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Time in quantum mechanics
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In the traditional formulation of quantum mechanics, time is not an
observable. Nevertheless it can be observed...
In the Schroedinger picture, the state is defined at fixed times,
which distinguishes the time. In this picture, time measurement
is difficult to discuss since the time at which a state is considered
is always sharp.
In the Heisenberg picture, time is simply a parameter in the
observables, and therefore also distinguished, but in a different way.
Parameters are in fact just continuous indices and not observables.
As 3 is not an observable while p_3 is one, so t is not an observable
but H(t) is one. Observables have at _each_ time an expected value;
the moment of time (''now'') is not modelled as observable.
But what can be modelled is a clock, i.e., a system with an observable
that changes with time in a predictable way. If the observable u(t)
of a system satisfies
ubar(t) := __ = u_0 + v (t - t_0) (v nonzero) (*)
with sufficient accuracy, one has a clock and can find out by means of
____ how much time
T = Delta t
passed between two observed data sets.
This is also the usual way we measure time in classical physics.
Of course, to be a meaningful time measurement, T must be large enough
compared with the intrinsic uncertainty
Sigma_T := |v^{-1}| sigma(u(t)).
Here
sigma(u(t)) = sqrt(<(u(t)-ubar(t))^2>)
is the standard deviation in the properly calibrated
(quantum mechanical) state <.>. If (*) has significant errors
then Sigma_T is of course correspondingly larger.
In relativistic quantum field theory (which in its covariant version
can only be formulated in the Heisenberg picture), the 1-dimensional
time t turns into the 4-dimensional space-time position x. Now x
is a vector parameter in the observables (fields), and hence is not an
observable. Space and time are now on the same level (allowing a
covariant point of view), but both as non-observables.
The observables are fields; positions and times of particles are
modelled by unsharp 1-dimensional world lines characterized by a high
density of the expectations of the corresponding fields.
(Think of the trace of a particle in a bubble chamber.)
For position and time measurement, one now needs a 4-vector field
u(x) with
____ = u_0 + V (x - x_0)
and a nonsingular 4x4 matrix V, and the intrinsic uncertainty
takes the form
Sigma_T := sigma(V^{-1}u(x))
with
sigma(a(x)) = sqrt(<(a(x)-abar(x))^*(a(x)-abar(x))>),
abar(x)=.
Conclusion: In nonrelativistic quantum mechanics, time is always
measured indirectly via the expectations of distinguished observables
of clocks in calibrated quantum mechanical states. In relativistic
quantum field theory, the same holds for both position and time.
However, this analysis works only when one assigns to single clocks
a well-defined state, hence assumes a version of the Copenhagen
interpretation.
From the point of view of the minimal statistical interpretation,
one needs in contrast a whole ensemble of identically prepared
clocks to measure time...
Note that in relativistic quantum mechanics, a single particle is
described (in the absence of an external field) by an irreducible
representation of the Poincare group. Here only the components of
4-momentum and the 4-angular momentum are observables. From these,
one can reconstruct an observer-dependent 3-dimensional (Newton-Wigner)
position operator satisfying canonical commutation rules, but not
a time operator.
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