1. Learning Conservation Principles in Particle Physics Oliver Schulte School of Computing Science Simon Fraser University oschulte@cs.sfu.ca

2. Learning Conservation Principles - UBC LCI2/30 The Scientific Problem 100s of known reactions Empirical questions: What are the laws of particle interaction? Are there particles we haven’t seen? What is the relationship between matter and antimatter? Valdes 1994, Kocabas 1991, Machine Learning

3. Learning Conservation Principles - UBC LCI3/30 Objectives Application: Support high-level knowledge discovery, scientific model construction, data analysis. Machine Learning New Algorithms for Learning in Linear Spaces Apply and illustrate Computational Learning Theory

4. Learning Conservation Principles - UBC LCI4/30 Outline Finding conserved quantities in particle reactions Algorithm Data Findings Learning-theoretic analysis Introducing extra particles to fit the data better A New Experiment

5. Learning Conservation Principles - UBC LCI5/30 Additive Conservation Principles = “Selection Rules”

6. Learning Conservation Principles - UBC LCI6/30 Basic Learning Principle: Disallow as much as you can Kenneth Ford (1965). “Everything that can happen without violating a conservation law does happen.” Nobel Laureate Leon Cooper (1970). “In the analysis of events among these new particles, where the forces are unknown and the dynamical analysis, if they were known, is almost impossibly difficult, one has tried by observing what does not happen to find selection rules, quantum numbers, and thus the symmetries of the interactions that are relevant.”

7. Learning Conservation Principles - UBC LCI7/30 How much can we rule out?  -   - + n  -   - +    -  e - +  + e n  e - + e + p p + p  p + p +  observed reactionsnot yet observed reactions n  e - + e p + p  p + p +  +  can’t rule out Hypothetical Scenario

8. Learning Conservation Principles - UBC LCI8/30 The Vector Representation for Reactions Fix n particles. Reaction  n-vector: list net occurrence of each particle.

9. Learning Conservation Principles - UBC LCI9/30 Conserved Quantities in Vector Space

10. Learning Conservation Principles - UBC LCI10/30 Conserved Quantities are in the Null Space of Observed Reactions Let q be the vector for a quantum number, r for a reaction. Then q is conserved in r  q  r = 0. Let Q be a matrix of quantities. Then Qr = 0  r is allowed by Q. So: if r 1, …, r k are allowed, so is any linear combination. F i = 1 k a k r k

11. Learning Conservation Principles - UBC LCI11/30 Maximally strict selection rules = basis for nullspace of observations Defn: A list of selection rules Q is maximally strict  nullspace(Q) = span(R). Proposition: Q is maximally strict  span(Q) = R . R linear combinations of R conserved quantities = R . = linear combinations of Q. Q = q1, q2, …, q k

12. Learning Conservation Principles - UBC LCI12/30 System for Finding a Maximally Strict Set of Selection Rules Read in Observed Reactions Convert to list of vectors R Compute basis Q for nullspace R  from database using conversion utility Maple function nullspace

13. Learning Conservation Principles - UBC LCI13/30 Database Conversion Utility

14. Learning Conservation Principles - UBC LCI14/30 The Data: Particles Particles from Review of Particle Physics Total 193 particles Separate entries for particle and anti- particles e.g., p, p = 2 entries

15. Learning Conservation Principles - UBC LCI15/30 The Data: Reactions At least one decay for each particle with a decay mode. 182 out of 193 particles have decay modes. Particle utility converts to vector representation.

16. Learning Conservation Principles - UBC LCI16/30 Why Decays? Wanted: linearly independent reactions. Proposition: Assuming Special Relativity, decays of distinct particles are linearly independent.

17. Learning Conservation Principles - UBC LCI17/30 Finding #1: Classifying Reactions {E. Charge, Baryon#, Muon#, Electron#, Tau#} is basis for nullspace of known reactions. 1.Output of Program is equivalent classifier to standard rules. 2.All absolutely conserved quantum numbers are linear combinations of {Baryon#, E. Charge, Muon#, Electron#, Tau#} e.g., Lepton# = Muon# + Electron# + Tau#

18. Learning Conservation Principles - UBC LCI18/30 Finding #2: Matter/Antimatter Observation: Physicists’ rules match particle-antiparticle pairings. On repeated runs, program always matches particle-antiparticle pairings. Proposition If there is any basis that matches particle- antiparticle pairings, then all bases match particle- antiparticle pairings.

