Naber G.L. The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity

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Chapter 3

The Theory of Spinors

3.1 Representations of the Lorentz Group

The concept of a “spinor” emerged from the work of E. Cartan on the repre-

sentations of simple Lie algebras. However, it was not until Dirac employed a

special case in the construction of his relativistically invariant equation for the

electron with “spin” that the notion acquired its present name or its current

stature in mathematical physics. In this chapter we present an elementary

introduction to the algebraic theory of spinors in Minkowski spacetime and

illustrate its utility in special relativity by recasting in spinor form much

of what we have learned about the structure of the electromagnetic ﬁeld in

Chapter 2. We shall not stray into quantum mechanics and, in particular,

will not discuss the Dirac equation (for this, see the encyclopedic monograph

[PR] of Penrose and Rindler). Since it is our belief that an intuitive appreci-

ation of the notion of a spinor is best acquired by approaching them by way

of group representations, we have devoted this ﬁrst section to an introduction

to these ideas and how they arise in special relativity. Since this section is

primarily motivational, we have not felt compelled to prove everything we say

and have, at several points, contented ourselves with a reference to a proof

in the literature.

A vector v in M (e.g., a world momentum) is an object that is decribed in

each admissible frame of reference by four numbers (components) with the

property that if v = v

=ˆv

ˆe

and [Λ

] is the Lorentz transformation

relating {e

} and {ˆe

} (i.e., e

=Λ

ˆe

), then the components v

and ˆv

are

related by the “transformation law”

ˆv

=Λ

,a=1, 2, 3, 4. (3.1.1)

A linear transformation L : M→M(e.g., an electromagnetic ﬁeld) is an-

other type of object that is again described in each admissible basis by a set

of numbers (the entries in its matrix relative to that basis) with the property

, : An Introduction

, Applied Mathematical Sciences 92,

G.L. Naber The Geometry of Minkowski Spacetime to the Mathematics

of the Special Theory of Relativity

DOI 10.1007/978-1-4419-7838-7_ , © Springer Science+Business Media, LLC 2012

135

136 3 The Theory of Spinors

that if [L

]and[

] are the matrices of L in {e

} and {ˆe

},then

=Λ

,a,b=1, 2, 3, 4, (3.1.2)

where





is the inverse of [Λ

], i.e., Λ

=Λ

= δ

((3.1.2)isjust

the familiar change of basis formula). As we found in Chapter 2, it is often

convenient to associate with such a linear transformation a corresponding

bilinear form

L : M×M→R deﬁned by

L(u, v)=u · Lv. Again,

L is

described in each admissible basis by its set of components L

L(e

)

and components in diﬀerent bases are related by a speciﬁc transformation law:

=Λ

αβ

,a,b=1, 2, 3, 4. (3.1.3)

Such bilinear forms can, of course, arise naturally of their own accord without

reference to any linear transformation. The Lorentz inner product is itself

such an example. Indeed, if we deﬁne g : M×M→R by

g(u, v)=u · v,

then, in all admissible bases, g

= g(e

)=e

·e

= η

= g(ˆe

, ˆe

)=ˆg

In this very special case the components are the same in all admissible bases,

but, nevertheless, (1.2.14) shows that the same transformation law is satisﬁed:

ˆg

=Λ

αβ

,a,b=1, 2, 3, 4.

