Journal of Modern Physics, 2018, *, *-*

http://www.scirp.org/journal/jmpISSN Online:****-****ISSN Print: ****-****

Algebra of Classical and Quantum BinaryMeasurements

Carl A. Brannen

Retired, 415 Max Ct, Henderson, Nevada 89011, USAEmail: carl.brannen@wsu.com

How to cite this paper: Bran-nen, C. A. (2018) Algebra of Classi-cal and Quantum Binary Measure-ments, Journal of Modern Physics,*, *-*.http://dx.doi.org/**.****/***.2018.*****Received: **** **, ***Accepted: **** **, ***Published: **** **, ***Copyright c© 2018 by author(s) andScientific Research Publishing Inc.This work is licensed under the Cre-ative Commons Attribution Inter-national License (CC BY 4.0).http://creativecommons.org/licenses/by/4.0/

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Fusce blandit. Aliquamerat volutpat. Aliquam euismod.Aenean vel lectus. Nunc imperdietjusto nec dolor.Etiam euismod. Fusce facilisislacinia dui. Suspendisse potenti. Inmi erat, cursus id, nonummy sed,ullamcorper eget, sapien. Praesentpretium, magna in eleifend eges-tas, pede pede pretium lorem, quisconsectetuer tortor sapien facilisismagna. Mauris quis magna var-ius nulla scelerisque imperdiet. Ali-quam non quam. Aliquam portti-tor quam a lacus. Praesent vel arcuut tortor cursus volutpat. In vi-tae pede quis diam bibendum plac-erat. Fusce elementum convallisneque. Sed dolor orci, scelerisqueac, dapibus nec, ultricies ut, mi.Duis nec dui quis leo sagittis com-modo.Aliquam lectus. Vivamus leo.Quisque ornare tellus ullamcorpernulla. Mauris porttitor pharetratortor. Sed fringilla justo sed mau-ris. Mauris tellus. Sed non leo.Nullam elementum, magna in cur-sus sodales, augue est scelerisquesapien, venenatis congue nulla arcuet pede. Ut suscipit enim velsapien. Donec congue. 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Vestibulum ante ip-sum primis in faucibus orci luctus etultrices posuere cubilia Curae; Ae-nean quis mauris sed elit commodoplacerat. Class aptent taciti so-ciosqu ad litora torquent per conu-bia nostra, per inceptos hymenaeos.Vivamus rhoncus tincidunt libero.Etiam elementum pretium justo.Vivamus est. Morbi a tellus egetpede tristique commodo. Nulla nisl.Vestibulum sed nisl eu sapien cur-sus rutrum.Nulla non mauris vitae wisi posuereconvallis. Sed eu nulla nec erosscelerisque pharetra. Nullam var-ius. Etiam dignissim elementummetus. Vestibulum faucibus, me-tus sit amet mattis rhoncus, sapiendui laoreet odio, nec ultricies nibhaugue a enim. Fusce in ligula.Quisque at magna et nulla com-modo consequat. Proin accumsanimperdiet sem. Nunc porta. Donecfeugiat mi at justo. Phasellus facil-isis ipsum quis ante. In ac elit egetipsum pharetra faucibus. Maecenasviverra nulla in massa.Nulla ac nisl. Nullam urna nulla,ullamcorper in, interdum sit amet,gravida ut, risus. Aenean ac enim.In luctus. 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Fusce auguevelit, scelerisque sollicitudin, dic-tum vitae, tempor et, pede. Donecwisi sapien, feugiat in, fermentumut, sollicitudin adipiscing, metus.Donec vel nibh ut felis consectetuerlaoreet. Donec pede. Sed id quamid wisi laoreet suscipit. Nulla lec-tus dolor, aliquam ac, fringilla eget,mollis ut, orci. In pellentesquejusto in ligula. Maecenas turpis.Donec eleifend leo at felis tinciduntconsequat. Aenean turpis metus,malesuada sed, condimentum sitamet, auctor a, wisi. Pellentesquesapien elit, bibendum ac, posuereet, congue eu, felis. Vestibulummattis libero quis metus scelerisqueultrices. Sed purus.Donec molestie, magna ut luctus ul-trices, tellus arcu nonummy velit,sit amet pulvinar elit justo et mau-ris. In pede. Maecenas euismod eliteu erat. Aliquam augue wisi, facil-isis congue, suscipit in, adipiscinget, ante. In justo. Cras lobortisneque ac ipsum. Nunc fermentummassa at ante. Donec orci tortor,egestas sit amet, ultrices eget, vene-natis eget, mi. 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Proiniaculis lectus.Sed mattis, erat sit amet gravidamalesuada, elit augue egestas diam,tempus scelerisque nunc nisl vi-tae libero. Sed consequat feugiatmassa. Nunc porta, eros in eleifendvarius, erat leo rutrum dui, nonconvallis lectus orci ut nibh. Sedlorem massa, nonummy quis, eges-tas id, condimentum at, nisl. Mae-cenas at nibh. Aliquam et augueat nunc pellentesque ullamcorper.Duis nisl nibh, laoreet suscipit, con-vallis ut, rutrum id, enim. Phasel-lus odio. Nulla nulla elit, molestienon, scelerisque at, vestibulum eu,nulla. Ut odio nisl, facilisis id, mol-lis et, scelerisque nec, enim. Ae-nean sem leo, pellentesque sit amet,scelerisque sit amet, vehicula pel-lentesque, sapien.Sed consequat tellus et tortor. Uttempor laoreet quam. Nullam idwisi a libero tristique semper. Nul-lam nisl massa, rutrum ut, eges-tas semper, mollis id, leo. Nullaac massa eu risus blandit mattis.Mauris ut nunc. In hac habitasseplatea dictumst. Aliquam eget tor-tor. Quisque dapibus pede in erat.Nunc enim. In dui nulla, com-modo at, consectetuer nec, male-suada nec, elit. Aliquam ornare tel-lus eu urna. Sed nec metus. Cumsociis natoque penatibus et mag-nis dis parturient montes, nasceturridiculus mus. Pellentesque habi-tant morbi tristique senectus et ne-tus et malesuada fames ac turpisegestas.Phasellus id magna. Duis male-suada interdum arcu. Integer me-tus. Morbi pulvinar pellentesquemi. Suspendisse sed est eu magnamolestie egestas. Quisque mi lorem,pulvinar eget, egestas quis, luctusat, ante. Proin auctor vehicula pu-rus. Fusce ac nisl aliquam ante hen-drerit pellentesque. Class aptenttaciti sociosqu ad litora torquentper conubia nostra, per inceptos hy-menaeos. Morbi wisi. Etiam arcumauris, facilisis sed, eleifend non,nonummy ut, pede. Cras ut lacustempor metus mollis placerat. Vi-vamus eu tortor vel metus interdummalesuada.Sed eleifend, eros sit amet fau-cibus elementum, urna sapien con-sectetuer mauris, quis egestas leojusto non risus. Morbi non felisac libero vulputate fringilla. Mau-ris libero eros, lacinia non, sodalesquis, dapibus porttitor, pede. Classaptent taciti sociosqu ad litoratorquent per conubia nostra, per in-ceptos hymenaeos. Morbi dapibusmauris condimentum nulla. Cumsociis natoque penatibus et mag-nis dis parturient montes, nasceturridiculus mus. Etiam sit amet erat.Nulla varius. Etiam tincidunt duivitae turpis. Donec leo. Morbivulputate convallis est. Integer ali-quet. Pellentesque aliquet sodalesurna.Nullam eleifend justo in nisl. In hachabitasse platea dictumst. Morbinonummy. Aliquam ut felis. Invelit leo, dictum vitae, posuere id,vulputate nec, ante. Maecenas vi-tae pede nec dui dignissim suscipit.Morbi magna. Vestibulum id puruseget velit laoreet laoreet. Praesentsed leo vel nibh convallis blandit.Ut rutrum. Donec nibh. Donec in-terdum. Fusce sed pede sit ametelit rhoncus ultrices. Nullam atenim vitae pede vehicula iaculis.Class aptent taciti sociosqu adlitora torquent per conubia nostra,per inceptos hymenaeos. Aeneannonummy turpis id odio. Inte-ger euismod imperdiet turpis. Utnec leo nec diam imperdiet lacinia.Etiam eget lacus eget mi ultriciesposuere. In placerat tristique tor-tor. Sed porta vestibulum me-tus. Nulla iaculis sollicitudin pede.Fusce luctus tellus in dolor. Cur-abitur auctor velit a sem. Morbisapien. Class aptent taciti sociosquad litora torquent per conubia nos-tra, per inceptos hymenaeos. Donecadipiscing urna vehicula nunc. Sedornare leo in leo. In rhoncus leo utdui. Aenean dolor quam, volutpatnec, fringilla id, consectetuer vel,pede.Nulla malesuada risus ut urna. Ae-nean pretium velit sit amet me-tus. Duis iaculis. In hac habitasseplatea dictumst. Nullam molestieturpis eget nisl. Duis a massaid pede dapibus ultricies. Sedeu leo. In at mauris sit amettortor bibendum varius. Phasel-lus justo risus, posuere in, sagittisac, varius vel, tortor. Quisque idenim. Phasellus consequat, liberopretium nonummy fringilla, tortorlacus vestibulum nunc, ut rhoncusligula neque id justo. Nullam ac-cumsan euismod nunc. Proin vi-tae ipsum ac metus dictum tem-pus. Nam ut wisi. Quisque tortorfelis, interdum ac, sodales a, sem-per a, sem. Curabitur in velit sitamet dui tristique sodales. Viva-mus mauris pede, lacinia eget, pel-lentesque quis, scelerisque eu, est.Aliquam risus. Quisque bibendumpede eu dolor.Donec tempus neque vitae est. Ae-nean egestas odio sed risus ul-lamcorper ullamcorper. Sed innulla a tortor tincidunt egestas.Nam sapien tortor, elementum sitamet, aliquam in, porttitor fau-cibus, enim. Nullam congue sus-cipit nibh. Quisque convallis.Praesent arcu nibh, vehicula eget,accumsan eu, tincidunt a, nibh.Suspendisse vulputate, tortor quisadipiscing viverra, lacus nibh dig-nissim tellus, eu suscipit risus antefringilla diam. Quisque a liberovel pede imperdiet aliquet. Pellen-tesque nunc nibh, eleifend a, con-sequat consequat, hendrerit nec,diam. Sed urna. Maecenas laoreeteleifend neque. Vivamus purusodio, eleifend non, iaculis a, ultri-ces sit amet, urna. Mauris faucibusodio vitae risus. In nisl. Praesentpurus. Integer iaculis, sem eu eges-tas lacinia, lacus pede scelerisqueaugue, in ullamcorper dolor eros aclacus. Nunc in libero.Fusce suscipit cursus sem. Viva-mus risus mi, egestas ac, imperdietvarius, faucibus quis, leo. Aeneantincidunt. Donec suscipit. Crasid justo quis nibh scelerisque dig-nissim. Aliquam sagittis elemen-tum dolor. Aenean consectetuerjusto in pede. Curabitur ullamcor-per ligula nec orci. Aliquam pu-rus turpis, aliquam id, ornare vitae,porttitor non, wisi. Maecenas luc-tus porta lorem. Donec vitae ligulaeu ante pretium varius. Proin tor-tor metus, convallis et, hendreritnon, scelerisque in, urna. Cras quislibero eu ligula bibendum tempor.Vivamus tellus quam, malesuadaeu, tempus sed, tempor sed, velit.Donec lacinia auctor libero.Praesent sed neque id pede mollisrutrum. Vestibulum iaculis risus.Pellentesque lacus. Ut quis nuncsed odio malesuada egestas. Duisa magna sit amet ligula tristiquepretium. Ut pharetra. Vestibulumimperdiet magna nec wisi. Maurisconvallis. Sed accumsan sollicitudinmassa. Sed id enim. Nunc pedeenim, lacinia ut, pulvinar quis, sus-cipit semper, elit. Cras accumsanerat vitae enim. Cras sollicitudin.Vestibulum rutrum blandit massa.Sed gravida lectus ut purus. Morbilaoreet magna. Pellentesque euwisi. Proin turpis. Integer sollici-tudin augue nec dui. Fusce lectus.Vivamus faucibus nulla nec lacus.Integer diam. Pellentesque sodales,enim feugiat cursus volutpat, semmauris dignissim mauris, quis con-sequat sem est fermentum ligula.Nullam justo lectus, condimentumsit amet, posuere a, fringilla mol-lis, felis. Morbi nulla nibh, pel-lentesque at, nonummy eu, sollic-itudin nec, ipsum. Cras neque.Nunc augue. Nullam vitae quamid quam pulvinar blandit. Nunc sitamet orci. Aliquam erat elit, phare-tra nec, aliquet a, gravida in, mi.Quisque urna enim, viverra quis,suscipit quis, tincidunt ut, sapien.Cras placerat consequat sem. Cur-abitur ac diam. Curabitur diamtortor, mollis et, viverra ac, tempusvel, metus.Curabitur ac lorem. Vivamus nonjusto in dui mattis posuere. Etiamaccumsan ligula id pede. Maecenastincidunt diam nec velit. Praesentconvallis sapien ac est. Aliquamullamcorper euismod nulla. Inte-ger mollis enim vel tortor. Nullasodales placerat nunc. Sed tem-pus rutrum wisi. Duis accumsangravida purus. Nunc nunc. Etiamfacilisis dui eu sem. Vestibulumsemper. Praesent eu eros. Vestibu-lum tellus nisl, dapibus id, vestibu-lum sit amet, placerat ac, mauris.Maecenas et elit ut erat placeratdictum. Nam feugiat, turpis et so-dales volutpat, wisi quam rhoncusneque, vitae aliquam ipsum sapienvel enim. Maecenas