<<

2 + 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 G3 over Euclidian E3 . The construction critically depends on generalization of formal, unspecified, complex 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 1S generated by arbitrary unit value 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 C of two dimensional complex vectors, can be 1 generalized to unit value elements   I S  of even subalgebra of over Euclidian space . I called such generalized qubits g -qubits [1].

Some minimal information about and is necessary. Algebraically, is linear space with canonical 1,e1,e2 ,e3,e2e3,e3e1,e1e2 ,e1e2e3, where 1 is unit value ,ei  are vectors in , eie j  are oriented, mutually orthogonal unit value areas (bivectors) spanned by ei  and e as edges, with orientation defined by to by ; 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,e e ,e e ,e e . Variables  and  in 2 3 3 1 1 2    IS G3 are scalars, I S is a unit size oriented area, , lying in an arbitrary given plane

S  E3 . Bivector is 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

It was explained in [4], [5] that elements   I S  only differ from what is traditionally called “complex ” by the fact that S  E3 is an arbitrary, though explicitly defined, plane in E3 . Putting it simply, are “complex numbers” depending on E2 embedded into . Traditional “” 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 “” with additional algebraic property i 2  1. That may be a source of ambiguities, as it happens in .

2 I will denote unit value oriented volume e1e2e3 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 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   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 is left (right) 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 g -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.

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 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  i2 

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  i3 

 z 3     i  and so,,S     1    3 ,i  B  3    3  z2  2  i1 

Bi i i So we have three different maps so, , S(z1, z2 ) defined by explicitly declared complex planes Bi satisfying (1.1):

   i   1 ,i  B so(, , S)     B   B   B    1 1 1 2 2 3 3  3  i2      1B1  (3  2B1)B3      i   2     ,i  B2 (2.2)   2B2  (1  3B2 )B1    1  i3     B  (   B )B     i3  3 3 2 1 3 2     ,i  B3  2  i1 

There exists infinite number of options to select the triple Bi . It means that to recover a g -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 so,,S depending on choosing of a triple of orthonormal bivectors B1, B2, B3 in 3D satisfying (1.1), and associating one 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 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 xM , is homeomorphic to standard fiber .

Bi i i We can think about the map so, , S(z1, z2 ) as a fiber bundle. In the current case, the fiber  2 2 bundle will have so(,,S)  g G3 : g 1 as total space and {z  (z1, z2 )C ; z 1} 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 B1, B2, B3 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 .

By some reasons that will be explained a bit later I will use complex plane associated with B3 , so by (2.2) the projection is:

   i   3   : so(, , S)    1B1  2B2  3B3    (3.1) 2  i1 

 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 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

5

  G | 3 All such rotations in G3 are also identified by elements of 3 S 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

 B3 2 G | 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) by elements of .

Now the explanation why B3 was taken as complex plane. As was shown in [1], [2] the variant of classical Hopf fibration

2 2 ~ ~ ~ ~ 2 2 C | 3  S Pr z , z  z z  z z ,i z z  z z , z  z S :  1 2   1 2 1 2  1 2 1 2  1 2 

can be received as one of the cases of generalized Hopf fibrations:

 B 2 B ~ 2 G | 3 S : 3 S so(,,S) so(,,S) Bso(,,S)  S .

The basis bivector B1 gives:

B1 so(, , S) (  IS )B1(  IS )  (  1B1  2B2  3B3)B1(  1B1  2B2  3B3)  2 2 2 2 (  1 )  (2  3 )B1  2(3  12 )B2  2(13 2 )B3  2 2 2 2 (  1 )  (2  3 ),23  12 ,213 2  (3.2) and for two other basis bivectors: so, , SB2 2   B   2   2   2   2 B  2    B  1 2 3 1 2 1 3 2 1 2 3 3 (3.3) 2 2 2 2 212 3 ,  2  1  3 ,21  23  so, , SB3 2    B  2   B   2   2   2   2 B  2 1 3 1 2 3 1 2 3 1 2 3 (3.4) 2 2 2 2 22  13 ,223 1,  3  1  2 

3 It is often convenient to write elements so,,S as exponents: so, , S eI S .

6

In the literature mostly the third or the first variants are called Hopf fibrations. For my considerations it does not matter which variant to choose. I take the case (3.4) with B3 as complex plane:

   i   3  so(,,S)   1B1  2B2  3B3   3B3  (2  1B3)B2   ,i  B3 2  i1 

4. Tangent .

We need to temporarily get back to the case of G2 - geometric algebra on a plane [4].

