The theory of groups and quantum mechanics

My desire to show how the concepts arising in the theory of groups find their application in physics by discussing certain of the more important examples has necessitated the inclusion o

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THE THEORY OF GROUPS AND

QUANTUM MECHANICS

D ,:nated' by

1,:rg Yenlll:Ua Bappu

t '

The Indian Institute of Astrophysics

from the personal collection

of

Dr M K V Bappu

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THE THEORY OF GROUPS AND QUANTUM MECHANICS

BY HERMANN WEYL

PROF ESSOR 01" MATHEMATICS IN THE UNIVE.RSITY OF GOTTINGEN

TRANSLATED FROM THE SECOND (REVISED)

Origillally jul11ished in German under the title

PRINTED IN THE U.S.A

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TO MY FRIEND

WALTER DALLENBACH

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FROM THE AUTHOR'S PREFACE TO

THE FIRST GERMAN EDITION

T HE importance of the standpoint afforded of groups for the discovery of the general laws of by the theory

quantum theory has of late become more and more apparent Since I have for some years been deeply concerned

has seemed to me appropriate and important to give an account

of the knowledge won by mathematicians working in this field

in a form suitable to the requirements of quantum physics An additional impetus is to be found in the fact that, from the purely mathematical standpoint, it is no longer justifiable to draw such sharp distinctions between finite and continuous groups in discussing the theory of their representations as has been done in the existing texts on the subject My desire to show how the concepts arising in the theory of groups find their application in physics by discussing certain of the more important examples has necessitated the inclusion of a short account of the foundations of quantum physics, for at the tiInc the manuscript was written there existed no treatment of the subject to which

I could refer the reader In brief this book, if it fulfills its purpose, should enable the reader to learn the essentials of the theory of groups and of quantum mechanics as well as the rela- tionships existing between these two subjects; the mathenlatical portions have been written with the physicist in mind, and vice versa I have particularly emphasized the "reciprocity" be- tween the representations of the symmetric permutation group and those of the complete linear group; this reciprocity has as yet been unduly neglected in the physi'calliterature, in spite of the fact that it follows most naturally from the conceptual structure of quantum mechanics

vii

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There exists, in my opinion, a plainly discernible paralleli s,

physics Occidental mathematics has in past centuries broke away from the Greek view and followed a course which seen

to have originated in India and which l1as been transmitte( with additions, to us by the Arabs; in it the concept of numb~

appears as logically prior to the concepts of geometry_ Th result of this has been that we have applied this systematicall: developed number concept to all branches, irrespective of whethe

it is most appropriate for these particular applications Bu

the present trend in mathematics is clear~y in the direction of ~

return to the Greek standpoint; we now look upon each branc}

of mathematics as determining its own characteristic domair

of quantities The algebraist of the present day considers thE continuum of real or complex numbers as merely one " field ,: among many; the recent axiomatic foundation of projective geometry may be considered as the geometric counterpart of

this view This newer mathematics, including the modern theory of groups and "abstract algebra," is clearly motivated

by a spirit different from that of " classical mathematics," which found its highest expression in the theory of functions of a complex variable The continuum of real numbers has retained its ancient prerogative in physics for the expression of physical measurements, but it can justly be maintained that the essence

of the new Heisenberg-Schrodinger-Dirac quantum mechanics is

to be found in the fact that there is associated with each physical system a set of quantities, constituting a non-commutative algebra in the technical mathematical sense, the elements of which are the physical quantities themselves

ZURICH, August, I928

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AUTHOR'S PREFACE TO THE SECOND GERMAN EDITION

D in mathematical physics in Princeton University The DRING the academic year 19 z8 -29 I held a professorship

lectures which I gave there and in other American insti tutions afforded me a much desired opportunity to present anew, and from an improved pedagogical standpoint, the connection

found its expression in this new edition, in which the subject has been treated from a more thoroughly elementary standpoint Transcendental methods, which are in group theory based on the calculus of group cha1~acteristics, have the advantage of offering a rapid view of the subject as a whole, but true under- standing of the relationships is to be obtained only by following

connection the derivation of the Clebsch~Gorda.n series, which is

of fundamental in1portancc for the whole of spectroscopy and for the applications of quantum theory to chemistry, the section

on the Jordan-Hold,er theorem and its analogues, and above all tIle careful investigation of the connection between the algebra

of symmetric transforrnations and the syn1metric permutation group The reciprocity laws expressing this connectionl which

