Relativistic Spin Operator with Observers in Motion

Transcription

1 EJTP 7, No Electronic Journal of Theoretical Physics Relativistic Spin Operator with Observers in Motion J P Singh Department of Management Studies, Indian Institute of Technology Roorkee, Roorkee 47667, INDIA Received 8 January 00, Accepted 0 February 00, Published 0 March 00 Abstract: We obtain transformation equations for the Bell basis states under an arbitrary Lorentz boost and compute the expectation values of the relativistic center of mass spin operator under each of these boosted states. We also obtain expectation values for spin projections along the axes. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Quantum Information; Entanglement; Bell Basis; Lorentz Transformations PACS 008: a; Ud; Bg; p; w. Introduction Ever since the publication of the EPR Paradox as exposed by Einstein, Podolsky & Rosen in their seminal article [], quantum entanglement has been mystifying physicists across the world. Even to this day, a complete explanation of this phenomenon defies human endeavors although its existence is, now, universally acknowledged. In fact, the focus of research has perceptibly shifted to exploiting this unique property of composite quantum systems for performing information processing tasks with unprecedented efficiency quantum entanglement is gradually being acknowledged as a resource for quantum computing and communication. It, has, therefore, become all the more necessary to examine the effect of relativistic transformations on mutually entangled subsystems. This essentially is the objective of this work. We obtain transformation equations for the Bell basis states under an arbitrary boost and compute the expectation values of the relativistic center of mass spin operator under each of these boosted states. Our work is based on Bohm s version of the paradox [,3] applying discrete spin states. Pioneering work in this direction was initiated by Czachor [4]. However, the paper confines itself to the averaging of the relativistic spin operator [5] that relates to the jatinfdm@iitr.ernet.in, Jatinder pal000@yahoo.com, jpsiitr@gmail.com

2 6 Electronic Journal of Theoretical Physics 7, No spatial components of the Pauli Lubanski vector over the spin singlet state in the laboratory frame. The work does not consider the boosting of the Bell states and hence, does not embrace the case of relative motion of an arbitrary observer frame. The paper does, however, establish that relativistic effects are seriously relevant to quantum computing operations. This was followed by a spurt of papers [6-8]. Each of these works, in essence, takes a certain specific case and examines its physical implications instead of solving the problem in its generality. For instance, boosts are always taken perpendicular to the momenta, Alsing [6] investigates the massless case in a specific gauge formulation wherein the photon polarization vectors are coplanar with the electric and magnetic fields, Gingrich [8] assumes a Gaussian momentum distribution, Pachos [7] looks at a relativistic formulation on the basis of a magnetic dipole-dipole interaction etc. The important point is that none of these papers attempts an averaging of the center of mass relativistic spin operator with boosted Bell states which could lead to relativistic correction in the Bell s inequalities for observers in arbitrary motion. Following exactly the same methodology as Alsing [6] i.e. using Wigner rotation & momentum eigenstates Terashima & Ueda [9-0] show that the spin singlet state undergoes a change in anti-correlation due to Lorentz transformation. However, while Alsing [6] represents the state in terms of 4 component Dirac spinors, Terashima & Ueda [9-0] use Hilbert space state vectors. While they do consider boosted states to accommodate observer frame, they do not use the center of mass relativistic spin operator. The spin observable considered in [] is different from the center of mass relativistic spin operator. Without explicitly introducing a functional form of the spin operator, they define it as an operator having the property based on the sameness of the expectation values of one particle spin measurement evaluated in two relative reference frames, one in the laboratory frame in which the particle has a velocity v and the observer is at rest, the other in the moving frame Lorentz boosted with v, in which the particle is at rest and the observer is moving with a velocity - v. However, the appropriateness of such an operator for a two/multi particle state is still an open question.. The Relativistic Spin Operator [5] The relativistic spin operator, S, has been an issue of considerable debate among physicists. The fundamental properties that need to be satisfied by an acceptable spin operator, S, are [5]:- a It must be expressible as some combination of the complete set of generators of the Poincare group i.e. P 0, P, J, K; b It must behave as a 3-vector under spatial rotations i.e. [S i,j j ]=iε ijk S k whence it must be linear in the generators J, K so that we can express it as S = f J f K f 3 P f 4 P.J P f 5 P.K P f 6 P Jf 7 P K where each f i f i P,P 0 ; c It should be independent of generators of translations i.e. [S i,p μ ]=0 d It should be equal to the difference of the total angular momentum and the orbital

