오실레이터 표현

수학에서 오실레이터 표현은 어빙 시걸, 데이비드 셰일, 안드레 웨일 등이 먼저 조사한 공통분모 집단의 투사적 단일 표현이다.표현의 자연스러운 확장은 1988년 로저 하우에 의해 발진기 세미그룹으로 소개된 수축기 세미그룹으로 이어진다.세미그룹들은 다른 수학자들과 물리학자들, 특히 1960년대에 Felix Berezin에 의해 이전에 연구되었다.한 차원에서는 가장 간단한 예가 SU(1,1)에 의해 제시된다.확장된 복잡한 평면에서 뫼비우스의 변형으로 작용하여 단위 원을 불변하게 한다.이 경우, 오실레이터 표현은 SU(1,1)의 이중 커버를 단일체로 표현한 것이며, 오실레이터 세미그룹은 단위 디스크를 자체로 가져가는 뫼비우스 변환에 해당하는 SL(2,C)의 세미그룹 수축 연산자에 의한 표현에 해당한다.

신호까지만 결정된 수축 연산자는 가우스 함수인 커널을 가지고 있다.최소 수준에서 세미그룹은 라이트 콘으로 식별할 수 있는 SU(1,1)의 리 대수에서 원뿔에 의해 설명된다.동일한 프레임워크는 무한한 차원의 아날로그를 포함하여 더 높은 차원의 공감대 그룹에 일반화된다.본 기사는 SU(1,1)에 대한 이론을 상세히 설명하고, 이론이 어떻게 확장될 수 있는지를 요약하고 있다.

과거 개요

베르너 하이젠베르크와 에르윈 슈뢰딩거에 의한 양자역학의 수학적 공식화는 원래 힐베르트 공간의 한없는 자기 적응 연산자들의 관점에서 이루어졌다.포지션과 모멘텀에 해당하는 기본 연산자는 하이젠베르크의 정류 관계를 만족시킨다.고조파 오실레이터를 포함하는 이러한 연산자의 2차 다항식도 정류자 아래에서 닫힌다.

1920년대와 1930년대에 많은 양의 연산자 이론이 개발되어 양자역학에 대한 엄격한 기초를 제공하였다.그 이론의 일부는 주로 헤르만 바일, 마샬 스톤, 존 폰 노이만의 기여를 통해, 운영자의 단일 집단의 관점에서 공식화되었다.결국 이러한 수학적 물리학의 결과는 1933년 노르베르트 비에너(Norbert Wiener)의 강의 노트로부터 시작하여 수학 분석에서 소분되었는데, 그는 화음 발진기에 열 알맹이를 사용하여 푸리에 변환의 특성을 도출했다.

스톤-본 노이만 정리에 공식화된 하이젠베르크 교감 관계의 고유성은 나중에 그룹 대표론, 특히 조지 맥키가 시작한 유도 대표론 내에서 해석되었다.2차 연산자는 그룹 SU(1,1)와 그것의 Lie 대수학의 투사적인 단일 표현으로 이해되었다.어빙 시걸과 데이비드 셰일(David Shale)은 이 구성을 유한하고 무한의 차원으로 하는 공감대 그룹에 일반화했다. 물리학에서는 이것을 흔히 보소닉 정량화(bosonic quantization)라고 부른다. 무한의 차원 공간의 대칭 대수학으로 구성된다.세갈과 셰일은 무한 차원 힐버트 공간의 외부 대수학으로 건설되는 페르미온 양자화 사례도 다루었다.1+1차원의 정합장 이론의 특별한 경우, 두 버전은 소위 "보손-페르미온 서신"을 통해 동등해진다.이는 보소닉과 페르미온 힐버트 공간 사이에 단일 연산자가 있는 분석뿐만 아니라 정점 연산자 알헤브라의 수학적 이론에도 적용된다.정점 연산자 자체는 1960년대 후반에 이론물리학, 특히 끈 이론에서 생겨났다.

안드레 웨일은 이후 p-adic Lie 그룹까지 건설을 확장하여, 특히 세타 함수와 이차적 상호주의에 대한 집단 이론적 설명을 하기 위해 그 사상이 숫자 이론에 어떻게 적용될 수 있는지를 보여주었다.여러 물리학자와 수학자들이 고조파 발진기에 해당하는 열 커널 연산자가 SU(1,1)의 복합화에 연관되어 있음을 관찰하였다. 이는 SL(2,C)의 전부가 아니라 자연 기하학적 조건에 의해 정의된 복잡한 세미그룹이었다.이 세미그룹의 대표 이론과 유한하고 무한한 차원의 일반화는 수학과 이론 물리학 둘 다에 응용이 있다.^[1]

SL(2,C)의 세미그룹

그룹:

${\displaystyle G=\operatorname {SU}(1,1)=\왼쪽\{\왼쪽).{\\cHB{pmatrix}\pmatrix}\\{\overline{\\\nd{pmatrix}}\right \i1}{pmatrix}}\right \}}$

G_c = SL(2,C)의 부분군이며, 결정인자가 1인 복합 2 x 2 행렬의 그룹이다.G₁ = SL(2,R)이면

${\displaystyle G=CG_{1}C^{-1},\qquad C={\begin{pmatrix}1&i\i&1\end{pmatrix}}.}$

이는 해당 뫼비우스 변환이 상반신 평면을 유닛 원반 위에, 실선을 유닛 원반 위에 운반하는 케이리 변환이기 때문이다.

그룹 SL(2,R)은 다음에 의해 추상 그룹으로 생성된다.

${\displaystyle J={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$

하위 삼각 행렬의 부분군

$\displaystyle \왼쪽\{\왼쪽.{\begin{pmatrix}a&0\\b&a^{-1}\end{pmatrix}\right a,b\in \mathbf {R},a>0\right\}.}$

실제로 벡터의 궤도는

${\displaystyle v={\pmatrix}0\\\1\end{pmatrix}}}$

이러한 행렬에 의해 생성된 부분군 아래는 R의² 전체로 쉽게 볼 수 있으며, G에서₁ V의 스태빌라이저가 이 부분군 내부에 있다.

$SU{\displaystyle \mathfrak{}}$(1$,$1)의$$ Lie $대수$ g {\displaystyle ${\mathfak{g}}$는 행렬로 구성된다.

${\displaystyle {\begin}ix&w\{\overline {w}&-ix\end{pmatrix},\quad x\in \mathbf {R}.}$

기간_c 2 자동형성 σ의 G

$\displaystyle \sigma(g)=M{\overline{g}M^{-1}$

와 함께

${\displaystyle M={\begin{pmatrix}0&1\\1&0\end{pmatrix},}$

이후 고정 점 부분군 G가 있음

${\displaystyle \pmatrix}a&b\c&d\end{pmatrix}={\\noverline{d}{d}\c}\{\overline{b}\{a}\overline{pmatrix}}}}.}$

이와 유사하게 동일한 공식은 미량 0의 복잡한 행렬인 G의_c Lie 대수 $g{\displaystyle {\mathfrak{}}_{}}$ ${\$의 기간 2 자동형성을 정의한다.C에 대한$$ $g{\displaystyle {\mathfrak{}}_{}}$ ${\$의 표준 기준은 다음과 같다.

${\displaystyle L_{0}={\begin{pmatrix}{1 \over 2}&0\\0&-{1 \over 2}\end{pmatrix}},\quad L_{-1}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad L_{1}={\begin{pmatrix}0&0\\-1&0\end{pmatrix}}.}$

따라서 -1 ≤ m, n ≤ 1의 경우

${\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}}$

직접 합이 분해되어 있다.

${\displaystyle {\mathfrak{g}_{c}={\mathfrak{g}\mathfrak {g}\mathfrak i{\mathfrak{g},}$

여기서$$ $g{\displaystyle \mathfrak{}}$ ${\$ {$g}$은(는) σ의 +1 eigenspace이고 $i{\displaystyle }$ ${\displaystyle g\mathfrak{}}$ ${\$ i${\mathfak {g}$은(는$$) –1 eigenspace이다.

$i{\displaystyle }$ ${\displaystyle g\mathfrak{}}$ ${\$의 행렬 X에 형식이 있음$$

${\displaystyle X={\begin{pmatrix}x&w\-{\\overline {w}-x\end{pmatrix}.}$

참고:

${\displaystyle -\det X=x^{2}- w ^{2}.}$

$i{\displaystyle }$ ${\displaystyle g\mathfrak{}}$ ${\$의 콘 C는 두 가지 조건으로 정의된다$$.첫 번째는 < ${\displaystyle \det X<0.}$ 정의상 이 조건은 G에 의한 결합에 의해 보존된다. G가 연결되어 있기 때문에 두 구성 요소는 x > 0과 x < 0 불변성으로 남게 된다.두 번째 조건은 < ${\displaystyle x<0.$$}$

G^c 그룹은 뫼비우스가 확장된 복합면에서 변형하여 행동한다.부분군 G는 단위 디스크 D의 자동화로 작용한다.Olshanskii(1981)에 의해 처음 고려된 G의^c 세미그룹 H는 기하학적 조건에 의해 정의될 수 있다.

${\displaystyle g({\overline {D})\subset D.}$

세미그룹은 원뿔 C:^[2]의 관점에서 명시적으로 설명할 수 있다.

${\displaystyle H=G\cdot \exp(C)=\exp(C)\cdot G.}$

사실 행렬 X는 G의 원소에 의해 행렬에 결합될 수 있다.

${\displaystyle Y={\begin{pmatrix}-y&0\\0&y\end{pmatrix}}$

와 함께

${\displaystyle y={\sqrt {x^{2}- w ^{2}}}0.}$

exp Y에 해당하는 뫼비우스 변환은 ez에 z를^−2y 보내기 때문에, 오른손은 sem그룹에 놓여 있는 것을 따른다.반대로 g가 H에 놓여 있으면 닫힌 유닛 디스크를 내부의 작은 닫힌 디스크로 운반한다.G의 원소에 의해 결합되면, 작은 원반은 중심 0을 가질 수 있다.그러나 적절한 y에 대해 $e{\displaystyle {}^{}}$- Y ${\displaystyle {}^{}}$ ${\displaystyle e^{-Y}g}$ 요소는 D를 스스로 운반하므로$$ G에 위치한다.

비슷한 주장은 역시 세미그룹인 H의 폐쇄가 다음과 같이 주어지는 것을 보여준다.

${\displaystyle {\overline{H}=\{g\in \operatorname {SL}(2,\mathbf {C}) gD\subseteq D\}=G\cdot \exp {\overline {C}\c}\cdot G.}$

부부관계에 관한 위의 진술에서, 그것은 다음과 같다.

${\displaystyle H=GA_{+}G,}$

, where

${\displaystyle A_{+}=\좌측\{\좌측.{\pmatrix}e^{-y}&#0\\0&e^{y}\end{pmatrix}\오른쪽 y>0\right\}.}$

만약

${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}\in H}$

그때

${\displaystyle {\begin{pmatrix}{\overline{a}&{b}\\\\overline{c}\d},\quad {\begin{pmatrix}a&c\-b&d\end{pmatrix}}}H,}$

후자는 transpose를 취하여 ±1 항목으로 대각 행렬을 이용하여 얻는다.따라서 H는 또한 다음을 포함한다.

${\displaystyle {\pmatrix}{\overline {a}-{\overline {c}\\\\overline {b}&gt;{d}\end{pmatrix}}}}.}$

원래 행렬이 SU(1,1)에 있을 경우 역행렬을 제공한다.

H의 모든 요소는 D의 한 점을 고정해야 하며, G의 요소와의 결합에 의해 0이 될 수 있다는 점에 유의함으로써 결합에 대한 추가적인 결과가 뒤따른다.그러면 H의 원소에는 형태가 있다.

${\displaystyle M={\begin{pmatrix}a&0\b&a^{-1}\end{pmatrix},\qquad a <1\quad {\text{and}}\quad b < ^{-1}- a .}$

그러한 하위 삼각형 행렬의 집합은 H의 하위 그룹 H를₀ 형성한다.

이후

${\displaystyle M{\begin{pmatrix}x&0\0\x^{-1}\end{pmatrix}}={\ba^{\1}(x-x^{-1})&x^{pmatrix}M,}$

H의₀ 모든 행렬은 H의₀ 행렬 M에 의해 대각 행렬에 결합된다.

마찬가지로 H의 모든 1-모수 세미그룹 S(t)는 D의 동일한 점을 고정하므로 G의 요소에 의해 H의₀ 1-모수 세미그룹에 결합된다.

H에는₀ M행렬이 있어서 다음과 같이 되어 있다.

$[\displaystyle MS(t)=S_{0}(t)M,}$

S₀(t) 대각선으로마찬가지로 H에는₀ 다음과 같은 행렬 N이 있다.

${\displaystyle S(t)N=NS_{0}(t),}$

세미그룹 H는₀ 결정인자 1을 갖는 복잡한 하위 삼각 행렬의 부분군 L을 생성한다(위의 공식은 given 0으로 표시).그것의 Lie 대수학은 형태의 행렬로 구성되어 있다.

${\displaystyle Z={\begin{pmatrix}z&0\\w&-z\end{pmatrix}.}$

특히 < ${\displaystyle \re z<0}$ 및 > 1 ${\displaystyle$ \$re$z > 1 w. {\displaystyle$\$re z $>{\tfrac{1}{$2}} w .}인 경우에만 하나의 매개 변수 sem그룹 exp tZ가 모든 t > 0에 대해₀ H에 놓여 있다.

이는 H에 대한 기준 또는 공식에서 직접 나온 것이다.

${\displaystyle \exp Z={\begin{pmatrix}e^{z}&0\\f(z)ww&e^{-z}\end{pmatrix}},\qquad f(z)={\sinh z \over z}.}$

지수지도는 이 경우 L그룹 전체에서 추월적이기는 하지만, 이 경우 추월적이지 않은 것으로 알려져 있다.이것은 Squaring 연산이 H에서 수행되지 않기 때문에 뒤따른다.실제로 원소의 사각형은 원소가 0을 고치는 경우에만 0을 고치기 때문에 H로₀ 이를 증명하기에 충분하다.α < 1과 함께 α를 취한다.

${\displaystyle \left \reft +\put ^{-1}\right < \put + \put ^{-1}.}$

a = α일 경우²

${\displaystyle b=(1-\cHB)(^{1}- a ),}$

와 함께

${\displaystyle (1-\property )^{2}={ \ft +\ft ^{-1} \over \ft + \ft ^{-1}}}$

그 다음 행렬

${\displaystyle {\pmatrix}a&0\\b&a^{-1}\end{pmatrix}}$

H에₀ 제곱근은 없다. 제곱근은 형태를 가질 것이기 때문이다.

