Wavefunctions are not physical waves and have no energy or momenta, but mathematically represent system properties.
Wavefunction is about infinite-dimensional abstract Hilbert space, in which wavefunction rotates as a unitary function and is deterministic.
energy and frequency
Because particle matter waves resonate in physical systems, wavefunctions have fundamental frequency and harmonics of fundamental frequency. System energy levels depend on wavefunction frequencies. System energy levels are discrete, and quanta separate energy levels. High-frequency waves have high energies. System boundary conditions set used or injected energy and wave fundamental frequency and harmonics of fundamental frequency.
amplitude, intensity, and probability
Wavefunction amplitudes are complex numbers that reflect physical-system position, time, energy, or momentum relations. Probabilities that particles are at locations depend on wavefunction amplitude for that location. Probabilities are linear and add, so probability of a set of states is sum of state probabilities. Wavefunction amplitudes can normalize, so sum of all state probabilities is one.
Intensity is absolute value of wavefunction-amplitude squared: wavefunction complex conjugate times position vector times wavefunction. Squared amplitude eliminates imaginary numbers and so is only real numbers. Absolute value makes only positive numbers. Intensities and energies are only discrete real positive numbers (eigenvalue). Amplitude squared absolute value relates to particle cross-section, collision frequency, and scattering-angle probabilities, and so to state probabilities.
wavelength
Waves have wavelength and so cannot be at a point but must spread over one wavelength. Particles have wave properties and can be at any point in region one-wavelength wide. Regions have wave amplitudes and so probabilities that particle is there.
resonance
In systems, reflected matter waves add constructively, and superpositions make standing-wave harmonic frequencies. Other frequencies cancel. Resonating fundamental wave has wavelength equal to system length or diameter and lowest-frequency. Fundamental-wave harmonic frequencies determine discrete possible particle energy levels.
deterministic
Wavefunctions are deterministic.
one particle
A one-particle system has a fundamental matter wave and its harmonics that determine possible particle positions and momenta. Harmonic wavefunctions are orthogonal/independent and linearly superpose. For particles with small momentum range and small position range moving along a straight line, wavefunctions are helices around the line with almost no amplitude at line ends and rising amplitude then falling amplitude near particle location.
one particle: definite momentum
For definite particle momentum along a straight line, position wavefunctions are helices around the line. If particle is at a well-defined position, helical waves have short wavelengths. If particle is at widespread positions, helical waves have long wavelengths.
one particle: no momentum
If particle has no momentum, momentum wavefunction is a straight line, and position wavefunction is constant.
one particle: definite position
For definite particle position along a straight line, momentum wavefunctions are helices around the line. If particle has high momentum, helical waves have short wavelengths. If particle has low momentum, helical waves have long wavelengths.
one particle: no position
If particle can be anywhere along a straight line, position wavefunction is a straight line, and momentum wavefunction is constant.
If energy times wavefunction, minus potential times wavefunction, is greater than zero, wavefunction oscillates {unbound state} {continuous spectrum}. Wavelength and quantum energy levels are too small to detect.
If energy times wavefunction, minus potential times wavefunction, is less than zero, wavefunction goes to zero {bound state} {discrete spectrum} only at special eigenvalues or else goes to infinity. At special eigenvalues, wavelength and quantum energy levels are large enough to detect.
Wavefunctions are complex-number functions with complex-number solutions, but intensity has positive real values {eigenvalue, quantum mechanics}.
Wavefunction amplitudes can adjust {renormalization, probability} {normalization, wavefunction}, so sum of all amplitude-square absolute values, or all energy-level probabilities, is 1 = 100%. Because systems have an infinite number of harmonic wavefunctions, without renormalization the sum of probabilities is infinite.
In quantum field theory, generalized wavefunctions {wavefunctional} are about higher spaces {field space}.
To find electronic-transition-series {series of electronic transitions} {electronic transition series} probability, multiply wavefunction complex-number amplitudes and then square product absolute value.
Abstract phase space describes system particle momenta and positions. Wavefunctions describe possible system particle positions and momenta states {state vector} {quantum state}. For example, in a system, a single particle has constant momentum and two possible positions. System has two (non-interacting) state wavefunctions, S1 and S2, with different probabilities depending on wave amplitude at the state, c1 and c2. Wavefunction W is sum of each state's amplitude times state wavefunction: W = c1 * S1 + c2 * S2. System wavefunction is a superposition of weighted state wavefunctions.
multiple particles
Particles have state wavefunctions at all possible positions and momenta. Particles can be independent or interact. If they are independent, particle wavefunctions multiply to make (linear) tensor products. Phase is not important for bosons, and tensor product commutes. Phase is important for fermions, and tensor product does not commute. If particles interact, system has entangled wavefunction.
Over time, Schrödinger-equation wavefunctions can change deterministically {unitary evolution, wavefunction}, as position, time, energy, or momentum change.
