On the calculation rule of probability of
relativistic free particle in quantum mechanics
T. Mei
(Department of Journal, Central China Normal University, Wuhan, Hubei PRO, People’s Republic of China E-Mail: [email protected] [email protected] )
Abstract: As is well known, in quantum mechanics, the calculation rule of the probability that an eigen-value an is observed when the physical quantity A is measured for a state described by the state vector Ψ 〉 is P(an ) = 〈Ψ An 〉〈 An Ψ 〉 . However, in Ref.[1], based on strict logical reasoning and mathematical calculation, it has been pointed out, replacing 〈Ψ An 〉〈 An Ψ 〉 , one should use a new rule to calculate P(an ) for particle satisfying the Dirac equation. In this paper, we first state some results given by Ref.[1]. And then, we present a proof for the new calculation rule of probability according to Dirac sea of negative energy particles, hole theory and the principle “the vacuum is not observable”. Finally, we discuss simply the case of particle satisfying the Klein-Gordon equation. Key words: the Dirac theory; calculation rule of probability; Dirac sea; the Klein-Gordon equation
1 Some results given by Ref.[1] 1.1 Definition and determination of some distribution functions
Definitions of the three distribution functions Λ(v) , f ( p) and ρ(t , x ; t0 , x0 ) are as follows: ① The velocity distribution function of a ponit particle Λ(v) : When the position of a point particle is fully measured at a moment, the probability that the value of the velocity of the point particle is situated between v and v + dv is Λ(v)d 3v . ② The momentum distribution function of a ponit particle f ( p) : When the position of a point particle is fully measured at a moment, the probability that the value of the momentum of the 3 point particle is situated between p and p + dp is f ( p)d p .
③ The position distribution function of a free ponit particle ρ(t , x ; t0 , x0 ) : If a free point particle is situated at position x0 at time t0 , then the probability that the free point particle is 3 situated between x and x + dx at time t is ρ(t , x ; t0 , x0 )d x , where t > t0 . Based on the principle the values of probability measured in different inertial frames are the same and some mathematical derivation, the above three distribution functions are determinated, and the concrete forms of nonrelativistic and relativistic cases are given by Tab. 1 and Tab. 2,
1 respectively. . Tab. 1 Various distribution density functions in the nonrelativistic case Physical quantity Distribution density function
velocity Λ1(v) = Λ1(0) = constant
1 momentum f ( p) = 1 (2π)3
1 m3 ρ (t , x ; t , x ) = position density 1 0 0 3 3 (2π) ()t − t0 Free particle 3 1 m (x − x0 ) current density j1(t , x) = 3 4 (2π) ()t − t0
Tab. 2 Various distribution density functions in the relativistic case Physical quantity Distribution density function
Λ (0) Λ (v) = 2 velocity 2 2 ()1− v2
1 1 f ( p) = momentum 2 3 (2π) 1+ p2 / m2
1 m3(t − t ) 0 , x − x < t − t ; position 3 2 0 0 (2π) 2 2 ρ2 (t , x ; t0 , x0 ) = []()t − t0 − (x − x0 ) density Free 0 , x − x0 ≥ t − t0 .
1 m3 (x − x ) particle 0 , x − x < t − t ; current 3 2 0 0 (2π) 2 2 j2 (t , x) = []()t − t0 − (x − x0 ) density 0 , x − x0 ≥ t − t0 .
In Tab. 1 and Tab. 2, the form of the current density function of a free point particle j(t , x) is given by
x − x0 ji (t , x) = ρi (t , x ; t0 , x0 )v = ρi (t , x ; t0 , x0 ) (i =1, 2) , t − t0 and from Tab. 1 and Tab. 2 we can verify easily the current conservation ∂ρ (t , x ; t , x ) i 0 0 + ∇ ⋅ j (t , x) = 0 (i =1, 2) . ∂t i 1.2 Comparing the corresponding conclusions in quantum mechanics with the results in Tab. 1 and Tab.2 In Ref. [1], it is point out that the conclusions about nonrelativistic particle in quantum
2 mechanics are in accord with Tab. 1, but the conclusions about relativistic particle in quantum mechanics are not in accord with Tab. 2.
At first, in quantum mechanics, when a point particle is situated at position x0 at time t0 , the state of the point particle is described by the state vector t 〉 = x 〉 ; and, no matter whether it 0 0 is nonrelativistic or relativistic case, if we measure momentum for this state, then the probability density f ( p) that momentum volue p is measured is
1 −ip⋅x0 1 ip⋅x0 1 f ( p) = 〈x0 p〉〈 p x0 〉 = e ⋅ e = . (2π)3 / 2 (2π)3 / 2 (2π)3
Comparing f ( p) with f1 ( p) in Tab. 1 and f 2 ( p) in Tab. 2, respectively, we see that
f ( p) = f1 ( p) in nonrelativistic case but f ( p) ≠ f 2 ( p) in relativistic case.
Secondly, in quantum mechanics, the physical meaning of the propagator K(t , x ; t0 , x0 ) of a particle is
+ K (t , x ; t0 , x0 ) ⋅ K(t , x ; t0 , x0 ) = ρ(t , x ; t0 , x0 ) .
And, further, the propagator of a nonrelativistic free particle is 3 2 2 p ()x− x0 3 −i ()t−t 2 i d p ip⋅()x−x 0 m t−t K (t , x ; t , x ) = e 0 e 2m = e 0 , 0 0 0 ∫ 3 (2π) 2πi()t − t0 we therefore have
3 + 1 m K0 (t , x ; t0 , x0 ) ⋅ K0 (t , x ; t0 , x0 ) = 3 3 = ρ1(t , x ; t0 , x0 ) , (2π) ()t − t0
where ρ1(t , x ; t0 , x0 ) is given by Tab. 1. We see that such result is in accord with Tab. 1.
On the other hand, in relativistic case, the propagator KD (t , x , t0 , x0 ) of a free particle
[2] determined by the Dirac Hamiltonian H D = α ⋅ pˆ + βm is
3 d p ip⋅()(x − x0 −i α⋅ p+βm )()t −t0 KD (t , x , t0 , x0 ) = e e ∫ (2π)3
3 d p ip⋅()x− x0 2 2 α ⋅ p + βm 2 2 = e cos p + m ()t − t0 − i sin p + m ()t − t0 ∫ (2π)3 2 2 p + m ∂ ∂ = ii + α ⋅ + βm∆()x − x0 , ∂t i∂x where[2]
2 2 3 sin p + m t J (ms) , x < t ; d p ip⋅x 1 ∂ 0 ∆(x) = − e = (1) ∫ 3 2 2 (2π) p + m 4πr ∂r 0 , x > t .
