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SU(3) Phase Operators: Solutions and Properties

Explore solutions and properties of SU(3) phase operators, including polar decomposition and complementarity. Learn about commuting and non-commuting phase operators in the context of classical harmonic oscillators and quantization methods. Discover the geometric aspects and weight space relations in SU(3) systems. Dive into the concepts of generalized discrete Weyl pairs and higher-dimensional cases. Find out about infinite-dimensional limits and commuting solutions in phase operator algebra.

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SU(3) Phase Operators: Solutions and Properties

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  1. SU(3) phase operators:some solutions and properties Hubert de Guise Lakehead University

  2. Collaborators: Luis Sanchez-Soto Andrei Klimov

  3. Summary • Polar decomposition: • can be easily generalized but many “free parameters” • Normally yields non-commuting phase operators • Complementarity: • cannot be easily generalized but no “free parameters” • Normally yields commuting phase operators

  4. The origin: the classical harmonic oscillator Classical harmonic oscillator: Use: Quantize:

  5. Two approaches

  6. Two approaches • write operator in polar form: • think of as the exponential of a hermitian phase operator

  7. Two approaches • write operator in polar form: • think of as the exponential of a hermitian phase operator • Use complementarity condition:

  8. What they have in common • Look at rather than

  9. What they have in common • Look at rather than • is assumed unitary: is hermitian

  10. What they have in common • Look at rather than • is assumed unitary: is hermitian • Must fix some “boundary” problems by hand

  11. SU(2) phase operator mod(2j+1)

  12. SU(2) phase operator mod(2j+1) Only one “boundary” condition

  13. An example: j=1 -2=1mod(3)

  14. An example: j=1 -2=1mod(3) -2=1mod(3)

  15. A short course on su(3) • There are eight elements in su(3)

  16. A short course on su(3) • There are eight elements in su(3) • There are now two relative phases

  17. A short course on su(3) • There are eight elements in su(3) • There are now two relative phases • States are of the form

  18. Commutation relations

  19. Commutation relations

  20. Commutation relations

  21. Commutation relations

  22. Commutation relations

  23. Geometry of weight space

  24. Geometry of weight space

  25. Geometry of weight space

  26. Geometry of weight space

  27. 3-dimensional case

  28. 3-dimensional case

  29. 3-dimensional case

  30. 3-dimensional case

  31. 3-dimensional case

  32. 3-dimensional case NOT an su(3) system

  33. SU(3) phase operators:polar decomposition

  34. Solution 1: commuting solution

  35. Solution 1: commuting solution

  36. Solution 1: commuting solution

  37. Complementaritry The matrices form generalized discrete Weyl pairs, in the sense

  38. Solution 2: the SU(2) solution

  39. Higher-dimensional cases • No commuting solutions • No complementarity

  40. Infinite dimensional limit • The edges are infinitely far • One can find commuting solutions: the phase operator commute, and have common eigenstates of zero uncertainty

  41. Summary • Polar decomposition: • can be easily generalized but many “free parameters” • Normally yields non-commuting phase operators • Complementarity: • cannot be easily generalized but no “free parameters” • Normally yields commuting phase operators

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