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

3

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.

The Dirac equation and reducing of him to the form

• f the Schrödinger equation

4

m x

α α α

ψ γ ψ ψ ∂ ⎛ ⎞ + Φ − = ⎜ ⎟ ∂ ⎝ ⎠

2g E

α β β α αβ

γ γ γ γ + = H H g

μ ν α β μν αβ

η =

;

1 4 H H S

ε μν α μ νε α

Φ = −

H

β α α β

γ γ =

2 E

β β αβ α α

γ γ γ γ η + =

( )

1 2

v v v

S μ

μ μ

γ γ γ γ = −

[ ]

diag 1,1,1,1

αβ

η = −

ˆ , i H t t ∂Ψ = Ψ ∂

( ) ( ) ( )

00 00 00

ˆ

k k k k

im i i H i x g g g γ γ γ γ γ ∂ = − + − Φ + Φ ∂ − − −

†

ˆ ˆ H H ≠

• time of infinitive distant observer

and depend on the choice of tetrad vectors

The Kerr solution

( solution of Einstein equations for an axially symmetrical stationary gravitational field generated by a rotating point body with mass and moment )

5

( ) ( )

00 3 00 3

1 2 ; 2 ; 2 ; 1 2 ; 1 2 ; 2 ; 2 .

kl l k mn mn mn k mn kl l mn mn

M J R M M g g g R R R M g R M J R M M g g g R R R δ δ δ δ ⎫ = − + = = + ⎪ ⎪ ⎪ − = + ⎬ ⎪ ⎪ = − − = = − ⎪ ⎭

The metric of the Kerr solution in the first-order approximation in “mass” with length dimensionality ( )

2

/ , / M GM c Mc → → J J

M

J

The systems of tetrad vectors

6

( )

3

1 ; 0; 2 ; 1

k m kl l k k mk

M J R M M H H H H R R R δ ⎡ ⎤ ⎡ ⎤ = + = = = − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

• (

)

3

1 ; 2 ; 0; 1

k m kl l k k mk

M J R M M H H H H R R R δ ⎡ ⎤ ⎡ ⎤ ′ ′ ′ ′ = + = − = = − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

( ) ( )

3 3

1 ; ; ; 1

k m kl l kl l k k mk

M J R M J R M M H H H H R R R R δ ⎡ ⎤ ⎡ ⎤ = + = − = = − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

• System of tetrad vectors in symmetric gauge:

System of tetrad vectors used in papers of Hehl, Ni, Obukhov, Silenko, Teryaev: The Killing system of tetrad vectors:

HAMILTONIANS FOR THE KERR SOLUTION

7

The Killing system of tetrad vectors:

( ) ( ) ( ) ( )

3 3 3 3 5 3 5

2 2 3 . 2 ˆ 2 2 2

k k k k k k kl l k k kl l ml l mk k l l k k k

H M M i MR im im i i R x R x R M J R i M J R M J R im i S R R x M J R R i M J R R R x γ γ γ γ γ γ γ γ γ γ γ γ ∂ − + − ∂ ⎧ ⎫ − − ⎨ = ∂ ∂ = − − + + + ∂ ∂ ∂ + − ⎬ ⎭ ∂ ⎩

HAMILTONIANS FOR THE KERR SOLUTION

8

System of tetrad vectors used in papers of Hehl, Ni, Obukhov, Silenko, Teryaev:

( ) ( )

3 3 5 3 5

ˆ 3 2 2 2 . 2

k k k k k k kl l l l k k k k

M M i MR H im im i i R x R x R M J R M J R R i M J R R R i x γ γ γ γ γ γ γ γ γ γ γ ∂ ∂ ′ = − − + + + ∂ ∂ ∂ + + ∂ ⎧ ⎫ + − ⎨ ⎬ ⎩ ⎭

HAMILTONIANS FOR THE KERR SOLUTION

9

System of tetrad vectors in symmetric gauge:

( ) ( ) ( )

3 3 3 3

ˆ 2 2 2 .

k k kl l m k k k k k l l l mk k l k k

M M i MR H im i im i x R R x R M J R M J i R x R M J R im i S R R x γ γ γ γ γ γ γ γ γ ∂ ∂ = − − + + + ∂ ∂ ∂ + − ∂ ∂ − ⋅ + ∂

