The Landscape of Theoretical Physics:
A Global View
From Point Particles to the Brane World and Beyond,
in Search of a Unifying principle
Kluwer Academic Publishers, 2001
Contents
(with links to sample pages in which misprints have been corrected)
Preface
Acknowledgments
Introduction
Part I Point Particles
1.The spinless point particle
1.1 Point particles versus worldlines
1.2 Classical theory
1.3 First quantization
Flat spacetime
Curved
spacetime
1.4 Second quantization
Classical
field theory with invariant evolution parameter
The canonical
quantization
Comparison
with the conventional relativistic quantum field theory 45
2. Point particles and Clifford algebra
2.1 Introduction to geometric calculus based on
Clifford algebra
2.2 Algebra of spacetime
2.3 Physical quantities as polyvectors
2.4 The unconstrained action from the polyvector
action
Free particle
Particle in a fixed background field
2.5 Quantization of the polyvector action
2.6 On the second quantization of the polyvector
action
2.7 Some further important consequences of Clifford
algebra
Relativity of signature
Grassmann numbers from Clifford numbers
2.8 The polyvector action and De Witt--Rovelli material
reference system
3. Harmonic oscillator in pseudo-Euclidean space
3.1 The 2-dimensional pseudo-Euclidean harmonic
oscillator
3.2 Harmonic oscillator in d-dimensional
pseudo-Euclidean space
3.3 A system of scalar fields
3.4}Conclusion
Part II Extended Objects
4. General principles of membrane kinematics and
dynamics
4.1 Membrane space M
4.2 Membrane dynamics
Membrane theory as a free fall in M-space
Membrane theory as a minimal surface in an embedding space
Membrane theory based on the geometric calculus in M-space
4.3 More about the interconnections among various
membrane actions
5. More about physics in M-space
5.1 Gauge fields in M-space
General considerations
A specific case
M-space
point of view again
A system
of many membranes
5.2 Dynamical metric field in M-space
Metric of VN from the metric of M-space
6. Extended objects and Clifford algebra
6.1 Mathematical preliminaries 168
Vectors in curved spaces 168
Vectors
in an infinite-dimensional space 177
6.2 Dynamical vector field
in M-space 180
Description with the vector field in spacetime 183
6.3 Full covariance in the space ot parameters $\phi
^A$ 188
Description in spacetime 188
Description in M-space 195
Description in the enlarged M-space 197
7. Quantization 203
7.1 The quantum theory of unconstrained membranes
203
The commutation relations and the Heisenberg equations of motion
204
The Schr\" odinger representation 205
The stationary Schr\" odinger equation for a membrane 210
Dimensional reduction of the Schr\" odinger equation 211
A particular solution to the covariant Schr\" odinger equation 212
The wave packet 217
The expectation values 220
Conclusion 222
7.2 Clifford algebra and quantization
223
Phase space 223
Wave function as a polyvector 225
Equations of motion for basis vectors 229
Quantization of the p-brane: a geometric approach 238
Part III Brane World
8. Spacetime as a membrane in a higher-dimensional
space 249
8.1 The brane in a curved embedding space
249
8.2 A system of many intersecting branes 256
The brane
interacting with itself 258
A system
of many branes creates the bulk and its metric 260
8.3 The origin of matter in the brane world
261
Matter from the intersection of our brane with other branes
261
Matter from the intersection of our brane with itself 261
8.4 Comparison with the Randall--Sundrum model
264
The metric around a brane in a higher-dimensional bulk 267
9. The Einstein-Hilbert action on the brane as the effective action
271
9.1 The classical model 272
9.2 The quantum model 274
9.3 Conclusion 281
10. On the resolution of time problem in quantum gravity
283
10.1 Space as a moving 3-dimensional membrane in
VN
285
10.2 Spacetime as a moving 4-dimensional membrane
in VN 287
General consideration 287
A physically interesting solution 289
Inclusion of sources 294
Part IV Beyond the Horizon
11. The landscape of theoretical physics: a global view 303
12. Nobody really understands quantum mechanics
315
12.1 The `I' intuitively understands quantum mechanics
318
12.2 Decoherence 323
12.3 On the problem of basis in the Everett interpretation
326
12.4 Brane world and brain world 328
12.5 Final discussion on quantum mechanics, and
conclusion 333
13. Final discussion 339
13.1 What is wrong with tachyons? 339
13.2 Is the electron indeed an event moving in spacetime?
340
13.3 Is our world indeed a single huge 4-dimensional
membrane? 342
13.4 How many dimensions are there?
343
13.5 Will it ever be possible to find solutions
to the classical and quantum brane equations of motion and make predictions?
344
13.6 Have we found a unifying principle?
345
Appendices
The dilatationally invariant system of units
347
Index
363
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