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No Individual Particles
ever_a_student20

A single point in quantum configuration space, is the product of multiple point positions per quantum field ...

An attempt at translation: each point in the quantum configuration space corresponds to a particular configuration of every field (e.g. electromagnetic, gluon, electron etc.), and the regularities in these fields tell us how many particles of each species do we have and where they are. Is this even approximately correct?

So... how many dimensions does the real quantum configuration space then have? Uncountably infinite?

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1Luke_A_Somers
Yes, that is correct. The fields that physicists actually use (Fock spaces or Hilbert spaces) are considerably more general than the configuration spaces Eliezer has been using. In Eliezer's case, each position in configuration space corresponds to a particular number of point excitations in each field, and you need to integrate these to reach a full quantum configuration. A single point in the Hilbert/Fock spaces that physicists use correspond to a complete quantum state; Eliezer's configuration spaces are each just a choice of basis vectors in that space. When you add them up to get the full quantum state, it's adding up the basis vectors to get the full vector. That's how you get from a full distribution in Eliezer's configuration spaces corresponding to a single point in a Hilbert/Fock space.
No Individual Particles
ever_a_student10

In quantum physics, the configuration space is the fundamental thing, and you get the appearance of an individual particle when the amplitude distribution factorizes...

Um, if individual particles are derived from the amplitude distribution on the configuration space, and the dimension of that space is related to the number of those particles, how do we then know how many dimensions should a particular configuration space have?

Concretely, how did we know that we have to draw a 2-d diagram (+2 dimensions for the amplitudes) up there? One spatial dimensio... (read more)

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1Luke_A_Somers
A full diagram in that style would be 1 dimension of graph for each dimension of space for each particle that happens to exist, and you have a separate diagram for each different combination of different numbers of particles. Obviously, he's simplified it, so the answer to your question is just, "That's how he chose to reduce it".