Pseudo-Hermitian quantum mechanics

10

1 †

ˆ ˆ H H ρ ρ − =

ˆ H - the Hamiltonian in

• representation

ˆ H i t ∂Ψ = Ψ ∂

ˆ i H t ψ ψ ∂ = ∂

ˆ ηψ Ψ =

( )

( )

3 †

, d x Φ Ψ = Φ Ψ

∫

( )

3 †

, d x

ρ

ϕ ψ ϕ ρψ = ∫

• the condition of pseudo-Hermiticity of the Hamiltonian

If , then and

†

ρ η η =

1

ˆ ˆ H H η η − =

†

ˆ ˆ H H =

η

• standard scalar product

for Hermitian Hamiltonians

• scalar product

for pseudo-Hermitian Hamiltonians

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System of tetrad vectors in symmetric gauge System used by Hehl, Ni, Obukhov, Silenko, Teryaev Killing system Operator Operator Tetrad vectors

( )

3

1 3 2

km m k

M J R M R R γ γ + +

( )

3

3 1 2

km m k

M J R M R R γ γ + +

1 3 M R +

3 1 2 M R +

( )

3

3 1 1 2 2

km m k

M J R M R R γ γ + +

ρ η

Operators and for three systems of system vectors

ρ

η

Table 1

( )

3

1 3

km m k

M J R M R R γ γ + +

1 †

ˆ ˆ H H ρ ρ − =

1 †

ˆ ˆ ˆ H H H η η − = =

• the same for any

choice of tetrad vectors

The Hermitian Hamiltonian operator

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( ) ( )

1 † 3 3 5 3 5

ˆ ˆ ˆ H H 2 2 3 . 2

k k k k k k kl l k l l k k k

H M M MR im im i i i R x R x R M J R i R x M J R R i M J R R η η γ γ γ γ γ γ γ γ γ γ γ

−

= = = ∂ ∂ = − − + − + ∂ ∂ ∂ + + ∂ ⎧ ⎫ + − ⎨ ⎬ ⎩ ⎭

ˆ H

is the same for any system of tetrad vectors

THE PARKER SCALAR PRODUCT

13

The matrix:

† †

,

α α α α

γ γ γ γ γ γ γ γ = =

( )

3 †

, dx g ϕ ψ ϕ γ γ ψ = −

∫

γ

For three system of tetrad vectors considered by us

gγ γ ρ − = , ,

ρ

ϕ ψ ϕ ψ =

that is

Hermiticity of Hamiltonian concerning Parker scalar product

14

ˆ ˆ ,( ) ( ), H H ϕ ψ ϕ ψ Δ ≡ −

1 2 3

Δ ≡ Δ + Δ + Δ

( ) ( )

3 † 1 00 † 3 00

im dx g g im dx g g ϕ γ γ γ ψ γ ϕ γ γ ψ ⎛ ⎞ ⎧ ⎫ ⎪ ⎪ Δ = − − − ⎜ ⎟ ⎨ ⎬ ⎜ ⎟ − ⎪ ⎪ ⎩ ⎭ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎧ ⎫ ⎪ ⎪ − − − ⎜ ⎟ ⎜ ⎟ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ − ⎪ ⎪ ⎩ ⎭ ⎝ ⎠ ⎝ ⎠

∫ ∫

{ }

( )

{ }

( )

( )

† 3 † 3 2

dx g i dx g i ϕ γ γ ψ ϕ γ γ ψ Δ = − − Φ − − − Φ

∫ ∫

( ) ( )

3 † 3 00 † 3 00

.

k k k k

i dx g g i dx g g ϕ γ γ γ γ ψ γ γ ϕ γ γ ψ ⎛ ⎞ ⎧ ⎫ ⎪ ⎪ Δ = − ∇ − ⎜ ⎟ ⎨ ⎬ ⎜ ⎟ − ⎪ ⎪ ⎩ ⎭ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎧ ⎫ ⎪ ⎪ − − ∇ ⎜ ⎟ ⎜ ⎟ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ − ⎪ ⎪ ⎩ ⎭ ⎝ ⎠ ⎝ ⎠

