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Philosophical Reflections in Physics and Math
Quantum Quandries - Part 4


The Necker cube-analog

Another manner of looking at the nature of the spectrum of ratios of constraints and degrees of freedom associated with the dynamics of dimensional dialectics is through a sort of Necker cube-analog. If one were to treat the complete structure of some given quantum entity as a set of ratios of constraints and degrees of freedom, then, particle-effects could be considered to be a subset of the overall set of constraints and degrees of freedom which constitutes the structural character of the quantum entity being considered. Wave-effects, on the other hand, might be considered to be another subset of the overall set of constraints and degrees of freedom. The Necker cube-analog aspect arises during the transition between the particle-effect and wave-effect arrangements of a structure's spectrum of ratios of constraints and degrees of freedom.

Alternatively, wave-effects could reflect the transitions between, or among different particle-effect phase states. The phase relationships between, or among, such states have a Necker cube-like character in which the structural character of a given sub-atomic entity is able to switch back and forth between, or oscillate among, different phase states. Wave-effects are due to the oscillatory character of the phase relationships that link the different ways in which a given structure gives manifestation to its phase states.

The cause of these phase transitions or shifts could be the result of spontaneous factors which are internal to the dialectic among the various subsets of a given quantum entity's structure. The phase shifts also might be the result of external factors which induce such a transition in the phase relationships which tie together different subsets of constraints and degrees of freedom in the quantum entity as a whole.

Necker-cube analog phase transitions can be characterized in the following way. They refer to any transition resulting in an individual or multiple change of the expression, orientation, engagement, or value of the phase relationships among various point-structures, as long as such transitions are not also accompanied by any change in the structural character of the point-structures being linked through the phase relationships.

The shifts which can be observed in the two dimensional Necker cube is a simple example of the kind of transition being alluded to in the foregoing. However, there are some important qualifiers which must be kept in mind.

The Necker cube illusion is generated by the way in which the human visual system engages a two-dimensional figure constructed in an appropriate fashion. Similarly, in the case of Necker cube analog phase transitions, such shifts arise in the context of the dialectic between a given hermeneutical system and the way that system dialectically engages phenomenological structures. Nonetheless, the transitions which arise out of this engagement are not necessarily an illusion.

The basic reason why the Necker cube is referred to as an illusion is because it helps give rise to an experience in which movement in the Necker cube is suggested without any movement actually having occurred in that structure. However, there is an important sense in which this illusion approach to the phenomenon misses an opportunity to focus on a key aspect of the process that is not illusory in the least. More specifically, the capacity of the human visual system to be able to look at something in a variety of ways, to get a variety of readings on the structural character of a given object, event, process, and so on, is an extremely fundamental and heuristically valuable tool that can be a source of important insights, perspectives and understanding.

Suppose one were to characterize the links between the nodes of a Necker cube as being phase relationships between different aspects of a hermeneutical or phenomenological structure, rather than merely as being lines connecting the vertices of a geometric structure. Given the foregoing, one might construe the shifts in perspective which can occur with respect to Necker cubes as probes of, for example, the inferential mapping component of the hermeneutical operator, or of congruence functions attempting to establish connections of a certain structural character between, or among, different point-structures of a given object, event, state, and so on.

Moreover, another feature to which attention is being drawn by means of the idea of Necker cube analog phase transitions, concerns the way such transitions seem to occur almost instantaneously, without going through any intermediary stage. This is consistent, or appears to be so, with how a structure gives expression, over time and under various circumstances, to different facets of its spectrum of ratios of constraints and degrees of freedom.

More specifically, previously, I have spoken of the switching on and off of various dimensional genes. This occurs with respect to different sets of phase relationships which are given expression through the dynamic dialectic that arises amidst the tension existing between the constraints and degrees of freedom that constitute the dimensions structural character. The switching on and off of such sets of phase relationships is somewhat like the change of perspective which occurs in relation to the visual engagement of the Necker cube.

