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GENERAL COMMENTS            It is well known that J. von Neumann, in his book Mathematical Foundations of Quantum Mechanics (Princeton, Princeton Univ. Press, 1955), showed that hidden variables can be excluded in quantum mechanics on the basis of some fundamental assumptions (in particular, five assumptions contained in pages 311, 313, 314 of the book). The most important with regard to our formulation appears in page 313 in the following manner:  - There corresponds to each observable of a quantum mechanical system a unique hypermaximal Hermitian operator in Hilbert-space. This correspondence is assumed to be one-one, i. e., each such operator corresponds to an observable (see pag. 313). Actually, von Neumann used the term Physical Quantity instead of the term Observable but with the same connotation.  In our work we have attempted to use a totally different approach to express all the foundations and results of the usual quantum mechanics. We have not used the standard quantum mechanical operator language in Hilbert space, but we have chosen instead to use a suitable algebra: we have chosen the algebra of the biquaternions and we have called Biquaternion Quantum Mechanics (BQM) our formulation. We are unable to mention here all the previous authors that in the past studied quantum mechanics using quaternions. Only we mention here that Birkoff and von Neumann, just for the first time in 1932, used quaternions for quantum mechanics. A lot of authors subsequently used quaternions to characterize themes and fields of quantum mechanical interest. However, our approach cannot be considered to enter in the framework of such previous important studies. For the first time we have used biquaternions and their algebra in a different manner and showing, first of all, the intrinsic indetermination character that such algebra evidences. In this manner we have found the way to express the intrinsic indetermination of quantum reality not through a mathematical formalism to which indetermination may be linked in the consequent interpretative framework, but through a proper algebra whose intrinsic character is to exhibit indetermination in itself. This appears to be an important point in the basic finality to realize a general quantum theory.  The second point has been the quantization. We have also shown that the quantization arises as consequence of the basic algebra of the biquaternions. It is intrinsic to the basic algebra of the biquaternions. Also this result appears to be of particular interest in the aim to construct a general quantum theory.  As third point, we have found the manner to reduce the biquaternions to simple numerical evaluations in the cases of interest. This is still of particular interest since we find that all the numerical fields may be reduced to be biquaternions with connected pure numerical expressions: natural, complex, octonion numbers may be reduced to be biquaternions and, when assuming numerical values, the biquaternions, starting with their algebra, generate each time a new subalgebra compatible with their given general algebra.  This is our basic result from a pure mathematical view point. Let us see in quantum mechanics. Using this biquaternion approach, step by step, we have re-derived all the standard results of quantum mechanics. We have found that actually all the usual quantum mechanics may be rederived using our approach of biquaternion algebra and subalgebra. We have re-obtained all the basic foundations of the theory (Schrödinger equation-time evolution, ....) and all the previous standard results of the theory (harmonic oscillator, hydrogen atom, and so on). In the book it is shown that all the standard physical quantities of the usual quantum mechanics are made by biquaternions and that all the usual quantum mechanics is no more that a biquaternion calculus.  Quantum Mechanics is a biquaternion calculus.  We repeat that the same quantization of quantum systems has been shown to be strongly connected with the basic algebraic features of the biquaternions. All the standard quantum mechanics has been re-derived only using the algebra of the biquaternions. Some basic differences, however, arise in BQM respect to the standard quantum mechanical formulation. In his operatorator formulation in Hilbert space, usual quantum mechanics, as any correct scientific theory, gives us the possibility to express accurate predictions. Given a well arranged experiment, this theory rightly predicts the result also assigning the correct probabilities for any possible result of the measurement.  