Is quantum mechanics unique?
Quantum mechanics describes nature perfectly and no experiment has ever detected violations of quantum mechanics predictions. Last time I have shown that there is no realistic interpretation possible for quantum mechanics and this fact flies in the face of our classical intuition. But this intuition was developed as part of evolution of species on Earth (a lion chasing a gazelle needs not solve Schrodinger's equation) and is simply irrelevant to modern science.
If quantum mechanics is not a realist theory, then perhaps there are other realistic theories possible. Special relativity replaces Newtonian ideas of absolute space and time, electromagnetism is part of the larger electroweak theory, etc. When we look at answering uniqueness questions, there are two approaches possible.
First you can prove no-go theorems. Bell famously said that no-go theorems only prove a lack of imagination for the author. When you hold dear to your heart a contrarian paradigm, no-go theorems will never convince you to change your point of view. I know this first hand from arguing with people who think they can beat Bell theorem in a locally realistic computer simulation although that is a mathematical impossibility.
Then there is a second approach possible: derive a physical theory from physical principles. Special theory of relativity has far fewer challengers today compared with quantum mechanics because it is much harder to argue with the principle of relativity. Can quantum mechanics be derived from physical principles? The answer is yes, the physical principle is the invariance of the laws of nature under composition:
If system A is described by quantum mechanics, and system B is described by quantum mechanics, then the composed system is described by quantum mechanics as well.
Physically this means that the Planck constant does not change when we add additional degrees of freedom. Mathematically quantum mechanics follows from using category theory arguments. But what other theories obey this invariance under composition principle?
It turns out that there are 3 such solutions possible:
- elliptic composition
- parabolic composition
- hyperbolic composition
Elliptic composition is quantum mechanics (and this is why this blog is called elliptic composability), parabolic composition is classical mechanics, but what is this hyperbolic case?
Formally, hyperbolic composability is nothing but quantum mechanics with \(\sqrt{-1}\) replaced by \(\sqrt{+1}\) and the resulting theory is known as hyperbolic quantum mechanics.
The first thing one notices is that in hyperbolic quantum mechanics one continues to add amplitudes just like in ordinary quantum mechanics, and therefore this is not a realistic theory either. As such realism is dead for good. But can this theory describe anything in nature? Nope, but nevertheless I'll explore the mathematics of this theory in the next post because the most valuable aspect of this theory is to act as a comparison backdrop against quantum mechanics and illuminate various properties of it.
I will not start today to dig into the mathematical aspects, and I want to discuss instead the meaning of lost realism in physics. We have seen that quantum mechanics is a probabilistic, and not a deterministic theory. The wavefunction cannot have an ontic interpretation for two main reasons: it does not carry energy or momentum, and it has several distinct representations. But perhaps realism is saved by an epistemic interpretation: what if the experiment simply reveals pre-existing values? This hope was kept alive by various classical toy models, but they were put to rest by the PBR theorem. So realism is not an option anymore, but is there a real world independent of us? Do we have to resort to solipsism or even worse to some sort of quantum cargo-cult religious new age babble based on the discredited ideas of Stapp and the crackpot new age guru Deepak Chopra? Does the observer play an active role in quantum mechanics?
Let's see what the math shows. Quantum mechanics reconstruction uses category theory, and category theory is known as "objects with arrows". The nature of the objects is completely irrelevant, all that matters are the arrows which represent the relationships. Originally category theory was introduced to map the similarities between topological and algebraic objects, and the higher abstraction of categorical proofs allowed the extraction of common behaviors in very different mathematical domains. Because quantum mechanics reconstruction is categorical in nature, this derivation is blind to any hypothetical underlying quantum ontology. The "true meaning" of the wavefunction simply does not matter. The question: what does quantum mechanics describe? is not a testable, meaningful scientific question.