In each case, the switch seems to go directly from a pre-transitional structural configuration to a post-transitional structural configuration. In other words, seemingly, one set of phase relationships has been switched off or de-emphasized (namely, the pre-transitional structural configuration) and another set of phase relationships has been switched on or emphasized (namely, the post-transitional structural configurations). In short, one set of phase relationships has been replaced by another set of phase relationships, and the switch over has not necessarily proceeded through any intermediate stage.1

The yin-yang principle of phase relationships


There is a further aspect, however, to the Necker cube analog phase relationship notion. In both the case of the Necker cube 'illusion', as well as in the case of the hermeneutical counterpart to the visual illusion, one of the features being emphasized is the way in which each of these phenomena is rooted in the process of engagement between two or more systems.

However, there is another possibility which should be mentioned as well. This further possibility involves the internal dialectic of a given structure independent of its being engaged by, or engaging, other structures.

Part of what is meant by the spontaneous, dynamical activity of a given structure is an expression of the Necker cube analog phase relationship idea which has been outlined above. The patterns of emphasis/de-emphasis, or switching on/switching off, that occur with different sets of phase relationships to which a structure's spectrum of ratios of constraints and degrees of freedom give expression, seem to reflect the various properties of the Necker cube analog phase relationships process.

One might suppose there are a number of fundamental thematic sets of phase relationships (i.e., the basic harmonics of the structure), together with an additional number of secondary, tertiary, and so on, sets of phase relationships that dialectically interact with one another and which constitute the internal dynamics of the structure. These secondary (etc.) themes are comparable to, but not exactly the same as, the second, third and forth harmonics of a given structure- the difference being that these additional thematic sets are not necessarily multiples of the primary harmonic themes as is the case in normal waveforms.

Conceivably, the various sets of phase relationship themes run through cyclical patterns (whether periodic or aperiodic) in which they induce one another to turn on or off, or to become modulated in one fashion or another. The continued, spontaneous oscillatory character of the dialectic would be responsible for providing the structure with many of the main themes of phase relationships which go into making the spectrum of ratios of constraints and degrees of freedom have the character it does for that structure.

Other secondary or tertiary (and so on) properties of the structural character would emerge during the process of engagement with other kinds of structures. These other properties would emerge as different aspects of the spectrum of ratios of constraints and degrees of freedom to which a given structure is capable of giving expression were induced into, or spontaneously generated, some given form of manifestation.

A structure (whether an object, event, process or state) consists of a spectrum of ratios of constraints and degrees of freedom. This spectrum of ratios is capable of expressing itself in a variety of ways under different circumstances.

At any given point in time, a particular ratio gives expression to one of the phase states of a structure's spectrum of ratios. The full structural character of the structure is capable of giving expression to other phase states. Thus, although the basic spectrum of ratios of constraints and degrees of freedom remains the same, different facets (i.e., ratios) of that spectrum become actively emphasized or switched on while other facets (i.e., ratios) are de-emphasized or switched off.

As one runs through the various phase state possibilities of a given structure, there is a transition or shifting from one pattern of switching on and off to another pattern of switching on and off. Or, said in a slightly different way, there is a transition from one ratio of constraints and degrees of freedom to another ratio of constraints and degrees of freedom.

The transition or shifting is effected through the phase relationships which tie together the various constraints and degrees of freedom of different ratios that are drawn from the structure's overall spectrum of ratios. Moreover, such phase relationships are an expression of the dialectic of the various ratios of constraints and degrees of freedom that constitute the structural character of a given object, process, event, and so on.

As the constraints and degrees of freedom which constitute a given object interact with, and play off against, one another, they give expression to, as well as generate, phase relationships. When the internal character of that dialectic changes its mode of expression, either spontaneously or through being induced to do so, phase shifts or phase transitions result. Among other things, these phase shifts or transitions involve various patterns of emphasis/de-emphasis, or switching on and off, involving different combinations and facets of the constraints and degrees of freedom which make up the structural character of the object, event or process.

Each constraint or degree of freedom of a particular structure's spectrum of ratios is a manifestation of different phase relationships which have been generated through the dialectic between, or among, a variety of dimensions. Moreover, each phase relationship of the dialectic between, or among, various dimensions has, in a sense, two themes. One theme concerns the property of a degree (or degrees) of freedom. The other property concerns the property of constraint. This is the yin-yang principle of phase relationships.

