The problem of free . . . Interval and fuzzy . . . What we plan to describe Symmetry How to Reconcile Symmetries and . . . Physical Theories with Case of a single particle Case of two particles the Idea of Free Will: Case of 3 particles: . . . General case: analysis From Analysis General conclusion of a Simple Model Discussion Acknowledgments to Interval and Fuzzy Title Page Approaches ◭◭ ◮◮ ◭ ◮ Julio C. Urenda 1 and Olga Kosheleva 2 Page 1 of 15 Departments of 1 Mathematical Sciences and 2 Teacher Education University of Texas, El Paso, TX 79968, USA Go Back emails jcurenda@miners.utep.edu, olgak@utep.edu Full Screen Close
The problem of free . . . Interval and fuzzy . . . 1. Introduction What we plan to describe • Free will: a natural idea. If we walk to a corner, then Symmetry we can turn right or cross the street. Symmetries and . . . Case of a single particle • Commonsense belief: it is not possible to predict be- forehand what exactly a person will do. Case of two particles Case of 3 particles: . . . • In classical physics: General case: analysis – once we know the positions and velocities of all the General conclusion particles, Discussion – we can uniquely predict the exact future locations Acknowledgments and velocities of all the particles. Title Page • Problem: can we reconcile physics with free will? ◭◭ ◮◮ • Clarification: with 10 23 particles, predictions are not ◭ ◮ practically possible. Page 2 of 15 • From the commonsense viewpoint, even a theoretical Go Back prediction probability is very disturbing. Full Screen Close
The problem of free . . . Interval and fuzzy . . . 2. Is quantum physics an answer? What we plan to describe • At first glance, it may look as if this problem disap- Symmetry pears in quantum physics. Symmetries and . . . Case of a single particle • Due to Heisenberg’s principle, we cannot exactly pre- Case of two particles dict both the location and the velocity. Case of 3 particles: . . . • Schroedinger’s equations describe how the state (“wave General case: analysis function”) ψ ( x, t ) changes with time t . General conclusion • These equations are deterministic Discussion Acknowledgments – once we know the original state ψ ( x, t 0 ), Title Page – we can uniquely determine the future state. ◭◭ ◮◮ • So, we can uniquely predict the probabilities. ◭ ◮ • In particular, we can predict (at least theoretically) the Page 3 of 15 probability that a person turns right. Go Back • This also contradicts to common sense. Full Screen Close
The problem of free . . . Interval and fuzzy . . . 3. The problem of free will in physics has been actively studied in philosophy of physics What we plan to describe Symmetry • Mainstream approach: Symmetries and . . . Case of a single particle – keep the physics as is; Case of two particles – commonsense intuition is faulty. Case of 3 particles: . . . • Argument: quantum mechanics showed that common- General case: analysis sense intuitions are only approximately correct. General conclusion • Alternative approach (Penrose et al.): we need to mod- Discussion ify our physical theories. Acknowledgments Title Page • Problem: no well-developed physical theory is fully ◭◭ ◮◮ consistent with our free will intuition. ◭ ◮ Page 4 of 15 Go Back Full Screen Close
The problem of free . . . Interval and fuzzy . . . 4. Interval and fuzzy approaches: towards reconcilia- tion between physics and free will What we plan to describe Symmetry • Traditional approach: differential equations. Symmetries and . . . • Idea: the rate of change is uniquely determined by the Case of a single particle state: d� v i Case of two particles dt = � F i ( � r 1 , . . . ,� r n ,� v 1 , . . . ,� v n ) . Case of 3 particles: . . . • Conclusion: no free will. General case: analysis • Corollary: to get free will, we must allow several pos- General conclusion sible values of rate of change. Discussion Acknowledgments • Natural idea: interval of possible values: Title Page dv ia dt ∈ [ F ia ( � r 1 , . . . ,� r n ,� v 1 , . . . ) , F ia ( � r 1 , . . . ,� r n ,� v 1 , . . . )] . ◭◭ ◮◮ • Alternative idea: several intervals corresponding to dif- ◭ ◮ ferent degrees of certainty. Page 5 of 15 • Such nested intervals can be viewed as α -cuts of a fuzzy Go Back set, so we get fuzzy differential inclusions. Full Screen Close
