Reciprocal Symmetry and its Relation to Einstein ’ s Postulate , Lorentz Transformation and Discreteness

Objectivity demands that kinematics should be independent of the quantities (velocities or their reciprocals) we define. This demand is translated as Reciprocal Symmetry (RS), which we have defined. We have shown that RS gives an upper bound velocity, the mathematical analogue of Einstein’s postulate, and Lorentz transformation. RS also requires discreteness. RS promises to bridge the gap between relativity and quantum mechanics.


Introduction
Velocity v is defined as the distance covered per unit time; v = x/t.We might as well define slowness v' as the reciprocal of v; v'=1/v = t/x.Physics is independent of the quantities we define.Therefore, objectivity demands that motion described in terms of slowness should be as valid as the description in terms of velocities.We postulate: Postulate: Kinematics is symmetric under reciprocal inversion of velocities.
The postulate demands that description of motion should remain invariant under inversion and vice-versa.Relative velocity or distance covered should be independent of the reciprocity of velocity.We shall see that relativistic relative velocity (Lorentz transformation) is reciprocal symmetric.

/v v 1 →
Reciprocal symmetry requires that velocity have an upper bound (like Einstein's postulate).The reciprocal (slowness) has a lower bound.Slowness must form a group; difference between two slownesses is slowness.Therefore, the difference must be bounded below.Therefore, slowness must be discrete [1].This relates reciprocal symmetry to discreteness.Since discreteness is in the heart of quantum mechanics, we expect to find a relation between relativity and quantum mechanics through reciprocal symmetry

Reciprocal Symmetry
Definition 1.A function f will be called reciprocal symmetric if it remains invariant when one (or more) of the arguments is replaced by its reciprocal, i.e.

(
) ( ) Definition 2. A function f will be called ant-symmetric (or reciprocal ant-symmetric) if it changes sign when one of the arguments is replaced by its reciprocal, i.e.
The two above relations may be combined into one , when m is an integer (2.5)

Reciprocal Symmetry and Upper Bound
Definition: Let ⊕ represent a reciprocal symmetry sum so that ( ) Theorem: (3.9) (3.9) is Einstein's postulate [1] if we assume that c is the speed of light.

Lorentz Transformation
A representation of ⊕ , which fulfills (3.1), is ( ) This is Lorentz transformation [2] for motion in one space dimension.

Galilean and Reciprocal Symmetric Velocities
To study symmetry properties of velocity we shall transform Galilean velocities to a form where symmetry properties come to surface.Consider the identity (5.1) We define [2] v/c = tanh(V/c) and call v the Lorentz algebraic (or reciprocal symmetric) velocity corresponding to Galilean velocity V. We may write as given by (4.1).

Lorentz-Algebraic Distance
To study symmetry properties if distance, we define, as in (5.2), reciprocal symmetric distance x corresponding to (familiar) Galilean distance X by L is a constant distance (corresponding to c of Eq. (5.2)).

Reciprocal Symmetric Multiplication
The rule of multiplication (6.3) has the following symmetry properties under reciprocal inversion For t/T = even integer For t/T = odd integer Relation (6.3) may be called "Lorentz multiplication" as it is consistent with (4.1) in the sense that it is distributive with respect to Lorentz-Einstein addition for t/T = odd integer (8.2) Let us consider the reciprocation operation Using the above relations (8.1) and (8.1) may be written in the compact form

Conclusion
It is shown that reciprocal symmetry yields Einstein's postulate and Lorentz transformation.The symmetry also gives discreteness.Reciprocal symmetry is, therefore, the ground where relativity and quantum mechanics may meet.
looking at the other part of (3.3), we have v/c w/c u/c ⊕ = (3.6)Short Communication Comparison between (3.5) and (3.6) shows 3) gives the distance traveled by a body in time t.A part of the contribution comes from velocity v/c.The other part comes from reciprocal velocity c/v.The time of travel must be an integral multiple of quantum of time T. As seen in (6.5), in the limit , we get the classical expression∞ → c vt x → .