사용자:Kobmuiv/로런츠 군의 표현과 물리학

Hendrik Antoon Lorentz (right) after whom the Lorentz group is named and Albert Einstein whose special theory of relativity is the main source of application. Photo taken by Paul Ehrenfest 1921.

로렌츠 군은 특수 상대성 이론의 시공간에서 대칭을 이루는 리 군이다. 이 군은 일부 힐베르트 공간에서 행렬, 선형 변환 또는 유니터리 연산자들의 집합으로 구현될 수 있으며, 다양한 표현을 가지고 있다.. 로렌츠 군의 무한 차원 유니터리 표현을 물리학에 적용하는 연구는 양자역학과 함께 특수 상대성 이론이 가장 철저하게 확립된 두 물리학 이론이기 때문에 이 군은 아주 중요하다. 이들은 주류 물리학에서 역사적 중요성을 가지고 있을 뿐만 아니라 더 사변적인 오늘날의 이론들과의 연관성도 가지고 있다.

발전

반단순 리 대수의 표현론의 일반적인 방식을 통해 로런츠 군의 리 대수의 유한 차원 표현에 대한 이론 전체가 얻어진다. 리 대응과 행렬 지수를 적용하여 로런츠 군 O(3; 1)의 연결 성분 ${\text{SO}}(3;1)^{+}$의 유한 차원 표현들을 얻는다. ${\text{SL}}(2,\mathbb {C} )$와 ${\mathfrak {sl}}(2,\mathbb {C} )$의 표현들에서 어떤 함수 공간에 작용하는 방식으로 ${\text{SO}}(3;1)^{+}$의 보편 덮개 ${\text{SL}}(2,\mathbb {C} )$의 유한 차원 표현들에 대한 전체 이론을 명식적으로 얻는다. 시간 역전과 장소 역전의 대표원들은 시간 역전과 장소 역전에서 얻어지고, 로런츠군 전체에 대한 유한차원 표현을 완성한다. (m,n) 표현들의 일반적 성질을 대략 설명 하겠다. 구면 조화 함수 공간이나 리만 P 함수 공간 위의 작용 같은 함수 공간 위의 작용이 고려된다. 무한 차원 기약 유니터리 표현들이 ${\text{SL}}(2,\mathbb {C} )$ principal series와 complementary series에 대해 얻어진다. 마지막으로, ${\text{SL}}(2,\mathbb {C} )$에 대한 플렌셰렐 공식이 주어지며, SO(3, 1)의 표현들이 분류된다.

군 표현론은 역사적으로 엘리 카르탕과 헤르만 바일을 필두로 한 반단순군의 일반적인 표현론의 발전을 따랐지만, 로렌츠 군은 물리학에서의 중요성 때문에 관심을 받았다. 로런츠 군 표현의 발전에서 수리 물리학자 E. P. 위그너와 수학자 발렌타인 바그만이 대표적인 공헌자들이다. 그들의 기여는 바그만-위그너 프로그램을 통해 잘 알려져 있다. 대략적으로, inhomogeneous 로렌츠 군의 모든 유니터리 표현들의 분류는 가능한 모든 상대론적 파동 방정식들의 분류와 같다. 로런츠 군의 무한 차원 기약 표현들의 분류는 1947년 폴 디랙의 이론물리학 박사과정 학생인 하리쉬 찬드라가 수학자로 전향하면서 확립되었다. $\mathrm {SL} (2,\mathbb {C} )$의 무한차원 기약 표현들의 분류는 바르그만, 그리고 겔판드 & 네이막이 독립적으로 같은 해에 발표 했다.

응용들

유한 차원과 무한 차원 표현을 포함하여, 로런츠 군의 표현들 중 많은 경우가 이론물리학에서 중요하다. 로런츠 군의 표현은 고전장론에서 가장 중요한 전자기장, 상대론적 양자역학에서 입자의 설명뿐만 아니라 양자장론에서 입자와 양자장, 끈이론과 그 너머에서 다양한 물체의 설명에도 나타난다. 로런츠 군의 표현은 또한 스핀이라는 개념에 대한 이론적 토대를 제공한다. 시공간의 충분히 작은 영역에서 물리학은 특수 상대성 이론이라는 의미에서, 로런츠 군의 표현은 일반 상대성 이론에도 적용 된다..

로런츠 군의 표현들 중 물리학과 직접적인 관련성을 갖는 것은 유니터리가 아닌 유한 차원 기약 표현과 비균질 로렌츠 군인 푸앵카레 군의 기약 무한 차원 유니터리 표현이다.

상대론적 양자역학과 양자장론의 힐베르트 공간에 작용하는 푸앵카레 군의 기약 무한 차원 유니터리 표현의 제한에 의해 로렌츠 군의 무한 차원 유니터리 표현이 나타난다. 이것들은 순수 수학적 관심의 대상이기도 하며, 단순한 제한의 역할이 아닌 다른 역할에서 물리학과 직접적인 관련성이 있을 수 있는 대상이기도 하다. 상대성 이론과 양자역학과 일치하는 추측성 이론(텐서와 스피너는 디랙의 expansor와 하리쉬 찬드라의 expinor에 무한히 대응한다)들도 있었지만, 검증된 물리적 응용을 발견하지 못했다. 현대의 추측성 이론들은 잠재적으로 다음과 비슷한 요소들을 가지고 있다.

고전장론

중력장과 함께 전자기장은 자연을 정확하게 기술하는 대표적인 고전 장이지만, 다른 종류의 고전 장들도 중요하다. 2차 양자화라고 불리는 양자장론 접근법에서 출발점은 하나 이상의 고전 장인데, 여기서 디랙 방정식을 푸는 파동함수는 2차 양자화 이전의 고전 장로 여겨진다. 2차 양자화와 그와 관련된 라그랑지안 형식주의는 양자장론의 기본적인 측면은 아니지만, 표준 모형을 포함하여 지금까지 모든 양자장론에 이러한 방식으로 접근할 수 있다. 이러한 경우에는 최소 작용의 원리를 사용하여 라그랑지안에서 유도된 오일러-라그랑지 방정식에서 나온 고전적인 형태의 장 방정식들이 있다. 장 방정식들은 상대론적으로 불변해야 하며, (아래 정의에 따라 상대론적 파동함수로 인정되는) 해들은 로렌츠 군의 어떤 표현 하에서 변형되어야 한다.

field configuration들이 이루는 공간에 대한 로렌츠 군의 작용(field configuration은 특정 해의 시공간 역사이다. 예를 들어, 모든 장소와 모든 시간에서의 전자기장은 하나의 field configuration이다.)은 교환자 괄호가 장론적 포아송 괄호로 대체된다는 점을 제외하고는 양자역학에서 힐베르트 공간에 대한 작용과 비슷하다

상대론적 양자역학

지금 목적에 따라 다음 정의를 도입한다.: 상대론적 파동 함수는, 임의의 적절한 로런츠 변환 Λ에 대해

$$\psi '^{\alpha }(x)=D{[\Lambda ]^{\alpha }}_{\beta }\psi ^{\beta }\left(\Lambda ^{-1}x\right)$$

로 변환되는 시공간(민코프스키 공간)에서 정의된 n개의 함수들 ψ^α의 집합이다. 여기서 D[Λ]는 아래에서 설명할 어떤 (m, n) 표현들의 직합 안에서 Λ를 표현한 n-차원 행렬이다.

하나의 입자에 대한 가장 유용한 상대론적 양자역학 이론은 클라인-고든 방정식과 디랙 방정식이다. 이 방정식들은 상대론적으로 불변이며 그 해들은 로런츠 변환에 대해 로런츠 스칼라들 ((m, n) = (0, 0))과 각각 bispinor ((0, 1/2) ⊕ (1/2, 0))처럼 변한다. 전자기장은 이 정의에서 사앧론적 파동함수이며 (1, 0) ⊕ (0, 1) 아래서 변한다.

무한차원 표현들은 산란을 분석하는데 쓸 수 있다.

양자장론

양자장론에서, S-행렬이 반드시 푸앵카레 불변이여야 한다는 조건에서 오는 많은 방식들 중에서 상대론적 불변성에 대한 요구가 생긴다. 이는 포크 공간에 작용하는 로런츠 군의 무한 차원 표현이 하나 이상 존재함을 함의한다.이러한 표현의 존재성을 보장하는 한 가지 방법은 표준 형식주의를 사용하는 계의 라그랑지안 묘사(적은 요구 사항이 부과됨)의 존재성을 보이는 것이며, 이로부터 로런츠 군의 생성원들에 대한 realization이 유도될 수 있다.

장 연산자의 변환은 로런츠 군의 유한 차원 표현과 푸앵카레 군의 무한 차원 유니터리 표현이 수학과 물리학 사이의 깊은 연관성을 목격하면서 수행하는 보완적 역할을 보여준다. 이를 대략 설명하기 위해 n-성분 장 연산자를 정의하자. 상대론적 장 연산자는, 민코프스키 공간에서 정의되고 적절한 푸앵카레 변환 (Λ, a)에 대해

$$\Psi ^{\alpha }(x)\to \Psi '^{\alpha }(x)=U[\Lambda ,a]\Psi ^{\alpha }(x)U^{-1}\left[\Lambda ,a\right]=D{\left[\Lambda ^{-1}\right]^{\alpha }}_{\beta }\Psi ^{\beta }(\Lambda x+a)$$

로 변환되는 연산자 값 함수들 n개의 집합이다. 여기서 U[Λ, a]는 Ψ가 정의된 힐베르트 공간 위에서 (Λ, a)를 나타내는 유니터리 연산자이다. D는 로런츠 군의 n-차원 표현이다. 이 변환 규칙은 양자장론의 와이트만 공리계의 두번째 공리이다.

질량 m이고 s인 단일 입자를 묘사하기 위해 장 연산자가 가져야할 미분 제한 조건을 고려하여

$$\Psi ^{\alpha }(x)=\sum _{\sigma }\int dp\left(a(\mathbf {p} ,\sigma )u^{\alpha }(\mathbf {p} ,\sigma )e^{ip\cdot x}+a^{\dagger }(\mathbf {p} ,\sigma )v^{\alpha }(\mathbf {p} ,\sigma )e^{-ip\cdot x}\right),$$

(X1)

를 얻는다. 여기서 a^†, a 들은 각각 생성 연산자와 소멸 연산자로 해석된다.. 생성 연산자 a^†는

$$a^{\dagger }(\mathbf {p} ,\sigma )\rightarrow a'^{\dagger }\left(\mathbf {p} ,\sigma \right)=U[\Lambda ]a^{\dagger }(\mathbf {p} ,\sigma )U\left[\Lambda ^{-1}\right]=a^{\dagger }(\Lambda \mathbf {p} ,\rho )D^{(s)}{\left[R(\Lambda ,\mathbf {p} )^{-1}\right]^{\rho }}_{\sigma },$$

과 같이 변환되며, 소멸 연산자도 비슷하게 변환된다. The point to be made is that 장 연산자는 로런츠 군의 유한차원 비 유니터리 표현에 따라 변환되며 the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, 생성 연산자는 푸앵카레 군의 무한차원 유니터리 표현에 따라 변환된다. while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin (m, s) of the particle. 이 둘의 연결점은 계수 함수라고도 불리는 파동함수

$$u^{\alpha }(\mathbf {p} ,\sigma )e^{ip\cdot x},\quad v^{\alpha }(\mathbf {p} ,\sigma )e^{-ip\cdot x}$$

이며 이들은 로런트 변환과 연관된 첨자 (x, α) 와 푸앵카레 변환과 연관된 첨자 (p, σ) 둘 다 가지고 있다. This may be called the Lorentz–Poincaré connection. To exhibit the connection, subject both sides of equation (X1) to a Lorentz transformation resulting in for e.g. u,

$${D[\Lambda ]^{\alpha }}_{\alpha '}u^{\alpha '}(\mathbf {p} ,\lambda )={D^{(s)}[R(\Lambda ,\mathbf {p} )]^{\lambda '}}_{\lambda }u^{\alpha }\left(\Lambda \mathbf {p} ,\lambda '\right),$$

where D is the non-unitary Lorentz group representative of Λ and D^(s) is a unitary representative of the so-called Wigner rotation R associated to Λ and p that derives from the representation of the Poincaré group, and s is the spin of the particle.

All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the (m, n) representation under which it is supposed to transform, and also that of the wave function, can be derived from group theoretical considerations alone once the frameworks of quantum mechanics and special relativity is given.

추측성 이론들

시공간이 D = 4차원을 넘을 수도 있는 물리학 이론에서는 그 차원에 적절한 일반화된 로런츠 군O(D − 1; 1)이 O(3; 1)를 대신한다.