\vere proved by transcenden tal nlethods in the first edi tion, as well

as the group-theoretic probleln arising from the existence of spin

have also been treated from the elementary standpoint Indeed,

readers, much too condensed' and more difficult to understand than the rest of the book-has been entirely re-written l"he

algebraic standpoint has been emphasized, in harmony' with the recent development of U abstract algebra," which has proved so

ix

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x THE THEORY OF GROUPS

impossible to avoid presenting the principal part of the theory

of representations twice; first in Chapter III, where the sentations are taken as given and their properties examined, and again in Chapter V, where the method of constructing the representations of a given group and of deducing their properties

repre-is developed Bu t I believe the reader will find threpre-is two-fold treatment an advantage rather than a hindrance

To come to the changes in the more physical portions, in Chapter IV'- the role of the group of virtual rotations of space

is more clear~y presented But above all several sections have been added which deal with the energy-momentum theorem of quantum physics and with the quantization of the wave equation

in accordance with the recent work of Heise'nberg and Pauli

This extension already leads so far away from the fundamental purpose of the book that I felt forced to omit the formulation

of the quantum laws in accordance with the general theory of relativity, as developed by V Fock and myself, in spite of its desirability for the deduction of the energy-momentum tensor The fundamental problem of the proton and the electron has been discussed in its relation to the symmetry properties of the quantum laws with respect to the interchange of right and left, past and future, and positive and negative electricity At present no solution of the problem seems in sight; I fear that the clouds hanging over this part of the subject will roll together

to form a new crisis in quantum physics I have intentionally presented the more difficult portions of these problems of spin and second quantization in considerable detail, as they have been for the most part either entirely ignored or but hastily indicated in the large number of texts which have now appeared

on quantum mechanics

It has been rumoured that the "group pest" is gradually being cut out of quantum physics This is certainly not true

in so far as the rotation and Lorentz groups are concerned;

as for the permutation group, it does indeed seem possible to avoid it with the aid of the Pauli exclusion principle Never-theless the theory must retain the representations of the per-mutation group as a natural tool in obtaining an understanding

of the relationships due to the introduction of spin, so long as its specific dynamic effect is-neglected I have here followed the

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PREFACE TO SECOND GERMAN EDITION Xl

trend of the times, as far as justifiable, in presenting the group theoretic portions in as elcn1cntary a forn1 as possible The calculations of perturbation theory arc widely separated fronl these general considerations; I have therefore restricted lnyself

to indicating the nlethod of attack without either going into details or n1entioning the many applications which have been

others

The constants c and h, the velocity of light and the quantunl

of action, have caused some trouble 'The insight into the significance of these constants, obtained by the theory of rela tivity on the one hand and quantum theory on the other, is most forcibly expressed by the fact that they do not occur in the laws of Nature in a thoroughly systen1atic development of these theories But physicists prefer to retain the usual c.g.s units-principally because they are of the order of magnitude of the physical quantities with which we deal in everyday life., Only a wavering conlpromise is possible between these practical' considerations and the ideal of the systematic theorist; I initially adopt, with SOine regret, the current physical usage, but in the course of Chapter IV the theorist gains the upper hand

An attempt has been made to increase the clarity of the exposition by numbering the forruulre in accordance \vith the sections to which they belong, by emphasizing the lllore inl"

them, and by lists of operational sYlnbols and of letters having

a fixed significance

ll WEYL

GOTTINGEN, Novernber, I930

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TRANSLATOR'S PREFACE

was acting as assistant to Professor Weyl in Princeton' Unforeseen delays prevented the completion of the manuscript

at that time, and as Professor Weyl decided shortly afterward

to undertake the revision outlined in the preface above it seemed

this manuscript the German has been followed as closely as

de-tract from the elegant and logical treatment which characterizes

to follow the more usual English terminology in general, this

knowledge which have in the past been so \videly separated as the theory of groups and quantum theory can be accomplished only by adapting the existing terminology of each to that of

in the fact that the development of " fields" and" algebras"

Winger of Union College for the assistance he' has rendered in

''*'

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I The n-dimensional Vector Space

