Three Roads To Quantum Gravity

More documents

Recommendations

Info

KNOTS, LINKS AND KINKS 139 satisfying that there is a complete correspondence between the classical and quantum pictures of geometry. In classical geometry the volumes of regions and the areas of the surfaces depend on the values of gravitational ®elds. They are coded in certain complicated collections of mathematical functions, known collectively as the metric tensor. On the other hand, in the quantum picture the geometry is coded in the choice of a spin network. These spin networks correspond to the classical description in that, given any classical geometry, one can ®nd a spin network which describes, to some level of approximation, the same geometry (Figure 27). In classical general relativity the geometry of space evolves in time. For example, when a gravitational wave passes a surface, the area of that surface will oscillate in time. There is an equivalent quantum picture in which the structures of the spin networks may evolve in time in response to the passage of a gravitational wave. Figure 28 shows some of the simple steps by which a spin network evolves in time. If we let a spin network evolve, we get a discrete spacetime structure. The events of this discrete spacetime are the processes by which changes of the form shown in Figure 28 occur. We can draw pictures of evolving spin networks; they look like Figures 29 to 31. An evolving spin network is very like a spacetime, but it is discrete rather than continuous. We can say what the causal relations are among the events, so it has light cones. But it also has more, for we can draw slices through it that correspond to moments of time. As in relativity theory, there are many different ways of slicing an evolving spin network, so as to see it as a succession of states evolving in time. Thus, the picture of spacetime given by loop quantum gravity agrees with the fundamental principle that in the theory of relativity there are no things, only processes. John Wheeler used to say that on the Planck scale spacetime would no longer be smooth, but would resemble a foam, which he called spacetime foam. In tribute to Wheeler, the mathematician John Baez has suggested that evolving spin networks be called spin foam. The study of spin foam has sprung up since the mid-1990s. There are several different versions presently under study, invented by Mike Reisenberger, by Louis Crane
140 THREE ROADS TO QUANTUM GRAVITY FIGURE 28 Simple stages by which a spin network can evolve in time. Each one is a quantum transition of the geometry of space. These are the quantum theoretic analogues of the Einstein equations. [From F. Markopoulou, `Dual formulation of spin network evolution', gr-qc/9704013. All the papers referenced here as gr-qc/xxxx are available at xxx.lanl.gov.] and John Barrett, and by Fotini Markopoulou-Kalamara. Carlo Rovelli, John Baez, Renate Loll and many of the other people who contributed to loop quantum gravity are now engaged in the study of spin foam. So this is presently a very lively area of research. Figure 32 shows a computer simulation of a world with one space and one time dimension, modelled upon ideas from spin foam theory. This is work of Jan Ambjùrn, Kostas Anagnastopoulos and Renate Loll. These universes are very small, each edge corresponding to one Planck length. They do not always evolve smoothly; instead, from time to time the size of the universe jumps suddenly. These are the quantum ¯uctuations of the geometry. After many years, we have here a real quantum theory of the geometry of spacetime.
Page 1:
“It would be hard to imagine a be
Page 5 and 6:
To my parents Pauline and Michael C
Page 7 and 8:
vi THREE ROADS TO QUANTUM GRAVITY P
Page 9 and 10:
viii THREE ROADS TO QUANTUM GRAVITY
Page 11 and 12:
2 THREE ROADS TO QUANTUM GRAVITY th
Page 13 and 14:
4 THREE ROADS TO QUANTUM GRAVITY Si
Page 15 and 16:
6 THREE ROADS TO QUANTUM GRAVITY co
Page 17 and 18:
