Quantum Gauge Theory - Part 2
Semiotic quanta isotopic-spin and hermeneutical tensors
In quantum
field theory, when one describes the interaction of two particles, the force which is
manifested during the course of that interaction is in the form of an exchange of virtual
particles. Consequently, from the perspective of quantum field theory, a force is
construed as that process which mediates various kinds of quantum interactions by means of
the exchange of virtual particles.
The mass of
the virtual particle being exchanged determines the range of the force manifested during
the transaction. Thus, because the graviton, which has been postulated to be
responsible for mediating gravitational effects, is believed to have a mass of zero, the
range of the gravitational force is considered to be infinite. The same is said to be true
of the range of the massless photon which is responsible for the electromagnetic force.
On the other hand, the massive, relatively speaking, W boson that helps mediate the
weak force has an effective range of approximately 10-15 centimeters, which is
exceedingly small.
The number
of different states that can be assumed by a field's force-carrying quantum determines the
number of components for that field. Moreover, the number of different states or
orientations which is possible for a field's quantum is a function of the spin angular
momentum of the different particles which make up a given field.
Spin angular
momentum can only have discrete integer or half integer values. Both the magnitude of the
spin, as well as its direction, are assigned these discrete integer or half integer
values.
The general
rule for any given field quantum is that the number of possible states for a quantum is
equivalent to two times its spin's magnitude, plus one. For example, the
electron, which has a spin magnitude of 1/2, will have, according to the above rule, two
spin states. The photon, on the other hand, has a spin magnitude of one, and, therefore,
will have three spin states.
The graviton
is postulated to have a spin magnitude of 2, which means that, according to the foregoing
rule, it has five spin states. However, the case of the graviton is complicated somewhat
by its massless nature.
According to
theory, the graviton is massless and, therefore, travels at the speed of light. This means
that, unlike quanta with finite masses, the graviton's transverse spin states will not be
observed (This also is true of the one transverse spin state of the massless photon).
Since the graviton is believed to have three transverse spin states, only two of the
graviton's spin states are capable of being detected.
The
gravitational field has 10 components. Not all of these components are independent from
one another. As a result of the non-independence of some of these components, the
mathematical techniques used to solve problems involving these components involve tensors.
Consequently, the gravitational field is referred to as a tensor field.
Similarly,
in the case of the hermeneutical field, not all of the components of that field are
independent from one another. They have covariant and contravariant relationships with one
another. Therefore, the hermeneutical field can be considered to be, given certain
qualifications, an n-component tensor field.
The kinds of
stress, tension or dialectical relationship which the components of the hermeneutical
field can have with one another may be more complex than can be expressed through the
ideas of covariance, contravariance and mixed tensors in the usual mathematical sense.
Nevertheless, the term "tensor" is retained in order to allude to the complexity
of the stresses, tensions and dialectical currents which are possible in the hermeneutical
field.
The various
aspects of the semiotic quantum (such as reflexive awareness, identifying reference,
characterization, etc.) are comparable to a sort of isotopic-spin.
What is meant here is similar to the structural character of the nucleon.
The proton
and neutron are alternative versions, states or expressions of a single particle known as
a nucleon. Depending on its internal spin characteristics, the nucleon sometimes manifests
itself as a proton, and at other times, the nucleon manifests itself as a neutron.
The semiotic
quantum, like the nucleon, also will manifest itself in different ways depending on its
internal spin characteristics. However, the internal spin characteristics of the semiotic
quantum are far more complex than is the case for the isotopic-spin of the nucleon. In
other words, rather than having only two alternative modes of expression as is the case
for the nucleon, the semiotic quantum has six distinct modes of expression, together with
an indefinite variety of dialectical combinations of these six basic modes.
The
character of hermeneutical isotopic-spin is like a tensor-matrix (a hermeneutical
tensor-matrix which has similarities to, but is quite different from, the mathematical
notions of either a tensor or matirx) in which the individual cells of the matrix weave
together covariant, contravariant and mixed currents from the other five orientations or
spin states of the semiotic quantum.
In addition,
the tensor character of the semiotic quantum's isotopic-spin takes into account what might
be referred to as transvariant currents. These sort of currents do not conform to
the largely linear characteristics of covariant tensors, contravariant tensors or mixed
tensors. Transvariant currents refer, instead, to multi-dimensional, non-linear tensions,
stresses, and dialectical activities that are capable of affecting the manner in which the
semiotic quantum gives expression to its property of isotopic-spin.
The
hermeneutical tensor process of weaving together different currents of the semiotic
quantum's complex isotopic-spin takes place in a context of specific experiences, ideas,
values, beliefs, actions, desires, emotions, motivations, needs, sensations, and so on.
