The vacuum as imaginary space. The unreasonable effectiveness of
complex numbers.
GIANFRANCO MINATI
Italian Systems Society http://www.airs.it
Via Pellegrino Rossi, 42
ITALY
http://www.gianfrancominati.net/
Abstract: - The background to the article is the classic and quantum understandings of the vacuum and the use of
imaginary numbers in quantum models. The purpose of the article is to outline the possible understanding of the
vacuum as imaginary space always coupled with the real space in the complex space of complex numbers. This
understanding relates to the duality real-potential, collapsed–collapsible, and superimpositions of waves-
phenomena as in quantum mechanics. The incomputability of the imaginary parts may represent the physical
meaning of the permanent potential pending nature of the vacuum. The presence of imaginary numbers in models
may be intended as warranty that it is not possible to compute definitive results, but it is possible to have pending
multiple equivalences and superimpositions as in quantum physics and emergent collective processes in
complexity. We consider how much the complexity (i.e., the study of emergence and chaos) may be considered
related to and represented by complex numbers (i.e., properties of their dual variables and their collapsibility in
real numbers). The usage of imaginary numbers may also be intended as the expression or manifestation of
something we do not understand yet, as it was for the indemonstrability of the fifth Euclidian postulate and the
unavailability of a distribution law for prime numbers. We conclude that a new global understanding is necessary
and capable of explaining what we understand as the unreasonable effectiveness of complex numbers.
Key-Words: - Collapse, Computability, Emergence, Field, Imaginary, Matter, Quantum, Vacuum.
Received: August 28, 2021. Revised: October 22, 2022. Accepted: November 27, 2022. Published: December 31, 2022.
1 Introduction
This article is finalized to consider concepts,
problems, characteristics, and particularly possible
representations of the vacuum in classical and
quantum physics having aspects of compatibility with
the complex space in mathematics using imaginary
numbers.
The classic naïve representation of the vacuum,
considered in section 2, reduces the vacuum to the
emptiness of something specific up to the possible
ideal lack of everything, which encounters conceptual
inconsistencies such as the problem of the boundaries
and coexistence between vacuum and non-vacuum.
In contrast with this supposed absolutely nothing, the
vacuum can be considered as relating to the non-
material presence of implicit potential properties
such as incompatibilities, metastability, and mutual
exclusivity of alternative properties such as the
uncertainty and complementary principles, remote
synchronization, and the theoretical incompleteness
necessary for processes of emergence. Emergence is
indeed intended, in short, to be a continuous,
irregular, undesigned, and unpredictable but also
coherent multiple processes of acquisition of
coherent, new -compared to those already owned-,
non-equivalent -that is, not linearly convertible one to
another- properties of complexity, such as
topological and behavioral (e.g., the properties
acquired by the climate system, collective behaviors
of flocks and swarms, and whirlpools) [1].
We consider the concept of vacuum domains as
having properties. Such properties are active
independently by the materiality observable: they are
pending properties ready to collapse/to decide
between equivalences such as in bifurcations
(changes in the topological structure of the system
and in the number or type of attractors due to small,
smooth changes in parameter values); to keep long-
range correlations and remote synchronizations
between elements without direct structural
connections or intermediate mediating entities; and in
quantum mechanics (QM) when the wave function
initially in a superposition of several eigenstates
collapses, it reduces to a single eigenstate as a
consequence of the interaction with the external
world. We may use the term materialize, and the
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vacuum is defined as the state with the lowest
possible energy, namely the zero-point energy (ZPE).
We consider the idea that under the usual naiveness
attributed in the classical understanding of the
vacuum, it may be possible to find problems,
characteristics, and representations allowing
conceptual forms of continuity between classic and
non-classic physics, considered here representable by
complex variables (see section 3) and relating to the
duality real-potential, collapsed–collapsible,
superimpositions of waves-phenomena, as in QM,
and imaginary-real.
One of these possible continuities considered in the
article relates to the representation of the vacuum as
imaginary space of complex variables connecting real
space with the usage of imaginary numbers in
quantum physics and the consideration of imaginary
time as in approaches to special relativity and QM.
Possible aspects of representations allowing
conceptual forms of continuity between classic and
non-classic physics may also be useful to represent
processes of emergence (i.e., the discontinuity
between acquired, non-equivalent properties) and
how emergence emerges [1, 2] and, in case, to
represent processes of phase transitions as considered
in section 4.
In section 5, we present the results and concluding
remarks on different issues such as how the presence
of imaginary numbers and their theoretical
incomputablity in models may be intended as
warranty that it is not possible to have objectivistic
final results, but it is possible to have multiple
pending multiple equivalences and superimpositions
as in quantum physics and emergent collective
processes in complexity. We may have
materialization only when there is collapse, as
represented when the imaginary numbers become
real through some computations and combinations.
We mention how the phenomenological collapse can
be represented but not reduced to computability,
intended only as the result of a computation, as in
correspondence with the similar incomputability of
processes of emergence.
We conclude by mentioning some related research
issues.
The purpose of the article is to outline the possible
understanding of the vacuum as imaginary space
always coupled with a non-imaginary space in the
complex space of complex numbers. The real space
as the real part of the dual complex space is composed
of the real and the unavoidable pending, implicit,
imaginary part representing the vacuum. This
representation relates to the duality real-potential,
collapsed–collapsible, and superimpositions of
waves-phenomena as in QM. The incomputability of
the imaginary part may represent the physical
meaning of the permanent potential pending nature of
the vacuum allowing superimpositions and potential
properties. Another purpose is to outline the
correspondence between quantum and emergence
processes, between pervasive vacuum and
environment. As a methodology we will start by
considering how the vacuum has been considered in
physics up to the modern quantum understanding.
Considerations on the reality of incomputable and
imaginary numbers follow. Finally, the vacuum is
considered as an imaginary space (not only
incomputable) in relation to the wide use of
imaginary numbers in quantum physics. This is
followed by a section of concluding remarks in which
resulting understandings and research issues are dealt
with.
2 A short overview of the classic
vacuum
In this opening section, we mention some of the
classic understanding of the vacuum.
The vacuum problem has been considered since
ancient times, as by Aristotle, Democritus, and
Leucippus.
The theme was then considered by Galilei (1564–
1642) and his school until the realization of
’Torricelli’s famous experiment in 1644.
We mention the corresponding idea of the existence
of the ether supposed as having the character of an
immovable material substance in absolute space. As
is well known, this idea was definitively refuted by
the experiments of J.C. Maxwell (1831–1879) and
A.A. Michelson (1852–1931).
We will then come to Rutherford (1871–1937) and
his famous experiment in 1909, relevant that the atom
is largely made of a vacuum and his introducing the
idea of matter as discontinuous.
We mention now some specific cases.
The vacuum represented by total absence is
conceptually fragile, raising questions such as
those related to the edges of vacuum, ignoring its
pervasiveness, and assuming and admitting its
possible locality, its observability, its relation with
implicit, potential, and pending properties (see
point 2.6) such as for metastability. How can the
vacuum coexist with the non-vacuum? There is
coexistence in first-order phase transitions (e.g.,
among phases such as liquid and gas phase of
matter), whereas there is not in second-order
phase transitions (e.g., among phases such as
ferromagnetic and magnetic). What coexistence,
if any, exists between the vacuum and non-
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vacuum? Where does one begin and the other end?
We should consider open intervals of the vacuum
and the non-vacuum, such as between irrational
numbers.
We mention the related concept of empty set
considered in mathematics. However, the concept
of the empty set suffers from paradoxes (e.g., the
empty set may contain another empty set and can
be introduced in different ways). For instance, by
using a semantic definition, a rule to determine
what the elements are and membership rules that
allow us to decide whether an element belongs to
the set or not (the rule can be even a list of possible
elements). Set theory may be defined by axioms;
however, according to ’Gödel’s incompleteness
theorems, it is not possible to demonstrate that
axiomatic set theory is free from paradox. The
empty set is usually denoted by the symbol
. It
is supposed to state that the sets we were
considering include an empty one of them. Then
we can use it in computations as the zero in
number theory. The empty set seems to be a
generic zero set unavoidably specified in any
collection of specific sets. It seems to make sense
if it is the zero of a collection of sets. The empty
set is both open and closed for any set and
topology. In fact, the empty set is open by
definition in any topological space because the
complement of an open set is closed. Moreover,
the closure of the empty set is empty.
In thermodynamics, in the nineteenth century it
was believed that a degree of temperature
corresponds to a well-defined amount of energy
supplied. It could reasonably be assumed that it
would always take the same amount of energy to
produce a one-degree change in temperature. The
German chemist W.H. Nernst (1864–1941),
thanks to the progress of cryogenics, in 1906
measured the specific heats of substances up to
temperatures of the order of 5 absolute degrees.
He noted that the specific heats, far from being
constant, became smaller and smaller as the
temperature decreased. Therefore, it had to be
explained how it was possible to make the same
energy leap (i.e., raising the temperature by one
degree) by supplying smaller and smaller
quantities of energy from the outside. It was
necessary to identify a physical subject, different
from the material bodies in question, capable of
supplying the missing supplement of energy.
Nernst affirmed that this subject was the vacuum,
not coinciding with nothing but intended as a
physical entity, not analyzable into atoms and not
separable from bodies, but, however, able to
influence the temperature.
In chemistry, we mention how subsequent
sequences of dilutions of chemical elements lead
to dilutions where the chemical initial element is
no longer present as stated by the Avogadro
(1776–1856) number. In relation to the diluted
chemical product, the vacuum, intended as the
total absence of the initial element, is reached. The
diluent passes from being gradually predominant
to being the only entity present and detectable.
The pervasiveness of the diluent corresponds
conceptually to that of the vacuum. “In chemistry,
the limit for high dilution is represented by
Avogadro’s number. However, there is an intense
debate about possible properties acquired by
elements diluted beyond Avogadro’s number,
studied by the physics of high dilutions” [3, pp.
342-349]. The topic, once accepted conceptually,
is explored by considering approaches of quantum
physics to explain the properties of high dilutions
of matter [4-6].
The appearance of the vacuum as a physical object
(i.e., a special unavoidable environment)
undermined the concept of isolated body at its
root, leading to the entanglement considered in
quantum physics. In this regard, we mention the
ideal correspondence between the vacuum and the
environment. “Regarding the separability of
systems from the environment, a simple example
of the inapplicability of this assumption is given
by ecosystems where the differentiation between
external and internal is unsuitable. In these cases,
the environment pervades the elements which
produce, in their turn, an active environment. This
environment, if we can still call it such, is active
and not an amorphous, abstract space-hosting
processes. It is interesting to consider eventual
conceptual correspondences with the quantum
vacuum pervading everything.” [7, p. 13]. This
description also corresponds to the concept of
multiple systems when the same components
establish at the same time different, superimposed
systems [7, pp. 161-170]. We also mention the
concept of systems propagation related, for
instance, to “synchronizations and remote
synchronizations occurring when nonadjacent
pairs of entities become substantially
synchronized in spite of the absence of direct
structural connections between them or
intermediate mediating entities such as in the
brain and networks [8, 9]; and those belonging to
the basin of an attractor.” [10]. Furthermore, we
may consider the exchange of information without
direct exchange between elements of collective
behaviors keeping coherences such as in swarms
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and flocks in long-range correlations [7, pp. 271-
272].
The Aristotelian horror vacui has long since been
abandoned, accepting the possible pervasive but also
local nature of the vacuum.
Another way to consider the vacuum is as an ideal
place of implicit potentialities ready to
materialize/collapse (see point 2.8) in non-
equivalent real material events. Examples are
given by meta-stabilities and phase transitions.
Furthermore, collective conditions of behavioral
adequacy, admissibility, compatibility,
equivalence, and interchangeability of agents
make possible, induce, and facilitate long-range
correlations and the emergence of coherent
collective behaviors. The occurrence of such
collective conditions establishes immaterial
domains influencing any entering entities and
consisting of real dynamic degrees of freedom,
however immaterially prescribed such as in
ecosystems and remote synchronization. We may
consider these domains as properties of active
vacuums, pervading the collective system. Pre-
established environmental conditions may be
considered as the implicit pre-existence of
domains, completely different, for instance, from
real, material fields of physics. This also relates to
singularities in catastrophe theory, high sensitivity
to initial conditions, and change of attractors in
chaos theory. The entering by an entity in these
vacuum domains, implicitly full of unlimited but
specific possibilities allowed by degrees of
freedom and constraints, involves it being
significantly affected. In short, we consider
vacuum immaterial domains [10].
It is possible to deduce and suppose the existence
of the vacuum as a strange physical entity [11] that
is difficult to detect and measure but nevertheless
indisputable as it is for quantum physics. In some
approaches, it seems that the vacuum is
considered as a physical entity lacking
measurability but allowed to have quasi-
localization (such as quasi-particles), no
significant edges, and low detectability and being
constant over time. We may consider issues such
as the local and general percentages of vacuum
versus the non-vacuum, the variability and the
changing nature of the balance and ways to change
it, the possible invariance of the vacuum or the
replacing dynamics between vacuum and non-
vacuum, the possibility of transferring the
vacuum, the possibility of differentiating between
vacuums, and how the vacuum takes place.
Should we consider only one kind of vacuum, that
is, are all vacuums equivalent? Is it possible to
differentiate and transform one kind of vacuum
from another? Is it possible to consider the quasi-
vacuum, a situation of unstable dynamics between
vacuum and non-vacuum?
As we know, most answers have come from quantum
physics.
We conclude this section by mentioning the concept
of collapse of the vacuum, that is, in short, the shift
from implicit, potential, pre-existing domains, pre-
phase transitions, configurations, and metastable
states into detectable and measurable effects. This is
in conceptual correspondence, in quantum physics,
with conceptual collapse, for instance, of a wave—in
a superposition of several eigenstates (possible values
of the observable)—that then collapses, reduces to a
single eigenstate. About the observable, we mention
that in classical physics, almost any quantity may be
considered observable (e.g., energy, mass, and
momentum), starting from the introduction of
electromagnetism the obviousness of the situation
changed. As a matter of fact, quantities considered in
electromagnetism (e.g., fields and potentials) are not
directly measurable. With the introduction of QM,
the concept of observable is further fuzzified because,
over and above the conceptual measurement limits
imposed by Heisenberg’s uncertainty principle, some
fundamental quantities introduced by QM are not
only not observable but are not even real quantities
and are described using complex numbers. The idea
is that matter/particles can be intended as excited
states of the underlying quantum vacuum [12].
Furthermore, the properties of matter can be intended
as vacuum fluctuations arising from interactions of
the zero-point field [13] and macroscopic
manifestations of quantum field theory (QFT) [14],
which differently from QM (see section 4.2)
considers particles as excited states of their
underlying quantum fields as in statistical field
theory. Furthermore, a quantum system and the
external environment may interact in such a way as
to destroy the quantum coherence. When the
decoherence (by which exposure to and entanglement
with any macroscopic environment converts quantum
information into classical information) time is short
enough, macroscopic coherence due only to QM
becomes unobservable. The states that are a
superposition of basic states can no longer exist,
because the interaction with the environment selects,
decides one particular basic state among all the
various possible ones, and then the system falls into
it with a probability equal to 1, and its dynamics loses
its quantum character. This limits the effectiveness of
QM to specific cases (e.g., the world of atoms and
molecules, very low temperatures, etc.) [15, pp 230-
239].
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3 On the reality of incomputable and
imaginary numbers
Reality in mathematics is mostly understood as
effective computability. Effective computable
numbers are intended to be real numbers [16] that can
be computed by a Turing machine terminating in a
finite amount of time and with arbitrary precision,
where the Turing machine is specified as a quadruple
T = (Q, Σ, s, δ) where Q is a finite set of states qi; Σ
is a finite set of symbols sj, (e.g., an alphabet); s is the
initial state s ⸦ Q, being Q the set of all the possible
states; and δ is a transition function that determines
the next acquired state occurring from computation
state qi to computation state qi+1 in finite time and
with arbitrary finite precision. Different versions of
the Turing machine are all computationally
equivalent [17].
3.1 Incomputability
Real numbers that are incomputable, due to the
unavailability of an algorithm that computes in finite
time and with arbitrary precision, include algebraical
irrational, trigonometric (based on Euler’s formula),
and transcendent numbers, endless processes of
convergence, and some special numbers, such as the
so-called ‘Chaitin omega number’ [18]. Here, we
term this incomputability as non-Turing
computability.
Furthermore, there are forms of non-explicit
computability, that is non-symbolic computability,
such as for artificial neural networks (ANN) and
cellular automata (CA) performing emergent
computation [19] and dealing with the reality of
incomputable real numbers [20].
We consider here the incomputability of imaginary
numbers as theoretical incomputability, for which no
resolutive procedure is conceivable/admissible, much
less a Turing machine. We may term this theoretical
incomputability as t-incomputability. This is matter
of numbers containing the imaginary unity i =
such as in physical equations, which are actually
anything but imaginary in the common sense. More
precisely,
In the case of non-Turing computability, the
existence of a hypothetical Turing machine is
admissible, but it is not effectively available
because at least one of the definitory requests
listed above is not satisfied (e.g., to end the
computation in finite time and with arbitrary finite
precision). This issue is related to irrational
numbers. The partial Turing computability is,
however, admitted, that is, admitting operative,
acceptable approximations occurring in finite time
(i.e., renouncing to arbitrary finite precision in
finite time).
In the case of non-explicit computability, the
computational processing is non-analytically
representable, and that is why it is called sub-
symbolic, whereas the computational process is
performed by an explicit computable algorithm.
The computational process’s whole set of weights
and levels used in ANN and transition rules used
in CA cannot be zipped [19] (i.e., analytically
represented into individual general formulae or
functions), being instead a dynamical
computational process to be subsequentially and
completely performed to reach the result.
In the case of t-incomputability, the existence of
whatsoever resolutive procedure is not
admissible, such as for i. However, the t-
incomputability also relates to physical
phenomena and the unpredictability,
indeterminism, and randomness of measurements
such as for ’Heisenberg’s uncertainty principle,
whose equations, coincidentally, use imaginary
numbers. This condition relates in general to
situations of quasi-ness and theoretical
incompleteness, incompletability, uniqueness,
and equivalences [3] when, for instance, a
collective system is not always a system, not
always the same system, and not only a system in
the dynamics of maintaining, losing, and resuming
variable levels of coherence.
3.2 Complex numbers
It is well known that every complex number, see
Figure 1, zi can be expressed in the form x + iy, where
x and y are real numbers. Each complex number can
be then represented by the couple (x, y) ⸦ R x R. C is
the set of complex numbers and is the plane R × R =
R2 equipped with complex addition and complex
multiplication making it the complex field.
The n-dimensional complex coordinate space is the
set of all ordered n-tuples of complex numbers. It is
denoted Cn, and is the n-fold Cartesian product of the
complex plane C with itself. Symbolically,
Cn = (z1, …, zn), where zi ⸦ C. (1)
In the Cartesian plane, the point (x, y) can also be
represented in polar coordinates such as the following
(x, y) = (r·cosθ, r·sinθ) (2)
where the module r and the phase θ are obtained from
the formulas
r = ; θ = arctan
. (3)
In mathematics, a first bridge between imaginary
numbers and their representations in the plane is
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allowed by the so-called Euler identity, which relates
e, i, π, 1, and 0:
ei π + 1 = 0. (4)
The geometric interpretation of the formula allows
complex numbers to be viewed as points in the plane.
Furthermore, Euler’s formula states that, for any real
number x, we have
eix = cos x + i sinx (5)
where e is the base of the natural logarithms, i is the
imaginary unit, and sine and cosine are trigonometric
functions.
C Complex numbers with the form x + iy, where x and y are
real numbers and i is the imaginary unit, imaginary solution
to the equation x2 = −1. C is not orderable.
Fig. 1. Complex numbers
We note that C and R have the same cardinality.
Therefore, a bijection f: C→R is possible.
We consider now the ‘collapse’ of complex numbers
into real numbers as the result of proper computations
prescribed by equations and formulas. Cases of such
collapsing, that is mutation of imaginary numbers
into real numbers, are given by
1. The squaring of complex variables, when (iy)k
with k = 2n, n real integer.
2. The effects of the conjugation of two complex
numbers, z and its conjugate . Let z= x + iy. Then
considering = x – iy, one is the conjugate of the
other. The equality of two conjugate complex
numbers x + iy = x – iy implies that iy = - iy (i.e.,
y = 0).
3. The effect given by the sum of two conjugate
complex numbers (x + iy) + (x – iy) = 2x.
4. The product between two conjugate complex
numbers (x + iy) · (x – iy) = x2 – i2y2 = x2 + y2.
A complex variable function is defined as a function
defined on a subset of the complex numbers having
values in this same set. In complex analysis, there is
the study of the theory of functions of several
complex variables.
We conclude this section by mentioning how the
collapsing of imaginary numbers into real numbers in
some way corresponds to the collapsing of some
Turing-incomputable real numbers into computable
ones. The latter is the case, for instance, for all non-
computable roots of numbers that are not exact
powers, that is, (
)2, where x is not a power of two,
as instead it is in the case, for instance, of
. Where
(
)2 and
are formally but not computationally
equivalent because (
)2 =
·
, as for
= 1.41421356237… ·
= 1.41421356237… =
1.99999999999… ≠
= 2.
Correspondingly, the computation of
is
impossible, and (
)2 is also impossible and
incomputable when x is not a power of two. When
considering this problem formally and not
computationally,
, then (
)2 = i2= -1.
Computation is mostly symbolic calculus that delays
as much as possible, as optional or after assigning
values to the variables, effective numerical
computation. In the case of imaginary numbers, this
last step is impossible/not feasible.
In analogy with the Schrödinger’s cat having
superimposed the states of being dead and alive,
computation may be like a metaphorical card game
that does not always end by seeing, but often, after
changing cards, by passing or raising. With
imaginary numbers, we can only play by passing,
raising, and never seeing. However, we still do play.
The possible mutation between incomputable, non-
Turing solvable to computable, for instance through
exponentiation, may be intended to have prevalent if
not only mathematical aspects, whereas the mutation
between imaginary to non-imaginary may be
intended to have a significant physical
meaning/interpretation due to its use in models and
physical equations, as mentioned below. In this
regard, we stress that we do not speak of the space of
non-Turing computable numbers, while we speak of
the space of t-incomputable complex numbers.
R Real numbers are not all algebraic (algebraic irrational
are obtained as solutions to polynomial equations with
integer coefficients, e.g., x2 – 2 = 0).
R = Q U I, R ⸦ C
U
Q Rational
numbers,
- expressible
as a ratio of
two integers,
that is, by
means of a
fraction.
I Irrational numbers,
- all and only algebraic,
non-periodic, unlimited,
decimal numbers,
therefore, not expressible
by means of a fraction;
- transcendent numbers are
irrational but not
algebraic numbers, i.e.,
they are not solutions to
any polynomial equation,
e.g., the Euler number e
and π.
Furthermore, irrational,
decimal unlimited numbers
are not computable (in a
finite amount of time).
