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TRANSCRIPT
Final project (McGill, 14/01/20) Marc-Antoine Fiset (260539607)
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MATH 742 β ADVANCED TOPICS IN MATHEMATICAL PHYSICS REPRESENTATION THEORY OF THE POINCARΓ GROUP
FINAL PROJECT 1 Introduction ........................................................................................................................................................1
2 The PoincarΓ© Group ............................................................................................................................................2
3 The PoincarΓ© Algebra .........................................................................................................................................4
3.1 Casimir elements of the PoincarΓ© algebra ........................................................................................................6
4 Representations of the PoincarΓ© Group ..............................................................................................................7
4.1 Hilbert spaces ...................................................................................................................................................7
4.2 Space of classical fields ..................................................................................................................................10
5 Acknowledgements ..........................................................................................................................................13
6 Appendix β 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 βthat
the world has the topology of β4β, the theory stems from two easy to grasp (although maybe hard to believe)
physical axioms [1]:
Axioms of Special Relativity:
β The speed of light is constant. It has the same value, π = 1 in some system of units, no
matter in which condition whoever measures it.
β 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 βall of
which being confirmed by experimentsβ 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 meβ¦ 2 A system of coordinates (or a frame of reference) is a set of four real numbers π₯ = (π₯0, π₯1, π₯2, π₯3), where π₯0 β‘ π‘ is a measure
of the time elapsed since a certain event and where π₯1, π₯2, π₯3 are usual distances from a chosen origin along three orthogonal
axes.
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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 PoincarΓ© group from first principles and some fundamental remarks
about it. We then study the PoincarΓ© 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 Wignerβs 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 PoincarΓ© 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 PoincarΓ©-related ideas than to prove very precise statements.
2 The PoincarΓ© Group Of course, several different systems of four real numbers π₯π , π β {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
Ξπ 2 β‘ β(Ξπ₯0)2 + β(Ξπ₯π)2
3
π=1
between two points π₯ and π₯ + Ξπ₯ 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 βgeneralized3 distance between π₯ and π¦β, it suggests to identify our spacetime with a βgeneralized
metric spaceβ. The latter is properly called Minkowski spacetime π΄. It is a space that has the topology of β4 and
its vector space structure along with the Minkowski metric (or quadratic form), a map such that
π βΆ π Γ π β (π₯, π¦) βΌ βπ₯0π¦0 + βπ₯ππ¦π
3
π=1
β β.
The output of this map is the inner product of π₯ and π¦. It is often denoted ππππ₯ππ¦π , where a sum on π, π is
understood (as always in the present text) and where
πππ β‘ diag(β1 1 1 1).
Note that the interval defined above is just the inner product of Ξπ₯ with itself.
Let us finally note that an isometry (or a symmetry) on π is defined just as in the case of conventional metric spaces:
it is a map from π to itself that preserves the distances. These preliminaries now allow us to define the PoincarΓ©
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 βgeneralized lengthβ) has the same value no matter in
which frame it is measured.
3 I use βgeneralizedβ because distances are usually assumed to be positive-definite in mathematics.
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PoincarΓ© group π·
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
π is a non-compact Lie group. [2]
Without loss of generality4 [3], an element π of the PoincarΓ© group is always assumed to be a linear transformation
of the form
π₯π βΆπ
π₯β²π β‘ Ξπππ₯
π + ππ , (1)
where Ξππ are the entries of a 4-matrix representing5 a so-called Lorentz transformation and where ππ are the
entries a constant 4-vector representing a translation in spacetime. Since the interval must be conserved, we have a
constraint on Ξππ (but not on ππ):
πππΞπ₯πΞπ₯π = πππΞπ₯β²πΞπ₯β²π = πππΞππΞ
ππΞπ₯πΞπ₯π β Ξπ₯
β πππΞππΞ
ππ = πππ . (2)
This mimics the fact that the PoincarΓ© group is the direct product [4] of the Lorentz group π³ = π(π, π) (the subgroup
of π leaving the origin fixed) and the group of translations β3,1:
Proposition 2
π = O(1,3) β β1,3. 6
As seen in problem set 2, O(1,3) can be separated into four connected components according to the determinant of
Ξππ, a matrix representing an element of O(1,3), and to the sign of Ξ0
0. Given the simple relation with π, the same
is true in the case of π:
π β {
π+β = πΏ+
β β β1,3 = ISOβ(1,3), Proper (det Ξ = 1) orthochronous (Ξ00 > 0) transf.
