# An Investigation on the Basic Conceptual Foundations of Quantum Mechanics by Using the Clifford Algebra

*Advanced Studies in Theoretical Physics*5 (11):485-544 (2011)

**Abstract**

We review our approach to quantum mechanics adding also some new interesting
results. We start by giving proof of two important theorems on the existence of
the A(Si) and i,±1 N Clifford algebras. This last algebra gives proof of the von
Neumann basic postulates on the quantum measurement explaining thus in an
algebraic manner the wave function collapse postulated in standard quantum
theory. In this manner we reach the objective to expose a self-consistent version
of quantum mechanics. In detail we realize a bare bone skeleton of quantum
mechanics recovering all the basic foundations of this theory on an algebraic
framework. We give proof of the quantum like Heisenberg uncertainty relations
using only the basic support of the Clifford algebra. In addition we demonstrate
the well known phenomenon of quantum Mach Zender interference using the
same algebraic framework, as well as we give algebraic proof of quantum
collapse in some cases of physical interest by direct application of the theorem
that we derive to elaborate the i,±1 N algebra. We also discuss the problem of time
evolution of quantum systems as well as the changes in space location, in
momentum and the linked invariance principles. We are also able to re-derive the
basic wave function of standard quantum mechanics by using only the Clifford
algebraic approach. In this manner we obtain a full exposition of standard
quantum mechanics using only the basic axioms of Clifford algebra.
We also discuss more advanced features of quantum mechanics. In detail, we give
demonstration of the Kocken-Specher theorem, and also we give an algebraic
formulation and explanation of the EPR paradox only using the Clifford algebra.
By using the same approach we also derive Bell inequalities. Our formulation is
strongly based on the use of idempotents that are contained in Clifford algebra.
Their counterpart in quantum mechanics is represented by the projection operators
that, as it is well known, are interpreted as logical statements, following the basic
von Neumann results. Von Neumann realized a matrix logic on the basis of
quantum mechanics. Using the Clifford algebra we are able to invert such result.
According to the results previously obtained by Orlov in 1994, we are able to give
proof that quantum mechanics derives from logic. We show that indeterminism
and quantum interference have their origin in the logic. Therefore, it seems that
we may conclude that quantum mechanics, as it appears when investigated by the
Clifford algebra, is a two-faced theory in the sense that it looks from one side to
“matter per se”, thus to objects but simultaneously also to conceptual entities.
We advance the basic conclusion of the paper: There are stages of our reality in
which we no more can separate the logic ( and thus cognition and thus
conceptual entity) from the features of “matter per se”. In quantum mechanics
the logic, and thus the cognition and thus the conceptual entity-cognitive
performance, assume the same importance as the features of what is being
described. We are at levels of reality in which the truths of logical statements
about dynamic variables become dynamic variables themselves so that a
profound link is established from its starting in this theory between physics and
conceptual entities.
Finally, in this approach there is not an absolute definition of logical truths.
Transformations , and thus … “redefinitions”…. of truth values are permitted
in such scheme as well as the well established invariance principles, clearly
indicate

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