Equations First Example: . . . First Example (cont-d) Without - PowerPoint PPT Presentation

Need to Formalize . . . Mathematician’s View . . . Physicists’ Explanation Equations First Example: . . . First Example (cont-d) Without Equations: Second Example: . . . Challenges on a Way Scalar Field: . . . Acknowledgments to a More Adequate Title Page Formalization of ◭◭ ◮◮ Reasoning in Physics ◭ ◮ Page 1 of 9 Roberto Araiza 1 and Vladik Kreinovich 1 , 2 Go Back 1 Bioinformatics Program 2 Department of Computer Science Full Screen University of Texas, El Paso, TX 79968, USA Close raraiza@utep.edu, vladik@utep.edu Quit

Need to Formalize . . . 1. Need to Formalize Reasoning in Physics Mathematician’s View . . . • Fact: in medicine, geophysics, etc., expert systems use Physicists’ Explanation automated expert reasoning to help the users. First Example: . . . First Example (cont-d) • Hope: similar systems may be helpful in general theo- Second Example: . . . retical physics as well. Scalar Field: . . . • What is needed: describe physicists’ reasoning in pre- Acknowledgments cise terms. Title Page • Reason: formalize this reasoning inside an automated ◭◭ ◮◮ computer system. ◭ ◮ • Formalized part of physicists’ reasoning: theories are Page 2 of 9 formulated in terms of PDEs (or ODEs) dx dt = F ( x ). Go Back • Meaning: these equations describe how the correspond- Full Screen ing fields (or quantities) x change with time t . Close Quit

Need to Formalize . . . 2. Mathematician’s View of Physics and Its Limita- Mathematician’s View . . . tions Physicists’ Explanation • Mathematician’s view: we know the initial conditions First Example: . . . x ( t 0 ) at some moment of time t 0 . First Example (cont-d) Second Example: . . . • We solve the corresponding Cauchy problem and find Scalar Field: . . . the values x ( t ) for all t . Acknowledgments • Limitation: not all solutions to the equation Title Page dx dt = F ( x ) are physically meaningful. ◭◭ ◮◮ • Example 1: when a cup breaks into pieces, the corre- ◭ ◮ sponding trajectories of molecules make physical sense. Page 3 of 9 • Example 2: when we reverse all the velocities, we get Go Back pieces assembling themselves into a cup. Full Screen • Fact: this is physically impossible. Close • Fact: the reverse process satisfies all the original Quit (T-invariant) equations.

Need to Formalize . . . 3. Physicists’ Explanation Mathematician’s View . . . • Reminder: not all solutions to the physical equation Physicists’ Explanation are physically meaningful. First Example: . . . First Example (cont-d) • Explanation: the “time-reversed” solution is non-physical Second Example: . . . because its initial conditions are “degenerate”. Scalar Field: . . . once we modify the initial conditions • Clarification: Acknowledgments even slightly, the pieces will no longer get together. Title Page • Conclusion: not only the equations must be satisfied, ◭◭ ◮◮ but also the initial conditions must be “non-degenerate”. ◭ ◮ • Two challenges in formalizing this idea: Page 4 of 9 – how to formalize “non-degenerate”; Go Back – the separation between equations and initial condi- tions depends on the way equations are presented. Full Screen • First challenge: can be resolved by using Kolmogorov Close complexity and randomness. Quit

Need to Formalize . . . 4. First Example: Schr¨ odinger’s Equation Mathematician’s View . . . • Example: Schr¨ odinger’s equation Physicists’ Explanation First Example: . . . ∂t = − � 2 i � · ∂ Ψ 2 m · ∇ 2 Ψ + V ( � r ) · Ψ . First Example (cont-d) Second Example: . . . • In this representation: the potential V is a part of the Scalar Field: . . . equation, and Ψ( � r, t 0 ) are initial conditions. Acknowledgments • Transformation: Title Page – we represent V ( � r ) as a function of Ψ and its deriva- ◭◭ ◮◮ tives, ◭ ◮ – differentiate the right-hand side by time, and Page 5 of 9 – equate the derivative w.r.t. time to 0. Go Back • Result: Full Screen ∂t + � 2 2 m · ∇ 2 Ψ � i � � ∂ Ψ · ∂ Ψ = 0 . Close Ψ ∂t Quit

