Geometry of Quantum Theory

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Springer Science & Business Media, Dec 3, 2007 - Science - 412 pages
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It was about four years ago that Springer-Verlag suggested that a revised edition in a single volume of my two-volume work may be worthwhile. I agreed enthusiastically but the project was delayed for many reasons, one of the most important of which was that I did not have at that time any clear idea as to how the revision was to be carried out. Eventually I decided to leave intact most ofthe original material, but make the current edition a little more up-to-date by adding, in the form of notes to the individual chapters, some recent references and occasional brief discussions of topics not treated in the original text. The only substantive change from the earlier work is in the treatment of projective geometry; Chapters II through V of the original Volume I have been condensed and streamlined into a single Chapter II. I wish to express my deep gratitude to Donald Babbitt for his generous advice that helped me in organizing this revision, and to Springer-Verlag for their patience and understanding that went beyond what one has a right to expect from a publisher. I suppose an author's feelings are always mixed when one of his books that is comparatively old is brought out once again.
 

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Contents

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE
1
2 THE LOGIC OF A CLASSICAL SYSTEM
6
3 BOOLEAN ALGEBRAS
8
4 FUNCTIONS
12
NOTES ON CHAPTER I
17
PROJECTIVE GEOMETRIES
18
2 ISOMORPHISMS OF PROJECTIVE GEOMETRIES SEMILINEAR TRANSFORMATIONS
20
3 DUALITIES AND POLARITIES
22
2 HILBERT SPACES OF VECTOR VALUED FUNCTIONS
208
3 FROM COCYCLES TO SYSTEMS OF IMPRIMITIVITY
213
4 PROJECTION VALUED MEASURES
217
5 FROM SYSTEMS OF IMPRIMITIVITY TO COCYCLES
219
6 TRANSITIVE SYSTEMS
222
7 EXAMPLES AND REMARKS
228
8 SEMIDIRECT PRODUCTS
236
NOTES ON CHAPTER VI
241

4 ORTHOCOMPLEMENTATIONS AND HILBERT SPACE STRUCTURES
26
5 COORDINATES IN PROJECTIVE AND GENERALIZED GEOMETRIES
28
NOTES ON CHAPTER II
38
THE LOGIC OF A QUANTUM MECHANICAL SYSTEM
42
2 OBSERVABLES
45
3 STATES
48
4 PURE STATES SUPERPOSITION PRINCIPLE
52
5 SIMULTANEOUS OBSERVABILITY
54
6 FUNCTIONS OF SEVERAL OBSERVABLES
62
7 THE CENTER OF A LOGIC
63
8 AUTOMORPHISMS
67
NOTES ON CHAPTER III
70
LOGICS ASSOCIATED WITH HILBERT SPACES
72
OBSERVABLES AND STATES
80
SYMMETRIES
104
4 LOGICS ASSOCIATED WITH VON NEUMANN ALGEBRAS
112
5 ISOMORPHISM AND IMBEDDING THEOREMS
114
NOTES ON CHAPTER IV
122
MEASURE THEORY ON GSPACES
148
2 LOCALLY COMPACT GROUPS HAAR MEASURE
156
3 GSPACES
158
4 TRANSITIVE GSPACES
164
5 COCYCLES AND COHOMOLOGY
174
6 BOREL GROUPS AND THE WEIL TOPOLOGY
191
NOTES ON CHAPTER V
200
SYSTEMS OF IMPRIMITIVITY
201
MULTIPLIERS
243
2 MULTIPLIERS AND PROJECTIVE REPRESENTATIONS
247
3 MULTIPLIERS AND GROUP EXTENSIONS
251
4 MULTIPLIERS FOR LIE GROUPS
259
5 EXAMPLES
275
NOTES ON CHAPTER VII
287
KINEMATICS AND DYNAMICS
288
2 COVARIANCE AND COMMUTATION RULES
293
3 THE SCHRODINGER REPRESENTATION
295
4 AFFINE CONFIGURATION SPACES
300
SPIN
303
6 PARTICLES
312
NOTES ON CHAPTER VIII
315
RELATIVISTIC FREE PARTICLES
322
2 THE LORENTZ GROUP
330
3 THE REPRESENTATIONS OF THE INHOMOGENEOUS LORENTZ GROUP
343
4 CLIFFORD ALGEBRAS
348
5 REPRESENTATIONS IN VECTOR BUNDLES AND WAVE EQUATIONS
356
6 INVARIANCE UNDER THE INVERSIONS
372
7 LOCALIZATION
377
8 GALILEAN RELATIVITY
391
NOTES ON CHAPTER IX
399
BIBLIOGRAPHY
400
INDEX
407
Copyright

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Page 405 - H. Weyl, The Theory of Groups and Quantum Mechanics (Dover Publications, Inc., New York, 1931), pp.
Page 1 - ... spatial coordinates are included in the meaning of the term); but they must be such that, when the values of the q's at any instant are given, the state of the system at that instant is completely fixed. If n is chosen to be as small as possible, consistently with this condition being fulfilled, it is called the number of degrees of freedom of the system. The motion, or change of state, of the system at the same instant is then completely described by the n quantities ql, ..., qn.
Page 405 - integration dans les groupes topologiques et ses applications, Hermann, Paris, 1938.

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