With the passage of time, there is a stream of differentiated semiotic quanta.
Individual
semiotic quanta are generated through focal/horizonal dialectical activity. Said in
another way, focal/horizonal dialectical activity is the gateway through which semiotic
quanta are emitted.
Focal/horizonal
dialectical activity is rooted in the phenomenology of the experiential field. In fact,
the hermeneutical field is embedded in the phenomenological field as a potential for
generating structure or curvature in that phenomenological field. This potential is
activated, or turned on, in one of two cases: (a) through inducement and (b)
spontaneously.
In the
former case, semiotic quanta are generated or released when certain thresholds of the
phenomenology of the experiential field are surpassed. This is somewhat akin to what
happens in the case of the photoelectric effect when in-coming photons cause electrons to
be emitted as a result of raising the energy level of those electrons engaged by the
photons.
Phenomenological
thresholds do not exist just with respect to sensory stimuli. They also exist in relation
to: motivation, memory, fantasy, interests, likes, dislikes understanding, beliefs,
values, and so on.
On the other
hand, when semiotic quanta are spontaneously generated or released, this is an expression
of underlying attractors involving, for example, beliefs, values, commitments, and
methodological frameworks which aperiodically release semiotic quanta. The spontaneous
release of these quanta can give expression to shifts in attention in relation to various
horizonal components.
In the case
of the spontaneous transition in the orientation of intentionality, once this sort of
semiotic quantum arises, an investment is made in a given horizonal attractor. The
selection of investment venue can be arbitrary, or it can be made on the basis of a series
of brief dialectical interludes (a sort of mini-sampling process) with different horizonal
attractor candidates (for example, as a result of general curiosity, interest, question,
or aptitude - which constitutes a natural inclination inherent in the individual).
The
dialectical activity of the semiotic quantum brings together a number of dimensions such
as: time, space, materiality, energy, consciousness, will,
and understanding. However, the primary contribution of the semiotic quantum
concerns its various modes of hermeneutical isotopic-spin, along with their concomitant
capacity to engage, and be engaged by, such dimensions across a variety of levels of scale.
Semiotic
quanta are discrete point-structures that are linked together into neighborhoods, lattices
and latticeworks through a network of phase relationships. These phase relationships are
bound together in the form of hermeneutical counterparts to strings, sheafs, fiber
bundles, and so on. In other words, hermeneutical structures are
generated and woven together in an attempt to 'cover', or account for, various aspects of
the phenomenological manifold to which they are experientially linked.
The
hermeneutical operator or semiotic quantum is an intrinsic part of the phenomenology of
the experiential field. Indeed, it gives expression to the "curvature" of the
different levels of scale of the n-dimensional character of the phenomenological manifold.
When the
hermeneutical operator generates a structure that accurately reflects some aspect of the
phenomenology of the experiential field or of some aspect of ontology which makes an
experiential field of such character possible, it has zero curvature - that is, it does
not distort what it reflects. When the structure which is generated does not accurately
reflect the structural character of that to which identifying reference is being made,
then, the curvature of the phenomenology of the experiential field, due to the presence of
such semiotic quanta, will be some non-zero quantitative and/or qualitative value. The
greater the degree of distortion, the greater will be the magnitude of the non-zero
curvature value.
Gauge fields in physics and hermeneutics
A field is a
region of space-time for which some variable quantity has been assigned to each point of
that region. In broad general terms, there are two kinds of fields which are possible -
namely, scalar and vector fields.
A scalar
field exists when a magnitude, without orientation, is assigned to each point of the
field. For instance, if one were to assign a temperature to every point of a given region
of space-time, this would constitute a scalar field.
A vector
field exists when one adds the property of orientation to the magnitude which is assigned
to every point of a given region of space-time. Thus, a vector field has a directed
magnitude assigned to every point of a given region of space-time. For example, if one to
were to describe a field in terms of the thermal currents which run through it, then, such
a field would be a vector field.
A gauge, in
field theory, refers to a standard of measurement which is capable of undergoing change as
a result of being transported to different points of the field. If the value
of measurement of the gauge changes during the process of transportation, such changes are
said to be due to the effect of the field on the gauge.
For example,
since a field gives expression to a vectoral quantity, the strength of the field has the
capacity to register on the gauge both with respect to magnitude of intensity as well as
with respect to orientation or direction of that intensity. Therefore, if one's
measurement gauge is a dial which contains a pointer, then, the pointer will take on
different orientations, depending on, say, the varying strength of the field, as the gauge
is moved about the field.
