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Philosophical Reflections in Physics and Math
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Quantum Gauge Theory - Part 3
Coupled oscillators and dimensionality
A simple
vibrating unit or element is referred to as an oscillator. One can construct
more complex systems of vibrating elements by coupling a number of such oscillators
together. The character of the medium through which the oscillators are linked together is
presumed to be elastic so that the form and energy of the vibration in one oscillator can
be transmitted to other oscillators in the coupled system.
A naturally
occurring example of such a coupled oscillatory system is the array of atoms in a crystal.
Each of the atoms of the crystal vibrates back and forth about its equilibrium position,
and all of these individual units of oscillation are coupled together by means of the
atomic forces which are active within the crystal. The whole system of coupled oscillators
gives expression to what is known as lattice vibration.
The
foregoing idea of lattice vibration may have counterparts in various phenomenological
systems. Hermeneutical structures within such systems form an array or latticework of
oscillators. These latticeworks are coupled together by a variety of dimensional forces
that are manifested through the phase relationships that are characteristic of such
systems
Thus,
dimensions which interact may act like coupled oscillators. The nature of the dialectic of
this dimensional interaction generates complex latticework patterns consisting of arrays
of point-structures. The parameters of these point-structures give expression to the
spectrum of ratios of constraints and degrees of freedom, together with accompanying phase
relationships, through which various structures or complex waveforms are manifested.
Dispersive and nondispersive mediums
Phase
velocity describes the rate at which any point on a wave is propagated through a given
medium. If one has a fixed medium, such as a string, one can increase phase
velocity by increasing the tension in the string.
In general,
phase velocity is functionally dependent on the square root of the strain which exists in
a given material, divided by the inertial mass of the material. The strength of the
restoring force is given expression through the strain component, whereas the kinetic
activity of the vibrator is given expression through the inertial mass of the medium.
In those
cases when phase velocity is not dependent on wave frequency, the medium is referred to as
being nondispersive. However, when phase velocity is dependent on wave
frequency, the mediums in which this occurs are described as being dispersive.42
An example
of the latter takes place when white light is separated into different colors by a prism
as a result of the interaction between the electromagnetic wave and electronic oscillators
in the prism material. This is known as optical dispersion.
In the
hermeneutical context, the rate at which phase shifts occur among a given spectrum of
ratios of constraints and degrees of freedom may serve as the counterpart to the idea of
phase velocity. However, the aspect of 'rate' may be somewhat problematic for it suggests
a purely quantitative value.
Yet, one may
not have any means of measuring the number of phase shifts occurring per given unit of
time. Consequently, in the hermeneutical context, phase velocity might be treated as a
purely heuristic device which enables one to have a way of making identifying reference to
certain aspects of the hermeneutical engagement of the phenomenology of the experiential
field. If this is the case, then, one might be able to use the idea of phase velocity in
conjunction with ideas such as dispersion, strain, and inertia in order to discuss the
sorts of factors which affect the rate at which phase shifts are propagated through
various kinds of hermeneutical medium.
One should
keep in mind that when, say, light enters a medium, the medium is made up of electrons
bound to atoms. Such bound electrons act like oscillators.
When these
oscillators are engaged by the wave motion of electromagnetic radiation, they begin to
vibrate as a result of that interaction. The vibration of these oscillators of the medium
generate, in turn, their own electromagnetic field which interacts with the light wave
radiation.
The
aforementioned dialectic affects the rate at which light will be propagated through the
medium. It is referred to as the refractive property of the medium. As such, the speed of
light through a given medium will be given by: v = c/n, where c is the speed of light in a
vacuum, and n is the refractive index of the given medium through which light is being
propagated.
In the same
way, one might speak of the refractive property of a given hermeneutical medium. This is
the capacity of a given hermeneutical medium to affect the velocity (or quality of
transmission) with which a certain kind of communication or understanding is propagated
through such a medium.
A
hermeneutical medium can be conceived of as being made up of a series of hermeneutical
operators which act like coupled oscillators. They are given a complex vibrational mode
through incoming hermeneutical, experiential, phenomenological, sensory, emotional and/or
spiritual waveforms. However, these hermeneutical oscillators also generate, in turn,
their own field which is capable of entering into a dialectic with the incoming waveforms.
This dialectic will affect the rate, as well as the qualitative manner, by which the
waveform is transmitted through the hermeneutical medium.
