MATH 742 A T M P REPRESENTATION THEORY OF THE POINCARÉ ... · The Poincaré algebra 𝖕 [is the Lie algebra associated with 𝑃. It is a vector space along with a bracket ⋅,⋅]∶

Final project (McGill, 14/01/20) Marc-Antoine Fiset (260539607)

1

MATH 742 – ADVANCED TOPICS IN MATHEMATICAL PHYSICS REPRESENTATION THEORY OF THE POINCARÉ GROUP

FINAL PROJECT 1 Introduction ........................................................................................................................................................1

2 The Poincaré Group ............................................................................................................................................2

3 The Poincaré Algebra .........................................................................................................................................4

3.1 Casimir elements of the Poincaré algebra ........................................................................................................6

4 Representations of the Poincaré Group ..............................................................................................................7

4.1 Hilbert spaces ...................................................................................................................................................7

4.2 Space of classical fields ..................................................................................................................................10

5 Acknowledgements ..........................................................................................................................................13

6 Appendix ‒ Rotation .........................................................................................................................................13

7 References ........................................................................................................................................................14

1 Introduction As far as common sense is concerned, the world surrounding humans has three infinite space dimensions and it is

evolving as time increases (whatever these quantities are precisely). When developing his Special Theory of

Relativity (SR), Albert Einstein considered time, which somehow stands apart based on intuitive knowledge, as

merely pèp

an additional dimension of reality (or spacetime more precisely).1 Apart from this preliminary assumption ―that

the world has the topology of ℝ4―, the theory stems from two easy to grasp (although maybe hard to believe)

physical axioms [1]:

Axioms of Special Relativity:

⋅ The speed of light is constant. It has the same value, 𝑐 = 1 in some system of units, no

matter in which condition whoever measures it.

⋅ The principle of relativity, which heuristically says that the result of an experiment

should not depend on the observer (or more properly on the system of coordinates2 used

by this observer).

When learning about SR, I have been surprised and seduced by the amount of interesting theoretical facts ―all of

which being confirmed by experiments― that can be deduced from these simple starting points using only a

generous amount of mathematical cleverness. This exemplifies what I find particularly appealing and exciting in

mathematical physics. It is a symbiotic association of ideas from physicists and mathematicians, people with

different objectives and background, that yields otherwise unachievable discoveries.

1 This is something I personally still find hard to accept. Time is so much different. Why does it appear to be going in one

direction while we can move back and forth in space? The typical argument of entropy is not really satisfying to me… 2 A system of coordinates (or a frame of reference) is a set of four real numbers 𝑥 = (𝑥0, 𝑥1, 𝑥2, 𝑥3), where 𝑥0 ≡ 𝑡 is a measure

of the time elapsed since a certain event and where 𝑥1, 𝑥2, 𝑥3 are usual distances from a chosen origin along three orthogonal

axes.


2

In this document, we push a step forward the mathematical deductions stemming out of these axioms using the tools

of representation theory of Lie groups. As we will see, the mathematical objects that transform under representations

of the isometry group of the SR spacetime are classical fields and elements of quantum Hilbert spaces. They are

used by physicists in Quantum Field Theories to model virtually every interactions in the universe.

The discussion starts with a definition of the Poincaré group from first principles and some fundamental remarks

about it. We then study the Poincaré algebra and its Casimir elements because of their importance in the

representation theory. In the mathematical process, many links with physics are being made. The representation

theory is finally tackled, culminating essentially in the Wigner’s classification and in the identification of types of

particles in Nature. An appendix also discuss questions related to the subject of rotation in the context of Lie groups.

While reading about it, I sincerely found the subject of the Poincaré group very rich and intellectually enlightening.

Because of its numerous ramifications, a choice has to be made between a long and detailed presentation and a

shorter but less explicit discussion. For concision needs, I choose the latter option, so our goal is rather to weave a

web between Poincaré-related ideas than to prove very precise statements.

2 The Poincaré Group Of course, several different systems of four real numbers 𝑥𝜇 , 𝜇 ∈ {0,1,2,3}, can be used to describe whatever

happens in spacetime. Physicists conscientiously define a special class of systems of coordinates that are called

inertial. It can be shown [1] using the axioms of SR that, no matter which system of inertial coordinates is chosen,

the interval

Δ𝑠2 ≡ −(Δ𝑥0)2 + ∑(Δ𝑥𝑖)2

3

𝑖=1

between two points 𝑥 and 𝑥 + Δ𝑥 in spacetime is invariant. (This property is sometimes taken as a definition of

inertial systems of coordinates, but I find this practise somewhat unsatisfying on physical grounds.) If this is

regarded as a “generalized3 distance between 𝑥 and 𝑦”, it suggests to identify our spacetime with a “generalized

metric space”. The latter is properly called Minkowski spacetime 𝑴. It is a space that has the topology of ℝ4 and

its vector space structure along with the Minkowski metric (or quadratic form), a map such that

𝜂 ∶ 𝑀 × 𝑀 ∋ (𝑥, 𝑦) ⟼ −𝑥0𝑦0 + ∑𝑥𝑖𝑦𝑖

3

𝑖=1

∈ ℝ.

The output of this map is the inner product of 𝑥 and 𝑦. It is often denoted 𝜂𝜇𝜈𝑥𝜇𝑦𝜈 , where a sum on 𝜇, 𝜈 is

understood (as always in the present text) and where

𝜂𝜇𝜈 ≡ diag(−1 1 1 1).

Note that the interval defined above is just the inner product of Δ𝑥 with itself.

Let us finally note that an isometry (or a symmetry) on 𝑀 is defined just as in the case of conventional metric spaces:

it is a map from 𝑀 to itself that preserves the distances. These preliminaries now allow us to define the Poincaré

group (see table below). We have two definitions that are equivalent because, as seen above, inertial frames of

references are characterized by the fact that the interval (the “generalized length”) has the same value no matter in

which frame it is measured.

3 I use “generalized” because distances are usually assumed to be positive-definite in mathematics.


3

Poincaré group 𝑷

Mathematical definition:

Group of isometries of Minkowski

spacetime.

Physical definition:

Group of conversions between

inertial frames of reference in

Minkowski spacetime.

Active point of view:

Actual transformation, same

coordinates.

Passive point of view:

Change of coordinates, no actual

transformation.

It turns out that the mathematical definition hides an active point of view of the transformations while the physical

definition is more passive in essence.