But what about the observer role? The observer does not cause the outcome. It that were true, you can use quantum mechanics to send signals faster than the speed of light. It is not the observer who is important, but the configuration of the measurement device. This is because non-commutative observables cannot be measured simultaneously. The observer only plays an indirect role in deciding what and how to measure. Here Andrei can argue along the lines of "free will is an illusion": the observer is part of nature, subject to quantum (or hypothetical sub quantum deterministic) laws as well. However, as an emergent phenomena, human consciousness is fundamentally different. Why? Because in quantum mechanics information is conserved, but people are born and later on die and there is no information conservation for the soul. Free will is a manifestation of this lack of information conservation.
To date, the best correct interpretation of quantum mechanics available is QBism. However I am not completely satisfied with it. If qubism is Copenhagen done right, I am working on "qbism done right" :) but more on this in future posts.
I will not start today to dig into the mathematical aspects, and I want to discuss instead the meaning of lost realism in physics. We have seen that quantum mechanics is a probabilistic, and not a deterministic theory. The wavefunction cannot have an ontic interpretation for two main reasons: it does not carry energy or momentum, and it has several distinct representations. But perhaps realism is saved by an epistemic interpretation: what if the experiment simply reveals pre-existing values? This hope was kept alive by various classical toy models, but they were put to rest by the PBR theorem. So realism is not an option anymore, but is there a real world independent of us? Do we have to resort to solipsism or even worse to some sort of quantum cargo-cult religious new age babble based on the discredited ideas of Stapp and the crackpot new age guru Deepak Chopra? Does the observer play an active role in quantum mechanics?
Let's see what the math shows. Quantum mechanics reconstruction uses category theory, and category theory is known as "objects with arrows". The nature of the objects is completely irrelevant, all that matters are the arrows which represent the relationships. Originally category theory was introduced to map the similarities between topological and algebraic objects, and the higher abstraction of categorical proofs allowed the extraction of common behaviors in very different mathematical domains. Because quantum mechanics reconstruction is categorical in nature, this derivation is blind to any hypothetical underlying quantum ontology. The "true meaning" of the wavefunction simply does not matter. The question: what does quantum mechanics describe? is not a testable, meaningful scientific question.
But what about the observer role? The observer does not cause the outcome. It that were true, you can use quantum mechanics to send signals faster than the speed of light. It is not the observer who is important, but the configuration of the measurement device. This is because non-commutative observables cannot be measured simultaneously. The observer only plays an indirect role in deciding what and how to measure. Here Andrei can argue along the lines of "free will is an illusion": the observer is part of nature, subject to quantum (or hypothetical sub quantum deterministic) laws as well. However, as an emergent phenomena, human consciousness is fundamentally different. Why? Because in quantum mechanics information is conserved, but people are born and later on die and there is no information conservation for the soul. Free will is a manifestation of this lack of information conservation.
To date, the best correct interpretation of quantum mechanics available is QBism. However I am not completely satisfied with it. If qubism is Copenhagen done right, I am working on "qbism done right" :) but more on this in future posts.
Is quantum mechanics universal? Quantum mechanics seems to be almost more than a physical theory and in a way a physical logic. I did this calculation a long time ago where I proposed a gauge potential to cancel out double relativity. This was an idea back in the late 90s about how the Planck scale cut off the Lorentz transformation in this rather ugly way. In effect Lorentz symmetry is ultimately broken. The thought occurred to me that this might also be at work to preserve the Heisenberg uncertainty principle as well. There is a so called generalized uncertainty principle that is due to the string scale, but this holds for string theory with
ReplyDeleteThis is
ΔpΔx = ħ + Cα'Δp^2.
Where C is a constant and α' is the string coupling parameter. Δp^2 is energy and this is related to force E = F(x)Δx.
This may then connect with the Verlinde idea of entropy gravity with Δp = F(x)Δt, and this uncertainty may be writted as
ΔpΔx = ħ + F(x)ΔxΔt.
