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Equivalence of manifestly covariant theories for spin 32
Nuclear Physics 58 (1964) 314--320; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
EQUIVALENCE OF MANIFESTLY COVARIANT T H E O R I E S FOR SPIN A. D. BRYDEN Department of Natural Philosophy, The University, Glasgow
Received 17 February 1964 Abstract: The transformations from the Bargmann-Wigner equations to the Proca equations for spin 1 and from the Bargmann-Wigner equations to the Rarita-Schwinger equations for spin
| are presented. The scalar products are shown to be invariant under these transformations. Some general statements of relationships among manifestly covariant wave functions are also derived. 1. Introduction One of the basic postulates of quantum field theory is that of relativistic covariance i.e. the equations describing the properties of elementary particles do not change their form under Lorentz transformations. We shall only be concerned with those equations in which the wave function may he expressed as ~A, where A represents a number of vector or spinor indices, so that ~a has well-known transformation properties under the Lorentz group. Equations of this type will he referred to as manifestly covariant equations. In general, the wave function $A does not describe a system with a unique spin and in order to achieve this we must restrict the number of independent components of $A by means of suitable (covariant) supplementary conditions. The best-known of these is the Lorentz condition for spin 1. For any particular spin we may have any number of different manifestly covariant equations, containing wave functions with different transformation properties, but with the same number of independent components. Pursey 1) has given a general formalism for the relationship between these manifestly covariant theories in terms of the inverse Foldy-Wouthuysen transformation 2) to the canonical representation of the inhomogeneous Lorentz group by Macfarlane 3). In this paper we deal explicitly with the case of spin ½ for which we consider two manifestly covariant equations, those of Rarita and Schwinger 4) and of Bargmann and Wigner s). In sect. 2 we derive some useful results from the Bargmann-Wigner equations for spin 1, and apply a generalization of these in sect. 3 to the problem of finding the transformation between the Rarita-Schwinger and Bargmann-Wigner theories for spin ½. In sect. 4 it is shown that the scalar product is invariant under this transfolmation. The spin ½ result differs from that of Johnson and Sudarshan 6) and this is discussed in sect. 5, where it is shown that the transformations for spin 1 and spin ½ are particular cases of some general rules for the equivalence of manifestly covariant wave functions describing arbitrary spin. 314
EQUIVALENCE OF MANIFESTLY COVARIANT THEORIES
315
2. Transformation for Spin 1
Throughout this paper, we shall use a metric g.V with components goo = - g l l
= --gz2 = - g 3 3 = 1, g ~ v = O , # # v,
and Dirac ?-matrices satisfying
7~'?"+?"?~'
= 2g~'".
The Bargmann-Wigner equations for spin 1 are
PI, Y~o~pP = m~=.8,
(2.1)
where ~ and/~ both take the values 1. . . . . 4 and ~b=pis symmetric in the interchange of • and ft. We can build a four-vector and an antisymmetric second-rank tensor from ~b=pby defining ~p" = (C?")=p~=p, (2.2)
F ~'" = (Ca~")=s~=p = ½i[C(7"?"-?"7~')1=,81,/,,,~,
(2.3)
summation over ~ and fl being understood, where C satisfies C?#C-I = _(?/I)T, cT=
--C,
C + ~-- C - 1
(C T is the transpose and C + the hermitian conjugate of C). Then it is easy to show that ~p",F ~* have the required transformation properties. It is convenient to express all manifestly covariant wave functions in terms of the basis vectors for the finite-dimensional representations of the homogeneous Lorentz group. From the generators J, N of infinitesimal rotations and Lorentz transformations, we define = 5(J+iN),
t. = 5 ( 1 - i N )
so that [K, L] = 0, and the components of K and of L b o t h satisfy the commutation rules of angular momentum components. Then the representations aie labelled by (k,/) and are denoted by ~kt(k, l = 0, 5, 1. . . . ) where the eigenvalue o f K 2 and L 2 are k ( k + 1) and l(l+ 1), respectively. A four-vector belongs to the (5, 5) representation, and a four-spinor to the direct sum representation (5, 0) E~ (0, 5)- Any manifestly covariant wave function b~.longs to a direct sum of irreducible representations. In particular ~,=p belongs to the symmetrized direct product of two spinor representations, which can be reduced to
(5,5) • (I,0) @ (0, 1). The vector (p~ thus defines those components of ~b~pbelonging to the (5,5) representation and F ~ consistsof the components of ~=p belonging to the (I,0) ~ (0, I) representation. It is shown in appendix I that ~, F "v satisfythe Proca equations 7)
316
A. D. BRYDEN
for spin 1
p~@V_p~q~ = imF~,,
(2.4)
p" F~, = im%,.
