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hiGlobal Journal of Science Frontier Research: F
Mathematics and Decision Sciences
Volume 22 Issue 3 Version 1.0 Year 2022
Type : Double Blind Peer Reviewed International Research Journal
Publisher: Global Journals
Online ISSN: 2249-4626 & Print ISSN: 0975-5896
Explaining Some Weird Quantum Mechanical Features in
Geometric Algebra Formalism
By Alexander Soiguine
Abstract- The Geometric Algebra formalism opens the door to developing a theory replacing
conventional quantum mechanics. Generalizations, stemming from implementation of complex
numbers as geometrically feasible objects in three dimensions, followed by unambiguous
definition of states, observables, measurements, bring into reality clear explanations of some
weird quantum mechanical features, particularly, the results of double-slit experiments where
particles create diffraction patterns inherent to a wave, or modeling atoms as a kind of solar
system.
Keywords: geometric algebra, states, observables, measurements.
GJSFR-F Classification: DDC Code: 530.12 LCC Code: QC174.3
ExplainingSomeWeirdQuantumMechanicalFeaturesinGeometricAlgebraFormalism
Strictly as per the compliance and regulations of:
Explaining Some Weird Quantum
Mechanical Features in Geometric Algebra
Formalism
Abstract-
The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum
mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in
three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear
explanations of some weird quantum mechanical features, particularly, the results of double-slit experiments where
particles create diffraction patterns inherent to a wave, or modeling atoms as a kind of solar system.
Keywords:
geometric algebra, states, observables, measurements.
I.
Introduction. States, Observables, Measurements
Complementarity principle in physics says that a complete knowledge of phenomena on
atomic dimensions requires a description of both wave and particle properties. The
principle was announced in 1928 by the Danish physicist Niels Bohr. His statement was
that depending on the experimental arrangement, the behavior of such phenomena as
light and electrons is sometimes wavelike and sometimes particle-like and that it is
impossible to observe both the wave and particle aspects simultaneously.
In the following it will be shown that actual weirdness of all conventional quantum
mechanics comes from logical inconsistence of what is meant in basi
c definitions and
has nothing to do with the phenomena scale and the attached artificial complementarity
principle.
It will be explained below that theory should speak not about complementarity but about
perfect splitting of measurement process into the operator (“state” in confusing
conventional terminology, though “wave function is a little better) and the operand
(observable) components.
Unambiguous definition of states and observables, does not matter are we in “classical”
or “quantum” frame, should follow general paradigm, [1], [2], [3]:
- Measurement of observable by state2is a map:
,
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Author: e-mail: alex@soiguine.com
a) General definitions
Notes
Alexander Soiguine
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Explaining Some Weird Quantum Mechanical Features in Geometric Algebra Formalism
where is an element of the set of observables. is element of,
generally though not necessarily, another set, set of states.
- The result (value) of a measurement of observable by state is a map
sequence: ,
where is a set of (Boolean) algebra subsets identifying possible results of
measurements.
Thus, state and observable are different things. Evolution of state should be considered
separately, and then action of modified state will be applied to observable in
measurement.
The importance of the above definitions becomes obvious even from trivial examples.
Take a point moving along straight line. The definitions are pictured as (see Fig.1.1):
States, observables, measurements on straight line
In this classical kinematic example, it does not formally matter do we consider evolution
of “state” or of “measurement of observable by the state” or of “the result of
measurement” because they differ only by an additive constant or the map of one-
dimensional vector to its length.
The above one-dimensional situation radically changes if the process entities become
belonging to a plane, that’s dimensionality of physical process increases, though we
continue watching results in one dimensional projection (see Fig.1.2):
2One should say “by a state”. State is operator acting on observable.
b) Classical kinematic illustration
Fig. 1.1:
Notes
Explaining Some Weird Quantum Mechanical Features in Geometric Algebra Formalism
States, observables, measurements on plane, projected on straight line
In a not deterministic evolution, the central point of randomness of observed values is
the fact that their probabilities are associated with partition of the space of states. Each
partition element is fiber (level set)3of each of the observable value under the action of
the state on observable. Probabilities are (relative) measures of those fibers (see
Fig.1.3):
Probabilities as measures of partition elements
The option to expand, to lift the space where physical processes are considered, may
have critical consequence to a theory. A kind of expanding is the core of the suggested
formulation aimed at the theory deeper than conventional quantum mechanics.
A theory that is an alternative to conventional quantum mechanics has been under
development for a while, see, [1], [2], [4], [5].
Its novel features are:
-
Replacing complex numbers by elements of even subalgebra of geometric
algebra in three dimensions, that’s by elements of the form “scalar plus bivector”.