πββ = πΏβ
β β β1,3, Improper (det Ξ = β1) orthochronous (Ξ00 > 0) transf.
π+β = πΏ+
β β β1,3, Proper (det Ξ = 1) non-orthochronous (Ξ00 < 0) transf.
πββ = πΏβ
β β β1,3, Improper (detΞ = β1) non-orthochronous (Ξ00 < 0) transf.
4 Suppose otherwise there were an order-two term Ξπ
πππ₯ππ₯π in the Taylor expansion of the transformation. Plugging in (2)
gives that Ξπππ = 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 π . 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.
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Of these subspaces, only π+β β β1,3 = ISOβ(1,3) is a subgroup because it
contains the identity of π. It can be shown that any element of the other
subspaces are obtainable from an element of SOβ(1,3) β β1,3 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 PoincarΓ© 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 π. A
rotation is a proper transformation that leaves the time fixed, so it is
orthochronous. Hence, SO(3) β πΏ+β . Similarly, translations in β3 form a
subgroup of β1,3.
3 The PoincarΓ© Algebra The PoincarΓ© algebra π is the Lie algebra associated with π. It is a vector space along with a bracket [β ,β ] βΆ π Γ π βπ, which can be specified by its action on a basis of vectors called the generators of π. 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 π½ππ, ππ π½π, πΎπ, ππ, π» πΏπ, π π, ππ, π»
Com
muta
tion r
elati
ons
[π½ππ , π½ππ] = π(ππππ½ππ + ππππ½ππ
β ππππ½ππ β ππππ½ππ)
[π½ππ , ππ] = π(πππππ β πππππ)
[ππ , ππ] = 0
[π½π, π½π] = ππππππ½π
[π½π, πΎπ] = ππππππΎπ
[πΎπ, πΎπ] = βππππππ½π
[π½π, ππ] = πππππππ
[πΎπ, ππ] = ππ»πΏππ
[ππ, ππ] = [ππ, π»] = 0
[π½π, π»] = 0
[πΎπ, π»] = πππ
[πΏπ, πΏπ] = ππππππΏπ
[π π, π π] = ππππππ π
[πΏπ, π π] = 0
[πΏπ, ππ] =1
2(ππππππ
π β π»πΏππ)
[π π, ππ] =1
2(ππππππ
π + π»πΏππ)
[πΏπ, π»] = βππ
2
[π π, π»] =ππ
2
No
tati
on
πππ β‘ πππ π½π β‘1
2πππππ½
ππ
πΎπ β‘ π½0π
ππππ is the sign of the
permutation of πππ
π, π, π β {1,2,3}
πΏπ β‘1
2(π½π + ππΎπ)
π π β‘1
2(π½π β ππΎπ)
Table 1 β Commutation relations of π expressed in three important sets of generators
7 This set is actually valid for the complexification π°π¬(1,3)β β‘ π°π¬(1,3) β β of the Lorentz algebra because πΏπ and π π will be
defined as complexified Lorentz generators.
π³+β
πΏββ
πΏββ πΏ+
β
ππ,π
π³ π·
Sketch of the most important
subspaces and subgroups (in
bold) of π
ππ(π)
ππ
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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 Ξππ = πΏπ
π + πππ (ππ
π βͺπΏπ
π) and (2) forces on it the following constraint:
πππ(πΏππ + ππ
π)(πΏππ + ππ
π) = πππ
πππ(πΏπππ
ππ + ππ
ππΏππ) = 0 to first order
πππ = βπππ where ππππππ β‘ πππ
The elements of the Lie algebra of πΏ 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 PoincarΓ© algebra has dimension 10. A generic element of π°π¬(1,3) and π―1,3 (Lie
algebras for the Lorentz and translations groups) is typically written as follows.
