- Email: [email protected]
A theory of particles of spin one-half
ANNALS
OF PHYSICS:
6,
96-106 (1968)
A Theory
of Particles
of Spin
One-Half*
A. 0. BARUT Department
of Physics,
Syracuse
University,
Syracuse, New York
From the two-component second order fermion theory of Feynman and GellMann a four component theory of spin-J5 particles in Hamiltonian form is derived. When a suitable inde6nit.e metric is introduced in the spin space by means of charge density the spin as well as the charge degrees of freedom may be separated. Spin and angular momentum are separately constants of the motion which is not the case for the Dirac equation. If states with positive norm describe particles the states of the corresponding charge conjugate particles have negative norm. The Hamiltonian is quadratic in the momenta which allows a straightforward transition to the nonrelativistic Pauli equation. The law of transformation of the wave function under the Lorentz transformations is given, and the relation and equivalence of the theory to Dirac and Foldy-Wouthuysen equations is established. 1. INTRODUCTION
There is a well-known single particle interpretation of the Klein-Gordon Equation for bosons first discussedby Sakata and Taketani (I), then by Heitler (2), Case (S), and recently by Feshbach and Villars (4). This theory employs a two component wave function with an indefinite metric in the two-dimensional space. The relativistic particle has an intrinsic charge degrees of freedom described by the norm (f 1) of the wave function. The main purpose of this paper is to show that a similar theory can be formulated for spin-34 particles in terms of a four component wave function with an indefinite metric in the four-dimensional space. The present formulation differs both in form and in physical interpretation from the usual Dirac equation. It is, however, equivalent to the latter as far as interactions with an external electromagnetic fields are concerned. For example, in the present formulation spin and angular momentum are separately constants of the motion, and both charge and spin degrees of freedom are explicit,ly separated. It amounts to a generalization of the Pauli theory of spin that includes all spin and relativistic effects exactly, i.e., in a manner equivalent to the Dirac equation. The transformation properties of the wave-function which makes the wave equation invariant under the Lorentz transformations is more complicated than * Supported
in part by the Air Force Office of Scientific 95
Research.
96
BARUT
the spinor transformations. It appears that the intimate linking of spin and angular momentum is not due to the requirement of relativistic invariawe, but tjo the use of the irreducible spin representations of the Lorentz group. Thus, the theory discussed in this paper will be useful, at present, mainly for purposes of physical interpretation of t’he relativistic theory and the passage to the nonr&~tivistic~ limit.. We shall use the Dirac m&ices satisfying +‘r’ + r”r” = L)$‘, \vhrrc $‘“: (1, - 1, - 1, - 1); y5 = ir”r’r”r”, (7’)’ = 1, and the st’andard representation \vith
The natural units with c = fi = 1 will he employed. The four vectors are indicated by a Greek sub- or superscript: s’, p, . The electjromagnetic potent’ials nw A”: (4, A) or A, : (4, -A),
2. THE
WAVE
EQUATION
We start from the second order Dime equation
(1) where
P, = p, - e.4, , In terms of the new quantities
0
x=
(D
=
(1
+
Y5k cp =
(a
( > $1
+
$3
$2
+
*4
*
(2)
Equation (1) splits into two identical two-component equations
(3) where
U
PY
11
O
- iua
=
0
=ii
I
iU2 -
iu, 0,
-(r
VI”
SPIN-34
97
PARTICLES
Feynman and Gell-Mann (5) have shown that there is a one-to-one correspondence between (3) and the first order Dirac equation. These authors have also used x instead of $ in writing down the universal Fermi interaction (without derivative coupling) which gives rise to an essentially unique form of the coupling (V and A in the language of P-decay) in agreement with experiments. This fact furnishes another motivation for the study of unusual forms of the wave equation, for it makes a difference in what form the coupling of two or more fields is written down. Let us introduce an independent two-component quantity cpodefined by Pw Then (3) can be written
= -mpo.
(4)
as
-mPgoo
- P2v -I f upvFMv(p- rn’p = 0.
Equations (4) and (5) together determine ing the linear combinations
a four-component
(5) equation.
Introduc-
0.3
01:
Ic = (r5- YO);; 0 PO
0
into (4) and (5), and by adding and subtracting tain an equation for XPin Hamiltonian form
the resulting
equations
po\E = H\E
we ob(7)
where H = &
1
P* - ; 2” Fpy ~“(1 + r5) - my0 + eA0
At this point we remark that (7) is not an admissible Hamiltonian with rest mass zero. In this case, instead of (4), we put
for particles
P&o = -(oo.
