Local Fields without Restrictions on the Spectrum of 4 ...

Local Fields without Restrictions on the Spectrum of 4-Momentum Operator and Relativistic Lindblad Equation2 Maxim Kurkov V.A. Franke 3 Department of high-energy physics Saint-Petersburg State University

International School of Subnuclear Physics Erice, 2009 2 3

based on arXiv: 0908.2415[hep-th] co-author and supervisor

Maxim Kurkov , V.A. Franke 4 (SPbSU)

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Outline

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Introduction: relativistic Lindblad equation.

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Fields with unconventional 4-momentum spectrum.

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Local Field Annihilating the Vacuum State

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Summary

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bibliography


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Introduction What is the Lindblad equation? Generalization for dissipative systems of the von Neumann equation. Nonrelativistic Lindblad equation has the following form X dρ = −i[H, ρ] + λn 2An ρA†n − A†n An ρ − ρA†n An dt n

(1)

In nonrelativistic QM it particulary permits to describe the evolution of quantum system and the measurement process in a unique way (by corresponding choice of the operators An ).

Why does one need its relativistic generalization? Description of the early Universe There are no any external devices, making measurements over the Universe Maxim Kurkov , V.A. Franke 6 (SPbSU)

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Relativistic Generalization of Lindblad equation gµν = diag {1, −1, −1, −1} One may use Tomonaga-Schwinger formalism in the interaction picture to construct relativistic Lindblad equation. δρ(σ) = −i [H1 (x), ρ(σ)] + λ 2ϕ(x)ρϕ† (x) − ϕ† (x)ϕ(x)ρ − ρϕ† (x)ϕ(x) δσ(x) (2) Resolvability condition of the equation (2) is δ 2 ρ(σ) δ 2 ρ(σ) = δσ(x1 )δσ(x2 ) δσ(x2 )δσ(x1 )

(3)

To fulfill the condition (3) it is sufficient to assume, that scalar field ϕ(x) is local i.e. if (x1 − x2 )2 < 0 h i [ϕ(x1 ), ϕ(x2 )] = 0, ϕ(x1 ), ϕ† (x2 ) = 0, [ϕ(x1 ), H1 (x)] = 0 (4)


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Relativistic Generalization of Lindblad equation: problems If ϕ(x) is ordinary scalar field one has irremovable u.v. divergence For the sake of simplicity assume that H1 (x) = 0, so δρ(σ) = +λ 2ϕ(x)ρϕ† (x) − ϕ† (x)ϕ(x)ρ − ρϕ† (x)ϕ(x) (5) δσ(x) Let us assume, in particular, that on the surface σ the state ρ(σ) is the vacuum of the interaction picture: ρ(σ) = |0ih0|. Under the variation δσ(x) of the surface σ in vicinity of the point x it appears δρ(σ) that ρ(σ + δσ) = |0ih0| + δσ(x) δσ(x). What is the probability of the vacuum state |0ih0| to remain unchanged? Sp {|0ih0|ρ (σ + δσ(x))} = 1 − 2λh0|ϕ† (x)ϕ(x)|0iδσ(x) {z } | +∞

One need local field annihilating the vacuum state: ϕ(x)|0i = 0. Maxim Kurkov , V.A. Franke 8 (SPbSU)

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Fields with unconventional spectrum of 4-momentum: notations. Axiomatic Quantum Field Theory: It is impossible to construct local field annihilating the vacuum state with conventional 4-momentum spectrum. Fields with unconventional 4-momentum spectrum are of special interest. One must assume for consistency with experiment that the interaction with conventional matter is extremely weak. Notations: u(x) ≡ h0|ϕ(x)ϕ† (0)|0i, w (x) ≡ h0|ϕ† (x)ϕ(0)|0i R 4 ikx R 4 −ikx 1 e = d xe ξ(x), e ξ(x) = (2π) ξ(k) d ke ξ(k) 4

