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Archive for February, 2016

Now that I have the Mittag-Leffler universal autocorrelation model for financial time series, it is worth investigating how this can help us in the relationship between gold and mining stocks.  In this case an equal-weighted mining stock index returns have a correlation of around 0.2 with daily gold returns.  The immediate feature of the mining index is that the autocorrelation is -0.09 while the autocorrelation of gold is 0.1.  Our mining companies are PG, PRU, NEM, GOLD, WAC, R, ROG, RPM, between 2013-07-01 and 2016-01-12.  Let’s start our analysis by looking at the volatility equalized difference.

X - Y \sigma_Y/\sigma_X

We can then look at more refined predictions of volatility using an AR(12) model using stochastic volatility.

Date Gold GoldStocks
2013-07-11 0.0111702994 0.032133729
2013-07-12 -0.0007811359 0.001677091
2013-07-15 0.0020009699 -0.0007413315
2013-07-16 0.0023284814 0.0054855717
2013-07-17 -0.0043634456 0.0018267313
2013-07-18 0.0022717568 0.0091715167
2013-07-19 0.0029322699 0.0123216809
2013-07-22 0.0142415227 0.0196691773
2013-07-23 -0.0004227977 0.0046006564
2013-07-24 -0.0049742659 -0.0154637656
2013-07-25 0.0030502249 0.005804416
2013-07-26 -0.0023858793 0.003639055
2013-07-29 0.002333583 -0.0061121949
2013-07-30 -0.0014406211 0.0051335932
2013-07-31 -0.003847537 0.0122947824
2013-08-01 -0.0005827324 0.0176734877
2013-08-02 -0.0001656917 -0.010361333
2013-08-05 -0.00269305 -0.0003282505
2013-08-06 -0.0065864554 -0.0252888846
2013-08-07 0.0009808454 -0.0079071999
2013-08-08 0.0082323657 0.0216122176
2013-08-09 0.0007286822 0.0014797749
2013-08-12 0.0072203572 0.0105851646
2013-08-13 -0.0044821931 -0.0102097458
2013-08-14 0.0042542939 0.013293179
2013-08-15 0.0088644472 -0.0057906519
2013-08-16 0.0031474208 -0.0026103989
2013-08-19 -0.0016821454 -0.0151049025
2013-08-20 0.0021886851 0.0108624159
2013-08-21 -0.0007917282 -0.0095597944
2013-08-22 0.0001901464 0.018090305
2013-08-23 0.007818548 0.0124746464
2013-08-26 -0.0008098444 -0.0002788575
2013-08-27 0.008337228 -0.0305134502
2013-08-28 -0.0004590043 -0.0092408109
2013-08-29 -0.0018101382 0.0080192986
2013-08-30 -0.0051960079 -0.0109788587
2013-09-03 0.0049192207 0.0055505934
2013-09-04 -0.0068526093 0.0170843872
2013-09-05 -0.0053458109 -0.0060948132
2013-09-06 0.0042819 0.0029344419
2013-09-09 6.26551947857656E-005 0.0066723547
2013-09-10 -0.0071697092 -0.0010108198
2013-09-11 -6.36982227462113E-005 0.0039846922
2013-09-12 -0.0107055512 -0.0183976294
2013-09-13 -0.0072090556 0.0050499916
2013-09-16 0.0031089678 0.0102221403
2013-09-17 -0.0027771666 0.0054541311
2013-09-18 -0.0005642128 0.024866961
2013-09-19 0.0200539129 -0.0080478883
2013-09-20 -0.0118622277 -0.0201378033
2013-09-23 -0.0018288861 -0.0080982845
2013-09-24 -0.0036150336 -0.0049136537
2013-09-25 0.0065375658 0.0027638221
2013-09-26 -0.0039710784 0.0023614777
2013-09-27 0.0049192774 -0.0065308004
2013-09-30 -0.0039034601 -0.002742566
2013-10-01 -0.0135634559 0.0030566506
2013-10-02 0.0115294396 0.0022455322
2013-10-03 -0.0010205917 -0.0137129795