In fact,
because the hermeneutical field is complex, there will be a differential distribution of
oscillator properties as one goes from one locus of hermeneutical activity in the field to
other loci of hermeneutical activity in that field. This is somewhat analogous to the way
in which an electromagnetic field often will manifest differential strengths of electrical
charge as one moves from point to point in that field.
As indicated
previously, in physics, dispersive mediums are described as those mediums in which phase
velocity will be affected by the frequency of the wave which is being propagated through
that medium. In a sense, there are aspects of almost everyone's understanding which are
dispersive in character.
One of the
tasks of an educator is to be sensitive to how the phase velocity of a given communication
or transmission can be affected by the potentially dispersive aspects of a recipient's
medium of understanding. There may be qualitative/quantitative levels of frequency,
intensity, wave-length, orientation and so on, in relation to a given communication, which
are able to facilitate a transmission's phase velocity.
Such
facilitating strategies have the capacity to either by-pass, or catalytically transform,
dispersive elements in the recipient's phenomenological/hermeneutical medium. As a result,
these sorts of strategies may help to bring about the emergence of nondispersive mediums
of understanding.
On the other
hand, there may be qualitative/quantitative levels of frequency, intensity, orientation,
etc., of communication which impede that communication's phase velocity through the
recipient's phenomenological/hermeneutical medium. This will prove detrimental to the
emergence of a nondispersive medium of understanding.
In the
hermeneutical context, dispersive mediums are unfocused, without direction or orientation.
Therefore, when a given complex waveform (i.e., communication - spoken, written,
behavioral, emotional) engages such a medium, the waveform becomes dispersed or
fragmented. Different facets of the waveform are separated out and treated as separate,
autonomous entities, as the prism does with a light wave.
In a
nondispersive medium, on the other hand, the waveform is treated as a whole. This is the
case even though the medium may engage that waveform primarily through only a select ratio
of constraints and degrees of freedom which is part of the spectrum of ratios making up
the structural character of the waveform. Under such circumstances, there are ratios of
constraints and degrees of freedom inherent in the medium which serve to vector phase
shifts along focused, heuristic lines.
In the case
of dispersive hermeneutical mediums, there is a relative absence of such inherent ratios.
As a result, hermeneutical activity tends to fragment the incoming waveform and break it
apart in ways that prove to have no focus or direction to them. Under such circumstances,
there tends to be an increased rate of phase shifts as a line or lines of focus is sought.
This increased frequency of phase shifts gives expression to dispersive tendencies as the
waveform is propagated through the recipient's phenomenological/hermeneutical medium.
One could
construe education as a process in which there is an attempt to take a normally dispersive
medium (namely, the undisciplined, unfocused, un-channeled hermeneutical operator at work
within various individuals) and generate circumstances which are conducive to the
transformation of that dispersive medium into a chaotic dynamic, out of which will emerge
a nondispersive medium capable of lending focus, direction, orientation, organization and
so on to waveforms encountered during the course of the hermeneutical operator's activity.
Looked at from another direction, the task of education might be construed as that of
guiding the transition from one kind of nondispersive medium (which may be constructed
around false beliefs, problematic values, weak or nonexistent congruence functions,
untenable mapping relationships, unproductive expressions of the interrogative imperative,
and so on) to a better focused, more stably oriented, more heuristically valuable
nondispersive medium.
One also
might note that keeping some sort of balance is important between: (a) the tendencies of a
system to focus, direct and orient the activity of the hermeneutical operator in the sense
of a nondispersive medium, and (b) the tendencies of a system to be open to proceeding and
exploring in fluid, flexible ways. After all, orientation and focus are, in their own
ways, biases that, sometimes, are capable of closing one off to certain heuristic
possibilities.
The
interrogative imperative may assume a large share of the responsibility in keeping alive
this dimension of openness. However, maintaining an appropriate sort of openness by means
of the interrogative imperative, involves a delicate balance.
On the one
hand, one should not ask so many questions that everything becomes unstable and
unreliable. Yet, on the other hand, one needs to retain a flexibility or receptivity -
through a questioning, inquiring probing - to a variety of possibilities.
Such
possibilities may help one improve one's present understanding, or they may take one in
directions that will enrich, deepen, broaden or inspire one's understanding with respect
to new horizons and new levels of scale. This suggests that developing educational
strategies which are designed to help the individual to develop balanced modes of inquiry
and questioning may be of fundamental importance.