This is a version of the uncertainty principle modified for string theory
ΔpΔx = ħ + μΔx^2,
where μΔx = F(x)Δx = F(x)Δt/c. Here I have replaced Δp^2 with (Δx/ ħ)^2 and absorbed the Planck constant into the string coupling parameter. This could also contain another Δp^2, but the string constants are small and this vanishes FAPP. Then the constant μ has been replaced with a force that depends upon distance or a region in space/spacetime. I have though a term F(x) that is written to indicate force. This leads to the spread in the width of the uncertainty
Δx = Δpc/2F(x) +/- c sqrt{Δp^2 – 4ħF(x)/c}/2F(x)
From ΔpΔx = ħ + F(x)ΔxΔt we can divide by Δt with Δx/Δt ~ c to get
F(x)Δx = Δpc - ħ/Δx
where the LHS is the work equivalent due to this force. We may write this according to ΔW = Δpc - ħ/Δx, so that the work is seen to increase with Δp and decrease with Δx. This may also be expressed to connect the momentum-position uncertainty with a “quantum work-time uncertainty” so that
ΔWΔt = ΔpΔx - ħ.
For F = 0 then ΔpΔx = ħ and the LHS is zero. This illustrates how ΔWΔt is associated with an increase in uncertainty.
If you think this looks similar to the entropy force of gravity results you are not too far off! The entropy force of gravity concerns the transverse displacement of a holographic screen. Here we are concerned with the stretching or longitudinal motion along the screen. The entropy force of gravity F = T∇S determines work as
ΔW = FΔx = T∇SΔx = TΔS.
We employ E = (1/2)NkT which equals Mc^2, We then see that temperature is T = Mc^2/Nk. This is inserted into the equation for the entropy force
FΔx = (2Mc^2/Nk)ΔS
The motion of the holographic screen by a distance Δx = ħ/mc results in
F = (Mmc^3/Nk ħ)ΔS
and we use N = A/4L_p^2 = πR^2/(Għ/c^3). We then have that
F = 2GMm/(kR^2)ΔS
Where we then assign the unit of entropy ΔS = k/2 to get the entropy force of gravity result.
Now let me fix the uncertainty principle. This is fairly ugly. The most elementary way is to gauge the momentum. So I let p → p + igA so that Δp → Δp + ig∇ xA*Δx' = Δp + F*Δx'. If this is equal in magnitude to the gravitational entropy force then we recover the original uncertainty principle. The gauge force here corresponds to they Yang-Mill quantum field on the holographic screen that is dual to gravitation.
Lawrence, I will not comment on string theory because I am not an expert in this. I will criticize Lubos each and every time he will say something incorrect about quantum mechanics and I will not create an opening by shooting from the hip to give him a chance to return the favor.
DeleteSo you want to avoid
Deletehttps://www.youtube.com/watch?v=YzTGlSp0oB8
Some of these are hilarious.
LC
Dear Florin,
ReplyDelete“When we look at answering uniqueness questions, there are two approaches possible.
“First you can prove no-go theorems. Bell famously said that no-go theorems only prove a lack of imagination for the author. When you hold dear to your heart a contrarian paradigm, no-go theorems will never convince you to change your point of view.”
The reason for dismissing Bell’s theorem as a valid argument against the existence of classical theories capable of reproducing the predictions of QM has nothing to do with how much love you may have for these theories but with basic logical reasoning. It should be obvious that the free will assumption and measurement independence assumption are necessarily wrong in a classical field theory. Free-will is incompatible with determinism, while the independence of distant systems is incompatible with the long-range interactions present in field theories.
Free-will is not a fundamental principle of physics. The only evidence for its existence is the personal experience of some people. You can find yourself here in the good company of those experiencing encounters with divine beings or alien abductions. I would even argue that the evidence for those encounters is even stronger than the one for free-will because you can at least imagine such an event. On the other hand how in the world would you expect to distinguish between a subconscious random process and a pseudorandom one? Even if you had the mind of a god you couldn’t say if a finite series of numbers are generated by a deterministic algorithm (like the decimal expansion of some irrational number) or by a genuinely random process. In other words you cannot even have the illusion of free-will, but the illusion of free-will is itself an illusion, in the words of Sam Harris.