(8)
Then by the same steps as before we obtain with m = 0 H = f
P2 - f 2”’ Fp
~“(1 + r6) + a ~“(1 - 7”) + eAo.
(9)
98
BARUT
The Hamiltonian (7) is gauge-invariant and me note the appearance of the (1/2m)P” term, so characteristically different from the Dirac Hamiltonian. We shall discuss the transformation law of q under the Lorentz group in Section 5. First we investigate t,he physical interpreta,tion of (7), or (9), which is basically t,he same. We also note that (7) has essentially the same form as the Hamiltonian in the spin 0 case (see for example Ref. 4) except the additional spin term Y’F,, and t,he y-matrices instead of Pauli-matrices. The Hamiltonian (7) is not Hermitian in the usual sense. But a consistent treatment is possible if use is made of an indefinite metric in the four-dimensional spin space. We arrive at. the form of t.he metric by considering the charge density. For free particles a current density can easily be obtained from (3) j’ = con&. ((bpl;p* - q*p’;p). Using successively
(10)
(4) and (6) we find for the charge density j* = const. **y*\k.
J” is not positive definite. However, l!*/12 then with
if we normalize
= $ q’*y’\k
this metric H becomes Hermitian, Y’HI;Y’
(11) according
to
d3x = fl,
(1%
i.e.,
= H,
(1%
where Hf indicates the usual transpose and Hermitian conjugation in the ,spin space. The rules of operations with the metric (12) have been discussed in Appendix I. In particular H has real eigenvalues and eigenvectors belonging to distinct eigenvalues are orthogonal in the metric (12). The fact that (12) is the correct normalization for the Hamiltonian (‘7) is ako seen from a consideration of the charge c0njugat.e wave function \k” to q. \k” satisfies a SchrGdinger equation ..C w = H%“, and, from (7) we find H” if e is replaced by -e, (p -t eAj2 -I- E Z” F,,” In the case of the Dirac
equation
1
~‘(1 -I- r5> - my0 - eAo .
the charge conjugate
wave funct’ion
(l-2) $” is
SPIN-45
99
PARTICLES
related to $ by $” = C+*. In the present case we find by comparing conjugate complex of (7) that
(14) and the
H” = - (Cr”)H*(C~O)-l and \k” = crO\E*
(15)
C has to anticommute with y” and satisfy the condition C’ = C-‘; a choice for C is C = i-r” as in the case of the Dirac equation. We can now easily prove the assertion that if \k is normalized to +l according to (12), then the corresponding @, as given by (15), is normalized to - 1. (W, yOP) = (cyO\k*, r°Cro**)
= - (*, To*).
The expectation value of the charge e is fe according to (12). Equation (7), thus, describes particles and charge conjugate particles simultaneously. Furthermore, in the limit e -+ 0 (free particles, or neutral particles) H” = H and \E and \k” are independent solutions of the free particle equation. It will be shown in the next section that the sign of the norm in Eq. (12) is identical with the sign of the energy. 3. PROPERTIES
OF THE
FREE
PARTICLE
EQUATION
Equation (7) reduces in this case to
Ho= 2
~‘(1
+ 7”) - my’.
The eigenvalues of Ho are +E, = f(p2 + rn’)l” since Ho2 = p2 + m2. In a state with definite momentum the matrix elements of Ho and hence the solutions are all real. If H\k = E,q then we find that H(Cr’\k) = -E,(Cr’\k); i.e., if the positive energy solution is normalized to + 1, then the negative energy solution is normalized to - 1. The expectation value of the energy is +E, for both states. For free particles the orbital angular momentum will be a constant of the motion [Ho, L,] = 0
(17)
[Ho, &I = 0,
08)
as well as the spin
where 2,
=
i-fly’
=
( ) fl3
0
0
03
100 has eigenvalues
BARUT
f 1. Then the four stationary 1 z,
=
solutions can be rlassified as
+1
Iz,
=
follows
-1
E = -j-E, Norm
cI!))
+ 1
E = -E, Norm
- 1
where
E, + m ip.,, a = 2(mE,)1’2 ’ for the Hamiltonian
m - E, p ” = ‘4mEp)U?