The point of all of this is that examples of this sort abound in geometry

and physics. In each case one has under consideration an “object” of geomet-

rical or physical signiﬁcance (an inner product, a world momentum vector,

an electromagnetic ﬁeld transformation, etc.) which is described in each ad-

missible basis by a set of numerical “components” and with the property that

components in diﬀerent bases are related by a speciﬁc linear transformation

law that depends on the Lorentz transformation relating the two bases. Dif-

ferent “types” of objects are distinguished by their number of components

in each basis and by the precise form of the transformation law. Classically,

such objects were called “world tensors” or “4-tensors” (we give the precise

deﬁnition shortly). World tensors are well suited to the task of expressing

“Lorentz invariant” relationships since, for example, a statement which as-

serts the equality, in some basis, of the components of two world tensors of

the same type necessarily implies that their components in any other basis

must also be equal (since the “transformation law” to the new basis compo-

nents is the same for both). This is entirely analogous to the use of 3-vectors

in classical physics and Euclidean geometry to express relationships that are

true in all Cartesian coordinate systems if they are true in any one. For

many years it was tacitly assumed that any valid Lorentz invariant state-

ment (in particular, any law of relativistic physics) should be expressible as

a world tensor equation. Dirac put an end to this in 1928 when he proposed

3.1 Representations of the Lorentz Group 137

a law (equation) to describe the relativistic electron with spin that was man-

ifestly Lorentz invariant, but not expressed in terms of world tensors. To

understand precisely what world tensors are and why they did not suﬃce for

Dirac’s purposes we must take a more careful look at “transformation laws”

in general.

Observe that if v is a vector with components v

and ˆv

in two admis-

sible bases and if we write these components as column vectors, then the

transformation law (3.1.1) can be written as a matrix product:

⎡

⎢

⎣

ˆv

⎤

⎥

⎦

⎡

⎢

⎣

⎤

⎥

⎦

⎡

⎢

⎣

⎤

⎥

⎦

By virtue of their linearity the same is true of (3.1.2)and(3.1.3). For example,

writing

the L

and

as column matrices, (3.1.2) can be written in terms

of the 16 × 16 matrix





⎡

⎢

⎣

⎤

⎥

⎦

⎡

⎢

⎣

··· Λ

⎤

⎥

⎦

⎡

⎢

⎣

⎤

⎥

⎦

Exercise 3.1.1 Write (3.1.3) as a matrix product.

this

y one can think of a transformation law as a rule which assigns

to each Λ ∈La certain matrix D

which transforms components in one

basis {e

} to those in another {ˆe

}, related to {e

} by Λ. Observe that, for

each of the examples we have considered thus far, these rules Λ → D

carry

the identity matrix in L onto the corresponding identity “transformation

matrix” (as is only fair since, if the basis is not changed, the components

of the “object” should not change). Moreover, if Λ

and Λ

are in L and

is their product (still in L), then Λ

→ D

= D

(this is

obvious for (3.1.1)sinceD

= Λ and follows for (3.1.2)and(3.1.3)either

from a rather dreary calculation or from standard facts about change of

basis matrices). This also makes sense, of course, since the components in

any basis are uniquely determined so that changing components from basis

#1 to basis #2 and then from basis #2 to basis #3 should give the same

result as changing directly from basis #1 to basis #3. In order to say all of

this more eﬃciently we introduce some terminology.

Let n be a positive integer. A matrix group of order n is a collection G of

n × n invertible matrices that is closed under the formation of products and

inverses (i.e., if G, G

,andG

are in G,thenG

−1

and G

are also in G).

We have seen numerous examples, e.g., the Lorentz group L is a matrix group

of order 4, whereas SL(2, C) is a matrix group of order 2. The collection of

138 3 The Theory of Spinors

all n ×n invertible matrices (with either real or complex entries) clearly also

constitutes a matrix group and is called the general linear group of order n

andwritteneitherGL(n, R)orGL(n, C ) depending on whether the entries are

real or complex. Observe that a matrix group of order n necessarily contains

the n × n identity matrix I

= I since, for any G in the group, GG

−1

= I.

If G is a matrix group and G



is a subset of G,thenG



is called a subgroup

of G if it is closed under the formation of products and inverses, i.e., if it is

itself a matrix group. For example, the set R of rotations in L is a subgroup

of L (Exercise 1.3.7), SU

is a subgroup of SL(2, C) (Exercise 1.7.6) and,

of course, any matrix group is a subgroup of some general linear group. A

homomorphism from one matrix group G to another H is a map D : G→H

that preserves matrix multiplication, i.e., satisﬁes D(G

)=D(G

)D(G

)

whenever G

and G

are in G. As is customary we shall often write the image

of G under D as D

rather than D(G) and denote the action of D on G by

G → D

.IfG has order n and H has order m,thenD necessarily carries

onto I

since D(I

)=D(I

)D(I

)sothatD(I

)(D(I

))

−1

D(I

)D(I

)(D(I

))

−1

and therefore I

= D(I

Exercise 3.1.2 Show that a homomorphism D : G→Hpreserves inverses,

i.e., that D(G

−1

)=(D(G))

−1

for all G in G.