The problem of free . . . Interval and fuzzy . . . 5. What we plan to describe What we plan to describe • Our objective is Symmetry Symmetries and . . . – to reasonably modify the equations of physics Case of a single particle – so that it will be possible to make the motion no Case of two particles longer uniquely predictable. Case of 3 particles: . . . • In plain terms: a physically explicit free will would General case: analysis mean that General conclusion – by simply exercising our will, Discussion Acknowledgments – we can actually change the motion of the physical Title Page particles. ◭◭ ◮◮ • We would like to check if this is indeed possible within a meaningful physical theory. ◭ ◮ Page 6 of 15 Go Back Full Screen Close
The problem of free . . . Interval and fuzzy . . . 6. Symmetry What we plan to describe • We need a theory which is consistent with free will. Symmetry Symmetries and . . . • We want this theory to be physically meaningful. Case of a single particle • In modern physics, one of the most important notions Case of two particles is the notion of symmetry . Case of 3 particles: . . . • The behavior of the physical particles must not change General case: analysis if we simply General conclusion Discussion – shift them to a different spatial location, Acknowledgments – or rotate the whole configuration, Title Page – or start the experiment at a later time moment. ◭◭ ◮◮ • Thus, a meaningful physical theory must be invariant ◭ ◮ w.r.t natural symmetries. Page 7 of 15 Go Back Full Screen Close
The problem of free . . . Interval and fuzzy . . . 7. Symmetries and conservation laws What we plan to describe • It is known that in physical equations, invariance with Symmetry respect to symmetries lead to conservation laws: Symmetries and . . . Case of a single particle – invariance w.r.t. shifts in time means that energy n 1 v i ) 2 must be preserved; Case of two particles E = � 2 · m i · ( � Case of 3 particles: . . . i =1 – invariance w.r.t spatial shifts means that the (lin- General case: analysis n General conclusion ear) momentum � p = � m i · � v i must be preserved; i =1 Discussion – invariance w.r.t. rotations means that the angular Acknowledgments n momentum � Title Page M = � m i · ( � v i × � r i ) must be preserved. i =1 ◭◭ ◮◮ • Thus, we require that these three quantities are pre- ◭ ◮ served in our physical theory. Page 8 of 15 Go Back Full Screen Close
The problem of free . . . Interval and fuzzy . . . 8. Case of a single particle What we plan to describe • Situation: let us start our analysis with the case of a Symmetry single particle. Symmetries and . . . Case of a single particle • Fact: for this particle, the momentum � p = m 1 · � v 1 is Case of two particles preserved. Case of 3 particles: . . . • Conclusion: the velocity � v 1 is also preserved. General case: analysis • Conclusion: no matter how much we exercise our will, General conclusion this particle will not be diverted from its inertial path. Discussion Acknowledgments • So, for a single particle, no “true free-will” theory is Title Page possible. ◭◭ ◮◮ ◭ ◮ Page 9 of 15 Go Back Full Screen Close
The problem of free . . . Interval and fuzzy . . . 9. Case of two particles What we plan to describe • Select the t = t 0 center of mass as the coordinates Symmetry origin: m 1 · � r 1 + m 2 · � r 2 r 2 = − m 1 = � 0, hence � · � Symmetries and . . . r 1 . m 1 + m 2 m 2 Case of a single particle • Take a system that (originally) moves with the center: Case of two particles v 2 = � m 1 · � v 1 + m 2 · � 0 . Case of 3 particles: . . . • Since the momentum � p = m 1 · � v 1 + m 2 · � v 2 is preserved, General case: analysis a 2 = − m 1 we have m 1 · � a 1 + m 2 · � a 2 = 0 hence � · � General conclusion a 1 . m 2 Discussion • Since the angular momentum is preserved, we get Acknowledgments r 1 = � m 1 · ( � a 1 × � r 1 ) + m 2 · ( � a 2 × � r 2 ) = 0, hence � a 1 × � 0. Title Page • Thus, � a 1 is collinear with � r 1 . ◭◭ ◮◮ v 2 • Since energy � � ◭ ◮ i 2 is preserved, we get � a 1 · � v 1 = 0 hence Page 10 of 15 r 1 · � v 1 = 0 – which is in general not true. � Go Back • Conclusion: no free will for 2-particle systems. Full Screen Close
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