(7), and a -
1 - E’P 2( Ep)l’z
(+P.X
h _ 1 + EP (p.x 2(E,P2
for the Hamiltonian (9). Consequently the new representation accomplishes for particles and antiparticles a definite separation of spin states which is not. possible in the Dirac form. Moreover, spin and angular momentum are not. coupled for the free part,icle and the absolute value of the spin (45) can only he asserted from the additional interaction term. It is also of interest to discuss the Heisenberg equation of the motion for t,he operators x, the so called Zitterhewegung. We find i = i[&)
xl = i yO(l + Y5).
ir has all eigenvalues equal to zero; however, usual relativistic value
the expectation
value of k has t.hrk
((i) = 1 for massless particle). although equivalent in observable effects, Since different representations, give different values for x and ir it is questionable which of these should be interpreted as the “position” and the “velocity” operator in the single particlt? interpretation. However, “velocity” and current density are related to each ot’her in each representation in the usual fashion. The diffkult,y is alreadv kunwn ill
SPIN->$
101
PARTICLES
the Foldy-Wouthuysen (6) form of the Dirac equation. A localized state in the Dirac representation correspond, for example, to a smeared out state of dimension rn-’ in the Foldy-Wouthuysen coordinates and vice versa. This points again to an essential limitation and incompleteness of all relativistic single particle theories as is well known. The expectation value of the momentum in the states (19) is given by (P> = +P
forE
= +E,,
(P) = -P
forE
= -E,.
In order to find the relation of this theory to the Dirac equation we first diagonalize the Hamiltonian (16) to get two two-component equations describing separately particles and charge conjugate particles. This can be done by a “unitary” transformation V in the sense of the metric (12), i.e., satisfying v-1 = yov+“/o*
(22)
The requirement V-‘HoV
= -y”Ep
(23)
leads then to
v=
l
2(mEp)1’2
v-1 =
[E, + m + (m -
Eph51, (24)
1 2(mEp)lj2
[E, + m - (m - EJY’I,
or, for the Hamiltonian (9) to
J&L2(Ep)“2
0 + Ep + (1 - -%>r5>, (24’)
v-1 = h2
(1 + E, - 11 - %h51. P
Thus we obtain again the Foldy-Wouthuysen i&
equation
= --~~Epyb,
(25)
where
Since the Eq. (25) can be transformed into the Dirac equation by a unitary transformation (4, 6)
’ =
[2J3p(E,1+
m)]‘/” (Ep - YoHp’p
102
BAHUT
where Ho is the Dirac Hamiltonian H, = a.p + vq3 we can transform the Dirac: wave function #D into the present one *D = lJ’#,
= ppp;
* = vc*u
= a’#,, .
(27)
Indeed, it is easily checked that the Dirac solutions go over into the solutions (19) with the correct norms under W. Hence TI’ is the transformation which srparates spin and charge states for the Dirao equation. It is a transformation between two spacesone with a positive definite, the other with an indefinite m&C. Since 1Y is nonsingular, the mapping between the two vector-spaces is one-t,o-one. 4. INTERACTION
WITH
AN
ESTERYAL
ELECTROMAGNETIC
FIELI,
To show the equivalence of the spec%rumof the Hamiltonian (7) with that, of the Dirac Hamiltonian we go back to l?q. (3) which for stationary problems ~11 he written as [(E - eAo)” - m*]p =
1
P” - % u”“F,,,, cp,
(28)
Once eigenvalues and solutions of this equation have been found, we (*RI) find cpcfrom (4) 1 (00= -% (E - eAo)q
(29)
cptogether with cpodetermine * according t,o (6). For example, for the spec%rum of the H-atom, Eq. (28) becomes
(30) The eigenvalue problem in t’he form (30) has been already solved by Hyllernas (7) giving the usual fine-st,ructure formula. Equation (28) is the most convenient starting point for the solution of practical problems. For example, for a free part,icle (28) is simply (p” + ,nl’Jcp= B”p with two independent solutions