Proposition 1

𝑃 is a non-compact Lie group. [2]

Without loss of generality4 [3], an element 𝑔 of the Poincaré group is always assumed to be a linear transformation

of the form

𝑥𝜇 ⟶𝑔

𝑥′𝜇 ≡ Λ𝜇𝜈𝑥

𝜈 + 𝑐𝜇 , (1)

where Λ𝜇𝜈 are the entries of a 4-matrix representing5 a so-called Lorentz transformation and where 𝑐𝜇 are the

entries a constant 4-vector representing a translation in spacetime. Since the interval must be conserved, we have a

constraint on Λ𝜇𝜈 (but not on 𝑐𝜇):

𝜂𝜌𝜎Δ𝑥𝜌Δ𝑥𝜎 = 𝜂𝜇𝜈Δ𝑥′𝜇Δ𝑥′𝜈 = 𝜂𝜇𝜈Λ𝜇𝜌Λ

𝜈𝜎Δ𝑥𝜌Δ𝑥𝜎 ∀ Δ𝑥

⇒ 𝜂𝜇𝜈Λ𝜇𝜌Λ

𝜈𝜎 = 𝜂𝜌𝜎 . (2)

This mimics the fact that the Poincaré group is the direct product [4] of the Lorentz group 𝑳 = 𝐎(𝟏, 𝟑) (the subgroup

of 𝑃 leaving the origin fixed) and the group of translations ℝ3,1:

Proposition 2

𝑃 = O(1,3) ⋉ ℝ1,3. 6

As seen in problem set 2, O(1,3) can be separated into four connected components according to the determinant of

Λ𝜇𝜈, a matrix representing an element of O(1,3), and to the sign of Λ0

0. Given the simple relation with 𝑃, the same

is true in the case of 𝑃:

𝑃 ⊃ {

𝑃+↑ = 𝐿+

↑ ⋉ ℝ1,3 = ISO↑(1,3), Proper (det Λ = 1) orthochronous (Λ00 > 0) transf.

𝑃−↑ = 𝐿−

↑ ⋉ ℝ1,3, Improper (det Λ = −1) orthochronous (Λ00 > 0) transf.

𝑃+↓ = 𝐿+

↓ ⋉ ℝ1,3, Proper (det Λ = 1) non-orthochronous (Λ00 < 0) transf.

𝑃−↓ = 𝐿−

↓ ⋉ ℝ1,3, Improper (detΛ = −1) non-orthochronous (Λ00 < 0) transf.

4 Suppose otherwise there were an order-two term Λ𝜇

𝑎𝑏𝑥𝑎𝑥𝑏 in the Taylor expansion of the transformation. Plugging in (2)

gives that Λ𝜇𝑎𝑏 = 0. We can believe the rest to be true for higher order terms.

5 By using 4-matrices and 4-vectors, we are already working in a particular 4-dimensional representation of 𝑃 . This is

uncomfortable since we want to treat representation theory latter, but it is inevitable (as far as I know) in order to understand

the group we are working on. We could call it the fundamental representation. 6 I could not find a satisfying proof of this decomposition.


4

Of these subspaces, only 𝑃+↑ ⋉ ℝ1,3 = ISO↑(1,3) is a subgroup because it

contains the identity of 𝑃. It can be shown that any element of the other

subspaces are obtainable from an element of SO↑(1,3) ⋉ ℝ1,3 via a time-

inversion and/or a space-inversion or (parity transformation).

As a concluding remark, note that the general form of (1) and our

understanding of the Poincaré group allow us to guess that some Lorentz

transformations should be rotations in 3-dimensional space. We will see this

more clearly latter, but let us record here this additional subgroup of 𝑃. A

rotation is a proper transformation that leaves the time fixed, so it is

orthochronous. Hence, SO(3) ⊂ 𝐿+↑ . Similarly, translations in ℝ3 form a

subgroup of ℝ1,3.

3 The Poincaré Algebra The Poincaré algebra 𝖕 is the Lie algebra associated with 𝑃. It is a vector space along with a bracket [⋅,⋅] ∶ 𝔭 × 𝔭 →𝔭, which can be specified by its action on a basis of vectors called the generators of 𝑃. We will obtain three common

and useful sets of generators. The commutation relations (specifying the bracket) for these sets will be the following:

First set Second set Third set7

Generators 𝐽𝜇𝜈, 𝑃𝜇 𝐽𝑖, 𝐾𝑖, 𝑃𝑖, 𝐻 𝐿𝑖, 𝑅𝑖, 𝑃𝑖, 𝐻

Com

muta

tion r

elati

ons

[𝐽𝜇𝜈 , 𝐽𝜌𝜎] = 𝑖(𝜂𝜈𝜎𝐽𝜇𝜌 + 𝜂𝜇𝜌𝐽𝜈𝜎

− 𝜂𝜇𝜎𝐽𝜈𝜌 − 𝜂𝜈𝜌𝐽𝜇𝜎)

[𝐽𝜇𝜈 , 𝑃𝜌] = 𝑖(𝜂𝜇𝜌𝑃𝜈 − 𝜂𝜈𝜌𝑃𝜇)

[𝑃𝜇 , 𝑃𝜈] = 0

[𝐽𝑖, 𝐽𝑗] = 𝑖𝜖𝑖𝑗𝑘𝐽𝑘

[𝐽𝑖, 𝐾𝑗] = 𝑖𝜖𝑖𝑗𝑘𝐾𝑘

[𝐾𝑖, 𝐾𝑗] = −𝑖𝜖𝑖𝑗𝑘𝐽𝑘

[𝐽𝑖, 𝑃𝑗] = 𝑖𝜖𝑖𝑗𝑘𝑃𝑘

[𝐾𝑖, 𝑃𝑗] = 𝑖𝐻𝛿𝑖𝑗

[𝑃𝑖, 𝑃𝑗] = [𝑃𝑖, 𝐻] = 0

[𝐽𝑖, 𝐻] = 0

[𝐾𝑖, 𝐻] = 𝑖𝑃𝑖

[𝐿𝑖, 𝐿𝑗] = 𝑖𝜖𝑖𝑗𝑘𝐿𝑘

[𝑅𝑖, 𝑅𝑗] = 𝑖𝜖𝑖𝑗𝑘𝑅𝑘

[𝐿𝑖, 𝑅𝑗] = 0

[𝐿𝑖, 𝑃𝑗] =1

2(𝑖𝜖𝑖𝑗𝑘𝑃

𝑘 − 𝐻𝛿𝑖𝑗)

[𝑅𝑖, 𝑃𝑗] =1

2(𝑖𝜖𝑖𝑗𝑘𝑃

𝑘 + 𝐻𝛿𝑖𝑗)

[𝐿𝑖, 𝐻] = −𝑃𝑖

2

[𝑅𝑖, 𝐻] =𝑃𝑖

2

No

tati

on

𝜂𝜇𝜈 ≡ 𝜂𝜇𝜈 𝐽𝑖 ≡1

2𝜖𝑖𝑗𝑘𝐽

𝑗𝑘

𝐾𝑖 ≡ 𝐽0𝑖

𝜖𝑖𝑗𝑘 is the sign of the

permutation of 𝑖𝑗𝑘

𝑖, 𝑗, 𝑘 ∈ {1,2,3}

𝐿𝑖 ≡1

2(𝐽𝑖 + 𝑖𝐾𝑖)

𝑅𝑖 ≡1

2(𝐽𝑖 − 𝑖𝐾𝑖)

Table 1 ‒ Commutation relations of 𝔭 expressed in three important sets of generators

7 This set is actually valid for the complexification 𝔰𝔬(1,3)ℂ ≡ 𝔰𝔬(1,3) ⊗ ℂ of the Lorentz algebra because 𝐿𝑖 and 𝑅𝑖 will be

defined as complexified Lorentz generators.

