# Fermions

Fermions

Consider a multiway graph representing the time evolution of some quantum states.

#### Case 1: the edges merge (bosons)

Case 1: the edges merge (bosons)

I.e. the path weights add, and you get a single quantum state out of these multiple inputs

#### Case 2: the edges don’t merge (fermions)

Case 2: the edges don’t merge (fermions)

If it is a pure tree, you have to go back to the beginning of the universe to get a merger: i.e. the states will be maximally separated in branchial space : i.e. they will have opposite phases

And these states can never get together: i.e. the “electrons” can never wind up in the same state.

And these states can never get together: i.e. the “electrons” can never wind up in the same state.

#### Essentially: the branchtime worldlines of different fermions do not converge, but they do for bosons.....

Essentially: the branchtime worldlines of different fermions do not converge, but they do for bosons.....

#### Total antisymmetry with n fermions: everything has to be at an opposite corner of branchial space

Total antisymmetry with n fermions: everything has to be at an opposite corner of branchial space

KaryTree[31]

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Minimal case: n independent paths.

Why exactly is it antisymmetrizing? I.e. are taking the various path weights, and combining

Array[p,5]

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{p[1],p[2],p[3],p[4],p[5]}

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Array[p,2]

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{p[1],p[2]}

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ResourceFunction["MultiwaySystem"][{"A""AB"},"AA",5,"BranchialGraph"]

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ResourceFunction["MultiwaySystem"][{"B""BA"},"BB",5,"BranchialGraph"]

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ResourceFunction["MultiwaySystem"][{"B""BA","A""AB"},{"AA","BB"},3,"BranchialGraph"]

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{mwGraph1=ResourceFunction["MultiwaySystem"]["A""AB","AA",8,"StatesGraphStructure"],mwGraph2=ResourceFunction["MultiwaySystem"]["A""BA","AA",8,"StatesGraphStructure"]}

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,

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{branchialGraph1=ResourceFunction["MultiwaySystem"]["A""AB","AA",8,"BranchialGraphStructure"],branchialGraph2=ResourceFunction["MultiwaySystem"]["A""BA","AA",8,"BranchialGraphStructure"]}

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{,}

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rulialGraph=ResourceFunction["MultiwaySystem"][{"A""AB","A""BA"},"AA",8,"StatesGraphStructure"]

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HighlightGraph[rulialGraph,{mwGraph1,mwGraph2}]

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rulialBranchialGraph=ResourceFunction["MultiwaySystem"][{"A""AB","A""BA"},"AA",8,"BranchialGraphStructure"]

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Potential claim: the fermion wave function is very “puffy” and fills out a volume that is essentially LeviCivita * basis vectors

Whereas the boson wave function is “branchially” just a point...

Whereas the boson wave function is “branchially” just a point...

## The next step: spin-statistics / spinors

The next step: spin-statistics / spinors

When projected to make a “spatial slice”, the boson case will be “two way” and the fermion case not....

## Next up: spin / spin quantization

Next up: spin / spin quantization

## Basic concept of fermions

Basic concept of fermions

[[Fermions exist because branchial space does not contract to a point. In other words, fermions “maintain quantum mechanics”. Analogous to the maintenance of space. ]]