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On quantum mechanics of identical particles. II
ANNALS
OF PHYSICS
65. I%34 (1971)
On Quantum
Mechanics Y.
Department of Physics,
of Identical
Particles.
II
OHNUKI Nagoya
University, Nagoya,
Japan
AND
S . KAMEFUCHI Department
of Physics,
Tokyo
University
of Education,
Tokyo,
Japan
Received November 17. 1970 The first-quantized formalism for paraparticles, which was developed in our previous papers, is investigated further with emphasis on permutation operators. Four different kinds of permutation operators are introduced, which all act on the arguments of wavefunctions. The correspondence is established between the first- and second-quantized formalisms, and on this basis the problem of observability of these permutation operators is discussed. An outline is also given of a first-quantized formalism for particles which do not obey the ordinary parastatistics.
1. INTRODUCTION In our previous papers [l] we have formulated a first-quantized theory of identical particles which obey parastatistics, including ordinary Bose and Fermi statistics as particular cases. The definition of identical particles we have adopted there is that the field quanta which result from second quantization of a given field 4 are all regarded as identical. In view of this our first-quantized formalism was founded on the basis of the corresponding second-quantized formalism, i.e., parafield theory of 4: We obtained the required formalism just by translating the results of the latter into the language of the former. It was found, however, that in order to attain the complete equivalence between both formalisms the usual framework of the first-quantized formalism, which is applicable to ordinary bosons and fermions, must be extended in certain respects. The characteristic features of the formalism thus obtained are as follows. In order to describe a system of n particles we need a multicomponent wavefunction @[(x).1 = {cp[(x),; 1, s]; I = 1,2,. . . , h), where (x), stands for the coordinates x1, x2,. . . , x, placed in a certain fixed order, ~[(x),; I, s] is an arbitrary component (s = sI, say) of the I-th irreducible representation (&dimensional) of 19
20
OHNUKI
AND
KAMEFUCHI
the symmetric group S, of the particle permutation operators n,(p) to act on xi, and h irreducible representations appearing here are all inequivalent. Corresponding to this, physical observables 9, including Hamiltonians .X, are, in general, of matrix-structure: The matrix elements 99, combining different components of @, do not necessarily vanish for certain 1 # 1’.Since the last property may apply, in particular, to interaction Hamiltonians, the Schrodinger equation becomes a set of simultaneous equations in which different components of @ are connected with each other via .Xy$. In this formalism we notice, first of all, that symmetry types of wavefunctions, being labeled by 1,are not always conserved. This is quite contrary to the postulate in the conventional formalism that all physical observables must be symmetric functions of the particle coordinates xi. Another important result to be noticed is that for the complete description of a given system we need only one, arbitrary component cp[(x),; 1, s,] for each irreducible representation 1. This implies that the particle permutation operations L!,(p), the role of which is to transform the labels s in cp among themselves, are of no physical meaning, and therefore that they are not physical observables. On the other hand, we find, in the literature, a seemingly contradictory statement that particle permutation operators are physical observables in the firstquantized formalism [2]. In their recent paper, Stolt and Taylor [3], agreeing with the authors of Ref. [2], have made an objection to our paper in this respect. We believe, however, that such a situation has been caused simply by misunderstanding due to careless use of the word “particle permutations”, and wish to show in the following that if the word is used more carefully, there exists actually no contradiction between the above authors and ourselves as far as unperturbed systems are concerned. Now, in so doing it becomes necessary to make clear the correspondence between the first- and second-quantized formalisms, and to establish this correspondence constitutes another purpose of the present paper. The basic standpoint which we adopt in what follows is the same as that of (A) and (B): We base all of our arguments on parafield theory. Thus, the problem, for example, of observability of particle permutation operators in the firstquantized formalism is investigated here as the corresponding problem in the second-quantized. We begin, in Section 2, with pointing out the necessity of distinguishing quantum numbers ki to specify state vectors from variables xi to label individual particles, both appearing as arguments in wavefunctions. Four different kinds of permutation operators can then be defined: Particle (place) permutation operators of the quantum numbers ki, Lr,(_p)(U,(p)), and particle (place) permutation operators of the variables xi, ZI,(p)(Lr,(p)). In Section 3 we establish the precise correspondence between the first- and second-quantized formalisms. This is done by defining the one-to-one correspondence between state vectors in the latter formalism and ket vectors in the former. rt is then found
IDENTICAL
21
PARTICLES
that the permutation operators II,( introduced in Section 2 correspond, in the first-quantized formalism, to the particle permutation operators P,(p)(@)) which act on the quantum numbers ki to specify ket vectors (on the subscripts i to specify Hilbert spaces of individual particles). The problem of observability of various kinds of permutation operators is discussed in Section 4. It is emphasized, though usually overlooked in the literature, that the situation differs considerably according as a given system of particles is unperturbed or in interaction. On the basis of field theory we conclude that in the case of unperturbed systems the operators U,(p) or P,(p) are physical observables, whereas the operators U,(p) or P(p) are not. When we distinguish between the two kinds of particle permutation operators ZZ,(p) and II,(p), our conclusion in this case is found to be consistent with that of the above-mentioned authors. However, in the case of interacting systems both n,(p)(P,(p)) and n,(p)(&)) are generally not physical observables. We note that the character operators C of the group S, remain physical observables in both cases. In Section 5 we briefly discuss a first-quantized formalism for identical particles which do not obey the ordinary parastatistics. Such a formalism is made possible by enlarging the Hilbert space of ket vectors in such a way as to take account of a new degree of freedom q which is characteristic of this kind of particles. 2. PARTICLE AND PLACE PERMUTATION
OPERATORS OF QUANTUM
NUMBERSANDOF~ARIABLES
As in our previous papers (A) and (B) we start with the quantum field theory of a nonrelativistic field 4(x) which has no internal degrees of freedom, and consider an unperturbed system .of n identical particles which result from second quantization of this field.’ We denote the state vector, in this formalism, for n such particles with momenta k, , k,, . . , k, by Ik,, k,, . . . >k,) E at(k
. . . at(k,)
(2.1)
where at(k) is the creation operator for a particle with momentum k, defined by at(k) = V- l/2 J #x$+(x) eikx. For simplicity, we assume that k takes discrete values and that all ki’s are different from each other. Corresponding to the state vector (2.1) we define the wavefunction in the xrepresentation by dxrl,
.xr2,.