로런츠 불변성을 요구하는 조건은 아마 끈이론에서 가장 극적인 효과를 나타낼 것이다. 고전적 상대론적 끈들은 남부-고토 작용을 사용하여 라그랑주 역학의 틀 안에서 다뤄질 수 있다. 임의의 차원의 시공간에서 이는 상대론적 불변량 이론으로 귀결된다. 그러나 밝혀진 바와 같이, 열린 또는 닫힌 보손 끈 이론(가장 단순한 끈 이론)은 배경 시공간의 차원이 26이 아니면 로런츠 군이 상태 공간(힐베르트 공간)에 나타나는 방식으로 양자화하는 것은 불가능하다. 초끈이론에서 이에 대응되는 결과는 다시 로런츠 불변성을 요구하여 유도되지만 이제 초대칭성도 갖고 있다. 이 이론들에서 푸앵카레 대수는 푸앵카레 대수를 확장하는 Z₂ 등급 리 대수인 초대칭 대수로 바뀐다. 이런 대수의 구조는 로런츠 불변성 조건으로 인해 상당히 한정된다. 특히, 페르미온 연산자들은 로런츠 리 대수의 (0, 1/2) 또는 (1/2, 0) 표현 공간에 속한다. 이런 이론들에서 가능한 시공간 차원은 오직 10차원 뿐이다.

유한차원 표현

일반적으로, 군의 표현론, 특히 리 군의 표현론은 아주 풍부한 내용을 가지고 있다. 로런츠 군은 표현론의 맥락에서 좋거나 까다로운 성질들을 가지고 있다; 로런츠 군은 단순하고, 따라서 반단순이지만, 연결이 아니고 어떤 연결 성분도 단일 연결이 아니다. 더욱이, 콤팩트가 아니다.

유한차원 표현에 대해, 로런츠 군이 반단순이므로, 이미 잘 발전된 반단순군들에 대한 표현론을 적용 할 수 있다. 더욱이, 모든 표현들은 기약 표현들로부터 구성된다. 왜냐하면, 리 대수는 완전 환원 성질을 가지고 있기 때문이다. 하지만, 로런츠 군이 콤팩트가 아님과 단일 연결이 아닌 점이 조합되어서, 콤팩트 단일 연결 리 군처럼 간단하게 다룰 수 없다. 연결 단순 리군이 콤팩트가 아니면, 자명하지 않은 유한차원 유니터리 표현이 존재하지 않는다. 또한 단일 연결이 아님은 스핀 표현을 야기한다. 로런츠 군 전체의 표현에서 로런츠 군이 연결이 아님은 시간 반전과 장소의 향 반전을 따로 다루어야 함을 의미한다.

역사

로런츠 군의 유한차원 표현론의 발전은 대체로 표현론의 발전을 따른다. 리 이론은 노르웨이 수학자 소푸스 리로부터 1873년에 시작되었다. 1888년에 빌헬름 킬링에 의해 단순 리대수의 분류가 완료되었다. 1913년에 엘리 카르탕이 단순 리 대수의 표현론에서 최고 가중치 정리를 완성하였다. 1935–38년의 기간 동안 리차드 브라우어는, 로런츠 리 대수의 스핀 표현들이 어떻게 클리포트 대수 안에 매장되는지 알려주는 바일-브라우어 행렬들의 발전에 크게 기여하였다. 로런츠 군은 또한 물리학에서의 중요성으로 인한 관심을 받아왔다. 수학자 헤르만 바일과 하리쉬-찬드라 그리고 물리학자 유진 위그너와 발렌타인 바르그만은 로런츠 군의 표현론에 지대한 기여를 하였다. 1928.년 물리학자 폴 디랙은 아마 로런츠 군의 표현론을 디랙 방정식과 함께 그 모든 것을 지속적으로 중요한 실용적인 응용과 짜맞춘 사람일 것이다.

리 대수

Wilhelm Killing, Independent discoverer of Lie algebras. The simple Lie algebras were first classified by him in 1888.

이 절에서는 로런츠 군의 리 대수 ${\mathfrak {so}}(3;1)$의 복소화 ${\mathfrak {so}}(3;1)_{\mathbb {C} }$의 기약 복소 선형 표현을 다룬다. ${\mathfrak {so}}(3;1)$에 대한 편한 기저는 회전의 세 생성원 J_i과 부스트들의 세 생성원 K_i이다. 이들은 규약과 리 대수 기저 절에서 명시적으로 소개할 것이다..

이 리 대수는 복소화되면서 기저가 이들이 두 이데알들

$$\mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}},\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}.$$

의 성분으로 바뀐다. A = (A₁, A₂, A₃)의 성분과 B = (B₁, B₂, B₃)의 성분은 각각 따로 리 대수 ${\mathfrak {su}}(2)$의 교환 관계를 만족한다. 더욱이, 이들은 서로 교환한다:

$$\left[A_{i},A_{j}\right]=i\varepsilon _{ijk}A_{k},\quad \left[B_{i},B_{j}\right]=i\varepsilon _{ijk}B_{k},\quad \left[A_{i},B_{j}\right]=0,$$

여기서 i, j, k 들은 1,2,3 값을 갖는 첨자이며 ε_ijk는 3차원 레비-치비타 기호이다. $\mathbf {A} _{\mathbb {C} }$와 $\mathbf {B} _{\mathbb {C} }$를 각각 A와 B의 복소 linear span이라 하자.

그러면 다음과 같은 동형관계들이 존재한다:

$${\begin{aligned}{\mathfrak {so}}(3;1)\hookrightarrow {\mathfrak {so}}(3;1)_{\mathbb {C} }&\cong \mathbf {A} _{\mathbb {C} }\oplus \mathbf {B} _{\mathbb {C} }\cong {\mathfrak {su}}(2)_{\mathbb {C} }\oplus {\mathfrak {su}}(2)_{\mathbb {C} }\\[5pt]&\cong {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )\\[5pt]&\cong {\mathfrak {sl}}(2,\mathbb {C} )\oplus i{\mathfrak {sl}}(2,\mathbb {C} )={\mathfrak {sl}}(2,\mathbb {C} )_{\mathbb {C} }\hookleftarrow {\mathfrak {sl}}(2,\mathbb {C} ),\end{aligned}}$$

(A1)

여기서 ${\mathfrak {sl}}(2,\mathbb {C} )$는 ${\mathfrak {su}}(2)\cong \mathbf {A} \cong \mathbf {B}$의 복소화이다.

${\mathfrak {su}}(2)$의 모든 기약 표현들, 즉, ${\mathfrak {sl}}(2,\mathbb {C} )$의 모든 복소 선형 기약 표현들이 알려져 있기 때문에, 이 동형관계들은 유용하다. ${\mathfrak {sl}}(2,\mathbb {C} )$의 복소 선형 기약 표현은 최대 가중치 표현들 중 하나와 동형이다. 이들은 ${\mathfrak {sl}}(2,\mathbb {C} )$의 복소 선형 기약 표현들로부터 명시적으로 주어진다.

유니터리 기법

Hermann Weyl, inventor of the unitarian trick. There are several concepts and formulas in representation theory named after Weyl, e.g. the Weyl group and the Weyl character formula.

Photo courtesy of ETH-Bibliothek Zürich, Bildarchiv

리 대수 ${\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )$는 ${\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} )$의 리 대수이다. 이는 콤팩트 부분군 ${\text{SU}}(2)\times {\text{SU}}(2)$를 포함하고, 이 부분군의 리 대수는 ${\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)$이다. 이 리 대수는 ${\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )$의 콤팩트 실 형태이다. 즉, 유니터리 기법의 첫번째 진술로부터, ${\text{SU}}(2)\times {\text{SU}}(2)$의 표현들은 ${\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} )$의 정칙 표현들과 일대일 대응한다.

${\text{SU}}(2)\times {\text{SU}}(2)$는 콤팩트 리 군이므로, 여기에 피터-바일 정리가 적용되며, 따라서 기약 특성의 직교성도 고려 할 수 있다. ${\text{SU}}(2)\times {\text{SU}}(2)$의 기약 유니터리 표현들은 정확하게 SU(2)의 유니터리 표현들의 텐서곱이다.

단일 연결성에 따라, 유니터리 기법의 두 번째 문장이 적용된다. 다음 목록의 표현들은 일대일 대응 관계에 있다:

• ${\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} )$의 정칙 표현들
• ${\text{SU}}(2)\times {\text{SU}}(2)$의 매끄러운 표현들
• ${\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)$의 실수 선형 표현들
• ${\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )$의 복소 선형 표현들

리 대수에서 표현들의 텐서곱들은

$${\begin{aligned}\pi _{1}\otimes \pi _{2}(X)&=\pi _{1}(X)\otimes \mathrm {Id} _{V}+\mathrm {Id} _{U}\otimes \pi _{2}(X)&&X\in {\mathfrak {g}}\\\pi _{1}\otimes \pi _{2}(X,Y)&=\pi _{1}(X)\otimes \mathrm {Id} _{V}+\mathrm {Id} _{U}\otimes \pi _{2}(Y)&&(X,Y)\in {\mathfrak {g}}\oplus {\mathfrak {g}}\end{aligned}}$$

(A0)

들 중 하나로 나타난다. 여기서 Id는 항등 연산자이다. Here, the latter interpretation, which follows from (G6), is intended. The highest weight representations of ${\mathfrak {sl}}(2,\mathbb {C} )$의 최고 가중치 표현들은 μ = 0, 1/2, 1, ...로 첨자를 붙였다. (최고 가중치들은 사실 2μ = 0, 1, 2, ...,이지만 여기서는 ${\mathfrak {so}}(3;1)$의 것을 사용했다.) The tensor products of two such complex linear factors 그러면 이러한 두 복소 선형 인자들의 텐서곱들은 ${\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )$의 기약 복소 선형 표현들을 이룬다.

마지막으로,왼쪽에 실수 형태들의 $\mathbb {R}$-선형 표현들representations of the real forms of the far left, ${\mathfrak {so}}(3;1)$, and the far right, ${\mathfrak {sl}}(2,\mathbb {C} ),$ in (A1) are obtained from the $\mathbb {C}$-linear representations of ${\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )$ characterized in the previous paragraph.

The (μ, ν)-representations of sl(2, C)

The complex linear representations of the complexification of ${\mathfrak {sl}}(2,\mathbb {C} ),{\mathfrak {sl}}(2,\mathbb {C} )_{\mathbb {C} },$ obtained via isomorphisms in (A1), stand in one-to-one correspondence with the real linear representations of ${\mathfrak {sl}}(2,\mathbb {C} ).$ The set of all real linear irreducible representations of ${\mathfrak {sl}}(2,\mathbb {C} )$ are thus indexed by a pair (μ, ν). The complex linear ones, corresponding precisely to the complexification of the real linear ${\mathfrak {su}}(2)$ representations, are of the form (μ, 0), while the conjugate linear ones are the (0, ν). All others are real linear only. The linearity properties follow from the canonical injection, the far right in (A1), of ${\mathfrak {sl}}(2,\mathbb {C} )$ into its complexification. Representations on the form (ν, ν) or (μ, ν) ⊕ (ν, μ) are given by real matrices (the latter are not irreducible). Explicitly, the real linear (μ, ν)-representations of ${\mathfrak {sl}}(2,\mathbb {C} )$ are

$$\varphi _{\mu ,\nu }(X)=\left(\varphi _{\mu }\otimes {\overline {\varphi _{\nu }}}\right)(X)=\varphi _{\mu }(X)\otimes \operatorname {Id} _{\nu +1}+\operatorname {Id} _{\mu +1}\otimes {\overline {\varphi _{\nu }(X)}},\qquad X\in {\mathfrak {sl}}(2,\mathbb {C} )$$

where ${\textstyle \varphi _{\mu },\mu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }$ are the complex linear irreducible representations of ${\mathfrak {sl}}(2,\mathbb {C} )$ and ${\overline {\varphi _{\nu }}},\nu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots$ their complex conjugate representations. (The labeling is usually in the mathematics literature 0, 1, 2, …, but half-integers are chosen here to conform with the labeling for the ${\mathfrak {so}}(3,1)$ Lie algebra.) Here the tensor product is interpreted in the former sense of (A0). These representations are concretely realized below.

The (m, n)-representations of so(3; 1)

Via the displayed isomorphisms in (A1) and knowledge of the complex linear irreducible representations of ${\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )$ upon solving for J and K, all irreducible representations of ${\mathfrak {so}}(3;1)_{\mathbb {C} },$ and, by restriction, those of ${\mathfrak {so}}(3;1)$ are obtained. The representations of ${\mathfrak {so}}(3;1)$ obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible. Since ${\mathfrak {so}}(3;1)$ is semisimple, all its representations can be built up as direct sums of the irreducible ones.

Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers m = μ and n = ν, conventionally written as one of

$$(m,n)\equiv \pi _{(m,n)}:{\mathfrak {so}}(3;1)\to {\mathfrak {gl}}(V),$$

where V is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given by

$$\pi _{(m,n)}(J_{i})=J_{i}^{(m)}\otimes 1_{(2n+1)}+1_{(2m+1)}\otimes J_{i}^{(n)}$$ $$\pi _{(m,n)}(K_{i})=-i\left(1_{(2m+1)}\otimes J_{i}^{(n)}-J_{i}^{(m)}\otimes 1_{(2n+1)}\right),$$

(A2)

where 1_n is the n-dimensional unit matrix and

$$\mathbf {J} ^{(n)}=\left(J_{1}^{(n)},J_{2}^{(n)},J_{3}^{(n)}\right)$$

are the (2n + 1)-dimensional irreducible representations of ${\mathfrak {so}}(3)\cong {\mathfrak {su}}(2)$ also termed spin matrices or angular momentum matrices. These are explicitly given as

$${\begin{aligned}\left(J_{1}^{(j)}\right)_{a'a}&={\frac {1}{2}}\left({\sqrt {(j-a)(j+a+1)}}\delta _{a',a+1}+{\sqrt {(j+a)(j-a+1)}}\delta _{a',a-1}\right)\\\left(J_{2}^{(j)}\right)_{a'a}&={\frac {1}{2i}}\left({\sqrt {(j-a)(j+a+1)}}\delta _{a',a+1}-{\sqrt {(j+a)(j-a+1)}}\delta _{a',a-1}\right)\\\left(J_{3}^{(j)}\right)_{a'a}&=a\delta _{a',a}\end{aligned}}$$

where δ denotes the Kronecker delta. In components, with −m ≤ a, a′ ≤ m, −n ≤ b, b′ ≤ n, the representations are given by

$${\begin{aligned}\left(\pi _{(m,n)}\left(J_{i}\right)\right)_{a'b',ab}&=\delta _{b'b}\left(J_{i}^{(m)}\right)_{a'a}+\delta _{a'a}\left(J_{i}^{(n)}\right)_{b'b}\\\left(\pi _{(m,n)}\left(K_{i}\right)\right)_{a'b',ab}&=-i\left(\delta _{a'a}\left(J_{i}^{(n)}\right)_{b'b}-\delta _{b'b}\left(J_{i}^{(m)}\right)_{a'a}\right)\end{aligned}}$$

Common representations

Irreducible representations for small (m, n). Dimension in parentheses.

	m = 0	1/2	1	3/2
n = 0	Scalar (1)	Left-handed Weyl spinor (2)	Self-dual 2-form (3)	(4)
1/2	Right-handed Weyl spinor (2)	4-vector (4)	(6)	(8)
1	Anti-self-dual 2-form (3)	(6)	Traceless symmetric tensor (9)	(12)
3/2	(4)	(8)	(12)	(16)

• The (0, 0) representation is the one-dimensional trivial representation and is carried by relativistic scalar field theories.
• Fermionic supersymmetry generators transform under one of the (0, 1/2) or (1/2, 0) representations (Weyl spinors).
• The four-momentum of a particle (either massless or massive) transforms under the (1/2, 1/2) representation, a four-vector.
• A physical example of a (1,1) traceless symmetric tensor field is the traceless part of the energy–momentum tensor T^μν.

Off-diagonal direct sums

Since for any irreducible representation for which m ≠ n it is essential to operate over the field of complex numbers, the direct sum of representations (m, n) and (n, m) have particular relevance to physics, since it permits to use linear operators over real numbers.

• (1/2, 0) ⊕ (0, 1/2) is the bispinor representation. See also Dirac spinor and Weyl spinors and bispinors below.
• (1, 1/2) ⊕ (1/2, 1) is the Rarita–Schwinger field representation.
• (3/2, 0) ⊕ (0, 3/2) would be the symmetry of the hypothesized gravitino. It can be obtained from the (1, 1/2) ⊕ (1/2, 1) representation.
• (1, 0) ⊕ (0, 1) is the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation.

The group

The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence. The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras. The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted $\exp :{\mathfrak {g}}\to G.$

If $\pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)$ for some vector space V is a representation, a representation Π of the connected component of G is defined by

$${\begin{aligned}\Pi (g=e^{iX})&\equiv e^{i\pi (X)},&&X\in {\mathfrak {g}},\quad g=e^{iX}\in \mathrm {im} (\exp ),\\\Pi (g=g_{1}g_{2}\cdots g_{n})&\equiv \Pi (g_{1})\Pi (g_{2})\cdots \Pi (g_{n}),&&g\notin \mathrm {im} (\exp ),\quad g_{1},g_{2},\ldots ,g_{n}\in \mathrm {im} (\exp ).\end{aligned}}$$

(G2)

This definition applies whether the resulting representation is projective or not.

Surjectiveness of exponential map for SO(3, 1)

From a practical point of view, it is important whether the first formula in (G2) can be used for all elements of the group. It holds for all $X\in {\mathfrak {g}}$, however, in the general case, e.g. for ${\text{SL}}(2,\mathbb {C} )$, not all g ∈ G are in the image of exp.

But $\exp :{\mathfrak {so}}(3;1)\to {\text{SO}}(3;1)^{+}$ is surjective. One way to show this is to make use of the isomorphism ${\text{SO}}(3;1)^{+}\cong {\text{PGL}}(2,\mathbb {C} ),$ the latter being the Möbius group. It is a quotient of ${\text{GL}}(n,\mathbb {C} )$ (see the linked article). The quotient map is denoted with $p:{\text{GL}}(n,\mathbb {C} )\to {\text{PGL}}(2,\mathbb {C} ).$ The map $\exp :{\mathfrak {gl}}(n,\mathbb {C} )\to {\text{GL}}(n,\mathbb {C} )$ is onto. Apply (Lie) with π being the differential of p at the identity. Then

$$\forall X\in {\mathfrak {gl}}(n,\mathbb {C} ):\quad p(\exp(iX))=\exp(i\pi (X)).$$

Since the left hand side is surjective (both exp and p are), the right hand side is surjective and hence $\exp :{\mathfrak {pgl}}(2,\mathbb {C} )\to {\text{PGL}}(2,\mathbb {C} )$ is surjective. Finally, recycle the argument once more, but now with the known isomorphism between SO(3; 1)⁺ and ${\text{PGL}}(2,\mathbb {C} )$ to find that exp is onto for the connected component of the Lorentz group.

Fundamental group

The Lorentz group is doubly connected, i. e. π₁(SO(3; 1)) is a group with two equivalence classes of loops as its elements. 틀:Math proof

Projective representations

Since π₁(SO(3; 1)⁺) has two elements, some representations of the Lie algebra will yield projective representations. Once it is known whether a representation is projective, formula (G2) applies to all group elements and all representations, including the projective ones — with the understanding that the representative of a group element will depend on which element in the Lie algebra (the X in (G2)) is used to represent the group element in the standard representation.

For the Lorentz group, the (m, n)-representation is projective when m + n is a half-integer. See § Spinors.

For a projective representation Π of SO(3; 1)⁺, it holds that

$$\left[\Pi (\Lambda _{1})\Pi (\Lambda _{2})\Pi ^{-1}(\Lambda _{1}\Lambda _{2})\right]^{2}=1\Rightarrow \Pi (\Lambda _{1}\Lambda _{2})=\pm \Pi (\Lambda _{1})\Pi (\Lambda _{2}),\quad \Lambda _{1},\Lambda _{2}\in \mathrm {SO} (3;1),$$

(G5)

since any loop in SO(3; 1)⁺ traversed twice, due to the double connectedness, is contractible to a point, so that its homotopy class is that of a constant map. It follows that Π is a double-valued function. It is not possible to consistently choose a sign to obtain a continuous representation of all of SO(3; 1)⁺, but this is possible locally around any point.

The covering group SL(2, C)

Consider ${\mathfrak {sl}}(2,\mathbb {C} )$ as a real Lie algebra with basis

$$\left({\frac {1}{2}}\sigma _{1},{\frac {1}{2}}\sigma _{2},{\frac {1}{2}}\sigma _{3},{\frac {i}{2}}\sigma _{1},{\frac {i}{2}}\sigma _{2},{\frac {i}{2}}\sigma _{3}\right)\equiv (j_{1},j_{2},j_{3},k_{1},k_{2},k_{3}),$$

where the sigmas are the Pauli matrices. From the relations

$$[\sigma _{i},\sigma _{j}]=2i\epsilon _{ijk}\sigma _{k}$$

(J1)

is obtained

$$[j_{i},j_{j}]=i\epsilon _{ijk}j_{k},\quad [j_{i},k_{j}]=i\epsilon _{ijk}k_{k},\quad [k_{i},k_{j}]=-i\epsilon _{ijk}j_{k},$$

(J2)

which are exactly on the form of the 3-dimensional version of the commutation relations for ${\mathfrak {so}}(3;1)$ (see conventions and Lie algebra bases below). Thus, the map J_i ↔ j_i, K_i ↔ k_i, extended by linearity is an isomorphism. Since ${\text{SL}}(2,\mathbb {C} )$ is simply connected, it is the universal covering group of SO(3; 1)⁺.

More on covering groups in general and the ${\text{SL}}(2,\mathbb {C} )$ covering of the Lorentz group in particular

A geometric view

E.P. Wigner investigated the Lorentz group in depth and is known for the Bargmann-Wigner equations. The realization of the covering group given here is from his 1939 paper.

Let p_g(t), 0 ≤ t ≤ 1 be a path from 1 ∈ SO(3; 1)⁺ to g ∈ SO(3; 1)⁺, denote its homotopy class by [p_g] and let π_g be the set of all such homotopy classes. Define the set

$$G=\{(g,[p_{g}]):g\in \mathrm {SO} (3;1)^{+},[p_{g}]\in \pi _{g}\}$$

(C1)

and endow it with the multiplication operation

$$(g_{1},[p_{1}])(g_{2},[p_{2}])=(g_{1}g_{2},[p_{12}]),$$

(C2)

where $p_{12}$ is the path multiplication of $p_{1}$ and $p_{2}$:

$$p_{12}(t)=(p_{1}*p_{2})(t)={\begin{cases}p_{1}(2t)&0\leqslant t\leqslant {\tfrac {1}{2}}\\p_{2}(2t-1)&{\tfrac {1}{2}}\leqslant t\leqslant 1\end{cases}}$$

With this multiplication, G becomes a group isomorphic to ${\text{SL}}(2,\mathbb {C} ),$ the universal covering group of SO(3; 1)⁺. Since each π_g has two elements, by the above construction, there is a 2:1 covering map p : G → SO(3; 1)⁺. According to covering group theory, the Lie algebras ${\mathfrak {so}}(3;1),{\mathfrak {sl}}(2,\mathbb {C} )$ and ${\mathfrak {g}}$ of G are all isomorphic. The covering map p : G → SO(3; 1)⁺ is simply given by p(g, [p_g]) = g.

An algebraic view

For an algebraic view of the universal covering group, let ${\text{SL}}(2,\mathbb {C} )$ act on the set of all Hermitian 2×2 matrices ${\mathfrak {h}}$ by the operation

$${\begin{cases}\mathbf {P} (A):{\mathfrak {h}}\to {\mathfrak {h}}\\X\mapsto A^{\dagger }XA\end{cases}}\qquad A\in \mathrm {SL} (2,\mathbb {C} )$$

(C3)

The action on ${\mathfrak {h}}$ is linear. An element of ${\mathfrak {h}}$ may be written in the form

$$X={\begin{pmatrix}\xi _{4}+\xi _{3}&\xi _{1}+i\xi _{2}\\\xi _{1}-i\xi _{2}&\xi _{4}-\xi _{3}\\\end{pmatrix}}\qquad \xi _{1},\ldots ,\xi _{4}\in \mathbb {R} .$$

(C4)

The map P is a group homomorphism into ${\text{GL}}({\mathfrak {h}})\subset {\text{End}}({\mathfrak {h}}).$ Thus $\mathbf {P} :{\text{SL}}(2,\mathbb {C} )\to {\text{GL}}({\mathfrak {h}})$ is a 4-dimensional representation of ${\text{SL}}(2,\mathbb {C} )$. Its kernel must in particular take the identity matrix to itself, A^†IA = A^†A = I and therefore A^† = A⁻¹. Thus AX = XA for A in the kernel so, by Schur's lemma, A is a multiple of the identity, which must be ±I since det A = 1. The space ${\mathfrak {h}}$ is mapped to Minkowski space M⁴, via

$$X=(\xi _{1},\xi _{2},\xi _{3},\xi _{4})\leftrightarrow {\overrightarrow {(\xi _{1},\xi _{2},\xi _{3},\xi _{4})}}=(x,y,z,t)={\overrightarrow {X}}.$$

(C5)

The action of P(A) on ${\mathfrak {h}}$ preserves determinants. The induced representation p of ${\text{SL}}(2,\mathbb {C} )$ on $\mathbb {R} ^{4},$ via the above isomorphism, given by

$$\mathbf {p} (A){\overrightarrow {X}}={\overrightarrow {AXA^{\dagger }}}$$

(C6)

preserves the Lorentz inner product since

$$-\det X=\xi _{1}^{2}+\xi _{2}^{2}+\xi _{3}^{2}-\xi _{4}^{2}=x^{2}+y^{2}+z^{2}-t^{2}.$$

This means that p(A) belongs to the full Lorentz group SO(3; 1). By the main theorem of connectedness, since ${\text{SL}}(2,\mathbb {C} )$ is connected, its image under p in SO(3; 1) is connected, and hence is contained in SO(3; 1)⁺.

