Jeevanjee N. An Introduction to Tensors and Group Theory for Physicists

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224 6 The Wigner–Eckart Theorem and Other Applications

where the ‘∗’ denotes complex conjugation (note that the positions of the right and

left-handed spinors are switched here). You can easily check that if ψ transforms

like a Dirac spinor, i.e. ψ →D(A)ψ, then

ψ transforms like

ψ −→



−1

0 A

†



ψD(A)

−1

. (6.25)

If we consider the associated column vector

, it transforms like

−→



−1T

0 A

∗



which you should recognize as the representation dual to the Dirac spinor, since

the matrices of dual representations are just the inverse transpose of the original

matrices. However, we know from Problem 5.2 that (

, 0) and (0,

) are equivalent

to their duals, so we conclude that

ψ transforms like a Dirac spinor as well.

With

ψ in hand, we can now deﬁne the Dirac bilinears

•

ψψ scalar

•

ψγ

ψ vector

•

ψγ

ψ, μ =ν antisymmetric 2-tensor

•

ψγ

ψ, μ =ν =ρ pseudovector

•

ψγ

ψ pseudoscalar

Each of these contains a product of two Dirac spinors, and can be seen as compo-

nents of tensors living in C

⊗C

. Note the similarities between the names of the

Dirac bilinears and the ﬁrst ﬁve entries of the table from Example 5.41. This makes

it seem like the Dirac bilinears should transform like the components of antisym-

metric tensors (of ranks 0 through 4). Is this true? Well, using (6.25), we ﬁnd that

the scalar transforms like

ψ −→

ψD(A)

−1

D(A)ψ =

ψψ

and so really does transform like a scalar. Similarly, using (6.23), we ﬁnd that the

vector transforms like

ψγ

ψ −→

ψD(A)

−1

D(A)ψ

= 

−1

ψγ

and so really does transform like a vector. Using the anticommutation relation

(6.24), you can similarly verify that the antisymmetric 2-tensor, pseudovector and

pseudoscalar transform like antisymmetric tensors of ranks 2, 3, and 4, respectively.

If we let 

∗

) denote the set of all antisymmetric tensor products of R

,i.e.



∗

≡



k=0



and let 

∗

denote its complexiﬁcation, then C

⊗C

contains a subspace equiv-

alent to 

∗

. However, one can check that both spaces have (complex) dimension

16, so as sl(2, C)

representations

6.4 Problems 225

⊗C



∗

, (6.26)

i.e. the tensor product of the Dirac spinor representation with itself is equiva-

lent to the space of all antisymmetric tensors!

Another way to obtain this same result is to use our tensor product decomposition

(5.95). The tensor product C

⊗C

of two Dirac spinors is given by



, 0



⊕





⊗



, 0



⊕







, 0



⊗



, 0



⊕



, 0



⊗





⊕





⊗



, 0



⊕





⊗





=(1, 0) ⊕(0, 0) ⊕





⊕





⊕(0, 1) ⊕(0, 0)

=(0, 0) ⊕





⊕(1, 0) ⊕(0, 1) ⊕





⊕(0, 0) (6.27)

where in the ﬁrst equality we used the fact that the tensor product distributes over

direct sums (see Problem 5.11), in the second equality we used (5.95), and in the

third equality we just rearranged the summands. However, from Example 5.41 we

know that as an so(3, 1) representation, 

∗

decomposes as



∗

(0, 0) ⊕





⊕(1, 0) ⊕(0, 1) ⊕





⊕(0, 0)

which is just (6.27)!

Exercise 6.4 Verify (6.25). Also verify that the antisymmetric 2-tensor, pseudovector and

pseudoscalar bilinears transform like the components of antisymmetric tensors of rank 2, 3

and 4.

6.4 Problems

6.1 In this problem we will establish a couple of the basic properties of the orthog-

onal projection operator. To this end, let H be a Hilbert space with inner product

(·|·) and let W be a ﬁnite-dimensional subspace of H.

(a) Show that for any v ∈H, there exists a unique vector P(v)∈W such that



P(v)|w







v|w





∀w



∈W.