Despite of the great importance and out come of this kind of theoretical approach, it remains that the usual quantum mechanics remains limited to only its predictive attitude. This is obtained by making the theory a basic distinction between a physical quantity at the quantum level of the physical reality (the operator « observable) and the possible values that such quantity may assume and may be predicted. Further, it must still considered that a strong correspondence results to be established  between the mathematical apparatus of the theory and the interpretation that is given of the foundations of quantum reality. Every mathematical object in the theory becomes quite meaningless from the pure physical viewpoint (see, i.e., the concept of wave function), it results to be only a mathematical instrument to give predictions on observables conceived by operators. However, in any physical condition, a theory must attempt to characterize and to describe the physical quantities that enter in the explorated physical dynamics, and, in particular in the case of a quantum theory, depending on what has been measured, these physical quantities must assume directly definite numerical values (and others do not) in the framework of the theory being not sufficient the prediction of such assumed values. Instead, the usual quantum mechanics remains substantially unable to express algebraic equations involving different physical quantities beyond its predictive attitude, and, in particular, it is unable to analyze the behaviour of such physical quantities when they must be considered and written in terms of their assumed numerical values. We retain that, out of the experimental confirmations, a correct physical theory must have this important feature. It cannot limit itself to only predict the possible numerical results of its physical quantities as consequence of the measurements, but it must be able also to write these physical quantities in terms of these assumed numerical values, and, thus, explaining in this manner to us what happens in the physical dinamics when such physical quantities assume such definite numerical values and others do not. Only this kind of theoretical approach gives us an understanding of the physical events and of their quantum reality as considered to be at the basis of the observations and of the measured results. This is precisely the quantum mechanical approach that has been developed by us by biquaternion quantum mechanics.   OTHER COMMENTS 1) Let me clear what we intend when using the terms algebra and subalgebra of the biquaternions. The existence of the biquaternions as hypercomplex number field is shown in the book in the most rigorous manner using only three basic anticommutative elements ei (i = 1, 2, 3) that are isomorphic to Pauli's well known matrices. We use only two basic axioms for biquaternion field:  a)  (i= 1, 2, 3) , ei different from 1;                                 (1.1)  b) eiej = - ejei (i = 1, 2, 3; j = 1, 2, 3; i different from j) (Basic Anticommutation Rule)  Biquaternions are hypercomplex expressions given by Z= ( m = 0, 1, 2, 3) (e0 =1), and xm real or complex numbers.  Let us show our approach with the aid of an example. Let us consider the following two biquaternions  A = 1 – e3 ;           B = e2i + e1 Let us examine the strange property of such two biquaternions. We have that AB = 0 and BA = 2B.  How does it arise such strange property of biquaternions to give AB = 0 and BA = 2B? Let us remember the two basic axioms of biquaternions: we have that eiej = - ejei (i = 1, 2, 3; j = 1, 2, 3; i different from  j) and  (i = 1, 2, 3). This time, the possibility of biquaternion bases to assume numerical values is contained itself in their algebra. Since the possibility for a basis to assume a numerical value corresponds from a physical view point, to the possibility for an observable of a quantum systems to assume a numerical value through a measurement, using biquaternion calculus in quantum mechanics we have this time the intrinsic possibility by such algebra also to characterize physical situations of measurement. This is an extraordinary result if examinated respect to the standard theory where the problem of the measurement and/or of wave packet reduction remained substantially unresolved. Let us return to consider AB = 0 and BA different from 0. To understand this biquaternion property, consider the trivial numerical expression: 1 – 1 = 0. In the biquaternion case it becomes that 1 - (i = 1, 2, 3). Consider i = 3 for brevity. We have that  1 - = (1 – e3) (1 + e3) = 0 = (1 + e3) (1 – e3) The previous relation is satisfied for e3 = + 1 or for e3 = - 1.  