This force has been measured, and is a striking example of an effect purely due to second quantization.

The observed charge quantization, namely the fact that all known elementary particles carry electric charges which appear to be exact multiples of of the " elementary " charge, has led to the idea that hypercharge interactions and possibly the strong and weak interactions might be embedded in one Grand Unified interaction described by a single, larger simple symmetry group containing the Standard Model.

However, the use of canonical quantization has left its mark on the language and interpretation of quantum field theory.

showed that we can use macroscopic oil droplets on a vibrating surface as a model of wave – particle duality — localized droplet creates periodical waves around and interaction with them leads to quantum-like phenomena: interference in double-slit experiment, unpredictable tunneling ( depending in complicated way on practically hidden state of field ) and orbit quantization ( that particle has to ' find a resonance ' with field perturbations it creates — after one orbit, its internal phase has to return to the initial state ).

Assuming a full-scale sine wave signal ( that is, the quantizer is designed such that it has the same minimum and maximum values as the input signal ), the quantization noise approximates a sawtooth wave with peak-to-peak amplitude of one quantization level and uniform distribution.

Due to the density matching property of vector quantization, the compressed data has errors that are inversely proportional to density.

The quantization of the Hall conductance has the important property of being incredibly precise.

This phenomenon, referred to as " exact quantization ", has been shown to be a subtle manifestation of the principle of gauge invariance.

DAT has the ability to record at higher, equal or lower sampling rates than a CD ( 48, 44. 1 or 32 kHz sampling rate respectively ) at 16 bits quantization.

This form of distortion, sometimes called granular or quantization distortion, has been pointed to as a fault of some digital systems and recordings ( Knee & Hawksford 1995, Stuart n. d .: 6 ).

Optimal use of dither ( triangular probability density function dither in PCM systems ) has the effect of making the rms quantization error independent of signal level ( Dunn 2003: 143 ), and allows signal information to be retained below the least significant bit of the digital system ( Stuart n. d .: 3 ).

Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals.

However, the concept of coherent states has been considerably generalized, to the extent that it has become a major topic in mathematical physics and in applied mathematics, with applications ranging from quantization to signal processing and image processing ( see Coherent states in mathematical physics ).

DSD is 1-bit, has a sampling rate of 2. 8224 MHz, and makes use of noise shaping quantization techniques in order to push 1-bit quantization noise up to inaudible ultrasonic frequencies.

It has been shown to be a valid model in cases of high resolution quantization ( small relative to the signal strength ) with smooth probability density functions.

The Poission bracket has a foundation in quantum mechanics: Dirac's canonical quantization replaces the Poisson bracket with the commutator of quantum operators, denoted by hats (^):

Also since a digital computer has finite precision ( See quantization ), extra care is needed to ensure the error in coefficients, A / D conversion, D / A conversion, etc.

" Almost 80 years have passed since that time, but the problem of space quantization has still not been completely solved, although from time to time some researchers return to the topic, which confirms the ongoing timeliness of the problem.

In Geometry of Batalin-Vilkovisky quantization, Albert Schwarz has proven that the integral of a function H over a Lagrangian submanifold L depends only on the cohomology class of H and on the homology class of the body of L in the body of the ambient supermanifold.

At the classical level, a mass gap appears while, at quantum level, one has a tower of excitations and this property of the theory is preserved after quantization in the limit of momenta going to zero.

This explains also why the choice of z-axis for the directional quantization of the angular momentum vector is immaterial: an orbital of given ℓ and m ′ obtained for another preferred axis z ′ can always be represented as a suitable superposition of the various states of different m ( but same l ) that have been obtained for z.

" Redshift quantization " has also been called redshift periodicity, redshift discretization, preferred redshifts, and redshift-magnitude bands.

The QSHJE can be demonstrated to imply the Schrödinger equation with square-summability of the wave function, and thus quantization of energy, due to continuity conditions of the quantum potential, without any assumption on the probabilistic interpretation of the wave function.

After the advent of quantum mechanics, work done by Landau in 1930 predicted the quantization of the Hall conductance for electrons confined in two dimensions.

If the integers generated by quantization are

quantization is used by several speech compression methods.

In the discretization stage, the space of signals is partitioned into equivalence classes and quantization is carried out by replacing the signal with representative signal of the corresponding equivalence class.

In the quantization stage the representative signal values are approximated by values from a finite set.

The digital control introduces additional distortion called quantization error caused by its conversion of the input signal to a digital value.

These equations of motion are then supplemented by commutation relations derived from the canonical quantization procedure described below, thereby incorporating quantum mechanical effects into the behavior of the field.

The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the single-particle states etc.

Second quantization indexes the field by enumerating the single-particle quantum states.

This is because the minimum possible noise level is the error caused by the quantization of the signal, sometimes called Quantization noise.

Each extra quantization bit increases the dynamic range by roughly 6 dB.

The more recently developed wavelet transform is also used extensively, followed by quantization and entropy coding.

Vector quantization is a classical quantization technique from signal processing which allows the modeling of probability density functions by the distribution of prototype vectors.

# Move the nearest quantization vector centroid towards this sample point, by a small fraction of the distance

The integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true.

The link between exact quantization and gauge invariance was subsequently found by Robert Laughlin.

This step was taken by Sommerfeld, who proposed the general quantization rule for an integrable system:

Luna 2 was instrumented with a three component fluxgate magnetometer, similar to that used on Luna 1, but with the dynamic range reduced by a factor of 4 to-750 to + 750 nanoteslas ( gammas ) so that the quantization uncertainty was-12 to + 12 nT.

The quantization noise at these high frequencies is removed or greatly attenuated by use of an analog low-pass filter at the output ( sometimes a simple RC low-pass circuit is sufficient ).

In a digital system, the number of quantization levels, in binary systems determined by and typically stated in terms of the number of bits, will have a bearing on the level of noise and distortion added to that signal.

The quantization noise level is directly determined by this number, decreasing exponentially as the resolution increases ( or linearly in dB units ), and with an adequate number of true bits of quantization, random noise from other sources will dominate and completely mask the quantization noise.