When one engages different levels of scale of ontology, one is, in actuality, engaging different levels of scale of dimensional dialectics. In this respect, wave-effects and particle-effects are just certain phase states which have Necker cube analog phase relationships, all of which have been established through the dialectic of various dimensions.

The structural character of quantum events


If one considers the ground state of a hydrogen atom, the wave function for the orbiting electron describes a sphere. The size of this sphere is approximately 2 x 10 larger than the size of the proton about which the electron orbits.

According to the Copenhagen school of quantum theory, as long as the electron is not being measured, the electron does not actually exist at any given point within the sphere of its orbit. The electron is said, instead, to be spread throughout the sphere.

Only when the electron is engaged by a process of measurement, does the sphere collapse down to the size of the point-like electron which is observed in the experimental context. What the electron does when it is not being measured or how the electron does whatever it does while occupying the sphere defined by its wave function is entirely unknown.

Another unknown is the precise nature of the collapse of the wave function. In other words, just how does the act of measurement cause the wave function to collapse? Just how does the process of measurement engage a cloud of probability or cloud of possibility in order to eliminate all possibilities but one: namely, the one which is observed at the time of measurement? Furthermore, given that the proxy wave or probability wave is not considered to be anything real, how does a real process such as the act of measurement engage an entirely fictional entity?

There are three possible excited energy states for the hydrogen atom. Such states last only a few billionths of a second.

The orthodox quantum theoretical position maintains that entirely random factors intrinsic to the ontology of the atom will determine when a photon will be emitted, thereby returning the excited atom to the ground state. Thus, if one takes exactly the same atom and excites it again, although another photon will be emitted, the time of emission will differ from the previous time of emission due to the intrinsic random factors which shape the of quantum phenomena.

Furthermore, in one of the excited states of the hydrogen atom, the character of the electron's wave function describes two disjoint proxy waves. Squaring the amplitude of this wave function indicates there is an equal probability (.50) that an electron will turn up in either section of the disjoint figure, but there is zero probability the electron will manifest itself anywhere between the two disjoint areas.

The perspective being proposed in this chapter would account for the wave function that represents the disjointed phase state of the hydrogen atom in the following way. The wave function actually describes portions of the spectrum of ratios of constraints and degrees of freedom which constitute the structural character of the electron.

Included in this description is a set of on/off patterns of phase relationship that are made possible by the dialectic of the ratios of constraints and degrees of freedom that constitutes the electron's structural character. Which pattern of on/off phase relationships is being manifested by the electron at the time of measurement will determine: (a) where the electron will show up at the time of measurement, as well as (b) the state of that electron when it is engaged during the measurement process.

Furthermore, the reason why there is zero probability of finding the electron in the zone separating the two disjointed areas described by the wave function is because Necker cube analog phase relationships are involved. In other words, in order for a possibility to exist, it has to be an expression of one of the ratios from among the spectrum of ratios that constitutes a structure's character. Since there is no ratio of constraints and degrees of freedom which corresponds to the zone between the disjointed regions described by the wave function for the electron of the hydrogen atom which is in a given energy state, there is no possibility for such a ratio to manifest itself.

On the other hand, there are ratios which correspond to the two disjointed possibilities described by the wave function. These possibilities will manifest themselves in Necker cube-like fashion according to the internal dynamics of the hydrogen atom in a given energy state.

In other words, while orbiting about the hydrogen nucleus, the electron is busily running through its spectrum of on/off patterns of phase relationships.

The transition from one pattern to the next is Necker-like in character. This means there are no intermediate stages between one pattern and the next.

However, although the individual states are discrete in character, the sequence or order in which the various states succeed one another is continuous. The sequence is continuous in the sense that as one pattern or ratio becomes de-emphasized, another pattern or ratio becomes emphasized, just as if one runner in a relay race had passed on the baton to the next runner.