Any field
which is capable of bringing about the foregoing sorts of changes in the gauge as it is
transported about the field is known as a gauge field. Moreover,
because a gauge field actually involves the dialectic between a measuring methodology and
a given ontological field, the gauge field incorporates a set of rules. These rules permit
one not only to describe, but keep track of, the transitions undergone by the gauge. This
rules-property of the gauge field enables one to make comparisons of the strength of the
field at different points in that field.
The
hermeneutical operator also satisfies the conditions for a gauge field. The following
points outline how the gauge field conditions are satisfied.
To begin
with, the hermeneutical operator is a standard of measurement. As is true in all cases of
measurement, the operator provides a methodological mode of engagement with that which is
to be measured. This mode of engagement is intended to provide a standard which can serve
as a uniform basis for comparison (either quantitatively or qualitatively) from engagement
to engagement.
Of course,
the idea of measurement in relation to the hermeneutical operator is considerably more
complex than normal modes of measurement. This is primarily because of the problems which
surround the establishing of a uniform basis of comparison both for a given individual, as
well as for a community of individuals.
To be sure,
the number and general structure of the components of the hermeneutical operator are the
same from individual to individual. In other words, there are six basic modes or
components in the hermeneutical operator activity of every human being capable of even
minimally intelligent behavior.
Moreover,
the general character of these components or modes is the same in everyone in the sense
that they involve: identifying reference, reflexive awareness, characterization, the
interrogative imperative, inferential mappings and congruence functions. In addition, the
hermeneutical operator always manifests itself in the context of a focal/horizonal
dialectic.
However,
despite such common themes in the character of the hermeneutical operator as it is
manifested from one person to the next, there are tremendous differences in the power,
sophistication, scope, and quality of the way the various components of the hermeneutical
operator are given expression as one goes from individual to individual, community to
community and historical period to historical period.
Nevertheless,
while the degree of difficulty of the kinds of problems encountered in the hermeneutical
search for a uniform basis of comparative measurement may be more complex than is the case
with many instances of physical measurement, such problems really are only variations on
the sorts of themes that arise regularly in the theory of measurement underlying the
physical sciences. Even in the, relatively speaking, less complex problems that surround
the issue of measurement in the physical sciences, there are a variety of sources of
contamination and/or fluctuation which affect the uniformity of measurement from one
situation to the next and from one individual to the next.
Furthermore,
like its counterparts in the physical sciences, the hermeneutical operator is a standard
of measurement capable of undergoing changes as a result of its being transported (due to
shifts in intentionality and the concomitant transitions in the focal/horizonal dialectic)
from place to place in the phenomenological field. This satisfies the conditions of a
gauge.
In addition,
when the hermeneutical operator gauge is transported from point to point in the
phenomenological field, it is capable of responding to, or being affected by, differences
in the strength of the field, at various points in that field. However, in the case of the
hermeneutical operator, although the strength of the field can be expressed as a vectoral quantity,
nonetheless, under appropriate circumstances, the strength of the field also can be
expressed as a vectoral or tensoral quality.
This means
the structural character of the orientation aspect of the hermeneutical vector field
cannot be restricted to purely quantitative issues. It will include, as well, qualitative
issues such as meaning, value, purpose, likes, dislikes, attitudes, beliefs, and so on.
Finally, the
hermeneutical operator's engagement of the phenomenology of the experiential field
generates a set of rules or principles that permit one to both describe, as well as keep
track of, the changes in the strength of the phenomenological field as the hermeneutical
gauge is moved about from point to point in the field. This set of rules or principles
consists of the field equations which give expression to the spectrum of ratios of
constraints and degrees of freedom that is characteristic of the dialectical activity of
the six components of the hermeneutical operator over time.
Thus, in
view of the foregoing considerations, the hermeneutical operator's dialectical engagement
of the phenomenology of the experiential field satisfies the conditions of a gauge field.
In short, the dialectics of this engagement involve a standard of measurement capable of
being affected by variations in the strength of the field through which the gauge is
moved. Moreover, this same hermeneutical gauge operates according to a set of rules or
principles that permit one to describe and keep track of changes in field strength as the
gauge is transported about the phenomenological field.
One of the
dynamic aspects of the hermeneutical gauge field, however, needs to be highlighted, to
some degree. While this aspect actually is present in all gauge fields, it's role tends to
be de-emphasized.
