- Email: [email protected]
The bra-ket formalism for free relativistic particles
OF PHYSICS:
46,
559-576 (1968)
The Bra-Ket
Formalism
for Free Relativistic
Particles
R. FONG Department of Physics, University of Durham, Durham, Engkmd AND
E. G. P. ROWE Department of Mathematics, University of Durham, Durham, England
Relativistic single particle theories are presented from the point of view of the bra-ket formalism of Dirac. Not only does this give a coherent and unified formulation of the theory but through it we are also able to understand clearly the relationship between and meaning of the different forms of amplitudes and of the position operator that exist in the literature.
I. INTRODUCTION
We wish, in this paper, to trace the development of free relativistic one-particle theories for elementary particles in a coherent and unified way. We do this by formulating it all in the bra-ket formalism. By so doing, we also resolve in a simple manner the confusion surrounding the different forms of amplitudes and of the position operator that exist in the literature. It is perhaps unlikely that Dirac would have found his equation if he had begun from this point of view. But if, in any case, he had rewritten the theory in this way, the long wait for the Foldy-Wouthuysen explanation (I) of the anomalies of “Zitterbewegung” and of nonconserved orbital angular momentum for the free electron would have been shortened. Foldy and Wouthuysen themselves would probably not then have introduced their famous prefix “mean”, which required a further ten years to be questioned (2) and which still has not yet completely disappeared from present usage. The bra-ket notation is useful, even essential, in keeping clear the various amplitudes that necessarily arise in a relativistic theory, in which the simplest transforming amplitude is different from the most easily interpretable one. The connection between these amplitudes (&p 1 F) and n(p I v) for spin-zero in the momentum representations, &x 1 tpt) and s(x I yt) for spin zero in the coordinate representations) is not a unitary transformation of the theory, i.e., the physical operators such as p and x and the physical kets 1y) remain unchanged, 559
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FONG
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ROWE
but the basic kets are altered (a matter only of normalization in momentum space, a change, however, from eigenstates of x to slightly spread out states in coordinate space). This matter is especially important in connection with the position operator, since the different possibilities for the Fourier transform of momentum amplitudes lead apparently to various position operators, e.g., the x and X of Foldy and Wouthuysen (I). The position operator long remained obscure because of an unjustified demand for “manifest covariance” and, for the spin-4 case, because of Dirac’s four-component amplitude which cannot be used in a simply way to expand the physical ket in terms of two spin eigenstates denying thus the use of his own elegant bra-ket formalism. We develop relativistic single-particle wave equations from a group-theoretic point of view, as it is the existence of the appropriate representation of the inhomogeneous Lorentz group with its associated Hilbert space that leads one naturally to the momentum representation. In Sec. II, we then present two momentum representations. One uses a straightforward normalization of the momentum states to a &function of three-momentum and the other is a “covariant” representation, the momentum eigenstates of the two being related by a factor of the square-root of the energy. We proceed from this to look simply for a conjugate realization of the physical system, i.e., we wish to set up a basic set of states labeled by “position”. This is, of course, familiar from nonrelativistic quantum mechanics, the nonrelativistic Schrbdinger equation being essentially just such a realization. However, it has not been completely clear how this is arrived at for relativistic wave equations. The wish for “manifestly covariant” wavefunctions, in fact, leads one to a coordinate-space realization quite different from what one is naturally lead to in seeking a conjugate representation from the momentum representation. This, of course, is due to the fact that pure Lorentz transformations must necessarily mix space and time in a highly nontrivial way. We consider thus, in Sec. III, the conjugate representation in that we seek eigenstates of an operator x which is such that
h ,131= ihi
and
[Xk , Xj] = 0
(we shall put & = c = 1 throughout this article). This then gives rise to a wavefunction which, though it satisfies the relativistic wave equation, transforms under pure Lorentz transformations in a rather complex manner. We can however derive from it the usual “covariant” wavefunction and we display explicitly its relationship to the Hilbert space by considering the bra-ket formalism for it. The operator x is then seen to be just the Newton-Wigner position operator (3) as one might have expected. Sets. II and III deal only with a spin-zero elementary particle. In Sec. IV, we present the formalism for nonzero spin. Though more complicated, it is basically the same as for spin zero. A discussion is given in Sec. V.
