A short note on Casimir force and radius stabilization in QFT with non-commutative target space
aa r X i v : . [ phy s i c s . g e n - ph ] N ov A short note on Casimir force and radius stabilization in QFT with non-commutativetarget space
Michal Demetrian ∗ Comenius UniversityMlynska Dolina F2, 842 48,Bratislava, Slovak Republic (Dated: November 12, 2020)Stable radius of cylindrical space due to additional repulsion caused by noncommutativity oftwo-component field values is found.
PACS numbers: 11.10.-, 11.10,Kk.
I. INTRODUCTION
Since the famous paper of Snyder [1] the idea to buildup quantum theory on noncommutative spaces has beenwidely studied by many authors. Quantum physics onnoncommutative space became one of the main trendsin modern physics as possible tool to study how space-time structure itself acts upon matter at very small scalesand how space-time itself can be understood as quan-tum object, or quantized. Appearance of Casimir forcein quantum fields under some nontrivial geometrical ortopological situations has been studied widely in bothcommutative and non-commutative cases, as for exam-ple in [3]. Special attention is often paid to the effect ofradius stabilization of suitable manifolds. In this shortnote we transfer the notion of non-commutativity intothe abstract space of values of quantized field in verysimple case of scalar field. The idea of this work followsclearly explained treatment of the work [4], where thereader can find references to other related papers.
II. THE COMMUTATIVE MODEL
Let us start with two (in fact, there is no way howto proceed with single-valued fields, on the other way,one may try to use the quaternions, see [2]) compo-nent real scalar ( ϕ , ϕ ) , ϕ a = ϕ a ( x, t ) defined overtwo dimensional cylinder, with spatial period equal to2 πR . This means we assume the periodic conditions: ϕ ( x, t ) = ϕ ( x + 2 πR, t ) , ∂ x ( x, t ) = ∂ x ( x + 2 πR, t ) hold.Such fields can be decomposed into its Fourier spatialcomponents as follows ϕ a ( x, t ) = X n ∈ Z e πiR nx φ an ( t ) , φ an = φ a⋆ − n , a = 1 , , where the last condition guarantees reality of ϕ a . Let usconsider ϕ a be a pair of massless free scalars, this means ∗ Electronic address: [email protected] the dynamics is given by the lagrangian L = 12 Z d x (cid:2) ( ∂ t ϕ ) + ( ∂ t ϕ ) − ( ∂ x ϕ ) − ( ∂ x ϕ ) (cid:3) = R X a,n " ˙ φ an ˙ φ a − n − (cid:18) πnR (cid:19) φ an φ a − n . (1)Defining momenta: π an = ∂L∂ ˙ φ an = R ˙ φ a − n = ( π a − n ) ⋆ one easily constructs the Hamiltonian of our system: H = X a ( π a ) R + 12 R X a,n =0 (cid:2) π an π a − n + (2 πn ) φ an φ a − n (cid:3) , (2)where[ φ am , φ bn ] = [ π am , π bn ] = 0 , [ φ am , π bn ] = iδ mn δ ab . III. THE MODEL WITH NONCOMMUTATIVEPLANE AS TARGET SPACE
Motivated by the work [4] we shall consider the follow-ing modification of the canonical commutation relations h ˆ φ an , ˆ φ bm i = iR ǫ ab θ ( n ) δ m + n, , (cid:2) ˆ π an , ˆ π bm (cid:3) = 0 , (3) h ˆ φ an , ˆ π bm i = iδ ab δ mn , where the smearing function θ depends upon index of themode: θ ( n ) = ϑe − π σ R n . (4)In this expression ϑ is the parameter of non-commutativity of dimension L . The commutation re-lations (3) (the first one considering coordinates) can berewritten for spatially dependent field coordinates as thefollowing equal-time commutation relation with Gaussiansmearing on the right-hand side: (cid:2) ˆ ϕ a ( x, t ) , ˆ ϕ b ( y, t ) (cid:3) = iǫ ab ϑ √ πσ e − ( x − y )22 σ . This reduces, in the limit σ →
0, to the local commuta-tion rule: (cid:2) ˆ ϕ a ( x, t ) , ˆ ϕ b ( y, t ) (cid:3) = iǫ ab ϑδ ( x − y ) , by means of the standard gaussian approximation ofDirac mapping. We shall consider, accordingly with [4],the system which dynamics is given by the followinghamiltonian: H = 12 gR X a,n h ˆ π an ˆ π a − n + (2 πn ) ˆ φ an ˆ φ a − n i . (5)Now, the idea is, that the commutation relations (3) canbe transformed to the canonical ones by the linear trans-formation [5]: ˆ φ an = φ an − R ǫ ab θ ( n ) π b − n , ˆ π an = π an . This transforms our hamiltonian (5) into the form: H = X a,n (" (cid:18) πnθ ( n ) R (cid:19) π an π a − n + (2 πn ) φ an φ a − n ) − X a,b,n (2 πn ) R ǫ ab θ ( n ) φ an π bn . (6)The second term in this hamiltonian is proportional tothe (1 ,