(2.5)
Eqs. (2.2) and (2.3) may now be inverted by making use of the relation between the Dirac v-matrices s) [I ]~p[I]~,p, + [?~]~p[?~]¢p, + ½[g~ ]~ [ ~ ]~,~, -I- [i~5~]~p[i~5)'~]~,~, + [?s]~[~s]~'p" = 46~B't~'p.
(2.6)
Since C, C~5~~, C~5 are antisymmetric, this gives
[(Cy~)~p(?~C-l)~,p,+½(C¢~')~p(cT~,,C-l)~,~,]~b~p
=
4¢~,p,,
(2.7)
and thus ¢~p = ¼[tp~(?~C- l)~+½F~,,(%,, C- l)~].
(2.8)
This equation expresses each component of ¢~p as a linear combination, with constant coefficients, of the components of tp~ and F ~'. By using eq. (2.4) or eq. (2.5), gc~pcan be written in terms oftp ~ or F ~ alone, but. then the coefficients are momentum dependent.. 3. Transformation for Spin 2s_ In this section the equivalence of the Bargmann-Wigner and Rarita-Schwinger equations for spin ½, is examined. The Rarita-Schwinger spin ½ equations are
(p~--m)~b" = O,
(3.1)
where ~b" is a vector-spinor i.e. it has both a vector and a spinor index (which will often be suppressed). It also satisfies the subsidiary condition ~ b ~ = 0.
(3.2)
p~b ~ = 0.
(3.3)
Eqs. (3.1) and (3.2) together imply The vector-spinor ~b~ belongs to the direct product representation (½, ½) ® [(½, 0) (0, ½)], which may be reduced to the direct sum ~ "" ~bl,½, ~ ~.t ~B ~/½,o ~ ~o,~,
(3.4)
where ~bk,z belongs to the (k, l) representation of the Lorentz group. Since ~ is a linear combination of the components of ~b~ and transforms as a spinor, we must have
~'.¢"
=
¢'~,o • ¢o,~,
(3.5)
and eq. (3.2) implies that ~b~,o ~ ~o,~ = 0. This is to be expected since it is impossible to construct from ~'½,o @ ~ko,½ an entity which transforms under t h e j = ½ representation of the rotation subgroup of the Lorentz group. Thus we need oaly consider
317
EQUIVALENCE OF MANIFESTLY COVARIANT THEORIES
those components of ~b" which belong to the (1, ½) ~ (½, 1) representation of the Lorentz group. The Bargmann-Wigner equations for spin ½ are p~, y,,p ~pp~, = m~,p~,
(3.6)
and two similar equations, where ¢~a~ is symmetric in the interchange of 0~,#, y and thus belongs to the symmetrized direct product of three spinor representations; this can be reduced to the direct sum (½, 0) @ (0, ½) @ (1, ½) @ (½, 1). In a similar manner to the spin 1 case, we can define quantities A~, B~v by A~ = B~" = ( c a " ' ) p ~ , ~
(3.7) = ½i[c(r"r'-
r'r")]p~¢~.
(3.8)
Then A~ is a vector-spinor and B~v is an antisymmetric tensor-spinor. The quantity B~v belongs to the direct product of the (1, 0) ~ (0, 1) and (½, 0) ~ (0, ½) representations, which cart be reduced to the direct sum
B"' ~ (~,0) ~ (0, ~) • (l,½) • (½, 1) ~ (½,0) ~ (0, ½). It is shown in appendix 2 that A~ satisfies the Rarita-Schwinger eqs. (3.1) and (3.2): (p~,y~'-m)A" = 0, (3.9) yj, A ~' = 0.
(3.10)
Also A~, B "~ satisfy equations similar to eqs. (2.4) and (2.5): p'B~,, = imam,,
(3.11)
p~' A " - p " A ~' = i m B ~".
