In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take the unit disk into itself.
The contraction operators, determined only up to a sign, have kernels that are Gaussian functions. On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and summarizes how the theory can be extended.
and 26 Related for: Oscillator representation information
In mathematics, the oscillatorrepresentation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David...
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually...
roughly as the tensor product of the group ring of L with the oscillatorrepresentation in n dimensions (which is itself isomorphic to a polynomial ring...
applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually denoted...
controlled oscillator (NCO) is a digital signal generator which creates a synchronous (i.e., clocked), discrete-time, discrete-valued representation of a waveform...
complex-valued function found to be the propagator of the quantum harmonic oscillator. Mehler (1866) defined a function E ( x , y ) = 1 1 − ρ 2 exp ( − ρ...
Generally speaking, radio-frequency engineers speak of the phase noise of an oscillator, whereas digital-system engineers work with the jitter of a clock. Historically...
shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for his developments in quantum...
CPU is normally determined by the frequency of an oscillator crystal. Typically a crystal oscillator produces a fixed sine wave—the frequency reference...
Drude particles are model oscillators used to simulate the effects of electronic polarizability in the context of a classical molecular mechanics force...
devices. The timekeeping element in every modern clock is a harmonic oscillator, a physical object (resonator) that vibrates or oscillates at a particular...
lattice. The ladder operators of the quantum harmonic oscillator or the "number representation" of second quantization are just special cases of this...
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum...
modeled as harmonic oscillators near their equilibria, a derivation of the resonant frequency for a driven, damped harmonic oscillator is shown. An RLC circuit...
sawtooth oscillator. The voltage on the grid controls the breakdown voltage and thus the period of oscillation. Thyratron relaxation oscillators were used...
over a maximal compact subgroup. OscillatorOscillatorrepresentation orbit orbit method, an approach to representation theory that uses tools from symplectic...
of chronobiology (the study of circadian rhythms), the dual circadian oscillator model refers to a model of entrainment (where rhythmic events in an organism...
harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was...
light modes, which are harmonic oscillators. Examples of Wigner-function time evolutions in a quantum harmonic oscillator A coherent state. Combined ground...
frequency ν {\displaystyle \nu } (in the harmonic oscillator approximation). See quantum harmonic oscillator for graphs of the first 5 wave functions, which...
described using a finite number of parameters Parametric oscillator, a harmonic oscillator whose parameters oscillate in time Parametric surface, a particular...
state variables. Simple pendulum, see picture (right). Simple harmonic oscillator where the phase portrait is made up of ellipses centred at the origin...
(2) 172 (2010), no. 2, 1407–1434. MR2680495 L-functions and the oscillatorrepresentation. Springer. 1987. ISBN 0691081565. with Stephen Gelbart and Ilya...
change. Relative strength index (RSI) – oscillator showing price strength. Relative Vigor Index (RVI) – oscillator measures the conviction of a recent price...
portmanteau of wobble and oscillator. A "wobbulator" (without capitalization) is a generic term for the swept-output RF oscillator described above, a frequency-modulated...
Global Information