Quantum Gauge Theory - Part 6
Sometimes
one is not able to make the necessary adjustments in one's theory, model or belief system
to compensate for the changes which have occurred in relation to some field variable. When
this occurs, one has lost the property of local gauge invariance.
Under such
circumstances, the understanding to which the theory, model or belief system gives
expression has encountered a crisis, lacunae, unanswered question or problem. As a result,
that understanding needs to be modified, changed or abandoned if symmetry is to be
preserved with respect to some aspect of ontology.
Furthermore,
when symmetry is lost, one may no longer be able to make meaningful, heuristically
valuable or defensible comparisons about the changing character of field strengths as one
moves about (or through) the phenomenology of the experiential field. This is so because
questions have been raised about, or problems have emerged in relation to, the
hermeneutical field equations which serve as the set of rules that permits one to
understand what the effects are of the process of transporting the gauge through or about
the phenomenological field. In other words, the very basis of comparison by means of a
given mode of hermeneutical gauge has been brought into question or become problematic.
A plaquette
consists of the square formed by four neighboring vertexes connected by a gauge field that
runs along the lattice links between the various vertexes. If one introduces a twist into
the gauge field running along one side of the plaquette, no matter what combination of
local gauge operations are performed, one will not be able to get rid of the twist in the
gauge field. One can relocate where the twist occurs, but one cannot eliminate the twist .
In addition,
if one transports an oriented arrow around a twisted plaquette, the orientation of the
arrow will not be the same after traveling through the twist as it was before being
transported through that twist. Such a plaquette (namely, one in which all the twists that
have been introduced cannot be removed and in which, therefore, there will be a transition
in orientation of a directed arrow which is transported about the plaquette) is referred
to, as indicated earlier, as a frustrated plaquette.
Frustrated
plaquettes are used to represent the locus of quantum energy fluctuations in a vacuum. The
degree of a plaquettes frustration is said to be an index of that plaquette's action.
Consequently, if one calculates the sum of the actions of all the frustrated plaquettes of
a given lattice, one will have an index for the action of the lattice as a whole.
Moreover,
the degree of the change of orientation of a vector quantity which is transported about
such a frustrated plaquette is known as the phase angle, and the phase angle can be used
as a reflection of the amount of quantum fluctuation which occurs at a given point in the
field. In a sense, a plaquette represents a twist or loop or knot in the fabric of
space-time.
Plaquettes
which are not frustrated, on the other hand, can be used to represent the classical
vacuum. In other words, there is supposed to be no energy in the classical
vacuum. Since the presence of energy can be represented in terms of twists in a plaquette
that cannot be eliminated or by arrows which undergo transitions in orientation, a
plaquette which does not exhibit either of these characteristics can considered to be
devoid of energy. Obviously, such a plaquette will have no action associated with it.
The general
structural properties of the lattice/gauge approach which is used in the quark confinement
paper may have application to certain aspects of the process of modeling language. For
example, instead of using the vertexes to represent particles which have different
state-characters, one could use the vertexes to represent words that have different
state-characters (either semantic or syntactic or both). Thus, the same word could assume
different state-characters depending on circumstances.
The links
running between vertexes could be used to represent the phase relationships which link
words together to form noun phrases, adverbial phrases, adjectival phrases, prepositional
phrases, gerunds, and so on. A plaquette or a group of plaquettes could represent various
kinds of propositions or sentences.
A frustrated
plaquette might be thought of as giving expression to a particular kind of orientation. In
other words, just as the word at the vertexes can assume different state-characters, so,
too, a plaquette can assume different field state-characters depending on the degree of
twisting which exists in the plaquette. However, unlike the limited dimensionality of
physical plaquettes, hermeneutical plaquettes can involve, or make reference to, many more
dimensional components.
The action
of a frustrated hermeneutical plaquette would be the total meaning or understanding to
which a plaquette or the lattice as a whole gives expression. One could determine the
action character of any given lattice or plaquette by summing up the individual plaquettes
which make up a given hermeneutical lattice. However, one would probably have to work in
some kind of dialectical interaction component to take into account the manner in which
different hermeneutical plaquettes are capable of playing off against one another.
Consequently,
in addition to the links which connect the vertexes of a given hermeneutical plaquette,
one will have to postulate links connecting different plaquettes. The identity of these
links between plaquettes may be a function of the hermeneutical gauge field itself. This
could be the result either of: (a) the individual generating the meaning structure; (b)
the individual interpreting the meaning structure, or (c) some combination of (a) and (b).