2013-10-04 -0.002545444 0.006652565
2013-10-07 0.0050105118 -0.0066297703
2013-10-08 -0.0001639033 -0.0187241246
2013-10-09 -0.0057427115 0.0032877764
2013-10-10 -0.0034355489 0.0177568131
2013-10-11 -0.0097187414 -0.0062220233
2013-10-14 0.0028670912 0.00610282
2013-10-15 -0.0011582101 -0.0016327226
2013-10-16 0.0030930123 0.0005707818
2013-10-17 0.0135702019 0.0174025423
2013-10-18 -0.0027662162 0.0059500023
2013-10-21 0.0003962541 0.0002666358
2013-10-22 0.0087567608 0.0194712545
2013-10-23 -0.0027908133 -0.0109764721
2013-10-24 0.0052744381 0.0110778244
2013-10-25 0.0007070061 0.0005461254
2013-10-28 -9.63421785078467E-005 0.0032960864
2013-10-29 -0.0021572295 -9.80796964737498E-005
2013-10-30 0.0012248183 0.0057611152
2013-10-31 -0.0083189515 -0.0118268382
2013-11-01 -0.0034586948 -0.010511605
2013-11-04 0.0004957889 0.0133826798
2013-11-05 -0.0021857173 -0.0041001124
2013-11-06 0.0032085576 0.0066034995
2013-11-07 -0.0030757762 -0.0155023982
2013-11-08 -0.0080058093 0.0084526856
2013-11-11 -0.0011848865 0.0038668926
2013-11-12 -0.003369147 -0.004033195
2013-11-13 -0.0009576508 0.008737251
2013-11-14 0.0060860363 0.0142540038
2013-11-15 0.0003712352 0.0021195769
2013-11-18 -0.0051239776 -0.0085312046
2013-11-19 0.0004094221 0.0029370159
2013-11-20 -0.0053183082 -0.010537335
2013-11-21 -0.0049999277 0.0065947972
2013-11-22 0.0001745767 0.0044890969
2013-11-25 -0.0009855322 -0.0055286193
2013-11-26 5.59766040479914E-005 -0.0049281437
2013-11-27 -0.0012542051 0.0031360518
2013-11-29 0.0043913283 0.0045756634
2013-12-02 -0.010027193 -0.016863737
2013-12-03 -0.0003911442 -0.0085950077
2013-12-04 0.0092915816 0.0092010933
2013-12-05 -0.005360647 -0.0099064387
2013-12-06 -0.0010235723 0.0060132829
2013-12-09 0.0018336592 0.0052111103
2013-12-10 0.0093639836 0.0044234149
2013-12-11 -0.0013451533 -0.016594431
2013-12-12 -0.0113037377 -0.0085827646
2013-12-13 0.003425638 -0.0003573337
2013-12-16 0.0034337298 0.0081894737
2013-12-17 -0.0050195839 -0.0032238049
2013-12-18 0.0017265391 0.0116108478
2013-12-19 -0.0148081475 -0.0101511515
2013-12-20 0.0036594504 -0.0057677677
2013-12-23 -0.0024241101 0.0070036347
2013-12-24 0.0022797663 0.0091853891
2013-12-26 0.0032361885 4.25891867326249E-005
2013-12-27 0.0006085816 9.89043585603745E-005
2013-12-30 -0.0036643478 -0.0068369945
2013-12-31 -0.0005414919 0.0039022161
2014-01-02 0.008194141 -0.0016897026
2014-01-03 0.0047240875 -0.0038944359
2014-01-06 -0.000210431 -0.0057417117
2014-01-07 -0.0029567902 0.0019492449
2014-01-08 -0.0014505389 -0.0038879878
2014-01-09 0.0013798932 -0.0020339541
2014-01-10 0.0061384162 0.006141816
2014-01-13 0.0014603992 -0.004793164
2014-01-14 -0.0019831627 0.0012455141
2014-01-15 -0.0024829885 0.0069208717
2014-01-16 0.0006658541 0.0007832107
2014-01-17 0.0040779124 0.0038094281
2014-01-21 -0.0035179839 0.0067859195
2014-01-22 -0.0011205798 -0.0050110814
2014-01-23 0.0082315065 -0.0022318291
2014-01-24 0.0006875557 -0.0155214693
2014-01-27 -0.0003092654 -0.0145655532
2014-01-28 -0.0043738333 0.0094055063
2014-01-29 0.0039611366 -0.0011847227
2014-01-30 -0.0068737244 -0.0031052941