, and one immediately obtains t$r (i) and (:‘) normalized solutions (19). Similarly the problem of the electron iu :1 c*onstnut, external magnet#ic field can he solved in R simple and straightforward m:mn(ht starting with (28). To find the nonrelativistic limit of (7) in the weak field approximation WC’note that \kl represent the two small components of * and q’z the two large components. Eliminating \kl , and passing to a nonrelativistic wa1.e funct,ion tiNR b! \kz = e-““$bxR ,
SPIN-34
we find the usual kinetic tion
103
PARTICLES
energy and spin correction
.= &R
2+ P2 + e& - 3;
P” - &
(d.B
terms as a first approxima-
- ideE)
(30) - &3 (B - iE)’ + . . .) &‘. 5. RELATIVISTIC
INVARIANCE
The relativistic invariance of the Dirac equation is established by defining a suitable transformation law for the wave function. The importance of the Dirac form of the theory of spin-35 particles lies in the fact that here one has to do with the irreducible “spin-representations” of the Lorentz-group. Similarly we can define a transformation law for the wave function 9 such that Eq. (7) is invariant under the Lorentz transformations. For this purpose we start from the transformation property of Eq. (25) as discussedby Foldy (8). Foldy has constructed ten operators, i.e., P, - r”G,
J
and -$
(XE, + E,x) + c+xE”
P
- tp,
which have the same commutation relations as the ten infinitesimal operators of the inhomogeneous Lorentz group. Let us denote the operators (31) collectively by L. Then the Eq. (25) is invariant under the transformation u = eieL,
* --$ *’ = w,
(32)
where E is an infinitesimal quantity. Now since Eq. (17) can be obtained from (25) by the operator V given by (24) the transformation law of \k under Lorentztransformations is
?I?’ = A*,
n = ei’L’
L' = PLV
(33)
The additional spin term (e/2) 2”“F,, is obviously Lorentz invariant. We would like to remark that this term could be easily modified to account for a possible anomalous magnetic moment term if such a generalization of the Dirac equation should be necessary. APPENDIX
In this Appendix we discuss briefly the rules of operations with the metric (12) which have been used in the text.
104
BARUT
The norm II * I]” = e, row is real if y” = definition of (JP’“, y’q@)). “Unitary”
C-k 1)
yO+. We have chosen an Hermitian, in fact a real y”. We extend the the norm to that of the scalar product of two vectors, i.e., Two vectors are “orthogonal” if t.his scalar product va,nishes. operators will preserve t,he scalar product. From (Vqp,
YoT/\k(?)) = (lp”‘, y”qP))
we find v+y”v Similarly
we must define a new operation (&p’,
which
= y”.
(A 3
of Hermitian
conjugat,ion
(d)
by
yo\k’2’) = (Q”), yo$\E’2’),
gives d = y”A+y”.
Hence an operator
A is “Hermitian”
if =1 = r0A +?‘I.
Expectation
value of an operat,or A in a state \k is given by (A) = (*, yOA\k),
which is equal to (A+, ~“9) if A is “Hermitian”. We shall prove that the eigenvalues of an “Hermitian” operator A are real if 7” = T”+ and if the corresponding eigenvectors have nonzero norm, and, that the eigenvectors belonging t’o distinct, cigenvalucs arc “orthogonal”. Let .4*, = a,*, , then pPm ) rOAW
= a,,pPm ) r”u,,j
= CL‘l*,, ) r”W
= a,(*?,
) 7°K)
hence, wm ) YOW = 0
if a,, # a, .
Furthermore, G&L,
Ye*,)
= (A*,
,7O*,j
= @,,
7OAW
= (~O.49~) \E,) = G*, or
, -y”+$h) = (Afin,
yo\kn) = real
SPIN-;5
PARTICLES
105
The Heisenberg equations of motion remain valid in the indefinite metric with the understanding that the time evolution operator U = eeiH1 is unitary in the new sense, i.e., U+ = eiH+t = y”U-l”lo. We further note that
(*, ‘YOdA dt 9 )- RECEIVED:
d(A) . (*, r’i[H, AN?) = $ (q, r”Aq) = dt
July 3, 1958 REFERENCES
1. SAKATA AND TAKETANI, Proc. Phys. Math. Sot. Japan 22, 757 (1940). 2. W. HEITLER, Proc. Roy. Irish Acad. 49, 1 (1943). 3. K. M. CASE, Phys. Rev. 96, 1323 (1954). 4. H. FESHBACH AND F. VILLARS, Revs. Modern Phys. 30, 24 (1958). 5. R. P. FEYNMAN AND M. GELL-MANN, Phys. Rev. 109, 193 (1958). 6. L. L. FOLDY AND S. A. WOUTHUYSEN, Phys. Rev. 73, 29 (1950). 7. E. HYLLERAAS, 2. Physik 140, 626 (1955). 8. L. L. FOLDY, Phys. Rev. 102, 568 (1956).