This deﬁnes the orthogonal projection map P : H → W which projects H

onto W . (Hint: there are a few ways to show that P(v) exists and is unique.

One route is to consider the map L : H → H

∗

given by L(v) = (v|·) and then

play around with restrictions to W .)

(b) Quickly show that P(w)=w for all w ∈W , and hence that P

=P .

226 6 The Wigner–Eckart Theorem and Other Applications

}be a (possibly inﬁnite) orthonormal basis for H where the ﬁrst k vectors

, i = 1,...,k are a basis for W . Show that if we expand an arbitrary v ∈ H

as v = v

where the implied sum is over all i (and where the sum may be

inﬁnite), then P takes the simple form

P(v)=P







i=1

Thus P can be thought of as ‘projecting out all the components orthogonal

to W ’.

(d) Let {e



} be an arbitrary ON basis for H. Show that in this case the action of P

is given by

P(e



) =



i=1



6.2 Let (, H) be a unitary representation of a group G on a Hilbert space H.In

this problem we will show that inequivalent irreducible subspaces of H are orthog-

onal.

(a) Let W ⊂H be a ﬁnite-dimensional irreducible subspace and P : H → W the

orthogonal projection operator onto W . Use the deﬁning property of P ,the

unitarity of , and the invariance of W to show that P is an intertwiner.

(b) Let V ⊂H be another irreducible subspace inequivalent to W . Restrict P to V

to get P |

:V →W , which is still an intertwiner. Use Schur’s lemma to deduce

that P |

=0, and conclude that V and W are orthogonal.

Appendix

Complexiﬁcations of Real Lie Algebras

and the Tensor Product Decomposition

of sl(2,C

Representations

The goal of this appendix is to prove Proposition 5.8 about the tensor product

decomposition of two sl(2, C)

representations. The proof is long but will intro-

duce some useful notions, like the direct sum and complexiﬁcation of a Lie al-

gebra. We will use these notions to show that the representations of sl(2, C)

are in 1–1 correspondence with certain representations of the complex Lie alge-

bra sl(2, C) ⊕ sl(2, C), and the fact that this complex Lie algebra is a direct sum

will imply certain properties about its representations, which in turn will allow us to

prove Proposition 5.8.

A.1 Direct Sums and Complexiﬁcations of Lie Algebras

In this text we have dealt only with real Lie algebras, as that is the case of greatest

interest for physicists. From a more mathematical point of view, however, it actually

simpliﬁes matters to focus on the complex case, and we will need that approach

to prove Proposition 5.8. With that in mind, we make the following deﬁnition (in

total analogy to the real case): A complex Lie algebra is a complex vector space

g equipped with a complex-linear Lie bracket [·,·] :g × g → g which satisﬁes the

usual axioms of antisymmetry

[X, Y ]=−[Y,X]∀X, Y ∈g,

and the Jacobi identity



[X, Y ],Z





[Y,Z],X





[Z,X],Y



=0 ∀X, Y, Z ∈g.

Examples of complex Lie algebras are M

n ×n matrices, and sl(n, C), the set of all complex, traceless n ×n matrices. In both

cases the bracket is just given by the commutator of matrices.

For our application we will be interested in turning real Lie algebras into complex

Lie algebras. We already know how to complexify vector spaces, so to turn a real Lie

algebra into a complex one we just have to extend the Lie bracket to the complexiﬁed

vector space. This is done in the obvious way: given a real Lie algebra g with bracket

N. Jeevanjee, An Introduction to Tensors and Group Theory for Physicists,

DOI 10.1007/978-0-8176-4715-5, © Springer Science+Business Media, LLC 2011

227

228 Complexiﬁcations of Real Lie Algebras

[·,·], we deﬁne its complexiﬁcation to be the complexiﬁed vector space g

=C⊗g

with Lie bracket [·,·]

deﬁned by

[1 ⊗X

+i ⊗X

, 1 ⊗Y

+i ⊗Y

]

≡1 ⊗[X

]−1 ⊗[X

]+i ⊗[X

]

where X

∈g,i=1, 2. If we abbreviate i ⊗X as iX and 1 ⊗X as X, this tidies

up and becomes

+iX

+iY

]

≡[X

]−[X

]+i



]+[X

]



This formula deﬁnes [·,·]

in terms of [·,·], and is also exactly what you would get

by naively using complex-linearity to expand the left hand side.