βπ
2ππππ½
ππ β π°π¬(1,3) β π βπππππ β π―1,3 β π
(πππ , ππ β β,
π½ππ = βπ½ππ and
πππ = βπππ)
(3)
Note that these are abstract vectors which do not depend on the defining representation βthe latter was only useful
to obtain the antisymmetry of πβ. The factor of a half is for later convenience and the π 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 πππ or ππ 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 PoincarΓ© algebra is well defined at this point but physicists prefer using the set of generators9
π½π β‘
1
2πππππ½
ππ, πΎπ β‘ π½0π, ππ , π» β‘ π0, (4)
as they can be given a physical interpretation. The π½π have the commutation relations of rotation-related Lie algebras
(see table 1 and especially appendix 1), so the interpretation is immediate. Similarly the ππ are quite obviously
generating translations in β3 (because of how they arise). Detailing convincingly the interpretation for the
generators and the group elements associated with πΎπ and π» would take us away from our main concern, but let us
nevertheless record in table 2 the important elements on the PoincarΓ© group along with their generator and their
physical interpretation. The following change of variables occurred:
8 This remark requires material to be introduced latter. ππ
π is usually denoted πππ even though the profound signification is
not the same. Here is why:
πππ = β
π
2πππ(π½ππ)π
π = βπ
2ππππ(ππππΏπ
π β ππππΏππ) =
1
2(ππ
π β πππ) = βππ
π.
9 The definition of π½π is equivalent to π½1 = π½23, π½2 = π½31, π½3 = π½12 because of the factor of a half.
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πππ = (
0 π1 π2 π3βπ1 0 π3 βπ2
βπ2 βπ3 0 π1
βπ3 π2 βπ1 0
) ππ = πππππ = (
π‘π1π2
π3
)
Group element Rotation πβποΏ½ββοΏ½ β π½ ,
π β β3
Boost πβποΏ½βοΏ½ β οΏ½ββοΏ½ ,
π β β3
Translation πβππ β οΏ½βοΏ½ ,
π β β3
Inverse10 time-evolution
πππ‘π»,
π‘ β β
Generator11 Angular
momentum π½ β Momentum οΏ½βοΏ½ Energy π»
Table 2 β 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
πΏπ β‘1
2(π½π + ππΎπ), π π β‘
1
2(π½π β ππΎπ), ππ , π».
Let us focus on the Lorentz algebra. Since we are using complexified vectors, the set is actually generating the
complexified version of π°π¬(1,3), which is π°π¬(1,3)β β‘ π°π¬(1,3) β β. 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
π°π¬(1,3)β β π°π²(2)β β π°π²(2)β
3.1 Casimir elements of the PoincarΓ© algebra The PoincarΓ© algebra π 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 π·π β‘ πππππππ
πΎπ = πππππππ
where ππ β‘1
2ππππππππ½ππ is the Pauli-Ljubanski pseudo-vector
Interpretation Mass Spin (and mass)
Table 3 β Casimir elements of the PoincarΓ© algebra
10 The time evolution operator πβππ‘π» 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 π½ β‘ (π½1 π½2 π½3); οΏ½ββοΏ½ β‘ (πΎ1 πΎ2 πΎ3); οΏ½βοΏ½ β‘ (π1 π2 π3). 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.
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Proof
[ππ , π2] = 0 is trivial
[π½ππ , π2] = π½πππππππππ β ππππππππ½ππ
= πππ(πππ½ππ + [π½ππ , ππ])ππ β ππππππππ½ππ
= πππ(πππ½ππ + π(πππππ β πππππ))ππ β ππππππππ½ππ
= ππππππ½ππππ + πππππππππππ β πππππππππππ β ππππππππ½ππ
= πππππ(πππ½ππ + [π½ππ , ππ]) + ππΏππππππ β ππΏππππππ β ππππππππ½ππ
= πππππ(πππ½ππ + π(πππππ β πππππ)) β ππππππππ½ππ
= ππππππππ½ππ + πππππππππππ β πππππππππππ β ππππππππ½ππ
= ππΏππππππ β ππΏππππππ
= 0
The proof that π2 is a Casimir is more tedious. It is detailed in [5].
β
Proposition 4
ππππ = 0
Proof
ππππ =1
2ππππππππ½ππππ
=1
2πππππππ(πππ½ππ + [π½ππ , ππ])
=1
2πππππππ(πππ½ππ + ππππππ β ππππππ)
=1
4ππππππππππ½ππ +
1
4ππππππππππ½ππ +
π
2ππππππππππππ β
π
2ππππππππππππ
=1
4ππππππππππ½ππ β
1
4ππππππππππ½ππ +
π
2ππππππππππππ β
π
2ππππππππππππ
= 0
β
4 Representations of the PoincarΓ© Group I was disappointed to find no reference trying to be absolutely exhaustive in describing the representations of π.