(3.12)
From eqs. (3.12) and (3.10) we have (3.13)
A ~ = iy~,B uv,
and therefore ~,~B"" = 0.
(3.14)
Therefore iy~,B ~'" form those components of B vv which belong to the representation (1, ½) @ (½, 1) @ (½, 0) @ (0, ½) and eq. (3.14) shows that the spinor part of B ~ is zero. Therefore B v" (with the condition (3.14)) belongs to the same representation as the Bargmann-Wigner wave function $~arAs before eqs. (3.7) and (3.8) may be inverted, giving 1
¢=Pr = i-T[(Y.C
-1
/./
-1
+~-a[(a.vC-'
=
p
-1
p
)prA=+ ( y v C ) ~ = Ap + ( y . C ) , ~ A v]
)p~A~,+ 2i--m
""
-'
""
-'~
"']
)#~(p A , , - p A~,)+ symmetric terms
(3.15) . (3.16)
318
A . D . BRYDEN
Also we could substitute eq. 0.13) irtto (3.15) and express ~k,#~in terms ofB~ v. This equation has constant coefficients while the coefficients in eq. (3.16) are momentum dependent. Eqs. (3.7) and (3.16) demonstrate the equivalence of the BargmannWigner and Rarita-Schwinger wave functions. 4. Scalar Product
In this section we show that the scalar product is invariant under the transformation (3.16) i.e. the transformation is unitary. The Bargmann-Wigner scalar product 5) for spin ½ is (¢, ~)~w = f ~k~,r,8,,, fl##' ,8~' ~k,,,,,, dO,
(4.1)
where dO --- d3p/po and the matrix ,8 satisfies the equations r~t = ,8yt,,8-I, ,8 = ,8+ = ,8-I.
(4.2)
Eq. (4.1) can be expressed in terms of the Rarita-Schwinger wave function A ~ by substitution of eq. (3.16), and after a rather tedious calculation gives (~/, ~b)ew = -- ½ f At~,8~, A~,dt2,
(4.3)
which is the expected form of the Rarita-Schwinger scalar product in momentum space. By a method similar to that in ref. 5), it may be shown that the scalar product - S A*',SA.~dI2 is positive definite. The scalar product (4.3) becomes
-- ) At',sA~d3p/po = - m ;At"A~
J
d 3-/j.
(4.4)
Substituting A ° = p" A/po into eq. (4.4),
_ P~mat,A" = P~m--[p~(At " A ) - ( p . A)t(p • A)] m
> ~o4 [p2(At "A)-(p" A)t(p • A)] _~ 0. Similar results can also be derived for the case of spin 1. 5. Discussion
In sects. 2 and 3, transformations were derived among manifestly covariant wave functions describing spins 1 and ½, and it was observed that some relations have coefficients which are momentum dependent, while others have constant coefficients. These results are all particular cases of the following generalization.
EQUIVALENCE OF MANIFESTLY COVARIANT THEORIES
319
Let v be a wave function belonging to the representation V, which can be reduced to a direct sum V = ~Vi and let u be a wave function belonging to the representation U. Then if the representation U (which may be reducible or irreducible) is contained in V i.e. in the direct sum ~V~, the components of u must be linear combinations with constant coefficients of the components of o. In matrix form u = Av,
(5.1)