Need to Formalize . . . 5. First Example (cont-d) Mathematician’s View . . . • Reminder: Physicists’ Explanation First Example: . . . ∂t + � 2 2 m · ∇ 2 Ψ � i � Ψ · ∂ Ψ � ∂ = 0 . First Example (cont-d) ∂t Ψ Second Example: . . . • Mathematically: the new equation (2nd order in time) Scalar Field: . . . is equivalent to the Schr¨ odinger’s equation: Acknowledgments – every solution of the Schr¨ odinger’s equation for any Title Page V ( � r ) satisfies this new equation, and ◭◭ ◮◮ – every solution of the new equation satisfies Sch¨ odinger’s ◭ ◮ equation for some V ( � r ). Page 6 of 9 • Observation: in the new equation, initial conditions, in Go Back effect, include V ( � r ). Full Screen • Conclusion: “non-degeneracy” (“randomness”) condi- Close tion must now include V ( � r ) as well. Quit

Need to Formalize . . . 6. Second Example: General Scalar Field Mathematician’s View . . . • Example: consider a scalar field ϕ with a generic La- Physicists’ Explanation def = ϕ ,i ϕ ,i . grange function L ( ϕ, a ), with a First Example: . . . First Example (cont-d) • Traditional formulation: every Lagrangian is possible, Second Example: . . . but initial conditions ϕ ( x, t 0 ) must be non-degenerate. Scalar Field: . . . • Euler equations: ∂L ∂L = L ,ϕ − ∂ i (2 L ,a · ϕ ,i ) = 0: Acknowledgments ∂ϕ − ∂ i ∂ϕ ,i Title Page L ,ϕ − 2 L ,a · � ϕ − 2 L ,aϕ · ( ϕ ,i ϕ ,i ) − 4 L ,aa · ϕ ,ij ϕ ,i ϕ ,j = 0 . ◭◭ ◮◮ ◭ ◮ • In general, on a 3-D Cauchy surface t = t 0 , we can find points with arbitrary combination of ( ϕ, ϕ ,i ϕ ,i , � ϕ ). Page 7 of 9 • Thus, by observing the evolution, we can find ϕ ,ij ϕ ,i ϕ ,j Go Back for all possible triples ( ϕ, ϕ ,i ϕ ,i , � ϕ ). Full Screen • So, we can predict future evolution – w/o knowing L . Close Quit

Need to Formalize . . . 7. Scalar Field: Discussion and Conclusions Mathematician’s View . . . • Observation: the new “equation” does not contain L Physicists’ Explanation at all. First Example: . . . First Example (cont-d) • Fact: a field ϕ satisfies the new equation ⇔ it satisfies Second Example: . . . the Euler-Lagrange equations for some L . Scalar Field: . . . • Observation: Acknowledgments – similarly to Wheeler’s cosmological “mass without Title Page mass” and “charge without charge”, ◭◭ ◮◮ – we now have “equations without equations”. ◭ ◮ • Conclusion: when formalizing physical equations: Page 8 of 9 – we must not only describe them in a mathematical Go Back form, Full Screen – we must also select one of the mathematically equiv- Close alent forms. Quit

8. Acknowledgments Need to Formalize . . . Mathematician’s View . . . This work was supported in part: Physicists’ Explanation First Example: . . . • by National Science Foundation grant HRD-0734825, First Example (cont-d) and EAR-0225670 and DMS-0532645 and Second Example: . . . Scalar Field: . . . • by Grant 1 T36 GM078000-01 from the National Insti- Acknowledgments tutes of Health. Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 9 Go Back Full Screen Close Quit