8 THREE ROADS TO QUANTUM GRAVITY th
Page 19 and 20:
10 THREE ROADS TO QUANTUM GRAVITY m
Page 21 and 22:
12 THREE ROADS TO QUANTUM GRAVITY I
Page 24:
I POINTS OF DEPARTURE
Page 27 and 28:
18 THREE ROADS TO QUANTUM GRAVITY p
Page 29 and 30:
20 THREE ROADS TO QUANTUM GRAVITY w
Page 31 and 32:
Page 33 and 34:
24 THREE ROADS TO QUANTUM GRAVITY i
Page 35 and 36:
CHAPTER 2 .........................
Page 37 and 38:
28 THREE ROADS TO QUANTUM GRAVITY d
Page 39 and 40:
30 THREE ROADS TO QUANTUM GRAVITY a
Page 41 and 42:
32 THREE ROADS TO QUANTUM GRAVITY s
Page 43 and 44:
34 THREE ROADS TO QUANTUM GRAVITY u
Page 45 and 46:
Page 47 and 48:
38 THREE ROADS TO QUANTUM GRAVITY T
Page 49 and 50:
40 THREE ROADS TO QUANTUM GRAVITY t
Page 51 and 52:
42 THREE ROADS TO QUANTUM GRAVITY a
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
52 THREE ROADS TO QUANTUM GRAVITY i
Page 63 and 64:
54 THREE ROADS TO QUANTUM GRAVITY c
Page 65 and 66:
56 THREE ROADS TO QUANTUM GRAVITY F
Page 67 and 68:
58 THREE ROADS TO QUANTUM GRAVITY f
Page 69 and 70:
Page 71 and 72:
62 THREE ROADS TO QUANTUM GRAVITY J
Page 73 and 74:
64 THREE ROADS TO QUANTUM GRAVITY m
Page 76:
II WHAT WE HAVE LEARNED
Page 79 and 80:
Page 81 and 82:
72 THREE ROADS TO QUANTUM GRAVITY o
Page 83 and 84:
Page 85 and 86:
76 THREE ROADS TO QUANTUM GRAVITY b
Page 87 and 88:
Page 89 and 90:
80 THREE ROADS TO QUANTUM GRAVITY c
Page 91 and 92:
82 THREE ROADS TO QUANTUM GRAVITY g
Page 93 and 94:
84 THREE ROADS TO QUANTUM GRAVITY T
Page 95 and 96:
Page 97 and 98: CHAPTER 7 .........................
Page 99 and 100: 90 THREE ROADS TO QUANTUM GRAVITY F
Page 101 and 102: 92 THREE ROADS TO QUANTUM GRAVITY a
Page 103 and 104: 94 THREE ROADS TO QUANTUM GRAVITY p
Page 105 and 106: 96 THREE ROADS TO QUANTUM GRAVITY d
Page 107 and 108: 98 THREE ROADS TO QUANTUM GRAVITY i
Page 109 and 110: 100 THREE ROADS TO QUANTUM GRAVITY
Page 115 and 116: CHAPTER 9 .........................
Page 147: 138 THREE ROADS TO QUANTUM GRAVITY
Page 155 and 156: CHAPTER 11 ........................
Page 178 and 179: CHAPTER 12 ........................
Page 180 and 181: THE HOLOGRAPHIC PRINCIPLE 171 gener
Page 182 and 183: THE HOLOGRAPHIC PRINCIPLE 173 FIGUR
Page 184 and 185: THE HOLOGRAPHIC PRINCIPLE 175 the g
Page 186 and 187: THE HOLOGRAPHIC PRINCIPLE 177 In th
Page 188 and 189: CHAPTER 13 ........................
Page 190 and 191: HOW TO WEAVE A STRING 181 thing!" I
Page 192 and 193: HOW TO WEAVE A STRING 183 other. Th
Page 194 and 195: HOW TO WEAVE A STRING 185 What is r
Page 196 and 197: HOW TO WEAVE A STRING 187 for a pic
Page 198 and 199:
HOW TO WEAVE A STRING 189 in Einste
Page 200 and 201:
HOW TO WEAVE A STRING 191 p in the
Page 202 and 203:
HOW TO WEAVE A STRING 193 may be th
Page 204 and 205:
WHAT CHOOSES THE LAWS OF NATURE? 19
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
EPILOGUE: .........................
Page 218 and 219:
EPILOGUE: 209 And quantum gravity i
Page 220 and 221:
EPILOGUE: 211 of systems containing
Page 222 and 223:
POSTSCRIPT 213 These cosmic rays ar
Page 224 and 225:
POSTSCRIPT 215 Three explanations h
Page 226 and 227:
POSTSCRIPT 217 light, which is the
Page 228 and 229:
POSTSCRIPT 219 This is good news in
Page 230 and 231:
POSTSCRIPT 221 But there is a secon
Page 232 and 233:
POSTSCRIPT 223 Chopin Soo and Marti
Page 234 and 235:
POSTSCRIPT 225 speed of light with
Page 236 and 237:
GLOSSARY 227 cannot be greater than
Page 238 and 239:
GLOSSARY 229 event In relativity th
Page 240 and 241:
GLOSSARY 231 Newton's gravitational
Page 242 and 243:
GLOSSARY 233 spin The angular momen
Page 244 and 245:
SUGGESTIONS FOR FURTHER READING ...
Page 246 and 247:
SUGGESTIONS FOR FURTHER READING 237
Page 248 and 249:
SUGGESTIONS FOR FURTHER READING 239
Page 250 and 251:
INDEX .............................
Page 252 and 253:
INDEX 243 black hole, 70, 73±4, 17
Page 254 and 255:
INDEX 245 black hole formation, 189
show all

Three Roads To Quantum Gravity

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?