Consider that we have also that          (1 – e3) (1 + e3) e1 = 0 that is to say: (1 – e3) (e1 + i e2) = 0  We have found the two biquaternions A and B and the reason to be AB = 0 and BA = 2B resides in an intrinsic property of B.  In fact, we have found that             (1 + e3) e1 = 0 if and only if e3 = -1. In other terms, when e3 = -1, in the biquaternions, the algebra changes and we have that e1 + i e2 = 0.  e3 = -1 gives e1 + i e2 = 0 on the right for (1 + e3).  Let us consider now           (1 + e3) (1 – e3) e1 = 0 we have that           (1 + e3) (e1 - i e2) = 0 with e3 = +1; e1 - i e2 = 0 on the right for (1 – e3).  Consider on the left        e1(1 – e3) (1 + e3) = 0        (e1 + i e2) (1 + e3) = 0 We have e1 + i e2 = 0 for e3 = 1;          e1(1 + e3) (1 - e3) = 0 ; we have          (e1 - i e2) (1 - e3) = 0 ; we have e1 - i e2 = 0 for e3 = - 1. What is the meaning of such our simple calculations? It is that we are to a radical turning-point in our studies on the foundations of quantum mechanics. Using biquaternion calculus in quantum mechanics, it emerges that the basic assumption that was given by von Neumann in pag. 313 of his book is probably correct in his basic operator approach of quantum mechanics but it cannot be accepted as valid when we construct a general quantum theory using an algebra for its realization. When we use the biquaternion algebra we have the basic rules  and eiej = - ejei when the basic unities ei do not assume a definite value. When such situation instead happens (i.e. e3 = +1 or e3 = -1), the basic algebra of e1, e2, e3 cannot remain to be the same as before the attribution of the numerical value. Instead we have a new algebra. A subalgebra of the biquaternions is generated having the subordinate that e3 is now + 1 or – 1. We have the new subalgebra  e11 = - i e2 and 1 e1 = i e2 In contrast with von Neumann’s basic assumption, e1 and e2 change in their algebra when e3 assumes a numerical value (i.e., e3 = + 1) respect to the general case in which e3 lives with its basic indetermination and possibility to assume a definite numerical value.  The same thing happens when e3 is considered to assume the value – 1. Similar results are obtained also when e1 or e2 are considered to assume definite numerical values. A new subalgebra is generated. Consider, in particular, that all this may be acknowledged to be intrinsic to the biquaternion algebra. As discussed in detail in the book, quantum mechanics started by a mistake when the founder fathers of the theory did not revise the opportunity to realize it by biquaternions while instead they had the physical and the cultural conditions to realize such result.  Von Neumann’s assumptions on quantum mechanics are wrong when considered under the perspective of a biquaternion formulation of quantum mechanics that instead appears to emerge as the correct basis for the formulation of the theory.  2) With the aid of a further example let us examine now how the superposition principle of quantum mechanics may be found in the biquaternion calculus of quantum mechanics. For brevity, let us consider a quantum physical quantity to be connected to e3. We have that = 1, thus e3 may assume numerical values +1 or – 1. Quantum states in Biquaternion Quantum Mechanics are found to be biquaternions. We search the biquaternion j1 giving e3 j 1 = j1, and the biquaternion j2 giving e3j 2 = - j2. We find that j1 = and j2 =  with j1j 2 = j2j 1 = 0 and j1j1= 1 (e3 = + 1), and j2j2= 1 (e3 = - 1). Let us consider the generic biquaternion Z = A + B e1 + C e2+ D e3 with A2 + D2 = 1/2.  