OF PHYSICS:
6,
96-106 (1968)
A Theory
of Particles
of Spin
One-Half*
A. 0. BARUT Department
of Physics,
Syracuse
University,
Syracuse, New York
From the two-component second order fermion theory of Feynman and GellMann a four component theory of spin-J5 particles in Hamiltonian form is derived. When a suitable inde6nit.e metric is introduced in the spin space by means of charge density the spin as well as the charge degrees of freedom may be separated. Spin and angular momentum are separately constants of the motion which is not the case for the Dirac equation. If states with positive norm describe particles the states of the corresponding charge conjugate particles have negative norm. The Hamiltonian is quadratic in the momenta which allows a straightforward transition to the nonrelativistic Pauli equation. The law of transformation of the wave function under the Lorentz transformations is given, and the relation and equivalence of the theory to Dirac and Foldy-Wouthuysen equations is established. 1. INTRODUCTION
There is a well-known single particle interpretation of the Klein-Gordon Equation for bosons first discussedby Sakata and Taketani (I), then by Heitler (2), Case (S), and recently by Feshbach and Villars (4). This theory employs a two component wave function with an indefinite metric in the two-dimensional space. The relativistic particle has an intrinsic charge degrees of freedom described by the norm (f 1) of the wave function. The main purpose of this paper is to show that a similar theory can be formulated for spin-34 particles in terms of a four component wave function with an indefinite metric in the four-dimensional space. The present formulation differs both in form and in physical interpretation from the usual Dirac equation. It is, however, equivalent to the latter as far as interactions with an external electromagnetic fields are concerned. For example, in the present formulation spin and angular momentum are separately constants of the motion, and both charge and spin degrees of freedom are explicit,ly separated. It amounts to a generalization of the Pauli theory of spin that includes all spin and relativistic effects exactly, i.e., in a manner equivalent to the Dirac equation. The transformation properties of the wave-function which makes the wave equation invariant under the Lorentz transformations is more complicated than * Supported
in part by the Air Force Office of Scientific 95
Research.
96
BARUT
the spinor transformations. It appears that the intimate linking of spin and angular momentum is not due to the requirement of relativistic invariawe, but tjo the use of the irreducible spin representations of the Lorentz group. Thus, the theory discussed in this paper will be useful, at present, mainly for purposes of physical interpretation of t’he relativistic theory and the passage to the nonr&~tivistic~ limit.. We shall use the Dirac m&ices satisfying +‘r’ + r”r” = L)$‘, \vhrrc $‘“: (1, - 1, - 1, - 1); y5 = ir”r’r”r”, (7’)’ = 1, and the st’andard representation \vith
The natural units with c = fi = 1 will he employed. The four vectors are indicated by a Greek sub- or superscript: s’, p, . The electjromagnetic potent’ials nw A”: (4, A) or A, : (4, -A),
2. THE
WAVE
EQUATION
We start from the second order Dime equation
(1) where
P, = p, - e.4, , In terms of the new quantities
0
x=
(D
=
(1
+
Y5k cp =
(a
( > $1
+
$3
$2
+
*4
*
(2)
Equation (1) splits into two identical two-component equations
(3) where
U
PY
11
O
- iua
=
0
=ii
I
iU2 -
iu, 0,
-(r
VI”
SPIN-34
97
PARTICLES
Feynman and Gell-Mann (5) have shown that there is a one-to-one correspondence between (3) and the first order Dirac equation. These authors have also used x instead of $ in writing down the universal Fermi interaction (without derivative coupling) which gives rise to an essentially unique form of the coupling (V and A in the language of P-decay) in agreement with experiments. This fact furnishes another motivation for the study of unusual forms of the wave equation, for it makes a difference in what form the coupling of two or more fields is written down. Let us introduce an independent two-component quantity cpodefined by Pw Then (3) can be written
= -mpo.
(4)
as
-mPgoo
- P2v -I f upvFMv(p- rn’p = 0.
Equations (4) and (5) together determine ing the linear combinations
a four-component
(5) equation.
Introduc-
0.3
01:
Ic = (r5- YO);; 0 PO
0
into (4) and (5), and by adding and subtracting tain an equation for XPin Hamiltonian form
the resulting
equations
po\E = H\E
we ob(7)
where H = &
1
P* - ; 2” Fpy ~“(1 + r5) - my0 + eA0
At this point we remark that (7) is not an admissible Hamiltonian with rest mass zero. In this case, instead of (4), we put
for particles
P&o = -(oo.