It can be shown that the Lie map of $\mathbf {p} :{\text{SL}}(2,\mathbb {C} )\to {\text{SO}}(3;1)^{+},$ is a Lie algebra isomorphism: $\pi :{\mathfrak {sl}}(2,\mathbb {C} )\to {\mathfrak {so}}(3;1).$ The map P is also onto.

Thus ${\text{SL}}(2,\mathbb {C} )$, since it is simply connected, is the universal covering group of SO(3; 1)⁺, isomorphic to the group G of above.

Non-surjectiveness of exponential mapping for SL(2, C)

This diagram shows the web of maps discussed in the text. Here V is a finite-dimensional vector space carrying representations of ${\mathfrak {sl}}(2,\mathbb {C} ),{\mathfrak {so}}(3;1),$ ${\text{SL}}(2,\mathbb {C} )$ and ${\text{SO}}(3;1)^{+}.$ $\exp$ is the exponential mapping, p is the covering map from ${\text{SL}}(2,\mathbb {C} ),$ onto SO(3; 1)⁺ and σ is the Lie algebra isomorphism induced by it. The maps Π, π and the two Φ are representations. the picture is only partially true when Π is projective.

The exponential mapping $\exp :{\mathfrak {sl}}(2,\mathbb {C} )\to {\text{SL}}(2,\mathbb {C} )$ is not onto. The matrix

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

(S6)

is in ${\text{SL}}(2,\mathbb {C} ),$ but there is no $Q\in {\mathfrak {sl}}(2,\mathbb {C} )$ such that q = exp(Q).

In general, if g is an element of a connected Lie group G with Lie algebra ${\mathfrak {g}},$ then, by (Lie),

$$g=\exp(X_{1})\cdots \exp(X_{n}),\qquad X_{1},\ldots X_{n}\in {\mathfrak {g}}.$$

(S7)

The matrix q can be written

$${\begin{aligned}&\exp(-X)\exp(i\pi H)\\{}={}&\exp \left({\begin{pmatrix}0&-1\\0&0\\\end{pmatrix}}\right)\exp \left(i\pi {\begin{pmatrix}1&0\\0&-1\\\end{pmatrix}}\right)\\[6pt]{}={}&{\begin{pmatrix}1&-1\\0&1\\\end{pmatrix}}{\begin{pmatrix}-1&0\\0&-1\\\end{pmatrix}}\\[6pt]{}={}&{\begin{pmatrix}-1&1\\0&-1\\\end{pmatrix}}\\{}={}&q.\end{aligned}}$$

(S8)

Realization of representations of SL(2, C) and sl(2, C) and their Lie algebras

The complex linear representations of ${\mathfrak {sl}}(2,\mathbb {C} )$ and ${\text{SL}}(2,\mathbb {C} )$ are more straightforward to obtain than the ${\mathfrak {so}}(3;1)^{+}$ representations. They can be (and usually are) written down from scratch. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are related to the complex linear Lie algebra representations by exponentiation. The real linear representations of ${\mathfrak {sl}}(2,\mathbb {C} )$ are exactly the (μ, ν)-representations. They can be exponentiated too. The (μ, 0)-representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here).

The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of i and there is no factor of i in the exponential mapping compared to the physics convention used elsewhere. Let the basis of ${\mathfrak {sl}}(2,\mathbb {C} )$ be

$$H={\begin{pmatrix}1&0\\0&-1\\\end{pmatrix}},\quad X={\begin{pmatrix}0&1\\0&0\\\end{pmatrix}},\quad Y={\begin{pmatrix}0&0\\1&0\\\end{pmatrix}}.$$

(S1)

This choice of basis, and the notation, is standard in the mathematical literature.

Complex linear representations

The irreducible holomorphic (n + 1)-dimensional representations ${\text{SL}}(2,\mathbb {C} ),n\geqslant 2,$ can be realized on the space of homogeneous polynomial of degree n in 2 variables $\mathbf {P} _{n}^{2},$ the elements of which are

$$P{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=c_{n}z_{1}^{n}+c_{n-1}z_{1}^{n-1}z_{2}+\cdots +c_{0}z_{2}^{n},\quad c_{0},c_{1},\ldots ,c_{n}\in \mathbb {Z} .$$

The action of ${\text{SL}}(2,\mathbb {C} )$ is given by

$$(\phi _{n}(g)P){\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=\left[\phi _{n}{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}P\right]{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=P\left({\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}^{-1}{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}\right),\qquad P\in \mathbf {P} _{n}^{2}.$$

(S2)

The associated ${\mathfrak {sl}}(2,\mathbb {C} )$-action is, using (G6) and the definition above, for the basis elements of ${\mathfrak {sl}}(2,\mathbb {C} ),$

$$\phi _{n}(H)=-z_{1}{\frac {\partial }{\partial z_{1}}}+z_{2}{\frac {\partial }{\partial z_{2}}},\quad \phi _{n}(X)=-z_{2}{\frac {\partial }{\partial z_{1}}},\quad \phi _{n}(Y)=-z_{1}{\frac {\partial }{\partial z_{2}}}.$$

(S5)

With a choice of basis for $P\in \mathbf {P} _{n}^{2}$, these representations become matrix Lie algebras.

Real linear representations

The (μ, ν)-representations are realized on a space of polynomials $\mathbf {P} _{\mu ,\nu }^{2}$ in $z_{1},{\overline {z_{1}}},z_{2},{\overline {z_{2}}},$ homogeneous of degree μ in $z_{1},z_{2}$ and homogeneous of degree ν in ${\overline {z_{1}}},{\overline {z_{2}}}.$ The representations are given by

$$(\phi _{\mu ,\nu }(g)P){\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=\left[\phi _{\mu ,\nu }{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}P\right]{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=P\left({\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}^{-1}{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}\right),\quad P\in \mathbf {P} _{\mu ,\nu }^{2}.$$

(S6)

By employing (G6) again it is found that

$${\begin{aligned}\phi _{\mu ,\nu }(E)P=&-{\frac {\partial P}{\partial z_{1}}}\left(E_{11}z_{1}+E_{12}z_{2}\right)-{\frac {\partial P}{\partial z_{2}}}\left(E_{21}z_{1}+E_{22}z_{2}\right)\\&-{\frac {\partial P}{\partial {\overline {z_{1}}}}}\left({\overline {E_{11}}}{\overline {z_{1}}}+{\overline {E_{12}}}{\overline {z_{2}}}\right)-{\frac {\partial P}{\partial {\overline {z_{2}}}}}\left({\overline {E_{21}}}{\overline {z_{1}}}+{\overline {E_{22}}}{\overline {z_{2}}}\right)\end{aligned}},\quad E\in {\mathfrak {sl}}(2,\mathbf {C} ).$$

(S7)

In particular for the basis elements,

$${\begin{aligned}\phi _{\mu ,\nu }(H)&=-z_{1}{\frac {\partial }{\partial z_{1}}}+z_{2}{\frac {\partial }{\partial z_{2}}}-{\overline {z_{1}}}{\frac {\partial }{\partial {\overline {z_{1}}}}}+{\overline {z_{2}}}{\frac {\partial }{\partial {\overline {z_{2}}}}}\\\phi _{\mu ,\nu }(X)&=-z_{2}{\frac {\partial }{\partial z_{1}}}-{\overline {z_{2}}}{\frac {\partial }{\partial {\overline {z_{1}}}}}\\\phi _{\mu ,\nu }(Y)&=-z_{1}{\frac {\partial }{\partial z_{2}}}-{\overline {z_{1}}}{\frac {\partial }{\partial {\overline {z_{2}}}}}\end{aligned}}$$

(S8)

Properties of the (m, n) representations

The (m, n) representations, defined above via (A1) (as restrictions to the real form ${\mathfrak {sl}}(3,1)$) of tensor products of irreducible complex linear representations π_{m = μ} and π_{n = ν} of ${\mathfrak {sl}}(2,\mathbb {C} ),$ are irreducible, and they are the only irreducible representations.

• Irreducibility follows from the unitarian trick and that a representation Π of SU(2) × SU(2) is irreducible if and only if Π = Π_μ ⊗ Π_ν, where Π_μ, Π_ν are irreducible representations of SU(2).
• Uniqueness follows from that the Π_m are the only irreducible representations of SU(2), which is one of the conclusions of the theorem of the highest weight.

Dimension

The (m, n) representations are (2m + 1)(2n + 1)-dimensional. This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of ${\text{SL}}(2,\mathbb {C} )$ and ${\mathfrak {sl}}(2,\mathbb {C} )$. For a Lie general algebra ${\mathfrak {g}}$ the Weyl dimension formula,

$$\dim \pi _{\rho }={\frac {\Pi _{\alpha \in R^{+}}\langle \alpha ,\rho +\delta \rangle }{\Pi _{\alpha \in R^{+}}\langle \alpha ,\delta \rangle }},$$

applies, where R⁺ is the set of positive roots, ρ is the highest weight, and δ is half the sum of the positive roots. The inner product $\langle \cdot ,\cdot \rangle$ is that of the Lie algebra ${\mathfrak {g}},$ invariant under the action of the Weyl group on ${\mathfrak {h}}\subset {\mathfrak {g}},$ the Cartan subalgebra. The roots (really elements of ${\mathfrak {h}}^{*}$ are via this inner product identified with elements of ${\mathfrak {h}}.$ For ${\mathfrak {sl}}(2,\mathbb {C} ),$ the formula reduces to dim π_μ = 2μ + 1 = 2m + 1, where the present notation must be taken into account. The highest weight is 2μ. By taking tensor products, the result follows.

Faithfulness

If a representation Π of a Lie group G is not faithful, then N = ker Π is a nontrivial normal subgroup. There are three relevant cases.

1. N is non-discrete and abelian.
2. N is non-discrete and non-abelian.
3. N is discrete. In this case N ⊂ Z, where Z is the center of G.

In the case of SO(3; 1)⁺, the first case is excluded since SO(3; 1)⁺ is semi-simple. The second case (and the first case) is excluded because SO(3; 1)⁺ is simple. For the third case, SO(3; 1)⁺ is isomorphic to the quotient ${\text{SL}}(2,\mathbb {C} )/\{\pm I\}.$ But $\{\pm I\}$ is the center of ${\text{SL}}(2,\mathbb {C} ).$ It follows that the center of SO(3; 1)⁺ is trivial, and this excludes the third case. The conclusion is that every representation Π : SO(3; 1)⁺ → GL(V) and every projective representation Π : SO(3; 1)⁺ → PGL(W) for V, W finite-dimensional vector spaces are faithful.

By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial non-discrete normal subgroups replaced by (one-dimensional) nontrivial ideals in the Lie algebra, and the center of SO(3; 1)⁺ replaced by the center of ${\mathfrak {sl}}(3;1)^{+}$The center of any semisimple Lie algebra is trivial and ${\mathfrak {so}}(3;1)$ is semi-simple and simple, and hence has no non-trivial ideals.

A related fact is that if the corresponding representation of ${\text{SL}}(2,\mathbb {C} )$ is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding ${\text{SL}}(2,\mathbb {C} )$ representation is not faithful, but is 2:1.

Non-unitarity

The (m, n) Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary. This is due to the non-compactness of the Lorentz group. In fact, a connected simple non-compact Lie group cannot have any nontrivial unitary finite-dimensional representations. There is a topological proof of this. Let u : G → GL(V), where V is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group G. Then u(G) ⊂ U(V) ⊂ GL(V) where U(V) is the compact subgroup of GL(V) consisting of unitary transformations of V. The kernel of u is a normal subgroup of G. Since G is simple, ker u is either all of G, in which case u is trivial, or ker u is trivial, in which case u is faithful. In the latter case u is a diffeomorphism onto its image, u(G) ≅ G and u(G) is a Lie group. This would mean that u(G) is an embedded non-compact Lie subgroup of the compact group U(V). This is impossible with the subspace topology on u(G) ⊂ U(V) since all embedded Lie subgroups of a Lie group are closed If u(G) were closed, it would be compact, and then G would be compact, contrary to assumption.

In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of A and B used in the construction are Hermitian. This means that J is Hermitian, but K is anti-Hermitian. The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.

Restriction to SO(3)

The (m, n) representation is, however, unitary when restricted to the rotation subgroup SO(3), but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an (m, n) representation have SO(3)-invariant subspaces of highest weight (spin) m + n, m + n − 1, ..., | m − n|, where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) j is (2j + 1)-dimensional. So for example, the (1/2, 1/2) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.

