Something you and the OP might find interesting is one of those things that is basically equivalent to a wavefunction, but represented in different mathematics is a Wigner function. It behaves almost exactly like a classical probability distribution, for example it integrates up to 1. Bayes rule updates it when you measure stuff. However, in order for it to “do quantum physics” it needs the ability to have small negative patches. So quantum physics can be modelled as a random stochastic process, if negative probabilities are allowed. (Incidentally, this is often used as a test of “quantumness”: do I need negative probabilities to model it with local stochastic stuff? If yes, then it is quantum).
If you are interested in a sketch of the maths. Take W to be a completely normal probability distribution, describing what you know about some isolated, classical ,1d system. And take H to be the classical Hamiltonian (IE just a function for the system’s energy). Then, the correct way of evolving your probability distribution (for an isolated classical, 1D system) is:
˙W=H(←−∂∂x−→∂∂p−←−∂∂p−→∂∂x)W Where the arrows on the derivatives have the obvious effect of firing them either at H or W. The first pair of derivatives in the bracket is Newton’s Second law (rate of change of Energy (H) with respect to X is going to turn potential’s into Forces, and the rate of change with momentum on W then changes the momentum in proportion to the force), the second term is the definition of momentum (position changes are proportional to momentum).
Instead of going to operators and wavefunctions in Hilbert space, it is possible to do quantum physics by replacing the previous equation with:
˙W=2Hℏsin(ℏ2(←−∂∂x−→∂∂p−←−∂∂p−→∂∂x))W
Where sin is understood from Taylor series, so the first term (after the hbars/2 cancel) is the same as the first term for classical physics. The higher order terms (where the hbars do not fully cancel) can result in W becoming negative in places even if it was initially all-positive. Which means that W is no longer exactly like a probability distribution, but is some similar but different animal. Just to mess with us the negative patches never get big enough or deep enough for any measurement we can make (limited by uncertainty principle) to have a negative probability of any observable outcome. H is still just a normal function of energy here.
Also, the OP is largely correct when they say “destructive interference is the only issue”. However, in the language of probability distributions dealing with that involves the negative probabilities above. And once they go negative they are not proper probabilities any more, but some new creature. This, for example, stops us from thinking of them as just our ignorance. (Although they certainly include our ignorance).
I’d expect Wigner functions to be less ontologically fundamental than wavefunctions because a wavefunction into a real function in this way introduces a ton of redundant parameters, since now it’s a function of phase space instead of configuration space. But they’re still pretty cool.
Imagine you have a machine that flicks a classical coin and then makes either one wavefunction or another based on the coin toss. Your ordinary ignorance of the coin toss, and the quantum stuff with the wavefunction can be rolled together into an object called a density matrix.
There is a one-to-one mapping between density matrices and Wigner functions. So, in fact there are zero redundant parameters when using Wigner functions. In this sense they do one-better than wavefunctions, where the global phase of the universe is a redundant variable. (Density matrices also don’t have global phase.)
That is not to say there are no issues at all with assuming that Wigner functions are ontologically fundamental. For one, while Wigner functions work great for continuous variables (eg. position, momentum), Wigner functions for discrete variables (eg. Qubits, or spin) are a mess. The normal approach can only deal with discrete systems in a prime number of dimensions (IE a particle with 3 possible spin states is fine, but 6 is not.). If the number of dimensions is not prime weird extra tricks are needed.
A second issue is that the Wigner function, being equivalent to a density matrix, combines both quantum stuff and the ignorance of the observer into one object. But the ignorance of the observer should be left behind if we were trying to raise it to being ontologically fundamental, which would require some change.
Another issue with “ontologising” the Wigner function is that you need some kind of idea of what those negatives “really mean”. I spent some time thinking about “If the many worlds interpretation comes from ontologising the wavefunction, what comes from doing that to the Wigner function?” a few years ago. I never got anywhere.
Another issue with “ontologising” the Wigner function is that you need some kind of idea of what those negatives “really mean”. I spent some time thinking about “If the many worlds interpretation comes from ontologising the wavefunction, what comes from doing that to the Wigner function?” a few years ago. I never got anywhere.
Wouldn’t it also be many worlds, just with a richer set of worlds? Because with wavefunctions, your basis has to pick between conjugate pairs of variables, so your “worlds” can’t e.g. have both positions and momentums, whereas Wigner functions tensor the conjugate pairs together, so their worlds contain both positions and momentums in one.
