Table Of ContentGEOMETRY OF COMPLEX DOMAINS
a seinlnar condticted by
PROFESSORS OSWALD VEBLEN and JOHN VON NEUMANN
193^-36
First fbenti lectures by Professor'Veblen
Sepond term lectures by Mri J. -W, Givens
I
Notes by Dr. A. H. Taub and Mt*. J. ¥. Givens
The Institute for Advanced Study
Princeton, New Jers^
. Reissued with corrections,
1955.
ccai TENTS
Chapter I. SPINORS AND FROJECTIVE GECfflEIRY
Page
. The Miikowski space Represented by Herlnitian Matrices
1 1-1
2. The Complex Projective Line---------------------------------- 1-7
3. The Lorentz Group Isomorphic to the Quadric Group
1-10
k. The Projective Group in P^ Isomorphic to the Proper Lorentz
Group
1-13
5. The Antiprojective Group in P
Isomorphic to the Lorentz
Groun------1---- ----------------i______________________ __
1-18
6. Coordinate -Transformations and Tensor Calculus----------- 1-21
7. The Alternating Numerical Tensors--------------------------------
1-23
8. Dual Coordinates in R^ -—-—— ------ ---——________ 1-25
9. The Spinor Calculus in P, -------------------- ----------------------
1-31
. Transformations of Coordinates in P^ ---------------------------
10 1-33
. Involutions in P]^----------------------- i___________________
11 1-37'
4. 12. Antiinvolutions in P^ ------------------------------------------------
1-itl
“ 13. Point-Place Reflections in R- ------------------------------------- l-it5
1 lU.
Line Reflections in R_--------------------------------- I4t6
Factorization of the Fundan^ntal Quadratic Form---------- - I4t8
Chapter II.
1. Underlying and Tangent Spaces —
2-1
2. Spin and Gauge Spaces ---------------
2-it
3. Definition of Spinors ---------------
2-7
^ it. Gauge Transformations---------------
2-11
^ 5. Spinors of Weight ------ --------
2-13
=： 6. Spinors of Qtiier Weights--------—
2-16
^ 7. Spinors of Indices l/O and J=0 —
2-18
^ 8. Differentiation of Spinors ---------
2-19
'9. Invariant Differential Equations
2-22
t 10. Dirac Equations -----------------------
2-26
11. " ” (Coiitihued)--------
2-27
12. ” (Continued)--------
2-29
13. Current Vector
2-31
Chapter III.
1. Covariant Differentiation —-------------------------------------
3-1
2. The Transformation Law of p |__________________________
3-it
3. Examples of Covariant Differentiation-------------------------
3-6
it. Covariant Differentiation of Spinors with Tensor Indices
3-10
^ i'; delations between P ^ and i 1
3-12
" 6. " " ■ »'
3-13
7. Extension to the General Theory of Relativity
3-15
8k Geodesic Spin Coordinates
3-17
9. Definition of the Covariant Derivative of a Spinor _____
3-21
10. Relations between the Curvature Tensor and the Curvature
Spinor__________________________________________
3-23
11, Dirac Equations___________________ _______ ____ ____
3-25
Chapter IV. PROJECTIVE GEOMETRY OF (k=l)-DIMENSICHIS Pagfe
1. Definition of Projective Spaces —---------------------------- -------- U-1
2. Hyperplanes —------------------------------------- ----------------------- h-3
3. Coordinates of Linear Subspaces -------------------------------------- k-$
ii, Antiprojective Group ---------—----------------------- -—------—---- k“9
Matrix Notation -—-----------------------------------------------
6. iCornmuting Transformations---------—--------------------------------- U-19
7. Involutions------ ----------------------————------------------— ii-23
8. Anti-Involutions-------------- --------------------------------------------- U-28
9. Polarities-----------------------—--------------------------—-------------- U-31
10. Antipolarities ------—---- -------—---------—----------------—------- i;-33
Chapter V. LINEAR FAMILIES OF REFLECTIONS
. Statement of the Problem ——----------------------------------- 5-1
1
. The Centered Euclidean Space E - -------------------------------- 5-2
2
3. ■Transformations of Linear Famines of Involutions----—— 5-5
U. Existence of Linear Families of Involutions ----------------- 5-8
9* Equivalence of T^sets ------------------—---------—-----— 5-Ï2
6. Algebraic Properties of -/-sets —-----------------------------— 5-13
7. Representation of Rotation Group .in by Collineations
5-16
8. The Case 'k»2 ——--------------------------------------------------- 5-19
9. Commutative and Anticommutative Involutions ------------------ 5-23
10. Equivalence of Pairs of Anticommuting Involutions ------r— 5-25
11. The Reguli Determined by Two Anticommuting Involutions — 5-28
. Algebraic Discussion of the Involutions ------------------------ 5-30
12
13. Brbof of Theorems (5.1) to (5.U) ---------------------------------- 5r33
Chapter VI. THE EXTENSION TO CORRELATIONS
1. The Dual Mapping of Egy , onto Itself ------------------------------ 6-1
2. Representation of Improper Orthogonal Transformations -------- 6-2
3. The Invariant Polarity —-----------------------——------- — 6-5
i;. Geometrical Fi’operties' of the Invariant-Polarity-------^------ 6-7
5. The (1-2) Matrix Representation of ------------------------ 6-9
