|Knots in j-space
28th November 2014
The role of topology in explaining the structure formation in the high-energy physics is becoming more and more prominent. It is required that we first establish a fundamental topological structure before developing the description in (t, x, y, z) physical space. The combination of time axis with Euclidean space to describe an event is equivalent to developing an associated metric for the topological structure. The metric has to follow topology and therefore if we know the topological properties we can explain some of the most basic issues such as the definitions of continuity and connectedness. The description in terms of energy and momentum can follow afterwards.
The knot theory is one of the instruments which allows us to visualize the possible topological states a system may take over a period of time. What started as a suggestion by Lord Kelvin and consequently an extensive work done by Tait on characterizing the various types of knots, has evolved into a very powerful mathematical field which has the potential to unlock the fundamental principles guiding the formation of the elementary structures.
Mathematically the knot structures are algebraically represented by knot-invariants, which are polynomials which remain unchanged for the equivalent knots. The earliest example of such polynomials is Alexander polynomial. The other well-known examples are Alexander-Conway polynomial, Jones polynomial, and HOMFLY polynomial. We must remember that these knot polynomials represent a description in three-dimensional space. Conventionally we develop a mathematical structure for a physical knot based on certain mathematical rules, determine the knot invariants and correlate them to the probability amplitudes. A desirable quality of these knot-invariants is to make distinction between the knots based on properties such as chirality as was shown by Jones polynomial. Clearly more are the properties distinguished by a knot-invariant more robust is the polynomial.
We can take a slightly different approach. Rather than assuming the existence of a knot, we form a knot made of step-by-step measurements in a space characterized by entropy. The polynomial is written in 2+1 dimensional measurement space instead of conventional R3 space. Please note that 2+1 dimensional space in the context of knots, is not the 2+1 dimensional Minkowski space. Also since the entropy is in picture, the order in which the moves are being made while forming the knot becomes important. The basic idea is shown below:
For a given event, an observer of very high capacity will measure it as a shallow-well problem or a zero-entropy problem and hence an unknot is formed. At the same time for an observer of infinitesimal capacity, the event is equivalent to an infinite well or Q-box problem. As a result the entropy is very high and a knot is formed. Therefore the nature of the knot is independent of the nature of the event and depends upon the observer's capacity. It is equivalent to saying that the poor man's blackhole is a rich man's pothole.
In the sections describing the formation of knots, we will describe the formation of Trefoil knot in j-space. Later we will add sections on Figure-8 knot and Unknot. We will also discuss the formation of composite knots because of the measurements made by a macroscopic observer and how they lead to higher order polynomials.
String Theory vs. Knot theory? It is reasonable to expect that the laws of nature will be metric independent, therefore may be we should sort out things in topological space first.
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Cauchy and Gaussian Distributions
Discrete Space, b-field and lower mass bound
The cat in box
The initial state and symmetries
Discrete measurement space
The frog in well