A new proposal to the extension of complex numbers
AA new proposal to the extension of complex numbers
Israel A. Gonz´alez Medina
Instituto Superior de Tecnologias y Ciencias Aplicadas.Universidad de la Habana.Cuba
Abstract
We propose the extension of the complex numbers to be the new domain where new concepts,like negative and imaginary probabilities, can be defined. The unit of the new space is definedas the solution of the unsolvable equation in the complex domain: | z | = z ∗ z = i . The existenceof the unsolvable equation in a closed domain as complex’s lead to the definition of a new typeof multiplication, for not violate the fundamental theorem of algebra. The definition of thenew space also requests the inclusion of a new mapping operation, so the absolute value of thenew extended number being real and positive. We study the properties of the vector space likepositive-definiteness, linearity, and conjugated symmetry. Keywords: hypercomplex number, vector space, inner product, complex number
Email address: [email protected] (Israel A. Gonz´alez Medina)
Preprint submitted to arXiv.org December 3, 2020 a r X i v : . [ phy s i c s . g e n - ph ] S e p . Introduction The accuracy of Quantum Mechanics (QM) in its realm is indisputable. QM works fineuntil it deals with relativistic particles, like photons that have rest mass zero and travel with aspeed of c . Quantum field theory (QFT) is the central mathematical framework on which aredeveloped most of the quantum mechanical models of subatomic particles to try these particlesin nowadays particle physics.The appearance of infinities is a significant problem of quantum field theory. The emergenceof infinities appeared when calculations of higher orders of the perturbation series led to infiniteoutcomes. One way to treat these infinities is by introducing formal techniques, known asRenormalization, that modify these quantities introducing effects of self-interactions. However,the various forms of infinities suggested the divergences are related to a more fundamentalissue instead of some failures for a specific calculation. Some physicists propose alternativeapproaches that included changes in the basic concepts. Among them, we found the assumptionof the existence of negative or incomplete probabilities.Paul Dirac introduced the concept of negative energies and negative probabilities, in 1942,when he quoted in the article ”The Physical Interpretation of Quantum Mechanics” (2) that:”Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money.” Also, Richard Feynman argued onits work ”Negative probabilities” (3) how not only negative probabilities but probabilities differ-ent from unity could be useful in probability calculations. Negative or incomplete probabilitieshave later been proposed to solve several problems and paradoxes.Although there is no mathematical reason for not defining the concepts of negative andincomplete probability, their physical interpretation is as complicated as diverse. In quantummechanics, the probability for a state being measured is the modulus squared of the wavefunction, which is a complex quantity and is usually represented like ψ . Then, mathematicallyspeaking, the assumption of a negative probability can be written as the existence in the theoryof an expression like ψ ∗ ψ <
0. The inclusion of the quantum concept of the negative probabilityimplies the mathematical need for extending the complex numbers for equations like z ∗ z < n -VMVF systems) and obtain two sets of Hamilton equations for solving the problem. Fromthe Hamilton theory, two canonical transformations are needed to evolve the system: one usingthe rectangular coordinates and another using the angulars. The bi-dimensional canonicaltransformation of the canonical variables of the system points out that the quantum theory for n -VMVF systems might be represented with two-component vectors, different than the complexHilbert space. This new space can be defined from the expansion of the complex vector space.The classical result, together with the possibility of including the negative probability, isthe main motivation and guide for defining the new space and expanding the complex numbers.In the second section, we briefly recall the obtaining process and the history of the complexnumbers as the starting point for their expansion to the new domain. Next, in section 3, wedefine the new domain’s unit and apply the vector space’s axioms to the new proposed innerproduct, which reveals the need to include new operations to the extended domain. In section4, we formally define the new operations and their properties. They are the bases, in section5, to analyze the algebraic properties of extended numbers like Associativity, Commutativity,Distributivity, and the Existence of identity and elements. In section 6, we define and studythe new domain’s inner product. In section 7, we study the division between two numbers onthe new domain by reviewing extraneous and missing solutions when the same factor multiplies2oth members of an equation. In sections 8, 9, and 10, we study some of the properties ofthe inner product and its Linearity and Conjugate symmetry. In section 11, we summarize themethod and the equations for computing the needed equations to define the new space. Finally,we present the conclusions in chapter 12.
2. History
Numbers are fundamentals in science. They are mathematical objects used to describequantity, order, or measure of concepts like time, space, matter, and fields. Physics theorieslike classical mechanics, electromagnetism, and quantum theory are developed using analyticentities like metrics space, tensors, fields, whose operations between them are based on theproperties of the numbers that the theory lays on. Numbers are classified as i ) natural, ii )integer, iv ) rational, v ) real, and vi ) complex. The Complex numbers are defined as thesolution of unsolved equations in the real domain, such as x + 1 = 0, from where the complexunit is defined as i = √− R , the complexes C , the quaternions Q , and the octonions O (6). LaterClifford develops what is known in the literature as Clifford algebra that generalizes the realnumbers, complex numbers, quaternions, and several other hypercomplex number systems andis connected with the theory of quadratic forms and orthogonal transformations.One of the most famous is the quaternions, which were first described by William RowanHamilton in 1843 as the quotient of two directed lines in a three-dimensional space or equiva-lently as the quotient of two vectors (7). A quaternion is usually represented as a + b i + c j + d k (2.1)where a, b, c , and d are real numbers, and i , j , and k are the fundamental quaternion units. Anotable property of a quaternion is that the multiplication of two quaternions is noncommuta-tive.Different from the work developed by Hamilton and others great mathematicians that de-velop the previous number systems, the objective of the present article is to propose a newnumber system that includes the arithmetical concept of the negative probability that can beused for the construction of the new Lorentz space on which can be developed a new proposalfor the quantum mechanic for the n -VMVF systems.
3. The complex extended unit k
We construct the new domain in the same way complex number was defined: as the solutionof an unsolvable equation in the domain about to be extended. The chosen unsolvable equation3n the domain of complex numbers C is | z | = i . (3.1)We can name the new set of numbers as the Extended Complex, because of the inclusion ofthe conjugated complex numbers, and we can represent it by E . The unit for this new set ofnumbers can be defined as | k | ≡ k ∗ k = i or k = ∗ √ i k ∈ E (3.2)being E , the set of the extended complex numbers and ∗ (cid:112) () is the conjugated square root of anumber x . The conjugated square root is the operation that results in a number y such thatits product over its complex conjugate is x : y ∗ y = x .It is very important to remark that this unsolvable equation is not unique, and the extendedunit’s definition may be different. We can set a general unsolvable equation for defining theextended unit as | z | = exp i θ ∀ < θ < π. (3.3)The definition of the extended unit in this proposal is established as above, k = ∗ √ i. However,this expression must be considered as replaceable in case of future improvements needed.We can express a general extended complex number as α = x k + y ∀ x, y ∈ C . (3.4)or replacing the complex numbers with their real components: α = a ik + b k + c i + d, ∀ a, b, c, d ∈ R . (3.5)As the absolute value of the imaginary unit i ∗ i = 1 and because of our extended unitdefinition, k ∗ k = i, we lost no generality by imposing that the absolute value for the extendedunit must be equal to 1. It implies the necessity to define a new map, besides the conjugatedmap, for the absolute value of the extended number being real. The new map, O new (k), appearswith the definition of the new domain. So when acting on a pure complex number, it must keepthe number invariant, the same way when the conjugated map, () ∗ , acts over real numbers.Then O new ( x ) = x ∀ x ∈ C . (3.6)If this new map, O new ( x ), is proven as a homomorphism, or the inverse of the previousassumption was satisfied, at least for the pure complex numbers, then an absolute value is aproduct that should have at least four factors. Indeed, if we proposed the absolute value forthe extended unit as1 = O new ( k )k ∗ k = O new ( k )i then O new ( k ) = i ∗ , (3.7)which, according to the invariant property of the new map over complex numbers, led to k = i ∗ .For the absolute value of the extended unit, we propose then a four term’s expression like O new (k ∗ ) O new (k) = i ∗ so O new (k ∗ ) O new (k) k ∗ k = i ∗ i = 1 (3.8)or O ∗ new (k) O new (k) = i ∗ so O ∗ new (k) O new (k) k ∗ k = i ∗ i = 1 , (3.9)4here we have subtly use the associativity property of the multiplication between pairs O ∗ new (k) O new (k)or O ∗ new (k) O new (k) and k ∗ k.We represent this operation O new (k) as k • , so we have the first definition:k ∗ k = i(k ∗ ) • k • = i ∗ (3.10)or k ∗ k = i(k • ) ∗ k • = i ∗ . (3.11)The operations k ∗ and k • should transform one point of the space into another point fromthe same space. The operations should have the general form:k ∗ = z k + w k • = z k + w (3.12)where z i and w i are complex numbers to be determined.However, this assumption and the definition of the extended unit lead to some inconsisten-cies. For example, from that definition, and replacing k ∗ = z k + w , we obtaink ∗ k = ( z k + w )k = i → k = − w z k + i z . (3.13)This expression constraints operation k , contradicting the independence that should existbetween such operations. If k and k ∗ k are related, then the unsolvable equation from wherethe extended unit was defined can be related to an expression containing k . In that case, wewill not have any unsolvable equation, because k is defined at all the complex domain. Fromthe second equation of any of the expressions 3.10 or 3.11, we can extract another independentequation for k . It relates the complex numbers z , z , w , w , and with them, the operationsk ∗ and k • , which should be independent between them.