With this biquaternion we reobtain the superposition principle of quantum mechanics. In fact, the biquaternion Z may be re-written in the following manner:  Z = aj1 + b (e1 + i e2 ) + g (e1 - i e2 ) + d j2            (2.1) That is the superposition of states in Biquaternion Quantum Mechanics with a + d = 2A; a - d = 2D; b + g = B; b - g = - i C, and a2 = Prob. (e3 = + 1); b2 = Prob. (e3 = -1). On the other hand, we have different possibilities to connect probabilities to Z for e3. Let us consider the case in which e3 assumes the value + 1. Remember that in this case a subalgebra of biquaternions is generated with e1 + i e2 = 0 on one hand, and e1 - i e2 = 0 on the other hand. Z is reduced to be a=a . Similarly, in the case of e3 equal to – 1; we find for Z the value d=d . All the standard results of the usual quantum mechanics are well reproduced but, in addition, we have in this case the possibility to attribute actual numerical values to e3 and thus to evaluate the induced results. Let us consider an explicit case: i.e. we have  Z= + 2 e1 + 3 e2 +  e3. We have that a =; d = ; b= ;  g = ; a2 = P1 = ; d2 = P2 = (a2 + d2 = 1), and the (2.1).  Consider two subsequent measurements of two noncommuting physical quantities, i.e. related to e3 and e2 (e3 e2 different from e2 e3). For e3 we have the two biquaternions states j1 and j2. Similarly for e2 the two possible biquaternion states will be obviously y1 = , and y2 = , again with y1y 2 = y2y 1 = 0; = 1(e2 = + 1); = 1(e2 = - 1). The execution of the first measurement will give, i.e., e3 = + 1 and thus Z will be reduced to be the biquaternion aj1 = a with a = . The subsequent measurement of e2 will induce the biquaternion Z to assume the form  Z = a1+ b1(e1 + i e3) + g1(e1 - i e3) + d1 With a= Prob. (e2 = + 1); d= Prob. (e2 = - 1); with = a ; a1 - d1 = 0; b1 + g1 = 0; i b1 – i g1 = ; a + d= 1. We will obtain that a1 = d1 = a . For e2 = + 1, the biquaternion Z will be reduced to a1= a1 y1 = a1, while in the case e2 = - 1, the same biquaternion will be reduced to be d1= d1y 2 = d1. The subalgebra of the biquaternions will induce e1 + i e3 = 0 and e1 - i e3 = 0 respectively on the left and on the right. All the standard results of the usual quantum will be reproduced. By biquaternion formulation it will be also reproduced another standard result of quantum mechanics, that is  Prob (e3 = + 1, e2 = - 1) different from Prob (e2 = - 1, e3 = + 1) As it is well known, this is a standard and an essential result of quantum mechanics to which the same Bell’s unequality may be regarded in the case of coumpound systems. Thus, also by simple considerations, we have here a direct confirmation about the intrinsic attitude of biquaternions to characterize quantum theory. To this purpose we may also recall another basic quantum result now expressed by biquaternions.  3) The basic statement of BQM is that all the physical quantities of the usual quantum mechanics may be expressed by biquaternions. Owing to linearity of these numbers, we are thus reconduced to consider biquaternions in their standard form. Consider the well known quantum mechanical expression a e1 + b e2 + g e3 with a2 + b2 + g2 = 1 as it was examinated also by Bohm and Bub just in 1966 and regarding von Neumann’s well known assumption of linearity. This is still a biquaternion. We have that Z = a e1 + b e2 + g e3; Z2 = (a e1 + b e2 + g e3) (a e1 + b e2 + g e3) = 1, thus Z may assume the numerical values + 1 or – 1. When Z assumes the values + 1 or – 1, a new subalgebra is generated in the algebra of the biquaternions. As previously, we may derive such subalgebra considering that, i.e., in the case a e1 + b e2 + g e3 = 1, we must have that  e1 (a e1 + b e2 + g e3 - 1) = 0  e2 (a e1 + b e2 + g e3 - 1) = 0          (2.4) e3 (a e1 + b e2 + g e3 - 1) = 0 and similar expressions with ei (i = 1, 2, 3) multiplied on the right. We have the new subalgebra:    e11 = a + i b e3 - i g e2 e21= - i a e3 + b + i g e1         (2.5) e31= i a e2 - i b e1+ g and a similar subalgebra by ei (i = 1, 2, 3) multiplied on the right. If we insert now the found expressions in Z, we correctly obtain that                     Z = a2 + iab e3 - iga e2 - iab e3 + b2 + igb e1 + iag e2 - ibg e2 + g2 = a2 + b2 + g2 = 1  as required.  This is the only manner in which it may be shown that the biquaternion Z, and correspondingly, in the usual quantum mechanics, the physical quantity Z, may assume the numerical value + 1. The same happens in the case Z = - 1. The obtained subalgebra, given in (2.5), results to the absolutely necessary in order to attribute numerical values to the physical quantity Z. Also in this case von Neumann’s original assumption results to be not valid.  