Quantum Gauge Theory - Part 5
Hermeneutical string, fiber bundles, sheafs and logic
In
analogical form, the strength of a unit of hermeneutical charge (i.e., the impact of a
given hermeneutical operator at a given point in the phenomenological field) will be
proportionate to the number of lines of force (i.e., the phase relationships) which pass
through a unit area of surface (i.e., the point-structure or neighborhood or latticework)
of the phenomenology of the experiential field. However, in addition to the number of
lines of force which pass through a given point of the phenomenology of the experiential
field, the qualitative character of those lines also becomes extremely important. In other
words, the qualitative character of the spectrum of the ratios of constraints and degrees
of freedom that give expression to the phase relationships linking the 'oppositely'
charged point-structures of the hermeneutical operator and the phenomenological field must
be taken into consideration when assessing the intensity or strength of the impact of the
hermeneutical operator on the phenomenological field.
From the
foregoing perspective, one might describe a hermeneutical string in the
following manner. Such strings are compressed bundles of phase relationships. In addition,
the ideas of sheaf and fiber bundle may have analogical implications for
development of the notion of a hermeneutical string. More specifically, in the context of
hermeneutical strings, the notions of sheaf and fiber bundle may help lend definition to
the idea of inference.
Inference is
not necessarily about truth. Inference is about the issue of continuity.
In other
words, inference is about what links one idea with another. Inference is about the way
such continuity manifests itself and to what degree continuity manifests itself.
Consequently, inference really is about the process of proposing or seeing mappings that
describe the structural character of the phase relationships between one hermeneutical
point-structure and other such points.
Hermeneutical
strings, sheafs, fiber bundles may all be different ways of referring to how inferential
mappings operate. A hermeneutical string, for example, may refer to the compressed or
focused character of the spectrum of ratios of constraints and degrees of freedom that
constitute the complex, dialectical relationship between focus and horizon.
Hermeneutical
fiber bundles, on the other hand, might be thought of as a group of such phase
relationships or hermeneutical strings which have a common focus or common set of
linkages. Thus, the hermeneutical fiber bundle represents a set of coupled, multiple
mappings which interact to strengthen, a proposed inference between one point-structure
(neighborhood or latticework) and other such point-structures (neighborhoods or
latticeworks). As such, a hermeneutical fiber bundle, under normal circumstances,
constitutes a stronger inferential argument than does a hermeneutical string.
Exceptions
to the foregoing contention would involve instances in which a hermeneutical string gives
expression to a better insight than does a given fiber bundle. Although hermeneutical
fiber bundles tend to represent powerfully coherent sets of phase relationships,
nonetheless, if this set is rooted in distorted hermeneutical transductions, then, such a
set of hermeneutical fiber bundles may offer less accurate insight than is provided by a
given hermeneutical string.
Finally, hermeneutical
sheafs might be construed as a way of organizing a variety of hermeneutical strings
and fiber bundles in order to 'cover, or account for, the structural character of a given
aspect of the manifold of the phenomenology of the experiential field. In this sense,
hermeneutical sheafs give expression to models or theories.
Hermeneutical
sheafs explore the way a model or theory is held together by a set of phase relationships
between, and among, a variety of point-structures, neighborhoods, and latticeworks. The
purpose of such exploration is to search for arrangements of hermeneutical strings and
fiber bundles that are capable of 'covering' (i.e., explaining or understanding) a given
aspect of the dialectic between phenomenological and ontological manifolds. Therefore,
hermeneutical sheafs place emphasis on the inferential mappings and congruence functions
which lend a theory or model its explanatory structural character vis-a-vis some aspect of
experience as well as the aspect(s) of ontology which helps make experience of such
structural character possible.
Seen from
the foregoing perspective, entailment exists when one can show that the structural
character of the continuity which links two (or more) point-structures, neighborhoods, or
latticeworks, has a particular kind of vectoral mapping character. More
specifically, in order for entailment to be present, one must be able to show: (a) the
structural character of, say, a given point-structure is largely shaped and determined by
the structure(s) with which it is linked through mapping; and (b) the reverse is not the
case.
