When the unitary evolution is derived from the collapse postulate in QM

[seratend]

If we treat space and time equally, in classical QM, by defining, formally, a time observable (and a conjugate observable), we can see that the postulates of QM have a simpler form: we can derive the unitary evolution from the measurment collapse postule.
(We use the units hbar=1).

Just take the classical QM wave function psi(x,t) which is the coordinates of the vector |psi(t)>= 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=H_time(x)H_space.
Where H_time is the Hilbert space spawned by the hermitian operators (t, E=d/idt):[t,d/idt]=i and H_space the usual quantum Hilbert space (spawned by (q,p=d/idq): [q,p]=i), in hbar=1 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= H_time(x)H_space we have the state vector of a quantum system defined by:

|psi>>=sum_t|psi(t)>|t>= sum_(xt) psi(x,t)|t>|x> or <<t,x|psi>>= 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=H^2 if we prefer to get a more rigorous definition of the operators space and a relativistic compatible formulation) to recover the Schrödinger equation:

(E+H)|psi>>=0 => <t|(E+H)|psi>>=0 (using partial projection on the |t> basis)

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

This is the Schrödinger equation (SE).

(E+H)|psi>>=0=> |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>>= sum_h f(h) |e=-h>|h> = sum_(ht) f(h).exp(-iht)|t>|h> is also solution of (E+H)|psi>>=0.

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

Where |psi(0)> is a vector of the subspace H_space.
Where H is the hamiltonian operator and E|e>=e|e> and H|h>=h|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)| (the "initial state" observable), 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: the initial state and the E+H measurements (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 => the system state is completely defined on the hilbert space H and therefore on H_space: <t|psi>>=|psi(t)>= sum_h f(h).exp(-iht)|h>= exp(-iHt).|psi(0)>.
If we invert the measurement order, we just have the known collapse postulate of 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), i.e. the unitary evolution.
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.

 

[vanesch]

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: the initial state and the E+H measurements (we need to apply the state projection one “after” the other, where the order is important like in any sequence of incompatible measurements).

This looks fun !
However, I have a difficulty: as this system is "timeless" (after all, time is just another observable), how do you specify a "time order" of the incompatible measurements ? I mean: suppose at 10h, I measure P, and at 11h I measure Q, and at 12h, I measure P again. How is this integrated in your formalism ?

cheers,
Patrick.

 

[seratend]

This looks fun !
However, I have a difficulty: as this system is "timeless" (after all, time is just another observable), how do you specify a "time order" of the incompatible measurements ? I mean: suppose at 10h, I measure P, and at 11h I measure Q, and at 12h, I measure P again. How is this integrated in your formalism ?

cheers,
Patrick.

Yes, this is fun. In addition, I think one should appreciate it especially with your mind approach of measurements.
In fact, I think this formulation may force someone to understand better (or may be ask some good questions) the measurement formalism: we have only logical statements, i.e. pure quantum logic without time. For example measurement A result “a” is true (=> state |a> is true, projector P_a), measurement B result “b” is true (=> state |b> is true) etc ...

Therefore, we have a logic order not a time order (i.e. time is treated as position: no bias in the treatment). The logic order is simply defined by the projectors order. When the measurements are compatible we can invert the logic order without changing the logic (the true statement concerning the system).

If I take your example, I get:
projector P1= |Po,11h><Po,11h|=|Po><Po|(x)|11h><11h| (measurement result po at 11h true)
projector P2= |Qo,12h><Qo,12h|=|Qo><Po|(x)|12h><12h| (measurement result qo at 12h true)

Therefore, I may define two true exclusive logical statements concerning the system:
P1 o P2 or exclusive P2oP1.
Both results are possible (from the set of the possible measurements we can make on this system) but only one may be true at a time (incompatible measurements) for that system.
If you go further on, you can see that the apparent time ordering is totally external to the model. We can just externally choose to remove the measurements of the form P1oP2. However, they are allowed. We just recover from the projection postulate that we can choose externally an arrow of time (but we can simply choose another arrow given by an entangled basis of position and time).

Seratend.

 

[masudr]

The non-LaTeX is painful on the eyes and the mind! What's the difference between |psi>> and |psi> ?

Edit: Sorry I understand now.