ON THE PROBLEM OF UNIQUENESS AND HERMITICITY OF HAMILTONIANS FOR - PowerPoint PPT Presentation

ON THE PROBLEM OF UNIQUENESS AND HERMITICITY OF HAMILTONIANS FOR DIRAC PARTICLES IN GRAVITATIONAL FIELDS V.P.Neznamov neznamov@vniief.ru Russian Federal Nuclear Center All-Russian Research Institute of Experimental Physics According to the paper of M.V.Gorbatenko, V.P.Neznamov arXiv: 1007.4631 [ gr-qc ]

The authors prove that the dynamics of spin 1/2 particles in stationary gravitational fields can be described using an approach, which builds upon the formalism of pseudo-Hermitian Hamiltonians. The proof consists in the analysis of three expressions for Hamiltonians, which are derived from the Dirac equation and describe the dynamics of spin 1/2 particles in the gravitational field of the Kerr solution. The Hamiltonians correspond to different choices of tetrad vectors and differ from each other. The differences between the Hamiltonians confirm the conclusion known from many studies that the Hamiltonians derived from the Dirac equation are non-unique. Application of standard pseudo-Hermitian quantum mechanics rules to each of these Hamiltonians produces the same Hermitian Hamiltonian. 2

The eigenvalue spectrum of the resulting Hamiltonian is the same as that of the Hamiltonians derived from the Dirac equation with any chosen system of tetrad vectors. For description of the dynamics of spin 1/2 particles in stationary gravitational fields can be used not only the formalism of pseudo-Hermitian Hamiltonians, but also an alternative approach, which employs the Parker scalar product. The authors show that the alternative approach is equivalent to the formalism of pseudo- Hermitian Hamiltonians. 3

The Dirac equation and reducing of him to the form of the Schrödinger equation ∂ ψ ⎛ ⎞ α γ + Φ ψ − ψ = ⎜ ⎟ m 0 α α ∂ ⎝ ⎠ x α β β α αβ γ γ + γ γ = 2 g E [ ] μ ν = η η = − H H g diag 1,1,1,1 α β μν αβ αβ β β αβ α α γ γ + γ γ = η 2 E ( ) 1 β γ α = α γ 1 S μ μ μ = γ γ − γ γ v v v Φ = − ε μν H 4 H H S β α μ νε α 2 ; ∂Ψ = Ψ ˆ i H , t - time of infinitive distant observer ∂ t ∂ im i i = − γ + γ γ − Φ + γ γ Φ ˆ 0 0 k 0 k H i ( ) ( ) ( ) ∂ − − − 0 k k 00 00 00 x g g g ˆ ≠ ˆ † H H and depend on the choice of tetrad vectors 4

The Kerr solution ( solution of Einstein equations for an axially symmetrical stationary M gravitational field generated by a rotating point body with mass and J moment ) The metric of the Kerr solution in the first-order approximation in → → 2 “mass” with length dimensionality ( M GM / c , J J / Mc ) ( ) ⎫ M J R M M = − + = = δ + δ kl l g 1 2 ; g 2 ; g 2 ; ⎪ 00 0 k mn mn mn 3 R R R ⎪ ⎪ M − = + ⎬ 1 2 ; g R ⎪ ( ) ⎪ M J R M M = − − = = δ − δ 00 0 k mn kl l ⎪ g 1 2 ; g 2 ; g 2 . mn mn ⎭ 3 R R R 5

