2 + Quantum state evolution in C and G3 Alexander M. SOIGUINE Abstract: It was shown, [1] - [3], that quantum mechanical qubit states as elements of C 2 can be generalized to elements of even subalgebra G3 of geometric algebra G3 over Euclidian space E3 . The construction critically depends on generalization of formal, unspecified, complex plane to arbitrary variable, but explicitly defined, planes in 3D, and of usual Hopf 3 2 B 2 fibration S S to maps g : g G3 , g 1S generated by arbitrary unit value bivectors B G3 . 2 2 Analysis of the structure of the map G3 C demonstrates that quantum state evolution in C gives only restricted information compared to that in . 1. Introduction 2 Qubits, unit value elements of the Hilbert space C of two dimensional complex vectors, can be 1 generalized to unit value elements I S of even subalgebra of geometric algebra over Euclidian space . I called such generalized qubits g -qubits [1]. Some minimal information about and is necessary. Algebraically, is linear space with canonical basis 1,e1,e2 ,e3,e2e3,e3e1,e1e2 ,e1e2e3, where 1 is unit value scalar,ei are orthonormal basis vectors in , eie j are oriented, mutually orthogonal unit value areas (bivectors) spanned by ei and e as edges, with orientation defined by rotation to by angle ; and e e e is unit value j 2 1 2 3 oriented volume spanned by ordered edges e1 , e2 and e3. Subalgebra is spanned by scalar 1 and basis bivectors: 1,e2e3,e3e1,e1e2 . Variables and in IS G3 are scalars, I S is a unit size oriented area, bivector, lying in an arbitrary given plane S E3 . Bivector is linear combination of basis bivectors ei e j , see Fig.1a. 1 This sum bears the sense “something and something”. It is not a sum of geometrically similar elements giving another element of the same type, see [3], [9]. 1 Fig.1a Fig.1b E3 It was explained in [4], [5] that elements I S only differ from what is traditionally called “complex numbers” by the fact that is an arbitrary, though explicitly defined, plane in . Putting it simply, are “complex numbers” depending on E2 embedded into . Traditional “imaginary unit” i is actually associated with some unspecified I S – everything is going on in one tacit fixed plane, not in 3D world. Usually i is considered just as a “number” with additional algebraic property i 2 1. That may be a source of ambiguities, as it happens in quantum mechanics. 2 I will denote unit value oriented volume by I 3 . It has the property I3 1. Actually, there always are two options to create oriented unit volume, depending on the order of basis vectors in the product. They correspond to the two types of handedness – left and right screw handedness. In Fig.1a the variant of right screw handedness is shown. It is geometrically obvious that eie j e jei (bivector flips, changes orientation to the opposite, in swapping the edge vectors). Then changing of handedness, for example by e1e2e3 e2e1e3 , gives e2e1e3 e1e2e3 I3 . Any basis vector is conjugate (conjugation means multiplication by ), or dual, to a basis bivector and inverse. For example I3e2e3 e2e3I3 e1 . By multiplying these equations from left or right by we get e2e3 I3e1 e1I3 . Such duality holds for arbitrary, not just basis, vectors and bivectors. Any vector e1e2e3 b is conjugate to some bivector: b I3B BI 3 and for any bivector B there exists such that B I3b bI3 . Good to remember that in vector and bivector are in handedness opposite to while in they are in the same handedness as . One can also think about as a right (left) single thread screw helix of the height one. In this way I3 isS left E (right)3 screw helix. All above remains true (see Fig.1b) if we replace basis bivectors e2e3,e3e1,e1e2 by an arbitrary triple of unit bivectors B1, B2, B3 satisfying the same multiplication rules which are valid for (see Fig.2, right screw handedness assumed): B1B2 B3 ; B1B3 B2 ; B2B3 B1 (1.1) 2 Fig.2 2 3 2. Parameterization of unit value elements in G3 and C by points of S Let’s take a -qubit: 2 2 IS b1B1 b2B2 b3B3 1B1 2B2 3B3,i bi , 1, 2 2 2 b1 b2 b3 1 I will use notation so,,S for such elements. They can be considered as points on unit