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In discussing Lorentz transformations, we adopt the passive viewpoint and use the Schrodinger method, in which the kets change, I v) -+ U I q~>, but the operators do not. That is, if a ket 1 y) represents a physical system in an inertial frame K, and x, p, etc., represent physical measurements in K, then U J q~) represents the same state viewed from K', and x, p, etc., represent the similar physical measurements in K’. It is as if one had a common computing center for processing the information from all inertial frames. Each inertial observer sends observations measured with his own coordinates; the computer accepts these subjective measurements and constructs from them, according to tied rules (e.g., for dealing with a measurement along an x-axis no matter whose axis it is), a ket for each frame. Thus, the difference between frames is distinguished by having different kets represent the same physical system but viewed from the various corresponding inertial frames. So it is only in discussing Lorentz transformations and in the change from the Heisenberg picture to the Schrijdinger picture that we ever contemplate transformations of the physical ket 1 v); the operators never change except in switching pictures, e.g., xf-) x(t). All other changes of amplitude then result from trivial basis alterations, in which case the scalar products (I/J 1 v), while taking different forms when expressed in terms of the different amplitudes, are by definition numerically independent of which amplitudes are used to evaluate them. II. MOMENTUM
REPRESENTATION
FOR A SPINLESS
PARTICLE
We use the representation space associated with the irreducible representation for spin zero and mass m of the inhomogeneous Lorentz group as given by Wigner (4). As usual, we consider it as “spanned” by the improper basis kets (vectors) I p)u which are defined via their behavior under translations WI
I 1013 = e+“‘a
where T(u) is the operator for translation p-a=
I P)H ,
(2.1)
through the four-vector a,
-p”ao+p*a
and p” = + (p p + mz)l12. l
(2.2)
We interpret JP>~ as the eigenstate of an elementary particle of mass m, with momentum p. These basis states can be normalized according to H(q
1 &I3
=
a@
-
Q),
(2.3)
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FONG
AND
ROWE
their completeness being then expressed as
s
~PlP>Iiri(Pl
= 1.
(2.4)
A general state 1 v) belonging to the Hilbert space may be expanded in this basis as (2.5)
the scalar product of two such states 1 $) and 1 v) being consequently
This permits the interpretation of ) pn(P)la as the momentum probability density. It might at this point be clarifying to note that it is this physical assignment, as given by Eq. (2. l), of the labels p to the improper kets 1P)n that is a momentum realization of the Hilbert space, making a physical interpretation possible. That is, without introducing something like Eq. (2.1), the abstract Hilbert space is, of course, void of any physical meaning. Also, as (p’ 1 y) is finite and positive, we see then that this representation of the Hilbert space gives just the space La, as expected. It should also be noted that, throughout this section, we are using the Heisenberg picture of quantum mechanics. This is most natural since the specification of an irreducible representation is made by the momentum-energy relation, Eq. (2.2), which, in fact, commits one to the momentum representation. (The full hyperboloid p2 + me = 0 can be distinguished in coordinate space by the operator a2 - m2, but the further necessary condition p” > 0 demands the use of momentum space.) Furthermore, we have the operator equation p(t) = &+20pe--iP~01e0 = p,
x0 = t,
