Time: Hitoshi Kitada's Home Page

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:


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


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


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).