Since the angular momentum operator is given by J = A + B, the highest spin in quantum mechanics of the rotation sub-representation will be (m + n)ℏ and the "usual" rules of addition of angular momenta and the formalism of 3-j symbols, 6-j symbols, etc. applies.

Spinors

It is the SO(3)-invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the (m, n) representation has spin if m + n is half-integer. The simplest are (1/2, 0) and (0, 1/2), the Weyl-spinors of dimension 2. Then, for example, (0, 3/2) and (1, 1/2) are a spin representations of dimensions 2⋅3/2 + 1 = 4 and (2 + 1)(2⋅1/2 + 1) = 6 respectively. According to the above paragraph, there are subspaces with spin both 3/2 and 1/2 in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under SO(3). It cannot be ruled out in general, however, that representations with multiple SO(3) subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out unphysical components, leaving only a single spin.

Construction of pure spin n/2 representations for any n (under SO(3)) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints.

Dual representations

The root system A₁ × A₁ of ${\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} ).$

The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation:

1. The set of weights of the dual representation of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation.
2. Two irreducible representations are isomorphic if and only if they have the same highest weight.
3. For each semisimple Lie algebra there exists a unique element w₀ of the Weyl group such that if μ is a dominant integral weight, then w₀ ⋅ (−μ) is again a dominant integral weight.
4. If $\pi _{\mu _{0}}$ is an irreducible representation with highest weight μ₀, then $\pi _{\mu _{0}}^{*}$ has highest weight w₀ ⋅ (−μ).

Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If −I is an element of the Weyl group of a semisimple Lie algebra, then w₀ = −I. In the case of ${\mathfrak {sl}}(2,\mathbb {C} ),$ the Weyl group is W = {I, −I}. It follows that each π_μ, μ = 0, 1, ... is isomorphic to its dual $\pi _{\mu }^{*}.$ The root system of ${\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )$ is shown in the figure to the right. The Weyl group is generated by $\{w_{\gamma }\}$ where $w_{\gamma }$ is reflection in the plane orthogonal to γ as γ ranges over all roots. Inspection shows that w_α ⋅ w_β = −I so −I ∈ W. Using the fact that if π, σ are Lie algebra representations and π ≅ σ, then Π ≅ Σ, the conclusion for SO(3; 1)⁺ is

$$\pi _{m,n}^{*}\cong \pi _{m,n},\quad \Pi _{m,n}^{*}\cong \Pi _{m,n},\quad 2m,2n\in \mathbf {N} .$$

Complex conjugate representations

If π is a representation of a Lie algebra, then ${\overline {\pi }}$ is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication. In general, every irreducible representation π of ${\mathfrak {sl}}(n,\mathbb {C} )$ can be written uniquely as π = π⁺ + π⁻, where

$$\pi ^{\pm }(X)={\frac {1}{2}}\left(\pi (X)\pm i\pi \left(i^{-1}X\right)\right),$$

with $\pi ^{+}$ holomorphic (complex linear) and $\pi ^{-}$ anti-holomorphic (conjugate linear). For ${\mathfrak {sl}}(2,\mathbb {C} ),$ since $\pi _{\mu }$ is holomorphic, ${\overline {\pi _{\mu }}}$ is anti-holomorphic. Direct examination of the explicit expressions for $\pi _{\mu ,0}$ and $\pi _{0,\nu }$ in equation (S8) below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression (S8) also allows for identification of $\pi ^{+}$ and $\pi ^{-}$ for $\pi _{\mu ,\nu }$ as

$$\pi _{\mu ,\nu }^{+}=\pi _{\mu }^{\oplus _{\nu +1}},\qquad \pi _{\mu ,\nu }^{-}={\overline {\pi _{\nu }^{\oplus _{\mu +1}}}}.$$

Using the above identities (interpreted as pointwise addition of functions), for SO(3; 1)⁺ yields

$${\begin{aligned}{\overline {\pi _{m,n}}}&={\overline {\pi _{m,n}^{+}+\pi _{m,n}^{-}}}={\overline {\pi _{m}^{\oplus _{2n+1}}}}+{\overline {{\overline {\pi _{n}}}^{\oplus _{2m+1}}}}\\&=\pi _{n}^{\oplus _{2m+1}}+{\overline {\pi _{m}}}^{\oplus _{2n+1}}=\pi _{n,m}^{+}+\pi _{n,m}^{-}=\pi _{n,m}\\&&&2m,2n\in \mathbb {N} \\{\overline {\Pi _{m,n}}}&=\Pi _{n,m}\end{aligned}}$$

where the statement for the group representations follow from exp(X) = exp(X). It follows that the irreducible representations (m, n) have real matrix representatives if and only if m = n. Reducible representations on the form (m, n) ⊕ (n, m) have real matrices too.

The adjoint representation, the Clifford algebra, and the Dirac spinor representation

Richard Brauer and wife Ilse 1970. Brauer generalized the spin representations of Lie algebras sitting inside Clifford algebras to spin higher than 1/2.

Photo courtesy of MFO.

In general representation theory, if (π, V) is a representation of a Lie algebra ${\mathfrak {g}},$ then there is an associated representation of ${\mathfrak {g}},$ on End(V), also denoted π, given by

$$\pi (X)(A)=[\pi (X),A],\qquad A\in \operatorname {End} (V),\ X\in {\mathfrak {g}}.$$

(I1)

Likewise, a representation (Π, V) of a group G yields a representation Π on End(V) of G, still denoted Π, given by

$$\Pi (g)(A)=\Pi (g)A\Pi (g)^{-1},\qquad A\in \operatorname {End} (V),\ g\in G.$$

(I2)

If π and Π are the standard representations on $\mathbb {R} ^{4}$ and if the action is restricted to ${\mathfrak {so}}(3,1)\subset {\text{End}}(\mathbb {R} ^{4}),$ then the two above representations are the adjoint representation of the Lie algebra and the adjoint representation of the group respectively. The corresponding representations (some $\mathbb {R} ^{n}$ or $\mathbb {C} ^{n}$) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular.

Applying this to the Lorentz group, if (Π, V) is a projective representation, then direct calculation using (G5) shows that the induced representation on End(V) is a proper representation, i.e. a representation without phase factors.

In quantum mechanics this means that if (π, H) or (Π, H) is a representation acting on some Hilbert space H, then the corresponding induced representation acts on the set of linear operators on H. As an example, the induced representation of the projective spin (1/2, 0) ⊕ (0, 1/2) representation on End(H) is the non-projective 4-vector (1/2, 1/2) representation.

For simplicity, consider only the "discrete part" of End(H), that is, given a basis for H, the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4-vector representation of above on this simplified End(H) has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, ${\mathcal {Cl}}_{3,1}(\mathbb {R} ),$ whose complexification is ${\text{M}}(4,\mathbb {C} ),$ generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the (0, 0), a pseudoscalar irrep, also the (0, 0), but with parity inversion eigenvalue −1, see the next section below, the already mentioned vector irrep, (1/2, 1/2), a pseudovector irrep, (1/2, 1/2) with parity inversion eigenvalue +1 (not −1), and a tensor irrep, (1, 0) ⊕ (0, 1). The dimensions add up to 1 + 1 + 4 + 4 + 6 = 16. In other words,

$${\mathcal {Cl}}_{3,1}(\mathbb {R} )=(0,0)\oplus \left({\frac {1}{2}},{\frac {1}{2}}\right)\oplus [(1,0)\oplus (0,1)]\oplus \left({\frac {1}{2}},{\frac {1}{2}}\right)_{p}\oplus (0,0)_{p},$$

(I3)

where, as is customary, a representation is confused with its representation space.

The (1/2, 0) ⊕ (0, 1/2) spin representation

The six-dimensional representation space of the tensor (1, 0) ⊕ (0, 1)-representation inside ${\mathcal {Cl}}_{3,1}(\mathbb {R} )$ has two roles. The

$$\sigma ^{\mu \nu }=-{\frac {i}{4}}\left[\gamma ^{\mu },\gamma ^{\nu }\right],$$

(I4)

where $\gamma ^{0},\ldots ,\gamma ^{3}\in {\mathcal {Cl}}_{3,1}(\mathbb {R} )$ are the gamma matrices, the sigmas, only 6 of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,

$$\left[\sigma ^{\mu \nu },\sigma ^{\rho \tau }\right]=i\left(\eta ^{\tau \mu }\sigma ^{\rho \nu }+\eta ^{\nu \tau }\sigma ^{\mu \rho }-\eta ^{\rho \mu }\sigma ^{\tau \nu }-\eta ^{\nu \rho }\sigma ^{\mu \tau }\right),$$

(I5)

and hence constitute a representation (in addition to spanning a representation space) sitting inside ${\mathcal {Cl}}_{3,1}(\mathbb {R} ),$ the (1/2, 0) ⊕ (0, 1/2) spin representation. For details, see bispinor and Dirac algebra.

The conclusion is that every element of the complexified ${\mathcal {Cl}}_{3,1}(\mathbb {R} )$ in End(H) (i.e. every complex 4×4 matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on $\mathbb {C} ^{4},$ making it a space of bispinors.

Reducible representations

There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained by taking direct sums, tensor products, and quotients of the irreducible representations. Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e.g. ${\text{GL}}(n,\mathbb {R} )$ and the Poincaré group. These representations are in general not irreducible.

The Lorentz group and its Lie algebra have the complete reducibility property. This means that every representation reduces to a direct sum of irreducible representations. The reducible representations will therefore not be discussed.

Space inversion and time reversal

The (possibly projective) (m, n) representation is irreducible as a representation SO(3; 1)⁺, the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If m = n it can be extended to a representation of all of O(3; 1), the full Lorentz group, including space parity inversion and time reversal. The representations (m, n) ⊕ (n, m) can be extended likewise.

Space parity inversion

For space parity inversion, the adjoint action Ad_P of P ∈ SO(3; 1) on ${\mathfrak {so}}(3;1)$ is considered, where P is the standard representative of space parity inversion, P = diag(1, −1, −1, −1), given by

$$\mathrm {Ad} _{P}(J_{i})=PJ_{i}P^{-1}=J_{i},\qquad \mathrm {Ad} _{P}(K_{i})=PK_{i}P^{-1}=-K_{i}.$$

(F1)

It is these properties of K and J under P that motivate the terms vector for K and pseudovector or axial vector for J. In a similar way, if π is any representation of ${\mathfrak {so}}(3;1)$ and Π is its associated group representation, then Π(SO(3; 1)⁺) acts on the representation of π by the adjoint action, π(X) ↦ Π(g) π(X) Π(g)⁻¹ for $X\in {\mathfrak {so}}(3;1),$ g ∈ SO(3; 1)⁺. If P is to be included in Π, then consistency with (F1) requires that

$$\Pi (P)\pi (B_{i})\Pi (P)^{-1}=\pi (A_{i})$$

(F2)

holds, where A and B are defined as in the first section. This can hold only if A_i and B_i have the same dimensions, i.e. only if m = n. When m ≠ n then (m, n) ⊕ (n, m) can be extended to an irreducible representation of SO(3; 1)⁺, the orthochronous Lorentz group. The parity reversal representative Π(P) does not come automatically with the general construction of the (m, n) representations. It must be specified separately. The matrix β = i γ⁰ (or a multiple of modulus −1 times it) may be used in the (1/2, 0) ⊕ (0, 1/2) representation.

If parity is included with a minus sign (the 1×1 matrix [−1]) in the (0,0) representation, it is called a pseudoscalar representation.

Time reversal

Time reversal T = diag(−1, 1, 1, 1), acts similarly on ${\mathfrak {so}}(3;1)$ by

$$\mathrm {Ad} _{T}(J_{i})=TJ_{i}T^{-1}=-J_{i},\qquad \mathrm {Ad} _{T}(K_{i})=TK_{i}T^{-1}=K_{i}.$$

(F3)

By explicitly including a representative for T, as well as one for P, a representation of the full Lorentz group O(3; 1) is obtained. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the P^μ, in addition to the Jⁱ and Kⁱ generate the group. These are interpreted as generators of translations. The time-component P⁰ is the Hamiltonian H. The operator T satisfies the relation

$$\mathrm {Ad} _{T}(iH)=TiHT^{-1}=-iH$$

(F4)

in analogy to the relations above with ${\mathfrak {so}}(3;1)$ replaced by the full Poincaré algebra. By just cancelling the i's, the result THT⁻¹ = −H would imply that for every state Ψ with positive energy E in a Hilbert space of quantum states with time-reversal invariance, there would be a state Π(T⁻¹)Ψ with negative energy −E. Such states do not exist. The operator Π(T) is therefore chosen antilinear and antiunitary, so that it anticommutes with i, resulting in THT⁻¹ = H, and its action on Hilbert space likewise becomes antilinear and antiunitary. It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix. This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, Π is not a representation.

When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, (1/2, 0) ⊕ (0, 1/2), is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.

The third discrete symmetry entering in the CPT theorem along with P and T, charge conjugation symmetry C, has nothing directly to do with Lorentz invariance.

