Spinors in Hilbert Space

96 Pages·1974·0.824 MB·English

by  P. A. M. Dirac (auth.)

Spinors in
Hilbert Space
ilbert
P. A. M. Dirac
Center for Theoretical Studies
University of Miami
Coral Gables, Florida
PLENUM PRESS • NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
Dirac. Paul Adrien Maurice, 1902-
Spinors in Hilbert space.
"A series of lectures given in 1969 and revised in 1910."
I. Spinor analysis. 2. Hilbert space. I. TItle.
QA4S3.DS 515'.13 74-18371
ISBN 978·1·4757-0036-7 IS8N 978·1·4757-(1034·3 (eBook)
DOl 10.1007/978-1·4757-0034-3
Research supported by the Air Force Office of Scientific Research
Grant Number AF AFOSR 1268-67
A series of lectures given in Miami and later revised
Cl I974 Plenum Pres.!;, New York.
Softcover reprint oftbe hardcover 1st edition 1974
A Division of Plenum Publishing Corporation
227 West 17th Street, New York., N.Y. 10011
United Kingdom edition published by Plenum Press. London
A Division of Plenum Publishing Company, Ltd.
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Contents
Index of Technical Terms .............. .
Introduction
I. Hilbert Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Spinors...................................... 4
Finite Number of Dimensions
3. Rotations in n Dimensions. . . . . . . . . . . . . . . . . . . . . . 5
4. Null Vectors and Null Planes. . . . . . . . . . . . . . . . . . . 7
5. The Independence Theorem. . . . . . . . . . . . . . . . . . . . . 8
6. Specification ofa Null Plane without Its Coordinates 9
7. Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
8. Expression of a Rotation in Terms of an Infinitesimal
Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9. Complex Rotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10. The N oncommutative Algebra . . . . . . . . . . . . . . . . . . 19
II. Rotation Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
12. Fixation of the Coefficients of Rotation Operators . 23
13. The Ambiguity of Sign. . . . . . . . . . . . . . . . . . . . . . . . . 26
14. Kets and Bras ................................ 28
15. Simple Kets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
v
vi CONTENTS
Even Number of Dimensions
16. The Ket Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
17. The Two-Ket-Matrix Theorem.................. 36
18. The Connection between Two Ket Matrices. . . . . . . 39
19. The Representation of Kets . . . . . . . . . . . . . . . . . . . . . 42
20. The Representative of a Simple Ket. General. . . . . . 45
21. The Representative of a Simple Ket. Special Cases. . 48
22. Fixation of the Coefficients of Simple Kets . . . . . . . . 50
23. The Scalar Product Formula. . . . . . . . . . . . . . . . . . . . 53
Infinite Number of Dimensions
24. The Need for Bounded Matrices. . . . . . . . . . . . . . . . . 57
25. The Infinite Ket Matrix . . . . . . . . . . . . . . . . . . . . . . . . 58
26. Passage from One Ket Matrix to Another. . . . . . . . . 62
27. The Various Kinds ofKet Matrices.............. 65
28. Failure of the Associative Law . . . . . . . . . . . . . . . . . . 66
29. The Fundamental Commutators. . . . . . . . . . . . . . . . . 70
30. Boson Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
31. Boson Emission and Absorption Operators. . . . . . . 75
32. Infinite Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 79
33. Validity of the Scalar Product Formula. . . . . . . . . . . 83
34. The Energy of a Boson. . . . . . . . . . . . . . . . . . . . . . . . . 88
35. Physical Application. . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Spinors in
Hilbert Space
Index of
Technical Terms
A- ................. 6 Orthogonal matrix ... 6, 58
<A>................ 20 Orthogonal vectors. . . 7
{A} ................ 6 Orthonormal ket
[IX, PJ + . . . . . . . . . . . . .. 20 matrix. . . . . . . . . . .. 34
[IX, fJJ _ . . . . . . . . . . . . .. 67 Orthonormal vectors. . 9
Bra. . . . . . . . . . . . . . . .. 28 Perpendicular vectors. 7
Bra matrix. . . . . . . . .. 36 Reverse operator. . . .. 23
Complete quarterturn. 12 Right matrix. . . . . . . .. 76
Inverse ket . . . . . . . . .. 44 Rotation ............ 5, 58
Ket. . . . . . . . . . . . . . . .. 28 Rotation operator. . .. 22
Ket matrix . . . . . . . . .. 33 Simple bra .. . . . . . . .. 32
Large matrix. . . . . . . . . 13 Simple ket. . . . . . . . . .. 31