so that it is evident that a basis of momentum eigenstates is well suited to the Heisenbcrg picture, since time is not involved anywhere. Also, in the Schrodinger picture, one would have the very difficult task of relating amplitudes on two hyperplanes, t = constant in one frame and t’ = some other constant in another frame; in momentum space, one would have the unnatural conjunction of variables p and t, while, in coordinate space, the complication of having the uncertainty principle coupled with Lorentz transformations would make the prospect of arriving at Eq. (3.13) rather dim. What we have done so far is familiar from nonrelativistic quantum mechanics. However, these amplitudes are not those introduced by Wigner (4), i.e., because of the “noncovariant” normalization of I p>n , Eq. (2.3), m(p) as defined by
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Eq. (2.5) does not transform in a “manifestly covariant” transformations. We can, in fact, write Wigner’s amplitudes WV@) = 94) We then obtain the usual “covariant”
way under Lorentz as (2.7)
= WY2 mw scalar product
(2.8)
Under a homogeneous Lore& transformation “manifestly covariant” manner, i.e.,
M4
WI@>= PAP?,
(1, r&p)
transforms
p’ = A-lp,
in a (2.9)
which also shows explicitly that U(A) is unitary as dp/p” is an invariant. We use the same notation V(A) for a transformation in a space of amplitudes as for the corresponding transformation in the linear ket space. These are just alternative ways of representing the same linear ket space and the operations to be performed in it corresponding to the physical operations in mind. This is made clear by the simple introduction of “covariantly” normalized momentum eigenstates,
I P>c = (P”.Y2 I Ph ,
(2.10)
c(Q I P>c = PO&P- Q),
(2.11)
with and completeness expressed as
s
(2.12)
g I P>cc
Then, Eqs. (2.5) and (2.7) show that (2.13) As we require,
WV> %I@) = CAPI W>l FDA
= J $$kdP~ IPn>c,
p” = Ap,
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FONG
under the Lorentz transformation
AND
ROWE
d. As, also
we must then have (2.14)
w-4 I P>c = I P”>c .
Conversely, the manifest covariance of Eq. (2.9) and the first of Eqs. (2.13) demand that 1p)c is covariantly normalized, which then enables us to write Eq. (2.7). The transformation laws for yn(p) and 1 p)n are somewhat more complicated: w>
I P)H = (p”“/~o)1’2 I P%i
(2.15)
H
= ($)“nplu(P~)
(2.16)
and
[WV wIl@) =
III. COORDINATE
REPRESENTATION
FOR A SPINLESS
PARTICLE
As in nonrelativistic quantum mechanics, we look also for a representation of the theory conjugate to the momentum representation, the coordinate representation We begin by finding the Schrodinger position operator since, being time independent, it can be fixed uniquely. We base this on the traditional requirements (5) [xi , pj] = iSij and
(3.1) [Xi ) Xj] = 0.
As this is generally taken to be the precise mathematical statement of the uncertainty principle, we take x to be the position operator. Dirac has shown that, by exploiting the freedom of the phase of I P)~ , we can arrange that Xj
= i@/+J
(3.2)
when applied to the amplitude vn(p) = n(p 1 q), since only for this amplitude is it Hermitian (6). For the Heisenberg operators, this arrangement is agreed to at t = 0. At other times, x(t) and p(t) must still satisfy
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but the phase factors are determined by the t = 0 condition x&)
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PARTICLES
and the equation
= eif150xk(0) trip’@ = x*(O) + x”pk/po.
(3.3)
The eigenstates of x, with the usual phase convention to make --iv tation of p, are I es = (2~r)+~ 1 dp e-iP*x 1P)~ ,
the represen(3.4)
with normalization s
=
6(x
Y)
(3.5)
= 1.
(3.6)
-
and completeness as i
dxIx),,(x)
We wish now to display explicitly the correspondence with the “covariant” coordinate representation that is usually given for a spinless particle. As this involves time, it is best to work now in the Schriidinger picture. The Heisenberg state is identied as the corresponding Schrijdinger state at time t = 0. The Schrijdinger state at time t = x0 associated with the Heisenberg state 1 v) is then 1 @) = e-fpOzO1 (s) and the Schrodinger amplitude &x)
(3.7)
(in the coordinate representation)
= &x 1 qd) = (2+3/Z
This amplitude can also be interpreted picture 1y) by writing
1 dp c&p) eipez.