${\displaystyle {\pmatrix}\reason &0\\\reason ^{-1}\end{pmatrix}.}$

다른 한편으로는

$\displaystyle \cHB ={b \over \cHB +\reads ^{-1}={ \cHB }- \cHB \over 1-\cHB }}} \cHB \cHB \cHB \cHB-{-1}$

닫힌 세미그룹 $'{\${\$overline{H}}$은(는) SL(2,C)에서 최대값이다.^{[3][4][5][6][7]}

이론 물리학에 의해 동기 부여된 연산을 사용하여, 페라라 외 연구진(1973)은 일련의 불평등을 통해 정의된$$세미그룹 $H{\displaystyle }$ ${\displaystyle H}$를 도입했다.압축 세미그룹으로$$ $식별$되지 않은 H ${\$$displaystyle$ $H$$}$을(를) 사용하여 H의 최대성을 설정했으며$,$ 압축 세미그룹으로 정의하면 새로운 부분 변환 $g{\displaystyle }$ ${\displaystyle g}$을(를) $H$의${\$)에$$ 추가하면 어떤 일이 발생하는지 확인하는 것으로 최대성이 감소한다${\displaystyle \stackrel{}{.}}$$isplaystyle {\overline{H$ 증거의 발상은 두 디스크 $g{\displaystyle (D}$ ${\displaystyle )}$ ${\displaystyle g(D)}$과 $D{\displaystyle }$ ${\displaystyle D}$의 위치를 고려하는 데 달려 있다$.$ 주요 사례에서 한 디스크는 다른 디스크를 포함하거나 분리한다.$가장{\displaystyle }$ 간단한 경우 g ${\displaystyle g}$은(는) 스케일 변환의 역행 또는$$ ${\displaystyle \mathrm{g}(}$ ${\displaystyle z}$)${\displaystyle }$=${\displaystyle }$- ${\displaystyle 1}$/ z ${\displaystyle$ g$(z)=-1/z$ 어느 $경우든{\displaystyle }$ g ${\displaystyle g}$과 $H{\displaystyle }$ ${\displaysty H}$은 1의 열린 인접성을 생성하므로$$ SL(2,C) 전체가 생성된다.

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 D^c, z > 1. In fact if ${\displaystyle x\in S\setminus {\overline {H}},}$ then there is a small disk D₁ in D such that xD₁ lies in D^c.그리고 H의 일부 H의 경우₁ D = HD.마찬가지로 yxD₁ = 일부 y의 경우 D^c. so g = yxh는 S에 있고 D를 D로^c 보낸다.따라서² g는 유닛 디스크 D를 고정하므로 SU(1,1)에 위치한다.그래서 g는⁻¹ S에 누워있다.만약 t가 H에 있다면 tgD는 gD를 포함한다.따라서 $g{\displaystyle {}^{}}$- ${\displaystyle {}^{}{}^{}}$ ${\displaystyle {}^{}}$ -${\displaystyle 1^{}}$ g ${\displaystyle \in }$ ${\displaystyle H의\stackrel{}{.}}$ ${\displaystyle$ g$^{-1}t^{-1}g\in${\$overline {H}.}$ so⁻¹ t는 S에 있으므로 S는 1의 개방된 인접성을 포함한다.따라서 S = SL(2,C)이다.

정확히 동일한 주장이 R과ⁿ 오픈 세미그룹에 대한 Möbius 변환에 작용하여 폐쇄된 단위 구 x ≤ 1을 오픈 단위 구 x < 1로 가져간다.폐쇄는 모든 뫼비우스 변혁의 그룹에서 가장 적절한 세미그룹이다.n = 1일 때, 폐쇄는 닫힌 간격[–1,1]을 자체로 가져가는 실선의 뫼비우스 변환에 해당한다.^[8]

세미그룹 H와 그것의 폐쇄는 G로부터 물려받은 추가적인 구조를 가지고 있는데, 즉, G에서의 뒤집힘은 H와 그것의 폐쇄의 반정형성으로 확장되며, 이것은 exp C와 그것의 폐쇄의 요소들을 고정시킨다.을 위해

${\displaystyle g={\pmatrix}a&b\c&d\end{pmatrix},}$

반동형성은 에 의해 주어진다.

${\displaystyle g^{+}={\pmatrix}{\\overline{a}-{c}\\-{\overline{b}&{d}\end{pmatrix}}}}}}$

그리고 SL(2,C)의 반동형성으로 확장된다.

유사하게 반정형주의

${\displaystyle g^{\pmatrix}={\noverline{d}-{b}\-{\overline{c}}{\overline{a}\end{pmatrix}}}}}}$

G₁ 불변성을 남기고 exp C와₁ 그 폐쇄에 원소를 고정하기 때문에 G에서₁ sem그룹에 대한 유사한 특성을 가지고 있다.

하이젠베르크와 웨일과의 교감 관계

$S{\displaystyle \mathcal{}}$ ${\$을(를) R에 있는 슈워츠 함수의 공간이 되도록$$ 한다.R의 사각형 통합 함수의 힐버트 공간² L(R)에 밀도가 높다.양자역학의 용어에 따라 "모멘텀" 연산자 P와 "위치" 연산자 Q를 S ${\$에 정의한다.

${\displaystyle Pf(x)=if',\qquad Qf(x)=xf(x).}$

그곳의 운영자들은 하이젠베르크 통신 관계를 만족시킨다.

$PQ-QP=I.}$

P와 Q는 모두² L(R)에서$$ $상속받은{\displaystyle \mathcal{}}$ S$${\$displaystyle${\$mathcal{S}$에 있는 내부 제품에 대한 자가 적응이다.

$S{\displaystyle \mathcal{}}$ {\$displaystyle {\mathcal {S}$ 및$$ L²(R)에 대해 다음과 같은 두 개의 단일 매개 변수 그룹 U와 V(t)를 정의할 수 있다$.$

${\displaystyle U(s)f(x)=f(x-s),\qquad V(t)f(x)=e^{ixt}f(x).}$

정의에 따라

${\d 디스플레이 스타일 {d \over ds}U=iPU(s)f,\qquad {d \over dt}V(t)f=iQV(t)f}$

$f{\displaystyle }$ ${\displaystyle \in }$ ${\displaystyle S\mathcal{}}$ ${\$에 대해$,$ 정식으로.

${\displaystyle U.s=e^{iPs}\qquad V(t)=e^{iQt}.}$

하나의 파라미터 그룹 U와 V가 Weyl commutation relation relation relationship을 만족한다는 정의에서 바로 나온다.

${\displaystyle U(s)V(t)=e^{-ist}V(t)U.}$

L²(R)에서 U와 V의 실현을 슈뢰딩거 대표라고 한다.

푸리에 변환

푸리에 변환은 $S{\displaystyle \mathcal{}}$ ${\$에서^[9] 정의되며

${\displaystyle {\widehat{f}(\xi )={1 \over {\sqrt {2\pi }}\int_{-\infit }^{\f(x)e^{-ix\x\x.}$

자연 위상에 대해 $S{\displaystyle \mathcal{}}$ ${\$의 연속 지도를 자체로$$ 정의한다.

등고선 통합에 의해 함수가 표시됨

${\displaystyle H_{0}(x)={e^{-x^{2}/2}} \\sqrt{2\pi }}}}}}}}$

푸리에의 변형이 그것만의 것이다.

한편, 부품별로 통합하거나 적분하에서의 차별화,

${\displaystyle {\widehat{Pf}}=-Q{\widehat {f},\qquad {\widehat {Qf}=P{\widehat {f}}.}$

$S{\displaystyle \mathcal{}}$ ${\$ {$S}$의 연산자가 다음에 의해$$ 정의됨

${\displaystyle Tf(x)={\widehat {f}}(-x)}$

Q(및 P)와 통근하다.다른 한편으로는

${\displaystyle TH_{0}=H_{0}}}$

그리고 그 이후로

${\displaystyle g(x)={f(x)-f(a)H_{0}(x)/H_{0}(a) \over x-a}}}$

$S{\displaystyle \mathcal{}}$ ${\$에 위치하며$,$ 그 뒤를 따른다.

${\displaystyle T(x-a)g _{x=a}=(x-a)Tg _{x=a}=0}$

그래서

$[\displaystyle Tf(a)=f(a)].}$

이는 푸리에 반전 공식을 의미한다.

${\displaystyle f(x)={1\over {\sqrt {2\pi }}\int_{-\infit }^{-\infit }{\widehat{f}(\xi )e^{ix\xi }\,d\xi }}$

그리고 푸리에 변환이 $S{\displaystyle \mathcal{}}$ ${\$의 이형성임을 보여준다.

푸비니의 정리로는

${\displaystyle \int _{-\infty }^{\infty }f(x){\widehat {g}}(x)\,dx={1 \over {\sqrt {2\pi }}}\iint f(x)g(\xi )e^{-ix\xi }\,dxd\xi =\int _{-\infty }^{\infty }{\widehat {f}}(\xi )g(\xi )\,d\xi .}$

반전 공식과 결합할 경우 이는 푸리에 변환이 내부 제품을 보존한다는 것을 의미한다.

${\displaystyle \왼쪽 \widehat {f},{\widehat {g}\오른쪽)=(f,g)}$

$따라서{\displaystyle \mathcal{}}$ S ${\$의 등위계를 자체로$$ 정의한다.

밀도에 의해 그것은 Planchrel의 정리에 의해 주장된 L²(R) 상의 단일 운영자로 확장된다.

스톤-본 노이만 정리

U(s)와 V(t)가 Weyl commutation 관계를 충족하는$$ Hilbert 공간 $H{\displaystyle \mathcal{}}$ ${\$ {$H}}$의 단일 매개 변수 그룹이라고 가정하십시오.

${\displaystyle U(s)V(t)=e^{-ist}V(t)U.}$

$F{\displaystyle }$(${\displaystyle s}$ ,${\displaystyle t\mathbf{}}$ ) ${\displaystyle \in }$ ${\displaystyle S}$( R${\displaystyle }$×${\displaystyle R\mathbf{}}$) , ${\displaystyle F(s,t)\in${\$mathcal {S}(\mathbf {R} \times \mathbf {R}),}$은(으)로$$^[10][11] 한다.

${\displaystyle F^{\ve }(x,y)={1 \over {\sqrt {2\pi }}\int_{-\infit }^{-\f(t,y)^{-itx}\,dt}$

그리고 다음$$ $기준$으로$$H {\$displaystyle${\$mathcal${H}에 대한 경계 연산자를 정의하십시오$.$

${\displaystyle T(F)=\iint F^{\vee }(x,x+y)U(x)V(y)\,dxdy.}$

그러면

${\displaystyle {\begin}T(F)T(G)&=T(F\star G)\\T(F)^{*}&=T(F^{*}}}\ended{aligned}}}}}}}}}}}$

어디에

${\displaystyle {\begin}(F\star G)(x,y)&=\int F(x,z)G(z,y)\,dz\\F^{*}(x,y)&={\overline {F(y,x)}}\end{aigned}}}$

연산자 T(F)는 중요한 비감소 특성을 가지고 있다. 모든 벡터 T(F)ξ의 선형 범위는 $H{\displaystyle \mathcal{}}$ ${\$에 밀도가 있다$.$

실제로 fds와 gdt가 콤팩트한 지원으로 확률측정을 정의한다면, 얼룩진 연산자는

${\displaystyle U(f)=\int U(s)f\,ds,\qquad V(g)=\int V(t)g(t)\,dt}$

만족시키다

$[\displaystyle \ U(f)\ ,\ V(g)\ \leq 1}$

측정값이 0으로 감소할 경우 강한 측정 시스템 위상에 수렴한다.

U(f)V(g)는 T(F) 형태를 가지기 때문에 비기존성이 따른다.

$H{\displaystyle \mathcal{}}$ ${\$이(가) L²(R)에 대한 슈뢰딩거 표현일$$ 때 연산자 T(F)는

$[\displaystyle T(F)f(x)=\int F(x,y)f(y)\,dy.}$

슈워츠 함수인 커널이 부여한 연산자에 대한 사실이기 때문에 U와 V가 슈뢰딩거 표현에 대해 이해할 수 없는 공동 작용을 하는 것은 이 공식에서 비롯된다.구체적인 설명은 선형 표준 변환에 의해 제공된다.

반대로 $H{\displaystyle \mathcal{}}$ ${\$에 대한 Weyl 정류 관계를 나타내는 것으로 볼 때$,$ 커널 연산자의 *-알지브라에 대한 비감소적 표현을 발생시킨다.그러나 그러한 모든 표현은 위와 같이 각 사본에 대한 작용과 함께² L(R) 사본의 직교적 직접 합계에 있다.이것은 N × N 행렬의 표현이 C에^N 대한 표준 표시의 직접 합계에 있다는 기본적인 사실의 간단한 일반화다.매트릭스 단위를 사용하는 증거는 무한 차원에서도 동등하게 잘 작동한다.

단일 매개 변수 그룹 U와 V는 각 구성 요소를 불변하게 하여 슈뢰딩거 표현에 대한 표준 작용을 유도한다.

특히 이것은 스톤-본 노이만(Stone-von Neumann)의 정리를 암시한다: 슈뢰딩거(Shrödinger) 표현은 힐버트 공간에 있는 Weyl commutation 관계의 독특한 불가해한 표현이다.

SL(2,R)의 오실레이터 표현

Weyl commutation 관계를 만족하는 U와 V를 고려하여 정의하십시오.

${\displaystyle W(x,y)=e^{\frac {ixy}{2}}U(x)V(y).}$

그러면

${\displaystyle W(x_{1},y_{1})W(x_{2},y_{2})=e^{i(x_{1}y_{2}-y_{1}-{1}x_{2}}}W(x_{1}+x_{2}, y_{1}+y_{2}),}$

그래서 W는 R의² 투사적인 단일 표현을 정의하고 cocycle은 다음과 같다.

${\displaystyle \omega(z_{1},z_{2})=e^{iB(z_{1},z_{2}}},}$

여기서 $z{\displaystyle }$= ${\displaystyle x}$+ i ${\displaystyle y}$=${\displaystyle }$( x${\displaystyle }$, ${\displaystyle y}$) ${\displaystyle z=x+iy=(x,y)}$ 및$$ B는 R에² 주어진 공통점 형식이다.

${\displaystyle B(z_{1},z_{2})=x_{1}y_{1}-y_{1}-{1}x_{2}=\Im z_{1}{1}{\overline {z_{2}}.}$

스톤-본 노이만 정리에 의해, 이 고치에 해당하는 독특한 불가해한 표현이 있다.

따라서 g가 형식 B, 즉 SL(2,R)의 요소를 보존하는 R의² 자동형이라면 L²(R)에 공분산 관계를 만족하는 단일 itary(g)가 있다.

${\displaystyle \pi (g)W(z)\pi (g)^{*}=W(g(z))}$

슈르의 보조정리자에 의해 단일하수체 π(g)는 ζ = 1의 스칼라 ζ에 의해 최대 곱셈까지 고유하므로, π은 SL(2,R)의 투영적 단일 표현을 정의한다.

