Quantum Quandries - Part 5
Phase states, Airy patterns and the two-slit experiment
The
foregoing discussion has focused on contexts which do not involve experimental set-ups
designed to generate interference phenomena. However, one could extend certain aspects of
that discussion to contexts which give rise to phenomena involving interference. For
example, consider the points below.
The
particular character of an Airy pattern ( in terms of the intensity of each light circle
and the diameter of each dark ring, as well as in terms of the size of the angular
diameter of the central solid spot- known as the Airy Disc) will be a
function of the dialectic between the diameter of the hole through which the electrons are
shot and the wave length of the electrons which are shot through the hole. The nature of
this dialectic could be construed in a variety of ways.
One possible
way to construe the above-mentioned dialectic is as follows. The electrons that form part
of the molecules that make up the material in which a hole has been placed, and through
which electrons from the heated filament are shot, give expression to a field that is
capable of interacting with the field of the electron from the gun. This interaction may
lead to the generation of interference patterns.
On the other
hand, one might wish to argue that the field associated with the electron shot from the
gun gets pinched or distorted as it goes through the hole. This pinching process disturbs
the shifting patterns of phase relationships in the internal dialectic of the electron,
thereby leading to the generation of interference patterns.
In effect,
the electron from the gun gets thrown-off its routine sequence of shifting arrangements of
ratios and phase relationship patterns. As a result, this disturbance generates a certain
amount of bifurcation in the ratio arrangements and phase relationship patterns.
The
bifurcation process causes different aspects of the internal dialectic of the electron to
go out of phase with one another. Therefore, interference activity ensues.
The Airy
interference patterns may also, of course, be a combination of the foregoing factors.
However, whatever the precise source of the interference may be, the end result is to
cause out-of-phase variations, either within the internal dialectic of the electron or
between the interacting fields of the electrons from the gun and the molecules making up
the material with the hole through which the former electrons are shot. These variations
give expression to interference phenomena which, in turn, show up on the screen.
The pattern
on the screen is generated by an interference process. Nonetheless, like its
non-interference counterparts discussed in the previous section of this chapter, the
interference pattern is constructed by a series of discrete engagements between the
phosphor molecules on the screen and the in-coming electrons.
Moreover,
like its non-interference counterparts, the interference pattern on the screen is a
time-lapsed portrait of structural character. However, in the case of the interference
process, the pattern is a reflection of the effect that interference has on the way
different electrons give expression to their internal dialectics under such circumstances.
Thus, the
pattern on the screen is not, strictly speaking, a direct portrait of interference. It is
an indirect portrait which is, at least, one step removed from the actual process of
interference occurring while the electron from the gun is on its way to the screen. As
such, the pattern serves as a characteristic indicator of the presence of interference
processes since the pattern constitutes the tell-tale effect-signature or imprint or trace
that interference leaves on the structural character of an electron's internal dialectic
at some point prior to impact with the screen.
In view of
the foregoing, although one is able to derive a number of patterns in different
experimental situations suggesting that interference phenomena are somehow involved, this
constitutes a misinterpretation of the meaning of the patterns which emerge in the
different contexts. In only one case, namely the Airy pattern, is interference present.
In both
cases, however, the structural character of the pattern on the screen is consistent with
the presence of oscillatory which is generated by the shifts in the arrangement of ratios
and phase relationship patterns in the electron's spectrum of constraints and degrees of
freedom during its journey from the gun to the screen. Differences in the character of the
patterns on the screen are reflections of the variations in the way in which the internal
dialectic of electrons manifest themselves in different circumstances .
The above
comments apply equally well to the two-slit experiment of Thomas Young. Like
the Airy pattern, the Young experiment involves elements of interference phenomena.
Moreover, like the Airy pattern, the Young experiment generates a set of discrete events
which give differential expression to light's spectrum of ratios of constraints and
degrees of freedom.
This
spectrum reflects the presence of interference. However, one of the basic differences
between the various electron gun experiments and Young's experiment lies with the
sensitivity of the detection equipment.
In the
simpler, two-slit experiment, the detection equipment is a piece of paper or cardboard.