The
interaction between individual and society occurring during the process of education is a
complex dialectic transpiring on a variety of levels of scale. For instance, this
dialectic involves a variety of institutional currents which are aimed at preserving
certain cultural principles of identity with respect to different kinds of historical,
social, religious, philosophical, economic, and political transformations. Nevertheless,
there are many currents, introduced by individual members of the community or culture,
that give expression to dissipative structures in relation to various cultural and/or
individual principles of identity.
Dynamic
hermeneutical equilibrium exists with respect to the aforementioned dialectic when the
ratio of unanswered questions to defensible congruences is low. In addition, in order for
hermeneutical equilibrium to exist, the character of those questions which are unanswered
must not be of a fundamental or essential nature. However, when the ratio of unanswered
questions to defensible congruences is high, and/or the character of the questions left
unanswered are of a fundamental or essential nature, then, a far from equilibrium
condition exists.
The
relationship between equilibrium and dissipation gives expression to the hermeneutical
remainder theorem. This theorem focuses on the quantitative and qualitative character
of the questions which are left over in a given hermeneutical context at a given time.
Dissipative
structures arise when, through a series of engagements by semiotic quanta, problems are
generated with respect to: characterization, identifying reference, the interrogative
imperative, inferential mappings and/or congruence functions, in relation to some given:
theme, event, issue, idea, value, understanding, belief, theory, model, and/or
methodology. Perhaps one of the primary modes of creating conditions conducive to the
generation of dissipative structures is through the interrogative imperative which has the
capacity to push a given hermeneutical context, which previously had exhibited dynamic
equilibrium, to far from equilibrium conditions. The impetus for the interrogative
imperative can come from both sides of the educational process - that is, the individual
as well as society.
Out of these
far from equilibrium conditions may arise a dissipative structure which serves as seed for
the development, construction, generation or emergence of a new hermeneutical attractor.
In fact, a form of catastrophe theory, adapted to the structural character of
hermeneutical gauge field theory, may be applicable to this issue of dissipative
structures.
Essentially,
such a theory of hermeneutical catastrophe would attempt to determine and grasp the
character of the dissipative structures which might arise out of far from equilibrium
conditions. In addition, this sort of theory would attempt to map out the kinds of problem
that might emerge under such conditions.
Part of
educational theory would be directed toward trying to show how to bring about such
catastrophes in the beliefs, theories, values, ideas, methodologies and understandings of
the student in as heuristically constructive a fashion as is possible. Obviously, one of
the dangers here involves the possibility for indoctrination in which the individual
becomes a passive and/or unwilling and/or unwitting participant in a catastrophic
transition process directed toward establishing certain kinds of beliefs, values, ideas
and so on in the understanding of the student.
Approached
from another perspective, hermeneutical catastrophe theory could be seen as an exploratory
journey into certain aspects of the creative process. In other words, the student is
introduced to a set of algorithms, strategies and methodologies designed to help the
individual to discover ways of resolving the tensions, stresses and so on which have been
generated during the transitions or phase shifts to far from equilibrium conditions.
Resolution
would result from the development of a new hermeneutical attractor capable of removing the
tensions, stresses, etc. which arose during the phase shift away from dynamic equilibrium.
Such hermeneutical attractors are characterized by giving expression to a low ratio of
unanswered questions to defensible congruences in line with the requirements of the
remainder theorem. Essentially, one is looking for an hermeneutical tensor-matrix of the
right structural character through which one can re-establish equilibrium and/or restore
various kinds of phenomenological, ontological or hermeneutical symmetries.
The other
side of the dialectic of the educational process involving individual and society concerns
the drive to preserve various kinds of hermeneutical symmetries involving cultural,
religious, political, epistemological, economic and individual identity orientations. Both
sides are fighting to preserve those sorts of symmetries that will permit a state of
hermeneutical equilibrium to be maintained.
Unfortunately,
the symmetries which each side is attempting to preserve are often in conflict with one
another. Furthermore, often times, neither side pays very much attention to the
fundamental role which needs to be played by the hermeneutical remainder theorem in the
dialectical engagement of individual and society during the educational process.
