Quantum Gauge Theory - Part 1
The idea of symmetry
The property
of symmetry is observed when one performs a transformation on some object,
structure or process, and a pattern or form that was initially manifested in such an
object, etc., remains invariant after the transformation. Thus, although other aspects of
the object, structure, and so on, may be altered as a result of a transformation which is
applied, as long as some given form or pattern remains invariant across the
transformation, then, symmetry is said to be preserved in that object, etc., with respect
to the given pattern, form, or aspect.
For example,
a circle is said to have continuous symmetry with respect to rotational transformations
since: (a) the properties of the circle remain invariant both before and after the
transformation; (b) the invariance is preserved independently of the extent of the
transformation (i.e., irrespective of the number of degrees involved in the rotation).
On the other
hand, a square shows positional symmetry only with respect to 90 degree rotational
transformations (or multiples thereof). In other words, one cannot discern any difference
in positional orientation of a square that has been moved through one or more rotations of
90 degrees.
However, any
rotational transformation which is less than 90 degrees, or more than 90 degrees but less
than 180 degrees, will display a positional orientation distinguishable from any sequence
of rotations of 90 degrees. Therefore, the symmetry of positional orientation will not
have been preserved under such circumstances.
The
foregoing examples of symmetry are geometrical. In fact, the idea of symmetry originally
arose in the context of geometrical investigations. Nonetheless, the concept has been
extended to non-geometrical contexts as well. For instance, if one changes the polarity of
the charges in an electromagnetic field whose strength is known, the character of the
forces acting between charges does not change.
Consequently,
whether one is discussing geometric or non-geometric contexts, the underlying principle
remains the same. If a given symmetry is to be preserved, then, some property must remain
invariant across the transformation that is being applied to an object, structure, process
or event.
One may have
to make a distinction between methodological symmetry and ontological symmetry. In
methodological symmetry no variance in a given property can be detected from the
perspective of the methodology being employed to determine whether some given feature
remains invariant. However, from some other methodological perspective, one may become
aware that a variance has occurred in relation to the feature which is being subjected to
a transformation of some sort and, therefore, symmetry has not been preserved.
In
ontological symmetry, on the other hand, there is an invariance of a structural, thematic
or essential character that is preserved independent of methodological considerations. For
example, suppose one were to place a small mark on the back of one of the corner angles of
a square, and, then,, one rotated the square through 90 degrees or 180 degrees or 270
degrees.
One would
not be able to detect any difference in positional orientation by looking at the front of
the square. Yet, by looking at the back of the rotated square, one would be able to see
that the ontology of positional symmetry had not been preserved. Furthermore, if one were
to perform the same kind of marking procedure with respect to the so-called continuous
symmetry of the circle, one also would discover that ontological symmetry had not been
preserved with respect to positional rotations .
In each of
the foregoing cases, one only would be able to see that positional symmetry had not been
preserved if one's methodology enabled one to take both sides of the rotated object into
consideration. Consequently, a variety of methodological approaches are capable of
arriving at different answers to the issue of the preservation of symmetry with respect to
some given property, principle, pattern and so on. On the other hand, when a variety of
different methodological perspectives all point in the direction of a given property
remaining invariant despite undergoing one or more transformations, then, this is strong
evidence - albeit not conclusive - that an ontological symmetry relationship of some sort
has been preserved.
Symmetry relationships identity and coupling constants
Another
feature to keep in mind with respect to symmetry relationships is that one is not
necessarily talking about an entire structure, object event, etc., when one speaks of the
preservation of symmetry. Usually, one is speaking of only a certain property or
relationship or a small set of such properties or relationships.
In the
context of a given structure's spectrum of ratios of constraints and degrees of freedom,
the foregoing point means only certain ratios from amongst such a spectrum need to remain
invariant across one or more transformations in order for one to say that a certain kind
of symmetry has been preserved. This point may be important when it comes to talking about
the identity of a given structure.