이것은 슈뢰딩거 표현에 대한 재확정성만을 사용하여 직접 설정할 수 있다.무reducibility는 운영자들이 그 사실에 대한 직접적인 결과였다.

${\displaystyle \iint K(x,y)U(x)V(y)\,dxdy,}$

K a Schwartz 함수는 정확히 Schwartz 함수를 가진 커널에 의해 주어진 연산자와 일치한다.

이들은 힐버트-슈미트 연산자의 공간에 밀집되어 있는데, 유한한 순위 연산자를 포함하고 있기 때문에 이해할 수 없는 행동을 한다.

π의 존재는 슈뢰딩거 대표성의 불가해성만을 사용하여 증명할 수 있다.운영자는 다음과 같은 표지에 따라 고유하다.

${\displaystyle \pi(gh)=\pm \pi(g)\pi(h),}$

SL(2,R)의 투영적 표현을 위한 2-Cocycle이 ±1의 값을 취하도록 한다.

실제로, 그룹 SL(2,R)은 양식의 행렬에 의해 생성된다.

${\displaystyle g_{1}={\begin{pmatrix}a&0\\0&a^{-1}\end{pmatrix}},\,\,g_{2}={\begin{pmatrix}1&0\\b&1\end{pmatrix}},\,\,g_{3}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}},}$

그리고 다음과 같은 연산자가 위의 공분산 관계를 만족하는지 직접 확인할 수 있다.

${\displaystyle \pi (g_{1})f(x)=\pm a^{-{\frac {1}{2}}}f(a^{-1}x),\,\,\pi (g_{2})f(x)=\pm e^{-ibx^{2}}f(x),\,\,\pi (g_{3})f(x)=\pm e^{\frac {i\pi }{8}}{\widehat {f}}(x).}$

발생기 g는_i 그룹 SL(2,R)을 고유하게 지정하는 다음과 같은 Bruhat 관계를 만족한다.^[12]

${\displaystyle g_{3}^{2}=g_{1}(-1),\,\,g_{3}g_{1}(a)g_{3}^{-1}=g_{1}(a^{-1}),\,\,g_{1}(a)g_{2}(b)g_{1}(a)^{-1}=g_{2}(a^{-2}b),\,\,g_{1}(a)=g_{3}g_{2}(a^{-1})g_{3}g_{2}(a)g_{3}g_{2}(a^{-1}).}$

이러한 관계는 해당 운영자가 표식까지 만족한다는 것을 직접 계산하여 확인할 수 있으며, 이는 코키클이 ±1 값을 취한다는 것을 확립한다.

SL(2,R)의 이중 커버로서 메타폴리트 집단의 명시적 구성을 이용한 보다 개념적인 설명이 있다.^[13]SL(2,R)은 상반면 H에서 Möbius 변환에 의해 작용한다. 더욱이 다음과 같은 경우

${\displaystyle g={\pmatrix}a&b\c&d\end{pmatrix},}$

그때

${\dg(z) \overdz}={1 \over(cz+d)^{2}}.}$

함수

${\displaystyle m(g,z)=cz+d}$

1과 1의 관계를 만족시키다.

$[\displaystyle m(gh,z)=m(g,hz)m(h,z)]}$

각 g에 대해, 함수 m(g,z)은 H에 비바니싱이므로 가능한 두 개의 홀로모르픽 제곱근을 가지고 있다.메타폴틱 그룹은 그룹으로 정의된다.

${\displaystyle \operatorname {Mp}(2,\mathbf {R})=\{{(g,G) G(z)^{2}=m(g,z)\}}$

정의상 SL(2,R)의 이중 커버로 연결된다.곱하기:

${\displaystyle(g,G)\cdot(h,H)=(gh,K),}$

어디에

$[\displaystyle K(z)=G(z)H(z)]}$

따라서 메타폴로지 그룹의 요소 g에 대해서는 1-코클 관계를 만족하는 고유하게 결정된 함수 m(g,z)^1/2이 있다.

> ${\displaystyle \Im$ z$>0}$인 경우

${\displaystyle f_{z}(x)=e^{izx^{2}/2}}$

L에² 거짓말을 하고 논리 정연한 상태라고 불린다.

이러한 함수는 에 의해 생성되는 SL(2,R)의 단일 궤도에 위치한다.

${\displaystyle f_{i}(x)=e^{-{\frac{x^{2}}},}$

sl(2,R)의 g에 대해 그 이후.

${\displaystyle \pi((g^{t})^{-1}f_{z}(x)=\pm m(g,z)^{-1/2}f_{gz}(x).}$

보다 구체적으로 g가 Mp(2,R)에 있는 경우

${\displaystyle \pi((g^{t})^{-1}f_{-1}f_{z}(x)=m(g,z)^{-1/2}f_{gz}(x)^.}$

실제로, 만약 이것이 g와 h를 유지한다면, 그것은 또한 그들의 제품을 유지한다.반면 g에^t g형식이_i 있고 이들이 발전기인지 쉽게 확인할 수 있다.

이것은 메타폴리트 집단의 일반적인 단일적 표현을 정의한다.

요소(1,–1)는 L²(R)에서 –1만큼 곱하기 역할을 하며, 여기서부터 SL(2,R)의 코코클은 ±1 값만 취한다.

마슬로프 지수

Lion & Vergne(1980년)에서 설명한 바와 같이, ±1 값을 취하면서 메타폴틱 표현과 관련된 SL(2,R)의 2-Cocycle은 Maslov 지수에 의해 결정된다.

평면에서 세 개의 0이 아닌 벡터 u, v, w가 주어질 때, 그들의 마슬로프 지수 $\tau {\displaystyle }$(${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$) ${\displaystyle \tau (u,v,w)}$은 정의한 R에서³ 2차 형태의 서명으로 정의된다$$.

$[\displaystyle Q(a,b,c)=abB(u,v)+bcB(v,w)+caB(w,u)]}$

Maslov 인덱스의 속성:

• 벡터에 의해 확장되는 1차원 하위 공간에 따라 다름
• 그것은 SL(2,R)에 따라 불변한다.
• 그것은 그 주장을 교대하고 있다. 즉, 두 개의 주장이 상호 교환될 경우 그 기호가 바뀐다.
• 두 개의 서브스페이스가 일치하면 사라진다.
• 값 –1, 0 및 +1: u와 v가 B(u,v) = 1과 w = au + bv를 만족하면 매슬로프 지수는 if ab = 0이며, 그렇지 않으면 ab의 부호를 뺀 값과 같다.
• ${\displaystyle \displaystyle {\tau(v,w,z)-\tau(u,v,z)+\tau(u,v,w)=0}$

0이 아닌 벡터 u를₀ 선택하면 함수가

${\displaystyle \Oomega(g,h)=\exp -{\pi i \over 4}\tau(u_{0}gu_{0}ghu_{0}}}}}$

SL(2,R)에 2-cocycle을 정의하고, 단결의 8번째 뿌리에 값을 부여한다.

2-Cocycle의 수정은 메타폴릭 코코클과 연결된 ±1의 값을 가진 2-Cocycle을 정의하는 데 사용될 수 있다.^[14]

실제로 0이 아닌 벡터 u, 평면에서 v가 주어진 경우 f(u,v)를 다음과 같이 정의한다.

• u와 v가 비례하지 않으면 b(u,v)의 부호를 곱한다.
• u = λv일 경우 λ의 기호

만약

${\displaystyle b(g)=f(u_{0},gu_{0}),}$

그때

${\displaystyle \Oomega(g,h)^{2}=b(gh)b(g)^{-1}b(h)^{-1}.}$

메타폴리트 표현에서 대표자 π(g)를 선택할 수 있도록 한다.

${\displaystyle \pi(gh)=\omega(g,h)\pi(g)\pi(h)}$

여기서 2-256 Ω은

$\displaystyle \omega(g,h)=\Oomega(g,h)\beta(gh)^{-1}\beta(g)\beta(h),}$

와 함께

${\displaystyle \property (g)^{2}=b(g).}$

홀로모르픽 포크 공간

Holomorphic Fock space (also known as the Segal–Bargmann space) is defined to be the vector space ${\displaystyle {\mathcal {F}}}$ of holomorphic functions f(z) on C with

${\displaystyle {1 \over \pi }\iint _{\mathbf {C} } f(z) ^{2}e^{- z ^{2}}\,dxdy}$

finite. It has inner product

${\displaystyle (f_{1},f_{2})={1 \over \pi }\iint _{\mathbf {C} }f_{1}(z){\overline {f_{2}(z)}}e^{- z ^{2}}\,dxdy.}$

${\displaystyle {\mathcal {F}}}$ is a Hilbert space with orthonormal basis

${\displaystyle e_{n}(z)={z^{n} \over {\sqrt {n!}}},\quad n\geq 0.}$

Moreover, the power series expansion of a holomorphic function in ${\displaystyle {\mathcal {F}}}$ gives its expansion with respect to this basis.^[15] Thus for z in C

${\displaystyle f(z) =\left \sum _{n\geq 0}a_{n}z^{n}\right \leq \ f\ e^{ z ^{2}/2},}$

so that evaluation at z is gives a continuous linear functional on ${\displaystyle {\mathcal {F}}.}$ In fact

${\displaystyle f(a)=(f,E_{a})}$

where^[16]

${\displaystyle E_{a}(z)=\sum _{n\geq 0}{(E_{a},e_{n})z^{n} \over {\sqrt {n!}}}=\sum _{n\geq 0}{z^{n}{\overline {a}}^{n} \over n!}=e^{z{\overline {a}}}.}$

Thus in particular ${\displaystyle {\mathcal {F}}}$ is a reproducing kernel Hilbert space.

For f in ${\displaystyle {\mathcal {F}}}$ and z in C define

${\displaystyle W_{\mathcal {F}}(z)f(w)=e^{- z ^{2}/2}e^{w{\overline {z}}}f(w-z).}$

Then

${\displaystyle W_{\mathcal {F}}(z_{1})W_{\mathcal {F}}(z_{2})=e^{-i\Im z_{1}{\overline {z_{2}}}}W_{\mathcal {F}}(z_{1}+z_{2}),}$

so this gives a unitary representation of the Weyl commutation relations.^[17] Now

${\displaystyle W_{\mathcal {F}}(a)E_{0}=e^{- a ^{2}/2}E_{a}.}$

It follows that the representation ${\displaystyle W_{\mathcal {F}}}$ is irreducible.

Indeed, any function orthogonal to all the E_a must vanish, so that their linear span is dense in ${\displaystyle {\mathcal {F}}}$.

If P is an orthogonal projection commuting with W(z), let f = PE₀. Then

${\displaystyle f(z)=(PE_{0},E_{z})=e^{ z ^{2}}(PE_{0},W_{\mathcal {F}}(z)E_{0})=(PE_{-z},E_{0})={\overline {f(-z)}}.}$

The only holomorphic function satisfying this condition is the constant function. So

${\displaystyle PE_{0}=\lambda E_{0},}$

with λ = 0 or 1. Since E₀ is cyclic, it follows that P = 0 or I.

By the Stone–von Neumann theorem there is a unitary operator ${\displaystyle {\mathcal {U}}}$ from L²(R) onto ${\displaystyle {\mathcal {F}}}$, unique up to multiplication by a scalar, intertwining the two representations of the Weyl commutation relations. By Schur's lemma and the Gelfand–Naimark construction, the matrix coefficient of any vector determines the vector up to a scalar multiple. Since the matrix coefficients of F = E₀ and f = H₀ are equal, it follows that the unitary ${\displaystyle {\mathcal {U}}}$ is uniquely determined by the properties

${\displaystyle W_{\mathcal {F}}(a){\mathcal {U}}={\mathcal {U}}W(a)}$

and

${\displaystyle {\mathcal {U}}H_{0}=E_{0}.}$

Hence for f in L²(R)

${\displaystyle {\mathcal {U}}f(z)=({\mathcal {U}}f,E_{z})=(f,{\mathcal {U}}^{*}E_{z})=e^{- z ^{2}}(f,{\mathcal {U}}^{*}W_{\mathcal {F}}(z)E_{0})=e^{- z ^{2}}(W(-z)f,H_{0}),}$

so that

${\displaystyle {\mathcal {U}}f(z)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{-(x^{2}+y^{2})}e^{-2ixy}f(t+x)e^{-t^{2}/2}\,dt={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }B(z,t)f(t)\,dt,}$

where

${\displaystyle B(z,t)=\exp[-z^{2}-t^{2}/2+zt].}$

The operator ${\displaystyle {\mathcal {U}}}$ is called the Segal–Bargmann transform^[18] and B is called the Bargmann kernel.^[19]

The adjoint of ${\displaystyle {\mathcal {U}}}$ is given by the formula:

${\displaystyle {\mathcal {U}}^{*}F(t)={1 \over \pi }\iint _{\mathbf {C} }B({\overline {z}},t)F(z)\,dxdy.}$

Fock model

The action of SU(1,1) on holomorphic Fock space was described by Bargmann (1970) and Itzykson (1967).

The metaplectic double cover of SU(1,1) can be constructed explicitly as pairs (g, γ) with

${\displaystyle g={\begin{pmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}}$

and

${\displaystyle \gamma ^{2}=\alpha .}$

If g = g₁g₂, then

${\displaystyle \gamma =\gamma _{1}\gamma _{2}\left(1+{\beta _{1}{\overline {\beta _{2}}} \over \alpha _{1}\alpha _{2}}\right)^{1/2},}$

using the power series expansion of (1 + z)^1/2 for z < 1.

The metaplectic representation is a unitary representation π(g, γ) of this group satisfying the covariance relations

${\displaystyle \pi (g,\gamma )W_{\mathcal {F}}(z)\pi (g,\gamma )^{*}=W_{\mathcal {F}}(g\cdot z),}$

where

${\displaystyle g\cdot z=\alpha z+\beta {\overline {z}}.}$

Since ${\displaystyle {\mathcal {F}}}$ is a reproducing kernel Hilbert space, any bounded operator T on it corresponds to a kernel given by a power series of its two arguments. In fact if

${\displaystyle K_{T}(a,b)=(TE_{\overline {b}},E_{a}),}$

and F in ${\displaystyle {\mathcal {F}}}$, then

${\displaystyle TF(a)=(TF,E_{a})=(F,T^{*}E_{a})={\frac {1}{\pi }}\iint _{\mathbf {C} }F(z){\overline {(T^{*}E_{a},E_{z})}}e^{- z ^{2}}\,dxdy={\frac {1}{\pi }}\iint _{\mathbf {C} }K_{T}(a,{\overline {z}})F(z)e^{- z ^{2}}\,dxdy.}$

The covariance relations and analyticity of the kernel imply that for S = π(g, γ),

${\displaystyle K_{S}(a,z)=C\cdot \exp \,{1 \over 2\alpha }({\overline {\beta }}z^{2}+2az-\beta a^{2})}$

for some constant C. Direct calculation shows that

${\displaystyle C=\gamma ^{-1}}$

leads to an ordinary representation of the double cover.^[20]

Coherent states can again be defined as the orbit of E₀ under the metaplectic group.