suscipit cursusmi.Quisque consectetuer. In suscipitmauris a dolor pellentesque con-sectetuer. Mauris convallis nequenon erat. In lacinia. Pellentesqueleo eros, sagittis quis, fermentumquis, tincidunt ut, sapien. Maece-nas sem. Curabitur eros odio, inter-dum eu, feugiat eu, porta ac, nisl.Curabitur nunc. Etiam fermentumconvallis velit. Pellentesque laoreetlacus. Quisque sed elit. Nam quistellus. Aliquam tellus arcu, adipisc-ing non, tincidunt eleifend, adipisc-ing quis, augue. Vivamus elemen-tum placerat enim. Suspendisse uttortor. Integer faucibus adipiscingfelis. Aenean consectetuer mattislectus. Morbi malesuada faucibusdolor. Nam lacus. Etiam arculibero, malesuada vitae, aliquam vi-tae, blandit tristique, nisl.Maecenas accumsan dapibussapien. Duis pretium iaculis arcu.Curabitur ut lacus. Aliquamvulputate. Suspendisse ut purussed sem tempor rhoncus. Utquam dui, fringilla at, dictum eget,ultricies quis, quam. Etiam semest, pharetra non, vulputate in,pretium at, ipsum. Nunc sempersagittis orci. Sed scelerisque sus-cipit diam. Ut volutpat, dolor atullamcorper tristique, eros purusmollis quam, sit amet ornare antenunc et enim.Phasellus fringilla, metus id feu-giat consectetuer, lacus wisi ul-trices tellus, quis lobortis nibhlorem quis tortor. Donec egestasornare nulla. Mauris mi tellus,porta faucibus, dictum vel, non-ummy in, est. Aliquam erat vo-lutpat. In tellus magna, portti-tor lacinia, molestie vitae, pellen-tesque eu, justo. Class aptent tac-iti sociosqu ad litora torquent perconubia nostra, per inceptos hy-menaeos. Sed orci nibh, scelerisquesit amet, suscipit sed, placerat vel,diam. Vestibulum nonummy vulpu-tate orci. Donec et velit ac arcu in-terdum semper. Morbi pede orci,cursus ac, elementum non, vehic-ula ut, lacus. Cras volutpat. Namvel wisi quis libero venenatis plac-erat. Aenean sed odio. Quisqueposuere purus ac orci. Vivamusodio. Vivamus varius, nulla sitamet semper viverra, odio mau-ris consequat lacus, at vestibulumneque arcu eu tortor. Donec iac-ulis tincidunt tellus. Aliquam eratvolutpat. Curabitur magna lorem,dignissim volutpat, viverra et, adip-iscing nec, dolor. Praesent lacusmauris, dapibus vitae, sollicitudinsit amet, nonummy eget, ligula.Cras egestas ipsum a nisl. Viva-mus varius dolor ut dolor. Fuscevel enim. Pellentesque accumsanligula et eros. Cras id lacus nontortor facilisis facilisis. Etiam nislelit, cursus sed, fringilla in, conguenec, urna. Cum sociis natoque pe-natibus et magnis dis parturientmontes, nascetur ridiculus mus. In-teger at turpis. Cum sociis na-toque penatibus et magnis dis par-turient montes, nascetur ridiculusmus. Duis fringilla, ligula sed portafringilla, ligula wisi commodo felis,ut adipiscing felis dui in enim. Sus-pendisse malesuada ultrices ante.Pellentesque scelerisque augue sitamet urna. Nulla volutpat ali-quet tortor. Cras aliquam, tellusat aliquet pellentesque, justo sapiencommodo leo, id rhoncus sapienquam at erat. Nulla commodo, wisieget sollicitudin pretium, orci orcialiquam orci, ut cursus turpis justoet lacus. Nulla vel tortor. Quisqueerat elit, viverra sit amet, sagittiseget, porta sit amet, lacus.In hac habitasse platea dictumst.Proin at est. Curabitur tempusvulputate elit. Pellentesque sem.Praesent eu sapien. Duis elitmagna, aliquet at, tempus sed, ve-hicula non, enim. Morbi viverraarcu nec purus. Vivamus fringilla,enim et commodo malesuada, tor-tor metus elementum ligula, nec ali-quet est sapien ut lectus. Aliquammi. Ut nec elit. Fusce euismod luc-tus tellus. Curabitur scelerisque.Nullam purus. Nam ultricies ac-cumsan magna. Morbi pulvinarlorem sit amet ipsum. Donecut justo vitae nibh mollis congue.Fusce quis diam. Praesent tempuseros ut quam.Donec in nisl. Fusce vitae est. Vi-vamus ante ante, mattis laoreet, po-suere eget, congue vel, nunc. Fuscesem. Nam vel orci eu eros viverraluctus. Pellentesque sit amet au-gue. Nunc sit amet ipsum et lacusvarius nonummy. Integer rutrumsem eget wisi. Aenean eu sapien.Quisque ornare dignissim mi. Duisa urna vel risus pharetra imperdiet.Suspendisse potenti.Morbi justo. Aenean nec dolor.In hac habitasse platea dictumst.Proin nonummy porttitor velit. Sedsit amet leo nec metus rhoncus var-ius. Cras ante. Vestibulum com-modo sem tincidunt massa. Namjusto. Aenean luctus, felis et condi-mentum lacinia, lectus enim pulv-inar purus, non porta velit nisl sederos. Suspendisse consequat. Mau-ris a dui et tortor mattis pretium.Sed nulla metus, volutpat id, ali-quam eget, ullamcorper ut, ipsum.Morbi eu nunc. Praesent pretium.Duis aliquam pulvinar ligula. Utblandit egestas justo. Quisque po-suere metus viverra pede.Vivamus sodales elementum neque.Vivamus dignissim accumsanneque. Sed at enim. Vestibulumnonummy interdum purus. Maurisornare velit id nibh pretium ul-tricies. Fusce tempor pellentesqueodio. Vivamus augue purus, laoreetin, scelerisque vel, commodo id,wisi. Duis enim. Nulla interdum,nunc eu semper eleifend, enim dolorpretium elit, ut commodo ligulanisl a est. Vivamus ante. Nulla leomassa, posuere nec, volutpat vitae,rhoncus eu, magna.Quisque facilisis auctor sapien. Pel-lentesque gravida hendrerit lec-tus. Mauris rutrum sodales sapien.Fusce hendrerit sem vel lorem. In-teger pellentesque massa vel au-gue. Integer elit tortor, feugiatquis, sagittis et, ornare non, la-cus. Vestibulum posuere pellen-tesque eros. Quisque venenatis ip-sum dictum nulla. Aliquam quisquam non metus eleifend interdum.Nam eget sapien ac mauris male-suada adipiscing. Etiam eleifendneque sed quam. Nulla facilisi.Proin a ligula. Sed id dui eunibh egestas tincidunt. Suspendissearcu.Maecenas dui. Aliquam volutpatauctor lorem. Cras placerat est vi-tae lectus. Curabitur massa lec-tus, rutrum euismod, dignissim ut,dapibus a, odio. Ut eros erat,vulputate ut, interdum non, portaeu, erat. Cras fermentum, felis inporta congue, velit leo facilisis odio,vitae consectetuer lorem quam vi-tae orci. Sed ultrices, pede euplacerat auctor, ante ligula rutrumtellus, vel posuere nibh lacus necnibh. Maecenas laoreet dolor atenim. Donec molestie dolor nec me-tus. Vestibulum libero. Sed quiserat. Sed tristique. Duis pede leo,fermentum quis, consectetuer eget,vulputate sit amet, erat.Donec vitae velit. Suspendisseporta fermentum mauris. Ut velnunc non mauris pharetra varius.Duis consequat libero quis urna.Maecenas at ante. Vivamus var-ius, wisi sed egestas tristique, odiowisi luctus nulla, lobortis dictumdolor ligula in lacus. Vivamus ali-quam, urna sed interdum porttitor,metus orci interdum odio, sit ameteuismod lectus felis et leo. Prae-sent ac wisi. Nam suscipit vestibu-lum sem. Praesent eu ipsum vi-tae pede cursus venenatis. Duis sedodio. Vestibulum eleifend. Nulla utmassa. Proin rutrum mattis sapien.Curabitur dictum gravida ante.Phasellus placerat vulputate quam.Maecenas at tellus. Pellentesqueneque diam, dignissim ac, venenatisvitae, consequat ut, lacus. Namnibh. Vestibulum fringilla arcumollis arcu. Sed et turpis. Donecsem tellus, volutpat et, varius eu,commodo sed, lectus. Lorem ip-sum dolor sit amet, consectetueradipiscing elit. Quisque enim arcu,suscipit nec, tempus at, imperdietvel, metus. Morbi volutpat purusat erat. Donec dignissim, sem idsemper tempus, nibh massa eleifendturpis, sed pellentesque wisi purussed libero. Nullam lobortis tor-tor vel risus. Pellentesque conse-quat nulla eu tellus. Donec velit.Aliquam fermentum, wisi ac rhon-cus iaculis, tellus nunc malesuadaorci, quis volutpat dui magna id mi.Nunc vel ante. Duis vitae lacus.Cras nec ipsum.Morbi nunc. Aliquam consectetuervarius nulla. Phasellus eros. Crasdapibus porttitor risus. Maece-nas ultrices mi sed diam. Prae-sent gravida velit at elit vehiculaporttitor. Phasellus nisl mi, sagit-tis ac, pulvinar id, gravida sit amet,erat. Vestibulum est. Lorem ipsumdolor sit amet, consectetuer adip-iscing elit. Curabitur id sem ele-mentum leo rutrum hendrerit. Utat mi. Donec tincidunt faucibusmassa. Sed turpis quam, sollici-tudin a, hendrerit eget, pretium ut,nisl. Duis hendrerit ligula. Nuncpulvinar congue urna.Nunc velit. Nullam elit sapien,eleifend eu, commodo nec, sem-per sit amet, elit. Nulla lectusrisus, condimentum ut, laoreet eget,viverra nec, odio. Proin lobor-tis. Curabitur dictum arcu velwisi. Cras id nulla venenatis tor-tor congue ultrices. Pellentesqueeget pede. Sed eleifend sagittiselit. Nam sed tellus sit amet lec-tus ullamcorper tristique. Maurisenim sem, tristique eu, accumsanat, scelerisque vulputate, neque.Quisque lacus. Donec et ipsum sitamet elit nonummy aliquet. Sedviverra nisl at sem. Nam diam.Mauris ut dolor. Curabitur ornaretortor cursus velit.Morbi tincidunt posuere arcu. Crasvenenatis est vitae dolor. Viva-mus scelerisque semper mi. Donecipsum arcu, consequat scelerisque,viverra id, dictum at, metus.Lorem ipsum dolor sit amet, con-sectetuer adipiscing elit. Ut pedesem, tempus ut, porttitor biben-dum, molestie eu, elit. Suspendissepotenti. Sed id lectus sit ametpurus faucibus vehicula. Praesentsed sem non dui pharetra interdum.Nam viverra ultrices magna.Aenean laoreet aliquam orci.Nunc interdum elementum urna.Quisque erat. Nullam temporneque. Maecenas velit nibh,scelerisque a, consequat ut, viverrain, enim. Duis magna. Donecodio neque, tristique et, tincidunteu, rhoncus ac, nunc. Maurismalesuada malesuada elit. Etiamlacus mauris, pretium vel, blanditin, ultricies id, libero. Phasellusbibendum erat ut diam. In congueimperdiet lectus.Aenean scelerisque. Fusce pretiumporttitor lorem. In hac habitasseplatea dictumst. Nulla sit amet nislat sapien egestas pretium. Nuncnon tellus. Vivamus aliquet. Namadipiscing euismod dolor. Ali-quam erat volutpat. Nulla ut ip-sum. Quisque tincidunt auctor au-gue. Nunc imperdiet ipsum egetelit. Aliquam quam leo, con-sectetuer non, ornare sit amet, tris-tique quis, felis. Vestibulum anteipsum primis in faucibus orci luc-tus et ultrices posuere cubilia Cu-rae; Pellentesque interdum quam sitamet mi. Pellentesque mauris dui,dictum a, adipiscing ac, fermentumsit amet, lorem.Ut quis wisi. Praesent quis massa.Vivamus egestas risus eget lacus.Nunc tincidunt, risus quis biben-dum facilisis, lorem purus rutrumneque, nec porta tortor urna quisorci. Aenean aliquet, libero sempervolutpat luctus, pede erat laciniaaugue, quis rutrum sem ipsum sitamet pede. Vestibulum aliquet,nibh sed iaculis sagittis, odio do-lor blandit augue, eget mollis urnatellus id tellus. Aenean aliquet ali-quam nunc. Nulla ultricies justoeget orci. Phasellus tristique fer-mentum leo. Sed massa metus,sagittis ut, semper ut, pharetra vel,erat. Aliquam quam turpis, eges-tas vel, elementum in, egestas sitamet, lorem. Duis convallis, wisi sitamet mollis molestie, libero maurisporta dui, vitae aliquam arcu turpisac sem. Aliquam aliquet dapibusmetus.Vivamus commodo eros eleifenddui. Vestibulum in leo eu erat tris-tique mattis. Cras at elit. Cras pel-lentesque. Nullam id lacus sit ametlibero aliquet hendrerit. Proin plac-erat, mi non elementum laoreet,eros elit tincidunt magna, a rhoncussem arcu id odio. Nulla eget leo aleo egestas facilisis. Curabitur quisvelit. Phasellus aliquam, tortor necornare rhoncus, purus urna posuerevelit, et commodo risus tellus quistellus. Vivamus leo turpis, tem-pus sit amet, tristique vitae, laoreetquis, odio. Proin scelerisque biben-dum ipsum. Etiam nisl. Praesentvel dolor. Pellentesque vel magna.Curabitur urna. Vivamus congueurna in velit. Etiam ullamcorper el-ementum dui. Praesent non urna.Sed placerat quam non mi. Pellen-tesque diam magna, ultricies eget,ultrices placerat, adipiscing rutrum,sem.