(8)
Then by the same steps as before we obtain with m = 0 H = f
P2 - f 2”’ Fp
~“(1 + r6) + a ~“(1 - 7”) + eAo.
(9)
98
BARUT
The Hamiltonian (7) is gauge-invariant and me note the appearance of the (1/2m)P” term, so characteristically different from the Dirac Hamiltonian. We shall discuss the transformation law of q under the Lorentz group in Section 5. First we investigate t,he physical interpreta,tion of (7), or (9), which is basically t,he same. We also note that (7) has essentially the same form as the Hamiltonian in the spin 0 case (see for example Ref. 4) except the additional spin term Y’F,, and t,he y-matrices instead of Pauli-matrices. The Hamiltonian (7) is not Hermitian in the usual sense. But a consistent treatment is possible if use is made of an indefinite metric in the four-dimensional spin space. We arrive at. the form of t.he metric by considering the charge density. For free particles a current density can easily be obtained from (3) j’ = con&. ((bpl;p* - q*p’;p). Using successively
(10)
(4) and (6) we find for the charge density j* = const. **y*\k.
J” is not positive definite. However, l!*/12 then with
if we normalize
= $ q’*y’\k
this metric H becomes Hermitian, Y’HI;Y’
(11) according
to
d3x = fl,
(1%
i.e.,
= H,
(1%
where Hf indicates the usual transpose and Hermitian conjugation in the ,spin space. The rules of operations with the metric (12) have been discussed in Appendix I. In particular H has real eigenvalues and eigenvectors belonging to distinct eigenvalues are orthogonal in the metric (12). The fact that (12) is the correct normalization for the Hamiltonian (‘7) is ako seen from a consideration of the charge c0njugat.e wave function \k” to q. \k” satisfies a SchrGdinger equation ..C w = H%“, and, from (7) we find H” if e is replaced by -e, (p -t eAj2 -I- E Z” F,,” In the case of the Dirac
equation
1
~‘(1 -I- r5> - my0 - eAo .
the charge conjugate
wave funct’ion
(l-2) $” is
SPIN-45
99
PARTICLES
related to $ by $” = C+*. In the present case we find by comparing conjugate complex of (7) that
(14) and the
H” = - (Cr”)H*(C~O)-l and \k” = crO\E*
(15)
C has to anticommute with y” and satisfy the condition C’ = C-‘; a choice for C is C = i-r” as in the case of the Dirac equation. We can now easily prove the assertion that if \k is normalized to +l according to (12), then the corresponding @, as given by (15), is normalized to - 1. (W, yOP) = (cyO\k*, r°Cro**)
= - (*, To*).
The expectation value of the charge e is fe according to (12). Equation (7), thus, describes particles and charge conjugate particles simultaneously. Furthermore, in the limit e -+ 0 (free particles, or neutral particles) H” = H and \E and \k” are independent solutions of the free particle equation. It will be shown in the next section that the sign of the norm in Eq. (12) is identical with the sign of the energy. 3. PROPERTIES
OF THE
FREE
PARTICLE
EQUATION
Equation (7) reduces in this case to
Ho= 2
~‘(1
+ 7”) - my’.
The eigenvalues of Ho are +E, = f(p2 + rn’)l” since Ho2 = p2 + m2. In a state with definite momentum the matrix elements of Ho and hence the solutions are all real. If H\k = E,q then we find that H(Cr’\k) = -E,(Cr’\k); i.e., if the positive energy solution is normalized to + 1, then the negative energy solution is normalized to - 1. The expectation value of the energy is +E, for both states. For free particles the orbital angular momentum will be a constant of the motion [Ho, L,] = 0
(17)
[Ho, &I = 0,
08)
as well as the spin
where 2,
=
i-fly’
=
( ) fl3
0
0
03
100 has eigenvalues
BARUT
f 1. Then the four stationary 1 z,
=
solutions can be rlassified as
+1
Iz,
=
follows
-1
E = -j-E, Norm
cI!))
+ 1
E = -E, Norm
- 1
where
E, + m ip.,, a = 2(mE,)1’2 ’ for the Hamiltonian
m - E, p ” = ‘4mEp)U?
