Addendum to Quantum Wave Function Collapse of a System Having Three anti Commuting Elements. Elio Conte School of Advanced International Studies on Applied Theoretical and Non Linear Methodologies of Physics, Bari, Italy. Abstract : We indicate a new way in the solution of the problem of the quantum measurement . In past papers we used the well-known formalism of the density matrix using an algebraic approach in a two states quantum spin system S, considering the particular case of three anticommuting elements. We demonstrated that, during the wave collapse, we have a transition from the standard Clifford algebra, structured in its space and metrics, to the new spatial structure of the Clifford dihedral algebra. This structured geometric transition, which occurs during the interaction of the S system with the macroscopic measurement system M, causes the destruction of the interferential factors. In the present paper we construct a detailed model of the (S+M) interaction evidencing the particular role of the Time Ordering in the (S+M) coupling since we have a time asymmetric interaction . We demonstrate that , during the measurement , the physical circumstance that the fermion creation and annihilation operators of the S system must be destroyed during such interaction has a fundamental role . corresponding author : elio.conte@fastwebnet.it key words : foundations of quantum mechanics, quantum collapse, Clifford algebra, fermion creation and destruction operators . 1. Introduction In ninety years since its beginnings quantum mechanics has had great functional and theoretical success leaving little reason to doubt its intrinsic validity. Nevertheless, we cannot ignore that some questions concerning the foundations of this theory remained unsolved, and historic debates among scientists who deeply influenced the early development of the theory remain. The first important question concerns the problem of the wave-function collapse by measurement. Its solution would be of relevant significance because it would provide us with a self-consistent formulation of the theory, which presently depends on the von Neumann postulates that have been added from the outside of the body of theory. For a complete examination of the actual problems that are involved, we refer the reader to the several reviews that may be found in pertinent literature [1-12]. Consider the measurement of a given observable F on a quantum-mechanical system S that is in a normalized superposition of states i i ic φψ ∑= ; 1;),( 2 == ∑ i iii cc ψφ ; (1) where iφ is a normalized eigenstate of F , relative to an eigenvalue iλ , iiiF φλφ = , ijji δφφ =),( . The probability of finding the eigenvalue iλ during the measurement is 2 ic , the corresponding eigenstate is iφ and during the measurement the wave function ψ is subjected to the transition iφψ → characterizing the completed collapse . The density matrix approach as it was initiated by von Neumann is k k kFSjij i j iS k cc φφρφφψψρ λ ∑∑∑ =→== ∗ , . (2) Usually, we consider a macroscopic measuring device M and we postulate that the states of M entangle with those of S )3()( 2 , tt kMkk k kMSMjij i j iMS ccc ρφφρρφφρρρ ⊗=→⊗=⊗= ∑∑∑ ∗ If the system is not destroyed by the measurement, and if the interaction fits into the so called measurement of the first kind, then the quantum state after the measurement will be the eigenstate associated with the measurement outcome, or more generally (to include degenerancies), the normalized projection of the original state onto the eigensubspace associated with the outcome. This rule is known as the projection postulate. It originated with Dirac and von Neumann [13], and was later formalized in degenerate cases by Luders and Ludwig [14,15] . Consider S to be a quantum two states system . The complete phase-damping by using projection postulate gives )4(11110000)( ρρρ +=D Generally speaking, we have a set of mutually orthogonal projectors ( ).,,........., 21 NPPP which complete to unity, jijji PPP δ= , 1=∑ i iP , the result i is