Left matrix. . . . . . . . .. 76 Small matrix. . . . . . . .. 13
L plan. . . . . . . . . . . . .. 77 Squared length of a
~atching . . . . . . . . . .. 58 complex vector. . . . . 7
Normalized vector. . . . 7 Transposed matrix. . . . 6
Null plane. . . . . . . . . . . 7 Well ordered. . . . . . . .. 43
Introduction
1. Hilbert Space
The words "Hilbert space" here will always denote what math
ematicians call a separable Hilbert space. It is composed of
vectors each with a denumerable infinity of coordinates ql'
q2' Q3, .... Usually the coordinates are considered to be complex
numbers and each vector has a squared length ~rIQrI2. This
squared length must converge in order that the q's may specify a
Hilbert vector.
Let us express qr in terms of real and imaginary parts,
qr = Xr + iYr' Then the squared length is l:.r(x; + y;). The x's
and y's may be looked upon as the coordinates of a vector. It is
again a Hilbert vector, but it is a real Hilbert vector, with only
real coordinates.
Thus a complex Hilbert vector uniquely determines a real
Hilbert vector. The second vector has, at first sight, twice as
many coordinates as the first one. But twice a denumerable in
finity is again a denumerable infinity, so the second vector has the
same number of coordinates as the first. Thus a complex Hilbert
vector is not a more general kind of quantity than a real one.
A real Hilbert space is the more elementary concept. A
complex Hilbert space should be looked upon as a real one in
which a certain structure is introduced, namely a pairing of the
coordinates, each pair being then considered as a complex
number. Changing the phase factors of these complex numbers
then provides a special kind of rotation in the Hilbert space.
3
4 SPINORS IN HILBERT SPACE
In a structureless real Hilbert space there are no special
linear transformations. All are on the same footing. This is
the most suitable basis for a general mathematical theory.
The existence of special transformations would complicate
the discussion of the fundamental ideas. We shall therefore
deal with a real Hilbert space, where the vectors have real
coordinates.
2. Spinors
Spinors, like tensors, are geometrical objects embedded in a
space and have components that transform linearly under
transformations of the coordinates of the space. Spinors differ
from tensors in that they change sign when one applies a com
plete revolution about an axis, while tensors are unchanged.
Spinors are thus always associated with an ambiguity of sign.
Spinors exist in a real Euclidean space with any number of
dimensions (greater than one). They can also exist in other spaces
in which the concept of perpendicularity has a meaning, for
example, the Minkowski space of physics. The extra time
dimension occurring here, as compared with a three-dimensional
Euclidean space, is not of great importance in influencing the
theory of the spinors. It is the dimensions that have the ordinary
perpendicularity of Euclidean space that are of dominating
importance.
Hilbert space is just a Euclidean space with an infinite
number of dimensions and which is made precise by a conver
gence condition imposed on the coordinates of the vectors in
it. We shall study spinors in Hilbert space by first studying
spinors in a Euclidean space of n dimensions and then making
n -+ 00.
There are various ways in which one may establish the
theory of spinors in n-dimensional Euclidean space. The way
that has been followed here has been chosen so as to facilitate
the later passage n -+ 00.
Finite Number of
Dimensions
3. Rotations in n Dimensions
Considering a Euclidean space of n dimensions, we have vectors
q with coordinates q, (r = 1,2, ... , n). For the present we shall
restrict the q's to be real, so that the qr are real numbers. The
vector q has the squared length qrq" a summation being under
stood over r. We also write it as the scalar product (q, q).
Consider a rotation of the vector space, each vector q being
changed to the vector q* with coordinates
(3.1)
the R being real numbers. The lengths of vectors are not changed
rs
by the rotation, nor the scalar product of two vectors q and p,
so
In order that this may hold for all vectors q and p we must have
(3.2)
The R may be looked upon as the elements of a matrix R.
rs
5