﻿ Unification

A Unification of Quantum Mechanics and Special Theory of Relativity

Now we are in a position to give a solution of unifying quantum mechanics (QM) and classical mechanics (CM) governed by special theory of relativity, which I have arrived at on February 15, 2001, whose precise formulation is given in Time IX paper.

The relation that I have announced above to be assumed between the inside and the outside of a local system is described as follows:

Let u be the internal velocity of the particles inside a local system and let v be the velocity of the (center of mass of the) local system relative to the observer. Then we impose the following two conditions on the relation between the two velocities:

Axiom 1.

|u|2+|v|2=c2, where c is the velocity of light in vacuum.

Axiom 2.

The magnitude of the internal momentum mu is constant independent of the velocity v relative to the observer:

m2|u|2=m02c2,

where m0 is the rest mass of the local system and m is the observed mass of the local system moving with velocity v relative to the observer.

These two axioms are an extension of Einstein's principle of the constancy of the velocity of light in vacuum and are a version of T. S. Natarajan's postulates IV and V in the paper Do Quantum Particles have a Structure? where Natarajan considers both of internal and external worlds are classical. We consider the internal world quantum mechanical. Thus the above axioms need a justification in order for these to have a consistent meaning, which discussion is given in Time IX paper and will be sketched below. This will then yield a unification of quantum mechanics and special theory of relativity.

The idea is simple. From Axioms 1 and 2, we first note

m02c2+m2|v|2=m2c2,

which yields

m = m0 /{1-(v/c)2}1/2 ( > or = m0 ).

As we have discussed above, we can define a clock inside each local system as the one that has an actual size. A best definition of a clock of a local system would be the evolution of the local system itself: exp(-2pi itH/h), where H is the quantum mechanical Hamiltonian of the system; pi is the ratio of the circumference of a circle to its diameter (Ludolphian number); i is the square root of -1; and h is the Planck's constant. Then the local time of the system is defined by the t on the exponent of the clock exp(-2pi itH/h). Then we can show that this t equals the time defined by using a clock of any part of the system up to the error that is consistent with the uncertainty principle. (See Time IX paper, etc.)

If the time of a local system is defined as such, we can express the motion of particles inside the system by the solution f (t)=exp(-2pi itH/h)f (0) of Shroedinger equation:

(h/(2pi i )) (d/dt)f (t) + Hf (t) = 0.

When we think a free Hamiltonian H of one particle with mass m :

H = P2/(2 m ) = - (h/(8 pi2 m)) Laplacian,

where P is a quantum mechanical momentum identified with (h/(2pi i ))(del/delx1 , del/delx2 , del/delx3 ), we can move, by using Fourier transformation, to the momentum space and we are able to identify H with a real number L/m . Likewise, a general Hamiltonian can be identified, by using spectral representation theorem for selfadjoint operators, with a real number L/m .

In this identification our axioms 1 and 2 become:

Axiom 1. 2L /m2 + |v|2 = c2.

Axiom 2. 2L = m02c2.

Thus the clock of a local system, when moved to spectral representation space, becomes

exp(- 2pi itL/(hm )).

Then the clock of a local system has a period:

p (v) = hm /L = 2hm /(m02c2),

which shows that our QM clock has the natural property that is expected for usual clocks. This period takes the minimum value when v = 0:

p (0) = hm0 /L = 2h /(m0c2).

We call this p (0) the least period of time (LPT) of the local system. This is thought as giving a sort of "quantization of time" of a local system. Now by the above relation between m and m0, we have

p (v) = hm /L = p (0) /{1-(v/c)2}1/2 ( > or = p (0) ).

Thus time, measured by our QM clock, of a local system moving with velocity v relative to the observer becomes slow with the rate

{1-(v/c)2}1/2,

which is exactly the same as the rate that the special theory of relativity gives.

These mean that the QM clock defined as the QM evolution of a local system follows the classical relativistic change of coordinates of space-time. Thus giving a consistent unification of QM and special relativistic CM.

We can also give a relation between Planck mass and Planck time by using LPT (see time IX paper).