=
. t x,,; &,I, ko2, . , k,,)
QW(x,1Mx,2)
’ Throughout the present paper definitions as those in (A) and (B).
... 4(x,&,, we adopt,
unless
3 kc,,, . . , km), otherwise
stated,
the same
(2.2) notations
and
22
OHNUKI
AND
KAMEFUCHI
where r and c are arbitrary elements of the symmetric group S, of order n! In (2.2) the coordinates xi, x2, . . . , x, are the quantities that correspond to dynamical variables in the first-quantized formalism. In our following discussions it is essential to make a clear distinction between the quantum numbers ki that specify state vectors and the variables xi that are introduced as coordinates of individual particles. Obviously, any physical quantities other than momentum and position may be chosen as quantum numbers and variables, respectively (when we retain the same notations k and x for them, most of the following relations remain unaltered).’ For the wavefunction (2.2) we can formally introduce the following four kinds of permutation operators, each corresponding to an element p E S, : ~xW(~(x,t,
xrz, . . . 3 x,,; k,t, ka2, . . . > km)
= dxprt,xpr2, ~,(p)&rt,
. . ..~~rn.ku’c,z,...
,kA
x,2, . . . , x,,; k,, , km2, . . . , k,,)
= cpk, 7 ~72, . . . > x,, ; &,.,I, kpa2, . . . 7 kpan), il,(ddx,,
,
fi,Wdx,,
, x,2, (Ph,
(2.4)
. . . , x,,; k,, , kc,,, . . . , km)
x,2,
(2.5)
=(~(~,~-1,,xr~-12,...,xr~-~n;k,t,k,z,...,k,,),
=
(2.3)
. . . , x,, x72,
; k,,
. . . 3 %I;
, k,, kc+t,
, . . . , km) +a,.
. , k+n).
(2.6)
These are all independent operations for general operands cp,and especially there hold the relations
[I,,
m41 = [fl,(P),m41 = 0.
(2.7)
We refer to ZZ,(p)(II,(p)) as particle permutation_ opera_tors of the variables xi (of the quantum numbers k,), whereas we refer to J7,(p)(A!,(p)) as place permutation operators of the variables xi (of the quantum numbers ki). As was discussed in (A), the place permutation operators fix(p) and n,(p) are not well-defined operations for the wavefunction (2.2) when the field operator 4(x) satisfies certain commutation relations such as those for parafields. In the following we therefore restrict ourselves to the particle permutation operators of both kinds.3 In the second-quantized formalism, there do not originally exist those dynamical variables that label individual particles. As far as we are concerned with identical 2 More generally, we may even choose k such that states specified by different values of k and not necessarily orthogonal. 3 In (A) and (B) both IT,(p) and II,(p) were simply called particle permutation operators. This seems to be the cause of the confusion mentioned in Section 1. As will be seen below, the two kinds of operators play quite different roles, especially in the first-quantized formalism.
IDENTICAL PARTICLES
23
particles there is no physical meaning in the statement such that xi is the variable that describes a certain particle in the state ki. We take this to be one of the implications of indistinguishability. Accordingly, it is expected that the particle permutation operators of variables II,(p) are not physical observables (in fact, this was proved in (B)). On the other hand, we are assuming in the second-quantized formalism that state vectors can be specified by a set of quantum numbers ki, so that the particle permutation operators of quantum numbers II,(p) are expected to be physical observables. This problem is examined more closely in Section 4. Now, the wavefunctions (p(xrl, x,~, . . . , x,,; kcl, ka2, . . . , k,,) can be decomposed into irreducible representations of the symmetric groups S, of II,(p) and of II,(p). The detailed procedure of this decomposition is described in Section 3.
3. CORRESPONDENCE BETWEENTHE FIRST- AND SECOND-QUANTIZED FORMALISMS
For definiteness let us assume that 4(x) is a parafermi (parabose) field of order (equal-time) paracommutation relations :
p, which satisfies the following
[4(x),[4+(x’),4W)lT1= 26’3’(x- x’ww’), [(b(X),[cj+(X'),q5+(X")]r] = 26(3)(x - x')c#J+(x")f 26(3)(x - xp)'(x'),
[4(x),[4(X’)> 4W)l TI = 0,
(3.1)
where the upper (lower) signs correspond to parafermi (parabose) cases. For the vacuum state IO) there holds the relation &x)c)+(x’)lo) = pP’(x
- x’)IO).