3 Electronic Journal of Theoretical Physics 7, No angular momentum i.e. S = J x P; e Being an angular momentum, it must satisfy the commutators [S i,s j ]=iε ijk S k. Application of a e yield [5]:- S = P 0 M J M P K P P.J M M P 0 The above expression for the relativistic spin operator is, in fact, the value of the total angular momentum operator of the particle in its rest system. This is easily seen. The relativistic transformation of an arbitrary 3-vector that has the value x in a system with momentum P to its rest frame is given by x = x P x.pmx0 whence, MMP 0 the expression for the spatial part of the Pauli Lubanski operator W μ = εαβχμ J αβ P χ W = P 0 J P K,W 0 =P.J in its rest frame becomes W = P 0 J P K P P.J. Defining the relativistic spin operator S by P 0 J = MJ = W = MS, we MP 0 obtain the above expression for S. In the canonical representation of the Poincare group characterized by the momentum operator being diagonal i.e. P μ = p μ and the infinitesimal generators of rotations and boosts taking the form J = ip p S and K = i p p S [], we have P.J =0 M whence S = P 0 J P K = W. In this representation, one can set S = s = σ M M M being the two dimensional representation D / of the little group SO 3 in the rest frame. Identifying P 0 J and P K respectively as being the components of spin M M parallel and perpendicular to direction of momentum p, and using the above explicit representation of J, K, we obtain S = S S = p0 M p p.s p s p p.s p In the rest frame, obviously, S = s = p p.s p. 3. The Lorentz Transformation of Single Particle States The effect of Lorentz boosts on single particle states is well documented and can be expressed in terms of the Wigner rotation R W p, Λ [3] as: U Λ p, s = Λp 0 s Dσs j R W p Λp,σ 3 σ=s Explicit expression for the Wigner rotation has been obtained by various approaches. Following Halpern [4], we can, making use of the SL, C representation of the Lorentz group, obtain an explicit representation of R W p, Λ for spin massive particles as: D / R W p, Λ = cos θ i σ.nsin θ 4

4 64 Electronic Journal of Theoretical Physics 7, No where cos n sin θ = θ = cosh α cosh β e.f sinh α sinh β [ cosh α cosh β e.f sinh α sinh β ] / 5 e f sinh α sinh β [ cosh α cosh β e.f sinh α sinh β cosh α =Λ 0 0 = γ =vb /,coshβ = p0, f = p M p direction of the boost Λ. 4. The Lorentz Transformation of Bell States ] / 6 and e is the unit vector in the The Bell state basis for a pair of spin particles with respective momenta p, p in the rest frame consists of the basis vectors Φ = p, p, p, p,, Φ = p, p, p, p,, Ψ = p, p, p, p, and Ψ = p, p, p, p,. The Lorentz transformation induced by Λ transforms these basis states as follows: i Φ transforms as: U Λ Φ = / σ,σ =/ Now, making use of D / R W p, Λ = cos Λp 0 D / R W p, p 0, Λ =cos we obtain D / σ,/ R W p Λp 0 D / σ,/ R W p θ i σ.nsin θ i σ.nsin Λp, 0 D / σ,/ p, Λp, σ Λp, σ Λp, 0 D / σ,/ p, Λp, σ Λp, σ θ cos θ = in 3 sin θ in n sin θ in n sin θ cos θ in 3 sin θ θ = cos θ in 3 sin θ in n sin θ in n sin θ cos θ in 3 sin θ U Λ Φ = Λp 0 = Λp 0 { [ ] Λp, n n in sin θ [ ] n sin θ cos θ in 3 sin θ Λp, Λp, [ n cos θ ] Λp, [ Λp, n sin θ cos θ in 3 sin θ Λp, Λp, ] Λp, Λp, [ n n in sin θ ] Λp, Λp, Λp, in n cos θ Λp, Λp, Λp, Λp, n sin θ Λp, Λp, Λp, Λp, in n 3 cos θ Λp, Λp, Λp, Λp,

