navstokgap

The tangent groupoid, its zoom, and whole trajectories

Markdown source · PDF

Result, 2026-09-27 (user’s question: what Connes’s tangent groupoid adds, beyond the user’s 1998 and 2003 readings, and whether it extends to a whole trajectory). Three statements.

  1. The zoom is the renormalization. Debord and Skandalis’s action of \(\mathbb R_+^*\) on the tangent groupoid, \((x,y,\varepsilon)\mapsto(x,y,\lambda^{-1}\varepsilon)\) and \((x,v,0)\mapsto(x,\lambda v,0)\), is the dilation family of the fifth-postulate note (§4, Theorem D) and the control transformation \(h\mapsto\mu h\) of the user’s 1998 note. By van Erp and Yuncken, pseudodifferential operators are exactly the distributions on the tangent groupoid that are homogeneous under the zoom up to smoothing terms; their restriction to \(\varepsilon=0\) is the principal symbol. What survives the passage \(\varepsilon\to0\) is therefore the principal symbol, a function of place and velocity together.
  2. Trajectories live on a time-graded groupoid. The tangent groupoid composes chords at one fixed \(\varepsilon\), and at \(\varepsilon=0\) it composes only vectors at one point. Refining a trajectory into more pieces is an operation on the groupoid \((x,y,u)\circ(y,z,v)=(x,z,u+v)\) that the user introduced in 2003 as the “elementary school” groupoid. Refinement consistency is a morphism property on it: classically \(\min_y[S(x,y,r)+S(y,z,u-r)]=S(x,z,u)\), satisfied by Hamilton’s principal function, the exact discrete Lagrangian of discrete mechanics, and for Galileo’s constant force by the corrected action \(\ell_h\) of the refinement note, eq. (2) (Proposition 1); quantum mechanically the convolution law \(K(u+v)=K(u)*K(v)\). The first is the \(\hbar\to0\) tropical shadow of the second (Maslov dequantization).
  3. Along a quantum trajectory the ultimate velocity fails. Quantum paths have Hausdorff dimension two: \(\langle\Delta x^2\rangle\sim\hbar\tau/M\) over a step \(\tau\), so \(\Delta x/\tau\) diverges like \(\tau^{-1/2}\). The tangent groupoid’s \(\varepsilon\to0\) fibre, where \((y-x)/\varepsilon\to v\), is reached by ballistic scaling; quantum trajectories need the diffusive scaling \(\Delta x^2\propto\tau\), the user’s 1998 remark that the path groupoid relates to Connes’s only after \((x,y,\varepsilon)\mapsto(x,y,\varepsilon^2)\), and his 2003 remark on the \(\sqrt t\) of Itô’s calculus. Newton’s velocitas ultima is the ballistic boundary fibre, which quantum paths do not reach.

The constructions are established; the identifications are readings, and Proposition 1 is elementary. The note gives the atlas’s Newton row a geometric home: one step is an arrow of the tangent groupoid, a trajectory is a factorization in the time-graded groupoid, the zoom is the renormalization, and the principal symbol is what survives.

1. Connes’s construction

For a manifold \(M\), the tangent groupoid is \(\mathbb G_M=(M\times M\times(0,1])\cup(TM\times\{0\})\), with the pair-groupoid law at each \(\varepsilon>0\), the fibrewise vector addition \((x,X)\circ(x,Y)=(x,X+Y)\) at \(\varepsilon=0\), and the topology in which \((x_n,y_n,\varepsilon_n)\to(x,v,0)\) iff \(x_n\to x\) and \((y_n-x_n)/\varepsilon_n\to v\) (in a chart). Its \(C^*\)-algebra is a continuous field whose fibre at \(\varepsilon>0\) is the compact operators on \(L^2(M)\) and at \(\varepsilon=0\) is \(C_0(T^*M)\) by Fourier transform; Connes used the deformation to prove the index theorem (A. Connes, Noncommutative Geometry, Academic Press 1994, §II.5, metadata), and the construction generalizes the Moyal rule (Cariñena, Clemente-Gallardo, Follana, Gracia-Bondía, Rivero and Várilly 1999, abstract; Landsman 2002, arXiv:math-ph/0208004, abstract). The \(\varepsilon=0\) fibre is where a chord has become a velocity at a place: Newton’s velocitas ultima, “certain and definite”, with place and velocity commuting. The fibre at \(\varepsilon>0\) is noncommutative.