(7), and a -
1 - E’P 2( Ep)l’z
(+P.X
h _ 1 + EP (p.x 2(E,P2
for the Hamiltonian (9). Consequently the new representation accomplishes for particles and antiparticles a definite separation of spin states which is not. possible in the Dirac form. Moreover, spin and angular momentum are not. coupled for the free part,icle and the absolute value of the spin (45) can only he asserted from the additional interaction term. It is also of interest to discuss the Heisenberg equation of the motion for t,he operators x, the so called Zitterhewegung. We find i = i[&)
xl = i yO(l + Y5).
ir has all eigenvalues equal to zero; however, usual relativistic value
the expectation
value of k has t.hrk
((i) = 1 for massless particle). although equivalent in observable effects, Since different representations, give different values for x and ir it is questionable which of these should be interpreted as the “position” and the “velocity” operator in the single particlt? interpretation. However, “velocity” and current density are related to each ot’her in each representation in the usual fashion. The diffkult,y is alreadv kunwn ill
SPIN->$
101
PARTICLES
the Foldy-Wouthuysen (6) form of the Dirac equation. A localized state in the Dirac representation correspond, for example, to a smeared out state of dimension rn-’ in the Foldy-Wouthuysen coordinates and vice versa. This points again to an essential limitation and incompleteness of all relativistic single particle theories as is well known. The expectation value of the momentum in the states (19) is given by (P> = +P
forE
= +E,,
(P) = -P
forE
= -E,.
In order to find the relation of this theory to the Dirac equation we first diagonalize the Hamiltonian (16) to get two two-component equations describing separately particles and charge conjugate particles. This can be done by a “unitary” transformation V in the sense of the metric (12), i.e., satisfying v-1 = yov+“/o*
(22)
The requirement V-‘HoV
= -y”Ep
(23)
leads then to
v=
l
2(mEp)1’2
v-1 =
[E, + m + (m -
Eph51, (24)
1 2(mEp)lj2
[E, + m - (m - EJY’I,
or, for the Hamiltonian (9) to
J&L2(Ep)“2
0 + Ep + (1 - -%>r5>, (24’)
v-1 = h2
(1 + E, - 11 - %h51. P
Thus we obtain again the Foldy-Wouthuysen i&
equation
= --~~Epyb,
(25)
where
Since the Eq. (25) can be transformed into the Dirac equation by a unitary transformation (4, 6)
’ =
[2J3p(E,1+
m)]‘/” (Ep - YoHp’p
102
BAHUT
where Ho is the Dirac Hamiltonian H, = a.p + vq3 we can transform the Dirac: wave function #D into the present one *D = lJ’#,
= ppp;
* = vc*u
= a’#,, .
(27)
Indeed, it is easily checked that the Dirac solutions go over into the solutions (19) with the correct norms under W. Hence TI’ is the transformation which srparates spin and charge states for the Dirao equation. It is a transformation between two spacesone with a positive definite, the other with an indefinite m&C. Since 1Y is nonsingular, the mapping between the two vector-spaces is one-t,o-one. 4. INTERACTION
WITH
AN
ESTERYAL
ELECTROMAGNETIC
FIELI,
To show the equivalence of the spec%rumof the Hamiltonian (7) with that, of the Dirac Hamiltonian we go back to l?q. (3) which for stationary problems ~11 he written as [(E - eAo)” - m*]p =
1
P” - % u”“F,,,, cp,
(28)
Once eigenvalues and solutions of this equation have been found, we (*RI) find cpcfrom (4) 1 (00= -% (E - eAo)q
(29)
cptogether with cpodetermine * according t,o (6). For example, for the spec%rum of the H-atom, Eq. (28) becomes
(30) The eigenvalue problem in t’he form (30) has been already solved by Hyllernas (7) giving the usual fine-st,ructure formula. Equation (28) is the most convenient starting point for the solution of practical problems. For example, for a free part,icle (28) is simply (p” + ,nl’Jcp= B”p with two independent solutions