The piece of paper or cardboard cannot show that the form of the shadows on the screen are
generated through individual electrons engaging the molecules and electrons of the screen
while they are manifesting specific phase states.
What one
sees on the paper or cardboard screen is an outline of the aggregate or collective form of
the individual electrons engaging the screen in a discretely continuous fashion. In this
sense, the two-slit paper/cardboard screen does not have the capacity for resolution of
its more sophisticated phosphor-coated counterpart.2
Intensity, probability and the dimension of energy
Quantum
physicists' interpretation of the squaring of a waveform's amplitude as a probability
index or measure is not entirely wrong, just incomplete and misleading if left by itself.
In point of fact, the probability distribution represented by the squaring of the
amplitude in the Schrodinger wave equation is really a sort of mathematical sketch of the
structural character of a given entity's ratio of constraints and degrees of freedom as
they express themselves over time in various arrangements of emphasis/de-emphasis or
on/off patterns of phase relationships.
The reason
why one can assign probability values is because the ratio of constraints and degrees of
freedom has an oscillatory character in which certain tendencies (i.e., certain ratio
arrangements of emphasis/de-emphasis or on/off patterns of phase relationships) are more
likely to be expressed under certain circumstances than are other such tendencies. These
tendencies assume a distribution pattern over time. Moreover, since the internal dialectic
is non-linear in nature, the probabilities only indicate tendencies rather than
determinate, fixed, self-same patterns of manifestation.
Therefore,
the probabilities are not drawn out of thin air as if there were no rhyme or reason as to
why different quanta have certain kinds of probability distributions associated with them.
In addition, the probability distributions have a referent beyond the of a given quantum.
In other
words, the of a quantum entity is not somehow autonomous as if that had no causal or
physical principles to which it gave expression or manifestation. Probability
distributions describe certain tendencies of a given quantum's structural character as a
function of the internal dialectic of shifts in emphasis/de-emphasis among the ratios and
phase relationships of the quantum entity's spectrum of constraints and degrees of freedom
which makes a structure of such character possible.
The
tendencies of a given ratio of constraints and degrees of freedom to manifest some
combinatorial sets of emphasis/de-emphasis or on/off patterns of phase relationships
rather than others, is not necessarily a function of the energy which is to be associated
with every point of a given quantum entity. In fact, the manner in which energy is
utilized to maintain structural integrity or to give expression to shifts in the way in
which the spectrum of ratios manifests itself is also regulated and shaped by various
ratios of constraints and degrees of freedom.
Indeed,
energy distribution and manifestation are themselves expressions of a multi-faceted
dimensional dialectic, of which energy is only one of the dimensional components
Schrodinger's wave equation is capable of providing information about certain aspects of
that dialectic in the form of, among other features, the amplitude of a given quantum
entity's waveform. Such information can be used as a basis for drawing a sketch of certain
aspects (e.g., its tendency to behave in one way rather than another) of the structural
character of the entity to which the wave equation is making identifying reference.
On the other
hand, there is, undoubtedly, some quantity of energy which is available to a given entity
and which forms part of the ratio of constraints and degrees of freedom that gives
expression to that entity's structural character. This seems to indicate that, in an
important way, the manner in which the ratio manifests itself over time will be a
reflection of, and reflected in, the modes through which energy is present and is expended
over time. Therefore, even if the square of the amplitude of the wave function does not
serve as a measure of the energy which can be assigned to every point of the quantum
entity with which the wave is associated, there is, nevertheless, an energy value, which
must be derived in some other way, that is associated with the ratio of constraints and
degrees of freedom which give expression to the structural character of the given entity
being investigated.
The
foregoing seems to indicate that energy, expressed as intensity, both is, and is not,
associated with every point of the structural character of a given quantum entity. Energy
is associated with every point in the sense that should any particular aspect of the
entity's spectrum of ratios of constraints and degrees of freedom be given expression, it
does so because it has access to, and is being "funded" by the energy supply
associated with the dimensional dialectic which generates that entity's structural
character.
