Instantly Propagating States

Full text

Journal of Applied Mathematics and Physics, 2021, 9, 468-475
https://www.scirp.org/journal/jamp
ISSN Online: 2327-4379
ISSN Print: 2327-4352
DOI:
10.4236/jamp.2021.93032 Mar. 25, 2021 468 Journal of Applied Mathematics and Physics
Instantly Propagating States
Alexander Soiguine
SOiGUINE Quantum Computing, Aliso Viejo, USA
Abstract
The current work considers the sprefield wave functions received as special
g-qubit solutions of Maxwell equations
in the terms of geometric algebra. I
will call such g-qubits spreons or sprefields
. The purpose of this article is to
analyze behavior of such wave functions in scattering and measurements. It is
shown that sprefields are defined through the whole three-
dimensional space
at all values of the time parameter. They instantly change all the
ir values
when get scattered, that is subjected to Clifford translation. In “measure-
ments”, when a sprefield acts on a static geometric algebra element through
the Hopf fibration, sprefield
collapses and new geometric algebra non static,
rotating element is thereby created.
Keywords
Wave Functions, Geometric Algebra, Measurements, Scattering,
Entanglement
1. Introduction
Usage of even subalgebra
3
G+
of geometric algebra
3
G
[1] [2] [3] stems from
generalization of complex numbers [3] [4]. The sprefield wave functions (states)
received as special
3
G+
solutions of Maxwell equations [5] [6].
In terms of geometric algebra
3
G+
, the electromagnetic Maxwell equation in
free space
( )
0
tF∂ +∇ =
, (1.1)
where
( )
0
exp
S
F F I t kr
ω
= −⋅


, has two linear independent solutions [3] [5] [6]:
( ) ( )
( )
( )
[ ]
( )
( ) ( )
( )
( )
[ ]
( )
0 30 3
0 30 3
0 30 3
0 30 3
exp
exp exp
exp
exp exp
SS
S SS
SS
S SS
F e Ih I t II r
e Ih I t I II r
F e Ih I t II r
e Ih I t I II r
ω
ω
ω
ω
+
−

=+ −⋅




=+ −⋅

 

=+ +⋅



=+⋅



(1.2)
How to cite this paper:
Soiguine, A.
(20
21) Instantly Propagating States.
Journal
of
Applied Mathematics and Physics
,
9,
468
-475.
https://doi.org/10.4236/jamp.2021.93032
Received:
January 6, 2021
Accepted:
March 22, 2021
Published:
March 25, 2021
Copyright © 20
21 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access

A. Soiguine
DOI:
10.4236/jamp.2021.93032 469 Journal of Applied Mathematics and Physics
For arbitrary scalars
λ
and
µ
:
( )
( ) ( )
()
33
0 30
ee e
SS SS
S
I II r I II r
It
F F e Ih
ω
λµ λ µ
 
−⋅ ⋅
 
+−
+=+ +
(1.3)
is also solution of (1.1). The item in the second parenthesis is weighted linear
combination of two states with the same phase in the same plane but opposite
sense of orientation. These states are strictly coupled because bivector plane
should be the same for both, does not matter what happens with that plane.
Formula (1.3) does not immediately look like an element of
3
G+
due to the
factor
( )
0 30
e Ih+
. But necessary transformations of the initial bivector basis
{ }
123
,,BBB
into triple of unit value orthonormal bivectors
{ }
00
,,
SB E
II I
where
S
I
is unit value bivector, dualto the propagation direction vector;unit value
0
B
I
is dual to initial vector of magnetic field
0
h
; unit value
0
E
I
is dual to ini-
tial vector of electric field
0
e
, change (1.3) into:
ee
Plane Plane
II
ϕϕ
λµ
++ −−
+
(1.4)
where
( )
( )
13
1
cos cos
2S
t II r
ϕω
±−


= ⋅




,
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
0
0
33
22
33
3
23
sin cos
1 sin 1 sin
sin
1 sin
SS
Plane S B
SS
S
E
S
t II r t II r
II I
t II r t II r
t II r
It II r
ωω
ωω
ω
ω
±
 
