Quantum Field Theory, Conformal Group Theory, Conformal Field Theory : Mathematical and Conceptual Foundations, Physical and Geometrical Applications

Mathematical and Conceptual Foundations, Physical and Geometrical Applications

By: R Mirman

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Hardcover | 1 December 2000

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Conformal groups illustrate and emphasise how rich group theory is, something usually not recognised, and they also emphasise how fundamental geometry is in physics (and conversely). Reasons, and implications for physics, are explored as a start to the study of the vast fields of mathematics and physics suggested and required. These groups have non-linear transformations, ones with singularities. Other aspects also show richness beyond what is usually realised, giving possibilities of many new insights and applications, to physics, mathematics, group theory, geometry, (non-linear) differential equations, special functions, and likely elsewhere, including optics, squeezing, quantum communication, quantum computation, cryptography, and so on. There are also suggestions about relations to the well-known elementary -- particle mass-level formula, which is reviewed. And representations of the groups for dimension greater than two have Clifford algebras as entries in their matrices. The formalism developed for quantum field theory, as a foundation for the study of conformal field theory, itself has important applications. For example while the obvious reason that the proton cannot decay -- baryon number thus lepton number must be conserved -- is almost trivial, the formalism allows a mathematical proof. Other insights provided by this and the conformal group can also be seen. How much of physics does geometry (through its transformation group) determine? As we see, a very large part. Might the geometry in which physics takes place, in which we exist, completely determine the physics within it, and conversely? These studies are motivated by that, and so attempt to raise questions, to show possibilities, to stimulate. These subjects are indeed stimulating.

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Non-Fiction Mathematics Groups & Group Theory Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory

Preface	p. i
Foundations, geometry, group theory, field theory	p. v
The unappreciated richness of group theory	p. vi
The conformal group, geometry and physics	p. vii
Acknowledgements	p. viii
Why Conformal Groups are Relevant	p. 1
Conformal invariance of geometry and physics	p. 1
The relevance of groups even without invariance	p. 2
What is transformed?	p. 3
The reason for the appearance of the rest mass	p. 3
Who makes measurements	p. 5
The conformal group	p. 8
Why the conformal group is fundamental	p. 9
Physics is dilatationally invariant	p. 9
Transversion transformations and geometry	p. 15
Why the conformal group is related to geometry	p. 17
Transversions and constant acceleration	p. 23
We need consider only two-dimensional space	p. 24
Derivation of the equilateral hyperbola as the path of constant acceleration	p. 24
Energy and momentum along the path	p. 28
Accelerated statefunctions	p. 30
Why constant acceleration is related to the conformal group action on space	p. 35
The conformal group and the Poincare representation	p. 36
The spin label	p. 36
The mass of an object seen by an accelerated observer	p. 38
The conformal group leaves the Poincare representation unchanged	p. 38
Consequences of constant acceleration come from the full theory	p. 42
Transversions as point-dependent dilatations	p. 43
Why does geometry require that physics be invariant under transversions?	p. 43
Why are these point-dependent dilatations and not accelerations?	p. 44
Transversions are dilatations that vary with space	p. 44
Invariance of angles makes the conformal group the largest physical group	p. 45
Transversions are unobservable	p. 46
Determining whether a line is straight	p. 49
What are the conformal group basis states?	p. 50
Is the singularity of the transformation relevant?	p. 52
Can points at finite and infinite distances from the origin be correlated?	p. 52
The conformal group and causality	p. 53
The conformal group and interactions	p. 55
Do physical objects have to produce gravitational fields?	p. 56
Nonlinear momentum operators	p. 58
How the transversion field is related to other objects	p. 60
How these relate to realistic interactions	p. 63
Moebius Groups	p. 66
Conformal groups, geometry, and realizations	p. 66
The conformal transformations in the complex plane	p. 67
Expressions for the conformal transformations	p. 68
Invariance of angles	p. 73
Why are there two special conformal transformations in the plane?	p. 74
The effect of transversions on the axes	p. 77
Transformations among lines and circles	p. 77
The equation for a circle	p. 78
How transversions act on circles	p. 79
Action of transformations using parametric forms for circles	p. 81
Transformations of hyperbolas	p. 82
How acceleration is changed by transversions	p. 88
When do straight lines go into straight lines?	p. 89
How special conformal transformations differ	p. 93
Elliptic coordinates	p. 96
The invariants in elliptic coordinates	p. 96
Dirac's equation in elliptic coordinates	p. 97
The Moebius group in terms of Clifford algebras	p. 98
The basis of a Clifford algebra	p. 98
Clifford algebras and orthogonal-group Lie algebras	p. 99
The generalization of involutions	p. 100
The conformal group over this space	p. 101
The Lie Algebra of the Vahlen Group	p. 102
The Dirac operator	p. 103
Conformal Groups	p. 106
The conformal group and physics	p. 106
Conformal groups in general spaces	p. 106
The relevant forms of the transformations of the conformal group	p. 107
The transformations of the conformal group	p. 107
Transversions as inversions and translations	p. 108
Realization of conformal generators with internal parts	p. 115
The number of commuting generators	p. 115
Labeling states and representations	p. 117
Using the Poincare group to label states of the conformal group	p. 118
The relationship between the Poincare and conformal algebras	p. 118
Groups with properties similar to those of conformal groups	p. 120
The su(1,1) algebra of D, P and K	p. 120
Relating SU(1,1) generators to those of the simple conformal group	p. 133
The other groups with fifteen parameters	p. 137
The SO(4,2) realization	p. 137
The su(3,1) algebra and its implications	p. 142
Relating coordinates of four and of six dimensions	p. 143
Why the spaces are related as they are	p. 147
Diagonalizing the operators	p. 154
Implications and applications	p. 161
The appearance of discreteness	p. 161
Why generators are nonlinear	p. 162
Using what has been suggested by these groups	p. 164
The unappreciated richness of group theory and possible implications	p. 164
Conformal Field Theory	p. 166
Defining field theory	p. 166
Interactions in quantum field theory	p. 167
Applications and extensions	p. 167
How we mathematically describe fields	p. 168
Where should interactions be inserted into theories?	p. 169
What are the states of a system?	p. 171
What is the meaning of decay?	p. 179
The general form of states	p. 181
Multiparticle states	p. 184
Why these are the expressions for the states	p. 187
Equations governing systems	p. 188
The Dirac equation and its consequences	p. 192
These discussions are purely formal	p. 193
Is quantum mechanics linear?	p. 193
Requirements from transformation groups	p. 194
The effect of the Poincare group	p. 195
Interactions and internal symmetry	p. 198
Restrictions due to dilatations	p. 201
Transformations of coupling constants under dilatations	p. 206
Transformation of the gravitational coupling constant and reasons why	p. 209
Accelerations in terms of the mass levels	p. 212
Differing Interactions Require Baryon and Lepton Conservation	p. 212
How interactions lead to conservation	p. 213
Mathematical analysis	p. 214
Thus leptons are conserved	p. 218
Why the argument for gravitation is different from that for conservation of baryons	p. 219
Questions and implications	p. 220
Physics is constrained, and laws have reasons	p. 221
Symmetry, transformations and physics	p. 221
Lorentz group representations and their meaning	p. 223
The Lorentz group as a subgroup	p. 223
The meaning of Lorentz representation labels	p. 224
Why, physically, is a second label necessary?	p. 224
Translation gives an infinite number of spherical harmonics	p. 225
Arbitrariness of the origin requires second eigenvalue	p. 227
How translations and boosts give matrix elements	p. 228
Types of eigenfunctions of representation-labeling operators	p. 229
The commuting Lorentz labeling operators	p. 229
Eigenfunctions of exponential type	p. 230
Eigenfunctions of rotation type	p. 231
Using recursion relations to find L[subscript zt] eigenfunctions	p. 232
Finite-dimensional Lorentz-group representations	p. 233
Matrix elements of boost generators from a realization	p. 234
Calculation of matrix elements from action on coordinates	p. 235
Matrix elements of rotation generators	p. 235
Boost matrix elements	p. 236
Comparison with the algebraic construction	p. 238
Representations whose basis states are plane waves	p. 239
Expansion of Poincare basis states in terms of Lorentz states	p. 240
Expanding a plane wave as a sum of spherical harmonics	p. 240
Spherical harmonics as sums over plane waves	p. 242
The SO(2,1) subgroup of the Lorentz group	p. 242
Noncompact groups and subgroups	p. 244
Nonlinear realizations and nonlinear differential equations	p. 244
Increasing complexity and richness	p. 245
What information do elementary particle masses give?	p. 246
The importance of spectroscopy and the elementary particle masses	p. 246
The table comparing experiment and the formula	p. 247
Disagreements and implications	p. 249
The formula must be relevant	p. 250
What is it telling us?	p. 250
There are two formulas	p. 251
The expression of Gell-Mann and Okubo	p. 251
Implications of the two formulas	p. 252
Masses and isospin	p. 253
Conclusion and implications	p. 254
Conclusion and implications	p. 255
Table for isospin mass differences in Mev	p. 268
References	p. 269
Index	p. 277
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Quantum Field Theory, Conformal Group Theory, Conformal Field Theory : Mathematical and Conceptual Foundations, Physical and Geometrical Applications

Mathematical and Conceptual Foundations, Physical and Geometrical Applications

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