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 π βΆ π β GL(π) a PoincarΓ© transformation π β π
on some vector space π. In one case, the space is a Hilbert space arising in Quantum Field Theory (βsee below for
clarificationsβ). In the other case, π 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 πππ, π β β) by an element |πβ© of a Hilbert space π³ (a certain complex vector space). The Hilbert space is endowed with an inner product β¨β | β β© βΆ β Γ β βΆ β with these properties [6]
β¨π|πβ© = β¨π|πβ©β
β¨π|ππ1 + ππ2β© = πβ¨π|π1β© + πβ¨π|π2β© β¨ππ1 + ππ2|πβ© = πββ¨π1|πβ© + πββ¨π2|πβ©
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β¨π|πβ© β₯ 0.
(There are other technical conditions for β to be a Hilbert space.) The probability for the system in state |πβ© to be
measured in a state |ππβ© is
π«(π β ππ) = |β¨ππ|πβ©|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 (Wignerβs theorem)
A PoincarΓ© transformations π is represented on the Hilbert space by an operator π = π(π) = exp(ππ) (π β π)
that is either
(i) Unitary, β¨π|πβ© = β¨ππ|ππβ©, and linear π(ππ + ππ) = ππ(π) + ππ(π) or
(ii) Antiunitary, β¨π|πβ©β = β¨ππ|ππβ©, and antilinear π(ππ + ππ) = πβπ(π) + πβπ(π)
According to Weinberg [6], any PoincarΓ© 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
π, which will give, via the exponential map, representations of π. Since we want π(π) to be a unitary operator on a
Hilbert space, the corresponding generator π has to be Hermitian:
1 = π(π)β π(π) = exp(ππ)β exp(ππ) = exp(ππ β ππβ ) β π = πβ if [π, πβ ] = 0
~ ~ ~
The rest of this subsection describes Wignerβs classification [6, 7] of irreducible infinite-dimensional unitary
representations of π. We first use the eigenstates |ππβ© of ππ (the generators defined in section 3) as a basis of the
Hilbert space13. We assume here that ππ|ππβ© = ππ|ππβ©, where ππ β β4 is regarded as a 4-vector.
A trick of the classification consists of writing ππ as a standard momentum ππ via a Lorentz transformation:
ππ = Ξπππ
π.
This simplifies the problem as, in a sense, we record only the βessential featuresβ of ππ by writing identicaly every
βsimilar enoughβ eigenvalues. It turns out that the only functions of ππ independent of Lorentz transformations are
π2 = πππππππ (obviously) and the sign of π0 in the case of ππππ
πππ β€ 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 ππ Little group Interpretation
βΆ (a) π2 < 0 π0 > 0 (π, 0,0,0) SO(3) Particle of mass π and spin π = 0, 1 2β , 1, β¦
(b) π2 < 0 π0 < 0 (βπ, 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 choiceβ¦
Final project (McGill, 14/01/20) Marc-Antoine Fiset (260539607)
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βΆ (c) π2 = 0 π0 > 0 (π , π , 0,0) ISO(2) Massless particle with unconstrained helicity
(d) π2 = 0 π0 < 0 (βπ , π , 0,0) ISO(2)
(e) π2 > 0 (0, π, 0,0) SO(1,2) Tachyon
βΆ (f) ππ = 0 (0,0,0,0) SO(1,3) Vacuum
Table 4 β Standard momentum and little groups [6]
Subgroups of the Lorentz group leaving ππ 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 π 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. Schurβs 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
π2|ππβ© = πππππππ|ππβ© = ππππ
πππ = βπ2,
which has the interpretation of a mass squared because of the interpretation given to ππ 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 ππππ = 0,
ππππ|ππβ© = ππππ|ππβ© = βππ0|ππβ© = 0,
so π0 = 0. The other components of ππ are
ππ =1
2ππ0πππ
0π½ππ
such that
π2 = πππβ²ππππβ² =1
4πππβ²ππ0ππππβ²0πβ²πβ²π0π½πππ0π½π
β²πβ²
=π2
4πππβ²π0ππππ0πβ²πβ²πβ²π½πππ½π
β²πβ²= π2 β(
1
2πππππ½
ππ)2
π
= π2|π½ |2,
using (4). If the Hilbert space β is written in the basis of eigenvectors |ππβ© of |π½ |2, then the representation theory of
π°π¬(3) β π°π²(2) (seen in class) yields
π2|ππβ© = π2|π½ |2|ππβ© = π2π (π + 1)|ππβ©
where π = 0, 1 2β , 1, 3 2β ,β¦ 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?
