Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
Подождите немного. Документ загружается.
This page intentionally left blankThis page intentionally left blank
Bibliography
[1]
R
Brauer
and
H.
Weyl: 'Spinors in
n
dimensions',
Am.
J.
Math.
57,425-
449 (1935).
[2]
G. Racah:
'Group
Theory
and
Spectroscopy',
Princeton
lectures,
1951,
reprinted
in
Ergebn. der.
Exakten
Naturwiss,
37,
28-84
(1965).
[3]
W.
Pauli:
'Continuous
Groups
in
Quantum
Mechanics,'
CERN
Lectures,
1956;
reprinted
in
Ergebn. der.
Exakten
Naturwiss.
37,
85-104 (1965).
[4]
L.
Pontrjagin:
Topological Groups,
Princeton
University
Press,
1958.
[5]
RE.
Behrends;
J.
Dreitlein, C. Fronsdal
and
B.W. Lee, "Simple
Groups
and
Strong
Interaction
Symmetries",
Revs. Mod. Phys.,
34,
1-40, 1962.
[6]
M. Hamermesh:
Group Theory and its Applications to Physical Problems,
Addison Wesley, 1962.
[7]
A. Salam:
"The
Formalism
of
Lie
Groups",
Trieste Lectures, 1963.
[8]
H. Boerner:
Representations
of
Groups with special consideration
for
the
needs
of
Modern Physics,
North
Holland, 1970.
[9]
E.C.G.
Sudarshan
and
N.
Mukunda:
Classical
Dynamics
- A Modern
Perspective,
Wiley, New York, 1974.
[10]
R
Gilmore:
Lie Groups,
Lie
Algebras and
Some
of
their Applications,
Wiley, New York, 1974.
[11]
B.G.
Wybourne:
Classical Groups for Physicists,
Wiley, New York, 1974.
[12]
R
Slansky:
'Group
Theory
for Unified Model Building',
?hys.
Rep.,
79C,
1,
1981.
[13]
P. Langacker:
'Grand
Unified Theories
and
Proton
Decay',
Phys. Rep.,
72C,
185, 1981.
[14]
J.
Wess:
'Introduction
to
Gauge
Theory',
Les Houches Lectures, 1981.
[15]
H. Georgi:
Lie Algebras
in
Particle Physics,
Benjamin/Cummings,
1982.
274 Bibliography
[16]
F.
Wilczek
and
A. Zee: 'Families from Spinors',
Phys. Rev.,
D25,
553,
1982.
[17]
P. van Nieuwenhuizen: "Simple
supergravity
and
the
Kaluza-Klein
Pro-
gram,"
Les Houches Lectures, 1983.
[18]
A.P.
Balachandran
and
C.G.
Trahern:
Lectures on Group Theory for Physi-
cists,
Bibliopolis, Napoli, 1984.
[19]
A.O.
Barut
and
R. Raczka:
"Theory
of
Group
Representations
and
Appli-
cations," World Scientific, Singapore, 1986.
Index
A
I
,205
BI,199
Goo
atlas, 43
Goo
compatibility, 43
G
I
,201
D
I
,195
SU(3)
root
system, 191
f
tensor,
75
7r-system, 192
2-Iorm, 58
affine connection,
73
antisymmetric
tensor
representations,
250, 258
arcwise connectedness,
22
associativity, 147
atlas, 43
atlas, maximal, 44
automorphism
of
groups, 130
barycentric
coordinates, 89
basis for a topology,
11
Betti
number,
98
Bianchi identity,
72
bijective, 6
Bohm-Aharonov
effect, 103
boundary
of a chain, 93, 94
boundary
of
a manifold,
81
boundary
of a set, 18
boundary
of
a simplex, 92
canonical coordinates
on
a Lie group,
149
Cartan
classification, 172
Cartan
subalgebra, 173
Cartan-
Weyl form
of
Lie algebra, 178
Cartesian
coordinates, 42
Cartesian
product,
6
chain, 92
chart,
42
closed form, 85,
100
closed interval, 7
closed set, 7
closed simplex, 89
closure, 13
commutator
subgroup, 126
compact
Lie group, 132
compactness,
15
compatibility
of
atlases, 44
compatibility
of
charts,
43
complement, 5
complexification
of
real
Lie algebra, 169
conjugate
elements, 124
connected spaces, 13
connection,
70
connectivity,
21
continuity, 5
continuous function, 17
contractible
space, 29, 114
coset, 125
cotangent
bundle, 111
cotangent
space, 56
covariant derivative,
70,
72
cover, 14
curvature
2-form,
72
cycle, 94
de
Rham
cohomology, 80, 100, 101
de
Rham's
theorem,
103
diffeomorphism, 48
differentiable fibre bundle, 112
differentiable
structure,
44
differential forms in electrodynamics,
60
276
differential forms,
integration
of, 77
differential
map,
54
differentiation
on
manifolds, 47
dimension
of
a
chart,
42
dimension of a space,
41
Dirac algebra, 244
direct
product
of
groups, 131
direct
product
of
representations, 144
direct
sum
of
Lie algebras, 161
discrete topology, 9
dual
vector space, 55
Dynkin
diagram, 192
element, 5
empty
set, 5
equivalence class,
25
equivalence relation, 24
Euler
characteristic, 98
Euler
relation, 99
exact
form, 85, 100
exact
sequence, 95
exterior
derivative, 57, 59
faces
of
a simplex, 89
factor algebra, 159
factor group, 126
fermions, 69
fibre bundle, 105,
108
first
homotopy
group, 26