2 Linear Correspondences Matrix Calculus

3 The Dual Vector Space

4 Unitary Geometry and Hermitian Forms

5 Transformation to Principal Axes

6 Infinitesimal Unitary Transformations

7 Remarks on oo·dimensional Space

2 The de Broglie Waves of a Particle 48

3 Schrodinger's Wave Equation The Harmonic Oscillator 54

4 Spherical Harmonics 60

5 Electron in Spherically Symmetric Field Directional

Quan-tiza tion 63

6 Collision Phenomena 70

7 The Conceptual Structure of Quantum Mechanics 74

8 The Dynamical Law Transition Probabilities 80

9 Perturbation Theory 86

10 The Problem of Several Bodies Product Space 89

I I Commutation Rules Canonical Transformatlons · · 93

12 Motion of a Particle in an Electro-magnetic Field Zeeman

13 Atom in Interaction with Radiation 102

I Transformation Groups I 10

2 Abstract Groups and their Realization • 113

3 Sub-groups and Conjugate Classes 116

4 Representation of Groups by Linear Transformations 120

5 Formal Processes Clebsch-Gordan Series I23

6 The Jordan-HOlder Theorem and its Analogues 13I

7 Unitary Representations 136

8 Rotation and Lorentz Groups 140

9 Character of a Representation 150

10 Schur's Lemma and Burnside's Theorem 152

I I Orthogonality Properties of Group Characters · 157

xv

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XVI TI1E THEOR Y OF GROUPS

'Aaa

12 Extension to Closed Continuous Groups 160

13 The Algebra of a Group 16 5

14 Invariants and Covariants 17 0

IS Remarks on Lie's Theory of Continuous Groups of Trans

16 Representation by Rotations of Ray Space 180

IV ApPLICATION OF THE THEORY OF GROUPS TO QUANTUM MECHANICS 18,5

A The Rotation Group

I The Representation Induced in System Space by the

Rota-tion Group 185

2 Simple States and Term Analysis Examples 191

3 Selection and Intensity Rules · 197

4 The Spinning Electron, Multiplet Structure and Anomalous

Zeeman Effect 202

B The LO'Yentz Group

S Relativistically Invariant Equations of Motion of an Electron 210

6'0 Energy and Momentum Remarks on the Intercbange of Past

7 Electron in Spherically Symmetric Field 227

8 Selection Rules Fine Structure • 232

C The Permutation Group

9 Resonance between Equivalent Individuals 238

10 The Pauli Exclusion Principle and the Structure of the

Periodic Table 242

I I The Problem of Several Bodies and the Quantization of

the Wave Equation • 246

12 Quantization of the Maxwell-Dirac Field Equations 253

13 The Energy and Momentum Laws of Quantum Physics

Relativistic Invariance 264

D Quantum Kinematics

14 Quantum Kinematics as an Abelian Group of Rotations • 272

IS Derivation of the Wave Equation from the Commutation

2 Symmetry Classes of Tensors 286

3 Invariant Sub-spaces in Group Space 291

4 Invariant Sub-spaces in Tensor Space 296

5 Fields and Algebras 302

6 Representations of Algebras • 3 0 4

7 Constructive Reduction of an Algebra into Simple Matrie

B E~snsion of the Theory and Physical Applications

8 The Characters of th~ Symmetric Group and Equivalence

Degeneracy in Quantum Mechanics 3 1 9

9 Relation between the Characters of the Symmetric Per

mutation and Affine Groups 3 26

10 Direct Product Sub-groups • 33 2

1 I Perturbation Theory for the Construction of Molecules 339

12 The Symmetry Problem of Quantum Theory • 347

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CON'TENTS

XVII

C Explicit Algebraic Construction

PAGE

13 Young's Symmetry Operators " 358

14 Irreducibility, Linear Independence, Inequivalence and

3 A THEOREM CONCERNING NON-DEGENERATE ANTI-SYMMETRIC

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INTRODUCTION

T liE quantum theory of atomic processes was proposed by

NIELS BOHR in the year 19I3, and was based on the

deduction of the Balmer series for the line spectrum of hydrogen

constituted its first convincing confirmation This theory gave

us the key to the understanding of the regularities observed in optical and X-ray spectra, and led to a deeper insight into the

