## Geometry of Quantum TheoryIt 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

1 | |

6 | |

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 |

400 | |

407 | |

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### Common terms and phrases

abelian analytic automorphism Boolean algebra Boolean subalgebra Borel function Borel group Borel map Borel measure Borel set Borel space Borel structure bounded central extension Chapter classical closed linear manifold cocycle commutes complex numbers Corollary corresponding countable defined denote division ring elements equation equivalence classes exists a Borel finite dimensional finite measure follows g e G G-space geometry Haar measure hence Hilbert space homomorphism identity implies induced inhomogeneous irreducible representation isomorphism lattice lcsc group lemma Lie algebra Lie group logic Lorentz group matrices multiplier Neumann nonnegative nonzero obtain obvious one-one open set orbit orthogonal particle physical projection valued measure prove Quantum Mechanics real number representation of G satisfied self-adjoint operator semisimple sequence shows simply connected spin stability subgroup standard Borel strict cocycle subspace Suppose symmetric system of imprimitivity theorem theory topology transformations trivial unitary operator vector space write

### Popular passages

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.