19. Learning Conservation Principles - UBC LCI19/30 Physicists’ Rules Match Matter/Antimatter Pairings

20. Learning Conservation Principles - UBC LCI20/30 Finding #3: Clustering + Simplicity = Standard Quantities Observation: Different runs often produce version of the lepton family laws Baryon#, Muon#, Electron#, Tau#. Is there something special about these laws? Williams (1997): “these laws have no basis in fundamental physical principles”.

21. Learning Conservation Principles - UBC LCI21/30 Conservation Principles classify reactions and cluster particles A particle p carries a quantity q if the value of q for p ≠ 0. Observation: The standard conservation principles have disjoint carriers. pn -- 00 e-e- e --  --  Baryon#Electron#Muon#Tau#

22. Learning Conservation Principles - UBC LCI22/30 Physicists’ Quantities Have Disjoint Carriers

23. Learning Conservation Principles - UBC LCI23/30 Clustering by Conservation Principles is Unique Theorem. Let q 1, q 2, q 3, q 4 be any quantities such that 1. {charge, q 1, q 2, q 3, q 4 } classify reactions as {charge, B#, E#, M#, T#} do, and 2. q 1, q 2, q 3, q 4 have disjoint carriers. Then the carriers of the q i are the same as the carriers of B#, E#, M#, T#.

24. Learning Conservation Principles - UBC LCI24/30 Clustering by Conservation Principles is Unique: Illustration pn -- 00 e-e- e --  --  Baryon#Electron#Muon#Tau# Carriers Quantum#1Quantum#2Quantum#3Quantum#4 Any alternative set of 4 Q#s with disjoint carriers

25. Learning Conservation Principles - UBC LCI25/30 Computational Search for Clustering Conservation Principles Take electric charge as given. Choose suitable objective function to encourage clustering. Minimizing objective function -> rediscovers standard principles. Work with Mark Drew.

26. Learning Conservation Principles - UBC LCI26/30 Learning-Theoretic Analysis The maximally strict learner is a PAC- learner. Given n particles,  tolerance, 1-  confidence, a sample of n/  x ln(n/  suffices. E.g. ≥ 90% accuracy, ≥ 90% confidence  14,600 data points. Proposition. The maximally strict learner is the only learner that identifies a correct set of conservation principles in the limit with at most n mind changes.

27. Learning Conservation Principles - UBC LCI27/30 More Particles can lead to stricter Conservation Principles Well-known example: if  e =  e, then n + n  p + p + e - + e - should be possible. Elliott and Engel (May 2004): “What aspects of still-unknown neutrino physics is it most important to explore? …it is clear that the absolute mass scale and whether the neutrino is a Majorana or Dirac particle are crucial issues.”

28. Learning Conservation Principles - UBC LCI28/30 When do more particles lead to stricter Conservation Principles? Theorem An extra particle yields stricter selection rules for a set of reactions R  there is a reaction r such that 1. r is a linear combination of R 2. but only with fractional coefficients.

29. Learning Conservation Principles - UBC LCI29/30 Critical Reaction for  e   e Discovered by Computer Finding if  e =  e, then the process Υ + Λ 0  p + e - cannot be ruled out with selection rules.

30. Learning Conservation Principles - UBC LCI30/30 Conclusions Program computes maximally strict set of selection rules. Good match with {Baryon#, Charge, Muon#, Electron#, Tau#} Classifies reactions as possible or impossible in exact agreement. Reproduces particle-antiparticle pairings Clustering particles given Charge leads to complete agreement. Extra particle: Computes a novel critical experiment to test if  e =  e.

31. Learning Conservation Principles - UBC LCI31/30 Polynomial Time Algorithm for Deciding if New Particle is Needed Theorem (Smith 1861). Let A be an integer matrix. Then there are matrices U,V,S such that A = USV S is diagonal (S = Smith Normal Form of A) U,V are unimodular. Theorem (Giesbrecht 2004). Let R be the matrix whose rows are the observed reactions. Then a new particle is needed  Smith Normal Form of R T has a diagonal entry outside of {0,1,-1}.