In relativistic physics a field is defined as a continuous physical quantity that has a value for each point in space and time while Relativistic Quantum Field Theory (QFT) defines particles as excited states of an underlying physical field.
However there is a conceptual discontinuity between QFT and its relativistic component because it is based on the abstract properties mathematics while the Einstein’s Theory of Relativity is based on the physical properties of space-time environment.
For example the Schrödinger wave equation that is used to mathematically define a particle in QFT does so in terms of a non-dimensional harmonic oscillator at each point in space but does not define what is physically oscillating while Einstein defines the relativistic properties of space and time in terms of a physical interaction of time with three-dimensional space. Additionally it is difficult to understand how non-dimensional point oscillation can be responsible for dimensional properties of a particle because by definition it cannot have those properties.
Yet one may be able resolve this issue if one views the relativistic properties of our universe in terms of four *spatial* dimensions instead of four dimensional space-time and define what physically happens to continuous properties of space and time when a particle is created.
(The reason will become obvious later.)
Einstein gave us the ability to do this when he used the velocity of light to define the geometric properties of space-time because it allows one to convert a unit of time in his four dimensional space-time universe to a unit of space identical to those of our three-dimensional space. Additionally because the velocity of light is constant it is possible to defined a one to one correspondence between his space-time universe and one made up of four *spatial* dimensions.
In other words by mathematically defining the geometric properties of time in his space-time universe in terms of the constant velocity of light he provided a qualitative and quantitative means of redefining it in terms of the geometry of four *spatial* dimensions.
The fact that one can use Einstein’s equations to qualitatively and quantitatively redefine the curvature in space-time he associated with energy in terms of the field properties of four *spatial* dimensions is one bases for assuming, as was done in the article “Defining energy?†Nov 27, 2007 that all forms of energy can be derived in terms of a spatial displacement in a “surface” of a three-dimensional space field with respect to a fourth *spatial* dimension.
However it also allows one to define the physicality of the harmonic oscillator QFT associates with a particle in terms of physical interaction of the field properties of three-dimensional space with a fourth spatial dimensional similar to how Einstein define gravity in terms of a physical interaction of time with three-dimensional space.
For example the article, “Why is energy/mass quantized?” Oct. 4, 2007 showed that one can explain and understand the physicality of the harmonic oscillator QFT associates with particles terms of the classical field properties of a wave by extrapolating the laws of resonance in a three-dimensional environment to a matter wave moving on “surface†of a three dimensional space manifold with respect to a fourth *spatial* dimension. It also explained why all energy must be quantized or exist in these discrete resonant oscillators when observed.
Briefly it showed the four conditions required for resonance to occur in a classical environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial would occur in a matter wave moving in four *spatial* dimensions.
The existence of four *spatial* dimensions would give a matter wave the ability to oscillate spatially on a “surface” between a third and fourth *spatial* dimensions thereby fulfilling one of the requirements for classical resonance to occur.
These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital. This would force the “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.
However, the oscillations caused by such an event would serve as forcing function allowing a resonant system or “structure” to be established in four *spatial* dimensions.
Observations of a three-dimensional environment show the energy associated with resonant system can only take on the incremental or discreet values associated with a fundamental or a harmonic of the fundamental frequency of its environment.
Similarly the energy associated with resonant systems in four *spatial* dimensions could only take on the incremental or discreet values associated a fundamental or a harmonic of the fundamental frequency of its environment.
Therefore these resonant systems in would be responsible incremental or discreet energy associated with quantum mechanical systems.
This allows one to define the physicality of the harmonic oscillators QFT associates with particles in terms field properties of either four dimensional space-time or four spatial dimensions because as was shown earlier they are equivalent.
However, one can also define its field properties of a particle in terms of the boundaries of its harmonic oscillator.
In classical physics, a point on the two-dimensional surface of paper is confined to that surface. However, that surface can oscillate up or down with respect to three-dimensional space.
Similarly an object occupying a volume of three-dimensional space would be confined to it however, it could, similar to the surface of the paper oscillate “up” or “down” with respect to a fourth *spatial* dimension.
The confinement of the “upward” and “downward” oscillations of a three-dimension volume with respect to a fourth *spatial* dimension is what defines the spatial boundaries of the harmonic resonator associated with a particle in the article “Why is energy/mass quantized?“
However as mentioned earlier it also defines the physical boundaries of the harmonic oscillator QFT associates with a particle in terms of the properties of a wave moving on a continuous field consisting of four *spatial* dimensions or four dimensional space-time because remember as was show earlier they are equivalent
This also provides the ability to understand the inseparability of the concepts of a field and particles in QFT because it clearly defines how one is depend on the other.
However it also explains why a field can display either the properties of a particle or the wave properties of a harmonic oscillator when measured because if one wants to measure the total energy contained in a given volume of space one will observe it as a particle while if one want to measure how it is propagated through space one must observe its wave properties.
Additionally it defines a classical reason why particles sometimes behave like oscillators and sometimes like particle and why it is impossible simultaneously observe these two different properties.
As shown earlier the energy contained in a quanta of space associated with a particle would be defined by the wavelength of its harmonic oscillator. In other words to observe or measure the particle properties of a given volume of space one has to sample all of its energy leaving nothing of its wave component to measure. Similarly if one wants to observe or measure fully the wave energy of a quantum of space one would have to sample all of its energy leaving none of its particle properties behind.
(If one does not want to observe all of the energy in a given volume of space then one would expect that the difference would be made up by the emission of the harmonic oscillator QFT associates with photon or other particle.)
The reason why one cannot simultaneously measure both its wave and particle of the harmonic oscillator it is because as mentioned the energy of a particle is defined by its wave properties. Since the energy that defines a particle is the smallest unit of its harmonic oscillator if one measures its particle properties there would be no wave energy left for measuring its wave proprieties while if someone measure its wave energy there would be no energy left to support its particle properties. Therefore making one of these measurements precludes the other.
This shows that one can integrate the abstract mathematical properties of Schrödinger wave equation or the foundation of Quantum field theory with the continuous physical of space and time In Einstein’s theory of relativity.
It should be remember Einstein’s genius allows us to chose whether to solve all problems in either a space-time environment or one consisting of four *spatial* dimension when he defined the geometry of space-time in terms of energy/mass and the constant velocity of light. This interchangeability broadens the environment encompassed by his theories by making them applicable to both the spatial as well as the time properties of our universe and gives us a new perspective on the integration of QFT with the relativistic properties of space and time.
Later Jeff
Copyright Jeffrey O’Callaghan 2014