Action on function spaces

If V is a vector space of functions of a finite number of variables n, then the action on a scalar function $f\in V$ given by

$$(\Pi (g)f)(x)=f\left(\Pi _{x}(g)^{-1}x\right),\qquad x\in \mathbb {R} ^{n},f\in V$$

(H1)

produces another function Πf ∈ V. Here Π_x is an n-dimensional representation, and Π is a possibly infinite-dimensional representation. A special case of this construction is when V is a space of functions defined on the a linear group G itself, viewed as a n-dimensional manifold embedded in $\mathbb {R} ^{m^{2}}$ (with m the dimension of the matrices). This is the setting in which the Peter–Weyl theorem and the Borel–Weil theorem are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations. The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. ${\text{SL}}(2,\mathbb {C} ).$

The following exemplifies action of the Lorentz group and the rotation subgroup on some function spaces.

Euclidean rotations

 이 부분의 본문은 Rotation group SO(3) § Spherical harmonics 및 Spherical harmonics입니다.

The subgroup SO(3) of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space

$$L^{2}\left(\mathbb {S} ^{2}\right)=\operatorname {span} \left\{Y_{m}^{l},l\in \mathbb {N} ^{+},-l\leqslant m\leqslant l\right\},$$

where $Y_{m}^{l}$ are the spherical harmonics. An arbitrary square integrable function f on the unit sphere can be expressed as

$$f(\theta ,\varphi )=\sum _{l=1}^{\infty }\sum _{m=-l}^{l}f_{lm}Y_{m}^{l}(\theta ,\varphi ),$$

(H2)

where the f_lm are generalized Fourier coefficients.

The Lorentz group action restricts to that of SO(3) and is expressed as

$${\begin{aligned}(\Pi (R)f)(\theta (x),\varphi (x))&=\sum _{l=1}^{\infty }\sum _{m,=-l}^{l}\sum _{m'=-l}^{l}D_{mm'}^{(l)}(R)f_{lm'}Y_{m}^{l}\left(\theta \left(R^{-1}x\right),\varphi \left(R^{-1}x\right)\right),\\[5pt]&R\in \mathrm {SO} (3),x\in \mathbb {S} ^{2},\end{aligned}}$$

(H4)

where the D^l are obtained from the representatives of odd dimension of the generators of rotation.

The Möbius group

 이 부분의 본문은 Möbius transformation 및 Lorentz group § Relation to the Möbius group입니다.

The identity component of the Lorentz group is isomorphic to the Möbius group M. This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection, the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere.

In the plane, a Möbius transformation characterized by the complex numbers a, b, c, d acts on the plane according to

$$f(z)={\frac {az+b}{cz+d}},\qquad ad-bc\neq 0$$ .

(M1)

and can be represented by complex matrices

$$\Pi _{f}={\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\lambda {\begin{pmatrix}a&b\\c&d\end{pmatrix}},\qquad \lambda \in \mathbb {C} -\{0\},\operatorname {det} \Pi _{f}=1,$$

(M2)

since multiplication by a nonzero complex scalar does not change f. These are elements of ${\text{SL}}(2,\mathbb {C} )$ and are unique up to a sign (since ±Π_f give the same f), hence ${\text{SL}}(2,\mathbb {C} )/\{\pm I\}\cong {\text{SO}}(3;1)^{+}.$

The Riemann P-functions

 이 부분의 본문은 Riemann's differential equation입니다.

The Riemann P-functions, solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as

$${\begin{aligned}w(z)&=P\left\{{\begin{matrix}a&b&c&\\\alpha &\beta &\gamma &\;z\\\alpha '&\beta '&\gamma '&\end{matrix}}\right\}\\&=\left({\frac {z-a}{z-b}}\right)^{\alpha }\left({\frac {z-c}{z-b}}\right)^{\gamma }P\left\{{\begin{matrix}0&\infty &1&\\0&\alpha +\beta +\gamma &0&\;{\frac {(z-a)(c-b)}{(z-b)(c-a)}}\\\alpha '-\alpha &\alpha +\beta '+\gamma &\gamma '-\gamma &\end{matrix}}\right\}\end{aligned}},$$

(T1)

where the a,  b,  c,  α,  β,  γ,  α′,  β′,  γ′ are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is

$$P\left\{{\begin{matrix}0&\infty &1&\\0&a&0&\;z\\1-c&b&c-a-b&\end{matrix}}\right\}={}_{2}F_{1}(a,\,b;\,c;\,z).$$

(T2)

The set of constants 0, ∞, 1 in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation. Its exponents, i. e. solutions of the indicial equation, for expansion around the singular point 0 are 0 and 1 − c ,corresponding to the two linearly independent solutions, and for expansion around the singular point 1 they are 0 and c − a − b. Similarly, the exponents for ∞ are a and b for the two solutions.

One has thus

$$w(z)=\left({\frac {z-a}{z-b}}\right)^{\alpha }\left({\frac {z-c}{z-b}}\right)^{\gamma }{}_{2}F_{1}\left(\alpha +\beta +\gamma ,\,\alpha +\beta '+\gamma ;\,1+\alpha -\alpha ';\,{\frac {(z-a)(c-b)}{(z-b)(c-a)}}\right),$$

(T3)

where the condition (sometimes called Riemann's identity)

$$\alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1$$

on the exponents of the solutions of Riemann's differential equation has been used to define γ′.

The first set of constants on the left hand side in (T1), a, b, c denotes the regular singular points of Riemann's differential equation. The second set, α, β, γ, are the corresponding exponents at a, b, c for one of the two linearly independent solutions, and, accordingly, α′, β′, γ′ are exponents at a, b, c for the second solution.

Define an action of the Lorentz group on the set of all Riemann P-functions by first setting

$$u(\Lambda )(z)={\frac {Az+B}{Cz+D}},$$

(T4)

where A,  B,  C,  D are the entries in

$$\lambda ={\begin{pmatrix}A&B\\C&D\end{pmatrix}}\in {\text{SL}}(2,\mathbb {C} ),$$

(T5)

for Λ = p(λ) ∈ SO(3; 1)⁺ a Lorentz transformation.

Define

$$[\Pi (\Lambda )P](z)=P[u(\Lambda )(z)],$$

(T6)

where P is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Möbius transformation of the argument is that of shifting the poles to new locations, hence changing the critical points, but there is no change in the exponents of the differential equation the new function satisfies. The new function is expressed as

$$[\Pi (\Lambda )P](u)=P\left\{{\begin{matrix}\eta &\zeta &\theta &\\\alpha &\beta &\gamma &\;u\\\alpha '&\beta '&\gamma '&\end{matrix}}\right\},$$

(T6)

where

$$\eta ={\frac {Aa+B}{Ca+D}}\quad {\text{ and }}\quad \zeta ={\frac {Ab+B}{Cb+D}}\quad {\text{ and }}\quad \theta ={\frac {Ac+B}{Cc+D}}.$$

(T7)

Infinite-dimensional unitary representations

History

The Lorentz group SO(3; 1)⁺ and its double cover ${\text{SL}}(2,\mathbb {C} )$ also have infinite dimensional unitary representations, studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) at the instigation of Paul Dirac. This trail of development begun with Dirac (1936) where he devised matrices U and B necessary for description of higher spin (compare Dirac matrices), elaborated upon by Fierz (1939), see also Fierz & Pauli (1939), and proposed precursors of the Bargmann-Wigner equations. In Dirac (1945) he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors. These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinite-dimensional generalization of spinors in his 1947 paper.

The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by Harish-Chandra (1951) and Gelfand & Graev (1953), based on an analogue for ${\text{SL}}(2,\mathbb {C} )$ of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in Rühl (1970) and Knapp (2001).

The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3-dimensional hyperbolic space sitting in Minkowski space is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on $\mathbb {R} .$ This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008).

Principal series for SL(2, C)

The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup B of $G={\text{SL}}(2,\mathbb {C} ).$ Since the one-dimensional representations of B correspond to the representations of the diagonal matrices, with non-zero complex entries z and z⁻¹, they thus have the form

$$\chi _{\nu ,k}{\begin{pmatrix}z&0\\c&z^{-1}\end{pmatrix}}=r^{i\nu }e^{ik\theta },$$

for k an integer, ν real and with z = re^iθ. The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when k is replaced by −k. By definition the representations are realized on L² sections of line bundles on $G/B=\mathbb {S} ^{2},$ which is isomorphic to the Riemann sphere. When k = 0, these representations constitute the so-called spherical principal series.

The restriction of a principal series to the maximal compact subgroup K = SU(2) of G can also be realized as an induced representation of K using the identification G/B = K/T, where T = B ∩ K is the maximal torus in K consisting of diagonal matrices with | z | = 1. It is the representation induced from the 1-dimensional representation z^kT, and is independent of ν. By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions |k| + 2m + 1 with m a non-negative integer.

Using the identification between the Riemann sphere minus a point and $\mathbb {C} ,$ the principal series can be defined directly on $L^{2}(\mathbb {C} )$ by the formula

$$\pi _{\nu ,k}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}^{-1}f(z)=|cz+d|^{-2-i\nu }\left({cz+d \over |cz+d|}\right)^{-k}f\left({az+b \over cz+d}\right).$$

Irreducibility can be checked in a variety of ways:

• The representation is already irreducible on B. This can be seen directly, but is also a special case of general results on irreducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition G = B ∪ BsB where s is the Weyl group element
  $${\begin{pmatrix}0&-1\\1&0\end{pmatrix}}$$
  .
• The action of the Lie algebra ${\mathfrak {g}}$ of G can be computed on the algebraic direct sum of the irreducible subspaces of K can be computed explicitly and the it can be verified directly that the lowest-dimensional subspace generates this direct sum as a ${\mathfrak {g}}$-module.

Complementary series for SL(2, C)

The for 0 < t < 2, the complementary series is defined on $L^{2}(\mathbb {C} )$ for the inner product

$$(f,g)_{t}=\iint {\frac {f(z){\overline {g(w)}}}{|z-w|^{2-t}}}\,dz\,dw,$$

with the action given by

$$\pi _{t}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}^{-1}f(z)=|cz+d|^{-2-t}f\left({az+b \over cz+d}\right).$$

The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of K, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of K = SU(2). Irreducibility can be proved by analyzing the action of ${\mathfrak {g}}$ on the algebraic sum of these subspaces or directly without using the Lie algebra.

Plancherel theorem for SL(2, C)

The only irreducible unitary representations of ${\text{SL}}(2,\mathbb {C} )$ are the principal series, the complementary series and the trivial representation. Since −I acts as (−1)^k on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even.

To decompose the left regular representation of G on $L^{2}(G)$ only the principal series are required. This immediately yields the decomposition on the subrepresentations $L^{2}(G/\{\pm I\}),$ the left regular representation of the Lorentz group, and $L^{2}(G/K),$ the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with k = 0.)

The left and right regular representation λ and ρ are defined on $L^{2}(G)$ by

$${\begin{aligned}(\lambda (g)f)(x)&=f\left(g^{-1}x\right)\\(\rho (g)f)(x)&=f(xg)\end{aligned}}$$

Now if f is an element of C_c(G), the operator $\pi _{\nu ,k}(f)$ defined by

$$\pi _{\nu ,k}(f)=\int _{G}f(g)\pi (g)\,dg$$

is Hilbert–Schmidt. Define a Hilbert space H by

$$H=\bigoplus _{k\geqslant 0}{\text{HS}}\left(L^{2}(\mathbb {C} )\right)\otimes L^{2}\left(\mathbb {R} ,c_{k}{\sqrt {\nu ^{2}+k^{2}}}d\nu \right),$$

where

$$c_{k}={\begin{cases}{\frac {1}{4\pi ^{3/2}}}&k=0\\{\frac {1}{(2\pi )^{3/2}}}&k\neq 0\end{cases}}$$

and ${\text{HS}}\left(L^{2}(\mathbb {C} )\right)$ denotes the Hilbert space of Hilbert–Schmidt operators on $L^{2}(\mathbb {C} ).$ Then the map U defined on C_c(G) by

$$U(f)(\nu ,k)=\pi _{\nu ,k}(f)$$

extends to a unitary of $L^{2}(G)$ onto H.

The map U satisfies the intertwining property

$$U(\lambda (x)\rho (y)f)(\nu ,k)=\pi _{\nu ,k}(x)^{-1}\pi _{\nu ,k}(f)\pi _{\nu ,k}(y).$$

If f₁, f₂ are in C_c(G) then by unitarity

$$(f_{1},f_{2})=\sum _{k\geqslant 0}c_{k}^{2}\int _{-\infty }^{\infty }\operatorname {Tr} \left(\pi _{\nu ,k}(f_{1})\pi _{\nu ,k}(f_{2})^{*}\right)\left(\nu ^{2}+k^{2}\right)\,d\nu .$$

Thus if $f=f_{1}*f_{2}^{*}$ denotes the convolution of $f_{1}$ and $f_{2}^{*},$ and $f_{2}^{*}(g)={\overline {f_{2}(g^{-1})}},$ then

$$f(1)=\sum _{k\geqslant 0}c_{k}^{2}\int _{-\infty }^{\infty }\operatorname {Tr} \left(\pi _{\nu ,k}(f)\right)\left(\nu ^{2}+k^{2}\right)\,d\nu .$$

The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively.