2 2 In the above formula, J0 (ms) is a Bseeel function, s = t − x . Hence, when x > t ,
3 + KD (t , x ; t0 , x0 ) ⋅ KD (t , x ; t0 , x0 ) = 0 ; however, when x < t , from (1) we can be prove
3 + 1 m (t − t0 ) KD (t , x ; t0 , x0 ) ⋅ KD (t , x ; t0 , x0 ) ≠ , (2π)3 2 2 2 []()t − t0 − (x − x0 ) hence,
+ KD (t , x ; t0 , x0 ) ⋅ KD (t , x ; t0 , x0 ) ≠ ρ2 (t , x ; t0 , x0 ) ,
where ρ 2 (t , x ; t0 , x0 ) is given by Tab. 2.
1.3 Three mathematical formulas The following three mathematical formulas have been given and proved in Ref.[1].
Defining a function I1(t , x) :
d3 p d3 p′ m ip⋅x 2 2 ip′⋅x ′2 2 I1(t , x) ≡ 3 e cos p + m t × 3 e cos p + m t ∫ (2π) ∫ (2π) p′2 + m2
d3 p d3 p′ m ip⋅x 2 2 ip′⋅x ′2 2 + 3 e sin p + m t × 3 e sin p + m t , ∫ (2π) ∫ (2π) p′2 + m2 the first formula is 1 m3t , x < t ; (2π)3 2 2 2 I1 (t , x) = ()t − x 0 , x ≥ t . Notice that
I1 (t − x , t0 − x0 ) = ρ 2 (t , x ; t0 , x0 ) .
And then, based on the following operators and functions
3 ˆ 1 α ⋅ p + βm d p ip⋅()x− x′ 1 α ⋅ p + βm M ± = p〉 1± 〈 p , M ± ()x , x′ = e 1± . ∑ 2 2 2 ∫ (2π)3 2 2 2 p p + m p + m
K (t , x ; t , x ) = M x , x′ d 3 x′K (t , x′ ; t , x ) , D± 0 0 ∫ ± ( ) D 0 0 we consider a function I2 (t , x ; t0 , x0 ) :
I (t , x ; t , x ) ≡ K + (t , x ; t , x ) ⋅ K (t , x ; t , x ) 2 0 0 D+ 0 0 D+ 0 0 + − K D− (t , x ; t0 , x0 ) ⋅ K D− (t , x ; t0 , x0 ) , the second formula is
4 I2 (t , x ; t0 , x0 ) = β I1(t − x , t0 − x0 ) = β ρ2 (t , x ; t0 , x0 ) 3 1 m ()t − t0 β , x − x0 < t − t0 ; (2π)3 2 2 2 = []()t − t0 − (x − x0 ) 0 , x − x0 ≥ t − t0 .
Finally, for a function I3 (t , x ; t0 , x0 ) :
I (t , x ; t , x ) ≡ K + (t , x ; t , x ) ⋅ α ⋅ K (t , x ; t , x ) 3 0 0 D+ 0 0 D+ 0 0 + − K D− (t , x ; t0 , x0 ) ⋅ α ⋅ K D− (t , x ; t0 , x0 ) , the third formula is
I3(t , x ; t0 , x0 ) = β j2 (t , x) 3 1 m (x − x0 ) β , x − x0 < t − t0 ; (2π)3 2 2 2 = []()t − t0 − (x − x0 ) 0 , x − x0 ≥ t − t0 . where j2 (t , x) is given by Tab. 2.
1.4 New calculation rule of probability Based on the above discussion, in Ref.[1], a new calculation rule of probability for a free particle satisfying the Dirac equation has been given as follows. ~ For an arbitrary state described by the state vector Ψ 〉 , we define
~ ˆ ~ Ψ ± 〉 = M ± Ψ 〉 ;
For a physical quantity A of which the eigenvalue equation reads ˆ AAn (x) = an An (x) , (2)
the probability P(an ) that an eigen-value an is observed when the physical quantity A is
~ measured for the state Ψ 〉 is:
~ ~ ~ ~ P(an ) = 〈Ψ + An 〉〈 An Ψ + 〉 − 〈Ψ − An 〉〈 An Ψ − 〉 (3) ~ ˆ ˆ ~ ~ ˆ ˆ ~ = 〈Ψ M + An 〉〈 An M + Ψ 〉 − 〈Ψ M − An 〉〈 An M − Ψ 〉 .
In Ref.[1], it has been proved that, the results obtained by the above calculation rule (3) are in accord with Tab. 2 for a free particle satisfying the Dirac equation. Especially, taking advantage of the above thirt formula, it has been proved that the three dimensioal current density ~ ~ ~ ~ 〈Ψ + x〉α〈xΨ + 〉 − 〈Ψ − x〉α〈xΨ − 〉 corresponding to position probability density ~ ~ ~ ~ 〈Ψ + x〉〈 xΨ + 〉 − 〈Ψ − x〉〈 xΨ − 〉 obtained by (3) for a free particle equals j2 (t , x) in Tab. 2, hence, so called “zitterbewegung”[2] does not appear. In this paper, we prove that the calculation rule of probability (3) can be derived from the
5 idea of Dirac sea of negative energy particles, hole theory and the principle “the vacuum is not observable”.