𝑳+↑

𝐿−↑

𝐿−↓ 𝐿+

↓

𝐑𝟏,𝟑

𝑳 𝑷

Sketch of the most important

subspaces and subgroups (in

bold) of 𝑃

𝐒𝐎(𝟑)

𝐑𝟑


5

As done during the semester, we now use a group element infinitesimally close to the identity to obtain information

about the Lie algebra. In the fundamental representation, the Lorentz transformation is8 Λ𝜇𝜈 = 𝛿𝜇

𝜈 + 𝜆𝜇𝜈 (𝜆𝜇

𝜈 ≪𝛿𝜇

𝜈) and (2) forces on it the following constraint:

𝜂𝜇𝜈(𝛿𝜇𝜌 + 𝜆𝜇

𝜌)(𝛿𝜇𝜎 + 𝜆𝜇

𝜎) = 𝜂𝜌𝜎

𝜂𝜇𝜈(𝛿𝜇𝜌𝜆

𝜈𝜎 + 𝜆𝜇

𝜌𝛿𝜈𝜎) = 0 to first order

𝜆𝜌𝜎 = −𝜆𝜎𝜌 where 𝜂𝜇𝜌𝜆𝜌𝜈 ≡ 𝜆𝜇𝜈

The elements of the Lie algebra of 𝐿 being 4 by 4 antisymmetric matrices in this representation (as just shown) and

because such a matrix has 6 degrees of freedom, the Lorentz algebra has dimension 6. Since there is no constraint

on the infinitesimal translations, the Poincaré algebra has dimension 10. A generic element of 𝔰𝔬(1,3) and 𝔯1,3 (Lie

algebras for the Lorentz and translations groups) is typically written as follows.

−𝑖

2𝜔𝜇𝜈𝐽

𝜇𝜈 ∈ 𝔰𝔬(1,3) ⊂ 𝔭 −𝑖𝑐𝜇𝑃𝜇 ∈ 𝔯1,3 ⊂ 𝔭

(𝜔𝜇𝜈 , 𝑐𝜇 ∈ ℝ,

𝐽𝜇𝜈 = −𝐽𝜈𝜇 and

𝜔𝜇𝜈 = −𝜔𝜈𝜇)

(3)

Note that these are abstract vectors which do not depend on the defining representation ―the latter was only useful

to obtain the antisymmetry of 𝜆―. The factor of a half is for later convenience and the 𝑖 will allow us eventually to

identify the generators with hermitian operators. Note that there is some freedom regarding the sign of these

expressions. This choice is irrelevant for most applications as we can simply redefine 𝜔𝜇𝜈 or 𝑐𝜇 to absorb the minus

sign. Some complications can however occur when considering the representations on the space of functions (see

section 4.2).

A derivation of one of the commutation relations of the first set in table 1 is detailed in [2]. It uses only material that

we developed so far. The other relations can be worked out similarly using the same technique.

~ ~ ~

The Poincaré algebra is well defined at this point but physicists prefer using the set of generators9

𝐽𝑖 ≡

1


𝑗𝑘, 𝐾𝑖 ≡ 𝐽0𝑖, 𝑃𝑖 , 𝐻 ≡ 𝑃0, (4)

as they can be given a physical interpretation. The 𝐽𝑖 have the commutation relations of rotation-related Lie algebras

(see table 1 and especially appendix 1), so the interpretation is immediate. Similarly the 𝑃𝑖 are quite obviously

generating translations in ℝ3 (because of how they arise). Detailing convincingly the interpretation for the

generators and the group elements associated with 𝐾𝑖 and 𝐻 would take us away from our main concern, but let us

nevertheless record in table 2 the important elements on the Poincaré group along with their generator and their

physical interpretation. The following change of variables occurred:

8 This remark requires material to be introduced latter. 𝜆𝜇

𝜈 is usually denoted 𝜔𝜇𝜈 even though the profound signification is

not the same. Here is why:

𝜆𝜇𝜈 = −

𝑖

2𝜔𝜌𝜎(𝐽𝜌𝜎)𝜇

𝜈 = −𝑖

2𝜔𝜌𝜎𝑖(𝜂𝜎𝜇𝛿𝜌

𝜈 − 𝜂𝜌𝜇𝛿𝜎𝜈) =

1

2(𝜔𝜈

𝜇 − 𝜔𝜇𝜈) = −𝜔𝜇

𝜈.

9 The definition of 𝐽𝑖 is equivalent to 𝐽1 = 𝐽23, 𝐽2 = 𝐽31, 𝐽3 = 𝐽12 because of the factor of a half.


6

𝜔𝜇𝜈 = (

0 𝜁1 𝜁2 𝜁3−𝜁1 0 𝜃3 −𝜃2

−𝜁2 −𝜃3 0 𝜃1

−𝜁3 𝜃2 −𝜃1 0

) 𝑐𝜇 = 𝜂𝜇𝜈𝑐𝜈 = (

𝑡𝑐1𝑐2

𝑐3

)

Group element Rotation 𝑒−𝑖�⃗⃗� ⋅𝐽 ,

𝜃 ∈ ℝ3

Boost 𝑒−𝑖�⃗� ⋅�⃗⃗� ,

𝜁 ∈ ℝ3

Translation 𝑒−𝑖𝑐 ⋅�⃗� ,

𝑐 ∈ ℝ3

Inverse10 time-evolution

𝑒𝑖𝑡𝐻,

𝑡 ∈ ℝ

Generator11 Angular

momentum 𝐽 ‒ Momentum �⃗� Energy 𝐻

Table 2 ‒ Generators, corresponding group element and their physical interpretation

The commutation relations from the first set of generators allow very straightforwardly to find the commutation

relations of this set (see table 1 again).

