Need to Formalize . . . First Example: . . . Equations Second Example: . . . Interesting Relation to . . . Without Equations: Acknowledgments Challenges on a Way Physicists Assume . . . to a More Adequate Title Page ◭◭ ◮◮ Formalization of ◭ ◮ Reasoning in Physics Page 1 of 22 Go Back Roberto Araiza, Vladik Kreinovich, and Juan Ferret Full Screen University of Texas, El Paso, TX 79968, USA raraiza@gmail.com, vladik@utep.edu Close Quit
1. Need to Formalize Reasoning in Physics Need to Formalize . . . First Example: . . . • Fact: in medicine, geophysics, etc., expert systems use Second Example: . . . automated expert reasoning to help the users. Interesting Relation to . . . • Hope: similar systems may be helpful in general theo- Acknowledgments retical physics as well. Physicists Assume . . . • What is needed: describe physicists’ reasoning in pre- Title Page cise terms. ◭◭ ◮◮ • Reason: formalize this reasoning inside an automated computer system. ◭ ◮ • Formalized part of physicists’ reasoning: theories are Page 2 of 22 formulated in terms of PDEs (or ODEs) dx dt = F ( x ). Go Back Full Screen • Meaning: these equations describe how the correspond- ing fields (or quantities) x change with time t . Close Quit
2. Mathematician’s View of Physics and Its Limita- Need to Formalize . . . tions First Example: . . . Second Example: . . . • Mathematician’s view: we know the initial conditions Interesting Relation to . . . x ( t 0 ) at some moment of time t 0 . Acknowledgments • We solve the corresponding Cauchy problem and find Physicists Assume . . . the values x ( t ) for all t . • 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 22 • 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.
3. Physicists’ Explanation Need to Formalize . . . First Example: . . . • Reminder: not all solutions to the physical equation Second Example: . . . are physically meaningful. Interesting Relation to . . . • Explanation: the “time-reversed” solution is non-physical Acknowledgments because its initial conditions are “degenerate”. Physicists Assume . . . • Clarification: once we modify the initial conditions 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 22 – how to formalize “non-degenerate”; Go Back – the separation between equations and initial condi- Full Screen tions depends on the way equations are presented. Close • First challenge: can be resolved by using Kolmogorov Quit complexity and randomness.
4. First Example: Schr¨ odinger’s Equation Need to Formalize . . . First Example: . . . • Example: Schr¨ odinger’s equation Second Example: . . . ∂t = − � 2 i � · ∂ Ψ 2 m · ∇ 2 Ψ + V ( � Interesting Relation to . . . r ) · Ψ . Acknowledgments • In this representation: the potential V is a part of the Physicists Assume . . . equation, and Ψ( � r, t 0 ) are initial conditions. Title Page • Transformation: ◭◭ ◮◮ – we represent V ( � r ) as a function of Ψ and its deriva- ◭ ◮ tives, Page 5 of 22 – differentiate the right-hand side by time, and – equate the derivative w.r.t. time to 0. Go Back Full Screen • Result: ∂t + � 2 2 m · ∇ 2 Ψ � i � � ∂ Ψ · ∂ Ψ Close = 0 . Ψ ∂t Quit
5. First Example (cont-d) Need to Formalize . . . First Example: . . . • Reminder: Second Example: . . . ∂t + � 2 2 m · ∇ 2 Ψ � i � Ψ · ∂ Ψ � ∂ = 0 . Interesting Relation to . . . ∂t Ψ Acknowledgments Physicists Assume . . . • Mathematically: the new equation (2nd order in time) is equivalent to the Schr¨ odinger’s equation: Title Page – every solution of the Schr¨ odinger’s equation for any ◭◭ ◮◮ 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 22 • 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