The Plancherel formula extends to all $f_{i}\in L^{2}(G).$ By a theorem of Jacques Dixmier and Paul Malliavin, every smooth compactly supported function on $G$ is a finite sum of convolutions of similar functions, the inversion formula holds for such f. It can be extended to much wider classes of functions satisfying mild differentiability conditions.

Classification of representations of SO(3, 1)

The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to assume they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinite-dimensional representation Π_H on a Hilbert space H of SO(3; 1)⁺ is at hand. Since SO(3) is a subgroup, Π_H is a representation of it as well. Each irreducible subrepresentation of SO(3) is finite-dimensional, and the SO(3) representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of SO(3) if Π_H is unitary.

The steps are the following:

1. Choose a suitable basis of common eigenvectors of J² and J₃.
2. Compute matrix elements of J₁, J₂, J₃ and K₁, K₂, K₃.
3. Enforce Lie algebra commutation relations.
4. Require unitarity together with orthonormality of the basis.

Step 1

One suitable choice of basis and labeling is given by

$$\left|j_{0}\,j_{1};j\,m\right\rangle .$$

If this were a finite-dimensional representation, then j₀ would correspond the lowest occurring eigenvalue j(j + 1) of J² in the representation, equal to |m − n|, and j₁ would correspond to the highest occurring eigenvalue, equal to m + n. In the infinite-dimensional case, j₀ ≥ 0 retains this meaning, but j₁ does not. For simplicity, it is assumed that a given j occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown that the assumption is possible to avoid (with a slightly more complicated calculation) with the same results.

Step 2

The next step is to compute the matrix elements of the operators J₁, J₂, J₃ and K₁, K₂, K₃ forming the basis of the Lie algebra of ${\mathfrak {so}}(3;1).$ The matrix elements of $J_{\pm }=J_{1}\pm iJ_{2}$ and $J_{3}$ (the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by

$${\begin{aligned}\left\langle j\,m\right|J_{+}\left|j\,m-1\right\rangle =\left\langle j\,m-1\right|J_{-}\left|j\,m\right\rangle &={\sqrt {(j+m)(j-m+1)}},\\\left\langle j\,m\right|J_{3}\left|j\,m\right\rangle &=m,\end{aligned}}$$

where the labels j₀ and j₁ have been dropped since they are the same for all basis vectors in the representation.

Due to the commutation relations

$$[J_{i},K_{j}]=i\epsilon _{ijk}K_{k},$$

the triple (K₁, K₂, K₃) ≡ K is a vector operator and the Wigner–Eckart theorem applies for computation of matrix elements between the states represented by the chosen basis. The matrix elements of

$${\begin{aligned}K_{0}^{(1)}&=K_{3},\\K_{\pm 1}^{(1)}&=\mp {\frac {1}{\sqrt {2}}}(K_{1}\pm iK_{2}),\end{aligned}}$$

where the superscript (1) signifies that the defined quantities are the components of a spherical tensor operator of rank k = 1 (which explains the factor √2 as well) and the subscripts 0, ±1 are referred to as q in formulas below, are given by

$${\begin{aligned}\left\langle j'm'\left|K_{0}^{(1)}\right|j\,m\right\rangle &=\left\langle j'\,m'\,k=1\,q=0|j\,m\right\rangle \left\langle j\left\|K^{(1)}\right\|j'\right\rangle ,\\\left\langle j'm'\left|K_{\pm 1}^{(1)}\right|j\,m\right\rangle &=\left\langle j'\,m'\,k=1\,q=\pm 1|j\,m\right\rangle \left\langle j\left\|K^{(1)}\right\|j'\right\rangle .\end{aligned}}$$

Here the first factors on the right hand sides are Clebsch–Gordan coefficients for coupling j′ with k to get j. The second factors are the reduced matrix elements. They do not depend on m, m′ or q, but depend on j, j′ and, of course, K. For a complete list of non-vanishing equations, see Harish-Chandra (1947, 375쪽).

Step 3

The next step is to demand that the Lie algebra relations hold, i.e. that

$$[K_{\pm },K_{3}]=\pm J_{\pm },\quad [K_{+},K_{-}]=-2J_{3}.$$

This results in a set of equations for which the solutions are

$${\begin{aligned}\left\langle j\left\|K^{(1)}\right\|j\right\rangle &=i{\frac {j_{1}j_{0}}{\sqrt {j(j+1)}}},\\\left\langle j\left\|K^{(1)}\right\|j-1\right\rangle &=-B_{j}\xi _{j}{\sqrt {j(2j-1)}},\\\left\langle j-1\left\|K^{(1)}\right\|j\right\rangle &=B_{j}\xi _{j}^{-1}{\sqrt {j(2j+1)}},\end{aligned}}$$

where

$$B_{j}={\sqrt {\frac {(j^{2}-j_{0}^{2})(j^{2}-j_{1}^{2})}{j^{2}(4j^{2}-1)}}},\quad j_{0}=0,{\tfrac {1}{2}},1,\ldots \quad {\text{and}}\quad j_{1},\xi _{j}\in \mathbb {C} .$$

Step 4

The imposition of the requirement of unitarity of the corresponding representation of the group restricts the possible values for the arbitrary complex numbers j₀ and ξ_j. Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning

$$K_{\pm }^{\dagger }=K_{\mp },\quad K_{3}^{\dagger }=K_{3}.$$

This translates to

$${\begin{aligned}\left\langle j\left\|K^{(1)}\right\|j\right\rangle &={\overline {\left\langle j\left\|K^{(1)}\right\|j\right\rangle }},\\\left\langle j\left\|K^{(1)}\right\|j-1\right\rangle &=-{\overline {\left\langle j-1\left\|K^{(1)}\right\|j\right\rangle }},\end{aligned}}$$

leading to

$${\begin{aligned}j_{0}\left(j_{1}+{\overline {j_{1}}}\right)&=0,\\\left|B_{j}\right|\left(\left|\xi _{j}\right|^{2}-e^{-2i\beta _{j}}\right)&=0,\end{aligned}}$$

where β_j is the angle of B_j on polar form. For |B_j| ≠ 0 follows $\left|\xi _{j}\right|^{2}=1$ and $\xi _{j}=1$ is chosen by convention. There are two possible cases:

• ${\underline {j_{1}+{\overline {j_{1}}}=0.}}$ In this case j₁ = − iν, ν real,
  $$\left\langle j\left\|K^{(1)}\right\|j\right\rangle ={\frac {\nu j_{0}}{j(j+1)}}\quad {\text{and}}\quad B_{j}={\sqrt {\frac {(j^{2}-j_{0}^{2})(j^{2}+\nu ^{2})}{4j^{2}-1}}}$$
  This is the principal series. Its elements are denoted $(j_{0},\nu ),2j_{0}\in \mathbb {N} ,\nu \in \mathbb {R} .$
• ${\underline {j_{0}=0.}}$ It follows:
  $$\left\langle j\left\|K^{(1)}\right\|j\right\rangle =0\quad {\text{and}}\quad B_{j}={\sqrt {\frac {j^{2}-\nu ^{2}}{4j^{2}-1}}}$$
  Since B₀ = B_j0, B틀:Supsub is real and positive for j = 1, 2, ..., leading to −1 ≤ ν ≤ 1. This is complementary series. Its elements are denoted (0, ν), −1 ≤ ν ≤ 1

This shows that the representations of above are all infinite-dimensional irreducible unitary representations.

Explicit formulas

Conventions and Lie algebra bases

The metric of choice is given by η = diag(−1, 1, 1, 1), and the physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of basis for the Lie algebra is, in the 4-vector representation, given by:

$${\begin{aligned}J_{1}=J^{23}=-J^{32}&=i{\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix}},&K_{1}=J^{01}=-J^{10}&=i{\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}},\\[8pt]J_{2}=J^{31}=-J^{13}&=i{\begin{pmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\end{pmatrix}},&K_{2}=J^{02}=-J^{20}&=i{\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\end{pmatrix}},\\[8pt]J_{3}=J^{12}=-J^{21}&=i{\begin{pmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{pmatrix}},&K_{3}=J^{03}=-J^{30}&=i{\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{pmatrix}}.\\[8pt]\end{aligned}}$$

The commutation relations of the Lie algebra ${\mathfrak {so}}(3;1)$ are:

$$\left[J^{\mu \nu },J^{\rho \sigma }\right]=i\left(\eta ^{\sigma \mu }J^{\rho \nu }+\eta ^{\nu \sigma }J^{\mu \rho }-\eta ^{\rho \mu }J^{\sigma \nu }-\eta ^{\nu \rho }J^{\mu \sigma }\right).$$

In three-dimensional notation, these are

$$\left[J_{i},J_{j}\right]=i\epsilon _{ijk}J_{k},\quad \left[J_{i},K_{j}\right]=i\epsilon _{ijk}K_{k},\quad \left[K_{i},K_{j}\right]=-i\epsilon _{ijk}J_{k}.$$

The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol J above and in the sequel should be observed.

For example, a typical boost and a typical rotation exponentiate as,

$$\exp(-i\xi K_{1})={\begin{pmatrix}\cosh \xi &\sinh \xi &0&0\\\sinh \xi &\cosh \xi &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},\qquad \exp(-i\theta J_{1})={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos \theta &-\sin \theta \\0&0&\sin \theta &\cos \theta \end{pmatrix}},$$

symmetric and orthogonal, respectively.

Weyl spinors and bispinors

Solutions to the Dirac equation transform under the (1/2, 0) ⊕ (0, 1/2)-representation. Dirac discovered the gamma matrices in his search for a relativistically invariant equation, then already known to mathematicians.

By taking, in turn, m = 1/2, n = 0 and m = 0, n = 1/2 and by setting

$$J_{i}^{\left({\frac {1}{2}}\right)}={\frac {1}{2}}\sigma _{i}$$

in the general expression (G1), and by using the trivial relations 1₁ = 1 and J⁽⁰⁾ = 0, it follows

$${\begin{aligned}\pi _{\left({\frac {1}{2}},0\right)}(J_{i})&={\frac {1}{2}}\left(\sigma _{i}\otimes 1_{(1)}+1_{(2)}\otimes J_{i}^{(0)}\right)={\frac {1}{2}}\sigma _{i}\\\pi _{\left({\frac {1}{2}},0\right)}(K_{i})&={\frac {-i}{2}}\left(1_{(2)}\otimes J_{i}^{(0)}-\sigma _{i}\otimes 1_{(1)}\right)={\frac {i}{2}}\sigma _{i}\\[6pt]\pi _{\left(0,{\frac {1}{2}}\right)}(J_{i})&={\frac {1}{2}}\left(J_{i}^{(0)}\otimes 1_{(2)}+1_{(1)}\otimes \sigma _{i}\right)={\frac {1}{2}}\sigma _{i}\\\pi _{\left(0,{\frac {1}{2}}\right)}(K_{i})&={\frac {-i}{2}}\left(1_{(1)}\otimes \sigma _{i}-J_{i}^{(0)}\otimes 1_{(2)}\right)={\frac {-i}{2}}\sigma _{i}\end{aligned}}$$

(W1)

These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) V_L and V_R, whose elements Ψ_L and Ψ_R are called left- and right-handed Weyl spinors respectively. Given

$$\left(\pi _{\left({\frac {1}{2}},0\right)},V_{\text{L}}\right)\quad {\text{and}}\quad \left(\pi _{\left(0,{\frac {1}{2}}\right)},V_{\text{R}}\right)$$

their direct sum as representations is formed,

$${\begin{aligned}\pi _{\left({\frac {1}{2}},0\right)\oplus \left(0,{\frac {1}{2}}\right)}\left(J_{i}\right)&={\frac {1}{2}}{\begin{pmatrix}\sigma _{i}&0\\0&\sigma _{i}\end{pmatrix}}\\[8pt]\pi _{\left({\frac {1}{2}},0\right)\oplus \left(0,{\frac {1}{2}}\right)}\left(K_{i}\right)&={\frac {i}{2}}{\begin{pmatrix}\sigma _{i}&0\\0&-\sigma _{i}\end{pmatrix}}\end{aligned}}$$

(D1)

This is, up to a similarity transformation, the (1/2,0) ⊕ (0,1/2) Dirac spinor representation of ${\mathfrak {so}}(3;1).$ It acts on the 4-component elements (Ψ_L, Ψ_R) of (V_L ⊕ V_R), called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of ${\mathfrak {so}}(3;1).$ Expressions for the group representations are obtained by exponentiation.

