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- Summary
- There is no Heisenberg Picture version of the Born Rule.

This is in reference to a question, never fully resolved, posed here:

The von Neumann postulates for Quantum Theory - Evolution (Schrödinger's equation) and Projection (Born's rule) are always framed in the Schödinger Picture. The Heisenberg Picture version of Evolution is the Heisenberg equations of motion. The essence of the original question is: what about the other postulate, Projection? What is the Heisenberg Picture version of Born rule?

Well, apparently someone fell asleep at the switch there! There is no formulation. It's a gap in the foundations that's never been addressed, and apparently never even been noticed. It looks like nobody's ever gotten around to actually writing down what the Born rule looks like in the Heisenberg Picture!

Do a literature search. You're not going to find anything on "Heisenberg Picture" + "Born Rule". For instance, here's a literature search on arXiv.

As of 2019 June, empty. A web search won't do much better.

A Google Scholar search is mostly empty. You may find this entry there:

which basically notes ... that the Heisenberg picture has no Born Rule (and tries to explain why it's missing). It's a review of one of Born and Heisenberg's early works coming out of Solvay 1927. In fact, you can find the Born-Heisenberg paper right here at Solvay

in the conference proceedings starting on page 143.

The treatment of the Born rule is the central focus of all divergence of opinion on what quantum theory means and lies at the center of the Measurement Problem. That underscores how serious this gap in the foundation actually is. And the presence of this gap also shows why there is a divergence of opinion in the first place. Einstein was right, Quantum Theory is incomplete, and this is the point where it's incomplete. No Heiseberg Picture-Born Rule exists.

The closest treatment to this that you get are treatments of open systems quantum theory (which hybridize classical and quantum dynamics) or - more generally - treatments that hybridize classical and quantum dynamics. Noteworthy is a recent one found on arXiv, which successfully hybridizes classical gravity with quantum theory

"A Post Quantum Theory of Classical Gravity" https://arxiv.org/abs/1811.03116

Needless to say, its main talking point is that a resolution of the Measurement Problem emerges for free from its hybrid dynamics.

In the Heisenberg Picture, states are timeless. The Born rule is the rule for wave function collapse. But if states are timeless, then what exactly is changing when a Born rule projection occurs? And when and where? So, you can already see there that there's a basic problem with consistency here. How are you going to get a Born or Lueder's projection to happen when Heisenberg states are timeless?

This also puts permanent No Go on all the endeavors to "explain away" the Born rule. For, any attempt to do so will, upon success, end up making everything timeless. If you have nothing but Evolution, then in the Heisenberg Picture this translates into nothing but timelessness; and nothing ever changes. A kind of Zeno's Paradox is waiting at the end of the tunnel for all the {Decoherence, Everett, Consistent Histories} people! And when they get there at the end of the tunnel, it will jump out and say "Hi! Surprised to see me?!"

Here's an attempt from within the Everett camp to address the question

The very thing people want to achieve by these alternate interpretations (removing the Born rule) is the very thing you want to avoid! The beauty of the question "what is the Heisenberg Picture Born Rule" is that it draws focus to the point where the goals conflict.

It gets even deeper than this. The disagreement that supposedly lies between General Relativity (GR) and Quantum Theory (QT), particularly on its treatment of time, is only a gap between General Relativity and the Schrödinger Picture. The Heisenberg Picture's treatment of time already meshes quite well with General Relativity's treatment of time. In particular, the Heisenberg equations when generalized to fields, become partial differential equations whose forms do not single out any dimension of spacetime. They treat time and space on an equal footing.

So, the gap widely held to lie between GR and QT has been misplaced: it's not QT vs. GR, but is actually (Schrödinger QT) vs. (Heisenberg QT & GR).

To try and resolve this discrepancy, you first need to use replace state vectors|ψ> by what may be more properly regarded as the state W =|ψ><ψ| / <ψ|ψ>; and generalize this further to include mixed states. The Schrödinger equation iħ d|ψ>/dt = H|ψ> then becomes iħ dW/dt = [H, W]. The Born rule produces a transition W → Σ_a P_a W P_a upon measure of a quantity A in state W, where A quantizes to the operator Â = Σ_a a P_a, with eigenvalues (a) and eigenspace projectors P_a.