Under these
circumstances, one would say the point-structure being shaped and determined is entailed
by the structure(s) which is doing the shaping and determining. Thus, entailment suggests
a vectoral component to the mapping process.
The last
four or five paragraphs all tend to point in the same general direction with respect to
the structural character of logic. In effect, logic is the study of continuity, structural
form, and mapping relationships.
Perturbative versus nonperturbative methodologies
The
Heisenberg uncertainty principle concerns the measured relationships between certain conjugate
pairs. One such conjugate pair involves momentum and position. Another conjugate pair
concerns time and energy.
In the
latter case, the uncertainty principle indicates that the product of the uncertainties
surrounding the measured values for energy and time will be greater than, or equal to, a
mathematical expression involving Planck's constant. Thus, if one fixes the measured value
for the time of an event at an extremely small interval, there will be a very large
uncertainty concerning the amount of energy which is associated with the event in
question. Similarly, if one fixes the measured value for the energy of an event, then,
there will be a large uncertainty concerning the precise time when the event occurred.
The
uncertainties concerning the energy which, within a small interval of time, is to be
associated with any given point of a field, give rise to variations in the measured energy
values for that point of the field. These variations are known as quantum fluctuations.
Normally
speaking, when one wishes to calculate the intensity of a certain aspect of the field
between two points of opposite charge, one uses Maxwell's classical field equations.
However, if one wishes to take quantum fluctuations into account in such calculations (as
one would wish to do when dealing with high-energy collisions), then, one must generate
values involving weighted probabilities. Essentially, this means one must obtain an
average of all the quantum fluctuations which are possible at a given point in the field,
after one has assigned each of these possibilities a probability according to the
likelihood of such a fluctuation occurring.
The standard
method for making corrections for quantum fluctuations is, first, to make calculations
concerning various field variables as if the field manifested itself in a classical
vacuum. Then,, one proceeds to introduce a series of successively more complex corrections
for quantum fluctuation.
The more
complex the correction that is introduced, the less likely such a fluctuation will occur
in the field. This aspect of the standard method which involves introducing successively
more complex corrections for quantum fluctuations is referred to as a perturbative
expansion.
The method
of purturbative expansion is only able to produce accurate results if relatively few
corrections for quantum fluctuations are needed to supplement the classical method of
field calculations. Moreover, the corrections being introduced should not be too complex
since this would involve both uncertainties of a larger magnitude, as well as modes of
fluctuation that are more improbable. The larger the magnitude of uncertainty and the more
improbable the magnitude of uncertainty which are introduced into field calculations, the
less probable will be the likelihood that the process of purturbative expansion can
produce a result showing convergence toward a given predicted or observational value.
Field
calculations that do not converge toward some given value, as a result of the size of the
magnitude of uncertainty and improbability which are introduced into those calculations,
are referred to as being nonperturbative. The compression of the lines of force
which is called for in the chromoelectric dynamic approach to quarks is an example of such
a nonperturbative phenomenon.
In a way,
one could treat attempts like Kant's Critique of Pure Reason as a sort of Newtonian
classical approach to epistemology in which the knower is considered to exist in a
largely, non-interactive, ontological vacuum. As a result, the ontological vacuum is not
believed to contribute much in the way of structural currents, transformational energy,
vectoral forces, and so on, to the individual's construction of an understanding. At best,
the ontological vacuum would be supposed to present a relatively static, non-interactive
background against which the largely internal or subjective activity of epistemological
construction takes place.
However, if
one permits ontology to be a quantum-like vacuum, then, ontology comes alive with
interactive potential. Consequently, in order to come up with an epistemological framework
which is capable of taking the effects of such a dialectical, interactive, dynamic
'vacuum' into account in one's epistemological framework, one is going to have to increase
the complexity of one's mode of solving various kinds of issues. In other words, if
ontology gives expression to a quantum-like vacuum that introduces a variety of
fluctuations which disturb, affect, shape, alter and transform various aspects of the
phenomenology of the experiential field, then, one is going to need to introduce
corrective factors into the classical Kantian model of epistemology in order to be able to
reflect the fluctuations of the noumena as they intrude into phenomenological and
hermeneutical space.
