Distributive Law in QM

[Bruce Kellett]

The distributive law is well known in algebra and number theory:

     a(b + c) = ab + ac

The product of one number with a sum of other numbers is the same as the sum of the individual products.

This is widely used in algebra and number theory, but the question is, 'Does this work for quantum theory?'

It is assumed in things like many-worlds theory, where we have

   (|1> + |0>)|e> = |1>|e> + |0>|e>

(ignoring normalization factors) where |1> and |0> are two eigenvalues in a superposition, and |e> is the environment, including the apparatus and the observer.

Under unitary evolution, the equation above proceeds to

  (|1> + |0>)|e> = |1>|e> + |0>|e> = |1>|e_1> + |0>|e_0>

where |e_1> represents the observer and environment recording outcome 1, and similarly for |e_0>.

But when we ask what this means physically we are faced with a problem. The final states in which we have environments recording 1 and 0 outcomes, respectively, are produced by decoherence from the |1> and |0> eigenfunctions respectively. But for this to happen, each component of the superposition must interact with an original environment. In other words, the distributive law applied to the tensor product of the initial superposition and the environment means that the original environment must be duplicated before any interaction takes place. It is very hard to make physical sense of this. What causes the environment to duplicate? How can this even be possible? If the environment is not duplicated before the interaction, then the only possible interpretation is that environmental decoherence records no more than the original superposition, and we know that this is not what happens physically.

The conclusion I draw is that the simple distributive law cannot apply in quantum mechanics and, consequently, Everettian many-worlds theory cannot get off the ground.

Bruce

[John Clark]

On Tue, Oct 1, 2024 at 9:19 PM Bruce Kellett <bhkel...@gmail.com> wrote:

> But when we ask what this means physically we are faced with a problem the distributive law applied to the tensor product of the initial superposition and the environment means that the original environment must be duplicated before any interaction takes place. It is very hard to make physical sense of this.

In Quantum Mechanics "measurement" is a non-commutative operator, the order in which measurements are performed can profoundly affect the outcome. And you're right, it is indeed hard to make physical sense out of that, it's weird but it's a fact, and physicists have been trying to figure it out for about a century now and there is still no consensus. It's especially hard for the Copenhagen people because, although it's an extremely important concept in their theory, they can't tell you exactly what a "measurement" is, nor can they explain why Quantum Mechanics doesn't seem to work on people or their measuring equipment and only works for the things that they're measuring.  

Pilot Wave theory does a little better but at the price of greatly increase complexity, in addition to Schrödinger's Equation it also needs another equation called the Pilot Wave Equation that is non-local and very complex and does nothing but go around erasing all those other annoying worlds that Schrodinger's Equation says is there. That's why some call the pilot wave interpretation Many Worlds theory in denial.  It's also very odd that the pilot wave can affect an electron's motion but the electron has absolutely no effect on the wave. Nothing else in physics is like that. For everything else in the universe if X effects Y then Y is going to have at least some effect on X, but not if X is a pilot wave.

 

> What causes the environment to duplicate? How can this even be possible? 

From the Many Worlds point of view, that's equivalent to asking why is there something rather than nothing, and no theory can give an entirely satisfactory answer to that question, at least not yet. Many Worlds is bare bones no nonsense Quantum Mechanics with no silly bells and whistles attached, such as the pilot wave equation. All Many Worlds says is that everything always obeys Schrodinger's Wave Equation, it never collapses, and the sum total of reality is the Universal Wave Function. The multitude of worlds that some people find so upsetting is just a consequence of that, if you want to get rid of them you're going to have to invent a disappearing worlds theory such as pilot wave or objective collapse theory, but that would enormously complicate things and personally I think quantum mechanics is complicated enough as it is.

  John K Clark    See what's on my new list at  Extropolis

5vv

[Bruce Kellett]

On Thu, Oct 3, 2024 at 5:28 AM John Clark <johnk...@gmail.com> wrote:

On Tue, Oct 1, 2024 at 9:19 PM Bruce Kellett <bhkel...@gmail.com> wrote:

> But when we ask what this means physically we are faced with a problem the distributive law applied to the tensor product of the initial superposition and the environment means that the original environment must be duplicated before any interaction takes place. It is very hard to make physical sense of this.

In Quantum Mechanics "measurement" is a non-commutative operator, the order in which measurements are performed can profoundly affect the outcome. And you're right, it is indeed hard to make physical sense out of that, it's weird but it's a fact, and physicists have been trying to figure it out for about a century now and there is still no consensus. It's especially hard for the Copenhagen people because, although it's an extremely important concept in their theory, they can't tell you exactly what a "measurement" is, nor can they explain why Quantum Mechanics doesn't seem to work on people or their measuring equipment and only works for the things that they're measuring.  

Irrelevant comment!

Pilot Wave theory does a little better but at the price of greatly increase complexity, in addition to Schrödinger's Equation it also needs another equation called the Pilot Wave Equation that is non-local and very complex and does nothing but go around erasing all those other annoying worlds that Schrodinger's Equation says is there. That's why some call the pilot wave interpretation Many Worlds theory in denial.  It's also very odd that the pilot wave can affect an electron's motion but the electron has absolutely no effect on the wave. Nothing else in physics is like that. For everything else in the universe if X effects Y then Y is going to have at least some effect on X, but not if X is a pilot wave.