Let an orthonormal basis e1,e2 is taken. It generates basis 1,e1,e2,e1e2 where e1e2  e1 e2  e1  e2 is usual geometric product of two vectors: first member is scalar product of the vectors and second member is oriented area (bivector) swept by rotating e1 to e2 by the angle which is less than  .

The basis vectors satisfy particularly the properties:

2 2 I2  e1e2   1, I2e1  e2 (clockwise rotation), e1I2  e2 (counterclockwise rotation), I2e2  e1 , e2 I2  e1 . I am using notation I 2 for unit bivector e1e2 .

For further convenience, let’s construct a basis isomorphic to . Commonly used  0  agreement will be that scalars are identified with scalar matrices, for example:     .  0  

For the case the second order matrices will suffice to get necessary . Let’s take 1 0  0 1     4 1    and  2    as corresponding to geometric basis vectors e1 and e2 . They satisfy 0 1 1 0 2 1 0  0 1 1 0  2   2 2 the properties mentioned above, for example: i    , I2  12        , 0 1 1 0  0 1  0 11 0   0 1 I 21          2 , etc. 1 00 1 1 0 

The representing scalar product gives:

4 I begin with ordinary real component matrices. That is why the order of Pauli matrices in my following definition is different from usual one. An analogue of “imaginary” unit should logically be the last one, after pure real items.

7

1 1  0 1 0 1 0 0                  1 2  1 2 2 1       (4.1) 2 2 1 0 1 0  0 0

as it should be. If we take two arbitrary vectors expanded in basis 1,2:

  0   0       1   2   1 2  a 11 22           0 1  2 0  2 1 

  0   0        1   2   1 2  b  11  22           0  1  2 0  2  1  then their product is:

      0   0        1 0  1 1 2 2   2 1 1 2  ab        (a b)   12 2 I2  0 11 22  12 21 0  0 1

The first member is usual scalar product and the second one is bivector of the value equal to the area of   parallelogram with the sides a and b .

Important thing to keep in mind is that multiplication of any vector by unit bivector I 2 rotates the  vector by  : 2

    0 1     1 2   2 1  aI2         21 12 (counterclockwise rotation) 2 1 1 0  1 2 

  0 1        1 2   2 1  I2a         21 12 (clockwise rotation) 1 02 1  1 2 

This remains valid for any unit bivector of the same orientation as . We can conclude that such multiplications give basis vectors of the tangent spaces to the original vectors.

8

This is identical to considering even elements 1  B2 corresponding to vectors and their multiplication by B : 1  B2 B  2  B1 . Elements and 2  B1 are orthogonal:   B ,  B    B    B   0 (index 0 means scalar part). 1 2 2 1 0 1 2 2 1 0

General element of G2 in matrix basis 1,1,2,12  I2, isomorphic to geometrical basis

1,e1,e2,e1e2, is:

1 0 1 0  0 1  0 1       0 1 2 3  g2  0  1  2  3     0 1 0 1 1 0 1 0 2 3 0 1 

This is arbitrary real valued matrix of the second order. Inversely, any matrix of second order can be uniquely mapped to the element of in the basis :

 x y x  t x  t y  z y  z     1  2  I2  z t  2 2 2 2

To upgrade to G3 , we need basis matrices of a higher order because a second order non-zero matrix orthogonal in the sense of scalar product (4.1) to both 1 and  2 does not exist.

It is easy to verify that the three, playing the role of three-dimensional orthonormal basis, necessary matrices can be taken as 4th-order block matrices:

 I 0  0 I   0  I   2  1   , 2   , 3    , 0  I   I 0  I2 0 

1 0  0 1     where I    - scalar matrix corresponding to scalar 1 , I2     1 2 . Easy to see that  1 0 1 1 0 and  2 are received from 1 and  2 by replacing scalars by corresponding scalar matrices.