of Naturwissenschaften, dedicated to BOHR and entitled "Die

cler Atome" (Vol 11, p 535 (1923)), gives a short account of the successes of the theory at its peak But about this time it began

a compromise between the old tt classical" physics and a new quantum physics which has been in the process of development

described the situation in an address on t( Atomic Theory and Mechanics" (appearing in Nature, 116, p 845 (1925)) in the words: It From these results it seems to follow that, in the general problem of the quantum theory, one is faced not with

a modification of the mechanical and electro dynamical theories describable in terms of the usual physical concepts, but with

an essential failure of the pictures in space and time on which the description of natural phenomena has hitherto been based." The rupture which led to a new stage of the theory was made

by HEISENBERG, who replaced Bohr's negative prophecy by a positive guiding principle

rrhe foundations of the new quantum physics! or at least its more important theoretical aspects, are to be treated in this

X1X

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xx THE THEORY OF GROUPS

which are urgently required, I name above all the fourth edition

of SOMMERFELD'S well-known" Atombau und Spektrallinien"

Structure and Spectral Lines" (London, 1923) of the third edition, together with the recent (1929) "Wellenmechanischer Erganzungsband " or its English translation "Wave Mechanics "

AND UREY, "Atoms, Molecules and Quanta" (New York, 1930), which appears in the " International Series in Physics," edited

Qut valuable survey" Experimentelle Grundlagen der Qua~ten­

theorie" (Braunschweig, 1921 ) The spectroscopic data, pre sented in accordance with the new quantum theory, together with complete references to the literature, are given in the following three volumes of the series "Struktur der Materie,"

FRANCK:-F HUND, "Linienspektren und periodisches System der Elemente" (1927);

E BACK AND A LANDE, "Zeemaneffekt und struktur der Spektrallinien" (1925);

Multiplett-W GROTRIAN, Ie Graphische Darstellung der Spektren von Atomen und Ionen mit ein, xwei und drei Valenzelektronen" (1928)

The spectroscopic aspects of the subject are also discussed

Spectra t7 (1930), which also appears in the "International Series in Physics l '

possible by the enormous refinement of experimental technique,

which has given us an almost direct insight into atomic processes If in the following Ii ttle is said concerning the experimental facts, it should not be attributed to the mathe-matical haughtiness of the author; to report on these things lies Qutside his 'fj.eld Allow me to express now, once and for all, my deep respect for the work of the experimenter and for his fight to wring significant facts from an inflexible' Nature, who says so distinctly U No n and, so indistinctly "Yes" to our theories

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INTRODUCTION

XXI

Our generation is witness to a development of physical knowledge such as has not been seen sin~e the days of KEPLER,

GALILEO AND NEWTON, and mathematics has scarcely ever

recompense, mathematics is less bound to the course of worldly

back only as far as 1900, the origin of the theory of groups

is lost in a past scarcely accessible to history; the earliest works of art show that the symmetry groups of plane figures were even then already known, although the theory of these was only given definite form in the latter part of the eighteenth

group concept· as most characteristic of nineteenth century

to natural science lay in the description of the symmetry of crystals, but it has recently been recognized that group theory

is of fundamental importance for quantum physics; it here reveals the essential features which are not contingent on a special form of the dynamical laws nor on special assumptions concerning the forces involved We may well expect that it is just this part of quantum physics which is most certain of a lasting place 'fWD groups, the group of rotations in 3-dimen sional spa.ce and the permutation group, play here the principal role, for the laws governing the possible electronic configurations grouped about the stationary nucleus of an atonl or an ion are spherically symmetric with respect to the nucleus, and since the various electrons of which the atom or ion is composed are identical, these possible configurations are invariant under a

theory oj the representation of groups by linear tra?,lsjor1nations,

and it is exactly this mathematically most important part which is necessary for an adequate description of the quantum mechanical relations .I1ll quantum numbers, ~e'ith the exception

of the so-called principal quantum nurnber, are indices 'izin,g representations of groups