4. Introduction to the extended numbers
The definition of the extended unit as k ∗ k = i might seem contradictory since the funda-mental theorem of algebra states that the set of complex numbers is algebraically closed for thesum and multiplication operations, so it should not exist an unsolvable equation for an n -degreepolynomial. Well, there are not. The unsolvable equation on a closed space only appears if weinclude the complex conjugated numbers in the polynomial. Indeed, from a polynomial like a n x n + a n − x n − + ... + a x + a x + a = (cid:88) n a n x n = 0 ∀ a n ∈ R (4.1)could never be extracted from the unsolvable equation used for defining the extended unit k.Instead, an equation like | x | = x ∗ x = i can be extracted from a different type of polynomial,such as: (cid:88) n,n (cid:48) a n,n (cid:48) ( x ∗ ) n x n (cid:48) = 0 ∀ a n,n (cid:48) ∈ R . (4.2)This kind of polynomials that includes the conjugated numbers will have unsolvable equationslike the one using for the new unit’s definition, signalizing the possibility of expansion withoutviolating the fundamental theorem in algebra. 5n Abstract, an algebraic structure is defined as the junction of a nonempty set A , a collec-tion of operations on A of finite arity (typically binary operations) and a finite set of identities,known as axioms, that the set and the operations must satisfy. For example, the ring of thecomplex numbers is algebraically represented as( C ; + , · ) . (4.3)The inclusion of conjugated complex numbers in the above polynomials 4.2 can be done byadding those numbers to the set of scalars or adding new mathematical operations that resemblethe existing operations and include the conjugated complex numbers. The operations arethe sum and multiplication with the complex numbers. The algebraic structures representingprevious cases are( E ∪ E ∗ ; + , · ) (4.4)and ( E ; + , · , ⊕ , (cid:12) ) (4.5)respectively. The development of any case should be in correspondence with the other. On thiswork, we study the new algebraic structure that adds two new operations to the structure asthe last case 4.5.We introduced the new sum operation of two complex numbers a, b , as the conjugatedsum. The operation is composed of two steps: i ) to transform the first added into its complexconjugated and ii ) sum the result with the second number like a ⊕ b ≡ a ∗ + b. (4.6)The new product of two complex numbers is designed as the conjugated product and is explicitlywritten like a (cid:12) b ≡ a ∗ b. (4.7)The operation is composed of two steps: i ) to transform the first number into its complexconjugated and ii ) multiply the result with the second number.From here and below, we must consider the absence of an operator in a product of twoextended numbers, αβ , as an erroneous expression, since the rules for the multiplication of theextended unit are undetermined. Nevertheless, the multiplication with no explicit operator isallowed once the rules are already defined and applied.The expansion of the conjugated product of two extended numbers that can be written as α (cid:12) β = α ∗ β = ( x k + y ) ∗ ( u k + v ) = ( x ∗ k ∗ + y ∗ )( u k + v ) , (4.8)includes the expression k ∗ k. As a general form, we consider that the conjugated mappingdepends on the number, α = x k + y , which is applied on. However, the extended unit’sdefinition, and with it, k ∗ k = i, should remain for all the extended numbers. That means thatthe extended unit’s definition needs to be replaced with the more general expression:k ∗ k → k ( α ) (cid:12) k = k ∗ ( α ) k = i (4.9)and conjugated product of two extended numbers is written as α (cid:12) β = α ∗ β = ( x k + y ) ∗ ( u k + v )= ( x ∗ k ∗ ( α ) + y ∗ )( u k + v )= x ∗ u k ∗ ( α ) k + x ∗ v k ∗ ( α ) + y ∗ u k + y ∗ v. (4.10)6f k ∗ ( α ) = z ( α )1 k + w ( α )1 and equation 4.9, the term k ∗ ( α ) k leads to the relationk ∗ ( α ) k = i → k = − w ( α )1 z ( α )1 k + i z ( α )1 (4.11)where z ( α )1 , w ( α )1 corresponds to the extended and the imaginary parts of the complex conjugatedmap of the extended number.On the other side, the standard product of the extended unit k · k must also remain invariantfor all sets of extended numbers. Different from the definition we gave to the extended unit, thesquare operation does not arise from an unsolvable equation. Nevertheless, it must be definedfor all extended numbers. We define the standard operation between the extended unit k · kas: k · k = k = z k + w ∀ E . (4.12)where z , w are two complex numbers to be defined. The definition set an independent expres-sion of k from 4.11. The standard multiplication of the extended unit, as on complex numbers,does not depend on the product’s factors. Hence the constant character of z , w and the ∀ E .Using this definition, the expressionk ∗ ( α ) · k = k ∗ ( α ) k = ( z ( α )1 k + w ( α )1 )k = ( z z ( α )1 + w ( α )1 )k + z ( α )1 w (cid:54) = i (4.13)Resuming, within the extended numbers, we define not one product but two: the standardand the conjugated product. They are just two common products following different rules forthe expressions k ∗ ( α ) (cid:12) k and k · k. The definitions that define those rules arek ∗ ( α ) (cid:12) k = i and k · k = z k + w . To compute these operations, we must compute the complex extended for the first factor ifneeded (only for the conjugated product), then establish the rules to be used for the expressionsinvolving the extended unit, k ∗ ( α ) k, and k · k = k , and finally apply them to the result.Being k ∗ ( α ) = z ( α )1 k + w ( α )1 , the expressions k and k ∗ ( α ) k, for the standard product mustcompute as equations 4.12 and 4.13, while for the conjugated product the expression have theform like reference 4.11. The table 1 resumes the different expressions for every case. Table 1: Expressions for k and k ∗ ( α ) k for each extended product α · β α (cid:12) β k z k + w − w ( α )1 z ( α )1 k + i z ( α )1 k ∗ ( α ) k ( z z ( α )1 + w ( α )1 )k + z ( α )1 w iAs ⊕ and (cid:12) are a type of sum and multiplication operations, the order of priority of theoperations then is similar. That means that any multiplication is granted higher precedencethan any addition. However, the order of priority between standard and conjugated operations,multiplication or addition, should be emphasized with parentheses ( ) or brackets [ ].The choice of expressing an extended number like x k+ y , where x and y are complex numbersand the standard product with the expression k · k = z k + w , responds to the prepositionthat state that “the Cartesian product of two rings is also a ring” as shown on chapter 2 two7f reference (8). From this preposition we can conclude that the set C × C = C is a ring withthe operations:( a, b ) + ( c, d ) = ( a + c, b + d )( a, b ) + ( c, d ) = ( ac, bd ) . (4.14)As shown below, the extended numbers set with the standard sum and multiplication alsobehave as rings.Lets now show both types of extended products between two extended numbers using bothrules: • The standard product of two extended is pretty straightforward, computing the productas usual and replacing 4.12. For the extended numbers α = x k + y and β = u k + v : α · β = ( x k + y ) · ( u k + v ) = xu k + ( xv + yu )k + yv = xu ( z k + w ) + ( xv + yu )k + yv = ( xuz + xv + yu )k + xuw + yv. (4.15)Let us verify that computation of the standard product is independent of choosing k ∗ ork expression. First, we express a complex extended number as the complex conjugate ofthe conjugated square root: α = ( ∗ √ α ) ∗ ≡ α (cid:48) ∗ . (4.16)We can relate their coefficients as: x k + y = ( x (cid:48) k + y (cid:48) ) ∗ = x (cid:48) ∗ z ( α (cid:48) )1 k + x (cid:48) ∗ w ( α (cid:48) )1 + y (cid:48) , (4.17)from where x (cid:48) = x ∗ z ∗ ( α (cid:48) )1 and y (cid:48) = y ∗ − x ∗ z ∗ ( α (cid:48) )1 w ∗ ( α (cid:48) )1 . (4.18)The standard product of α (cid:48) ∗ and β can be found using any of the expressions of the table1. We replace the map α (cid:48) ∗ and then we proceed with the standard product: α (cid:48) ∗ · β = ( x (cid:48) k + y (cid:48) ) ∗ · ( u k + v ) = ( x (cid:48) ∗ k ∗ ( α ) + y (cid:48) ∗ ) · ( u k + v ) (4.19)then we replace k ∗ ( α (cid:48) ) and compute the standard product in the usual way withk = z k + w ( x (cid:48) ∗ k ∗ ( α (cid:48) ) + y (cid:48) ∗ ) · ( u k + v ) = [ x (cid:48) ∗ ( z ( α (cid:48) )1 k + w ( α (cid:48) )1 ) + y (cid:48) ∗ ] · ( u k + v )= x (cid:48) ∗ uz ( α (cid:48) )1 k + ( x (cid:48) ∗ vz ( α (cid:48) )1 + x (cid:48) ∗ uw ( α (cid:48) )1 + y (cid:48) ∗ u )k + x (cid:48) ∗ vw ( α (cid:48) )1 + y (cid:48) ∗ v =( x (cid:48) ∗ uz z ( α (cid:48) )1 + x (cid:48) ∗ vz ( α (cid:48) )1 + x (cid:48) ∗ uw ( α (cid:48) )1 + y (cid:48) ∗ u )k + x (cid:48) ∗ uw z ( α (cid:48) )1 + x (cid:48) ∗ vw ( α (cid:48) )1 + y (cid:48) ∗ v. (4.20)Replacing equations 4.18, we obtain α (cid:48) ∗ · β = ( xuz + xv + xu w ( α (cid:48) )1 z ( α (cid:48) )1 + uy − xu w ( α (cid:48) )1 z ( α (cid:48) )1 )k + xuw + xv w ( α (cid:48) )1 z ( α (cid:48) )1 + yv − xv w ( α (cid:48) )1 z ( α (cid:48) )1 = ( xuz + xv + uy )k + xuw + yv (4.21)8hich is the same result as the standard product α · β . Now, we compute the same product α (cid:48) ∗ · β , replacing this time the expression k ∗ ( α (cid:48) ) k in equation 4.19 withk ∗ ( α (cid:48) ) k = ( z z ( α (cid:48) )1 + w ( α (cid:48) )1 )k + z ( α ) (cid:48) w . In that case, we have:( x (cid:48) ∗ k ∗ ( α (cid:48) ) + y (cid:48) ∗ ) · ( u k + v ) = x (cid:48) ∗ u k ∗ ( α (cid:48) ) k + x (cid:48) ∗ v k ∗ ( α (cid:48) ) + y (cid:48) ∗ u k + y (cid:48) ∗ v = ( x (cid:48) ∗ uz z ( α (cid:48) )1 + x (cid:48) ∗ vz ( α (cid:48) )1 + x (cid:48) ∗ uw ( α (cid:48) )1 + y (cid:48) ∗ u )k + x (cid:48) ∗ uw z ( α ) (cid:48) + x (cid:48) ∗ vw ( α (cid:48) )1 + y (cid:48) ∗ v, (4.22)which is