⋅⋅
 
= +
 
+⋅+⋅
 

⋅

+
+⋅





The expression (1.4) is linear combination of two geometric algebra states,
g-qubits, which are elements of the form
Plane
I
αβ
+
with some arbitrary unit
value bivector
Plane
I
in three dimensions.
Let us calculate
ee
Plane Plane
II
ϕϕ
λµ
++ −−
+
with
1
λµ
= =
(the
2
is included
for normalization to further write the expression as exponent):
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
00
00
33
33
33
33
ee
11
cos sin
22
11
cos sin
22
11
cos sin
22
11
cos sin
22
2
2
Plane Plane
II
SS S
B SE S
SS S
B SE S
t II r I t II r
I t II r I t II r
t II r I t II r
I t II r I t II r
ϕϕ
ωω
ωω
ωω
ωω
++ −−
+
 
= −⋅+ −⋅
 
 
+ −⋅+ −⋅
 
 
+ +⋅+ +⋅
 
 
+ +⋅+ +⋅
 
=




( )
( )
( )
00
3
cos cos sin cos sin
S SB E
II r tI tI tI t
ω ωω ω ω

⋅ ++ +


(1.5)
I will call such g-qubits
spreons
or
sprefields
. The purpose of this article is to
analyze behavior of such wave functions in scattering and measurements.

A. Soiguine
DOI:
10.4236/jamp.2021.93032 470 Journal of Applied Mathematics and Physics
2. Scattering of Sprefields
The sprefield wave function:
( )
( )
( )
( )
123
31 1 2 3
, ,, , ,
2 cos cos sin cos sin
Sp r t B B B
IBr tB tB tB t
ω
ω ωω ωω

= ⋅ ++ +



(2.1)
can be written, following multiplication rules of basis bivectors, as:
( ) ( )
( )
11
12 1 2
,,, , ,, e e
Bt Bt
Sp r t B B R r B B
ωω
ωω
−
= +

(2.2)
where
( )
1
,,R rB
ω

is a scalar valued function.
The specifics of (2.1)-(2.2) is that it is a non-local field object instantly
spreading its modifications, caused by Clifford translations or by Hopf fibrations
(“measurements”), through the whole 3D and time parameter values. In the
Hopf fibration new element is created, not static one, opposite to the measured
3
G
+
element, with stable rotation characteristics depending on the sprefield
wave function parameters.
The scheme suggested in the current text is based on manipulation and trans-
ferring of quantum states (wave functions) as operators acting on observables,
both formulated in terms of geometrical algebra. Wave functionsact in the cur-
rent context on static
3
G
+
elements through measurements, creating “particles”.
Normalized wave functions as elements of
3
G
+
are naturally mapped onto
unit sphere
3

.
Two-state system is then just a couple of points on
3

,
{ }
123
1 11 11 11
,,,bbb
αβ β β
and
{ }
123
2 22 22 22
,,,bbb
αβ β β
, corresponding to wave func-
tions:
1
11
12 3
1 1 1 11 1 11 2 11 3
e
S
IS
I bB bB bB
ϕ
α βαβ β β
=+=+ + +
2
22
12 3
2 2 2 22 1 22 2 22 3
eS
IS
I bB bB bB
ϕ
α βαβ β β
=+=+ + +
with
( ) ( )
( ) ( ) ( )
()
() ( )
222
22 22
123
11111 11
1bbb
αβ αβ
+ ++ =+=
( ) ( )
( ) ( ) ( )
()
( ) ( )
222
22 22
123
22222 22
1bbb
αβ αβ
+ ++ = + =
in some bivector basis
123
1BBB =
, with multiplication rules
12 3
BB B= −
,
13 2
BB B=
,
23 1
BB B= −
.
Then it follows that two wave functions of an arbitrary two-function system
are, in any case, connected by the Clifford translation1:
( )
( )
2 21 1 1
2 211 1
2 211
e e e e ,,, e
S SSS S
I III I
Cl S S
ϕ ϕϕ ϕ ϕ
ϕϕ
−
= ≡
(2.3)
The product of exponents
21
21
ee
SS
II
ϕϕ
−
is trivial in the case
12
SSS= ≡
(the
case of geometrically unspecified imaginary unit plane in conventional quantum
mechanics)
( )
21 21
21
ee e
SS S
II I
ϕϕ ϕϕ
−−
=
. Though in general case we have more com-
plicated result:
1
It is universally possible due to the hedgehog (hairy ball) theorem which says that there exists non-
vanishing continuous tangent vector field on odd-dimensional sphere
3

.