, and one immediately obtains t$r (i) and (:‘) normalized solutions (19). Similarly the problem of the electron iu :1 c*onstnut, external magnet#ic field can he solved in R simple and straightforward m:mn(ht starting with (28). To find the nonrelativistic limit of (7) in the weak field approximation WC’note that \kl represent the two small components of * and q’z the two large components. Eliminating \kl , and passing to a nonrelativistic wa1.e funct,ion tiNR b! \kz = e-““$bxR ,
SPIN-34
we find the usual kinetic tion
103
PARTICLES
energy and spin correction
.= &R
2+ P2 + e& - 3;
P” - &
(d.B
terms as a first approxima-
- ideE)
(30) - &3 (B - iE)’ + . . .) &‘. 5. RELATIVISTIC
INVARIANCE
The relativistic invariance of the Dirac equation is established by defining a suitable transformation law for the wave function. The importance of the Dirac form of the theory of spin-35 particles lies in the fact that here one has to do with the irreducible “spin-representations” of the Lorentz-group. Similarly we can define a transformation law for the wave function 9 such that Eq. (7) is invariant under the Lorentz transformations. For this purpose we start from the transformation property of Eq. (25) as discussedby Foldy (8). Foldy has constructed ten operators, i.e., P, - r”G,
J
and -$
(XE, + E,x) + c+xE”
P
- tp,
which have the same commutation relations as the ten infinitesimal operators of the inhomogeneous Lorentz group. Let us denote the operators (31) collectively by L. Then the Eq. (25) is invariant under the transformation u = eieL,
* --$ *’ = w,
(32)
where E is an infinitesimal quantity. Now since Eq. (17) can be obtained from (25) by the operator V given by (24) the transformation law of \k under Lorentztransformations is
?I?’ = A*,
n = ei’L’
L' = PLV
(33)
The additional spin term (e/2) 2”“F,, is obviously Lorentz invariant. We would like to remark that this term could be easily modified to account for a possible anomalous magnetic moment term if such a generalization of the Dirac equation should be necessary. APPENDIX
In this Appendix we discuss briefly the rules of operations with the metric (12) which have been used in the text.
104
BARUT
The norm II * I]” = e, row is real if y” = definition of (JP’“, y’q@)). “Unitary”
C-k 1)
yO+. We have chosen an Hermitian, in fact a real y”. We extend the the norm to that of the scalar product of two vectors, i.e., Two vectors are “orthogonal” if t.his scalar product va,nishes. operators will preserve t,he scalar product. From (Vqp,
YoT/\k(?)) = (lp”‘, y”qP))
we find v+y”v Similarly
we must define a new operation (&p’,
which
= y”.
(A 3
of Hermitian
conjugat,ion
(d)
by
yo\k’2’) = (Q”), yo$\E’2’),
gives d = y”A+y”.
Hence an operator
A is “Hermitian”
if =1 = r0A +?‘I.
Expectation
value of an operat,or A in a state \k is given by (A) = (*, yOA\k),
which is equal to (A+, ~“9) if A is “Hermitian”. We shall prove that the eigenvalues of an “Hermitian” operator A are real if 7” = T”+ and if the corresponding eigenvectors have nonzero norm, and, that the eigenvectors belonging t’o distinct, cigenvalucs arc “orthogonal”. Let .4*, = a,*, , then pPm ) rOAW
= a,,pPm ) r”u,,j
= CL‘l*,, ) r”W
= a,(*?,
) 7°K)
hence, wm ) YOW = 0
if a,, # a, .
Furthermore, G&L,
Ye*,)
= (A*,
,7O*,j
= @,,
7OAW
= (~O.49~) \E,) = G*, or
, -y”+$h) = (Afin,
yo\kn) = real
SPIN-;5
PARTICLES
105
The Heisenberg equations of motion remain valid in the indefinite metric with the understanding that the time evolution operator U = eeiH1 is unitary in the new sense, i.e., U+ = eiH+t = y”U-l”lo. We further note that
(*, ‘YOdA dt 9 )- RECEIVED:
d(A) . (*, r’i[H, AN?) = $ (q, r”Aq) = dt
July 3, 1958 REFERENCES
1. SAKATA AND TAKETANI, Proc. Phys. Math. Sot. Japan 22, 757 (1940). 2. W. HEITLER, Proc. Roy. Irish Acad. 49, 1 (1943). 3. K. M. CASE, Phys. Rev. 96, 1323 (1954). 4. H. FESHBACH AND F. VILLARS, Revs. Modern Phys. 30, 24 (1958). 5. R. P. FEYNMAN AND M. GELL-MANN, Phys. Rev. 109, 193 (1958). 6. L. L. FOLDY AND S. A. WOUTHUYSEN, Phys. Rev. 73, 29 (1950). 7. E. HYLLERAAS, 2. Physik 140, 626 (1955). 8. L. L. FOLDY, Phys. Rev. 102, 568 (1956).