the expression as result 4.20, showing the consistency of the two rules for thestandard product. • The conjugate product can be computed spanning the product using the distributivelaw, applying the complex conjugated map, and using the definition k ∗ ( α ) k = i. For theextended numbers α = x k + y and β = u k + v we have α (cid:12) β = ( x k + y ) (cid:12) ( u k + v ) = x ∗ u k ∗ ( α ) k + x ∗ v k ∗ ( α ) + y ∗ u k + y ∗ v = x ∗ u i + x ∗ v ( z ( α )1 k + w ( α )1 ) + y ∗ u k + y ∗ v = ( x ∗ vz ( α )1 + y ∗ u )k + x ∗ u i + x ∗ vw ( α )1 + y ∗ v. (4.23)The expression can also be computed, applying first the complex conjugated map andthen the distributive law. In this case, the quantity k must have the definitions accordingto table 1:k = − w ( α )1 z ( α )1 k + i z ( α )1 . (4.24)The conjugate product then have the expression( x k + y ) (cid:12) ( u k + v ) = ( x ∗ k ∗ + y ∗ )( u k + v )= ( x ∗ z ( α )1 k + x ∗ w ( α )1 + y ∗ )( u k + v )= x ∗ uz ( α )1 k + ( x ∗ vz ( α )1 + x ∗ uw ( α )1 + y ∗ u )k + x ∗ vw ( α )1 + y ∗ v = ( x ∗ vz ( α )1 + x ∗ uw ( α )1 + y ∗ u − x ∗ uw ( α )1 )k + x ∗ vw ( α )1 + y ∗ v + x ∗ u i= ( x ∗ vz ( α )1 + y ∗ u )k + x ∗ u i + x ∗ vw ( α )1 + y ∗ v, (4.25)which match the result computed with rule k ∗ ( α ) k = i.The standard and the conjugated sum of two extended number are α + β = ( x k + y ) + ( x k + y ) = ( x + u )k + ( y + v ) (4.26)and α ⊕ β = ( x k + y ) ⊕ ( u k + v )= ( x k + y ) ∗ + ( u k + v ) = ( x ∗ k ∗ ( α ) + y ∗ ) + ( u k + v )= (cid:2) x ∗ ( z ( α )1 k + w ( α )1 ) + y ∗ (cid:3) + ( u k + v ) = ( x ∗ z ( α )1 k + x ∗ w ( α )1 + y ∗ ) + ( u k + v )= ( x ∗ z ( α )1 + u )k + ( x ∗ w ( α )1 + y ∗ + v ) (4.27)respectively. 9 . Algebraic properties of extended numbers In Abstract Algebra theory, the classification of any set of numbers plus the sum andmultiplication operations are described by the compliance or not of the following properties:1. Associativity of addition and multiplication2. Commutativity of addition and multiplication3. Existence of additive and multiplicative identity elements4. Existence of additive inverses and multiplicative inverses5. Distributivity of multiplication over additionWe start the study of the properties of the standard and conjugated sum and product of theextended numbers, using their definitions and taking into account that values like z , w for thestandard multiplication operation are parameters still to be determined. The standard summation “ + ”
1. Associativity: For all α = x k + y , α = x k + y , α = x k + y ∈ E then( α + α ) + α = (cid:2) ( x k + y ) + ( x k + y ) (cid:3) + ( x k + y )= (cid:2) ( x + x )k + ( y + y ) (cid:3) + ( x k + y )= ( x + x + x )k + ( y + y + y ) = x k + y + (cid:2) ( x + x )k + ( y + y ) (cid:3) = α + ( α + α ) (5.1)2. Commutativity For all α = x k + y , α = x k + y ∈ E then α + α = ( x + x )k + ( y + y ) = α + α ∀ α , α ∈ E (5.2)3. Existence of additive identity element 0 E in E such that α + 0 E = ( x + 0)k + ( y + 0) = 0 E + α = α (5.3)4. Existence of additive inverse such that for every α ∈ E , there exists an element − α ∈ E ,such that α + ( − α ) = 0. If α = x k + y then − α = − x k − y . It can be verified that y k + y + ( − x k − y ) = ( x − x )k + ( y − y ) = 0 . (5.4) The standard product “ · ”
1. Associativity:Standard product is associative if the extended numbers α , α , α ∈ E satisfy α · ( α · α ) = ( α · α ) · α . (5.5)Being α = x k + y , α = x k + y and α = x k + y . The left member of the aboveequation is( x k + y ) · (cid:2) ( x k + y ) · ( x k + y ) (cid:3) = ( x k + y ) · (cid:2) ( x x z + x y + y x )k + x x w + y y (cid:3) = ( x x x z + x x y + x y x )k + ( x x x w + x y y + y x x z + y x y + y y x )k+ y x x w + y y y = (cid:2) x x x z + ( x x y + x y x + y x x ) z + x x x w + x y y + y x y + y y x (cid:3) k+ x x x w z + ( x x y + x y x + y x x ) w + y y y . (5.6)10he right member of the axiomatic equation is (cid:2) ( x k + y ) · ( x k + y ) (cid:3) ( · x k + y ) = (cid:2) ( x x z + x y + y x )k + x x w + y y (cid:3) · ( x k + y )= ( x x x z + x y x + y x x )k + ( x x y z + x y y + y x y + x x x w + y y x )k+ x x y w + y y y = (cid:2) x x x z + ( x x y + x y x + y x x ) z + x x x w + x y y + y x y + y y x (cid:3) k+ x x x w z + ( x x y + x y x + y x x ) w + y y y . (5.7)which is the same result as above.2. Distributivity of the standard multiplication over the standard additionThe standard extended inner product is distributive over the standard addition if for all α , α , α ∈ E it is satisfied the relation α · ( α + α ) = α · α + α · α . (5.8)If the numbers α i = x i k + y i , for all i = 1 ,
2, the right member of the preposition is α · ( α + α ) = ( x k + y ) · (cid:2) ( x k + y ) + ( x k + y ) (cid:3) = ( x + k y ) (cid:2) ( x + x )k + y + y ) (cid:3) = (cid:2) x ( x + x ) z + x ( y + y ) + y ( x + x ) (cid:3) k + x ( x + x ) w + y ( y + y )= (cid:2) ( x x z + x y + y x )k + x x w + y y (cid:3) + (cid:2) ( x x z + x y + y x )k+ x x w + y y (cid:3) = ( x k + y ) · ( x k + y ) + ( x k + y ) · ( x k + y ) = α · α + α · α , (5.9)as stated before.3. Commutativity:The standard extended product is commutative if relation α α = α α is satisfied forall α , α ∈ E . The replacement of numbers α i = x i k + y i , for all i = 1 ,
2, set standardproduct has the form: α · α = ( x k + y ) · ( x k + y ) = ( x x z + x y + x y )k + x x w + y y = ( x k + y ) · ( x k + y ) = α · α (5.10)4. Zero-product propertyIn algebra, the zero-product property states that the product of two nonzero elements isnonzero. In other words, if: α · β = 0 , only if α = 0 or β = 0 . (5.11)The standard product of these two extended numbers, where α = x k + y and β = u k + v ,show the existence of nontrivial zero-divisors on the standard product. Setting zero thestandard product of two extended numbers we have α · β = ( x k + y ) · ( u k + v ) = (cid:2) x ( uz + v ) + yu (cid:3) k + xuw + yv = 0 (5.12)which led to two complex equations x ( uz + v ) + yu = 0 , xuw + yv = 0 (5.13)Without lost generality, we can analyze this set of equations for different cases of β number: 11a) u = 0 , v (cid:54) = 0. In this case, the equations are xv = 0 , yv = 0 (5.14)which its satisfied if x = y = 0.(b) u (cid:54) = 0 , v = 0. In this case, the equation 5.13 take the form( xz + y ) u = 0 , xuw = 0 (5.15)which its satisfied if x = y = 0, for w (cid:54) = 0. If w = 0, then a nontrivial root fromthe equation xz + y = 0 is included.(c) u (cid:54) = 0 , v (cid:54) = 0. In this case, multiplying first equation 5.13 by v , the second one by u and subtracting one from the other we obtain x ( v + uvz − u w ) = 0 , (5.16)which introduce the nontrivial root coming from the equation: v + uvz − u w = 0 (5.17)5. Existence of multiplicative identity element:For every α ∈ E , exist a number 1 E = 1 in E such that α · · α = α. (5.18)On the other side, the conjugate sum nor product is not associative, commutative, and hasno additive identity nor inverse element. The following discussion is referred to the extendednumbers whose maps z , w , w , and w are defined. That means we exclude purely extendedand complex numbers. For these cases, we should proceed using the extended unit definition. The conjugated sum “ ⊕ ”
1. AssociativityThe associativity property implies that( α ⊕ α ) ⊕ α = α ⊕ ( α ⊕ α ) . (5.19)The left member of the last relation( α ⊕ α ) ⊕ α = (cid:2) ( x k + y ) ⊕ ( x k + y ) (cid:3) ⊕ ( x k + y )= (cid:2) ( x ∗ z ( α )1 + x )k + x ∗ w ( α )1 + y ∗ + y (cid:3) ⊕ ( x k + y )= (cid:2) ( x z ∗ ( α )1 + x ∗ ) z ( α ⊕ α )1 + x (cid:3) k + ( x z ∗ ( α )1 + x ∗ ) w ( α ⊕ α )1 + x ∗ w ( α )1 + y ∗ + y (5.20)is not equal to the right member since α ⊕ ( α ⊕ α ) = ( x k + y ) ⊕ (cid:2) ( x k + y ) ⊕ ( x k + y ) (cid:3) = ( x k + y ) ⊕ (cid:2) ( x ∗ z ( α )1 + x )k + x ∗ w ( α )1 + y ∗ + y (cid:3) = (cid:2) x ∗ z ( α )1 + x ∗ z ( α )1 + x (cid:3) k + x ∗ w ( α )1 + x ∗ w ( α )1 + y ∗ + y ∗ + y (5.21)12. CommutativityThe expression α ⊕ α = ( x k + y ) ⊕ ( x k + y )= (cid:2) x ∗ z ( α )1 + x (cid:3) k + x ∗ w ( α )1 + y ∗ + y (5.22)while α ⊕ α = ( x k + y ) ⊕ ( x k + y )= (cid:2) x + x ∗ z ( α )1 (cid:3) k + x ∗ w ( α )1 + y + y ∗ , (5.23)which means that the conjugated sum does not satisfy the commutative property.3. There is no conjugated additive identity element 0 E in E because0 E ⊕ α = x k + y = α (5.24)while α ⊕ E = x ∗ z ( α )1 k + x ∗ w ( α )1 + y (cid:54) = α (5.25)4. The conjugated additive inverse element also does not exist. That can be probed bystraight substitution and by noting that the property will not be satisfied if the conjugatedsum is non-commutative. The conjugate product “ (cid:12) ”
1. AssociativityThe associativity property implies that α (cid:12) ( α (cid:12) α ) = ( α (cid:12) α ) (cid:12) α . (5.26)Computing the left member we have α (cid:12) ( α (cid:12) α ) = ( x k + y ) (cid:12) (cid:2) ( x k + y ) (cid:12) ( x k + y ) (cid:3) = ( x k + y ) (cid:12) (cid:2) ( x ∗ y z ( α )1 + x y ∗ )k + i x ∗ x + x ∗ y w ( α )1 + y ∗ y (cid:3) = (cid:2) x ∗ (i x ∗ x + x ∗ y w ( α )1 + y ∗ y ) z ( α )1 + y ∗ ( x ∗ y z ( α )1 + x y ∗ ) (cid:3) k+ i x ∗ ( x ∗ y z ( α )1 + x y ∗ ) + x ∗ (i x ∗ x + x ∗ y w ( α )1 + y ∗ y ) w ( α )1 + y ∗ (i x ∗ x + x ∗ y w ( α )1 + y ∗ y )= (cid:2) i x ∗ x ∗ x z ( α )1 + x ∗ x ∗ y z ( α )1 w ( α )1 + x ∗ y ∗ y z ( α )1 + x ∗ y ∗ y z ( α )1 + x y ∗ y ∗ (cid:3) k+ x ∗ x ∗ y z ( α )1 + x ∗ x y ∗ + x ∗ i x ∗ x w ( α )1 + x ∗ x ∗ y w ( α )1 w ( α )1 + x ∗ y ∗ y w ( α )1 + i x ∗ x y ∗ + x ∗ y ∗ y w ( α )1 + y ∗ y ∗ y . (5.27)while the computation of the right member is( α (cid:12) α ) (cid:12) α = (cid:2) ( x k + y ) (cid:12) ( x k + y ) (cid:3) (cid:12) ( x k + y )= (cid:2) ( x ∗ y z ( α )1 + x y ∗ )k + i x ∗ x + x ∗ y w ( α )1 + y ∗ y (cid:3) (cid:12) ( x k + y )= (cid:2) ( x ∗ y z ( α )1 + x y ∗ ) ∗ y z ( α (cid:12) α )1 + (i x ∗ x + x ∗ y w ( α )1 + y ∗ y ) ∗ x (cid:3) k+ i( x ∗ y z ( α )1 + x y ∗ ) ∗ x + ( x ∗ y z ( α )1 + x y ∗ ) ∗ y w ( α (cid:12) α )1 + (i x ∗ x + x ∗ y w ( α )1 + y ∗ y ) ∗ y = (cid:2) x y ∗ y z ∗ ( α )1 z ( α (cid:12) α )1 + x ∗ y y z ( α (cid:12) α )1 − i x x ∗ x + x x y ∗ w ∗ ( α )1 + x y y ∗ (cid:3) k+ i x x y ∗ z ∗ ( α )1 + i x ∗ x y + x y ∗ y z ∗ ( α )1 w ( α (cid:12) α )1 + x ∗ y y w ( α (cid:12) α )1 i x x ∗ y + x y ∗ y w ∗ ( α )1 + y y ∗ y , (5.28)13howing that α (cid:12) ( α (cid:12) α ) (cid:54) = ( α (cid:12) α ) (cid:12) α , (5.29)or what is the same, it not comply with the associative property.2. CommutativityThe commutative property, α (cid:12) α = α (cid:12) α , ∀ α , α ∈ E is not satisfied. For numbers α i = x i k + y i , where i = 1 ,