∫ ∫

Δ =

Comparison of quantum mechanics treatment of dynamics of Dirac particle in stationary gravitational field (I)

15

Yes Yes No Dependence of Hamiltonian type

• n choice of

system of tetrad vectors Hamiltonian Approach based on Parker scalar product Pseudo-Hermitian quantum mechanics in initial representation Hermitian quantum mechanics in - representation C B A Methods of quantum mechanics treatment

H H† ˆ ˆ =

1 †

ˆ ˆ H H ρ ρ

−

=

( ) ( )

1 †

ˆ H ˆ g H g γ γ γ γ

−

= = − −

η

Table 2

Comparison of quantum mechanics treatment of dynamics of Dirac particle in stationary gravitational field (II)

16

Connection between wave function Connection between scalar products Connection between Hamiltonians With an weight operator Standard scalar product for Hilbert space: Scalar product Approach based on Parker scalar product Pseudo-Hermitian quantum mechanics in initial representation Hermitian quantum mechanics in

• representation

C B A Methods of quantum mechanics treatment

η

( )

( )

3 †

, d x Φ Ψ = Φ Ψ

∫

( )

3 †

, d x

ρ

ϕ ψ ϕ ρψ = ∫

( )

†

g ρ η η γ γ = = −

( )

3 †

, d x g ϕ ψ ϕ γ γ ψ = = −

∫

† :

ρ η η =

1

ˆ ˆ H= H η η −

( )

, , ,

ρ

ϕ ψ ϕ ψ Φ Ψ = =

ηψ Ψ =

Table 3

Conclusions

The results obtained in this study make it possible to look at the description of quantum mechanics of spin 1/2 particles in stationary gravitational fields from a new point of view. For example, before this study, the problem of non-uniqueness of Hamiltonians and their sensitivity to the choice of tetrad vectors in our

• pinion was unresolved. The apparatus of pseudo-Hermitian quantum

mechanics used in this study allowed us to resolve this issue at least as applied to the Schwarzschild and the Kerr solution. It was proven that the resulting Hamiltonian does not depend on the choice of tetrad vectors and is Hermitian. The scalar products also do not dependent on the choice of tetrad vectors, if they were calculated using

• perators shown in Table 1.

ρ

17

1

ˆ ˆ H H η η − =

In our opinion, the uniqueness of the Hamiltonian is

• nontrivial. Indeed, all of the three initial expressions for Hamiltonians,

first, differ from each other and, second, are non-Hermitian concerning standard scalar product in Hilbert space. After applying procedures for the transformion of initial Hamiltonians into their Hermitian expressions , the latter could in principle differ in some Hermitian

• summands. It is not the case, however, and the expressions for

are the same in all the three cases. Such a coincidence means that whatever the choice of tetrad vectors in a gravitational field there will always exist a single Hermitian Hamiltonian , which has the same spectrum of energy levels as any of the starting operators .

ˆ H ˆ H ˆ H

ˆ H ˆ H

ˆ H

18

Upon transition to the Hamiltonian , one can use the quantum mechanics apparatus in its standard form. In particular, the left member

• f the Schrödinger equation will contain the operator , in which

the time coordinate is understood to be the time of an infinitely distant observer. Interestingly, the expression derived in this study for the Hamiltonian is the same as the expression proposed Obukhov, Silenko, Teryaev for the weak Kerr field. The formalism of pseudo- Hermitian Hamiltonians in fact validates the expressions for Hermitian Hamiltonians used Obukhov, Silenko, Teryaev .

ˆ H ( )

i t ∂ ∂

t

ˆ H

19

The results of this study allow us to claim that pseudo- Hermitian Hamiltonians provides for the application of the relativistic quantum mechanics formalism practically in its standard form. The expression for the operator in - representation allows to get rid

• f “ambiguity” connected with different type of the initial Hamiltonians

at the use of different system of tetrad vectors. We think that this feature of the pseudo-Hermitian method makes it preferable as applied to the problems, in which gravitational effects are resolved and their quantitative characteristics are analyzed.

ˆ H

η

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Thank you for your attention