By Abhay Ashtekar
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21) < dζ± dζ± >= ±(D + D∗ )dt; 2mD = = α + iβ The complex time derivative operator becomes (δ/dt) = ∂t +v·∇−(i/2)(D+D∗ )∇2 . Writing again ψ = exp[iS/2mD] = exp(iS/ ) one obtains v = −2iD∇log(ψ). The NLSE is then obtained (via the Newton law) via the relation −∇U = m(δ/dt)v = −2imD(δ/dt)∇log(ψ). 22) ∇U = 2im[D∂t ∇log(ψ) − 2iD2 (∇log(ψ) · ∇)(∇log(ψ)− i − (D + D∗ )D∇2 (∇log(ψ)] 2 Now using the identities (i) ∇∇2 = ∇2 ∇, (ii) 2(∇log(ψ)·∇)(∇log(ψ) = ∇(∇log(ψ))2 and (iii) ∇2 log(ψ) = ∇2 ψ/ψ − (∇log(ψ))2 leads to a NLSE with nonlinear (kinematic pressure) potential, namely 2 i ∂t ψ = − α 2 β (∇log(ψ))2 ψ 2m 2m Note the crucial minus sign in front of the kinematic pressure term and also that = α + iβ = 2mD is complex.
1. THE KLEIN-GORDON AND DIRAC EQUATIONS Before embarking on further discussion of QM it is necessary to describe some aspects of quantum ﬁeld theory (QFT) and in particular to give some foundation for the Klein-Gordon (KG) and Dirac equations. For QFT we rely on [120, 457, 528, 764, 827, 1015] and concentrate on aspects of general quantum theory that are expressed through such equations. We alternate between signature (−, +, +, +) and (+, −, −, −) in Minkowski space, depending on the source. It is hard to avoid using units = c = 1 when sketching theoretical matters (which is personally repugnant) but we will set = c = 1 and shift to the general notation whenever any real meaning is desired.
Thus Ω consists of one open square of size 1/3, 2 open squares of size 1/9, 4 open squares of size 1/27, etc. (see  for pictures and explanations). Then the spectral zeta function for the Dirichlet Laplacian on the square ∞ is ζB (s) = n1 ,n2 =1 (n21 + n22 )s/2 and the spectral zeta function of the spray is ζν (s) = ζCS (s) · ζB (s). Now E∞ is composed of an inﬁnite hierarchy of sets E(j) with dimension (1 + φ)j−1 = 1/φj−1 (j = 0, ±1, ±2, · · · ) and these sets correspond 3. REMARKS ON FRACTAL SPACETIME 27 to a special case of boundaries ∂Ω for fractal sprays Ω whose scaling ratios are j−1 suitable binary powers of 2−φ .
2D Quantum Gravity and SC at high Tc by Abhay Ashtekar