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Matej Pavsic

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

References     353

Index             363
 

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