Isolated wavefunctions deterministically calculate future possible states. However, observing a particle measures particle position or momentum, putting particle into a definite phase-space state, and so cancels particle wavefunction {wavefunction collapse} {collapse of wavefunction}| {reduction of wave packet} {wave-packet reduction} {collapse of the wavefunction} {state vector reduction}. Wavefunction collapse is a discontinuity in physics. Collapse is time asymmetric. After observation, particle again has a wavefunction, until the nect observation.
observation and measurement
Observers and measuring instruments are too large to have observable wavefunctions, matter-wave wavelengths, matter-wave frequencies, or energy quanta. Observing and measuring cause particle interaction with a macroscopic system and make a new macroscopic system that includes the particle. Observers and instruments put particle wavefunctions into definite phase-space states {state preparation}, ready for measuring. Macroscopic systems have definite object positions and momenta.
Measuring requires that observer or instrument has definite phase-space state, and particle has definite phase-space state. Observers and instruments measure along one direction and detect particle position, time, momentum, angular momentum, or energy. Therefore, position, time, momentum, angular momentum, or energy observation/measurement operates on particle complex-number wavefunction and transforms it into a position, time, momentum, angular momentum, or energy real positive value. The value is any one of the set of possible different-probability quantum values (operator eigenfunction) described by the observer/instrument/particle wavefunction. State selection is completely random. Measurement results in a single value, not value superpositions or multiple values. The observer/instrument/particle wavefunction collapses to zero {measurement problem}. At measurement, particle phase-space state no longer exists, because particle wavefunction no longer exists.
operators
Measuring wavefunctions mathematically uses linear differential Hermitean operators.
causes
Measurements, absorptions, collisions, electromagnetic forces, and gravitational forces collapse particle wavefunctions. Gravitational effects can be gravitational waves, mass separation changes, gravitational self-energy changes, or fixed-star gravitational-field disturbances. Perhaps, measuring equipment is large and so affects wavefunction drastically (Bohr). Perhaps, collapse is large information gain (Heisenberg).
Perhaps, wavefunction collapse is due to particle and wavefunction properties. Perhaps, previous states have lingering wavefunctions that affect later wavefunctions. Perhaps, Gaussian wavefunction distributions coincide at random. Perhaps, wavefunctions have continual operators. Perhaps, wavefunctions are unstable every billion years {Ghirardi-Rimini-Weber} (GRW), so large masses collapse immediately (Giancarlo Ghirardi, Alberto Rimini, Tullio Weber).
Perhaps, wavefunction collapse is due to quantum mechanics. Perhaps, quantum fluctuations average {quantum averaging} to make definite energy states and space and time. Perhaps, cosmic inflation caused macroscopic-size quantum uncertainty and fluctuations {quantum uncertainty}.
wavefunctions and reality
Are wavefunctions just calculating devices, or do they exist in physical reality? Why do physical laws follow mathematical laws? How does perception relate to physical laws, mathematical laws, and material world? How does wavefunction collapse relate to physical laws, mathematical laws, and material world? How does wavefunction collapse relate to wavefunction time and space changes? How can observation/measurement and wavefunctions unify into a continuous explanation, rather than a discontinuous one?
alternatives: real wavefunctions
Perhaps, classical potential and quantum-mechanical potential both exist, so wavefunction is real. Measuring real wavefunction releases energy, starts wave fluctuations, and collapses wavefunction.
alternatives: undefined and defined states
Perhaps, particles have no wavefunction, so there is no collapse. Instead of wavefunctions, particles have only defined and undefined states. Undefined states can become one defined state. For example, particle density matrices represent possible different-probability physical states. Particle moves from undefined states to one state on the matrix diagonal. However, particles can be in superposed states, which matrices cannot represent. Particles can have only one or two possible states, which matrices cannot represent.
alternatives: subquanta
Perhaps, quantum levels involve even smaller properties, or quantities that cause them. However, particles have no hidden variables and so no subquanta.
alternatives: larger whole
Perhaps, physics has another conservation law about a larger whole. Observers and instruments measure only observable parts, while other parts are not observable. Whole system, observable and not observable, is deterministic, continuous, and time symmetric. For example, objects always travel at light speed, but some are time-like, and some are space-like. However, particles have no hidden variables and so no larger whole.
alternatives: two state vectors
Perhaps, quantum states have two phase-space state vectors, one starting from last wavefunction collapse and going forward in time and the other starting from next wavefunction collapse and going backward in time (Yakir Aharonov, Lev Vaidman, Costa de Beauregard, Paul Werbos) [1989]. Before and after phase-spaces are different. At events, forward-state vector happens first, and then backward-state vector happens. Their vector product makes density matrices, allowing smooth transitions between wavefunctions and collapses. This theory gives same results as quantum mechanics with one state vector. Forward and backward effects allow consistency with general relativity. However, time cannot flow backward, by general relativity.
alternatives: positivism
Perhaps, only measured results count, and wavefunctions are non-measurable things. However, experiments involving primitive measurements demonstrate that quantum state is deterministic and unique, so wavefunctions seem to have reality.
