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Normal Order & Graphs

Normal Order & Graphs. Paweł Błasiak & Andrzej Horzela . Combinatorial graffiti. Institute of Nuclear Physic s, Polish Academy of Sciences. Collaboration:. Karol A. Penson , Université Paris VI Allan I. Solomon, Open Univ ersity ( UK ) Gérard Duchamp, Université de Paris-Nord.

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Normal Order & Graphs

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  1. Normal Order & Graphs Paweł Błasiak & Andrzej Horzela Combinatorial graffiti Institute of Nuclear Physics,Polish Academy of Sciences Collaboration: Karol A. Penson, Université Paris VI Allan I. Solomon, Open University(UK) Gérard Duchamp, Universitéde Paris-Nord

  2. Content • Theme: See operators as graphs • Introduction • Operators in Quantum Mechanics • Occupationnumber representation • Graphs • Definitions and Examples • Algebra of graph composition • Operators vs Graphs • Equivalence and calculus

  3. Occupation number representation • Number operator Nnnn • Basis in Fock space 0, 1, 2, …… , n1, n, n+1, …… annihilation creation • Operators [a+,N]a+ ladder structure a+n n+1n+1 annihilationoperator [a,N]a creationoperator a nnn1 Heisenberg algebra [a,a+]1 ICSSUR 2007, Bradford

  4. Commutation • Creation and annihilation operators do notcommute • a a+ a + a1 [a,a+]1 • Order is important normalform • a a+ a + a+ 1 • a a+ a + a normalordering • Normal order • a a+ aaa+ a • a + 2a4+4a + a3+ 2 a2 unique [a,a+]1 • a +- to the left a - to the right ICSSUR 2007, Bradford

  5. Operator representations • A - bounded operator • Matrixrepresentation • Coherent state representation (Bargmann – Segal repr.) • Normallyordered form ICSSUR 2007, Bradford

  6. Graph representations • A- algebra of operators • G - algebra of graphs 1:1 •   • a+ k a l • - basis inA • - basis in G •   k l 1:1 k l • What about the product inA ? • Can be transfered on graphs and is consistent • with the natural graph composition !!! ICSSUR 2007, Bradford

  7. Graphs: Basic concepts I • G – graph : vertices & lines + some rules • Basic buildingblocks: • output • The vertex has two kinds of lines k outgoing lines l ingoing lines • All lines are distinguishable • between each other • input • Picture of a process ICSSUR 2007, Bradford

  8. Graphs: Basic concepts II • Composition rules: • CORRECT • Vertices may be connected by joining • outgoing and ingoing lines, • i.e. lines keep their direction • WRONG • , • & • & • & • 4 • 2 • 2 • 1 ICSSUR 2007, Bradford

  9. Graphs: Basic concepts II • Labeling: • Vertices are numbered with consecutive integers 1,2,3, … • and followdirection of the lines 8 3 5 3 1 2 7 2 1 2 3 4 1 6 1 2 3 • WRONG • CORRECT • WRONG ICSSUR 2007, Bradford

  10. Graphs: Basic concepts II • Labeling: • Vertices are numbered with consecutive integers 1,2,3, … • and followdirection of the lines 8 5 2 7 3 4 1 6 • Processes grow step by step ICSSUR 2007, Bradford

  11. Graphs: Basic concepts III • Weights: • Vertices may be attached weights • making up the overall weight of the graph • &       • , • ,     • *  2   2  2  • *  5 3 ICSSUR 2007, Bradford

  12. Graphs: Equivalence relation G1 G2 No. of and in G1andG2 in are equal, resp.     / , , … , , …    4lines 5lines 3lines 2lines ICSSUR 2007, Bradford

  13. Graphs: Example • , • No connection (disconnected) 1 2 1 2 1 2 1 2 2 2 2 2 • , • One connection 2 2 4 x x x 1 1 1 1 • All two vertex graphs made of : • Twoconnections 2 2 2 2 x x • & 1 1 ICSSUR 2007, Bradford