The systems of tetrad vectors The Killing system of tetrad vectors: ( ) ⎡ ⎤ ⎡ ⎤ M J R M M � � � � = + = = = δ − 0 k 0 kl l m H 1 ; H 0; H 2 ; H 1 ⎢ ⎥ ⎢ ⎥ 0 0 k k mk ⎣ ⎦ ⎣ ⎦ 3 R R R System of tetrad vectors used in papers of Hehl, Ni, Obukhov, Silenko, Teryaev: ( ) ⎡ ⎤ ⎡ ⎤ M J R M M ′ ′ ′ ′ = + = − = = δ − 0 k kl l 0 m H 1 ; H 2 ; H 0; H 1 ⎢ ⎥ ⎢ ⎥ 0 ⎣ ⎦ 0 k k mk ⎣ ⎦ 3 R R R System of tetrad vectors in symmetric gauge: ( ) ( ) ⎡ ⎤ ⎡ ⎤ M J R M J R M M = + = − = = δ − 0 k 0 m kl l kl l 1 ; ; ; 1 H H H H ⎢ ⎥ ⎢ ⎥ 0 0 k k mk ⎣ ⎦ 3 3 ⎣ ⎦ R R R R � 6

HAMILTONIANS FOR THE KERR SOLUTION The Killing system of tetrad vectors: ˆ � = H ∂ ∂ M M i MR = γ − γ − γ γ + γ γ + γ γ + k k k k im im i 2 i ∂ ∂ 0 0 0 0 0 k k 3 R x R x 2 R ( ) ∂ M J R + − kl l 2 i ∂ 3 k R x ( ) ( ) ∂ M J R M J R − γ + − k kl l ml l 2 im 2 i S ∂ mk 3 3 k R R x ( ) ⎧ ⎫ M J R R i M J − − γ γ γ ⎨ l l k ⎬ 3 . k 5 0 k 3 5 ⎩ ⎭ 2 R R 7

HAMILTONIANS FOR THE KERR SOLUTION System of tetrad vectors used in papers of Hehl, Ni, Obukhov, Silenko, Teryaev: ∂ ∂ M M i MR ′ = γ − γ − γ γ + γ γ + γ γ + ˆ k k k H im im i 2 i ∂ ∂ 0 0 0 k 0 0 k k 3 2 R x R x R ( ) ∂ M J R + + kl l 2 i ∂ 3 k R x ( ) ⎧ ⎫ M J R R i M J + − γ γ γ ⎨ l l k ⎬ 3 . k 5 0 k 3 5 ⎩ ⎭ 2 R R 8

HAMILTONIANS FOR THE KERR SOLUTION System of tetrad vectors in symmetric gauge: ∂ ∂ M M i MR = γ − γ γ − γ + γ γ + γ γ + ˆ k k k H im i im 2 i ∂ ∂ 0 0 k 0 0 0 k k 3 x R R x 2 R ( ) ∂ M J R + − k l l 2 i ∂ 3 k R x ( ) ( ) ∂ M J R M J R − ⋅ γ + kl l m l l im i S . ∂ k mk 3 3 k R R x 9

Pseudo-Hermitian quantum mechanics ρ − = ρ ˆ ˆ 1 † H H - the condition of pseudo-Hermiticity of the Hamiltonian ρ = η η η − = η = ˆ ˆ ˆ ˆ † 1 † H H H H If , then and η ˆ H - the Hamiltonian in -representation ( ) ∂Ψ = ( ) ∫ Φ Ψ = Φ Ψ 3 † Ψ ˆ , d x - standard scalar product i H ∂ t for Hermitian Hamiltonians Ψ = ηψ ˆ ∂ ψ = ψ = ∫ ˆ ( ) i H ϕ ψ ϕ ρψ ∂ 3 † , d x t - scalar product ρ for pseudo-Hermitian Hamiltonians 10

η ρ Operators and for three systems of system vectors Table 1 ρ η Tetrad vectors Operator Operator ( ) ( ) M J R Killing system 3 M M J R M + + γ γ + + γ γ km m km m 1 1 3 2 0 k 3 0 k 3 2 R R R R 1 3 M 3 M System used by Hehl, Ni, + + 1 Obukhov, Silenko, Teryaev R 2 R ( ) ( ) System of tetrad vectors in M J R M J R M 3 M 1 + + γ γ + + γ γ km m km m 1 1 3 symmetric gauge 0 k 0 k 3 3 2 R 2 R R R ρ − = - the same for any η − = η = ˆ ˆ ˆ 1 † ρ H H H ˆ ˆ 1 † H H choice of tetrad vectors 11