AbstractThe simplest measurements in physics are binary; that is, theyhave only two possible results. An example is a beam split-ter. One can take the output of a beam splitter and use itas the input of another beam splitter. The compound mea-surement is described by the product of the Hermitian matri-ces that describe the beam splitters. In the classical case theHermitian matrices commute (are diagonal) and the measurementscan be taken in any order. The general quantum situationwas described by Julian Schwinger with what is now known as“Schwinger’s Measurement Algebra”. We simplify his results by re-striction to binary measurements and extend it to include classi-cal as well as imperfect and thermal beam splitters. We use el-ementary methods to introduce advanced subjects such as geomet-ric phase, Berry-Pancharatnam phase, superselection sectors, symme-tries and applications to the identities of the Standard Model fermions.

KeywordsQuantum Mechanics; Schwinger Measurement Algebra; Quantum Thermo-dynamics.

1. Introduction

In 1955, Julian Schwinger began work on the foundations of quantum fieldtheory while employed at Harvard. The result was what is now known as“Schwinger’s Measurement Algebra”. The algebra was described in fourof his 1959-1960 papers: [7, 8, 10, 9]. Schwinger used his algebra to teachintroductory quantum mechanics. He joined the faculty at the Universityof California, Los Angeles in 1972 and his lecture notes from there resultedin two textbooks [12, 11] that many of his students then used when they

DOI: 10.4236/***.2018.***** **** **, 2018

C. A. Brannen

taught introductory quantum mechanics. These textbooks cover the usualsubjects in a standard quantum mechanics class. Where they are distinctis in their introduction to the subject, the algebra that this paper expands.