  14. Graphs: Example • All two vertex graphs made of : • & ICSSUR 2007, Bradford

  15. Graph algebra • Graph algebra: • Addition: k l • Multiplication: • Quotient graph algebra: ICSSUR 2007, Bradford

  16. Graph algebra: Examples • 1  • 2  • 1 2  2 1  , • 2 • 2 • = • + • 4 • + • 2 • * • 2 • 1 • 1 • 1 • 2 • = • + • * • 2 • 2 • 1 • 1 • Noncommutative !! ICSSUR 2007, Bradford

  17. Operators vs Graphs I • Graph • Operator k outputs a+ a+ a+ a+ • a+ ka l 1:1 a a a l inputs • Basic quantum process and its picture. ICSSUR 2007, Bradford

  18. Operators vs Graphs II • A- algebra of operators • G - algebra of graphs 1:1 • - basis inG •   • a+ k a l • - basis inA •   k l 1:1 k l • G •  •  1:1 • A • G ICSSUR 2007, Bradford

  19. ‘Proof’ • Commutator: : rki 2 i connections • = • * r k l s 1 : sli ICSSUR 2007, Bradford

  20. Example I: Product • & • Product in A : • Product in G : • 2 • 2 • = • + • 4 • + • 2 • * • 2 • 1 • 1 • 1 ICSSUR 2007, Bradford

  21. Example II: Powers  2 • =a + 4a4+ 2 a + 4a3+ a + 4a2+ 4 a + 3a3+ • + 6 a + 3a2+ 2 a + 3a+ 2 a + 2a2+ 2 a + 2a • a + 2a2+a + 2 a • + • + • + • + • =  2 2 1 1 2 2 1 1 2 2 • + • + 2 • + • + 2 2 2 4 2 2 2 4 4 6 1 1 1 1 • + • + 2 2 2 2 2 2 1 1 ICSSUR 2007, Bradford

  22. Conclusion and outlook • G • Summary: •  •  1:1 • A • G • a+ k a l k l • Conclusion: Operatorscan be seenas graphs • Outlook: • Seeoperators as processes • Quantum Mechanicsingraphrepresentation • AlternativeLanguage & Interpretation ICSSUR 2007, Bradford

  23. Example III: Powers  2 • =a + 4a4+ 2 a + 4a3+ a + 4a2+ 4 a + 3a3+ • + 6 a + 3a2+ 2 a + 3a+ 2 a + 2a2+ 2 a + 2a • a + 2a2+a + 2 a • + • + • = • +  2 • + 1 1 2 2 1 1 2 2 • + • + • + 2 • + 2 2 2 2 2 4 1 1 1 1 • + • + 2 2 2 2 1 1 ICSSUR 2007, Bradford

  24. Graphs: Basic concepts I k l • Compositionrules: Vertex & Lines • Verticesmay be connected by joining • lines keepingtheirdirection • CORRECT • & • WRONG ICSSUR 2007, Bradford

  25. Graphs: Basic concepts II • Compositionrules: • CORRECT • Verticesmay be connected by joining • outgoing and ingoing lines, • i.e. lines keeptheirdirection • WRONG • & ICSSUR 2007, Bradford

  26. Graphs: Basic concepts II • Labeling: • Verticesarenumberedwithconsecutive • integers 1,2,3, … inaccordancewiththe • direction of lines 3 3 1 2 1 2 1 2 3 • WRONG • CORRECT • WRONG ICSSUR 2007, Bradford

  27. Graphs: Basic concepts II • & • Compositionrules: • Verticesmay be connected by joining • outgoing and ingoing lines, • i.e. lines keeptheirdirection • 4 • 1 • WRONG • CORRECT • 2 • 2 ICSSUR 2007, Bradford

  28. Graphs: Equivalence relation G1 G2 No. of and in G1and G2 in are equal, resp.     / , , 8 5 2 7  10 lines 7 lines 3 4 1 6 ICSSUR 2007, Bradford

  29. Graphs: Basic concepts II • Labeling: • Vertices are numbered with consecutive • integers 1,2,3, … in accordance with the • direction of lines 8 5 2 7 3 4 1 6 • Processes grow step by step ICSSUR 2007, Bradford

  30. Example I: Product • & • Product in A : • Product in G : • + • + • 2 • = • 4 • * ICSSUR 2007, Bradford

  31. Example II: String • & • Product in A : • Product in G : • 2 • 2 • = • + • 4 • + • 2 • * • 2 • 1 • 1 • 1 ICSSUR 2007, Bradford

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