On the other
hand, the reason why one cannot say the intensity of the quantum wave is an index of the
energy at every point of the probability wave is because, from the perspective of physics,
to say the less likely possibilities have the same amount of energy associated with them
as do the more likely possibilities is problematic, or makes no sense. After all, if all
the possibilities which are assigned various probability values have the same energy
associated with them, then, why aren't all these possibilities equally likely? What is
constraining them? Indeed, the idea of constraints suggests, perhaps, more structure in
relation to point-particles than quantum physicists feel comfortable with.
Consequently,
the assumption is made, apparently, that the intensity of the quantum wave cannot refer-
as is the case with 'normal, everyday' varieties of waveforms- to the energy that is to be
associated with each point of the quantum entity with which the wave function is
associated. The intensity function is interpreted, instead, in terms of a probability
distribution concerning the likelihood of a certain kind of property or set of properties
being manifested at a given point in measured time and space. However, such an
interpretation tends to gloss over the underlying dimensional dialectic in which
probability distributions are rooted and out of which they emerge during the course of the
measurement process.
As indicated
previously, intensity provides a sketch of the structural character of the internal
dialectics of the ratio of constraints and degrees of freedom as expressed in terms of
combinatorial sets of emphasis/de-emphasis or on/off patterns of phase relationships.
Furthermore, as also was indicated earlier, the intensity value provides, at the same
time, a measure of the energy which is available to each of the sets of phase
relationships should they manifest themselves.
Manifestation
occurs when some threshold (which is set by one or more of the ratio components making up
the spectrum of constraints and degrees of freedom) is exceeded. When this threshold is
exceeded, it opens a dimensional gate permitting shifts in ratio arrangements and phase
relationships to proceed.
However, as
indicated previously, the energy which is available to a given quantum entity is not
necessarily to be found at every point of that entity's structure. The availability of
energy, which is a dimension distinct from the dimensions of space and time, is via the
mediation of a complex of phase relationships that shift, in accordance with transitions
in the phase states of the internal dialectic. These shifts and transitions lead, in turn,
to the opening and closing of dimensional gates which permit (or prevent) energy to be
(from being) expended through the form of a particular mode of manifestation.
So, under
appropriate circumstances, every aspect of a given quantum entity's structural character
can have access to the energy which forms part of the entity's spectrum of ratios of
constraints and degrees of freedom. Yet, this energy value does not exist at every point
of the structure. The energy exists as another dimension which manifests itself in a
particular form, as well as at a particular locus and time, when the dialectic of
dimensions generates appropriate combinatorial arrangements of ratios of
emphasis/de-emphasis or on/off phase relationship patterns to mediate the interaction of
such dimensions.
Consequently,
energy is not stored or housed in any spatial or material sense. Energy exists as a
separate dimension altogether. However, energy can be expressed in spatial/material
contexts in terms of Planck's constant, according to the manner in which the shifting of
phase relationships opens and closes gates linking the material, spatial, and energy
dimensions, as well as any other dimensions which may be affected by, or involved in, such
a dimensional dialectic.
In fact, Planck's
constant is an index of the presence of phase relationship activity between the energy
dimension and various aspects of, for example, the spatial and material dimensions. The
rate, location, orientation, and intensity of the flow (in the aforementioned sense of
discrete continuity) of the bundles of energy that are described by Planck's constant will
be a function of dimensional dialectics. Indeed, the quantum of action, in which
Planck's constant has a prominent place, is an attempt to sum up, at least from the
physical side of things, the structural character of a given instance of dimensional
dialectics involving space. energy and materiality.
Generally
speaking, there may be only trace amounts of energy (and not enough to create the
self-energy problem) constantly available to a particle. This energy might trickle through
the dimensional gates and permit certain kinds of minimally necessary shifts in ratio
arrangements as well as transitions in phase relationship activity to take place.
In this
sense, a minimal energy state (the so-called ground state) is a sort of idling state which
is capable of sending signals (through phase relationships) that the dimensional gates
should be operating differently as circumstances change- for example, when external forces
impinge on the structure through the exchange of various kinds of boson vectors. When a
particle engages a given boson, the material side of the structure breaks down (i.e., the
phase relationships connecting a given structure with the material dimension are
temporarily ruptured) leaving an energy component that will reassemble (by re-establishing
a new set of phase relationships with the material dimension) the residue into a
structural configuration or configurations that can, in a stable manner, express the new
information which has been passed on by the boson through the mediation of phase
relationships.
