Re: Time, Parameter, Operator



Ali wrote: 
Dear Members,

on p.68 of Sakurai's "Modern Quantum Mechanics" we read "... Time
is just a parameter in quantum mechanics, not an operator. In
particular, Time is not an observable in the language of the previous
chapte." What is the deference between a "parameter" and " operator"
from a physical point of view?


You put your finger on the central problem of modern theoretical
physics.

Consider the difference between "time" and "position"
in quantum mechanics. You can talk about "position of a particle",
so position is an observable that can be measured on
(is a characteristic
of) a physical system (e.g., particle). The measured position of a
particle depends on the state in which you find the particle.
In QM such observables as position are described by Hermitian
operators, and the
measured values are described by matrix elements of these operators.

On the other hand, you cannot measure "time of a particle".
Time is not a property of the particle, time does not depend on
the state of the particle. In other words, time is not an
observable. Then what is time?

In each laboratory we have a classical device called "clock"
that gives us a numerical parameter called "time". The time label
is attached to all measurements performed in the laboratory.
So, time is just a classical numerical parameter in quantum mechanics.

This is very different from the way time and position are treated
in special (and general) relativity. Einstein's relativity declares
that time and position are just coordinates in the 4D space-time
manifold. From the point of view of different observers these
coordinates are interchangeable. This is reflected in the way
time and position form components of a 4-vector in the Einstein's
 theory.

As we saw above, this "equivalence" of time and position has no analog
in quantum mechanics. I believe, this contradiction is the main
obstacle for all current attempts to "quantize gravity". There are
many reviews regarding the "problem of time" on arxiv.org.
My personal opinion is that the only way to reconcile the principle
of relativity with quantum mechanics is to reject the 4D space-time
"unification" of space and time. This idea is, actually, not so scary.
I explored its consequences in physics/0504062 and the resulting theory
looks good, though non-conventional.

Eugene.




.

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