Exercise 3.1.3 Show that if D : G→His a homomorphism, then its image

D(G)={D(G):G ∈G}is a subgroup of H.

A homomorphism of one matrix group G into another H is also called a (ﬁnite

dimensional) representation of G. For reasons that will become clear shortly,

we will be particularly concerned with the representations of L and SL(2, C).

If H is of order m and V

is an m-dimensional vector space (over C if the

entries in H are complex, but otherwise arbitrary), then the elements of H

can, by selecting a basis for V

, be regarded as linear transformations or,

equivalently, as change of basis matrices on V

. In this case the elements of

are called carriers of the representation. M itself may be regarded as a

space of carriers for the representation D : L→GL(4, R )ofL correspond-

ing to (3.1.1), i.e., the identity representation Λ → D

= Λ. Similarly, the

vector space of linear transformations from M to M and that of bilinear

forms on M act as carriers for the representations [Λ

] →





and

[Λ

] →





corresponding to (3.1.2)and(3.1.3), respectively. It is

rather incon

venien

t, however, to have diﬀerent representations of L acting on

carriers of such diverse type (vectors, linear transformations, bilinear forms)

and we shall see presently that this can be avoided.

The picture we see emerging here from these few examples is really quite

general. Suppose that we have under consideration some geometrical or phys-

ical quantity that is described in each admissible basis/frame by m uniquely

determined numbers and suppose furthermore that these sets of numbers cor-

responding to diﬀerent bases are related by linear transformation laws that

depend on the Lorentz transformation relating the bases (there are objects

3.1 Representations of the Lorentz Group 139

of interest that do not satisfy this linearity requirement, but we shall have no

occasion to consider them). In each basis we may write the m numbers that

describe our object as a column matrix T =col[T

···T

]. Then, associated

with every Λ ∈Lthere will be an m×m matrix D

whose entries depend on

those of Λ and with the property that

T = D

T if {e

} and {ˆe

} are related

by Λ. Since the numbers describing the object in each basis are uniquely de-

termined, the association Λ → D

must carry the identity onto the identity

and satisfy Λ

→ D

= D

, i.e., must be a representation of the

Lorentz group. Thus, the representations of the Lorentz group are precisely

the (linear) transformation laws relating the components of physical and ge-

ometrical objects of interest in Minkowski spacetime. The objects themselves

are the carriers of these representations. Of course, an m × m matrix can be

thought of as acting on any m-dimensional vector space so the precise math-

ematical nature of these carriers is, to a large extent, arbitrary. We shall ﬁnd

next, however, that one particularly natural choice recommends itself.

We denote by M

∗

the dual of the vector space M, i.e., the set of all real-

valued linear functionals on M.Thus,M

∗

= {f : M→R : f (αu + βv)=

αf(u)+βf(v) ∀ u, v ∈Mand α, β ∈ R}. The elements of M

∗

are called

covectors. The vector space structure of M

∗

is deﬁned in the obvious way,

i.e., if f and g are in M

∗

and α and β are in R,thenαf + βg is deﬁned

by (αf + βg)(u)=αf(u)+βg(u). If {e

} is an admissible basis for M,its

dual basis {e

} for M

∗

is deﬁned by the requirement that e

)=δ

for

a, b =1, 2, 3, 4. Let {ˆe

} be another admissible basis for M and {ˆe

} its dual

basis. If Λ is the element of L relating {e

} and {ˆe

},then

ˆe

=Λ

,a=1, 2, 3, 4, (3.1.4)

and

ˆe

=Λ

,a=1, 2, 3, 4. (3.1.5)

We prove (3.1.5) by showing that the left- and right-hand sides agree on

the ba

sis {ˆe

} ((3.1.4)isjust(1.2.15)). Of course, ˆe

(ˆe

)=δ

.Butalso

(ˆe

)=Λ





=Λ

)=Λ

=Λ

= δ

since [Λ

]and





are inverses.