2 Discussion in coordinate representation 2.1 Some well known characteristics of the Dirac theory 1 1 We considering the complete set of eigenstates U ( p)〉 , V ( p)〉 , r = − , of the Dirac r r 2 2
Hamilton H D = α ⋅ pˆ + βm , which satisfy
α ⋅ pˆ + βm U ( p)〉 = p2 + m2 U ( p)〉 , α ⋅ pˆ + βm V ( p)〉 = − p2 + m2 V ( p)〉 , (4) ()r r ()r r
〈U ( p)U ( p)〉 = δ , 〈V ( p)V ( p)〉 = δ , 〈U ( p)V ( p)〉 = 0 , 〈V ( p)U ( p)〉 = 0 , (5) r s rs r s rs r s r s
2 2 1 α ⋅ pˆ + βm 1 α ⋅ pˆ + βm U ( p)〉〈U r ( p) = 1+ , V ( p)〉〈Vr ( p) = 1− . (6) ∑ r 2 2 2 ∑ r 2 2 2 r=1 p + m r=1 p + m
The corresponding wave functions U r ( p , x) (r =1, 2) and Vr ( p , x) (r =1, 2) are:
eip⋅x eip⋅x U r ( p , x) = 〈xU r ( p)〉 = U r ( p) , Vr ( p , x) = 〈xVr ( p)〉 = Vr ( p) . (7) (2π)3 / 2 (2π)3 / 2
~ ~ The wave function Ψ (t , x) of a state Ψ (t)〉 can be written to the form
2 ~ ~ 3 Ψ (t , x) = 〈xΨ 〉 = ∫ d p∑()Ur ( p , x)Ar (t , p) +Vr ( p , x)Br (t , p) , (8) r =1
~ where Ψ (t)〉 describes a “one-particle” state (In fact, it should be replaced by a state vector of many particles system after considering Dirac sea, see the discussion of (14) below); and ~ 3 + Ar (t , p) = 〈U r ( p)Ψ (t)〉 = d xU r ( p , x)Ψ (t , x) , ∫ (9) ~ 3 + Br (t , p) = 〈Vr ( p)Ψ (t)〉 = ∫ d xVr ( p , x)Ψ (t , x) ,
2 ~ ~ 3 ~+ ~ 3 ∗ ∗ 〈Ψ (t)Ψ (t)〉 = ∫ d xΨ (t , x)Ψ (t , x) = ∫ d p∑()Ar ( p , x)Ar (t , p) + Br ( p , x)Br (t , p) =1. (10) r =1 2.2 The wave function of the vacuum when Dirac sea is as the vacuum We now assume that the momentum of particle take quantized discrete values. According to the model of Dirac sea and the theory of many particles system, we regard the vacuum as the state that the whole negative energy states are filled up, and the wave function of such vacuum Ψvacuum (x1 , x2 , L , xi , L) is:
6 Ψ vacuum (x1 , x2 , L , xi , L) δ x , x , , x , 1 ()i1 i2 L is′ L = (−1) V 1 ( p1 , xi )V1 ( p1 , xi )V 1 ( p2 , xi )V1 ( p2 , xi ) ∑ − 1 2 − 3 4 L N! x , x , , x , 2 2 2 2 ()i1 i2 L is′ L (11) ×V 1 ( ps−1 , xi )V1 ( ps−1 , xi )V 1 ( ps , xi )V1 ( ps , xi ) − 2(s−1)−1 2(s−1) − 2s−1 2s 2 2 2 2 ×V 1 ( ps+1 , xi )V1 ( ps+1 , xi ) , − 2(s+1)−1 2(s+1) L 2 2 where the elements in the set ( p1 , p2 , L , ps , L) traverse the whole values of momentums, N is the total number of the negative energy particles, (x , x , , x , ) is a permutation of i1 i2 L is′ L
()x1 , x2 , L , xs′ , L , the sum ∑ traverse all permutations, x , x , , x , ()i1 i2 L is′ L δ (x , x , , x , ) i1 i2 L is′ L
1, when ()x , x , , x , is even permutation of ()x , x , , x ′ , ; i1 i2 L is′ L 1 2 L s L = −1, when x , x , , x , is odd permutation of ()x , x , , x , . ()i1 i2 L is′ L 1 2 L s′ L 2.3 The wave function of a particle when Dirac sea is as the vacuum
At first, for a particle situated positive energy state U ( p)〉 , because there is now Dirac sea, r we must use the wave function (+) Ψ r ( p ; p1 , p2 , L , pi′ , L ; x1 , x2 , L , xi , L) δ x , x , , x , 1 ()i1 i2 L is′ L = (−1) Ur ( p1 , xi )V 1 ( p1 , xi )V1 ( p1 , xi ) ∑ 1 − 2 3 N! x , x , , x , 2 2 ()i1 i2 L is′ L (12) ×V 1 ( p2 , xi )V1 ( p2 , xi ) V 1 ( ps−1 , xi )V1 ( ps−1 , xi )V 1 ( ps , xi )V1 ( ps , xi ) − 4 5 L − 2(s−1) 2(s−1)+1 − 2s 2s+1 2 2 2 2 2 2 ×V 1 ( ps+1 , xi )V1 ( ps+1 , xi ) − 2(s+1) 2(s+1)+1 L 2 2 to describe such system in which there is a positive energy particle and Dirac sea. Secondly, according to the Dirac hold theory, Dirac sea can be observed only when it lacks some negative energy state particles, hence, we can track formally the change of negative energy state by observing the motion of the hold (i.e., the motion of antiparticle). The concrete method has been given by Ref. [3]: If Dirac sea changes from the state lacking V ( p)〉 at time t to r another state lacking V ( p′)〉 at time t+dt, then, formally, we can regard this process as that a r′ negative energy particle moves from the state V ( p′)〉 at time t+dt to the state V ( p)〉 at time t. r′ r Hence, when we are tracking formally a negative energy state V ( p)〉 , what we are actually r observing is a many particles system described by the wave function
7 (−) Ψ r ( p ; p1 , p2 , L , pi′ , L ; x1 , x2 , L , xi , L) δ x , x , , x , δ ()r , p +1 1 ()i1 i2 L is′ L = (−1) (−1) V 1 ( p1 , xi )V1 ( p1 , xi )V 1 ( p2 , xi )V1 ( p2 , xi ) ∑ − 1 2 − 3 4 L N! x , x , , x , 2 2 2 2 ()i1 i2 L is′ L ×V 1 ( ps−1 , xi )V1 ( ps−1 , xi )V−r ( p , xi )V 1 ( ps+1 , xi )V1 ( ps+1 , xi ) ; (13) − 2(s−1)−1 2(s−1) 2s−1 − 2s 2s+1 L 2 2 2 2
The above expression lacks the term V ( p , x ) (we assume p = p ), δ (r, p) is the "situation r is′′′ s number" of V ( p , x ) situated in the expression r is′′′
V 1 ( p1 , xi )V1 ( p1 , xi )V 1 ( p2 , xi )V1 ( p2 , xi ) − 1 2 − 3 4 L 2 2 2 2
×V 1 ( ps−1 , xi )V1 ( ps−1 , xi )V 1 ( p , xi )V1 ( p , xi ) − 2(s−1)−1 2(s−1) − 2s−1 2s 2 2 2 2
×V 1 ( ps+1 , xi )V1 ( ps+1 , xi ) , − 2(s+1)−1 2(s+1) L 2 2 namely, V ( p , x ) is situated on δ (r, p) -th situation in the above expression. r is′′′
Based on the above discussion, for a “one-particle” system described by the wave function
(8), we must replace it by a wave function Ψ (t ; x1 , x2 , L , xi , L) of many particles system
(1) (2) Ψ (t ; x1 , x2 , L , xi , L) =Ψ (t ; x1 , x2 , L , xi , L) +Ψ (t ; x1 , x2 , L , xi , L) (14) to describe this system, where