6. Linear Families of Correlations ———------ 6-10
7. The Representation of Collineations in ------ -
Chapter VII. TENSOR COORDINATES OF LINEAR SPACES
1. Introduction -----— -----—------------ -—-— ----------------- 7-1
2. Definition of the Coordinate Tensors--------------------------—— 7-2
3. The Quadratic Identities------ -------------------------------------— 7-i^
ij.. Joins and Intersections —------ —------ -—-————-------—- 7-7
5. The Quadratic Form ———------------ 7-11
6, Linear Spaces on a Quadric---------------------———7-I6
Chapter VIII. REPRESENTATION OF LINEAR SPACES OH A QUADRIC IN P«^ , Page
1, The Projective Space P2y„i ------------------------------------------Zi-
8-1
j 2, The Correspopdence between Tensor Sets and Matrices ---------— 8-3
3. The Collineations Corresponding to Linear Spaces on the
Quadric —-------------- ----------------—----------------------------- 8-^
ii. Spaces in P2>^„i Determined by Spaces on the Quadric in P» . 8-7
5. Piroperties of R« and Na —---------------------------------------- -— 8-9
6. Geometry of a Generalization of the Pluecker-Klein Cor
respondence —~—---------------------------------------------- - 8-lii
7. Collineation Representatipn of for >'> 2 --------------------- 8-1$
8. Matrix Representation of H^v/ fdr^^ a/ > 2 ---------------------
8-20
9. Representations of and -------------------------------------------- 8-2^
Chapter IX. THE LORENTZ GROUPS
1. Definition of the Lorentz Groups--------------------------------- 9-1
2. Definition and Spinor Representation of the Antiorthogonal
Group A.----------------------------------------------------------- 9-3
3. The Invariant Antiinvolution and 'Antipblarity —-—---------- 9-6
li. Reality of the Spinor Representation of Lpy ——-----— 9-10
5>. -Sets in which' Eabh Matrix is Real or P№e’imaginary —
9^12
6. Spatial and Tei^oral Signatures of Lorentz Matrices —— 9-13
7. The Invariant Spinors Associated with L〇 , ------------------- 9-17
¿vjs
1-1
Chapter I
SPINCES MB PROJECTIVE GEOMETRY
THE MIHKOi^KI SPACE REPRESffl'pD BY HEEMITIAH MATRICES
1. If a fixed origin is specified in the Minkowski space of the
special theory of Relativity it becomes a real vector (centered affine) space, R]^,
of foxir dimensions which contains a special invariant locus, the light cone, with
vertex at the origin. The points of R|^ may be put in a (1-1) correspondence with
the two-row Hermitian matrices. To do this we recall that an Hermitian matrix
is defined ty the conditions
(A, B = 1, 2)
(1.1)
where thenar d^otes the complex conjugate, and hence and are real
while 'i'21 congjlex conjugates of one another. Every Hermitian
matrix of order two is therefore expressible in terms of four real parameters
(X^, X^, X^, X^) 137 means of the equations
^12
(1.2)
'i'22
This correspondence is (1-1), the inverse equations being
The light-cone is the locus of those points of Rj^ which correspond to
singular Hermitian matrices. For,
and hence a matrix\(icar which ^ corresponds under (1.3)
I
1
1-2
to a t>qint of Rj^ which lies on the cone
(1.^) g-..xV = - (X^)^ - (X^)^ - (X^)^ + (X^)^ = 0. ；
J-J
In the abhreryiated expression for the invariant quadratic form in (1^5) we are
using the summation convention which is customary in relativity theory and de
fining tbe numberp g.. by the matrix equation
-1 0 0 0
0 -1 0 0
<1^6)
0 a -1 0
0 0 0 +1
We also use the convention that small Latin letters i, etc., take on the values
1, 2, 3, h.
The rofws of a singular matrix are proportional so that
condition (l.l) implies for a non-zero matrix that
~ real and positive. I'^iting ^ we have the re
sult that every two-row singular Hermitian matrix is of one of the forms
Hi Ml ^1^2
(1.7) + ^2^1 ^2^2 9
•v>
the zero matrix being the only one which can be written in both foi*ras.
The points of the light cone, exclusive of the vertex (O, 0, 0, O),
thus fall into two distinct elapses corresponding to -the two possible signs in
(1.7). The points of one class, those constituting the futtpe branch of the
light-cone, are parameterized by the equations
(^1^2 ~
the other branch being given by changing the sign of the right member in each
1-3
of these equations. The points of each branch clearly form a continuous fainily
while the two branches are connected only through the vertex (O, 0, 0, Q)^
Points on the futxire branch of the light-cone are characterized in terms of
(X^, X^, X^) by the conditions
(1-9) > 0,
and those on the past branch by
(1.10) < Oc
Equations (1.8) give a correspondence^
(l.U)
^ point on future branch of light cone^
between pairs of cor^lex numbers, not both zero, and points on the future branch of
the light-cone 9 This correspondence is not (1-1) for (e^®v^^, ®
real, cdrresponds to the same point as does correspondence
i© i@
(1.12) (e e ^ point on future branch of light cone.
between families of pairs of complex numbers and points on the future (or the
past) branch of the light-cone is (1-1).
