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.
Historical overview 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. The fundamental operators corresponding to position and momentum satisfy the Heisenberg commutation relations. Quadratic polynomials in these operators, which include the harmonic oscillator, are also closed under taking commutators. A large amount of operator theory was developed in the 1920s and 1930s to provide a rigorous foundation for quantum mechanics. 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. In turn these results in mathematical physics were subsumed within mathematical analysis, starting with the 1933 lecture notes of Norbert Wiener, who used the heat kernel for the harmonic oscillator to derive the properties of the Fourier transform. The uniqueness of the Heisenberg commutation relations, as formulated in the Stone–von Neumann theorem, was later interpreted within group representation theory, in particular the theory of induced representations initiated by George Mackey. The quadratic operators were understood in terms of a projective unitary representation of the group SU(1,1) and its Lie algebra. Irving Segal and David Shale generalized this construction to the symplectic group in finite and infinite dimensions—in physics, this is often referred to as bosonic quantization: it is constructed as the symmetric algebra of an infinite-dimensional space. Segal and Shale have also treated the case of fermionic quantization, which is constructed as the exterior algebra of an infinite-dimensional Hilbert space. In the special case of conformal field theory in 1+1 dimensions, the two versions become equivalent via the so-called "boson-fermion correspondence." Not only does this apply in analysis where there are unitary operators between bosonic and fermionic Hilbert spaces, but also in the mathematical theory of vertex operator algebras. Vertex operators themselves originally arose in the late 1960s in theoretical physics, particularly in string theory. André Weil later extended the construction to p-adic Lie groups, showing how the ideas could be applied in number theory, in particular to give a group theoretic explanation of theta functions and quadratic reciprocity. 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. The representation theory of this semigroup, and its generalizations in finite and infinite dimensions, has applications both in mathematics and theoretical physics.
Semigroups in SL(2,C) The group
G = SU ( 1 , 1 ) = { ( α β β ¯ α ¯ ) | | α | 2 − | β | 2 = 1 } {\displaystyle G=\operatorname {SU} (1,1)=\left\{\left.{\begin{pmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}\right||\alpha |^{2}-|\beta |^{2}=1\right\}}
is a subgroup of
G c = SL ( 2 , C ) {\displaystyle G_{c}=\operatorname {SL} (2,\mathbb {C} )} , the group of complex 2 × 2 matrices with determinant 1. The real subgroup of G c {\displaystyle G_{c}}
G 1 = S L ( 2 , R ) {\displaystyle G_{1}=\operatorname {SL(2,\mathbb {R} )} }
is conjugate to G {\displaystyle G}
G = C G 1 C − 1 , C = ( 1 i i 1 ) , {\displaystyle G=CG_{1}C^{-1},\quad C={\begin{pmatrix}1&i\\i&1\end{pmatrix}},}
as matrix multiplication shows.
G 1 {\displaystyle G_{1}} is generated by
J = ( 0 1 − 1 0 ) {\displaystyle J={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}
and the subgroup P {\displaystyle P} of lower triangular matrices
P = { ( a 0 b a − 1 ) | a , b ∈ R , a > 0 } . {\displaystyle P=\left\{\left.{\begin{pmatrix}a&0\\b&a^{-1}\end{pmatrix}}\right|a,b\in \mathbf {R} ,a>0\right\}.}
The orbit of the vector
v = ( 0 1 ) {\displaystyle v={\begin{pmatrix}0\\1\end{pmatrix}}}
under the subgroup generated by these matrices is easily seen to be R 2 − 0 {\displaystyle \mathbb {R} ^{2}-0} and the stabilizer of v in G1 lies in inside this subgroup. The Lie algebra g {\displaystyle {\mathfrak {g}}} of SU(1,1) consists of matrices
( i x w w ¯ − i x ) , x ∈ R . {\displaystyle {\begin{pmatrix}ix&w\\{\overline {w}}&-ix\end{pmatrix}},\quad x\in \mathbf {R} .}
The period 2 automorphism σ of Gc
σ ( g ) = M g ¯ M − 1 , {\displaystyle \sigma (g)=M{\overline {g}}M^{-1},}
