Chaos Theory and Nonsqueezed-State Boson-Field Mechanics in Quantum Gravity and Statistical Optics Kevin Daley April 25, 2009
Abstract: Previous work, resulting in an entropy-based quantum field theory for electrodynamics and candidate quantum gravity formalism, provides exactly soluble methods basically incomplete without a similarly convenient optical theory. Our answer arises from the known logarithmic divergence of the entropy given a Spin-Boson Hamiltonian with a unitary coupling constant. This fact is used as a generalization of the maximum-entropy principle in an uneven gauge field. In such a manner we justify the Bayesian axiomatic quantization of optical phenomena. A straightforward numerical algorithm is obtained that converges to universally applicable, algebraic formula for the optical density as a function of space and excitation mode. One last result involves the resulting complex dynamics of turbulent continuous bodies. We propose that the acceleration associated with a given excitation mode obeys P n a 0 (z), P being equivalent to a frequency-normalized quadratic map so that the turbulent region is merely a sum of Julia sets. The correspondence limit is expressed as a series of gradual bifurcations of the logistic map. Keywords: statistical mechanics, quantum optics, maximum-entropy, quantum gravity, axiomatic quantization, stochastic dynamical systems, Clifford algebras, chaos, turbulence. 1
1 Introduction Our recent developments in the field of axiomatic quantization have reduced stochastic QED and quantized Gauge-theory gravity to a thermodynamic model. This new theory consists merely of a geodesic-domain fourier series over the excitation modes of a given system. Its positivist, Bayesian approach to quantum effects renders it a chaotic and entropic but deterministic model, rendering it suitable for currently unwieldy mathematical and numerical predictions in many-body theory.[1] Our axiomatic approach, developed in our previous paper, involves differential optimization of the Shannon entropy with respect to the geodesic; in other words, it is theorized that quantum effects are observed because, in the presence of all possible excitation modes in equal number, or none whatsoever, a particle s entropy will be maximized or minimized, respectively. We derived heisenberg s uncertainty principle and the stochasticity of fermion states as a fourier duality between position and energy, formulated as a normalized, finiteseries Fock-state convolution of the Lagrangian. We extend this method directly to gravitation, employing Clifford algebras. The integrity of such methods, however, relies on approximate knowledge of the photon number and partition densities at each point. This can be unwieldy and highly dynamic information, unless we employ a similar quantization scheme to the boson field. We begin with our axiomatic technique, this time applied to the field containing all photons of a given frequency. It is known that each constant-frequency mode will be confined to the coherent state, represented by a polarization vector and the scalar value of a Poisson function. It is also known that the entropy in a Spin-Boson Hamiltonian system with a unitary coupling constant diverges logarithmically. [7] Our hypothesis is then a simple application of Bayesian logic: each possible optical excitation mode in an electrodynamic system corrects the entropy slightly until it reaches its critical point. In an electrodynamic system, the dynamics of a system if only a subset of the possible modes exist at a given point-event, however this can be adjusted for via a time-dependent analysis of the photon field through similar means. The delta-values corresponding to each mode can be calculated by applying the simpler, maximum-entropy case to the gauge field. Newton s method leads us to a straightforward, versatile and convergent numerical method for these calculations. Our last novelty involves the resulting complex dynamics of turbulent continuous bodies. We propose that the acceleration associated with a given excitation mode obeys Pa n 0 (z), P being equivalent to a frequency-normalized quadratic map so that the turbulent region is merely a sum of Julia sets. The correspondence limit is expressed as a series of gradual bifurcations of the logistic map. 2
