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Applied Science and Innovative Research
ISSN 2474-4972 (Print) ISSN 2474-4980 (Online)
Vol. 6, No. 1, 2022
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46
Original Paper
Explaining Double Split Experiment with Geometrical Algebra
Formalism
Alexander Soiguine1*
1 SOiGUINE Quantum Computing, 31 Aurora, Aliso Viejo, CA, USA
* E-mail: alex@soiguine.com
Received: February 8, 2022 Accepted: February 15, 2022 Online Published: February 18, 2022
doi:10.22158/asir.v6n1p46 URL: http://doi.org/10.22158/asir.v6n1p46
Abstract
The Geometric Algebra formalism opens the door to developing a theory upgrading conventional
quantum mechanics. Generalizations, stemming from implementation of complex numbers as
geometrically feasible objects in three dimensions; unambiguous definition of states, observables,
measurements bring into reality clear explanations of conventional weird quantum mechanical features,
particularly the results of double split experiments where particles create diffraction patterns inherent
to wave diffraction. This weirdness of the double split experiment is milestone of all further difficulties
in interpretation of quantum mechanics.
Keywords
geometric algebra, quantum states, observables, measurements
1. Introduction. Working with G-qubits Instead of Qubits
Theory of upgrading conventional quantum mechanics has been under development (Soiguine, 1996,
2014, 2015, 2020).
The main novel features are:
- Replacing complex numbers with elements of even subalgebra of geometric algebra in three
dimensions, that’s by elements of the form “scalar plus bivector”.
- The objects identifying physical media are of the same structure: explicitly defined plane
along with angle of rotation in that plane.
- Operators acting on the objects are operators of rotation having the same structure: scalar
plus bivector. That is the measurement operation.
- Mapping of operator to the result of measurement, that is collapse, creates a “particle”.
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- The operators can be identified by points on the three-sphereand are connected, due to
hedgehog theorem, by Clifford translations. Modifying the operators due to Clifford
translation is identified by the generalization of Schrodinger equation containing unit
bivectors in three dimensions instead of formal imaginary unit
Qubits, identifying states in conventional quantum mechanics, mathematically are elements of
two-dimensional complex spaces:
, unit value element of.
Imaginary unitis used formally, without geometrical identification in three dimensions. In another
accepted notations a qubit is:
In the suggested formalism complex numbersare replaced with elements of, subalgebra
of.
Theelements of the formwhereis some unit bivector arbitrary placed in
three-dimensional space, comprise so called even subalgebra of algebra. This subalgebra is denoted
by (Soiguine, 2015, 2020). Elements ofcan be depict as in Figure 1.
Figure 1. An Element of
Unit value elements of,, will be called g-qubits. The wave functions (states)
implemented as g-qubits store much more information than qubits, see Figure 2.
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Figure 2. Geomectrically Picted Qubits and G-qubits
2. Lift of Qubits to G-qubits, Fiber Bundles and Probabilities
Take right-hand screw oriented basisof unit value bivectors, with the multiplication
rules ,,,(or equivalently ),
whereis oriented unit value volume in three dimensions named also pseudoscalar.
Quantum mechanical qubit state, , is linear combination of two basis
statesand. In theterms these two states correspond to the following classes of equivalence
in, depending on which basis bivector is selected as complex plane:
Ifis taken as complex plane, then
- Statehas fiber (level set) of theelements(0-typestates):
• ,
- Statehas fiber of theelements(1-typestates):
• ,
Ifis taken as complex plane, then
- Statehas fiber (level set) of theelements(0-typestates):
• ,
- Statehas fiber of theelements(1-typestates):
• ,
Ifis taken as complex plane, then
- Statehas fiber (level set) of theelements(0-typestates):
• ,
- Statehas fiber of theelements(1-typestates):
• ,
General definition of measurement in the suggested approach includes:
- the set of observables to be,
- the set of states to be elements of(g-qubits up to some scalar factor),
- measurement of an observable
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by g-qubit (wave function)
is defined as
with the result:
Probabilities of observed values are relative measures of the g-qubit fibers for each observable value
received by the action of the states on observable.
