Polarization of a scalar field vacuum on a manifold conformally equivalent to the manifold R⊗G

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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“…The EMT for a scalar field on the space-timeM is shown to be related to the initial EMT on M as [19]:T…”

Section: Sor For a Scalar Fieldmentioning

confidence: 99%

“…In this regard, we note that influence of space topology on the Casimir effect in the geometry of two parallel plates and in cylindrical carbon nanotubes was studied in [18]. The vacuum polarization effect on a Lie group with a stationary right-invariant metric was considered in detail in [19][20][21].…”

Section: Introductionmentioning

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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Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups

Бреев

Mosman

2016

Russ Phys J

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Integration the relativistic wave equations in Bianchi IX cosmology model

Бреев¹,

Шаповалов²,

Kozlov³

2016

CRM

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We consider integration Clein-Gordon and Dirac equations in Bianchi IX cosmology model. Using the noncommutative integration method we found the new exact solutions for Taub universe.Noncommutative integration method for Bianchi IX model is based on the use of the special infinite-dimensional holomorphic representation of the rotation group, which is based on the nondegenerate orbit adjoint representation, and complex polarization of degenerate covector. The matrix elements of the representation of form a complete and orthogonal set and allow you to use the generalized Fourier transform. Casimir operator for rotation group under this transformation becomes constant. And the symmetry operators generated by the Killing vector fields in the linear differential operators of the first order from one dependent variable. Thus, the relativistic wave equation on the rotation group allow non-commutative reduction to ordinary differential equations. In contrast to the well-known method of separation of variables, noncommutative integration method takes into account the non-Abelian algebra of symmetry operators and provides solutions that carry information about the non-commutative symmetry of the task. Such solutions can be useful for measuring the vacuum quantum effects and the calculation of the Green's functions by the splitting-point method.The work for the Taub model compared the solutions obtained with the known, which are obtained by separation of variables. It is shown that the non-commutative solutions are expressed in terms of elementary functions, while the known solutions are defined by the Wigner function. And commutative reduced by the Klein-Gordon equation for Taub model coincides with the equation, reduced by separation of variables. A commutative reduced by the Dirac equation is equivalent to the reduced equation obtained by separation of variables.

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Polarization of a scalar field vacuum on a manifold conformally equivalent to the manifold R⊗G

Cited by 3 publications

References 4 publications

Vacuum quantum effects on Lie groups with bi-invariant metrics

Vacuum quantum effects on Lie groups with bi-invariant metrics

Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups

Integration the relativistic wave equations in Bianchi IX cosmology model

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