The bottom
line with respect to the above suggestions is that meaning in language and
understanding in hermeneutics are being made analogs for the notion of action In physics, Just
as the energy characteristics of a given physical system can be summarized by calculating
the action for that system, so too, the hermeneutical characteristics of a given meaning
system or system of understanding can be summarized by calculating (in a qualitative
sense) the hermeneutical analog for the action of that system. The action of a system,
whether physical or hermeneutical, establishes the spectrum of ratios of constraints and
degrees of currents. themes, properties, waveforms and so on, which manifest themselves -
actively and potentially - in such a system.
The spectral character of structure
Phase angle
rotations constitute the basic group operation of quantum electrodynamics. Moreover, since
the phase angle rotation operations of QED form an Abelian group, the order in which the
rotations occur does not effect the final phase state which results from those rotations.
Consequently, the rotations of phase angles generates the phase space of quantum
fluctuations for the phenomena of QED.
Phase angle
rotations also constitute the basic group operation of quantum chromodynamics. However,
there are fundamental differences between the group operations of QCD and QED.
To begin
with, instead of transporting a gauge with one vectoral arrow (which represents electric
charge) as is the case in QED, the gauge fields of QCD involve transporting a gauge with
three vectoral arrows - one arrow for each of the color charges which is possible.
Secondly, unlike QED, where the order in which a sequence of rotations takes place does
not matter to the character of the resultant phase state, in QCD, the order in which a
sequence of phase angle rotations takes place is important. The phase angle rotations of
QCD are not commutative, and, consequently, the gauge fields of QCD are non-Abelian.
Because the
gauge fields of QCD incorporate more degrees of freedom than do the gauge fields of QED,
it opens up the possibility that the plaquettes of the lattices representing QCD gauge
fields can manifest more varieties of twists than are exhibited in the plaquettes of QED
gauge fields. In other words, there may be more kinds of frustrated plaquettes which are
possible in QCD than are possible in QED.
Rebbi
believes one may be able to trace the confinement property of the quarks within hadrons to
two factors. First, the greater range of frustrated plaquettes that are postulated for
QCD. Secondly, the fact that QCD involve non-Abelian gauge fields.
When one
measures the strength of a chromodynamic field, one is actually calculating an average for
all the various phase state fluctuation configurations which are possible for the field in
question. However, since these possible configurations do not all contribute in the same
way to shaping the structural character of the average for the field, one has to weight
these configurations.
The process
of weighting is usually accomplished by multiplying each configuration by the probability
that such a configuration will actually manifest itself in a given field. However, since,
in point of fact, there are too many configurations to take into consideration in even a
very small volume of phase space, one must employ some form of statistical sampling in
order to come up with a quantum expectation value for the strength
of a chromodynaimc field.
The
weighting factor which is to be associated with any given configuration is a function of
the action of that configuration. More specifically, the greater the action
of a configuration, the less that configuration will be weighted during the process of
determining the average for the quantum expectation value in a given field.
In
hermeneutical gauge field theory the various ratios of constraints and degrees of freedom
that make up a structure's spectrum constitute configurations. However, unlike the
configurations of quantum theory which are infinite in number (and this is a reflection of
the methodology of quantum mechanics rather than a reflection of the ontology of the
quantum phenomena which are being modeled by such methodology), the configurations or
ratios of constraints and degrees of freedom which give expression to a structure on a
given level of scale are finite since they are generated in finite periods of time and by
means of delimited hermeneutical operations.
The ratios
of constraints and degrees of freedom that constitute a structure's spectral character
represent the attractor themes, currents, or principles on different levels of scale of
that structure. These attractors can be designated as primary, secondary, tertiary and so
on, in relation to whatever level of scale one is currently engaging.
Not all
levels of scale will necessarily be equally important when considering a particular
structural issue, problem, or event, on some level of scale currently being engaged.
Therefore, the levels of scale one considers to be secondary, tertiary and so on will
depend on the individual's purposes, needs, goals, desires, values and so on.
In other
words, even if the character of the ratios of a structure may vary, these ratios will
never be zero as long as the structure of which they are a part remains intact. Such
themes or principles are the configurations of a given level of scale of the structure,
and the range of values which such configurations may have refers to the phase states of
that configuration. These different phase states are the result of various transitions in
the phase relationships that govern or shape or organize the ratio of constraints and
degrees of freedom that constitute the theme or principle in question on a given level of
scale.
When engaged
on a certain level of scale, each structure consists of a spectrum of thematic ratios of
constraints and degrees of freedom. This spectrum sets the parameters within which, and
through which, the structure as a whole will manifest its character on that level of
scale.
When one
changes the level of scale of one's engagement of a given structure, one finds, in turn, a
further spectrum of ratios of constraints and degrees of freedom which establish the
parameters through which any given theme or principle on the new level of scale will
manifest itself. In addition, themes and principles encountered on previous levels of
scale may or may not manifest themselves on the new level of scale. Yet, if such
previously engaged themes/principles do manifest themselves, they will do so as an
expression of one or more of the ratios of the spectrum of constraints and degrees of
freedom on the new level of scale.
