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If you have a quantum-mechanical system that you describe by specifying a Hamiltonian, I can always write down an exactly equivalent path integral. A failure o

by evanb 4y ago

If you have a quantum-mechanical system that you describe by specifying a Hamiltonian, I can always write down an exactly equivalent path integral. A failure of equivalence between these two descriptions would be extremely exciting, as it'd be evidence of some paradigm beyond quantum mechanics.

ImHereToVote 4y ago
Can you do that for two systems that interfere? I think that is where the multiverse stuff comes from.
- evanb 4y ago
  Well, it depends on what you mean by "two systems". If they are described by one Hamiltonian the result is one path integral. In a relativistic quantum field theory, for example, one may describe systems of any number of particles. The path integral captures all the interference effects.
skissane 4y ago
The mathematical equivalence of Hamiltonians and path integrals, by itself, doesn't tell us anything about a "multiverse", however. Since the mid-20th century, many philosophers have come to analyse everyday talk about possibility in terms of "possible worlds" or "possible universes". However, those who invented that language (such as Carnap and Kripke) weren't claiming that all those possible worlds were just as actual as this one – that claim only arose later (David Lewis). Similarly, the originators of the path integral formulation of QM (Dirac and Feynman) weren't claiming that all those paths were "equally real". By contrast, the majority of proponents of "multiverse" theories, are claiming those other "universes" are ultimately just as real as this one, as as opposed to merely being some theoretical construct, a mathematical metaphor. You can understand these "paths" in two ways – either as just a useful mathematical tool for analysing past observations and experimental results, and predicting future ones, but passing no judgement on their ultimate ontological status; or, as a claim that in some sense, all these paths "really exist". And, if you adopt the later reading, you can then argue whether they are all "equally real", or maybe one of them is "actual" and the others "non-actual", or even just maybe that some are "more actual" than others. Nothing wrong with those debates – but when we start engaging in them, we are no longer doing physics proper, we are doing the philosophy of physics.
- evanb 4y ago
  Agree with your first paragraph and the first half of your second paragraph. But if you listen to Feynman talk about the path integral, he emphasizes that all the paths are real, are in fact critical to describing experimental results, and that it is simple to construct experiments to get those paths to coherently add to produce classically-unexpected results! I don't know about Dirac's "philosophy"; what you say of his stance may be correct. I don't think Feynman would say any path is actual or non-actual. It's not the case that ONE of them "really" happens and the others are a calculational trick. It's that the thing that happens is a sum of alternatives. The sum is real, full stop. If you leave out terms in the sum you get the wrong answer. Is that enough to mark them as "real"? THAT is philosophy. Certainly you may find another description for the same sum which does not discuss a sum over paths at all. Perhaps that means that the terms in the sum are not real? But we can all agree that the sum itself comes out right, whichever picture of the world you want.
  - skissane 4y ago
    > But if you listen to Feynman talk about the path integral, he emphasizes that all the paths are real, are in fact critical to describing experimental results, and that it is simple to construct experiments to get those paths to coherently add to produce classically-unexpected results! Okay, but is he claiming that the paths are "real" in the exact same sense as the actual experimental results are? We (in principle) add up an infinite number of paths, and that infinite sum produces a probability amplitude, which gives us a probability, and then we expect that if we repeat the experiment enough times, and sufficiently control the sources of experimental error, the observed probability should converge to the calculated one. Even if he says that all those paths are "real", I don't think he is claiming that they are "real" in quite the same sense as the actual experimental results are "real". Whereas, a proper "multiverse theory" would be claiming something like that.
    - evanb 4y ago
      OK, I think I appreciate the issue: “multiverse” is overloaded. When people talk about the many worlds of quantum mechanics, or alternate histories, they’re talking about the terms in the path integral sum. But multiverse can also mean, separately, at least two additional things. First, different bubble universes that may be a part of cosmology. Second, alternate quantum field theories (with different actions, or fundamental parameters, or different gauge groups, or whatever). The standard quantum-mechanical path integral is a sum over histories / a many-world theory in the sense that the different histories / parts of the wave function can be detected via interference experiments. Other meanings of multiverse are speculative (though maybe a cosmologist will come and tell me that bubble universes are a direct prediction of inflationary cosmology and are not optional either).

