In a separate article titled 'What ails the Fundamental Research in Physics', it has been brought out that 'all is not well with Fundamental Physics' which has inadvertently come under overbearing dominance of Mathematics during the 20th century. In another article titled 'Permittivity and Permeability Constants of vacuum', the role of physical concepts and dimensions associated with all physical quantities has been discussed. These discussions also highlighted the fact that in mathematics we generally deal with dimensionless numbers and mathematicians are therefore tempted to underplay the role of physical dimensions in sophisticated mathematical analysis. But in physics, the essential linkage between mathematical analysis and the physical reality is provided through the dimensions of relevant physical quantities and the associated physical concepts. Some mathematicians vehemently argue for the equivalence of space and time dimensions [L] and [T], and sincerely believe the velocity of light c to be dimensionless. This sort of viewpoint causes immense damage to the well-established physical concepts, to the mental grasp of physical reality and appears to be at the root of the crisis.
In view of this, let us examine the circumstances, constraints and internal contradictions of Fundamental Physics, which tend to damage the well-established physical concepts and undermine the mental grasp of the physical phenomenon. For this, let us first distinguish between a Physical Theory and Mathematical Models.
In applied sciences and engineering applications, mathematical models stand out as clear, unambiguous and distinct entities and are valued as such. In these cases, conceptual understanding of the associated physical phenomenon is a prerequisite for the development of a mathematical model. Generally, for the development of a required mathematical model, we need to
a) Begin with a qualitative physical description of the phenomenon
b) Identify the physical quantities or parameters that influence the system.
c) Define a clear objective.
d)
Specify the inter-relationships
and inter-dependence of physical parameters.
e)
Specify the initial conditions, end
constraints and the supporting assumptions.
f)
Provide at least a limited quantitative
input.
g)
Develop the model through 'trial &
error', feedback and 'fine-tuning'.
The resulting mathematical model becomes invaluable tool at the hands of concerned scientists and engineers. The mathematical models too have to be rigorously tested over the whole range of their applicability before they could be relied upon.
However, in theoretical physics the demarcation or distinction between physical theories and mathematical models is quite blurred and ambiguous. As it is, theoretical physics is concerned with grasping, understanding and explaining all aspects of basic physical phenomenon through physical theories. A physical theory will therefore consist of
a) A clear-cut unambiguous definition and explanation of associated physical concepts.
b) A qualitative description of the physical phenomenon in terms of well defined physical concepts, which can be fully grasped by the human brain.
c) A mathematical model to provide quantitative description of the physical phenomenon in terms of well defined physical quantities.
In this regard, a mathematical model is generally a built-in element of a physical theory. A physical theory must be able to specify its operational range, supporting assumptions and the end constraints. Above all a physical theory must be based on a minimum possible number of supporting assumptions that are plausible and compatible with well-established physical concepts. As an essential end constraint, a physical theory must broaden and enlarge the human perception of physical phenomenon and not weaken it through introduction of weird new ‘physical’ concepts, which are beyond the grasp of human brain.
A physical theory may be developed ab-initio through in-depth understanding of the physical phenomenon, inter-relationships of associated physical quantities and through mental insight, contemplation and relevant mathematical model based on plausible assumptions. The only guidance for such development of a physical theory is that it must account for the known physical observations, it must be consistent and compatible with well-established physical concepts. This development approach is quite difficult and hence adopted rarely.
On the other hand, a physical theory may be developed through an in-depth analysis of a large amount of data of relevant physical observations. In this approach a mathematical model is developed first to account for known physical observations and to make predictions. If the predictions don't get fully verified then the model is readjusted and fine-tuned to improve the predictions to the desired accuracy. A major weakness of this development approach is that not only the mathematical model is based on crucial assumptions, even the verification of 'predictions' through physical observations has to be based on crucial assumptions since the physical observation (including its interpretation) at ultra-microscopic scale is quite an intricate and complex process.