A second measurement of, say, a quantity C would then produce a projection Σ_a P_a W P_a → Σ_{ac} P'_c P_a W P_a P'_c where the P'_c are the projectors for C. The dependence on time-ordering can be encapsulated by using the time-ordering pseudo-operator T[] and its reversal T'[] to write this as Σ_{ac} T'[P_a P'_c] W T[P_a P'_c]. All of this will survive intact upon conversion to the Heisenberg picture, except that the operators, projectors will now have the time dependence and W will be time-independent; and the Schrödinger equation on states will be replaced by the Heisenberg equation on the quantized operators that correspond to physical quantities.

This generalizes to space-time form, where the time dependence of the projectors and operators now becomes dependence on space-time points. The most important feature to note is that the space-time coordinates are now on an equal footing, and the only vestige of causal ordering that remains is in the appearance of the T[] and T'[] brackets.

This introduces an effective time dependence on the timeless Heisenberg states. But, unlike the case of the Schrödinger picture, there is no real time dependency remaining in the evolution postulate - at least not for the states. They remain timeless. Instead, the time-dependency has moved over to the other projection postulate. To make this work consistently, you need the following structure:

(1) a point cloud that contains all the points at which applications of the Born rule will occur

(2) a network of Heisenberg states, written as mixed states W in general

(3) associated with each state, a partition of the point cloud into a "before" set and "after" set. Their causal ordering should be consistent so that no point in the after set has a timeline or null curve that leads to a point in the before set.

(4) a transition between any 2 states whose before/after sets agree on all but a finite number of points; the transition being given in a form similar to what described above.

Each state, despite being timeless, encodes a "now" by virtue of how it partitions the point cloud into a before and after set. So, each application of the Born rule bumps up the "now" by a notch. So, in effect, a flow of time is introduced, even though the operators themselves and dynamics do not single out any "now".

This actually gets pretty close to what Smolin described and call out for here in this lecture

Lee Smolin Public Lecture: Time Reborn

in which he presents some of the key talking points of his "Time Reborn" book

### Interpretation of the Heisenberg picture in QM

I was always a bit puzzled by the Heisenberg picture (not mathematically, I'm fine with that, but conceptually) - if a "state" describes a system, how can it not be time-dependent, if the system changes? I just found an alternative way of looking at it which seems to make sense to me, but I'm...

www.physicsforums.com

The von Neumann postulates for Quantum Theory - Evolution (Schrödinger's equation) and Projection (Born's rule) are always framed in the Schödinger Picture. The Heisenberg Picture version of Evolution is the Heisenberg equations of motion. The essence of the original question is: what about the other postulate, Projection? What is the Heisenberg Picture version of Born rule?

Well, apparently someone fell asleep at the switch there! There is no formulation. It's a gap in the foundations that's never been addressed, and apparently never even been noticed. It looks like nobody's ever gotten around to actually writing down what the Born rule looks like in the Heisenberg Picture!

Do a literature search. You're not going to find anything on "Heisenberg Picture" + "Born Rule". For instance, here's a literature search on arXiv.

### Search | arXiv e-print repository

arxiv.org

A Google Scholar search is mostly empty. You may find this entry there:

### The Statistical Interpretation according to Born and Heisenberg

At the 1927 Solvay conference Born and Heisenberg presented a joint report on quantum mechanics. I suggest that the signi cance of this report lies in that it contains a ' nal' formulation of the statistical interpretation of quantum mechanics that goes beyond Born's original proposal. In...

halshs.archives-ouvertes.fr

which basically notes ... that the Heisenberg picture has no Born Rule (and tries to explain why it's missing). It's a review of one of Born and Heisenberg's early works coming out of Solvay 1927. In fact, you can find the Born-Heisenberg paper right here at Solvay

in the conference proceedings starting on page 143.

The treatment of the Born rule is the central focus of all divergence of opinion on what quantum theory means and lies at the center of the Measurement Problem. That underscores how serious this gap in the foundation actually is. And the presence of this gap also shows why there is a divergence of opinion in the first place. Einstein was right, Quantum Theory is incomplete, and this is the point where it's incomplete. No Heiseberg Picture-Born Rule exists.

The closest treatment to this that you get are treatments of open systems quantum theory (which hybridize classical and quantum dynamics) or - more generally - treatments that hybridize classical and quantum dynamics. Noteworthy is a recent one found on arXiv, which successfully hybridizes classical gravity with quantum theory

"A Post Quantum Theory of Classical Gravity" https://arxiv.org/abs/1811.03116

Needless to say, its main talking point is that a resolution of the Measurement Problem emerges for free from its hybrid dynamics.