2. The zoom action and what survives

Debord and Skandalis (2014) (metadata) define the action of \(\mathbb R_+^*\) on the adiabatic (tangent) groupoid displayed in the summary and identify the crossed product with an ideal of pseudodifferential operators; van Erp and Yuncken (2019) (metadata) characterize pseudodifferential operators as the distributions on the tangent groupoid, properly supported and transversal, that are homogeneous under the zoom modulo smooth ones. Three identifications follow (readings):

3. Whole trajectories: the time-graded groupoid

The user’s 2003 note (Rivero, “Flashes of noncommutativity”, arXiv:math/0302285, full read) introduces the groupoid over configuration space with arrows \((x,y,u)\), \(u\) an elapsed time, and law \((x,y,u)\circ(y,z,v)=(x,z,u+v)\). Its convolution algebra has product \((AB)(x,z,t)=\int\!\!\int A(x,y,r)\,B(y,z,t-r)\,dy\,dr\); Fourier transformation in \(t\) gives, with \(\varepsilon\sim1/\hat t\), the product of the \(\varepsilon>0\) part of the tangent groupoid, so the latter’s algebra sits inside the former’s (the 2003 note’s second flash). Refining a trajectory is factoring an arrow \((x,z,u)\) through intermediate points and times, which is Newton’s insertion of \(t_{2.6}\).

Proposition 1. (a) A function \(S(x,y,u)\) satisfies \(\min_y[S(x,y,r)+S(y,z,u-r)]=S(x,z,u)\) for all \(0<r<u\) exactly when it is a morphism of the time-graded groupoid into the \((\min,+)\) semiring. For Galileo’s constant force \(F\) on a mass \(M\) this holds for \[S(x,y,u)=\frac{M(y-x)^2}{2u}+\frac{Fu}2(x+y)-\frac{F^2u^3}{24M},\] the corrected action \(\ell_u\) of the refinement note, eq. (2), which is the action along the true path. (b) A kernel \(K(x,y,u)\) satisfies \(\int K(x,y,r)K(y,z,u-r)\,dy=K(x,z,u)\) exactly when it is a morphism into the convolution algebra; the constant-force propagator \(U_u\) of the refinement note, eq. (3), does.

Proof. (a) and (b) restate the morphism property. The displayed \(S\) is the classical action of the path \(q(s)=x+(y-x)s/u+\frac F{2M}s(s-u)\), which is the refinement note’s \(\ell_u\); its composition law is eq. (2) there, and \(U_u\)’s is eq. (3). \(\square\)

The displayed \(S\) is the exact discrete Lagrangian of discrete mechanics (Marsden and West 2001, metadata), and discrete Lagrangian mechanics on groupoids is Weinstein’s (1996) (metadata). The kick–drift–kick cell \(S_h=K_{Fh/2}D_hK_{Fh/2}\) of the refinement note is the variational (Störmer–Verlet) integrator of the trapezoidal discrete Lagrangian \(L_h\), and the cubic counterterm turns \(L_h\) into the exact one. The two laws of Proposition 1 are related by Maslov dequantization: in Euclidean form, \(-\hbar\log\int e^{-S/\hbar}\to\min S\) as \(\hbar\to0\) (Litvinov 2005, metadata). So refinement consistency of a whole trajectory is the morphism property, tropical in classical mechanics and convolutional in quantum mechanics.

4. Quantum trajectories miss the ultimate velocity

In a path integral the typical increment over a step \(\tau\) has \(\langle\Delta x^2\rangle\sim\hbar\tau/M\), so the paths have Hausdorff dimension two (Abbott and Wise 1981, metadata), and the difference quotient \(\Delta x/\tau\sim(\hbar/M\tau)^{1/2}\) diverges. The tangent groupoid reaches its \(\varepsilon=0\) fibre along ballistic sequences, \((y-x)/\varepsilon\to v\). Quantum trajectories approach along diffusive ones, \(\Delta x^2\propto\tau\): the user’s 1998 remark that Connes’s groupoid relates to the groupoid of paths only after the rescaling \((x,y,\varepsilon)\mapsto(x,y,\varepsilon^2)\), and his 2003 remark on the \(\sqrt t\) of Itô’s calculus, are this fact. The dimensionless ratio \(M\,\Delta x^2/(\hbar\tau)\) is the action per step in units of \(\hbar\): ballistic refinement sends it to zero (the classical limit, Newton’s ultimate velocity), diffusive refinement holds it fixed (the dimension ladder: refining makes each Newtonian cell quantum). The failure of joint determinacy of the fifth-postulate note is, along a trajectory, the failure of the ballistic limit.

5. Consequence for STATE

The atlas’s Newton row gains a geometric home: one step is an arrow of the tangent groupoid, a trajectory is a factorization in the time-graded groupoid, refinement consistency is the morphism property of Proposition 1, the zoom action is the renormalization, and the principal symbol is what survives \(\varepsilon\to0\). A natural next question for the gauge atlas is the corresponding groupoid of the lattice: the holonomy groupoid for two dimensions, where series moves are compositions, and a higher (double) groupoid for the parallel moves of three and four dimensions.