extended logical negation should incorporate classical and quantum features
equivalently.
Summarizing, one finds that the classical contradiction or paradox related to the
Gödel proposition, choosing P ¼ G, where G is Gödel’s famous arithmetical
proposition so constructed that neither G nor :G are provable within the given set
of axioms, is formulated as a well-defined singularity in the extended logical
description presented above. Hence (4.16) tells that simultaneously G is not true
and :G is not false neither G nor :G are false. Nevertheless classical truth-values,
being undecidable, will not reveal any singular behaviour since B
2
¼ 0, while the
canonical vectors in (4.14) or (4.14′) replicates inclusive uncertainty with respect to
the input information.
The classical inconsistency or paradox has been represented by a matrix
degeneracy, reproducing the truth functional proposition calculus by “quantumlike” states, and allowing the case p = ½ to be expressed as a quantum transition
between truth and falsity. Of course Gödel used the terminology “provable” and
“not provable”, which is vital in the logical proofs, but nonetheless will not
essentially influence our present argument.
Finally one can directly compare (4.8) and (4.13) or (4.8′) and (4.13′) translating
the bias matrix B (or B
0 ) into Eqs. (3.6) and (3.7) displaying a precise relation with
the theory of general gravity, identifying pðrÞ ¼ ð1 À jðrÞÞ.
3 The self-referential
aspects of the present argument translates into a fundamental law, which will
advance important generalizations to biotic organizations in complex enough systems giving way to biological order and communication. In particular one notes that
decoherence into a quantum or classical state is impossible due to Gödel’s inconsistency theorem, here portraying the paradox as a crossing phenomenon, e.g. as a
genuine bifurcation, authentically prohibiting or code forbidding any state collapse.
5 Non-Hermitian Quantum Mechanics
The scope of this presentation will not permit portrayals of all the microscopically
relevant formulations that can be deduced from the general operator array algebra of
previous sections, see Ref. [15] for more details. Instead we will focus on what is
essential in order to appreciate the consequences of the arguments needed to formulate ZEUS, the Zero-Energy Universe Scenario, as an all-inclusive concept that
in addition incorporates the paradigm of evolution.
As is well known it is usually resolved that the many-body Schrödinger equation
in particular and quantum mechanics in general portray reality to unmatched perfection. This anticipation acquiesced Löwdin [16] to establish the foundation of a
new journal, the International Journal of Quantum Chemistry. With the emergence
3 Note that the probability function/operator p in this paragraph should not be confused with the
absolute value of the momentum variable of previous sections.
A Zero Energy Universe Scenario: From Unstable Chemical …
259
equivalently.
Summarizing, one finds that the classical contradiction or paradox related to the
Gödel proposition, choosing P ¼ G, where G is Gödel’s famous arithmetical
proposition so constructed that neither G nor :G are provable within the given set
of axioms, is formulated as a well-defined singularity in the extended logical
description presented above. Hence (4.16) tells that simultaneously G is not true
and :G is not false neither G nor :G are false. Nevertheless classical truth-values,
being undecidable, will not reveal any singular behaviour since B
2
¼ 0, while the
canonical vectors in (4.14) or (4.14′) replicates inclusive uncertainty with respect to
the input information.
The classical inconsistency or paradox has been represented by a matrix
degeneracy, reproducing the truth functional proposition calculus by “quantumlike” states, and allowing the case p = ½ to be expressed as a quantum transition
between truth and falsity. Of course Gödel used the terminology “provable” and
“not provable”, which is vital in the logical proofs, but nonetheless will not
essentially influence our present argument.
Finally one can directly compare (4.8) and (4.13) or (4.8′) and (4.13′) translating
the bias matrix B (or B
0 ) into Eqs. (3.6) and (3.7) displaying a precise relation with
the theory of general gravity, identifying pðrÞ ¼ ð1 À jðrÞÞ.
3 The self-referential
aspects of the present argument translates into a fundamental law, which will
advance important generalizations to biotic organizations in complex enough systems giving way to biological order and communication. In particular one notes that
decoherence into a quantum or classical state is impossible due to Gödel’s inconsistency theorem, here portraying the paradox as a crossing phenomenon, e.g. as a
genuine bifurcation, authentically prohibiting or code forbidding any state collapse.
5 Non-Hermitian Quantum Mechanics
The scope of this presentation will not permit portrayals of all the microscopically
relevant formulations that can be deduced from the general operator array algebra of
previous sections, see Ref. [15] for more details. Instead we will focus on what is
essential in order to appreciate the consequences of the arguments needed to formulate ZEUS, the Zero-Energy Universe Scenario, as an all-inclusive concept that
in addition incorporates the paradigm of evolution.
As is well known it is usually resolved that the many-body Schrödinger equation
in particular and quantum mechanics in general portray reality to unmatched perfection. This anticipation acquiesced Löwdin [16] to establish the foundation of a
new journal, the International Journal of Quantum Chemistry. With the emergence
3 Note that the probability function/operator p in this paragraph should not be confused with the
absolute value of the momentum variable of previous sections.
A Zero Energy Universe Scenario: From Unstable Chemical …
259