A. Soiguine
DOI:
10.4236/jamp.2021.93032 471 Journal of Applied Mathematics and Physics
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
21
21
2 211
1 212123212
31 2 1 3 2 1 1 2
,,, e e
cos cos sin sin cos sin
cos sin sin sin
SS
II
Cl S S
s s Is
Is I s s
ϕϕ
ϕϕ
ϕϕ ϕϕ ϕϕ
ϕϕ ϕϕ
−
≡
= +⋅ +
+ +×
(2.4)
where
1
s
and
2
s
are vectors dual to planes
1
S
and
2
S
matching orientation
of
3
I
.
The result of Clifford translation (2.4) is a
3
G+
element. From knowing Clif-
ford translation connecting any two wave functions as points on
3

it follows
that the
result of measurement of any observable C by wave function
1
1
eS
I
ϕ
, for
example
( )
11
11
11
ee ,
SS
II
C CS
ϕϕ
ϕ
−
≡
, immediately gives the result of (not made)
measurement by
2
2
e
S
I
ϕ
:
( )
( ) ( ) ( )
2 2 2 11 11 2
2 2 2 11 11 2
21 1 2
21 12
11
2 211 11 2 2 11
e e eee eee
e e ,ee
,,, , ,,,
S S S SS SS S
S S SS
I I I II II I
I I II
CC
CS
Cl S S C S Cl S S
ϕ ϕ ϕ ϕϕ ϕϕ ϕ
ϕϕ ϕϕ
ϕ
ϕϕ ϕ ϕϕ
− − −−
−−
=
=
=
2
This is geometrically clear and unambiguous explanation of strict connectivity
of the results of measurements instead of “entanglement” in conventional quan-
tum mechanics.
Take the spreon (1.5):
()
()
()
000
00
3
, ,, , ,
11
2cos cos sin
22
11
cos sin
22
SBE
SS
BE
Sp r t I I I
II r t I t
I tI t
ω
ω ωω
ωω


= ⋅+




++




By redifining for reading formulas easier
1S
IB≡
,
0
2B
IB≡
,
0
3
E
IB≡
we
have the following:
Sprefield when scattered by a
3
G+
element
( )
11 2 2 3 3
cos sin e
B
BBB
γ
γ γγ γ γ
+ ++ =
,
11 2 2 3 3
B B BB
γγ γ
++≡
, unit value bivec-
tor if
222
123
1
γγγ
++=
, becomes:
( ) ( )
( )
( )
( )
11
11
12 1 2
12
e ,,, , e ,, e e
,, ee ee
Bt Bt
BB
Bt Bt
BB
Sp r t B B R r B B
R rB B
ωω
γγ
ωω
γγ
ωω
ω
−
−
= +
= +


Let us use again a general formula for the product of two geometric algebra
exponents:
( )
( )
12
1 2 31
32 3 1 2
e e cos cos sin sin cos sin
cos sin sin sin
SS
II
s s Is
Is I s s
αβ
α β αβ βα
αβ αβ
= −⋅ +
+ −×
where
1
s
and
2
s
are vectors dual correspondingly to bivectors
1
S
I
and
2
S
I
.
In
1
ee
Bt
B
ω
γ
we have
cos cos
αγ
=
,
( )
1 123
,,s
γγγ
=
,
cos cos t
βω
=
,
( )
21,0,0
s=
and in
1
ee
Bt
B
ω
γ
−
,
( )
21,0,0s= −
. Thus,
2Difference in exponent signs from usual measurement definition is made just for some conven
i-
ence. It means that the angle has opposite sign or can be thought that the bivector plane was flipped.

A. Soiguine
DOI:
10.4236/jamp.2021.93032 472 Journal of Applied Mathematics and Physics
( )
( )
1
1 11 2 2 3 3
1 32 23
e e cos cos sin sin sin cos
cos sin sin sin
Bt
B
t tBBB t
B tBB t
ω
γ
γωγ γωγ γ γ γω
γωγ γ γω
= − + ++
+ +−
()
()
1
1 11 2 2 3 3
1 32 23
e e cos cos sin sin sin cos
cos sin sin sin
Bt
B
t tBBB t
B t BB t
ω
γ
γωγ γωγ γ γ γω
γω γ γ γω
−
= − + ++
− +− +
( )
( )
1
2 2 21 1 3 2 3 1
3 3 21
e e cos cos sin sin sin cos
cos sin sin sin
Bt
B
B B tB t B B t
B tB t
ω
γ
γω γγω γ γγ γω
γωγγ γω
−
= − +− − +
+ ++
Then it follows that the result of scattering is:
( )
( )
( )
( )
( )
( )
( )
( )
1 2 31
31 2 1
2 31 2
31 2 3
, , cos cos sin cos sin sin
sin cos sin sin cos sin
sin cos sin sin cos cos
sin cos sin sin cos sin
R rB t t t
t t tB
t t tB
t t tB
ω γωγ γωγγ γω
γγ γ ωγ γω γω
γ γωγγ γω γω
γγ γ ωγ γω γω
− +−