e(k) ≡ α(k 2 , θ(k 2 )sgn(k 0 )) = (2π)4 h0|ϕ(0)|kihk|ϕ† (0)|0i≥ 0 u

e (k) ≡ β(k 2 , θ(k 2 )sgn(k 0 )) = (2π)4 h0|ϕ† (0)|kihk|ϕ(0)|0i≥ 0 w


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Locality and Analyticity

locality in x-space ⇒ analyticity in k-space F (x) = ih0|[ϕ(x), ϕ† (0)]|0i, Fr (x) = θ(x 0 )F (x), Fa (x) = −θ(−x 0 )F (x) | {z } =0, at x 2 0 expression for the propagator of ϕ(x) (from (7)) p i −k 2 − iǫ sgn(k 0 ) 1 1 1 e (k) = G + 2 2 0 2 4π m − k − iǫ sgn(k ) 4πm (m − k 2 − iǫ sgn(k 0 )) (10) Maxim Kurkov , V.A. Franke 14 (SPbSU)

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Construction of free local field annihilating the vacuum state with given Wightman functions (and consequently propagator) in terms of creation and annihilation operators. 0 We assume that: β(k 2 , θ(k 2 )sgn(k √ )) ≡ 0 −k 2 1 α(k 2 , θ(k 2 )sgn(k 0 )) = θ(−k 2 ) + θ(k0 )δ(k 2 − m2 ) {z } 2πm (m2 − k 2 ) | {z } | α(k 2 ,+1) α(k 2 ,0)

free field ϕ isRdefined by the following relation: 1 d 4 k{α(k))e −ikx a(k)} ϕ(x) ≡ (2π) 2 creation and annihilation operators are defined in the support of α 2 2 )θ(−k 2 ) √ k 2 < 0 : [a(k), a† (k ′ )] ≡ δ 4 (k − k ′ ) (m −k −k 2 p k 2 = m2 > 0 : [a(~k), a† (k~′ )] = 2k 0 δ 3 (~k − k~′ ), k 0 = ~k 2 + m2 4-momentum operator R Pµ ≡ d 4 k kµ {α(k)a† (k)a(k)} ϕ(x) = e iPx ϕ(0)e −iPx ,

Maxim Kurkov , V.A. Franke

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(SPbSU)

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Pµ a† (k)|0i = kµ a† (k)|0i

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5. Difficulties caused by interaction between the field ϕ annihilating the vacuum and ordinary fields. interaction picture is used ξ(x) is ordinary Hermitian scalar field with mass M Gξ (k) = M 2 −k1 2 −iǫ interaction Hamiltonian Hint (x) = g ξ(x)ϕ† (x)ϕ(x) The following diagram has u.v. diverges particulary at p10 + ... + pj0 > |~p1 + ... + ~pj |

this theory is nonrenormalizable Maxim Kurkov , V.A. Franke 16 (SPbSU)

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Summary

Summary Kallen-Lehmann representation for the propagator of local scalar field with arbitrary spectrum of 4-momentum operator is established. Restrictions on Whiteman functions of fields that violate CPT -invariance are found. Local scalar field that annihilates vacuum state and violate CPT invariance is constructed in this scope (nonconventional spectrum of 4-momentum). Such field is composite: it may have terms which correspond to positive square of mass and positive or negative sign of energy, but it must contain continues tachyonic spectrum of mass. Using such field we succeeded to write a correct (in particular without u.v. divergences) local relativistic generalization of Lindblad equation for statistical operator in the case of no interaction with other fields. Difficulties arising when the field annihilating the vacuum interacts with ordinary fields are discussed. Maxim Kurkov , V.A. Franke 17 (SPbSU)

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bibliography

bibliography P.Pearl, Relativistic Collapse Model With Tachyonic Features, Physical Review, A59, 80 (1999) A.Bassi, G .C .Ghirardi. Physics Reports, v.379, p.257, (2003). (quant-ph/0302164). R.F .Streater , A.S.Wightman. PCT, Spin and Statistics and All that. W.A.Benjamin, INC, New York - Amsterdam,1964. G .Lindblad. Commun.Math.Phys., v.48, p.119 (1976). V .A.Franke. On the general form of the dynamical transformation of density matrices, Theor. Math. Phys., v.27, p.172 (1976) M.A.Kurkov , V .A.Franke. Local Fields without Restrictions on the Spectrum of 4-Momentum Operator and Relativistic Lindblad Equation arXiv: 0908.2415[hep-th]; sent to ”Foundations of Physics” (August 2009) Maxim Kurkov , V.A. Franke 18 (SPbSU)

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