What does this process yield in familiar cases? For su(2) we deﬁne a map

φ :su(2)

→sl(2, C)

1 ⊗X

+i ⊗X

→X



i 0

0 i



∈su(2).

(A.1)

You will show below that this is a Lie algebra isomorphism, and hence su(2)

complexiﬁes to become sl(2, C). You will also use similar maps to show that

u(n)

gl(n, R)

gl(n, C). If we complexify the real Lie algebra sl(2, C)

,we

also get something nice. The complexiﬁed Lie algebra (sl(2, C)

)

has complex

basis

≡

(1 ⊗S

−i ⊗

), i =1, 2, 3

≡

(1 ⊗S

+i ⊗

), i =1, 2, 3

(A.2)

which we can again abbreviate

≡

−i

), i =1, 2, 3

≡

), i =1, 2, 3.

These expressions are very similar to ones found in our discussion of sl(2, C)

rep-

resentations, and using the bracket on (sl(2, C)

)

one can verify that the analogs

of (5.83) hold, i.e. that

, N

]

, M

]



k=1



ij k

, N

]



k=1



ij k

(A.3)

Careful here! When we write i

,thisisnot to be interpreted as i times the matrix

,asthis

would make the

identically zero (check!); it is merely shorthand for i ⊗

A.2 Representations of Complexiﬁed Lie Algebras 229

Notice that both Span{M

} and Span{N

} (over the complex numbers) are Lie sub-

algebras of (sl(2, C)

)

and that both are isomorphic to sl(2, C), since

Span{M

}Span{N

}su(2)

sl(2, C).

Furthermore, the bracket between an element of Span{M

} and an element of

Span{N

}is 0, by (A.3). Also, as a (complex) vector space (sl(2, C)

)

is the direct

sum Span{M

}⊕Span{N

}. When a Lie algebra g can be written as a direct sum

of subspaces W

and W

, where the W

are each subalgebras and [w

]=0for

all w

∈W

, w

∈W

, we say that the original Lie algebra g is a Lie algebra direct

sum of W

and W

, and we write g = W

⊕ W

Thus, we have the Lie algebra

direct sum decomposition



sl(2, C)



sl(2, C) ⊕sl(2, C). (A.4)

This decomposition will be crucial in our proof of Proposition 5.8.

Exercise A.1 Prove that (A.1) is a Lie algebra isomorphism. Remember, this consists of

showing that φ is a vector space isomorphism, and then showing that φ preserves brackets.

Then ﬁnd similar Lie algebra isomorphisms to prove that

u(n)

gl(n, C)

gl(n, R)

gl(n, C).

A.2 Representations of Complexiﬁed Lie Algebras and

the Tensor Product Decomposition of sl(2,C

Representations

In order for (A.4) to be of any use, we must know how the representations of a

real Lie algebra relate to the representations of its complexiﬁcation. First off, we

should clarify that when we speak of a representation of a complex Lie algebra

g we are ignoring the complex vector space structure of g; in particular, the Lie

algebra homomorphism π : g → gl(V ) is only required to be real-linear,inthe

sense that π(cX) =cπ(X) for all real numbers c.Ifg is a complex Lie algebra, the

representation space V is complex and π(cX) =cπ(X) for all complex numbers c,

then we say that π is complex-linear. Not all complex representations of complex

Lie algebras are complex-linear. For instance, all of the sl(2, C)

representations

described in Sect. 5.10 can be thought of as representations of the complex Lie

algebra sl(2, C), but only those of the form (j, 0) are complex-linear, as you will

show below.

Notice that this notations is ambiguous, since it could mean either that g is the direct sum of W

and W

as vector spaces (which would then tell you nothing about how the direct sum decomposi-

tion interacts with the Lie bracket), or it could mean Lie algebra direct sum. We will be explicit if

there is any possibility of confusion.