5 Electronic Journal of Theoretical Physics 7, No = Λp0 { [ n cos θ ] } Φ Λ in n cos θ Φ Λ n sin θψ Λ in n 3 cos θ Ψ Λ 7 ii Φ transforms as: U Λ Φ = / { = Λp 0 σ,σ =/ Λp 0 D / σ,/ R W p Λp 0 D / σ,/ R W p [n n in cos θ ] Λp, Λp, Λp, 0 D / σ,/ p, Λp, σ Λp, σ Λp, 0 D / σ,/ p, Λp, σ Λp, σ [in sin θ n n 3 cos θ] Λp, Λp, [in sin θ n n 3 cos θ] Λp, Λp, [ n n in cos θ] Λp, Λp, = Λp 0 [ n cos θ] Λp, Λp, Λp, Λp, in n cos θ Λp, Λp, Λp, Λp, in sin θ Λp, Λp, Λp, Λp, n n 3 cos θ Λp, Λp, Λp, Λp, } = Λp0 { in n cos θ Φ Λ [ n cos θ ] } Φ Λ in sin θψ Λ n n 3 cos θ Ψ Λ iii For Ψ,wehave 8 U Λ Ψ = / = Λp 0 σ,σ =/ Λp 0 D / σ,/ R W p Λp 0 D / σ,/ R W p Λp, 0 D / σ,/ p, Λp, σ Λp, σ Λp, 0 D / σ,/ p, Λp, σ Λp, σ { [n 3 n in cos θ] Λp, Λp, [ in3 sin θ n 3 cos θ] Λp, Λp, [ in 3 sin θ n 3 cos θ] Λp, Λp, [n3 n in cos θ] Λp, Λp, = Λp 0 in n 3 cos θ Λp, Λp, Λp, Λp, n n 3 cos θ Λp, Λp, Λp, Λp, [ n 3 cos θ] Λp, Λp, Λp, Λp, in 3 sin θ Λp, Λp, Λp, Λp, } = Λp0 { in n 3 cos θ Φ Λ n n 3 cos θ Φ Λ [ n 3 cos θ ] } Ψ Λ in 3 sin θψ Λ iv Transformation of Ψ is as follows: 9 U Λ Ψ = / σ,σ =/ Λp 0 D / σ,/ R W p Λp 0 D / σ,/ R W p Λp, 0 D / σ,/ p, Λp, σ Λp, σ Λp, 0 D / σ,/ p, Λp, σ Λp, σ

6 66 Electronic Journal of Theoretical Physics 7, No { = Λp 0 = Λp 0 in n sin θ Λp, Λp, cos θ in3 sin θ Λp, Λp, cos θ in 3 sin θ Λp, Λp, in n sin θ Λp, Λp, n sin θ Λp, Λp, Λp, Λp, in sin θ Λp, Λp, Λp, Λp, in 3 sin θ Λp, Λp, Λp, Λp, cos θ Λp, Λp, Λp, Λp, } = Λp0 { } n sin θφ Λ in sin θφ Λ in 3 sin θψ Λ cos θψ Λ 0 5. The Expectation of the Relativistic Spin Observable in Arbitrarily Boosted Bell States Given a unit vectorm, the projection of spin S in the direction of m is obtained as m.s = [ m.s ] p 0 p M m.pp.s with a magnitude /M. We can accordingly, define normalized spin observable as S m = m.sνm.pp.s /M = m.σνm.pp.σ /M p whereν =. Identifying unit vectors M a, b in relation to the two particles constituting the paired state, the spin observable for the paired state is obtained as S ab = S a S b. Using the abbreviationsa i = a i ν a.p p i, B i = b i ν b.p p i, i =,, 3, we can express the action of this observable on the various basis states as:- S a S b Λp, Λp, = M A 3 B 3 Λp, Λp, A3 B ib Λp, Λp, A ia B 3 Λp, Λp, A ia B ib Λp, Λp, A 3 B ib Λp, Λp, A3 B 3 Λp, S a S b Λp, Λp, = M Λp, A ia B ib Λp, Λp, A ia B 3 Λp, Λp, 3 A ia B 3 Λp, Λp, A ia B ib S a S b Λp, Λp, = M Λp, Λp, A3 B 3Λp, Λp, A 3 B ib Λp, Λp, 4