For w complex, set

${\displaystyle F_{w}(z)=e^{wz^{2}/2}.}$

Then ${\displaystyle F_{w}\in {\mathcal {F}}}$ if and only if w < 1. In particular F₀ = 1 = E₀. Moreover,

${\displaystyle \pi (g,\gamma )F_{w}=({\overline {\alpha }}+{\overline {\beta }}w)^{-{\frac {1}{2}}}F_{gw}={\frac {1}{\overline {\gamma }}}\left(1+{{\overline {\beta }} \over {\overline {\alpha }}}w\right)^{-1/2}F_{gw},}$

where

${\displaystyle gw={\alpha w+\beta \over {\overline {\beta }}w+{\overline {\alpha }}}.}$

Similarly the functions zF_w lie in ${\displaystyle {\mathcal {F}}}$ and form an orbit of the metaplectic group:

${\displaystyle \pi (g,\gamma )[zF_{w}](z)=({\overline {\alpha }}+{\overline {\beta }}w)^{-3/2}zF_{gw}(z).}$

Since (F_w, E₀) = 1, the matrix coefficient of the function E₀ = 1 is given by^[21]

${\displaystyle (\pi (g,\gamma )1,1)=\gamma ^{-1}.}$

Disk model

The projective representation of SL(2,R) on L²(R) or on ${\displaystyle {\mathcal {F}}}$ break up as a direct sum of two irreducible representations, corresponding to even and odd functions of x or z. 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.^[22][23]

The even functions correspond to holomorphic functions F₊ for which

${\displaystyle {1 \over 2\pi }\iint F_{+}(z) ^{2}(1- z ^{2})^{-1/2}\,dxdy+{2 \over \pi }\iint F'_{+}(z) ^{2}(1- z ^{2})^{\frac {1}{2}}\,dxdy}$

is finite; and the odd functions to holomorphic functions F_– for which

${\displaystyle {1 \over 2\pi }\iint F_{-}(z) ^{2}(1- z ^{2})^{-1/2}\,dxdy}$

is finite. The polarized forms of these expressions define the inner products.

The action of the metaplectic group is given by

${\displaystyle {\begin{aligned}\pi _{\pm }(g^{-1})F_{\pm }(z)&=\left({\overline {\beta }}z+{\overline {\alpha }}\right)^{-1\pm {\frac {1}{2}}}F_{\pm }(gz)\\&=\left(-{\overline {\beta }}z+\alpha \right)^{-1\pm {\frac {1}{2}}}F_{\pm }\left({{\overline {\alpha }}z-\beta \over -{\overline {\beta }}z+\alpha }\right)&&g={\begin{pmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}\end{aligned}}}$

Irreducibility of these representations is established in a standard way.^[24] Each representation breaks up as a direct sum of one dimensional eigenspaces of the rotation group each of which is generated by a C^∞ vector for the whole group. It follows that any closed invariant subspace is generated by the algebraic direct sum of eigenspaces it contains and that this sum is invariant under the infinitesimal action of the Lie algebra ${\displaystyle {\mathfrak {g}}}$. On the other hand, that action is irreducible.

The isomorphism with even and odd functions in ${\displaystyle {\mathcal {F}}}$ can be proved using the Gelfand–Naimark construction since the matrix coefficients associated to 1 and z in the corresponding representations are proportional. Itzykson (1967) gave another method starting from the maps

${\displaystyle U_{+}(F)(w)={\frac {1}{\pi }}\iint _{\mathbf {C} }F(z)e^{{\frac {1}{2}}w{\overline {z}}^{2}}e^{- z ^{2}}\,dxdy,}$

${\displaystyle U_{-}(F)(w)={\frac {1}{\pi }}\iint _{\mathbf {C} }F(z){\overline {z}}e^{{\frac {1}{2}}w{\overline {z}}^{2}}e^{- z ^{2}}\,dxdy,}$

from the even and odd parts to functions on the unit disk. These maps intertwine the actions of the metaplectic group given above and send zⁿ to a multiple of wⁿ. Stipulating that U_± should be unitary determines the inner products on functions on the disk, which can expressed in the form above.^[25]

Although in these representations the operator L₀ has positive spectrum—the feature that distinguishes the holomorphic discrete series representations of SU(1,1)—the representations do not lie in the discrete series of the metaplectic group. Indeed, Kashiwara & Vergne (1978) noted that the matrix coefficients are not square integrable, although their third power is.^[26]

Harmonic oscillator and Hermite functions

Consider the following subspace of L²(R):

${\displaystyle {\mathcal {H}}=\left\{f\in L^{2}(\mathbf {R} )\left f(x)=p(x)e^{-{\frac {x^{2}}{2}}},p(x)\in \mathbf {R} [x]\right.\right\}.}$

The operators

${\displaystyle {\begin{aligned}X&=Q-iP={d \over dx}+x\\Y&=Q+iP=-{d \over dx}+x\end{aligned}}}$

act on ${\displaystyle {\mathcal {H}}.}$ X is called the annihilation operator and Y the creation operator. They satisfy

${\displaystyle {\begin{aligned}X&=Y^{*}\\XY&=D+I&&D=-{d^{2} \over dx^{2}}+x^{2}\\XY-YX&=2I\\XY^{n}-Y^{n}X&=2nY^{n-1}&&{\text{by induction}}\end{aligned}}}$

Define the functions

${\displaystyle F_{n}(x)=Y^{n}e^{-{\frac {x^{2}}{2}}}}$

We claim they are the eigenfunctions of the harmonic oscillator, D. To prove this we use the commutation relations above:

${\displaystyle {\begin{aligned}DF_{n}&=DY^{n}F_{0}\\&=(XY-I)Y^{n}F_{0}\\&=\left(XY^{n+1}-Y^{n}\right)F_{0}\\&=\left(\left((2n+2)Y^{n}+Y^{n+1}X\right)-Y^{n}\right)F_{0}\\&=\left((2n+1)Y^{n}+Y^{n+1}X\right)F_{0}\\&=(2n+1)Y^{n}F_{0}+Y^{n+1}XF_{0}\\&=(2n+1)F_{n}&&XF_{0}=0\end{aligned}}}$

Next we have:

${\displaystyle \ F_{n}\ _{2}^{2}=2^{n}n!{\sqrt {\pi }}.}$

This is known for n = 0 and the commutation relation above yields

${\displaystyle (F_{n},F_{n})=\left(XY^{n}F_{0},Y^{n-1}F_{0}\right)=2n(F_{n-1},F_{n-1}).}$

The nth Hermite function is defined by

${\displaystyle H_{n}(x)=\ F_{n}\ ^{-1}F_{n}(x)=p_{n}(x)e^{-{\frac {x^{2}}{2}}}.}$

p_n is called the nth Hermite polynomial.

Let

${\displaystyle {\begin{aligned}A&={1 \over {\sqrt {2}}}Y={1 \over {\sqrt {2}}}\left(-{d \over dx}+x\right)\\A^{*}&={1 \over {\sqrt {2}}}X={1 \over {\sqrt {2}}}\left({d \over dx}+x\right)\end{aligned}}}$

Thus

${\displaystyle AA^{*}-A^{*}A=I.}$

The operators P, Q or equivalently A, A* act irreducibly on ${\displaystyle {\mathcal {H}}}$ by a standard argument.^[27][28]

Indeed, under the unitary isomorphism with holomorphic Fock space ${\displaystyle {\mathcal {H}}}$ can be identified with C[z], the space of polynomials in z, with

${\displaystyle A={\frac {\partial }{\partial z}},\qquad A^{*}=z.}$

If a subspace invariant under A and A* contains a non-zero polynomial p(z), then, applying a power of A*, it contains a non-zero constant; applying then a power of A, it contains all zⁿ.

Under the isomorphism F_n is sent to a multiple of zⁿ and the operator D is given by

${\displaystyle D=2A^{*}A+I.}$

Let

${\displaystyle L_{0}={1 \over 2}(A^{*}A+{1 \over 2})={1 \over 2}(z{\partial \over \partial z}+{1 \over 2})}$

so that

${\displaystyle L_{0}z^{n}={1 \over 2}(n+{1 \over 2})z^{n}.}$

In the terminology of physics A, A* give a single boson and L₀ is the energy operator. It is diagonalizable with eigenvalues 1/2, 1, 3/2, ...., each of multiplicity one. Such a representation is called a positive energy representation.

Moreover,

${\displaystyle [L_{0},A]=-{1 \over 2}A,[L_{0},A^{*}]={1 \over 2}A^{*},}$

so that the Lie bracket with L₀ defines a derivation of the Lie algebra spanned by A, A* and I. Adjoining L₀ gives the semidirect product. The infinitesimal version of the Stone–von Neumann theorem states that the above representation on C[z] is the unique irreducible positive energy representation of this Lie algebra with L₀ = A*A + 1/2. For A lowers energy and A* raises energy. 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.

Let

${\displaystyle L_{-1}={1 \over 2}A^{2},L_{1}={1 \over 2}A^{*2},}$

so that

${\displaystyle [L_{-1},A]=0,\,\,\,[L_{-1},A^{*}]=A,\,\,\,[L_{1},A]=-A^{*},\,\,\,[L_{1},A^{*}]=0.}$

These operators satisfy:

${\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}}$

and act by derivations on the Lie algebra spanned by A, A* and I.

They are the infinitesimal operators corresponding to the metaplectic representation of SU(1,1).

The functions F_n are defined by

${\displaystyle F_{n}(x)=\left(x-{d \over dx}\right)^{n}e^{-{\frac {x^{2}}{2}}}=(-1)^{n}e^{\frac {x^{2}}{2}}{d^{n} \over dx^{n}}\left(e^{-x^{2}}\right)=\left(2^{n}x^{n}+\cdots \right)e^{-{\frac {x^{2}}{2}}}.}$

It follows that the Hermite functions are the orthonormal basis obtained by applying the Gram-Schmidt orthonormalization process to the basis xⁿ exp -x²/2 of ${\displaystyle {\mathcal {H}}}$.

The completeness of the Hermite functions follows from the fact that the Bargmann transform is unitary and carries the orthonormal basis e_n(z) of holomorphic Fock space onto the H_n(x).

The heat operator for the harmonic oscillator is the operator on L²(R) defined as the diagonal operator

${\displaystyle e^{-Dt}H_{n}=e^{-(2n+1)t}H_{n}.}$

It corresponds to the heat kernel given by Mehler's formula:

${\displaystyle K_{t}(x,y)\equiv \sum _{n\geq 0}e^{-(2n+1)t}H_{n}(x)H_{n}(y)=(4\pi t)^{-{1 \over 2}}\left({2t \over \sinh 2t}\right)^{1 \over 2}\exp \left(-{1 \over 4t}\left[{2t \over \tanh 2t}(x^{2}+y^{2})-{2t \over \sinh 2t}(2xy)\right]\right).}$

This follows from the formula

${\displaystyle \sum _{n\geq 0}s^{n}H_{n}(x)H_{n}(y)={1 \over {\sqrt {\pi (1-s^{2})}}}\exp {4xys-(1+s^{2})(x^{2}+y^{2}) \over 2(1-s^{2})}.}$

To prove this formula note that if s = σ², then by Taylor's formula

${\displaystyle F_{\sigma ,x}(z)\equiv \sum _{n\geq 0}\sigma ^{n}e_{n}(z)H_{n}(x)=\pi ^{-{1 \over 4}}e^{-{\frac {x^{2}}{2}}}\sum _{n\geq 0}{(-z)^{n}\sigma ^{n} \over 2^{n}n!}{d^{n}e^{x^{2}} \over dx^{n}}=\pi ^{-{\frac {1}{4}}}\exp \left(-{x^{2} \over 2}+{\sqrt {2}}xz\sigma -{z^{2}\sigma ^{2} \over 2}\right).}$

Thus F_σ,x lies in holomorphic Fock space and

${\displaystyle \sum _{n\geq 0}s^{n}H_{n}(x)H_{n}(y)=(F_{\sigma ,x},F_{\sigma ,y})_{\mathcal {F}},}$

an inner product that can be computed directly.

Wiener (1933, pp. 51–67) establishes Mehler's formula directly and uses a classical argument to prove that

${\displaystyle \int K_{t}(x,y)f(y)\,dy}$

tends to f in L²(R) as t decreases to 0. This shows the completeness of the Hermite functions and also, since

${\displaystyle {\widehat {H_{n}}}=(-i)^{n}H_{n},}$

can be used to derive the properties of the Fourier transform.

There are other elementary methods for proving the completeness of the Hermite functions, for example using Fourier series.^[29]

Sobolev spaces

The Sobolev spaces H_s, sometimes called Hermite-Sobolev spaces, are defined to be the completions of ${\displaystyle {\mathcal {S}}}$ with respect to the norms

${\displaystyle \ f\ _{(s)}^{2}=\sum _{n\geq 0} a_{n} ^{2}(1+2n)^{s},}$

where

${\displaystyle f=\sum a_{n}H_{n}}$

is the expansion of f in Hermite functions.^[30]

Thus

${\displaystyle \ f\ _{(s)}^{2}=(D^{s}f,f),\qquad (f_{1},f_{2})_{(s)}=(D^{s}f_{1},f_{2}).}$

The Sobolev spaces are Hilbert spaces. Moreover, H_s and H_–s are in duality under the pairing

${\displaystyle \langle f_{1},f_{2}\rangle =\int f_{1}f_{2}\,dx.}$

For s ≥ 0,

${\displaystyle \ (aP+bQ)f\ _{(s)}\leq ( a + b )C_{s}\ f\ _{\left(s+{1 \over 2}\right)}}$

for some positive constant C_s.

Indeed, such an inequality can be checked for creation and annihilation operators acting on Hermite functions H_n and this implies the general inequality.^[31]

It follows for arbitrary s by duality.

Consequently, for a quadratic polynomial R in P and Q

${\displaystyle \ Rf\ _{(s)}\leq C'_{s}\ f\ _{(s+1)}.}$

The Sobolev inequality holds for f in H_s with s > 1/2:

${\displaystyle f(x) \leq C_{s,k}\ f\ _{(s+k)}(1+x^{2})^{-k}}$

for any k ≥ 0.

Indeed, the result for general k follows from the case k = 0 applied to Q^kf.