Another irrelevant comment.

> What causes the environment to duplicate? How can this even be possible? 

From the Many Worlds point of view, that's equivalent to asking why is there something rather than nothing, and no theory can give an entirely satisfactory answer to that question, at least not yet. Many Worlds is bare bones no nonsense Quantum Mechanics with no silly bells and whistles attached, such as the pilot wave equation. All Many Worlds says is that everything always obeys Schrodinger's Wave Equation, it never collapses, and the sum total of reality is the Universal Wave Function. The multitude of worlds that some people find so upsetting is just a consequence of that, if you want to get rid of them you're going to have to invent a disappearing worlds theory such as pilot wave or objective collapse theory, but that would enormously complicate things and personally I think quantum mechanics is complicated enough as it is.

The multitude of worlds is not the issue. The naive application of the distributive law is the problem. I question whether this rule is applicable in QM. If it is not, then MWI collapses for that reason alone.

Bruce

[Brent Meeker]

On 10/2/2024 12:28 PM, John Clark wrote:

All Many Worlds says is that everything always obeys Schrodinger's Wave Equation, it never collapses,

That's right.  It never says where the Born rule comes from.  But I guess that's just one of those silly bells and whistles.

Brent

[John Clark]

On Wed, Oct 2, 2024 at 7:03 PM Brent Meeker <meeke...@gmail.com> wrote:

>> All Many Worlds says is that everything always obeys Schrodinger's Wave Equation, it never collapses,

> That's right.  It never says where the Born rule comes from. 

1) Many worlds is the only quantum interpretation that even tries to derive the Born Rule, the others just assume it's true.  

2) Gleason's theorem mathematically proves that in dimensions of 3 or greater and if all probabilities are required to be non-negative, and add up to exactly 1, then the only consistent way to assign probabilities is the squared amplitudes of the wavefunction, provided you also insist that any combination of two valid quantum states is also a valid quantum state.  

So the real question Many Worlds needs to answer is not why is probability the squared amplitudes of the wavefunction but rather why is probability necessary at all given the fact that Schrodinger's wave equation is 100% deterministic? The answer is because of self locating uncertainty. In the instant after the split but before an observer has registered the outcome of a measurement there is only one rational way to apportion credence as to which branch of the wave function he is on and that is the Born Rule.

Sean Carroll and Charles Sebens go into much more detail here: 

Many Worlds, the Born Rule, and Self-Locating Uncertainty

John K Clark    See what's on my new list at  Extropolis

slu

 

[PGC]

When a quantum system interacts with an environment the combined system develops into an entangled state where system and environment become entangled through their interaction, resulting in the superposition of correlated states. This does not require the environment to physically duplicate. This naturally emerges from the unitary evolution governed by the Schrödinger equation.

The “branching” into multiple outcomes is a way to conceptualize the resulting entangled superposition, with each “branch” corresponding to a different outcome, with the system and environment in correlated states.

Also, I thought the tensor product is bilinear (linear in each of its arguments). So for quantum states, the following would be a standard operation:

(∣a⟩+∣b⟩)⊗∣e⟩=∣a⟩⊗∣e⟩+∣b⟩⊗∣e⟩

The distributive law applies due to properties of tensor products. Yes, we have nonlinear functions but linear operators. There’s potential for confusion here, as the Hamiltonian includes terms that are nonlinear functions of position and momentum. 

Iirc these functions are however constructed from linear operators and, when properly defined, result in linear operators themselves. Linearity for me refers to the way operators act on quantum states within a Hilbert space. Obviously this is essential for the superposition principle. I’m a tourist here, but nonlinear operators seem exotic. Linearity of QM is essential for consistency of the theory/accurate predictions.

[Bruce Kellett]

On Fri, Oct 4, 2024 at 9:21 AM PGC <multipl...@gmail.com> wrote:

When a quantum system interacts with an environment the combined system develops into an entangled state where system and environment become entangled through their interaction, resulting in the superposition of correlated states. This does not require the environment to physically duplicate. This naturally emerges from the unitary evolution governed by the Schrödinger equation.

It does require environment duplication if you want the different outcomes to each entangle separately.

The “branching” into multiple outcomes is a way to conceptualize the resulting entangled superposition, with each “branch” corresponding to a different outcome, with the system and environment in correlated states.

Also, I thought the tensor product is bilinear (linear in each of its arguments). So for quantum states, the following would be a standard operation:

(∣a⟩+∣b⟩)⊗∣e⟩=∣a⟩⊗∣e⟩+∣b⟩⊗∣e⟩

The distributive law applies due to properties of tensor products. Yes, we have nonlinear functions but linear operators. There’s potential for confusion here, as the Hamiltonian includes terms that are nonlinear functions of position and momentum. 

But does the distributive law apply to quantum states? We can represent the situation as a tensor product, but a tensor product of what? The things we write down may not be physical reality.