Recall that each v ∈Mgives rise, via the Lorentz inner product, to a

∗

∈M

∗

deﬁned by v

∗

(u)=v · u for all u ∈M.Moreover,ifv = v

,then

∗

= v

,wherev

= η

aα

since v

= v

∗

)=v·e

=(v

)·e

= v

)=η

aα

. Moreover, relative to another basis, v

∗

=ˆv

ˆe

=ˆv

(Λ

ˆv

) e

so v

=Λ

ˆv

and, applying the inverse, ˆv

=Λ

With this we can show that all of the representations of L considered thus

far can, in a very natural way, be regarded as acting on vector spaces of

multilinear functionals (deﬁned shortly). Consider ﬁrst the collection T

bilinear forms L : M×M→R on M.IfL, T ∈T

and α ∈ R,then

the deﬁnitions (L + T )(u, v)=L(u, v)+T (u, v)and(αL)(u, v)=αL(u, v)

are easily seen to give T

the structure of a real vector space. For any two

140 3 The Theory of Spinors

elements f and g in M

∗

we deﬁne their tensor product f ⊗g : M×M→R

by f ⊗ g(u, v)=f(u)g(v). Then f ⊗ g ∈T

Exercise 3.1.4 Show that, if {e

} is the dual of an admissible basis, then

⊗ e

: a, b =1, 2, 3, 4} is a basis for T

and that, for any L ∈T

L = L(e

⊗ e

= L

⊗ e

. (3.1.6)

Now, in another basis, L(ˆe

, ˆe

)=L



, Λ



=Λ

L(e

)so

=Λ

αβ

. (3.1.7)

Thus, components relative to bases of the form {e

⊗e

: a, b =1, 2, 3, 4} for

transform under the representation [Λ

] →





of (3.1.3)andwe

may

ther

efore regard the bilinear forms in T

as the carriers of this represen-

tation. Elements of T

are called world tensors of contravariant rank 0 and

covariant rank 2 (we will discuss the terminology shortly).

Next we consider the representation [Λ

] →





of L appropriate to

(3.1.2). Let T

denote the set of all real-valued functions L : M

∗

×M→R

that are linear in each variable, i.e., satisfy L(αf + βg, u)=αL(f, u)+

βL(g,u)andL(f,αu + βv)=αL(f, u)+βL(f,v) whenever α, β ∈ R,f,g∈

∗

and u, v ∈M. The vector space structure of T

is deﬁned in the obvious

way: If L, T ∈T

and α, β ∈ R,thenαL + βT ∈T

is deﬁned by (αL +

βT)(f, u)=αL(f, u)+βT(f, u). For u ∈Mand f ∈M

∗

we deﬁne u ⊗ f :

∗

×M→R by u ⊗ f (g, v)=g(u)f (v). Again, it is easy to see that

u ⊗ f ∈T

,that{e

⊗ e

: a, b =1, 2, 3, 4} is a basis for T

and that, for

any L in T

L = L(e

⊗ e

= L

⊗ e

. (3.1.8)

In another basis, L(ˆe

, ˆe

)=L



, Λ



=Λ

L(e

=Λ

. (3.1.9)

Thus, components relative to bases of the form {e

⊗ e

: a, b =1, 2, 3, 4}

transform under the representation [Λ

] →





of (3.1.2) so that the

elements

of T

are a natural choice for the carriers of this representation. The

elements of T

are called world tensors of contravariant rank 1 and covariant

rank 1.

