Experiment 3: Scaling

4 Experiment 3: Scaling

Python: scaling_1.py, scaling_2.py

We scale to longer chains (L = 4,6,8,12,16) and more frames, testing whether momentum quantisation k = 2πm∕L holds and whether the boson properties depend on system size.

The key result — analytically exact

For any single fermion hop from site 0 to site d, regardless of L or d:

1 ψ = √--(e0 + ied). 2

The boson matrix has exactly two non-zero entries:

i i B0,d = + 2, Bd,0 = − 2.

For the nearest-neighbour hop (d = 1, L = 4) this reads explicitly:

( i ) 0 + 2 0 0 || − i 0 0 0|| B = || 2 || . ( 0 0 0 0) 0 0 0 0

The same 2×2 antisymmetric block

0	+i∕2
−i∕2	0

is embedded at positions (0,d) and (d,0) for any hop distance d and any chain length L; all other entries are zero. The matrix is always purely antisymmetric, always has eigenvalues ±1∕2, and the phase of B_0,d is always −π∕2, giving:

k ⋅d = − π = ⇒ k = − -π-. 2 2d

This is not a plane-wave momentum but a phase relationship between the two fermion sites: the boson encodes exactly one number, the π∕2 phase shift between where the fermion was and where it went. A quarter-turn in phase space is the defining property of a propagator. This is what a virtual photon does in QED.

Conclusion: The compression residual of a single fermion hop is an antisymmetric, purely imaginary, ±1∕2-eigenvalued object carrying a universal π∕2 phase, independent of chain length and hop distance.

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