2014-01-31 -0.0008818023 -0.0157102845
2014-02-03 0.0069634268 -0.023142558
2014-02-04 -0.0030093402 0.0047156301
2014-02-05 0.0019739905 -0.0046720372
2014-02-06 0.0001036461 -0.0001529414
2014-02-07 0.0019645909 0.0166254715
2014-02-10 0.0040390226 0.0044159024
2014-02-11 0.0051143868 0.0118292091
2014-02-12 0.0017473958 -0.0052097001
2014-02-13 0.0017069899 0.0113294692
2014-02-14 0.0061363131 0.0082519193
2014-02-18 0.0019061007 -0.0013440481
2014-02-19 -0.0013136564 -0.0118425592
2014-02-20 -0.001152718 0.0088767095
2014-02-21 0.0022039608 0.0017661712
2014-02-24 0.0046993549 0.014262357
2014-02-25 0.0015228755 0.0046695712
2014-02-26 -0.0047809139 -0.0034030331
2014-02-27 0.0012409355 -0.0059659954
2014-02-28 -0.0033389806 -0.0014212159
2014-03-03 0.0093302377 -0.0004893844
2014-03-04 -0.0040066139 0.0207918987
2014-03-05 0.0007783639 0.0081523472
2014-03-06 0.0037104245 0.0138414445
2014-03-07 -0.0043914166 -0.0083270451
2014-03-10 0.0010696515 -0.0034503623
2014-03-11 0.0016801831 -0.0071980419
2014-03-12 0.0076081798 0.0101988741
2014-03-13 0.0006016695 -0.0077633682
2014-03-14 0.0020835568 0.0137636318
2014-03-17 -0.0019253612 0.0005107366
2014-03-18 -0.0044194483 0.0060448252
2014-03-19 -0.0056935321 -0.0135529812
2014-03-20 -0.0035110457 0.0016771407
2014-03-21 0.0017915793 -0.0014639189
2014-03-24 -0.0081375176 -0.0132024549
2014-03-25 6.62387680541165E-005 0.0025295691
2014-03-26 -0.0026574627 -0.023569954
2014-03-27 -0.0028951513 0.0011393335
2014-03-28 -0.0002147282 0.0070107841
2014-03-31 -0.0034975298 0.014660494
2014-04-01 -0.0012603378 0.0042615905
2014-04-02 0.0036489871 0.0068468274
2014-04-03 -0.0020910391 0.0023871143
2014-04-04 0.0063431178 -0.0115666799
2014-04-07 -0.0017359784 -0.0131497162
2014-04-08 0.0035977658 0.0115453593
2014-04-09 -0.001062901 0.0093319685
2014-04-10 0.0048284837 -0.0175496731
2014-04-11 -0.00049361 -0.0169669707
2014-04-14 0.0027897342 0.014153618
2014-04-15 -0.0089909672 -0.0002668614
2014-04-16 0.001067473 0.0048765907
2014-04-17 -0.0032103226 0.0017634317
2014-04-21 -0.00181629 0.006888539
2014-04-22 -0.0025013917 0.0048588701
2014-04-23 0.0011848865 0.0050847757
2014-04-24 0.0020237431 -0.0089398891
2014-04-25 0.0034188675 -0.0045592543
2014-04-28 -0.0006013772 -0.0051984632
2014-04-29 -0.00090363 0.006399854
2014-04-30 -0.0001340312 0.0005158489
2014-05-01 -0.0042094549 -0.0069889083
2014-05-02 0.0065490491 0.0033044274
2014-05-05 0.0021280838 -0.0063190459
2014-05-06 -0.0002322519 -0.0018668006
2014-05-07 -0.0065876923 -0.001661807
2014-05-08 -0.000404528 0.0262908231
2014-05-09 -3.37276808233256E-005 0.0096305102
2014-05-12 0.0027570076 -0.0016545832
2014-05-13 -0.0003352849 -0.0144387435
2014-05-14 0.0037072311 -0.0082514002
2014-05-15 -0.004109915 -0.009267104
2014-05-16 -6.71502872466867E-005 0.0015738773
2014-05-19 0.0001342902 -0.0018833065
2014-05-20 0.0002684559 0.002077129
2014-05-21 -0.0021860225 0.0053622681
2014-05-22 0.0023201882 0.0074845417
2014-05-23 -0.0011081088 -0.0040165377
2014-05-27 -0.0088857831 -0.0004413401
2014-05-28 -0.0020914975 0.0013223962
2014-05-29 -0.0009251182 0.0048879356
2014-05-30 -0.0037691035 0.0005909602