More
specifically, part of the problem of identity is the need to be able to justify saying
that a structure has remained essentially the same despite obvious alterations in some of
the ratios of constraints and degrees of freedom of that structure's spectral character.
In fact, one might speak of a law of structural identity in which the character of
a given structure retains its identifiability as the structure with which one began,
despite undergoing a sequence of transformations.
The spectrum
of ratios, together with the accompanying set of coupled phase relationships, constitute
the superpositional expression of a structure. Various aspects of the ratios of
constraints and degrees of freedom to which the phase relationships help give expression
are sometimes induced to manifest themselves (and, sometimes, do so spontaneously) as the
result of undergoing transformations of one form or another. As long as these induced, or
spontaneously manifested, transitions in phase relationships do not violate the coupling
constant basis of the structure's integrity, this would be an instance of preserving
symmetry with respect to structural identity.
The hermeneutical
coupling constant is an index of: (a) the way a given structure's spectrum of ratios
of constraints and degrees of freedom holds together as an integral unit; (b) the way a
given structure's spectrum of ratios either spontaneously can manifest different aspects
of its spectrum of ratios, or can be induced to manifest different aspects of its spectrum
of ratios. Each structure has its own, unique coupling constant, which differentiates that
structure's spectrum of ratios from all other spectrums of ratios of constraints and
degrees of freedom.
The coupling
constant gives expression to the set of phase relationships binding a spectrum of ratios
together over time and across a variety of circumstances. The coupling constant is an
index of the identity of a structure. It determines the manner in which a structure's
spectrum of ratios of constraints and degrees of freedom will be manifested (either
spontaneously or as a result of field induction) under a given set of circumstances.
Furthermore,
the coupling constant specifies the principles which govern the attractor basins that are
operative in a given structure. In addition, it sets the mode(s), current(s) or theme(s)
of phase quanta exchange which will occur in and through a given structure. Moreover, the
coupling constant is an index of a structure's action character, and, therefore, the
coupling constant is an index of a structure's dialectical potential. The coupling
constant marks the presence of the order-field which makes possible a structure of given
character with its concomitant aspects of action and dialectical potential.
If a given
structure loses its coupling constant quality, the integrity of that structure is
violated, and it will no longer manifest itself in characteristic ways. A structure whose
coupling constant has been disrupted will no longer manifest itself in terms of the
spectrum of ratios which normally establish the set of parameters within which, through
which, and by which that structure's character is given expression.
One might
use the foregoing perspective in order to approach the old philosophical problem of
identity posed by Hume. The problem concerns a ship which leaves port, and during the
voyage, each of the planks of the ship is removed and replaced by another plank. The
question is: is the ship which returns back to port the same ship that left port
originally?
Conceivably,
one could answer as follows. Despite the transformations which have been applied to the
ship, nonetheless, one might still wish to argue the general form and structural character
of the ship have remained invariant. Indeed, if the replacement process is done correctly,
each new plank must be introduced in accordance with the phase relationships that the old
plank had with the other planks of the ship. Thus, like the 90 degree rotation of a
square, or the n-degree rotation of a circle, or the 60 degree rotation of a snowflake,
there will be no detectable variance, as far as the general structure of the ship
is concerned, if one were to compare the pre-voyage ship with the post-voyage ship.
Gauage theory in relation to global and local symmetry
In physics,
symmetry comes in two varieties: global and local. Global symmetry occurs
whenever a certain property remains invariant across one or more transformations that are
applied simultaneously everywhere in a given framework. Some form of global symmetry is
present in virtually every kind of physical theory.
Local
symmetry, on the other hand, requires some property of a field to remain invariant despite
the fact that different transformations may be occurring at every point in the field.
Obviously, the conditions which must be satisfied in order for local symmetry to be
preserved are considerably more rigorous than is the case with respect to global symmetry.
In modern
physics, gauge theories are intimately linked with the concept of symmetry. A gauge theory
is a mathematical model which, among other things, refers to a particular class of quantum
field theory. It contains within it aspects of both the special theory of relativity as
well as quantum mechanics.