For k = 0 the Fourier inversion formula

${\displaystyle f(x)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f}}(t)e^{itx}\,dt}$

implies

${\displaystyle f(x) \leq C\left(\int \left {\widehat {f}}(t)\right ^{2}(1+t^{2})^{s}\,dt\right)^{1 \over 2}=C\left(\left(I+Q^{2}\right)^{s}{\widehat {f}},{\widehat {f}}\right)^{1 \over 2}\leq C'\left\ {\widehat {f}}\right\ _{(s)}=C'\ f\ _{(s)}.}$

If s < t, the diagonal form of D, shows that the inclusion of H_t in H_s is compact (Rellich's lemma).

It follows from Sobolev's inequality that the intersection of the spaces H_s is ${\displaystyle {\mathcal {S}}}$. Functions in ${\displaystyle {\mathcal {S}}}$ are characterized by the rapid decay of their Hermite coefficients a_n.

Standard arguments show that each Sobolev space is invariant under the operators W(z) and the metaplectic group.^[32] Indeed, it is enough to check invariance when g is sufficiently close to the identity. In that case

${\displaystyle gDg^{-1}=D+A}$

with D + A an isomorphism from ${\displaystyle H_{t+2}}$ to ${\displaystyle H_{t}.}$

It follows that

${\displaystyle \ \pi (g)f\ _{(s)}^{2}=\left ((D+A)^{s}f,f)\right \leq \left\ (D+A)^{s}f\right\ _{(-s)}\cdot \ f\ _{(s)}\leq C\ f\ _{(s)}^{2}.}$

If ${\displaystyle f\in H_{s},}$ then

${\displaystyle {d \over ds}U(s)f=iPU(s)f,\qquad {d \over dt}V(t)f=iQV(t)f,}$

where the derivatives lie in ${\displaystyle H_{s-1/2}.}$

Similarly the partial derivatives of total degree k of U(s)V(t)f lie in Sobolev spaces of order s–k/2.

Consequently, a monomial in P and Q of order 2k applied to f lies in H_s–k and can be expressed as a linear combination of partial derivatives of U(s)V(t)f of degree ≤ 2k evaluated at 0.

Smooth vectors

The smooth vectors for the Weyl commutation relations are those u in L²(R) such that the map

${\displaystyle \Phi (z)=W(z)u}$

is smooth. By the uniform boundedness theorem, this is equivalent to the requirement that each matrix coefficient (W(z)u,v) be smooth.

A vector is smooth if and only it lies in ${\displaystyle {\mathcal {S}}}$.^[33] Sufficiency is clear. For necessity, smoothness implies that the partial derivatives of W(z)u lie in L²(R) and hence also D^ku for all positive k. Hence u lies in the intersection of the H_k, so in ${\displaystyle {\mathcal {S}}}$.

It follows that smooth vectors are also smooth for the metaplectic group.

Moreover, a vector is in ${\displaystyle {\mathcal {S}}}$ if and only if it is a smooth vector for the rotation subgroup of SU(1,1).

Analytic vectors

If Π(t) is a one parameter unitary group and for f in ${\displaystyle {\mathcal {S}}}$

${\displaystyle \Pi (f)=\int _{-\infty }^{\infty }f(t)\Pi (t)\,dt,}$

then the vectors Π(f)ξ form a dense set of smooth vectors for Π.

In fact taking

${\displaystyle f_{\varepsilon }(x)={1 \over {\sqrt {2\pi \varepsilon }}}e^{-x^{2}/2\varepsilon }}$

the vectors v = Π(f_ε)ξ converge to ξ as ε decreases to 0 and

${\displaystyle \Phi (t)=\Pi (t)v}$

is an analytic function of t that extends to an entire function on C.

The vector is called an entire vector for Π.

The wave operator associated to the harmonic oscillator is defined by

${\displaystyle \Pi (t)=e^{it{\sqrt {D}}}.}$

The operator is diagonal with the Hermite functions H_n as eigenfunctions:

${\displaystyle \Pi (t)H_{n}=e^{i(2n+1)^{1 \over 2}t}H_{n}.}$

Since it commutes with D, it preserves the Sobolev spaces.

The analytic vectors constructed above can be rewritten in terms of the Hermite semigroup as

${\displaystyle v=e^{-\varepsilon D}\xi .}$

The fact that v is an entire vector for Π is equivalent to the summability condition

${\displaystyle \sum _{n\geq 0}{r^{n}\ D^{n \over 2}v\ \over n!}<\infty }$

for all r > 0.

Any such vector is also an entire vector for U(s)V(t), that is the map

${\displaystyle F(s,t)=U(s)V(t)v}$

defined on R² extends to an analytic map on C².

This reduces to the power series estimate

${\displaystyle \left\ \sum _{m,n\geq 0}{1 \over m!n!}z^{m}w^{n}P^{m}Q^{n}v\right\ \leq C\sum _{k\geq 0}{( z + w )^{k} \over k!}\ D^{k \over 2}v\ <\infty .}$

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.

Let

${\displaystyle W(z,w)=e^{-izw/2}U(z)V(w)}$

be the analytic continuation of the operators W(x,y) from R² to C² such that

${\displaystyle e^{-izw/2}F(z,w)=W(z,w)v.}$

Then W leaves the space of entire vectors invariant and satisfies

${\displaystyle W(z_{1},w_{1})W(z_{2},w_{2})=e^{i(z_{1}w_{2}-w_{1}z_{2})}W(z_{1}+z_{2},w_{1}+w_{2}).}$

Moreover, for g in SL(2,R)

${\displaystyle \pi (g)W(u)\pi (g)^{*}=W(gu),}$

using the natural action of SL(2,R) on C².

Formally

${\displaystyle W(z,w)^{*}=W(-{\overline {z}},-{\overline {w}}).}$

Oscillator semigroup

There is a natural double cover of the Olshanski semigroup H, and its closure ${\displaystyle {\overline {H}}}$ that extends the double cover of SU(1,1) corresponding to the metaplectic group. It is given by pairs (g, γ) where g is an element of H or its closure

${\displaystyle g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$

and γ is a square root of a.

Such a choice determines a unique branch of

${\displaystyle \left(-{\overline {b}}z+{\overline {d}}\right)^{1 \over 2}}$

for z < 1.

The unitary operators π(g) for g in SL(2,R) satisfy

${\displaystyle \pi (g)W(u)=W(g\cdot u)\pi (g),\,\,\,\pi (g)^{*}W(u)=W(g^{-1}\cdot u)\pi (g)^{*}}$

for u in C².

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

${\displaystyle TW(u)=W(g\cdot u)T,\,\,\,T^{*}W(u)=W(g^{\dagger }\cdot u)T^{*}}$

for u in C². The implementing operator T is uniquely determined up to multiplication by a non-zero scalar.

The implementable elements form a semigroup, containing SL(2,R). Since the representation has positive energy, the bounded compact self-adjoint operators

${\displaystyle S_{0}(t)=e^{-tL_{0}}}$

for t > 0 implement the group elements in exp C₁.

It follows that all elements of the Olshanski semigroup and its closure are implemented.

Maximality of the Olshanki semigroup implies that no other elements of SL(2,C) are implemented. Indeed, otherwise every element of SL(2,C) would be implemented by a bounded operator, which would contradict the non-invertibility of the operators S₀(t) for t > 0.

In the Schrödinger representation the operators S₀(t) for t > 0 are given by Mehler's formula. They are contraction operators, positive and in every Schatten class. Moreover, they leave invariant each of the Sobolev spaces. The same formula is true for ${\displaystyle \Re \,t>0}$ by analytic continuation.

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. The corresponding semigroup of contraction operators is called the oscillator semigroup. The extended oscillator semigroup is obtained by taking the semidirect product with the operators W(u). These operators lie in every Schatten class and leave invariant the Sobolev spaces and the space of entire vectors for W.

The decomposition

${\displaystyle {\overline {H}}=G\cdot \exp {\overline {C}}}$

corresponds at the operator level to the polar decomposition of bounded operators.

Moreover, since any matrix in H is conjugate to a diagonal matrix by elements in H or H⁻¹, every operator in the oscillator semigroup is quasi-similar to an operator S₀(t) with ${\displaystyle \Re t>0}$. In particular it has the same spectrum consisting of simple eigenvalues.

In the Fock model, if the element g of the Olshanki semigroup H corresponds to the matrix

${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}},}$

the corresponding operator is given by

${\displaystyle \pi (g,\gamma )f(w)={1 \over \pi }\iint _{\mathbf {C} }K(w,{\overline {z}})f(z)e^{- z ^{2}}\,dxdy,}$

where

${\displaystyle K(w,z)=\gamma ^{-1}\cdot \exp \,{1 \over 2a}(cz^{2}+2wz-bw^{2})}$

and γ is a square root of a. Operators π(g,γ) for g in the semigroup H are exactly those that are Hilbert–Schmidt operators and correspond to kernels of the form

${\displaystyle K(w,z)=C\cdot \exp \,{1 \over 2}(pz^{2}+2qwz+rw^{2})}$

for which the complex symmetric matrix

${\displaystyle {\begin{pmatrix}p&q\\q&r\end{pmatrix}}}$

has operator norm strictly less than one.

Operators in the extended oscillator semigroup are given by similar expressions with additional linear terms in z and w appearing in the exponential.

In the disk model for the two irreducible components of the metaplectic representation, the corresponding operators are given by

${\displaystyle \pi _{\pm }(g)F_{\pm }(z)=(-{\overline {b}}z+{\overline {d}})^{-1\pm 1/2}F_{\pm }\left({{\overline {a}}z-{\overline {c}} \over -{\overline {b}}z+{\overline {d}}}\right).}$

It is also possible to give an explicit formula for the contraction operators corresponding to g in H in the Schrödinger representation, It was by this formula that Howe (1988) introduced the oscillator semigroup as an explicit family of operators on L²(R).^[34]

In fact consider the Siegel upper half plane consisting of symmetric complex 2x2 matrices with positive definite real part:

${\displaystyle Z={\begin{pmatrix}A&B\\B&D\end{pmatrix}}}$

and define the kernel

${\displaystyle K_{Z}(x,y)=e^{-(Ax^{2}+2Bxy+Dy^{2})}.}$

with corresponding operator

${\displaystyle T_{Z}f(x)=\int _{-\infty }^{\infty }K_{Z}(x,y)f(y)\,dy}$

for f in L²(R).

Then direct computation gives

${\displaystyle T_{Z_{1}}T_{Z_{2}}=(D_{1}+A_{2})^{-1/2}T_{Z_{3}}}$

where

${\displaystyle Z_{3}={\begin{pmatrix}A_{1}-B_{1}^{2}(D_{1}+A_{2})^{-1}&-B_{1}B_{2}(D_{1}+A_{2})^{-1}\\-B_{1}B_{2}(D_{1}+A_{2})^{-1}&D_{2}-B_{2}^{2}(D_{1}+A_{2})^{-1}\end{pmatrix}}.}$

Moreover,

${\displaystyle T_{Z}^{*}=T_{Z^{+}}}$

where

${\displaystyle Z^{+}={\begin{pmatrix}{\overline {D}}&{\overline {B}}\\{\overline {B}}&{\overline {A}}\end{pmatrix}}.}$

By Mehler's formula for ${\displaystyle \Re \,t>0}$

${\displaystyle e^{-t(P^{2}+Q^{2})}=(\mathrm {cosech} \,2t)^{1 \over 2}\cdot T_{Z(t)}}$

with

${\displaystyle Z(t)={\begin{pmatrix}\coth 2t&-\mathrm {cosech} \,2t\\-\mathrm {cosech} \,2t&\coth 2t\end{pmatrix}}.}$

The oscillator semigroup is obtained by taking only matrices with B ≠ 0. From the above, this condition is closed under composition.

A normalized operator can be defined by

${\displaystyle S_{Z}=B^{1 \over 2}\cdot T_{Z}.}$

The choice of a square root determines a double cover.

In this case S_Z corresponds to the element

${\displaystyle g={\begin{pmatrix}-DB^{-1}&DAB^{-1}-B\\B^{-1}&-AB^{-1}\end{pmatrix}}}$

of the Olshankii semigroup H.

Moreover, S_Z is a strict contraction:

${\displaystyle \ S_{Z}\ <1.}$

It follows also that

${\displaystyle S_{Z_{1}}S_{Z_{2}}=\pm S_{Z_{3}}.}$

Weyl calculus

For a function a(x,y) on R² = C, let

${\displaystyle \psi (a)={1 \over 2\pi }\int {\widehat {a}}(x,y)W(x,y)\,dxdy.}$

So

${\displaystyle \psi (a)f(x)=\int K(x,y)f(y)\,dy,}$

where

${\displaystyle K(x,y)=\int a(t,{x+y \over 2})e^{i(x-y)t}\,dt.}$

Defining in general

${\displaystyle W(F)={1 \over 2\pi }\int F(z)W(z)\,dxdy,}$

the product of two such operators is given by the formula

${\displaystyle W(F)W(G)=W(F\star G),}$

where the twisted convolution or Moyal product is given by

${\displaystyle F\star G(z)={1 \over 2\pi }\int F(z_{1})G(z_{2}-z_{1})e^{i(x_{1}y_{2}-y_{1}x_{2})}\,dx_{1}dy_{1}.}$

The smoothing operators correspond to W(F) or ψ(a) with F or a Schwartz functions on R². The corresponding operators T have kernels that are Schwartz functions. They carry each Sobolev space into the Schwartz functions. Moreover, every bounded operator on L² (R) having this property has this form.

For the operators ψ(a) the Moyal product translates into the Weyl symbolic calculus. Indeed, if the Fourier transforms of a and b have compact support than

${\displaystyle \psi (a)\psi (b)=\psi (a\circ b),}$

where

${\displaystyle a\circ b=\sum _{n\geq 0}{i^{n} \over n!}\left({\partial ^{2} \over \partial x_{1}\partial y_{2}}-{\partial ^{2} \over \partial y_{1}\partial x_{2}}\right)^{n}a\otimes b _{\mathrm {diagonal} }.}$

This follows because in this case b must extend to an entire function on C² by the Paley-Wiener theorem.

This calculus can be extended to a broad class of symbols, but the simplest corresponds to convolution by a class of functions or distributions that all have the form T + S where T is a distribution of compact with singular support concentrated at 0 and where S is a Schwartz function. This class contains the operators P, Q as well as D^1/2 and D^−1/2 where D is the harmonic oscillator.

The mth order symbols S^m are given by smooth functions a satisfying

${\displaystyle \partial ^{\alpha }a(z) \leq C_{\alpha }(1+ z )^{m- \alpha }}$

for all α and Ψ^m consists of all operators ψ(a) for such a.

If a is in S^m and χ is a smooth function of compact support equal to 1 near 0, then

${\displaystyle {\widehat {a}}=\chi {\widehat {a}}+(1-\chi ){\widehat {a}}=T+S,}$

with T and S as above.