The
foregoing perspective lays down a basis for dealing with the self-energy problem
which has haunted the corridors of physics throughout the 20th century. By treating energy
as a dimension that is separate from, but capable of closely interacting with, the spatial
and material dimensions, one has provided a potential means of eliminating the anomalies
which arise when one supposes that energy is inherent in the material dimension and must
somehow be housed within the confines of a point-structure. Energy is 'housed' or stored
in a separate dimension and only manifests itself at material and spatial loci as the
circumstances of the dimensional dialectic require.
The
foregoing approach would keep intact such laws as: E = mc2. All that is being
altered is the interpretation of these sort of laws.
Thus, on the
basis of the perspective outlined above, the equivalency of mass and energy, which is
given expression in Einstein's equation, means the dimensional dialectic, that is set in
motion by the order-field, permits a spectrum of phase relationships which can translate
the character of material dimensional structures into equivalent energy dimensional
structures, and vice versa. In short, Einstein's equation is a way of showing the
existence of a translational equivalency between certain aspects of the structural
character of the dimensions of matter and energy. This translation is mediated by the
exchange of phase information which gives expression to structural equivalencies in the
respective dimensions.
Expanding the horizons of the concept of dimensionality
Although the
idea of a multiplicity of dimensions is fairly well established in mathematics and, to a
lesser extent, perhaps, in certain aspects of science, there seems to have been little
consideration given to just how qualitatively different dimensions interact with one
another. One suspects the reason why mathematicians and scientists have focused on the
idea of a multiplicity of spatial or spatial-like dimensions is due, either
consciously or unconsciously, to a desire to avoid the problems which emerge when one is
thinking about the dynamics or dialectics of qualitatively different dimensions.
Traditionally,
the problems arising in relation to the interaction of qualitatively different dimensions
are usually sidestepped by merely restricting attention to the geometry of 4-space, or the
algebraic representation of 4-space. One, then, proceeds to treat time as if it were
merely another kind of space which is amenable to being described as part of a coordinate
system consisting of the right number of axes and whose ordered n-tuples are expressed in
terms of the real number or complex number systems.
Furthermore,
the tendency has been to suppose that the relationship between (or among) any two (or
more) given dimensions will be somewhat similar to the relationship that exists among the
more familiar three spatial dimensions which always have at least one dimensional boundary
in common. However, when one begins to think about the interaction between, what very
likely are, qualitatively different dimensions, such as time and space, one cannot
necessarily reason by analogy from the relationships among the so-called three spatial
dimensions.
One cannot
continue to sweep problems beneath the coordinate or n-tuple carpet. One cannot continue
to assume that because one has a means of representation, therefore, such a mode of
representation accurately reflects the structural character of either the dimension of
space or time which the mathematical framework is being used to describe.
For
instance, just to mention one facet of such modes of representation that has been a source
of constant aggravation, one should consider the manner in which the infinite character of
the real number and complex number systems has introduced paradoxes and difficulties
galore into all manner of calculations involving space and time. The result has been to
create a lot of confusion and distortion concerning the character of the relationship
between the system of representation and that which is being represented. In effect, the
structural character of methodology is often presumed to be the ontology of that to which
the methodology is making identifying reference.
Of course,
along the way, various individuals have attempted to resolve such difficulties. For
example, Karl Weierstrass's epsilon/delta technique provides a way around
some of the difficulties involving infinities which arose in relation to the calculus.
However, Weierstrass' epsilon/delta technique does not solve the problems alluded to
above, as much as it allows one to proceed, or get on with the job of making useable, and
within limits, accurate calculations.
Furthermore,
when one comes to an issue like the problem of continuity and what is meant by continuity,
Weierstrass' technique is of no value because it cannot answer the questions which are at
the heart of the continuity issue. Indeed, Weierstrass' approach is designed to avoid
precisely the sorts of problems which are introduced by, among other things, the issue of
continuity.
Another
example of an attempt to get around certain problems involving infinities is
re-normalization theory. By finding ways of getting the positive and negative
infinities, which arise during the process of calculation, to cancel one another, thereby
leaving a finite solution, one, sometimes, can come up with a satisfactory mathematical
technique for dealing with the problem of infinities in certain aspects of particle
physics.