Let us examine in detail the case in which e3 assumes the value e3 = + 1. We obtain that  a e1 = a2 + iba - iga e2 b e2 = - iab + b2 + igb e1           (2.7) g e3 = iag e2 - ibg e1 + g2 ; e3 = 1 We have still that a e1 + b e2 + g e3 = a2 + b2 + g2 = 1 and a similar corresponding numerical expression we obtain in the case for e3 = - 1, or, equivalently, for e2 = + or - 1 or e1 = + or - 1. Let us examine the mean values. We have that    <a e1>= a2 + iba - iga <e2> <b e2> = - iab + b2 + igb <e1> <g e3 >= iag <e2 >- ibg <e1>+ g2 Consequently, <a e1 >+ <b e2 >+ <g e3> = a2 + b2 + g2 = <a e1 + b e2 + g e3> = 1. As it may be seen, it is re-found one of the basic statements of Von Neumann formulation of the standard quantum mechanics.  In conclusion, all the basic statements of the standard quantum mechanics are re-found by using BQM but this time we have in detail the possibility to attribute numerical values to the basic biquaternion unities and thus to resolve in this manner some unresolved questions of the standard quantum mechanical formulation and, in particular, the problem of the quantum measurements and of the wave packet reduction.  4) The moment in which BQM gives direct indication that for the first time the old and basic problems of quantum mechanics may be correctly solved in BQM, emerges when we consider composite quantum systems in BQM. This analysis immediately still gives violation of the basic von Neumann’s assumption regarding the unique hermitian operator to be connected to an observable in the standard quantum mechanical formulation by operators. It is seen that composite quantum systems require biquaternions not at order n = 2 but with order n = 4. At order n = 4 we have biquaternion basic unities E0i, Ei0, Eii, and Eij (i = 1, 2, 3; j = 1, 2, 3; i different from j). All these unities are required to describe physical phenomena at quantum level in BQM; in particular, the basic unities Eii and Eij result to be unexpressed in the usual formulation of the standard quantum mechanics while instead they show all their fundamental importance in order to realize a quantum theory and, in particular, to explain non separability and non locality of EPR systems. In this manner all the formulation of quantum mechanics follows but with a new light. Bell’s unequality is reobtained and it is found that it may be also generalized, the same old problematics regarding the existence of hidden variables now becomes evident and concrete to be analyzed in detail, and the basic approaches regarding non locality and non separability assume for the first time a new and definitive characterization.  By BQM it results, in particular, very clear the unreconducibility of composite systems to the partial analysis of the single components of a given quantum system. To this regard, it emerges the particular role explained from the Eii and Eij basic unities, ignored in the usual quantum mechanics.  5) In conclusion, we see that the analysis of quantum theory by biquaternions, as performed in the book Biquaternion Quantum Mechanics, finally gives strong indication for the basic unsolved problems on the foundations of quantum mechanics. By these comments, we have given here only some preliminar indications on the manner in which the problems are analyzed in detail in the book. We repeat here one of the results of the book: we have found that few basic axioms are required in order to realize a general quantum theory. They are also required for relativity. They have been given by us in (1.1) as the basic axioms of the biquaternions. We have also expressed quantum theory by an algebra that this time expresses by itself the basic indetermination that lives at quantum level, and, in addition, enables us to analyze quantum measurements that instead constituted a problem in the standard formulation of quantum mechanics. In any case we have found results that are in agreement with the standard formulation of quantum mechanics. However, our formulation has also indicated the necessary way that we must follow to acknowledge the limits of the standard theory and the channels leading to its necessary biquaternion revision.  