The
hermeneutical counterpart to the idea of perturbative expansion would mean one has a
methodology which only is capable of handling relatively small fluctuating intrusions of
the noumena into one's hermeneutical framework. Furthermore, hermeneutical perturbative
expansion methods would tend to assume that progressively more complex treatments would
converge on a single answer which becomes more clear cut and refined as one proceeds.
Hermeneutical
nonperturbative methods, on the other hand, would refer to approaches permitting a rather
free interplay between noumena and phenomena and which do not necessarily converge toward
one answer. This does not mean there is no correct answer or that things are arbitrarily
relative.
A
nonperturbative approach means, instead, that the structural character of any given aspect
of reality tends to be nonlinear in the sense that one will observe self-similar behavior
but not self-same behavior. Therefore, there may be a limit to the convergence aspect of
the methodology.
There is a
second way in which the convergence of a hermeneutical counterpart to a perturbative
expansion approach could breakdown. This could occur when a given aspect of reality had a
number or multiplicity of significances, all of which had to be taken into account in
order to have a proper grasp of the structural character of the phenomenon under
consideration.
In other
words, structural character consists of a spectrum of ratios of constraints and degrees of
freedom. One's methodology cannot converge on any one of these ratios if one hopes to
reflect the character of the phenomenon in an accurate fashion.
In addition,
any given point-structure is a gateway or window to further levels of scale which place
the point-structure in a broader, more complex, less convergent hermeneutical,
phenomenological and/or ontological framework. Therefore, under such circumstances, one
might have difficulty generating a hermeneutical counterpart to perturbative expansion
that resulted in convergence toward a single answer or solution.
Lattice gauge theory in relation to hermeneutical theory
In his
paper: "The Lattice Theory of Quark Confinement", Claudio Rebbi proposes a lattice
gauge theory which he believes is capable of handling the problem of quark confinement
outlined previously. His theory is based on the earlier work of Kenneth Wilson who, in the
mid-1970s, had proposed that quantum chromodynamics be construed in terms of a cubic
lattice.
A cubic
lattice is a methodological device for separating time and space into sets of discrete
points. As such, it provides one with a means of representing a variety of events which
occur in time and space in a way that permits one to make calculations that could not be
made otherwise.
In general
terms, as the mesh of this lattice is refined over time, the calculated values for
different, physical variables which have been defined in terms of the structural character
of the lattice, will approach, more and more closely, those values that would be predicted
by QCD in the context of continuous time and space (as opposed to the discrete
representation of space and time of cubic lattice theory). Rebbi contends that if one
makes the mesh of the lattice sufficiently fine, one can demonstrate that the property of
confinement is a function of quantum chromodynamics. Rebbi refers to the methodology which
is employed to refine the lattice and to make calculations under those conditions as lattice
gauge theory.
Lattice
gauge field works on the assumption that there exists a field that gives expression to the
confinement property. This field is known as a chromoelectric field.
Mathematically, it exhibits the characteristics of a gauge field.
In the
context of Rebbi's paper, a gauge provides a means of comparing the values for
various physical quantities at different points in a given lattice. The gauge is like a
measuring device capable of determining, for example, the orientation or length of
different variables at different locations in the lattice. The gauge uses these measured
values as a basis for comparison.
As
previously indicated, one of the peculiar properties of a gauge is that the structural
character of the measuring device to which the gauge gives expression can change as it is
moved about from one location to another location. Thus, the orientation of the gauge may
be altered with movement, and, so too, movement of the gauge may alter the character of
the gauge's ruler-like aspect.
Usually,
when lattice theory is applied to QCD, the variables that are defined on the lattice
constitute various states of certain particles in which one is interested.
For instance, a particle might be able to have several possible orientations. Thus,
descriptions of a given variable might indicate that the particle has orientation
‘A’ at a given point in the lattice and orientation ‘B’ at some other
point in the lattice.