Entangled particle wavefunctions depend on each other, maintain phase relations, and have coherence. In isolated systems with entangled particle wavefunctions, system wavefunction continues to evolve deterministically. In non-isolated systems with entangled particle wavefunctions, measurements, absorptions, collisions, electromagnetic forces, and gravitational forces disturb particles and cause entangled superposed particle states to become independent {decoherence}|. System wavefunctions become non-coherent, and particle waves no longer interfere with each other {decoherent histories}, though observers only know this afterwards. System-state phase-space vector reduces to zero. Each particle is independent and has one position and one momentum.
Non-local large-scale gravitational processes eventually collapse all system wavefunctions {objective reduction}. Particle systems cannot remain isolated, because universe gravitation is at all space points.
For macroscopic systems without observers, macroscopic observation can separate states, so system states are distinct {state distinction principle} {principle of state distinction}.
Entangled particles stay in immediate and direct contact, by sharing the same system wavefunction, over any-size space or time interval {non-locality}|. Changes in one particle immediately affect the other particle, seemingly sending information faster than light speed. Conservation laws hold, because particle travels as fast as information, and same particle can go to both detectors. Perhaps, non-locality is due to quantum-mechanical space and time being discrete, foam-like, and looping.
Particles, energies, fields, and quanta are always in space-time. Physical objects and events happen only in space-time.
Wavefunctions are abstract non-physical mathematical objects that describe possible particle or system states and their probabilities. Particle and particle-system wavefunctions are not physical forces, are not energy exchanges, and are not objects in space-time. Wavefunctions describe all space-time points simultaneously. Waves have wavelength, and so are not about only one point, but all wave points at once. Wavefunctions account for and connect all space points, and so appear infinitely long.
As particles interact (and so form an interacting-particle system), the particle wavefunctions superpose to make a system wavefunction, in which all particle states depend on each other. Because wavefunctions connect all space, particles separated by arbitrary distances have states that affect each other. If one particle changes state, the other particle instantaneously changes state, no matter how far apart in space the particles are, because the system wavefunction (and all waves) collapse at all points simultaneously. Experiments that measure energy and time differences, or momentum and position differences, show that particles can remain entangled over far distances and long times, and that wavefunction collapse immediately affects all system particles, fields, and points, no matter how distant. (Because later times involve new wavefunctions, wavefunction collapse never changes particles at same place at different times.) State-vector reduction seemingly violates the principle that all physical effects must be local interactions, because coordinated changes happen simultaneously at different places.
Particle and system wavefunctions are about particles in indefinite states. Observation of one particle's definite state instantaneously collapses the system wavefunction and puts all system particles in definite states, no matter how far apart they are. No physical force or energy at the other particle causes the definite state, but the no-definite-state simultaneously changes to definite state {action at a distance}. The cause seemingly travels faster than light speed to make an effect. Therefore, the cause is non-physical.
Physical causes and effects must occur at one event in space-time. All physical communications, forces, and energies require local interactions through field-carrying particle exchanges in space-time. Physical interactions can have no action at a distance.
theories
Perhaps, wavefunctions reflect something physical that can account for action at a distance. Perhaps, particles can travel backward in time, from measured position to previous position, to make cause and effect at same space-time point. Perhaps, wavefunctions have retrograde wave components, so particles are always interacting at same space-time point. For example, in double-slit experiments, backward-flowing waves (from detectors to incoming particles) determine particle paths and explain whether wave or particle phenomena appear. However, general relativity does not allow time to flow backward. Furthermore, space-time points cannot have different times simultaneously.
theories: no-space-time
Perhaps, every space-time point touches an abstract outside-space-time structure. Perhaps, quantum foam has no-space-time in it. Perhaps, just as all sphere points touch sphere interior, all space-time points touch a no-space-time interior. By whatever method, every space-time point communicates with all others through no-space-time. No-space-time has no distances or time intervals, so space and time do not matter, and action at a distance can occur.
No-space-time is an abstract mathematical object, just as are quantum-mechanical waves. Perhaps, no-space-time carries quantum-mechanical waves.
Before measurement, particles can be said to be everywhere {Copenhagen interpretation}|, not necessarily close to the observed position. Because particle is everywhere, measured particle is always adjacent to other system particles, so there is no non-locality.