7 Electronic Journal of Theoretical Physics 7, No A ia B ib Λp, Λp, A ia B 3 S a S b Λp, Λp, = M Λp, Λp, A3 B ib Λp, Λp, A3 B 3 Λp, Λp, 5 Furthermore, the expectation values of S ab states are given by: = S a S b in the various boosted Bell Φ Λ SabΦ M Λ = N c in c A 3 B 3 N c in c A3 B ib N s in 3 c A ia B 3 N s in3 c A ia B ib N c in c N s in 3 c A 3 B ib N c in c A3 B 3 N s in3 c A ia B ib N s in 3 c A ia B 3 N c in c N s in3 c A ia B 3 N c in c A ia B ib N s in 3 c A 3 B 3 N s in3 c A3 B ib N c in c N c in c A ia B ib N c in c A ia B 3 N s in c 3 A 3 B ib N s in 3 c A3 B 3 N c in c = M [ A 3 B 3 N c ] [ N c A B A B N c ] N c 4A B A B [ N c N c N A3 B s ] [ 3 N c N 3 A B A B s ] N c 3 4A B A B N s [ ] N c 3 4A3 N c B N s B N c 3 4A3 N c [ ] [ ] [ ] B N s B N c 3 4B3 N c A N s A N c 3 4B3 N c A N s A N c 3 6 Φ Λ SabΦ M Λ = N c in c A 3 B 3 N c in c A3 B ib in s N 3 c A ia B 3 in s N3 c A ia B ib N c in c in s N3 c A 3 B ib N c in c A3 B 3 in s N3 c A ia B ib in s N 3 c A ia B 3 N c in c in s N3 c A ia B 3 N c in c A ia B ib in s N 3 c A 3 B 3 in s N3 c A3 B ib N c in c N c in c A ia B ib N c in c A ia B 3 in s N3 c A 3 B ib in s N 3 c A3 B 3 N c in c = M [ A 3 B 3 N c ] [ N c A B A B N c ] N c 4A B A B [ N c N c N A3 B c ] [ 3 3 N s N A B A B c ] 3 N s 4A B A B N c [ ] 3 N s 4A3 N c B N c 3 B N s 4A3 N c [ ] [ ] [ ] B N c 3 B N s 4B3 N c A N c 3 A N s 4B3 N c A N c 3 A N s 7

8 68 Electronic Journal of Theoretical Physics 7, No Ψ Λ SabΨ M Λ = N3 c in 3 c A 3 B 3 N c 3 in3 c A3 B ib N33 c in s 3 A ia B 3 N c 33 in3 s A ia B ib N3 c in 3 c N c 33 in3 s A 3 B ib N3 c in 3 c A3 B 3 N c 33 in3 s A ia B ib N33 c in 3 s A ia B 3 N c 3 in3 c N c 33 in3 s A ia B 3 N c 3 in3 c A ia B ib N33 c in 3 s A 3 B 3 N c 33 in3 s A3 B ib N3 c in 3 c N c 3 in3 c A ia B ib N3 c in c 3 A ia B 3 N c 33 in s 3 A 3 B ib N33 c in 3 s A3 B 3 N c 3 in3 c [ N A 3 B c ] [ 3 3 N c N 3 A B A B c ] 3 N c 3 4 N c 3 N c 3 A B A B [ = M A 3 B 3 N c ] [ 33 N s 3 A B A B N c ] 33 N s 3 4A B A B [ ] [ ] N c 33 N s 3 4A3 N c 33 B N c 3 B N c 3 4A3 N s 3 B N c 3 B N c 3 8 [ ] [ ] 4B 3 N c 33 A N c 3 A N c 3 4B3 N s 3 A N c 3 A N c 3 Ψ Λ SabΨ M Λ = N s in s A 3 B 3 N s in s A3 B ib cos θ in s 3 A ia B 3 cos θ in s 3 A ia B ib N s in s A 3 B ib N cos θ in s s in s A3 B 3 cos θ in s 3 3 A ia B ib cos θ in3 s A ia B 3 N s in s A ia B 3 N s cos θ in s in s A ia B ib cos θ in s 3 3 A 3 B 3 cos θ in s 3 A3 B ib N s in s N s in s A ia B ib N s in s A ia B 3 cos θ in s 3 A 3 B ib cos θ in3 s A B N s in s = M [ N A 3 B s ] [ 3 N s N A B A B s ] N s A B A B A 3 B 3 [cos θ N3 s ] A B A B 4 N s N s [ cos θ N3 s ] 4 A B A B N s 3 cos θ 4A3 cos θ [ ] B N s B N s 4A3 N s 3 [ ] B N s B N s 4B3 cos θ [ ] [ ] A N s A N s 4B3 N s 3 A N s A N s 9 where N s i = n i sin θ, N s ij = n i n j sin θ, N c i = n i cos θ and N c ij = n i n j cos θ. 6. Expectation Values of Spin Projections along the axes Using the above expressions, we can, now, compute the expectation values of the projection of the spin operator S ab = S a S b along the various set of axes. However, the