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This book, which is to set forth the connection between groups and quanta, consists of five chapters The first of these is concerned with unitary geom.etry It is somewhat distressing that the theory of linear algebras must again and again be developed from the beginning, for the fundamental concepts

of this branch of mathematics crop up everywhere in matics and physics, and, a knowledge of them should be as widely disseminated as the clements of differential calculus

mathe-In this chapter many details will be introduced with an eye

to future use in the applications; it is to be hoped that in spite of this the simple thread of the argument has remained plainly visible Chapter I I is devoted to preparation on the physical side; only that has been given ",·hich seemed to me indispensable for an understanding of the meaning and methods

of quantum theory A multitude of physical phenomena, which have already been dealt with by quantum theory, have been omitted Chapter III develops the elementary portions of the theory of representations of groups and Chapter IV applies them

to quantum ph)lsics Thus mathematics and physics alternate

in the first four chapters, but in Chapter V the two are fused together, showing how completely the mathematical theory is adapted to the requirements of quantum physics In this last chapter the perntutation group and its representations, together with the groups of linear transformations in an affine or unitary space of an arbitary number of dimensions, will be subjected to

a thorough going study

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THE THEORY OF GROUPS AND

QUANTUM MECHANICS

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CHAPTER I

as well as of the theory of the representations of groups,

is the multi· dimensional affine or unitary space The axiomatic method of developing the geometry of such a space

is no doubt the most appropriate, but for the sake of clearness

I shaH at first proceed along purely algebraic lines I begin

linear space 9t = 9tn is a set of n ordered numbers (Xl' X 2 • , %n) ;

vector analysis is the calculus of such ordered sets The two

multiplica-tion of a vector ~ by a number a and the addition of t~eJO vectors!

and~ On introducing the notation

~ = (Xl' x 2 , • • , x".), t) = (Yl, )'27 • • " Yn)

these operations are defined by the equations

a~ = (axl' ax!, · " ax,,), ~ + ~ = (Xl + Yl, X2 + )'2, • • ,

3 a and c being any t1.VO vectors, there exists one and only one

vector ~ for which a + ~ = c It is called the difference c - a of ( and a (possibility of subtraction)

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2 UNITARY GEOMETRY

1 (a + b)~ = (ã) + (b~) (first distributive law)

2 ăbl) = (ab)~ (associative law)

3 It = l·

4 ẵ + t)) = (at) + (at)) (second distributive la1.f})

The existence of a vector 0 = (0,0, · • " 0) with the property

~+O=Ỡ=~

need not be postulated separately as it follows from the axioms

which are defined in terms of the two fundamental operations with which the axioms (Cl) and (~) are concerned; we mention

a few of the most important A number of vectors aI, a2 , • • , aIL

are said to be linearly independent if there exists between them

no homogeneous linear relation

c1a1 + cZa2 + · · · + c"a" = 0 except the trivial one with coefficients

C 1 = 0, C1 = 0, " ' , c,,~ 0

h such vectors are said to span an h-dimensional (li1'tear)

sub-space ffí consisting of all vectors of the form

where the ,'s are arbitrary numbers It follows from the fundamental theorem on homogeneous linear equations that there exists a non-trivial homogeneous relation between any

h + I vectors of mị The dimensionality h of al' can therefore

be characterized independently of the basis: every h + I vectors

in ffí are linearly dependent, but there exist in it h linearly independent 'vectors Any such system of h independent vectors al , a2, • • , ah in at' can be used as a co-ordinate system

or basis in m'; the coefficients '1, '2,· · , ~1t in the representation

system (aÍ a2, · • , tl1t.)

The entire space m is n-dimensional, and the vectors

THE n-DIMENSIONAL VECTOR SPACE 3

agree \vith the" absolute components" Xi :

t = rIel + .t"2e2 + · +

xne'n-From the standpoint of affine geometry, however, the" absolute co-ordinate system" (1.2) has no·preference over any other which

previous axioms, which did not concern themselves with the

(y) The maximum number of linearly independent vectors in m

must in particular have a =1= 0, and consequently any vector ~

can be expressed as a linear combination

and we arrive at the definitions from which we started The only-but important-difference between the arithmetic and the axiomatic treatment is that in the former the absolute co-ordinate system (1.2) is given the preference over any other,

Given any system of vectors, all vectors ~ which are obtained,

as (1.1), by linear combinations of a finite number of vectors

aI, a2, · · , Q7J, of the system constitute a (linear) sub-space-the su'b space " spanned" by the vectors Q

ffi is said to be decomposed or reduced into two linear spaces ffi', 9t" (Ut = m' + fft") if an arbitrary vector ~ can be expressed uniquely as the sum of a vector ~' of al' and a vector

sub-,,&" of ffi" A co-ordinate system in at' and a co-ordinate system

in 9l" constitute together a co ordinate system for the entire space 9t; this co-ordinate system in ffi is "adapted" to the decomposition 9t' + m" The sum n' + n" of the dimension

alities of 2}t' and 9t" is equal to n, the dimensionality of iR

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1 !