There is still another feature of BQM that we intend to outline here. As Bell stated just in 1987: "….. a problem is brought into focus. I think any sharp formulation of quantum mechanics has a very surprising feature: the consequences of events at one place propagate to other places faster than light …". For me this is the real problem with quantum theory: the apparently essential conflict between any sharp formulation and fundamental relativity.  We see that such a conflict could also start to lessen considering BQM. On one hand we have a right formulation of quantum mechanics by biquaternions. Indeed, in our opinion, this is the correct formulation of quantum theory. On the other hand, biquaternions have an intrinsic relativistic character. As discussed in detail in the book, the norm of a biquaternion fully reproduces the basic invariant of special relativity. As first it was considered by K. Imaeda, we may consider the biquaternion space consisting of all the biquaternions with a quadratic metric defined by the norm of the biquaternions. For a single particle, i.e., to a given biquaternion Z=x0 +x1e1+x2e2+x3e3, it corresponds the representative point P(xm ) (m = 0, 1, 2, 3) in the four dimensional space (x0,x1,x2,x3) and viceversa, where the norm of Z coincides with the quadratic metric of Minkowski space and Lorentz transformations result to be biquaternion transformations. Thus, it is the biquaternion space that has actually the representation of the physical events, and it is in correspondence with the Minkowski space that we use in relativity. Such space is thus regulated by the algebra and the subalgebra of the biquaternions. Note, in particular, that if we consider a quantum system composed by two components, we have the biquaternion Z1=x0 +x1E01+x2E02 +x3E03 for one component of the system, and the biquaternion Z2= x0’+ x1’ E10+x2’E20 +x3’ E30 for the other component of the system. Here, the space properties are regulated by the algebra and subalgebra of the two sets of basic unities E0i and Ei0 (i = 1, 2, 3) that result to be connected by the basic unities Eii and Eij, i.e., E0i = EiiEi0. Thus, a modification on the algebra of E0i, i.e., (thus on one component of the system in the space) immediately gives a modification of the algebra of Ei0 regarding the second component of the system in the space by the new relation of interconnection E i0 = EiiE0i. Not the spatial coordinates xm and xm’ (m = 0, 1, 2, 3) are involved in the correlated EPR systems but the new biquaternion space of the events where the measurement of one of the spin coordinates on one component regarding, i.e., E03 of E0i, immediately involves E30 = E33E03 (E33= -1) of the other component since a new biquaternion space configuration is realized as consequence of the measurement with a new subalgebra as induced by the measurement of E03. Thus, Minkowski space is only a "partial" representation of the true space that actually is the biquaternion space with connected basic unities ei (i = 1, 2, 3) for a single particle, and E0i, Ei0, Eii, Eij in the case of a compound system of particles as in EPR.  Thus, the conflicting condition correctly claimed by Bell, seems to vanish. Finally, we have the elements to intend non separability and non locality by the biquaternions Eii and Eij that operate as a bridge between E0i and Ei0 , E0i = Ei0Eii. 6) Finally, we intend to mention also that in the book a number of results are reached by using BQM and with regard to the application of BQM in particle physics and in biophysics. For the first time, i.e., are analyzed the quarks and also some basic aspects regarding the living matter are analyzed. I.e., it is shown that the basic universal macromolecules of living matter are described by BQM. It is found that BQM opens new perspectives also in these various fields of its possible applications.  With purpose we have not developed in detail these themes that we have introduced for particle physics as well as for biophysics and theoretical biology since our purpose was to give only the basic indications that will be required to the researchers to use BQM in developing such themes and other programs in their future activity of research. Therefore, we invite any researcher to read and to analyze the proposed results of this book. REFERENCES 1) J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, 1955 2) K. Imaeda, Nuovo Cimento, 32 B, 138, 1976 3) D. Bohm, J. Bub, Rev. Mod. Phys., 38, 3, 453, 1966