## Quantum vs. Classical Mechanics

### The search for a distinguishing principle

This is the last post discussing http://arxiv.org/abs/1407.7610 before I'll resume my prior math series.

After boiling down the essentials of quantum and classical mechanics and extracting the common algebraic structure, the question becomes "what is quantum"?

The standard answer from Dirac is that in quantum mechanics we add amplitudes, not probabilities. Even earlier, Schrodinger identified superposition. More modern takes on this starting from Hardy is that pure states are linked by continuous transformations.

A pure state is a state which cannot be decomposed into a sum of other states. Because state spaces are convex spaces, this means that pure states reside on the boundary of the state space. In classical physics pure states form a discrete set while in the quantum world pure states form a continuous surface. What does this mean? It means that a measurement in classical physics reveals an intrinsic property of the system, but in quantum mechanics even pure states can collapse from one into another.

But is this intuitive? Can we really claim that we understand the distinction between the classical and the quantum world? No, No, No.

Because quantum and classical physics are completely separated domains, first one cannot explain one in terms of the other, and second, there is no outside bird's eye view to introduce the concepts needed to explain them.

For the first part, imagine a world of triangles trying to grasp the concept of a circle. This is basically what various quantum interpretations actually attempt to do: explain the weirdness of the quantum world in terms of classical concepts: a futile approach. Each interpretation has intuitive parts, but also craziness baked in.

For the second part, to intuitively grasp the distinction between quantum and classical physics, you need to extract yourself from this quantum universe and explain both quantum and classical physics in terms of the laws of a meta-universe where both quantum and classical mechanics are valid. No such thing exists.

A comparison with special theory of relativity is helpful here. To really understand Lorenz transformations, one first needs to free himself/herself from the concept of aether. One does not attempt to understand the constant speed of light using notions of unbounded speeds in a Galilean framework. Just consider how silly a theory of relativity "interpretation" would be along those lines:

Light appears to have a constant propagation speed because there is a "Lorenzian potential" which acts contextually in a particular reference frame measurement. However, in reality light does not have a constant propagation speed.

Now if this is silly, why is Bohmian interpretation not silly?

To really understand quantum mechanics weirdness we need to let go of our classical prejudices. Relativity gave up the concept of aether. Nature is quantum mechanical, no ifs, ends, and buts. Isn't time for quantum mechanics to give up attempts to search for a natural distinguishing principle? It is a futile attempt.

1. According to the following paper:

http://arxiv-web3.library.cornell.edu/pdf/1308.2022.pdf

a resolution of double-slit experiment paradox is proposed by using Feynman path integral formalism by summing up all random walks.

That should allow us to take into account NOT ONLY classical particles trajectories, but even non classical paths by computing a quantum amplitude.

Thus, it seems that the EPR paradox could be eventually solved without resorting to nonlocality.

According to the formula related to the "cat paradox":

psi = 1 / sqrt(2) * ( | 00...0 \ + |11...1 \)

a particle could be "everywhere" before a measurement. However, if the conclusions stated in the above paper are true, than I wonder if a particle can be still considered as a "spooky" superposition of states before its measurement.

Moreover, which implication does the above paper have on quantum entanglement?

May be, after all, that the "hidden variables" mentioned by Albert Einstein are represented by such particle random walks. Please share your authoritative viewpoint and forgive my naive interpetation of such topics.

Best Regards

D.C.

2. Dear D.C, thank you for making me aware of this paper. Please give me a day or two to read and understand it.

3. Dear D.C, I read the paper, and is pure junk. Originally seems to have been submitted to Nature where it got rejected because it was not "sexy enough" for the journal, and then was submitted to the next best thing, Phys Rev Lett where got admitted by incompetent referees (nobody is perfect).

Here is why the paper is wrong. It has a logical error of the kind: A=true, A implies B, B implies that A is false. Here A = "ψAB = ψA + ψB" B=" Feynman path integral formalism". The paper claims that Feynman path integral formalism implies that ψAB = ψA + ψB is false. However from ψAB = ψA + ψB one gets the Feynman path integral formalism as follows:

I am citing from the excellent Quantum Field Theory in a nutshell by Zee, chapter I.2:

4. "The professor's nightmare: a wise guy in the class

Long ago, in a QM class, the professor droned on and on about the double-slit experiment, giving the standard treatment. A particle emitted from a source S at time t=0 passes through one or the other of two holes, A1 and A2, drilled in a screen and is detected at time t=T by a detector located at O. The amplitude for detection is given by the fundamental postulate of quantum mechanics, the superposition principle, as the sum of of the amplitude for the particle to propagate from the source S through the hole A1 and then onward to the point O and the amplitude for the particle to propagate from the source S through the hole A2, and then onward to the point O.

Suddenly, a very bright student, let us call him Feynman, asked: "Professor, what if we drill a third hole in the screen?" The professor replied, "Clearly, the amplitude to be detected at the point O is now given by the sum of three amplitudes, the amplitude for the particle to propagate from the source S through the hole A1, and then onward to the point O, the amplitude for the particle to propagate from the source S through the hole A2, and then onward to the point O, and the amplitude for the particle to propagate from the source S through the hole A3, and then onward to the point O."

