# Non-Hermitian operator for superposition

It is said that no Hermitian operator gives a time evolution where "I observed the spin to be both up and down" is a possible result. If you use non-Hermitian operator.. then it's possible.. and what operator is that where it is possible in principle where "I observed the spin to be both up and down"?

PeterDonis
Mentor
2020 Award
no Hermitian operator gives a time evolution
Do you mean "Hamiltonian operator"? That's the operator that describes time evolution.

where "I observed the spin to be both up and down" is a possible result
Or do you mean Hermitian operator describing an observable with this property? Observables are not the same thing as time evolution.

If you use non-Hermitian operator
Then you are not talking about an observable, because a non-Hermitian operator has eigenvalues that are not real numbers, and any result of a measurement must be a real number.

what operator is that where it is possible in principle where "I observed the spin to be both up and down"?
There isn't one.

Do you mean "Hamiltonian operator"? That's the operator that describes time evolution.

Or do you mean Hermitian operator describing an observable with this property? Observables are not the same thing as time evolution.
in thread https://www.physicsforums.com/threads/how-do-you-understand-mwi.908805/page-2 message #24, you yourself used the language "There is no Hermitian operator that gives a time evolution where "I observed the spin to be both up and down" is a possible result."" as in:

"Mathematically, the operation of a measuring device is modeled as a Hermitian operator that describes the device. Applying that operator to the initial state, before the measurement, is how you mathematically compute the state after the measurement; that's how you would derive the time evolution I gave in post #4. There is no Hermitian operator that gives a time evolution where "I observed the spin to be both up and down" is a possible result."

Is that a typo. Why didn't you use "There is no Hamiltonian operator that gives a time evolution where "I observed the spin to be both up and down" is a possible result." What did you use the word Hermitian to refer to time evolution which you just state now is related to observable and not time evolution?

Then you are not talking about an observable, because a non-Hermitian operator has eigenvalues that are not real numbers, and any result of a measurement must be a real number.

There isn't one.

PeterDonis
Mentor
2020 Award
This is better; you should always give a reference if you are referring to something specific like this.

In that post, I was using "time evolution" in a somewhat nonstandard sense, to indicate the operation of entangling the state of a measured system with the state of a measuring device as it is usually modeled. Strictly speaking, in order to model this mathematically, you would have to construct a Hamiltonian for the entire system (measured system plus measuring device) that included the interaction that produces the entanglement. But this is complicated and doesn't really add anything useful to the analysis, so often the whole process is described as I did in post #4 of the thread you linked to, by simply writing down the state of the measured system in the basis of eigenstates of a Hermitian operator (the one that describes the measurement you're making), and describing how each of those eigenstates gets entangled with an appropriate state of the measuring device by the measurement process. This simpler description isn't, strictly speaking, a "time evolution", because we haven't even tried to write down a Hamiltonian; but it describes the process of going from the state of measured system plus measuring device before the measurement, to the state of measured system plus measuring device after the measurement, which is a process that happens in time (though typically a very short amount of time).

None of this changes the final answer I gave in post #2.

• Blue Scallop
I inquired: "what operator is that where it is possible in principle where "I observed the spin to be both up and down"?"
You replied: "There isn't one."

Is the reason there isn't one is because no one has observed superposition or is it because it's not possible mathematically? Or is it possible mathematically? How?

PeterDonis
Mentor
2020 Award
Is the reason there isn't one is because no one has observed superposition or is it because it's not possible mathematically?
Both. We construct mathematical models in order to match what we observe.

Both. We construct mathematical models in order to match what we observe.
Let's say there is a workshop at the Perimeter Institute or other physics schools about creating mathematical models just for creative exercise even if it doesn't match what we observe.. and one is tasked to create operators that can produce "I observed the spin to be both up and down".. how do you write such math or just describe how such math can be attempted.. this exercise can also make one get more understanding and familiariry with orthodox QM from looking it from all angles and sides ands point of view.

PeterDonis
Mentor
2020 Award
Let's say there is a workshop at the Perimeter Institute or other physics schools about creating mathematical models just for creative exercise even if it doesn't match what we observe.. and one is tasked to create operators that can produce "I observed the spin to be both up and down".. how do you write such math or just describe how such math can be attempted..
This is all well outside the PF rules. Thread closed.