~ ~ ~

A third set of generators is defined by

𝐿𝑖 ≡1

2(𝐽𝑖 + 𝑖𝐾𝑖), 𝑅𝑖 ≡

1

2(𝐽𝑖 − 𝑖𝐾𝑖), 𝑃𝑖 , 𝐻.

Let us focus on the Lorentz algebra. Since we are using complexified vectors, the set is actually generating the

complexified version of 𝔰𝔬(1,3), which is 𝔰𝔬(1,3)ℂ ≡ 𝔰𝔬(1,3) ⊗ ℂ. We see from table 1 that it contains two

commuting sub Lie algebras obeying the commutation relations associated with rotation (see the appendix). This

gives the next decomposition.

Proposition 3

𝔰𝔬(1,3)ℂ ≅ 𝔰𝔲(2)ℂ ⊕ 𝔰𝔲(2)ℂ

3.1 Casimir elements of the Poincaré algebra The Poincaré algebra 𝔭 has two12 Casimir element, vectors commuting with the generators of the algebra. The table

below gives them as well as a physical interpretation that will be explained in section 4.

Casimir element 𝑷𝟐 ≡ 𝜂𝜇𝜈𝑃𝜇𝑃𝜈

𝑾𝟐 = 𝜂𝜇𝜈𝑊𝜇𝑊𝜈

where 𝑊𝜇 ≡1

2𝜖𝜇𝜈𝜌𝜎𝑃𝜈𝐽𝜌𝜎 is the Pauli-Ljubanski pseudo-vector

Interpretation Mass Spin (and mass)

Table 3 – Casimir elements of the Poincaré algebra

10 The time evolution operator 𝑒−𝑖𝑡𝐻 familiar from Quantum Mechanics corresponds to a passive time translation, which

explains that from our (active) perspective we get an inverse time evolution. When letting time evolve, we indeed actively push

everything backwards. 11 𝐽 ≡ (𝐽1 𝐽2 𝐽3); �⃗⃗� ≡ (𝐾1 𝐾2 𝐾3); �⃗� ≡ (𝑃1 𝑃2 𝑃3). Additional note: I would be able to justify the first line of my

table 2, but I could not find any reason why we should give these interpretation to the generators. Maybe we define the angular

momentum, the momentum and the energy as being represented by these operators… 12 I did not find any proof that the Poincaré algebra had only two Casimir elements.


7

Proof

[𝑃𝜇 , 𝑃2] = 0 is trivial

[𝐽𝜇𝜈 , 𝑃2] = 𝐽𝜇𝜈𝜂𝜌𝜎𝑃𝜌𝑃𝜎 − 𝜂𝜌𝜎𝑃𝜌𝑃𝜎𝐽𝜇𝜈

= 𝜂𝜌𝜎(𝑃𝜌𝐽𝜇𝜈 + [𝐽𝜇𝜈 , 𝑃𝜌])𝑃𝜎 − 𝜂𝜌𝜎𝑃𝜌𝑃𝜎𝐽𝜇𝜈

= 𝜂𝜌𝜎(𝑃𝜌𝐽𝜇𝜈 + 𝑖(𝜂𝜇𝜌𝑃𝜈 − 𝜂𝜈𝜌𝑃𝜇))𝑃𝜎 − 𝜂𝜌𝜎𝑃𝜌𝑃𝜎𝐽𝜇𝜈

= 𝜂𝜌𝜎𝑃𝜌𝐽𝜇𝜈𝑃𝜎 + 𝑖𝜂𝜌𝜎𝜂𝜇𝜌𝑃𝜈𝑃𝜎 − 𝑖𝜂𝜌𝜎𝜂𝜈𝜌𝑃𝜇𝑃𝜎 − 𝜂𝜌𝜎𝑃𝜌𝑃𝜎𝐽𝜇𝜈

= 𝜂𝜌𝜎𝑃𝜌(𝑃𝜎𝐽𝜇𝜈 + [𝐽𝜇𝜈 , 𝑃𝜎]) + 𝑖𝛿𝜇𝜎𝑃𝜈𝑃𝜎 − 𝑖𝛿𝜈𝜎𝑃𝜇𝑃𝜎 − 𝜂𝜌𝜎𝑃𝜌𝑃𝜎𝐽𝜇𝜈

= 𝜂𝜌𝜎𝑃𝜌(𝑃𝜎𝐽𝜇𝜈 + 𝑖(𝜂𝜇𝜎𝑃𝜈 − 𝜂𝜈𝜎𝑃𝜇)) − 𝜂𝜌𝜎𝑃𝜌𝑃𝜎𝐽𝜇𝜈

= 𝜂𝜌𝜎𝑃𝜌𝑃𝜎𝐽𝜇𝜈 + 𝑖𝜂𝜌𝜎𝑃𝜌𝜂𝜇𝜎𝑃𝜈 − 𝑖𝜂𝜌𝜎𝑃𝜌𝜂𝜈𝜎𝑃𝜇 − 𝜂𝜌𝜎𝑃𝜌𝑃𝜎𝐽𝜇𝜈

= 𝑖𝛿𝜇𝜌𝑃𝜌𝑃𝜈 − 𝑖𝛿𝜈𝜌𝑃𝜌𝑃𝜇

= 0

The proof that 𝑊2 is a Casimir is more tedious. It is detailed in [5].

∎

Proposition 4

𝑊𝜇𝑃𝜇 = 0

Proof

𝑊𝜇𝑃𝜇 =1

2𝜖𝜇𝜈𝜌𝜎𝑃𝜈𝐽𝜌𝜎𝑃𝜇

=1

2𝜖𝜇𝜈𝜌𝜎𝑃𝜈(𝑃𝜇𝐽𝜌𝜎 + [𝐽𝜌𝜎 , 𝑃𝜇])

=1

2𝜖𝜇𝜈𝜌𝜎𝑃𝜈(𝑃𝜇𝐽𝜌𝜎 + 𝑖𝜂𝜌𝜇𝑃𝜎 − 𝑖𝜂𝜇𝜎𝑃𝜌)

=1

4𝜖𝜇𝜈𝜌𝜎𝑃𝜈𝑃𝜇𝐽𝜌𝜎 +

1

4𝜖𝜈𝜇𝜌𝜎𝑃𝜇𝑃𝜈𝐽𝜌𝜎 +

𝑖

2𝜖𝜇𝜈𝜌𝜎𝜂𝜌𝜇𝑃𝜈𝑃𝜎 −

𝑖

2𝜖𝜇𝜈𝜌𝜎𝜂𝜇𝜎𝑃𝜈𝑃𝜌

=1

4𝜖𝜇𝜈𝜌𝜎𝑃𝜈𝑃𝜇𝐽𝜌𝜎 −

1

4𝜖𝜇𝜈𝜌𝜎𝑃𝜈𝑃𝜇𝐽𝜌𝜎 +

𝑖

2𝜖𝜇𝜈𝜌𝜎𝜂𝜌𝜇𝑃𝜈𝑃𝜎 −

𝑖

2𝜖𝜌𝜈𝜎𝜇𝜂𝜌𝜇𝑃𝜈𝑃𝜎

= 0

∎

4 Representations of the Poincaré Group I was disappointed to find no reference trying to be absolutely exhaustive in describing the representations of 𝑃.