(3.2)
As discussed in (A) the dual state vectors (O[~(X,~)&X,~) 1. . 4(x,,) and state vectors Ik,, , k,, , . . . , k,,) can be decomposed into irreducible representations of the symmetric groups S, of II,(p) and of II,(p), respectively. According to the theorem of (A) there appears, in such decompositions, once and only once each irreducible representation whose Young diagram has no more than p columns (rows). For the dual state vectors (O(C$(X,,)$J(X,,) . . . 4(x,,,) we denote by ((x),; 1,sI the s-th basis vector of the I-th such irreducible representation (dimension: d,) of the group S, of n,(p), where s = 1,2, . . . , d,. Similarly, for the state vectors lkal, 42, . . . , k,,) we denote by I(k),; t, S) the corresponding basis vector for the group S, of II,(p), where S = 1,2, . . . , di. The wavefunction (2.2) can thus be written as a linear combination of ((x),; 1, sl(k),; i, S). When both the dual and ordinary state vectors are suitably normalized, the inner products ((x).; 1, sl(k),; 1, S) are given by (3.20) of (B), and become nonvanishing only for 1 = t. As the wavefunctions with definite transformation properties under both II,(p) and ZZ,(p)
24
OHNUKI
AND
KAMEFUCHI
we have thus the following:
where at/(p) is the (s, @-element of the representation matrix, for an element p E S,,, of the l-th irreducible representation, and (pL(x) is the one-particle wavefunction corresponding to the eigenvalue k : (3.4)
(P&X)= +WW~t(WO.
From the way of construction
we see that the wavefunction (3.3) transforms as
~x(~)cp'~W,,s; (k),,51= ? d?W')cp"'[M,
t; (k),, $1,
(3.5)
t=1
and
k(~W~‘[(4,~s; (k),731= : &Wcp”‘[(x),7s; W, >ll.
(3.6)
f=l
We are now in a position to introduce the correspondence between the firstand second-quantized formalisms. In this connection we first note that the right side of (3.3) contains the product of n one-particle wavefunctions, so that through the intermediary of this relation we can construct the ket vectors, in the firstquantized formalism, that correspond to the wavefunctions #‘[(x)~, s: (k),; S] in the second-quantized.4 To this end let us assign a subscript i to all ket vectors 4 In fact, number
the case p 2 n we can
of possible
inequivalent simple
for
#n’s
which
x1(p), such that
we obtain
respect
to [Il,W{
i,
vRkrW)].
since
the
1 df = n!, where 4 is the number of all the possible ,=1 of S, By using a corollary of the second orthogonality theorem concerning i d&r) ,=I
fi e&4 i=l from
(3.3) with
in such a case equals
representations
characters
solve
= n! for p = e (unit
= ,gl jl
(2)
[ ii=,k~xdl n,(p)
fi
cp .( .)
This
“2~‘i’[b)..
element)
s; (4,
and 0 otherwise
[4], we find
sl,
does not mean, however,
i=L
that we may use f1 cpk,(xi)
instead of #r’s as wavefunction of an n-particle system, because, as emphasized of @r’s with different I’s does not in general represent pure states.
in (B), superposition
IDENTICAL
PARTICLES
25
in the Hilbert space of one-particle states described by the variable xi in the firstquantized formalism, and assume the following correspondence between the second- and first-quantized formalisms : V/ci(xoi)c* IkJoi 3
(3.7)
where (ki)Oi denotes the ket vector, in the first-quantized formalism, which is defined as the operand of the variable xgi and which corresponds to an eigenvalue ki of the quantum number under consideration.’ Thus, for the ket vector I(k),, $4’ with the correspondence
cp”‘lM, s; W, ,sl ++IW,,,9!‘,
(3.8)
we have, in view of (3.3) and (3.7), the following expression :
IW,Ai” = ; Ii2 & af’,‘(p)[P(p)(lk,),Ik,), +.. lkJ,H~ i .I
(3.9)
where the operators P(p), which may be called particle permutation operators of the subscripts, act on the subscripts attached to ket vectors in the following way:
and in particular for cr = e,
P’(d~lkAlk,), . . . lk,)Jl = [PAP-‘Hlk,W,h . . . lkJ,~J
(3.10’)
Here, we have also introduced, in the first-quantized formalism, particle and place permutation operators of the quantum numbers ki, P,(p), and p,Jp), which act on ki in the same way as in (2.4) and (2.6), respectively.6 It is to be noted that although in (3.10’) [(3.10)] the operators P(p) and P,(p- ‘) [P(p) and ii,(p)] are related to each other for the particular operand therein, they are independent operations for general operands such as those appearing in (3.11) and (3.12) below.’ As can be easily seen, the ket vector (3.9) has the following transformation properties under P(p) and P,(p): P(p)l(k),, s,p’ = 2 a%-
‘)I(&, s):“,
(3.11)
t=1
5 State vectors in the first- and second-quantized formalisms are denoted by 1 ) and 1 ), respectively. 6 Our definitions of particle and place permutation operators, P&J) and pk(p), are exactly the same as those of Landshoff and Stapp [2]. The operators P(p) were called label permutation operators by Stolt and Taylor [3]. ’ This is due to the fact that the df quantities j(k),, S):) (s, S = 1,2, , d,) are all linearly independent, since a slight generalization of (3.15) gives “Ii!(( S’l(k),, i):” = 6,1.bJs.~jj..