Nevertheless,
one should not be too quick to assume that what one has done mathematically has an
ontological counterpart. Indeed, such mathematical techniques introduce elements of
arbitrariness, ad hocness and aesthetic messiness into physics that has left many
scientists feeling extremely uncomfortable. For instance, the name of Paul Dirac, one of
the leading architects of modern quantum theory, comes readily to mind as one of the many
who felt unhappy with re-normalization theory despite the fact that, at least on paper, it
was able, some of the time, to eliminate embarrassing problems in physics.
In any
event, aside from whatever problems are introduced into science by using coordinate
systems, along with real or complex number systems to represent various dimensions, such
methods also tend not to address what is meant ontologically when qualitatively different
dimensions interact. In other words, to say a given point in a coordinate system can be
represented as an ordered n-tuple of the intersection of n-axes really says nothing about
the character of the dialectic or dynamic of the dimensions which are supposedly being
given representational expression through the intersection of axes or the n-tuple of
ordered points.
For example,
the so-called marriage of space and time into space-time which was suggested by Minkowski
is really a very static concept in which two ideas are juxtaposed without any real
exploration into the possible ontological meaning of the marriage dynamics which have been
proposed. The only dynamics or dialectic such a proposed marriage permits is that which is
allowed by the assumptions, postulates and so on of mathematics. However, such assumptions
may have little, or nothing, to say about how one is to translate such quantitatively
mathematical dialectics into qualitative aspects of ontological or dimensional dialectics.
As outlined
in the early part of this essay, the ontology underlying quantum theory (so-called
"orthodox ontology") makes a number of assumptions. In addition to the
postulates of ontological identity (with respect to the fundamental particles of a given
'species') and intrinsic randomness, the orthodox ontology also assumes the fundamental
quantum entities are mathematical point structures (i.e., having position but no size).
All of the
foregoing assumptions are at odds with the sort of position which is being advanced in the
perspective underlying the present chapter. However, only the first assumption, concerning
the ontological identity of all fundamental particles of a given species, will be
discussed in the following pages.
From the
view of the perspective being presented in this chapter, quantum entities which are
represented by the same wave function are not necessarily identical in all respects. More
specifically, when a given wave function is supplied with the appropriate values for
different variables, although such a wave function may be able to describe something of
the spectrum of ratios of constraints and degrees of freedom giving expression to a
particular quantum entity at a given point in time and under certain conditions of
measurement, the wave function into which specific values have been substituted is but a
sampling of the quantum entity's overall structural character.
Suppose one
were to measure the values for electrons as they leave the electron gun. Let us further
suppose that all these values are the same and, therefore, they can be represented by the
same wave function. Despite these givens, there is no guarantee that, as the structural
character of the various electrons unfolds over time, the spectrum of ratios of
constraints and degrees of freedom of the different electrons will manifest identical
shifts in ratio arrangements of emphasis/de-emphasis or on/off patterns of phase
relationships.
Conceivably,
one could have instances in which the general wave functions for, say, two electrons are
identical, but there may be a variety of arrangements of ratios and phase relationship
patterns that are capable of generating phenomena capable of being described by the same
wave function. In other words although the values which are measured by the wave function
may remain constant between the point of release and the target, the wave function does
not necessarily exhaustively describe the structural character of the electron. It
describes only what present modes of methodology are capable of engaging.
In short,
the differences in that are observed in relation to particles which, according to the
values given by the wave function, are identical, may arise from the realm of dimensional
dialectics. This dimensional dialectic is expressed through the shifts in arrangements of
emphasis/de-emphasis or on/off patterns of phase relationships among the ratio components
of the spectrum of constraints and degrees of freedom that constitutes the particle's
structural character.
Superstring
theory has begun to investigate, at least mathematically, certain features of
the role which dimensionality may play in the way the fundamental forces are related to
one another. Of course, finding ways to experimentally verify the mathematics of various
versions of superstring theory is quite another matter and seems, on the basis of current
technology, very unlikely in the foreseeable future.