- Elementary physical objects follow the structure: position in space plus explicitly
defined object as the 𝐺3, geometric algebra in three dimensions, elements.
3Recall that fiber of a point
y
in
Y
under a function
YXf →:
is the inverse image of
}{y
under
f
:
( )
})(:
{}{
1yxfXxyf ==
−
Probability to get result of measurement in interval
dr around r(making no sense to say “find system in
state r” as in conventional quantum mechanics) is
the integral of probability density of states over the
strip ds.
- Operators acting on those objects are identified as direct sums of position
translation and points on the three-sphere 𝕊3defining rotations. Those points are
connected, due to hedgehog theorem, by parallel (Clifford) translations.
- Evolution of the 𝕊3part of operators by Clifford translations is governed by
generalization of the Schrodinger equation with unit bivectors in three dimensions
instead of formal imaginary unit.
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Fig. 1.2:
Fig. 1.3:
II. Workingwith G-Qubits Insteadof Qubits
Notes
Explaining Some Weird Quantum Mechanical Features in Geometric Algebra Formalism
In the following the part of the operators will only be considered.
Qubits, identifying states in conventional quantum mechanics, mathematically are
elements of the two-dimensional complex spaces:
conditioned by , that is unit value elements of .
Imaginary unit is used formally with the property . In another accepted
notations a qubit is:
In the suggested formalism complex numbers are replaced with elements of even
subalgebra of – geometric algebra in three dimensions.
Even subalgebra is subalgebra of elements of the form where and
are (real)4scalars and is some unit bivector arbitrary placed in three-dimensional
space. Elements of can be depict as in Fig. 2.1.
An element of
Unit value elements of , when , will be called g-qubits. The wave
functions (states in the suggested approach) implemented as g-qubits store much more
information than qubits, see Fig 2.2.
4In the current formalism scalars can only be real numbers. “Complex” scalars make no sense anymore,
see, for example, [2],[5].
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Fig. 2.1:
Notes
Geomectrically picted qubits and g-qubits
Take right - hand screw oriented basis of unit value bivectors, with the
multiplication rules , , , (or
equivalently ), where is oriented unit value volume, pseudoscalar, in three
dimensions, see Fig.3.1.
Basis of bivectors, dual vectors and unit value pseudoscalar
The quantum mechanical qubit state, , is linear combination of two
basis states and . In the terms these two states correspond to the following
classes of equivalence in , depending particularly on which basis bivector is selected
as complex plane:
•If is taken as complex plane, then
- State has fiber (level set) of the elements (0-type states):
,
- State has fiber of the elements (1-type states):
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,
Fig. 2.2:
Fig. 3.1:
Notes
III. Lift of Qubits to G-Qubits
a) Lift of quantum mechanical qubit states to g-qubits
•If is taken as complex plane, then
- State has fiber (level set) of the elements (0-type states):
,
- State has fiber of the elements (1-type states):
,
•If is taken as complex plane, then
- State has fiber (level set) of the elements (0-type states):
,
- State has fiber of the elements (1-type states):
,
General definition of measurement in the suggested approach is based on:
- the set of observables, particularly elements of ,
- the set of states, normalized elements of , g-qubits,
- special case of measurement of a observable by g-
qubit (wave function) is defined as
with the result:
(3.1)
Since g-qubit (state, wave function) is normalized, the measurement can be written in
exponential form:
where .
The lift from to needs a reference 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 .
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b) Implementation of definitions 1.1 in the g-qubit state case
Notes
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 to in the down mapping from to
. Thus, we have clear explanation of common quantum mechanics wisdom
on “probability of finding system in state ”.
Similar calculations explain correspondence of to in the qubit
when the component in measurement just got flipped.
Any arbitrary state can be rewritten either as 0-
type state or 1-type state:
,
where
, 0-type,
or
,
where
, 1-type.
All that means that any state measuring observable
does not change the observable projection onto plane of
and just flips the observable projection onto plane
.
Measurement of observable by a state is defined as . Evolution of a state
is its movement on surface of .
Consider necessary formalism.
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IV. Evolution of G-Qubit States
Notes
Multiplication of two geometric algebra exponents reads, see Sec.1.2 of [5]:
It follows from the formula for bivector multiplication:
with vectors to which the unit bivectors and are duals: , .
In the current case , , ,,
and we get above formula for .
The product of two exponents is again an exponent, because generally
and , see Sec.1.3 of [5].
Multiplication of an exponent by another exponent is often called Clifford translation.
Using the term translation follows from the fact that Clifford translation does not change
distances between the exponents it acts upon when we identify exponents as points on
unit sphere :
This result follows again from :
Assume the angle in Clifford translation is a variable one. Then in the case
:
If is dual to some unit vector , (this is the case of the matrix Hamiltonian
map to ,see [3] ), then and
that is obviously Geometric Algebra generalization of the Schrodinger equation.