Schwinger’s first paper “The Algebra of Microscopic Measurements”[7]describes his measurement algebra. His measurements can be thought ofas beam splitters where one is concerned with only one of the exits. Thealgebra relates to how one models a complex beam splitter that consists ofa series of beam splitters connected together by arranging for the outputof one beam splitter to be used as the input of the next. The reader ofSchwinger’s paper will note that while his notation is different, the prop-erties of the elements of his algebra are similar to those of mixed densitymatrices. We will use density matrix notation in this paper. We expandSchwinger’s results to include the classical situation as well as imperfectand thermal beam splitters. To simplify the discussion, we will mostlyconsider binary measurements.

There are two reasons for reading this paper. The first is thatSchwinger’s beam splitter model provides the most direct method of pass-ing from the classical to the quantum domain. Since Schwinger’s methodis completely general for quantum mechanics and quantum field theory,this provides an immediate connection between the classical and quantumsituations and may provide an improved understanding of the foundationsof quantum mechanics for students. In addition, binary measurements arearguably the easiest introduction to quantum physics and we introduce sub-jects that usually require much more preparation such as geometric phase,Berry-Pancharatnam phase and quantum statistics. The second reason isthat some problems in quantum mechanics are far easier to understandin one formulation than in the others.[15] Schwinger’s formulation is par-ticularly useful in putting the symmetries of quantum mechanics on analgebraic basis. We demonstrate this by introducing a Schwinger algebramodel of the elementary fermions.

1.1. Beam Splitters

The physical apparatus we’re considering is a beam splitter. The beamsplitter has a single entrance and two exits. Entering particles must takeone of the two exits. Suppose that the particles are all identical of type “1”and that they do not influence one another and that they arrive at a beamsplitter with a rate |A1|2. We use |A1|2 here (instead of A1) for our ratesin order to match the notation in the physics literature for beam splittersdescribed with quantum mechanics. Our A1 is the square root of a rate soits units are

√particles/second. In general, A1 may be complex.

Suppose that a particle has a probability p1 of exiting the upper exitso that the probability of exiting the lower exit is 1 − p1. Then the rateof particles in the upper exit beam is |B1|2 = p1|A1|2 and the rate at theother exit is (1− p1)|A1|2 = |B1|2 − |A1|2. See Figure 1

If we know the input rate and the upper exit output rate, then wecan obtain the lower exit output rate by subtraction. Accordingly, for theremainder of this paper, we will ignore the lower exit and concentrate onlyon the upper exit. For clarity, our drawings will continue to include thelower exit but we will not write formulas for it.

2

C. A. Brannen

Trivial Beam Splitter

Identical particles

1|A1|2

|B1|2 = p1|A1|2

|A1|2 − |B1|2 = (1− p1)|A1|2

p1

1− p1

Figure 1: Trivial beam splitter splits a beam of type 1 particles arriving at a rate |A1|2. Particles take theupper exit with probability p1 so the output rate for the upper exit is |B1|2 = p1|A1|2 and the output ratefor the lower exit is |A1|2 − |B1|2 = (1 − p1)|A1|2. We’ve written the equations in terms of amplitudes Ajrather than rates |Aj |2 in order to match the standard quantum beam splitter notation.

1.2. Outline

Schwinger noted that what makes quantum measurements different fromclassical is that a quantum measurement of one property (say spin in the +zdirection) can disturb the system so that the results of a previous quantummeasurement (of spin in another direction), no longer applies. Schwingerconsidered compound beam splitters obtained by putting the output of afirst beam splitter into the input of a second beam splitter. He showedthat these experiments can be represented by Hermitian matrices and thata compound beam splitter is represented by the product of its constituentbeam splitter matrices.

In general, Hermitian matrices do not commute. For the Schwingermeasurement algebra this means that changing the order of measurementscan have a physical effect. If in addition to being Hermitian the matricesare also diagonal, then they will commute and the measurement order doesnot matter. In Section 2, we consider classical beam splitter experimentswith imperfections or at temperature. We show that these experiments canbe modeled with diagonal Hermitian matrices, that is, with real diagonalmatrices.

Section 3 continues the analysis to the quantum case by allowing fornonzero off diagonal entries in the matrices. For simplicity, we specialize tospin-1/2 Stern-Gerlach experiments. We find that the quantum situationis a natural extension of the classical situation. We show why quantummechanics uses complex numbers and amplitudes instead of probabilities,and we introduce the ideas of geometric phase, Berry-Pancharatnam phase,superselection sectors, quantum symmetries and quantum statistics.

As a unique formulation of quantum mechanics, Schwinger’s measure-ment algebra can be expected to provide unique applications to Nature.Our analysis of superselection sectors concludes that their algebras areblock diagonal in form. This suggests that we reverse the process. We

3

C. A. Brannen

Two Connected Beam Splitters

Identical particles

1|A1|2

p1|A1|2

|B1|2 = p1q1|A1|2

p1

q1

Figure 2: Two trivial beam splitters connected together. Particles of type 1 arrive at rate |A1|2. The beamsplitters have probability p1 and q1 so the overall probability is the product p1q1 and the output beam hasa rate of |B1|2 = p1q1|A1|2.

can start with an algebra and from it derive the particle content. Weinclude Section 4 as a speculation on the nature of the Standard Modelfermions and dark matter. The paper concludes with a conclusion andacknowledgements.

2. Classical Beam Splitters

The two outputs of the beam splitter of Figure 1 have intensities |B1|2 and|A1|2−|B1|2. These intensities add up to |A1|2, the intensity of the originalbeam. If we combine the two exits back together to obtain an exit beamwith intensity |A1|2 and expect that this beam will be indistinguishablefrom the original beam.

The beam splitter of Figure 1 is described with only a single number, theprobability of a particle leaving the top exit, p1. Suppose we have anotherbeam splitter, this one with the corresponding probability q1. Assumingindependence, a particle will make it through both beam splitter’s top exitswith probability q1p1. See Figure 2 This process can be continued with anynumber of consecutive beam splitters. If we have n beam splitters withprobability p1, the probability of a particle making it through all of themis (p1)n.

2.1. Classical Thermal Beam Splitters

To introduce thermal effects, let’s suppose that a trivial beam splitter hasits two probabilities p1 and 1 − p1, depend on temperature T accordingto a positive energy difference ∆E1 and temperature T via the Boltzmannfactor

p1 ∝ exp(−∆E1kbT

),

1− p1 ∝ exp(+∆E1kbT

).(1)

The proportionality can be determined by the requirement that p1 and1− p1 add to 1. So we divide by the sum of the right hand side to get the

4

C. A. Brannen

probabilities:

p1 = exp(−∆E1kbT

)/(exp(−∆E1kbT

) + exp(+∆E1kbT

),

1− p1 = exp(+∆E1kbT

)/(exp(−∆E1kbT

) + exp(+∆E1kbT

).(2)

In the high temperature limit kbT >> ∆E1, the probabilities approachp1 = 1− p1 = 1/2.

At the low temperature limit, the probabilities go to p1 = 1, p0 = 0and near zero, they are approximately

p1 ≈ exp(−2∆E1kbT

),

p2 ≈ 1− exp(−2∆E1kbT

).(3)

In this cold limit, putting n beam splitters together by taking the exit portsto the input port of the next beam splitter will give a final upper exit portprobability of pn1 and this corresponds to decreasing the temperature fromT to T/n:

(p1)n ≈(

exp(−2∆E1

kbT)

)n= exp

(−2∆E1

kbT/n

), (4)

Thus connecting identical thermal beam splitters has the effect of reducingthe temperature. This temperature effect will also work in the quantumsituation.

2.2. Classical Two Particle Beam Splitters

The reader may be relieved to read that now we consider beam splittersthat act on a particle beam with two kinds of particles, type 1 and type2. The two particles arrive with rates |A1|2 and |A2|2 and the probabilitiesthat they leave via the upper exits are p1 and p2. If the output of thebeam splitter is sent to another identical beam splitter, the probabilitieswill square so that

|B1|2 = p1p1|A1|2 = (p1)2 |A1|2,|B2|2 = p2p2|A2|2 = (p2)2 |A2|2.

(5)

We can rewrite the above equations in 2× 2 diagonal matrix form:(|B1|2 0

0 |B2|2)

=

(p1 00 p2

)(p1 00 p2

)(|A1|2 0

0 |A2|2)

(6)

See Figure 3.The reader may notice that we could have put the |Bj |2 and |Aj |2 num-

bers into vectors instead of matrices. We’re doing it with matrices so thatour presentation will be compatible with the density matrices of quantummechanics. For arguments supporting density matrices as a superior wayof representing quantum states see [16].

With the probabilities for the two particle types being p1 and p2, thecorresponding probabilities for the lower exit will be 1−p1 and 1−p2. Thisis represented by a matrix with those numbers on the diagonal. So if werecombine the lower and upper exit ports the resulting experiment will berepresented by the sum of the two matrices which is the unit matrix:(

p1 00 p2

)+

(1− p1 0

0 1− p2

)=

(1 00 1

)= 1̂. (7)

The unit matrix has no effect on particle rates so, as expected, the effectof recombining the two outputs of a beam splitter is to give back a beamidentical to the original.

5

C. A. Brannen

Two Identical, Classical Beam Splitters

Identical particles

12

|A1|2

|A2|2

p1|A1|2

p2|A2|2

|B1|2 = (p1)2|A1|2

|B2|2 = (p2)2|A2|2(p1 00 p2

)(p1 00 p2

)

Figure 3: Two identical, classical beam splitters connected together. Particles of type 1 and 2 arrive mixedtogether in a single beam with rates |A1|2 and |A2|2. The beam splitters have probability p1 and p2 for thetwo types so their overall probabilities are (p1)2 and (p2)2. This can be expressed with matrix multiplication.

3. Quantum Beam Splitters

In the classical situation, particles do not interfere with each other so wecan use rates such as |Aj |2 to completely describe a mixture. For such asituation, the rates add the same way that probabilities do. For example,if I have two beams each with the same rate |A1|2, combining the beamswill give a beam with twice that rate 2 |A1|2.