where A has constant elements. Any covariant wave function u which can be formed in this way belongs to a representation contained in the direct sum ~ Vi. If eq. (5.1) can be inverted so that v may be expressed linearly with constant coefficients in terms o f u, then the representation V is contained in the direct sum for the representation U and therefore the representations V and U are identical. If v cannot be expressed with c o n s t a n t coefficients in terms o f u, then V = ~ V i has representations which are not contained in U. In the case of spin ½, it is seen that A ~, which belongs to the (1, ½) ~ (½, 1) representation, can be expressed in terms of ~k~pr,which belongs to the (1, ½) ~ (½, 1) (½, 0) ~ (0, ~) representation, by a matrix whose coefficients are constant. However, in the inverse transformation (3.16) the matrix coefficients are momentum dependent, representing the expression of the (½, 0) ~ (0, ½) components in terms of the (1, ½) ~ (½, 1) components. Also ~k~p~ and B ~" {with the condition (3.14)} belong to the same representation and it has been shown that each can be written linearly with constant coefficients in terms of the other. The equation obtained by substituting eqs. (3.13)and (3.12)into eq. (3.16)shows that the Bargmann-Wigner wave function ~k~ar, symmetrized in the spinor indices, is equivalent to the direct product of the Rarita-Schwinger wave function ~ (with V~k~ = 0) and the momentum vector p~, this product being antisyrmnetrical in the vector indices. Similar results hold for spin 1, where the Bargmann-Wigner wave function ~,,p can be written in terms of the wave function {q~, F ~'} with constant coefficients, but the expression of ~k~pin terms of q~ or F ~" alone is momentum dependent. These general relations have apparently not been realized before. For example, Johnson and Sudarshan 4) write down the same form of transformation from the multispinor to the vector-spinor formalisms as in eq. (3.7) but then they assume a form with constant coefficients for the inverse transformation i.e. eq. (3.16) with only the first term on the right-hand side. Finally we remark that eqs. (3.7) and (3.16) provide a method of obtaining the transformation from the Rarita-Schwinger spin ] equations to the Foldy canonical form 9), as an alternative to that of ref. 1o). The transformation to the BargmannWigner equations is first performed, and then the result of Pursey 11) in the particular case of spin ~ is used to obtain the Foldy expression. The author is indebted to Dr. D. L. Pursey for his guidance throughout this work. He also acknowledges the receipt of a D.S.I.R. studentship.
320
A.D. BRYDEN
Appendix 1 The proof of eq. (2.5) is immediate, involving only eq. (2.1) and the antisymmetry of C. For eq. (2.4) we wish to show
p"(CT~)~p~k~#-p*(Cy~)~d/~#
= -½m{C(y~y~-y'y~)}~p~#.
(A.1)
Now the right hand side of eq. (A.1) is equal to
- ½{c(:r'-~:~,"):p,}~#~,~#, which is zero if p # v or p # g since Cy~y'y# is antisymmetric if the indices are all different: - ½
[c(r"~,'-~'r"):p,],~ ~,~#
= - ½{C(y" I~'- 1~"Yu)(Y,P" + Y,P')}~# ~b~#,
(A.2)
where no summation is implied over p, v. Since y'y~ = 1, v = 0, 1, 2, 3 (no summation), eq. (A.1) follows. Appendix 2 The proof of eq. (3.9) is obvious from the definition of A" and eq. (3.6). To show eq. (3.10) it is more convenient to consider Cv~'A~, i . e . (C7~')p~(C~,#)~p~,~#r and investigate the symmetry of (Cy~')p~(C~:')~# in the imerchange of fl, 7. To do this, we use the result of G o o d s) that J 2 - J 4 is antisymmetrical in the interchange of wave functions ~'2, ~/4, where ']2 = ( ~ 1 ~ / 2 ) ( ~ 3 ~ 4 ) ,
(A.3)
J4. =
(A.4)
-- (~1 ~ 5
~.¢2)(~3 ~/~5 ~4.).