230 Complexiﬁcations of Real Lie Algebras

Exercise A.2 Consider all the representations of sl(2, C)

as representations of sl(2, C) as

well. Show directly that the fundamental representation (

, 0) of sl(2, C) is complex-linear,

and use this to prove that all sl(2, C) representations of the form (j, 0) are complex-linear.

Furthermore, by considering the operators N

deﬁned in (5.82), show that these are the only

complex-linear representations of sl(2, C).

Now let g be a real Lie algebra and let (π, V ) be a complex representation of g.

Then we can extend (π, V ) to a complex-linear representation of the complexiﬁca-

tion g

in the obvious way, by setting

π(X

+iX

) ≡π(X

) +iπ(X

), X

∈g. (A.5)

(Notice that this representation is complex-linear by deﬁnition, and that the operator

iπ(X

) is only well-deﬁned because V is a complex vector space.) Furthermore,

this extension operation is reversible: that is, given a complex-linear representation

(π, V ) of g

, we can get a representation of g by simply restricting π to the subspace

{1 ⊗X +i ⊗0 |X ∈g}g, and this restriction reverses the extension just deﬁned.

Furthermore, you will show below that (π, V ) is an irrep of g

if and only if it

corresponds to an irrep of g. We thus have

Proposition A.1 The irreducible complex representations of a real Lie algebra g

are in one-to-one correspondence with the irreducible complex-linear representa-

tions of its complexiﬁcation g

This means that we can identify the irreducible complex representations of g with

the irreducible complex-linear representations of g

, and we will freely make this

identiﬁcation from now on. Note the contrast between what we are doing here and

what we did in Sect. 5.9; there, we complexiﬁed real representations to get complex

representations which we could then classify; here, the representation space is ﬁxed

(and is always complex!) and we are complexifying the Lie algebra itself, to get a

representation of a complex Lie algebra on the same representation space we started

with.

Exercise A.3 Let (π, V ) be a complex representation of a real Lie algebra g, and extend it

to a complex-linear representation of g

. Show that (π, V ) is irreducible as a representation

of g if and only if it is irreducible as a representation of g

Example A.1 The complex-linear irreducible representations of sl(2, C)

As a ﬁrst application of Proposition A.1, consider the complex Lie algebra sl(2, C).

Since sl(2, C)  su(2)

, we conclude that its complex-linear irreps are just the ir-

reps (π

) of su(2)! In fact, you can easily show directly that the complex-linear

sl(2, C) representation corresponding to (π

) is just (j, 0). 

As a second application, note that by (A.4) the complex-linear irreps of

sl(2, C) ⊕sl(2, C) are just the representations (j

, j

) coming from sl(2, C)

. Since

sl(2, C) ⊕ sl(2, C) is a direct sum, however, there is another way to construct

complex-linear irreps. Take two complex-linear irreps of sl(2, C),say(π

)

A.2 Representations of Complexiﬁed Lie Algebras 231

and (π

). We can take a modiﬁed tensor product of these representations such

that the resulting representation is not of sl(2, C) but rather of sl(2, C) ⊕sl(2, C).

We denote this representation by (π

⊗π

⊗V

) and deﬁne it by

(π

⊗π

)(X

) ≡π

) ⊗I +I ⊗π

) ∈L(V

⊗V

∈sl(2, C). (A.6)

Note that we have written the tensor product in π

⊗π

as ‘

⊗’ rather than ‘⊗’;

this is to distinguish this tensor product of representations from the tensor product

of representations deﬁned in Sect. 5.3. In the earlier deﬁnition, we took a tensor

product of two g representations and produced a third g representation given by

(π

⊗π

)(X) =π

(X) ⊗ I +1 ⊗π

(X), where the same element X ∈ g gets fed

into both π

and π

; here, we take a tensor product of two g representations and

produce a representation of g ⊕g, where two different elements X

∈g get fed

into π

and π

Now, one might wonder if the representation (π

⊗π

⊗V

) of sl(2, C) ⊕

sl(2, C) deﬁned above is equivalent to (j

, j

); this is in fact the case! To prove this,

recall the following notation: the representation space V

⊗V

is spanned by vec-

tors of the form v

⊗v

, k

=0,...,2j

, i =1, 2 where the v

are characterized

by (5.76). Similarly, the representation space V

)

of (j

, j

) is spanned by vectors

of the form v

, k

= 0,...,2j

, i = 1, 2, where these vectors are characterized

by (5.87). We can thus deﬁne the obvious intertwiner

φ :V

⊗V

→V

)