2014-06-02 -0.000669777 0.0075404977
2014-06-03 0.0001745206 0.0060515515
2014-06-04 -6.97998204497452E-005 0.0063738981
2014-06-05 0.0031299385 0.009695758
2014-06-06 -0.0002773051 -0.0068344643
2014-06-09 0.0004851678 7.80067253961252E-005
2014-06-10 0.0021421092 -0.004057938
2014-06-11 0.0003789505 -0.0016262105
2014-06-12 0.0043854659 0.0057527043
2014-06-13 3.40877110107662E-005 0.0007687073
2014-06-16 0.000408844 0.0037370415
2014-06-17 -0.0011252484 0.0100617477
2014-06-18 0.0002389328 0.005329258
2014-06-19 0.0139023712 0.0087066802
2014-06-20 0.0008254355 0.0006536793
2014-06-23 0.0005933436 -0.0067033562
2014-06-24 0.0009542406 0.0023302088
2014-06-25 0.0004270834 0.0022123483
2014-06-26 -0.0018427434 0.0046955632
2014-06-27 0.0009881562 0.0014492189
2014-06-30 0.0006575239 0.0013635286
2014-07-01 0.0015085378 -0.0038631243
2014-07-02 0.0014054321 0.0040243667
2014-07-03 -0.0033741321 -0.0053336646
2014-07-07 -0.001185518 -0.0071596716
2014-07-08 -0.0001649115 0.0063092447
2014-07-09 0.0025655156 0.0004978802
2014-07-10 0.0048590616 -0.0008405111
2014-07-11 -0.0005841218 0.0043894564
2014-07-14 -0.0100854278 -0.0051634446
2014-07-15 -0.0032024318 -0.0045805871
2014-07-16 0.0009030733 -0.0034400199
2014-07-17 0.005676265 0.0075205423
2014-07-18 -0.0024804611 -0.0030857875
2014-07-21 0.001489976 0.0056660454
2014-07-22 -0.0025193858 -0.00440542
2014-07-23 -0.0005322645 0.0030824132
2014-07-24 -0.0046517057 -0.0002420839
2014-07-25 0.0041854385 -0.0030457177
2014-07-28 8.66303258857393E-005 -0.0126673579
2014-07-29 -0.001608809 0.0019266323
2014-07-30 -0.0010847993 -0.0138525237
2014-07-31 -0.0046239264 -0.0010411501
2014-08-01 0.0041543463 0.0026001898
2014-08-04 -0.0019508304 -0.0030552173
2014-08-05 -0.0012490891 0.0110442413
2014-08-06 0.007640563 0.0015219342
2014-08-07 0.0013929274 0.0035699032
2014-08-08 -0.0004969614 -0.0151606938
2014-08-11 -0.0001326191 0.0039062173
2014-08-12 3.31585785158239E-005 0.004045596
2014-08-13 0.0012912138 0.0068288551
2014-08-14 0.0003965255 -0.0024906885
2014-08-15 -0.003149117 0.007355372
2014-08-18 -0.0023016544 0.0005855924
2014-08-19 -0.0008704641 -0.0004436749
2014-08-20 -0.0005365387 0.0055805654
2014-08-21 -0.0066951114 -0.0039098396
2014-08-22 0.0016325601 -0.0005720557
2014-08-25 -0.0004415461 0.0033748459
2014-08-26 0.0021356287 -0.0003838454
2014-08-27 -0.0006091088 0.0011506343
2014-08-28 0.0023639727 0.003071228
2014-08-29 -0.0010115559 0.0022134042
2014-09-02 -0.0075596526 -0.0081111009
2014-09-03 0.001748356 -0.0051794493
2014-09-04 -0.0013020278 -0.0063412396
2014-09-05 0.0002401407 -0.0055479344
2014-09-08 -0.0045162207 -0.007636588
2014-09-09 -0.0020494639 0.0009178495
2014-09-10 -0.0011156323 -0.0013296393
2014-09-11 -0.0021697271 5.29169213250001E-006
2014-09-12 -0.0026392405 -0.0024755909
2014-09-15 0.0013040299 0.003814224
2014-09-16 0.0005626944 -0.0017925433
2014-09-17 -0.000281256 -0.0053668429
2014-09-18 -0.0031767456 -0.0085156536
2014-09-19 -0.0037000819 -0.0113522446
2014-09-22 0.0004642379 -0.0079867411
2014-09-23 0.0014964629 0.0078549857
2014-09-24 -0.0008544902 -0.0119126049