Some time a major problem arises here. After fine-tuning a mathematical model, it is automatically elevated to the status of a physical theory. What is missed in the process is the in-depth understanding of the associated physical phenomenon in terms of well-defined physical concepts, which can be fully grasped by the human brain. Often abstract mathematical concepts, introduced during development of the mathematical model, are passed on as physical concepts even if they are beyond the grasp of human brain (that is, even if the human brain fails to establish one to one correlation between such concepts and the physical reality). In the process of elevating a mathematical model to the status of a physical theory with the crutches of abstract mathematical concepts, the physical reality is made to appear far too complex, weird and beyond the grasp of human brain; thus defeating the very objective of a physical theory. Often it is claimed that the human brain is incapable of perceiving such abstract mathematical concepts as physical concepts because of certain unspecified biological deficiency in its development over the ages. Believe me, if you may, neither the physical reality is actually so complex and weird as it is made out to be, nor there is any biological deficiency in the development of human brain as claimed to be. In fact the human brain is one of the most wonderful achievements of the Universal evolution over the ages.
Typical examples of abstract mathematical concepts that have been perforce thrust into the realm of physics, are 'curvature of space' introduced in GR; 'infinite mass density' associated with the notions of point mass elementary particles, black holes and the big bang; equivalence of physical dimensions of length [L] and time [T] in SR; and the ψ wave function of QM representing a physical quantity devoid of any physical concept whatsoever. Out of these, the concept of 'infinite mass density' violates the well established and well defined physical concepts of mass, space point & volume and hence must be declared as physically invalid. We may discuss the mathematical concept of 'curvature of space' in some detail to show its physical invalidity.
Dimensions of Radius of Curvature of Space
If we consider a one-dimensional straight line, it can be curved in an infinitesimal local region around point P, only when the line is extended into two-dimensional plane. The radius of curvature at point P, of this line, will extend into the second dimension perpendicular to the original (linear) dimension of the line. If we consider a two dimensional plane, it can be curved in an infinitesimal local region around point P, only when the plane surface is extended into three dimensional space. The radius of curvature at point P, (of any curve on this surface) will extend into the third (space) dimension perpendicular to the original surface. If we consider a three dimensional volume of space, it 'may' be curved in an infinitesimal local region around point P, only when the three dimensional space volume is extended into four dimensional space-time manifold. (Obviously the four dimensional space-time cannot be curved because there is no fifth dimension to accommodate this curvature). The radius of curvature at point P, will extend into the fourth [Time] dimension perpendicular to the original three-dimensional space. Thus the radius of curvature of space will have the dimension of time [T] which contradicts the well established physical concept of a radius having spatial dimension [L]. Probably to reconcile this contradiction the time dimension [T] in the four-dimensional space-time manifold is perforce considered equivalent to the spatial dimension [L]. This forced equivalence of spatial dimension [L] with the time dimension [T], shatters the physical concepts of both and hence must be declared physically invalid.
From the foregoing discussion, it may be concluded that even a fine-tuned mathematical model should not be elevated to the status of a physical theory, if it distorts or damages the well established physical concepts. Therefore, we must accept that GR is only a fine-tuned mathematical model of gravitation and the mathematical concept of space curvature must not be treated as a physical concept. For the GR to be elevated to a physical theory, it must explain as to why and how does a mass particle produce or gives rise to the gravitational field.
Similarly it can be argued that the QM and the 'Standard Model' are also mathematical models and any attempt to elevate them to the status of a physical theory, shatters some or the other physical concept in the process. For QM to qualify for the status of a physical theory, it must explain the physical concept associated with the 'ψ' wave function. Any physical theory of elementary particles must explain the why and how of their existence or occurrence in 'vacuum', as well as the how and why of their mutual interactions. Hopefully a future fundamental physical theory should be able to accommodate all the existing mathematical models in its overall framework.
Other
Important Articles
§
Ether, Vacuum or the
Elastic Continuum
§
What Ails the Fundamental Research in Physics
§
Permittivity and Permeability Constants of
Vacuum
§
What If the Permittivity and Permeability of
Vacuum were Zero
§ GTR is founded on a Conceptual Mistake
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