Is Quantum Mechanics Infinitely More Complex than Classical Mechanics?

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It's widely believed that BQP (including the simulation of general quantum systems) requires exponential time and classical physics simulations only take polynomial time. If this is true (and it may very well not be), then doesn't the gap between the complexity of quantum and classical systems become infinite (specifically uncountably infinite) as the complexity of the underlying system goes to infinity?
 
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user30 said:
I take it you mean partial differential equations?
No, the formal structure. Especially the Hamilton-Jacobi formulation. There are many mathematical formalisms that are borrowed and modified in Quantum Mechanics. For instance, Sakurai, Revised edition page 50 shows how the classical Poisson brackets are transformed into the quantum Commutators by the multiplication of ih. The math necessary for formal Quantum Theory was there for the tweaking in the 1920's but Sakurai makes the important point that while Classical mechanics can be derived from Quantum Mechanics, the reverse is not true. It was the formal framework to begin with and became the limit as h goes to zero. This was also true for the development of Quantum Field Theory. The Classical concepts of generators and Canonical transformations morphed into concepts like Feynman's propagators.
 
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Botsina said:
It's widely believed that BQP (including the simulation of general quantum systems) requires exponential time and classical physics simulations only take polynomial time. If this is true (and it may very well not be), then doesn't the gap between the complexity of quantum and classical systems become infinite (specifically uncountably infinite) as the complexity of the underlying system goes to infinity?
It just means you need a quantum computer to solve quantum systems.
 
user30 said:
Is it really fair to state that all models of classical mechanics are approximations? That wasn't the view traditionally, was it?
With a view of qt, classical mechanics is a useful approximation. Even in qt, classical mechanics emerges as an approximation.

Are you saying that classical mechanics is fundamental and qt emerges from classicality? How would you explain blackbody radiation, stability of matter, quantum tunneling, sharing of electrons in covalent bonds, etc. processes that agree to extraordinary precision given the successes of qt?
 
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Well, the statement in the paper isn't "accurate" whatever that means.

If cardinality is a measure of complexity then I would argue QM is simpler. The number of states in a Hilbert space are countable (I think it's called separability?) whereas the points in classical phase space are uncountable.
 
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Paul Colby said:
The number of states in a Hilbert space are countable

No, they aren't. Where are you getting that from?

Paul Colby said:
I think it's called separability?

Separability means the space has a countable, dense subset. It does not mean the space itself is countable.

For example, the real numbers are separable because they have a countable, dense subset (the rational numbers). But the real numbers are not countable.
 
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Paul Colby said:
Well, the statement in the paper isn't "accurate" whatever that means.

If carnality is a measure of complexity then I would argue QM is simpler. The number of states in a Hilbert space are countable (I think it's called separability?) whereas the points in classical phase space are uncountable.
Carnality? What am I missing?
 
I read some of the article. The statement was made in the introduction. Even in research papers a certain amount of "poetic license" is permitted in the introduction and it is not meant to be taken quite as literal as some may take it. The point of the introduction is to build up why the reader is reading the article
I doubt if the paper's referees read this and would ask the author to justify the statement in the literal sense.

Apparently Heisenberg (the QM expert himself) did not necessarily think classical physics was so easy. He is quoted as saying when he dies, he would ask god 2 questions, Why relativity, and why turbulence (based on classical mechanics), he said god might have and answer for the first one.

If I were to write a classical mechanics paper, I might put in the introduction, the following:

Classical mechanics is far more complicated than quantum mechanics. In classical mechanics, you need to know (at least) the position and velocity as a function of tine in all phase space. In quantum mechanics, all you need is the wave function. You hit it with the momentum operator to get the momentum, you hit it with the position operator to get the position, you hit it with the energy operator to get the energy. All information is in the wavefunction.

Probably, the paper's referees would allow for this build up and not look to judge this literally.
 
But quantum mechanics dispenses with trajectory altogether, in the ordinary sense of the word.

Lawrence Krauss:

"No area of physics stimulates more nonsense in the public arena than quantum mechanics—and with good reason. No one intuitively understands quantum mechanics because all of our experience involves a world of classical phenomena where, for example, a baseball thrown from pitcher to catcher seems to take just one path, the one described by Newton’s laws of motion. Yet at a microscopic level, the universe behaves quite differently. Electrons traveling from one place to another do not take any single path but instead, as Feynman first demonstrated, take every possible path at the same time.

