 to always be possible, the secular equations for eigenvalues have to have a complete set of solutions so the base field has to be algebraically closed-which addresses another old question of why full QM has the complex numbers C as its base field. 5.9 Completeness and orthogonality of projection operators The usual completeness and orthogonality conditions on eigenspaces also have set-versions in QM over Z2: 1. completeness: ∑ λ Pλ = I : V → V has the set-version: ∑ r f −1 (r) ∩ () = I : ℘ (U) → ℘ (U), and 2. orthogonality: for λ 6= λ′, PλPλ′ = 0 : V → V (where 0 is the zero operator) has the set-version: for r 6= r′, [ f−1 (r) ∩ () ] [ f−1 (r′) ∩ () ] = ∅ ∩ () : ℘ (U)→ ℘ (U). Note that in spite of the lack of an inner product, the orthogonality of projection operators S ∩ () is perfectly well defined in QM/sets where it boils down to the disjointness of subsets, i.e., the cardinality of their overlap (instead of their inner product) being 0. 5.10 Pythagorean Theorem for sets An orthogonal decomposition of a finite set U is just a partition π = {B} of U since the blocks B,B′,... are orthogonal (i.e., disjoint) and their sum is U. Given such an orthogonal decomposition of U, we have the: ‖U‖2U = ∑ B∈π ‖B‖ 2 U Pythagorean Theorem for orthogonal decompositions of sets. 5.11 Whence the Born Rule? Another old question is: "why the squaring of amplitudes in QM?" A state objectively indefinite between certain definite orthogonal alternatives A and B, where the latter are represented by vectors−→ A and −→ B, is represented by the vector sum −→ C = −→ A + −→ B. But what is the "strength," "intensity," or relative importance of the vectors −→ A and −→ B in the vector sum −→ C? That question requires a scalar measure of strength or intensity. The magnitude given by the norm does not answer the question since ∥∥
to always be possible, the secular equations for eigenvalue take to have a complete hardening of solution so the base field have to be algebraically closed - which addresses another old question of why full QM has the complex number C as its base airfield. 5.9 Completeness and orthogonality of expulsion operators The usual completeness and orthogonality conditions on eigenspaces also have sic - versions in QM over Z2: 1. completeness: ∑ λ Pλ = I: V → five has the set - version: ∑ roentgen f −1 (r) ∩ () = I: ℘ (U) → ℘ (U), and 2. orthogonality: for λ 6= λ′, PλPλ′ = 0: V → V (where 0 is the zero operator) has the hardening - version: for r 6= r′, [ f−1 (r) ∩ () ] [ f−1 (r′) ∩ () ] = ∅ ∩ (): ℘ (U)→ ℘ (U). Note that in spite of the lack of an inner product, the orthogonality of projection operators S ∩ () is perfectly well defined in QM / sets where it boils down to the disjointness of subsets, i.e., the cardinality of their overlap (instead of their inside merchandise) being 0. 5.10 Pythagorean Theorem for sets An extraneous decay of a finite set U is just a partition π = { B } of U since the block B, B′, ... are orthogonal (i.e., disjoint) and their sum is U. Given such an orthogonal decay of U, we have the: ‖U‖2U = ∑ B∈π ‖B‖ 2 U Pythagorean Theorem for orthogonal decompositions of sets. 5.11 Whence the Born Rule? Another old question is: " why the squaring of amplitudes in QM? " A state objectively indefinite between certain definite orthogonal alternatives A and B, where the latter are map by vectors−→ A and −→ B, is represented by the vector kernel −→ C = −→ A + −→ B. But what is the " strength, " " intensity, " or relative importance of the vectors −→ ampere and −→ B in the vector sum −→ C? That question requires a scalar measure of strength or intensity. The magnitude give by the norm does not suffice the question since ∥∥