9 Electronic Journal of Theoretical Physics 7, No calculations are elaborate and tedious. We can simplify them to some extent without loss of generality by setting the coordinate axes such that the pair momenta is oriented along the x direction i.e. p p, 0, 0 so that n 0,n,n 3 and A B i = a b i νp δ,i = p a b i [ 0 δ M,i ]. Combinations of unit vectors along the three axes and the corresponding expectation values in the various Bell states are tabulated below: State Φ Λ a, 0, 0 0,, 0 0, 0, b, 0, 0 N c 4 M N N s N3 c p N s 3 c 4 M N 0 p N c s 0 0,, 0 4 M N s p N3 c M N c 4 M N 0 N s N3 c N c 3 c 0, 0, 4 M N c p N s 4 M N c 0 N3 c M N c N c N s N3 c State Φ Λ a, 0, 0 0,, 0 0, 0, b, 0, ,, 0 0 M 0 0, 0, 0 0 M

10 70 Electronic Journal of Theoretical Physics 7, No State Ψ Λ a, 0, 0 0,, 0 0, 0, b, 0, 0 N 33 c 4 M N N3 s N3 c p 33N c 3 s 4 M N 0 p 3N s 3 c 0 0,, 0 4 M N c p 33N3 s M N 33 c 4 M N 0 N3 s N3 c 33N c 3 c 0, 0, 4 M N c p N s 4 M N c 0 33N3 c M N 3 c N3 s N33 c State Ψ Λ a, 0, 0 0,, 0 0, 0, b, 0, 0 N s N3 s 4 M N cos p 3cosθ s 4 M N 0 p cosθ s 0 θ 0,, 0 4 M N s p 3cosθ M N 3 s N s 4 M N 0 cos p θ 0 N s 3 s 0, 0, 4 M N s p cosθ 4 M N s 0 3N s M N s N3 s cos θ The important point to be noted here is that, unlike in the nonrelativistic case, the relativistic case becomes momentum dependent. Furthermore, the composition of the projection axes that lead to maximal violation of Bell s inequalities in the relativistic case is still an open question. We employ momentum eigenstates in the above formulation as is the case in [6-8]. Mathematically, momentum eigenstates are not square-integrable functions and thus are outside the L Hilbert space. Physically, they represent an idealization of the absolutely sharp momentum. Any real state is represented by a wave packet, which is most conveniently decomposed as a continuous superposition of the momentum eigenstates. Those superpositions are square-integrable, i.e. normalizable. The convention of using momentum eigenstates is an approximation. It is strictly true only as a limiting case. Nevertheless, it is a very good approximation in many cases, e.g. in the scatteringexperiments of a high-energy physics. In the momentum eigenstates representation, Lorentz boost on a pair state operates

11 Electronic Journal of Theoretical Physics 7, No through local unitary operators independently on the constituent particle states thereby preserving entanglement and not mixing spin-momentum entanglement so that the spin reduced density matrix is covariant. But when the momenta of the particles is not sharply defined then spin-momentum entanglement as well as spin-spin entanglement becomes frame dependent [5]. References [] A Einstein, B Podolsky, & N Rosen, Phys. Rev. 47, 777, 935. [] A Bohm, Quantum Mechanics 3rd Edition, Springer-Verlag, New York, 993. [3] J S Bell, Speakables & Unspeakables in Quantum Mechanics, Cambridge University Press, 004. [4] M. Czachor 003 arxiv: quant-ph/96090v. [5] R. A. Berg, J. Math. Physics, 4, No., [6] P. M. Alsing and G. J. Milburn 00 arxiv: quant-ph/00305v. [7] J. Pachos and E. Solano 00 arxiv: quant-ph/003065v. [8] R. M. Gingrich and C. Adami 00 arxiv: quant-ph/00579v4. [9] H.Terashima and M. Ueda 00 arxiv: quant-ph/00438v4. [0] H. Terashima and M. Ueda 00 arxiv: quant-ph/077v. [] D. Lee and EE Chang-Young 004 arxiv: quant-ph/030856v. [] Y Ohnuki, Unitary Representations of the Poincare Group & Relativistic Wave Equations, World Scientific, 976. [3] S Weinberg, The Quantum theory of Fields, Cambridge University Press, 995. [4] F. R.Halpern,Special Relativity and Quantum Mechanics, Prentice Hall, 968. [5] A Peres & D R Terno, arxiv:quant-ph

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