Tensor Methods

Tensors are great tools for doing practical calculations in SU(3) and in many other groups as well. As you will see, the idea of tensors is closely related to the idea of a wave function in quantum mechanics.

10.1 lower and upper indices

The idea starts by labeling the states of the 3 as

11/2, v'3/6) = ¹¹⁾

l-1/2, v'3/6) = ¹²⁾

IO, -1/../3) = ¹³⁾

^(10.1)

The 1, 2 and 3 are to remind you of the fact that the eigenvectors of the H1

and H2 matrices corresponding to these weights are vectors with a single non-zero entry in the first, second and third position. We have also written the indices below the line for a reason that will become clear shortly.

If we define a set of matrices with one upper and one lower index, as

follows . 1

[Ta]j

=

2[-\a]ij

^(10.2)

then the triplet of states, li), transforms under the algebra as

(10.3) The important thing to notice is that the sum over j involves one upper and one lower index.

138 DOI: 10.1201/9780429499210-11

10.2. TENSOR COMPONENTS AND WAVE FUNCTIONS ¹³⁹ Label the states of the 3 as

Then

l-1/2, -VJ/6) =

¹¹⁾

ll/2, -VJ/6) =

¹²⁾

IO, 1/v'3) =

^{13 )}

(10.4)

(10.5) This is true because the 3 is the complex conjugate representation, generated by _-

[T*]j -

a i - - a i - - aj

[TT]j - [T. ]i

(10.6)

Now I can define, as usual, a state in the tensor product of n 3s and m 3s.

It transforms as

li.l···i:m)=lil)···lim)l·)···l·)

)l"')n )1 )n

T.

Iii "·im)

a )l"')n

_ "'""' lil"'im

n

) [T. ]k

- L.. h .. ·h-lkje+l'"jn a je

l=l m

_"'""' lii"'i!-lkif+l"'im) [T.]ie

_L..

)l'")n a k

l=l

(10.7)

(10.8)

This distinction between upper and lower indices is useful because SU ( 3) has two different kinds of 3-dimensional representations - the 3 and 3. We need some way to distinguish them. Raising and lowering the indices is just a handy notational device.

10.2 Tensor components and wave functions

Now consider an arbitrary state in this tensor product space

(10.9) v is called a tensor. A tensor is just a "wave-function", because we can find v by taking the matrix element of lv) with the tensor product state.

(10.10)

140 CHAPTER 10. TENSOR METHODS The correspondence here is exactly like the relation between the space wave function of a particle in quantum mechanics and the state describing the par-ticle in the Hilbert space.

1/J(x)

=

(xi'I/J) ^(10.11) The tensor

v

is characterized by its tensor components,

vf;::j;:,.

^Now

we can think of the action of the generators on lv) as an action on the tensor components, as follows:

This defines the action of the generators on an arbitary tensor!

10.3 Irreducible representations and symmetry

(10.12)

(10.13)

We can now use the highest weight procedure to pick out the states in the ten-sor product corresponding to the irreducible representation ( n, m). Because

11} is the highest weight of the ^(1,0) representation, and ^12} is the highest weight of the (0, 1) representation, the state with highest weight in ( n, m) is

1222···}

111···

It corresponds to the tensor v ^H with components

VH~l _t1···tm ···in

= N 0

_Jl

·

1 · · •

0

_Jn

·

¹

0

₁₁

·

^{2 · · ·}

0

_1m

·

²

(10.14)

(10.15) Now we can construct all the states in (n, m) by acting on the tensor ^VH with lowering operators. The important point is that v ^H has two properties that are preserved by the transformation VH-+ TavH.

1. v ^H is symmetric in the upper indices, and symmetric in the lower in-dices.

2. v ^H satisfies

10.4. INVARIANTTENSORS 141 (10.16) The first is preserved because the generators act the same way on all the upper indices and the same way on all the lower indices. The second is preserved because of the minus sign in (10.13) and the tracelessness of theTas. All the states in (n, m) therefore correspond to tensors of this form, symmetric in both upper and lower indices, and traceless (which just means that it satisfies ( 1 0.16) which is a generalization of the condition of tracelessness for a ma-trix). It turns out that the correspondence also goes the other way. Every such tensor gives a state in ( n,

m).

10.4 Invariant tensors

The oJ~ is called an invariant tensor. An invariant tensor is one that does not change under an

SU(3)

transformation. The change in a tensor is propor-tional to the action of some linear combination of generators on it, but

(10.17) thus oJ~ doesn't change under any

SU(3)

transformation. There are two other invariant tensors in

SU(3)

-the completely antisymmetric tensors, ^f.ijk and

Eij k. These are invariant because of the tracelessness of Ta. Consider (10.18) This is completely antisymmetric, so we can look at the 123 component

-[Ta

f.l123

= [Tal}

^/23

+[Tali

^{f.l£3 +[Tal~ f.12£}

= [Tali

^{f.123 +}

[Tag

^{f.123 +}

[Tag

f.123

=

0 Thus ^f.ijk is invariant.

10.5 Clebsch-Gordan decomposition

(10.19)

We can use tensors to decompose tensor products explicitly. Suppose that u is a tensor with n upper indices and m lower indices, and v is a tensor with p upper indices and q lower indices. Then it follows from the definition of a tensor that the product, u ® v defined by the product of the tensor components (10.20)

142 CHAPTER 10. TENSOR METHODS is the tensor that describes the tensor product state

I ^u)

^®

I ^v).

Thus we can analyze tensor products by manipulating tensors -the wave-functions of the corresponding states. The general strategy for doing these decompositions is to make irreducible representations out of the product of tensors, and then express the original product as a sum of terms proportional to various irre-ducible combinations. The advantage of this procedure is that we are directly manipulating wave functions, which is often what we want to know about.

Consider, for example, 3 ® 3. If we have two 3s, ui and vi, we can write the product as

(10.21) The first term,

~

( ui vi

+

^{ui vi) (10.22)}

transforms like a 6. This contains the highest weight state u¹ v1. We have added the ui vi term to make it completely symmetric in the two upper in-dices, and thus irreducible. The lower index object

(10.23) having only one lower index, transforms like a

3.

Thus we have explicitly decomposed the tensor product into a sum of 6 and a

3.

Not only does this show that

(10.24) or

(1, 0)

®

(1, 0)

=

(2, 0) EB (0, 1)

^(10.25) it shows us how to actually build wave functions with the required symmetry properties. Later, we will see how this makes some kinds of calculations easy.

Note also how as in (10.23) whenever a tensor with more than one upper index is not completely symmetric, we can trade two upper indices for one lower index using the E tensor.

Next, let's look at 3 ®

3,

a product of ui (a 3) and vi (a

3).

We can write (10.26) The first term in parentheses is traceless, and transforms like the 8, while the tensor with no indices, ukvk, transforms like the trivial representation, (0,0), or 1. Thus

(10.27)

10.6. TRIALITY 143

or

(1, 0)

®

(0, 1) = (1, 1) EB (0, 0)

(10.28) Notice the role of the invariant tensor,

8;.

One way of understanding why only traceless tensors are irreducible is that if a tensor is not traceless, the

8

tensor can be used to construct a non-zero tensor with fewer indices, and thus explicitly reduce it.

One more example-ui (a 3) times v{ (an 8).

uiv{ =

~

(ui v1

+

^{uj vi -}

~5i

ue v1 -

~5 ¹

ue vi)

2 k k 4 k e 4 k e

+ 4E

^{1 ijl (} Eemn u vk Ekmn u Ve ^m

ⁿ +

^m

n)

(10.29)

+~ (38tuev~- 8{uev~)

where the first term on the right hand side is a (2,1) (or 15-it's 15 dimen-sional), the second term is a

6,

and the last is a 3.

3

®

8

=

15 EB 6 EB 3.

(10.30) or

(1, 0)

®

(1, 1) = (2, 1) EB (0, 2) EB (1, 0).

(10.31)

10.6 Triality

Notice that ( n-m) mod 3 is conserved in these tensor product decomposi-tions. This is true because the invariant tensors all have ( n-m) mod 3

=

0, so there is no way to change this quantity. It is called triality.

10.7 Matrix elements and operators

The bra state (vI is

( I _

V-V:: ^il··-Jn .

*

^{(i1 ···im} . .

I

~1···~m )l" .. Jn (10.32) The bra transforms under the algebra with an extra minus sign. For example, the triplet

(i I

transforms as

(10.33)

144 CHAPTER 10. TENSOR METHODS In words, this says that the bra with a lower index transforms as if it had an upper index. This is because of the complex conjugation involved in going from a bra to a ket. Similarly, the bra with an upper index transforms as if it has a lower index. This suggests that we define the tensor corresponding to a bra state with the upper and lower indices interchanged so that the contraction of indices works in (10.33). Thus we say that the tensor components of the bra tensor (vI are

so that (10.32) becomes

( I _

_{V -V·} -i1 ···im (i1 ···im JI···Jn JI···Jn _{· · ·}

I

(10.34)

(10.35) Then when the state (vi is transformed by

-(viTa.

the tensor vis transformed

byTav.

For example, consider the matrix element

(10.36)

The indices are all repeated and summed over (contracted for short), which they must be because the matrix element is invariant.

10.8 Normalization

A corollary to (10.36) is that if the state

lv}

is normalized, satisfying

(v

v} = 1, then the tensor components must satisfy a normalization condition

i1···im

h···}n

(10.37)

For example, the tensor ^VH in (10.15) satisfies (10.37), because only a single term contributes in the sum. But a tensor of the form

(10.38) describes a state with norm

j=k=l j=k=2 j=k=3

INI

²

~)2 6{ ^6k- 6~6~- 6~6~) ² = INI

^{2 (}

---:4' ⁺ '"!' + '"!' ) ^{= 6INI}

²

J,k (10.39)

10.9. TENSOR OPERATORS 145

10.9 Tensor operators

We can extend the concept of tensors to include the coefficients of tensor operators in an obvious way. For example, if a set of operators, 0, transform according to the (n, m) representation of

SU(3),

we can form the general linear combination

(10.40) then if

W

transforms by commutation with the generators, the coefficients,

wf; :::1;;,,

transform like ( n, m), (1 0.13).

10.10 The dimension of ( n, m)

A very simple application of tensor methods is the calculation of the dimen-sion of the irreducible representation ( n, m). The dimension of the repre-sentation is the number of independent components in the tensor. We know that the tensor has n upper and m lower indices, and is separately symmetric in each. The number of independent components of a object symmetric in n indices each of which run from 1 to 3 is equal to the number of way of separating n identical objects with two identical partitions - which is

( n+2)

=

^(n+2)!

=

(n+2)(n+1)

2 n!2! 2 ^(10.41)

Thus if there were no other constraint, the number of independent coefficients would be

B(n ) = (n+2)(n+

1) (m+2)(m+1)

,m

2 2

^(10.42)

However, the tensor is also required to be traceless. This says that the object we get by taking the trace vanishes, and it is symmetric in n-1 upper indices and m-llower indices. Thus this imposes

B (

n-1, m-1) constraints, so the total is

146 CHAPTER 10. TENSOR METHODS You can check that this formula works for the small irreducible representa-tions that we have discussed.

Im Dokument Lie Algebras in Particle Physics (Seite 157-165)