Hermeneutical gauge field theory
When one can
preserve various kinds of symmetries in a field despite subjecting the quanta of that
field to different sorts of phase transformation, one has what physicists refer to as a
gauge symmetry. While one does not need to know the absolute phase value of the quanta of
a field in order to be able to make measurements or run experiments with respect to that
field, one does, nonetheless, have to select a gauge convention in
order to be able to specify differences in phase value. Usually, the gauge convention
which is chosen permits one to measure phase differences in terms of angles relative to
some given point of reference.
The concept
of gauge symmetry was first developed by Hermann Weyl in the early 1920s. This idea arose
during the course of Weyl's attempts to marry Einstein's general theory of relativity to
electromagnetic phenomena. At a certain point in his theoretical
deliberations, Weyl needed to provide a means of deploying separate standards of time and
length for every point of space-time in order to be able to preserve invariance in the
face of, for example, spatial contractions or dilations.
The means
Weyl had in mind for realizing his scheme of separate standards of time and length for the
various points of space-time was akin to a method used by machinists. Machinists used
different polished 'gage' steel blocks to establish standards of length in various
circumstances. Like the machinists, Weyl proposed that measurements in physics required
the selection of standards which were capable of being varied from one circumstance or
point to another. This procedure of measurement is sometimes known as a gauge or
calibration invariance.
In modern
gauge theory certain changes and additions have been made in relation to the foundations
laid by Weyl nearly 70 years ago. Among the most fundamental of these changes has been the
substitution of phase angles for lengths as the basis for selecting a gauge or standard of
measurement.
To say that
an electromagnetic field conveys forces between, or among, the charged particles of that
field, is to indicate that the transmission of force is capable of altering, in various
ways, the character of the particles which come under the influence of these forces. From
the perspective of quantum field theory, the phase of an electron field is shifted
whenever an electron of the field absorbs or emits a photon, and such a shift in the phase
of the electron field is considered to be one of the ways in which the transmission of
forces (that occurs through the exchange of photons which are the carriers of force in the
electromagnetic field) is capable of altering the character of particles in that field.
From the
perspective of hermeneutical gauge field theory, the phase of the hermeneutical field is
shifted whenever a semiotic quantum of the field is absorbed or emitted by other semiotic
quanta of the field. In this sense it is like a photon-photon interaction. Therefore,
since the semiotic quantum is the carrier of force of the hermeneutical field, any
semiotic-semiotic quantum interaction will give expression to a complex vector/tensor
dialectic of phase transformations.
The shifting
of phase relationships alters the structural character of that which engages or is engaged
by the presence of such a quantum event. This results in the shifting of the ratio of
constraints and degrees of freedom of the point-structure, or neighborhood or latticework
which is involved in the transaction.
Obviously,
in the light of the foregoing comments, a fundamental aspect of the educational process
will revolve around the kinds of phase shifts which occur as a result of the packages of
semiotic quanta that are directed at the student. These packages of semiotic quanta come
in the form of: attitudes, beliefs, values, ideas, goals, interests, motivations, fears
and so on, which are being transmitted through: the teachers, the text materials, the
officials of the school, other students, the rules of the school, the surrounding
community, and so on.
In other
words, the educational process can be construed as a force field capable of altering or
generating spectra of ratios of constraints and degrees of freedom by means of, among
other things, causing shifts in the phase character or orientation or phase relationships
of the point-structures, neighborhoods, and latticeworks of the students. The carrier or
vector boson of the educational force field is manifested in the form of the exchange of
semiotic quanta.
The phase of
an electron wave is firmly established in a given field if one assigns determinate values
both to the electric and magnetic vector potentials of that field. Moreover, since the
electromagnetic field gives expression to local gauge symmetry, one can adopt different
values for these vector potentials at every point of the field.
If this
happens, the phase of the electron field also will vary at every point of the field.
However, the phase which is fixed at any given point will always be a reflection of the
convention which is used to set the values of the electric and magnetic vector potential
for the various points of the field.
In order to
establish the phase character of a given point of the hermeneutical field, one must assign
determinate values for the vector/tensor potentials of each of the six components of the
semiotic quantum in relation to a given focal/horizonal context. Once these values have
been assigned, the phase relationships will have been fixed for that semiotic quantum in a
given focal/horizonal context.
Moreover,
just as one can assign different electric and magnetic vector potentials at each point of
the field, so too, one can assign different vector/tensor potentials for each of the six
isotopic-spin states of the semiotic quanta at each point of the hermeneutical field,
relative to some given focal/horizonal point(s) of reference. This means the phase
relationships which are fixed at each point of such a hermeneutical field will be a
function of the assigned values of the isotopic-spin states of the semiotic quanta at each
of the points.