More
specifically, the hermeneutical gauge is not just a passive recorder of fluctuations of
the phenomenological field. The hermeneutical gauge also is capable of actively operating
on that field and generating interpretations of the significance or meaning of the changes
in field strength which are registered. Consequently, as is the case with any mode of
measurement (but especially in light of the active, interpretive, projective character of
the hermeneutical operator), the hermeneutical operator is capable of distorting the
structural character of that which is being measured.
Phase and orientation
Both the
magnetic and the electric aspects of the electromagnetic field are vector quantities. This
is because each point of the region of space-time which characterizes the field has a
directional component as well a magnitude associated with it.
According to
Maxwell's theory, the distribution of electric charges around a given point of the field
gives expression to the strength of the field at that point. In practice, however, people
who use the theory often speak in terms of the potential or voltage that exists in a given
region of the field. This potential also is rooted in the charge distribution. More
specifically, the potential is construed in terms of charge density for a given region of
the field. The higher (lower) the charge density, the higher (lower) the potential.
Similarly,
the value of a hermeneutical field at any juncture is determined, ultimately, by the
density, together with the qualitative orientation and phase relationships, of the
semiotic quanta of that field. These components of density, orientation and character of
the phase relationships determine the vectoral/tensoral 'charge' potential of the
hermeneutical field.
This charge
potential is expressed through the dialectic of focus and horizon. The vectoral direction
of the hermeneutical charge potential can go either: from the focus to the horizon; from
the horizon to the focus; or, both ways simultaneously.
If an
electrical field is kept stationary, it will not generate a magnetic field, and,
therefore, the field will be a "pure" electric field. If one were to lower (or
raise) the potential of the entire field, there will be no detectable difference in the
general characteristics of the field (aside, of course, from the decrease/increase in
potential) in any measurements which are taken before and after the change in potential.
Thus, one
observes a case of global gauge symmetry with respect to the transformation of the field's
potential. In other words, since the electric field's characteristics are a function of
differences in potential, rather than absolute potential, as long as there are no
differences of potential introduced into the field, the general characteristics of the
field will be preserved, and global gauge symmetry will be observed.
If the
aforementioned stationary electrical field is moved, it will generate a magnetic field.
While, in terms of Maxwell's theory, the magnetic field is the result of the moving
electric charges, the general practice is to speak in terms of a magnetic potential
(similar to the idea of electric potential) as the cause of the magnetic field. If, in
turn, the magnetic field is moving, it generates an electrical field.
The
dialectic between moving electric and magnetic fields allows one to establish local gauge
symmetries in the electromagnetic field with respect to various kinds of transformations.
This is because every local transformation of the electric field is compensated for by a
corresponding change in the associated magnetic field. The reverse is also the case.
Therefore, despite local transformations, the general characteristics of the
electromagnetic field remain invariant, and, as a result, local gauge symmetry is
preserved.
In the
quantum interpretation of electromagnetic phenomena, fluctuations in the field's
electrical potential involve shifts in the phase character of the wave to which the
electron gives expression. Because the electron has two spin states, the
field to which it gives expression also is described in terms of two components.
However, the
mathematical means of representing the two wave components involves complex numbers. This
means one of the two components will be expressed as a real number, while the other of the
two components will be expressed as an imaginary number.
From the
quantum perspective, an electric field consists of a collection of quantum wave packets.
The different amplitude values of these wave packets are reflected by changes in the
magnitude of both the imaginary, as well as, the real components of the complex numbers
used to represent the wave packets.
To be
precise, the complex numbers used to describe oscillations in amplitude values of the wave
packets do not actually describe a given electron's field. When the real and imaginary
components of the complex number are squared, they describe the probability of finding an
electron of a particular spin character at a particular juncture of space-time in the
field.
If one
wishes to determine a complete description of the oscillatory character of an electron
wave packet, one needs to work out certain equivalencies. For instance, the wavelength of
the wave-packet's oscillatory character is proportional to the electron's momentum,
whereas the frequency of the oscillation of the wave packet is proportional to the energy
of the electron. In addition, one needs to take into consideration the phase character of
the oscillation.
Phase refers
to the degree of displacement of some aspect of an oscillation relative to a certain point
of reference. This point of reference usually is selected arbitrarily.
Generally,
phase is measured in terms of an angle.33 Moreover, the phase angle of the real
component of the wave has an inverse relationship to the phase angle of the imaginary
component of the wave. Consequently, whenever one complex component of the wave has a zero
value, the other complex component has a maximum value.
The phase of
the oscillatory character of the electron's wave packet is a function of the relationship
of the two components of the complex number. This is the case not only with respect to
some arbitrarily chosen reference point, but also with respect to one another.