People usually focus instead on some interesting cases. I distinguished two important classes of representations

motivated by the needs of theoretical physics. Both implement via 𝜌 ∶ 𝑃 → GL(𝑉) a Poincaré transformation 𝑔 ∈ 𝑃

on some vector space 𝑉. In one case, the space is a Hilbert space arising in Quantum Field Theory (―see below for

clarifications―). In the other case, 𝑉 is a space of classical fields (tensor valued functions of spacetime). We look

at the two cases separately.

4.1 Hilbert spaces In Quantum Mechanics, the physical state of a system is described (up to a factor of 𝑒𝑖𝜃, 𝜃 ∈ ℝ) by an element |𝜓⟩ of a Hilbert space 𝕳 (a certain complex vector space). The Hilbert space is endowed with an inner product ⟨⋅ | ⋅⟩ ∶ ℋ × ℋ ⟶ ℂ with these properties [6]

⟨𝜙|𝜓⟩ = ⟨𝜓|𝜙⟩∗

⟨𝜙|𝑎𝜓1 + 𝑏𝜓2⟩ = 𝑎⟨𝜙|𝜓1⟩ + 𝑏⟨𝜙|𝜓2⟩ ⟨𝑎𝜙1 + 𝑏𝜙2|𝜓⟩ = 𝑎∗⟨𝜙1|𝜓⟩ + 𝑏∗⟨𝜙2|𝜓⟩


8

⟨𝜓|𝜓⟩ ≥ 0.

(There are other technical conditions for ℌ to be a Hilbert space.) The probability for the system in state |𝜓⟩ to be

measured in a state |𝜓𝑖⟩ is

𝒫(𝜓 → 𝜓𝑖) = |⟨𝜓𝑖|𝜓⟩|2.

Just as in our previous discussion in section 2, the main physical features of a system should not depend on the

inertial frame of reference. In particular, the above probability should not depend on the inertial system of

coordinates used to measure it. Here is a theorem from Eugene Wigner.

Proposition 5 (Wigner’s theorem)

A Poincaré transformations 𝑔 is represented on the Hilbert space by an operator 𝑈 = 𝜌(𝑔) = exp(𝑖𝑎) (𝑎 ∈ 𝔭)

that is either

(i) Unitary, ⟨𝜙|𝜓⟩ = ⟨𝑈𝜙|𝑈𝜓⟩, and linear 𝑈(𝑎𝜙 + 𝑏𝜓) = 𝑎𝑈(𝜙) + 𝑏𝑈(𝜓) or

(ii) Antiunitary, ⟨𝜙|𝜓⟩∗ = ⟨𝑈𝜙|𝑈𝜓⟩, and antilinear 𝑈(𝑎𝜙 + 𝑏𝜓) = 𝑎∗𝑈(𝜙) + 𝑏∗𝑈(𝜓)

According to Weinberg [6], any Poincaré transformation that can be made trivial by a continuous change of a

parameter (all of them, then) is represented by a unitary transformation. We also have the following [4].

Proposition 6

Non-compact groups do not have finite-dimensional unitary representations

Therefore (see proposition 1), we are looking here for irreducible infinite-dimensional unitary representations of

𝖕, which will give, via the exponential map, representations of 𝑃. Since we want 𝜌(𝑔) to be a unitary operator on a

Hilbert space, the corresponding generator 𝑎 has to be Hermitian:

1 = 𝜌(𝑔)†𝜌(𝑔) = exp(𝑖𝑎)† exp(𝑖𝑎) = exp(𝑖𝑎 − 𝑖𝑎†) ⇒ 𝑎 = 𝑎† if [𝑎, 𝑎†] = 0

~ ~ ~

The rest of this subsection describes Wigner’s classification [6, 7] of irreducible infinite-dimensional unitary

representations of 𝑃. We first use the eigenstates |𝜓𝑝⟩ of 𝑃𝜇 (the generators defined in section 3) as a basis of the

Hilbert space13. We assume here that 𝑃𝜇|𝜓𝑝⟩ = 𝑝𝜇|𝜓𝑝⟩, where 𝑝𝜇 ∈ ℝ4 is regarded as a 4-vector.

A trick of the classification consists of writing 𝑝𝜇 as a standard momentum 𝑘𝜇 via a Lorentz transformation:

𝑝𝜇 = Λ𝜇𝜈𝑘

𝜈.

This simplifies the problem as, in a sense, we record only the “essential features” of 𝑝𝜇 by writing identicaly every

“similar enough” eigenvalues. It turns out that the only functions of 𝑝𝜇 independent of Lorentz transformations are

𝑝2 = 𝜂𝜇𝜈𝑝𝜇𝑝𝜈 (obviously) and the sign of 𝑝0 in the case of 𝜂𝜇𝜈𝑝

𝜇𝑝𝜈 ≤ 0 (this is well-known to physicists but I did

not reproduce an argument here for mathematicians). There are thus 6 classes of standard momentum (table 4).

Standard 𝑘𝜇 Little group Interpretation

⟶ (a) 𝑝2 < 0 𝑝0 > 0 (𝑚, 0,0,0) SO(3) Particle of mass 𝑚 and spin 𝑠 = 0, 1 2⁄ , 1, …

(b) 𝑝2 < 0 𝑝0 < 0 (−𝑚, 0,0,0) SO(3)

13 This makes sense on a physical basis, but I confess that I can provide no compelling mathematical reasons for this choice…


9

⟶ (c) 𝑝2 = 0 𝑝0 > 0 (𝜅, 𝜅, 0,0) ISO(2) Massless particle with unconstrained helicity

(d) 𝑝2 = 0 𝑝0 < 0 (−𝜅, 𝜅, 0,0) ISO(2)

(e) 𝑝2 > 0 (0, 𝑛, 0,0) SO(1,2) Tachyon

⟶ (f) 𝑝𝜇 = 0 (0,0,0,0) SO(1,3) Vacuum

Table 4 – Standard momentum and little groups [6]

Subgroups of the Lorentz group leaving 𝑘𝜇 unchanged are called little groups. They are given in table 4. [6] gives

a few comments about how they are obtained and argues that the representations of 𝑃 can be found from

representations of the little groups via the method of induced representations.

