Fouldy-Wouthuysen tranformation for non-Hermitian Hamiltonians Jean - - PDF document

Fouldy-Wouthuysen tranformation for non-Hermitian Hamiltonians

Jean Alexandre King’s College London, Department of Physics

• 1. Motivation
• 2. Equation of motion
• 3. Unitarity
• 4. Fouldy-Wouthuysen tranformation
• 5. Lorentz-symmetry violating model

JA and C. Bender aXiv:1412 DISCRETE 2014 - KCL - 2nd December 1

1 Motivation

Parity-violating mass term

• C. Bender, H. Jones, R. Rivers - 2005

L = ψ ( i/ ∂ − m − µγ5) ψ Non-Hermitian but PT-symmetric Hamiltonian density H = ψ ( i⃗ γ · ⃗ ∇ + m + µγ5) ψ Dispersion relation ω2 = m2 + p2 − µ2 − → real energies for all momenta if m2 ≥ µ2 − → look for a mapping to a Hermitian Hamiltonian 2

Fouldy-Wouthuysen tranformation Original Hamiltonian density and equation of motion H = ψHψ − → i∂0ψ = Hψ Mapping (non-unitary) χ ≡ Uψ , and UHU −1 ≡ ωγ0 − → Equation of motion for χ i∂0χ = ωγ0χ − → Original Hamiltonian mapped onto the Hermitian Hamiltonian density HFW = ωχχ 3

2 Equation of motion

Non-Hermitian Lagrangian δS δψ = 0 not equivalent with (δS δψ )† = 0 Original derivation of the equation of motion S = ∫ d4x ψOψ = ∫ d4x(ϕa − iχa)[γ0O]ab(ϕb + iχb) Variational principle δS δϕa = 0 , δS δχa = 0 We find γ0 (1 2 δS δϕ + i 2 δS δχ ) = δS δψ − → equivalent with δS/δψ = 0 when ψ and ψ are considered independent. 4

3 Unitarity

Naive probability density ψ†ψ not conserved since, for eigen modes ψ†(t, ⃗ p)ψ(t, ⃗ p) = ψ†(0, ⃗ p) exp(iH†t) exp(−iHt)ψ(0, ⃗ p) ̸= ψ†(0, ⃗ p)ψ(0, ⃗ p) − → need for a new definition of current to be conserved Look for jν = ψγνAψ , with A = a + bγ5 − → conserved current jν = ψγν ( 1 + µ mγ5) ψ Probability density ρ ≥ 0 for µ2 ≤ m2 ρ = ψ† ( 1 + µ mγ5) ψ = ( 1 + µ m ) |ψR|2 + ( 1 − µ m ) |ψL|2 5

4 Fouldy-Wouthuysen transformation

Equation of motion i∂0ψ = γ0 [ ⃗ p · ⃗ γ + m + µγ5] ψ Look for a mapping χ ≡ Uψ such that i∂0χ = √ m2 + p2 − µ2 γ0χ By analogy with the case of the Dirac equation U ≡ exp ( θ ⃗ p · ⃗ γ + µγ5 √ p2 − µ2 ) = cos θ + ⃗ p · ⃗ γ + µγ5 √ p2 − µ2 sin θ − → Necessarily tan(2θ) = √ p2 − µ2 m 6

Alternative ansatz U ′ = exp ( θ′ ⃗ p · ⃗ γ + µγ5 √ µ2 − p2 ) = cosh θ′ + ⃗ p · ⃗ γ + µγ5 √ µ2 − p2 sinh θ′ Necessarily tanh(2θ′) = √ µ2 − p2 m − → equivalent with the previous mapping if θ = iθ′ UV regime p2 ≥ µ2 swapped with IR regime µ2 ≥ p2 Non-singular mappings lim

p2→µ2 U = lim p2→µ2 U ′ = 1 + ⃗

p · ⃗ γ + µγ5 2m 7

5 Lorentz-symmetry violating model

Lorentz-symmetry violating + non-Hermitian Lagrangian L = ψ ( i/ ∂ − i/ b − m ) ψ CPT even and PT odd Motivated by gravity induced effective Lorentz-symmetry violation (real vev)

• G. Shore - 2005

bµ ∝ α m2∂µR For motion perpendicular to ⃗ b: usual conserved current; dispersion relation ω2 = m2 + p2 − b2 Fouldy-Wouthuysen transformation U = exp ( θ (⃗ p − i⃗ b) · ⃗ γ √ p2 − b2 ) , with tan(2θ) = √ p2 − b2 m 8

6 Conclusion

Consistent non-Hermitian Fermionic Hamiltonian Mapping to a Hermitian Hamiltonian in the parameter space when energies are real Relevance to cancellation of chiral anomaly? Neutino kinematics? 9