Spin-zero-particle decay can make two entangled coupled spin-1/2 particles, one +1/2 and one -1/2, which have one coherent system wavefunction {Einstein-Podolsky-Rosen experiment} {EPR experiment}. After particle-pair production, one particle always has spin opposite to the first, by conservation of angular momentum, but observation has not yet determined which particle has which spin. If an instrument detects one particle's spin direction and collapses the system wavefunction, the other particle immediately has the opposite spin, even over long distances. Einstein, Podolsky, and Rosen said instantaneous information transmission was impossible, so particles changed to the measured spins when the particles separated. Experiments showed that both particles have no definite spin until measured, so particles had superposed states until measured. By quantum mechanics, neither particle has definite spin-axis direction, so particles have superposition of +1/2 and -1/2 states until measured.
Experimenters must choose direction around which to measure spin and can measure in any direction. If they measure opposite direction, they can observe opposite spin. Therefore, particle production alone does not determine measured spin, and realism does not happen. Measuring system and particle together, as a new system, determine measured spin.
spin detection
If two spin-1/2 particles are in singlet state, three detectors oriented at -120, 0, and +120 degree angles perpendicular to moving-particle path can measure one particle's spin. Probability that both spins have opposite values is cos^2(A/2), where A is angle.
Named things have unique values {nominal level} {level of measurement} {measurement level} {absolute, measurement}. Name and value have one-to-one correspondence. Origin and units do not matter.
Different named things have value differences {interval level}. Affine linear transformations, such as t(m) = c * m + d, where m is value and c and d are constants, maintain differences.
For many named things, values have positions {ordinal level} in order. Monotone increasing transformations maintain order.
Values have ratios {ratio level} {log-interval level}. Power transformations, such as t(m) = c * m^d, where m is value and c and d are constants, maintain ratios. Linear transformations maintain ratio relations.
Interaction with matter collapses wavefunctions {measurement postulate}.
How do wavefunctions, such as electron fields, collapse everywhere simultaneously {quantum mechanical measurement problem}. Collapse is absolute, with no relativity.
Measurements can map directly to object properties {scale, measurement} {measurement scale}. Measurement relations can map directly to object-property relations.
Perhaps, measurement theory needs special prohibitions {superselection rules} on measurements.
For objective measurement, events {observable} must be independent of where or when they happen. Objective measurements cannot be functions of space or time coordinates.
Measurements need reference points, such as x=0, and measurement units, such as meter. By relativity, objective measurements cannot be functions of reference points or units.
Measured state is orthogonal to all other possible states, because if one state happens, others do not. Measured state can be along coordinate {primitive measurement}.
Measurements in systems with no waves, or with waves with no phase differences, can have any order {commuting measurement}. Primitive measurements commute, because they are not about phase, only about yes or no. Measurements in systems with waves and phase differences depend on sequence {non-commuting measurement}. Most measurements do not commute, because they find value or probability.
subjective measurement
In quantum mechanics, time and space are not continuous but have quanta. In phase space, momenta relate to positions, and energies relate to times, so events are functions of space and time coordinates. Because positions and lengths relate to momenta, events are functions of reference points and units. Objective measurement is not possible. Quantum mechanics has only subjective measurement.
interaction
To measure particle size, light must have wavelength less than particle diameter and so high frequency and energy. High energy can change particle momentum. Higher energy increases momentum uncertainty.
To measure particle momentum, light must have low energy, to avoid deflecting particle, and so long wavelength. Longer wavelength increases location uncertainty.
Measuring position requires different-frequency light wave than measuring momentum, so experiments cannot find both position and momentum simultaneously (uncertainty principle).
wavefunction collapse
Measuring disturbs particle and creates a new system of observer, instrument, and particle, with a new wavefunction. At actual measurement, the new system wavefunction collapses to zero. Measuring allows observing only one particle property.
operator
Momentum, energy, angular momentum, space, or time functions {operator, wavefunction} operate on wavefunction to find discrete positive real values (eigenvalue) of momentum, energy, angular momentum, space, or time, which are all possible outcomes, each with probability. Direct measurements project onto space or time coordinate or energy or momentum vector.
Projection operators operate on wavefunction and project onto space or time coordinate, or energy or momentum vector, to give discrete positive real values (eigenvalue) {direct measurement}, which are all possible outcomes, each with probability. Experimenters can know possible measured values and predict probabilities. However, values may be less than quantum sizes and so not measurable. Operators have same dimensions as particle.
Alternatively, experimenters can prepare a quantum system in a known initial state, have particle interact with prepared quantum system, separate particle and prepared quantum system, and then measure quantum-system state {indirect measurement}. Indirect measurements require entangling particle and prepared quantum system, to couple their states. Wavefunction collapse puts quantum system into a state that indirectly determines particle state. Quantum system can have same or more dimensions as particle.
Operators on wavefunctions produce discrete positive real values (eigenvalue) {positive-operator-valued measure} {positive operator-valued measure} (POVM).
Operators {projection operator, measurement} on wavefunctions can project values onto measurement axis.
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}.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225