The number
of levels of scale which exist in relation to any given structure may be indefinite, but
they are not necessarily infinite in character. In any event, one does not have to take
into consideration all the phase states of all configurations on all levels of scale in
order to be able to grasp the general character of the manner in which a given structure
is manifested on a given level of scale. Obviously, the more detail one wants, then, the
more data one is likely to seek in relation to various phase state configurations on
different levels of scale.
In a sense,
one seeks as much information and understanding as is necessary to meet one's needs or
solve one's problems or satisfy one's interests or resolve various issues under a given
set of circumstances. Thus, like the methodological techniques of quantum chromodynamics,
hermeneutical gauge field uses a process of sampling to select data from one or more
levels of scale. Unlike the methods of QCD, however, hermeneutical gauge field theory does
not presuppose that either the spectrum of ratios or the configurations or the phase
states or the levels of scale are infinite in number.
More
importantly, even if one were to suppose that there were infinite configurations or phase
states or levels of scale associated with a given structure's spectrum of ratios of
constraints and degrees of freedom, hermeneutical gauge field is able to work on a sort of
need-to-know basis, taking into account only what is believed to be necessary to get on
with things in a given set of circumstances. The accuracy, competency, proficiency,
efficiency or aesthetics of how one decides to get on with things will depend on the
individual and the circumstances. Therefore, different circumstances may require one to
employ different methods of weighting configurations in order to grasp the character of
the way in which a given structure's spectrum of ratios of constraints and degrees of
freedom manifests itself on a given level of scale.
Fundamental forces: physical and hermeneutical
A worthwhile
exercise, at this point, may be to develop, in analog fashion, the parallels between, on
the one hand, various collections, groupings or categories of semiotic quanta, and, on the
other hand, the four physical forces: namely, electromagnetism, gravitation, as well as
the strong and weak forces. For example, one might consider the realm of dialectical
reactivity involving phase relationships to be the hermeneutical counterpart to
electromagnetism.
Like the
electrons of atoms and molecules, such dialectically reactive groupings of semiotic quanta
might determine the structural character of the kinds of hermeneutical reactions
(comparable to chemical reactions) which are possible between, or among, various
hermeneutical reactants or structures. One might even work out a hermeneutical counterpart
to thermodynamics in terms of hermeneutical stability, equilibrium, dissipative structures
and so on.
The
hermeneutical counterpart to gravitational forces, on the other hand, might focus on the
way certain groupings or arrangements of semiotic quanta form attractors which have
spheres of influence comparable to gravitational pull. One might even suppose there is an
inverse square law concerning the strength of such attractors across emotional,
experiential, phenomenological or hermeneutical 'distance'.
The
hermeneutical counterpart to the strong force might involve the spectrum of ratios of
constraints and degrees of freedom which set the tone, so to speak, for the character of a
given structure. This grouping of semiotic quanta would be comparable to the combination
of neutrons and protons in the atomic nucleus. Consequently, such groupings would
establish the parameters of phase relationship activity within a given structure.
In addition,
these hermeneutical nucleon groupings also would serve as a countervailing force to the
hermeneutical counterpart to electromagnetic dialectical reactivity. In other words, the
former groupings might play a fundamental role in maintaining the integrity of a
structure's identity over time, despite the shifts and transitions in phase relationships
and ratio arrangements that occur as a result of the structure's exchange of semiotic
quanta with other structures during their dialectical interaction.
Finally, the
hermeneutical counterpart to the weak force might concern either of two possibilities. One
possibility could be the tendency of an organized grouping of semiotic quanta (e.g.,
beliefs, values, theories, models, systems, networks and so on) to disintegrate or
dissipate, over time, due to the weaknesses of certain ratios of constraints and degrees
of freedom. As these ratios unravel, so to speak, and become less capable of giving
expression to a normal complement of phase transitions or phase shifts, the belief, or
theory or whatever, decays with time.
This
suggests, at least in the hermeneutical context, there may be an intrinsic relationship of
tension between the strong and weak forces present in any given organized grouping of
semiotic quanta. In other words, the central binding force (i.e., the hermeneutical strong
force analog) or coupling constant giving expression to the basic ratios of constraints
and degrees of freedom that constitutes a given phenomenological or hermeneutical object's
or event's structural identity may be engaged constantly in a dialectical relationship
with a force (i.e., the hermeneutical weak force) that undermines or weakens the
hermeneutical counterpart to the strong force.
Presumably,
some hermeneutical structures have a higher tendency toward dissolution or dissipation
than do other such structures (e.g., theories or hypotheses that are quickly proven to be
problematic), just as different elements have different rates of radioactive decay.