2 Ψ (1) (t ; x , x , , x , ) = d3 p A (t , p)Ψ (+) ( p ; p , p , , p , ; x , x , , x , ) , (15) 1 2 L i L ∫ ∑ r r 1 2 L i′ L 1 2 L i L r =1
2 Ψ (2) (t ; x , x , , x , ) = d3 p B∗ (t , p)Ψ (−) ( p ; p , p , , p , ; x , x , , x , ) . (16) 1 2 L i L ∫ ∑ r r 1 2 L i′ L 1 2 L i L r=1
∗ We emphasize that we must use Br (t , p) but not Br (t , p) in (16), or we cannot obtain correct result (See the calculation process in (44) below); on the other hand, this process is also in accord with the Feynman’s theory of positrons[3], we explain this point in slightly detail. ~ ~ We use Ψ r ( p)〉electron and Ψ r ( p)〉positron denote the state of electron and positron with momentum p and spin r, respectively; if the initial and final state of a electron are ~ ~ ~ ~ Ψ i 〉electron = Ψ r ( p)〉electron and Ψ f 〉electron = Ψ r′ ( p′)〉electron , respectively, then the transition ~ ~ amplitude between Ψ i 〉electron and Ψ f 〉electron is 〈U r′ ( p′) S U r ( p)〉 , where S is S-matrix; ~ ~ however, if the initial and final state of a positron are Ψ i 〉positron = Ψ r ( p)〉positron and ~ ~ Ψ f 〉 positron = Ψ r′ ( p′)〉 positron , respectively, then according to the Feynman’s theory of positrons, the ~ ~ transition amplitude between Ψ i 〉positron and Ψ f 〉positron is 〈V−r (− p) S V−r′ (− p′)〉 , notice that it seems formally as if a negative energy particle transits from the final state V−r′ (− p′)〉 to the initial state V−r (− p)〉 in inverse time direction. And, further, if the initial and final state of an electron are
8 ~ ~ ~ Ψ 〉 = A Ψ ( p )〉 + A Ψ ( p )〉 , i electron 1 r1 1 electron 2 r2 2 electron
~ ~ ~ Ψ 〉 = A′Ψ ( p′)〉 + A′ Ψ ( p′ )〉 , f electron 1 r1′ 1 electron 2 r2′ 2 electron
∗ ∗ ∗ ∗ respectively, where A1 A1 + A2 A2 = 1 , A1′ A1′ + A2′ A2′ = 1, then the transition amplitude between ~ ~ Ψ i 〉electron and Ψ f 〉electron is
∗ ∗ (A1′ 〈U r′( p1′) + A2′ 〈U r′ ( p2′ ) ) S(A1 U r ( p1)〉 + A2 U r ( p2 )〉); (17) 1 2 1 2 however, if the initial and final state of a positron are ~ ~ ~ Ψ 〉 = B Ψ ( p )〉 + B Ψ ( p )〉 , i positron 1 r1 1 positron 2 r2 2 positron
~ ~ ~ Ψ 〉 = B′Ψ ( p′)〉 + B′ Ψ ( p′ )〉 , f positron 1 r1′ 1 positron 2 r2′ 2 positron
∗ ∗ ∗ ∗ respectively, where B1 B1 + B2 B2 = 1 , B1′ B1′ + B2′ B2′ = 1, then the transition amplitude between ~ ~ Ψ i 〉positron and Ψ f 〉positron is
∗ ∗ ()B1 〈V−r (− p1) + B2 〈V−r (− p2 ) S(B1′V−r′ (− p1′)〉 + B2′ V−r′ (− p′2 )〉). (18) 1 2 1 2
We see that, it seems formally as if a negative energy particle transits from the final state B′ V (− p′)〉 + B′ V (− p′ )〉 to the initial state B V (− p )〉 + B V (− p )〉 in inverse time 1 −r1′ 1 2 −r2′ 2 1 −r1 1 2 −r2 2 direction, hence, the forms appeared in (18) of the coefficients B1 and B2 in the initial state ~ ∗ ∗ Ψ i 〉 positron of a positron are B1 and B2 , respectively; this is differ from the case of the ~ coefficients A1 and A2 in the initial state Ψi 〉electron of an electron in (17). 2.4 Derivation of the calculation rule of probability given by (3) According to the calculation rule of probability of many particles system, if a physical quantity A of which the eigenvalue equation is given by (2) is measured for a state of many particles system described by a wave function Ψ (t ; x1 , x2 , L , xi , L) , then the probability
Psystem (an ) that an eigen-value an is obtained is:
3 3 3 + 3 Psystem (an ) = N d x2d x3 Ld xiL [ Ψ (t ; x1′ , x2 , L , xi , L)d x1′An (x1′) ∫ ∫ (19) + 3 × ∫ An (x1)d x1Ψ (t ; x1 , x2 , L , xi , L)] .
However, if we substitute the wave function (14) to (19), then what result we obtain from (19) includes the observation results about Dirac sea, which is now as the vacuum. On the other hand, the vacuum is not observable; thus, an actual probability P(an ) should be the result after removing the contribution from the vacuum:
P(an ) = Psystem (an ) − Pvacuum (an ) . (20)
In (20), Pvacuum (an ) is the probability that the eigen-value an is obtained when the physical quantity A is measured for the vacuum described by the wave function Ψvacuum (x1 , x2 , L , xi , L) given by (11):
9 3 3 3 + 3 Pvacuum (an ) = N d x2d x3 Ld xiL [ Ψ vacuum (x1′ , x2 , L , xi , L)d x1′An (x1′) ∫ ∫ (21) + 3 × ∫ An (x1)d x1Ψ vacuum (x1 , x2 , L , xi , L)] . The process removing the contribution from the vacuum in (19), and, thus, obtaining (20), is in accord to the spirit of the Dirac hold theory. According to the Dirac hold theory, the total energy and the total charges of the vacuum are − 2∑ p2 + m2 and −2Ne , respectively; if Dirac sea p lacks one negative energy state particle with the momentum value p0, then the total energy and the
2 2 2 2 total charges of the vacuum change to − 2 p + m − − p + m and − 2Ne − e , ∑ 0 p respectively. However, what we obverse actually are the results after removing the contribution from the vacuum, hence, what energy and charge we obverse actually are 2 2 2 2 2 2 2 2 − 2 p + m − − p + m − − 2 p + m = p + m and −2Ne − e − (−2Ne) = −e , ∑ 0 ∑ 0 p p
2 2 respectively; namely, what we obverse actually is an antiparticle with energy p0 + m and charge − e . For the wave function (14), using (4) ~ (6), we can prove that (20) is just (3), but the calculation process is lengthy; in occupation number representation, this calculation process can be greatly predigested.