By multiplications in full 4 4 form one can easily proof that block-wise multiplications are correct, the basis 1, 2,3 is orthonormal, anticommutative, and  2 3 ,  31 and 1 2 satisfy requirements (1.1).

 I 0   I 0   I 0   2   2    Taking I3  1 2 3    , multiplications give:  23     I3   I31 ,  0 I2   0  I2  0  I 

 0 I  0 I   0 I   0  I   2       2  2 31     I3   I3 2 , 1 2     I3   I3 3 . Then, since I3  1,  I2 0   I 0  I 0  I2 0  we easily get:

9

 23 31 1 2 ,  23 1 2 31 , 311 2   23

that is exactly (1.1) with basis bivectors B1   2 3 , B2 31 and B3 1 2 .

Following all that we can write general element g3 of algebra G3 expanded in formal matrix basis

1,1, 2,3,23,31,1 2,1 23 1,1,2,3, I31, I3 2, I33, I3 as:

g3  I 11 2 2 3 3  1 2 3  2 31  31 2  01 2 3 

  1  I2 0  1 2  3  I2 2 3    (4.2) 2  3  I2 2 3   1  I2 0  1  

 0 I    This is arbitrary “complex” valued matrix where “imaginary unit” is matrix I 2    ,  I 0 corresponding to bivector expanded in the first two basis vectors 1 ,  2 , that geometrically is bivector

B3 . Inversely, any “complex” valued matrix can be written as element in matrix basis

1,1, 2,3, I31, I3 2, I33, I3 isomorphic to geometrical basis 1,e1,e2 ,e3,e2e3,e3e1,e1e2 ,e1e2e3:

 s  I t u  I v  s  z s  z u  x y  v t  w y  v u  x t  w  2 2      1   2   3   2 3   31  1 2  I3  x  I2 y z  I2w 2 2 2 2 2 2 2 2  If belongs to even subalgebra G3 then, by identifying  2 3  B1 ,  31  B2 , 1 2  B3, we have from (4.2) the correspondence:

   I    I     2 1 3 2 2  g3   1B1  2B2  3B3    (4.3)  3  I22   I21 

In the same way as in the G2 case, multiplications of so(,1,2 ,3 )    1B1  2 B2  3B3 by basis bivectors Bi give basis bivectors of tangent spaces to original bivectors:

T1  so(, 1, 2 , 3 )B1    1B1  2B2  3B3 B1  1 B1  3B2  2 B3

T2  so(,1,2,3)B2    1B1  2B2  3B3 B2  2  3B1 B2  1B3

T3  so(,1,2,3)B3    1B1  2B2  3B3 B3  3  2B1  1B2 B3

 G | 3 These 3 S elements are orthogonal to   1B1  2 B2  3B3 and to each other, and are the tangent space basis elements at points .

2 T C | 3 Projections of i onto S are:

10

   i     i     i  1 2 2 1  3   (T1 )    ,  (T2 )    ,  (T3)      3  i     i3   1  i2 

2 2 C | 3 C These elements of S are orthogonal, in the sense of Euclidean scalar product in :

~ ~    i3  z, w  Rez1w1  z2w2 , to each other and to the projection  (so(, 1, 2 , 3 )    of 2  i1   G | 3 the original state in 3 S . They are the tangent space basis elements in at points

 z1     i3       .  z2  2  i1 

5. Hamiltonians in and their lifts in G3

Take self-adjoint (Hamiltonian) linear operator in C 2 in its most general form:

 a  b c  I 2d  H  aI  b1  c 2  d 3    . Its corresponding G3 element, see (4.3), is c  I 2d a  b  a  I3bB1  cB2  dB3  and does not belong to , though the result of its action on an element from (qubit), has  1 preimage in . Let’s prove that. Take a qubit and a Hamiltonian in :

 x  iy   r ei1  x y z   1 1    1  , r  x2  y 2 , cos  k , sin  k , k  1,2   i k k k k k x  iy r e 2  2 2 2 2  2 2   2  xk  yk xk  yk

 a  b Rei  c d H    , R  c2  d 2 , cos  , sin   i  Re a  b c2  d 2 c2  d 2

Then:

 a  b Rei  rei 1  Hz    1    i  i 2  Re a  br2e  (5.1) (a  b)r1 cos1  Rr2 cos(  2 )  i(a  b)r1 sin1  Rr2 sin(  2 )    (a  b)r2 cos 2  Rr1 cos( 1)  i(a  b)r2 sin 2  Rr1 sin( 1)