2014-09-25 0.0009754071 0.0015658838
2014-09-26 -0.0023961529 -0.0104297473
2014-09-29 0.0013066867 -0.0086908077
2014-09-30 -0.0024667155 -0.0128249733
2014-10-01 0.001474564 -0.0015127973
2014-10-02 -0.0001429423 0.0008001391
2014-10-03 -0.0080079823 -0.0163060513
2014-10-06 0.005211162 -0.0001921631
2014-10-07 0.0018307271 -0.0081061774
2014-10-08 -0.0022986199 -0.0075073197
2014-10-09 0.0068951257 -0.0115218451
2014-10-10 -0.0012778596 0.0118162267
2014-10-13 0.0029405376 -0.0027466762
2014-10-14 0.0015156176 0.0141983567
2014-10-15 0.0036788507 0.0073005373
2014-10-16 -0.0012578127 0.0006885913
2014-10-17 -0.0007704605 0.0205862435
2014-10-20 0.0019933832 -0.0161014378
2014-10-21 0.0024355627 -0.0018235897
2014-10-22 -0.0021565204 0.0222896093
2014-10-23 -0.0057565132 -0.0028208268
2014-10-24 0.0009529811 0.0104756479
2014-10-27 -0.0008823181 0.0026918816
2014-10-28 3.53271633093399E-005 -0.0037379027
2014-10-29 -0.0015925741 -0.0050180928
2014-10-30 -0.0094263613 0.0007248454
2014-10-31 -0.0098949104 0.0084988671
2014-11-03 -0.000667746 -0.013333137
2014-11-04 -0.0007803368 -0.0106616677
2014-11-05 -0.0082603671 0.0042537848
2014-11-06 -0.0011766934 -0.0049904922
2014-11-07 0.0102173973 0.0037667201
2014-11-10 -0.0037285126 -0.0101314426
2014-11-11 0.0011966103 0.0033641544
2014-11-12 -0.0014588089 0.0063590673
2014-11-13 0.0008983083 0.0075410576
2014-11-14 0.0089189765 0.0072078194
2014-11-17 -0.0007699284 0.0025085811
2014-11-18 0.0049621685 0.0062615602
2014-11-19 -0.0011624786 -0.0014778771
2014-11-20 -0.0010926569 0.0035156406
2014-11-21 0.0024727543 -0.0114655363
2014-11-24 -0.0006604468 0.0017102975
2014-11-25 0.000544397 -0.0077465392
2014-11-26 -8.70576783706945E-005 0.0025786923
2014-11-28 -0.0079444765 0.00120874
2014-12-01 0.0154824776 0.0050587431
2014-12-02 -0.0067188987 0.000819313
2014-12-03 0.0033544776 -0.0164066574
2014-12-04 -0.0003594558 -0.0108861366
2014-12-05 -0.0062661481 -0.0130490711
2014-12-08 0.0016386429 0.0217725366
2014-12-09 0.0132791467 0.0302423285
2014-12-10 -0.000917499 0.0168157472
2014-12-11 -0.0013444565 -0.0079672967
2014-12-12 -0.0010998845 -0.0138079148
2014-12-15 -0.0052898015 0.0051798811
2014-12-16 -0.004845634 0.0077570258
2014-12-17 7.27217821365755E-005 0.0039989058
2014-12-18 0.0001090598 -0.0025367545
2014-12-19 0.0004359657 -0.0088328154
2014-12-22 -0.0059227878 -0.0059327999
2014-12-23 -0.0006631014 -0.0083309577
2014-12-24 -0.0016621965 -0.0066525391
2014-12-26 0.0079938256 0.0087344198
2014-12-29 -0.0048961869 -0.0057866807
2014-12-30 0.0067452541 -0.0103588194
2014-12-31 -0.0059376056 0.0022249852
2015-01-02 0.0007695386 0.0228191324
2015-01-05 0.0064685672 0.0138198423
2015-01-06 0.0055197038 0.0052171189
2015-01-07 -0.0031096483 -0.0088034105
2015-01-08 -0.0007898877 0.0114396853
2015-01-09 0.0027226337 -0.0148276954
2015-01-12 0.0059233372 -0.0085225967
2015-01-13 0.0005632873 -0.0058142525
2015-01-14 3.5181212859392E-005 0.0320037108
2015-01-15 0.0105307627 0.0101534176
2015-01-16 0.00413503 -0.0131205215
2015-01-20 0.0058445085 0.0053244822
2015-01-21 -0.0001678173 0.0096983064
2015-01-22 0.0023435621 -0.0050619253