obtained with probability ψψ ii Pp = and the state collapses to ψi i P p 1 . It is known that quantum mechanics has some peculiar features that are missing in the counterpart of classical physics. Two basic features are quantum interference and the collapse . Starting with 2009 [16,17,18] our tentative approach was to use the Clifford algebra with the aim to construct a bare bone skeleton of quantum mechanics but giving collapse. We will deepen here some basic features evidencing in particular that in order the −ψ collapse to be obtained, the fermion creation and destruction operators must be realized in our considered system. . 2. Theoretical Elaboration Let us start with a proper definition of the 3-D space Clifford algebra 3Cl . It is an associative algebra generated by three abstract algebraic elements ,e,e 21 and 3e that satisfy the orthonormality relation jkjkkj eeee δ2=+ for [ ]321 ,,,k,j ∈λ (5) That is 12 =λe and jkkj eeee −= for kj ≠ The algebra holds about only two postulates that are a) it exists the scalar square for each basic element: 111 kee = , 222 kee = , 333 kee = with R∈ik . (6) In particular we have also the unit element, 0e , such that 100 =ee , and 00 eeee ii= (7) b) The basic elements ie are anticommuting elements 1221 eeee −= , 2332 eeee −= , 3113 eeee −= . (8) Following Ilamed and Salingaros [19] we may give proof of two theorems. Theorem n.1. Assuming the two postulates given in (a) and (b) with 1=ik , the following commutation relations hold for such algebra : 31221 ieeeee =−= ; 12332 ieeeee =−= ; 23113 ieeeee =−= ; 321 eeei = , ( 1 2 3 2 2 2 1 === eee ) (9) They characterize the Clifford iS algebra. We will call it the algebra )( iSA Theorem n.2a . Assuming the postulates given in (a) and (b) with 11 =k , 12 =k , 13 −=k , the following commutation rules hold for such new algebra: 1 2 2 2 1 == ee ; 1 2 −=i ; iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (10) They characterize the Clifford iN algebra. We will call it the algebra 1,+iN Theorem n.2b .Assuming the postulates given in (a) and (b) with 11 =k , 12 =k , 13 −=k , the following commutation rules hold for such new algebra 1 2 2 2 1 == ee ; 1 2 −=i ; iee −=21 , iee =12 , 12 eie = , 12 eie −= , 21 eie −= , 21 eie = (11) They characterize the Clifford Ni algebra. We will call it the algebra 1,−iN The algebra 1,±iN is the well known Clifford Dihedral algebra . The demonstration of the two theorems as well as the construction of a bare bone skeleton of quantum mechanics were given by us in previous papers ( 16,17,18 ). Let us evidence an important feature of Clifford algebra )( iSA . In Clifford algebra )( iSA we have idempotents , two of such idempotents are 2 1 3 1 e+ =ψ and 2 1 3 2 e− =ψ (12) 1 2 1 ψψ = and 2 2 2 ψψ = . Let us examine now the following algebraic relations: 13113 )1( ψψψ +== ee (13) 23223 )1( ψψψ −== ee (14) Similar relations hold in the case of 1e or 2e . The inspection of ( 13) and (14) reveals a net analogy with a two states z-spin system constructed in their proper Hilbert space in quantum mechanics . Of course the analogy between the three basic elements ie and quantum spin operators is trivial since we have ii eS 2 h = and ie relating the well known spin Pauli matrices. Consider the previous two states system S with its proper representation in Hilbert space . The complex coefficients ic ( )2,1=i are the well known probability amplitudes for the considered quantum state       = 2 1 c c ψ and 1 2 2 2 1 =+ cc (15) For a pure state in quantum mechanics it is ρρ =2 . We have a corresponding algebraic member that in )( iSA is given in the following manner 321 decebeaS +++=ρ (16) with 2 2 2 1 22 cc a += , 2 2121 ∗∗ + = cccc b , 2 )( 2121 ccccic ∗∗ − = , 2 2 2 2 1 cc d − = In our scheme a theorem may be demonstrated in Clifford algebra [20,21] . It is that ↔= SS ρρ 2 2 1 =a and 2222 dcba ++= and Tr( 1) =ρ (17) Let us write the state of the two state quantum system S with connected quantum observable 3S relating 3e of )( iSA . We have 2211 φφψ cc += ,       = 0 1 1φ ,       = 1 0 2φ (18) and 1 2 2 2 1 =+ cc As we know, the density matrix of such system is easily written 321 decebea +++=ρ (19) with 2 2 2 2 1 cc a + = , 2 * 212 * 1 ccccb + = , 2 )( 2 * 1 * 21 ccccic − = , 2 2 2 2 1 cc d − = (20) where in matrix notation, 1e , 2e , and 3e are the well known Pauli matrices       = 01 10 1e ,       − = 0 0 2 i i e ,       − = 10 01 3e (21) The ( 17) and (19) coincide . To examine the consequences, starting with the algebraic element (16), write it in the two equivalent algebraic forms that are obviously still in the algebra A(Si). 