non-hermitian hamiltonians of lie algebraic type paulo eduardo goncalves de assis city university...
TRANSCRIPT
Non-Hermitian Hamiltonians of Non-Hermitian Hamiltonians of Lie algebraic typeLie algebraic type
Paulo Eduardo Goncalves de AssisPaulo Eduardo Goncalves de Assis
City University LondonCity University London
Non Hermitian Hamiltonians
Real spectra ?
Hermiticity: sufficient but not necessary
Non Hermitian Hamiltonians
- W.Heisenberg, Quantum theory of fields and elementary particles, Rev.Mod.Phys. 29 (1957) 269.
- J.L.Cardy and R.L.Sugar, Reggeon field theory on a lattice, Phys.rev. D12 (1975) 2514.
- F.G.Scholtz, H.B.Geyer and F. Hahne, Quasi-Hermitian operators in Quantum Mechanics and the variational principle, Ann. Phys. 213 (1992) 74.
- T.Hollowood, Solitons in affine Toda field theory, Nucl.Phys. B384 (1992) 523.
- D.I.Olive, N.Turok, and J.W.R.Underwood, Solitons and the energy momentum tensor for affine Toda theory, Nucl.Phys. B401 (1993) 663.
- C.M.Bender and S.Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetries, Phys.Rev.Lett. 80 (1998) 5243.
- C.Korff and R.A.Weston, PT Symmetry on the Lattice: The Quantum Group invariant XXZ spin-chain, J.Phys. A40 (2007) 8845.
- A.K.Das, A.Melikyan and V.O.Rivelles, The S-Matrix of the Faddeev-Reshetikhin model, diagonalizability and PT-symmetry, J.H.E.P. 09 (2007) 104.
Real spectra ?
Hermiticity: sufficient but not necessary
When is the spectrum real?
• PT-symmetry: Invariance under parity and time-reversal
Anti-unitarity:
When is the spectrum real?
• PT-symmetry: Invariance under parity and time-reversal
Anti-unitarity:
• Unbroken PT :
Not only the Hamiltonian but also the eigenstates are invariant under PT
When is the spectrum real?
• PT-symmetry: Invariance under parity and time-reversal
Anti-unitarity:
• Unbroken PT :
Not only the Hamiltonian but also the eigenstates are invariant under PT
When is the spectrum real?
• PT-symmetry: Invariance under parity and time-reversal
Anti-unitarity:
• Unbroken PT :
Not only the Hamiltonian but also the eigenstates are invariant under PT
Hermitian Hamiltonian real eigenvalues
complex eigenvalues Non-Hermitian Hamiltonian
Hermitian Hamiltonian real eigenvalues
complex eigenvalues Non-Hermitian Hamiltonian
Hermitian Hamiltonian real eigenvalues
complex eigenvalues Non-Hermitian Hamiltonian
IF
Hermitian Hamiltonian real eigenvalues
complex eigenvalues Non-Hermitian Hamiltonian
IF
Isospectral transformation:
non-Hermitian Hamiltonian Hermitian counterparts.
map
orthogonal
basis
bi-orthogonal
basis
Eigenstates of h and H are essentially different:
orthogonal
basis
bi-orthogonal
basis
Eigenstates of h and H are essentially different:
orthogonal
basis
bi-orthogonal
basis
Eigenstates of h and H are essentially different:
bi-orthogonality as non trivial metric
Similarity transformation as a change in the metric
Pseudo-Hermiticity
H is Hermitian with respect to the new metric.
H is Hermitian with respect to the new metric.
all observables transform
H is Hermitian with respect to the new metric.
all observables transform
non-Hermitian Hamiltonian X ambiguous physics
What is being studied?
• Non-Hermitian Hamiltonians of Lie algebraic type,
P.E.G.Assis and A.Fring, in preparation.
• non Hermitian Hamiltonian with real eigenvalues:
constraints, metrics, Hermitian counterparts.
• eigenvalues and eigenfunctions when possible.
What is being studied?
• Hamiltonians are formulated in terms of Lie algebras.
– General approach different models
– Successful framework for integrable or solvable models
• Non-Hermitian Hamiltonians of Lie algebraic type,
P.E.G.Assis and A.Fring, in preparation.
• non Hermitian Hamiltonian with real eigenvalues:
constraints, metrics, Hermitian counterparts.