These operators preserve the Schwartz functions and satisfy;

${\displaystyle \Psi ^{m}\cdot \Psi ^{m}\subseteq \Psi ^{m+n},\,\,\,\,[\Psi ^{m},\Psi ^{n}]\subseteq \Psi ^{m+n-2}.}$

The operators P and Q lie in Ψ¹ and D lies in Ψ².

Properties:

• A zeroth order symbol defines a bounded operator on L²(R).
• D⁻¹ lies in Ψ⁻²
• If R = R* is smoothing, then D + R has a complete set of eigenvectors f_n in ${\displaystyle {\mathcal {S}}}$ with (D + R)f_n = λ_nf_n and λ_n tends to ≈ as n tends to ≈.
• D^1/2 lies in Ψ¹ and hence D^−1/2 lies in Ψ⁻¹, since D^−1/2 = D^1/2 ·D⁻¹
• Ψ⁻¹ consists of compact operators, Ψ^−s consists of trace-class operators for s > 1 and Ψ^k carries H_m into H_m–k.
• ${\displaystyle \mathrm {Tr} \,\psi (a)=\int a}$

The proof of boundedness of Howe (1980) is particularly simple: if

${\displaystyle T_{a,b}v=(v,b)a,}$

then

${\displaystyle T_{W(z)a,b}=e^{ z ^{2}/2}[W(z)T_{a,E_{0}}W(z)^{-1}T_{E_{0},b}],}$

where the bracketed operator has norm less than ${\displaystyle \ a\ \cdot \ b\ }$. So if F is supported in z ≤ R, then

${\displaystyle \ W(F)\ \leq e^{R^{2}/2}\ {\widehat {F}}\ _{\infty }.}$

The property of D⁻¹ is proved by taking

${\displaystyle S=\psi (a)}$

with

${\displaystyle a(z)={1 \over z ^{2}+1}.}$

Then R = I – DS lies in Ψ⁻¹, so that

${\displaystyle A\sim S+SR+SR^{2}+\cdots }$

lies in Ψ⁻² and T = DA – I is smoothing. Hence

${\displaystyle D^{-1}=A-D^{-1}T}$

lies in Ψ⁻² since D⁻¹ T is smoothing.

The property for D^1/2 is established similarly by constructing B in Ψ^1/2 with real symbol such that D – B⁴ is a smoothing operator. Using the holomorphic functional calculus it can be checked that D^1/2 – B² is a smoothing operator.

The boundedness result above was used by Howe (1980) to establish the more general inequality of Alberto Calderón and Remi Vaillancourt for pseudodifferential operators. An alternative proof that applies more generally to Fourier integral operators was given by Howe (1988). He showed that such operators can be expressed as integrals over the oscillator semigroup and then estimated using the Cotlar-Stein lemma.^[35]

Applications and generalizations

Theory for finite abelian groups

Weil (1964) noted that the formalism of the Stone–von Neumann theorem and the oscillator representation of the symplectic group extends from the real numbers R to any locally compact abelian group. A particularly simple example is provided by finite abelian groups, where the proofs are either elementary or simplifications of the proofs for R.^[36][37]

Let A be a finite abelian group, written additively, and let Q be a non-degenerate quadratic form on A with values in T. Thus

${\displaystyle (a,b)=Q(a)Q(b)Q(a+b)^{-1}}$

is a symmetric bilinear form on A that is non-degenerate, so permits an identification between A and its dual group A* = Hom (A, T).

Let ${\displaystyle V=\ell ^{2}(A)}$ be the space of complex-valued functions on A with inner product

${\displaystyle (f,g)=\sum _{x\in A}f(x){\overline {g(x)}}.}$

Define operators on V by

${\displaystyle U(x)f(t)=f(t-x),\,\,\,V(y)f(t)=(y,t)f(t)}$

for x, y in A. Then U(x) and V(y) are unitary representations of A on V satisfying the commutation relations

${\displaystyle U(x)V(y)=(x,y)V(y)U(x).}$

This action is irreducible and is the unique such irreducible representation of these relations.

Let G = A × A and for z = (x, y) in G set

${\displaystyle W(z)=U(x)V(y).}$

Then

${\displaystyle W(z_{1})W(z_{2})=B(z_{1},z_{2})W(z_{2})W(z_{1}),}$

where

${\displaystyle B(z_{1},z_{2})=(x_{1},y_{2})(x_{2},y_{1})^{-1},}$

a non-degenerate alternating bilinear form on G. The uniqueness result above implies that if W'(z) is another family of unitaries giving a projective representation of G such that

${\displaystyle W'(z_{1})W'(z_{2})=B(z_{1},z_{2})W'(z_{2})W'(z_{1}),}$

then there is a unitary U, unique up to a phase, such that

${\displaystyle W'(z)=\lambda (z)UW(z)U^{*},}$

for some λ(z) in T.

In particular if g is an automorphism of G preserving B, then there is an essentially unique unitary π(g) such that

${\displaystyle W(gz)=\lambda _{g}(z)\pi (g)W(z)\pi (g)^{*}.}$

The group of all such automorphisms is called the symplectic group for B and π gives a projective representation of G on V.

The group SL(2.Z) naturally acts on G = A x A by symplectic automorphisms. It is generated by the matrices

${\displaystyle S={\begin{pmatrix}0&1\\-1&0\end{pmatrix}},\qquad R={\begin{pmatrix}1&0\\1&1\end{pmatrix}}.}$

If Z = –I, then Z is central and

${\displaystyle {S^{2}=Z,\,\,\,(SR)^{3}=Z,\,\,\,Z^{2}=I.}}$

These automorphisms of G are implemented on V by the following operators:

${\displaystyle {\begin{aligned}\pi (S)f(t)&= A ^{-{\frac {1}{2}}}\sum _{x\in A}(-x,t)f(x)&&{\text{the Fourier transform for }}A\\\pi (Z)f(t)&=f(-t)\\\pi (R)f(t)&=Q(t)^{-1}f(t)\\\end{aligned}}}$

It follows that

${\displaystyle (\pi (S)\pi (R))^{3}=\mu \pi (Z),}$

where μ lies in T. Direct calculation shows that μ is given by the Gauss sum

${\displaystyle \mu = A ^{-{\frac {1}{2}}}\sum _{x\in A}Q(x).}$

Transformation laws for theta functions

The metaplectic group was defined as the group

${\displaystyle \operatorname {Mp} (2,\mathbf {R} )=\left\{\left(\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}},G\right)\right G(\tau )^{2}=c\tau +d,\tau \in \mathbf {H} \right\},}$

The coherent state

${\displaystyle f_{\tau }(x)=e^{{\frac {1}{2}}i\tau x^{2}}}$

defines a holomorphic map of H into L²(R) satisfying

${\displaystyle \pi ((g^{t})^{-1})f_{\tau }=(c\tau +d)^{-{\frac {1}{2}}}f_{g\tau }.}$

This is in fact a holomorphic map into each Sobolev space H_k and hence also ${\displaystyle H_{\approx }={\mathcal {S}}}$.

On the other hand, in ${\displaystyle H_{-\approx }={\mathcal {S}}'}$ (in fact in H_–1) there is a finite-dimensional space of distributions invariant under SL(2,Z) and isomorphic to the N-dimensional oscillator representation on ${\displaystyle \ell ^{2}(A)}$ where A = Z/NZ.

In fact let m > 0 and set N = 2m. Let

${\displaystyle M={\sqrt {2\pi m}}\cdot \mathbf {Z} .}$

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

${\displaystyle \Psi _{b}=\sum _{x\in M}\delta _{x+b}}$

with b in M₁, where

${\displaystyle M_{1}={1 \over 2m}M\supset M.}$

The sum defining Ψ_b converges in ${\displaystyle H_{-1}\subset {\mathcal {S}}'}$ and depends only on the class of b in M₁/M. On the other hand, the operators U(x) and V(y) with 'x, y in M₁ commute with all the corresponding operators for M. So M₁ leaves the subspace V₀ spanned by the Ψ_b invariant. Hence the group A = M₁ acts on V₀. This action can immediately be identified with the action on V for the N-dimensional oscillator representation associated with A, since

${\displaystyle U(b)\Psi _{b'}=\Psi _{b+b'},\qquad V(b)\Psi _{b'}=e^{-imbb'}\Psi _{b'}.}$

Since the operators π(R) and π(S) normalise the two sets of operators U and V corresponding to M and M₁, it follows that they leave V₀ invariant and on V₀ must be constant multiples of the operators associated with the oscillator representation of A. In fact they coincide. From R this is immediate from the definitions, which show that

${\displaystyle R(\Psi _{b})=e^{\pi imb^{2}}\Psi _{b}.}$

For S it follows from the Poisson summation formula and the commutation properties with the operators U)x) and V(y). The Poisson summation is proved classically as follows.^[38]

For a > 0 and f in ${\displaystyle {\mathcal {S}}}$ let

${\displaystyle F(t)=\sum _{x\in M}f(x+t).}$

F is a smooth function on R with period a:

${\displaystyle F(t+a)=F(t).}$

The theory of Fourier series shows that

${\displaystyle F(0)=\sum _{n\in \mathbf {Z} }c_{n}}$

with the sum absolutely convergent and the Fourier coefficients given by

${\displaystyle c_{n}=a^{-1}\int _{0}^{a}F(t)e^{-{\frac {2\pi int}{a}}}\,dt=a^{-1}\int _{-\infty }^{\infty }f(t)e^{-{\frac {2\pi int}{a}}}\,dt={{\sqrt {2\pi }} \over a}{\widehat {f}}\left({\tfrac {2\pi n}{a}}\right).}$

Hence

${\displaystyle \sum _{n\in \mathbf {Z} }f(na)={\frac {\sqrt {2\pi }}{a}}\sum _{n\in \mathbf {Z} }{\widehat {f}}\left({\tfrac {2\pi n}{a}}\right),}$

the usual Poisson summation formula.

This formula shows that S acts as follows

${\displaystyle S(\Psi _{b})=(2m)^{-{\frac {1}{2}}}\sum _{b'\in M_{1}/M}e^{-imbb'}\Psi _{b'},}$

and so agrees exactly with formula for the oscillator representation on A.

Identifying A with Z/2mZ, with

${\displaystyle b(n)={\frac {{\sqrt {2\pi }}n}{2m}}}$

assigned to an integer n modulo 2m, the theta functions can be defined directly as matrix coefficients:^[39]

${\displaystyle \Theta _{m,n}(\tau ,z)=(W(z)f_{\tau },\Psi _{b(n)}).}$

For τ in H and z in C set

${\displaystyle q=e^{2\pi i\tau },\qquad u=e^{\pi iz}}$

so that q < 1. The theta functions agree with the standard classical formulas for the Jacobi-Riemann theta functions:

${\displaystyle \Theta _{n,m}(\tau ,z)=\sum _{k\in {\frac {n}{2m}}+\mathbf {Z} }q^{mk^{2}}u^{2mk}.}$

By definition they define holomorphic functions on H × C. The covariance properties of the function f_τ and the distribution Ψ_b lead immediately to the following transformation laws:

${\displaystyle {\begin{aligned}\Theta _{n,m}(\tau ,z+a)&=\Theta _{n,m}(\tau ,z)&&a\in \mathbf {Z} \\\Theta _{n,m}(\tau ,z+b\tau )&=q^{-b^{2}}u^{-b}\Theta _{n,m}(\tau ,z)&&b\in \mathbf {Z} \\\Theta _{n,m}(\tau +1,z)&=e^{\frac {\pi in^{2}}{m}}\Theta _{n,m}(\tau ,z)\\\Theta _{n,m}(-{\tfrac {1}{\tau }},{\tfrac {z}{\tau }})&=\tau ^{\frac {1}{2}}e^{-{\frac {i\pi }{8}}}(2m)^{-{\frac {1}{2}}}\sum _{n'\in \mathbf {Z} /2m\mathbf {Z} }e^{-{\frac {\pi inn'}{m}}}\Theta _{n',m}(\tau ,z)\end{aligned}}}$

Derivation of law of quadratic reciprocity

Because the operators π(S), π (R) and π(J) on L²(R) restrict to the corresponding operators on V₀ for any choice of m, signs of cocycles can be determined by taking m = 1. In this case the representation is 2-dimensional and the relation

${\displaystyle {(\pi (S)\pi (R))^{3}=\pi (J)}}$

on L²(R) can be checked directly on V₀.

But in this case

${\displaystyle \mu ={\frac {1}{\sqrt {2}}}\left(e^{\frac {i\pi }{4}}+e^{-{\frac {i\pi }{4}}}\right)=1.}$

The relation can also be checked directly by applying both sides to the ground state exp -x²/2.

Consequently, it follows that for m ≥ 1 the Gauss sum can be evaluated:^[40]

${\displaystyle \sum _{x\in \mathbf {Z} /2m\mathbf {Z} }e^{\pi ix^{2}/2m}={\sqrt {m}}(1+i).}$

For m odd, define

${\displaystyle {G(c,m)=\sum _{x\in \mathbf {Z} /m\mathbf {Z} }e^{2\pi icx^{2}/m}.}}$

If m is odd, then, splitting the previous sum up into two parts, it follows that G(1,m) equals m^1/2 if m is congruent to 1 mod 4 and equals i m^1/2 otherwise. If p is an odd prime and c is not divisible by p, this implies

${\displaystyle {G(c,p)=\left({c \over p}\right)G(1,p)}}$

where ${\displaystyle \left({c \over p}\right)}$ is the Legendre symbol equal to 1 if c is a square mod p and –1 otherwise. Moreover, if p and q are distinct odd primes, then

${\displaystyle {G(1,pq)/G(1,p)G(1,q)=\left({p \over q}\right)\left({q \over p}\right)}.}$

From the formula for G(1,p) and this relation, the law of quadratic reciprocity follows:

${\displaystyle {\left({p \over q}\right)\left({q \over p}\right)=(-1)^{\frac {(p-1)(q-1)}{4}}.}}$

Theory in higher dimensions

The theory of the oscillator representation can be extended from R to Rⁿ with the group SL(2,R) replaced by the symplectic group Sp(2n,R). The results can be proved either by straightforward generalisations from the one-dimensional case as in Folland (1989) or by using the fact that the n-dimensional case is a tensor product of n one-dimensional cases, reflecting the decomposition:

${\displaystyle L^{2}({\mathbf {R} }^{n})=L^{2}({\mathbf {R} })^{\otimes n}.}$

Let ${\displaystyle {\mathcal {S}}}$ be the space of Schwartz functions on Rⁿ, a dense subspace of L²(Rⁿ). For s, t in Rⁿ, define U(s) and V(t) on ${\displaystyle {\mathcal {S}}}$ and L²(R) by

${\displaystyle U(s)f(x)=f(x-s),\qquad V(t)f(tx)=e^{ix\cdot t}f(x).}$

From the definition U and V satisfy the Weyl commutation relation

${\displaystyle U(s)V(t)=e^{-is\cdot t}V(t)U(s).}$

As before this is called the Schrödinger representation.