2 Information Entropy and Quantum Theory Recently, our previous publication broke significant new ground in applied information theory [1]. The model was based on the principle of maximized Shannon/Gibbs entropy. That is, the entropy, x = D [s] pe is (1) is optimized with respect to the geodesic; x = hk B Lk δ k s D [s] 2 e ilks, (2) The stochastic quantization method equates the thermodynamic distribution with the path-integral density[4] (for more information, see [1]). Our key observation was that (2) forms an orthonormal basis for the thermodynamic ensemble along any such path: x = hk B Lk δ k s D [s] 2 e ilks, (3) where we substitute the Schroedinger momentum and exploit the invariance of energy over a given plane-wave term. We finally reduced the series to an algebraic expression: we obtained x = i h c 2 k B Lk δ k e il ks. (4) This essentially gave us a Fourier series for the canonical displacement given the Lagrangian. Extension of the series to the Geometric Calculus of Hestenes and others and incorporation of quantum gravitational effects allowed a nearprecise calculation of the cosmological constant. The resulting equation was then, x = i h c 2 k k B B e L(k,m)s 2 L(k, m)δ k,m e k B L(k,m)s 2, (5) where our Lagrangian is, L = (ω + T ) k C k B Ω. (6) and where spectral data of the substance determines the excitation modes. 3 Maximum-Entropy quantization of the optical field Now, the density of an optical basis wave is bound to the Glauber-Sudarshan P-representation of the coherent state; optimizing the logarithmic entropy per basis term, 3
0 = 1 π k B s ν ( [k] [k1 ]) log ( α [k 1 ] α [k]), (7) where the ν and are frequency and polarization vectors, respectively and the k 1 form a diagonal expansion of the coherent state. and, it is trivial to show by differentiating the Poisson distribution with a time-linear expected value (as in Poisson-noise), this amounts to 0 = 1 2π k B ν ( ) ( [k] [k1 ] λ[k 1 ] λ[k] + k 1 λ[k 1 ]s + k ). (8) We want to employ Newton s method but must be sure the derivative is nonzero for positive k-values (this is not the case if the expectation value happens to be zero at a given state). We introduce the basis into the path-integral: x = ct = i h c k B ν Rearranging: 0 = 1 i h c 2 k B ν ( ) ( D[s]ν[k] [k] [k1 ] λ[k 1 ] λ[k] + k 1 λ[k 1 ]s + k ). (9) ( ) ( ν[k] [k] [k1 ] λ[k 1 ] λ[k] + k 1 λ[k 1 ]s + k ), so that for any small perturbation of the system we must adjust by or, simplifying, k + δk = δk = i h c 2 k B ν[k] ( 1 i h c k 2 B ν[k] [k] i h c 2 k B ν[k] ( [k] [k 1] (10) ) ( ) [k 1] λ[k] + k ) ( ), (11) ( [k] [k 1] 1 ) i h c 2 k Bν[k] [k] [k 1 ] λ[k] 2 s, (12) and subsequently adjust the rate parameter to equal the expectation value: λ = 1 t It is easy to see that this converges. (kδ + (1 δ)) e λt λtk (k)!. (13) 4
4 Polynomial Maps and Chaos Theory Decomposing the displacement operator once again into its Fourier modes, x = i h c 2 k B Lk δ k e il ks. (14) Taking the laplacian gives us the well-known hyperbolic wave equation at each band; applying the chain rule and substituting a relativistic debroglie phase velocity, 2 x = 2 µx (15), so that 2 x = v2 c 4 2 t x (16) 2 t µ 2 µδx = v2 c 4 ( 2 t x)( 2 t µ) (17) 2 t µ 2 µδx = v2 c 4 ( 2 t x) 2 (18) 2 t µ 2 µδx = 2 t δx (19) t 2 x = v2 c 4 ( 2 t x) 2 + a 0. (20) Here we see a quadratic map. One must simply take the union of the Julia sets of all significant energy bands to find the turbulent regions. Because the bifurcation locus is the Mandelbrot set, whose bifurcations map perfectly to the logistic family s bifurcation diagram, we see that in the quantum limit the solution set becomes very large. [6] This growing complexity, it is proposed, is the turbulence of a system. 5 Conclusion Having reduced the inordinate problem of optical calculations to a single, deterministic, convergent numerical method general to all non-squeezed cases involving bosons, we have shown the versatility of a general scheme for physically accurate quantization built robustly atop Bayesian statistical mechanics. The Poisson distribution being a highly versatile model, as is the quadratic map, it is believed that general statistical inference may benefit greatly from these results. 5
References [1] Daley, Kevin M. Quantum-Gravity Thermodynamics, Incorporating the Theory of Exactly Soluble Active Stochastic Processes, with Applications. IJTP (2009); DOI: 10.1007/s10773-009-0027-9 [2] Glueck, Alexander and Hueffler, H. Nonlinear Brownian Motion and Higgs mechanism. Physics Letters B. Volume 659, Issues 1-2, 17 January 2008, Pages 447-451. [3] Hestenes, David. Real Spinor Fields. Journal of Mathematical Physics 8, 798 (1967). [4] Hueffel, Helmuth and Kelnhofer, Gerald. QED Revisited: Proving Equivalence Between Path Integral and Stochastic Quantization. Physics Letters B. Volume 588, Issues 1-2, 20; May 2004, Pages 145-150. [5] Lasenby, Anthony; Doran, Christopher; and Gull, Stephen. Gravity, Gauge Theories, and Geometric Algebra. Phil. Trans. R. Soc. Lond. A 356, 487-582 (1998). [6] Lennart Carleson and Theodore W. Gamelin. Complex Dynamics: Springer 1993. [7] N. Lambert, C. Emary, and T. Brandes. Entanglement and Entropy in a Spin-Boson Quantum Phase Transition. Physical Review A, vol. 71, Issue 5 (2008). 6