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Note that it is not entirely surprising that π°π¬(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 π½1 here as can be deduced from [8].) It coincides here with the definition of the helicity operator of the particle
β =π½ β οΏ½βοΏ½
|οΏ½βοΏ½ |
(because π1 = π 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 PoincarΓ© 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 π:
π βΆ π β π₯ βΌ π(π₯),
where π(π₯) is in general a finite-dimensional tensor even though we focus here on vector fields ππ(π₯). Under a
PoincarΓ© transformation π (active transformation), a vector field is expected to transform as
ππ(π₯) βΆπ
ππππ
π(πβ1(π₯)), πππ β Mat4, πβ1 βΆ π β π
i.e. the transformed field at π₯π depends linearly on the initial field evaluated at the untransformed point15. The matrix
πππ quite obviously represents a PoincarΓ© transformation (ππ
π = π(π) for a certain representation map π), so we
just described a physical reason for finding the irreducible finite-dimensional representations of π, especially on
the space of vectors.
What about the function πβ1? In a certain system of coordinates, π₯ is actually a 4-vector π₯π, so πβ1 can be thought
of as a map from β4 to itself. It is also a finite-dimensional representation of the same PoincarΓ© transformation π.
However, π₯ is only the argument of a function and we would prefer to have a representation π(π) acting on the
function itself, i.e.
ππ(π₯) βΆπ
π(π)π(π)[ππ(π₯)] β‘ π(π)ππ(πβ1(π₯)) . (5)
15 To understand the appearance of the inverse PoincarΓ© transformation in the argument, it is useful to consider the example of
a rotation by π 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: π(π₯) βΆ πβ²(π₯). However, the new field is really just the old field at the
untransformed coordinate: π(π₯) βΆ πβ²(π₯) = π(πβ1(π₯))
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This motivates the search for infinite-dimensional representations of π, 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 βfundamental representationβ as we used it to understand better
the Lorentz group in section 3. In particular, we found that the elements of π°π¬(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
(π½ππ)ππ = π(ππππΏπ
π β ππππΏππ). (6)
This is interesting and very important, but conceptually slightly oversimplified. A more systematic way to obtain
this representation βand other significant onesβ would be to use the decomposition from proposition 3:
π°π¬(1,3)β β π°π²(2)β β π°π²(2)β.
From the representation theory of π°π²(2)β, this tells us that the finite-dimensional irreducible representations of
π°π¬(1,3)β are labeled by a pair (π 1, π 2), π 1, π 2 = 0, 1 2β , 1, 3 2β ,β¦ of numbers that again have the interpretation of
a spin (of the field here). They are representations of dimension 2π 1 + 1 + 2π 2 + 1 = 2(π 1 + π 2 + 1). Table 5 gives
the most important special cases.
The generators of π°π²(2)β 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 (π½ππ)ππ = 01Γ1 β 01Γ1 = 01Γ1
π is called a scalar field
or a Lorentz scalar if it is
constant over spacetime.
Spinorial (1
2, 0) 2
πΏπ =ππ
2β 11Γ1 =
ππ
2,
π π = 12Γ2 β 01Γ1 = 02Γ2
ππ are called left-handed
Weyl spinors
Spinorial (0,1
2) 2
πΏπ = 01Γ1 β 12Γ2 = 02Γ2,
π π = 11Γ1 βππ
2=
ππ
2
ππ are called right-
handed Weyl spinors
Fundamental (1
2,1
2) 4
πΏπ =ππ
2β 12Γ2,
π π = 12Γ2 βππ
2
or
(π½ππ)ππ = π(ππππΏπ
π β ππππΏππ)
Table 5 β Important finite-dimensional representations of the Lorentz group. ππ 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
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found, the group elements are obtainable via exponentiation as usual. We use (1) to express πβ1 in the fundamental
representation,
π(πβ1(π₯)) = π ((Ξβ1)ππ(π₯π β ππ)),
and (3) gives
π(πβ1(π₯)) = π ((exp+π
2ππππ½ππ)
π
π(π₯π β ππ))
= π ((πΏππ
+π
2πππ(π½ππ)π
π + β―)(π₯π β ππ))
= π (π₯π β ππ +π
2πππ(π½ππ)π
ππ₯π + β―).
A Taylor expansion of π about π₯π yields
π(πβ1(π₯)) β π(π₯π) β πππππ(π₯π) +π
2πππ(π½ππ)π
ππ₯ππππ(π₯π) +
1
2(ππππ)
2π(π₯π) + β―
= exp (βππππ +π
2πππ(π½ππ)π
ππ₯πππ)π(π₯π)
so, using (5) and (6), we can readily identify the operator representation of ππ and π½ππ:
βππππ«π β‘ βππππ β π«π β‘ βππππππ.