More
specifically, the essential element of a gauge theory is that it should encompass a group
of transformations, referred to as gauge transformations, which are performed on
the various variables of a quantum field and, yet, which leaves unchanged the physics of
that field. The aspect of the theory that preserves the basic physics of the field is
known as gauge invariance.
The quality
of gauge invariance places constraints on the character of the group transformation
equations used to describe the quantum field. Such constraints mean, in turn, that the
field is not free to engage, or interact with, other quantum fields in just any way. A
given field must interact with other fields in accordance with the restrictions inherent
in the structural character of the set of group transformations which preserve that
field's symmetry.
Although the
idea of a gauge transformation was not introduced until the early part of the 20th century,
Maxwell's electromagnetic field theory is considered to be an excellent exemplar of a
gauge theory.
In the
theory of the electromagnetic field, the strengths of the electrical and magnetic fields
are the fundamental field variables. These variables are often expressed in terms of
vector and scalar potentials. When, in accordance with certain gauge transformations
appropriate to the electromagnetic field, the values for the field variables or potentials
are altered, the basic physics of the electric and magnetic fields is preserved through
gauge invariance.
Einstein's
general theory of relativity is another example of a theory displaying gauge symmetry.
However, in the general theory of relativity, the gauge symmetry involves a coordinate
system representing space-time rather than a matter field. In other words, instead of
assigning values to each point-particle of a field, values are assigned to each point of
the coordinate system giving expression to the structural character of space-time.
Each point
of the coordinate system is assigned four numbers, three of which are spatial coordinates
and one of which is a temporal coordinate. The origin of the coordinate system marks the
point where all four numbers have the value of zero.
The point of
space-time one chooses to be the origin is entirely a matter of convention. However, once
this point is established, all of the other coordinates will be assigned numerical values
relative to that reference point.
When one
performs transformations - such as rotations, mirror reflections or translations - on a
given coordinate system as a whole, all the laws involving relationships between or among
various points of the system will remain invariant. In other words, the laws of the
coordinate system have symmetry with respect to the transformation operations of rotation,
mirror reflections and translations.
All of the
invariances outlined in the previous paragraph are instances of global symmetry.
In effect, each of the transformation operations shows one how to generate a new set of
coordinates by performing certain kinds of operations on the old set of coordinate values,
provided that the transformations are applied to every point in the old coordinate system.
The general
theory of relativity, however, provides a means of preserving the laws of nature despite
allowing for transformations that may vary from point to point in the coordinate system.
The theory accomplishes this by introducing a gravitational field which has properties
able to establish local gauge symmetry. Thus, just as is the case in relation to
electrodynamics, gauge symmetry is established by introducing a field which has the
appropriate vectoral potential characteristics.
Virtual particles and field quanta
Irrespective
of whether one is dealing with global or local gauge symmetries, one needs to introduce
the concept of a force. This is necessary for two reasons: (a) in order to account for how
influences are propagated from point to point in a given field; (b) to be able to account
for how symmetries are preserved.
In one
sense, forces in a field are not exerted directly by one particle on another. Each
particle is considered to be a local generator of field properties.
Consequently,
any given particle in a field interacts with the fields which are generated by the other
particles of that field. Indeed, the field, taken as a whole, is really the dialectical
product of all the locally generated fields.
However, in
another sense, and this is largely for the sake of conceptual convenience, the interacting
fields of any two given particles are construed in terms of a virtual particle
which is exchanged between the particles. Such exchanged particles are the field's
quanta.
These field
quanta are said to be virtual because there existence is transitory. In fact, their
existence is so ephemeral they cannot be experimentally detected. Moreover, the larger the
amount of energy being transmitted by these virtual particles, the shorter the duration of
their postulated existence.
According to
quantum theory, and this comes largely from Heisenberg's uncertainty principle, there is
said to be a conjugate relationship between the energy of the virtual particle and the
amount of time the particle exists. Supposedly, a virtual particle must steal
energy from the cloud of uncertainty surrounding any given particle and, then,, replace
that energy before any laws of nature are violated.