3 Discussion in occupation number representation 3.1 Some conclusions of many particles system[4]
Considering a set of complete states of orthogonality and normalization ϕ 〉 and { i } introducing the creation and annihilation operator cˆ+ and cˆ of the state ϕ 〉 , we have i i i
+ + + { cˆi , cˆ j }= δij , { cˆi , cˆ j }= { cˆi , cˆ j }= 0 , cˆi 0〉 = 0 .
Defining ˆ ˆ ˆ ˆ + ˆ+ ∗ ˆ+ ψ (x) = ∑〈xϕi 〉ci = ∑ϕi (x) ci , ψ (x) = ∑〈ϕi x〉ci = ∑ϕi (x) ci , i i i i
1 + + + x1 , x2 , L , xN 〉 = ψˆ (x1)ψˆ (x2 )Lψˆ (xN ) | 0〉 , N!
The state vector that N particles are situated N states ϕ 〉 , ϕ 〉 , , ϕ 〉 is j1 j2 L jN
Ψ (t)〉 = c+ c+ c+ 0〉 , j1 j2 L j N the corresponding wave function Ψ (t ; x1 , x2 , L , xN ) is:
10 Ψ (t ; x1 , x2 , L , xN ) = 〈x1 , x2 , L , xN Ψ (t)〉 1 δ ()x , x , , x (22) = (−1) i1 i2 L iN ϕ ( x )ϕ (x ) ϕ (x ) . ∑ j1 i1 j2 i2 L jN iN N! x , x , , x ()i1 i2 L iN
The probability P(x1 , x2 , L , xN ) that there is one situated x1 , one situated x2 ,…, one situated xN in N particles is
P(x , x , , x ) = Ψ ∗(t ; x , x , , x ) Ψ (t ; x , x , , x ) 1 2 L N 1 2 L N 1 2 L N = 〈Ψ (t) x1 , x2 , L , xN 〉〈 x1 , x2 , L , xN Ψ (t)〉 , and, further, the probability P(x) that there is one situated x in the N particles is
3 3 3 ∗ P(x) = N ∫ d x2d x3 Ld xNΨ (t ; x , x2 , L , xN )Ψ (t ; x , x2 , L , xN ) = N d3x d3x d3x 〈Ψ (t) x , x , , x 〉〈 x , x , , x Ψ (t)〉 ∫ 2 3 L N 2 L N 2 L N = 〈Ψ (t)ψˆ + (x)ψˆ(x)Ψ (t)〉 .
For a physical quantity A of which the eigenvalue equation is given by (2), we define ˆ ˆ + + ψ ()An = ∑〈An ϕi 〉ci , ψ (An ) = ∑〈ϕi An 〉ci , i i
1 + + + An , An , L , An 〉 = ψˆ (An ) ψˆ (An )Lψˆ (An )| 0〉 , 1 2 N N! 1 2 N
Ψ ()t ; A , A , , A = 〈A , A , , A Ψ (t)〉 ; n1 n2 L nN n1 n2 L nN
The probability P(A , A , , A ) that there is one situated the state A 〉 , one situated n1 n2 L nN n1
A 〉 ,…, one situated A 〉 in N particles is n2 nN
∗ P(An , An , L , An )= Ψ (t ; An , An , L , An ) Ψ (t ; An , An , L , An ) 1 2 N 1 2 N 1 2 N (23) = 〈Ψ (t) A , A , , A 〉〈 A , A , , A Ψ (t)〉 , n1 n2 L nN n1 n2 L nN
and, further, the probability P()An that there is one situated the state An 〉 in the N particles is
P(A ) = N Ψ ∗ (t ; A , A , , A ) Ψ (t ; A , A , , A ) n ∑∑L n n2 L nN n n2 L nN nn2 N = N 〈Ψ (t) A , A , , A 〉〈 A , A , , A Ψ (t)〉 (24) ∑∑L n n2 L nN n n2 L nN nn2 N + = 〈Ψ (t)ψˆ ()()An ψˆ An Ψ (t)〉 . 3.2 The case of the Dirac theory
Introducing the creation and annihilation operator aˆ + ( p) and aˆ ( p) of the state U ( p)〉 , r r r and the creation and annihilation operator bˆ+ ( p) and bˆ ( p) of the state V ( p)〉 , where the r r r
1 1 states U ( p)〉 , V ( p)〉 , r = − , satisfy (4) ~ (6), the anticommutation relations among r r 2 2
+ ˆ ˆ+ aˆr ( p) , aˆr ( p) , br ( p) and br ( p) are
11 aˆ ( p) , aˆ + ( p′) = δ δ 3(p − p′) , bˆ ( p) , bˆ+ ( p′) = δ δ 3(p − p′) , {}r r′ rr′ { r r′ } rr′ (25) Others equal zero.
According to the idea of Dirac sea, the state of the vacuum can be described by + + + + + + 0〉 = b 1 ( p1)b1 ( p1)b 1 ( p2 )b1 ( p2 )Lb 1 ( pi )b1 ( pi )L bare vacuum〉 , (26) − − − 2 2 2 2 2 2 where bare vacuum〉 is the state vector of “the bare vacuum” in which there is not any particle possessing positive or negative energy. Defining
ˆ ˆ+ ˆ + ˆ dr ( p) = br ( p) , dr ( p) = br ( p) , (27)
+ ˆ ˆ + we can obtain the anticommutation relations among aˆr ( p) , aˆr ( p) , dr ( p) and dr ( p) from
(25):
aˆ ( p) , aˆ + ( p′) = δ δ 3(p − p′) , dˆ ( p) , dˆ + ( p′) = δ δ 3(p − p′) , {}r r′ rr′ { r r′ } rr′ (28) Others equal zero. and we have ˆ aˆr ( p) 0〉 = 0 , dr ( p) 0〉 = 0 . (29)
According to some conclusions given by §3.1, we define
2 ˆ 3 ˆ ˆ ψ ()x = ∫ d p∑ (〈xU r ( p)〉ar ( p) + 〈xVr ( p)〉br ( p)) r =1 (30) 2 3 ˆ ˆ + = ∫ d p∑ ()U r ( p , x)ar ( p) +Vr ( p , x)dr ( p) , r =1
2 ˆ + 3 ˆ + ˆ+ ψ ()x = ∫ d p∑ (〈U r ( p) x〉ar ( p) + 〈Vr ( p) x〉br ( p)) r =1 (31) 2 3 ∗ ˆ + ∗ ˆ = ∫ d p∑ ()U r ( p , x)ar ( p) +U r ( p , x)dr ( p) , r =1
1 + + + x1 , x2 , L , xi , L〉 = ψˆ (x1)ψˆ (x2 )Lψˆ (xi )L | bare vacuum〉 ; (32) N!
