Isospectral deformations of the Dirac operator (original) (raw)

We give more details about an integrable system [26] in which the Dirac operator D = d + d * on a graph G or manifold M is deformed using a Hamiltonian system D = [B, h(D)] with B = d − d * + βib. The deformed operator D(t) = d(t) + b(t) + d(t) * defines a new exterior derivative d(t) and a new Dirac operator C(t) = d(t) + d(t) * and Laplacian M (t) = C(t) 2 and so a new distance on G or a new metric on M. For β = 0, the operator D(t) stays real for all t. While L = M (t) + V (t) does not change, the new Laplacian M (t) = C(t) 2 and the emerging potential V (t) = b 2 do evolve. The operators M, V are always real and commute. The cohomology defined by the deformed exterior derivative d(t) is the same as for d = d(0) as we can explicitly deform cocycles and coboundaries. The new Dirac operator C(t) defines a new metric, so that the isospectral flow is an evolution which deforms the geometry as defined by zero forms. If U = BU is the associated unitary, then the McKean-Singer formula str(U(t)) = χ(G) still holds. While super partners f, D(t)f span the same plane at all times, observable super symmetry fades: if f is an eigenvector of L which is a fermion-an eigen-2k + 1-form of L for some integer k-then D(t)f is only bosonic at t = 0 and the angle between the fermionic subspace and D(t)f goes to zero exponentially fast. The coordinate system has changed so that the original superpartner D(0)f is now far away for the new geometry. The linear relativistic wave equation u(t) = −Lu(t) and its solution u(t) = cos(Dt)u(0) + sin(D(t)t)D −1 u (0) = e iD (u(0) − iD −1 u (0) with fixed D is not affected by the symmetry since only L and not D enters in the solution formula. But the preperation of the initial velocity, the nonlinear solution u(t) = cos(D(t)t) + sin(D(t)t)D(0) −1 u (0) of the wave equation with time dependent D or the unitary evolution U (t) defined by the deformation depends on D. The evolution has a geometric effect: with d(t) as a new exterior derivative, the property d(t) → 0 for |t| → ∞ implies that space expands, with a fast inflationary start. The inflation rate can be tuned by scaling D. Instead of solutions to the wave equation, the nonlinear evolution has more soliton-and particle-like solutions which feature interaction with adjacent forms. In the limit t → ±∞, the operator D becomes block diagonal b + = −b − with b 2 ± = L, leading to linear solutions cos(bt)u(0)+sin(bt)b −1 u (0) of the wave equation which leaves each space Ωp of p-forms invariant. For G = K 2 , explicit formulas illustrate the inflation. We also look at the circle case, where already the 0-form and 1-form spaces can be joint by solutions of the wave equation. The nonlinear Dirac wave equation uses the entire geometric space but asymptotically, we get linear wave equation which preserve each p-form subspace and which is relativistic quantum mechanics and classical Riemannian geometry. Geometry alone can lead to interesting nonlinear wave dynamics, the emergence of new dimensions, complex structures, inflation and to a geometric toy model featuring Riemannian or graph geometry in which super symmetry is present but difficult to measure.