Bell's inequality with Dirac particles

0.07 MB·English

by  Shahpoor Moradi

Bell’s inequality with Dirac particles
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0 Shahpoor Moradi1,
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a January 4, 2012
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p 1 Department of Physics, Razi University, Kermanshah, IRAN
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n Tel: [phone], Fax: [phone]
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Abstract
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v We study Bell’s inequality using the Bell states constructed from four component
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0 Dirac spinors. Spin operator is related to the Pauli-Lubanski pseudo vector which is
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relativistic invariant operator. By using Lorentz transformation, in both Bell states and
.
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0 spin operator, weobtain an observer independentBell’s inequality, so thatit is maximally
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violated as long as it is violated maximally in the rest frame.
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r Proposed PACS number: 03.65.Ud
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Keywords: Bell’s inequality, Lorentz transformation
∗e-mail: [email protected]
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Relativistic entanglement and quantum nonlocality is investigated by many authors [1-28].
M.Czachor [7], investigated Einstein-Podolsky-Rosen experiment withrelativistic massive spin-
1 particles. The degree of violation of the Bell’s inequality is shown to depend on the velocity of
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the pair of particles with respect to the laboratory. He considered the spin singlet of two spin-1
2
massive particles moving in the same direction. He introduced the concept of a relativistic spin
observable using the relativistic center-of-mass operator. For two observers in the lab frame
measuring the spin component of each particle in the same direction, the expectation value of
the joint spin measurement, i.e., the expectation value of the tensor product of the relativistic
spinobservableofeachconstituent particle, depends ontheboostvelocity. Onlywhentheboost
speed reaches that of light, or when the direction of the spin measurements is perpendicular to
the boost direction, the results seem to agree with the EPR correlation. In the present article
we use a relativistic spin operator for spin-1 particles which is constructed from Pauli-Lubanski
2
pseudo vector. Also we use the four components Dirac spinors for constructing Bell states.
With this spin operator the Bell’s inequality is maximally violated and Bell observable does
not depend on velocity of particles.
The Pauli-Lubanski pseudo vector is
i
Wµ = ǫµνρσσ ∂ , (1)
νρ σ
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where σ = i[γ ,γ ] and in the Dirac representation the γ-matrices are
µν 2 µ ν
1 0 0 σi 0 1
γ0 =  , γi =  , γ5 =  . (2)
0 1 σi 0 1 0
 −   −   
     
     
The invariant spin observable for Dirac particle is
2W sµ 1
µ
sˆ= = γ s p = γ s, (3)
5 5
m ±m 6 6 6
2
where sµ is space-like normalized four-vector orthogonal to p. The pervious equation holds for
plane wave solutions, upper(lower) sign is related to positive(negative) energy solutions. For
rest particles the vector sµ becomes (0,n), so that
sˆ= γ γ n γ = Σ n. (4)
5 0
· ·
If we choose s along z axis, we see that following spinors are eigenstates of Σ
z
1 0
   
1 0 1 1
   
u(0, ) =  , u(0, ) =  , (5)
2   −2  
 0   0 
   
   
   
 0   0 
   
   
   
0 0
   
1 0 1 0
   
v(0, ) =  , v(0, ) =  , (6)
2   −2  
 1   0 
   
   
   
 0   1 
   
   
   