However, irrespective of such intrinsic rates, when the strength of the weak force,
relative to the strength of the strong force, is greater, then, there will be an
accelerated trend toward complete breakdown of the given grouping of semiotic quanta.
Another
possibility concerning the hermeneutical counterpart to the weak force has to do with the
idea of commitment. In other words, over time there may be a lessening of commitment to
some give organized grouping of semiotic quanta (such as a belief or idea or value, etc.).
This tendency for commitment to spontaneously decay or disintegrate over time may be due
to the nature of the structural character of the set of semiotic quanta that give
expression to such commitment and which shape how that mode of commitment dialectically
interacts with a given belief or value structure.
One of the
most salient features of the weak force involves its extremely limited range
(approximately 10-15 centimeters which is, roughly, 1/100 of the size of a
proton's radius). The shortness of the range of the weak force suggests the boson
or force carrying particle is probably quite massive, with current calculations
putting the mass of this particle at around 100 times the proton's mass.
In order to
extend the analogy of the hermeneutical counterpart to the weak force, one would have to
postulate that the hermeneutical weak force is extremely limited in its range and that the
semiotic quantum which is responsible for carrying the hermeneutical weak force may be
quite large.
One
possibility which suggests itself in this respect is that, for the most part, beliefs,
values and theories tend to be fairly resistant to dissolution, dissipation or
disintegration. One of the reasons for this is due to the numerous, reinforcing phase
relationships which exist within the neighborhood or latticework that gives expression to
such a belief, value or theory.
Altering a
few, or even a sizable number, of these phase relationships may do little to cause the
neighborhood or latticework to breakdown. This means, in effect, the tendency toward, or
force of, disintegration will be relatively small when one considers how the hermeneutical
weak force tends to manifest itself through, relatively speaking, only a few phase
relationships compared to the far greater number and strength of the surrounding
manifestations of the hermeneutical strong force.
Moreover,
the range of the hermeneutical weak force might be- in many, if not most, cases-quite
limited since it would tend to be restricted (although there will, undoubtedly, be
exceptions to this) to, or affect, only those phase relationships which are sensitive to,
or receptive, to its decay character. Thus, even though any given phase relationship is
intertwined with a variety of other reinforcing phase relationships, the spontaneous decay
of a particular phase relationship wouldn't necessarily affect these other phase
relationships with which it is linked since the other phase relationships may be
stabilized, to a certain extent, by the way they are rooted in the neighborhood or
latticework of s set of beliefs considered as a whole.
Of course,
such an explanation would raise, in turn, a question about why any phase relationship
would decay if it exists in the midst of such a stabilizing environment. The only chance
one would have of answering this question is to take a look at the specific phase
relationship which decayed and attempt to determine what permitted it to break loose from
the support network. In principle, however, there is nothing to prevent isolated cases of
phase relationship decay despite the presence of a supporting network of phase
relationships. Indeed, the isolated, anomalous character of such decay events conforms to
the most salient characteristic of the weak force- namely, its limited range.
On the other
hand, if the decay of phase relationships occurs at a high rate, then, the size of the set
of phase relationships which serves to carry this force becomes increasingly massive. So,
when the rate of decay becomes large it is because more phase relationships are becoming
involved. Although the range of any given phase relationship expression of the weak force
may still be relatively limited, the combined effect of a set of decaying relationships
makes for a fairly massive source of the hermeneutical weak force.
Under such
circumstances, the structure would have a hermeneutically "radioactive"
character. Conceivably, each kind of hermeneutical structure has its own unique,
radioactive (or decay) signature.
Thus, the
size of the carrier of the hermeneutical weak force will range all the way from a single
phase relationship up to one or more latticeworks. However, the range of the weak force
will still be very limited, no matter what the size of the carrier is, because the weak
force is communicated or conveyed or transmitted only through individual phase
relationships in the context of integrated neighborhoods and latticeworks that are coupled
together by manifestations of the hermeneutical strong force.
In a sense,
the foregoing provides for a unified approach to hermeneutical gauge field theory. In
other words, the carrier of hermeneutic force across all levels of scale is the semiotic
quantum. However, although the general structural character of all semiotic quanta is the
same (in terms of the six components of the hermeneutical operator), nonetheless, semiotic
quanta are carriers of variable force.
In other
words, because the various combinatorial possibilities of isotopic-spin states of the
semiotic quantum are huge, the structural character of the force carried by a given
instance of semiotic quantum in a given set of circumstances can assume an indefinite
variety of gradations of strength, intensity, orientation, shape and so on. Consequently,
both hermeneutical unity as well as hermeneutical multiplicity are capable of being given
expression through the way semiotic quanta dialectically engage, and are engaged by, the
phenomenology of the experiential field, together with the aspects of ontology that make
an experiential field of such structural character possible.
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