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I was reading this lecture notes "Black holes from A to Z" by Andrew Strominger. In the first chapter Introduction the following statement is made:

"If we know the present, there are laws that determine the future and can be used to predict the past. This is true in Classical Physics. It is also true in Quantum Physics, where we determine the state, instead of classical variables, in the future or the past. We refer to this as the Unitarity of Quantum Mechanical systems."

I haven't understood how unitarity is related to the system being deterministic? How unitarity is related to time evolution of the system?

The link to the lecture notes is "https://scholar.harvard.edu/files/yshi/files/physics_211r_-_black_holes_final.pdf"

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In Quantum Mechanics if you know the state of the system at a given time, then the quantum equations of motion tells you what the state is for any future as well as past time.

The state of the system at instant $t_0$ is the vector $|\psi(t_0)\rangle$ and we postulate that it evolves in time through a linear operator $U(t-t_0)$ (the time evolution operator), $$|\psi(t)\rangle = U(t-t_0)|\psi(t_0)\rangle.$$ You can think of $U(t-t_0)$ as a matrix depending on the parameter $t$. If you know the state of the system at any instant $t_0$, then you act with that matrix on the vector state and you obtain the vector state at time $t$. Totally deterministic!

Notice that this operator being unitary is not necessary to a deterministic time evolution of a vector $|\psi\rangle$. In fact, what makes sure that time evolution is deterministic is the existence of a time evolution operator satisfying the above equation. The fact that $U$ is unitary is necessary in QM in order that the relation between states is preserved though. For instance, if $\langle\psi(0)|\phi(0)\rangle=0$, then $\langle\psi(t)|\phi(t)\rangle=\langle\psi(0)U(t)^\dagger U(t)|\phi(0)\rangle=\langle\psi(0)|\phi(0)\rangle=0$.

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  • $\begingroup$ This is good, but it's also work mentioning that is important that U is invertible. Otherwise we can't determine the past from the present. But of course all unitaries are invertible. $\endgroup$ – Jahan Claes Nov 19 '17 at 17:13
  • $\begingroup$ @JahanClaes Totally correct! $\endgroup$ – Diracology Nov 19 '17 at 17:23

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