Conversely, if the sub-spaces 91', ~" .have no ,,:e~tor except 0

in common, and if the sum of theIr dlmenSlonahtles IS n, then

m = 9l' + gt"

m' being an n-dimensional sub-space, two vectors ~ and ~ are

said to be congruent modulo 9i' :

straction, which we call projection with respect to »i', the n-dimensional space m gives rise to an (n - n') dimensional space \I Ul is also a vector space, for from

~1 == t2J ~l == ~2 (mod ~')

foll~w the relations

The operations of multiplication by a number and addition can therefore be considered ones which operate directly on the vectors! of Ol All vectors t of m which are congruent mod al' giv"'e rise to the same vector ~ of tJl If ill' is one-dimensional and is spanned by e the above process is the fami1iar one of parallel projection in the direction of e; it is not necessary to give an (n - I)-dimensional sub-space of m on to which the

projection is made

If a is a non-null vector, all vectors ~ which arise by plying (l by a number are said to lie on the same ray as Q Two non-null vectors determine the same ray when, and only when, one is a multiple of the other In a given co-ordinate system the vector a is characterized by its components al) aI, · · , aft wllereas the ray a is characterized by their ratios al : as : • • : an;

multi-these ratios have meaning only when the components of a do not all vanish, i.e only when Ct =1= o

The transition from one co-ordinate system et to another e/ is accomplished by expressing the new co ordinate vectors e/ in terms of the old :

e.,c' = E al1c et•

i-1

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on transition to the new co-ordinate system e/ and are said to transform cogrediently

The formula (1.4) can, however, be otherwise interpreted;

mapping of the space ffi: on itself But for this purpose it

accented and the unaccented co-ordinates On employing a definite co-ordinate system ei , the equation

(2.1)

associates with an arbitrary vector ~ with components x a vector

!' with components X/I I'fhis correspondence A : ~ -? !' of ffi on itself can be characterized as linear by the two assertions: if

t, t) go over into ~'; t)', then a~ goes over into a,&' and l + ~ into

~' + t)' Linear correspondences therefore leave all affine rela tions unaltered; hence their prominence in the theory of affine

determine the linear correspondence (2.1), consider the following:

if a correspondence A which satisfies these conditions sends the

6 UNITARY GEOMETRY

customary in quantum ph)TSICS to call the hnear correspondences

vector ~ of iR

Let A B be two linear correspondences, the first of which

sends "&' into ,,&" = B.~' = B~A"&) "T~c rcsu~tant corre.spondence

C which carries ~ directly In to "& , IS also hnear and IS denoted

by (BA) (to be read from right to left /) :

(BA)~ = B(A~)

This "multiplication " satisfies laws which are similar to those

as-sociative law

C(BA) = (CB)A

Ai = lA = A

distinct ones The algebraic condition for this is the

AA-l = A-1A == 1

oet (BA) = net B • det A

also" add" them This concept of add1.·tio17~ arises quite

(A + B)~ = A~ + B,&_

and multiplication by a number obey the same laws as the

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LINEAR CORRESPONDENCES 7 analogous operations on vectors Addition is commutative,

multiplication:

(A + B)C = AC + Be, C(A + B) = CA + CB,

(aA)C = a(AC), C(aA) = a(CA)

