A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons

<table class="fixed-width table-group" id="tab2"><tr><td><table class="table"><colgroup><col style="width:14.65em"/><col style="width:25.80em"/><col style="width:11.31em"/></colgroup><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left">Continuum integration</td><td class="align_center">Discretized integration</td><td class="align_center">PC matrices</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left"> Heuristic Lebesgue measure</td><td class="align_center">Lebesgue measure (finite dimension, Whitney discretization)</td><td class="align_center">?</td></tr><tr><td class="align_left">Integral to be studied</td><td class="align_center">Product measure on <svg height="12.4646pt" id="M155" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -12.2582 43.8124 12.4646" width="43.8124pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g 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710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path></g><g transform="matrix(.0091,0,0,-0.0091,34.718,-5.741)"><path d="M381 704H317L48 -163H112L381 704Z" id="g50-48"></path></g><g transform="matrix(.0091,0,0,-0.0091,38.626,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg> (<svg height="8.8423pt" id="M156" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 9.02496 8.8423" width="9.02496pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path></g></svg> compact), on the discretization by holonomies</td><td class="align_center">Measures on <svg height="7.35473pt" id="M157" style="vertical-align:-0.3499303pt" version="1.1" viewbox="-0.0498162 -7.0048 26.6711 7.35473" width="26.6711pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" id="g113-111"></path></g><g transform="matrix(.013,0,0,-0.013,9.432,0)"><path d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z" id="g117-42"></path></g><g transform="matrix(.013,0,0,-0.013,19.968,0)"><path d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" id="g113-111"></path></g></svg> PC matrices</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Advances in Mathematical Physics

tab2

Table 2

Table 2: A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons 