categorical quantum LM- logic algebras

categorical quantum LM- logic algebras

\xyoption

curve

0.1

Quantum LM-algebraic Logic

0.1.1

Fundamental concepts of space and time in quantum theory

vs.

space-times in general relativity

A notable feature of

current

21-st century physical thought involves a close examination of the

validity

of the classical model of space-time as a

4

–dimensional manifold equipped with a Lorentz metric. The

expectation

of the earlier approaches to

quantum gravity

(QG) was to cope with microscopic length scales where a traditional manifold

structure

(in the conventional sense) needs to be forsaken (for

instance

, at the Planck length

L

p

=

(

G

⁢

ℏ

c

3

)

1

2

≈

10

-

35

⁢

m

). Whereas Newton, Riemann, Einstein, Weyl, Hawking, Penrose,
Weinberg and many other exceptionally creative theoreticians
regarded physical space as represented by a

continuum

,
there is an increasing number of proponents for a

discrete

,
‘quantized’

structure of space–time, since space itself is
considered as discrete on the Planck scale. Like most radical
theories, the latter view carries its own set of problems. The
biggest problem arises from the fact that any discrete,
‘point-set’ (or

discrete topology

), view of physical space–time
is not only in immediate conflict with Einstein’s General
Relativity

representation

of space–time as a

continuous

Riemann

space, but it also conflicts with the fundamental
impossibility of carrying out quantum measurements that would
localize precisely either quantum events or masses at ‘singular points’ (in
the sense of

disconnected

, or

isolated

, sharply defined,
geometric points) in space–time. Since GR seems to break down at the Planck scale,

space–time may no longer be describable by a smooth manifold
structure

such as a Riemann

metric tensor

. While not neglecting the large scale classical model,
one needs to propose a structure of ‘ideal observations’ as manifest
in a limit, in some sense, of ‘discrete’, or at least

separable

,
measurements, where in such a limit it also encompasses the classical event.
Further details are given in our recent, related paper

[

1

]

.

0.2

Quantum Logics (QL) and Logic Lattice Algebras (LA):
Operational Quantum Logic (OQL) and Łukasiewicz Quantum Logics (LQL)

0.2.1

Quantum Fields, General Relativity and Symmetries

As the experimental findings in high-energy physics–coupled
with theoretical studies– have revealed the presence of new fields
and

symmetries

, there appeared the need in modern physics to develop
systematic procedures for generalizing space–time and

Quantum State
Space

(QSS) representations in order to

reflect

these new

concepts

. In the General Relativity (GR) formulation, the local structure of
space–time, characterized by its various tensors (of
energy–momentum,

torsion

, curvature, etc.), incorporates the
gravitational fields surrounding various masses. In Einstein’s own
representation, the physical space–time of GR has the structure
of a Riemannian

R

4

space over large

distances

, although the
detailed local structure of space–time – as Einstein perceived
it – is likely to be significantly different. On the other hand,
there is a growing consensus in theoretical physics that a valid
theory of quantum gravity (QG) requires a much
deeper understanding of the small(est)–scale structure of quantum
space–time (QST) than currently developed. In Einstein’s GR
theory and his subsequent attempts at developing a unified field
theory (as in the space concept advocated by Leibnitz), space-time
does

not

have an

independent

existence

from objects,
matter or fields, but is instead an entity generated by the

continuous

transformations

of fields. Hence, the continuous
nature of space–time was adopted in GR and Einstein’s subsequent
field theoretical

developments

. Furthermore, the quantum, or
‘quantized’, versions of space-time, QST, are operationally
defined through local quantum measurements in general reference
frames that are prescribed by GR theory. Such a

definition

is
therefore subject to the

postulates

of both GR theory and the
axioms of local quantum physics (LQP). We must emphasize, however, that
this is

not

the usual definition of position and time
observables in ‘standard’ QM. The general reference
frame positioning in QST is itself subject to the Heisenberg
uncertainty principle, and therefore it acquires through quantum
measurements, a certain ‘fuzziness’ at the Planck scale which is
intrinsic to all microphysical

quantum systems

. Such systems with
fuzziness include

spin networks

that change in time thus giving birth
to

spin foam

.

0.2.2

Operational Quantum Logic (OQL) and Łukasiewicz Quantum Logic (LQL)

As pointed out by von Neumann and Birkhoff in 1930, a

logical foundation

of quantum mechanics

consistent

with

quantum algebra

is essential for both the completeness and mathematical validity of the theory. The development of quantum mechanics from its very beginnings both inspired and required the consideration of specialized logics

compatible

with a new theory of measurements for microphysical systems. Such a specialized logic was initially formulated by von Neumann and Birkhoff (1932) and called ‘

quantum logic

’. Subsequent research on MV-logics and quantum logics (Chang, 1958;
Genoutti, 1968; Dalla Chiara, 1968, 2004) resulted in several approaches that involve several types of non-distributive

lattice

(

algebra

) for

n

–valued quantum logics. Thus, modifications of the Łukasiewicz logic algebras that were introduced in the context of

algebraic categories

by Georgescu and Vraciu in 1970

[

2

]

, also recently reviewed and expanded by Georgescu in 2006, can provide an appropriate

framework

for representing quantum systems, or– in their unmodified form- for describing the activities of complex networks in

categories

of Łukasiewicz logic algebras

[

3

]

.

0.2.3

Lattices and Von Neumann-Birkhoff (VNB) Quantum Logic: Definitions and Basic Logical Properties

We commence here by giving the

set-based definition of a Lattice

. An

s–lattice

𝐋

, or a ‘set-based’ lattice, is defined as a

partially ordered set

that has all
binary

products

(defined by the

s

–lattice

operation

“

⋀

”) and

coproducts

(defined by the

s

–lattice
operation “

⋁

”), with the ”

partial ordering

” between two

elements

X and Y belonging to the

s

–lattice being written as
“

X

⪯

Y

”. The

partial order

defined by

⪯

holds in

L

as

X

⪯

Y

if and only if

X

=

X

⁢

⋀

Y

(or equivalently,

Y

=

X

⁢

⋁

Y

Eq.(3.1)(p. 49 of Mac Lane and Moerdijk, 1992).

0.2.4

Łukasiewicz-Moisil (LM) Quantum Logic (LQL) and Algebras

With all truth ’nuances’ or assertions of the type

≪

system A

is excitable
to the

i

-th level and system B is excitable to the

j

-th level

≫

one can define
a special type of lattice which is subject to the axioms introduced by Georgescu and Vraciu in 1970

[

2

]

which then becomes a

n

-valued Łukasiewicz-Moisil, or LM-algebra

. Further algebraic and logic details are provided in ref.

[

4

]

and also in

[

1

]

. In order to have the

n

-valued Łukasiewicz logic algebra

represent

correctly the basic behaviour of quantum systems (that is, as observed through measurements that involve a quantum system interactions with a measuring instrument –which is a macroscopic object), several of these axioms have to be significantly changed so that the resulting lattice becomes

non-distributive

and also (possibly)

non–

associative

(Dalla Chiara, 2004), in

addition

to being

non-commutative

. With an appropriately defined quantum logic of events one can proceed to define Hilbert, or ‘nuclear’/Frechet, spaces in order to be able to utilize the ‘standard’ procedures of

quantum theories

.

0.2.5

Fundamental concepts of algebraic topology with potential application to ontology levels theory and
space-time structures

We shall consider briefly the potential impact of novel

algebraic topology

concepts, methods and results on the problems of defining and classifying rigorously quantum space-times. With the advent of

Quantum Groupoids

–generalizing

quantum groups

, quantum algebra and quantum algebraic topology, several fundamental concepts and new

theorems

of algebraic topology may also acquire an enhanced importance through their potential applications to current problems in theoretical and mathematical physics, such as those described in

[

6

]

, and also in subsequent publications

[

1

]

and

[

7

]

.

Now, if quantum mechanics is to reject the notion of a continuum,
then it must also reject the notion of the

real line

and the notion
of a path. How then is one to

construct

a

homotopy theory

?
One possibility is to take the route signalled by Čech, and which
later developed in the hands of Borsuk into ‘shape theory’. Thus, a quite general space is studied by means of its approximation by

open covers

.

A few fundamental concepts of algebraic topology and

category theory

are summarized here that have an extremely wide range of applicability to the higher complexity levels of reality as well as to the fundamental, quantum level(s). Technical details are omitted in this

section

in order to focus only on the ontologically-relevant aspects; full mathematical details are however also available in a recent paper by Brown et al (2007) that focuses on a mathematical/conceptual framework for a completely formal approach to categorical ontology and the theory of levels.

0.3

Local–to–global (LG) Construction Principles Consistent with AQFT

A novel approach to QST construction in algebraic/axiomatic QFT (AQFT) involves the use
of generalized fundamental theorems of algebraic topology from
specialized, ‘globally well-behaved’

topological spaces

, to
arbitrary ones (Baianu et al, 2007c). In this category, are the generalized,

Higher
Homotopy van Kampen theorems (HHvKT)

of algebraic topology with
novel and unique

non-Abelian

applications. Such theorems greatly aid
the calculation of higher homotopy of topological spaces. R. Brown and coworkers (1999, 2004a,b,c) generalized the

van Kampen theorem

, at first to

fundamental groupoids

on a set of

base points

(Brown,1967), and
then, to

higher dimensional algebras

involving, for example,

homotopy double groupoids

and

2-categories

(Brown, 2004a). The more
sensitive

algebraic invariant

of topological spaces seems to
be, however, captured only by

cohomology

theory through an
algebraic

ring

structure that is not

accessible

either in

homology

theory, or in the existing homotopy theory. Thus, two
arbitrary topological spaces that have

isomorphic

homology groups

may not have isomorphic cohomological ring structures, and may also
not be

homeomorphic

, even if they are of the same

homotopy type

.
Furthermore, several

non-Abelian

results in algebraic topology could only be derived from the Generalized van Kampen theorem (

viz

. Brown, 2004a), so that one may find links of such results to the expected

‘non-commutative

geometrical’ structure of quantized space–time
(Connes, 1994). In this context, the important algebraic–topological concept of a

Fundamental

Homotopy Groupoid

(FHG) is applied to a quantum topological space (QTS)

as a “partial classifier” of the

invariant

topological propertiesof quantum spaces of any

dimension

; quantum topological
spaces are then linked together in a

crossed complex over a
quantum groupoid

(Baianu, Brown and Glazebrook, 2006), thus
suggesting the construction of global topological structures from
local ones with

well-defined

quantum homotopy groupoids. The latter
theme is then further pursued through defining locally

topological
groupoids

that can be globally characterized by applying the
globalization theorem, which involves the

unique

construction
of the holonomy groupoid. How such concepts might be applied in the context of algebraic or axiomatic quantum
field theories (AQFT) will be separately considered in order to provide a local-to-global construction of quantum space-times which would still be valid in the presence of intense gravitational fields without generating singularities as in GR. The result of such a construction is a

quantum holonomy groupoid

, (QHG) which is unique up to an

isomorphism

.

0.3.1

Physical and Mathematical Theories: Physical Axioms, Principles, Postulates and Laws

The Greeks devised

the axiomatic method

, but thought of it
in a different manner to that we do today. One can imagine that
the way Euclid’s ‘

Geometry

’ evolved was simply through the
delivering of a course

covering

the established facts of the time.
In delivering such a course, it is natural to formalize the
starting points, and so arranging a sensible structure. These
starting points came to be called

postulates, definitions
and axioms

, and they were thought to deal with real, or even
ideal, objects, named points, lines, distance and so on. The
modern view, initiated by the discovery of

non-Euclidean geometry

,
is that the words points, lines, etc. should be taken as undefined
terms, and that axioms give the

relations

between these.
This allows the axioms to apply to many other instances, and has
led to the power of modern geometry and algebra. Clarifying the meaning to be ascribed to ‘concept’, ‘percept’, ‘thought’, ‘emotion’, etc., and above all the

relations

between these words, is clearly a fundamental but
time–consuming step. Although relations–in their turn–can be, and
were, defined in terms of sets, their axiomatic/

categorical

introduction

greatly expands their range of applicability well beyond that of set-relations.
Ultimately, one deals with

relations among relations

and relations of higher order.

The more

rigorous

scientific theories, including those founded in
logics and mathematics, proceed at a fundamental level from axioms
and principles, followed in the case of ‘natural sciences’ by laws
of nature that are valid in specific contexts or well-defined
situations. Whereas the hierarchical theory of levels provides a powerful,
systemic approach through categorical ontology, the foundation
of science involves

universal

models and theories
pertaining to different levels of reality. Such theories are based
on axioms, principles, postulates and laws operating on distinct
levels of reality with a specific degree of complexity.

Because of such distinctions, inter-level principles or laws are rare and
over-simplified principles abound. As relevant examples, consider
the chemical/biochemical thermodynamics, physical biochemistry
and molecular biology fields which have developed a rich structure
of specific-level laws and principles, however, without ‘breaking
through’ to the higher, emergent/integrative level of organismic
biology. This does not detract of course from their usefulness, it
simply renders them incomplete as theories of biological reality.
With the possible exceptions of evolution and genetic principles
or laws, biology has until recently lacked other universal principles
for highly complex dynamics in organisms, populations and species,
as it will be shown in the following sections. One
can therefore consider biology to be at an almost ‘pre–Newtonian’
stage by comparison with either physics or chemistry.

Whereas axioms are rarely invoked in the natural
sciences perhaps because of their abstract and exacting

attributes

, (as well as their coming into existence through
elaborate processes of repeated abstraction and

refinement

),
postulates are ‘

obvious

assumptions

’ of extreme generality that do
not require proof but just like axioms are accepted on the basis
of their very numerous, valid

consequences

. Principles and laws, even though quite strict, may not apply under certain exceptional, or ‘

singular

’ situations. Natural laws are applicable to well-defined zones or levels of reality, and are thus less general, or universal, than principles. Unlike physical laws that are often expressed through
mathematical

equations

, principles are instead often explained in words,
and tend to have the most general form attainable/acceptable in an
established theory. It is interesting to note that in Greek, and later
Roman

antiquity

, both philosophers and orators did link philosophy and logic; moreover,
in medieval time, first Francis Bacon, then Newton opted for quite precise formulations
of “natural philosophy” and a logical approach to ‘objective’ reality. In Newton’s
approach, the

logical and

precise formulation of such “natural principles” demanded the
development of mathematical concepts suitable for the exact determination and quantification
of the rate of a change in the “state of motion” of any mechanical body, or system.
Later philosophical developments have strayed from such precise formulations and, indeed,
mathematical developments seem to have lost their appeal in ‘natural philosophy’.

On the other hand, it would seem natural to expect that theories aimed at different ontological levels of reality should have different principles. Furthermore, one may ontologically, address the question of why such distinct levels of reality originated in the first place, and then developed, or emerged, both in space and time. Without reverting to any form of Newtonian or quantum-mechanical determinism, we are also pointing out in this essay the need for developing precise but nevertheless ‘flexible’ concepts and novel mathematical representations suitable for understanding the emergence of the higher complexity levels of reality.

It is also in this context that the ‘local-to-global’ model approach becomes relevant, as in the case of generalized van Kampen theorems.

0.3.2

Symmetry, Commutativity and Abelian Structures

The hierarchy constructed above, up to level 3, can be
further extended to higher,

n

-levels, always in a consistent,
natural manner, that is using

commutative diagrams

. Let us see
therefore a few simple examples or specific instances of
commutative properties. The type of global, natural hierarchy of
items inspired by the mathematical TC-FNT has a kind of

internal symmetry

because at all levels, the link

compositions

are

natural

, that is, if

f

:

x

⟶

y

and

g

:

y

⟶

z

⟹

h

:

x

⟶

z

, then the composition of

morphism

g

with

f

is
given by another unique morphism

h

=

g

∘

f

. This general property involving the

equality

of such link composition chains or diagrams comprising any number of sequential links between the same beginning and ending objects is called

commutativity

(see for example Samuel and Zarisky, 1957), and is often expressed as a

naturality condition for diagrams

. This key mathematical property also includes the mirror-like symmetry

x

⋆

y

=

y

⋆

x

; when

x

and

y

are

operators

and the symbol ’

⋆

’ represents the
operator

multiplication

. Then, the equality of

x

⋆

y

with

y

⋆

x

defines the statement that ”the

x

and

y

operators

commute

”; in physical terms, this translates
into a sharing of the same set of eigenvalues by the two commuting
operators, thus leading to ‘

equivalent

’ numerical results i.e.,
up to a multiplication

constant

); furthermore, the observations
X and Y corresponding, respectively, to these two operators
would yield the same result if X is performed before Y in time,
or if Y is performed first followed by X. This property, when present,
is very convenient for both mathematical and physical applications (such as those encountered in quantum mechanics). When commutativity is global in a structure, as in an

Abelian

(or

commutative

) group, commutative

groupoid

, commutative ring, etc., such a structure that is commutative throughout is usually called