In the classical situation, we used real diagonal matrices to representthe beam splitters and the product of two real diagonal matrices is also realdiagonal so real diagonal matrices were sufficient to represent a single beamsplitters as well as a connected series of beam splitters. For the quantumcase, an individual beam splitter will be represented by an Hermitian matrixand products of these matrices are more general than Hermitian.

3.1. Pauli Spin Matrices

Most of this paper is restricted to the subject of beams with only twoparticle types. For quantum mechanics the standard example is su(2) spin-1/2 where the two particle types are typically taken to be spin-up whichwe will label with +z and spin-down which we will label as −z. Therebeing nothing special about the z direction, we can also consider spin-1/2measured in other directions such as ±x, ±y or for spin measured in ageneral direction u.

A matrix ρ+z for measuring spin-1/2 in the +z direction will give spin-up in the upper exit. The reverse measurement ρ−z, for spin in the −zdirection will give spin-down in the upper exit. The (classical or quantum)matrices for these measurements are:

ρ+z =

(1 00 0

),

ρ−z =

(0 00 1

).

(8)

6

C. A. Brannen

The above matrices are idempotents, that is, they are not changed by squar-ing (ρ±z)

2 = ρ±z. This is an important feature of spin measurements; re-peating the measurement gives the same result. And the two matrices sumto 1̂; that is, spin-up and spin-down account for all the possible measure-ments (exit ports). We require that the matrices for spin-1/2 measurementsin any other direction have these features.

Consider a general idempotent 2 × 2 complex matrix where we’ve in-cluded a factor of 1/2 for later convenience:

ρ(a, b, c, d) =1

2

(a bc d

), where ρ2 = ρ. (9)

This gives four equations:

a/2 = (a2 + bc)/4,b/2 = b(a+ d)/4,c/2 = c(a+ d)/4,d/2 = (d2 + bc)/4.

(10)

If we let b = c = 0, we would have the classical case. Dividing the middletwo of the above four equations by b or c implies that a+ d = 2. Applyingthis to the sum of the first and fourth equation we get 1 = (a2 +d2 +2bc)/4.

Now consider spin measurements on the z = 0 plane, that is, the equa-tor of the sphere (cos(θ), sin(θ), 0). These directions are equidistant fromthe ±z = (0, 0,±1) directions so we expect to get equal particle rates whenwe insert one of these measurements between two ρ+z or two ρ−z measure-ments. Computing we find

ρ+z ρ(a, b, c, d) ρ+z = (a/2) ρ+z,ρ−z ρ(a, b, c, d) ρ−z = (d/2) ρ−z.

(11)

And so for ρ(a, b, c, d) to have the same effect on spin-up as it does on spin-down, we must have that a = d. Together with a + d = 2 this shows thata = d = 1 giving

ρ+z ρ(a, b, c, d) ρ+z = (1/2) ρ+z,ρ−z ρ(a, b, c, d) ρ−z = (1/2) ρ−z.

(12)

Having solved for a and d we find that c = 1/b. This gives the generalsolution for matrices for spin measurements on the z = 0 plane as

ρ(b) =1

2

(1 bb−1 1

). (13)

We need different b values for spin-1/2 measurements in the +x and +ydirections. Call these bx and by.

Consider a sequence of measurements that being with spin in the +xdirection, then +y, and finally +x. This complex measurement will end upas a (1/2) multiple of ρ+x just as we found in Equation 12. Calculating wehave

ρ+x ρ+y ρ+x = (1/2) ρ+x,

12

(1 bxb−1x 1

)12

(1 byb−1y 1

)12

(1 bxb−1x 1

)= (1/2)1

2

(1 bxb−1x 1

).

(14)

7

C. A. Brannen

Multiplying out the matrices gives four equations, each of which is equiv-alent to 2 = 2 + bx/by + by/bx or

b2x + b2y = 0, or

bx = iby.(15)

This shows that in order to model 3 dimensions, we will need complexnumbers.

So for any complex number b = bx = iby, with |b| not zero, we obtainthree projection operators;

ρ±x = 12

(1 ±b±b−1 1

),

ρ±y = 12

(1 ∓ib

±ib−1 1

),

ρ±z = 12

(1± 1 0

0 1∓ 1

).

(16)

The above projection operators are of the form ρ±u = 0.5(1̂±σu(b)) whereσu(b) are three matrices:

σx(b) =

(0 bb−1 0

),

σy(b) =

(0 −ib

ib−1 0

),

σz(b) =

(1 00 −1

).

(17)

These matrices each square to 1̂ and anti-commute:

σx(b)2 = σy(b)2 = σz(b)

2 = 1,σx(b)σy(b) = −σy(b)σx(b),σy(b)σz(b) = −σz(b)σy(b),σz(b)σx(b) = −σx(b)σz(b).

(18)

With the choice b = 1, they are called the “Pauli spin matrices” and areHermitian. From here on we will use them:

σx =

(0 11 0

),

σy =

(0 −ii 0

),

σz =

(1 00 −1

).

(19)

Any Hermitian 2 × 2 matrix H can be written using 1̂ and the Pauli spinmatrices as a basis over the real numbers:

H = h0

(1 00 1

)+ h1

(0 11 0

)+ h2

(0 −ii 0

)+ h3

(1 00 −1

),

= h01̂ + h1σx + h2σy + h3σz.(20)

where hj are four real numbers.

8

C. A. Brannen

With the Pauli spin matrices, the projection operators for spin in the±x, ±y and ±z directions are defined as:

ρ±x = (1̂± σx)/2,

ρ±y = (1̂± σy)/2,ρ±z = (1̂± σz)/2.

(21)

One can also define a spin-1/2 matrix for spin in other directions. Given aunit vector u = (ux, uy, uz), the spin matrix is:

σu = uxσx + uyσy + uzσz,

=

(uz ux − iuy

ux + iuy −uz

).

(22)

Like the usual Pauli spin matrices, σu squares to 1̂ and can be used to definethe projection operator for spin-1/2 in the u direction by ρ+u = (1̂+σu)/2.

The assumption that observables are Hermitian is often included as anaxiom of quantum mechanics. The justification is that observables shouldbe real numbers and, since Hermitian matrices have real eigenvalues, theyare a natural choice. As hinted at in Equation 17, one can modify Her-mitian matrices by multiplying a column by a nonzero complex number band the corresponding row by b−1. This transformation is a isomorphismin that it preserves matrix multiplication and addition. The transforma-tion, in general, takes Hermitian matrices to non Hermitian matrices butleaves their eigenvalues unchanged and only slightly modifies their eigen-vectors. So we see that while it is convenient for quantum mechanics touse Hermitian matrices it is not necessary.

Two Hermitian matrices H and K have a product HK that is notin general Hermitian. The product will be Hermitian if and only if thetwo matrices commute. We can write HK = (HK + KH)/2 + (HK −KH)/2 and split the product into an Hermitian part (HK + KH)/2 andan anti-Hermitian part (HK − KH)/2. Multiplying the anti-Hermitianpart by i gives the Hermitian matrix i(HK − KH)/2. For matrices thatrepresent beam splitters, the non commutative case corresponds to thephysical situation where the order of measurements has an effect as wediscuss in Subsection 3.2

A perfect beam splitter for spin-1/2 in the +u direction is representedby the matrix ρ+u = (1 + σu)/2. The particles that take the lower exitwill be spin-1/2 in the −u direction so that port will be represented by thematrix ρ−u = (1− σu)/2. These two matrices sum to unity:

ρ+u + ρ−u = (1 + σu)/2 + (1− σu))/2,

= 1̂,(23)

so, as in the classical case, combining the outputs of a single perfect beamsplitter results in an experiment that has no effect on the beam.

Recombining the two exits has the effect of canceling the measurement.This behavior may seem contradictory to the concept that measurementsaffect quantum systems. Perhaps beam splitters can be better describedas devices that split waves rather than devices that make measurementson particles. The traditional notation we use in this paper promotes the“separation fallacy” well described in [2]. To make a measurement requiressomething to absorb the particle and be permanently altered such as a pho-tographic plate. Beam splitters do not measure, they allow measurements.

9

C. A. Brannen

Until measurement the particle is just a wave and can be recombined thesame way that the electric and magnetic fields of a light wave can be re-combined.

3.2. Geometric Phase / Perfect Beam Splitter Calculations

Suppose we have three perfect quantum beam splitters, two for spin-1/2in the +x direction and one for spin-1/2 in the +z direction. We connectthem up so that a particle is measured first for spin in +x direction, thenspin in the +z direction and finally spin in the +x direction. The matrixcorresponding to the three consecutive measurements is given by the prod-uct of the matrices. Computing with the above matrix definitions we have:

ρ+x+z+x = ρ+xρ+zρ+x,= 1

2ρ+x,

=(√

1/2)2

ρ+x.

(24)

Compared with a single measurement of spin-1/2 in the +x direction, the

compound measurement is decreased by a factor(√

1/2)2

. In the second

line, the 1/2 indicates that the compound measurement has a reductionin the particle rate of 1/2, i.e. |B1|2 = (1/2)|A1|2. The rate reduction isthe result of particles having to navigate the change in the measurementsbetween +z and +x. But there are two such navigation changes so each ishaving an effect of

√1/2 as shown in the third line.

