Vacuum polarization of a scalar field on lie groups and homogeneous spaces

Бреев, А. И.; Широков, И. В.; Магазев, А. А.

doi:10.1007/s11232-011-0035-9

Cited by 11 publications

(16 citation statements)

References 13 publications

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“…Substituting ( 59) into (58), we obtain (55). Thus, relations (19) and ( 20) are satisfied and the lemma 1 holds. It follows that the Dirac operator D M (x, ∂ x ) can be obtained by projection of the operator B a η a + B where B a and B are given by ( 54), and we come to (53).…”

Section: Dirac Equation In Homogeneous Spacementioning

confidence: 83%

“…In what follows, we will need the Christoffel symbols of the Levi-Civita connection with respect to a G-invariant metric g M given by [19,33]…”

Section: Invariant Metric On a Homogeneous Spacementioning

confidence: 99%

“…Projection ( 53) is determined if the coefficients B a and B of the form (54) satisfy Equations ( 19) and (20). From (49), it follows that the commutator of Λ α andγ a satisfies the first condition in (19). In this case, the condition ( 20) is reduced to…”

Section: Dirac Equation In Homogeneous Spacementioning

confidence: 99%

“…Therefore, the problem arises of constructing exact solutions to the wave equation in the case when it has symmetry operators, but they do not form a complete set and separation of variables can not be carried out. We consider the non-commutative integration method (NCIM) based on non-commutative algebras of symmetry operators admitted by the equation [16][17][18][19][20]. This method can be thought as a generalization of the method of SoV.…”

Section: Introductionmentioning

confidence: 99%

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Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Бреев

Шаповалов

2020

Symmetry

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We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

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Section: Dirac Equation In Homogeneous Spacementioning

confidence: 83%

“…In what follows, we will need the Christoffel symbols of the Levi-Civita connection with respect to a G-invariant metric g M given by [19,33]…”

Section: Invariant Metric On a Homogeneous Spacementioning

confidence: 99%

Section: Dirac Equation In Homogeneous Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Бреев

Шаповалов

2020

Symmetry

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“….1) it followsθ J = (−1) m−n , J = {−m, −n, j}.The complex polarization p = {e 1 , e 2 +ie 3 } of the covector λ(j) corresponds to the operators of λ-representation (3.3)[21] …”

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confidence: 99%

Vacuum quantum effects on Lie groups with bi-invariant metrics

Бреев

Шаповалов

2019

Int. J. Geom. Methods Mod. Phys.

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We consider the effects of vacuum polarization and particle creation of a scalar field on Lie groups with a non-stationary bi-invariant metric of the Robertson-Walker type. The vacuum expectation values of the energy momentum tensor for a scalar field determined by the group representation are found using the noncommutative integration method for the field equations instead of separation of variables. The results obtained are illustrated by the example of the three-dimensional rotation group.The Hamiltonian H(τ ) in terms of the creation and annihilation operators readsImpose the initial conditions on the functions f λ (τ ):Then F J (τ 0 ) = 0 and the Hamiltonian (6.6) is diagonal at the initial moment τ = τ 0 with respect to the operators a (±) J and a (±) † J . To diagonalize the Hamiltonian at an arbitrary instant τ , we introduce the operators c (±) J and c (±) † J related to the operators a (±) J and a (±) † J by the Bogolyubov canonical transformation:For the adjoint operators, we have, respectively:The condition F J (τ ) = 0 of diagonalization of the Hamiltonian at the moment τ with respect to the operators {c} is compatible with the normalization condition (4.6) only if ω 2 (τ ) > 0. This is equivalent to the requirement k < 0.From equation F J (τ ) = 0, we obtainwhere χ J (τ ) = χ J (τ ) is an arbitrary complex function such that |χ J (τ )| = 1. It is convenient to modify the operators {c (±) J , c (±) † J } by d (+) J = χ J (τ )c (+) J , d (−) J = χ J (τ )c (−) J , where the operators {d (+) J , d (−) J } satisfy the same commutation relations as the original operators {c (±) J , c (±) † J }. Then the Hamiltonian H(η) is diagonal with respect to the operators {d (+) J , d (−) J }:Suppose the quantized scalar field at the initial moment η 0 is in the state | 0 , which is annihilated by the operators {a (±) J , a (±) † J }. At the moment τ > τ 0 the vacuum state is defined as follows: d (−) J | 0 τ = d (−) † J | 0 τ = 0.In the Heisenberg picture, the state | 0 is not vacuum one subject to τ > τ 0 . Using the inverse transformations (6.7), we can easily find that in each mode J this state contains n J (τ ) pairs of quasiparticles with quantum numbers J and J, where n J (τ ) = 0 | d (+) † J d (−) J | 0 = 0 | d (+) J d (−) † J | 0 = |β J (τ )| 2 .

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