radius sphere in the sense that any point ,1,2 ,3 parameterizes a -qubit. g A pure qubit state in terms of conventional quantum mechanics is two dimensional unit value vector with complex components: z 1 0 1 2 2 ~ ~ 1 2 z1 z2 , z1 z2 z1z1 z2 z2 1, zk zk iz k ,k 1,2 z2 0 1 We can also think about such pure qubit states as points on S 3 because: 3 1 2 1 2 2 2 ~ ~ 1 2 2 2 1 2 2 2 S z1 , z1 , z2 , z2 {z (z1, z2 )C ; z z1z1 z2 z2 z1 z1 z2 z2 1} (2.1) Explicit relations, based on their parameterization by points, between elements and elements so,,S can be established through the following construction2. Let basis bivector B1 is chosen as defining the complex plane S , then we have (see multiplication rules (1.1)): 2 Though both and can be parametrized by points of , it is not correct to say that 2 2 {z (z1, z2 )C ; z 1} is sphere or -qubit is sphere . 3 so(,,S) ( 1B1) 2B1B3 3B3 ( 1B1) (3 2B1)B3 Hence we get the map: z1 i so,,S 1 1 , where i B 1 1 z2 3 i2 In the similar way, for two other selections of complex plane we get: z2 i so, , S 1 2 ,i B , 2 2 z2 1 i3 z 3 i and so,,S 1 3 ,i B 3 3 z2 2 i1 Bi i i So we have three different maps so, , S(z1, z2 ) defined by explicitly declared complex planes Bi satisfying (1.1): so,,S i 1 ,i B so(, , S) B B B 1 1 1 2 2 3 3 3 i2 g 1B1 (3 2B1)B3 i 2 ,i B2 (2.2) 2B2 (1 3B2 )B1 1 i3 B ( B )B i3 3 3 2 1 3 2 ,i B3 2 i1 There exists infinite number of options to select the triple Bi . It means that to recover a -qubit z so,,S in 3D associated with 1 it is necessary, firstly, to define which bivector B in 3D i1 z2 should be taken as defining “complex” plane and then to choose another bivector B , orthogonal to i2 B . The third bivector B , orthogonal to both B and B , is then defined by the first two by i1 i3 i1 i2 orientation (handedness, right screw in the used case ): I B I B I B I . 3 i1 3 i2 3 i3 3 The conclusion is that to each single element (2.1) there corresponds infinite number of elements depending on choosing of a triple of orthonormal bivectors in 3D satisfying B , B , B (1.1), and associating1 2 one3 of them with complex plane. This allows constructing map, fibration 2 G3 C restricted to unit value elements. 4 3. Fiber bundle Take general definition of fiber bundle as a set E, M,,G, F where E is bundle (or total) space; M - the base space; F - standard fiber; G - Lie group which acts effectively on ; - bundle projection: 1 : E M , such that each space Fx (x) , fiber at xM , is homeomorphic to standard fiber . We can think about the map as a fiber bundle. In the current case, the fiber bundle will have so(,,S) g G3 : g 1 as total space and as base 2 2 G | 3 C | 3 :G | 3 C | 3 space. I will denote them as 3 S and S respectively. The projection 3 S S depends on which particular Bi is taken from an arbitrary triple satisfying (1.1) as associated complex plane of complex vectors of C 2 and explicitly given by (2.2), so we should write Bi 2 :G | 3 C | 3 3 S S . Bi i i By some reasons that will be explainedso, a bit, S later I will(z 1use, z2 )complex plane associated with B3 , so by (2.2) the projection is: i 3 : so(, , S) 1B1 2B2 3B3 (3.1) 2 i1 x1 iy1 2 Then for any z C | 3 the fiber in is comprised of all elements S x2 iy2 Fz x1 y2B1 x2B2 y1B3 with an arbitrary triple of orthonormal bivectors in 3D satisfying (1.1). That particularly means that standard fiber is equivalent to the group of rotations of the triple y2B1, x2B2 , y1B3 as a whole: Fig 3a. Fig 3b. Two example sections of a fiber. 3b received from 3a by 90 deg. counterclockwise rotation around vertical axis B1, B2, B3 2 2 {z (z1, z2 )C ; z 1} 5 All such rotations in are also identified by elements of since for any bivector B the result of its rotation is 3 (see, for example [6], [7]): ~ ~ ~ so( ,, S) Bso( ,,S) , where so( ,, S) ( 1B1 2B2 3B3) 1B1 2B2 3B3 So, standard fiber is identified as and composition of rotations is: I I I I I I ~ I I e S2 e S1 Be S1 e S2 e S1 e S2 Be S1 e S2 B G3 | 3 3 2 SG | 3 C | 3 All that means that the fibration 3 S S is principal fiber bundle with standard fiber and the group acting on it is the group of (right) multiplications by elements of .