Suppose we have four perfect quantum beam splitters, two for spin-1/2 in the +z direction (i.e. ρ+z) and one each for spin in the +x and+y direction (i.e. ρ+x and ρ+y). We connect the beam splitters so aparticle first enters one of the +z spin-1/2 measurements, then if it reachesthe upper exit it is made to enter a beam splitter for the +x spin-1/2measurement, then the +y spin-1/2 measurement and finally it enters thesecond +z spin-1/2 measurement. Computing, we find:

ρ+z+x+y+z = ρ+zρ+yρ+xρ+z,= 1−i

4 ρ+z,

= ei(1/2) (−π/2)(√

1/2)3

ρ+z

(25)

The final line in the above shows that the results of connecting the fourbeam splitters is a complex multiple of the last (or first) beam splitter andso is not Hermitian. There are a number of observations to make here.The first is that since the initial and final beam splitters are identical, thecompound beam splitter is a complex multiple of that initial / final beamsplitter. This is a general fact about products of primitive idempotentmatrices that begin and end with the same primitive idempotent matrix.

A second observation is the factor(√

1/2)3

. As with Equation 24, this

comes from the particles having to navigate three changes in measurement:+z to +x, then from +x to +y, and finally from +y to +z. More generally, iftwo perfect quantum beam splitters have directions that differ by θ degrees,combining them will reduce the amplitude by

√(1 + cos(θ))/2. This can

be verified by putting u = (sin(θ), 0, cos(θ)) and computing

ρ+z+u+z = ρ+zρ+uρ+z =

(√1 + cos(θ)

2

)2

ρ+z. (26)

10

C. A. Brannen

That is, there are two transitions, +z to +x and then to +z so the factorappears squared.

The last observation on Equation 25 is in regard to the complex phaseexp(i(1/2) (−π/2)). This is called a “geometric phase” as it depends onthe geometric area included in the path. Our path goes from +z to +x to+y and then back to +z. This is one octant of the surface of the sphere.The total surface of the sphere is 4π steradians so an octant is an eighth ofthis which gives the factor 4π/8 = π/2. The area is oriented and our pathis traversed in the negative direction so the π/2 takes a minus sign giving(−π/2). The remaining factor of 1/2 has to do with this being a path inthe spin of a spin-1/2 particle. The factor 1/2 implies that when the pathis taken around the equator (or any great circle), the quantum phase willbe (1/2)(2π) = π so the particle will get eiπ = −1. This is the −1 oneobtains by rotating a fermion through 360◦. A good exercise for studentsis to find the “remaining factor” for the 1, 0 and -1 cases of spin-1. For oneof these cases, they may find it useful to read Subsection 3.4.

Together, these three observations allow quick computation of the ma-trix corresponding to compound perfect quantum measurements in spin-1/2. For example, suppose the path includes one face of an icosahedron.Letting θ ≈ 65.435◦ be the angle between two adjacent points on an icosahe-dron, the product of the matrices will include a factor of ((1+cos(θ))/2)3/2

for the three changes in spin direction. Since the icosahedron has 20 faces,the geometric phase will be exp(i(1/2)(±4π/20) where the sign depends onwhich direction is taken.

3.3. Quantum Amplitudes

We now rewrite our particle rates |Aj |2 in terms of amplitudes Aj as istraditional for quantum mechanics. We begin with a classical beam split-ter acting on a beam with two particle types. From Subsection 2.2, theincoming particle rates |Aj |2 are related to the exit particle rates |Bk|2 by(

|B1|2 00 |B2|2

)=

(q1 00 q2

)(|A1|2 0

0 |A2|2)

(27)

where qj are probabilities for the passage of the two particle types. Thetwo particle rate matrices are treated the same way; we’ll discuss the |Aj |2matrix. We replace the matrices as follows:(

|A1|2 00 |A2|2

)=

(A1

0

)(A∗1 0

)+

(0A2

)(0 A∗2

). (28)

This factors an entry in the rate matrix into a product of two matrices, one2× 1 and the other 1× 2. These are called bras and kets. The kets are:(

A1

0

)= A1

(10

)= A1|+ z〉,(

0A2

)= A2

(01

)= A2| − z〉,

(29)

while the bras are:(A∗1 0

)= A∗1

(1 0

)= A∗1〈+z|,(

0 A∗2)

= A∗2(

0 1)

= A∗2〈−z|.(30)

11

C. A. Brannen

In most quantum mechanics textbooks, our notation |± z〉 is replaced with| ↑〉 and | ↓〉. We will use the ±z notation as we are concerned with otherspin measurements in other directions besides z.

In the bra ket notation, the particle rates are a product of bras andkets:

|A1|2 = A1〈+z| A∗1|+ z〉 = |A1|2 〈+z| + z〉,|A2|2 = A2〈−z| A∗2| − z〉 = |A2|2 〈−z| − z〉.

(31)

The above multiplications give a calculation which is bracketed “〈 〉”.Hence the 〈 | is called a bra and the | 〉 is called a ket. In this notation,the matrix of particle rates in Equation 28 becomes:(

|A1|2 00 |A2|2

)= A1|+ z〉 A∗1〈+z|+A2| − z〉 A∗2〈−z|,

= |A1|2 |+ z〉〈+z| + |A2|2 | − z〉〈−z|.(32)

Splitting the rate matrix into +z and −z components allows us to rewriteEquation 27 which deals with the particle rates |Bj |2 and |Aj |2 into a pairof equations that relate the amplitudes Aj |+ z〉 and Bj |+ z〉:

B1|+ z〉 =

(q1 00 q2

)A1|+ z〉,

B2| − z〉 =

(q1 00 q2

)A2| − z〉,

(33)

This shows that classical beams, like the quantum case, can naturally beanalyzed in terms of amplitudes instead of probabilities.

3.4. Berry-Pancharatnam Phases

Geometric phases are related to the “Berry-Pancharatnam phases” that aquantum state receives when it is adiabatically sent through a cycle. Inthe last example of Subsection 3.2, the cycle is from spin-1/2 in the +zdirection, through a sequence of spin measurements in other directions andthen back to the +z direction.

To make the cycle adiabatic, we insert intermediate spin measurementsbetween two consecutive pairs of measurements. For example, between the+z and +x measurement, we would insert a large number N of spin-1/2measurements in the directions u(n) = (sin((n/N)π/2), 0, cos((n/N)π/2))so that consecutive measurements differ in direction by only (1/N)π/2 radi-ans. The intermediate measurements are on the great circle route betweenthe +z and +x directions. If we deviated from this path, the area enclosedby the cycle could change and this would change the geometric phase.

Any two points on the surface of the sphere define a great circle routeunless those two points are on opposing ends of a diameter (antipodal). Fora unit vector u, the points u and −u are on the opposite ends of a diameterand we have

ρu ρ−u = 0. (34)

That is, the projection operators for opposite spin measurements annihilateeach other and there is no unique great circle route to make the transitionadiabatic.

The usual introduction to spin-1/2 uses raising and lowering operators.For the spin-1/2 case, these are operators that negate the direction of a

12

C. A. Brannen

quantum state (amplitude), for example the raising operator for spin-1/2takes the spin-down state to a spin-up state. More general raising andlowering operators change a state from one spin state to an orthogonal spinstate and again the projection operators annihilate.

The absence of a great circle route between annihilating spin projectionoperators means that there is no natural phase that can be assigned forsuch an operator. For example, the raising operator for spin-1/2 can bewritten as:

ρ+(θ) =

(0 exp(−iθ)0 0

)(35)

where θ is any angle. In the standard presentation of raising operators, θis chosen as zero. We can obtain the general case by making a choice forthe intermediate measurement along the equator of the sphere at the pointu(θ) = (cos(θ), sin(θ), 0) so that

ρ+u(θ) = (1 + cos(θ)σx + sin(θ)σy)/2. (36)

Then the arbitrary phased raising operator can be written as a product ofprojection operators:(

0 exp(−iθ)0 0

)= 2ρ+zρ+u(θ)ρ−z (37)

where the factor of 2 is included due to the√

1/2 loss in amplitude in thetwo transitions. The standard raising operator is obtained by choosing theintermediate measurement as ρ+x = (1 + σx)/2.

The method described here generalizes. A complete set of observablescan have the phases of their raising and lowering operators determinedby choosing a state like the u above, whose projection operator does notannihilate any of the basis states which one wishes to raise and lower.

3.5. Quantum Temperatures

Temperatures work the same in the quantum situation as they do in theclassical. Using the classical numbers from Equation 1, a quantum beamsplitter acting on a beam with only a single particle type will be identicalto the classical case. To make it more interesting we can consider such adevice acting on a beam with two particle types. Consider a measurementof spin-1/2 in the +z direction. A perfect measurement (temperature zero)is given by (1 +σz)/2. At finite temperatures some of the spin-up particlesend up in the lower exit and some of the spin-down particles take the upperexit. Using +∆E for the first particle type and −∆E for the second wehave a matrix for the upper exit port:

ρ(T ) =

(exp(+∆E

kb T) 0

0 exp(−∆Ekb T

)

)/(

exp(+∆EkbT

) + exp(−∆EkbT

))

(38)

In the low temperature limit, and choosing to keep the sum of the proba-bilities 1, these become

ρ(T ) ≈

(1− exp(−2∆E

kb T) 0

0 exp(−2∆Ekb T

)

), (39)

13

C. A. Brannen

similar to the classical result in Equation 4.

We can chain n of these experiments together; the composite experimentis modeled by the n power of the matrix ρ(T ):

(ρ(T ))n ≈

(1− exp(−2∆E

kb T/n) 0

0 exp(−2∆Ekb T/n

)

). (40)

The above is diagonal, as would be obtained for a classical thermal twoparticle beam splitter. To get off diagonal elements we need to considerspin in directions other than ±z. Our calculations have been for spin-1/2in the +z direction but that is not a special direction so we can generalizeto the +u direction. To do that, we need to rewrite the above in the Paulispin matrix basis. We find

(ρ(T ))n ≈(

1̂ +(

1− exp(−2∆Ekb T/n

))σ+u

)/2. (41)

where σz has been replaced with σ+u to give the general spin-1/2 case.