26
OHNUKI
AND
KAMEFUCHI
(3.12) The correspondence (3.8) indicates that between the permutation operators in the second- and first-quantized formalisms there exists the following correspondence :
fl,(P) 4-bP,(P).
(3.13)
We see from (3.12) above and (3.4) of(B) that the ket vector defined by (3.9) and I(k),; 1,S) are transformed in the same way under P,(p) and II,(p), respectively. However, the correspondence
I(k),; 199 4-bl(k),34!)
(3.14)
is not unique, because the subscript s can take any of the values s = 1,2, . . . , d,. This arbitrariness in the choice of s implies that in the second-quantized formalism particle permutation operators of variables are not physical observables, or in other words, that any state is represented by a generalized ray in the sense of Messiah and Greenberg [5], i.e., J(k),, S):” with s being fixed and s being arbitrary. Since the ket vectors I(k),, S):’ with different subscripts s are transformed among themselves under the operations P(p) as in (3.1 l), we can eliminate the generalized ray by choosing a particular value s = sl, say, thereby obtaining an ordinary ray. This is essentially what we have done in (B). We come back to this problem in the next section. The inner products of the ket vectors I(k),, S)!’ can be obtained from (3.9) and the orthogonality relation of&p) (cf. (3.8) of(B)) as follows : ‘?((k’)n3S’l(k)n7s)y’ = C a$)s(P)Jj 6(ki 3&i), PES”
(3.15)
which is independent of s. Bearing in mind the relation (4.2) of (B), in which xi, xi and the Dirac delta function are replaced by kj, ki and the Kronecker delta, respectively, we further find : “j((k’),, S/l(k),, s)f’ = ((k’),; l,S’l(k),; I, s).
(3.16)
Superposition of two or more ket vectors in the first-quantized formalism can also be determined by referring to the second-quantized. Let there be the correspondence
l(k),; 1,s) - IN,, s>:‘! [(k’),; I’, S’) +-+I(k’),, S’):!“.
(3.17)
IDENTICAL
Then, in the case of 1 = 1’ we have, for an arbitrary cl(k), ; 1,S) + c’l(k’),, 1,a>:-
27
PARTICLES
superposition,
cl(k),, sp + C’l(k’)“, sf’,
(3.18)
where it should be noticed that two ket vectors on the right side have the same subscript s. In the case of 1 # I’, however, it is not possible to consider a similar superposition cl(k), , s)f’ + c’l(k’), , S’)$“, corresponding to cl(k), ; I, s) + c’l(k’), ; I’, s’), because the latter does not represent a pure state owing to the superselection rule concerning symmetry types which holds true for an unperturbed system. This can also be understood from the fact that for I # 1’ the above ket vectors I(k),, s)B) and I(E),, S’)$” are to be represented by different components of a multicomponent wavefunction Qi, which it is therefore meaningless to superpose. The situation is quite different, however, for interacting systems. As shown in Sections 2 and 4 of (B), we may introduce, in the second-quantized formalism, those observables that have non-vanishing matrix elements between states with different l’s satisfying a certain condition. In such cases it is of physical meaning to consider superposition of states with different 1’s. We take up this problem again in the following section.
4. OBSERVABILITY
OF PERMUTATION
OPERATORS
We have seen in the preceding section that within the framework of the firstquantized formalism we can introduce mathematically two kinds of particle permutation operators P,(p) and P(p), which correspond, in the second-quantized formalism, to particle permutation operators of quantum numbers and of variables, respectively. The problem we now wish to examine is whether or not these permutation operators are physical observables. Since our first-quantized formalism has been so formulated in (B) as to be completely equivalent to the contents of the second-quantized formalism, the above problem is reduced to that of whether particle permutation operators of both kinds, U,(p) and U,(p), are physical observables in field theory. Let us begin with the case of an unperturbed system of n particles. State vectors in this case are of the form (2.1), Jk,, , k,, , . . . , kp,). For a given set of the quantum numbers k,, k,, . . . , k,, these state vectors are in general independent, and should correspond to different physical situation. Hence, there must exist, in field theory, some operators with which to distinguish them. We have seen in (A) and (B) that the particle permutation operators of quantum numbers n,(p), play the required role, and so they are to be regarded as physical observables. Another theoretical basis supporting this statement lies in the fact that the operators II,(p) can be expressed in terms of the field operators (cf, Appendix A of(B)), and be shown to meet the requirement of physical observables which we have discussed
28
OHNUKI
AND
KAMEFUCHI
in Section 4 of(B). The character operators C, which determine symmetry types of state vectors or the label I, are clearly physical observables, as they are given as functions of n,(p). In this connection we make a remark that field-theoretical expressions for U,(p) are not unique owing to the circumstance that the state vectors Ik,,,k,z, . . . , k,,) are not enough to span the entire state-vector space for n particles. One of such expressions, U(ki) kj), given for IZ,(i, j) in Appendix A of (B), reads U(k,, kj) = NijNji
- Nii,
(4.1)
where Nij z *[u+(kJ, u(kj)]3 f p&k,, kj). We note that U(ki, kj) is Hermitian. Another expression U’(k,, kj) for h’,(i,j) is given by U’(ki, kj) = exp
-iF(Nii
+
Njj)
+ Nji) 1, ] exp [ Zz(Nij ‘71
W’)
which is unitary and satisfies the relations U’(ki, kj)at(ki)U’(ki, k,)+ = a’(kj), U’(ki, kj)ut(kj)U’(ki, k,)+ = u’(k,).