• eigenvalues and eigenfunctions when possible.
sl2(R)-Hamiltonians
sl2(R)-Hamiltonians
sl2(R)-Hamiltonians
Representation:
invariant
Quasi-exactly solvable Turbiner et al
sl2(R)-Hamiltonians
Representation:
invariant
Quasi-exactly solvable Turbiner et al
sl2(R)-Hamiltonians
PT-symmetrize
Hermitian conjugates of J’s cannot be written in terms of them
Representation:
invariant
Quasi-exactly solvable Turbiner et al
su(1,1)-Hamiltonians
su(1,1)-Hamiltonians
su(1,1)-Hamiltonians
C.Quesne, J.Phys A40, (2007) F745.
su(1,1)-Hamiltonians
C.Quesne, J.Phys A40, (2007) F745.
su(1,1)-Hamiltonians
C.Quesne, J.Phys A40, (2007) F745.
su(1,1)-Hamiltonians
C.Quesne, J.Phys A40, (2007) F745.
D.P.Musumbu, H.B.Geyer, W.D.Heiss, J.Phys A39, (2007) F75.
Swanson Hamiltonian
su(1,1)-Hamiltonians
Holstein-Primakoff Two-mode
su(1,1)-Hamiltonians
Hermitian partners
• Metric Ansatz:
Hermitian partners
• Metric Ansatz:
constraints
Hermitian partners
• Metric Ansatz:
constraints
exact action of the metric on the generators
Hermitian partners
• Metric Ansatz:
constraints
exact action of the metric on the generators
Recall:
Recall:
Recall:
• Different possible subcases, e.g., purely linear or purely bilinear.
Recall:
• Different possible subcases, e.g., purely linear or purely bilinear.
• Large variety of models may be mapped onto a Hermitian couterpart.
• Metric depends either only on momentum or coordinate operators.
Recall:
• Different possible subcases, e.g., purely linear or purely bilinear.
• Large variety of models may be mapped onto a Hermitian couterpart.
Reducible Hamiltonian
Constraints
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative: µ- = µ-- = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-positive: µ+ = µ++ = µ+0 = 0non-negative: µ- = µ-- = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
non-negative: µ- = µ-- = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and µ+0 = µ0- = 0
non-negative: µ- = µ-- = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and µ+0 = µ0- = 0
non-negative: µ- = µ-- = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and µ+0 = µ0- = 0
eigenvalues and eigenstates
non-negative: µ- = µ-- = µ0- = 0
Non-reducible Hamiltonian
Constraints
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-positive: µ+ = µ++ = µ+0 = 0non-negative: µ- = µ-- = µ0- = 0
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and µ+0 = µ0- = 0
non-negative: µ- = µ-- = µ0- = 0
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and µ+0 = µ0- = 0
non-negative: µ- = µ-- = µ0- = 0
Eigenstates and Eigenvalues?
• So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra.
Diagonalization Hamiltonian in Harmonic Oscillator form
Eigenstates and Eigenvalues?
• So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra.
Diagonalization Hamiltonian in Harmonic Oscillator form
M.S.Swanson, J.Math.Phys 45, (2004) 585.Generalized Bogoliubov:
New Operators
Eigenstates and Eigenvalues?
• So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra.
Diagonalization Hamiltonian in Harmonic Oscillator form
M.S.Swanson, J.Math.Phys 45, (2004) 585.Generalized Bogoliubov:
New Operators
vacuum
eigenstates and eigenvalues of
Eigenstates and Eigenvalues?
• So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra.
Diagonalization Hamiltonian in Harmonic Oscillator form
M.S.Swanson, J.Math.Phys 45, (2004) 585.Generalized Bogoliubov:
New Operators
vacuum
eigenstates and eigenvalues of
More constraints for Ñ dependent Hamiltonian
More constraints for Ñ dependent Hamiltonian
metric-constraints and solvability-constraints combined ?
Exact: eigenvalues, eigenstates and suitable metric
More constraints for Ñ dependent Hamiltonian
metric-constraints and solvability-constraints combined ?
Exact: eigenvalues, eigenstates and suitable metric
Transition amplitudes
Conclusions
• Calculated conditions and appropriate metrics with respect to which a large class of non Hermitian Hamiltonians bilinear in su(1,1) generators can be considered Hermitian.
• The same non Hermitian Hamiltonians could be diagonalized and it was shown, whithout metrics, that although being non Hermitian real eigenvalues do occur.
• Possibility to have complete knowledge of spectra, eigenstates (both of Hermitian and non Hermitian Hamiltonians) and meaningful metrics.
• Hamiltonians explored are very general, allowing interesting models as sub cases.
• Other algebras may be employed.