Something you and the OP might find interesting is one of those things that is basically equivalent to a wavefunction, but represented in different mathematics is a Wigner function. It behaves almost exactly like a classical probability distribution, for example it integrates up to 1. Bayes rule updates it when you measure stuff. However, in order for it to “do quantum physics” it needs the ability to have small negative patches. So quantum physics can be modelled as a random stochastic process, if negative probabilities are allowed. (Incidentally, this is often used as a test of “quantumness”: do I need negative probabilities to model it with local stochastic stuff? If yes, then it is quantum).
If you are interested in a sketch of the maths. Take W to be a completely normal probability distribution, describing what you know about some isolated, classical ,1d system. And take H to be the classical Hamiltonian (IE just a function for the system’s energy). Then, the correct way of evolving your probability distribution (for an isolated classical, 1D system) is:
˙W=H(←−∂∂x−→∂∂p−←−∂∂p−→∂∂x)W
Where the arrows on the derivatives have the obvious effect of firing them either at H or W. The first pair of derivatives in the bracket is Newton’s Second law (rate of change of Energy (H) with respect to X is going to turn potential’s into Forces, and the rate of change with momentum on W then changes the momentum in proportion to the force), the second term is the definition of momentum (position changes are proportional to momentum).
Instead of going to operators and wavefunctions in Hilbert space, it is possible to do quantum physics by replacing the previous equation with:
˙W=2Hℏsin(ℏ2(←−∂∂x−→∂∂p−←−∂∂p−→∂∂x))W
Where sin is understood from Taylor series, so the first term (after the hbars/2 cancel) is the same as the first term for classical physics. The higher order terms (where the hbars do not fully cancel) can result in W becoming negative in places even if it was initially all-positive. Which means that W is no longer exactly like a probability distribution, but is some similar but different animal. Just to mess with us the negative patches never get big enough or deep enough for any measurement we can make (limited by uncertainty principle) to have a negative probability of any observable outcome. H is still just a normal function of energy here.
(Wikipedia is terrible for this topic. Way too much maths stuff for my taste: https://en.wikipedia.org/wiki/Moyal_bracket)
Also, the OP is largely correct when they say “destructive interference is the only issue”. However, in the language of probability distributions dealing with that involves the negative probabilities above. And once they go negative they are not proper probabilities any more, but some new creature. This, for example, stops us from thinking of them as just our ignorance. (Although they certainly include our ignorance).
Neat!
I’d expect Wigner functions to be less ontologically fundamental than wavefunctions because a wavefunction into a real function in this way introduces a ton of redundant parameters, since now it’s a function of phase space instead of configuration space. But they’re still pretty cool.
Imagine you have a machine that flicks a classical coin and then makes either one wavefunction or another based on the coin toss. Your ordinary ignorance of the coin toss, and the quantum stuff with the wavefunction can be rolled together into an object called a density matrix.
There is a one-to-one mapping between density matrices and Wigner functions. So, in fact there are zero redundant parameters when using Wigner functions. In this sense they do one-better than wavefunctions, where the global phase of the universe is a redundant variable. (Density matrices also don’t have global phase.)
That is not to say there are no issues at all with assuming that Wigner functions are ontologically fundamental. For one, while Wigner functions work great for continuous variables (eg. position, momentum), Wigner functions for discrete variables (eg. Qubits, or spin) are a mess. The normal approach can only deal with discrete systems in a prime number of dimensions (IE a particle with 3 possible spin states is fine, but 6 is not.). If the number of dimensions is not prime weird extra tricks are needed.
A second issue is that the Wigner function, being equivalent to a density matrix, combines both quantum stuff and the ignorance of the observer into one object. But the ignorance of the observer should be left behind if we were trying to raise it to being ontologically fundamental, which would require some change.
Another issue with “ontologising” the Wigner function is that you need some kind of idea of what those negatives “really mean”. I spent some time thinking about “If the many worlds interpretation comes from ontologising the wavefunction, what comes from doing that to the Wigner function?” a few years ago. I never got anywhere.
Wouldn’t it also be many worlds, just with a richer set of worlds? Because with wavefunctions, your basis has to pick between conjugate pairs of variables, so your “worlds” can’t e.g. have both positions and momentums, whereas Wigner functions tensor the conjugate pairs together, so their worlds contain both positions and momentums in one.