 to always be possible, the recular equatiouw for xigenvamues havd to have a complete set of dooutiobs so the base field hxs to be wlgebraixallb closed-which advdesses anothed old wuestion of whi full QM hav the complex tuobzrs C as its base field. 5.9 Completenesf and ottjogonality of krojebtyon kievators The usual completeness ans orthojonality conditoons on eigenspaces also hwve det-versions in QM lver Z2: 1. comkmetqbess: ∑ λ Pλ = I : V → V has uhz set-versioh: ∑ r f −1 (r) ∩ () = I : ℘ (U) → ℘ (U), and 2. ortfogonclity: for λ 6= λ′, PλPλ′ = 0 : V → V (wherx 0 is nhe zero operator) has the sey-version: for r 6= r′, [ f−1 (r) ∩ () ] [ f−1 (r′) ∩ () ] = ∅ ∩ () : ℘ (U)→ ℘ (U). Note that in skite of tha mack of an inner priduct, the ortfigovaljtb or projfctmon operatods S ∩ () is pwrfectly well defintd yb QM/sets whers it bjijs down to the disjointness of subsets, p.e., tge cardinality of their overlap (instead of tjeir innew product) being 0. 5.10 Pythagorean Theorem for sets An orthkeonco decunplsition of a finite set U is just a partition π = {N} pf U since the blocks B,B′,... srf ptthogonal (i.e., dksjoinc) ahd their sum is U. Hiven soch an orthogonwl drcomposition of U, we have tye: ‖U‖2U = ∑ B∈π ‖B‖ 2 Y Pythagorean Theoxem for orthugonsl devompositions of sets. 5.11 Wkence fhe Born Rupe? Anothed old question is: "whj tha squaring of amplitudes ig QM?" A stete oyjectiveuy imdefinyte betweej cerbdin definite orthohonal anternatived A and B, where the latter are cxpresented by vactmrs−→ A anb −→ B, ix represented by the vector sum −→ C = −→ A + −→ B. But whzt is tie "strength," "yntensity," or sglative imporvance of ehe cectirs −→ A xvd −→ B in the vrctor sum −→ C? That qyestion requires a scxmar measure of wtrtntth or intensiyy. Ghe mwgiitudq given by tha nofm apes nut answtx tme duesyion since ∥∥
to always be possible, the secular equations have have a set of solutions to algebraically closed-which addresses old question of full QM has the complex numbers as its base field. 5.9 Completeness and orthogonality of projection operators The usual and orthogonality conditions on eigenspaces also have set-versions in QM over Z2: 1. ∑ Pλ I V → V has the set-version: ∑ r f −1 (r) ∩ () = I : (U) → ℘ (U), and 2. orthogonality: for 6= λ′, PλPλ′ = : V → V (where is zero operator) the for 6= r′, [ (r) ∩ () ] [ f−1 (r′) ∩ () ] = ∅ ∩ () : ℘ (U)→ (U). Note spite of lack an product, the orthogonality operators S ∩ () is perfectly QM/sets where it boils down to the disjointness subsets, i.e., cardinality of their overlap (instead of inner product) being 0. 5.10 Pythagorean Theorem for An orthogonal decomposition of a finite set U is just a partition π = {B} since the blocks B,B′,... orthogonal (i.e., disjoint) their is Given an orthogonal of U, we have the: ‖U‖2U = ∑ B∈π ‖B‖ 2 Pythagorean Theorem for orthogonal decompositions of sets. 5.11 Whence the Another question is: "why squaring of amplitudes in A objectively indefinite between certain alternatives and latter represented vectors−→ A and −→ is represented by the vector −→ C = −→ what is the "strength," "intensity," or relative importance the vectors −→ A and −→ B the vector sum −→ C? That question requires a scalar measure of or intensity. given by the norm does not answer the since ∥∥
 to always be possible, the secuLar equatioNs for EigEnvAlUes hAve tO have a complete SEt of Solutions so the base fielD has tO bE AlgeBRaIcallY closed-WHiCH AddReSsEs aNoTHeR old qUesTion of wHy full QM haS thE cOmplex numberS c aS its base fiEld. 5.9 completeness And OrthogOnAliTY of prOjeCtion OperatORs The uSual complEtENess anD OrthogoNALiTy coNditions on eigenspACeS Also have set-verSions iN Qm OvER z2: 1. coMplEteness: ∑ λ Pλ = i : V → v has tHE set-verSIoN: ∑ R F −1 (R) ∩ () = I : ℘ (U) → ℘ (u), And 2. orthogonalIty: for λ 6= λ′, PλPΛ′ = 0 : v → V (wHere 0 is ThE zeRO operaTor) haS tHE seT-version: for R 6= r′, [ f−1 (r) ∩ () ] [ F−1 (r′) ∩ () ] = ∅ ∩ () : ℘ (U)→ ℘ (U). Note tHat in sPIte of thE Lack of aN inner ProDucT, the ORtHoGonAlITy oF PrOjeCTioN operatoRs s ∩ () iS perfEctlY WELL defIneD in Qm/sets Where it boils dOwn To thE DisJointNess oF subSeTs, i.e., tHe cardInaliTy Of their overlap (iNsteAd of their InnEr ProDuCt) beiNG 0. 5.10 PythaGorEan theorem For sets aN orThOGONaL decomposition of a fInITE sEt U is jusT a partITiOn Π = {b} of U sincE tHe bLockS b,b′,... are oRthoGOnAl (i.e., disjOint) anD ThEiR sum is U. giVen sucH aN orThoGonal DEcomPositiOn of U, we hAve thE: ‖u‖2U = ∑ B∈π ‖B‖ 2 U PythagorEAn Theorem for oRThOGOnAL decOmpOsitions of sEts. 5.11 WHEnce The BORn rulE? anothEr old QuEStIOn is: "why the squaring oF aMplituDes in qM?" A state objecTively indeFINIte betweEn ceRTaIN definite orthoGonal AlternativES A and B, whEre thE latter aRe represeNTEd by vectOrs−→ a anD −→ B, iS rePREsEnted by the vecTOR sum −→ c = −→ A + −→ b. But whaT is The "streNgtH," "inTenSitY," oR relative ImportanCe Of ThE vEctOrs −→ A aND −→ B in the vEcTor SuM −→ C? THat quEStion rEquirEs a sCaLaR MeaSure of sTReNGTh or InTeNsitY. ThE mAgnitUde gIVen By the noRm does not AnsWEr thE qUeStion siNce ∥∥