3 2 2 2 1212 * 121 * 21 2 2 2 1 )( 2 1 ))(( 2 1 ))(( 2 1 )( 2 1 eccieeccieeccccS −+−++++=ρ (22) and 3 2 2 2 1212 * 121 * 21 2 2 2 1 )( 2 1 ))(( 2 1 ))(( 2 1 )( 2 1 eccieeccieeccccS −+−++++=ρ (23) The ( 22) and (23) coincide , and both such expressions contain the following interference terms. ))(( 2 1 ))(( 2 1 212 * 121 * 21 ieeccieecc −++ (24) and ))(( 2 1 ))(( 2 1 212 * 121 * 21 ieeccieecc −++ (25) We know that they represent the hard problem in a theory of quantum collapse . We may write (23) in the following terms .int,1 SSS ρρρ += (26) where =S1ρ 3 2 2 2 1 2 2 2 1 )( 2 1 )( 2 1 eccccS −++=ρ (27) and =int,Sρ ))(( 2 1 ))(( 2 1 212 * 121 * 21 ieeccieecc −++ (28) or equivalently int,Sρ = ))(( 2 1 ))(( 2 1 212 * 121 * 21 ieeccieecc −++ (29) The mechanism that induces the collapse of the wave function is now evident. During the interaction of the system S with the macroscopic apparatus M the previous interference terms are destroyed. It never can happen until we assume in algebraic terms that the )(SiA algebra is acting in the ( )MS + interaction. and that ,during such coupling ( )MS + , the system undergoes a transition from the Clifford algebra )( iSA to the dihedral algebra 1,±iN . If , probabilistically speaking, the macroscopic instrument reads 2 3 h +=S , it means that the algebra 1,+iN has prevailed . If instead the macroscopic instrument reads 2 3 h −=S , it means that the algebra 1,−iN has prevailed. In the first case the basic commutation rules that hold are those given in theorem 2a , iee =21 , iee −=12 , (30) 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (31) The density matrix becomes .int,11, SSS ρρρ +=+ (32) with =int,Sρ 0))(( 2 1 ))(( 2 1 212 * 121 * 21 =−++ ieeccieecc (33) In the second case the basic commutation rules that hold are those given in theorem 2b, iee −=21 , iee =12 , 12 eie = , 12 eie −= , 21 eie −= , 21 eie = (34) The density matrix becomes .int,11, SSS ρρρ +=− (35) with int,Sρ = 0))(( 2 1 ))(( 2 1 212 * 121 * 21 =−++ ieeccieecc (36) The macroscopic apparatus has the task to differentiate 1,+S ρ from 1,−S ρ on the basis of its dihedral algebra, destroying interference . There is another important feature in such mechanism . The basic matrix density expression , written previously in equivalent manner in the (22) and (23) and valid only in the A(Si) algebra, ( this is to say before S interacts with M) contains two algebraic elements that in quantum mechanics relate the Fermion annihilation and creation operators . In fact they are explicitly expressed in such basic matrix density expression 3 2 2 2 1212 * 121 * 21 2 2 2 1 )( 2 1 ))(( 2 1 ))(( 2 1 )( 2 1 eccieeccieeccccS −+−++++=ρ (37) They act in )( iSA before of the interaction of S with M . The new key is here : when the system S interacts with M , the new commutation relations , given previously for the dihedral algebra in theorems 2a and 2b, or the (31) or the (34) , act and they completely cancel the presence of the algebraic terms corresponding to the two fermion creation and annihilation operators. Quantum collapse requires the cancellation of such two operators and it happens during the transition from an )( iSA to 1,±iN dihedral Clifford algebra in its algebraic , geometric and metric structure . This represents the basic mechanism of the ( )MS + interaction. We have the counterpart using the Hamiltonian and the evolution operator in quantum mechanics . Consider the quantum system S and indicate by 0ψ the state at the