If one
wishes to compare, say, the orientations of two variables in the lattice, then, the
representation for one of the two variables will have to be transported or moved next to
the variable with which one wishes to compare it. Consequently, one will have to define a
set of rules capable of describing, and keeping track of, whatever changes occur during
the process of transportation ( in the present case, these changes involve orientation).
This set of rules gives expression to what is known as a gauge field.
Lattices can
be conceived of as being built up from the edges and vertexes of a set of cubes that are
stacked tightly together. In physics, each point of a lattice is considered to give
representational expression to spatial and temporal coordinates.
Consequently,
a lattice is a four-dimensional array consisting of three spatial coordinates and one
temporal coordinate. A link exists between any two neighboring vertexes of
the lattice.
A square
area that is bounded by four links is known as a plaquette. The links and
plaquettes of a lattice are considered to be discrete in the sense that they do not
consist of an infinite set of points between the vertexes which establish boundary points
for both links and plaquettes.
In the
context of hermeneutical gauge field theory, a plaquette gives expression to a certain
spectrum of ratios of constraints and degrees of freedom. Depending on the level of
scale on which one engages a given latticework, the plaquette could refer to: (1) a
point-structure (which consists of a single ratio of constraints and degrees of freedom);
or, (2) a neighborhood (which consists of n-ratios of constraints and degrees of freedom,
depending on how many points are in the neighborhood); or, (3) a lattice (which consists
of a spectrum of the ratios of constraints and degrees of freedom of the neighborhoods and
point-structures that give expression to the lattice); or, (4) a latticework (which
consists of a set of lattices linked together to form a perspective, theory, model, value
system, methodology, etc.).
The
hermeneutical counterpart to the links which tie together the vertexes that, together,
give expression to a particular plaquette are phase relationships. Although the lattice
described in Rebbi's paper is fairly linear in character, with the links being relatively
simple, the hermeneutical latticework is oftentimes (though not always or necessarily so)
nonlinear in character, and the links or phase relationships can be quite complex. As
such, the idea of a link can refer either to a single phase relationship or a set or
fiber-bundle or sheaf of such phase relationships.
Given that a
plaquette is the structure which is manifested as a function of the way neighboring
vertexes of the lattice are tied together by links, when one transposes this idea to the
hermeneutical context, a plaquette may not be (and, usually, will not be) just a
four-sided figure. A hermeneutical plaquette will be, instead, a complex manifold of
irregular, dimensional shape.
The
vertexes, links and plaquettes which give expression to the structural character of a
lattice are abstract, discrete representations of some ontological counterpart which is
assumed to be concrete (i.e., it has actual spatial and temporal properties) and
continuous. In the terminology of theoretical physicists, a lattice is an instance of a
regularization. This is a mathematical construct which is designed to provide
a means of making certain kinds of calculations that could not be made if such a
regularization were not introduced to model some aspect of reality.
In the
chromoelectric field approach to quark interactions, the vertexes of the lattice are used
to represent the probability that a given particle will be found at such points. The links
which run between neighboring vertexes are used to give representational expression to the
strength or intensity of a field between neighboring vertexes (i.e, particles).
Fields are
manifestations of forces which have magnitude as well as orientation. Consequently, they
satisfy the conditions for being vector quantities. However, the vector quantities which
constitute a field can point in any number of directions. Thus, the links of the lattice
representing the strength of a field between any two given neighboring vertexes can assume
a orientation value that is directed toward either vertex.
The idea
behind the lattice method is this: as one makes the lattice mesh smaller and smaller, by
moving the vertexes of the lattice closer and closer together, one will begin to approach
the continuous conditions which are assumed to characterize the structure of ontological
space and time. Presumably, at the limit value for reducing the size of the
lattice mesh, one will have derived a mean value for the strength of the field.
This mean
value would take into consideration all the possible quantum fluctuations of such a field.
Consequently, the mean value would be capable of reflecting predicted or observed values
for an actual chromodynamic field.