2015-01-23 -0.0027129887 -0.0011221545
2015-01-26 -0.0044578052 9.08155600900003E-005
2015-01-27 0.0042359983 0.0057358949
2015-01-28 -0.0018595613 0.025429028
2015-01-29 -0.010607578 0.0162665188
2015-01-30 0.008115712 -0.0148797531
2015-02-02 -0.0007340305 0.0028922079
2015-02-03 -0.0056829507 -0.0029324207
2015-02-04 0.0014448974 0.0050876981
2015-02-05 -0.0006186532 0.0130345937
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2015-08-06 0.0017970006 -0.033932576
2015-08-07 0.0015911155 0.0016980401
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2015-08-13 -0.0031040567 -0.0235570811
2015-08-14 -0.001130722 0.0012651117
2015-08-17 0.0022196673 -0.0052979278
2015-08-18 -0.000583024 -0.0093586271
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2015-08-20 0.0095989006 -0.000827417
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2015-09-01 0.0027916705 -0.0029484845
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2015-09-03 -0.0035019283 -0.0120208004
2015-09-04 -0.0011994421 0.0043449116
2015-09-08 -7.74972310950694E-005 0.0141822467
2015-09-09 -0.0073077774 -0.0216484907
2015-09-10 0.0028674185 -0.0081078821
2015-09-11 -0.0023553948 0.0061383912
2015-09-14 0.0016501167 -0.0066002351
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2015-11-10 0.0001596231 0.0113573963
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2015-11-13 -4.0177111079398E-005 -0.0127396245
2015-11-16 0.0010834795 0.0190920666
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2015-11-18 4.06395435565976E-005 -0.0179039371
2015-11-19 0.0037226625 -0.0008123114
2015-11-20 -0.0006451316 0.0010652423
2015-11-23 -0.0038503333 -0.0172389087
2015-11-24 0.0027999461 -0.0138467871
2015-11-25 -0.0015478841 0.0169314456
2015-11-27 -0.0055889059 0.0152981025
2015-11-30 0.0037665293 -0.0278892243
2015-12-01 -0.0007751982 -0.0093508307
2015-12-02 -0.0039793 0.007108268
2015-12-03 0.0030390471 0.0184839349
2015-12-04 0.0092721031 0.0189750641
2015-12-07 -0.0035800887 0.001828575
2015-12-08 4.03900936727375E-005 -0.0058443481
2015-12-09 0.0004843884 0.0077223893
2015-12-10 -0.0018192488 -0.0155892772
2015-12-11 0.0014963833 -0.0038147903
2015-12-14 -0.0049945126 -0.0101712444
2015-12-15 -0.0007357461 -0.0012600175
2015-12-16 0.006174137 -0.0157212023
2015-12-17 -0.0111112248 -0.0136212811
2015-12-18 0.0063257857 -0.0098920918
2015-12-21 0.0063153554 -0.0095892859
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2015-12-23 -0.0023514882 -0.0251067403
2015-12-24 0.0030786791 -0.0056760446
2015-12-28 -0.0030786791 -0.0115389922
2015-12-29 -0.0001219757 -0.0268610556
2015-12-30 -0.0033473375 0.007854001
2015-12-31 0.0001638847 0.0023829975
2016-01-04 0.0061014558 0.0149319526
2016-01-05 0.0012906235 -0.0268887474
2016-01-06 0.0054029868 0.028706203
2016-01-07 0.0062784948 0.0055588426
2016-01-08 -0.0038985757 0.0003114799
2016-01-11 -0.0006729873 0.0291156789
2016-01-12 -0.0043800126 0.0057870711
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One can extract from Jonah Miller’s Kaluza-Klein paper (jonah-miller-kaluza-klein-theory) the following result:

If the metric for a four (space) dimensional space can be written as:

ds^2 = e^{2 \alpha \phi} g_{\mu\nu} dx^\mu dx^\nu + e^{2 \beta\phi}(dx^4 + A_{\mu} dx^{\mu})^2

then the components of the curvature 2-form are:

R^{zz} = \beta^2 e^{-2\alpha\phi} \box \phi + \frac{1}{4}e^{2(\beta - 2\alpha)\phi}F^2

and other terms and the scalar curvature in the snapshot:

miller-curvature-kaluza-klein

Now we can consider this a generic theorem for four space-dimensional manifolds and specialize to four-sphere hypersurfaces with the following special A_\mu dx^\mu: we take the second fundamental form 3×3 matrix and diagonalize it, setting A_{\mu} = h_{\mu\mu}.  Miller’s arguments for unification of electromagnetism and gravitation is unchanged given that the vector potential is given in this special form.

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Both the Dirac and Schroedinger equation of quantum mechanics are wave equations with potentials.  A natural conjecture is that all such potentials arise as second fundamental form terms from a hypersurface of S^4(1/h).  The formula for Dirac operator for a hypersurface is well known and presented by Hijazi (hijazi-hypersurface-dirac) for example:

hijazi-hypersurface-dirac

 

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Suppose M is a submanifold of S^4(1/h) and we diagonalize the second fundamental form h_{ij} to obtain three real-valued functions \lambda_1(x), \lambda_2(x), \lambda_3(x).  Now use the eigenvectors on a coframe adapted to M, i.e. \omega_1(x), \omega_2(x), \omega_3(x), \omega_4(x) = N(x) the last being the dual of the unit normal.  If the \lambda_i never vanish, then by the Poincare-Hopf theorem the Euler characteristic is zero for M; this is automatic since M is three dimensional.  Any orientable closed three-manifold is parallelizable, so this is not a giant restriction.  Now consider cohomology H^1(M,\mathbf{R}).  If this is zero, we have a homology sphere which is diffeomorphic to S^3 by Poincare Conjecture.  In this case, we can look at the Hodge decomposition of 2-forms \Omega^1(M) = d\Omega^1(M) \oplus \delta \Omega^3(M).  There is a unique 3-form G such that d*d A = d\delta G.  Now consider the heat flow (\partial_t - d\delta)  u = 0 starting at u(0,x) = G.  This will lead to a harmonic solution as t\rightarrow \infty.

