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entropy1

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In summary: No. The infinite dimensions in the Hilbert space are already there before the measurement, and the wave function is a function on the same Hilbert space after the measurement as before. In the MWI, "measurement" is just another unitary evolution process, and unitary evolution doesn't "create" anything; it's an information-preserving reversible process.

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entropy1

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You're not even wrong!entropy1 said:

Your post mixes up measurement outcomes with states. States are vectors and may be orthogonal to each other. Measurement outcomes are real numbers.

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entropy1

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Yes, the outcomes are eigenvalues and those are paired with eigenvectors, so you can map the measurement outcome (eigenvalue) onto the eigenvector on which the wavefunction collapsed, right? I mean to indicate that the math is already implicating a single outcome, except if you assume all outcomes simultaneously, like in MWI.PeroK said:Your post mixes up measurement outcomes with states. States are vectors and may be orthogonal to each other. Measurement outcomes are real numbers.

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The maths doesn't impose a single outcome, in the sense that we chose that mathematics because we only see a single outcome.entropy1 said:Yes, the outcomes are eigenvalues and those are paired with eigenvectors, so you can map the measurement outcome (eigenvalue) onto the eigenvector on which the wavefunction collapsed, right? I mean to indicate that the math is already imposing a single outcome, except if you assume all outcomes simultaneously, like in MWI.

In general the wave function collapses onto an eigenspace, which may be spanned by several orthogonal eigenstates. So, although we get one result, we don't necessarily have a 1D eigenstate as the result of a single measurement.

Only by observing all relevant compatible observables do you get a definite eigenvector.

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entropy1

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So I think you are saying that we don't necessarily have collapse in the math, which would suit for instance MWI, am I right?PeroK said:The maths doesn't impose a single outcome, in the sense that we chose that mathematics because we only see a single outcome.

In general the wave function collapses onto an eigenspace, which may be spanned by several orthogonal eigenstates. So, although we get one result, we don't necessarily have a 1D eigenstate as the result of a single measurement.

Only by observing all relevant compatible observables do you get a definite eigenvector.

Is it just the way you look at it?

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WWGD

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I assume you mean infinite-dimensional? I don't think there are cardinality-wise finite Hilbert spaces.entropy1 said:

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Yes, collapse is Copenhagen, for example. In many ways it's the simplest interpretation.entropy1 said:So I think you are saying that we don't necessarily have collapse in the math, which would suit for instance MWI, am I right?

Collapse is part of the interpretation of the maths.

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entropy1

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PeterDonis

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entropy1 said:I think you are saying that we don't necessarily have collapse in the math

We have a thing in the math that is called "collapse", but the math itself, the machinery that calculates predictions for experimental results, doesn't tell you whether this "collapse" is a real physical process or just a mathematical technique that let's you make correct predictions.

The mathematical technique that is sometimes called "collapse" is step 7 in the following Insights article:

https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/

PeroK said:Collapse is part of the interpretation of the maths.

In the sense that "collapse" refers to something more than just the mathematical technique that let's you make correct predictions, yes. See above.

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PeterDonis

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Moderator's note: Moved thread to QM interpretations and foundations forum.

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PeterDonis

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entropy1 said:

No. The infinite dimensions in the Hilbert space are already there before the measurement, and the wave function is a function on the same Hilbert space after the measurement as before. In the MWI, "measurement" is just another unitary evolution process, and unitary evolution doesn't "create" anything; it's an information-preserving reversible process.

Orthogonal eigenvectors are important in measurement because they represent the directions along which a system's properties are independent of each other. This means that measurements along these directions will not affect each other, allowing for more accurate and reliable data.

In quantum mechanics, orthogonal eigenvectors are used to represent the possible states of a quantum system. These states are then used to calculate the probabilities of different outcomes when the system is measured.

No, by definition, orthogonal eigenvectors must be perpendicular to each other. If they were not, then they would not be considered orthogonal.

Orthogonal eigenvectors can be determined experimentally by performing measurements in different directions and observing the results. If the measurements are independent of each other, then the corresponding eigenvectors are orthogonal.

Orthogonal eigenvectors and eigenvalues are closely related. The eigenvalues represent the magnitude of the change in the system's properties when measured along the corresponding eigenvector. Orthogonal eigenvectors ensure that these changes are independent of each other.

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