+ Based on above ψˆ()x , ψˆ ()x and x1 , x2 , L , xi , L〉 , according to (22) in §3.1, (12), (13) and
(14) can be written to the form
(+) (+) Ψ r ( p ; p1 , p2 , L , pi′ , L ; x1 , x2 , L , xi , L) = 〈x1 , x2 , L , xi , LΨ r ( p)〉 , (33)
(−) (−) Ψ r ( p ; p1 , p2 , L , pi′ , L ; x1 , x2 , L , xi , L) = 〈x1 , x2 , L , xi , LΨ r ( p)〉 , (34)
Ψ (t ; x1 , x2 , L , xi , L) = 〈x1 , x2 , L , xi , LΨ (t)〉 . (35) where
12 (+) (−) Ψ ( p)〉 = aˆ + ( p) 0〉 , Ψ ( p)〉 = bˆ ( p) 0〉 = dˆ + ( p) 0〉 , (36) r r r r r
2 3 ˆ + ∗ ˆ Ψ (t)〉 = ∫ d p∑ ()Ar (t , p)ar ( p) 0〉 + Br (t , p)br ( p) 0〉 r =1 (37) 2 ~ ~ 3 ˆ + ˆ + = ∫ d p∑ ()〈U r ( p)Ψ (t)〉ar ( p) 0〉 + 〈Ψ (t)Vr ( p)〉dr ( p) 0〉 . r =1
In (37), we have used (9). Notice that the state vector Ψ (t)〉 of many particles system is differ ~ from the state vector Ψ (t)〉 of “one-particle” of which the correcponding wave function is given by (8). For a physical quantity A of which the eigenvalue equation is given by (2), we define
2 ˆ 3 ˆ ˆ ψ ()An = ∫d p∑(〈An U r ( p)〉ar ( p) + 〈An Vr ( p)〉br ( p)) r=1 (38) 2 3 ˆ ˆ + = ∫d p∑()〈An U r ( p)〉ar ( p) + 〈An Vr ( p)〉dr ( p) , r=1
2 ψˆ + ()A = d3 p 〈U ( p) A 〉aˆ + ( p) + 〈V ( p) A 〉bˆ+ ( p) n ∫ ∑( r n r r n r ) r =1 (39) 2 = d3 p 〈U ( p) A 〉aˆ + ( p) + 〈V ( p) A 〉dˆ ( p) , ∫ ∑ ()r n r r n r r =1
1 + + + An , An , L , An , L〉 = ψˆ (An ) ψˆ (An )Lψˆ (An )L | bare vacuum〉 . (40) 1 2 i N! 1 2 i The form of (23) in §3.1 reads now
∗ Psystem (An ) = Ψ (t ; An , An , L , An ,L) Ψ (t ; An , An , L , An ,L) 2 i 2 i (41) = 〈Ψ (t) A , A , , A , 〉〈 A , A , , A , Ψ (t)〉 . n n2 L ni L n n2 L ni L According to (24) in §3.1, (19) and (21) now can be written to the forms P (A ) = N Ψ ∗ (t ; A , A , , A , ) Ψ (t ; A , A , , A , ) system n ∑∑L n n2 L ni L n n2 L ni L nn2 N = N 〈Ψ (t) A , A , , A , 〉〈 A , A , , A , Ψ (t)〉 (42) ∑∑L n n2 L ni L n n2 L ni L nn2 N + = 〈Ψ (t)ψˆ ()()An ψˆ An Ψ (t)〉 . P (A ) = N Ψ ∗ (t ; A , A , , A , ) Ψ (t ; A , A , , A , ) vacuum n ∑∑L vacuum n n2 L ni L vacuum n n2 L ni L nn2 N = N 〈0 A , A , , A , 〉〈 A , A , , A , 0〉 (43) ∑∑L n n2 L ni L n n2 L ni L nn2 N + = 〈0ψˆ ()()An ψˆ An 0〉 . Substituting (37), (30) and (31) to (42) and (43), using the characteristics given by (6), (9), (10), (28) and (29), we obtain
13 2 2 3 ∗ 3 ′ ′ ′ Psystem (An ) = ∫ d p∑ ()Ar (t , p)〈U r ( p) An 〉 ⋅ ∫ d p ∑ ()〈An U r′ ( p )〉 Ar′ (t , p ) r =1 r′=1 2 2 3 3 ′ ′ ∗ ′ − ∫ d p∑ ()Br (t , p)〈An Vr ( p)〉 ⋅ ∫ d p ∑()〈Vr′ ( p ) An 〉Br′ (t , p ) r =1 r′=1 2 2 3 ∗ ∗ 3 ′ ′ ′ + ∫ d p∑()Ar (t , p)Ar (t , p) + Br (t , p)Br (t , p) ⋅ ∫ d p ∑()〈An Vr′ ( p )〉〈Vr′ ( p ) An 〉 r =1 r′=1 2 ~ 2 ~ = d3 p 〈Ψ (t)U ( p)〉〈U ( p) A 〉 ⋅ d3 p′ 〈A U ( p′)〉〈U ( p′)Ψ (t)〉 (44) ∫ ∑()r r n ∫ ∑ ()n r′ r′ r =1 r′=1 2 ~ 2 ~ 3 3 ′ ′ ′ − ∫ d p∑ ()〈An Vr ( p)〉〈Vr ( p)Ψ (t)〉 ⋅ ∫ d p ∑()〈Ψ (t)Vr′ ( p )〉〈Vr′ ( p ) An 〉 r =1 r′=1 2 3 + ∫ d p∑ ()〈An Vr ( p)〉〈Vr ( p) An 〉 r =1 ~ ˆ ˆ ~ ~ ˆ ˆ ~ ˆ = 〈Ψ M + An 〉〈 An M + Ψ 〉 − 〈Ψ M − An 〉〈 An M − Ψ 〉 + 〈An M − An 〉 ,
+ ˆ Pvacuum (An ) = 〈0ψˆ (An ) ψˆ(An ) 0〉 = 〈An M − An 〉 ; (45)
Finally, we obtain ˆ ˆ ˆ ˆ P(An ) = Psystem (An ) − Pvacuum (An ) = 〈Ψ M + An 〉〈 An M + Ψ 〉 − 〈Ψ M − An 〉〈 An M − Ψ 〉 . (46)
This is just (3). 3.3 Three notes ① It is obvious that the calculation result of the rule (3) may be negative. For example, if we use (3) to calculate the the probability that a negative energy particle lacks at point x, e.g., an antiparticle situated x, we obtain a “negative probability”. Hence, what result obtained by (3) should be as “charge density” but not “particle number density”. In fact, for relativistic case, we can define local charge state but can not define local particle number state [5]. On the other hand, if a particle is situated in an area that is less than the Compton wavelength of the particle, then maybe the corresponding antiparticles are created. Hence, we can only study local charge density but can not study local particle number density for relativistic case. ② The calculation rule of probability (3) can be generalized to many particles case by using the method given by §3.1. For example, according to Ref. [4], the probability P(x1 ， x2 ) that there is one situated x1 and other situated x2 in the N particles is
3 3 Psystem (x1 , x2 ) = N(N −1) d x3 Ld xi L〈Ψ (t) x1 , x2 , x3 , L , xi , L〉〈x1 , x2 , x3 , L , xi , L Ψ (t)〉 ∫ + + = 〈Ψ (t)ψˆ (x2 )ψˆ (x1)ψˆ(x1)ψˆ(x2 )Ψ (t)〉 ， after removing the correcspongding contribution from the vacuum, the actual probability
P(x1 , x2 ) is
+ + + + P(x1 , x2 ) = 〈Ψ (t)ψˆ (x2 )ψˆ (x1)ψˆ(x1)ψˆ(x2 )Ψ (t)〉 − 〈0ψˆ (x2 )ψˆ (x1)ψˆ(x1)ψˆ(x2 ) 0〉 .