as a representative
is (3.8)
of the Heisenberg-
%W = I& I F>,
(3.9)
( x)H = eipozO( x&
(3.10)
where
is the Heisenberg state for a particle localized at x at time x0. The transformation from the Heisenberg picture to the Schrijdinger one is unitary, so that (3.11)
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FONG
AND
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The important point about this form of the scalar product is that it permits the interpretation of I ys la as a position probability density, associated with the expectation value of the projection operator
The transformation properties of I&X) under a Lore& best be deduced from those of &p):
= cw-a’2
transformation
A can
J A2 [U(A)q&p) e6D.2
= Gw-3’2 1o&i (5“0) &p)&P. A-lx = ,(-$y2tps] (A-lx), l/2
(3.13)
where p” = AJJ (considered as an operator equation for the last expression). Though this transformation is evidently local for rotations, it is nonlocal for a true Lore& transformation, in the sense that the value of the amplitude U(A) vs at a given spacetime point is determined by the values of ‘ps everywhere and not just at the same space-time point. We are now in a position to introduce the usual “manifestly covariant” coordinate representative of the ket 1 v). It is simply
(3.14)
which, indeed, transforms locally, i.e., “manifestly
covariantly”,
as
(3.15) In bra-ket form, we can write (3.16)
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PARTICLES
where (3.17) and (3.18) It can be seen that 1x)c transforms as W) either from its definition
or from U+ = U-l and
c(x I W)l qJ>= cpcw4 We can understand the effect of the two state for a particle to spread the state
(3.19)
I $22 = I mc
= d-lx
I 9J>.
(3.20)
the meaning of these nicely transforming states (7) by separating operators in its detinition. The state er@ 1x)s is the Heisenberg localized at x at time x0. The effect of the operator (PO)-‘@s out over a distance mm-l:
s(x 1Y)~ = (27r)-8/a J AZ cc= ( r )
eiD-(x-y) (3.21)
514
&kimr),
r=
Ix-yl.
Neither the 1x)c nor the I x)c = I xO)o are orthogonal: co I x>c = w
J$
eiP-(Z-V) = -2i d+(y - x)
(3.22)
By inverting the expression for q&x) in terms of q&p), one can write the general state 1 q~) in terms of q~c(x) (3.23)
I v> = i j do ~owAx) I X>C
an expression which is independent of x0, as indeed it must be. The expression
<+ I FJ>= j cfxdy ~oYwJ9Q&>,
0 I x>c
for the scalar product can be re-arranged, using the Herkiticity operator (pO)l/z, to give its more familiar form
property of the
(3.24)
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FONG
AND
ROWE
This complex form does not admit the interpretation of the integrand as a coordinate probability density as Eq. (3.11) does, because &dV&gOvc cannot be written as the expectation value of a projection operator. The position operator, which is introduced most simply in conjunction with amplitudes whose transformations are not manifestly covariant, takes other forms when used with other amplitudes. Its expressions in terms of the four amplitudes in the Schrijdinger picture are
s(x
(3.25)
H(P
(3.26)
c
(3.27)
c
(3.28)
and, as an example of the Heisenberg operator x(t), we have H(P
1 x(t)
I v,>
=
i $
‘?&I(P)
+
$$
vH(p).
(3.29)
These expressions all refer to the same operator, namely the Newton-Wigner position operator (3), though only the forms with C&X) and vc@) were given in their paper. This omission left unclear the very simple foundation to the expressions with &) and &P, t).
IV. PARTICLES
WITH
SPlN
As far as the use of bras and kets is concerned, much of the theory of particles with spin is similar to that of spin-zero particles. We start immediately with Wigner’s result (4). The spin-s, mass-m representation of the proper inhomogeneous Lore&z group is realized by the 2s + 1 functions ~(p, 5) on the surface p2+m2=0,po
>O:
with the scalar product (4.2)
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569
The latter expression, (#w , qw) = (+ 1 q~), arises from a decomposition (4.3) in terms of basis kets 1p, LJw with normalization
and transformation
w(a, 7 I PY 0w = P06(P - s) a,,
(4.4)
u(a, 4 I P, 5>w = I P”, +w QWP,A),, .
