**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Tue, 9 Nov 1999 01:36:11 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Koichiro Matsuno: "[time 979] RE: [time 978] a fundamental question on QM time"**Previous message:**Matti Pitkänen: "[time 977] Re: Physics and prime numbers (!)"**Next in thread:**Koichiro Matsuno: "[time 979] RE: [time 978] a fundamental question on QM time"

Dear All,

Let me try to propose a question which seems to be a fundamental

question on quantum mechanical time and seems not to have been paid an

appropriate attention in conventional physical theories.

In classical Newtonian mechanics, one can define (mean) velocity v by

v=x/t of a particle that starts from the origin at time t=0 and arrives

at position x at time t, if we assume that the coordinates of space and

time are given in an a priori sense. This definition of velocity and

hence that of momentum do not produce any problems, which assures that

in classical regime there is no problem in the notion of space-time.

Also in classical relativistic view, this seems to be valid insofar as

we discuss the motion of a particle in the coordinates of the

observer's.

Let us consider quantum mechanical case where the space-time coordinates

are given a priori. Then the velocity of a particle should be defined as

something like v=x/t. This is a definition, so this must hold in exact

sense if the definition works at all. However in QM case, the

uncertainty principle prohibits the position and momentum from taking

exact values simultaneously. That principle is based on the notion of

position and momentum operators that satisfy the canonical commutation

relation. In the point that the space and momentum are given by

operators, the definition of velocity v above does not apply to QM.

Further the uncertainty principle tells that there is a minimum value

for the product of the variances of the position and momentum from their

expected values, and thus tells that there is an absolute independence

between the notions of position and momentum. I.e. the principle does

not say anything about the relation like x=tv, but instead just tells

that they have to be away from their expectation values.

This observation seems to suggest that, if given a pair of a priori

space and time coordinates, QM becomes contradictory, and that the

independent quantities, space and momentum operators, have to be taken

as the fundamental quantities of quantum mechanics. As time t can be

defined as a ratio x/v in this view, time is a redundant notion that

should not be given a role independent of space and momentum.

This argument seems to give another support to the view that an a priori

notion of time is not a basic notion but the notion of independent space

and momentum operators are basic ones. In this view there can be found a

relation like x=tv as an approximate relation that holds to the extent

that the relation does not contradict the uncertainty principle.

(http://www.kitada.com/time_I.tex)

The quantum jumps that are assumed as an axiom on observation in usual

QM theory may arise from the classical nature of time that determines

the position and momentum in precise sense simultaneously. This nature

of time may urge one to think jumps must occur and consequently one has

to observe definite eigenstates. In actuality what one is able to

observe is scattering process, but not the eigenstates as the final

states of the process. Namely jumps and eigenstates are ghosts arising

based on the passed classical notion of time. Or in more exact words,

the usual QM theory is an overdetermined system that involves too many

independent variables: space, momentum, and time, and in that framework

time is not free from the classical image that velocity is defined by

v=x/t.

Best wishes,

Hitoshi

**Next message:**Koichiro Matsuno: "[time 979] RE: [time 978] a fundamental question on QM time"**Previous message:**Matti Pitkänen: "[time 977] Re: Physics and prime numbers (!)"**Next in thread:**Koichiro Matsuno: "[time 979] RE: [time 978] a fundamental question on QM time"

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