The more
energy that is borrowed, the more quickly the energy must be returned, and, therefore, the
more ephemeral the existence of the virtual particle. Furthermore, the shorter the
duration of the virtual particle's existence, the shorter will be the range of the force
carried by the virtual particle, since the ephemeral duration translates into a restricted
travel distance during the course of the virtual particle's life span.
There are a
number of aspects concerning the time-energy relationship of virtual particles in the
context of Heisenberg's uncertainty principle that seem problematic. For instance, one
wonders how a virtual particle, which, presumably, only exists by virtue of the energy
that is borrowed from the cloud of uncertainty, presupposes itself in order to be able to
borrow such energy? In other words, just what is it that is doing the borrowing here?
If it
already exists in order to direct the borrowing operation, then, what is its form of
existence and what makes it possible? Moreover, one wonders 'what' it is that 'knows: how
much to borrow, and where to borrow it from, or when to borrow it, as well as when to
return what has been borrowed, and where to deliver what has been borrowed.
No matter
which way one goes, the idea of a virtual particle encounters problems. Either one is
faced with a virtual particle which presupposes itself, or one has something at work
during the process of the exchange of quanta which has a structural character quite
different from the way in which the virtual particle has been described - but which makes
possible the structural character of the phenomenon for which the idea of a virtual
particle has been theoretically invented as an explanation.
Gauge symmetries, phase shift transformations and hermeneutics
There are
certain gauge symmetries occurring in relation to phase shift transformations which are
preserved in an electromagnetic field. For instance, in the two-slit experiment, when the
waves of the electromagnetic field pass through the slits and form interference patterns
on the other side of those slits, phase accounts for where the peaks and nodes form in the
interference patterns.
More
specifically, wherever the peaks of constructive interference occur, the waves of the
field are in phase. On the other hand, wherever the nodes of destructive interference
occur, the waves of the field are out of phase with one another.
If one were
to subject the field to a phase shift, although this transformation would alter the
field's form significantly, the interference pattern created in the two-slit would remain
unaltered. Therefore, interference patterns remain invariant with respect to phase shift
transformations.
As indicated
above, subjecting a field to a phase shift transformation alters the field's configuration
or form. This means there would be a different pattern of waves manifesting itself in the
field after the phase shift transformation relative to the pre-shift phase state of the
field.
For example,
where, prior to a phase shift, there had been, say, a peak, after the phase shift, there
would be a trough or node. Moreover, where, prior to a phase shift, there had been a
trough, after the phase shift, there would be a crest or peak.
However, as
far as interference patterns are concerned, nothing really would be affected. One still
would have the same set of waveforms coming together to generate the interference pattern.
The only difference would be that the waveforms would be coming together in a sort of
mirror image manner relative to how the waveforms interacted prior to the wave shift.
In order for
interference patterns to show symmetry across phase shift transformations, the
transformations must be applied globally to the electromagnetic field. Moreover, because
the interference patterns remain invariant throughout the transformations, the phase
shifts are not detectable. Since the phase shifts are not detectable, they are not
measurable.
In a sense,
the ideas of simple constructive and destructive interference may be too static and
two-dimensional to be of much use in helping one to grasp the structural character of the
dialectic of waveforms in the hermeneutical field. One needs a more sophisticated and
nuanced version of constructive and destructive interference.
The
hermeneutical counterpart to the crests and troughs of normal waveforms, are the patterns
of emphasis and de-emphasis that occur in relation to the manner in which the spectrum of
ratios of constraints and degrees of freedom give expression to various kinds of phase
shifts. A spectrum of ratios of constraints and degrees of freedom is not, in and of
itself, enough to give expression to structural character.
One also
needs to take into consideration transitions in phase relationships that, among other
things, show shifts in the character of the way different combinations of ratios manifest
themselves. More specifically, one needs to consider the degree of intensity with which
phase relationships manifest themselves in various phase state patterns.
