PIATRA . INSTITUTE

bordism to action

comparing classical mechanics with fully local TQFT

Classical Inclined Plane

forces & snapshots

pos: 0.0 m
vel: 0.0 m/s

a = g \sin\theta - \mu\, g \cos\theta

a = 4.05 m/s²

Euler integration of the differential equation yields the trajectory.

Fully Local TQFT

shapes & categories

Z = 1.000 + 0.000i

spin j0.50

Z(S³)0.5000

braid phase0.000 rad

|amplitude|1.0000

target category:

\text{Rep}(U_q(\mathfrak{sl}_2)),\; q = e^{2\pi i / 4}

The Path

Classical: A specific trajectory x(t) that minimizes action.

TQFT: A worldline shape. Only the topology (braids) matters.

Locality

Classical: Differential — look at t+dt.

TQFT: Fully local — cut spacetime into points, total = product of values.

The Answer

Classical: A number with units (meters, m/s).

TQFT: A dimensionless complex number (amplitude).

Classical Mechanics

On the left side, a block slides down an inclined plane under Newton's second law. The state of the world is defined by a snapshot: if you know the position and velocity right now, you can predict the next millisecond.

F = ma = mg\sin\theta - \mu\, mg\cos\theta

The simulation loop integrates this differential equation frame by frame, producing a specific trajectory through space.

Chern-Simons Theory

On the right side, we stop looking at snapshots and start looking at bordisms — the “shape” of time. In TQFT, we don't care about velocity at a given moment. We care about the worldline — the 1D string a particle leaves behind in 3D space.

Z(S^3) = \sqrt{\frac{2}{k+2}}\;\sin\frac{\pi}{k+2}

Every time two worldlines cross, the universe picks up a complex phase determined by the R-matrix. The conformal weight $h = \frac{3/4}{k+2}$ governs the braid eigenvalue:

\text{amplitude} = e^{2\pi i\, h \cdot \text{braids}}

The Rosetta Stone

The playground is a mathematical Rosetta Stone. It translates between the language of Calculus (how things move through space) and Category Theory (how things are connected).

Variable	Classical	TQFT
Angle / Level (k)	Changes the force of gravity	Resolution of the quantum space
Mass / Spin (j)	Inertia against air drag	Representation dimension; scales conformal weight
Friction / Braids	A force that drains energy	A topological twist that rotates the quantum state
Trajectory / Bordism	The line the block must follow	The shape of the spacetime container

Notes

This is a toy model. The classical side is a faithful Euler integration; the TQFT side is a simplified illustration of Chern-Simons invariants.
Inspired by Dan Freed's lectures on fully extended topological quantum field theories and the cobordism hypothesis.
Setting $k = 24$ is significant: this is the level where certain anomalies cancel, allowing the theory to be defined on simpler types of manifolds.
The R-matrix in this playground is the mathematical version of a quantum gate — braiding anyons to create logic gates is the basis of topological quantum computing.