Re: Time observables (but not time operators) DO exist in QM

a student wrote:
> meyr2@xxxxxx (meyr2@xxxxxx) wrote
> >
> > Thus to have discrete time, you'd need
> > something like a "time operator" with a discrete spectrum. As we
know,
> > in quantum mechanics there is no such time operator.
> >
>
> One doesn't need a 'time operator' per se (in the sense of a
Hermitian
> operator), only a 'time observable' - and these DO exist in QM, in a
> number of interesting cases (such as the free particle and the
> harmonic oscillator).
>

We do not need positive operator tools to define formally a time
operator (and "time" observable). In addition, we can see that the
postulates of QM have a simpler form when we consider these additional
observables (i.e when we treat space and time equally).
(We use the units hbar=3D1).
Just take the classical QM wave function psi(x,t) which is the
coordinates of the vector |psi(t)>=3D sum_x psi(x,t)|x> in the usual
Hilbert space H_space of QM.
Now formally enlarge the hilbert space into a space time hilbert space
H=3DH_time(x)H_space.
Where H_time is the Hilbert space spawned by the hermitian operators
(t, E=3Dd/idt):[t,d/idt]=3Di and H_space the usual quantum Hilbert space
(spawned by (q,p=3Dd/idq): [q,p]=3Di), in hbar=3D1 units (space-time flat
geometry).

Formally, we call t the time observable and E is the conjugate
observable.
(as we call q the position observable).

In this new hilbert space, H=3D H_time(x)H_space we have the state vector
of a quantum system defined by:

|psi>>=3Dsum_t|psi(t)>|t>=3D sum_(xt) psi(x,t)|t>|x> or <<t,x|psi>>=3D
psi(x,t)

(where |t> is the time eigenbasis of t in the H_time Hilbert space).
(we use the notation |a>> on the tensor product Hilbert space H and |a>
on one of the subspace H_time or H_space)

In this new hilbert space, we formally define the observable E+H (or
E^2=3DH^2 if we prefer to get a more rigorous definition of the operators
space and a relativistic compatible formulation) to recover the
Schr=F6dinger equation:

(E+H)|psi>>=3D0 =3D> <t|(E+H)|psi>>=3D0 (using partial projection on the =
|t>
basis)

=3D> -d/idt|psi(t)>=3D -<t|E|psi>=3D <t|H|psi>=3DH|psi(t)>
<=3D> -d/idt|psi(t)> =3DH|psi(t)>

This is the Schr=F6dinger equation (SE).

(E+H)|psi>>=3D0 =3D> |psi>>, solution of the SE, is a peculiar eigenstate
of the hermitian operator E+H (eigenvalue 0). Moreover, this eigenvalue
is degenerated on the Hilbert space H:

|psi>>=3D sum_h f(h) |e=3D-h>|h> =3D sum_(ht) f(h).exp(-iht)|t>|h> is als=
o
solution of (E+H)|psi>>=3D0.

=3D> <t|psi>>=3D|psi(t)>=3D sum_h f(h).exp(-iht)|h>=3D exp(-iHt).|psi(0)>

Where |psi(0)> is a vector of the subspace H_space.
Where H is the hamiltonian operator and E|e>=3De|e> and H|h>=3Dh|h>.
We have used the basis transformation between |e> and |t> eigenvectors
of the hermitian operators E and t.

Therefore, the SE may be viewed, formally, as a measurement of the
observable E+H in this enlarged Hilbert space H. The projected state
|psi>> is on the eigenspace (the SE) associated to the eigenvalue of
the (E+H) measurement. The peculiar eigenvalue of this measurement is
not significant as H is defined up to a constant (i.e. equivalence of
the eigenspaces of the observable E+H).

The initial condition |psi(0)> (element of the Hilbert subspace
H_space) defines completely the measurement state |psi>>. However the
operator Id(x)|psi(0)><psi(0)|, does not necessarily commute with
Id(x)H and therefore with E+H.
We just recover the incompatibility of the projection postulate and the
SE evolution in a simpler form (when the observables do not commute):
we just have 2 incompatible measurements results (we need to apply the
state projection one "after" the other, where the order is
important like in any sequence of incompatible measurements).

For example, to get the usual state evolution, we just have a first
measurement of the observable Id(x)|psi(0)><psi(0)| (measured value 1)
and after a second measurement of the observable E+H =3D> the system
state is completely defined on H and therefore H_space:
<t|psi>>=3D|psi(t)>=3D sum_h f(h).exp(-iht)|h>=3D exp(-iHt).|psi(0)>.
If we invert the measurement order, we just have the known collapse
postulate of the evolution of the wave function.

The apparent incompatibility between the projection postulate and the
SE has now a simple formulation in this enlarged Hilbert space: we just
speak about measurement on observables (compatible or incompatible). In
other words, in this enlarged Hilbert space we just have quantum logic
of observables.


CONCLUSION: in this enlarged Hilbert space, we simply have a time
observable (t), its conjugate observable (E) as we have the q and P
observables. The SE solutions are now simply the eigenspace of a
measurement of the peculiar observable E+H: we just need the
measurement postulates (collapse and born rules).
In other words, and for the adepts of QM measurements/Quantum logic,
the evolution of a quantum system is now the measurement result of the
hermitian operator E+H: From the collapse postulate, we just recover
the set of SE solutions (the eigenspace).
This formal result can be easily extended to relativistic QM/QFT (where
the hermitian operator E+H has to be changed in order to take into
account the Lorentz symmetries of the space time).

Seratend.

.

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