11

 x1  iy1  2  For each single qubit z   C | 3 its fiber (full preimage) in G | 3 is   S 3 S  x2  iy2 

Fz  x1  y2B1  x2B2  y1B3 with an arbitrary triple of orthonormal bivectors B1, B2, B3 in 3D satisfying (1.1). In exponential form:

 y x y  F  x  y B  x B  y B  x  1 x2  2 B  2 B  1 B  z 1 2 1 2 2 1 3 1 1  2 1 2 2 2 3   1 x1 1 x1 1 x1 

I S  cos  sinb1B1  b2B2  b3B3  e , (5.2) where

cos  x ,sin  1  x 2  1 1  y x y  b  2 ,b  2 ,b  1 (5.3) 1 2 2 2 3 2  1 x1 1  x1 1 x1   I S  b1B1  b2 B2  b3B3 

Using (5.2) we get from (5.3) the fiber at Hz :

IS ( H )(H ) FHz  e , (5.4)

where

cos(H )  (a  b)r1 cos1  Rr2 cos(  2 );  I  b (H )B  b (H )B  b (H )B ;  S (H ) 1 1 2 2 3 3  (a  b)r sin  Rr sin(  ) (a  b)r cos  Rr cos(  )  b (H )  2 2 1 1 ,b (H )  2 2 1 1 , 1 2 2 2  (5.5) 1 cos (H ) 1 cos (H )  (a  b)r sin  Rr sin(  )  b (H )  1 1 2 2  3 2 1 cos (H ) 

1 We can also explicitly write the element g3(H) which, acting (from the right) on  (z) gives  1(Hz) :

1 1  (z)g3(H)  (Hz)

From (5.1) and (5.4):

I S I S ( H )(H ) I S I S ( H )(H ) e g3(H)  e  g3(H)  e e

12 where IS ,  , IS (H ) and (H) are defined by (5.3) and (5.5).

6. Clifford translations

2 i C | 3 Cl z  e z Suppose, action on z in S is multiplication by an exponent:    , often called Clifford translation5, that’s:

i1 i  r1e   r1 cos( 1 )  ir1 sin( 1 )  Cl (z)  e       i 2    r2e  r2 cos(  2 )  ir2 sin(  2 )

 G | 3 Then the fiber of Cl (z) in 3 S is:

F  r cos(  )  r sin(  )B  r cos(  )B  r sin(  )B  Cl ( z) 1 1 2 2 1 2 2 2 1 1 3

r1 cos(  1 )  r1 sin(  1 )B3  r2 cos(  2 )  r2 sin(  2 )B3 B2  (6.1)

B3 B31 B3 B3 2 B3 r1e e  r2e e B2  e Fz

In other words, Clifford translations in are equivalent to multiplications of fibers in by standard fiber elements with bivector part B3 (which is associated with formal imaginary unit i ) and the same phase  .

In commonly used quantum mechanics all states from the set Cl z are considered as identical to the state , they have the same values of observables. That is not commonly true for g -qubits. Only in the

2 6 case when all the observables common plane, say IS , (unspecified in the C model ) is the same as the

IS I S plane associated with complex plane in , the -qubit e Fz leaves any such observable e unchanged:

~ I S I S I S ~ I S Fze e e Fz  Fze Fz

Suppose angle  in is varying. Then the tangent to Clifford orbit is

  Cl z  ei z  ie i z  iCl z.    

5 It is called “” because does not change between elements. 6   In the G3 model observables are also elements of G3 , see [1], [2]

13

2 ~ ~ It is orthogonal to Cl z in the sense of usual scalar product in C : u,v  Reu1v1  u2v2 , and

2 C | 3 remains on S : iCl (z)  1 (translational velocity of Clifford translation is one).

~  z1  2  z2  2 For any z    C | 3 define w    C | 3 . The elements iz , w and iw are orthogonal to   S  ~  S  z2   z1  z and pairwise orthogonal. That means they span tangent space and the plane spanned by w,iw is orthogonal to the Clifford orbit plane spanned by z,iz.

