Is unitarity necessary for the probabilistic interpretation?

[Demystifier]

It is usually said that unitarity is necessary for the consistent probabilistic interpretation. But is that really so? Suppose that $|\psi(t)\rangle$ does not evolve unitarily with time, so that $\langle\psi(t)|\psi(t)\rangle$ changes with time. Even then one can propose that probability $p_k$ is given by the formula
$$p_k(t)=\frac{\langle\psi(t)|\hat{\pi}_k|\psi(t)\rangle}{\langle\psi(t)|\psi(t)\rangle}$$
where $\hat{\pi}_k$ is a projector. Indeed, if the projectors obey $\sum_k \hat{\pi}_k =1$ (which does not depend on unitarity of the time evolution), then the sum of probabilities obeys
$$\sum_k p_k(t) =1$$
despite nonunitarity. So is unitarity really necessary, and if it is, why exactly is it so?

 

[atyy]

It probably won't work in relativity?

 

[HomogenousCow]

Canonical commutation relations would become time dependent if we had non-unitary time evolution, that sounds bad. I don't think that would reduce to the right classical limit.

 

[Demystifier]

It probably won't work in relativity?

I think it works in relativity too. For instance, in the Tomonaga-Schwinger approach one replaces the dependence on time $t$ with the dependence on the spacelike hypersurface $\Sigma$, and all formulas above with $t\to\Sigma$ work again.

 

[Demystifier]

Canonical commutation relations would become time dependent if we had non-unitary time evolution, that sounds bad. I don't think that would reduce to the right classical limit.

How about an average such as
$$\frac{\langle\psi(t)|[\hat{x},\hat{p}]|\psi(t)\rangle}{\langle\psi(t)|\psi(t)\rangle} ?$$
It seems that it leads to the correct classical limit.

 

[atyy]

I think it works in relativity too. For instance, in the Tomonaga-Schwinger approach one replaces the dependence on time $t$ with the dependence on the spacelike hypersurface $\Sigma$, and all formulas above with $t\to\Sigma$ work again.

But will you get Lorentz invariance? Or will there be a preferred frame?

 

[HomogenousCow]

How about an average such as
$$\frac{\langle\psi(t)|[\hat{x},\hat{p}]|\psi(t)\rangle}{\langle\psi(t)|\psi(t)\rangle} ?$$
It seems that it leads to the correct classical limit.

Now that you mention it Ehrenfest's theorem wouldn't hold either since you have a time dependent denominator. I feel like the problem is more to do with the physics, a quantum theory with non-unitarity time evolution just ruins all the good properties we're used to.

Edit: I think you also get all sorts of non-local effects because of the denominator

 

[atyy]

https://www.sciencedirect.com/science/article/abs/pii/037596019090786N
https://www.scottaaronson.com/papers/island.pdf

 

[Demystifier]

But will you get Lorentz invariance? Or will there be a preferred frame?

A priori, I don't see why violation of unitarity should violate Lorentz invariance. Can you support it with some equations?

 

[HomogenousCow]

Consider a particle in the state $\Phi(x) = \psi_1(x) + \psi_2(x)$, where $\psi_1$ and $\psi_2$ are infinitely separated, the position probability density function at a point $x$ where $\psi_1$ is appreciable would be approximately $$\frac{|\psi_1(x,t)|^2}{\int dy|\psi_1(y,t)|^2+|\psi_2(y,t)|^2}$$. This looks very non-local and screwed up.

 

[Demystifier]

Consider a particle in the state $\Phi(x) = \psi_1(x) + \psi_2(x)$, where $\psi_1$ and $\psi_2$ are infinitely separated, the position probability density function at a point $x$ where $\psi_1$ is appreciable would be approximately $$\frac{|\psi_1(x,t)|^2}{\int dy|\psi_1(y,t)|^2+|\psi_2(y,t)|^2}$$. This looks very non-local and screwed up.

Excellent point! And directly related to probability.

 

[A. Neumaier]

It is usually said that unitarity is necessary for the consistent probabilistic interpretation. But is that really so?

Lindblad equations are not unitary but consistent with a probabilistic interpretation. It is just that they are usually not regarded to be fundamental. (But GRW models are special Lindblad equations proposed as fundamental.)

 

[atyy]

A priori, I don't see why violation of unitarity should violate Lorentz invariance. Can you support it with some equations?

Yes, I guess you are right. Collapse is non-unitary, but still consistent with relativity.

 

[Demystifier]

But GRW models are special Lindblad equations ...

I don't think it's true. The Lindblad equation is a deterministic equation for a mixed state, while GRW model is a stochastic equation for a pure state. Unless you have in mind a more general notion of Lindblad equations and/or GRW models.

 

[A. Neumaier]

I don't think it's true. The Lindblad equation is a deterministic equation for a mixed state, while GRW model is a stochastic equation for a pure state. Unless you have in mind a more general notion of Lindblad equations and/or GRW models.

TheLindblad equation for the GRW model is displayed in the Wikipedia article on GRW (equation before the heading 'Examples'). Every Lindblad equation has an associated stochastic process for pure states, called its unravelling. It is not unique but exactly recovers the Lindblad equation upon averaging.

 

[Demystifier]

Every Lindblad equation has an associated stochastic process for pure states, called its unravelling. It is not unique but exacly recovers the Lindblad equation upon averaging.

Can you recommend some textbook treatment or review of unraveling?

 

[A. Neumaier]

Can you recommend some textbook treatment or review of unraveling?

For example:

Wiseman and Vaccaro,Maximally Robust Unravelings of Quantum Master Equations,1998

 

[atyy]

Belavkin https://arxiv.org/abs/math-ph/0512069 makes a similar comment as @A. Neumaier's post #12
" As extended to nondemolition observations continual in time [9]–[15], this approach consists in using the quantum filtering method for the derivation of nonunitary stochastic wave equations describing the quantum dynamics under the observation. Since a particular type of such equations has been taken as a postulate in the phenomenological theory of continuous reduction and spontaneous localization [16]–[20], the question arises whether it is possible to obtain this equation from an appropriate Schroedinger equation."

Diosi makes a similar comment in http://philsci-archive.pitt.edu/14072/1/howtoteachcollapse-27-10-17.pdf
"Finally in the nineteen-nineties I got rid of my ignorance and learned that unsharp measurements and my time-continuous measurement (monitoring) could have equally been derived from standard quantum theory [15, 16]. That was disappointing [1]."

Todd Brun https://arxiv.org/abs/quant-ph/9710021
"Quantum open systems are described in the Markovian limit by master equations in Lindblad form. I argue that common "quantum trajectory" techniques corresponding to continuous measurement schemes, which solve the master equation by unraveling its evolution into stochastic trajectories in Hilbert space, correspond closely to particular sets of decoherent (or consistent) histories. This is illustrated by a simple model of photon counting. An equivalence is shown for these models between standard quantum jumps and the orthogonal jumps of Diósi, which have already been shown to correspond to decoherent histories."