In any
event, the premise on which almost all versions of superstring theory are operating,
treats dimensionality almost exclusively in terms of spatial terms. Indeed, the
exploration and development of various approaches to compactification theory
is an attempt to find a mathematical way of allowing the extra dimensions that are being
proposed in many versions of superstring theory to fold up and remain hidden from the
three-space coordinate system that seems to describe the spatial character of the 'normal'
world so well.
Presumably,
these extra dimensions are construed as being spatial in character and, therefore,
inconsistent with the spatial structure of our everyday experience. Otherwise, one fails
to see why compactification theorists seem to feel compelled to find a plausible means of
eliminating the extra dimensions in such a spatialized manner.
FOOTNOTE
2.) Sheldon
Glashow in his book Interactions (see pages 53 - 54) describes a variation on the
two-slit experiment which he contends is capable of demonstrating "both the wavelike
and particle-like nature of the electron at the same time". According to Glashow,
each complementary facet of the electron can be exhibited within the experiment just by
moving the detector to different points in the experimental set-up.
As the
current discussion in the present essay is attempting to suggest, there is another way of
interpreting the foregoing experiment. More specifically, one is being asked to think of
the electron as the expression of an internal dialectic of dimensions.
This
dialectic establishes a spectrum of ratios of constraints and degrees of freedom through
which the "electron" manifests, in Necker-like analog fashion, its structural
character. From the foregoing perspective, the differential results produced by moving the
detector around in the experimental set-up can be interpreted to represent alternative
modes of methodologically engaging or sampling the spectrum of ratios of constraints and
degrees of freedom that are generated by the underlying dialectic of dimensions.
More
specifically, the internal dynamics or dialectic of dimensions has an oscillatory
character (albeit it is of a discretely continuous nature). These oscillations of the
internal dynamics are analogs for wave phenomena in the sense that the Necker-like
oscillations preserve the structural properties of waveforms but do so through a
non-waveform medium. When these analogs for wave phenomena are methodologically engaged in
certain ways by placing the detector at specific locations in the experimental set-up,
there will be interference effects which are produced.
These
effects are generated through a process which is not a function of waves. The interference
effects are, instead, a function of the way the discretely continuous oscillatory
character of the internal dynamics of the electrons is thrown out of its normal
arrangement of phase relationships. In effect, the double-slit set-up pushes the spectrum
of ratios of constraints and degrees of freedom of the various electrons into a chaotic
transition state on the detector side of the slits.
This
condition of chaos can reflect either the internal dynamics of an individual electron or
the dynamics of a number of interacting electrons or both together. In any of these cases,
the transition state marks the manner in which phase relationships generate a cascade of
bifurcations as the internal dynamics of the particle(s) 'seek' to re-establish the set of
phase relationships which characterized its (their) pre-slit state of in-phase stability.
However, the
interference-like pattern also is due to the change in the angle through which the
detector engages the incoming, altered (i.e., post-slit) spectrum of ratios of constraints
and degrees of freedom of the various electrons as a result of the manner in which the
detector is moved. The change in the detector's position in the experimental set-up brings
about a mode of sampling which engages a different facet of the altered spectrum of ratios
than will be the case when the detector is placed at other positions within the
experimental set-up. In a sense, the detector only can 'see' or detect the cascade of
bifurcations that produces the interference-like pattern from certain angles. From other
angles, the cascade of bifurcations which marks the chaotic state of transition is not
'visible' to the detector screen and, as a result, one observes just a particle-like
effect.
Thus,
certain changes in the detector's angle of engagement cause the detector to emphasize, in
the samples taken, patterns of on/off or emphasis/de-emphasis states which exhibit
interference-analog effects. Other changes in the detector's angle of engagement cause the
detector to emphasize or feature, in the samples taken, patterns of on/off or
emphasis/de-emphasis states which exhibit particle-like effects.
One must
keep in mind that chaotic states are not random states. The cascade of bifurcations of
phase relationships which occur during the interference-like process take place within a
set of determinate parameters. Consequently, how one samples such a state may affect the
structural character of the observed results- producing interference-like patterns when
taken from one direction, while generating particle-like patterns when taken from another
direction.[Return to Essay]
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