Identities: Basis in Language

Language is formed by Identity. Identity forms Language. Identity is defined as the source of Language. Identity takes various forms. Since Identity is the origin, a relative Identity which gives rise to specific Languages within the given Language is called Base. When there are multiple Identities of same degree, they are called Basis. For example:

  • In Euclidean Plane, Origin (\(0\)) is the Fundamental Identity or the Base. At the higher degree, circles form Basis, these can be projected along cones to get Ellipse, Parabola, Lines, Hyperbola as a basis at various degrees.

  • In the Cartesian Plane, Origin (\(0\)) is the Fundamental Identity or the Base. The Coordinate Axes form the Base at higher degree.

Translation is Change of Basis is Renaming Identity

Lattices (Crystals) are formed by Translation symmetry. This is Change of Basis by translating the origin (Fundamental Identity) to different Lattices Points to act as new Identities (New Basis).

  • Start with a point in the plane. This is the Fundamental Identity (Origin). Other points (Identities) may be obtained by shifting (translating) left or right along a line by a constant length. This is how translation creates new identities.

  • Newton’s \(1^{st}\) Law of Motion is a Change of Basis (Identity): The Fundamental Identity is named as \(0\) (Origin) which is the Rest state of the body. This Identity is named Position (\(x\)) in Physics Language. If this Identity is translated, a new Identity emerges. The new Identity may be named as (\(1\)). In Physics Language, this new identity is named as Velocity (\(\dot{x}\)). When composed with Mass (\(m\), the Identity of matter), it is renamed as Momentum (\(p = \dot{x} \,\,m\)). A body in Translation stays in its state of uniform motion is a characterization of this new Identity. That is, for a body in translation, Momentum stays conserved. Moreover the connection of the new Identity Momentum (\(1\)) back to the Fundamental Identity (\(0\)) is named as Force (\(F\)). In Mathematics Language:

\[\begin{eqnarray} x \,\, =\,\,0 \quad \textrm{(Position is Fundamental Identity)}\\ Translation \,\, of \,\, Identity \,\, = 0 \to 1 \quad \textrm{(Translation gives New Identity)}\\ \dot{x}\,\, =\,\, 1 \quad or \quad \dot{x}\,\,(m)\,\, =\,\, 1 \\ F = 1 \to 0 \\ (\,\,\,F\,\,)\,\,= (\,\,\dot{x} \,\,m \,\,) - 0 \end{eqnarray}\]

So, Force (\(1 \to 0\)) is seen as the dual (opposite) of Translation (\(0 \to 1\)). Hence, Identifying Force with translation (displacement) gives Energy ( \(E\), Identity of Distance). This forms the Energy cycle. That is :

\[\left(\,\, F \,\,\cdot\,\, \left(\,\,x_1 \,\,-\,\,x_0\,\,\right)\,\,\right)\,\, =\,\, E\]
  • Thus, I see Force is a form of translation. Force and Translation are inverse to each other and can be identified by placing them together. This completes the cycle.

Translating \(1\) to a new Identity say \(2\) is same as the Translation of \(0\) to \(1\). This can be seen by simply renaming \(1\) to be \(0\) and \(2\) to be \(1\). This is called isomorphism in Mathematics Languae. Hence, it suffices to see the Translation of Identity from \(0\) to \(1\) and then reproduce all other forms as composition of this. This is what makes Newton’s Laws of Motion fundamental. In the Third Law, Newton Identifies the Ultimate Symmetry by Identifying that every action has an equal and oppsosite reaction, thus showing the Existence of Identity formed by placing these inverse forces together as a group.

  • Time (\(t\)) comes in as a “Normalization” to rescale things back with respect to the Fundamental Scalar Identity \(1\) (Time scale is uniform scalar metric). That is, if we identify points on a uniform scalar scale by the name \(t\), then \(I\,\, = \,\, \int_0^1 \,\,1\,\, dt\) which gives \(I\,\, =\,\, t_1 -t_0\), which is simply the connection of points on the uniform scale where the action ended and started. Hence every expression on the right hand side can be rescaled with respect to time. This means dividing the expressions on the right hand side by \(t_1 - t_0\). This renormalization gives time as a base scale in which other actions can be compared. In calculus Language, this takes the differential form \(d\,\,\cdot \,\,t\). In some fancy Language this is called Renormalization group where instead of the scalar group, several other groups can be used to rescale. In one way, this write up is to show that all theories can be written as a composition of relationship between two scales, each with their own identity. I name this process of relating the two scales as Cyclic Translation.

Translation of Languages in the usual sense also involves a similar process in which one has to translate the words from one Language (Former Basis) to words in another Language (new Basis). Knowledge Conservation in different Languages is a statement about Translated Identites. For example:

  • Knowledge stays conserved in all Languages translates to Laws of Nature stay the same in all Inertial Frames of Reference (Special Relativity).

In Phase Transformations, Condensation and Decondensation act as the two opposites to go from one Phase to the other just like .

Rotation is Change of Basis

As a special case of rotation, Reflection is change of Basis. On the other hand, Rotations can be obtained by composition of reflections. Hence, rotation is composed of change of Basis, which in the finite case gives an effective change of Basis.

Linear Algebra is Change of Basis

The Axes in Cartesian Plane form a Basis. A new basis may be obtained by rotation of these Coordinate Axes. For example, instead of the usual “\(+\)” shaped Horizontal and Vertical coordinate Axes, we may rotate them by half of right angle to obtain Equally inclined Axes to give a \(\times\) shape of the axes. This is a conical shape.

Change of Scale is Change of Basis

Changing the scale (metric or the measure) is change if basis.

Phase Transformations is Change of Basis

Going from one Phase to another involves Change of basis. For example:

  • In solids, discrete symmetries are the basis (boundaries which form the structure). Whereas,

General Relativity is Change of Basis

Changing the Axes from Horizontal (Time) and Vertical (Space) to make it equally inclined by rotating the Basis gives the Light Cone and the corresponding algebraic metric as (\(+,+,+,-\)).