Multiplying and by the same real number r multiplies each of
1 2
the coordinates X ly r so that the pairs of complex numbers (^y^j ^
where ^ is a con5)lex parameter, correspond to the points of a ray (i.e. a half
line) on the futiire branch of the cone. Since each generator of the cone con
sists of two collinear rays through the vertex and is determined by either 〇f
them, there is a (l-l) correspondence.
(1-13) ^ line on the light-cone,
between the sets of numbers (lines of the Ught-cone.
1-U
The points of Rj^ which do' not lie on the light-cone fall into three
disconnected sets, the absolute future, the absolute past, and the absolute
elsewhere, which are characterized algebraically by the conditions
(l.lU) 2 >0, X^ > 0,
(1.15) 2 1 = g^jXV >0, X^ < 0,
and
(1.16) = g^.xV < 0,
respectively# '
If a point in the absolute future is joined to a point in the absolute
past by a continuous curve there'must be at least one point on it for which
X^ = 0. For such a point g^^X^X^ < 0 and the equality sign holds only for
the vertex pf the cone, (O, 0, 0, O). Hence the cturve either passes through
y'
the vertex or through a point in the absolute elsewhere. A schematic repre
sentation of Rj^ is given by the figure:
Figure 1.
t
1-5
The division qf into regions is easily described in terms of the
Hemitian matrices isrhich correspond to the points of Rj^. The condition (1®!)
♦ \
implies that the Hermitian form
(1.17) ^¿3
1 2
is real for all values of the complex-variables Z , Z 9 In particular, iiie
point (0, 0,■〇, 1) of R|^ lies in the absolute future and corresponds under (1.2)
to the Hermitian form (2^^ + Z^Z^) which is always > 0〇 if we agree to
vi2 12'
exclude (0, o) as a possible pair of values for (Z , Z ). Similarly, the point
(0, 0, 1, 1) lies on the future branch of the light-cone and corresponds to the
form yf2 Z^^ which is always > Oj the point (0, 0, 1, O) lies in the absolute
elsewhere and corresponds to the form —i- -2 2
Z Z ) which assumes all real
^ 1,1
-
valuesj the point (0, 0, -1, -1) corresponds to the form - /2 which is
X -1 1 —P P
always < Oj ^d the point (0, 0, 0, -1) corresponds to---- (-Z Z - Z Z ) which
is always <0.
If an Hermitian form assumes one positive value it assumes all positive
1 2
values since multiplying Z and Z by a real number r multiplies the form ly the
2
positive number r . A similar result holds for negative values and hence the
range of values of an arbitrary Hermitian form is identical with the range of
values of one and only one of the typical foanns
(ipi8) zV^ -z^^, zh^ + Z^Z^з -Z^^ - Z^Z^, Z^^ - Z^Z^.
Hence an Hermitian form in two variables falls into one of five distinct classes
according as its range of values is > 0, < 0, >0,<0, or = 0.
The range of values assumed ly an Hermitian form is unchanged if we
make the linear substitution
A
(1.19) Z"^ =
B
1-6
with ^ 0. On making this substitution in (1.17) we find that
'i'ÀB ■ ?ÂB
where
(1.20) icD'
To determine a substitution (1.19) which will reduce (1.17) to one of
the typical forms (1,18), we proceed as follows. Choose such that
*^1 ^1 ^/^1 ^ which is possible since we are assuming ^ 〇» Then
let O) be a solution of the single equation = 0 (and hence
not a multiple of ) and write Qg Qg ^ P 2’ trans
formation (1?19) with ^ and P^^ = will reduce to
wV or -W^V'e If ^2 ^ transformation (1.19) with P^'^ =
p2^ = P2^^2 reduce Z^Z® to + (iîV + W^) or + (wV* - wV).
¥e do not need to include both of the forms and -w\?^ + in our
list of canonical forms since interchanging thé variables carries one into the other.
It follows that an arbitrary Hermltian matrix || || 0) can be
transformed under (1.20) into one and only one of the matrices
1 0 -1 0 1 〇i -1 0 1 0
(1,21)
3 9 > 9
0 0 0 0 0 11 0 -1 0 -1
and the matrix is then said to be of signature (+), (-), (++), (—), or (+-)g re
spectively. From (1.20)
1?Ав1 " I^A I I^B I I'^CdI ’
and hence a matrix of signature (+-) is characterized hy the condition | < 〇•
Referring to (I.I6) we see that the'absolute elsewhere is the locus of points
corresponding to matrices of signature (+-)« Since (1.7) with the plus sign is