⊗v

→v

Of course, we must check that this map actually is an intertwiner, i.e. that

φ ◦



(π

⊗π

)(X

)



=π

)

) ◦φ ∀X

∈sl(2, C). (A.7)

Since sl(2, C) ⊕sl(2, C) is of (complex) dimension six, it sufﬁces to check this for

the six basis vectors

(iM

, 0)

(0,iN

)

, 0) ≡(iM

−M

, 0)

−

, 0) ≡(iM

, 0)

(0, N

) ≡(0,iN

−N

)

(0, N

−

) ≡(0,iN

where the M

and N

were deﬁned in (A.2). We now check (A.7)for(iM

, 0)

on an arbitrary basis vector v

⊗ v

, and leave the veriﬁcation for the other ﬁve

sl(2, C) ⊕ sl(2, C) basis vectors to you. The left hand of (A.7) gives (careful with

all the parentheses!)

232 Complexiﬁcations of Real Lie Algebras



(π

⊗π

)(iM

, 0)(v

⊗v

)



=φ



⊗I )(v

⊗v

)



=φ



−k

⊗v



=(j

−k

while the right hand side is

)

(iM

, 0)



φ(v

⊗v

)





)

(iM

, 0)



)

=iM

=(j

−k

and so they agree. The veriﬁcation for the other ﬁve sl(2, C) ⊕sl(2, C) basis vectors

proceeds similarly. This proves the following proposition:

Proposition A.2 Let (j

, j

) denote both the usual sl(2, C)

irrep as well as

its extension to a complex-linear irrep of (sl(2, C)

)

 sl(2, C) ⊕ sl(2, C). Let

(π

⊗π

⊗V

) be the representation of sl(2, C) ⊕sl(2, C) deﬁned in (A.6).

Then

, j

) (π

⊗π

⊗V

). (A.8)

Exercise A.4 Verify (A.7) for the other ﬁve sl(2, C) ⊕ sl(2, C) basis vectors, and thus

complete the proof of Proposition A.2.

We will now use Proposition A.2 to compute tensor products of the (sl(2, C)

)

irreps (j

, j

), which by Proposition A.1 will give us the tensor products of the

, j

) as sl(2, C)

irreps, which was what we wanted! In the following computa-

tion we will use the fact that

(π

⊗π

) ⊗(π

⊗π

) (π

⊗π

)

⊗(π

⊗π

), (A.9)

which you will prove below (note which tensor product symbols are ‘barred’ and

which are not). With this in hand, and using the su(2)

 sl(2, C) tensor product

decomposition (5.68), we have (omitting the representation spaces in the computa-

tion)

, j

) ⊗(k

, k

) (π

⊗π

) ⊗(π

⊗π

)

(π

⊗π

)

⊗(π

⊗π

)





=|j

−k



⊗





=|j

−k







)

⊗π

where |j

−k

|≤l

≤j

,i=1, 2





)

, l

) where |j

−k

|≤l

≤j

,i=1, 2.

This gives the tensor product decomposition of complex-linear (sl(2, C)

)

irreps.

However, these irreps are just the extensions of the irreps (j

, j

) of sl(2, C)

, and

A.2 Representations of Complexiﬁed Lie Algebras 233

you can check that the process of extending an irrep to the complexiﬁcation of a Lie

algebra commutes with taking tensor products, so that the extension of a product is

the product of an extension. From this we conclude the following:

Proposition A.3 The decomposition into irreps of the tensor product of two

sl(2, C)

irreps (j

, j

) and (k

, k

) is given by

, j

) ⊗(k

, k

) =



)

where |j

−k

|≤l

≤j

, |j

−k

|≤l

≤j

which is just Proposition 5.8.

Exercise A.5 Prove (A.9) by referring to the deﬁnitions of both kinds of tensor prod-

uct representations and by evaluating both sides of the equation on an arbitrary vector

) ∈sl(2, C) ⊕sl(2, C).