Replacing if1, i~3 by tpx C, tp2 C in eqs. (A.3) and (A.4), we obtain
(c:)~#(c~,)p~ + (cr"rs)~#(cr, r5),~ = - (c~"),,(cr~)~#- (cr"rs),~(cr, rD~#. (A.5) Therefore (C~)~#(C),u)p~,~# ~ = 0 since C~y5 is antisymmetrie. Since C is nonsingular, then ~A~ = 0.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
D. L. Pursey, to be published. L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78 (1950) 29 A. J. Macfarlane, J. Math. Phys. 4 (1963) 490 W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61 V. Bargmann and E. P. Wigner, Proc. U.S. Nat. Acad. Sci. 34 (1948) 211 K. Johnson and E. C. G. Sudarshan, Ann. of Phys. 13 (1961) 126 W. Pauli, Revs. Mod. Phys. 13 (1941) 203 R. H. Good, Revs. Mod. Phys. 27 (1955) 187 L. L. Foldy, Phys. Rev. 102 (1956) 568 A. D. Bryden, Nuclear Physics 53 (1964) 165 D. L. Pursey, Nuclear Physics 53 (1964) 174
EQUIVALENCE OF MANIFESTLY COVARIANT T H E O R I E S FOR SPIN A. D. BRYDEN Department of Natural Philosophy, The University, Glasgow
Received 17 February 1964 Abstract: The transformations from the Bargmann-Wigner equations to the Proca equations for spin 1 and from the Bargmann-Wigner equations to the Rarita-Schwinger equations for spin
| are presented. The scalar products are shown to be invariant under these transformations. Some general statements of relationships among manifestly covariant wave functions are also derived. 1. Introduction One of the basic postulates of quantum field theory is that of relativistic covariance i.e. the equations describing the properties of elementary particles do not change their form under Lorentz transformations. We shall only be concerned with those equations in which the wave function may he expressed as ~A, where A represents a number of vector or spinor indices, so that ~a has well-known transformation properties under the Lorentz group. Equations of this type will he referred to as manifestly covariant equations. In general, the wave function $A does not describe a system with a unique spin and in order to achieve this we must restrict the number of independent components of $A by means of suitable (covariant) supplementary conditions. The best-known of these is the Lorentz condition for spin 1. For any particular spin we may have any number of different manifestly covariant equations, containing wave functions with different transformation properties, but with the same number of independent components. Pursey 1) has given a general formalism for the relationship between these manifestly covariant theories in terms of the inverse Foldy-Wouthuysen transformation 2) to the canonical representation of the inhomogeneous Lorentz group by Macfarlane 3). In this paper we deal explicitly with the case of spin ½ for which we consider two manifestly covariant equations, those of Rarita and Schwinger 4) and of Bargmann and Wigner s). In sect. 2 we derive some useful results from the Bargmann-Wigner equations for spin 1, and apply a generalization of these in sect. 3 to the problem of finding the transformation between the Rarita-Schwinger and Bargmann-Wigner theories for spin ½. In sect. 4 it is shown that the scalar product is invariant under this transfolmation. The spin ½ result differs from that of Johnson and Sudarshan 6) and this is discussed in sect. 5, where it is shown that the transformations for spin 1 and spin ½ are particular cases of some general rules for the equivalence of manifestly covariant wave functions describing arbitrary spin. 314
EQUIVALENCE OF MANIFESTLY COVARIANT THEORIES
315
2. Transformation for Spin 1
Throughout this paper, we shall use a metric g.V with components goo = - g l l
= --gz2 = - g 3 3 = 1, g ~ v = O , # # v,
and Dirac ?-matrices satisfying
7~'?"+?"?~'
= 2g~'".
The Bargmann-Wigner equations for spin 1 are
PI, Y~o~pP = m~=.8,
(2.1)
where ~ and/~ both take the values 1. . . . . 4 and ~b=pis symmetric in the interchange of • and ft. We can build a four-vector and an antisymmetric second-rank tensor from ~b=pby defining ~p" = (C?")=p~=p, (2.2)
F ~'" = (Ca~")=s~=p = ½i[C(7"?"-?"7~')1=,81,/,,,~,
(2.3)
summation over ~ and fl being understood, where C satisfies C?#C-I = _(?/I)T, cT=
--C,
C + ~-- C - 1
(C T is the transpose and C + the hermitian conjugate of C). Then it is easy to show that ~p",F ~* have the required transformation properties. It is convenient to express all manifestly covariant wave functions in terms of the basis vectors for the finite-dimensional representations of the homogeneous Lorentz group. From the generators J, N of infinitesimal rotations and Lorentz transformations, we define = 5(J+iN),
t. = 5 ( 1 - i N )
so that [K, L] = 0, and the components of K and of L b o t h satisfy the commutation rules of angular momentum components. Then the representations aie labelled by (k,/) and are denoted by ~kt(k, l = 0, 5, 1. . . . ) where the eigenvalue o f K 2 and L 2 are k ( k + 1) and l(l+ 1), respectively. A four-vector belongs to the (5, 5) representation, and a four-spinor to the direct sum representation (5, 0) E~ (0, 5)- Any manifestly covariant wave function b~.longs to a direct sum of irreducible representations. In particular ~,=p belongs to the symmetrized direct product of two spinor representations, which can be reduced to
(5,5) • (I,0) @ (0, 1). The vector (p~ thus defines those components of ~b~pbelonging to the (5,5) representation and F ~ consistsof the components of ~=p belonging to the (I,0) ~ (0, I) representation. It is shown in appendix I that ~, F "v satisfythe Proca equations 7)
316
A. D. BRYDEN
for spin 1
p~@V_p~q~ = imF~,,
(2.4)
p" F~, = im%,.
