# Do conservation laws contradict quantum mechanics? [closed]

Take for example the double-slit experiment interpreted in the Copenhagen sense.

The particle leaves as an object with mass, yet passes through the slits as a massless wave, only to collapse again as a particle. We can consider this example as a generalisation of the principle of anti-realism posited by Bohr. Where does the energy "go" when the particle begins acting as a wave, and from where does it "come" when it reverts to acting as a particle?

Here's another example, this time assuming many-worlds.

Say two possible future worlds de-cohere from our world. These two-worlds must be assumed to be equally real if we are to maintain determinism. Surely there is now twice as much energy within the larger system (two worlds rather than one). I understand that energy is conserved WITHIN each world respectively, but how is this relevant? The only resolution I can see is to assert that the number of possible future worlds at the moment of the big bang is >/= to the number of actual identical worlds at the moment of the big bang. But this implies that the number of different future worlds determines the number of identical past worlds, which contradicts linear causality?

• A particle isn't a massless wave. In circumstances where it behaves as a wave it behaves as a massive wave i.e. a wave that produces a mass flux. – John Rennie May 24 '16 at 18:02
• That's what I said isn't it? – Bonj May 24 '16 at 18:20
• You said, and I quote, passes through the slits as a massless wave. The particle passes through the slits as a massive wave. The mass does not disappear because the particle is behaving as a wave. – John Rennie May 24 '16 at 19:00
• You are giving a very good example of how badly "collapse of the wavefunction" messes up minds. For one thing, there are no particles in quantum mechanics, there are only quanta. Quanta are the results of the measurements we perform on quantum fields. They are not little balls that are flying around in space. Neither can quanta change into waves and back by what you think the collapse of the wavefunction means (it never meant that). Quanta are properties of the fields under strong measurements. Leave the measurements away, and you don't even have to talk about quanta. – CuriousOne May 24 '16 at 20:36
• Ah yes you are right. Thanks John, so the wave does have mass. That's what I was after. The reason this confused me is that the CI does not profess the ontological realness of the wave function. Thus, I assumed that it could not be representing a physically real object (in this case but not necessarily, one that has mass)? – Bonj May 25 '16 at 14:18

Short answer: no. I'll give some context with the details of the simplest examples.

In the context of conservation laws, "energy" refers to the Hamiltonian. In classical mechanics, a quantity without explicit time dependence is conserved iff its Poisson bracket with the Hamiltonian is 0. In quantum mechanics, quantities are promoted to operators on a Hilbert space. The mean of a quantity without explicit time dependence is then conserved iff that quantity’s operator commutes with the Hamiltonian. (Quantities that commute with the Hamiltonian are called “good quantum numbers”.) Classical conservation laws thereby naturally survive quantisation; in particular, energy is conserved in both theories.

In the many-worlds interpretation, each universe uses its own “copy” of the Hamiltonian that is unrelated to what may happen in other universes. From the perspective of a physicist in either daughter universe, that universe existed before the wavefunction collapse, and quantities that classical mechanics predicts will be conserved are conserved in the above senses. And the total energy content of the parent universe may well be 0 anyway, in which case duplicating that universe would conserve even a multiverse energy measure. (Ditto with all other conserved quantities. For example, to within experimental error the universe has zero electric charge,)

According to particle-wave duality, an entity that is in some experiments a “particle” of energy $E$ and momentum $\mathbf{p}$ is in other experiments a “wave” of circular frequency $\omega=\tfrac{E}{\hbar}$ and wavevector $\mathbf{k}=\tfrac{\mathbf{p}}{\hbar}$. In truth neither of these classical interpretations are the whole story. The plane wave $\exp \text{i}\left(\mathbf{k}\cdot\mathbf{x}-\omega t\right)$ with $\text{i}^2=-1$ motivates the operator promotions $E=\text{i}\hbar\partial_t,\,\mathbf{p}=-\text{i}\hbar\boldsymbol{\nabla}$. (The former is the quantum-mechanical Hamiltonian discussed above.) In Newtonian mechanics, $E\psi=\tfrac{p^2}{2m}\psi +V\psi$ obtains the Schrödinger equation; in special relativity, $E^2\psi-c^2p^2\psi=m_0^2c^4\psi$ obtains the Klein-Gordon equation. (Including a potential energy term in the latter takes a bit more work.) In either case, the classical relation between energy, momentum and mass and associated conservation laws carry over naturally to a wave equation, solved by the above plane wave in the simplest cases.

• Okay, so are you saying that the total energy of the parent universe must necessarily be 0 to conserve energy in a multiverse measurement? – Bonj May 24 '16 at 18:48
• @Bonj we can't measure these universes; it's one of many interpretations of quantum mechanics, not known physical fact. But if you somehow could measure energy in each, then tally them up, "total" energy would be conserved iff individual universes have energy 0. However, there's no reason why we "need" such unempirical conservation laws, but it is interesting the universe's energy is 0 to within experimental error (which would be expected if it came from nothing). The point is conservation laws follow from analysing Hamiltonians (or Lagrangians); energy conservation isn't an axiom. – J.G. May 25 '16 at 5:45

Take for example the double-slit experiment interpreted in the Copenhagen sense.

The particle leaves as an object with mass, yet passes through the slits as a massless wave, only to collapse again as a particle. We can consider this example as a generalisation of the principle of anti-realism posited by Bohr. Where does the energy "go" when the particle begins acting as a wave, and from where does it "come" when it reverts to acting as a particle?

My understanding is that in the Copenhagen interpretation, a system collapses into one of the eigenstates of the measured observable. As long as the relevant eigenstate has the same energy as the original, energy is conserved.

Here's another example, this time assuming many-worlds.

Say two possible future worlds de-cohere from our world. These two-worlds must be assumed to be equally real if we are to maintain determinism. Surely there is now twice as much energy within the larger system (two worlds rather than one). I understand that energy is conserved WITHIN each world respectively, but how is this relevant? The only resolution I can see is to assert that the number of possible future worlds at the moment of the big bang is >/= to the number of actual identical worlds at the moment of the big bang. But this implies that the number of different future worlds determines the number of identical past worlds, which contradicts linear causality?

In the Schrodinger picture the state may evolve like so $|a\rangle\to\sum\alpha_i|a_i\rangle)$. There are two quantities in this expression that can be measured. One is the eigenvalue $a_i$ of each of the possible states. As long as the Hamiltonian respects energy conservation, the energy of each of those states will be the same as that of $|a\rangle$. The other is the set of square amplitudes $|\alpha_i|^2$, which add up to 1: this is the only way of "counting worlds" that makes much sense if you're going to add up worlds at all:

This measure over the set of worlds doesn't change over time, so if you were going to add up energy across worlds, then the only physical way of doing it says that the energy is conserved.