with
M = ( 0 1 1 0 ) , {\displaystyle M={\begin{pmatrix}0&1\\1&0\end{pmatrix}},}
has fixed point subgroup G since
σ ( a b c d ) = ( d ¯ c ¯ b ¯ a ¯ ) . {\displaystyle \sigma {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{pmatrix}{\overline {d}}&{\overline {c}}\\{\overline {b}}&{\overline {a}}\end{pmatrix}}.}
Similarly the same formula defines a period two automorphism σ of the Lie algebra g c {\displaystyle {\mathfrak {g}}_{c}} of Gc, the complex matrices with trace zero. A standard basis of g c {\displaystyle {\mathfrak {g}}_{c}} over C is given by
L 0 = ( 1 2 0 0 − 1 2 ) , L − 1 = ( 0 1 0 0 ) , L 1 = ( 0 0 − 1 0 ) . {\displaystyle L_{0}={\begin{pmatrix}{1 \over 2}&0\\0&-{1 \over 2}\end{pmatrix}},\quad L_{-1}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad L_{1}={\begin{pmatrix}0&0\\-1&0\end{pmatrix}}.}
They satisfy the Witt algebra
[ L m , L n ] = ( m − n ) L m + n , m , n ∈ { 1 , 0 , − 1 } . {\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n},\quad m,n\in \{1,0,-1\}.}
The Lie algebra of G c {\displaystyle G_{c}} is a direct sum
g c = g ⊕ i g , {\displaystyle {\mathfrak {g}}_{c}={\mathfrak {g}}\oplus i{\mathfrak {g}},}
where g {\displaystyle {\mathfrak {g}}} is the +1 eigenspace of σ. The –1 eigenspace i g {\displaystyle i{\mathfrak {g}}}
is the Lie algebra of SU ( 2 ) {\displaystyle \operatorname {SU} (2)} , a maximal compact subgroup of G c {\displaystyle G_{c}} . The matrices X in i g {\displaystyle i{\mathfrak {g}}} have the form
X = ( x w − w ¯ − x ) . {\displaystyle X={\begin{pmatrix}x&w\\-{\overline {w}}&-x\end{pmatrix}}.}
Note that
− det X = x 2 − | w | 2 . {\displaystyle -\det X=x^{2}-|w|^{2}.}
The cone C in i g {\displaystyle i{\mathfrak {g}}} is defined by two conditions. The first is det X < 0. {\displaystyle \det X<0.} By definition this condition is preserved under conjugation by G. Since G is connected it leaves the two components with x > 0 and x < 0 invariant. The second condition is x < 0. {\displaystyle x<0.}
The group Gc acts by Möbius transformations on the extended complex plane. The subgroup G acts as automorphisms of the unit disk D. A semigroup H of Gc, first considered by Olshanskii (1981), can be defined by the geometric condition:
g ( D ¯ ) ⊂ D . {\displaystyle g({\overline {D}})\subset D.}
The semigroup can be described explicitly in terms of the cone C:
H = G ⋅ exp ( C ) = exp ( C ) ⋅ G . {\displaystyle H=G\cdot \exp(C)=\exp(C)\cdot G.}
In fact the matrix X can be conjugated by an element of G to the matrix
Y = ( − y 0 0 y ) {\displaystyle Y={\begin{pmatrix}-y&0\\0&y\end{pmatrix}}}
with
y = x 2 − | w | 2 > 0. {\displaystyle y={\sqrt {x^{2}-|w|^{2}}}>0.}
Since the Möbius transformation corresponding to exp Y sends z to e−2yz, it follows that the right hand side lies in the semigroup. Conversely if g lies in H it carries the closed unit disk onto a smaller closed disk in its interior. Conjugating by an element of G, the smaller disk can be taken to have centre 0. But then for appropriate y, the element e − Y g {\displaystyle e^{-Y}g} carries D onto itself so lies in G. A similar argument shows that the closure of H, also a semigroup, is given by
H ¯ = { g ∈ SL ( 2 , C ) | g D ⊆ D } = G ⋅ exp C ¯ = exp C ¯ ⋅ G . {\displaystyle {\overline {H}}=\{g\in \operatorname {SL} (2,\mathbf {C} )|gD\subseteq D\}=G\cdot \exp {\overline {C}}=\exp {\overline {C}}\cdot G.}
From the above statement on conjugacy, it follows that
H = G A + G , {\displaystyle H=GA_{+}G,}
where
A + = { ( e − y 0 0 e y ) | y > 0 } . {\displaystyle A_{+}=\left\{\left.{\begin{pmatrix}e^{-y}&0\\0&e^{y}\end{pmatrix}}\right|y>0\right\}.}
If
( a b c d ) ∈ H {\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in H}
then
( a ¯ b ¯ c ¯ d ¯ ) , ( a − c − b d ) ∈ H , {\displaystyle {\begin{pmatrix}{\overline {a}}&{\overline {b}}\\{\overline {c}}&{\overline {d}}\end{pmatrix}},\quad {\begin{pmatrix}a&-c\\-b&d\end{pmatrix}}\in H,}
since the latter is obtained by taking the transpose and conjugating by the diagonal matrix with entries ±1. Hence H also contains
( a ¯ − c ¯ − b ¯ d ¯ ) . {\displaystyle {\begin{pmatrix}{\overline {a}}&-{\overline {c}}\\-{\overline {b}}&{\overline {d}}\end{pmatrix}}.}