(4.3)
The corresponding expression for general operators D,(p) can be made up of U(ki, kj) or U’(k,, kj) according to the recipe described in Appendix A of(B). It is easy to see that both U(k,, kj) and U’(k,, kj) play the role of particle permutation (transposition of ki and kj) when acting on Ik,, , kp2, . . . , kpn), but cease doing so when acting on arbitrary state vectors such as given below: They even lead to different results.8 We note further that both U(k,, kj) and U’(ki, kj) commute with N,,, a result in favor of our assertion, to be made below, that these operators play the required role only for an unperturbed system with a given set of the quantum numbers k,, k,, . . , k,. We now turn to the particle permutation operators of variables II,(p). First of all it must be recalled that the variables xi are not the quantities of primary importance in the second-quantized formalism, but have been introduced merely as mathematical auxiliaries in order to set up its correspondence with the firstquantized formalism. This makes us suspect that U,(p) might not be physical observables. That this in fact is the case can be seen from the arguments of Sections 3 and 4 of(B). We have shown there that any state vectors and matrix elements of physical observables can be expressed in terms only of such components of the wavefunction as #)[(x)~, s ; (k),, S] with 1 = 1,2, . . . , h, and s being fixed for each 1,i.e., s = s,, say. As mentioned in Section 3 this enables us to eliminate generalized U’(k,,
a For k,)lk;.
example, suppose k;, k;,. k;, , k:) = Ik;. k;, , k;).
,k:, # ki, k,.
Then,
U(ki,
k,)lk;,
k;,
, k:)
= 0,
and
IDENTICAL
29
PARTICLES
rays. Since, on the other hand, Jl,(p))s are, by definition, such operators as to transform the labels s among themselves, we arrive at the conclusion mentioned above. So far we have been concerned with an unperturbed system of n particles, and paid attention only to the states in which those particles have a fixed set of quantum numbers k, , k,, . . . , k,. The situation for an interacting system of particles is very different from the above case. For simplicity let us assume that the particle number is conserved. Then, a general state vector If) for n particles can be written in field theory as If> = C Cf~‘(k,,k,,...,k,)l(k),;I,S),
(4.4)
(k)mLJ
or
= C C 4f:f ‘(k, 7 k,, . . . , k,W,; (kh 1
(4.4’)
k s,>,
where c means the respective summations over all possible values of k, , k, , . . . , k,. Wn
The wavefunction for this state vector in the x-representation following components :
consists of the
l/2
4yw,, sl = 2 . (Cd,; 1,a->, i I
I=
1,2 )...) h.
Bearing in mind the correspondence (3.8) we see that the ket vector in the firstquantized formalism that corresponds to each component # in (4.5) is given by 1’2 c Cf$“(k,, Wn s
k,, . . . , k,) c al’s(p)lk,-,,),lk,-,,), P-=%
... lk,-,,),,(4.6)
or =
-5; 1’2 ~4f~‘(k,, k,, . . . , k,,) 1 al’,i(p)lk,-1,),Ik,-1,), . . . I+n)n. (4.6’) 0 Wn P& Now, since we are summing over all possible values of k, , k,, . . . , k, on the right sides of (4.4), (4.4’), (4.6), and (4.6’), it is clear that the particle permutation operators n,(p) or P,(p), defined for certain fixed values of kls, do no longer play the role of permutation when applied to If) or If)!*‘. Hence, we cannot regard them as physical observables with the required role in the present case. Quite contrary, the particle permutation operators of variables n,(p) or P(p) still retain the role of permutation in the mathematical sense, when applied to #[(x),, s] or If):‘. However, for the same reason as explained above these operators cannot in general be regarded as physical observables. At this point we hasten to add that even in the present case the character operators C remain physical observables. In fact, we have shown in (B) that it is
30
OHNUKI
AND
KAMEFUCHI
possible to construct, in field theory, the operators C as functions of the field operator 4 and that such operators play the required role when applied to general state vectors such as (4.4). Now, according to Lemma A, of (A), the characters of the group S, of U,(p) are shown to take the same values as those of the group S, of H,(p) as far as nonvanishing wavefunctions are concerned. (The same result applies to-the characters of the groups S, of P,(p) and of P(p): This can be seen from (3.9) and (4.6).) Consequently, we can say that the character operators C of the groups &, of ~,(~)(f’&)) and of ~,(PM’(PN are physical observables in both formalisms. Lastly, we wish to make a comment on the conventional form of the indistinguishability condition in the first-quantized formalism, that any physical observable % must commute with P(p): P(p - ‘)%P(p) = 8.
