Casimir effect on nontrivial topology spaces in Krein space quantization
aa r X i v : . [ g r- q c ] A p r Casimir effect on nontrivial topology spacesin Krein space quantization
M. Naseri , , S. Rouhani , M.V. Takook ∗ October 30, 2018 Islamic Azad University, Kermanshah Branch, Kermanshah, IRAN Department of Physics, Razi University, Kermanshah, IRAN Plasma Physics Research Centre, Islamic Azad University,P.O.BOX 14835-157, Tehran, IRAN
Abstract
Casimir effect of a topologically nontrivial two-dimensional space-time, through Kreinspace quantization [1, 2], has been calculated. In other words, auxiliary negative normstates have been utilized here. Presence of negative norm states play the role of anautomatic renormalization device for the theory. The negative norm states (which do notinteract with the physical world) could be chosen in two perspective. In the first case ourmethod results in zero or vanishing values for energy. In the second case, however, theresult are the same as the renormalization procedure.
Proposed PACS numbers : 04.62.+v, 03.70+k, 11.10.Cd, 98.80.H ∗ e-mail: [email protected] Introduction
Consideration of the negative norm states was proposed by Dirac in 1942. In 1950, Guptaapplied this idea in QED. The presence of higher derivatives in the lagrangian also led toghosts states with negative norms. In order to preserve the covariance principle in the gaugetheory, the auxiliary negative norms states were utilized. In previous paper [1], it was shownthat consideration of the negative norm states is necessary for a fully covariant quantizationof the minimally coupled scalar field in de Sitter space (Krein QFT). We have shown that forphysical states (positive norm states) the energy is positive, whereas, for the negative normstates (so called un-physical states) the energy is negative. It was also shown that the effect ofthese un-physical states merely appears in the physics of the problem as a tool for an automaticrenormalization of the theory in one-loop approximation [1, 2, 3, 4, 5, 6, 7].In a previous paper [8], the Casimir effect in Krein QFT has been studied as well. Onceagain it is found that the theory is automatically renormalized. This method is once againreexamined here by analysis of Casimir effect in space-time with nontrivial topology. Thepaper is organized as follows. The next section presents a brief review of ordinary Casimireffect in two-dimensional space-time with nontrivial topology. Section 3 is devoted to study ofthe vacuum energy density of scalar field in two-dimensional space-time with R × S topology.Finally, the results are discussed and analyzed in Section 5. Consider a real scalar field ϕ ( t, x ) defined on an interval 0 < x < a in an one-dimensional spacewith S topology. In this case, the boundary conditions can be written as ϕ ( t,
0) = ϕ ( t, a ) , ∂ x ϕ ( t,
0) = ∂ x ϕ ( t, a ) . (1)The scalar field equation is ( ✷ + m ) ϕ ( t, x ) = 0 . (2)The scalar product associated with this equation is( f, g ) = i Z t = const. dx ( f ∗ ∂ g − g∂ f ∗ ) , (3)where f and g are solutions of the eq. (2). It can be seen that the positive- and negative-frequency solutions of eq. (2) are u p ( k, t, x ) = e ikx − iwt q (2 π )2 w , u n ( k, t, x ) = e − ikx + iwt q (2 π )2 w . (4)These modes are orthonormalized by the following relations:( u p ( k, x, t ) , u p ( k ′ , x, t )) = δ ( k − k ′ ) , ( u n ( k, x, t ) , u n ( k ′ , x, t )) = − δ ( k − k ′ ) , ( u p ( k, x, t ) , u n ( k ′ , x, t )) = 0 . (5)2 p modes are positive norm states and the u n ’s are negative norm states. By imposing theboundary conditions (1) into (4) the positive- and negative frequency solutions can be obtainedas follows: ϕ ± N ( t, x ) = 1 q (2 aω N ) exp[ ± i ( ω N t − k N x )] , (6)where ω N = ( m + k N ) , k N = 2 πNa , N = 0 , ± , ± , .... Now the standard quantization of the field is performed by means of the expansion φ ( t, x ) = ∞ X N = −∞ [ ϕ (+) N ( t, x ) a N + ϕ ( − ) N ( t, x ) a † N ] . (7)The energy density operator is given by the 00-component of the energy-momentum tensor T = 12 { ( ∂ t φ ( t, x )) + ( ∂ x φ ( t, x )) } . (8)These the vacuum energy density of a scalar field on S can be calculated as follow [15] h | T | i = 12 a ∞ X N = −∞ ω N . (9)The total vacuum energy is E ( a, m ) = Z a h | T | i dx = 12 ∞ X N = −∞ ω N = ∞ X N =0 ω N − m . (10)The renormalization of this infinite quantity is performed by subtracting the contributionof the Minkowski space E ( a, m ) = [ ∞ X N =0 ω N − a π Z ∞ ω ( k ) dk ] − m . (11)Substituting A = am π and t = ak π , one can obtain E ( a, m ) = 2 πa [ ∞ X N =0 √ A + N − a π Z ∞ √ A + t dt ] − m . (12)Using the Abel-Plana formula [ ? ] X F ( N ) − Z ∞ F ( t ) dt = 12 F (0) + i Z ∞ dte πt − F ( it ) − F ( − it )] , (13)where F ( t ) = √ A + N , we finally obtain, E ren ( a, m ) = − πa Z ∞ A s t − A e πt − dt = − πa Z ∞ µ √ ξ − µ e ξ − dξ. (14)3here ξ = 2 πt , µ = 2 πA . In the massless case ( µ = 0) we have E ren ( a,