In our case, let us just mention that only the cases (a), (c) and (f) have a physical interpretation. To see it, we use

the Casimir elements (or operator in the current context) found previously. Schur’s lemma implies that the Casimir

operators are proportional to the identity operator in irreducible representations. They just act as multiplicative

constants, so they are typically used as labels for the irreducible representations. We have

𝑃2|𝜓𝑝⟩ = 𝜂𝜇𝜈𝑃𝜇𝑃𝜈|𝜓𝑝⟩ = 𝜂𝜇𝜈𝑝

𝜇𝑝𝜈 = −𝑚2,

which has the interpretation of a mass squared because of the interpretation given to 𝑃𝜇 and because of the

relativistic equation of energy (again well-known to physicists). Let us continue our investigation by studying the

cases (a), (c) and (f) separately.

Case of (a) This is an irreducible representation corresponding to something (we usually say a particle) with positive mass.

Let us work out what the other invariant gives. Since 𝑊𝜇𝑃𝜇 = 0,

𝑊𝜇𝑃𝜇|𝜓𝑝⟩ = 𝑘𝜇𝑊𝜇|𝜓𝑝⟩ = −𝑚𝑊0|𝜓𝑝⟩ = 0,

so 𝑊0 = 0. The other components of 𝑊𝜇 are

𝑊𝑖 =1

2𝜖𝑖0𝑗𝑘𝑃

0𝐽𝑗𝑘

such that

𝑊2 = 𝜂𝑖𝑖′𝑊𝑖𝑊𝑖′ =1

4𝜂𝑖𝑖′𝜖𝑖0𝑗𝑘𝜖𝑖′0𝑗′𝑘′𝑃0𝐽𝑗𝑘𝑃0𝐽𝑗

′𝑘′

=𝑚2

4𝜂𝑖𝑖′𝜖0𝑖𝑗𝑘𝜖0𝑖′𝑗′𝑘′𝐽𝑗𝑘𝐽𝑗

′𝑘′= 𝑚2 ∑(

1


𝑗𝑘)2

𝑖

= 𝑚2|𝐽 |2,

using (4). If the Hilbert space ℌ is written in the basis of eigenvectors |𝜓𝑗⟩ of |𝐽 |2, then the representation theory of

𝔰𝔬(3) ≅ 𝔰𝔲(2) (seen in class) yields

𝑊2|𝜓𝑗⟩ = 𝑚2|𝐽 |2|𝜓𝑗⟩ = 𝑚2𝑠(𝑠 + 1)|𝜓𝑗⟩

where 𝑠 = 0, 1 2⁄ , 1, 3 2⁄ ,… is the spin14 of the particle. All massive particles in nature correspond to this paradigm.

This is probably one of the most important and powerful result in the present text.

14 I am not convinced that this should be identified as the spin. Why is it not an angular momentum for example?


10

Note that it is not entirely surprising that 𝔰𝔬(3) came over the scene since it is the Lie algebra associated with the

little group of (a).

Case of (c) Here the particle is massless but similarly the irreducible representations of the little group can be used to complete

the derivation. The little group ISO(2) is the group of rotations and translations in a plane. It is not compact as

SO(3) was, so we do not expect unitary representation because of proposition 6. The only way to get unitarity is to

set to zero the non-compact generators, which leaves only the rotation generator orthogonal to the plane. (It happens

to be 𝐽1 here as can be deduced from [8].) It coincides here with the definition of the helicity operator of the particle

ℎ =𝐽 ⋅ �⃗�

|�⃗� |

(because 𝑝1 = 𝜅 is the only non-zero component). The helicity is quantized in the real world but, unlike the spin,

this cannot be obtained from the representation theory of the Poincaré group. It only becomes apparent upon

quantization of the fields in Quantum Field Theory [8].

Case of (f) It describes the vacuum.

4.2 Space of classical fields [9] A field is a function of Minkowski spacetime 𝑀:

𝜙 ∶ 𝑀 ∋ 𝑥 ⟼ 𝜙(𝑥),

where 𝜙(𝑥) is in general a finite-dimensional tensor even though we focus here on vector fields 𝜙𝑎(𝑥). Under a

Poincaré transformation 𝑔 (active transformation), a vector field is expected to transform as

𝜙𝑎(𝑥) ⟶𝑔

𝑚𝑎𝑏𝜙

𝑏(𝑚−1(𝑥)), 𝑚𝑎𝑏 ∈ Mat4, 𝑚−1 ∶ 𝑀 → 𝑀

i.e. the transformed field at 𝑥𝜇 depends linearly on the initial field evaluated at the untransformed point15. The matrix

𝑚𝑎𝑏 quite obviously represents a Poincaré transformation (𝑚𝑎

𝑏 = 𝜌(𝑔) for a certain representation map 𝜌), so we

just described a physical reason for finding the irreducible finite-dimensional representations of 𝖕, especially on

the space of vectors.

What about the function 𝑚−1? In a certain system of coordinates, 𝑥 is actually a 4-vector 𝑥𝜇, so 𝑚−1 can be thought

of as a map from ℝ4 to itself. It is also a finite-dimensional representation of the same Poincaré transformation 𝑔.

However, 𝑥 is only the argument of a function and we would prefer to have a representation 𝜚(𝑔) acting on the

function itself, i.e.


𝜌(𝑔)𝜚(𝑔)[𝜙𝑏(𝑥)] ≡ 𝜌(𝑔)𝜙𝑏(𝑚−1(𝑥)) . (5)

15 To understand the appearance of the inverse Poincaré transformation in the argument, it is useful to consider the example of

a rotation by 𝜃 of a scalar field (like temperature for example). Suppose there is a hotspot at some place to ease visualization.

The coordinate stay the same but the field changes: 𝜙(𝑥) ⟶ 𝜙′(𝑥). However, the new field is really just the old field at the

untransformed coordinate: 𝜙(𝑥) ⟶ 𝜙′(𝑥) = 𝜙(𝑚−1(𝑥))


11

This motivates the search for infinite-dimensional representations of 𝖕, especially on the space of functions.

Finite-dimensional representations, space of vectors In this context, the generators are matrices. For a reason that I have not seen explained anywhere, the finite-

dimensional representations are always discussed relatively to the Lorentz group; the translations are left aside.

