Pope C. Methods of theoretical physics 1

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Now , let us consider the vector triple product of any three 3-vectors

B and

C. This

is deﬁned as the vector

D, given by

D =

A × (

B ×

C) . (7.65)

From (7.62), we see that we can write the components of

D as

= ε

ijm

mkℓ

ℓ

. (7.66)

Note that

D is an ordinary vector, and not a pseudo-vector. This is because it involves

two epsilon pseudo-tensors in its deﬁnition (7.66) in terms of the vectors

B and

C, and

so the two det M factors that will arise when checking the transformation rule for

D will

multiply and give (+1), even under reﬂections.

The f act that

D is a tr ue vector is also evident from (7.60), which show s that the product

of the two eps ilon pseudo-tensors can be re-expressed in terms of products of Kronecker delta

tensors. In fact using (7.60), we can re-express (7.65) in terms of scalar products. First,

it is worth noting that because ε

ijk

has an odd number of indices, any rearrangement of

the indices that is achieved by a cyclic permutation implies an even number of index-pair

exch anges, and so it leaves the s ign of the epsilon tensor unchanged. In other words

ijk

= ε

jki

= ε

kij

. (7.67)

Therefore, it follows that we can cycle ε

mkℓ

to ε

kℓm

in (7.66) with no sign change, and then,

using (7.60), we get

= (δ

jℓ

− δ

iℓ

) A

ℓ

. (7.68)

Using the index-replacement rules for the Kronecker delta tensor, this implies

= B

− C

. (7.69)

Thus, writing it back in 3-vector notation, we have

D ≡

A × (

B ×

C) =

B (

A ·

C) −

C (

A ·

B) . (7.70)

There are many other examples of vector expressions that can be simpliﬁed using the

basic identities (7.60), or (7.58) for the epsilon tensor. The rule is that when ever an ex-

pression involves two or more vector p roduct symbols “×”, then they can be eliminated

pairwise, being replaced by scalar products. Once one is familiar with the basic structure

of (7.60), m ost expressions capable of such simpliﬁcations can be handled. It is so s imple

to derive th e results “as needed” that it is n o longer worth taking the trouble to remember

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a formula such as (7.70); it is easier to derive it as and when needed. Memorising (7.60) is

itself very simple; with the contracted index being “3’rd with 3’rd” on the epsilon tensors,

the right-hand side is the product of “1’st with 1’st” and “2’nd with 2’nd” Kronecker deltas,

minus “1’st with 2’nd” and “2’nd with 1’st.”

7.4.4 Hodge dualisation

The notation in three-dimensional Cartesian vector analysis of constructing a vector

from the vector product

C ≡

A ×

B of two vectors

A and

B is such a commonplace that

it sometimes surprises people to learn that it works only in three dimensions. The crucial

quantity involved in the constru ction of the vector product is the 3-index epsilon tensor

ijk

, and it has three indices precisely because of being in th ree dimen sions.

The notation that does generalise to an arbitrary dimension is that from any pair of

vectors A and B we can form an antisymmetric rank-2 tensor W whose components W

are deﬁned by

= A

−A

. (7.71)

In three d imen sions, we can map back and forth between W

and the vector C

deﬁned

above, by making use of the 3-index epsilon tensor:

ijk

= ε

ijk

= ε

ijk

. (7.72)

Note that this ability to map both ways can be seen using (7.60). Thus, given C

ijk

we calculate

ijk

kℓm

ℓm

(δ

iℓ

− δ

jℓ

) W

ℓm

= W

. (7.73)

So in th ree dimensions, having a 2-index antisymmetric tensor is essentially equivalent to

having a vector, since we can map freely backwards and forwards. (It is essential, of course,

that W

itself be antisymmetric in order for this invertible mapping to work.)

In higher dimensions, the nature of the mapping is diﬀerent. For example, in four

dimensions we have a 4-index epsilon tensor, and so from W

we can make another 2-index

antisymmetric tensor:

≡ ε

ijkℓ

kℓ

. (7.74)

This is again invertible, and in fact from (7.59) one can prove that

ijkℓ

kℓ

. (7.75)

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It is not so immediately obvious in four dimensions what the point of mapping fr om one

2-index antisymmetric tensor into the other would be, since one has not achieved any

reduction of the number of indices. Actually, it turns out that there are important uses for

this procedur e, and in fact a special s igniﬁ cance is attached to 2-index tensors that have

the property of mapping into themselves under this transformation.

The mapping process is kn own as Hodge Du alisation. To make the combinatorics work

nicely, it is better to put in a factorial coeﬃcient. The Hodge dual of a rank-2 antisymmetric

tensor W

in four dimensions is denoted by W

∗

, and deﬁned by

∗

ijkℓ

kℓ

. (7.76)

From this, one can show using (7.59) that

ijkℓ

∗

kℓ

. (7.77)

If a tensor happens to satisfy W

= ±W

∗

, it is called self-dual or anti-self-dual respectively.