In the Heisenberg Picture, states are timeless. The Born rule is the rule for wave function collapse. But if states are timeless, then what exactly is changing when a Born rule projection occurs? And when and where? So, you can already see there that there's a basic problem with consistency here. How are you going to get a Born or Lueder's projection to happen when Heisenberg states are timeless?

This also puts permanent No Go on all the endeavors to "explain away" the Born rule. For, any attempt to do so will, upon success, end up making everything timeless. If you have nothing but Evolution, then in the Heisenberg Picture this translates into nothing but timelessness; and nothing ever changes. A kind of Zeno's Paradox is waiting at the end of the tunnel for all the {Decoherence, Everett, Consistent Histories} people! And when they get there at the end of the tunnel, it will jump out and say "Hi! Surprised to see me?!"

Here's an attempt from within the Everett camp to address the question

The very thing people want to achieve by these alternate interpretations (removing the Born rule) is the very thing you want to avoid! The beauty of the question "what is the Heisenberg Picture Born Rule" is that it draws focus to the point where the goals conflict.

It gets even deeper than this. The disagreement that supposedly lies between General Relativity (GR) and Quantum Theory (QT), particularly on its treatment of time, is only a gap between General Relativity and the Schrödinger Picture. The Heisenberg Picture's treatment of time already meshes quite well with General Relativity's treatment of time. In particular, the Heisenberg equations when generalized to fields, become partial differential equations whose forms do not single out any dimension of spacetime. They treat time and space on an equal footing.

So, the gap widely held to lie between GR and QT has been misplaced: it's not QT vs. GR, but is actually (Schrödinger QT) vs. (Heisenberg QT & GR).

To try and resolve this discrepancy, you first need to use replace state vectors|ψ> by what may be more properly regarded as the state W =|ψ><ψ| / <ψ|ψ>; and generalize this further to include mixed states. The Schrödinger equation iħ d|ψ>/dt = H|ψ> then becomes iħ dW/dt = [H, W]. The Born rule produces a transition W → Σ_a P_a W P_a upon measure of a quantity A in state W, where A quantizes to the operator Â = Σ_a a P_a, with eigenvalues (a) and eigenspace projectors P_a.

A second measurement of, say, a quantity C would then produce a projection Σ_a P_a W P_a → Σ_{ac} P'_c P_a W P_a P'_c where the P'_c are the projectors for C. The dependence on time-ordering can be encapsulated by using the time-ordering pseudo-operator T[] and its reversal T'[] to write this as Σ_{ac} T'[P_a P'_c] W T[P_a P'_c]. All of this will survive intact upon conversion to the Heisenberg picture, except that the operators, projectors will now have the time dependence and W will be time-independent; and the Schrödinger equation on states will be replaced by the Heisenberg equation on the quantized operators that correspond to physical quantities.

This generalizes to space-time form, where the time dependence of the projectors and operators now becomes dependence on space-time points. The most important feature to note is that the space-time coordinates are now on an equal footing, and the only vestige of causal ordering that remains is in the appearance of the T[] and T'[] brackets.

This introduces an effective time dependence on the timeless Heisenberg states. But, unlike the case of the Schrödinger picture, there is no real time dependency remaining in the evolution postulate - at least not for the states. They remain timeless. Instead, the time-dependency has moved over to the other projection postulate. To make this work consistently, you need the following structure:

(1) a point cloud that contains all the points at which applications of the Born rule will occur

(2) a network of Heisenberg states, written as mixed states W in general

(3) associated with each state, a partition of the point cloud into a "before" set and "after" set. Their causal ordering should be consistent so that no point in the after set has a timeline or null curve that leads to a point in the before set.

(4) a transition between any 2 states whose before/after sets agree on all but a finite number of points; the transition being given in a form similar to what described above.

Each state, despite being timeless, encodes a "now" by virtue of how it partitions the point cloud into a before and after set. So, each application of the Born rule bumps up the "now" by a notch. So, in effect, a flow of time is introduced, even though the operators themselves and dynamics do not single out any "now".

This actually gets pretty close to what Smolin described and call out for here in this lecture

Lee Smolin Public Lecture: Time Reborn

in which he presents some of the key talking points of his "Time Reborn" book

### Time Reborn - Wikipedia

en.wikipedia.org