Before proceeding to the arithmetical expression of these

natural generalization We can map an m-dimensional vector

plished when with each vector ~ of ffi a vector t) of \E is associated

in such a way ~ -+ ~ that from ~l -+ ~l' !a -+ ~2 it follows that

system in the space ffi: and Yl, · · , Yrt have the corresponding

all a12 • • • aIm

A = II aki II and B = II b ki II

then

aA = 11 a · aki II, A + B = " aki + b ki II·

If we have a third (p dimensional) vector space st, the consec

u tive application of the correspondences A : ~ -+ ~ of ffi on (5 and

B : t) -+ & of ~ on ~ gives rise to the correspondence C = BA : ~ -+ 3

of 91 on~ This composition is expressed in terms of nlatrix

,

(2.4) from the components in the ztb row of B and the ith column

space as m; A is then a correspondence of m on 5, B of (S on 9l

Already here concepts of the theory of groups play an important

groups, the reader should return to the matter here discussed

as an illustration

The matrix calculus allows us to express the formulre for

a linear correspondence, such as (2.3), in an abbreviated form

for y In accordance with the rule (2.4) for the composition of

matrices, equations (2.3) can be written

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LINEAR CORRESPONDENCES 9 This form is particularly useful in examining the effect on the matrix A of a linear correspondence of a space m on a space <5

when the original co-ordinate systems are replaced by ne,v ones

If this change of co-ordinates is effected by the transformations

expressed by the matrix

I Iet us now return to the linear correspondence A of a space

at on to itself If ffi' is a linear n' -din1ensional sub-space of m'

we say that A leaves m' invariant if it carries any vector of m',

over into a vector of m' If the co-ordinate system is so chc)sen

that the first n' fundamental vectors lie in m', the n1atrix ()f

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10 UNITARY GEOMETRY

two square matrices arranged along the principal diagonal as

situated in these portions are all zero

sub-spaces Btl + ffi2 -+- • • , ma having the dimensionality noc; n is

lie in the sub-spaces m1 , al2 , • • •• The association l -+ ~ac is

correspond-ence A : l ~ t' of Ut on to itself, we consider that linear

into the component 'f.a.' in mIX of ~'" We call [A]cx~ the portion of

the matrix, is broken up into segments of lengths nex (ex = 1, 2, · · )

[A]t.l1J in which the a,th set of rows intersects the foth set of columns,

and which consist of nCl • nfJ elements

If A is the matrix of a correspondence of iR on to itself in

system obtained from the first by means of the reversible

may be formulated algebraically: to find expressions which

are so formed from the components of an arbitrary matrix that

they assume the same value for equivalent matrices, i.e for

,',

'"

, ,

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LINEAR CORRESPONDENCES 11

a mUltiple A! of itself under the influence of A The column x

of the components of ! must then satisfy the equation

Ax = Ax, or (A1 - A)x = o

But n linear homogeneous equations in n unknowns have a non-vanishing solutio'n only if their determinant vanishes; the

multiplier A is therefore necessarily a root of the " characteristic

polynomial' ,

from (2.7) or SA' = AS it follows that

S(A1 - A') = ("1 - A)S,

whence by the theorem concerning the multiplication of minants

deter-det S · det (Ai - A') = det (A1 - A) · det S

Since the determinant of the reversible transformation S cannot vanish, we can divide by it and obtain the required identity

I A1 - A'l == 1,\1 - A I

The characteristic polynomial is of degree n in A :

f(> ) = An - SI,\n-1 + · · · ± Sn

whose coefficients, certain integral functions of the elements

a ik , are invariants of the correspondence A The" norm" Sri

is nlerely the determinant of A The first coefficient SI, the

trace

Sl = all + a22 + · · + a n n = trA (2.9)

is of more importance, as it depends linearly on the aik :

tr(A 1 + A 2) = trAl + trA 2•

If A is a linear correspondence of the m-dimensional vector space ffi: on the n-dimensional space (S, and B is conversely a linear correspondence of ~ on fft, then we can build the corre spondences BA of m on to itself and AB of ~ on to itself 'rhese

two correspondences have the same trace

12 UNITARY GEOMETRY

where i runs from 1 to m and k from 1 to n The special

in which A and B are both correspondences of m on to i

naturally deserves particular consideratipn

§ 8 The Dual Vector Space

A function L(,&) of the arbitrary vector ~ of the form

IXI XI + !X#a + · · + CX"X n

is called a linear form This concept is invariant in the sens affine geometry: it can be defined by means of the functic properties

L(<<"1J = ex • L(l), L(~ + ~) = L(~) + L(t))