initial time in Hilbert space . The state at any time t will be given by 0)()( ψψ tUt = and )0(0 == tψψ (38) An Hamiltonian H must be constructed such that the evolution operator U(t), that must be unitary, gives iHt etU −=)( . It is well known that, given a finite N-level quantum system described by the state )(tψ , its evolution is regulated according to the time dependent Schrödinger equation )()( )( ttH dt td i ψ ψ =h with 0)0( ψψ = . (39) Let us introduce a model for the hamiltonian H(t). We indicate by H0 the hamiltonian of the system S, and we add to H0 an external time varying hamiltonian, H1(t), representing the coupling to which the system S is subjected by action of the measuring apparatus. We write the total hamiltonian as H(t) = H0 + H1(t) (40) so that the time evolution will be given by the following Schrödinger equation [ ] )()()( 10 ttHH dt td i ψ ψ +=h (40a) We have that [ ] )()()()()( 10 tUtHHtUtH dt tdU i +==h and U(0)=I (41) where U(t) pertains to the special group SU(N). Let A1,A2,........,An , (n=N 2 -1), are skew-hermitean matrices forming a basis of Lie algebra SU(N). In this manner one arrives to write the explicit expression of the hamiltonian H(t). It is given in the following manner [ ] MApparatusnHamiltoniaSSystemnHamiltoniaAbAatHHitiH j n j jj n j j +=+=+−=− ∑∑ == 11 10 )()( (42) where aj and bj = bj(t) are respectively the constant components of the hamiltonian of S and the timevarying control parameters characterizing the action of the measuring apparatus M . In order to continue such discussion we have to introduce the operator T, the time ordering parameter (for details see reff. [9,10,11]), in order to correctly describe the time (S+M) interaction , being in this case M a macroscopic apparatus marked from strong irreversibility . We have )))((exp())(exp()( 0 0 ττττ dAbaiTdHiTtU jj t t j +−=−= ∫ ∫ (43) that is the well known Magnus expansion. Consequently, U(t) may be expressed by exponential terms as it follows )........exp()( 2211 nn AAAtU γγγ +++= (44) on the basis of the Wein-Norman formula             + + + =             Ξ nnn n ba ba ba ...... ),......,,( 22 11 2 1 21 γ γ γ γγγ & & & (45) with Ξ n x n matrix, analytic in the variables iγ . We have 0)0( =iγ and I=Ξ )0( , and thus it is invertible. We obtain             + + + Ξ=             − nnn ba ba ba ...... 22 11 12 1 γ γ γ & & & (46) Consider a simple case based on the superposition of only two states. We have [ ]Tyy 21 ,=ψ and 1 2 2 2 1 =+ yy (47) We have here an SU(2) unitary transformation, selecting the skew symmetric basis for SU(2). We will have that       = 01 10 1e ,       − = 0 0 2 i i e ,       − = 10 01 3e (48) The following matrices are given jj e i A 2 = , j = 1,2,3 (49) The reader may now ascertain that the previously developed formalism is moving in direct correspondence with our Clifford algebra A(Si). We are now in the condition to express H(t) and U(t) in our case of interest. The most simple situation we may examine is that one of fixed and constant control parameters bj. In this condition the hamiltonian H will become fully linear time invariant and its exponential solution will take the following form       ++= ∑ ∑ = + = 3 1 ))(( )() 2 ( 2 ) 2 cos( 3 1 j jjj Abat Abat k sen k It k e j jjj (50) with 2 33 2 22 2 11 )()()( bababak +++++= . In matrix form it will result [ ] [ ] ( )           +−++−− +++++ = 331122 112233 22 cos)( 2 1 )( 2 1 )( 22 cos )( bat k sen k i t k baibat k sen k baibat k sen k bat k sen k i t k tU (51) and, obviously, it will result to be unimodular as required. Starting with this matrix representation of time evolution operator U(t), we may deduce promptly the dynamic time evolution of quantum state at any time t writing 0)()( ψψ tUt = (52) assuming that we have for 0ψ the following expression       = false true c c 0ψ (53) having adopted for the true and false states (or yes-not states, +1 and –1 corresponding eigenvalues of such states) the following matrix expressions       = 0 1 trueφ and Finally, one obtains the expression of the state )(tψ at any time [ ] [ ] falsefalsetrue truefalsetrue )ba(t k sen k i t k cosc)ba()ba(it k sen k c )ba(i)ba(t k sen k c)ba(t k sen k i t k cosc)t( φ φψ           +−+    +−+ +          ++++    ++= 332211 112233 222 1 2 1 22 (54) As consequence, the two probabilities Ptrue(t) and Pfalse(t), will be given at any time t by the following expressions )()( 2 1 2 cos)()( 222 2 222 BQAP k senkt QPt k sen k t k BAtPtrue +++++= (55) and )()( 2 1 2 cos)()( 222 2 222 DSRC k senkt RSt k sen k t k DCtPfalse +++++= (56) where A= Re ctrue , B=Im ctrue, C=Re cfalse , D=Im cfalse , P=-D(a1+b1)+C(a2+b2)-B(a3+b3), Q=C(a1+b1)+D(a2+b2)+A(a3+b3), R=-B(a1+b1)-A(a2+b2)+D(a3+b3), S=A(a1+b1)-B(a2+b2)-C(a3+b3) Until here we have developed only standard quantum mechanics. The reason to have developed here such formalism has been to evidence that at each step it has its corresponding counterpart in Clifford algebraic framework A(Si), and thus we may apply to it the two theorems previously demonstrated, passing from the algebra )(SiA to 1,±iN . In fact, to this purpose, it is sufficient to multiply the (50) by the (53) to obtain the final forms of )(tctrue and )(tc false       = 1 0 falseφ In the final state we have that       = )( )( tc tc false true tψ (57) We may now write the density matrix that will result to have the same structure of the previously case given in the (22-23) . In the Clifford algebraic framework it will pertain still to the Clifford algebra A( Si). In order to describe the wave-function collapse we have to repeat the same procedure that we developed previously from the (22) to the (37), considering that, in accord to our criterium, we have to pass from the algebra A(Si) to 1,±iN , and obtaining 2 )(tctrueM =→ ρρ (58) in the case 1,+iN and 2 )(tc falseM =→ ρρ (59) in the case 1,−iN , as required in the collapse. Using Clifford algebra, we have given now a complete theoretical elaboration of the problem of wave function reduction in quantum mechanics also considering the process under the profile of the time dynamics. A time value of the collapse may be also obtained by ai and bj. To avoid difficulties that could arise to have considered only an n=2 dimensional situation , we may also consider now the explicit case of the ( )MS + interaction in their corresponding tensor product . Clifford )(SiA algebra at order n=4 [16,17,18] is E0 i = I 1 ⊗ e i ; Ei 0 = e i ⊗ I 2 (60) The notation ⊗ denotes direct product of matrices, and I i is the ith 2x2 unit matrix. We have two distinct sets of Clifford basic unities, E0 i and Ei 0, with 120 =iE ; 1 2 0 =iE , i = 1, 2, 3; (61) E0 i E0 j = i E0 k ; Ei 0 Ej 0 = i Ek 0 , j = 1, 2, 3; i ≠ j and Ei0 E0 j =E0 j Ei 0 (62) with (i, j, k) cyclic permutation of (1, 2, 3). Let us examine now the following result (I 1 ⊗ ei) (ej ⊗ I 2) = E0 i Ej 0 =Ej i (63) We have E0 i Ej0 = Ej i with i = 1, 2, 3 and j=1, 2, 3, E j i 2 = 1, Ei j Ek m ≠ Ek m Ei j, and Ei j Ek m = Ep q where p results from the cyclic permutation (i, k, p) of (1, 2, 3) and q results from the cyclic permutation (j, m, q) of (1, 2, 3). In the case n = 4 we have two distinct basic set of unities E0 i , Ei 0 and, in addition, basic sets of unities (Ei j , Ei p , E0 m) with ( j, p, m) basic permutation of (1, 2, 3). We may now give the explicit expressions of E0 i, Ei 0, and Ei j . E0i refers to the measured system S while Ej0 refers to the measuring apparatus . Eij characterizes instead the coupling . E01 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 =             ; E i i i i 02 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E03 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 = − −             (64) E10 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 =             ; E i i i i 20 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E30 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 = − −             ; E11 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 =             ; E22 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 = − −             ; E33 