The key to
theories involving local gauge symmetry is to find a field capable of generating, or
giving expression to, units that carry force in a particular way. This particular way must
permit one to perform transformation operations of different sorts from point to point in
the field while preserving symmetry of the laws operative in the field.
The
structural character of the unit which is the carrier of force in the field is made up of
a spectrum of ratios of constraints and degrees of freedom. Shifts or transitions in this
spectrum of ratios is the means by which transformations are propagated or communicated
throughout the field.
In effect,
the carrier of force must be able to transmit the character of the transformation
accurately but in such a way that the laws of the field are preserved. If one selects a
carrier of the field force which has an inappropriate structural character, then, either:
(a) the laws of the field will not remain invariant; (b) the transformation operations
which are to be performed will not be transmitted accurately, or (c) both (a) and (b) will
occur.
Obviously,
the 'trick' in hermeneutical local gauge symmetry theory is to find a semiotic quantum
with the right kind of isotopic-spin characteristics. However, in order to do this, the
tensor-matrix of the isotopic-spin character of the semiotic quantum must be able to
preserve certain kinds of symmetries of structural character that exist in the aspect of
ontology to which the transformational operations of hermeneutical transduction are being
applied, and to which such operations are giving identifying reference.
The
symmetries to be preserved concern various laws of structural identity concerning the
character of different aspects of ontology. The initial transformations which are applied
to the ontological field are various kinds of sensory transduction processes.
The semiotic
quantum gives rise to a field which has to be introduced in order to establish local gauge
symmetry with respect to the ontological field being transduced by sensory processes.
However, in order for the field generated by the semiotic quantum to accomplish this task,
the field must be assigned an appropriate hermeneutical tensor-matrix. This tensor-matrix
summarizes the dialectical shaping influences of the individual hermeneutical components
of the semiotic quantum in a way that is capable of reflecting (that is, preserving) the
invariances or symmetries inherent in the structural character of the ontological field.
Each
individual cell of the hermeneutical tensor-matrix is determined by a process akin to the
process of differentiation. In fact, the values for any given cell of the matrix is
determined by taking a second derivative of the sensory transduction process which is the
first derivative value.
By noting
changes or transitions, through the process of hermeneutical differentiation, in the rate,
shape, character, orientation, and so on, of the structural character of a given
transduction process, one constructs a point-structure whose structural character provides
an interpretation of, or reflection (if correct) of, what makes a sensory transduction
process of such structural character possible. By repeating this process of taking
hermeneutical second derivatives of different first derivative sensory transduction
structures, one is in a position to generate or construct neighborhoods, lattices and
latticeworks.
This process
of hermeneutical differentiation takes into account the structural character of the way
different semiotic quanta at various points in the phenomenological field carry the
hermeneutical force. Among other things, the carrier of hermeneutical force transmits
phase information relative to various contexts of focal/horizonal dialectic.
The
structural character of the hermeneutic mode of taking second derivatives is a far more
complex process than is the mathematical process of taking a second derivative. This is
the case because the tensor-matrix of the semiotic quantum allows for a lot more
components, dimensions, degrees of freedom and dialectical interplay (the counterpart to
the idea of rates of change) to be worked into the qualitative hermeneutical calculation
than is the case with respect to usual mathematical instances of taking a second
derivative in relation to rates of change involving certain quantitative variables.
When one
wishes to key in on the contribution of any particular component of the hermeneutical
operator, it seems to be akin to the process of doing partial differential equations. In
other words, one holds constant the contributions of all the other shaping components,
and, then, one proceeds to explore what happens to structural character when one
manipulates one component across a variety of transformations and conditions.
In a sense,
the cells of the hermeneutical tensor-matrix of a given semiotic quantum represent a sort
of collective equation which has to be solved in order to determine if there is a means of
reflecting (i.e., preserving symmetry) the structural character of a given aspect of
phenomenology and/or ontology across the transformational currents which are impinging on
each cell of the semiotic quantum. Solving the equation means coming up with a set of
assignment values of hermeneutical isotopic-spin which are capable of preserving certain
kinds of invariance with respect to: (a) the structural character of a given sensory
transduction, as well as (b) the structural character of that (i.e., a given aspect of
ontology) which helps make possible a sensory transduction (i.e., experience) of such
structural character.
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