③ If we want to calculate the probability observing some particles situated some eigenstates + + ˆ + of momentum, e.g., aˆr ( p) 0〉 , aˆr ( p)dr ( p) 0〉 , etc, notice that according to (26), 0〉 is a many particle system, then we must use (41). For example, if we calculate the probability that a system
14 + described by state vector Ψ (t)〉 is situated aˆr ( p) 0〉 , then we cannot use (42) but must use (41), + the result is Psystem = 〈Ψ (t) aˆr ( p) 0〉〈0 aˆr ( p)Ψ (t)〉 ; of course, we must remove the correcspongding + contribution from the vacuum, however, from (29) we have Pvacuum = 〈0 aˆr ( p) 0〉〈0 aˆr ( p) 0〉 = 0 . We see that, for the case of calculating the probability observing some particles situated some eigenstates of momentum, what results we obtain are the same as that of the quantum theory of field.
4 The case of particle satisfying the Klein-Gordon equation As is well known, if the wave function ϕ(t , x) of a particle satisfies the Klein-Gordon equation: ∂2ϕ(t , x) − + ∇2ϕ(t , x) − m2ϕ(t , x) = 0 , (47) ∂2t then position probability density and three dimentional current density are
∗ i ∗ ∂ϕ(t , x) ∂ϕ (t , x) ρ(t , x) = ϕ (t , x) −ϕ(t , x) , (48) 2m ∂t ∂t
i j(t , x) = − (ϕ ∗ (t , x)∇ϕ(t , x) −ϕ(t , x)∇ϕ ∗(t , x)). (49) 2m We consider the following wave function saticfying the Klein-Gordon equation:
3 2 2 d k m −i k +m ()(t −t0 ik⋅ x− x0 ) ϕ(t , x) = 3 e e , (50) ∫ (2π) k 2 + m2 for which according to (48) we have
3 3 i d k m i k 2 +m 2 ()t −t −ik⋅ (x − x ) d k′ −i k′2 +m 2 ()t −t ik′⋅ (x − x ) ρ(t , x) = e 0 e 0 ⋅ (−im)e 0 e 0 3 2 2 3 2m ∫ (2π) k + m ∫ (2π) (51) 3 3 d k m −i k 2 +m 2 ()(t −t ik⋅ x− x ) d k′ i k′2 +m 2 ()t −t −ik′⋅ (x − x ) − e 0 e 0 ⋅ (im)e 0 e 0 , 3 3 ∫ (2π) k 2 + m2 ∫ (2π) and we can prove easily that
ρ(t , x) = I1(t − x , t0 − x0 ) = ρ2 (t , x ; t0 , x0 ) , (52) we see that such ρ(t , x) is in accord with Tab. 2. Substituting (50) to (49) we have 3 3 i d k m i k 2 +m 2 ()t −t −ik⋅ (x − x ) d k′ m −i k′2 +m2 ()t −t ik′⋅ (x − x ) j(t , x) = − e 0 e 0 ∇ e 0 e 0 ∫ 3 2 2 ∫ 3 2 2 2m (2π) k + m (2π) k′ + m 3 3 d k m −i k 2 +m 2 ()(t −t ik⋅ x− x ) d k′ m i k′2 +m 2 ()t −t −ik′⋅ (x − x ) − e 0 e 0 ∇ e 0 e 0 , ∫ 3 2 2 ∫ 3 2 2 (2π) k + m (2π) k′ + m (53) we can prove that (See the Appendix of this paper)
j(t , x) = j2 (t , x) . (54) Hence, using (48) and (49), for the wave function (50), we can obtaion the conclusions being in accord with Tab. 2.