Squaring a low temperature matrix ρ(T ) approximately gives the ma-trix for an even lower temperature ρ(T/2). For general temperatures, thesquaring does not necessarily divide the temperature by two but (otherthan the high temperature limit) it does reduce the temperature. The hightemperature limit has probabilities 1/2:

ρ(T >> ∆E/kB) ≈(

1/2 00 1/2

). (42)

Since the matrices for the upper and lower exits must sum to 1̂, the lowerexit matrix must be identical to the above. Squaring the above matrixgives a matrix with 1/4s on the diagonal instead of 1/2s but the matrixmust still correspond to the same high temperature limit so we need tomultiply the matrix by 2. This “renormalization” was (approximately)unnecessary at very low temperatures. At higher temperatures, one canmaintain the trace as one by dividing the matrix by the trace after squaring.The trace of the above matrix is 1/2 so dividing by 1/2 will renormalizeit. Density matrices are beyond the scope of this paper however we notethat requiring our matrices to have trace 1 makes them mathematicallyidentical to “mixed density matrices” and when density matrices are usedin statistical physics, squaring and renormalization is a method used toreduce their temperature.[14]

3.6. Superselection Sectors

Suppose we have a beam of spin-up electrons and we split it with a beamsplitter measuring spin-1/2 in the u = (ux, uy, uz) direction. The enteringbeam is represented by a ket that is pure spin-up:

|+ z〉 =

(10

), (43)

and the matrix representing the measurement is

ρ+u =1

2

(1 + uz ux − iuyux + iuy 1− uz

). (44)

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C. A. Brannen

Performing the matrix multiplication ρ+u |+ z〉, the exiting beam is repre-sented by a ket for spin-1/2 in the +u direction:

|+ u〉 = 12

(1 + uzux + iuy

),

= 1+uz2 |+ z〉 +

ux+iuy2 | − z〉.

(45)

The above shows that it is physically possible to begin with a beam ofpure spin-up and from it make a beam that is a “linear superposition” ofspin-up and spin-down. This is accomplished by arranging for the matrixρ+u to have non zero off diagonal elements. That is, we will get a linearsuperposition provided ux or uy is non zero.

It’s natural for students to imagine that it’s possible to create a superpo-sition of any quantum states, given that physicists talk about superpositionof states consisting of a live and dead cat. But in fact, not all quantum su-perpositions can be created; superpositions are limited by “superselectionsectors” and this is what we will discuss in this subsection. Superselectionsectors can be attributed to two subjects beyond the scope of this paper,symmetry and decoherence so we will provide only an outline of the ideas.The subject is explained in [4]; we will explore the subject from the pointof view of binary measurements.

Our starting point is to assume, as Julian Schwinger did, that given a“complete set of commuting observables” to define a quantum state, it ispossible to define a beam splitter whose upper exit passes only that par-ticular state, for example, spin-up. The spin-1/2 symmetry defines statesfor all the possible directions u so we can use intermediate states to ob-tain a superposition between spin-up and spin-down as was illustrated inEquation 45.

We will now consider a binary measurement on a particle beam thatcontains electrons and neutrinos. There are no intermediate quantum statesbetween the electron and neutrino so there is no way for us to use our spin-1/2 example to convert an electron beam into a beam that is a superpositionof electron and neutrino. This is the most common case for “internal”symmetries, however, the particle “generations” are an exception and infact, the weak force converts an electron into a linear superposition ofneutrinos from different generations.

Since it’s not possible to create linear superpositions of electrons andneutrinos, it is only possible to make statistical mixtures of these particles.This is the same case as the classical beams we considered in the previoussection. Classical beams are represented by matrices of particle rates thatare diagonal as in Equation 6. Before we split the particle rates into brasand kets, we represented spin-1/2 particles with 2× 2 Hermitian matrices.

By reconsidering our splitting we can create a 4 × 4 matrix that rep-resents a statistical mixture of electrons and neutrinos, each of which isa quantum superposition of spin-up and spin-down. Suppose our beam is40% electrons with spin-1/2 in the +u direction and 60% neutrinos with

15

C. A. Brannen

spin-1/2 in the +v direction. Then the matrix representation is:

ρ(T ) = 0.4|+ u, e〉〈+u, e| + 0.6|+ v, ν〉〈+v, ν| = 0.4ρ+u,e + 0.6ρ+v,ν

=

0.2 + 0.2uz 0.2ux − 0.2iuy

0.2ux + 0.2iuy 0.2− 0.2uz0.3 + 0.3vz 0.3vx − 0.3ivy

0.3vx + 0.3ivy 0.3− 0.3vz

(46)

where T is some arbitrary temperature and, for the sake of clarity, we’veleft the forbidden matrix entries blank instead of zero. This matrix is inthe form of a mixed density matrix subject to the requirement that its brasand kets not cross the superselection sector boundary. That is, the kets aresplit into two halves and only one of the halves can be non zero. If it isthe top half, then the ket is an electron ket while the bottom half definesa neutrino ket.

We’ve just shown that the particle rate matrix for beams that includemore than one superselection sector must be in block diagonal form. Thesplitting of two superselection sectors is a classical beam splitter and thiscan always be done. After reducing a beam to a single superselection sectorwe can do quantum measurements on it with the methods discussed above.We can do this to both the particle types and then reassemble the twobeams into one. This way we can design a beam splitter whose outputstream will have the properties of 40% electron and 60% neutrino beamdiscussed above, at least given an input stream with both electrons andneutrinos.

We can also consider thermal measurements of beams composed of par-ticles from two different superselection sectors. As before with Equation42, the high temperature limit for each superselection sector will be a mul-tiple of the unit matrix for that sector. If we balance the sectors to havethe same multiple, the high temperature limit will be a multiple of unity.Setting the trace to be 1, the high temperature limit for the electron /neutrino block diagonal matrix will be:

1

4

1 00 1

1 00 1

, (47)

where again we’ve left the forbidden entries blank rather than zero.We can reduce the temperature of the above matrix ρ(T ) in Equation

46 by squaring and renormalizing to keep the trace 1. This is easily done inthe form 0.4ρ+u,e + 0.6ρ+v,ν as ρ+u,e and ρ+v,ν are idempotent, annihilateeach other and have unit trace. We find:

ρ(T )n / tr(ρ(T )n) = ((0.4)nρ+u,e + (0.6)nρ+v,ν) / ((0.4)n+(0.6)n) (48)

In the limit as n→∞ the (0.6)n terms dominate giving a limit

ρ(0) = ρ+v,ν . (49)

It should be clear that this is a general attribute of quantum beam splittersthat cross superselection sectors. That is, their T = 0 limit falls into asingle sector.

16

C. A. Brannen

If we are able to make calculations in an algebra and want to know whatparticles it contains this implies an algorithm. We begin with the hightemperature limit of Equation 47 and add a small Hermitian modificationto it. Then repeatedly square and renormalize the trace to unity. It willapproach the primitive idempotent for a particle.

For our example of the electron / neutrino, the operator for electriccharge is zero in the neutrino part and −1̂ = −11̂ in the electron part:

Q =

−1 00 −1

0 00 0

. (50)

This matrix commutes with any element of the measurement algebra forthe electron / neutrino beam. In fact, the definition of a charge that cre-ates superselection sectors is that the symmetry must commute with everypossible measurement (observable).

Given an algebra, we can define the possible charges that define itssuperselection sectors. Each diagonal block can take a different charge.For the electron / neutrino algebra, the possible charges are:

Q(qe, qν) =

qe 00 qe

qν 00 qν

. (51)

where qe and qν are the charges for the electron and neutrino blocks. Thisobservation is trivial for block diagonal algebras but becomes interestingwhen the algebra is defined more subtly.

4. Standard Model / Dark Matter

The Standard Model of elementary particles is built around symmetries.This is a natural consequence of the experiments; humans look for patternsin the experimental results and the nicest way to define patterns is withsymmetry. So the quantum mechanics that models the elementary parti-cles is defined using the symmetries that are observed in the experiments.While this method is clearly the easiest way of obtaining a model thatmatches experiment, it should also be clear that the seeking of symmetriesis a human attribute and not necessarily an indication of how Nature ismost easily understood. Here we explore an alternative way of describingelementary particles.

The Standard Model of elementary particles consists of representativesof SU(3)xSU(2)xU(1) symmetry. Why did nature choose this symmetryrather than, for example, SU(4)xE4xU(1)xU(1)? The Standard Model pro-vides no explanation for the arbitrariness of the choice of symmetry. On theother hand, what is interesting here is that SU(3)xSU(2)xU(1) is a blockdiagonal symmetry.

It is not enough to choose the symmetry SU(3)xSU(2)xU(1). The par-ticles are taken as irreducible representations of this symmetry so one mustalso choose these irreps. To define an irrep of SU(3)xSU(2)xU(1) onechooses an irrep of SU(3), one of SU(2) and one of U(1). Each of these

17

C. A. Brannen

has an infinite number of choices; the ones used by the fermions in theStandard Model are:

SU(3) SU(2) U(1) Particles

1 1 −2 right-handed electron1 1 0 right-handed neutrino1 2 −1 left-handed leptons3 1 4/3 right-handed up quark3 1 −2/3 right-handed down quark3 2 1/3 left handed quarks.

(52)

Why did Nature choose these six representations? And why do they appearin three generations identical except for mass and the weak force? TheStandard Model provides no explanations for the arbitrariness of the abovechoice of symmetry representations.

What we would like is a single mathematical object that is natural,and reproduces the arbitrary choices of the Standard Model. To do that,we can assume that the symmetry and representations both arise from asingle choice of algebra instead of a choice of symmetry. One problem withthis idea is that SU(3)xSU(2)xU(1) is a gauge symmetry and the tool thispaper develops is the particle content. In addition, the SU(2) portion ofthe Standard Model symmetry is unbroken only in the high temperaturelimit. This paper looks at putting several particles into a single densitymatrix due to the consequences of block diagonal form. Another way ofputting multiple representations into mixed density matrices is explored in[16].

