Download Non-Hermitian Hamiltonians of Lie algebraic type

Non-Hermitian Hamiltonians of
Lie algebraic type
Paulo Eduardo Goncalves de Assis
City University London
Non Hermitian Hamiltonians
Real spectra ?
Hermiticity: sufficient but not necessary
Non Hermitian Hamiltonians
Real spectra ?
Hermiticity: sufficient but not necessary
-
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 FaddeevReshetikhin model, diagonalizability and PT-symmetry, J.H.E.P. 09 (2007) 104.
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
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:
orthogonal
basis
bi-orthogonal
basis
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?
• 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.
• Hamiltonians are formulated in terms of Lie algebras.
– General approach
different models
– Successful framework for integrable or solvable models
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
Representation:
invariant
Quasi-exactly solvable
Turbiner et al
PT-symmetrize
Hermitian conjugates of J’s cannot be written in terms of them
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
Swanson Hamiltonian
C.Quesne, J.Phys A40, (2007) F745.
D.P.Musumbu, H.B.Geyer, W.D.Heiss, J.Phys A39, (2007) F75.
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.
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.
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-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative:
µ- = µ-- = µ0- = 0
purely bilinear: µ+ = µ- = 0
non-positive: µ+ = µ++ = µ+0 = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and
µ+0 = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and
µ+0 = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative:
µ- = µ-- = µ0- = 0
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-reducible Hamiltonian
Constraints
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and
µ+0 = µ0- = 0
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and
µ+0 = 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
Generalized Bogoliubov:
New Operators
M.S.Swanson, J.Math.Phys 45, (2004) 585.
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
Generalized Bogoliubov:
New Operators
eigenstates and eigenvalues of
vacuum
M.S.Swanson, J.Math.Phys 45, (2004) 585.
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
Generalized Bogoliubov:
New Operators
eigenstates and eigenvalues of
vacuum
M.S.Swanson, J.Math.Phys 45, (2004) 585.
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.