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4 The vacuum as imaginary space
How can the imaginary space Cn be related to the
vacuum and the collapsing of Cn in Rn related to the
collapsing of the vacuum into detectable and
measurable effects?
In the quantum physics literature, the quantum
vacuum is intended as an entity that precedes matter,
so it must also precede space and time.
This situation is related to models of quantum physics
considering that it is the quantum vacuum giving
properties to matter, such as that of being always
connected, and not a lack of matter being the vacuum
[12, 13]. Imaginary, complex variables are regularly
used in quantum models (see below). However, it
seems there are no fully developed theoretical
reasons for this usage.
An initial possible generalizing idea may be to
represent the vacuum as a general t-incomputable
domain of possibilities specified by the imaginary
space, with complex variables, imaginary models
(i.e., models using imaginary variables), and
collapsing mechanisms that may be represented by
suitable symbolic collapse to turn imaginary numbers
into real ones (see point ‘a’ in section 5.1).
Furthermore, non-imaginary models may be intended
as particular complex models having the imaginary
part equal to zero.
As mentioned at the end of the previous section, we
focus here on incomputability as t-incomputability.
The t-incomputability identifies a space compatible
with research approaches considering/implementing
imaginary and non-imaginary models for the classic
vacuum, for instance imaginary attractors (see, for
instance [21-23]) considered in chaos theory; t-
incomputable dependence from initial conditions;
multiplicity of domains in ecosystems with effects of
systems propagation, remote synchronizations, and
long-range correlation; metastability; phenomena
represented with complex random probabilities;
conditions for the occurrence of symmetry-breaking
(when a symmetry transformation leaves invariant
the form of the evolution equations but changes the
form of their solutions) in phase transitions; and
complex variables as in statistics.
For instance, regarding the last case, complex random
variables considered in statistics and probability
theory generalize real-valued random variables to
complex numbers, that is, the possible values of
complex random variables may take complex
numbers. Complex random variables can then be
considered as pairs of real random variables
corresponding to their real and imaginary parts.
Accordingly, the distribution of one complex random
variable is intended as the joint distribution of two
real random variables. Complex random variables are
used in digital signal processing (e.g., biomedical,
and information theory) [24-26].
On the other hand, we may consider phenomena and
processes of complexity by which there is an
acquisition by systems of properties non-equivalent
to those already possessed such as in self-
organization and emergence (i.e., conceptual
mutations). We may ask how much the science of
complexity (i.e., the study of emergence and the
emergence of emergence) [1] and chaos [27] may be
considered to be related to and representable by
complex numbers (i.e., properties of their dual
variables and their collapsibility into real numbers)
[28-31]. How much is emergence non-Turing
computable and t-incomputable? Is complexity non-
Turing computable or t-incomputable? Probably, we
should consider a mix of possibilities, such as in the
occurrence of superconductivity and superfluidity,
transitions from paramagnetic to ferromagnetic
phases, and some order-disorder transitions, which
have a mix of very complicated transient dynamics
and classical and quantum aspects [32, 33].
4.1 The quantum case
In QM [34, 35], the crucial idea is of non-
commutativity, when the position and velocity of a
particle (at the subatomic scale) are non-commuting.
QM operates with manifolds of quantities, such as
matrices. As for imaginary models in this conceptual
context, we may refer to their large usage in quantum
physics.
As is well known, matrix mechanics is a formulation
of QM interpreting the physical properties of
particles as matrices that evolve in time. This is an
alternative to using usual scalar values and then
replacing the classic continuity with discretization
(i.e., possible admissible values).
In matrix mechanics intended as a formulation of QM
observables, when considering pairs of observables
an important quantity is the commutator. For
instance, for a pair of operators  and
, one defines
their commutator as
. (6)
In the case of position and momentum considered by
Heisenberg’s uncertainty principle, the commutator
is the canonical commutation relation using i:
= . (7)
We may figure out the physical meaning of the non-
commutativity when considering the effect of the
commutator on the position and
momentum eigenstates.
For instance, in a simplified, conceptual version of
(7),
PM-MP = , where (8)
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- P matrix – is the possible, admissible positions of
a particle;
- M matrix – is the possible, admissible momentum
of the particle, equal to its mass times its velocity
(PM ≠ MP because they are matrices);
- i – imaginary number,
;
- – is the Planck constant
.
Examples of other usages of i relate to imaginary time
considered in approaches, for instance, based on
special relativity.
We mention that in QM, the wave function Ψ (x, t) is
a complex variable function, as in the Schrödinger
equation using imaginary numbers.
Furthermore, there are several kinds of relationships
between traditional and non-traditional models (see
Table 1).
4.2 Quantum versus stochastic
Different approaches are available in the literature
dealing with the role of complex numbers in
modeling quantum versus stochastic processes, such
as [36]. In this regard, we mention the case of the so-
called Fokker–Planck equation (FPE), [37] initially
introduced for the statistical description of the
Brownian motion of a particle in a fluid [38]. The
FPE is a partial differential equation in the unknown
function P. Such an equation describes the time
evolution of probability in the presence of random
noise.We may consider P(x, t) as the probability of
transition of an element from an initial value x0,
present at time t = 0, to a value x at time t. This
represents the probability that the stochastic process
x(t) gives rise, at time t, just to the value x. It has been
shown that, if the stochastic process obeys the
stochastic differential equation:
dx/dt = g(x) + h(x)
(t) (9)
where
(t) is a Gaussian white noise process
(stationary stochastic process whose spectral power
density is constant over all frequencies considered for
mathematical convenience), then the associated
transition probability P (x, t) is represented by the so-
called Fokker-Planck equation (FPE) considered in [39;
7, pp. 273-279]:
t P = -
x [g(x) P] + (
2/2)
x {h(x)
x [h(x) P]} (10)
where:
•
t =
/
t,
x =
/
x;
• 2 is the noise intensity;
• The FPE is a partial differential equation in the
unknown function P.