“But perhaps realism is saved by an epistemic interpretation: what if the experiment simply reveals pre-existing values? This hope was kept alive by various classical toy models, but they were put to rest by the PBR theorem.”
Oh, realy? Let’s examine the assumptions used in this theorem:
“The argument depends on few assumptions. One is that a system has a “real physical state" - not necessarily completely described by quantum theory, but objective and independent of the observer.”
“The other main assumption is that systems that are prepared independently have independent physical states.”
In a classical field theory the second assumption fails. So much for this no-go theorem.
Andrei
Andrei, sorry for the delay, I was busy writing the next post. If free will is an illusion, how do you explain the justice system? If there is no free will, then there is no personal responsibility, and there is no right or wrong.
ReplyDeleteDear Florin,
Delete"If free will is an illusion, how do you explain the justice system? If there is no free will, then there is no personal responsibility, and there is no right or wrong. "
This is not actually an argument. It might be the case that there is no right or wrong, or personal responsibility, etc. This doesn't imply that classical determinism is wrong.
On the other hand I disagree with the above implications. If someone is a sociopath, he should be removed from society for our own protection. The fact that he didn't have a choice doesn't change the fact that he is dangerous.
The notions of right or wrong do not have anything to do with free will. If you make someone suffer it is wrong, regardless of the reasons behind that act.
Andrei
Dear Andrei, interesting arguments. I would equate suffering with the idea of loss. Ultimately it is an information loss. But the laws of physics prevent information loss, and because of this consciousness (which is ultimately responsible for acts of kindness or evil) cannot be reduced to chemistry and physics. Consciousness is an emergent phenomena. For an analogy you can think about physics and chemistry as the hardware, and consciousness as the software. If you pull the plug on the hardware, the software stops working.
DeleteIn a software program variables are created and initialized out of thin air, and their values and instances can be destroyed. Information is not conserved. The link with hardware and thermodynamics is given by Landauer's principle.
The software has a reality of its own: you can design, write the code, test, and fix bugs without ever having to consider the hardware. This is a counterexample to a reductionism (materialistic) approach.
Your elaboration of reality into the three classes of quantum, classical, and hyperbolic is very interesting. It would seem to me that all three are hierarchies of what reality is and so there is no need to choose one over the other.
ReplyDeleteSo it would seem like your very lovely three classes all represent the different parts or "functors" of the same reality, not really an exclusive single reality.
Clearly quantum is first up since it includes phase coherence and reality has phase coherence. The Schrodinger equation is the basis for action and discrete matter and time delay are the conjugates, not p and q.
The next up to bat is classical, which are made up of the norms of quantum matter waves. Classical Hamilton-Jacobi clearly comes from the norms or squares of quantum action.
Finally comes the hyperbolic solutions, which seem to be unphysical. Since antimatter and antiverses are both unphysical, the hyberbolic class would seem to represent these kinds of very exotic and unstable antimatter.
You do seem to have a very nice path to unification here...
Thank you Steve,. this unification originated from Bohr himself and I am only the last person to work on it. The three classes have no overlap and nature must pick one. Which one can only be determined by experimental means.
ReplyDeletePerhaps I misspoke. The choice of quantum is still supported by measurement. However, the classical algebra of Hamilton-Jacobi all derives from quantum norms, right? Therefore with any substantial objects, the uncertainty principle is overwhelmed by the chaos of complexity and the classical approach of H-J relativity follows but only for a limited portion of reality.
ReplyDeleteThe hyperbolic quantum solution is simply that of matter's complement, antimatter, and is also therefore limited to a portion of reality. The only thing left to do is quantum gravity. An approach for quantum gravity that seems to work is to bond each particle of matter to the universe matter as a gravity atom by the Schrodinger equation. So gravity is just the bonding of gravity atoms just like an atom is the bonding of charge and matter is the bonding of charge atoms.
This allows relativity to work just fine in its limited domain and still allow quantum gravity to rule over all of reality...except of course antimatter.