If one collectively describes all of the patterns or ratios that are manifested by the electron over time, one will have given expression to the shape of the wave function for the electron in such an excited state. In this sense, the wave function is like a time lapsed photograph. However, if one engages the electron by means of measurement while it is orbiting the nucleus, then, one will be catching the electron in a given phase relationship pattern.

In a way, the manner in which the electron (or any particle really) runs through its spectrum of phase relationship patterns is sort of like Feynman's sum-over-histories method. The major difference is that rather than describing all the possible paths (with cancellations of some occurring as a result of being in opposite phase states), one is describing only the set of phase state patterns which are capable of arising from the dialectic of the electron's spectrum of ratios of constraints and degrees of freedom.

Consequently, Feynman's sum-over-histories is actually a description of the effects of the underlying phase relationship dialectics. As such, his approach is at least one level of scale removed from the internal dialectic that makes possible paths which have the structural character that Feynman is describing through his method.

An obvious question concerning the spectrum of ratios of constraints and degrees of freedom notion is this: what is being constrained or given degrees of freedom? Is it a substance or materiality of some sort? The answer depends on what dimensions are dialectically engaging one another.

Ultimately, what is being constrained or given degrees of freedom, is the autonomous, self-regulating order-field. By its very nature, it spontaneously places constraints on itself and permits itself various degrees of freedom, across a variety of levels of scale.

This spontaneous activity of the order-field ranges from the generation of dimensions, to their dialectical activity, to the phase waveforms which are the effects of such dialectical activity. So, on whatever level of scale one wishes to consider, the phase relationships, which are manifested on that level and which give expression to the constraints and degrees of freedom on that level, mark the presence and activity of the order-field as it dialectically interacts with itself.

Consequently, from the perspective of the foregoing position, there are no fundamental particles in any classical sense of a structure which contains or gives rise to certain static and dynamic properties. What is treated as a particle is a spectrum of ratios of constraints and degrees of freedom which arise as a result of the dialectical interaction of different dimensions that have been generated and set in motion by the underlying order-field.

The point of intersection for this dialectic of dimensions takes place not in space but in the temporal dimension. This means the point of intersection is not spatial in character, but rooted in phase relationships.

However, because information is transmittable to the other dimensions by means of phase relationships, certain kinds of localized effects will be manifested in, for example, the spatial dimension. These effects may be observed to radiate out from the spatial side of the intersection, but the source of the radiation is not really in space.

Thus, one has a spectrum of ratios of constraints and degrees of freedom which is expressed in terms of phase relationships that are not particles, but which can produce particle-like effects. Moreover, when these phase relationships oscillate in some periodic or aperiodic fashion, they are capable of giving rise to waveforms.

On the above view, what hits the phosphor molecule of the detector screen is, in a sense, a ratio of constraints and degrees of freedom. This is the part of a complex phase state generated by a dialectic of dimensions that spills into the E3 space of everyday sensory experience.

This part of the phase state gives expression to the dynamic and static properties of the "particle" that are, to some degree, measurable. These properties are manifested in such a way that they have, when observed from a certain level of scale, the appearance of a traditional/classical particle in as much as the properties which are given expression are localized. Moreover, part of the reason why the 'particle' is localized is that the effects will be tied to the spatial-material-energy aspects of the dialectic of dimensions, all of which help to determine the context of the spatial locus or loci through which phase relationships will be translated.

One is reminded of some of the drawings in Rudy Rucker's book, The Fourth Dimension, in which a higher dimension interacts with a lower dimension. Only those aspects of the higher dimension which are compatible with the lower dimension are observable in the lower dimension. Furthermore, the way that the aspects of the higher dimension are observable in the lower dimension cannot be made sense of in terms of the physics or mathematics that is limited to the structural character of the lower dimension.

 Similarly, the way in which other dimensions make their presence felt in a given dimension cannot be explained in terms of hermeneutical systems which are strictly bounded by the dimension into which the influences of other dimensions have been introduced by means of appropriately translated phase relationships. In other words, although certain aspects of other dimensions make their presence known through the aspects of phase information which are compatible with the dimension(s) in which the influence or effects is being manifested, much of the structural character of the dialectical activity giving rise to that influence or effect will be 'out of sight' of the dimensional context through which the effect or influence is being manifested.