Such equation describes the time evolution of
probability in the presence of random noise.
Furthermore, it has been observed that the FPE [15,
pp. 273-279; 39] has a form strongly resembling that
of the Schrödinger equation used in QM. It is well
known that the two can be made identical by
introducing an imaginary time given by = i·t into
the Schrödinger equation, which immediately
becomes a sort of FPE such as in [40, 41, pp. 252-
253] and in equations (42–43) in [42].
This transformation is a mathematical trick, but
physical interpretations are necessary. We mention
the possibility of considering as a physical
interpretation a hypothesis introduced in the 1970s as
in [37]. It has been mathematically shown how the
formalism of QM could be used to describe a system
using classical mechanics and embed it into a
stochastic, noisy background. The probabilistic
features of QM can be then intended as a consequence
of the fact that the ground state of the universe is
merely a noisy state, and this prevents the existence
of truly deterministic phenomena [15, pp. 279-281].
Furthermore, it has been demonstrated how a
stochastic many-body system could be modeled by
using the formalism of the so-called second
quantization [43], one of the main technical tools of
QFT. Due to the probabilistic features of QM and
QFT, the latter, contrarily to what happens in QM,
assumes that the main physical entities are fields (of
force) and not particles [7, p. 36; 11, pp. 230-239].
From this line of research arose the so-called
statistical field theory [44-47].
At this point, we notice the central probabilistic
character of —especially quantum— so-called
physical ‘laws’, when considering implicit, unaware,
but effective, approximations and interpolations [48].
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“Emergence could be intended as coming first, as a
property of pre-matter, of the vacuum. The quantum
vacuum could thus be intended as a kind of field of
potentialities ready to collapse but always pervasive
as are the probabilistic features of QM…” [7, p. 130].
5 Concluding remarks
Could the imaginary units introduced by René
Descartes (1596–1650) in the treatise La Dioptrique,
Les Météores, et La Géométrie first published in
1637, subsequently elaborated by mathematicians
such as Euler, Gauss, and Cardano (the first to
introduce complex numbers into algebra) represent
the conceptual and physical distance between the
classic and non-classic, quantum representations?
If i were not available, several equations would be
impossible to write and symbolically solve,
intractable, and meaningless.
When dealing with real, non-Turing computable
numbers, the strategy is to process them symbolically
and postpone as much as possible the computation
requiring necessary levels of acceptable
approximation.
This is not suitable for t-incomputable imaginary
numbers only symbolically treatable ones. Complex
numbers may then be partially computable, only for
their real part.
Going back to the vacuum, we may consider it as
imaginary space always coupled with the non-
imaginary space in the complex space of complex
numbers. In other words, the real space is a real part
of the dual complex space composed of real and
imaginary parts (particular cases occur when the
imaginary part is zero or when the real part is zero).
This is intendable related to the duality real-
potential, collapsed–collapsible, and
superimpositions of something such as
/waves/phenomena/planes/states/spaces] as in QM.
However, by considering the pervasiveness of the
vacuum in physical models, as introduced above, the
first possibility (i.e., imaginary part as zero) would be
considered unlikely.
Furthermore, this intrinsic duality could represent the
perennial, intrinsic state of implicit interaction and
emergence of the so-called ‘matter’ ready to collapse
into entities acquiring properties such as the mass
(mass intended as vacuum collapsed). This relates,
for instance, to the fact that in QFT [49]
“It is the quantum vacuum giving properties to
matter, such as that of being always connected, and
not a lack of matter being the vacuum. The approach
based on considering material entities as fields (of
force) and not as particles has a long tradition in
physics, from Faraday and Maxwell, and onwards to
general relativity. Within this conceptual framework,
the concept of particle is considered to denote regions
of space where a field has a particularly high
intensity. The subject of such matter considered as a
condensation of emergent properties acquired by the
quantum vacuum will be considered below. Higher
levels of emergence acquire properties, …, such as
dimensionality, weight, volume and mass.” [7, p. 54].
Results as the ones mentioned above support the idea that,
from a formal point of view involving or not imaginary
numbers and complex variables, a number of stochastic
models could be suitably reformulated in such a way to
transform in QFT-based models (see below for the
difference between QM and QFT). Moreover, this requires
to redefine in a suitable way the Planck constant of the
system. Once the presence of the three fundamentals
ingredients, i.e., non-linearity, spatial extension and
fluctuations, for the occurring of radical emergence, i.e.,
when the new system’s phase requires a new description
level for its behaviors, such as for the protein folding,
acquisition of superconductivity, and superfluidity, has been
granted, then the theories can be found equivalent to one
another, at least with regard to their formal structure. This
possibly considering imaginary variables such as imaginary
time, imaginary mass and imaginary dimension.
Accordingly, a non-ideal model endowed with noisy
fluctuations, should have good probabilities of being
equivalent to a QFT model, without the need for quantizing
it [7, pp. 245-279].
Ideal models
characterized by a top-down
structure, based on general
assumptions assumed to be
largely valid and then covering
the widest possible spectrum of
phenomena. This feature
allows to deduce
particular consequences and to
forecast only if suitable
mathematical tools are
available.
Non-ideal
characterized by
a bottom-up
approaches,
based on opposite
assumptions
considering
‘lucky’ choices
and studied
through computer
simulations.
Examples
Examples
Chaos
Dissipative
structures
Noise-induced phase
transitions
Cellular
Automata
Spontaneous Symmetry
Breaking in Quantum
Field Theory
Agent-based
models
Network Science (ideal scale-
free networks)
Artificial Life
We distinguish between
Homogeneity-based models neglecting any
differences between the components and
treat them all being equivalent to one another
Heterogeneity-based models when
components are distinguishable
Table. 1 - Some hints at the relationship between
traditional and non-traditional, ideal and non-ideal
models.
We conclude this section by summing up how
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Based on the discussion above, we conclude that it is
possible to consider the usage of the mathematics of
complex variables not only in (typically quantum)
physics but also to consider and represent the implicit
properties of matter and the dualities between classic
and non-classic approaches.
5.1 Resulting understandings
It is interesting to compare the possible physical
meanings of the two corresponding collapsing
processes from imaginary, t-incomputable to real and
from non-Turing computable to Turing computable.
We considered how these cases occur particularly in
equations of quantum physics. The symmetric
reverse processes (i.e., from real to t-incomputable
imaginary and from computable to non-Turing
computable) seem less promising of physical
significance but have prevalent computational
aspects instead.
- The presence of imaginary numbers and their t-
incomputablity in models may be intended as
guarantee that it is not possible to see (as in an
imaginary card game with Nature), but it is
possible to have pending multiple equivalences
and superimpositions as in quantum physics and
emergent collective processes in complexity. We
can see only when there is collapse (see the
following point 5.1.2), as represented in simple
cases when the imaginary number becomes real
through some computations and combinations
such as in the occurrences 1–4 considered in
section 3.
- We consider the phenomenological nature of the
quantum collapsing of a wave function as a
superposition of several eigenstates, reducing to a
single one as an effect of the interaction with the
external world. We say phenomenological only
ideally computable, a contrast that is, incidentally,
also valid for the occurrence of processes of
emergence. The occurrence of phenomenological
incomputability is mostly represented by the t-
incomputablity of models using imaginary
numbers. On the other hand, the
phenomenological collapsing is represented,
corresponding even if not modeled by the
reducing of t-incomputabilities into computability
due, for instance, to the occurrence of
computational combinations with complex
random variables when modeling
unpredictability, indeterminism, and randomness
of measurements of physical phenomena [50]. A
similar situation occurs with processes of
emergence intrinsically, theoretically
incomputable and requiring incompleteness until
interactions with the environment lead to the
collapse of incompleteness, understood as
equivalences, to the establishment of
configurations endowed with variable and
dynamic coherences [1, 7, 19]. However, the
incomputability considered does not affect the
possibilities of simulation, at a level of description
under the responsibility of the researcher,
considering parametrically definable cases and
configurations.
5.2 Research issues
Interdisciplinary research could occur within the
same discipline (i.e., within physics) allowing
interchanged usage of approaches, redefinitions of
variables, meanings of constants, elaborate analogies,
and correspondences between non-quantum and
quantum modeling as considered in Table 1. This
could contribute to approaches and concepts allowing
forms of unification resulting in a unified, more
general theory. However, multiple, non-equivalent
superimposed representations is a feature of QM, as
in the duality real-potential and collapsed–
collapsible, conceptually transposable to complexity
for processes of emergence and multiple, pending, or
actual systems of ecosystems.
It may be interesting to research the complexity of
models of classical physics generalizable through the
usage of complex variables as in the cases mentioned
and for the FPE. We may research how much the
complexity (i.e., the study of emergence and the
emergence of emergence and chaos) may be
considered as being related to and representable by
complex numbers (i.e., properties of their dual
variables and their collapsibility in real numbers).
How much emergence is non-Turing computable
and/or t-incomputable?
Properties of complex numbers and of mechanisms of
mutation in real numbers may be intended to
conceptually correspond closely to models of
collapse in physics. The mathematical trick often
used to replace variables with i, as for the FPE,
considering complex variables and imaginary time,
should be given physical interpretations, probably all
related to probability and superimposed dualities
intrinsically open to collapse and the impossibility to
see.
- However, even the possible or limited possibility
of finding physical meanings should be
interpreted (i.e., it is curious that the issue comes
after modeling, as result, and not before as an
assumption: we do not start by considering
complex numbers).
- We may also consider the Turing-incomputability
of real numbers [51], as such or of real parts of
complex numbers, and the t-incomputability as
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representations of quasi-ness, continuous
negotiation between extremes, and of theoretical
incompleteness, as necessary ingredients for the
occurrence of equivalences to be collapsed in
emergence and complexity, as in [3, 7, 52].
However, it is always a matter of the usage of
incomputable, real, or complex results.
Paraphrasing ‘the unreasonable effectiveness of
mathematics’ introduced by Wigner [53] and
elaborated, for instance, in [54], we may consider the
unreasonable effectiveness of complex numbers. In
this regard, we mention how difficult it usually is to
model complex phenomena. In some cases, as for the
usage of imaginary, complex numbers, the case
seems to have reverse aspects. We use imaginary,
complex numbers to modify models, sometimes as a
mathematical trick, or we accept models implying
imaginary, complex numbers, and then we look for
the physical meaning.
- It may be intended as if the usage of imaginary,
complex numbers is not only guarantee that it is
not possible to compute definitive results, having
pending multiple equivalences and
superimpositions only phenomenologically
solvable as in quantum physics and emergent
collective processes in complexity, but also the
expression, manifestation of something we do not
understand yet. It may be related to the
impossibility in mathematics to demonstrate
inconsistent issues however initially assumed to
be admissible, if not evident. Such a situation is
considered as evidence of an intrinsic
consistency/logical robustness, contrasting with
the issues assumed to be evident but, however,
resistant to any attempt of demonstration. In such
cases, the issue to be considered is the
impossibility to demonstrate the supposed
evident, as signal indeed that something
unexpected must be realized, incompatible with
the demonstration pursued. There are also
significant methodological aspects. The typical
example is the impossibility to demonstrate that in
Euclidian geometry only one line parallel to
another one passes through the same point (the
fifth postulate). Such an impossibility opened the
doors to non-Euclidian geometry revealing that
what was supposed to be evident and then
provable was inconsistent/incoherent.
Another case is the impossibility of finding a
complete general explicit distribution law of prime
numbers, only asymptotically approximated in the
prime number theorem (PNT) proved independently
by Hadamard and Poussin in 1896 using ideas
introduced by Riemann, that is, his zeta function.
Such impossibility has been considered when
considering what is computable/decidable and what
is not. Indeed, the possibility of finding such a
distribution was intended to be equivalent to the
availability of an impossible algorithm…
“…able to compute the general properties of the
presumed primes’ distribution law without computing
such distribution. The link between the conceptual
availability of a distribution law of primes and
decidability is given by considering how to decide
whether a number is prime without computing.
Factorial properties of numbers, such as their
property of primality, require their factorization (or
equivalent, e.g., the sieves), that is, effective
computing. However, we have factorization
techniques available, but there are no (non-quantum)
known algorithms that can effectively factor arbitrary
large integers. Then, factorization is undecidable. We
consider the theoretical unavailability of a
distribution law for factorial properties, as being
prime, equivalent to its non-
computability/undecidability.” [55].
The solvability of the problem of finding the
hypothetical law of a distribution of prime numbers
is then considered as undecidable (i.e., non-Turing
computable) and equivalent to the availability of a
general algorithm of factorization. When the numbers
are sufficiently large, no efficient integer
factorization algorithm is known. Such non-efficiency
in the face of unlimited large numbers is then de facto
equivalent to the non-Turing computability, because
it admits computability in finite but, however, non-
limited time. This actually rules out effective
computability even though it has not been proven that
such an efficient integer factorization algorithm does
not exist.
However, there are new approaches and results that
may allow us to consider the problem from new
points of view such as by considering the existence of
bounded gaps between primes [56] proving that
lim (pn+1 − pn) < 7 × 107, where pn is the n-th prime
inf n→∞ (11)
and primes in tuples (i.e., proving that consecutive
primes exist that are closer than any arbitrarily small
multiple of the average spacing) [57, 58].
Furthermore, works on the still unreached proof of
the Riemann hypothesis can demonstrate, in the
absence of a regular cadence, the existence or
otherwise of a logic in the distribution of prime
numbers. This could have important effects on
cryptography using integers whose factorization into
prime numbers cannot be calculated in acceptable
times. Such knowledge of the distribution of prime
numbers could facilitate the factorization. The
alternative of using quantum cryptography for the
moment seems unassailable [59].
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6 Conclusions
Following the considerations introduced above, we
can conclude by stressing some themes correlating
the concept of the vacuum, imaginary numbers, and
their t-incomputability; traditional and non-
traditional, ideal and non-ideal models; and research
topics. In particular,
- The vacuum as a general t-incomputable domain
of possibilities specified by the imaginary space,
with complex variables, imaginary models (i.e.,
models using imaginary variables), and collapsing
mechanisms that may be represented by a suitable
symbolic computational collapse.
The vacuum as imaginary space always coupled with
the non-imaginary space in the complex space of
complex numbers. In other words, the real space as
the real part of the dual complex space is composed
of real and imaginary parts. This representation
relates to the duality real-potential, collapsed–
collapsible, and superimpositions of something such
as /waves/phenomena/planes/states/spaces as in QM.
- The use of imaginary, complex numbers, and their
t-incomputability may be intended as guarantee
and representation that it is not possible to
compute definitive results, but it is possible to
have pending multiple equivalences and
superimpositions as in quantum physics and
emergent collective processes in complexity.
- The phenomenological nature of the quantum
collapsing of a wave function in a superposition of
several eigenstates, reducing to a single one as an
effect of the interaction with the external world.
We say phenomenological only ideally
computable, when the phenomenological collapse
cannot be reduced to be a result of a computational
process. This contrast is in correspondence with
the phenomenological nature of processes of
emergence not reducible to computational, as in
the science of complexity. It is represented (not
reduced to) by the non-Turing computability, non-
explicit computability, and t-incomputability of
models using imaginary numbers. Furthermore,
we may consider the role of the incomputability to
represent systemic irreducibility such as between
systemic and non-systemic properties,
incoherences as the manifestation of non-
equivalences, irreducible multidimensionality,
incompleteness, and complexity of the world only
approximated by non-linearity as in simulations.
From a formal point of view, a number of stochastic
models could be suitably reformulated in such a way
as to transform into QFT-based models, which
sometimes involves the use of imaginary numbers as
a trick, but require, however, a physical
interpretation.
The effectiveness of imaginary numbers may then be,
moreover, in their role of ensuring the t-
incomputability of phenomenological collapsing and
in reformulating stochastic models as QFT-based
models.
Examples of related research lines to be considered
are
- The usage of the mathematics of complex
variables not only in (typically quantum) physics
but also to consider and represent the implicit
pending properties of matter; dualities between
classic and non-classic approaches; and
theoretically incomplete and incomputable
processes of emergence from the predominance of
collective multiple remote synchronization effects
[60, 61];
- The relations between physical, biological, and
mathematical processes of reduction of t-
incomputabilities into computability. Such
processes of reduction include the meaning of the
fact that imaginary t-incomputability disappears
as soon as we properly symbolically compute and
combine complex numbers;
- The meaning of i as theoretically incomputable
and, however, expressed by singularly
computable expressions as in (7).
- The use of imaginary, complex numbers as the
expression/manifestation of something we do not
understand yet, as was for the indemonstrability of
the fifth postulate of Euclid, an indemonstrability
that covered for centuries the non-Euclidean
geometries.
The usage of imaginary, complex numbers, complex
variables, their t-incomputability, and their
unreasonable effectiveness in physic modes, would
be an outstanding interdisciplinary project between
mathematics, physics, and philosophy of science.
Proper modeling is necessary, probably based on new
approaches such as the so-called complex-valued
neural networks (CVNN) as in [62] and on the so-
called sub-symbolic, emergent computation [18].
The present article is dedicated to the memory of
Professor Eliano Pessa with whom these issues were
under study and to celebrate his valuable
interdisciplinary contribution and expertise in the
science of complexity.
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Funding: This research received no
external funding.
Competing Interests: The author
declares no conflicts of interest.
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