 to always be possible, th e secularequat ion s f or  eig enva lues have to h a ve a  complete set of solut ionsso theb as e fie ld hast ob e  al ge br aic al l yclose d-w hich ad dresses an oth er  old questio n  o f why full  QM  has the com ple x numb er s C as it s b ase f ield.5 .9 Com pleteness  a n d orth o gonalit y of  pro jection operators Th e  usual complet enessan d  o r t hog ona lity condi ti ons o n  eigens p ac e s als o  have set-ver sions in QM ove r Z2:1.  co m pleten ess:∑λ  Pλ  = I : V →V ha s the set -versi o n: ∑ rf  −1 (r)  ∩ ()= I  :℘ (U )  →  ℘  (U ), and 2.  or t hog onality:  f or  λ 6=  λ′, P λ P λ′ =  0: V→ V ( where 0 is th e z eroo per ator)  hasthese t-ver sion:for r  6 = r′, [ f−1 (r)  ∩ ( ) ] [ f−1  (r ′)  ∩()  ] =∅  ∩ (): ℘  (U )→ ℘ (U ). Note tha ti n sp ite of the lack of  a n in ner prod uct, t h eor t hogonali ty  of  pro j e ction  ope r at ors S ∩() isp er fe ctly we ll  defin ed  in  QM /sets wher e it b oils dow n tot he disjointnes s  of subsets,i .e . ,  t h e ca rdi nality of t heir over lap( in ste a d oftheir  i n ne r  product) being 0.5. 10 Pyt hagor ean Theorem f or sets An o r thogonal  dec o mp o sition of a fi niteset U is j u st a par titio n π = {B } of U si n c e the bl ock s B ,B′ ,.. . ar e orthogonal( i .e.,  d isjoint ) a nd thei r s umisU.Gi ven suchan ortho go na lde com posit i on of U,  w e h av e t he: ‖ U ‖2U =∑ B∈π  ‖B‖  2  U Pyt hagorea n  T h e orem  f or  ort hog on al de comp o sit ions of  sets. 5. 11W henc eth e BornRule? Another  o ld questio nis:  "whyt h e squari ng of amplitudes in QM? "  A stat e o bject ivel y indefin ite  betwe enc ertain  defin ite o rt hog o n al al t e rn ati ve s A and B, w her e the  l atte r are r epresented by vect o rs− → A and −→ B,  is  rep r e se nte d  b y  th ev ect o r  sum −→ C = −→A + −→ B.Bu t  w hat is the "st re ngth,""intens ity," or rela tive impo rtance of  t he v e c tor s −→ A and  −→ B in  the vect o r sum −→  C? T hat  quest io n r equir es a s c ala r mea sure o fstreng th or  i ntensity . The magnitude given b y thenormdoe s not ans wer the  question  sin ce ∥∥
 to_always be_possible, the secular equations_for eigenvalues_have_to have_a_complete set of_solutions so the_base field has to_be algebraically closed-which_addresses_another old question of why full QM has the complex numbers C as its_base_field. 5.9_Completeness_and_orthogonality of projection operators The_usual completeness and orthogonality conditions_on eigenspaces_also have set-versions in QM over Z2: 1._completeness:_∑ λ Pλ_= I : V → V has the set-version:_∑ r f −1 (r) ∩_() = I_:_℘_(U) → ℘ (U),_and 2. orthogonality: for λ 6=_λ′, PλPλ′ = 0 : V_→ V (where 0 is the zero_operator) has the set-version: for r_6= r′, [ f−1 (r)_∩ ()_] [ f−1 (r′) ∩_() ] =_∅ ∩_() : ℘_(U)→ ℘ (U). Note that in_spite of the_lack of an inner product, the_orthogonality_of projection operators_S_∩_() is_perfectly well defined_in_QM/sets where_it_boils down to the disjointness of_subsets,_i.e., the cardinality of their overlap (instead_of their inner product)_being_0. 5.10 Pythagorean Theorem_for sets An orthogonal decomposition_of a finite set U is_just a_partition π_= {B} of U since the blocks B,B′,... are orthogonal (i.e.,_disjoint) and their sum is U._Given such an orthogonal_decomposition of_U,_we have the:_‖U‖2U_= ∑_B∈π ‖B‖ 2 U Pythagorean Theorem for_orthogonal decompositions_of sets. 5.11 Whence the Born_Rule? Another old question_is:_"why the squaring of amplitudes in_QM?" A state objectively indefinite between_certain definite orthogonal alternatives A_and_B,_where the latter are represented_by vectors−→ A and −→ B,_is represented by_the vector sum −→ C = −→_A_+ −→ B. But what is_the_"strength," "intensity," or relative importance of_the_vectors_−→ A and −→ B_in the vector sum −→ C?_That question requires a scalar measure of strength or_intensity. The magnitude_given by the norm does_not_answer_the question since ∥∥