(2.5)
Eqs. (2.2) and (2.3) may now be inverted by making use of the relation between the Dirac v-matrices s) [I ]~p[I]~,p, + [?~]~p[?~]¢p, + ½[g~ ]~ [ ~ ]~,~, -I- [i~5~]~p[i~5)'~]~,~, + [?s]~[~s]~'p" = 46~B't~'p.
(2.6)
Since C, C~5~~, C~5 are antisymmetric, this gives
[(Cy~)~p(?~C-l)~,p,+½(C¢~')~p(cT~,,C-l)~,~,]~b~p
=
4¢~,p,,
(2.7)
and thus ¢~p = ¼[tp~(?~C- l)~+½F~,,(%,, C- l)~].
(2.8)
This equation expresses each component of ¢~p as a linear combination, with constant coefficients, of the components of tp~ and F ~'. By using eq. (2.4) or eq. (2.5), gc~pcan be written in terms oftp ~ or F ~ alone, but. then the coefficients are momentum dependent.. 3. Transformation for Spin 2s_ In this section the equivalence of the Bargmann-Wigner and Rarita-Schwinger equations for spin ½, is examined. The Rarita-Schwinger spin ½ equations are
(p~--m)~b" = O,
(3.1)
where ~b" is a vector-spinor i.e. it has both a vector and a spinor index (which will often be suppressed). It also satisfies the subsidiary condition ~ b ~ = 0.
(3.2)
p~b ~ = 0.
(3.3)
Eqs. (3.1) and (3.2) together imply The vector-spinor ~b~ belongs to the direct product representation (½, ½) ® [(½, 0) (0, ½)], which may be reduced to the direct sum ~ "" ~bl,½, ~ ~.t ~B ~/½,o ~ ~o,~,
(3.4)
where ~bk,z belongs to the (k, l) representation of the Lorentz group. Since ~ is a linear combination of the components of ~b~ and transforms as a spinor, we must have
~'.¢"
=
¢'~,o • ¢o,~,
(3.5)
and eq. (3.2) implies that ~b~,o ~ ~o,~ = 0. This is to be expected since it is impossible to construct from ~'½,o @ ~ko,½ an entity which transforms under t h e j = ½ representation of the rotation subgroup of the Lorentz group. Thus we need oaly consider
317
EQUIVALENCE OF MANIFESTLY COVARIANT THEORIES
those components of ~b" which belong to the (1, ½) ~ (½, 1) representation of the Lorentz group. The Bargmann-Wigner equations for spin ½ are p~, y,,p ~pp~, = m~,p~,
(3.6)
and two similar equations, where ¢~a~ is symmetric in the interchange of 0~,#, y and thus belongs to the symmetrized direct product of three spinor representations; this can be reduced to the direct sum (½, 0) @ (0, ½) @ (1, ½) @ (½, 1). In a similar manner to the spin 1 case, we can define quantities A~, B~v by A~ = B~" = ( c a " ' ) p ~ , ~
(3.7) = ½i[c(r"r'-
r'r")]p~¢~.
(3.8)
Then A~ is a vector-spinor and B~v is an antisymmetric tensor-spinor. The quantity B~v belongs to the direct product of the (1, 0) ~ (0, 1) and (½, 0) ~ (0, ½) representations, which cart be reduced to the direct sum
B"' ~ (~,0) ~ (0, ~) • (l,½) • (½, 1) ~ (½,0) ~ (0, ½). It is shown in appendix 2 that A~ satisfies the Rarita-Schwinger eqs. (3.1) and (3.2): (p~,y~'-m)A" = 0, (3.9) yj, A ~' = 0.
(3.10)
Also A~, B "~ satisfy equations similar to eqs. (2.4) and (2.5): p'B~,, = imam,,
(3.11)
p~' A " - p " A ~' = i m B ~".