The appropriate generalization of these ideas should by now be clear. Let

r ≥ 0ands ≥ 0 be integers. Denote by T

the set of all real-valued functions

deﬁned on

∗

×···×M

∗

1 23 4

r factors

×M×···×M

23 4

s factors

that are linear in each variable separately (these are called multilinear

functionals). T

is made into a real vector space by the obvious pointwise

3.1 Representations of the Lorentz Group 141

deﬁnitions of addition and scalar multiplication. If u

,...,u

∈Mand

,...,f

∈M

∗

one deﬁnes u

⊗···⊗u

⊗ f

⊗···⊗f

in T

⊗···⊗u

⊗ f

⊗···⊗f

,...,g

,...,v

)

= g

) ···g

) ·f

) ···f

)

and ﬁnds that the set of e

⊗···⊗e

⊗e

⊗···⊗e

,...,a

=1, 2, 3, 4

and b

,...,b

=1, 2, 3, 4, form a basis for T

.Moreover,ifL ∈T

,then

L = L(e

,...,e

⊗···⊗e

⊗ e

⊗···⊗e

= L

···a

···b

⊗···⊗e

⊗ e

⊗···⊗e

(3.1.10)

Relative to another basis,

L(ˆe

,...,ˆe

, ˆe

,...,ˆe

)

= L



,...,Λ

, Λ

,...,Λ



=Λ

···Λ

L(e

,...,e

)

···a

···b

=Λ

···Λ

···α

···β

. (3.1.11)

The elements of T

are called world tensors (or 4-tensors) of contravariant

rank r and covariant rank s. “Contravariant rank r” refers to the r indices

,...,a

that are written as superscripts in the expression for the compo-

nents and which appear in the transformation law attached to an entry in Λ

(rather than Λ

−1

). Covariant indices are written as subscripts in the compo-

nents and transform under Λ

−1

.AnelementofT

has 4

r+s

components and

if these are written as a column matrix, then the transformation law (3.1.11)

written as a matrix product thus giving rise to an assignment

[Λ

] →

···Λ

to each element of L of a 4

r+s

× 4

r+s

matrix which can be shown to be a

representation of L andiscalledtheworld tensor (or 4-tensor) representation

of contravariant rank r and covariant rank s. Notice that even the identity

representation of L corresponding to (3.1.1) is included in this scheme (with

r =1

nds =

0). The carriers, however, are now viewed as linear functionals

on M

∗

, i.e., we are employing the standard isomorphism of M onto M

∗∗

(x ∈M→x

∗∗

∈M

∗∗

deﬁned by x

∗∗

(f)=f(x) for all f ∈M

∗

). The

elements of T

are sometimes called contravariant vectors, whereas those of

are covariant vectors or covectors.

World tensors were introduced by Minkowski in 1908 as a language in

which to express Lorentz invariant relationships. Any assertion that two world

142 3 The Theory of Spinors

tensors L and T are equal would be checked in a given admissible basis/frame

by comparing their components L

···a

···b

and T

···a

···b

in that basis

and, if these are indeed found to be equal in one basis, then the components

in any other basis must necessarily also be equal since they both transform

to the new basis under (3.1.11). World tensor equations are true in all admis-

sible

mes if and only if they are true in any one admissible frame, i.e., they

are Lorentz invariant. World tensors were introduced, in analogy with the

3-vectors of classical mechanics, to serve as the basic “building blocks” from

which to construct the laws of relativistic (i.e., Lorentz invariant) physics. So

admirably suited were they to this task that it was not until attempts got

under way to reconcile the principles of relativistic and quantum mechanics

that it was found that there were not enough “building blocks”. The reason

for this can be traced to the fact that the underlying physically signiﬁcant

quantities in quantum mechanics (e.g., wave functions) are described by com-

plex numbers ψ, whereas the result of a speciﬁc measurement carried out on

a quantum mechanical system is a real number that depends only on quan-

tities of the form ψ

ψ and these last quantities are insensitive to changes in

sign, i.e., (−ψ)(

−ψ)=ψ

ψ.Consequently,ψ and −ψ give rise to precisely the

same predictions as to the result of any experiment and so must represent the

same state of the system. As a result, transforming the state’s description in

one admissible frame to that in another (related to it by Λ) can be accom-

plished by either one of two matrices ±D

. As we shall see in Section 3.5 this

ambiguity in the sign is often an essential feature of the situation and cannot

be consistently removed by making one choice or the other. This fact leads

directly to the notion of what Penrose [PR] has called a “spinorial object”

d w

hich we shall discuss in some detail in Appendix B. For the present we

will only take these remarks as motivation for introducing what are called

“two-valued representations” of the Lorentz group (intuitively, assignments

Λ →±D

of two component transformation matrices, diﬀering only by sign,

to each Λ ∈L).