2, we have: α (cid:12) α = ( x ∗ y z ( α )1 + x y ∗ )k + x ∗ y w ( α )1 + i x ∗ x + y ∗ y (5.30)while α (cid:12) α = ( x ∗ y z ( α )1 + x y ∗ )k + x ∗ y w ( α )1 + i x ∗ x + y y ∗ (5.31)Due to the noncommutative property of the conjugated product, it’s convenient to specifythe order of the conjugated multiplication. Then, we can assume that the left conjugatemultiplication of an extended number α by other extended β stands for α (cid:12) β the rightmultiplication of α by β means β (cid:12) α .3. Distributivity of the conjugated multiplication over the standard additionWe study the distributive property for the right and left conjugated multiplication of thestandard sum. In the first case, this property is satisfied if: α (cid:12) ( α + α ) = α (cid:12) α + α (cid:12) α . (5.32)Computing the left member we have α (cid:12) ( α + α ) = ( x k + y ) (cid:12) (cid:2) ( x k + y ) + ( x k + y ) (cid:3) = (cid:2) x ∗ ( y + y ) z ( α )1 + ( x + x ) y ∗ (cid:3) k + x ∗ ( y + y ) w ( α )1 + i x ∗ ( x + x ) + y ∗ ( y + y )= (cid:2) x ∗ y z ( α )1 + x y ∗ (cid:3) k + x ∗ y w ( α )1 + i x ∗ x + y ∗ y + (cid:2) x ∗ y z ( α )1 + x y ∗ (cid:3) k + x ∗ y w ( α )1 + i x ∗ x + y ∗ y = α (cid:12) α + α (cid:12) α , (5.33)showing that the right conjugated product is distributive. Instead, we cannot arrive atthe same conclusion for the left conjugated multiplication of the sum. The distributivityproperty for the left conjugated multiplication of the sum is verified if:( α + α ) (cid:12) α = α (cid:12) α + α (cid:12) α . (5.34)The left member of the previous expression( α + α ) (cid:12) α = (cid:2) ( x k + y ) + ( x k + y ) (cid:3) (cid:12) ( x k + y )= (cid:2) ( x + x )k + ( y + y ) (cid:3) (cid:12) ( x k + y ) (cid:3) = (cid:2) ( x ∗ + x ∗ ) y z ( α + α )1 + x ( y ∗ + y ∗ ) (cid:3) k + ( x ∗ + x ∗ ) y w ( α + α )1 + i( x ∗ + x ∗ ) x + ( y ∗ + y ∗ ) y (5.35)is different from the right’s α (cid:12) α + α (cid:12) α = ( x k + y ) (cid:12) ( x k + y ) + ( x k + y ) (cid:12) ( x k + y )= (cid:2) ( x ∗ z ( α )1 + x ∗ z ( α )1 ) y + x ( y ∗ + y ∗ ) (cid:3) k + ( x ∗ w ( α )1 + x ∗ w ( α )1 ) y + i( x ∗ + x ∗ ) x + ( y ∗ + y ∗ ) y . (5.36)14he distributive property is then satisfied if : D ( α + α ) ( α , α , α ) ≡ (cid:2) ( x ∗ + x ∗ ) y z ( α + α )1 − ( x ∗ z ( α )1 + x ∗ z ( α )1 ) y (cid:3) k+( x ∗ + x ∗ ) y w ( α + α )1 − ( x ∗ w ( α )1 + x ∗ w ( α )1 ) y = 0 (5.37)is zero. The function D ( α + α ) ( α , α , α ) measure the magnitude of the differences be-tween the factors of the maps. Subtracting the explicit product of both members, thedistribution law can be expressed as:( α + α ) (cid:12) α = α (cid:12) α + α (cid:12) α + D ( α + α ) ( α , α , α ) . (5.38)The function D ( α + α ) ( α , α , α ) is null if • the maps for α and α numbers satisfy: z ( α )1 = z ( α )1 = z ( α + α )1 , w ( α )1 = w ( α )1 = w ( α + α )1 (5.39) • α and α are both pure complex numbers, e.i. x = x = 0 • α is a pure complex numbers, e.i. y = 0It can also be verified that D ( α + α ) ( α , α , α ) + D ( α + α ) ( α , α , α ) = D ( α + α ) ( α , α , α + α ) . (5.40)4. Existence of the multiplicative identity element.If the conjugate product has an identity element 1 ( ⊕ ) E , it must satisfy α (cid:12) ( ⊕ ) E = 1 ( ⊕ ) E (cid:12) α = α. (5.41)From the noncommutative property, we can see that the identity element does not exist.We do not attempt in here to classify the set of the extended numbers; however, accordingto Abstract Algebra, they behave like a commutative ring with the existence of nontrivial zerodivisors. We note that, same as extended numbers that satisfy the Associativity, Commutativ-ity, and Distributivity axioms for the standard sum and product, the pair α (cid:12) β will also satisfythe same properties. That means, for example, that pairs satisfy the distribution property:( α (cid:12) β ) · (cid:2) ( α (cid:12) β ) + ( α (cid:12) β ) (cid:3) = ( α (cid:12) β ) · ( α (cid:12) β ) + ( α (cid:12) β ) · ( α (cid:12) β ) . (5.42)Also, a product like( α • (cid:12) β • ) · ( α (cid:12) β ) (5.43)satisfies the distributive properties( α • (cid:12) β • ) · (cid:2) ( α (cid:12) β ) + ( α (cid:12) β ) (cid:3) = ( α • (cid:12) β • ) · ( α (cid:12) β ) + ( α • (cid:12) β • ) · ( α (cid:12) β ) (5.44)and (cid:2) ( α • (cid:12) β • ) + ( α • (cid:12) β • ) (cid:3) · ( α (cid:12) β ) = ( α • (cid:12) β • ) · ( α (cid:12) β ) + ( α • (cid:12) β • ) · ( α (cid:12) β ) . (5.45)The properties for quantities ( α (cid:12) β ), together with the intuition of the form of a quantumoperator, act over a two-component quantum state for n -VMVF systems, give us a hint forfinding the form of the inner product in the extended domain.15 . The extended inner product. We are now able to propose the absolute value for an extended number using the newoperations.The inner product should be a set of operations applied on four extended numbers, accordingto the first section of this chapter, and also connected with the two types of products. The moststraightforward possible definitions for the inner product for the extended numbers α, β, γ, δ are, regardless of the order of priority,1 . α • · β • · γ · δ . α • (cid:12) β • · γ · δ . α • · β • · γ (cid:12) δ . α • (cid:12) β • · γ (cid:12) δ . α • · β • (cid:12) γ · δ . α • (cid:12) β • (cid:12) γ · δ . α • · β • (cid:12) γ (cid:12) δ . α • (cid:12) β • (cid:12) γ (cid:12) δ, (6.1)where we include the extended conjugate map () • .As mentioned in the sections “Introduction”, the primary motivation of this work is theconstruction of a Hilbert space over a new domain of numbers that includes negative proba-bilities and, most important, that fits the new structure obtained in the classical theory. TheHamilton theory shows that two canonical transformations are needed to evolve the system,one using the rectangular coordinates and another using the angular’s (4). A point in thecanonical space and with it, a state of the system has two components: the rectangular andthe angular. The extended ket should have then two components in correspondence with thecanonical space, which, together with an extended bra , that should also have two components,define the inner product for obtaining real measurements. That points out that there shouldexist some pair-pair symmetry in the inner product. The cases 1,3,6 and 8 are the only ones whohave that kind of symmetry. The superposition principle also points out that the state vectorsmust satisfy the Associativity, Commutativity, and Distributivity properties. In that case, theproduct operation that satisfies those properties is the standard multiplication, retaining onlycases 1 and 6. The first proposition for the inner product, do not include the complex productor the new map () • . Because of that, according to the discussed above, it will not result in areal number.Based on this weak explanation embedded with intuition and the need to adjust the innerproduct to the expected behavior for the quantum theory, we propose the definition of the innerproduct of four extended numbers α, β, γ, δ as: (cid:104) α, β, γ, δ (cid:105) ≡ ( α • (cid:12) β • ) · ( γ (cid:12) δ ) . (6.2)The four power of the absolute value for an extended number α = x k + y can be then writtenas: | α | = ( α • (cid:12) α • ) · ( α (cid:12) α ) = (6.3)= (cid:2) ( x k + y ) •∗ ( x k + y ) • (cid:3) · (cid:2) ( x k + y ) ∗ ( x k + y ) (cid:3) , ∀ x, y ∈ C . (6.4)for x = 1 and y = 0 we rewrite the first equation for the absolute value of the extended unit,which we supposed is equal to 1: (cid:0) k • (cid:12) k • (cid:1) · (cid:0) k (cid:12) k (cid:1) = 1 (6.5)We proposed that the extended unit definition stands for all the extended numbers, e.i. k (cid:12) k = i. This proposition, with the absolute value of the extended unit equal to 1, set aconstraint:k • (cid:12) k • = i ∗ . (6.6)16e can consider this equation to stand with no loss of generality for all the extended numberswhen we have a product like: α • (cid:12) β • = ( α E k + α I ) • (cid:12) ( β E k + β I ) • = ( α E k ( α ) • + α I ) (cid:12) ( β E k ( β ) • + β I ) (6.7)In the case of α • (cid:12) α • , we have:k ( α ) • (cid:12) k ( α ) • = ( z ( α )2 k + w ( α )2 ) (cid:12) ( z ( α )2 k + w ( α )2 )= ( z ( α • )1 z ( α )2 ∗ w ( α )2 + z ( α )2 w ( α )2 ∗ )k + i | z ( α )2 | + z ( α )2 ∗ w ( α • )1 w ( α )2 + | w ( α )2 | = i ∗ (6.8)which lead to the relations between the coefficients z ( α • )1 z ( α )2 ∗ w ( α )2 + z ( α )2 w ( α )2 ∗ = 0 −| z ( α )2 | + i z ( α )2 ∗ w ( α • )1 w ( α )2 + i | w ( α )2 | = 1 . (6.9)Note the appearance of coefficients related to the number α • .Following the rules of the standard and conjugated product, we have the expression of theabsolute value power four: | α | = Γ E k + Γ I , (6.10)whereΓ E = | x | (cid:16) i z ( α • )1 z ( α )2 ∗ w ( α )2 + i z ( α )2 w ( α )2 ∗ (cid:17) + | x | | y | (cid:16) z z ( α )1 z ( α )2 + z z ( α • )1 z ( α )2 ∗ + z ( α • )1 z ( α )2 ∗ w ( α )2 + z ( α )2 w ( α )2 ∗ + z ( α )2 w ( α )1 + z ( α )1 w ( α )2 + w ( α )2 ∗ + z ( α )2 ∗ w ( α • )1 (cid:17) + | x | x ∗ y (cid:16) z z ( α • )1 z ( α )1 ∗ z ( α )2 w ( α )2 + z z ( α )1 z ( α )2 w ( α )2 ∗ + i z ( α • )1 z ( α )2 ∗ + z ( α • )1 z ( α )2 ∗ w ( α )1 w ( α )2 + z ( α )2 w ( α )1 w ( α )2 ∗ + i z ( α )1 | z ( α )2 | + z ( α )1 | w ( α )2 | + z ( α )1 z ( α )2 ∗ w ( α • )1 (cid:17) + | x | xy ∗ (cid:16) z z ( α • )1 z ( α )2 ∗ w ( α )2 + z z ( α )2 w ( α )2 ∗ + i z ( α )2 + i | z ( α )2 | + | w ( α )2 | + z ( α )2 ∗ w ( α • )1 w ( α )2 (cid:17) + | y | x ∗ y (cid:16) z ( α • )1 z ( α )2 ∗ + z ( α )1 (cid:17) + | y | xy ∗ (cid:16) z ( α )2 + 1 (cid:17) + ( xy ∗ ) (cid:16) z z ( α )2 + w ( α )2 (cid:17) + ( x ∗ y ) (cid:16) z z ( α • )1 z ( α )1 z ( α )2 ∗ + z ∗ ( α • )1 z ( α )2 ∗ w ( α )1 + z ∗ ( α )1 z ( α )2 ∗ w ( α • )1 + z ∗ ( α )1 w ( α )2 ∗ (cid:17) (6.11)17nd Γ I = | x | (cid:16) − | z ( α )2 | + i z ( α )2 ∗ w ( α • )1 w ( α )2 + i | w ( α )2 | (cid:17) + | y | + | x | | y | (cid:16) z ( α )1 z ( α )2 w + z ( α • )1 z ( α )2 ∗ w + i + i | z ( α )2 | + | w ( α )2 | + z ( α )2 ∗ w ( α • )1 w ( α )2 + w ( α )1 w ( α )2 (cid:17) + | x | x ∗ y (cid:16) z ( α • )1 z ( α )1 ∗ z ( α )2 w ( α )2 w + z ( α )1 z ( α )2 w ( α )2 ∗ w + i w ( α )2 ∗ + i z ( α )2 ∗ w ∗ ( α • )1 + i | z ( α )2 | w ∗ ( α )1 + w ( α )1 | w ( α )2 | + z ( α )2 ∗ w ( α )1 w ( α • )1 w ( α )2 (cid:17) + | x | xy ∗ (cid:16) z ( α • )1 z ( α )2 ∗ w ( α )2 w + z ( α )2 w ( α )2 ∗ w + i w ( α )2 (cid:17) + | y | x ∗ y (cid:16) z ( α )2 ∗ w ( α • )1 + w ( α )2 ∗ + w ( α )1 (cid:17) + | y | xy ∗ (cid:16) w ( α )2 (cid:17) + ( x ∗ y ) (cid:16) z ( α • )1 z ( α )1 z ( α )2 ∗ w + z ( α )2 ∗ w ( α • )1 w ( α )1 + w ( α )2 ∗ (cid:17) + ( xy ∗ ) (cid:16) z ( α )2 w (cid:17) . (6.12)We can explicitly replace the first line of expression for the extended part and the imaginarypart of the absolute value power four, using the equations 6.9, like: | x | (cid:16) i z ( α • )1 z ( α )2 ∗ w ( α )2 + i z ( α )2 w ( α )2 ∗ (cid:17) = 0 | x | (cid:16) − | z ( α )2 | + i z ( α )2 ∗ w ( α • )1 w ( α )2 + i | w ( α )2 | (cid:17) + | y | = | x | + | y | , (6.13)respectively. We can also group the terms of the result using the parameters φ = | x || y | and θ = θ x − θ y , where | x | , θ x , | y | and θ y are the absolute values and angles of complex numbers x and y , respectively. The following expressions can be written as:( x ∗ y ) = | x | | y | e − θ ( xy ∗ ) = | x | | y | e θ | x | x ∗ y = | x | | y | φe − i θ | x | xy ∗ = | x | | y | φe i θ | y | x ∗ y = | x | | y | φ − e − i θ | y | xy ∗ = | x | | y | φ − e i θ . (6.14)Replacing them, we obtainΓ E = | x | | y | (cid:104) z z ( α )1 z ( α )2 + z z ( α • )1 z ( α )2 ∗ + z ( α • )1 z ( α )2 ∗ w ( α )2 + z ( α )2 w ( α )2 ∗ + z ( α )2 w ( α )1 + z ( α )1 w ( α )2 + w ( α )2 ∗ + z ( α )2 ∗ w ( α • )1 + φe − i θ (cid:16) z z ( α • )1 z ( α )1 ∗ z ( α )2 w ( α )2 + z z ( α )1 z ( α )2 w ( α )2 ∗ + i z ( α • )1 z ( α )2 ∗ + z ( α • )1 z ( α )2 ∗ w ( α )1 w ( α )2 + z ( α )2 w ( α )1 w ( α )2 ∗ + i z ( α )1 | z ( α )2 | + z ( α )1 | w ( α )2 | + z ( α )1 z ( α )2 ∗ w ( α • )1 (cid:17) + φe i θ (cid:16) z z ( α • )1 z ( α )2 ∗ w ( α )2 + z z ( α )2 w ( α )2 ∗ + i z ( α )2 + i | z ( α )2 | + | w ( α )2 | + z ( α )2 ∗ w ( α • )1 w ( α )2 (cid:17) + φ − e − i θ (cid:16) z ( α • )1 z ( α )2 ∗ + z ( α )1 (cid:17) + φ − e i θ (cid:16) z ( α )2 + 1 (cid:17) + e θ (cid:16) z z ( α )2 + w ( α )2 (cid:17) + e − θ (cid:16) z z ( α • )1 z ( α )1 z ( α )2 ∗ + z ∗ ( α • )1 z ( α )2 ∗ w ( α )1 + z ∗ ( α )1 z ( α )2 ∗ w ( α • )1 + z ∗ ( α )1 w ( α )2 ∗ (cid:17)(cid:105) (6.15)18nd Γ I = | x | + | y | + | x | | y | (cid:104) z ( α )1 z ( α )2 w + z ( α • )1 z ( α )2 ∗ w + i + i | z ( α )2 | + | w ( α )2 | + z ( α )2 ∗ w ( α • )1 w ( α )2 + w ( α )1 w ( α )2 + φe − i θ (cid:16) z ( α • )1 z ( α )1 ∗ z ( α )2 w ( α )2 w + z ( α )1 z ( α )2 w ( α )2 ∗ w + i w ( α )2 ∗ + i z ( α )2 ∗ w ∗ ( α • )1 + i | z ( α )2 | w ∗ ( α )1 + w ( α )1 | w ( α )2 | + z ( α )2 ∗ w ( α )1 w ( α • )1 w ( α )2 (cid:17) + φe i θ (cid:16) z ( α • )1 z ( α )2 ∗ w ( α )2 w + z ( α )2 w ( α )2 ∗ w + i w ( α )2 (cid:17) + φ − e − i θ (cid:16) z ( α )2 ∗ w ( α • )1 + w ( α )2 ∗ + w ( α )1 (cid:17) + φ − e i θ (cid:16) w ( α )2 (cid:17) + e − θ (cid:16) z ( α • )1 z ( α )1 z ( α )2 ∗ w + z ( α )2 ∗ w ( α • )1 w ( α )1 + w ( α )2 ∗ (cid:17) + e θ (cid:16) z ( α )2 w (cid:17)(cid:105) . (6.16)The positive-definiteness condition of the absolute value state that the extended part ofthe extended equation 6.10 must be zero, and the imaginary part must be real and greaterthan zero. Applying that conjecture to our results and putting, together with relations 6.9, weobtain the set of equations that relate the coefficients z ( α )1 , w ( α )1 and z ( α )2 , w ( α )2 : . z ( α • )1 z ( α )2 ∗ w ( α )2 + z ( α )2 w ( α )2 ∗ = 0 . − | z ( α )2 | + i z ( α )2 ∗ w ( α • )1 w ( α )2 + i | w ( α )2 | = 1 . z z ( α )1 z ( α )2 + z z ( α • )1 z ( α )2 ∗ + z ( α • )1 z ( α )2 ∗ w ( α )2 + z ( α )2 w ( α )2 ∗ + z ( α )2 w ( α )1 + z ( α )1 w ( α )2 + w ( α )2 ∗ + z ( α )2 ∗ w ( α • )1 + φe − i θ (cid:16) z z ( α • )1 z ( α )1 ∗ z ( α )2 w ( α )2 + z z ( α )1 z ( α )2 w ( α )2 ∗ + i z ( α • )1 z ( α )2 ∗ + z ( α • )1 z ( α )2 ∗ w ( α )1 w ( α )2 + z ( α )2 w ( α )1 w ( α )2 ∗ + i z ( α )1 | z ( α )2 | + z ( α )1 | w ( α )2 | + z ( α )1 z ( α )2 ∗ w ( α • )1 (cid:17) + φe i θ (cid:16) z z ( α • )1 z ( α )2 ∗ w ( α )2 + z z ( α )2 w ( α )2 ∗ + i z ( α )2 + i | z ( α )2 | + | w ( α )2 | + z ( α )2 ∗ w ( α • )1 w ( α )2 (cid:17) + φ − e − i θ (cid:16) z ( α • )1 z ( α )2 ∗ + z ( α )1 (cid:17) + φ − e i θ (cid:16) z ( α )2 + 1 (cid:17) + e θ (cid:16) z z ( α )2 + w ( α )2 (cid:17) + e − θ (cid:16) z z ( α • )1 z ( α )1 z ( α )2 ∗ + z ∗ ( α • )1 z ( α )2 ∗ w ( α )1 + z ∗ ( α )1 z ( α )2 ∗ w ( α • )1 + z ∗ ( α )1 w ( α )2 ∗ (cid:17) = 0 . z ( α )1 z ( α )2 w + z ( α • )1 z ( α )2 ∗ w + i + i | z ( α )2 | + | w ( α )2 | + z ( α )2 ∗ w ( α • )1 w ( α )2 + w ( α )1 w ( α )2 + φe − i θ (cid:16) z ( α • )1 z ( α )1 ∗ z ( α )2 w ( α )2 w + z ( α )1 z ( α )2 w ( α )2 ∗ w + i w ( α )2 ∗ + i z ( α )2 ∗ w ∗ ( α • )1 + i | z ( α )2 | w ∗ ( α )1 + w ( α )1 | w ( α )2 | + z ( α )2 ∗ w ( α )1 w ( α • )1 w ( α )2 (cid:17) + φe i θ (cid:16) z ( α • )1 z ( α )2 ∗ w ( α )2 w + z ( α )2 w ( α )2 ∗ w + i w ( α )2 (cid:17) + φ − e − i θ (cid:16) z ( α )2 ∗ w ( α • )1 + w ( α )2 ∗ + w ( α )1 (cid:17) + φ − e i θ (cid:16) w ( α )2 (cid:17) + e − θ (cid:16) z ( α • )1 z ( α )1 z ( α )2 ∗ w + z ( α )2 ∗ w ( α • )1 w ( α )1 + w ( α )2 ∗ (cid:17) + e θ (cid:16) z ( α )2 w (cid:17) = R. (6.17) R is a non-negative real number. The expression of the four power of the absolute value ofextended numbers is then: | α | = ( α • (cid:12) α • ) · ( α (cid:12) α ) = | x | + | y | + R | x | | y | (6.18)whose maps are determined by equations 6.17, which depends on the parameters φ , θ , and R .Unfortunately, the four equations are not enough to define the maps’ coefficient related tonumber α , since they include coefficients of the number α • z ( α • )1 , and w ( α • )1 . We can then writethe equations for the absolute value of the number α • : | α • | = (( α • ) • (cid:12) ( α • ) • ) · ( α • (cid:12) α • ) = | x α • | + | y α • | + R | x α • | | y α • | . (6.19)We will obtain a similar set of equations like 6.17 replacing the complex numbers x, y bythe extended and imaginary parts of the number α • and the coefficients by z ( α • )1 , w ( α • )1 and19 ( α • )2 , w ( α • )2 . However, we still have coefficients related to the number ( α • ) • . One way to solvethis issue is to impose a closure condition. In this case, we can propose the condition( α • ) • = α, (6.20)or, using the extended and complex parts: . z ( α )2 w ( α • )2 + w ( α )2 = 0 . z ( α )2 z ( α • )2 − | α • | = (( α • ) • (cid:12) ( α • ) • ) · ( α • (cid:12) α • ) = ( α (cid:12) α ) · ( α • (cid:12) α • ) = | α | . (6.22)The third and fourth equations of the set of equations 6.17 are reduced to the third and fourthequations of equations 6.17. The first and second equations of the equations for the absolutevalue of the number α • , obtained from equations 6.8 to the product ( α • ) • (cid:12) ( α • ) • : z ( α •• )1 z ( α • )2 ∗ w ( α • )2 + z ( α • )2 w ( α • )2 ∗ = 0 −| z ( α • )2 | + i z ( α • )2 ∗ w ( α •• )1 w ( α • )2 + i | w ( α • )2 | = 1 (6.23)are modified using z ( α •• )1 = z ( α )1 and w ( α •• )1 = w ( α )1 according to the closure condition, like z ( α )1 z ( α • )2 ∗ w ( α • )2 + z ( α • )2 w ( α • )2 ∗ = 0 −| z ( α • )2 | + i z ( α • )2 ∗ w ( α )1 w ( α • )2 + i | w ( α • )2 | = 1 . (6.24)The previous equations are not independent of the set of equations 6.17 and 6.21 becausethe result of applying the closure condition on the absolute value expression for the number α • is the same as for α . Nevertheless, it explicitly shows the dependency of the equation systemof the quantities z ( α • )2 and w ( α • )2 .According to the above results, we have eight variables: z ( α )1 , w ( α )1 , z ( α )2 , w ( α )2 , z ( α • )1 , w ( α • )1 , z ( α • )2 ,and w ( α • )2 and six equations. That means we are two complex or one extended equation short.These equations will be added once the property of the Conjugate symmetry are discussedbelow. The set of extended complex numbers E can be written as a + ib +k c + i k d , where a, b, c, d ∈ R .The parameter R can be proposed in analogy with the isotropic property for linear spaces. Theisotropic property states that the length of a vector remains invariant under an axis rotation.If we look at the complex vector space, all numbers laying on the same centered circle, such asthe complex number z = a + i b , have the same absolute value, equal to | z | = √ a + b . Someof these numbers lying on the sphere are z = a + i b , z = − a + i b , z = a − i b , z = − a − i b , z = b + i a , z = − b + i a , z = b − i a and z = − b − i a .The absolute value of extended number α = a + i b + k c + ik d , where a, b, c, d ∈ R must,then, remain constant when the axis is rotated, which means that numbers: α = ± a ± i b ± k c ± ik dα = ± b ± i c ± k d ± ik aα = ± c ± i d ± k a ± ik bα = ± d ± i a ± k b ± ik c α .The proposed absolute value for an extended number raised to the fourth power, as shownin equations 6.18, is | α | = | x | + | y | + R | x | | y | x, y ∈ C and R ∈ R , R ≥ x and y the extended and imaginary part of the extended number. If we substitute x = a i + b and y = c i + d , being a, b, c, d ∈ R , we have | α | =( a + b ) + ( c + d ) + R ( a + b )( c + d )= a + b + c + d + 2 a b + 2 c d + R ( a c + a d + b c + b d ) . (6.26) R = 2 is the only possible value for the absolute value | α | = a + b + c + d + 2( a b + a c + a d + b c + b d + c d ) , (6.27)remain constant under any permutation of a, b, c, d with any combination of ± sign.The final form for the absolute value of an extended number α = x k + y is then | α | = | x | + | y | + 2 | x | | y | . (6.28)