The vector of translational velocity rotates while moving in the orbit plane. Both and also rotates   with the same unit value angular velocity since Cl w  ei w  ei iw  Cl iw and       Cl iw  ei iw  ie i iw  Cl w .    

To make the planes of rotations clearly identified, that is difficult to do with formal imaginary unit, let’s  G | 3 lift Clifford translation to 3 S using (6.1). Translational velocity, similar to the case, is   F  eB3 F   B F and is orthogonal to F :  Cl (z)  z 3 Cl (z) Cl (z)

~ ~ ~ FCl (z) , B3FCl (z)  FCl (z) FCl (z) B3   B3 0  0   0   0

Two other components of the tangent space, orthogonal to and B F at any point of the 3 Cl (z) orbit, are B F and B F 7. Their velocities while moving along Clifford orbit are: 1 Cl (z) 2 Cl (z)

  B3 B1FCl (z)  B1e Fz  B1B3FCl (z)  B2FCl (z) (derivative of is orthogonal to      and looking in the direction )

  B3 B2FCl (z)  B2e Fz  B2B3FCl (z)  B1FCl (z) (derivative of B F is orthogonal to      2 Cl (z) and looking in the direction  B F ) 1 Cl (z)

These two equations explicitly show that the two tangents, orthogonal to Clifford translation velocity, rotate in moving plane B1FCl (z) , B2FCl (z) with the same unit value rotational velocity. Interesting to   notice that the triple of the translational velocity and two rotational velocities has orientation opposite

7 Clearly, the three B F are identical to earlier considered tangents T i Cl (z) i

14 to the triple of tangents: if the tangents B1FCl (z) , B2 FCl (z) , B3FCl (z)  have right screw orientation,    the speed triple  B1FCl (z), B2FCl (z), B3FCl (z) is left screw.   

If a fiber, g -qubit, makes full circle in Clifford translation: F  F  eB3 F , 0   2 , both z Cl (z) z B F and B F also make full rotation in their own plane by 2 . This is special case of the - 1 Cl (z) 2 Cl (z) qubit Berry phase incrementing in a closed curve quantum state path.

7. Conclusions

 Evolution of a quantum state described in terms of G3 gives more detailed information about two state system compared to the C 2 Hilbert space model. It confirms the idea that distinctions between “quantum” and “classical” states become less deep if a more appropriate mathematical formalism is used. The paradigm spreads from trivial phenomena like tossed coin experiment [3] to recent results on entanglement and Bell theorem [8] where the former was demonstrated as not exclusively quantum property.

Works Cited

[1] A. Soiguine, "Geometric Algebra, Qubits, Geometric Evolution, and All That," January 2015. [Online]. Available: http://arxiv.org/abs/1502.02169.

[2] A. Soiguine, "What quantum "state" really is?," June 2014. [Online]. Available: http://arxiv.org/abs/1406.3751.

[3] A. Soiguine, "A tossed coin as quantum mechanical object," September 2013. [Online]. Available: http://arxiv.org/abs/1309.5002.

[4] A. Soiguine, Vector Algebra in Applied Problems, Leningrad: Naval Academy, 1990 (in Russian).

[5] A. Soiguine, "Complex Conjugation - Realtive to What?," in Clifford with Numeric and Symbolic Computations, Boston, Birkhauser, 1996, pp. 285-294.

[6] D. Hestenes, New Foundations of Classical Mechanics, Dordrecht/Boston/London: Kluwer Academic Publishers, 1999.

15

[7] C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge: Cambridge University Press, 2010.

[8] X.-F. Qian, B. Little, J. C. Howell and J. H. Eberly, "Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields," Optica, pp. 611-615, 20 July 2015.

[9] J. W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory, John Wiley & Sons, 2011.

[10] D. Chruscinski and A. Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics, Boston: Birkhauser, 2004.

16