Abelian

. However, in the case of category theory, this concept of Abelian structure has been extended to a special class of categories that have meta-properties formally

similar

to those of the category of commutative groups,

Ab

-

G

; the necessary and

sufficient

conditions for such ‘Abelianness’ of categories other than that of

Abelian groups

were expressed as three axioms

Ab1

to

Ab3

and their duals (Freyd, 1964; see also the details in Baianu et al 2007b and Brown et al 2007). A first step towards re-gaining something like the ‘global commutativity’ of an Abelian group is to require that all classes of morphisms

[

A

,

B

]

or

H

⁢

o

⁢

m

⁢

(

A

,

B

)

have the structure of commutative groups; subject to a few other general conditions such categories are called

additive

. Then, some kind of global commutativity is assured for all morphisms of

additive

categories. However, further conditions are needed to make

additive categories

‘Abelian’, and additional properties were also posited for

Abelian categories

in order to extend the applications of Abelian category theory to other fields of modern mathematics (Grothendieck, 1957; Grothendieck and Dieudoné 1960; Oberst 1969; Popescu 1973.) A Homotopy theory was also formulated in Abelian categories (Kleisli, 1962). The

equivalence

of Abelian categories was reported by Roux, and important imbedding theorems were proved by Mitchell (1964) and by Lubkin (1960); a

characterization

of Abelian categories with generators and exact limits was presented by Gabriel and Popescu (1964). As one can see from both earlier and recent literature, Abelian categories have been studied in great detail, even though one cannot say that all their properties have been already found.

However, not all quantum operators ‘commute’, and not all categorical diagrams or mathematical structures are, or need be, commutative. Non-commutativity may therefore appear as a result of ‘breaking’ the ‘internal symmetry’ represented by commutativity. As a physical

analogy

, this might be considered a kind of

‘symmetry breaking’

which is thought to be responsible for our expanding

universe

and CPT violation, as well as many other physical phenomena such as

phase transitions

and superconductivity (Weinberg, 2003).

References

1

Baianu, I.C., R. Brown and J. F. Glazebrook.(2007), A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity,

Axiomathes

,

17

: 169-225.

2

Georgescu, G. and C. Vraciu (1970). On the Characterization of Łukasiewicz Algebras.,

J. Algebra

,

16

(4), 486-495.

3

Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory.

Bulletin of

Mathematical Biophysics

,

39

: 249-258.

4

Georgescu, George. (2006).

N-Valued Logics

.

Axiomathes

.

17

: 251-262.

5

Baianu, I.C. and R. Poli. (2008), From Simple to Complex and

Ultra-Complex Systems

:
A Paradigm Shift Towards Non-Abelian

Systems Dynamics

., In:

Theory and Applications of Ontology

,
vol.

1

, R. Poli, Ed., Springer: Berlin.

6

Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook.(2006), Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic

Networks

,

Axiomathes

,

16

Nos. 1–2: 65–122.

7

Brown, R., Glazebrook, J. F. and I.C. Baianu.(2007), A Conceptual, Categorical and Higher Dimensional Algebra Framework of Universal Ontology and the Theory of Levels for Highly

Complex Structures

and Dynamics.,

Axiomathes

(17): 321–379.

8

Poli, R. (1998), Levels,

Axiomathes

,

9

, 1-2, pp. 197-211.

9

Poli, R. (2001a), The Basic Problem of the Theory of Levels of Reality,

Axiomathes

,

12

, 3-4, pp. 261-283.

10

Poli, R. (2001b), Alwis.

Ontology for Knowledge Engineers

. PhD Thesis, Univ. of Utrecht.

Title

categorical quantum LM- logic algebras

Canonical

name

CategoricalQuantumLMLogicAlgebras

Date of creation

2013-03-22 18:11:04

Last modified on

2013-03-22 18:11:04

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

83

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55U99

Classification

msc 55U40

Classification

msc 55U35

Classification

msc 55U30

Classification

msc 81T05

Classification

msc 18A40

Classification

msc 18A15

Classification

msc 18A05

Classification

msc 18A99

Synonym

formal logics

Synonym

classification of existential categories

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UltraComplexSystems

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ComplexSystemsBiology

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RosettaGroupoids

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NonAbelianTheories

Related topic

NonAbelianStructures

Related topic

CommutativeVsNonCommutativeDynamicModelingDiagrams

Related topic

GeneralizedToposesTopoiWithManyValuedLogicSubobjectClassifiers

Related topic

TopicEntryOnFoundationsOfMathematics

Defines

quantum logics

Defines

categorical

Defines

categorical ontology

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