In the
context of the lattice approach to chromoelectric fields, re-normalization refers
to the process of making the lattice mesh progressively smaller until one is able to
eliminate the lattice altogether, thereby recovering the supposed continuity (in the sense
of an infinite set of points) of space-time. Rebbi admits that reducing the lattice mesh
to zero is not really possible, and, therefore, complete renormalization has not been
achieved at the present time.
When the
characteristics of a gauge field are explored by means of the lattice method,
investigators usually employ what is known as a Monte Carlo simulation.
Rebbi and others have shown that when they employed Monte Carlo simulation to study QED in
order to determine if the property of confinement (observed at large values of the
coupling constant in such systems) disappears as one approaches the limit of the lattice
mesh, they discovered that deconfinement occurs. That is, at a certain point in the
reduction of the lattice mesh toward its limit, the lines of force which previously had
been confined to the links between two vertexes on the lattice, undergo a phase transition
and spread out beyond the links to which they had been confined.
Rebbi also
notes that when the same sort of Monte Carlo simulation was done in relation to a quantum
chromodynamic system, the investigators who did the study found that the system undergoes
no phase transition as the value of the coupling constant is lowered during the process of
reducing the size of the lattice mesh toward its limit. In other words, the charges in the
chromoelectric gauge field remained confined in accordance with the observed properties of
quarks within hadrons.
The idea of
making a lattice mesh progressively smaller as one approaches the limit reminds me, in a
way, of the manner in which Galois used the idea of groups. In each case, it
seems that one is working toward a limit by refining the methodological mesh one is using
in order to isolate the solution to a problem.
Similarly,
part of the purpose underlying hermeneutical methodology is to generate a progressively
refined meshwork to work toward generating or isolating a resolution to a given
hermeneutical/phenomenological problem or issue. The other purpose underlying
hermeneutical methodology is to provide a cohomological analog. Through this
sort of analog one gathers together a variety of different lattice-like solutions and
arranges them in a way that 'covers' the phenomenological manifold in the sense that the
arrangement might be hermeneutically capable of accounting for why various aspects of the
phenomenology of the experiential field have the structural character they do.
In the
lattice model, the vertexes are said to represent the state of a given particle at a
certain point in the field. Usually, some sort of simplifying assumptions are made, such
as: at any given vertex, a particle can have one of two possible states.
If one
further assumes the pointer on a gauge can form any angle, across a continuous range of
angles, with respect to a vertical reference point, then, one can compare the angles which
are formed under the influence of the field at different points of that field. However,
the angle of rotation takes place in the complex plane which means there is an imaginary
component, as well as a real component, involved in the rotation of the pointer of the
gauge.
Unlike the
electromagnetic gauge field which can be represented by a gauge consisting of but a single
pointer, the chromodynamic gauge field must be represented by a gauge consisting of three
pointers. This is necessary in order to be able to describe, and keep track
of, the three color properties believed to be responsible for generating the chromodynamic
field by means of their interaction.
In the
context of hermeneutical gauge field theory, the mode of gauge measurement to be employed
will have to consist of 6 pointers in order to be able to describe, and keep track of, the
various components of the semiotic quantum which is the carrier of the
hermeneutical force in the phenomenology of the experiential field. Thus, the
hermeneutical gauge pointer will be a complex sort of matrix in which each cell is the
dialectic product of one of the hermeneutical components taken as focus, and the other
five hermeneutical components are taken as horizonal components. The focal/horizonal
arrangement will change from cell to cell, with each of the components of the
hermeneutical operator taking, in turn, the role of focus.
In Rebbi's
lattice model, if the pointer on the gauge does not return to its initial angle of
orientation after being transported about a given plaquette (one should remember that the
gauge is moving not only through space but through time as well), the difference in the
angle reading taken at the beginning of the transport process and the angle reading taken
at the end of the transport process (i.e., when it has arrived back at the point where the
first reading was taken) is known as the phase angle. The phase angle provides an
indication of what is referred to as the frustration of the plaquette in question.
The frustration of a plaquette gives representational expression to the
directional-strength of the field (i.e., its vectoral property) at a given point in that
field.