(4.7)
As repeatedly emphasized above, the variables xi are not essential quantities in field theory, and we do not believe that it is possible to derive (4.7) from any other basic postulates in quantum field theory (contrary to some authors [5] who claim that (4.7) can somehow be proved). In Section 4 of(B) we have discussed in detail general properties of physical observables. There we have required that physical observables be given as functionals of the field operator $.and satisfy the locality condition (4.5) of (B). Further, we have shown that the observables in the firstquantized formalism, that correspond to those satisfying the above requirements in field theory, can be classified into two categories, observables of the first and of the second kinds : The former satisfy the condition (4.7) whereas the latter do not necessarily satisfy the same. It should be noted, incidentally, that this classification of physical observables is only specific to the first-quantized formalism and not of any essential significance in the second-quantized. In fact, the example given in Section 2 of(B) shows that one and the same operator in field theory can give rise to an observable of the first kind for n = 2, and to that of the second kind for n = 3. Now, Theorem 2 of(B) tells us that ifan observable F in field theory depends on 4 only through the combination [4’, 41, then the corresponding observables % in the first-quantized formalism are always (for all n) of the first kind. We conclude, therefore, that unless there can be found some reasonable physical conditions to be further imposed on quantum field theory, which enable us, for example, to restrict physical observables to functionals only of [4+, 41, (4.7) cannot be regarded as a condition of universal validity. Whether or not the condition (4.7) holds true, the first-quantized formalism we have constructed in (B) is completely equivalent to the corresponding second-quantized. Summarizing the results obtained above, we may say as follows: For the case of an unperturbed system of n identical particles, the particle permutation operators of quantum numbers U,(p) or P,(p) are physical observables, whereas
IDENTICAL PARTICLES
31
the particle permutation operators of variables II,(p) or P(p) are not. For the case of an interacting system, both kinds of permutation operators are not, in general, physical observables. In either of the above cases the character operators C remain physical observables.
5. FIRST-QUANTIZED APPROACH TO PARTICLESNOT OBEYING PARASTATISTICS
In our previous papers [6,1], we have briefly discussed, within the framework of the second-quantized formalism, a theoretical possibility of identical particles which do not obey the usual parastatistics. One of the most characteristic features of this kind of particles is that when n-particle state vectors (dual vectors) are decomposed into irreducible representations of the group S,, of II,(p)(II,(p)), the multiplicities g of certain irreducible representations become greater than unity, g > 1. (This is to be contrasted with g = 1 for all possible irreducible representations in the case of paraparticles; cf., the theorem of (A).) In the present section we wish to investigate whether and how a system of such particles can be described in terms of the first-quantized formalism. In order to study general properties of an unperturbed system of n such particles let us accept, for the time being, the conventional assumption in the firstquantized formalism that all the ket vectors necessary to describe n particles with momenta k,, k,, . . . , km can be obtained by applying to lk,),l/& .I- Ik,), the particle permutation operators P,(p) or P(p), as defined in Section 3. It is to be noted that owing to (3.10’) the ket vectors generated by applying P,(p) are not independent of those generated by applying P(p). Thus, in order to obtain the set of all n! independent ket vectors we have only to consider ket vectors generated by either of the above operations. Let us first pay attention to P(p) and construct the basis vectors for irreducible representations of the group S, of P(p) as linear combinations of (k,),,lk,),, . I. Ik,),, . According to Appendix B of(B), they are given in the form (apart from normalization) :
[A’% W4lk,h
. . . lk,),)l,
(5.1)
where
Now, from the relation (B.lO) of(B) we see that
P(p)A")(s,S)= 1 ay;
(5.3)
32
OHNUKI
AND
KAMEFUCHI
This implies that for a given 4 the ket vectors (5.1) transform, under P(p), as basis vectors for the I-th irreducible representation, and therefore that with (5.1) we have altogether d, equivalent representations corresponding to the Z-th Young diagram, since S = 1,2, . . . , d,. In order to see the meaning of the label S, let us now apply P,(p) to (5.1). By use of the relation [P,(p), P(a)] = 0 and (3.10’) we obtain MPW’h
~HlklMk,h
. . . IU”H
= [A”‘(%w-
‘Hl~1hl~2h. . . IWI.
(5.4)
On the other hand, (B.lO) of(B) gives us A”‘(S, s)P(p- ‘) = 1 u$‘,(p)A’“(s, S’). 3’
Combining
(5.5)
(5.4) and (5.5) we have
which implies that S is the label to specify the basis vectors for the I-th irreducible representation of the group S, of P,(p). We see, therefore, that when fixing s in (5.1), we are thereby selecting those basis vectors under the operations P,(p), which altogether form just one of the d, irreducible representations, for the group S, of P(p), corresponding to the I-th Young diagram. A similar situation occurs when the roles of P,(p) and P(p) are interchanged in the above argument. Such a result appears, at first sight, to imply that it is not possible to construct a theory of identical particles under consideration within the framework of the first-quantized formalism since as already mentioned above, the multiplicities g of equivalent representations of the group S, of P,(p) or of P(p) should in general be greater than unity. In other words we may say that the Hilbert space consisting of those ket vectors that we can obtain by applying P,(p) or P(p) to (k,),(k,), . -. Ik,), is not large enough to accommodate our particles. In order to construct a firstquantized formalism we have, therefore, to enlarge the Hilbert space. We study this problem by referring to the corresponding situation in field theory. Let at(k) be the creation operator for a particle with momentum k, which, together with u(k), does not necessarily satisfy the paracommutation relations such as (3.1). By taking suitable linear combinations of those state vectors that are generated by applying n,(p) to at(k . . . at(kn) we can construct the basis vectors J(k),; I, S,g) for the I-th irreducible representation of the group S, ofn,(p),whereg(= 1,2,. . . , g1 5 d,) is the label to specify one of the g1 equivalent representations corresponding to the I-th Young diagram. For these basis vectors we have the following relations:
(0%; I’,s’,g’lM,; 4s,s> = 4,4~~~,,~,
(5.7)
IDENTICAL
33
PARTICLES
and ok.