Moreover, although the underlying laws of quantum mechanics are completely deterministic—I need to repeat this, they are completely deterministic—the results of measurements can only be described probabilistically. This inherent uncertainty, enshrined most in the famous Heisenberg uncertainty principle, implies that various combinations of physical quantities can never be measured with absolute accuracy at the same time"

https://www.scientificamerican.com/article/a-year-of-living-dangerously/
 
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I guess the bigger picture is that when someone asks: What is quantum mechanics? You would have to give two answers:

Process 1: The discontinuous change brought about by the observation of a quantity with eigenstates in which the state will be changed to the state with a determined probability amplitude.

Process 2: The continuous, determined change of state of the (isolated) system with time according to the wave equation

I prefer to label process 2 as 1 but textbooks order it this way.

Whereas when someone asks: What is classical mechanics? One answer is sufficient.
 
user30 said:
Electrons traveling from one place to another do not take any single path but instead, as Feynman first demonstrated, take every possible path at the same time.
This is a completely wrong oversimplification of the meaning of Feynman's path integral. Electrons don"t take every possible path but a single fuzzy path, described quite well in a semiclassical way when the conditions of geometric optics are satisfied.
 
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A. Neumaier said:
This is a completely wrong oversimplification of the meaning of Feynman's path integral. Electrons don"t take every possible path but a single fuzzy path, described quite well in a semiclassical way when the conditions of geometric optics are satisfied.

I wasn't the one who wrote it but to make it clear that we are not down to semantics here, how does that functionally differ from Krauss's account?
 
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If an electron goes through both slits, how could it not take every possible path? I either goes through A or B, or both. What other path is there?:smile:
 
A. Neumaier said:
Electrons don"t take every possible path but a single fuzzy path, described quite well in a semiclassical way when the conditions of geometric optics are satisfied.

"Fuzzy" is not be the word I would use since it entails one of the following: "difficult to perceive; indistinct or vague". None of that applies to the electrons path, though it is not classical (which is fine).
 
user30 said:
"Fuzzy" is not be the word I would use since it entails one of the following: "difficult to perceive; indistinct or vague". None of that applies to the electrons path, though it is not classical (which is fine).
The definition of an electron path is limited in precision by the Heisenberg uncertainty relation, hence is similarly vague as the path traveled by a human - in the sense that it cannot be pinned down to arbitrary precision.
user30 said:
If an electron goes through both slits, how could it not take every possible path? I either goes through A or B, or both. What other path is there?
The electron field goes through both slits, and is noticed at the screen as a particle. A path is not involved at all - except one so vague that it covers both slits and the position of the detector event.
 
A. Neumaier said:
The definition of an electron path is limited in precision by the Heisenberg uncertainty relation, hence is similarly vague as the path traveled by a human - in the sense that it cannot be pinned down to arbitrary precision.

Why is the probability distribution deemed a mystery (measurement problem) when it is a direct consequence of Heisenbergs uncertainty principle?

If a principle (law) states that whenever you interact with a system, your knowledge of that system is undermined in one of two ways, then simply view that law the same as any causal interaction in the universe?

It works exactly the same as causality and is just as dependable.
 
user30 said:
If an electron goes through both slits, how could it not take every possible path? I either goes through A or B, or both. What other path is there?:smile:
There would be infinite possible paths through each slit as I see it.
 
bob012345 said:
There would be infinite possible paths through each slit as I see it.

I think Krauss meant that it takes both entries A and B at the same time.
 
user30 said:
I think Krauss meant that it takes both entries A and B at the same time.
Nice, but how would it distinguish that?
 
bob012345 said:
Nice, but how would it distinguish that?

Why would it need to?
 
user30 said:
Why would it need to?
'It' doesn't need to but the theory describing it does. I mean, if there are infinite possible paths and it does not take them all but takes some including through slit A and B, how does that happen? How many paths through A and B? Which paths through A and B? How many paths are enough?
 
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bob012345 said:
'It' doesn't need to but the theory describing it does. I mean, if there are infinite possible paths and it does not take them all but takes some including through slit A and B, how does that happen? How many paths through A and B? Which paths through A and B? How many paths are enough?

It happens the way it does every single time because it's a deterministic system, just as every single time you interact with it, you get a probability amplitude. If this exhibited pattern were to break, then our scientific inquiry goes to pieces, and so would we (we are sensitive to the laws of physics as you are well aware). But don't hold your breath :wink:
 
I think it is time to close this thread, which is reaching diminished returns.

My point of view is similar to @Vanadium 50: this quote should not be taken out of context. As someone who is working in computational physics, I will also defend the statement in this context that quantum mechanics is infinitely more complex than classical mechanics.

Thread closed.