The professor was just about ready to continue when Feynman interjected again, "What if we drill a fourth and a fifth hole in the screen?" Now the professor is visibly losing his patience: "All right wise guy, I think it is obvious to the whole class that we just sum over all the holes"

But Feynman persisted, "What if we now add another screen with some holes drilled in it?" The professor was really losing patience: "Look, can't you see that you just take the amplitude to go from the source S to the hole Ai in the first screen, then to the hole Bj in the second screen, then to the detector at O, and then sum over all i and j?"

Feynman continued to pester, "What if I put a third screen, a fourth screen, eh? What if I put in a screen and drill an infinite number of holes in it so that the screen is no longer there?" The professor sighed, "Let's move on; there is a lot of material to cover in this course."

But dear reader, surely you see what that wise guy Feynman was driving at. I especially enjoyed his observation that if you put in a screen and drill an infinite number of holes into it, then the screen is not really there. Very Zen! What Feynman showed is that even if there were just empty space between the source and the detector, the amplitude for the particle to propagate from the source to the detector is the sum of the amplitudes for the particle to go through each one of the holes in each one of the (nonexistent) screens. In other words, we have to sum over the amplitudes for the particle to propagate from the source to the detector following all possible paths between the source and the detector."

5. So now I think is clear that ψAB = ψA + ψB demands the Feynman path integral formalism. The paper claims by some convoluted computation that the Feynman path integral formalism implies that ψAB = ψA + ψB + "small correction". This is pure nonsense, most likely their computation is faulty and no referee bothered to double check the math which must have an error inside.

6. "That should allow us to take into account NOT ONLY classical particles trajectories, but even non classical paths by computing a quantum amplitude."

Indeed, this is correct, and it is what Feynman showed in his path integral formalism.

"However, if the conclusions stated in the above paper are true, than I wonder if a particle can be still considered as a "spooky" superposition of states before its measurement. "
The paper is incorrect.

"Moreover, which implication does the above paper have on quantum entanglement?"
None. Entangelment also follows from Feynman path formulation. Feynman path formulation is an equivalent description of the usual standard quantum mechanics: each one implies the other.

"May be, after all, that the "hidden variables" mentioned by Albert Einstein are represented by such particle random walks."
Nope. Hidden variables were conclusively rejected by Bell's theorem.

7. Dear Dr. moldoveanu,

thank you very much for your commendable review. It is sad to find out that even prized journals may sometime make the mistake to accept junk publication.

Fortunately, it is still possible to account for prepared minds like yours.

also, thank you for the fun anecdotal tale about Feynman. I enjoyed it very much.

In the paper entitled " A Sum-over-histories Account of an EPR(B) Experiment" , Sinha and Sorkin claim that the "sum over histories" interpretation explains the EPR paradox without resorting to nonlocality. What do you think about?

Best regards

D.C.

8. Hi D.C, Thank you for your kind works, please call me Florin. I am not infallible and I make mistakes myself. I am sure that referees just made an honest mistake here. There is some kernel of truth in the paper, and is coming from the Feynman treatment of the Bohm Aharonov effect. However, there the winding clockwise and counter-clockwise paths are not equivalent due the the boundary condition imposed by the magnetic field which ultimately has a topological origin. In this case there is no such thing and any "winding path" through the slits is countered by a counter-winding path if computed correctly. The finite width of the slits does contribute to a small departure from the usual interference pattern but this has nothing to do with non-classical paths.

I am not aware of the Sinha and Sorkin paper. Do you have a link I can use to read it? About "interpretation explains the EPR paradox without resorting to nonlocality" I would not read too much into it. QM violates local realism. There are realistic non-local models (Bohmian interpretation) which recovers completely the QM predictions (in the absence of spin). Also there are local non-realistic models like consistent history or like many-worlds.

Each QM interpretation has its special kind of "dirt under the rug" and it is not enough to like only some parts of the interpretation like in this case the lack on non-locality, but to accept their "pink elephant in the room" too. Different interpretations have different "elephants".

9. Hi Florin,

Suddendly, I can't provide you the Sinha and Sorkin paper. However, you can find a clear explanation of such interpretation by reading the following paper:

www.mdpi.com/2073-8994/3/3/524/pdf

Thank you.

Sincerely,

D. C.

10. Hi D.C. I met Ken Wharton and I blogged in the past about his approach: http://fmoldove.blogspot.com/2013/06/new-directions-in-foundations-of_5.html What he attempts to build is a lagrangean foundation for physics without using time. I don't think this will succeed because QM is best described in the Hamiltonian formalism and in there time is absolutely essential. However I wish him luck. If successful it will be a big deal.

11. Well, if time is an emergent phenomenon based on entanglement as demonstrated experimentally, then we'll see:

http://arxiv.org/abs/1310.4691

Thank you very much for your emergent phenomenon.

Best wishes

D. C.

12. I use to think that I understood time pretty well, but my arrogance bubble was burst by my own research. Time is really very tricky to get a grasp on. Space and time are actually the hard concepts, not quantum mechanics. I am not talking about understanding general relativity (which is a relatively simple thing to fully understand-simpler than quantum mechanics), but understanding why there is space and time in the first place. There are many hints that space and time are actually emergent concepts. Connes made some recent progress along those ideas and I'll explain some of his approach in subsequent posts.