If vector varies in time we get, assuming :
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Notes
with, generally,
.
Assume again constant and its unit length, . We see that displacement with
along big circle, intersection of the unit sphere by plane , rotates
lying on by angle in that plane.
Let us take two planes orthogonal to the plane of and comprising right-hand screw
with it: and . Right-handedness means:
,
and
(See the earlier definition of the right- hand oriented triple of basis bivectors.) Then the
three above formulas mean that the planes and rotate synchronically with
, correspondingly in planes and . Thus, the triple of planes ro tates as
solid while moving along big circle on .
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 in consists of all elements
with an arbitrary triple of orthonormal bivectors satisfying
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 fibers and .
Let we have a smooth curve
,
,
connecting points
and
,
on three-dimensional sphere such that and
.
The easiest way to define parallel transport is
.
For convenience purposes let us write and as exponents:
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V. Double-Slit Experiment
Notes
,
where
,
.
Angle is not uniquely defined since it can be any of
, ,
where
is, by definition, taken from interval . The angle
will be
denoted as .
,
where ,
.
As above, , The angle will be denoted as
.
Measurement of an observable
,
where ,
,
,
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Notes
by the wave function is:
Measurement by is:
Measurement by any intermediate parallel transport wave function
then reads:
Let us make natural for double slit 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 is when and when .
Consider the following simplified scenario.
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 angle
and is rotated in by angle .
Without loss of generality suppose that the angles and are equal by values
but opposite in sign: , ,
Then it follows that in Clifford translations the projection rotates in additionally
by , , , and projection rotates in
additionally by , , .
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Notes
Explaining Some Weird Quantum Mechanical Features in Geometric Algebra Formalism
Thus, in addition to and , we get infinite number of copies of
and multiplied every time by and separated by along the big circle of
intersection of plane with the sphere , see Fig. 5.1.
Multiple results of measurements when both slits get opened
Let the state has the Hamiltonian type form:
(6.1)
where is vector in three dimensions. An observable it will act upon is something of
a torsion kind, . Thus, at instant of time we have the following result of action of
state (6.1):
(6.2)
The Hamiltonian type wave function (6.1) bears its origin from proton, while the
observable represents electron.
The geometric algebra existence of the hydrogen atom can only follow from stable
sequence of measurement results (6.2) with appropriate combination(s) of and .
Let
Then
, bivector part of (6.1) is
and the scalar part of the wave function (6.1) is .
If initial bivector plane of observable is , , scalar part
then is , thus .
-
-
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Fig. 5.1:
VI. Model of Hydrogen Atom
Notes
Let us denote the plane
. Then the sequence of transformations (6.2)
reads:
If and assuming that does not depend on time. we get:
Angular velocity should be synchronized with Hamiltonian rotation by 5, though it
can be integer times greater than .
Now assume that . Thus, the result of (6.2) is:
The vector of length rotates in plane with angular velocity while element
rotates in plane . Again, for stability, angular velocity should be integer times
greater than .
Take the general formula (3.1) and substitute , ,
, , where are components of in the basis , and
, , are components of in the basis :
. The result of measurement after multiple transformations reads:
(6.3)
Formula (6.3) gives stable rotation of observable
(electron) due to action of the state
(proton.)
5Rotation by the double of the exponential is known from rotational rules in three-dimensional geometric
algebra, see, for example [3].
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Notes
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-slit experiment results with
diffraction patterns inherent to wave diffraction. This weirdness of the double - slit
experiment is milestone of all further difficulties in interpretation of conventional
quantum mechanics. The approach also allows elimination of the Bohr’s planetary
model of the hydrogen atom.
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VII. Conclusions
1. Soiguine, "What quantum "state" really is?," June 2014. [Online]. Available:
http://arxiv.org/abs/1406.3751.
2. A. Soiguine, "Geometric Algebra, Qubits, Geometric Evolution, and All That,"
January 2015. [Online]. Available: http://arxiv.org/abs/1502.02169.
3. A. Soiguine, Geometric Phase in Geometric Algebra Qubit Formalism,
Saarbrucken: LAMBERT Academic Publishing, 2015.
4. A. M. Soiguine, "Complex Conjugation -Relative to What?," in Clifford Algebras
with Numeric and Symbolic Computations, Boston, Birkhauser, 1996, pp. 284-294.
5. A. Soiguine, The Geometric Algebra Lift of Qubits and Beyond, LAMBERT
Academic Publishing, 2020.
Notes
References Références Referencias