(4.5)
law
In order to make the connection with our familiar concepts of momentum, spin and position, we first of all introduce the “physical” Heisenberg states
l I PY5>w I P, &I = (po>1/2
(4.6)
and amplitudes RdP, n = (p;‘,e mvb
09
(4.7)
in terms of which (4.8) and (4.9) Provided we properly exploit the freedom available in the momentum dependent matrices QCJ, A), the states I p, [)n can be interpreted as eigenstates of p and sg , the z-component of the spin operator s. Then, from the form of the scalar product, we can interpret I C&P, [)I” as the momentum probability density for a particle with z-component 5 of spin. To establish the previous statement, we recall Wigner’s little group construction (4) of the Q@, A) in terms of the unitary representations of the rotation group and a fairly arbitrary set of boost matrices CL@)satisfying a(P) Pm = P,
where pm = (m, 0).
(4.10)
One finds that
Q(P,4 = W9
(4.11)
570
FONG
AND
ROWE
where p = a(p)-’ Aa@‘)
(4.12)
and PCs> is a unitary (2s + l)-dimensional representative of the three-dimensional rotation j3, the Wigner rotation. In general, /3 depends on both p and A. Some p-dependence of j3 is unavoidable (otherwise one would have a finite dimensional unitary representation of the Lorentz group, which is impossible), but the p-dependence of /?(p, R), where R is a three-dimensional rotation, is avoidable. In fact, by choosing (8) (9), the boost a(p) corresponding to the SL(2,C) transformation [~y(p)]~~(~,~) = exp [q
tanh-l -$&I,
one has, as can be checked, B(P, R) = R
so that
Q(p, R) = P(R),
(4.14)
and our states can be chosen to be eigenstates of sg . This interpretation is strengthened by considering the spin-position eigenstates. The position variable can be introduced with little further complication than in the spin-zero case. For application on q&p, 0 the condition [xi , pr] = iSii and the Hermiticity of x imply that (41~ = i(Wh)
%, + h.h(P),
(4.15)
where fi,rl is real. Since the si are irreducible, the condition [xi, ~$1= 0 reduces J;:.h to fib> %I 3 which can be eliminated, using [xxi , xi] = 0, by an appropriate choice of the phases of the ] p, &‘)n’s. We thus get x = i@pp)
(4.16)
as the Schrodinger position operator acting on m(p, J). Its eigenstates
I x, 5)~ =
P7)-9/2 j 4 e+X
I P,5)~
(4.17)
transform under rotations as (4.18) Together with the scalar product (4.19)
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571
5> = SC& 5 I cpo,
(4.20)
RELKIWLSTIC
where dT
this permits the interpretation of [ x, [)s as the position state localized at x with z-component 5 of spin, if we choose S, to be diagonal. We wish now to find an amplitude cpc(x) which transforms locally (manifestly covariantly); i.e., (4.21)
wm WI(x) = wo ~CWW,
where L is a numerical, and not momentum-dependent, matrix and where we have assembled the components vc(x, 0 into a column vector. The simplest way of doing this (8), (9) is to use one of the two extensions of the (2s + l)dimensional unitary representation of the rotation group P(R) to a (2,s + l)dimensional nonunitary representation of the homogeneous Lorentz group. These are defined, for infinitesimal Lore& transformations, by
Using the notation B”(A) to indicate we have the factorization
either one of these two representations,
D”(p) = D’(a(p)-1) D”(A) Dqol(p’))
(4.23)
This allows a general ket I v) to be rewritten as (using a matrix notation for the sum over spin states) I+=/$
I P>w VW(P) = I$
I P>w W4PY)
~8(4PN
%dP> (4.24)
=
dP s 0P I P>c w4PY)
Y&P)
where (4.25)
and
I P>c = I P>c = I P>w w4PN
(4.26)
with normalization
cc4 IP>c
= PO&
- Q) D8(c&QB).
