I argue that counting branches is not well-behaved with the Hilbert space structure and unitary time evolution, and instead assigning a measure to branches (the ‘dilution’ argument) is the proper way to handle this. (See Wallace’s decision-theory ‘proof’ of the Born rule for more).
The quantum state is a vector in a Hilbert space. Hilbert spaces have an inner product structure. That inner product structure is important for a lot of derivations/proofs of the Born rule, but in particular the inner product induces a norm. Norms let us do a lot of things. One of the more important things is we can define continuous functions. The short version is, for a continuous function, arbitrarily small changes to the input should produce arbitrarily small changes to the output. Another thing commonly used for vector spaces is linear operators, which are a kind of function that maps vectors to other vectors in a way that respects scalar multiplication and vector addition. We can combine the notion of continuous functions with linear operators and we get bounded linear operators.
While quantum mechanics contains a lot of unbounded operators representing observables (position, momentum, energy, etc.), bounded operators are still important. In particular, projection operators are bounded, and every self-adjoint operator, whether bounded or unbounded, has projection-valued measures. Projection-valued measures go hand-in-hand with the Born rule, and they are used to give the probability of a measurement falling on some set of values. There’s an analogy with probability distributions. Sampling from an arbitrary distribution can in principle give an arbitrarily large number, and many distributions even lack a finite average. However, the probability of a sample from an arbitrary distribution falling in the interval [a,b] will always be a number between 0 and 1.
If we are careful to ask only about probabilities instead of averages, or even just to only ask about averages when the quantity is bounded, we can do practically everything in quantum mechanics with bounded linear operators. The expectation values of bounded linear operators are continuous functions of the quantum state. And so now we get to the core issue: arbitrarily small changes to the quantum state produce arbitrarily small changes to the expectation value of any bounded operator, and in particular to any Born rule probability.
So what about branch counting? Let’s assume for sake of discussion that we have a preferred basis for counting in, which is its own can of worms. For a toy model, if we have a vector like (1, 0, 0, 0, 0, 0, ….) that we count as having 1 branch and a vector like (1, x, x, x, 0, 0, ….) that we’re going to count as 4 branches if x is an arbitrarily small but nonzero number, this branch counting is not a continuous function of the state. If you don’t know the state with infinite precision, you can’t distinguish whether a coefficient is actually zero or just some really small positive number. Thus, you can’t actually practically count the branches: there might be 1, there might be 4, there might be an infinite number of branches. On the other hand, the Born rule measure changes continuously with any small change to the state, so knowing the state with finite precision also gives finite precision on any Born rule measure.
In short, arbitrarily small changes to the quantum state can result in arbitrarily large changes to branch counting.
I argue that counting branches is not well-behaved with the Hilbert space structure and unitary time evolution, and instead assigning a measure to branches (the ‘dilution’ argument) is the proper way to handle this. (See Wallace’s decision-theory ‘proof’ of the Born rule for more).
The quantum state is a vector in a Hilbert space. Hilbert spaces have an inner product structure. That inner product structure is important for a lot of derivations/proofs of the Born rule, but in particular the inner product induces a norm. Norms let us do a lot of things. One of the more important things is we can define continuous functions. The short version is, for a continuous function, arbitrarily small changes to the input should produce arbitrarily small changes to the output. Another thing commonly used for vector spaces is linear operators, which are a kind of function that maps vectors to other vectors in a way that respects scalar multiplication and vector addition. We can combine the notion of continuous functions with linear operators and we get bounded linear operators.
While quantum mechanics contains a lot of unbounded operators representing observables (position, momentum, energy, etc.), bounded operators are still important. In particular, projection operators are bounded, and every self-adjoint operator, whether bounded or unbounded, has projection-valued measures. Projection-valued measures go hand-in-hand with the Born rule, and they are used to give the probability of a measurement falling on some set of values. There’s an analogy with probability distributions. Sampling from an arbitrary distribution can in principle give an arbitrarily large number, and many distributions even lack a finite average. However, the probability of a sample from an arbitrary distribution falling in the interval [a,b] will always be a number between 0 and 1.
If we are careful to ask only about probabilities instead of averages, or even just to only ask about averages when the quantity is bounded, we can do practically everything in quantum mechanics with bounded linear operators. The expectation values of bounded linear operators are continuous functions of the quantum state. And so now we get to the core issue: arbitrarily small changes to the quantum state produce arbitrarily small changes to the expectation value of any bounded operator, and in particular to any Born rule probability.
So what about branch counting? Let’s assume for sake of discussion that we have a preferred basis for counting in, which is its own can of worms. For a toy model, if we have a vector like (1, 0, 0, 0, 0, 0, ….) that we count as having 1 branch and a vector like (1, x, x, x, 0, 0, ….) that we’re going to count as 4 branches if x is an arbitrarily small but nonzero number, this branch counting is not a continuous function of the state. If you don’t know the state with infinite precision, you can’t distinguish whether a coefficient is actually zero or just some really small positive number. Thus, you can’t actually practically count the branches: there might be 1, there might be 4, there might be an infinite number of branches. On the other hand, the Born rule measure changes continuously with any small change to the state, so knowing the state with finite precision also gives finite precision on any Born rule measure.
In short, arbitrarily small changes to the quantum state can result in arbitrarily large changes to branch counting.