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.
5-Physics-Quantum Mechanics-Quantum
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225