(4.27)
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FONG
AND
The advantage of this choice of amplitude law:
ROWE
is the simplicity
of its transformation
Kwo WI(P) = c
(4.28)
The peculiarity (4.29) while (4.30) is explained by the form of the scalar product
(4 I qJ>= j $f Y&'(P) m&)-3 9%(P)
(4.31)
and the resolution of the unit operator 1 = j $ I P>c W4PF2)
c
which gives the following two forms for U(A): (4.33a) =
4 s jlP,C
D”(a(p)
D”(A)
&l-~p
1.
(4.33b)
In the above manipulations, the fact that OS is Hermitian (in the strict matrix sense) for pure Lorentz transformations, such as a(p), although unitary for pure rotations has been used repeatedly; this fact is clear from Eq. (4.22). The complexity of the scalar product, Eq. (4.31), is necessary to ensure that the transformation (4.28) is unitary, despite the nonunitary nature of D(A). The amplitude in coordinate space that transforms locally is
(4.34) where (4.35)
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Although vc transforms locally under proper inhomogeneous Lorentz transformations, its transformation under spatial reflection is nonlocal. To see this, we use the linear unitary reflection operator lT defined by its effect on the physical states 1p, 5)n or, equivalently, 1p, &v (4.36)
n I P, 5>w = I -P, 5>w
(in a one-particle theory, we can always arrange that the arbitrary phase factor which could arise here is unity). Then
WJ>=j$ =sdP
I P>w D?a(-P)-l)
%(-P) (4.37)
I P>c WZ(-P), -5 P
since a(-~)-’
= a(p) as is clear from Eq. (4.22). Consequently, (4.38)
rnolcKP> = Wor(PY) W(-P),
which is momentum dependent and thus nonlocal. To achieve an amplitude which transforms locally even under reflections, we must use both of the representations in Eq. (4.22) and a slightly more elaborate notation. Putting ww
= ~‘“*“‘(~(P))
WYP) = ~‘“*“‘MP)) and using D(S*o)(ol(p)) = lW”)(a(-p)), q$yp)
= D’ya(p)2)
VW(P),
(4.39a)
Wv@),
(4.39b)
we have I&.
(4.40)
[~w21@>= %‘(-P).
(4.41)
qlcl = D’S*O’(a(-p)2)
Therefore Eq. (4.38) can be rewritten
[~WW The “doubled”
= W”(-P),
amplitude (4.42)
is in reduced form for proper transformations and flips with /3 = (f i) under reflections. It satisfies the generalized (8) Dirac equation which is equivalent to Eq. (4.40). Its rather complicated form, using representatives of ( y) with
574
FONG AND ROWE
two different sets of basis states 1p)cl and 1p)c2, makes the lack of a bra-ket interpretation of the Dirac equation less surprising. The form of the position operator acting on the covariant representatives can be found as in the spin-zero case:
CAPI m
v> = J-3 c
- &
i $ DT4---P))
(4.43)
+ $]9Jw@) + i $
- &
+ $1
ApI.
For the special case of spin-$, taking
Dc8~oY4P))= b(PhLC,,C~ = p;;p:++,sp;
,
(4.44)
we get (IO)
References (8) and (9) provide the expressions for D* for higher spins, from which the more and more complicated forms for x(t) can be calculated.
V. DISCUSSION
By rewriting the theory of relativistic free particles in bra-ket form, we have made clear several points where confusion has arisen in the past: first, that transformations such as that of Foldy and Wouthuysen (I) are not unitary transformations in the ket space, but are transformations of the amplitude resulting from changes of the basis; second, that the various forms of position operator given on the right-hand sides of Eqs. (3.25)-(3.28) are all representatives of the results of the same ket x I CJJ) but in different base+; that the important Foldy-Wouthuysen paper, that [IS, s] = [H, L] = 0 and x = p/pO, can be obtained cleanly in the bra-ket formalism without the use of “doubled” amplitudes whose interpretation has been obscure. We have discussed the Fourier transforms of momentum-space amplitudes. 1 Two of the representatives, Eqs. (3.27) and (3.28), are associated with the names Newton and Wigner, who have given an analysis applicable to all bases (3), but the operator x itself is the same as the SchrMinger operator of nonrelativistic quantum mechanics.