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I don’t have a clear answer yet — of course this is a question that led to many failed unified theories both by Weyl and Kaluza-Klein so there is no embarrassment in confusion.  So if we take the analogy to Kaluza-Klein, for example the nice paper of Jonah Miller (jonah-miller-kaluza-klein-theorykaluza-elementaryWeyl-Gravitation_and_Electricity), we find that the metric in 5D spacetime is (Greek indices go in his notaion between 0 and 3 with 0 representing time)

ds^2 = g_{\mu\nu} dx^\mu dx^\nu + (dx^4 + A_\mu dx^\mu)

What would be the analogue for four-sphere?  It would have to be the diagonalized entries of the second fundamental form as A_\mu.

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I have a fairly simple picture of how our actual universe ought to work.  I know that the universe has four macroscopic space dimensions and it is embedded in a large radius (radius 1/h) sphere.  So then I ask myself, is the electromagnetic tensor really the curvature of a U(1) connection?  There’s a physical circle normal to every point in the physical universe in my view.  A nice theorem tells me that the second fundamental form is globally diagonalizable for a hypersurface of S^4.  Then I look at the Gauss and Codazzi equations for the hypersurface
R_{kjih} = c( g_{kh}g_{ji} - g_{ki}g_{jh}) + H_{kh} H_{ji} - H_{ki} H_{jh}

\nabla_k H_{ji} = \nabla_j H_{ki}

Now these H_{ki} are analogous to the vector potential of electromagnetism.  Recall that in the latex U(1) curvature picture one looks at vector potentials \Phi_a and considers F = [\nabla_a + ie \Phi_a, \nabla_b + i e \Phi_b].  In submanifold geometry the second fundamental form terms give us A_i = \sum H_{ij} e_j as a modification to the Levi-Civita connection on the hypersurface and formally are identical in form to the electromagnetic potential.  From this point of view, a natural candidate for the electromagnetic tensor should be from here but this is standard:

hypersurface-gauss

There is thus a natural 2-form that occurs and we want to know whether this could be a good candidate for the electromagnetic 2-form.

 

The easy part of Maxwell’s equation \nabla_{[a} G_{b c]} = 0 follows from the Bianchi identity.https://wordpress.com/post/zulfahmed.wordpress.com/8477

Let \Omega_{abcd} = H_{ad}H_{bc} - H_{ac} H_{bd}.  Then check that

\Omega_{ab[cd;e]} =0 by using the product rule and the Codazzi equation by collecting terms by the non-differentiated factor, i.e.

\Omega_{ab[cd;e]} = \nabla_e H_{ad} H_{bc} + \nabla_c H_{ae}H_{bd} + \nabla_d H_{ac} H_{be} - \nabla_e H_{ac} H_{bd} - \nabla_c H_{ad} H_{be} - \nabla_d H_{ae} H_{bc}

then for example the term with H_{bc} is H_{bc} ( \nabla_e H_{ad} - \nabla_d H_{ae}).

This is still sloppy.  However, the comparison is to the gravitational field equations with the electromagnetic stress-energy tensor.  This reads

R_{ab} - 1/2 R g_{ab} = C T^{ab}

with T^{ab} = (1/\mu_0) [ F^{a k} F^b_k - 1/4 g^{ab} F_{jk}F^{jk}].  I am not sure if this formula has experimental support but it seems strange because it is a function of F rather than it’s potential.  I wonder if the relationship between gravity and electromagnetism is simpler if one considers the electromagnetic potential directly as a real physical object which would make geometric sense.

 

 

 

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A great article on Weyl’s development of electromagnetism is here:  varadarajan-weyl-em.  He introduced the notion of an electromagnetic field being the curvature of a connection on a U(1)-bundle.  Formally, one can consider the standard Levi-Civita connection for a hypersurface M\subset S^4(1/h) and write the connection:

\nabla^{^4}_X Y = \nabla^M_X Y + h(X,Y)$

where the second fundamental form term A=h(X,Y) acts just like the electromagnetic potential in the established connection-on-principal bundle approach and yields F_{\mu\nu} = A_{\mu;\nu} - A_{\nu;\mu} + [A_{\mu},A_{\nu}].  In subkmanifold geometry, this is a known issue (see CriticalPointTheory Chapter 2).  Thus formally, the principal bundle approach is identical to the normal connection approach where the vector potential is simply the shape operator.

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