More generally, if we are in n dimensions and we have a rank-p antisymmetric tensor

···i

, then its Hodge dual is a rank-(n − p) tensor with components T

∗

···i

n−p

given by

∗

···i

n−p

≡

···i

n−p

···j

. (7.78)

The procedure of m aking a vector

C =

A ×

B out of two vectors

A and

B in three

dimensions can n ow be und ers tood as a special case, in which one takes the Hodge dual of

the 2-index antisymmetric tensor with components A

−A

Notice, by the way, that one of our familiar concepts in three dimensions is that rotations

occur around axes. This is a very special feature of three dimensions, for precisely the

reasons we have been discussing. Think of the angular momentum vector,

L = ~r × ~p , (7.79)

for example, which, in components, would be written

= ε

ijk

. (7.80)

In a general dimension, we would instead simply view the angular momentum as a 2-index

antisymmetric tensor,

= x

−x

. (7.81)

Thus in a general dimension, a rotation occurs in a 2-plane, which is speciﬁed as the plane

in which the position vector ~r and the linear momentum vector ~p lie. It is a “coincidence”

of living in three sp atial dimensions that instead of saying “a rotation in the (x, y) plane,”

we can say “a rotation around the z axis.”

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7.5 Cartesian Tensor Calculus

The basic diﬀerential operator in vector and tensor calculus is the gradient operator ∇.

This is the vector-valued operator whose “components” are the set of partial derivatives

with respect to the Cartesian coordinates x

. For brevity, let us d eﬁ ne

∂

≡

∂

∂x

. (7.82)

Then we shall have

∇ = (∂

, ∂

, . . . , ∂

) (7.83)

in n dimensions.

We can easily see that ∇ is indeed a vector; the proof is the usual one, of s how ing that

its components transform as a vector under rotations of the Cartesian coordinates. Thus in

a rotated co ordinate system x

′

, for which, by deﬁnition, we have ∂

′

= ∂/∂x

′

, we ﬁnd, using

the chain rule,

∂

′

∂x

′

∂

. (7.84)

Now we have x

′

= M

under th e coordinate rotations, and so, multiplying by M

and

using (7.29), we have M

′

= x

. Diﬀerentiating (take care of the index choices!) we ﬁnd

∂x

′

= M

, and so we conclude that

∂

′

= M

∂

. (7.85)

This proves that ∂

transforms exactly as a vector sh ould, under rotations of the Cartesian

axes.

It is now str aightforward to see that if ∇ acts on any scalar ﬁeld φ, it will give a vector,

∇φ. In fact more generally, if ∇ acts on any rank-p tensor T , it will give a rank-(p + 1)

tensor S, with components given by

···j

= ∂

···j

. (7.86)

The proof is the usual one, of showing that S

···j

transforms with the proper tensor

transformation law (7.34) under rotations of the Cartesian coordinates. Of course, having

established that ∂

···j

is a tensor, all the usual r ules follow. In particular, for example,

it follows that we can take a divergence of the tensor T

···j

, by contracting the index i on

the derivative in (7.86) with one of the indices on T

···j

, and thereby get a rank-(p − 1)

tensor. (In general, if T

···j

has no special symmetry properties on its indices, there will

be p diﬀerent divergences that we can make, depending on which of th e j indices we choose

to contract with th e i index.)

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A special case of the above is to form the scalar quantity ∂

as the divergence of the

vector V

Note that the Laplacian operator

∇

≡ ∂

∂

∂x

+ ··· +

∂

∂x

(7.87)

is manifestly a scalar operator, and so if φ is a scalar, then so is ∇

φ.

There is a special s igniﬁcance in tensor calculus to antisymmetrised derivatives of tensors.

The most familiar example, in three dimensions, involves the antisymmetrised derivative of

a vector, ∂

− ∂

. As in our discussion of the vector pr oduct, it is then convenient to

take the Hodge dual of this, to obtain a vector. Thus one deﬁnes the curl operation, with

the curl of a vector

V being another vector (actually, of course, a pseudo-vector)

X, given

X ≡

∇ ×

V . (7.88)

In components, this is just

= ε

ijk

∂

. (7.89)

In index notation one can easily prove various 3-dimensional identities, based on the

fact that partial derivatives commute, ∂

∂

= ∂

∂

, such as

∇ ×

∇φ = 0 ,

∇ · (

∇ ×

V ) = 0 , (7.90)

for any scalar φ and any vector

V . One can also immediately see from (7.60) that

∇ × (

∇ ×

V ) =

∇(

∇ ·

V ) − ∇

V . (7.91)

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