; 1,s, s> = 1 &P)l(~)” ; 1,f, s>. t
(5.8)
As in the case of paraparticles, we introduce, by means of the transformation function (p,Jx), the state vectors Itx)n; k soS> - 1 lj Cp~(xi)l(k)n; I, % S>, (k).
(5.9)
i= 1
for which we then have
The corresponding, (3.3), by
nonvanishing
wavefunctions are defined, in a way similar to
which are then transformed in the same way as in (3.5) and (3.6) under II,(p) and 17,(p), respectively. It should be noticed here that the right side of (5.11) does not depend on g. In constructing the corresponding first-quantized formalism, we cannot, however, adopt (n!/dl)“2[A(“(s, S){lkl)llk2)2 . .. lk,),}] as the ket vector corresponding to the wavefunction (5.11). This is because as already mentioned above, such ket vectors with s being fixed lead only to the I-th irreducible representation, for the group S, of P(p), of multiplicity equal to unity. In order to introduce a new degree of freedom corresponding to g we must therefore assume that the Hilbert space 4$, for n-particle states is given as a direct sum of those subspaces !$‘7g’ that consist of the ket vectors corresponding to the wavefunction &rSg’: sj, = c @ !p). (5.12) L&T Thus, @n,g)consist of the ket vector l(k),, S),(Lo)which has the correspondence q+‘*“‘[(x) “Ys *7 (k),) S] c* I(k)“7 Sps P and which can be written in the form
(5.13)
34
OHNUKI
AND
KAMEFUCHI
where the subscript g is to indicate that the ket vector concerned belongs to @‘*g). In the case of paraparticles g = 1 for all 1, and (5.12) becomes a single summation c @ $3”*“. From the correspondence (5.13) it is expected that the quantities I and Q (also S for an unperturbed system) are physical observables, but s is not.9 As in the case of paraparticles any state belonging to !$‘*g) can be described by a generalized ray l(k), , ~)i’.g)with S being fixed and s being arbitrary. For a system of interacting particles ket vectors belonging to sj(‘,g) take the form If):‘*“’ = c f(‘*g)(kI, k,, . . . , k&k),,
S)f*g).
(5.15)
(k)n
Thus, most general states can be described in terms of a set of ket vectors {Ifp’; 1 = 1,2, . . . , h;g= L2,...,g,), and physical observables show the corresponding matrix-structure when acting on such sets of ket vectors. Superposition, inner product, etc. of two sets of ket vectors { Ifi)tsg’} and { 1fz)il*g)} can be defined in a way analogous to the case of paraparticles.
REFERENCES AND S. KAMEFUCHI, Ann. Phys. New York 51(1969), 3377358, to be referred to as (A) ; 57 (1970), 543-578, to be referred to as (B). 2. P. V. LANDSHOFF AND H. P. STAPP, Ann. Phys. New York 45 (1967). 72-92. 3. R. H. STOLT AND J. R. TAYLOR, Nucl. Phys. B19 (1970), I-19. 4. See for example, H. BOERNER, “Representations of Groups”, Chap. III, Section 7, p. 78, North Holland, Amsterdam, 1963. 5. A. M. L. MBSIAH AND 0. W. GREENBERG, Phys. Rev. 136 (1964), B248-B267. 6. Y. OHNUKI, Progr. Theorei. Phys. (Kyoto) Suppl., 37,38 (1966), 285-296. 1. Y. OHNUKI
spond
’ We are assuming here that there exists, in field theory, an operator to 9 = 1,2, , gr. We do not know, however, its explicit expression
G whose eigenvalues correin terms of field operators.
OF PHYSICS
65. I%34 (1971)
On Quantum
Mechanics Y.
Department of Physics,
of Identical
Particles.
II
OHNUKI Nagoya
University, Nagoya,
Japan
AND
S . KAMEFUCHI Department
of Physics,
Tokyo
University
of Education,
Tokyo,
Japan
Received November 17. 1970 The first-quantized formalism for paraparticles, which was developed in our previous papers, is investigated further with emphasis on permutation operators. Four different kinds of permutation operators are introduced, which all act on the arguments of wavefunctions. The correspondence is established between the first- and second-quantized formalisms, and on this basis the problem of observability of these permutation operators is discussed. An outline is also given of a first-quantized formalism for particles which do not obey the ordinary parastatistics.