We already know in some details the 4-dimensional “fundamental representation” as we used it to understand better

the Lorentz group in section 3. In particular, we found that the elements of 𝔰𝔬(1,3) are 4 by 4 antisymmetric matrices

in this representation. This gives a hint on what the actual matrices should look like. The commutation relations

would help us finding pretty straightforwardly that

(𝐽𝜌𝜎)𝜇𝜈 = 𝑖(𝜂𝜎𝜇𝛿𝜌

𝜈 − 𝜂𝜌𝜇𝛿𝜎𝜈). (6)

This is interesting and very important, but conceptually slightly oversimplified. A more systematic way to obtain

this representation ―and other significant ones― would be to use the decomposition from proposition 3:

𝔰𝔬(1,3)ℂ ≅ 𝔰𝔲(2)ℂ ⊕ 𝔰𝔲(2)ℂ.

From the representation theory of 𝔰𝔲(2)ℂ, this tells us that the finite-dimensional irreducible representations of

𝔰𝔬(1,3)ℂ are labeled by a pair (𝑠1, 𝑠2), 𝑠1, 𝑠2 = 0, 1 2⁄ , 1, 3 2⁄ ,… of numbers that again have the interpretation of

a spin (of the field here). They are representations of dimension 2𝑠1 + 1 + 2𝑠2 + 1 = 2(𝑠1 + 𝑠2 + 1). Table 5 gives

the most important special cases.

The generators of 𝔰𝔲(2)ℂ and the relations between the different basis of generators introduced in section 3 can be

used to get the generators in any basis. This would be a straightforward way to obtain (6) for example.

Name of the repr. Label Dim. Generators Comments

Trivial (0,0) 1 (𝐽𝜌𝜎)𝜇𝜈 = 01×1 ⊗ 01×1 = 01×1

𝜙 is called a scalar field

or a Lorentz scalar if it is

constant over spacetime.

Spinorial (1

2, 0) 2

𝐿𝑖 =𝜎𝑖

2⊗ 11×1 =

𝜎𝑖

2,

𝑅𝑖 = 12×2 ⊗ 01×1 = 02×2

𝜙𝑎 are called left-handed

Weyl spinors

Spinorial (0,1

2) 2

𝐿𝑖 = 01×1 ⊗ 12×2 = 02×2,

𝑅𝑖 = 11×1 ⊗𝜎𝑖

2=

𝜎𝑖

2

𝜙𝑎 are called right-

handed Weyl spinors

Fundamental (1

2,1

2) 4

𝐿𝑖 =𝜎𝑖

2⊗ 12×2,

𝑅𝑖 = 12×2 ⊗𝜎𝑖

2

or

(𝐽𝜌𝜎)𝜇𝜈 = 𝑖(𝜂𝜎𝜇𝛿𝜌

𝜈 − 𝜂𝜌𝜇𝛿𝜎𝜈)

Table 5 – Important finite-dimensional representations of the Lorentz group. 𝜎𝑖 are the Pauli matrices (see appendix).

Other important representations (Majorana, Dirac) can be obtained from the spinorial representations.

Infinite-dimensional representations, space of functions Here, the generators need to be operators (we will use script letters to distinguish from the generators of finite

dimensional representations). A trick can be used to obtain the expression of the generators. When the latter are


12

found, the group elements are obtainable via exponentiation as usual. We use (1) to express 𝑚−1 in the fundamental

representation,

𝜙(𝑚−1(𝑥)) = 𝜙 ((Λ−1)𝜇𝜈(𝑥𝜈 − 𝑐𝜈)),

and (3) gives

𝜙(𝑚−1(𝑥)) = 𝜙 ((exp+𝑖

2𝜔𝜌𝜎𝐽𝜌𝜎)

𝜇

𝜈(𝑥𝜈 − 𝑐𝜈))

= 𝜙 ((𝛿𝜈𝜇

+𝑖


𝜈 + ⋯)(𝑥𝜈 − 𝑐𝜈))

= 𝜙 (𝑥𝜇 − 𝑐𝜇 +𝑖


𝜈𝑥𝜈 + ⋯).

A Taylor expansion of 𝜙 about 𝑥𝜇 yields

𝜙(𝑚−1(𝑥)) ≈ 𝜙(𝑥𝜇) − 𝑐𝜇𝜕𝜇𝜙(𝑥𝜇) +𝑖


𝜈𝑥𝜈𝜕𝜇𝜙(𝑥𝜇) +

1

2(𝑐𝜇𝜕𝜇)

2𝜙(𝑥𝜇) + ⋯

= exp (−𝑐𝜇𝜕𝜇 +𝑖


𝜈𝑥𝜈𝜕𝜇)𝜙(𝑥𝜇)

so, using (5) and (6), we can readily identify the operator representation of 𝑃𝜇 and 𝐽𝜇𝜈:

−𝑖𝑐𝜇𝒫𝜇 ≡ −𝑐𝜇𝜕𝜇 ⇒ 𝒫𝜇 ≡ −𝑖𝜂𝜇𝜌𝜕𝜌.

−𝑖

2𝜔𝜇𝜈𝒥

𝜇𝜈 ≡𝑖


𝜈𝑥𝜈𝜕𝜇 ⇒ 𝒥𝜌𝜎 ≡ −(𝐽𝜌𝜎)𝜇

𝜈𝑥𝜈𝜕𝜇 = −𝑖(𝑥𝜌𝜂𝜎𝜇𝜕𝜇 − 𝑥𝜎𝜂𝜌𝜇𝜕𝜇)

Note that the 𝑐𝜇 introduced with 𝒫𝜇 corresponds exactly to the 𝑐𝜇 that we had before; hence the same notation. This

is because they both correspond to the Poincaré transformation 𝑔 and because of the 4-vector interpretation given

to the coefficient of the generators (see just before table 2).

We could verify that the commutation relations (table 1) are respected. Table 6 collects our new results.

Translations Lorentz transformations

𝒫𝜇 = −𝑖𝜂𝜇𝜈𝜕𝜈 = −𝑖𝜕𝜇 𝒥𝜇𝜈 = 𝑥𝜇𝒫𝜈 − 𝑥𝜈𝒫𝜇

Table 6 – Generators of an important infinite-dimensional representations of the Poincaré group

~ ~ ~

Before concluding, let us notice that representations of the Lorentz group appeared in both the finite and infinite

cases. Renaming 𝐽𝜇𝜈 ≡ 𝑆𝜇𝜈 and 𝒥𝜇𝜈 ≡ ℒ𝜇𝜈 and reusing our notations from the beginning of this section, we have,

for a Lorentz transformation 𝑔,

𝜌(𝑔) = 𝑒−𝑖

2𝜔𝜇𝜈𝑆𝜇𝜈

and 𝜚(𝑔) = 𝑒−𝑖

2𝜔𝜇𝜈ℒ𝜇𝜈

, so


𝜌(𝑔)𝜚(𝑔)[𝜙𝑏(𝑥)] = 𝑒−𝑖2𝜔𝜇𝜈𝔍𝜇𝜈

[𝜙𝑏(𝑥)], where

𝔍𝜇𝜈 ≡ 𝑆𝜇𝜈 + ℒ𝜇𝜈 .