Open problems

The classification and characterization of the representation theory of the Lorentz group was completed in 1947. But in association with the Bargmann–Wigner programme, there are yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations.

The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the little group of the Poincaré group of spacelike vectors in higher spacetime dimension. The corresponding infinite-dimensional unitary representations of the (generalized) Poincaré group are the so-called tachyonic representations. Tachyons appear in the spectrum of bosonic strings and are associated with instability of the vacuum. Even though tachyons may not be realized in nature, these representations must be mathematically understood in order to understand string theory. This is so since tachyon states turn out to appear in superstring theories too in attempts to create realistic models.

One open problem is the completion of the Bargmann–Wigner programme for the isometry group SO(D − 2, 1) of the de Sitter spacetime dS_D−2. Ideally, the physical components of wave functions would be realized on the hyperboloid dS_D−2 of radius μ > 0 embedded in $\mathbb {R} ^{D-2,1}$ and the corresponding O(D−2, 1) covariant wave equations of the infinite-dimensional unitary representation to be known.

See also

• Bargmann–Wigner equations
• Dirac algebra
• Gamma matrices
• Lorentz group
• Möbius transformation
• Poincaré group
• Representation theory of the Poincaré group
• Symmetry in quantum mechanics
• Wigner's classification

Freely available online references

• Bekaert, X.; Boulanger, N. (2006). “The unitary representations of the Poincare group in any spacetime dimension”. arXiv:hep-th/0611263.  Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006).
• Curtright, T L; Fairlie, D B; Zachos, C K (2014), “A compact formula for rotations as spin matrix polynomials”, 《SIGMA》 10: 084, arXiv:1402.3541, Bibcode:2014SIGMA..10..084C, doi:10.3842/SIGMA.2014.084, S2CID 18776942  Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group.

References

• Abramowitz, M.; Stegun, I. A. (1965). 《Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables》. Dover Books on Mathematics. New York: Dover Publications. ISBN 978-0486612720. 
• Bargmann, V. (1947), “Irreducible unitary representations of the Lorenz group”, 《Ann. of Math.》 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129  (the representation theory of SO(2,1) and SL(2, R); the second part on SO(3; 1) and SL(2, C), described in the introduction, was never published).
• Bargmann, V.; Wigner, E. P. (1948), “Group theoretical discussion of relativistic wave equations”, 《Proc. Natl. Acad. Sci. USA》 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292 
• Bourbaki, N. (1998). 《Lie Groups and Lie Algebras: Chapters 1-3》. Springer. ISBN 978-3-540-64242-8. 
• Brauer, R.; Weyl, H. (1935), “Spinors in n dimensions”, 《Amer. J. Math.》 57 (2): 425–449, doi:10.2307/2371218, JSTOR 2371218 
• Bäuerle, G.G.A; de Kerf, E.A. (1990). A. van Groesen; E.M. de Jager, 편집. 《Finite and infinite dimensional Lie algebras and their application in physics》. Studies in mathematical physics 1. North-Holland. ISBN 978-0-444-88776-4. 
• Bäuerle, G.G.A; de Kerf, E.A.; ten Kroode, A.P.E. (1997). A. van Groesen; E.M. de Jager, 편집. 《Finite and infinite dimensional Lie algebras and their application in physics》. Studies in mathematical physics 7. North-Holland. ISBN 978-0-444-82836-1 – ScienceDirect 경유. 
• Cartan, Élie (1913), “Les groupes projectifs qui ne laissant invariante aucun multiplicité plane”, 《Bull. Soc. Math. Fr.》 (프랑스어) 41: 53–96, doi:10.24033/bsmf.916 
• Churchill, R. V.; Brown, J. W. (2014) [1948]. 《Complex Variables and Applications》 9판. New York: McGraw–Hill. ISBN 978-0073-383-170. 
• Coleman, A. J. (1989). “The Greatest Mathematical Paper of All Time”. 《The Mathematical Intelligencer》 11 (3): 29–38. doi:10.1007/BF03025189. ISSN 0343-6993. S2CID 35487310. 
• Dalitz, R. H.; Peierls, Rudolf (1986). “Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984”. 《Biogr. Mem. Fellows R. Soc.》 32: 138–185. doi:10.1098/rsbm.1986.0006. S2CID 74547263. 
• Delbourgo, R.; Salam, A.; Strathdee, J. (1967). “Harmonic analysis in terms of the homogeneous Lorentz group”. 《Physics Letters B》 25 (3): 230–32. Bibcode:1967PhLB...25..230D. doi:10.1016/0370-2693(67)90050-0. 
• Dirac, P. A. M. (1928), “The Quantum Theory of the Electron”, 《Proc. R. Soc. A》 117 (778): 610–624, Bibcode:1928RSPSA.117..610D, doi:10.1098/rspa.1928.0023  (free access)
• Dirac, P. A. M. (1936), “Relativistic wave equations”, 《Proc. R. Soc. A》 155 (886): 447–459, Bibcode:1936RSPSA.155..447D, doi:10.1098/rspa.1936.0111 
• Dirac, P. A. M. (1945), “Unitary representations of the Lorentz group”, 《Proc. R. Soc. A》 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171 
• Dixmier, J.; Malliavin, P. (1978), “Factorisations de fonctions et de vecteurs indéfiniment différentiables”, 《Bull. Sci. Math.》 (프랑스어) 102: 305–330 
• Fierz, M. (1939), “Über die relativistische theorie Kräftefreier teilchen mit beliebigem spin”, 《Helv. Phys. Acta》 (독일어) 12 (1): 3–37, Bibcode:1939AcHPh..12....3F, doi:10.5169/seals-110930(pdf download available) 
• Fierz, M.; Pauli, W. (1939), “On relativistic wave equations for particles of arbitrary spin in an electromagnetic field”, 《Proc. R. Soc. A》 173 (953): 211–232, Bibcode:1939RSPSA.173..211F, doi:10.1098/rspa.1939.0140 
• Folland, G. (2015). 《A Course in Abstract Harmonic Analysis》 2판. CRC Press. ISBN 978-1498727136. 
• Fulton, W.; Harris, J. (1991). 《Representation theory. A first course》. Graduate Texts in Mathematics 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249. 
• Gelfand, I. M.; Graev, M. I. (1953), “On a general method of decomposition of the regular representation of a Lie group into irreducible representations”, 《Doklady Akademii Nauk SSSR》 92: 221–224 
• Gelfand, I. M.; Graev, M. I.; Vilenkin, N. Ya. (1966), 〈Harmonic analysis on the group of complex unimodular matrices in two dimensions〉, 《Generalized functions. Vol. 5: Integral geometry and representation theory》, 번역 Eugene Saletan, Academic Press, 202–267쪽, ISBN 978-1-4832-2975-1 
• Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969), 《Representation theory and automorphic functions》, Academic Press, ISBN 978-0-12-279506-0 
• Gelfand, I.M.; Minlos, R.A.; Shapiro, Z. Ya. (1963), 《Representations of the Rotation and Lorentz Groups and their Applications》, New York: Pergamon Press 
• Gelfand, I. M.; Naimark, M. A. (1947), “Unitary representations of the Lorentz group” (PDF), 《Izvestiya Akad. Nauk SSSR. Ser. Mat.》 (러시아어) 11 (5): 411–504, 2014년 12월 15일에 확인함(Pdf from Math.net.ru) 
• Green, J. A. (1998). 〈Richard Dagobert Brauer〉 (PDF). 《Biographical Memoirs》 75. National Academy Press. 70–95쪽. ISBN 978-0309062954. 
• Greiner, W.; Müller, B. (1994). 《Quantum Mechanics: Symmetries》 2판. Springer. ISBN 978-3540580805. 
• Greiner, W.; Reinhardt, J. (1996), 《Field Quantization》, Springer, ISBN 978-3-540-59179-5 
• Harish-Chandra (1947), “Infinite irreducible representations of the Lorentz group”, 《Proc. R. Soc. A》 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518 
• Harish-Chandra (1951), “Plancherel formula for complex semi-simple Lie groups”, 《Proc. Natl. Acad. Sci. U.S.A.》 37 (12): 813–818, Bibcode:1951PNAS...37..813H, doi:10.1073/pnas.37.12.813, PMC 1063477, PMID 16589034 
• Hall, Brian C. (2003), 《Lie Groups, Lie Algebras, and Representations: An Elementary Introduction》, Graduate Texts in Mathematics 222 1판, Springer, ISBN 978-0-387-40122-5 
• Hall, Brian C. (2015), 《Lie groups, Lie algebras, and Representations: An Elementary Introduction》, Graduate Texts in Mathematics 222 2판, Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 
• Helgason, S. (1968), 《Lie groups and symmetric spaces》, Battelle Rencontres, Benjamin, 1–71쪽  (a general introduction for physicists)
• Helgason, S. (2000), 《Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions (corrected reprint of the 1984 original)》, Mathematical Surveys and Monographs 83, American Mathematical Society, ISBN 978-0-8218-2673-7 
• Jorgenson, J.; Lang, S. (2008), 《The heat kernel and theta inversion on SL(2,C)》, Springer Monographs in Mathematics, Springer, ISBN 978-0-387-38031-5 
• Killing, Wilhelm (1888), “Die Zusammensetzung der stetigen/endlichen Transformationsgruppen”, 《Mathematische Annalen》 (독일어) 31 (2 (June)): 252–290, doi:10.1007/bf01211904, S2CID 120501356 
• Kirillov, A. (2008). 《An Introduction to Lie Groups and Lie Algebras》. Cambridge Studies in Advanced Mathematics 113. Cambridge University Press. ISBN 978-0521889698. 
• Klauder, J. R. (1999). 〈Valentine Bargmann〉 (PDF). 《Biographical Memoirs》 76. National Academy Press. 37–50쪽. ISBN 978-0-309-06434-7. 
• Knapp, Anthony W. (2001), 《Representation theory of semisimple groups. An overview based on examples.》, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 978-0-691-09089-4  (elementary treatment for SL(2,C))
• Langlands, R. P. (1985). “Harish-Chandra”. 《Biogr. Mem. Fellows R. Soc.》 31: 198–225. doi:10.1098/rsbm.1985.0008. S2CID 61332822. 
• Lee, J. M. (2003), 《Introduction to Smooth manifolds》, Springer Graduate Texts in Mathematics 218, ISBN 978-0-387-95448-6 
• Lie, Sophus (1888), 《Theorie der Transformationsgruppen I(1888), II(1890), III(1893)》 (독일어) 
• Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), 《Gravitation》, W. H. Freeman, ISBN 978-0-7167-0344-0 
• Naimark, M.A. (1964), 《Linear representations of the Lorentz group (translated from the Russian original by Ann Swinfen and O. J. Marstrand)》, Macmillan 
• Rossmann, Wulf (2002), 《Lie Groups – An Introduction Through Linear Groups》, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9 
• Rühl, W. (1970), 《The Lorentz group and harmonic analysis》, Benjamin  (a detailed account for physicists)
• Simmons, G. F. (1972). 《Differential Equations with Applications and historical Notes》 T M H판. New Dheli: Tata McGra–Hill Publishing Company Ltd. ISBN 978-0-07-099572-7. 
• Stein, Elias M. (1970), “Analytic continuation of group representations”, 《Advances in Mathematics》 4 (2): 172–207, doi:10.1016/0001-8708(70)90022-8  (James K. Whittemore Lectures in Mathematics given at Yale University, 1967)
• Takahashi, R. (1963), “Sur les représentations unitaires des groupes de Lorentz généralisés”, 《Bull. Soc. Math. France》 (프랑스어) 91: 289–433, doi:10.24033/bsmf.1598 
• Taylor, M. E. (1986), 《Noncommutative harmonic analysis》, Mathematical Surveys and Monographs 22, American Mathematical Society, ISBN 978-0-8218-1523-6 , Chapter 9, SL(2, C) and more general Lorentz groups
• Tung, Wu-Ki (1985). 《Group Theory in Physics》 1판. New Jersey·London·Singapore·Hong Kong: World Scientific. ISBN 978-9971966577. 
• Varadarajan, V. S. (1989). 《An Introduction to Harmonic Analysis on Semisimple Lie Groups》. Cambridge University Press. ISBN 978-0521663625. 
• Weinberg, S. (2002) [1995], 《Foundations》, The Quantum Theory of Fields 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7 
• Weinberg, S. (2000). 《Supersymmetry》. The Quantum Theory of Fields 3 1판. Cambridge: Cambridge University Press. ISBN 978-0521670555. 
• Weyl, H. (1939), 《The Classical Groups. Their Invariants and Representations》, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255 
• Weyl, H. (1931), 《The Theory of Groups and Quantum Mechanics》, Dover, ISBN 978-0-486-60269-1 
• Wigner, E. P. (1939), “On unitary representations of the inhomogeneous Lorentz group”, 《Annals of Mathematics》 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411 .
• Zwiebach, B. (2004). 《A First Course in String Theory》. Cambridge University Press. ISBN 0-521-83143-1.