by evanb 4y ago

ImHereToVote 4y ago

Can you do that for two systems that interfere? I think that is where the multiverse stuff comes from.

evanb 4y ago
Well, it depends on what you mean by "two systems". If they are described by one Hamiltonian the result is one path integral. In a relativistic quantum field theory, for example, one may describe systems of any number of particles. The path integral captures all the interference effects.

skissane 4y ago

The mathematical equivalence of Hamiltonians and path integrals, by itself, doesn't tell us anything about a "multiverse", however. Since the mid-20th century, many philosophers have come to analyse everyday talk about possibility in terms of "possible worlds" or "possible universes". However, those who invented that language (such as Carnap and Kripke) weren't claiming that all those possible worlds were just as actual as this one – that claim only arose later (David Lewis). Similarly, the originators of the path integral formulation of QM (Dirac and Feynman) weren't claiming that all those paths were "equally real". By contrast, the majority of proponents of "multiverse" theories, are claiming those other "universes" are ultimately just as real as this one, as as opposed to merely being some theoretical construct, a mathematical metaphor. You can understand these "paths" in two ways – either as just a useful mathematical tool for analysing past observations and experimental results, and predicting future ones, but passing no judgement on their ultimate ontological status; or, as a claim that in some sense, all these paths "really exist". And, if you adopt the later reading, you can then argue whether they are all "equally real", or maybe one of them is "actual" and the others "non-actual", or even just maybe that some are "more actual" than others. Nothing wrong with those debates – but when we start engaging in them, we are no longer doing physics proper, we are doing the philosophy of physics.

evanb 4y ago
Agree with your first paragraph and the first half of your second paragraph. But if you listen to Feynman talk about the path integral, he emphasizes that all the paths are real, are in fact critical to describing experimental results, and that it is simple to construct experiments to get those paths to coherently add to produce classically-unexpected results! I don't know about Dirac's "philosophy"; what you say of his stance may be correct. I don't think Feynman would say any path is actual or non-actual. It's not the case that ONE of them "really" happens and the others are a calculational trick. It's that the thing that happens is a sum of alternatives. The sum is real, full stop. If you leave out terms in the sum you get the wrong answer. Is that enough to mark them as "real"? THAT is philosophy. Certainly you may find another description for the same sum which does not discuss a sum over paths at all. Perhaps that means that the terms in the sum are not real? But we can all agree that the sum itself comes out right, whichever picture of the world you want.
- skissane 4y ago
  > But if you listen to Feynman talk about the path integral, he emphasizes that all the paths are real, are in fact critical to describing experimental results, and that it is simple to construct experiments to get those paths to coherently add to produce classically-unexpected results! Okay, but is he claiming that the paths are "real" in the exact same sense as the actual experimental results are? We (in principle) add up an infinite number of paths, and that infinite sum produces a probability amplitude, which gives us a probability, and then we expect that if we repeat the experiment enough times, and sufficiently control the sources of experimental error, the observed probability should converge to the calculated one. Even if he says that all those paths are "real", I don't think he is claiming that they are "real" in quite the same sense as the actual experimental results are "real". Whereas, a proper "multiverse theory" would be claiming something like that.
  - evanb 4y ago
    OK, I think I appreciate the issue: “multiverse” is overloaded. When people talk about the many worlds of quantum mechanics, or alternate histories, they’re talking about the terms in the path integral sum. But multiverse can also mean, separately, at least two additional things. First, different bubble universes that may be a part of cosmology. Second, alternate quantum field theories (with different actions, or fundamental parameters, or different gauge groups, or whatever). The standard quantum-mechanical path integral is a sum over histories / a many-world theory in the sense that the different histories / parts of the wave function can be detected via interference experiments. Other meanings of multiverse are speculative (though maybe a cosmologist will come and tell me that bubble universes are a direct prediction of inflationary cosmology and are not optional either).