Short-run and long-run effects of dimensional dialectics


In the previous account quanta have been described as being neither waves nor particles. They have been characterized in terms of the spectrum of ratios of constraints and degrees of freedom that are capable of generating effects which have wave and/or particle properties.

Moreover, the spectrum of ratios supposedly gives expression to shifting patterns of emphasis and de-emphasis with respect to phase relationships. These on/off patterns create, in turn, oscillating systems.

In view of the foregoing sorts of contention, an obvious question to ask is this: under circumstances in which there is an absence of conditions conducive to both interference and diffraction, why don't electrons being shot at a phosphor coated screen show wave characteristics? Shouldn't the wave properties which are capable of being produced by the oscillating internal dynamics of the electron register at some point during the experiment?

In the short run (i.e., focusing on engagements between, say, "particular electrons" and "particular phosphor molecules"), there will be no wave properties which manifest themselves on the screen. The basic reason for this is as follows.

Measurement processes which do not involve interference or diffraction phenomena as part of the experimental set-up tend to engage quantum phenomena while the latter are in a particular phase state. Such states are a function of a set of phase relationships that are generated by the dynamic tension between, or among, the constraints and degrees of freedom being manifested by a structure at a given point in time.

Although each phase state constitutes a slice of the overall oscillatory character of the quantum entity's structure, the character of the engagement process to which short-run measurement gives expression is not geared to be sensitive to, or reflective of, that oscillatory character. Under the conditions previously stipulated, the nature of the short-run measurement process is geared to be sensitive to, and reflective of, a series of isolated phase states drawn from the spectrum of possible ratios of a number of different electrons.

In other words, the nature of the short-run methodology is such that it is incapable of tapping into, and displaying, the oscillatory character of the on/off or emphasis/de-emphasis transitions in phase relationships that lead to, as well as give expression to, shifts in the way different ratios of a given structure's spectral character manifest themselves. All the short-term methodology can accomplish is to take samples of a given electron in a given phase state as the electron encounters one of the phosphor molecules coating the screen at particular points in time.

On the other hand, in the long-run, effects will register on the screen that seem to indicate the presence of wave characteristics. Indeed, experiments have been done (initially by accident) in which an electron gun was left on for a number of days, and the results showed a wave pattern had formed on the screen.

The wave pattern that emerged in the foregoing set of circumstances was not an Airy pattern. However, as is the case with Airy patterns, although the collective form on the screen had the characteristics of a waveform, the collective pattern was produced as a result of an indefinite number of discrete engagements between electrons and the phosphor molecules coating the detector screen.

The reason there are wave-like effects which show up during the long-run measurement process is due to the way an electron's spectrum of constraints and degrees of freedom unfolds across time to give expression to various ratios and combinations of phase relationships. These manifested ratios and phase relationship values will conform to a distribution pattern that will have a wave-like character. In point of fact, the distribution pattern represents a sort of time-lapsed record of various aspects (such as primary, secondary and tertiary themes) of the structural character of the electron's spectrum of ratios of constraints and degrees of freedom manifesting themselves over time.

A distribution pattern for a long-run measurement process has the character it does (i.e., it is wave-like) because the internal dialectic of the spectrum of ratios is undergoing transitions from the time an electron leaves the heated metal filament until it reaches the phosphor coated screen. Even in the case in which certain phase states are preserved over a period of time, there is an oscillation of phase relationships maintaining those states.

However, the character of these oscillatory systems tend to produce self-similar rather than self-same. As a result, there will be variability in, among other things, the character of the location of where the electrons will engage the phosphor molecules of the coated screen.

What shows up on the detection screen is, in a sense, a time-lapsed reflection of an individual electron's structural character writ large as a distribution pattern. More specifically, over time, different phase relationship patterns of emphasis/de-emphasis will be manifested as a result of the electron's own internal dialectic involving the various ratios of the spectrum of constraints and degrees of freedom which constitute the structural character of the electron.