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To maintain a physical interpretation of these, one must know in exactly what way they are a representative of the physical states. The amplitudes s(x I vt) and & J q) are interpretable because the bras s(x 1and c(x I are known in terms of eigenstates of x, but an amplitude C&Z) which is known only to transform locally has no immediate interpretation. In any event, amplitudes such as &x) = s(x 1 vt) which do not transform manifestly covariantly are prefectly suitable if inconvenient for relativistic discussions. The position operator is at the heart of the interpretation of the scheme: only when we have constructed states I 0 = x, {,\s localized at the origin and which transform according to the spin-s representation of the rotation group, do we feel confident of the spin and position identifications. And the operator x provides the means of constructing and interpreting the projection operator P(d V) which yields 1 vs(x)12 as a probability density and not I t&x)12. The point about interpreting vs and not vc has been strongly made in Marshak and Sudarshan’s little book (Zl). Other “position” operators have been proposed (12). The situation, though perhaps mathematically clear, is not at all clear physically. The question of other physical definitions for a “position” operator is outside the aims of this paper. It must evidently depend on an idea of what the position operator refers to, i.e., whether it refers to mass or charge or some other particle property. In other words, it must depend on an idea of the measurement of position. We remark further that the definitions based on finding an energy density in coordinate space seem in principle unrealistic, if this energy density is to have physical meaning rather than just a formalistic one. Definitions of currents and energy densities in coordinate space both require simultaneous knowledge of position and momentum, which conflicts with the uncertainty principle. The density obtained from the scalar product written in terms of C&X) has an appealing interpretation, so long as it is not pressed too far. The density &i~~ ?&c has the advantage of being the fourth component of a conserved four-vector. It is naturally used for electromagnetic interactions, but it is not identical with the previous density and its interpretation is unclear. We are unhappy that we have not found an interpretable form using the covariant kets 1x)c , and that it does not have its own position operator associated with it, a charge position perhaps. Foldy has given a discussion of the connection between these densities (23). We have given an interpretation of the “doubled” Dirac-like amplitudes for spin-s in terms of two different basis ket systems. The doubling was made to depend on a locality demand for transformations under reflections. This is curious. One might have expected antiparticles to produce the doubling. RECEIVED: August 1, 1967
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REFERENCES 1. 2.
3. 4. 5. 6. 7. 8. 9.
IO. II. 12. 13.
L. L. FOL.DY AND S. A. WOUTHUYSEN, Phys. Rev. 78,29 (1950). R. ACHARYA AND E. C. G. SUDARSHAN, J. Math. Phys. 1, 532 (1960). T. D. NEWTON AND E. P. WIGNER, Rev. Mod. Phys. 21, 400 (1949). E. P. WIGNER, Ann. Math. 40, 149 (1939). There is no need to alter the original nonrelativistic argument, as in the “accidental” derivation of the position operator for ~c(p) given by A. S. WIGHTMAN AND S. SCHWEBER, Phys. Rev. 98, 812 (1955), by taking the Hermitian part of ia/ap. P. A. M. DIRAC, “The Principles of Quantum Mechanics.” Oxford University Press, London, 1958. These states are precisely those created by the field-theoretic operator v*(x). See S. WEINBERG, Phys. Rev. 133, B1318 (1964), for not only this case but for any spin. S. WEINBERG, reference in (7). R. SHAW, Nuovo Cimento 33,1074 (1964). A. CHAKRABARTI, Unpublished doctoral dissertation, Paris (1965). R. E. MAR~HAK AND E. C. G. SIJDARSHAN, “Introduction to Elementary Particle Physics.” Interscience, New York, 1961. See, e.g., G. N. FLEMING, Phys. Rev. 137, B188 (1965), and references contained therein. L. L. FOLDY, Phys. Rev. 102, 568 (1956).