By Volker Dietrich, Klaus Habetha, Gerhard Jank

Clifford Algebras is still a fast-growing self-discipline, with ever-increasing functions in lots of medical fields. This quantity includes the lectures given on the Fourth convention on Clifford Algebras and their purposes in Mathematical Physics, held at RWTH Aachen in may well 1996. The papers signify an first-class survey of the latest advancements round Clifford research and its purposes to theoretical physics. viewers: This booklet may still attract physicists and mathematicians operating in components related to capabilities of complicated variables, associative jewelry and algebras, indispensable transforms, operational calculus, partial differential equations, and the arithmetic of physics.

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27) ∇± f (t) = ∂f ✷ x ∂f ✷f = + + ✷t ∂t ✷t ∂x n 2 1 ∂j f a ,j (t) j j! ∂x j−1 + o( 1/n ) We sketch now the derivation of a SE in the spirit of Nottale but with more mathematical polish. 28) f + (t) = 1 2 t+ 1 2 t+ f (s)ds; t− f (s)ds; f − (t) = t 1 2 t f (s)ds t− One considers quantum paths a` la Feynman so that limt→t [X(t) − X(t )]2 /(t − t ) exists. 1 for example this means dimH X[a, b] = 1/2. 29) Q(t) = t; Q(x(t)) = X(t); Q(v(t)) = V(t); Q df dt =Q d dt · Q(f ) where Q(d/dt) = d/dt if Q(f )(t) is differentiable and Q(d/dt) = ✷ /✷t where (x, h) > > 0 if Q(f )(t) is nondifferentiable.

10) cn ξn (x)e−iEn t/ Ψt (x : N ) = 1 in the limit N → ∞ (cf. 5)) where the ξn (x) are eigenvectors with eigenvalues En of the corresponding Hamiltonian. g. [324, 325, 326, 327, 328, 329, 415, 416, 418, 419, 420] where one modern version of BM is being developed. 5. 2. A numerical example is given in [844] and we only mention a few features here. 4) and x1 = 0 with x2 = L. 12) Ψt (x) = √ e−iE1 t/ n π L n odd where λn,1 = (En −E1 )/ (in the numerical calculations one uses L = m = = 1). 11)). g. ) and graphs are displayed - we omit any further discussion here.

5) < d >= ∞ 1 (0) n(dc )n−1 ∞ (0) n 1 (dc ) = 1 (0) dc (1 (0) − dc ) (n) This is contrived of course (and cannot represent E(dc ) since one is computing (n) reciprocals (n/dc ) but we could think of computing an expected ignorance and identifying this with the reciprocal of dimension. 1. We note that the normalized probability N = Ω(1) /Z0 = (0) (0) (0) Ω(1) (1 − Ω(1) ) = 1/ < d > for any dc . Further if < d >= 4 = 1/dc (1 − dc ) one (0) (0) has dc = 1/2 while < n >= 3 < 4 =< d >. One sees also that dc = 1/2 is the (0) minimum (where d < d > /d(dc ) = 0).

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