Since these shifting patterns are oscillatory in character (although they are aperiodic oscillations), the general form of the pattern which shows up on the screen will be wave-like because the electron-phosphor molecule engagement occurs at different stages of the oscillatory cycle of the internal dialectic of the electron, resulting in slightly different values from one engagement to the next. Eventually, there will be a sampling from nearly every phase state of the aperiodic, cyclical character of the electron's internal dialectic of phase relationships as this dialectic gives expression to shifting ratios of constraints and degrees of freedom.

However, the wave-like effects which show up on the detector screen during the long-run measurement process should not be confused with the oscillatory character of the electron's internal dialectic of phase relationships and shifting ratios of constraints and degrees of freedom. The former effect is, in fact, indirect evidence for, and made possible by, the latter phenomenon.

The long-term measurement process is, in a sense, more sensitive to the presence of oscillatory properties in the electron's structural character than in the case with respect to the short-run measurement process. This enhanced sensitivity is largely due to the capacity of the former measurement process to provide a sampling technique, unintended though it may have been originally, which engages the internal dialectic of the electron at different phases of its cyclical character. As a result, one is able to develop a better portrait of the electron as it runs through the various ratios which constitute the spectrum of constraints and degrees of freedom that gives expression to the electron's structural character.

Nonetheless, the portrait of the structural character of the electron derived through the long-run measurement process is not really a wave-phenomenon. It is an artifact of the sampling character of the measurement process. If one assumes all electrons have, more or less, the same basic structural character, then, by engaging a lot of different electrons during different phases of their oscillatory , one, in effect, can construct a representation of what, very likely, goes on in any given electron over time.

On the other hand, although the result of the long-run measurement process is not itself a wave phenomenon per se, it does provide evidence which is consistent with an interpretation that construes the internal dialectic of the electron in terms of oscillatory properties. However, one should keep in mind that the nature of such oscillatory properties is a function of Necker-cube analog phase relationship transitions. These transitions are manifestations of different ratios of constraints and degrees of freedom that replace one another in a continuously discrete or discretely continuous fashion, such as occurs in a relay race.

Thus, after being shot from the electron gun, the interim period of the electron's flight to the phosphor coated screen is spent manifesting a variety of phase states in an oscillatory manner. This means the electron is capable of giving expression to wave while in transit, even though it will engage the phosphor molecule in just one phase state and, therefore, appear to have a particulate character at the point of impact.


FOOTNOTE


1.) One further possibility connected with the idea of the Necker cube analog concerns a property of particle spin which has been something of a puzzle for quantum theorists since the introduction of the spin feature into the quantum model. Although all of the basic characteristics of spin can be quantitatively described, an explanation for what actually is occurring in relation to the phenomenon of spin has eluded theorists.

The problem is this. Experimental evidence indicates that a particle must go through what amounts to two "revolutions" before it is able to return to its starting point.

Consequently, the following question arises: What sort of process could account for such an effect? One would expect that just one 'revolution' should be sufficient to return any "normal" or conventional (i.e., conforming to everyday sorts of experiences) particle to its starting point.

In line with the perspective of the current article, the above problem might be resolved in the following fashion. Suppose one were to propose that particles were not like mathematical points. In other words, let us suppose that all so-called elementary particles had an internal structure that was somewhat analogous to a more complex n-dimensional version of a Necker cube. Thus, instead of being limited to the several degrees of freedom of a Necker cube, particles are to be characterized by an n-dimensional, m-degrees of freedom internal structure.

Furthermore, suppose that in order for such a particle to make a complete circuit of its internal states, the particle must 'turn-on' or run through a certain number or subset of the aforementioned m-degrees of freedom. If one were to assume that the process of running through this subset of the particle's internal structure of m-degrees of freedom is continuous (although discretely so in the manner of a relay race), then, the particle's internal dynamics serves as an analog for the manner in which, say, a sphere revolves. However, rather than requiring only 360 degrees to make a complete circuit, as is the case with a normal sphere, particles actually run through a sufficient number of degrees of freedom to generate an analog process for 720 degrees of rotation in a normal sphere before the particle returns to its starting point state.[Return to Essay]



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