The Fourier transform is defined on ${\displaystyle {\mathcal {S}}}$ by

${\displaystyle {{\widehat {f}}(t)={1 \over (2\pi )^{n/2}}\int _{{\mathbf {R} }^{n}}f(x)e^{-ix\cdot t}\,dx.}}$

The Fourier inversion formula

${\displaystyle {f(x)={1 \over (2\pi )^{n/2}}\int _{{\mathbf {R} }^{n}}{\widehat {f}}(t)e^{ix\cdot t}\,dt}}$

shows that the Fourier transform is an isomorphism of ${\displaystyle {\mathcal {S}}}$ onto itself extending to a unitary mapping of L²(Rⁿ) onto itself (Plancherel's theorem).

The Stone–von Neumann theorem asserts that the Schrödinger representation is irreducible and is the unique irreducible representation of the commutation relations: any other representation is a direct sum of copies of this representation.

If U and V satisfying the Weyl commutation relations, define

${\displaystyle {W(x,y)=e^{ix\cdot y/2}U(x)V(y).}}$

Then

${\displaystyle {W(x_{1},y_{1})W(x_{2},y_{2})=e^{i(x_{1}\cdot y_{2}-y_{1}\cdot x_{2})}W(x_{1}+x_{2},y_{1}+y_{2}),}}$

so that W defines a projective unitary representation of R²ⁿ with cocycle given by

${\displaystyle \omega (z_{1},z_{2})=e^{iB(z_{1},z_{2})},}$

where ${\displaystyle z=x+iy=(x,y)}$ and B is the symplectic form on R²ⁿ given by

${\displaystyle B(z_{1},z_{2})=x_{1}\cdot y_{2}-y_{1}\cdot x_{2}=\Im \,z_{1}\cdot {\overline {z_{2}}}.}$

The symplectic group Sp (2n,R) is defined to be group of automorphisms g of R²ⁿ preserving the form B. It follows from the Stone–von Neumann theorem that for each such g there is a unitary π(g) on L²(R) satisfying the covariance relation

${\displaystyle \pi (g)W(z)\pi (g)^{*}=W(g(z)).}$

By Schur's lemma the unitary π(g) is unique up to multiplication by a scalar ζ with ζ = 1, so that π defines a projective unitary representation of Sp(n). Representatives can be chosen for π(g), unique up to a sign, which show that the 2-cocycle for the projective representation of Sp(2n,R) takes values ±1. In fact elements of the group Sp(n,R) are given by 2n × 2n real matrices g satisfying

${\displaystyle {gJg^{t}=J,}}$

where

${\displaystyle {J={\begin{pmatrix}0&-I\\I&0\end{pmatrix}}.}}$

Sp(2n,R) is generated by matrices of the form

${\displaystyle g_{1}={\begin{pmatrix}A&0\\0&(A^{t})^{-1}\end{pmatrix}},\,\,g_{2}={\begin{pmatrix}I&0\\B&I\end{pmatrix}},\,\,g_{3}={\begin{pmatrix}0&I\\-I&0\end{pmatrix}},}$

and the operators

${\displaystyle {\pi (g_{1})f(x)=\pm \det(A)^{-{\frac {1}{2}}}f(A^{-1}x),\,\,\pi (g_{2})f(x)=\pm e^{-ix^{t}Bx}f(x),\,\,\pi (g_{3})f(x)=\pm e^{in\pi /8}{\widehat {f}}(x)}}$

satisfy the covariance relations above. This gives an ordinary unitary representation of the metaplectic group, a double cover of Sp(2n,R). Indeed, Sp(n,R) acts by Möbius transformations on the generalised Siegel upper half plane H_n consisting of symmetric complex n × n matrices Z with strictly imaginary part by

${\displaystyle {gZ=(AZ+B)(CZ+D)^{-1}}}$

if

${\displaystyle {g={\begin{pmatrix}A&B\\C&D\end{pmatrix}}.}}$

The function

${\displaystyle {m(g,z)=\det(CZ+D)}}$

satisfies the 1-cocycle relation

${\displaystyle {m(gh,Z)=m(g,hZ)m(h,Z).}}$

The metaplectic group Mp(2n,R) is defined as the group

${\displaystyle {Mp(2,\mathbf {R} )=\{(g,G):\,G(Z)^{2}=m(g,Z)\}}}$

and is a connected double covering group of Sp(2n,R).

If ${\displaystyle \Im Z>0}$, then it defines a coherent state

${\displaystyle {f_{z}(x)=e^{ix^{t}Zx/2}}}$

in L², lying in a single orbit of Sp(2n) generated by

${\displaystyle {f_{iI}(x)=e^{-x\cdot x/2}.}}$

If g lies in Mp(2n,R) then

${\displaystyle {\pi ((g^{t})^{-1})f_{Z}(x)=m(g,Z)^{-1/2}f_{gZ}(x)}}$

defines an ordinary unitary representation of the metaplectic group, from which it follows that the cocycle on Sp(2n,R) takes only values ±1.

Holomorphic Fock space is the Hilbert space ${\displaystyle {\mathcal {F}}_{n}}$ of holomorphic functions f(z) on C_n with finite norm

${\displaystyle {{1 \over \pi ^{n}}\int _{{\mathbf {C} }^{n}} f(z) ^{2}e^{- z ^{2}}\,dx\cdot dy}}$

inner product

${\displaystyle {(f_{1},f_{2})={1 \over \pi ^{n}}\int _{{\mathbf {C} }^{n}}f_{1}(z){\overline {f_{2}(z)}}e^{- z ^{2}}\,dx\cdot dy.}}$

and orthonormal basis

${\displaystyle {e_{\alpha }(z)={z^{\alpha } \over {\sqrt {\alpha !}}}}}$

for α a multinomial. For f in ${\displaystyle {\mathcal {F}}_{n}}$ and z in Cⁿ, the operators

${\displaystyle {W_{{\mathcal {F}}_{n}}(z)f(w)=e^{- z ^{2}}e^{w{\overline {z}}}f(w-z).}}$

define an irreducible unitary representation of the Weyl commutation relations. By the Stone–von Neumann theorem there is a unitary operator ${\displaystyle {\mathcal {U}}}$ from L²(Rⁿ) onto ${\displaystyle {\mathcal {F}}_{n}}$ intertwining the two representations. It is given by the Bargmann transform

${\displaystyle {{\mathcal {U}}f(z)={1 \over (2\pi )^{n/2}}\int B(z,t)f(t)\,dt,}}$

where

${\displaystyle B(z,t)=\exp[-z\cdot z-t\cdot t/2+z\cdot t].}$

Its adjoint ${\displaystyle {\mathcal {U}}^{*}}$ is given by the formula:

${\displaystyle {{\mathcal {U}}^{*}F(t)={1 \over \pi ^{n}}\int _{{\mathbf {C} }^{n}}B({\overline {z}},t)F(z)\,dx\cdot dy.}}$

Sobolev spaces, smooth and analytic vectors can be defined as in the one-dimensional case using the sum of n copies of the harmonic oscillator

${\displaystyle \Delta _{n}=\sum _{i=1}^{n}-{\partial ^{2} \over \partial x_{i}^{2}}+x_{i}^{2}.}$

The Weyl calculus similarly extends to the n-dimensional case.

The complexification Sp(2n,C) of the symplectic group is defined by the same relation, but allowing the matrices A, B, C and D to be complex. The subsemigroup of group elements that take the Siegel upper half plane into itself has a natural double cover. The representations of Mp(2n,R) on L²(Rⁿ) and ${\displaystyle {\mathcal {F}}_{n}}$ extend naturally to a representation of this semigroup by contraction operators defined by kernels, which generalise the one-dimensional case (taking determinants where necessary). The action of Mp(2n,R) on coherent states applies equally well to operators in this larger semigroup.^[41]

As in the 1-dimensional case, where the group SL(2,R) has a counterpart SU(1,1) through the Cayley transform with the upper half plane replaced by the unit disc, the symplectic group has a complex counterpart. Indeed, if C is the unitary matrix

${\displaystyle {C={1 \over {\sqrt {2}}}{\begin{pmatrix}I&iI\\I&-iI\end{pmatrix}}}}$

then C Sp(2n) C⁻¹ is the group of all matrices

${\displaystyle {g={\begin{pmatrix}A&B\\{\overline {B}}&{\overline {A}}\end{pmatrix}}}}$

such that

${\displaystyle {AA^{*}-BB^{*}=I,\,\,\,AB^{t}=BA^{t};}}$

or equivalently

${\displaystyle gKg^{*}=K,}$

where

${\displaystyle {K={\begin{pmatrix}I&0\\0&-I\end{pmatrix}}.}}$

The Siegel generalized disk D_n is defined as the set of complex symmetric n x n matrices W with operator norm less than 1.

It consist precisely of Cayley transforms of points Z in the Siegel generalized upper half plane:

${\displaystyle {W=(Z-iI)(Z+iI)^{-1}.}}$

Elements g act on D_n

${\displaystyle {gW=(AW+B)({\overline {B}}W+{\overline {A}})^{-1}}}$

and, as in the one dimensional case this action is transitive. The stabilizer subgroup of 0 consists of matrices with A unitary and B = 0.

For W in D_n the metaplectic coherent states in holomorphic Fock space are defined by

${\displaystyle {f_{W}(z)=e^{z^{t}Wz/2}.}}$

The inner product of two such states is given by

${\displaystyle {(f_{W_{1}},f_{W_{2}})=\det(1-W_{1}{\overline {W_{2}}})^{-1/2}.}}$

Moreover, the metaplectic representation π satisfies

${\displaystyle {\pi (g)f_{W}=\det({\overline {A}}+{\overline {B}}W)^{-1/2}f_{gW}.}}$

The closed linear span of these states gives the even part of holomorphic Fock space ${\displaystyle {\mathcal {F}}_{n}^{+}}$. The embedding of Sp(2n) in Sp(2(n+1)) and the compatible identification

${\displaystyle {\mathcal {F}}_{n+1}^{+}={\mathcal {F}}_{n}^{+}\oplus {\mathcal {F}}_{n}^{-}}$

lead to an action on the whole of ${\displaystyle {\mathcal {F}}_{n}}$. It can be verified directly that it is compatible with the action of the operators W(z).^[42]

Since the complex semigroup has as Shilov boundary the symplectic group, the fact that this representation has a well-defined contractive extension to the semigroup follows from the maximum modulus principle and the fact that the semigroup operators are closed under adjoints. Indeed, it suffices to check, for two such operators S, T and vectors v_i proportional to metaplectic coherent states, that

${\displaystyle \left \sum _{i,j}(STv_{i},v_{j})\right \leq \ \sum _{i}v_{i}\ ^{2},}$

which follows because the sum depends holomorphically on S and T, which are unitary on the boundary.

Index theorems for Toeplitz operators

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, ..., z_n. Let P be the projection onto Hardy space. It is known that if m(f) denotes multiplication by a continuous function f on S, then the commutator [P,m(f)] is compact. Consequently, defining the Toeplitz operator by

${\displaystyle {T(f)=Pm(f)P}}$

on Hardy space, it follows that T(fg) – T(f)T(g) is compact for continuous f and g. The same holds if f and g are matrix-valued functions (so that the corresponding Toeplitz operators are matrices of operators on H²(S)). In particular if f is a function on S taking values in invertible matrices, then

${\displaystyle {T(f)T(f^{-1})-I,\qquad T(f^{-1})T(f)-I}}$

are compact and hence T(f) is a Fredholm operator with an index defined as

${\displaystyle \operatorname {ind} T(f)=\dim \ker T(f)-\dim \ker T(f)^{*}.}$

The index has been computed using the methods of K-theory by Coburn (1973) and coincides up to a sign with the degree of f as a continuous mapping from S into the general linear group.

Helton & Howe (1975) gave an analytic way to establish this index theorem, simplied later by Howe. Their proof relies on the fact if f is smooth then the index is given by the formula of McKean and Singer:^[43]

${\displaystyle \operatorname {ind} T(f)=\operatorname {Tr} (I-T(f^{-1})T(f))^{n}-\operatorname {Tr} (I-T(f)T(f^{-1}))^{n}.}$

Howe (1980) noticed that there was a natural unitary isomorphism between H²(S) and L²(Rⁿ) carrying the Toeplitz operators

${\displaystyle {T_{j}=T(z_{j})}}$

onto the operators

${\displaystyle {(P_{j}+iQ_{j})\Delta ^{-1/2}.}}$

These are examples of zeroth order operators constructed within the Weyl calculus. The traces in the McKean-Singer formula can be computed directly using the Weyl calculus, leading to another proof of the index theorem.^[44] This method of proving index theorems was generalised by Alain Connes within the framework of cyclic cohomology.^[45]

Theory in infinite dimensions

The theory of the oscillator representation in infinite dimensions is due to Irving Segal and David Shale.^[46] Graeme Segal used it to give a mathematically rigorous construction of projective representations of loop groups and the group of diffeomorphisms of the circle. At an infinitesimal level the construction of the representations of the Lie algebras, in this case the affine Kac–Moody algebra and the Virasoro algebra, was already known to physicists, through dual resonance theory and later string theory. 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 representation of commutation on Fock space is generalized to infinite dimensions by replacing Cⁿ (or its dual space) by an arbitrary complex Hilbert space H. The symmetric group S_k acts on H^⊗k. S^k(H) is defined to be the fixed point subspace of S_k and the symmetric algebra is the algebraic direct sum

${\displaystyle {\bigoplus _{k\geq 0}S^{k}(H).}}$

It has a natural inner product inherited from H^⊗k:

${\displaystyle {(x_{1}\otimes \cdots \otimes x_{k},y_{1}\otimes \cdots \otimes y_{k})=k!\cdot \prod _{i=1}^{k}(x_{i},y_{i}).}}$

Taking the components S^k(H) to be mutually orthogonal, the symmetric Fock space S(H) is defined to be the Hilbert space completion of this direct sum.

For ξ in H define the coherent state e^ξ by

${\displaystyle {e^{\xi }=\sum _{k\geq 0}(k!)^{-1}\xi ^{\otimes k}.}}$

It follows that their linear span is dense in S(H), that the coherent states corresponding to n distinct vectors are linearly independent and that

${\displaystyle {(e^{\xi },e^{\eta })=e^{(\xi ,\eta )}.}}$

When H is finite-dimensional, S(H) can naturally be identified with holomorphic Fock space for H*, since in the standard way S^k(H) are just homogeneous polynomials of degree k on H* and the inner products match up. Moreover, S(H) has functorial properties. Most importantly

${\displaystyle S(H_{1}\oplus H_{2})=S(H_{1})\otimes S(H_{2}),\qquad e^{x_{1}\oplus x_{2}}=e^{x_{1}}\otimes e^{x_{2}}.}$

A similar result hold for finite orthogonal direct sums and extends to infinite orthogonal direct sums, using von Neumman's definition of the infinite tensor product with 1 the reference unit vector in S⁰(H_i). Any contraction operator between Hilbert spaces induces a contraction operator between the corresponding symmetric Fock spaces in a functorial way.