++ + +
+ +− +

+− − + 

(2.5)
This scattered sprefield is defined in all points
r

of three-dimensional space
and time parameter values
t
and is obviously independent of when scattering
took place.
In some special cases of the scattering element, we get the following:
If sprefield is scattered by
1
cos sin B
γγ
+
the result is:
( ) ( ) ( ) ( ) ( )
1 1 23
, , cos sin cos sinRrB t tB tB tB
ω ωγ ωγ ωγ ωγ
++++++−



If sprefield is scattered by
2
cos sin B
γγ
+
the result is:
( )
11
23
2 , , cos sin sin cos
44
cos cos sin sin
44
R rB t t B
t Bt B
ω ωγ ω γ
ωγ ωγ
π π
 
−+ −
 
 

π π
 
+ −+ −
 

 


If sprefield is scattered by
3
cos sin B
γγ
+
the result is:
( ) ( ) ( ) ( ) ( )
1 123
, , cos sin cos sinRrB t tB tB tB
ω ωγ ωγ ωγ ωγ
−+++−++



All these g-qubits are defined for all values of
t
and
r

, in other words the re-
sult of Clifford translation by spreon (1.5) instantly spreads through the whole
three-dimensions for all values of time.
The resulting state (10) is simultaneously redefined for all values of
t
. We par-
ticularly have changing of state backward in time. That is obvious demonstration
that the suggested theory allows indefinite event casual order. In that way the
very notion of the concept of cause and effect disappears, thus we might not
perceive time.
3. Measurements by Sprefields
The Hopf fibration, measurement of any observable
0 11 2 2 3 3
C CB CB CB++ +
in
the current formalism is [3]:

A. Soiguine
DOI:
10.4236/jamp.2021.93032 473 Journal of Applied Mathematics and Physics
( ) ( )
()( )
( )
( )
( ) ()
( )
( )
( ) ( )
( ) ()
( )
11 2 2 3 3
0 11 2 2 3 3
22 22
0 1 1 2 3 2 12 3 3 2 13 1
22 22
1 3 12 2 2 1 3 3 23 1 2
22 22
1 13 2 2 1 23 3 3 1 2 3
22
22
22
BBB
C CB CB CB
CC C C B
C C CB
C CC B
αβ β β
α β β β β β αβ αβ β β
αβ β β α β β β β β αβ
β β αβ αβ β β α β β β
++ +
++ +

+ +−+ + − + +


+ + + +−+ + −


+

− + + + +−+
→




(3.1)
Apply this formula for measurement by spreon (1.5):
( )
( )
00
3
11 1 1
2cos cos sin cos sin
22 2 2
S SB E
IIr tItI tI t
ω ωω ω ω


⋅ ++ +




,
that is use:
1S
BI=
,
0
2B
BI
=
,
0
3E
BI
=
( )
3
1
2cos cos
2
S
II r t
αω ω

= ⋅


;
( )
13
1
2cos sin
2
S
II r t
βω ω

= ⋅


( )
23
1
2cos cos
2
S
II r t
βω ω

= ⋅


;
( )
33
1
2cos sin
2
S
II r t
βω ω

= ⋅


The result of measurement is:
( )
( )
( )
( )
( )
00
0
0
0123
2
3 03 1 2
21
, , , , , , , ,,
4 cos sin 2 cos2
sin 2 cos2
SB E
SS B
E
OC C C C I I I tr
II r C CI C t C t I
C t C tI
ω
ω ωω
ωω

= ⋅ ++ +



+− 



(3.2)
Geometrically, this result means that the observable bivector plane rotates by
2
π
around vector
2
e
, such that the
C
3 component becomes lying in plane
S
I
.
Two other components lying in planes orthogonal to
S
I
rotate around normal
to
S
I
with angular velocity
2t
ω
. Both scalar and bivector parts get scalar fac-
tor
()
( )
2
3
4 cos
S
II r
ω

