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We learn from quantum mechanics courses that one recovers classical mechanics in the limit when Planck's constant goes to zero. This can be seen in the path integral formulation. This is why macroscopic objects mostly obey laws of classical mechanics and that quantum effects are only seen at small scales.

So let's say we start with a quantum mechanical system that evolves at first with a value of zero for Planck's constant (In this case it at first obeys classical mechanics). Then we slowly increase it over time. The objects in the system will still look like they obey classical mechanics when the constant is very small, but not zero. (They all look like macroscopic objects)

How does branching in the many-worlds start to occur if you keep the constant very small like this (if it does)?

And suppose there are many worlds, branches, what happens to the universe if you let the constant go to zero?

Edit : As more indication of the question: take the value of Planck's constant $h(t)$ to follow the bump function over time; and consider the beginning and the end of the bump. $$ h(t) = \begin{cases} e^{-\frac{1}{1-t^2}} & \text{for } -1 < t < 1, \\ 0 & \text{otherwise}. \end{cases} $$

(I know it's funny to call it a constant if you move it around.. but consider it as a Gedankenexperiment)


It would be fun to have an answer using the path integral formalism; since I like it a lot (it's mostly the only thing I am using now). And I don't mind a very math-heavy answer since I have a good background in mathematical physics.

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We learn from quantum mechanics courses that one recovers classical mechanics in the limit when Planck's constant goes to zero. This can be seen in the path integral formulation. This is why macroscopic objects mostly obey laws of classical mechanics and that quantum effects are only seen at small scales.

The claim that quantum theory tends to classical physics in the limit $\hbar\to 0$ is common, but it is false if quantum mechanics is an accurate description of how reality works. Even an object the size of Saturn's moon Hyperion would undergo quantum interference on large enough timescales (about one month) if it was isolated from interactions with all external systems:

https://arxiv.org/abs/quant-ph/9802054

https://arxiv.org/abs/quant-ph/0605249

This would happen regardless of the fact that $\hbar$ is small compared to macroscopic quantities in the same units. But if a system interacts with external systems to produce a record of the interfering observables, then interference is suppressed: this effect is called decoherence. Decoherence causes the world to look classical, not the size of $\hbar$. For a review see

https://arxiv.org/abs/1911.06282

For a path integral based approach see

https://arxiv.org/abs/quant-ph/0208026

Another problem with your question is that since $\hbar$ has units, you can change its numerical value to whatever non-zero value you want. As for a world in which it is close to zero on scales we are interested in, that looks like the world we live in now. Some observables evolve in a more-or-less classical way that looks a like a collection of parallel universes on the scales of everyday life:

https://arxiv.org/abs/quant-ph/0104033

https://arxiv.org/abs/1111.2189

This means you have to do precise experiments to see effects like quantum interference or entanglement, although even macroscopic objects carry locally inaccessible quantum information:

https://arxiv.org/abs/quant-ph/9906007

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    $\begingroup$ but a zero is zero at all scales $\endgroup$
    – hyportnex
    Commented Mar 18 at 10:23
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    $\begingroup$ "in the limit when Planck's constant goes to zero" is not the same as just setting it to zero $\endgroup$
    – alanf
    Commented Mar 18 at 10:52
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    $\begingroup$ It is not the same if one assumes discontinuity in said limit; either way, continuity or discontinuity, is an assumption. $\endgroup$
    – hyportnex
    Commented Mar 18 at 15:20
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    $\begingroup$ @GuillaumeLaporte $\hbar\to 0$ isn't the classical limit. It is neither necessary nor sufficient to produce classical behaviour as I explained above. For more discussion of the $\hbar\to 0$ limit see arxiv.org/abs/1201.0150 $\endgroup$
    – alanf
    Commented Mar 18 at 22:00
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    $\begingroup$ "This shows that classical mechanics cannot be regarded as emerging from quantum mechanics-at least in this sense-upon straightforward application of the limit $\hbar\to 0$."Quote from the abstract. I have explained the actual classical limit in my answer. $\endgroup$
    – alanf
    Commented Mar 19 at 7:53

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