A unitary operator on S(H) is uniquely determined by it values on coherent states. Moreover, for any assignment v_ξ such that

${\displaystyle {(v_{\xi },v_{\eta })=e^{(\xi ,\eta )}}}$

there is a unique unitary operator U on S(H) such that

${\displaystyle {v_{\xi }=U(e^{\xi }).}}$

As in the finite-dimensional case, this allows the unitary operators W(x) to be defined for x in H:

${\displaystyle {W(x)e^{y}=e^{-\ x\ ^{2}/2}e^{-(x,y)}e^{x+y}.}}$

It follows immediately from the finite-dimensional case that these operators are unitary and satisfy

${\displaystyle {W(x)W(y)=e^{-{i \over 2}\Im (x,y)}W(x+y).}}$

In particular the Weyl commutation relations are satisfied:

${\displaystyle {W(x)W(y)=e^{-i\Im (x,y)}W(y)W(x).}}$

Taking an orthonormal basis e_n of H, S(H) can be written as an infinite tensor product of the S(C e_n). The irreducibility of W on each of these spaces implies the irreducibility of W on the whole of S(H). W is called the complex wave representation.

To define the symplectic group in infinite dimensions let H_R be the underlying real vector space of H with the symplectic form

${\displaystyle {B(x,y)=-\Im (x,y)}}$

and real inner product

${\displaystyle {(x,y)_{\mathbf {R} }=\Re (x,y).}}$

The complex structure is then defined by the orthogonal operator

${\displaystyle {J(x)=ix}}$

so that

${\displaystyle {B(x,y)=-(Jx,y)_{\mathbf {R} }.}}$

A bounded invertible operator real linear operator T on H_R lies in the symplectic group if it and its inverse preserve B. This is equivalent to the conditions:

${\displaystyle {TJT^{t}=J=T^{t}JT.}}$

The operator T is said to be implementable on S(H) provided there is a unitary π(T) such that

${\displaystyle \pi (T)W(x)\pi (T)^{*}=W(Tx).}$

The implementable operators form a subgroup of the symplectic group, the restricted symplectic group. By Schur's lemma, π(T) is uniquely determined up to a scalar in T, so π gives a projective unitary representation of this subgroup.

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.

Unlike the finite-dimensional case where a lifting π could be chosen so that it was multiplicative up to a sign, this is not possible in the infinite-dimensional case. (This can be seen directly using the example of the projective representation of the diffeomorphism group of the circle constructed below.)

The projective representation of the restricted symplectic group can be constructed directly on coherent states as in the finite-dimensional case.^[47]

In fact, choosing a real Hilbert subspace of H of which H is a complexification, for any operator T on H a complex conjugate of T is also defined. Then the infinite-dimensional analogue of SU(1,1) consists of invertible bounded operators

${\displaystyle {g={\begin{pmatrix}A&B\\{\overline {B}}&{\overline {A}}\end{pmatrix}}}}$

satisfying gKg* = K (or equivalently the same relations as in the finite-dimensional case). These belong to the restricted symplectic group if and only if B is a Hilbert–Schmidt operator. This group acts transitively on the infinite-dimensional analogue D_≈ of the Seigel generalized unit disk consisting of Hilbert–Schmidt operators W that are symmetric with operator norm less than 1 via the formula

${\displaystyle {gZ=(AW+B)({\overline {B}}W+{\overline {A}})^{-1}.}}$

Again the stabilizer subgroup of 0 consists of g with A unitary and B = 0. The metaplectic coherent states f_W can be defined as before and their inner product is given by the same formula, using the Fredholm determinant:

${\displaystyle {(f_{W_{1}},f_{W_{2}})=\det(I-W_{2}^{*}W_{1})^{-{\frac {1}{2}}}.}}$

Define unit vectors by

${\displaystyle {e_{W}=\det(I-W^{*}W)^{1/4}f_{W}}}$

and set

${\displaystyle {\pi (g)e_{W}=\mu (\det(I+{\overline {A}}^{-1}{\overline {B}}W)^{-{\frac {1}{2}}})e_{gW},}}$

where μ(ζ) = ζ/ ζ . As before this defines a projective representation and, if g₃ = g₁g₂, the cocycle is given by

${\displaystyle {\omega (g_{1},g_{2})=\mu [\det(A_{3}(A_{1}A_{2})^{-1})^{-{\frac {1}{2}}}].}}$

This representation extends by analytic continuation to define contraction operators for the complex semigroup by the same analytic continuation argument as in the finite-dimensional case. It can also be shown that they are strict contractions.

Example Let H_R be the real Hilbert space consisting of real-valued functions on the circle with mean 0

${\displaystyle f(\theta )=\sum _{n\neq 0}a_{n}e^{in\theta }}$

and for which

${\displaystyle {\sum _{n\neq 0} n a_{n} ^{2}<\infty .}}$

The inner product is given by

${\displaystyle \left(\sum a_{n}e^{in\theta },\sum b_{m}e^{im\theta }\right)=\sum _{n\neq 0} n a_{n}{\overline {b_{n}}}.}$

An orthogonal basis is given by the function sin(nθ) and cos(nθ) for n > 0. The Hilbert transform on the circle defined by

${\displaystyle J\sin(n\theta )=\cos(n\theta ),\qquad J\cos(n\theta )=-\sin(n\theta )}$

defines a complex structure on H_R. J can also be written

${\displaystyle J\sum _{n\neq 0}a_{n}e^{in\theta }=\sum _{n\neq 0}i\operatorname {sign} (n)a_{n}e^{in\theta },}$

where sign n = ±1 denotes the sign of n. The corresponding symplectic form is proportional to

${\displaystyle B(f,g)=\int _{S^{1}}fdg.}$

In particular if φ is an orientation-preserving diffeomorphism of the circle and

${\displaystyle {T_{\varphi }f(\theta )=f(\varphi ^{-1}(\theta ))-{1 \over 2\pi }\int _{0}^{2\pi }f(\varphi ^{-1}(\theta ))\,d\theta ,}}$

then T_φ is implementable.^[48]

The operators W(f) with f smooth correspond to a subgroup of the loop group LT invariant under the diffeomorphism group of the circle. The infinitesimal operators corresponding to the vector fields

${\displaystyle {L_{n}=-\pi \left(ie^{in\theta }{d \over d\theta }\right)}}$

can be computed explicitly. They satisfy the Virasoro relations

${\displaystyle {[L_{m},L_{n}]=(m-n)L_{m+n}+{m^{3}-m \over 12}\delta _{m+n,0}.}}$

In particular they cannor be adjusted by addition of scalar operators to remove the second term on the right hand side. This shows that the cocycle on the restricted symplectic group is not equivalent to one taking only the values ±1.

See also

• Metaplectic group
• Invariant convex cone

Notes

References

• Baez, J. C.; Segal, I. E.; Zhou, Z.-F.; Kon, Mark A. (1992), "Introduction to algebraic and constructive quantum field theory", Physics Today, Princeton University Press, 46 (12): 43, Bibcode:1993PhT....46l..43B, doi:10.1063/1.2809125, ISBN 0-691-08546-3, S2CID 120693408
• Bargmann, V. (1970), Group representations on Hilbert spaces of analytic functions, Analytic methods in mathematical physics, Gordon and Breach, pp. 27–63
• Berg, M. C. (2000), The Fourier-analytic proof of quadratic reciprocity, Pure and Applied Mathematics, Wiley-Interscience, ISBN 0-471-35830-4
• Brunet, M.; Kramer, P. (1980), "Complex extension of the representation of the symplectic group associated with the canonical commutation relations", Reports on Mathematical Physics, 17 (2): 205–215, Bibcode:1980RpMP...17..205B, doi:10.1016/0034-4877(80)90063-4
• Đokovic, D. Z.; Hofmann, K.-H. (1997), "The surjectivity question for the exponential function of real Lie groups: a status report", Journal of Lie Theory, 7: 171–199
• Coburn, L. A. (1973), "Singular integral operators and Toeplitz operators on odd spheres", Indiana University Mathematics Journal, 23 (5): 433–439, doi:10.1512/iumj.1974.23.23036
• Connes, A. (1990), Géométrie non commutative, InterEditions, ISBN 2-7296-0284-4
• Ferrara, S.; Mattiolia, G.; Rossic, G.; Toller, M. (1973), "Semi-group approach to multiperipheral dynamics", Nuclear Physics B, 53 (2): 366–394, Bibcode:1973NuPhB..53..366F, doi:10.1016/0550-3213(73)90451-3
• Folland, G. B. (1989), Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, ISBN 9780691085289
• Goddard, Peter; Olive, David (1988), Kac-Moody and Virasoro Algebras: A Reprint Volume for Physicists, Advanced series in mathematical physics, vol. 3, World Scientific, ISBN 9789971504205
• Goodman, R. (1969), "Analytic and entire vectors for representations of Lie groups", Transactions of the American Mathematical Society, 143: 55–76, doi:10.1090/s0002-9947-1969-0248285-6
• Goodman, R.; Wallach, N. R. (1984), "Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle", Journal für die Reine und Angewandte Mathematik, 347: 69–133
• Hall, B. C. (2013), Quantum Theory for Mathematicians, Springer
• He, H. (2007), "Functions on symmetric spaces and oscillator representation", Journal of Functional Analysis, 244 (2): 536–564, doi:10.1016/j.jfa.2006.11.008
• Helton, J. W.; Howe, R. E. (1975), "Traces of commutators of integral operators", Acta Mathematica, 135: 271–305, doi:10.1007/bf02392022
• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 978-0-8218-2848-9
• Hilgert, J.; Neeb, K.-H. (1993), Lie semigroups and their applications, Lecture Notes in Mathematics, vol. 1552, Springer-Verlag, ISBN 0387569545
• Hille, E. (1940), "Contributions to the theory of Hermitian series. II. The representation problem", Transactions of the American Mathematical Society, 47: 80–94, doi:10.1090/s0002-9947-1940-0000871-3
• Hörmander, Lars (1983), Analysis of Partial Differential Operators I, Springer-Verlag, ISBN 3-540-12104-8
• Hörmander, Lars (1985), Analysis of Partial Differential Operators III, Springer-Verlag, ISBN 3-540-13828-5
• Howe, R. (1980), "Quantum mechanics and partial differential equations", Journal of Functional Analysis, 38 (2): 188–254, doi:10.1016/0022-1236(80)90064-6
• Howe, R. (1988), "The Oscillator Semigroup", Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 48: 61–132, doi:10.1090/pspum/048/974332, ISBN 9780821814826
• Howe, R.; Tan, Eng-Chye (1992), Non-abelian harmonic analysis: applications of SL(2,R), Universitext, Springer-Verlag, ISBN 0387977686
• Igusa, J. (1972), Theta functions, Die Grundlehren der mathematischen Wissenschaften, vol. 194, Springer-Verlag
• Itzykson, C. (1967), "Remarks on boson commutation rules", Communications in Mathematical Physics, 4 (2): 92–122, Bibcode:1967CMaPh...4...92I, doi:10.1007/bf01645755, S2CID 121928828
• Kashiwara, M.; Vergne, M. (1978), "On the Segal–Shale–Weil representations and harmonic polynomials", Inventiones Mathematicae, 44: 1–47, Bibcode:1978InMat..44....1K, doi:10.1007/bf01389900, S2CID 121545402
• Kac, V.G.; Raina, A.K. (1987), Bombay lectures on highest weight representations, World Scientific, ISBN 9971503956
• Kac, V.G. (1990), Infinite dimensional Lie algebras (3rd ed.), Cambridge University Press, ISBN 0521466938
• Kramer, P.; Moshinsky, M.; Seligman, T. H. (1975), Complex extensions of canonical transformations and quantum mechanics, Group Theory and its Applications, vol. 3, Academic Press
• Lang, S. (1985), SL₂(R), Graduate Texts in Mathematics, vol. 105, Springer-Verlag, ISBN 0-387-96198-4
• Lawson, J. D. (1998), "Semigroups in Möbius and Lorentzian geometry", Geometriae Dedicata, 70 (2): 139–180, doi:10.1023/a:1004906126006, S2CID 116687780
• Lawson, J. D. (2011), "Semigroups of Olshanski type", in Hofmann, K. H.; Lawson, J. D.; Vinberg, E. B. (eds.), Semigroups in Algebra, Geometry and Analysis, Walter de Gruyter, pp. 121–158, ISBN 9783110885583
• Lion, G.; Vergne, M. (1980), The Weil representation, Maslov index and theta series, Progress in Mathematics, vol. 6, Birkhäuser, ISBN 3-7643-3007-4
• Mackey, G. W. (1989), Unitary group representations in physics, probability, and number theory (2nd ed.), Addison-Wesley, ISBN 0-201-51009-X
• Mumford, D.; Nori, M.; Norman, P. (2006), Tata Lectures on Theta III, Progress in mathematics, Springer, ISBN 0817645705
• Neretin, Y. A. (1996), Categories of symmetries and infinite-dimensional groups, London Mathematical Society Monographs, vol. 16, Oxford University Press, ISBN 0-19-851186-8
• von Neumann, J. (1932), "Ueber Einen Satz Von Herrn M. H. Stone", Annals of Mathematics, 33 (3): 567–573, doi:10.2307/1968535, JSTOR 1968535
• Olshanskii, G. I. (1981), "Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series", Functional Analysis and Its Applications, 15 (4): 275–285, doi:10.1007/bf01106156, S2CID 121254166
• Pressley, A.; Segal, G. B. (1986), Loop groups, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-853535-X
• Segal, G. B. (1981), "Unitary representations of some infinite-dimensional groups", Communications in Mathematical Physics, 80 (3): 301–342, Bibcode:1981CMaPh..80..301S, doi:10.1007/bf01208274, S2CID 121367853
• Sohrab, H. H. (1981), "The C∗-algebra of the n-dimensional harmonic oscillator", Manuscripta Mathematica, 34: 45–70, doi:10.1007/bf01168709, S2CID 119837229
• Thangavelu, S. (1993), Lectures on Hermite and Laguerre expansions, Mathematical Notes, vol. 42, Princeton University Press, ISBN 0-691-00048-4
• Weil, A. (1964), "Sur certains groupes d'opérateurs unitaires", Acta Mathematica, 111: 143–211, doi:10.1007/BF02391012
• Wiener, N. (1933), The Fourier integral and certain of its applications (1988 reprint of the 1933 edition), Cambridge University Press, ISBN 0-521-35884-1
• Yoshida, H. (1992), "Remarks on metaplectic representations of SL(2)", Journal of the Mathematical Society of Japan, 44 (3): 351–373, doi:10.2969/jmsj/04430351