It is obvious that the expression (3.1) has these properties, conversely, on introducing a co-ordinate system ei and set1

l = EXiei, it follows that

On going over to another co-ordinate system such that components Xi of an arbitrary vector ~ undergo the transforl tion (1.4), the linear form becomes

The coefficients (J.i of a linear form are said to transforn1 contJ

It is, however, not necessary to consider the oc as consta and the Xl as variables When the (J.i do not all vanish the eql

tion L(,&) = 0 defines a "plane," i e an (n - 1) -dimensio: sub-space; a vector llies in the plane if its components sati this equation But on the other hand we can ask for the equati

of all planes which pass through a given non-vanishing vector; the Xi = Xio are then constants and the cx, variables It is the fore most appropriate to consider the two sets (Xl' .%27 • • , ;t (eXt, (X2, • • • J cx,,) in parallel

We therefore introduce in addition to the space in a seeD

n-dimensional vector space, the dual space P From the co, ponents (ell ~J' • • , en) of a vector ~ of P and a vec1

(Xl' Xa, • • " X'ta) of m we can construct the inner or scalar prodj

'1 Xl + '2 X2 + · · · + ,,,X, (3

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THE DUAL VECTOR SPACE 13

This product has, by definition, an invariantive significance, for when ffi is referred to a new co-ordinate system by means of

a transformation of the Xi the variables 'i of the dual space P undergo the contragredient transformation This dual space is

in fact introduced in order to enable us to associate a contra gredient transformation with each one-to one transformation

To repeat, two linear reversible transformations

are contragredient with respect to each other if they leave (3.2) unaltered:

~lXl + e2X 2 + · · + ~nxn = ~l'Xl' + '2'X2' + · · + '1&'x,/ (3.4)

A vector! of ffi and a vector, of P are said to be in involution

when their product (3.2) vanishes A ray in Ut determines a

plane in P, Le the plane consisting of the vectors which are in involution with the given ray, and conversely Duality is

a reciprocal relationship.t

The dual or transposed matrix 4 of a matrix A = lIa1cilf

is obtained by interchanging the rows and columns of A

A* = Ila~1J is therefore defined by a~ = aki, and has m rows and n columns We shall always employ the asterisk to in-dicate this process And what is its geometrical interpretation? Let Ut be an m dimensional, @5 an n-dimensional, vector space;

A : ~ -+ ~ a linear correspondence of m on 6, specified in terms

of given co-ordinate systems in ffi and @5 by the matrix A :

Yk = Eaki Xi,

i

and let P, E be the dual spaces The product

where '1 is an arbitrary vector of E with components 'YJk' has then

an invariantive significance A bilinear form which depends linearly on a v·ector 1] of E and a vector ~ of at is therefore in variantively associated with a linear correspondence of iR on <5,

and conversely This gives rise, as the expression of the bi linear form given in parentheses shows, to a correspondence

TJ ~ e : ei = .Ea 1ci 'Y}k

Ie

of E on P, i.e the dual A· of A The reciprocal relation existing between the correspondence A and its dual A* may be expressed

t In the theory of rel~tivity it is usual to call vectors in at and P

contra-variant and covat'iant fJIJctOYS, respectively_

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14 UNITARY GEOMETRY

the linear laws

(AI + A2)* = AI* + A2*' (aA)* = a · A*

If A is a correspondence of m on (5 and B a correspondence of

(BA)* = A*B" (3.5)

of ~ on the dual P of gt

Xl, X2, • • , Xu of components of a vector ~ as a column; the

transformations (3.3), from the first of which it follows that

A*A = 1 or A = (A*)-l, (3.6)

contra.-gredient transformation

vectors of P which are in involution with the totality of vectors

sional sub·space P' of P And from this we are led illlmediately

to the result that if a correspondence A of ffi on its~lf leaves the sub-space ~' t"nvar1:ant, then the dual correspondence A * oj P on itself leaves the associated sub-space P' invariant

ffi1 + ffi2 + · · · of dimensionalities n17 ns, · · , and let the

dimension-ality of which is also nI" Defining P ~b P 3 analogously, we arrive

latter statement, we note that if the sum is 0 then the first summand belongs to PI as well as to Ps + Pa + · · " i.e it is

in involution with all the vectors of ~I + 9ls + · · · as well as

to ml , for if ~ is an arbitrary vector in 911 and 7J a vector in P

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