7. Division between extended numbers
In the complex number domain, the division of two complex numbers can be accomplishedby multiplying the numerator and denominator by the complex conjugate of the denominator.To do a similar procedure for the division between extended numbers, we must first review theexistence of extraneous and missing solutions when the same factor multiplies both members ofan extended equation. The extraneous solution (or spurious solution) emerges from solving theproblem while a missing solution is a valid solution that of the original problem, but disappearedalong with the solution. We study the necessity and sufficiency for the equality of an extendedequation, before and after the multiplication.Let us consider the extended equation α = β ∀ α, β ∈ E . (7.1)If we multiply both members of the equation 7.1 by a third extended factor γ , will the equalityof the new equation still hold? Moreover, if it does, would it introduce or eliminate solutionsto the original equations?Let us analyze different cases: • The standard multiplication of both members by an extended number α · γ ? = β · γ ∀ α, β, γ ∈ E (7.2)The equation α = β means that their extended and imaginary parts are equals, respec-tively, like α E = β E and α I = β I . The members of the equation 7.2 have the form: α · γ = (cid:0) α E γ E z + α E γ I + α I γ E (cid:1) k + α E γ E w + α I γ I β · γ = (cid:0) β E γ E z + β E γ I + β I γ E (cid:1) k + β E γ E w + β I γ I , (7.3)which led to the equations( α E − β E )( γ E z + γ I ) + ( α I − β I ) γ E = 0( α E − β E ) γ E w + ( α I − β I ) γ I = 0 . (7.4)21f α E = β E and α I = β I , the sufficiency of the statement is probed. However, thenecessity for the inverted proposition is not true. Indeed, from equations 7.4, we cannotextract the initial equality α = β , and that is because of the appearances of extraneoussolutions. Multiplying first complex equation of 7.4 by γ I , the second by γ E and subtractone equation from another we obtain:( α E − β E )( γ I + γ E γ I z − γ E w ) = 0 , (7.5)which indicate the presence of a new solution related to the case γ I + γ E γ I z − γ E w = 0 . (7.6)The necessity can be easily probed for some specific cases like ( ∀ γ E = 0 , γ I (cid:54) = 0) or( ∀ γ E (cid:54) = 0 , γ I = 0 , w (cid:54) = 0). • The left conjugated multiplication of an extended number, γ (cid:12) α ? = γ (cid:12) β, (7.7)the equation for every member is γ (cid:12) α = (cid:0) γ ∗ E α I z ( γ )1 + γ ∗ I α E (cid:1) k + γ ∗ E α I w ( γ )1 + i γ ∗ E α E + γ ∗ I α I γ (cid:12) β = (cid:0) γ ∗ E β I z ( γ )1 + γ ∗ I β E (cid:1) k + γ ∗ E β I w ( γ )1 + i γ ∗ E β E + γ ∗ I β I . (7.8)Grouping the extended and imaginary part and setting equal to zero, we have γ ∗ E z ( γ )1 ( α I − β I ) + γ ∗ I ( α E − β E ) = 0( γ ∗ E w ( γ )1 + γ ∗ I )( α I − β I ) + i γ ∗ E ( α E − β E ) = 0 (7.9)which probe the sufficiency of the statement. After multiplying the first complex equationby Equation 7.9, the second by γ ∗ I and subtract one equation from another, head to theequation (cid:2) ( γ ∗ I ) + γ ∗ E γ ∗ I w ( γ )1 − i( γ ∗ E ) z ( γ )1 (cid:3) ( α I − β I ) = 0 (7.10)which shows the inclusion of a new solution from the extended number γ that satisfy theequation( γ ∗ I ) + γ ∗ E γ ∗ I w ( γ )1 − i( γ ∗ E ) z ( γ )1 = 0 (7.11) • The right conjugated multiplication of an extended number, α (cid:12) γ ? = β (cid:12) γ, (7.12)led to equations: ( α ∗ E z ( α )1 − β ∗ E z ( β )1 ) γ I + ( α ∗ I − β ∗ I ) γ E = 0( α ∗ E w ( α )1 − β ∗ E w ( β )1 + α ∗ I − β ∗ I ) γ I + i( α ∗ E − β ∗ E ) γ E = 0 . (7.13)In this case, the sufficiency of the statement can be verified because if numbers α, β areequals, then also their maps. We can not verify the necessity for this statement due tothe impossibility of factorizing the terms with z and w .22e can now define the division between extended numbers. Our first proposal is similar tothe division between complex numbers, where the numerator and denominator are multipliedby the extended complex conjugates of the denominator. Follow this reasoning; our initialproposition is first, conjugately left multiply both numerator and denominator by the numberlike ( α (cid:12) α ) and then standard multiply by the expression α • (cid:12) α • like λα = ( α (cid:12) λ )( α (cid:12) α ) = ( α (cid:12) λ ) · ( α • (cid:12) α • )( α (cid:12) α ) · ( α • (cid:12) α • ) = ( α (cid:12) λ ) · ( α • (cid:12) α • ) | α | . (7.14)This proposal is incorrect because we cannot obtain α with an inverse process from the num-ber resulting from the division. That means that we cannot get the above number by anymultiplication: λα (cid:12) α = α (cid:12) ( α (cid:12) λ ) · ( α • (cid:12) α • ) | α | (cid:54) = λ or λα · α = α · ( α (cid:12) λ ) · ( α • (cid:12) α • ) | α | (cid:54) = λ (7.15)We propose the standard multiplication of both numerator and denominator by the expression( ∗ √ α ) • (cid:12) ( ∗ √ α ) • , where we remember that the conjugated root is the inverse operation of theconjugated product. Indeed, any extended number can be expressed as the conjugated productof its conjugated root, like α = ∗ √ α (cid:12) ∗ √ α . The new proposal for the division between extendednumbers is then: λα ≡ λ ( ∗ √ α (cid:12) ∗ √ α ) = λ · [( ∗ √ α ) • (cid:12) ( ∗ √ α ) • ][ ∗ √ α (cid:12) ∗ √ α ] · [( ∗ √ α ) • (cid:12) ( ∗ √ α ) • ] = λ · [( ∗ √ α ) • (cid:12) ( ∗ √ α ) • ] | ∗ √ α | . (7.16)Now we can verify that: λα · α = α · λ · [( ∗ √ α ) • (cid:12) ( ∗ √ α ) • ] | ∗ √ α | = λ ( ∗ √ α (cid:12) ∗ √ α ) · [( ∗ √ α ) • (cid:12) ( ∗ √ α ) • ] | ∗ √ α | = λ | ∗ √ α | | ∗ √ α | = λ (7.17)For λ = 1, we can probe the existence of multiplicative inverse α − as one of the propertiesthat the extended numbers satisfy. The multiplicative inverse has the form α − = 1 α = 1( ∗ √ α (cid:12) ∗ √ α ) = [( ∗ √ α ) • (cid:12) ( ∗ √ α ) • ][ ∗ √ α (cid:12) ∗ √ α ] · [( ∗ √ α ) • (cid:12) ( ∗ √ α ) • ] = [( ∗ √ α ) • (cid:12) ( ∗ √ α ) • ] | ∗ √ α | (7.18)There is another possibility for the division between extended numbers since we can alsodefine the extended conjugated root • √ α as the inverse operation of the conjugated product ofthe () • -map like α = ( • √ α ) • (cid:12) ( • √ α ) • . (7.19)In this case, we can propose other division like λα = λ ( • √ α ) • (cid:12) ( • √ α ) • = λ · [ • √ α (cid:12) • √ α ][ • √ α (cid:12) • √ α ] · [( • √ α ) • (cid:12) ( • √ α ) • ] = λ · [ • √ α (cid:12) • √ α ] | • √ α | (7.20)and the inverse like α − = [ • √ α (cid:12) • √ α ] | • √ α | (7.21)Rest now to interpret the double existence of two divisions, and with it, the existence of twoinverse for every extended number. 23 . Extended domain properties. Absolute value. The dependency of the maps k , k ∗ , and k • on parameters φ , θ and R , provide the extendeddomain with some properties: • The absolute value raised to the fourth power of an extended number is: (cid:104) α, α, α, α (cid:105) = | x | + | y | + 2 | x | | y | x, y ∈ C , (8.1)and the absolute value is defined as: | α | = (cid:112) | x | + | y | + 2 | x | | y | . (8.2) • The absolute value of an extended number is zero if the extended and the imaginary partof the number are also zero. • It is straightforward to prove the property of the extended numbers that:( cα ) ∗ = c ∗ α ∗ and ( cα ) • = cα • ∀ c ∈ C , α ∈ E . • The relations for obtaining the extended and conjugated maps for pure extended andpure complex numbers result in undetermined values for numbers z , z , w and w . Thatis because of the dependency of equations 6.14 on the quantities φ and φ − , whose valuesturn zero or infinity for | x | and/or | y | being zero. However, both cases are trivial, sincetaking α = x k + y , for y = 0 we have x k (cid:12) x k = | x | i and [( x k) • (cid:12) ( x k) • ][( x k) (cid:12) ( x k)] = | x | , (8.3)while for x = 0, we have the typical operations between complex numbers. • The extended numbers α ≡ x k + y and cα = c ( x k + y ) c ∈ C have equaled values of θ and φ . From their definitions, we have: φ cα = | cx || cy | | c || x || c || y | = | x || y | ≡ φ α θ cα = ( θ x + θ c ) − ( θ y + θ c ) = θ x − θ y ≡ θ α . (8.4) • From the expression of the absolute value in Eq. 6.18 and the invariant character of θ and φ for extended numbers number α and cα where c ∈ C , as shown in Eq. 8.4, theabsolute value of cα have the expression: | cα | = (cid:112) [( cα • ) (cid:12) ( cα • )] · [( cα (cid:12) cα )]= (cid:112) c ∗ c c ∗ c [( α • (cid:12) α • · ( α (cid:12) α )]= (cid:112) | c | [( α • (cid:12) α • · ( α (cid:12) α )]= | c || α | (8.5)
9. Linearity