with eigenvalues +1 for u(0, 1) and v(0, 1), and 1 for u(0, 1) and v(0, 1). Entangled Bell
2 2 − −2 −2
state corresponding to positive energy spinors u(0, 1) is defined as
±2
1 1 1 1 1
Φ = u(0, ) u(0, ) u(0, ) u(0, ) . (7)
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√2 2 ⊗ −2 − −2 ⊗ 2
(cid:18) (cid:19)
Thetwoparticlestatesinvolveonlypositiveenergyspinorsuandnotthenegativeenergyspinors
v. This occurs because the positive energy states transformed among themselves separately and
do not mix with each other under Lorentz transformation [2]. We consider measurements of
the spins Σ and Σ along different directions performed on the correlated particles 1 and 2.
1 2
For particle 1, we measure either aˆ or aˆ′ where aˆ = a Σ and aˆ′ = a′ Σ . Measured values
1 1
· ·
3
of aˆ, aˆ′ are 1. Similarly, for particle 2, the quantity ˆb or ˆb′ is measured, where ˆb = b Σ and
2
± ·
ˆb′ = b′ Σ . One of the form of Bell’s inequality is
2
·
aˆ ˆb + aˆ′ ˆb + aˆ ˆb′ aˆ′ ˆb′ 2. (8)
|h ⊗ i h ⊗ i h ⊗ i−h ⊗ i| ≤
This inequality is violated by the quantum mechanical results for, say, the singlet state (7)
where aˆ ˆb = a b. It’s easy to check that the maximally violation of Bell’s inequality is
h ⊗ i − ·
equal to 2√2.
To construct the spinors in an arbitrary Lorentz frame we boost the rest frame spinors by
the standard boost from the rest frame to a frame with momentum p. Then the entangled Bell
states (7) takes the form
1 1 1 1 1
Ψ = u(p, ) u(p, ) u(p, ) u(p, ) . (9)
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√2 2 ⊗ −2 − −2 ⊗ 2
(cid:18) (cid:19)
Where the normalized positive energy spinors u(p, 1) are
±2
1 E +m ϕ(±)
u(p, ) = p  , (10)
±2 s 2Ep σ·p ϕ(±)
 Ep+m 
 
with σ ϕ(±) = ϕ(±). Now the Lorentz transformed spinors u(p, 1) are eigenstates of γ s
3 ± ±2 56
where s is the transform of s . In the frame in which the electron has momentum p the
z
polarization vector is obtained by applying the Lorentz boost
p n (p n)p
sµ = (s ,s) = · ,n+ · . (11)
0
m m(Ep +m)!
Expectation value of the spin projection γ s in state u(p) is
5
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E +m ϕ†(σ p)(σ s)(σ p)ϕ
γ s = p ϕ†(σ s)ϕ · · · . (12)
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h 6 i 2Ep · − (Ep +m)2 !
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Using the following identity
(σ p)(σ s)(σ p) = p 2(s σ s σ s σ ), (13)
z z y y x x
· · · | | − −
we have
1 1
u†(p, )(γ s)u(p, ) = γ−1s , (14)
5 z
2 6 2
1 1
u†(p, )(γ s)u(p, ) = γ−1s , (15)
5 z
−2 6 −2 −
1 1
u†(p, )(γ s)u(p, ) = s is , (16)
5 x y
2 6 −2 −
1 1
u†(p, )(γ s)u(p, ) = s +is , (17)
5 x y
−2 6 2
where γ = E/m = (1 u2)−1/2. Without loss of generality we assume that p = pzˆ, then after
−
some algebra the average of Bell operator on state Ψ to be
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Ψ aˆ ˆb Ψ = a b, (18)
11 11
h | ⊗ | i − ·
which results the Lorentz invariant Bell’s inequality. Here we compare our results with Czachor
work [7]. He considered to spin singlet of two spin-1 massive particles moving in the same direc-
2
tion. The normalized operator corresponding to the spin projection along arbitrary direction
a(a2 = 1) is [7]
(√1 β2a +a ).σ
⊥ k
aˆ = − , (19)
1 (a u)2
− ×
q
where the subscripts and denote the components which are perpendicular and parallel
⊥ k
to the boost speed u. In this case the average of the relativistic EPR operator to be
a b u2a b
ˆ ⊥ ⊥
Ψ aˆ b Ψ = · − · , (20)
11 11
h | ⊗ | i − 1 (a u)2 1 (b u)2
− × − ×
q q
Then against our results the expectation value is not Lorentz invariant and consequently the
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Bell’s inequality is speed dependent.
In conclusion we have discussed EPR-type experiment with Dirac particles. For obtaining
relativistic invariant Bell’s inequality the spin operator for Dirac particles should be Lorentz
transformed. Then in opposition of Czachor’s results[7], since the degree of violation does not
depend on momentums of particles, we have the relativistic invariant Bell’s inequality.
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