In Section 1.7 we constructed a mapping of SL(2, C)ontoL called the

spinor map which we now designate

Spin : SL(2, C) →L.

Spin was a homomorphism of the matrix group SL(2, C)ontothematrix

group L that mapped the unitary subgroup SU

of SL(2, C)ontotherota-

tion subgroup R of L and was precisely two-to-one, carrying ±G in SL(2, C)

onto the same element of L (which we denote either Spin(G) = Spin(−G)or

=Λ

−G

). Next we observe that any representation

D : L→H“lifts” to

arepresentationofSL(2, C). More precisely, we deﬁne D : SL(2, C) →H

by D =

D ◦ Spin. Of course, D has the property that, for every G ∈

SL(2, C),D

−G

D(Spin(−G)) =

D(Spin(G)) = D

. Conversely, suppose

3.1 Representations of the Lorentz Group 143

D : SL(2, C) →His a representation of SL(2, C) with the property that

−G

= D

for every G ∈ SL(2, C). We deﬁne

D : L→Has follows:

Let Λ ∈L. Then there exists a G ∈ SL(2, C) such that Λ

= Λ. Deﬁne

D(Λ) =

D(Λ

)=D

.Then

D is a representation of L since

D(Λ

)=D

= D

D(Λ

)

D(Λ

). Thus, there

is a one-to-one correspondence between the representations of L and the rep-

resentations of SL(2, C) that satisfy D

−G

= D

for all G ∈ SL(2, C ).

Before proceeding with the discussion of those representations of SL(2, C)

for which D

−G

= D

we introduce a few more deﬁnitions. Thus, we let G

and H be arbitrary matrix groups and D : G→HarepresentationofG.

If the order of H is m,weletV

stand for any space of carriers for D.A

subspace S of V

is said to be invariant under D if each D

, thought of as

a linear transformation of V

, carries S into itself, i.e., satisﬁes D

S ⊆ S.

For example, V

itself and the trivial subspace {0} of V

are obviously

invariant under any D.If{0} and V

are the only subspaces of V

that are

invariant under D,thenD is said to be irreducible;otherwise,D is reducible.

It can be shown (see [GMS]) that all of the representations of SL(2, C)ca

constructed from those that are irreducible. Finally, two representations

(1)

: G→H

and D

(2)

: G→H

,whereH

and H

have the same order,

are said to be equivalent if there exists an invertible matrix P such that

(2)

= P

−1

(1)

for all G ∈G. This is clearly equivalent to the requirement that, if V

is a

space of carriers for both D

(1)

and D

(2)

, then there exist bases {v

(1)

} and

(2)

} for V

such that, for every G ∈G, the linear transformation whose

matrix relative to {v

(1)

} is D

(1)

has matrix D

(2)

relative to {v

(2)

Theorem 3.1.1 (Schur’s Lemma) Let G and H be matrix groups of order

n and m respectively and D : G→Han irreducible representation of G.If

Aisanm × m matrix which commutes with every D

, i.e., AD

= D

for every G ∈G, then A is a multiple of the identity matrix, i.e., A = λI for

some (in general, complex) number λ.

Proof: We select a space V

of carriers and regard A and all the D

linear transformations on V

.LetS =kerA.ThenS is a subspace of

.Foreachs ∈ S, As = 0 implies A(D

s)=D

(As)=D

(0) = 0 so

s ∈ S, i.e., S is invariant under D.SinceD is irreducible, either S = V

or S = {0}.IfS = V

,thenA =0=0· I and we are done. If S = {0},

then A is invertible and so has a nonzero (complex) eigenvalue λ.Noticethat

(A − λI)D

= AD

− (λI)D

= D

A − D

(λI)=D

(A − λI)soA − λI

commutes with every D

. The argument given above shows that A − λI is

either 0 or invertible. But λ is an eigenvalue of A so A − λI is not invertible

and therefore A − λI =0asrequired. 