At normal temperatures, the weak SU(2) symmetry that relates the lefthanded electron and neutrino is broken. In fact, left handed fermions travelat speed c and so do not appear at all at zero temperature. This impliesthat the zero temperature particle content needs to deal with the electron,neutrino, up-quark and down-quark rather than their left and right handedcomponents. Under this assumption, there are 4 Standard Model fermions.Ignoring spin and anti particles, the electron and neutrino will appear astwo 1 × 1 blocks. The up and down quarks are color SU(3) triplet statesso they will appear as two 3× 3 blocks.

If we have a finite group G, we can create an algebra from it that iscalled C(G), the “complex group algebra”. A complex group algebra canbe put into block diagonal form.[5] We can then read the particle contentoff by noting the size of the blocks. Again, note that the particle contentis not the same as a gauge symmetry.

In Subsection 3.2 we showed that consecutive measurements in the +z,+x, +y and +z directions resulted in the beam picking up a geometricphase of −π/4. On the other hand, any complex plane wave picks up aphase through translation in space or time according to exp(ikx−ωt). Wecannot distinguish these two effects; either one will give a phase change toa beam that we can detect by seeing changes in an interference pattern. IfNature were truly simple, these two causes of phase changes would be thesame. For that to happen, the movement of spin-1/2 particles would haveto be accompanied by changes in their spin direction.[1]

The left and right handed fermions of Equation 52 are massless andtravel at speed c. A stationary electron with spin-1/2 in the +z directionis produced by combining a right handed electron with speed c in the +z

18

C. A. Brannen

direction and a left handed electron with speed c in the −z direction. The“Feynman checkerboard model” of the electron is a 1-dimensional modelof the Dirac equation based on sums over discrete paths on the corners ofa checkerboard.[3] The axes on the checkerboard correspond to position zand time t. Each step in a path the electron moves one unit forward intime by ε and steps by a distance ±εc in z. So the electron is moving upthe checkerboard on diagonal lines, the way a checker moves in a game ofcheckers or draughts. Each time the electron changes direction the pathreceives a factor of −iεmc2/~. In the ε→ 0 limit, the sum of all paths givesa propagator that satisfies the 1-d Dirac equation.

The 1-dimensional Feynman checkerboard paths have been extendedto 3-dimensional.[6, 13] The particle paths in space are on the points of acubic lattice. The possible symmetries of a cubic lattice are defined by theisometric crystallographic point groups:

Group: T Th O Td OhSize: 12 24 24 24 48

(53)

in Schoenflies notation.

The block diagonal structure of a complex group algebra can be readoff of the group character table. Each irreducible character corresponds toa block on the diagonal and the size of the block is given by the irreduciblecharacter of the identity E.[5] For example, the character table for T isgiven by

T E C2 C3 C ′3Size: 1 3 4 4

A 1 1 1 1E 1 1 ω ω2

E′ 1 1 ω2 ωT 3 −1 0 0

(54)

where ω = exp(2iπ/3) and “Size” is the number of group elements in thatclass. Each row corresponds to an irreducible representation of the symme-try. There are four: A,E,E′, T and each irrep will appear in the algebraas a block on the diagonal. So there are four blocks in C(T ); three of size1× 1 and one of size 3× 3.

A general element of the C(T ) algebra, in block diagonal form, lookslike:

αη

η′

τ11 τ12 τ13

τ12 τ22 τ23

τ31 τ32 τ33,

(55)

where α, η, η′ and τjk are complex numbers. For the Standard Modelfermions we need two 1 × 1 blocks for the leptons and two 3 × 3 blocksfor the quarks so T symmetry will not suffice. As we can count from Equa-tion 55, the total number of complex degrees of freedom in the C(T ) algebrais 12 + 12 + 12 + 32 = 12. This is the same as the size of the finite group T .In other words, putting the algebra into block diagonal form is just a wayof rewriting its degrees of freedom.

19

C. A. Brannen

Examining the character tables of the five isometric crystallographicpoint groups we find that their particle content consists of:

T : 3 1 1 1Th : 3 3 1 1 1 1 1 1 1Td : 3 3 2 1 1O : 3 3 2 1 1Oh : 3 3 3 3 2 2 1 1 1 1

(56)

and we find that all but the T group include the 3 3 1 1 particle content ofthe Standard Model. Of these five groups, Th and Oh include an inversioni or point reflection. Such a symmetry transforms an object into its mirrorimage. In the Standard Model the weak interaction is not symmetric underparity so we will reject those groups.

This leaves Td and O which share the same characters:

O : E C3 C2 C ′2 C4

Td : E C3 C2 σd S4

Size: 1 8 3 6 6

A1 : 1 1 1 1 1A2 : 1 1 1 −1 −1E : 2 −1 2 0 0T1 : 3 0 −1 −1 1T2 : 3 0 −1 1 −1

. (57)

In the above table, we’ve included an extra vertical line between the firstthree conjugacy classes and the last two. This is to note that E, C3 and C2

are even while the two right classes are odd. Even and odd are determinedby the A2 or “sign” irrep. We assign the Aj pair to the leptons and the Tjpair to the quarks. These pairs differ only in their odd characters so we pre-sume that it is the odd characters that define their weak isospin and weakhypercharge quantum numbers of the handed Standard Model fermions.The remaining irrep E is distinctive in that it has no odd characters. Sowe assume that it corresponds to a particle with no weak hypercharge orweak isospin and assign it to dark matter.

5. Conclusion

We’ve shown that binary measurements are quite similar whether theyare classical or quantum. Since binary measurements form a foundationfor quantum mechanics as demonstrated in the Schwinger measurementalgebra textbook introductions to quantum mechanics [12, 11], this showsthat this formulation of quantum mechanics is more closely aligned with ourclassical intuition than the other formulations. This makes this formulationimportant for pedagogy as well as the foundations of quantum mechanics.

The paper shows a new way of introducing quantum mechanics to stu-dents from the point of view of classical mechanics. This allows students tohave an intuitive understanding of how quantum mechanics works. We’veshown how the use of complex numbers and amplitudes instead of proba-bilities is natural to quantum mechanics.

Some of the subjects discussed in this paper including quantum phase,Berry-Pancharatnam phase, superselection sectors and quantum statistics

20

C. A. Brannen

are introduced in this paper at an elementary level. These subjects provideinteresting subjects to study for students in their first lectures on quantummechanics with interesting problems available for assignment.

Finally, we’ve illustrated the possible uses of this new formulation ofclassical and quantum mechanics in explorations of the nature of the el-ementary fermions. We show that the choice of an algebra defines bothsymmetry and particle representations. This may be a method of wrap-ping the symmetries of the Standard Model into an algebraic descriptionthat will be much more tightly determined than the arbitrary choice ofsymmetry and representations found in the Standard Model.

Acknowledgement

Thanks to the faculty at Washington State University for their encourage-ment and assistance; to Mark Mollo and Joseph. P. Brannen for financialsupport, and to the grad students at WSU for their fellowship.

References

[1] Brannen, C.A.: Spin path integrals and generations. Found. Phys. 40,1681–1699 (2010). DOI 10.1007/s10701-010-9465-8

[2] Ellerman, D.: A Very Common Fallacy in Quantum Mechanics: Su-perposition, Delayed Choice, Quantum Erasers, Retrocausality, andAll That. ArXiv e-prints (2011)

[3] Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals.New York: McGraw-Hill (1965)

[4] Giulini, D., Kiefer, C., Zeh, H.D.: Symmetries, superselection rules,and decoherence. Physics Letters A 199, 291–298 (1995). DOI 10.1016/0375-9601(95)00128-P

[5] Hammermesh, M.: Group Theory and its Application to PhysicalProblems. Dover (1962)

[6] Jacobson, T.: Feynman’s checkerboard and other games, pp. 386–395.Springer Berlin Heidelberg, Berlin, Heidelberg (1985). DOI 10.1007/3-540-15213-X 88. URL http://dx.doi.org/10.1007/3-540-15213-X 88

[7] Schwinger, J.: The Algebra of Microscopic Measurement. Proceedingsof the National Academy of Science 45, 1542–1553 (1959). DOI 10.1073/pnas.45.10.1542

[8] Schwinger, J.: The Geometry of Quantum States. Proceedings of theNational Academy of Science 46, 257–265 (1960). DOI 10.1073/pnas.46.2.257

[9] Schwinger, J.: The Special Canonical Group. Proceedings of the Na-tional Academy of Science 46, 1401–1415 (1960). DOI 10.1073/pnas.46.10.1401

[10] Schwinger, J.: Unitary Transformations and the Action Principle. Pro-ceedings of the National Academy of Science 46, 883–897 (1960). DOI10.1073/pnas.46.6.883

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[11] Schwinger, J.: Quantum Kinematics And Dynamic. Advanced BooksClassics. Avalon Publishing (2000)

[12] Schwinger, J., Englert, B.: Quantum Mechanics: Symbolism of AtomicMeasurements. Physics and astronomy online library. Springer (2001)

[13] Smith Jr., F.D.: Hyperdiamond Feynman checkerboard in four-dimensional space-time (1995)

[14] Storer, R.G.: PathIntegral Calculation of the Quantum - StatisticalDensity Matrix for Attractive Coulomb Forces. Journal of Mathe-matical Physics 9(6), 964–970 (1968). DOI 10.1063/1.1664666. URLhttps://doi.org/10.1063/1.1664666

[15] Styer, D.F., Balkin, M.S., Becker, K.M., Burns, M.R., Dudley, C.E.,Forth, S.T., Gaumer, J.S., Kramer, M.A., Oertel, D.C., Park, L.H.,Rinkoski, M.T., Smith, C.T., Wotherspoon, T.D.: Nine formulationsof quantum mechanics. American Journal of Physics 70(3), 288–297 (2002). DOI 10.1119/1.1445404. URL https://doi.org/10.1119/1.1445404

[16] Weinberg, S.: Quantum mechanics without state vectors. PRA 90(4),042102 (2014). DOI 10.1103/PhysRevA.90.042102

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