frame, 67
frame bundle, 112
function
on
a manifold, 50
fundamental
group,
26
fundamental
UIR, 231
generators
of
a representation, 163
global section, 113
group, abelian, 126
group,
commutative,
126
group, definition, 123
group, nilpotent, 128
group, semisimple, 128
group, simple, 128
group, solvable, 127
group, topological, 132
groups, exceptional, 222
harmonic
form, 85, 103
Hausdorff space,
18
Heine-Borel theorem, 14
Index
Hodge decomposition theorem, 85
Hodge dual,
76
homeomorphism, 17
homology group, 95
homomorphism, 27
homomorphism of
groups, 129
homotopy,
21
homotopy
group, higher, 34
homotopy
of
maps,
28
homotopy
type, 29
ideal, 159
image, 6
image of a
map,
95
indiscrete topology, 8
injective, 6
inner
automorphism,
130
integer, 5
integrability conditions, 152
integral
of
a form,
79
integration
of forms,
77
interior
of
a set, 18
interior points
of
a manifold,
81
intersection, 5
invariant subalgebra, 159
invariant subgroup, 125
inverse function,
15
isometry,
77
isomorphism, 27
isomorphism
of
groups, 130
isomorphism
of
Lie algebras, 158
Jacobi
identity, 155
kernel
of
a
map,
95
Laplacian, 84
Levi
splitting
theorem,
171
Levi-Civita
spin
connection,
73
Lie algebra, 155
Lie algebra, abelian, 159
Lie algebra, semisimple,
160
Lie algebra, simple, 160
Index
Lie algebras, exceptional, 213
Lie bracket, 63
Lie group, 132
limit point,
13
linear functional, 55
local coordinates in a Lie group, 145
local section, 112
locally finite cover,
79
locally finite refinement,
79
loop,
21
loop,
constant,
24
loops, based, 23
loops, homotopic, 24
loops, multiplication of,
23
Mobius
strip,
48
manifold, differentiable, 41, 44
manifold, orientable, 48
manifolds, diffeomorphic, 48
manifolds, isometric, 77
metric, 9, 66
metric
spaces, 9
metric
tensor, 66
metric
topology, 10
multiplicities, 225
multiply
connected space, 29
nondegenerate
roots, 173
normal
space, 19
normal
subalgebra, 159
normal
subgroup, 125
one-parameter
subgroup,
149
one-to-one, 6
onto, 6
open
cover, 14
open
disc, 10
open
interval, 7
open
set, 6
open
simplex, 89
orient ability, 48
oriented
simplex,
91
orthonormal
frame, 67
outer
automorphism,
130
paracompact
space,
79
parallelisability, 115
partition
of
unity
,
79
path,
22
path
connectedness,
22
Poincare's
lemma, 103
277
polyhedron
of
a simplicial complex,
91
positive
root,
187
principal bundle,
110, 112
product
manifold, 105
product
space, 105
proper
subgroup, 124
pseudo-orthogonal groups, 259
rational
number, 5
real number, 5
reflexive relation,
24
regular domain,
81
representation, 135
representation
of a Lie algebra, 163
representation, adjoint,
140
representation, complex conjugate, 140
representation,
contragredient,
140
representation, decomposable, 136
representation, fully irreducible, 138
representation, indecomposable, 136
representation, irreducible, 136
representation,
non-unitary, 140
representation,
orthogonal
, 140
representation, unitary, 139
representations, equivalent, 138
Riemannian
manifold, 67
Riemannian
metric,
66
Riemannian
volume form,
75
root
vector, 174
roots, 173
scalar density,
78
Schur's
lemma
, 139
semidirect
product,
131
semidirect sum, 161
separability,
18
set, 5
simple root, 187
simplex, 87
simplicial complex, 87,
90
simplicial homology, 87
278
simply connected space, 29
SO(2l),
195
SO(2l
+
1), 199
solvability, 171
spin connection,
70
spinor representation, 233, 243, 246,
255
spinor, Dirac, 264
spinor,
Majorana,
264, 265
spinor, Majorana-Weyl, 268
spinor, Weyl, 264
Stokes'
theorem
, 80
structure
constants, 152
SU(l
+
1),205
subalgebra, 159
subgroup, 124
subset, 5
surjective, 6
symmetric relation, 24
tangent
bundle,
111
tangent
space,
52
tangent
vector,
51
tensor
field, 53
topological space, 6, 8
topology, 6, 8
torsion 2-form,
71
torsion of affine connection,
73
transitive relation,
24
triangulation, 87
trivial bundle, 110
trivial subgroup, 124
UIR,
140
union, 5
universal covering group, 158
USp(2l), 201
vector
bundle
, 112
vector field, 53
vertices
of
a simplex, 89
vielbein, 67
volume form,
75
wedge
product,
58
weight, dominant, 228
weight, highest, 228
weights, 181, 225
Weyl
group
, 226
Index
This page intentionally left blankThis page intentionally left blank
