The mathematical formulation of quantum mechanics by Werner Heisenberg and Erwin Schrödinger was originally in terms of unbounded self-adjoint operators on a Hilbert space.
Part of the theory was formulated in terms of unitary groups of operators, largely through the contributions of Hermann Weyl, Marshall Stone and John von Neumann.
Segal and Shale have also treated the case of fermionic quantization, which is constructed as the exterior algebra of an infinite-dimensional Hilbert space.
Several physicists and mathematicians observed the heat kernel operators corresponding to the harmonic oscillator were associated to a complexification of SU(1,1): this was not the whole of SL(2,C), but instead a complex semigroup defined by a natural geometric condition.
carries D onto itself so lies in G. A similar argument shows that the closure of H, also a semigroup, is given by From the above statement on conjugacy, it follows that where If then since the latter is obtained by taking the transpose and conjugating by the diagonal matrix with entries ±1.
generate an open neighbourhood of 1 and hence the whole of SL(2,C) Later Lawson (1998) gave another more direct way to prove maximality by first showing that there is a g in S sending D onto the disk Dc, |z| > 1.
, so that formally It is immediate from the definition that the one parameter groups U and V satisfy the Weyl commutation relation The realization of U and V on L2(R) is called the Schrödinger representation.
By Fubini's theorem When combined with the inversion formula this implies that the Fourier transform preserves the inner product so defines an isometry of
, then The covariance relations and analyticity of the kernel imply that for S = π(g, γ), for some constant C. Direct calculation shows that leads to an ordinary representation of the double cover.
The two representations can be realized on Hilbert spaces of holomorphic functions on the unit disk; or, using the Cayley transform, on the upper half plane.
So any lowest energy vector v is annihilated by A and the module is exhausted by the powers of A* applied to v. It is thus a non-zero quotient of C[z] and hence can be identified with it by irreducibility.
The completeness of the Hermite functions follows from the fact that the Bargmann transform is unitary and carries the orthonormal basis en(z) of holomorphic Fock space onto the Hn(x).
For k = 0 the Fourier inversion formula implies If s < t, the diagonal form of D, shows that the inclusion of Ht in Hs is compact (Rellich's lemma).
Consequently, a monomial in P and Q of order 2k applied to f lies in Hs–k and can be expressed as a linear combination of partial derivatives of U(s)V(t)f of degree ≤ 2k evaluated at 0.
This reduces to the power series estimate So these form a dense set of entire vectors for U(s)V(t); this can also be checked directly using Mehler's formula.
The spaces of smooth and entire vectors for U(s)V(t) are each by definition invariant under the action of the metaplectic group as well as the Hermite semigroup.
An element g of the complexification SL(2,C) is said to implementable if there is a bounded operator T such that it and its adjoint leave the space of entire vectors for W invariant, both have dense images and satisfy the covariance relations for u in C2.
It can be seen directly in the Fock model that the implementing operators can be chosen so that they define an ordinary representation of the double cover of H constructed above.
[34] In fact consider the Siegel upper half plane consisting of symmetric complex 2x2 matrices with positive definite real part: and define the kernel with corresponding operator for f in L2(R).
The mth order symbols Sm are given by smooth functions a satisfying for all α and Ψm consists of all operators ψ(a) for such a.
It is generated by the matrices If Z = –I, then Z is central and These automorphisms of G are implemented on V by the following operators: It follows that where μ lies in T. Direct calculation shows that μ is given by the Gauss sum The metaplectic group was defined as the group The coherent state defines a holomorphic map of H into L2(R) satisfying This is in fact a holomorphic map into each Sobolev space Hk and hence also
Let The operators U(x), V(y) with x and y in M all commute and have a finite-dimensional subspace of fixed vectors formed by the distributions with b in M1, where The sum defining Ψb converges in
extend naturally to a representation of this semigroup by contraction operators defined by kernels, which generalise the one-dimensional case (taking determinants where necessary).
It consist precisely of Cayley transforms of points Z in the Siegel generalized upper half plane: Elements g act on Dn and, as in the one dimensional case this action is transitive.
Indeed, it suffices to check, for two such operators S, T and vectors vi proportional to metaplectic coherent states, that which follows because the sum depends holomorphically on S and T, which are unitary on the boundary.
Let S denote the unit sphere in Cn and define the Hardy space H2(S) be the closure in L2(S) of the restriction of polynomials in the coordinates z1, ..., zn.
Only the simplest case will be considered here, involving the loop group LU(1) of smooth maps of the circle into U(1) = T. The oscillator semigroup, developed independently by Neretin and Segal, allows contraction operators to be defined for the semigroup of univalent holomorphic maps of the unit disc into itself, extending the unitary operators corresponding to diffeomorphisms of the circle.
When applied to the subgroup SU(1,1) of the diffeomorphism group, this gives a generalization of the oscillator representation on L2(R) and its extension to the Olshanskii semigroup.
The Segal-Shale quantization criterion states that T is implementable, i.e. lies in the restricted symplectic group, if and only if the commutator TJ – JT is a Hilbert–Schmidt operator.
Then the infinite-dimensional analogue of SU(1,1) consists of invertible bounded operators satisfying gKg* = K (or equivalently the same relations as in the finite-dimensional case).