6. Towards 2nd Example: General Physical Theories Need to Formalize . . . First Example: . . . • Traditional description of physical theories: in terms Second Example: . . . of differential equations. Interesting Relation to . . . • Example (17 cent.): Newton’s mechanics m · d 2 x dt 2 = F. Acknowledgments Physicists Assume . . . • Important discovery (18 cent.): most physical theories can be reformulated as S → min for “action” S . Title Page • Example: Newton’s mechanics is equivalent to ◭◭ ◮◮ L dt → min , where L = 1 x 2 + V ( x ) . � S = 2 · m · ˙ ◭ ◮ • For functions f ( x 1 , . . . , x n ) : minimum when Page 7 of 22 f ( x + dx ) ≈ f ( x ), so ∂f = 0 for all i . Go Back ∂x i Full Screen • For functions of functions (“functionals”): minimum when S ( f + δf ) ≈ S ( f ), so δS Close δf ( x ) = 0 for all x . Quit
7. Euler-Lagrange Equations Need to Formalize . . . First Example: . . . • Reminder: physical theories can be formulated in terms Second Example: . . . of the minimal action principle S → min. Interesting Relation to . . . � • Here, S = L dx for a “Lagrange” f-n L that depends Acknowledgments = ∂ϕ def on the fields ϕ , . . . , and their derivatives ϕ ,i . Physicists Assume . . . ∂x i � • Euler-Lagrange equations: when S = L dx , Title Page � ∂L � δS δf = ∂L ∂f − ∂ ◭◭ ◮◮ = 0 . ∂x i ∂f ,i ◭ ◮ • Comment: we use “Einstein’s rule” that repeated in- Page 8 of 22 dices mean summation: e.g., f ,i f ,i means � f ,i f ,i . Go Back i • For a single scalar field ϕ : Full Screen � ∂L ∂L ∂ϕ − ∂ � Close = 0 . ∂x i ∂ϕ ,i Quit
8. Second Example: General Scalar Field Need to Formalize . . . First Example: . . . • General scalar theory: L = L ( ϕ, ϕ ,i ). Second Example: . . . • 3-D case: it is reasonable to consider rotation-invariant Interesting Relation to . . . Lagrangian functions L . Acknowledgments • Conclusion: L depends only on the length ϕ ,i ϕ ,i of the Physicists Assume . . . vector ϕ ,i ,not on its orientation. Title Page • 4-D case: L should be invariant w.r.t. Lorentz trans- ◭◭ ◮◮ formations (4-D “rotations”). ◭ ◮ def = ϕ ,i ϕ ,i . • Conclusion: L = L ( ϕ, a ), where a Page 9 of 22 • Traditional formulation: every Lagrangian is possible, but initial conditions ϕ ( x, t 0 ) must be non-degenerate. Go Back Full Screen • Our result: there exists a 3rd order equation such that: Close ϕ satisfies this equation ⇔ ϕ satisfies Euler-Lagrange equation for some L . Quit
9. Scalar Field: Proof Need to Formalize . . . First Example: . . . def = ϕ ,i ϕ ,i . • Reminder: L = L ( ϕ, a ), where a Second Example: . . . • Euler-Lagrange equations: ∂L ∂L ∂ϕ − ∂ i = 0. Interesting Relation to . . . ∂ϕ ,i Acknowledgments • Using chain rule: ∂L ( ϕ, a ) = ∂L ∂a · ∂a = ∂L Physicists Assume . . . ∂a · 2 ϕ ,i . ∂ϕ ,i ∂ϕ ,i Title Page • Conclusion: L ,ϕ − ∂ i (2 L ,a · ϕ ,i ) = 0. ◭◭ ◮◮ • Using chain rule again, we get L ,ϕ − 2 L ,a · � ϕ − 2 L ,aϕ · ( ϕ ,i ϕ ,i ) − 4 L ,aa · ϕ ,ij ϕ ,i ϕ ,j = 0 , ◭ ◮ Page 10 of 22 def = ϕ ,i where � ϕ ,i . Go Back • Conclusion: Full Screen – if at two points, we have the same values of Close ϕ , ϕ ,i ϕ ,i , and � ϕ , – then we have same values of ϕ ,ij ϕ ,i ϕ ,j . Quit
10. Scalar Field: Proof (cont-d) Need to Formalize . . . First Example: . . . • Reminder: if at two points, we have the same values of Second Example: . . . def ϕ , a = ϕ ,i ϕ ,i , and b = � ϕ , then we have same values Interesting Relation to . . . def = ϕ ,ij ϕ ,i ϕ ,j . of c Acknowledgments • Particular case: if we have dx k for which ϕ ,k · dx k = 0, Physicists Assume . . . a ,k · dx k = 0, and b ,k · dx k = 0, then c ,k · dx k = 0. • In geom. terms: if dx k ⊥ ϕ ,k , dx k ⊥ a ,k , and dx k ⊥ b ,k , Title Page then dx k ⊥ c ,k . ◭◭ ◮◮ • Conclusion: ϕ ,k , a ,k , b ,k , and c ,k lie in the same 3-plane. ◭ ◮ Page 11 of 22 • In algebraic terms: the determinant is 0: Go Back ε ijkl · ϕ ,i · a ,j · b ,k · c ,l = 0 , Full Screen where ε ijkl = 0 if some indices are equal and is ± 1 else. Close • We get a 3-rd order equation; so, we can predict future Quit evolution – w/o knowing L .
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