The lift fromtoneeds somereference frame of unit value bivectors. This frame, as a
solid, can be arbitrary rotated in three dimensions. In that sense we have principal fiber bundle
with the standard fiber as group of rotations which is also effectively identified by elements of.
Suppose we are interested in the probability of the result of measurement in which the observable
component does not change. This is relative measure of states
in the measurements:
That measure is equal to, that is equal toin the down mapping fromto.
Thus, we have clear explanation of common quantum mechanics wisdom on “probability of finding
system in state”.
Similar calculations explain correspondence of to in when the
componentin measurement just got flipped.
Any arbitrarystatecan be rewritten either as 0-type state
or 1-type state:
,
where
, 0-type,
or
,
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where
, 1-type.
All that means that anystatemeasuring arbitrary observable
does not change the observable projection onto plane of
and just flips the observable projection onto plane
.
3. Double Slit Experiment
Taking the set of g-qubits and projection of them onto:, we get fiber bundle. The
projection depends on which basis bivector plane is selected as corresponding to formal imaginary unit
plane. If we take, for example, the projection is:
Then for any
the fiber inconsists of all elements
with an arbitrary triple of orthonormal bivectorssatisfying multiplication rules. That
particularly means that the standard fiber is group of rotations of basis bivectors in the standard fiber.
Thus, the fiber bundle is principal fiber bundle.
Let one first slit is only open, and the fiber, wave function, is some
.
For the only open second slit the fiber is different:. When both slits
are open the corresponding fiber is defined by connection, parallel transport anywhere between
fibersand.
Let we have a smooth curve,, connecting points
and
, on three-dimensional spheresuch thatand. The
easiest way to define parallel transport is.
For convenience purposes let us writeandas exponents:
,
where,
.
Angleis not uniquely defined since it can be any of, , where
is, by definition, taken from interval. The anglewill be denoted as.
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,
where,
.
As above,,The anglewill be denoted as.
Measurement of an observable
,
where,
,
,
by the wave functionis:
Measurement byis:
Measurement by any intermediate parallel transport wave functionthen reads:
Let us make natural for double split experiment assumption(that is the two wave
functions, measuring states, are of 0-type with identical bivector planes.) Then we get the measurement
result by the intermediate parallel transport wave function:
It is easily seen that the result of measurement iswhenandwhen.
Consider the following simplified scenario.
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Assume we are only interested in the projections of and onto the plane of their
rotations,,and. Then from the general formula
we get that up to some factors is rotated in by
angleandisrotated inby angle.
Without loss of generality suppose that the anglesandare equal by values but opposite in
sign:
,,
Then it follows that in Clifford translations the projectionrotates inadditionally by
,,, and projection rotates in additionally
by,,.
Thus, we get infinite number of copies ofandwith varying values depending every
time on uniformly distributed, by assumption, value of,, and separated byalong the big
circle of intersection of planewith the sphere.
4. Conclusions
It was demonstrated that the geometric algebra formalism along with generalization of complex
numbers and subsequent lift of the two-dimensional Hilbert space valued qubits to geometrically
feasible elements of even subalgebra of geometric algebra in three dimensions allows, particularly, to
resolve the double split experiment results with diffraction patterns inherent to wave diffraction. This
weirdness of the double split experiment is milestone of all further difficulties in interpretation of
conventional quantum mechanics.
References
Soiguine, A. (2014). What quantum “state” really is? Retrieved from http://arxiv.org/abs/1406.3751
Soiguine, A. (2015). Geometric Algebra, Qubits, Geometric Evolution, and All That. Retrieved from
http://arxiv.org/abs/1502.02169
Soiguine, A. (2020). The Geometric Algebra Lift of Qubits and Beyond. s.l.:LAMBERT Academic
Publishing.
Soiguine, A. M. (1996). Complex Conjugation - Relative to What? In Clifford Algebras with Numeric
and Symbolic Computations (pp. 284-294). Boston: Birkhauser.