Bruce

[Bruce Kellett]

On Fri, Oct 4, 2024 at 5:16 AM John Clark <johnk...@gmail.com> wrote:

On Wed, Oct 2, 2024 at 7:03 PM Brent Meeker <meeke...@gmail.com> wrote:

>> All Many Worlds says is that everything always obeys Schrodinger's Wave Equation, it never collapses,

> That's right.  It never says where the Born rule comes from. 

1) Many worlds is the only quantum interpretation that even tries to derive the Born Rule, the others just assume it's true.  

2) Gleason's theorem mathematically proves that in dimensions of 3 or greater and if all probabilities are required to be non-negative, and add up to exactly 1, then the only consistent way to assign probabilities is the squared amplitudes of the wavefunction, provided you also insist that any combination of two valid quantum states is also a valid quantum state.  

So the real question Many Worlds needs to answer is not why is probability the squared amplitudes of the wavefunction but rather why is probability necessary at all given the fact that Schrodinger's wave equation is 100% deterministic? The answer is because of self locating uncertainty.

Self-locating uncertainty is not the answer. It requires branches to be split in ratios according to the Born rule. And no non-circular theory has ever been devised that can give this result. Besides, self-locating uncertainty, like any probability arising from ignorance, assumes some prior notion of randomness, or probability. It is also inherently dualist, since there is an unspoken assumption that only one of the copies on the different branches is really you.

Bruce

[Brent Meeker]

On 10/3/2024 12:16 PM, John Clark wrote:

On Wed, Oct 2, 2024 at 7:03 PM Brent Meeker <meeke...@gmail.com> wrote:

>> All Many Worlds says is that everything always obeys Schrodinger's Wave Equation, it never collapses,

> That's right.  It never says where the Born rule comes from. 

1) Many worlds is the only quantum interpretation that even tries to derive the Born Rule, the others just assume it's true.  

2) Gleason's theorem mathematically proves that in dimensions of 3 or greater and if all probabilities are required to be non-negative, and add up to exactly 1, then the only consistent way to assign probabilities is the squared amplitudes of the wavefunction, provided you also insist that any combination of two valid quantum states is also a valid quantum state.  

So the real question Many Worlds needs to answer is not why is probability the squared amplitudes of the wavefunction but rather why is probability necessary at all given the fact that Schrodinger's wave equation is 100% deterministic? The answer is because of self locating uncertainty. In the instant after the split but before an observer has registered the outcome of a measurement there is only one rational way to apportion credence as to which branch of the wave function he is on and that is the Born Rule.

But how many branches are there and how does the evolution of the Schroedinger equation produce those numbers?  It would be equally rational, and empirically consistent, to suppose that the Born rule assigned a probability weight to one branch representing each possible result.  Many Worlds has no answer to these questions except to point out that they give the right answer...the same answer as Copenhagen's collapse rule gives.

Brent

Sean Carroll and Charles Sebens go into much more detail here: 

Many Worlds, the Born Rule, and Self-Locating Uncertainty

John K Clark    See what's on my new list at  Extropolis

slu

 

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[John Clark]

On Thu, Oct 3, 2024 at 7:52 PM Bruce Kellett <bhkel...@gmail.com> wrote:

> Self-locating uncertainty is not the answer. It requires branches to be split in ratios according to the Born rule. And no non-circular theory has ever been devised that can give this result.

It's reasonable to assume that branches with equal quantum amplitude would have equal probability, and if you wish to generalize that so it covers cases with unequal probability then the Born Rule is the only way to do it, provided you wish  probability  to always be non-negative and  always add up to exactly 1.  

The way I like to think about it (I don't claim this is actually true, it's just an analogy) is that when the universe splits into two they are not necessarily of the same thickness, and so after the split but before you've looked at your measuring equipment if you had to bet on which universe you were in and we're rational then you would play the odds and bet you were in the thicker one. And that's why we need probability even though Schrodinger's Equation is 100% deterministic and the Universal Wave Function is the sum total of the entire Multiverse. If we were infinite beings and could observe and comprehend that entire wave function we wouldn't even need to talk about "worlds" we would just talk about that wave function, but unfortunately we are not infinite so talking about worlds is the best we can do.    

 

> Besides, self-locating uncertainty, like any probability arising from ignorance, assumes some prior notion of randomness, or probability.

Yes, and until I look at my measuring equipment I am ignorant, I don't know if I'm in the universe where the electron is spin up or spin down.  

 

> It is also inherently dualist, since there is an unspoken assumption that only one of the copies on the different branches is really you.

 

Now Bruce Kellett sounds like Bruno Marchal and Bruno Marchal never made one bit of sense to John Clark. It's RIDICULOUS to claim only one of the copies on the different branches is really you!  And when discussing matters of this sort personal pronouns should NEVER be used because there is no clear referent.

John K Clark    See what's on my new list at  Extropolis

mns

[Quentin Anciaux]

Bruno never ever claimed this... you did it on purpose, that's a straw man.

And when discussing matters of this sort personal pronouns should NEVER be used because there is no clear referent.

John K Clark    See what's on my new list at  Extropolis

mns

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