If the phase
angle is zero after being transported through a complete cycle of the plaquette, the
strength of the field is considered to be zero. With each increase in the phase angle, the
strength of the field in a given spatial direction is correspondingly greater.
From the
perspective of hermeneutical gauge field theory, phase angle refers to the differences in
hermeneutical orientation readings between some arbitrary initial starting point
and some terminal point arrived at after making a complete circuit (i.e., one has run
through a spectrum of ratios of constraints and degrees of freedom) of some given
structural character or plaquette. In other words, a given attitude, idea, view, belief,
or value, constitutes a hermeneutical orientation. Each of these also represents an
initial reading of the hermeneutical gauge field as a function of the dialectic of the six
components of the hermeneutical operator.
When one
explores, reviews or analyzes this attitude, etc., over time (which is comparable to
making a circuit around a lattice's plaquette), this represents a second reading of the
hermeneutical gauge field as a function of the dialectic of the six component carrier of
the hermeneutical force. By comparing the differences in orientation between the first and
second readings of the hermeneutical gauge field, one has an index of the hermeneutical
counterpart to the idea of a phase angle.
If the
hermeneutical phase angle is zero after making a circuit of a given plaquette, then, there
has been no change in the strength or intensity properties of the hermeneutical field. As
a result, the structural character of the attitude, idea, etc., with which one initially
started remains the same.
If, on the
other hand, there is a phase angle difference, then, the bigger the change in the angle,
the greater is the frustration, tension or stress in the hermeneutical counterpart to the
plaquette. Therefore, the strength of the vector or tensor force of the hermeneutical
field which is operating through that plaquette over time also will be proportionately
greater.
As
previously indicated, in a gauge field, as long as one, simultaneously, changes the
rest of the character of the gauge field in a corresponding manner, one can arbitrarily
change the state of a particle without this affecting one's physical description of the
field. Moreover, if both the state of a given particle and the rest of the field are
altered in an appropriate fashion, one can continue to make comparisons of the strength of
the field at different points without affecting the legitimacy of those comparisons. The
quality of a gauge field which enables one to compensate for changes in, say, the
orientation of a given field variable which is being transported, for purposes of
comparison with some other point in a given field, is known as local gauge invariance.
In
hermeneutical gauge field theory, the idea of local gauge invariance may have a variety of
possible applications. For example, there is a sense in which local gauge invariance
refers to the property of reversibility, comparable to the sorts of operations about which
Piaget spoke. One can reverse the character of a given field variable as long as one
introduces appropriate adjustments into the rest of the gauge field in order to compensate
for the reversal of that field variable.
Another
possibility for applying the local gauge invariance idea to the context of hermeneutics
concerns the relationship among ontology, phenomenology and hermeneutics. The process of
transduction alters the state-character of some given aspect of ontology.
By making
appropriate hermeneutical adjustments in the gauge field through activation of the
hermeneutical operator, one can compensate for the changes introduced by transduction and,
thereby, not necessarily lose any information concerning the structural character of what
makes a transduction of such character possible. Furthermore, one can continue to make
comparisons of changes in the strength of the field at different points in the field as a
means of deriving information on phase angle or orientation. This would permit one to gain
understanding of certain aspects of the structural character of the ontology that make
such phase angles or transitions in orientation possible.
Quite
conceivably, theories, models and belief systems may operate on the basis of some sort of
principle of local gauge invariance. More specifically, when one encounters an aspect of
the phenomenology of the experiential field which changes, one attempts to account for
this change by selecting an appropriate ratio from amongst the spectrum of ratios of
constraints and degrees of freedom that make up the theory or model or belief system.
The ratio(s)
which is(are) selected attempts to compensate for the changes in the phenomenology of the
experiential field without altering the character of one's understanding of how that field
variable fits into the scheme of one's theory or model or belief system. Moreover, the
compensations which are made, are done so in a way that will permit one to continue to
make comparisons about changing field strengths as one moves one's hermeneutical gauge
from point to point in the phenomenological field or as one encounters a succession of
points of differential field strength during the course of experience.
|