2) component of the ”angular momentum” oper-ator corresponding to the rotations in field plane. Thisterm can be interpreted as an effective interaction in fieldplane with auxiliary magnetic-like field. The hamiltoniancan be simply diagonalised with help of properly chosenannihilation and creation operators. In order to do this,let us define some useful notations: ω n = 2 π | n | R , Ω n = 1 + (cid:18) πnθ ( n ) R (cid:19) . We introduce the operators: a an = r ∆ n (cid:18) φ an + i π a − n ∆ n (cid:19) , a a † n = r ∆ n (cid:18) φ a − n − i π an ∆ n (cid:19) , where ∆ n = Rω n Ω n = 2 π | n | Ω n . This pair of operators obeys the canonical commutationrelations: (cid:2) a an , a bm (cid:3) = (cid:2) a a † n , a b † m (cid:3) = 0 , (cid:2) a an , a b † m (cid:3) = δ abmn , and our hamiltonian (6) is given by H = X a ( π a ) R + X a,n =0 ω n Ω n a a † n a an + i X a,b,n =0 θ ( n ) ω n ǫ ab a a † n a bn . (7)The following canonical transformation A n = 1 √ (cid:0) a n − ia n (cid:1) , A n = 1 √ (cid:0) a n + ia n (cid:1) transforms our hamiltonian into the final diagonal form: H = X n =0 ω n (cid:2) Λ n A n A † n + Λ n A n A † n (cid:3) , (8)where we have introduced the notation:Λ n = Ω n − π | n | θ ( n ) R , Λ n = Ω n + π | n | θ ( n ) R .
This result tells us that the magnetic like term causessplitting of the energy levels of the free-field hamiltonian.
IV. CASIMIR ENERGY
We shall rewrite the hamiltonian (8) into a more suit-able form: H = X n =0 (cid:8) ω n Ω n (cid:2) A n A † n + A n A † n (cid:3) + π | n | θ ( n ) R (cid:2) A n A † n − A n A † n (cid:3)(cid:27) . (9)We are interested in vacuum expectation value of this op-erator. It is evident that the 1 ↔ E C = h | H | i = X n =0 ω n Ω n (cid:10) (cid:2) A n A † n + A n A † n (cid:3) (cid:11) = X n =0 ω n (cid:10) (cid:2) A n A † n + A n A † n (cid:3) (cid:11) + π g R X n =0 ω n n θ ( n ) (cid:10) (cid:2) A n A † n + A n A † n (cid:3) (cid:11) ≡ E C + E C . The term E C that does not contain noncommutativityparameter is the standard one and its value (after neces-sary regularization) can be find in e.g. [6], or in an ex-haustive explanation of methods of computation is givenin the paper [7]. E C reads E renC = − R .
The ϑ -dependent contribution to the Casimir energy isalready regularized by σ in (4) and reads E C ( σ ) = 4 π ϑ R ∞ X n =1 n e − π σ R n . Renormalizing it we obtain E renC = 4 π ϑ R ζ ( −
3) = π ϑ R . Finally, we have the Casimir energy given by E C = − R + π ϑ R . (10) V. DISCUSSION If ϑ = 0 then the Casimir energy (10) is not amonotonous function of the radius R . This fact makes the formula for the Casimir energy in our situation dif-ferent from the standard result that is obtained letting ϑ = 0. There is a radius R at which the Casimir energyattains its minimal value, namely R = (cid:18) π (cid:19) / ϑ. (11)We see that the non-locality of the field field commutatorsgenerates in (10) the repulsive force at small distances R < R , and effectively this force can stabilize the radiusof the space. Acknowledgments
This work was supported by the grant scheme VEGA1/0785/19 of the Ministry of Education, Science, andSport of the Slovak Republic. [1] H.S. Snyder,
Quantized space-time , Phys. Rev. , 38(1947).[2] S. L. Adler, Quaternionic Quantum Mechanics and Quan-tum Fields, Oxford Univ. Press (1995).[3] S. Nam, Casimir force in compact noncommutative extradimensions and radius stabilization , JHEP 2000 (10).[4] A.P. Balachandran, A.R. Queiroz, A.M. Marquesand P. Teotonio-Sobrinho,
Quantum fields with non-commutative target spaces , Phys.Rev.D77:105032,2008,arXiv:0706.0021v1. [5] M. Demetrian and D. Kochan,
Quantum mechanics onnoncommutative plane , Acta Phys. Slovaca, 52 (2002), 1.[6] N.D. Birell and P.C.W. Davies,
Quantum fields in curvedspace (Cambridge Monographs in Mathematical Physics),Cambridge Univ. Press (1984).[7] M. Bordag, U. Mohideen and V.M. Mostepanenko,