Similarly,
the phase of the oscillatory character of the semiotic quantum's waveform structure is a
function of the relationship of the six components of the hermeneutical operator, not only
with respect to a given focal/horizonal point of reference, but also with respect to one
another. Like its physical/mathematical counterpart, the focal/horizonal point of
reference that is used to study phase properties can be chosen arbitrarily. Nonetheless,
the arbitrary choice of focal/horizonal reference point can assume great importance when
one is attempting to interpret the possible ontological significance of phase
relationships in a given context.
The
hermeneutical counterpart to using an angle as the measurement index of phase, is
orientation. The orientation of a given aspect of the hermeneutical operator is measured
(in a qualitative sense) in terms of not only how that aspect relates to the current
focal/horizonal point of reference but, also, how that aspect relates to other instances
of hermeneutical operator activity. These other instances of hermeneutical operator
activity may involve one's own past or future operator activity, as well as the
hermeneutical operator activity of other individuals.
This is
where intra-personal and intersubjective networks of phase relationships arise. These
complex phase relationships can be given expression as some form of tensor-matrix -
whether covariant, contravariant or transvariant.
Furthermore,
just as when, say, the real component of a complex number has a maximal phase value, then,
the imaginary component of the complex number will have a zero phase value, so too, when
any of the components of the hermeneutical operator (with one exception to be mentioned
shortly) has a maximal phase value, all of the other components of the hermeneutical
operator will have a zero phase value. Thus, when, for example, identifying reference has
a maximal phase value relative to a given focal/horizonal point of reference, all of the
other components of that semiotic quantum will have a zero phase value.
Therefore,
the structural character of the given semiotic quantum's oscillatory waveform will be
dominated by the phase character of the identifying reference component. However, when
none of the hermeneutical isotopic-spin components of the semiotic quanta have a maximal
phase value, then, the other components will have some non-zero phase value.
From the
perspective of hermeneutical field theory, the hermeneutical field (and, therefore, the
phenomenological field in which it is rooted and for which it is the source of curvature)
consists of a stream of, or series of currents of, semiotic quanta, all of which have a
phase relationship with the on-going focal/horizonal context. This means all semiotic
quanta which are not a part of the focal aspect of such a context will be part of the
horizonal aspect of that context. As horizon, they are stored in the form of a phase
relationship of a given tensor-matrix character (i.e., a memory).
One cannot
determine the phase of an electron field because, in order to accomplish this, one would
need to be able to establish the individual contributions of both the real and imaginary
components of the complex representation of the wave packet. However, these two components
are so inextricably intertwined in the mathematical representation of the wave packet that
their individual contributions cannot be separated out or distinguished through the
mathematical means employed to methodologically treat these components. All that can be
measured with respect to phase are phase differences between various aspects of the field.
As far as
the problem of separating out the character of various contributing components to phase
structure is concerned, one may be in a somewhat better position in the context of
hermeneutical fields than is the case with physical fields. This is so because the
different components of the hermeneutical operator tend to leave a characteristic
signature when they are present. Even if an absolute, precise quantitative determination
of the contributions made by different aspects of the hermeneutical operator cannot be
made, one, nonetheless, can detect the relative contributions of the shaping activity of
the six different components of the operator with respect to phase structure.
Because a
semiotic quantum's inner spin states of identifying reference, characterization, reflexive
awareness, the interrogative imperative, inferential mapping and congruence functions all
have different orientations to a given focal/horizon context, each of these components
leaves a distinct 'phase signature imprint' on the structural character of the semiotic
quantum. The different orientations or phase relationships which the various isotopic-spin
states have with the focal/horizonal dialectic can be cited as reasons why any given
semiotic quantum gives expression to an identifiable ratio of constraints and degrees of
freedom with a characteristic phase signature imprint.
As a result,
unlike the case with phase in the electron field, the phase character of a hermeneutical
field can have an effect on the manner in which one assigns meaning, value, purpose,
significance, or orientation to a given point-structure, neighborhood or lattice. One
cannot add or subtract any phase angle to the hermeneutical field and expect the symmetry
of the field to be preserved as is the case with respect to the adding or subtracting of
phase angles to the electron field.
However, the
foregoing comments notwithstanding, phase difference between any two given points of the
hermeneutical field also plays an important role. In other words, differences in
orientation with respect to a given focal/horizonal point of reference form an important
source of information concerning the structural character of certain aspects of
phenomenological experience and/or hermeneutical understanding. These differences of phase
orientation (which are given expression through phase relationships) lead to, among other
things, constructive and destructive patterns of interference that frequently, are
expressed in terms of covariant, contravariant and transvariant tensor patterns of
dialectical interaction.
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