(3.12)
From eqs. (3.12) and (3.10) we have (3.13)
A ~ = iy~,B uv,
and therefore ~,~B"" = 0.
(3.14)
Therefore iy~,B ~'" form those components of B vv which belong to the representation (1, ½) @ (½, 1) @ (½, 0) @ (0, ½) and eq. (3.14) shows that the spinor part of B ~ is zero. Therefore B v" (with the condition (3.14)) belongs to the same representation as the Bargmann-Wigner wave function $~arAs before eqs. (3.7) and (3.8) may be inverted, giving 1
¢=Pr = i-T[(Y.C
-1
/./
-1
+~-a[(a.vC-'
=
p
-1
p
)prA=+ ( y v C ) ~ = Ap + ( y . C ) , ~ A v]
)p~A~,+ 2i--m
""
-'
""
-'~
"']
)#~(p A , , - p A~,)+ symmetric terms
(3.15) . (3.16)
318
A . D . BRYDEN
Also we could substitute eq. 0.13) irtto (3.15) and express ~k,#~in terms ofB~ v. This equation has constant coefficients while the coefficients in eq. (3.16) are momentum dependent. Eqs. (3.7) and (3.16) demonstrate the equivalence of the BargmannWigner and Rarita-Schwinger wave functions. 4. Scalar Product
In this section we show that the scalar product is invariant under the transformation (3.16) i.e. the transformation is unitary. The Bargmann-Wigner scalar product 5) for spin ½ is (¢, ~)~w = f ~k~,r,8,,, fl##' ,8~' ~k,,,,,, dO,
(4.1)
where dO --- d3p/po and the matrix ,8 satisfies the equations r~t = ,8yt,,8-I, ,8 = ,8+ = ,8-I.
(4.2)
Eq. (4.1) can be expressed in terms of the Rarita-Schwinger wave function A ~ by substitution of eq. (3.16), and after a rather tedious calculation gives (~/, ~b)ew = -- ½ f At~,8~, A~,dt2,
(4.3)
which is the expected form of the Rarita-Schwinger scalar product in momentum space. By a method similar to that in ref. 5), it may be shown that the scalar product - S A*',SA.~dI2 is positive definite. The scalar product (4.3) becomes
-- ) At',sA~d3p/po = - m ;At"A~
J
d 3-/j.
(4.4)
Substituting A ° = p" A/po into eq. (4.4),
_ P~mat,A" = P~m--[p~(At " A ) - ( p . A)t(p • A)] m
> ~o4 [p2(At "A)-(p" A)t(p • A)] _~ 0. Similar results can also be derived for the case of spin 1. 5. Discussion
In sects. 2 and 3, transformations were derived among manifestly covariant wave functions describing spins 1 and ½, and it was observed that some relations have coefficients which are momentum dependent, while others have constant coefficients. These results are all particular cases of the following generalization.
EQUIVALENCE OF MANIFESTLY COVARIANT THEORIES
319
Let v be a wave function belonging to the representation V, which can be reduced to a direct sum V = ~Vi and let u be a wave function belonging to the representation U. Then if the representation U (which may be reducible or irreducible) is contained in V i.e. in the direct sum ~V~, the components of u must be linear combinations with constant coefficients of the components of o. In matrix form u = Av,
(5.1)
where A has constant elements. Any covariant wave function u which can be formed in this way belongs to a representation contained in the direct sum ~ Vi. If eq. (5.1) can be inverted so that v may be expressed linearly with constant coefficients in terms o f u, then the representation V is contained in the direct sum for the representation U and therefore the representations V and U are identical. If v cannot be expressed with c o n s t a n t coefficients in terms o f u, then V = ~ V i has representations which are not contained in U. In the case of spin ½, it is seen that A ~, which belongs to the (1, ½) ~ (½, 1) representation, can be expressed in terms of ~k~pr,which belongs to the (1, ½) ~ (½, 1) (½, 0) ~ (0, ~) representation, by a matrix whose coefficients are constant. However, in the inverse transformation (3.16) the matrix coefficients are momentum dependent, representing