Linearity is another property that the inner product must satisfy since it is the base prop-erty of the superposition principle. From the algebraic properties of the extended numbers,specifically those related to the standard product, we found that the complex products α (cid:12) β satisfy the distribution and associative property like shown on equation 5.42:( α (cid:12) β ) · (cid:2) ( α (cid:12) β ) + ( α (cid:12) β ) (cid:3) = ( α (cid:12) β ) · ( α (cid:12) β ) + ( α (cid:12) β ) · ( α (cid:12) β ) . α • (cid:12) β • ) · ( α (cid:12) β ) satisfies the distributive properties of equation 5.44( α • (cid:12) β • ) · (cid:2) ( α (cid:12) β ) + ( α (cid:12) β ) (cid:3) = ( α • (cid:12) β • ) · ( α (cid:12) β ) + ( α • (cid:12) β • ) · ( α (cid:12) β )Even when the two-dimensional vector satisfies Linearity’s property, we also study whatconditions the addends of the factors of the inner product should satisfy for the product remainlinear.For simplicity, we express the inner product of the equation 6.2 in the form of a two-rowmatrix (cid:20) α γβ δ (cid:21) ≡ ( α • (cid:12) β • ) · ( γ (cid:12) δ ) . The Linearity Property of the internal factors implies that: (cid:20) α + α γβ + β δ (cid:21) = (cid:20) α γβ δ (cid:21) + (cid:20) α γβ δ (cid:21)(cid:20) α γ + γ β δ + δ (cid:21) = (cid:20) α γ β δ (cid:21) + (cid:20) α γ β δ (cid:21) (9.1)and also (cid:20) α · α γβ · β δ (cid:21) = ( α • (cid:12) β • ) (cid:20) α γβ δ (cid:21) = ( α • (cid:12) β • ) (cid:20) α γβ δ (cid:21) . (cid:20) α γ · γ β δ · δ (cid:21) = ( γ (cid:12) δ ) (cid:20) α γ β δ (cid:21) = ( γ (cid:12) δ ) (cid:20) α γ β δ (cid:21) . (9.2)The conditions for the Linearity of inner products of equations 9.1 and 9.2 being satisfiedcan be computed using the algebraic properties of the operations defined above. We do notanalyze the Linearity for the internal factors since it is out of the scope of this work, eventhis property can be the base for the explanation of phenomena using a theory developed onextended numbers. We point out, nevertheless, the existence of two types of Properties ofLinearity and also that the proposed inner product satisfied the Linearity for the extendedproduct pair. That is essential to ensure the superposition principle for the quantum statevectors for n -VMVF systems.
10. Conjugate symmetry
The main objective of this works to propose a new Lorentz space where a new QuantumTheory for n -VMVF systems can be developed. That is the guide when proposing new proper-ties of the studied space. Indeed, from the classical theory obtained for n -VMVF systems, weproposed the extended unit definition and also an inner product that satisfies the superpositionprinciple for a pair of extended numbers. We proposed that the conjugated product of twoextended numbers as a representation of the quantum state of the system. The necessity ofincluding the “ bra ” state, demands the vector space to have a symmetric bilinear form. Forthat reason, we need to impose the Conjugate symmetry axiom for the pairs composing theinner product.In the space of complex vectors, the property of the conjugate symmetry has the form: (cid:104) u, v (cid:105) = (cid:104) v, u (cid:105) ∀ u, v ∈ C (10.1)where () is referred to as the map J : V → V ∗ . Is very well known that the conjugate symmetryensures that the complex inner product that satisfies: (cid:104) u, v + w (cid:105) = (cid:104) u, v (cid:105) + (cid:104) u, w (cid:105) and (cid:104) u, wv (cid:105) = w (cid:104) u, v (cid:105) (cid:104) u + w, v (cid:105) = (cid:104) u, v (cid:105) + (cid:104) w, v (cid:105) and (cid:104) wu, v (cid:105) = w ∗ (cid:104) u, v (cid:105) In the extended case, the expression for the inner product (cid:20) α γβ δ (cid:21) ≡ ( α • (cid:12) β • ) · ( γ (cid:12) δ ) , show us the existence of the map J : V → V (cid:48) , V, V (cid:48) ∈ E that J ( α (cid:12) β ) → ( α • (cid:12) β • ). We referto this map as the extended pair conjugate and denote it as J ≡ () (cid:13)• . () (cid:13)• map Unlike the complex numbers, the properties of the map J : V → V (cid:48) can not be extractedfrom the positive-definiteness axiom or the coefficients z ( α )1 , w ( α )1 , z ( α )2 , w ( α )2 , z ( α • )1 , w ( α • )1 , z ( α • )2 , and w ( α • )2 . Instead, with no loss of generality, we can impose the properties to this map and relateit then with the above coefficients. The properties that this map must satisfy are • Uniqueness. Being the extended pair conjugate an operation applied on a complex prod-uct, it must ensure that the map is unique; otherwise, we will have conflicts for a numberexpressed as different complex products. • Associativity for the Sum. The bi-linear form of the inner product demands that:( α + β ) (cid:13)• = ( α ) (cid:13)• + ( β ) (cid:13)• (10.2) • Associativity for the Standard Product. Same as before, the bi-linear form of the innerproduct demands that:( α · β ) (cid:13)• = ( α ) (cid:13)• · ( β ) (cid:13)• (10.3)As the expressions for the parameters z ( α ) i , w ( α ) i are unknown, the map’s uniqueness can beestablished by imposing the map being a function of the extended number resulting from theextended product, e.i. :if α (cid:12) β = γ then ( α (cid:12) β ) (cid:13)• = α • (cid:12) β • ≡ F ( γ ) ≡ F E ( γ E , γ I )k + F I ( γ E , γ I ) , (10.4)being γ E , γ I , the extended and the imaginary part of the extended number γ and F E , F I thesame quantities for the extended function F .We can also impose that the last two properties of associativity for the function F ( γ ): F ( α ) + F ( β ) = F ( α + β ) (10.5) F ( α ) · F ( β ) = F ( α · β ) (10.6)The Associativity for the Sum is satisfied if the proposed function is linear. Let us expressfunction F ( α ) with the linear expression: F ( α ) = F E ( α E , α I )k + F I ( α E , α I ) ≡ ( a α E + a α I )k + a α E + a α I , (10.7)where a , a , a , a are complex numbers. Replacing this expression on the sum of the maps ofthe extended numbers α and β we obtain: F ( α ) + F ( β ) =( a α E + a α I )k + a α E + a α I + ( a β E + a β I )k + a β E + a β I =[ a ( α E + β E ) + a ( α I + β I )]k + a ( α E + β E ) + a ( α I + β I )= F ( α + β ) , (10.8)26hich proves the Associativity for the Sum for the function 10.7.The Associativity for the Standard Product can not be proved because the coefficients forthe standard product are not defined. However, we can obtain their values from this property.From the properties of () (cid:13)• map, the Associativity for the Standard Product, equation 10.3,demands that: F ( α · β ) = F ( α ) · F ( β ) . Replacing the linear expressions 10.7, we can find relations between the coefficients a , a , a , a and z , w . Indeed, the extended and the imaginary part of the expression F ( α · β ) = F (cid:0) ( z α E β E + α E β I + α I β E )k + w α E β E + α I β I (cid:1) (10.9)are F E ( α · β ) = a ( z α E β E + α E β I + α I β E ) + a ( w α E β E + α I β I ) F I ( α · β ) = a ( z α E β E + α E β I + α I β E ) + a ( w α E β E + α I β I ) . (10.10)The extended and the imaginary part of the expression F ( α ) · F ( β ) =[ z F E ( α ) F E ( β ) + F E ( α ) F I ( β ) + F I ( α ) F E ( β )]k+ w F E ( α ) F E ( β ) + F I ( α ) F I ( β ) (10.11)are: [ F ( α ) · F ( β )] E = z ( a α E + a α I )( a β + a β I )+ ( a α E + a α I )( a β + a β I )+ ( a α E + a α I )( a β + a β I ) (10.12)[ F ( α ) · F ( β )] I = w ( a α E + a α I )( a β + a β I )+ ( a α E + a α I )( a β + a β I ) (10.13)Expanding the equations 10.10 and 10.13, and setting equals its extended and imaginary parts,we obtain: α E β E ( z a + 2 a a − z a − w a )+ α E β I ( z a a + a a + a a − a )+ α I β E ( z a a + a a + a a − a )+ α I β I ( z a + 2 a a − a ) = 0 (10.14)and α E β E ( w a + a − z a − w a )+ α E β I ( w a a + a a − a )+ α I β E ( w a a + a a − a )+ α I β I ( w a + a − a ) = 0 . (10.15)Being α and β any two extended numbers, this equation holds if all the coefficients of α i , β j are zero. Note that α E β I and α I β E have the same coefficient for the extended and imaginary27art, respectively. The independent set of equations for the equation 10.3 hold for any pair ofextended numbers is z a + 2 a a − z a − w a = 0 z a a + a a + a a − a = 0 z a + 2 a a − a = 0 w a + a − z a − w a = 0 w a a + a a − a = 0 w a + a − a = 0 (10.16)We obtain six equations for the same numbers of variables: a , a , a , a and z , w . This resultshows us that we can indeed express the () (cid:13)• map as a linear form like 10.7 and that the values z , w are extracted from equations 10.16. The fulfillment of the Associativity property for the Standard Product depends on thechoice of parameters z , w . Nothing has been imposed on the latest parameters, being theseproperties the constraints they must satisfy. For example, the most straight forward solutionis the one applied to the complex product, where: z = 0 , w = − , a = − a = 0 a = 0 a = 1 . (10.17)These values lead to the linear function F E ( γ E , γ I )k + F I ( γ E , γ I ) = − γ E k + γ I , (10.18)which we can recognize as the function for the complex conjugate operator () ∗ . The choicek = − = −