⋅


.
Formula (3.2) shows that only component of the result of measurement lying
in plane
S
I
does not depend on the value of time parameter.
We know that any two observables can be connected through Clifford transla-
tion. If we are concerned only in the
S
I
component of the result of measure-
ment then with placing another observable value
3
new
C
in (3.1) the latter can be
written, in assumption that the
S
I
old observable component is not zero, as:
( )
( )
( )
[ ]
2
3
33 3 3
3
4 cos
new
new SS
C
sign C sign C I I r C I
C
ω




⋅ ++






Thus, all the
S
I
components of any observable do simultaneously exist
whilst we only made measurement of one observable. All the other observables
values are calculated at values
3
3
new
C
rC

.
Consider more complicated way to get component of the result of measure-
ment not depending on time parameter.

A. Soiguine
DOI:
10.4236/jamp.2021.93032 474 Journal of Applied Mathematics and Physics
Assume the spreon is scattered by some state
( )
00
12 3
cos sin
SB E
II I
γ γγ γ γ
+ ++
before the measurement. The result of the
measurement in general case is a bit tedious. Let us take as the first example the
bivector components with
21
γ
=
,
13
0
γγ
= =
. In that case the result of mea-
surement, from (3.1), of
00
01 2 3SB E
C CI CI CI++ +
can be calculated as its mea-
surement by
0
cos sin
B
I
γγ
+
:
() ( )
0
00
00
cos sin
01 2 3
01 3 2 3 1
cos2 sin 2 cos2 sin2
B
I
SB E
SB E
C CI CI CI
C C C I CI C C I
γγ
γγ γγ
+
++ +
+

+−
→
++
followed by measurement by
( )
( )
00
3
11 1 1
2cos cos sin cos sin
22 2 2
S SB E
IIr tItI tI t
ω ωω ω ω


⋅ ++ +




that gives, using (3.2):
( )
( )
( )
( )
( )
( )
( )
( )
( )
0
0
0
2
3 03 1
13 2
2 13
4 cos cos2 sin 2 cos 2 sin 2
cos2 sin2 sin2 cos 2
sin 2 cos2 sin 2 cos2 cos2 sin 2
S SE
B
ES
II r C C C I I
C C t C tI
C tC C t I I
ω γ γγ γ
γ γω ω
ω γ γω γ γ


⋅+ − −


++ +

+ −+ +


If we take new orthonormal bivector basis:
{ } { }
0 00
12
cos2 sin2 ,sin 2 cos2 , , ,
S E S SB B
I I I II I I I
γγ
γ γγ γ
− +≡
the result of measurement reads:
( )
( )
()
( )
( )
()
( )
0
2
3 03 1 1
2 13
2 13 2
4 cos cos2 sin 2
cos2 cos2 sin 2 sin 2
sin 2 cos2 sin 2 cos2
S
B
II r C C C I
C t C C tI
C t C C tI
γ
γ
ω γγ
ω γ γω
ω γ γω


⋅+ −


+ ++

+ −+ 

that has constant value in plane
1
I
γ
plus rotation in planes
0
B
I
and
2
I
γ
with
angular velocity
2
ω
.
In that way we particularly get
r

-dependent variety of constant components
of the results of measurements:
( )
( )
( )
2
3 03 1 1
4 cos cos2 sin2
S
II r C C C I
γ
ω γγ


⋅+ −



Similar results are for other cases of scattering state:
11
γ
=
,
23
0
γγ
= =
and
3
1
γ
=
,
12
0
γγ
= =
.
4. Conclusions
All measured observable values are instantly spread through the whole set of
three-dimension points and time parameter values. If the measuring results
represent a function value, the values are available altogether, not through eva-
luating one by one.
The current approach transcends qubit entangled computational schemes
since the latter have tough problems of creating large sets of communicating qu-
bits. All the efforts today in building “quantum” computers are in implementa-

A. Soiguine
DOI:
10.4236/jamp.2021.93032 475 Journal of Applied Mathematics and Physics
tion of qubits (in various physically possible variants) effectively talking to each
other, thus emulating entanglement.
In the current scheme any observable can be placed into continuum of the
( )
,tr

dependent values of the sprefield. All other observables’ measurement
results are particularly connected by Clifford translations thus giving any num-
ber of values
( )
00
0123
, , , , , , , ,,
SB E
OC C C C I I I tr
ω

, spread over three-dimensions
and at all instants of time not generally following cause/effect ordering.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this pa-
per.
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