15 On the other hand, at time t0 , (50) becomes
3 d k m ik⋅()x− x0 ϕ(t0 , x) = 3 e ; (55) ∫ (2π) k 2 + m2
3 The form of (55) is different from the wave function Ψ (x) = δ (x − x0 ) that describes a particle situated at x0 . However, if we use (48) to calculate the probability ρ(t0 , x) that a particle described by the wave function (55) is situated x at time t0 , according to (51) we have
ρ(t0 , x) = lim ρ(t , x) t→t0 3 3 i d k m −ik⋅()x− x d k′ ik′⋅()x− x = e 0 ⋅ (−im)e 0 ∫ 3 2 2 ∫ 3 2m (2π) k + m (2π) 3 3 (56) d k m ik⋅()x− x d k′ −ik′⋅()x − x − e 0 ⋅ (im)e 0 3 3 ∫ (2π) k 2 + m2 ∫ (2π) 3 d k m 3 3 3 = 3 δ ()()()x − x0 = δ 0 δ x − x0 . ∫ (2π) k 2 + m2
In the last step of the above calculation process, we have used
3 ′ 3 ′ 3 3 3 d k ik′⋅()x − x0 d k d k m δ ()0 = lim δ (x − x0 )= lim e = = , x→ x x→ x ∫ 3 ∫ 3 ∫ 3 2 2 0 0 (2π) (2π) (2π) k + m
d [K(k)k]3 3mk 2 where k′ = K(k)k , K(k) satisfies { }= . dk k 2 + m2
The result of (56) is in accord with the form of “position probability density”. Because, according to the normal calculation rule of probability P(an ) = 〈Ψ An 〉〈 An Ψ 〉 , the “position 3 probability density” that a particle described by the wave function Ψ (x) = δ ()x − x0 and + 3 3 3 3 situated x is Ψ (x)Ψ (x) = δ ()x − x0 δ (x − x0 ) = δ (0)δ (x − x0 ) . Hence, under the calculation rule of position probability density (48), the result (56) shows that the wave function
(55) can describe the state of particle satisfying the Klein-Gordon equation and situated x0 at time t0 . We can generalize the formula (48) of calculating position probability density to the case of arbitrary physical quantity A and obtain a calculation rule of the probability that an eigen-value an is observed when the physical quantity A of which the eigenvalue equation is given by (2) is measured for a state described by the state vector ϕ(t)〉 satisfying the Klein-Gordon equation:
∂〈A ϕ(t)〉 ∂〈ϕ(t) A 〉 i n n ρ(t , an ) = 〈ϕ(t) An 〉 − 〈An ϕ(t)〉 , (57) 2m ∂t ∂t where
〈A ϕ(t)〉 = d3x〈A x〉〈 xϕ(t)〉 = d3xA∗(x)ϕ(t , x) . (58) n ∫ n ∫ n
16 Especially, for momentum, (57) and (58) become
∗ i ∗ ∂ϕ(t , p) ∂ϕ (t , p) ρ(t , p) = ϕ (t , p) −ϕ(t , p) , (59) 2m ∂t ∂t
1 ϕ(t , p) = d3x e−ip⋅xϕ(t , x) . (60) ∫ (2π)3/2 For the wave function (50), substituting it to (60) we have
2 2 1 m −i p +m ()t −t0 −ip⋅x0 ϕ(t , p) = 3/2 e e ; (61) (2π) p2 + m2 and, further, substituting (61) to (59), we obtain
1 m ρ(t , p) = 3 = f2 ( p) . (61) (2π) p2 + m2
According to the above discussion and the conclusions (61), (52) and (54), we see that if we use the calculation rule of probability (57) for particle satisfying the Klein-Gordon equation, then we can obtain conclusions being in accord with Tab. 2. Of course, as is well known, what we obtain from (48) should be as “local charge density”.
Appendix We use the method given in Ref. [1] that prove the three mathematical formulas in §1.3 to prove (50). For the sake of brevity, we rewrite (53) to the from
j(t , x) = J(t − t0 , x − x0 ) , (A-1) where
3 3 i d k m 2 2 d k′ m 2 2 J(t, x) = − ei k +m te−ik⋅x ∇ e−i k′ +m teik′⋅x ∫ 3 2 2 ∫ 3 2 2 2m (2π) k + m (2π) k′ + m 3 3 d k m 2 2 d k′ m 2 2 − e−i k +m teik⋅x ∇ ei k′ +m te−ik′⋅x ∫ 3 2 2 ∫ 3 2 2 (2π) k + m (2π) k′ + m 3 3 1 d k m 2 2 −ik⋅x d k′ m 2 2 ik′⋅x = − cos k + m t e ∇ sin k′ + m t e (A-2) ∫ 3 2 2 ∫ 3 2 2 m (2π) k + m (2π) k′ + m 3 3 d k m 2 2 ik⋅x d k′ m 2 2 −ik′⋅x − sin k + m t e ∇ cos k′ + m t e ∫ 3 2 2 ∫ 3 2 2 (2π) k + m (2π) k′ + m
= m()∆1(x)∇∆(x) − ∆(x)∇∆1(x) ,
[2] where ∆(x) is given by (1), ∆1(x) is
2 2 3 cos p + m t Y0 (ms) , x < t ; d p ip⋅x 1 ∂ ∆1(x) = e = (A-3) ∫ 3 2 2 2 (2π) p + m 4πr ∂r − K0 (ms1) , x > t . π
2 2 In the above formula, both Y0 (ms) and K0 (ms) are Bseeel functions, s = t − x ,
17 2 2 s1 = x − t .
From (1), when x > t , ∆(x) = 0 , ∇∆(x) = 0 , we have
J(t, x) = m(∆1(x)∇∆(x) − ∆(x)∇∆1(x)) = 0 ; (A-4)
From (1) and (A-3), when x < t , we have
J(t, x) = m()∆1(x)∇∆(x) − ∆(x)∇∆1(x)
1 ∂Y0 (ms) 1 ∂J0 (ms) 1 ∂J0 (ms) 1 ∂Y0 (ms) (A-5) = m ∇ − ∇ , 4πr ∂r 4πr ∂r 4πr ∂r 4πr ∂r [6] using one formula about Bseeel functions Jν (z) and Yν (z) :
m d −ν m −ν −m []z Zν (z) = (−1) z Zν +m (z) , zdz where Zν (z) represents the Bseeel functions Jν (z) or Yν (z) , (A-5) becomes m4 x J(t, x) = ()J 2 (ms)Y1(ms) − Y2 (ms)J1(ms) , 16π2 s3 and again, using another formula about Jν (z) and Yν (z) : 2 J (z)Y (z) − Y (z)J (z) = , ν ν −1 ν ν −1 πz (A-5) becomes
m4 x 2 1 m3 x J(t, x) = = . (A-6) 2 3 3 2 16π s πms (2π) ()t 2 − x 2 Combining (A-4) and (A-6), we have
1 m3 x , x < t ; 3 2 J (t , x) = (2π) ()t 2 − x2 (A-7) 0 , x ≥ t . From (A-1) and (A-7) we obtain (54).
Reference [1] T. Mei, Quantum Theory of the Free Particle, Physics Essays, 12(2): 332~339 (1999). [2] J. D. Bjorken, S. D. Drell, Relativistic Quantum Mechanics, McGraw Hill Book Company, 1964. [3] R. P. Feynman, The theory of positrons. Phys. Rev. 76, 749-759 (1949). [4] Yukawa Hideki (Ed.), Quantum Mechanics (I), 2nd edition (In Japanese), Iwanami Shoten, Tokyo, 1978. [5] D. Lurie, Particles and Fields, John Wiley & Sons, New York (1968). [6] G. N. Watson, Theory of Bessel functions, 2nd edition, Cambridge University Press, London (1922).
18