ˆ H The Hermitian Hamiltonian operator − = η η = = ˆ ˆ ˆ 1 † H H H is the same for any system of tetrad vectors ∂ ∂ M M MR = γ − γ − γ γ + γ γ − γ γ + k k k im im i 2 i i ∂ ∂ 0 0 0 k 0 0 k k 3 R x R x R ( ) ∂ M J R + + kl l 2 i ∂ 3 k R x ( ) ⎧ ⎫ M J R R i M J + − γ γ γ l l k ⎨ ⎬ 3 . k 5 0 k 3 5 ⎩ ⎭ 2 R R 12

THE PARKER SCALAR PRODUCT γ γ = γ γ γ γ = γ γ γ † † , The matrix: α α α α 0 0 0 0 0 ( ) ∫ ϕ ψ = − ϕ γ γ ψ 3 † 0 , dx g 0 For three system of tetrad vectors considered by us − g γ γ = ρ 0 0 that is ϕ ψ = ϕ ψ , , ρ 13

Hermiticity of Hamiltonian concerning Parker scalar product Δ ≡ Δ + Δ + Δ Δ ≡ ϕ ψ − ϕ ψ ˆ ˆ ,( H ) ( H ), 1 2 3 ⎛ ⎞ ⎧ ⎫ ⎪ ⎪ im ∫ Δ = − ϕ γ γ − γ ψ − 3 † 0 0 ⎜ ⎟ dx g ⎨ ( ) ⎬ ⎜ ⎟ − 1 0 00 ⎪ g ⎪ ⎩ ⎭ ⎝ ⎠ ⎛ ⎞ † ⎛ ⎞ ⎧ ⎫ ⎪ ⎪ im ∫ − − ⎜ − γ ϕ γ γ ψ ⎟ 3 0 0 dx g ⎜ ⎟ ⎨ ( ) ⎬ ⎜ ⎟ ⎜ − 0 ⎟ 00 ⎪ g ⎪ ⎩ ⎭ ⎝ ⎠ ⎝ ⎠ ( ) ( ) ( ) { } { } ∫ ∫ † Δ = − ϕ γ γ − Φ ψ − − − Φ ϕ γ γ ψ 3 † 0 3 0 dx g i dx g i 2 0 0 0 0 ⎛ ⎞ ⎧ ⎫ ⎪ ⎪ i ∫ Δ = − ϕ γ γ γ γ ∇ ψ − 3 † 0 0 k ⎜ ⎟ dx g ⎨ ( ) ⎬ ⎜ ⎟ − 3 0 k 00 ⎪ g ⎪ ⎩ ⎭ ⎝ ⎠ Δ = 0 ⎛ ⎞ † ⎛ ⎞ ⎧ ⎫ ⎪ ⎪ i ∫ − − ⎜ γ γ ∇ ϕ γ γ ψ ⎟ 3 0 k 0 ⎜ ⎟ dx g . ⎨ ( ) ⎬ ⎜ ⎟ ⎜ − ⎟ k 0 00 ⎪ g ⎪ ⎩ ⎭ ⎝ ⎠ ⎝ ⎠ 14

Comparison of quantum mechanics treatment of dynamics of Dirac particle in stationary gravitational field (I) Table 2 Methods of quantum mechanics treatment A B C Hermitian quantum Pseudo-Hermitian η Approach based on Parker mechanics in - quantum mechanics in scalar product representation initial representation = ˆ H = = ρ − ρ ( ) ( ) ˆ ˆ ˆ ˆ H † 1 † − H H H 1 Hamiltonian = − γ γ − γ γ ˆ 0 † 0 g H g 0 0 Dependence of Hamiltonian type on choice of No Yes Yes system of tetrad vectors 15