If the
degree of intensity is high, the phase relationship is being emphasized in the structure,
and it is like a peak or crest in a waveform. If the intensity of a phase relationship is
low, it is being de-emphasized in the structural character.
This
corresponds to a trough or node of a wave. The total set or spectrum of ratios of
constraints and degrees of freedom, together with the shifting patterns of emphasis and
de-emphasis in various combinations of ratios, constitute the structural character of a
given object, event, state, or process.
In addition,
there may be an aspect of hermeneutical phase relationships which is somewhat like the
notion of left-handed and right-handed optical isomers in organic molecules. These are
interpretive variations on a given phase relationship theme by the same individual. In
away, they are like different angles of hermeneutical engagement of one-and-the-same
structure.
If the
foregoing is the case, then, the hermeneutical structural character of a particular ratio
of constraints and degrees of freedom depends, in part, on the pattern of the orientation
of the phase relationships which are coupled together to help give expression to a given
ratio of constraints and degrees of freedom. However, in any given set of circumstances,
there may be a multiplicity of orientation configurations which are possible, and,
therefore, the number of hermeneutical isomers will be more varied than the
left-handed and the right-handed optical isomers of organic molecules.
In light of
the foregoing discussion, hermeneutical structural character is a function of the
following elements or aspects: (a) the spectrum of ratios of constraints and degrees of
freedom; (b) the emphasis/de-emphasis pattern of phase relationships that help give
expression to shifts in the way various combinations of ratios of constraints and degrees
of freedom manifest themselves; (c) the orientation of phase relationships as an
expression of the property of hermeneutical isomerism; (d) the coupling constant which
brings together, and maintains, the components of (a), (b) and (c) as a spectral character
of one sort, rather than another. The coupling constant is a function of the dimensional
dialectic which has been set in motion by the order-field.
As indicated
previously, local gauge symmetries identify themes of invariance under conditions of
transformation that vary from point to point in a given field. This sort of symmetry is
likely to take on fundamental importance in the development of understanding.
Especially
important in this respect, will be those symmetries or invariances which are preserved in
relation to transduction transformations along with the hermeneutical operator's
transformation of these transduction transformations. These kinds of invariance open the
possibility that one might be gaining insight into the structural character of some aspect
of reality or ontology. Therefore, one of the tasks of hermeneutical field theory will be
to identify those spectra of ratios of constraints and degrees of freedom that, despite
undergoing a variety of local gauge transformations, nonetheless, remain 'largely'
invariant with respect to structural character.
The term
"largely" has been used in the foregoing sentence to serve as a reminder that a
given structure need not remain invariant in every respect in order to be considered the
same structure. Some ratios of a particular spectrum will undergo phase shifts or phase
transitions. These phase shifts will alter the character of the ratio of which they are
apart.
However,
despite these phase transitions and despite the concomitant alteration in some of the
ratios of the spectrum being considered, the structural character to which the spectrum
gives expression remains intact. As such, these spectrums conform to the law of structural
identity in the sense that one can identify the post-transformational structure as being,
effectively, the same structure as existed prior to the transformation.
A further
aspect to be considered is this. The more complex a structure is, the more allowances one
has to make for the degrees of freedom which are exhibited by the structure, under various
circumstances, as a result of either spontaneous activity or induced activity or the
dialectic between spontaneous and induced activity.
Considered
from the foregoing perspective, the fact certain phase relationships or ratios of
constraints and degrees of freedom are not preserved across transformations (whether
spontaneous, induced or dialectical) is not evidence that symmetry with respect to
structural character has not been preserved. In fact, just the opposite may be the case.
Such alterations in ratios may be part of the fluidity or flexibility of a given
structure's character.
Consequently,
part of the task of hermeneutical field theory is to differentiate between critical
instances of symmetry failure and non-critical instances of symmetry failure within
conceptual and interpretive systems. In a sense, a given conceptual structure can go
through a multiplicity of states as various ratios and phase relationships undergo
transitions. As long as these phase transitions are of the non-critical variety, then,
symmetry is preserved with respect to the structure's coupling constant character.
|