βπ
2ππππ₯
ππ β‘π
2πππ(π½ππ)π
ππ₯πππ β π₯ππ β‘ β(π½ππ)π
ππ₯πππ = βπ(π₯ππππππ β π₯ππππππ)
Note that the ππ introduced with π«π corresponds exactly to the ππ that we had before; hence the same notation. This
is because they both correspond to the PoincarΓ© transformation π 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
π«π = βππππππ = βπππ π₯ππ = π₯ππ«π β π₯ππ«π
Table 6 β Generators of an important infinite-dimensional representations of the PoincarΓ© group
~ ~ ~
Before concluding, let us notice that representations of the Lorentz group appeared in both the finite and infinite
cases. Renaming π½ππ β‘ πππ and π₯ππ β‘ βππ and reusing our notations from the beginning of this section, we have,
for a Lorentz transformation π,
π(π) = πβπ
2ππππππ
and π(π) = πβπ
2πππβππ
, so
ππ(π₯) βΆπ
π(π)π(π)[ππ(π₯)] = πβπ2ππππππ
[ππ(π₯)], where
πππ β‘ πππ + βππ .
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.
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5 Acknowledgements As a concluding remark, I want to thank Johannes Walcher for encouraging me to work on the PoincarΓ© group. This
project allowed me to organize much more neatly many PoincarΓ©-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 β Rotation We review here how the notion of rotation in β3 connects to Lie groups and Lie algebra. This is a key discussion
for understanding the representation theory of the PoincarΓ© group.
SO(3) and SU(2)
The most natural Lie group representing rotations in β3 is the group of proper operations preserving the usual length
of β3; SO(3). In the canonical basis of β3,
SO(3) β {π β Mat3(β) | πππ = 1, det π = 1}.
Using a π = 1 + ππ infinitesimally close to the identity ( π β β, π βͺ 1 ), we obtain constraints on a matrix
representation of π°π¬(3) β π:
1 = πππ = (1 + πππ)(1 + ππ) = 1 + π(π + ππ) β ππ = βπ
1 = det π = det(1 + ππ) = 1 + π tr(π) β tr(π) = 0
π°π¬(3) β {π β Mat3(β) | ππ = βπ, tr π = 0}
A choice of generators so that πΊ β π = exp(βππππ½π) is thus
π½1 = (0 0 00 0 βπ0 π 0
) π½2 = (0 0 π0 0 0βπ 0 0
) π½3 = (0 βπ 0π 0 00 0 0
)
which obey
[π½π, π½π] = ππππππ½π.
An analogous reasoning on
SU(2) β {π β Mat2(β) | πβ π = 1, det π = 1} gives
π°π²(2) β {π β Mat2(β) | πβ = βπ, tr π = 0} = {(π3 π1 β ππ2
π1 + ππ2 βπ3) | ππ β β}
A set of generator is (the Pauli matrices)
π1 = (0 11 0
) π2 = (0 βππ 0
) π3 = (1 00 β1
)
which obey
[ππ, ππ] = 2πππππππ
Final project (McGill, 14/01/20) Marc-Antoine Fiset (260539607)
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The identification π½π β·ππ
2 shows that
Proposition 7
π°π²(2) β π°π¬(3)
SU(2) and SL(2,β) The group
SL(2, β) β {π β Mat2(β) | detπ = 1} has Lie algebra
π°π©(2, β) β {π β Mat2(β) | tr π = 0} = {(π΄ + ππ΅ πΆ + ππ·πΈ + ππΉ βπ΄ β ππ΅
) |π΄, π΅, πΆ, π·, πΈ, πΉ β β}.
Meanwhile,
π°π²(2)β β‘ π°π²(2) β β β {(π + ππΌ π β ππ + π(π½ β ππΎ)
π + ππ + π(π½ + ππΎ) βπ β ππΌ) | π, π, π, πΌ, π½, πΎ β β}
β {(π + ππΌ (π + πΎ) + π(π½ β π)
(π β πΎ) + π(π½ + π) βπ β ππΌ) | π, π, π, π, π, π β β}
So there is an obvious equality
Proposition 8
π°π©(2, β) = π°π²(2)β
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] MΓΌller-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 PoincarΓ© 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