the expression of the (½, 0) ~ (0, ½) components in terms of the (1, ½) ~ (½, 1) components. Also ~k~p~ and B ~" {with the condition (3.14)} belong to the same representation and it has been shown that each can be written linearly with constant coefficients in terms of the other. The equation obtained by substituting eqs. (3.13)and (3.12)into eq. (3.16)shows that the Bargmann-Wigner wave function ~k~ar, symmetrized in the spinor indices, is equivalent to the direct product of the Rarita-Schwinger wave function ~ (with V~k~ = 0) and the momentum vector p~, this product being antisyrmnetrical in the vector indices. Similar results hold for spin 1, where the Bargmann-Wigner wave function ~,,p can be written in terms of the wave function {q~, F ~'} with constant coefficients, but the expression of ~k~pin terms of q~ or F ~" alone is momentum dependent. These general relations have apparently not been realized before. For example, Johnson and Sudarshan 4) write down the same form of transformation from the multispinor to the vector-spinor formalisms as in eq. (3.7) but then they assume a form with constant coefficients for the inverse transformation i.e. eq. (3.16) with only the first term on the right-hand side. Finally we remark that eqs. (3.7) and (3.16) provide a method of obtaining the transformation from the Rarita-Schwinger spin ] equations to the Foldy canonical form 9), as an alternative to that of ref. 1o). The transformation to the BargmannWigner equations is first performed, and then the result of Pursey 11) in the particular case of spin ~ is used to obtain the Foldy expression. The author is indebted to Dr. D. L. Pursey for his guidance throughout this work. He also acknowledges the receipt of a D.S.I.R. studentship.
320
A.D. BRYDEN
Appendix 1 The proof of eq. (2.5) is immediate, involving only eq. (2.1) and the antisymmetry of C. For eq. (2.4) we wish to show
p"(CT~)~p~k~#-p*(Cy~)~d/~#
= -½m{C(y~y~-y'y~)}~p~#.
(A.1)
Now the right hand side of eq. (A.1) is equal to
- ½{c(:r'-~:~,"):p,}~#~,~#, which is zero if p # v or p # g since Cy~y'y# is antisymmetric if the indices are all different: - ½
[c(r"~,'-~'r"):p,],~ ~,~#
= - ½{C(y" I~'- 1~"Yu)(Y,P" + Y,P')}~# ~b~#,
(A.2)
where no summation is implied over p, v. Since y'y~ = 1, v = 0, 1, 2, 3 (no summation), eq. (A.1) follows. Appendix 2 The proof of eq. (3.9) is obvious from the definition of A" and eq. (3.6). To show eq. (3.10) it is more convenient to consider Cv~'A~, i . e . (C7~')p~(C~,#)~p~,~#r and investigate the symmetry of (Cy~')p~(C~:')~# in the imerchange of fl, 7. To do this, we use the result of G o o d s) that J 2 - J 4 is antisymmetrical in the interchange of wave functions ~'2, ~/4, where ']2 = ( ~ 1 ~ / 2 ) ( ~ 3 ~ 4 ) ,
(A.3)
J4. =
(A.4)
-- (~1 ~ 5
~.¢2)(~3 ~/~5 ~4.).
Replacing if1, i~3 by tpx C, tp2 C in eqs. (A.3) and (A.4), we obtain
(c:)~#(c~,)p~ + (cr"rs)~#(cr, r5),~ = - (c~"),,(cr~)~#- (cr"rs),~(cr, rD~#. (A.5) Therefore (C~)~#(C),u)p~,~# ~ = 0 since C~y5 is antisymmetrie. Since C is nonsingular, then ~A~ = 0.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
D. L. Pursey, to be published. L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78 (1950) 29 A. J. Macfarlane, J. Math. Phys. 4 (1963) 490 W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61 V. Bargmann and E. P. Wigner, Proc. U.S. Nat. Acad. Sci. 34 (1948) 211 K. Johnson and E. C. G. Sudarshan, Ann. of Phys. 13 (1961) 126 W. Pauli, Revs. Mod. Phys. 13 (1941) 203 R. H. Good, Revs. Mod. Phys. 27 (1955) 187 L. L. Foldy, Phys. Rev. 102 (1956) 568 A. D. Bryden, Nuclear Physics 53 (1964) 165 D. L. Pursey, Nuclear Physics 53 (1964) 174