1. Because of that, for theextended numbers, the parameter z can not be zero.The above equations 10.16 have multiple solutions. However, according to the main objec-tive of this work, we need to analyze and choose the solutions that best fit the new proposalfor the quantum mechanics that includes variable masses. We already show that z (cid:54) = 0, butalso it is essential to verify the solution that has a more useful meaning for future theory. The“useful” feature is related to the equation: α = α (cid:13)• , (10.19)which is automatically satisfied by the inner product (cid:20) α αβ β (cid:21) , as we will show in the nextsections. This quantity is related to quantum stationary states and also with fundamentalconcepts like measurement and the normalization condition. Indeed, the equation z = z ∗ ,where z ∈ C , lead to the result that z is a real number. This result is the key to obtainingthe stationary equation for any physical observable in Quantum Mechanics. In our case, theclassical theory (4) establish equations that have complex solutions. We can look for thesolutions of equations 10.16, whose values lead to pure complex numbers as the result of theequation 10.19.With the linear form of equation (cid:13)• map, the equations 10.19 can be rewritten as: α E = a α E + a α I α I = a α E + a α I . (10.20)28f we set a = 0 and a = 1, the equations have the form α E = a α E a α E , (10.21)that lead to α E = 0 , ∀ a (cid:54) = 1 || a (cid:54) = 0, or what is the same, that all extended numbers thatsatisfied equation 10.19 are pure complex numbers. Replacing a = 0 and a = 1 on the set ofequations 10.16, we obtain: z ( a − a ) + 2 a a = 0 w ( a − − z a + a = 0 . (10.22)The choice to set the variables a = 0 and a = 1 reduce the equations 10.19 to a set oftwo equations with four variables a , a , z , and w . That means we have two unconstrainedvariables that can have any value. The variables z and w are also parameters for the equations6.17. We can propose any values for z and w that simplified the referred equations, as longas they satisfy z (cid:54) = 0 and a (cid:54) = 1 || a (cid:54) = 0. We let this analysis and the exact computation offunction F ( α ) or the coefficients a , a and z , w for future works. Even we define the F ( α ) for the () (cid:13)• map, finding the functions that satisfy the propertiesdescribed above, the referred map is related to the quantity ( α • (cid:12) β • ) e.i. with the coefficients z ( α )1 , w ( α )1 , z ( α )2 , w ( α )2 , z ( α • )1 , w ( α • )1 , z ( α • )2 , and w ( α • )2 . Then, once defined the function F ( α ), thecoefficients must satisfy: F ( α (cid:12) β ) = F E (( α (cid:12) β ) E , ( α (cid:12) β ) I )k + F I (( α (cid:12) β ) E , ( α (cid:12) β ) I ) = α • (cid:12) β • , (10.23)of explicitly: F E ( α, β, z ( α )1 , w ( α )1 ) = z ( α • )1 z ( α ) ∗ w ( β )2 α ∗ E β E + z ( α • )1 z ( α ) ∗ α ∗ E β I + z ( β )2 w ( α ) ∗ α ∗ E β E + z ( β )2 α ∗ I β E F I ( α, β, z ( α )1 , w ( α )1 ) = i z ( α ) ∗ z ( β )2 α ∗ E β E + z ( α ) ∗ w ( α • )1 w ( β )2 α ∗ E β E + z ( α ) ∗ w ( α • )1 α ∗ E β I + w ( α ) ∗ w ( β )2 α ∗ E β E + w ( α ) ∗ α ∗ E β I + w ( β )2 α ∗ I β E + α ∗ I β I , (10.24)where we replace F i (( α (cid:12) β ) E , ( α (cid:12) β ) I ) ≡ F i ( α, β, z ( α )1 , w ( α )1 ) ∀ i = E, I , to show the z , z , w , w dependency of the arguments.From the inner product chapter 6, and using the Positive definiteness axiom of the number α , we were able to identify only six complex equations from the eight needed for computing theeight later coefficients z ( α )1 , w ( α )1 , z ( α )2 , w ( α )2 , z ( α • )1 , w ( α • )1 , z ( α • )2 and w ( α • )2 . We need one extendedor two complex equations more for fully compute the coefficients. On the other side, theequation set 10.24 involves the coefficient of the extended numbers α and β . That means thatwe also need to find the unknown coefficients z ( β )1 , w ( β )1 , z ( β )2 , w ( β )2 , z ( β • )1 , w ( β • )1 , z ( β • )2 and w ( β • )2 ofthe extended number β . Adding its respective coefficients and the set of equations 6.24 forthe number β have 16 coefficients to determine and two sets of equations 6.24 for a total of12 equations. The previous equations 10.24 can be added to the full set of equations lettingonly one extended equation to find. Also, since α and β are two aleatory numbers, they shouldsatisfy the same equations. As equations 10.24 are not symmetric for those numbers, we needto include the same equations 10.24, this time for the expression ( β · α ): F ( β (cid:12) α ) = F E (( β (cid:12) α ) E , ( β (cid:12) α ) I )k + F I (( β (cid:12) α ) E , ( β (cid:12) α ) I ) = β • (cid:12) α • , (10.25)29f explicitly: F E ( β, α, z ( β )1 , w ( β )1 ) = z ( β • )1 z ( β ) ∗ w ( α )2 β ∗ E α E + z ( β • )1 z ( β ) ∗ β ∗ E α I + z ( α )2 w ( β ) ∗ β ∗ E α E + z ( α )2 β ∗ I α E F I ( β, α, z ( β )1 , w ( β )1 ) = i z ( β ) ∗ z ( α )2 β ∗ E α E + z ( β ) ∗ w ( β • )1 w ( α )2 β ∗ E α E + z ( β ) ∗ w ( β • )1 β ∗ E α I + w ( β ) ∗ w ( α )2 β ∗ E α E + w ( β ) ∗ β ∗ E α I + w ( α )2 β ∗ I α E + β ∗ I α I , (10.26) Once the Associativity for the Sum and the Standard Product is satisfied by the operator,() (cid:13)• , we can show that the inner product satisfies the extended Pair Conjugate symmetry.Indeed, the Extended Pair Conjugate of the extended product of four extended numbers: (cid:20) α γβ δ (cid:21) (cid:13)• ≡ [( α • (cid:12) β • ) · ( γ (cid:12) δ )] (cid:13)• = ( α • (cid:12) β • ) (cid:13)• · ( γ (cid:12) δ ) (cid:13)• , (10.27)applying the Associative property of the standard product. On the other side, the applicationof the map( α • (cid:12) β • ) (cid:13)• · ( γ (cid:12) δ ) (cid:13)• = (( α • ) • (cid:12) ( β • ) • ) · ( γ • (cid:12) δ • ) , (10.28)can be modified using the closure condition 6.20, which state that ( α • ) • = α and obtain:(( α • ) • (cid:12) ( β • ) • ) · ( γ • (cid:12) δ • ) = ( α (cid:12) β ) · ( γ • (cid:12) δ • ) ≡ (cid:20) γ αδ β (cid:21) , (10.29)proving our initial statement and providing the space with a bi-linear form.
11. Final equations to determine coefficients z i , w i Along the develop of this work we has obtained some sets of equations for computing thecoefficients z ( α )1 , w ( α )1 , z ( α )2 , w ( α )2 related to the unknown complex product () (cid:12) () and the map() • . Some of the wanted equations has arrive from the axioms that the extended inner productsmust satisfy, according our main objective. Also, some of the parameter included on suchequations are proposed from logical analysis. This section resumes all the equations and statesthe method for computing the wanted coefficients.From the positive-definiteness, we obtain a set of six complex equations, equations 6.17 and6.21, relating the coefficients z , w , z , w for the extended numbers α and α • . We representthis set of equation as D n ( α, α • ) = 0 , ∀ ≤ n ≤ z , w , and R . The R -parameter was chosen to have value R = 2 and the z , w , are determined by the equations 10.16 proposed from the properties of the () (cid:13)• map. Also,the map () (cid:13)• is related with the complex product α (cid:12) β and β (cid:12) α as showed in the previoussection.For obtaining the coefficients z i , w i of any extended number, the general method is: firstobtaining the z , w , parameters from the equations 10.22. That means that the linear formof the function F , equation 10.7 for the () (cid:13)• map, is fully determined. We can then, imposethe equations 10.24 and 10.26. As these equations are related to the complex product of two30xtended numbers, we need to add the equations 11.1 for both numbers. The set of complexequations from equations 11.1, 10.24, and 10.26 for the described situation is D n ( α, α • ) = 0 D n ( β, β • ) = 0( α • (cid:12) β • ) E = F E (( α (cid:12) β ) E , ( α (cid:12) β ) I )( α • (cid:12) β • ) I = F I (( α (cid:12) β ) E , ( α (cid:12) β ) I )( β • (cid:12) α • ) E = F E (( β (cid:12) α ) E , ( β (cid:12) α ) I )( β • (cid:12) α • ) I = F I (( β (cid:12) α ) E , ( β (cid:12) α ) I ) , (11.2)summing the 16 equations needed for determining the 16 coefficients z i , w i of numbers α, β, α • , β • .However, following the main goal of this work, we set some values for the coefficients of thelinear expression of the () (cid:13)• map, as seen in section 10.2. The choice of the values, ensures thatthe inner product (cid:20) α αβ β (cid:21) ∈ C . This lead to two unconstrained coefficients that can have any values with some exceptions. Inthis case, we can follow other method and first check the D n ( α, α • ) and propose values for z and w that simplified those equations, then compute the others coefficients a and a , findingthe final form of function F . We set then the final equations 11.2 for computing the coefficients z i , w i for numbers α, α • , β and β • . Replacing the linear representation of function F , equation10.7, the final set of equations is D n ( α, α • ) = 0 D n ( β, β • ) = 0( α • (cid:12) β • ) E = a ( α (cid:12) β ) E ( α • (cid:12) β • ) I = a ( α (cid:12) β ) E + ( α (cid:12) β ) I ( β • (cid:12) α • ) E = a ( β (cid:12) α ) E ( β • (cid:12) α • ) I = a ( β (cid:12) α ) E + ( β (cid:12) α ) I , (11.3)where coefficients a , a satisfy equations 10.22 z ( a − a ) + 2 a a = 0 w ( a − − z a + a = 0 , and also z (cid:54) = 0 and a (cid:54) = 1 || a (cid:54) = 0.The solution of these equations provide a two components vector linear space with a bi-dimensional form. The proposed equations are numerous and have a high complexity. We letthis problem to be solved on future works.
12. Conclusions
In this work, we proposed a new vector space that includes negative probabilities. A newdomain of numbers is proposed from the definition of a new algebraic unit from an unsolvableequation in the complex domain. We identified that the new set of numbers defined withsuch a unit includes two new types of operations for not violate the fundamental theoremof algebra. The operations include the sum and the multiplication of conjugated numbers,31hose form was expressed as coefficients. Because of the positive-definiteness, we define anew map and also expressed with unknown coefficients. The vector space is provided with thepositive-definiteness, pair-pair linearity, and a pair-pair conjugated symmetry that provided theindependent set of equations needed for computing the coefficients from the new operations.That is our starting point, which we try to apply it to the already known algebraic concepts.The present proposal is just a preliminary study that should be enhanced.
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