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 = − −             ; E i i i i 12 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E13 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 = − −             ; E i i i i 21 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E31 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 = − −             ; E i i i i 23 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E i i i i 32 0 0 0 0 0 0 0 0 0 0 0 0 = − −             . We have now some different sets of Clifford algebras )(SiA ),,( 131201 EEE , ),,( 232201 EEE ,( ),, 333201 EEE , ),,( 131102 EEE , ),,( 232102 EEE ,( ),, 333102 EEE , ),,( 121103 EEE ,( ),, 222103 EEE , ),,( 323103 EEE ,( ),, 332310 EEE , 322210 ,,( EEE ), ),,( 312110 EEE ),,( 331320 EEE , ),,( 321220 EEE ,( ),, 311120 EEE ,( ),, 231330 EEE ,( ),, 221230 EEE ,( ),, 211130 EEE (65) We may apply the theorems n.1 and n.2 to each of such sets and consider the )( iSA and 1,1 ±N algebras that we used in the previous case of application. Fixed such algebraic premises, we have to extend the previous elaboration considering explicitly the presence of the measurement apparatus M obtaining tkMtkk k ktMSMjij j i i MS ccc ),( 2 ,, ρφφρρφφρρρ ⊗><=→⊗><=⊗= ∑∑∑ ∗ (66) We have connected the set iE0 to the quantum system S to be measured, and the set 0iE to the measuring apparatus M . The basic set ijE couples S with M . The resulting density matrix ρ is             −−− +−− ++− +++ = sittiqqidd itthifficc iqqiffeibb iddiccibba 212121 212121 212121 212121 ρ (67) ρ of the (67) is still a member of the Clifford algebra )( iSA . ) 4 () 4 ( 3303300033300300 EEEE e EEEE a −−+ + +++ =ρ + ) 4 ( 33300300 EEEE h −−+ + ) 4 ( 33300300 EEEE s +−− +     + − + ) 2 () 2 ( 32022 3101 1 EE b EE b +     + − + 2 () 2 ( 20232 1310 1 EE c EE c +     + − − ) 2 () 2 ( 21122 2211 1 EE d EE d +     − + + ) 2 () 2 ( 21122 2211 1 EE f EE f +             − + − 2 ) 2 ( 20232 1310 1 EE q EE q +     − + − ) 2 () 2 ( 02322 3101 1 EE t EE t (68) We must now pass from A(Si) to 1,±iN . Consider that , during ( )MS + interaction, the )( iSA algebra is vanishing, leaving the place to 1,±iN . In such transition 33E is now assuming numerical value +1 and this is to say that 03E , 30E during the transition ( measurement in 1,±iN ) are assuming or 13003 +== EE or 13003 −== EE . By inspection of the (68), it is seen that terms with e and h go to zero. It remains the term with a for 13003 +== EE and the term with s for 13003 −== EE . All the terms containing ib , ic , id , if , iq , it ( 2,1=i ) go to zero and the wave function collapse has happened. Let us explain as example as the term 2 3202 EE + (69) pertaining to 2b , goes to zero. Owing ( )MS + interaction, transition )( iSA -> 1,±iN is happening , 33E is becoming +1. By inspection of the (65), it is seen that the basic algebraic set )(SiA in which 33E enters is ( ),, 333201 EEE . Passing from the algebra )(SiA to the algebra 1,+iN (in fact we have attributed to 33E the numerical value +1) we obtain the new commutation rule that iEE =3201 . (70) On the other hand, considering in )( iSA the set ( 030201 ,, EEE ) of the (65) with attribution to 03E the numerical value -1, we have the new commutation rule in iN that iEE −=0201 (71) In conclusion we have that iEE 0132 = (72) and 2 3202 EE + = 0 22 01010102 = +− = + iEiEiEE (73) Following the same procedure, one obtains that also the other interference terms are erased and in conclusion, passing from the algebra )( iSA to 1,±iN , the density matrix ρ , given in (68), is reduced to be ) EEEE (a 4 33300300 +++=ρ + ) EEEE (s 4 33300300 +−− (74) where in the new application of the 1,±iN algebra, we may have or 13003 +== EE ( )133 +=E (75) and thus aM =→ ρρ (76) or 13003 −== EE ( )133 +=E (77) and thus sM =→ ρρ (78) and the collapse has happened. 3. Conclusion We have given a first solution to the problem of quantum collapse in quantum mechanics but using a quantum system having only three anticommuting elements. 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