which gives the inverse matrix if the original matrix lies in SU(1,1). A further result on conjugacy follows by noting that every element of H must fix a point in D, which by conjugation with an element of G can be taken to be 0. Then the element of H has the form
M = ( a 0 b a − 1 ) , | a | < 1 and | b | < | a | − 1 − | a | . {\displaystyle M={\begin{pmatrix}a&0\\b&a^{-1}\end{pmatrix}},\qquad |a|<1\quad {\text{and}}\quad |b|<|a|^{-1}-|a|.}
The set of such lower triangular matrices forms a subsemigroup H0 of H. Since
M ( x 0 0 x − 1 ) = ( x 0 b a − 1 ( x − x − 1 ) x − 1 ) M , {\displaystyle M{\begin{pmatrix}x&0\\0&x^{-1}\end{pmatrix}}={\begin{pmatrix}x&0\\ba^{-1}(x-x^{-1})&x^{-1}\end{pmatrix}}M,}
every matrix in H0 is conjugate to a diagonal matrix by a matrix M in H0. Similarly every one-parameter semigroup S(t) in H fixes the same point in D so is conjugate by an element of G to a one-parameter semigroup in H0. It follows that there is a matrix M in H0 such that
M S ( t ) = S 0 ( t ) M , {\displaystyle MS(t)=S_{0}(t)M,}
with S0(t) diagonal. Similarly there is a matrix N in H0 such that
S ( t ) N = N S 0 ( t ) , {\displaystyle S(t)N=NS_{0}(t),}
The semigroup H0 generates the subgroup L of complex lower triangular matrices with determinant 1 (given by the above formula with a ≠ 0). Its Lie algebra consists of matrices of the form
Z = ( z 0 w − z ) . {\displaystyle Z={\begin{pmatrix}z&0\\w&-z\end{pmatrix}}.}
In particular the one parameter semigroup exp tZ lies in H0 for all t > 0 if and only if ℜ z < 0 {\displaystyle \Re z<0} and | ℜ z | > 1 2 | w | . {\displaystyle |\Re z|>{\tfrac {1}{2}}|w|.}
This follows from the criterion for H or directly from the formula
exp Z = ( e z 0 f ( z ) w e − z ) , f ( z ) = sinh z z . {\displaystyle \exp Z={\begin{pmatrix}e^{z}&0\\f(z)w&e^{-z}\end{pmatrix}},\qquad f(z)={\sinh z \over z}.}
The exponential map is known not to be surjective in this case, even though it is surjective on the whole group L. This follows because the squaring operation is not surjective in H. Indeed, since the square of an element fixes 0 only if the original element fixes 0, it suffices to prove this in H0. Take α with |α| < 1 and
| α + α − 1 | < | α | + | α − 1 | . {\displaystyle \left|\alpha +\alpha ^{-1}\right|<|\alpha |+|\alpha ^{-1}|.}
If a = α2 and
b = ( 1 − δ ) ( | a | − 1 − | a | ) , {\displaystyle b=(1-\delta )(|a|^{-1}-|a|),}
with
( 1 − δ ) 2 = | α + α − 1 | | α | + | α − 1 | , {\displaystyle (1-\delta )^{2}={|\alpha +\alpha ^{-1}| \over |\alpha |+|\alpha ^{-1}|},}
then the matrix
( a 0 b a − 1 ) {\displaystyle {\begin{pmatrix}a&0\\b&a^{-1}\end{pmatrix}}}
has no square root in H0. For a square root would have the form
( α 0 β α − 1 ) . {\displaystyle {\begin{pmatrix}\alpha &0\\\beta &\alpha ^{-1}\end{pmatrix}}.}
On the other hand,
| β | = b | α + α − 1 | = | α | − 1 − | α | 1 − δ > | α | − 1 − | α | . {\displaystyle |\beta |={b \over |\alpha +\alpha ^{-1}|}={|\alpha |^{-1}-|\alpha | \over 1-\delta }>|\alpha |^{-1}-|\alpha |.}
The closed semigroup H ¯ {\displaystyle {\overline {H}}} is maximal in SL(2,C): any larger semigroup must be the whole of SL(2,C). Using computations motivated by theoretical physics, Ferrara et al. (1973) introduced the semigroup H {\displaystyle H} , defined through a set of inequalities. Without identification H {\displaystyle H} as a compression semigroup, they established the maximality of H ¯ {\displaystyle {\overline {H}}} . Using the definition as a compression semigroup, maximality reduces to checking what happens when adding a new fractional transformation g {\displaystyle g} to H ¯ {\displaystyle {\overline {H}}} . The idea of the proof depends on considering the positions of the two discs g ( D ) {\displaystyle g(D)} and D {\displaystyle D} . In the key cases, either one disc contains the other or they are disjoint. In the simplest cases, g {\displaystyle g} is the inverse of a scaling transformation or g ( z ) = − 1 / z {\displaystyle g(z)=-1/z} . In either case g {\displaystyle g} and H {\displaystyle H} 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