0) = − πa Z ∞ ξ exp( ξ ) − dξ = − π a . (15)For µ ≪ E ren ( a, m ) ≈ − √ µ √ πa e − µ , (16)i.e., the vacuum energy of the massive field is exponentially small. In the previous paper [2], we present the free field operator in the Krein space quantization φ ( t, x ) = φ p ( t, x ) + φ n ( t, x ) , (17)where φ p ( t, x ) = Z dk [ a ( k ) ϕ p ( k, x, t ) + a † ( k ) ϕ ∗ p ( k, x, t )] ,φ n ( t, x ) = Z dk [ b ( k ) ϕ n ( k, x, t ) + b † ( k ) ϕ ∗ n ( k, x, t )] , and a ( k ) and b ( k ) are two independent operators. Creation and annihilation operators areconstrained to obey the following commutation rules[ a ( k ) , a ( k ′ )] = 0 , [ a † ( k ) , a † ( k ′ )] = 0 , , [ a ( k ) , a † ( k ′ )] = δ ( k − k ′ ) , (18)[ b ( k ) , b ( k ′ )] = 0 , [ b † ( k ) , b † ( k ′ )] = 0 , , [ b ( k ) , b † ( k ′ )] = − δ ( k − k ′ ) , (19)[ a ( k ) , b ( k ′ )] = 0 , [ a † ( k ) , b † ( k ′ )] = 0 , , [ a ( k ) , b † ( k ′ )] = 0 , [ a † ( k ) , b ( k ′ )] = 0 . (20)The vacuum state | Ω > is then defined by a † ( k ) | Ω > = | k > ; a ( k ) | Ω > = 0 , (21) b † ( k ) | Ω > = | ¯1 k > ; b ( k ) | Ω > = 0 , (22) b ( k ) | k > = 0; a ( k ) | ¯1 k > = 0 , (23)where | k > is called a one particle state and | ¯1 k > is called a one “unparticle state”.We are now in a position to calculate the vacuum energy of two-dimensional space-timewith nontrivial topology in Krein space. The field operator in Krein space is build by joiningtwo possible solutions of field equation, positive and negative norms. The negative norm states,which do not interact with the physical world could be constructed with two perspective. Theseperspectives lead us to build two possible field operators.4 .1 First perspective In this case the boundary conditions (1) are intrinsic properties of the space-time. Consequentlyboth positive and negative energy basis are affected by these conditions. So the scalar fieldoperator in such this perspective (through Krein quantization method) can be written as: φ ( t, x ) = ∞ X N = −∞ [ ϕ (+) N ( t, x ) a N + ϕ ( − ) N ( t, x ) a † N ] + ∞ X N = −∞ [ ϕ ( − ) N ( t, x ) b N + ϕ (+) N ( t, x ) b † N ] , (24)where ϕ ( ± ) N ( t, x ) is defined in (5). The vacuum energy density of a scalar field on S can becalculated as follows h Ω | T Kre | Ω i = 12 a ∞ X N = −∞ ω N − a ∞ X N = −∞ ω N = 0 . (25)Therefore the vacuum energy is automatically renormalized and it is equal to zero. In thisperspective the structure of space-time is not affected by the vacuum energy. In this perspective, the gravitational field appears as an external phenomena imposed on thestructure of space-time. This is due to the fact that the boundary conditions (1) are onlyimposed on positive norm states.By imposing the above boundary conditions, the field operator in Krein QFT can be writtenas follows: φ ( t, x ) = ∞ X N = −∞ [ ϕ (+) N ( t, x ) a N + ϕ ( − ) N ( t, x ) a † N ] + Z dk [ b ( ~k ) u n ( k, x, t ) + b † ( ~k ) u ∗ n ( k, x, t )] (26)where ϕ ( ± ) N ( t, x ) and u n ( k, x, t ) are defined in (6) and (4) respectively. By Substituting theabove field operator in (7) and using Eqs (24),(25) and (26), one can easily obtain: h Ω | T Kre | Ω i = 12 a ∞ X N = −∞ ω N − π Z ∞ ω ( k ) dk. (27)The total vacuum energy in Krein QFT is E Kre ( a ) = Z a h Ω | T Kre | Ω i dx = [ ∞ X N = −∞ ω N − a π Z ∞ ω ( k ) dk ] − m . (28)It is clearly seen that once again the previous result (10) has attained. It should be notedthat this energy can not be detected locally. In the semiclassical treatment of gravitional field,however, it affects the curvature globally through R µν − Rg µν = 8 πG h T µν i , (29) i.e. it changes the structure of the space-time. In this perspective we have the quantuminstability of the space-time topology. 5 Conclusion
The Casimir effect is the simplest example of interaction field. The zero point energy of quan-tum fields in this case, i.e. Casimir energy, can be alternatively calculated as an interactionlagrangian without reference to zero point energies [9]. The goal of the presentation of Kreinspace quantization is to eliminate the singularity which appears in the interaction field. Inthe present paper, Casimir energy for two-dimensional space-time with nontrivial topology, hasbeen calculated through the Krein space quantization. Once again it is found that the theoryis automatically renormalized. The zero point energy of vacuum is found to be zero if we sup-pose that the structure of space-time is dependent on the matter it contains. If we take theperspective that the structure of space-time is not independent of matter and only boundaryconditions are imposed as the external interaction, we obtain the regular results found by otherauthors but we have the quantum instability of the space-time.
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