This is mostly a symbolic expression, but it links beautifully to the notion of total angular momentum from

Quantum Mechanics, which is the sum of the spin and orbital angular momentums.


13

5 Acknowledgements As a concluding remark, I want to thank Johannes Walcher for encouraging me to work on the Poincaré group. This

project allowed me to organize much more neatly many Poincaré-related ideas and gave rise to some new questions

(many of which being disseminated in this text) that will need to be answered in the upcoming months. I also thank

him for helping me understanding the rotation-related Lie groups (in the appendix).

6 Appendix ‒ Rotation We review here how the notion of rotation in ℝ3 connects to Lie groups and Lie algebra. This is a key discussion

for understanding the representation theory of the Poincaré group.

SO(3) and SU(2)

The most natural Lie group representing rotations in ℝ3 is the group of proper operations preserving the usual length

of ℝ3; SO(3). In the canonical basis of ℝ3,

SO(3) ≅ {𝑔 ∈ Mat3(ℝ) | 𝑔𝑇𝑔 = 1, det 𝑔 = 1}.

Using a 𝑔 = 1 + 𝜖𝑎 infinitesimally close to the identity ( 𝜖 ∈ ℝ, 𝜖 ≪ 1 ), we obtain constraints on a matrix

representation of 𝔰𝔬(3) ∋ 𝑎:

1 = 𝑔𝑇𝑔 = (1 + 𝜖𝑎𝑇)(1 + 𝜖𝑎) = 1 + 𝜖(𝑎 + 𝑎𝑇) ⇒ 𝑎𝑇 = −𝑎

1 = det 𝑎 = det(1 + 𝜖𝑎) = 1 + 𝜖 tr(𝑎) ⇒ tr(𝑎) = 0

𝔰𝔬(3) ≅ {𝑎 ∈ Mat3(ℝ) | 𝑎𝑇 = −𝑎, tr 𝑎 = 0}

A choice of generators so that 𝐺 ∋ 𝑔 = exp(−𝑖𝜃𝑖𝐽𝑖) is thus

𝐽1 = (0 0 00 0 −𝑖0 𝑖 0

) 𝐽2 = (0 0 𝑖0 0 0−𝑖 0 0

) 𝐽3 = (0 −𝑖 0𝑖 0 00 0 0

)

which obey

[𝐽𝑖, 𝐽𝑗] = 𝑖𝜖𝑖𝑗𝑘𝐽𝑘.

An analogous reasoning on

SU(2) ≅ {𝑔 ∈ Mat2(ℂ) | 𝑔†𝑔 = 1, det 𝑔 = 1} gives

𝔰𝔲(2) ≅ {𝑎 ∈ Mat2(ℂ) | 𝑎† = −𝑎, tr 𝑎 = 0} = {(𝜃3 𝜃1 − 𝑖𝜃2

𝜃1 + 𝑖𝜃2 −𝜃3) | 𝜃𝑖 ∈ ℝ}

A set of generator is (the Pauli matrices)

𝜎1 = (0 11 0

) 𝜎2 = (0 −𝑖𝑖 0

) 𝜎3 = (1 00 −1

)

which obey

[𝜎𝑖, 𝜎𝑗] = 2𝑖𝜖𝑖𝑗𝑘𝜎𝑘


14

The identification 𝐽𝑖 ⟷𝜎𝑖

2 shows that

Proposition 7

𝔰𝔲(2) ≅ 𝔰𝔬(3)

SU(2) and SL(2,ℂ) The group

SL(2, ℂ) ≅ {𝑔 ∈ Mat2(ℂ) | det𝑔 = 1} has Lie algebra

𝔰𝔩(2, ℂ) ≅ {𝑔 ∈ Mat2(ℂ) | tr 𝑎 = 0} = {(𝐴 + 𝑖𝐵 𝐶 + 𝑖𝐷𝐸 + 𝑖𝐹 −𝐴 − 𝑖𝐵

) |𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ∈ ℝ}.

Meanwhile,

𝔰𝔲(2)ℂ ≡ 𝔰𝔲(2) ⊗ ℂ ≅ {(𝑎 + 𝑖𝛼 𝑏 − 𝑖𝑐 + 𝑖(𝛽 − 𝑖𝛾)

𝑏 + 𝑖𝑐 + 𝑖(𝛽 + 𝑖𝛾) −𝑎 − 𝑖𝛼) | 𝑎, 𝑏, 𝑐, 𝛼, 𝛽, 𝛾 ∈ ℝ}

≅ {(𝑎 + 𝑖𝛼 (𝑏 + 𝛾) + 𝑖(𝛽 − 𝑐)

(𝑏 − 𝛾) + 𝑖(𝛽 + 𝑐) −𝑎 − 𝑖𝛼) | 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑑 ∈ ℝ}

So there is an obvious equality

Proposition 8

𝔰𝔩(2, ℂ) = 𝔰𝔲(2)ℂ

7 References [1] Schutz, B., A First Course in General Relativity, 2nd edition, Cambridge University Press, Cambridge, 2009.

[2] Moore, G. and Burgess Cliff, The Standard Model: A primer, Cambridge University Press, Cambridge, 2007.

[3] Peskin, M. and Schroeder, D., An introduction to Quantum Field Theory, Westview, 1995.

[4] Drake, K. et al., Representations of the Symmetry Group of Spacetime, URL: http://pages.cs.wisc.edu/~guild/symmetrycompsproject.pdf

[5] Müller-Kirsten, H. and Wiedemann, A., Supersymmetry, World Scientific, Singapore, 1987.

[6] Weinberg, S., Quantum Theory of Fields, Vol. 1, Cambridge University Press, Cambridge, 1995.

[7] Wigner, E., On unitary representations of the inhomogeneous Lorentz group, Ann. of Math., Vol. 40, No. 1, 1939.

[8] Murayama, H., 232A Lecture Notes: Representation Theory of Lorentz Group, URL: http://hitoshi.berkeley.edu/232A/

[9] Maciejko, J., Representations of Lorentz and Poincaré groups, URL: http://einrichtungen.ph.tum.de/T30f/lec/QFT/groups.pdf

[10] Pal, P., Dirac, Majorana and Weyl fermions, arXiv:1006.1718v2 [hep-ph], 2010

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MATH 742 A T M P REPRESENTATION THEORY OF THE POINCARÉ ... · The Poincaré algebra 𝖕 [is the Lie algebra associated with 𝑃. It is a vector space along with a bracket ⋅,⋅]∶