1. INTRODUCTION In our previous papers [l] we have formulated a first-quantized theory of identical particles which obey parastatistics, including ordinary Bose and Fermi statistics as particular cases. The definition of identical particles we have adopted there is that the field quanta which result from second quantization of a given field 4 are all regarded as identical. In view of this our first-quantized formalism was founded on the basis of the corresponding second-quantized formalism, i.e., parafield theory of 4: We obtained the required formalism just by translating the results of the latter into the language of the former. It was found, however, that in order to attain the complete equivalence between both formalisms the usual framework of the first-quantized formalism, which is applicable to ordinary bosons and fermions, must be extended in certain respects. The characteristic features of the formalism thus obtained are as follows. In order to describe a system of n particles we need a multicomponent wavefunction @[(x).1 = {cp[(x),; 1, s]; I = 1,2,. . . , h), where (x), stands for the coordinates x1, x2,. . . , x, placed in a certain fixed order, ~[(x),; I, s] is an arbitrary component (s = sI, say) of the I-th irreducible representation (&dimensional) of 19
20
OHNUKI
AND
KAMEFUCHI
the symmetric group S, of the particle permutation operators n,(p) to act on xi, and h irreducible representations appearing here are all inequivalent. Corresponding to this, physical observables 9, including Hamiltonians .X, are, in general, of matrix-structure: The matrix elements 99, combining different components of @, do not necessarily vanish for certain 1 # 1’.Since the last property may apply, in particular, to interaction Hamiltonians, the Schrodinger equation becomes a set of simultaneous equations in which different components of @ are connected with each other via .Xy$. In this formalism we notice, first of all, that symmetry types of wavefunctions, being labeled by 1,are not always conserved. This is quite contrary to the postulate in the conventional formalism that all physical observables must be symmetric functions of the particle coordinates xi. Another important result to be noticed is that for the complete description of a given system we need only one, arbitrary component cp[(x),; 1, s,] for each irreducible representation 1. This implies that the particle permutation operations L!,(p), the role of which is to transform the labels s in cp among themselves, are of no physical meaning, and therefore that they are not physical observables. On the other hand, we find, in the literature, a seemingly contradictory statement that particle permutation operators are physical observables in the firstquantized formalism [2]. In their recent paper, Stolt and Taylor [3], agreeing with the authors of Ref. [2], have made an objection to our paper in this respect. We believe, however, that such a situation has been caused simply by misunderstanding due to careless use of the word “particle permutations”, and wish to show in the following that if the word is used more carefully, there exists actually no contradiction between the above authors and ourselves as far as unperturbed systems are concerned. Now, in so doing it becomes necessary to make clear the correspondence between the first- and second-quantized formalisms, and to establish this correspondence constitutes another purpose of the present paper. The basic standpoint which we adopt in what follows is the same as that of (A) and (B): We base all of our arguments on parafield theory. Thus, the problem, for example, of observability of particle permutation operators in the firstquantized formalism is investigated here as the corresponding problem in the second-quantized. We begin, in Section 2, with pointing out the necessity of distinguishing quantum numbers ki to specify state vectors from variables xi to label individual particles, both appearing as arguments in wavefunctions. Four different kinds of permutation operators can then be defined: Particle (place) permutation operators of the quantum numbers ki, Lr,(_p)(U,(p)), and particle (place) permutation operators of the variables xi, ZI,(p)(Lr,(p)). In Section 3 we establish the precise correspondence between the first- and second-quantized formalisms. This is done by defining the one-to-one correspondence between state vectors in the latter formalism and ket vectors in the former. rt is then found
IDENTICAL
21
PARTICLES
that the permutation operators II,( introduced in Section 2 correspond, in the first-quantized formalism, to the particle permutation operators P,(p)(@)) which act on the quantum numbers ki to specify ket vectors (on the subscripts i to specify Hilbert spaces of individual particles). The problem of observability of various kinds of permutation operators is discussed in Section 4. It is emphasized, though usually overlooked in the literature, that the situation differs considerably according as a given system of particles is unperturbed or in interaction. On the basis of field theory we conclude that in the case of unperturbed systems the operators U,(p) or P,(p) are physical observables, whereas the operators U,(p) or P(p) are not. When we distinguish between the two kinds of particle permutation operators ZZ,(p) and II,(p), our conclusion in this case is found to be consistent with that of the above-mentioned authors. However, in the case of interacting systems both n,(p)(P,(p)) and n,(p)(&)) are generally not physical observables. We note that the character operators C of the group S, remain physical observables in both cases. In Section 5 we briefly discuss a first-quantized formalism for identical particles which do not obey the ordinary parastatistics. Such a formalism is made possible by enlarging the Hilbert space of ket vectors in such a way as to take account of a new degree of freedom q which is characteristic of this kind of particles. 2. PARTICLE AND PLACE PERMUTATION
OPERATORS OF QUANTUM
NUMBERSANDOF~ARIABLES
As in our previous papers (A) and (B) we start with the quantum field theory of a nonrelativistic field 4(x) which has no internal degrees of freedom, and consider an unperturbed system .of n identical particles which result from second quantization of this field.’ We denote the state vector, in this formalism, for n such particles with momenta k, , k,, . . , k, by Ik,, k,, . . . >k,) E at(k
. . . at(k,)
(2.1)
where at(k) is the creation operator for a particle with momentum k, defined by at(k) = V- l/2 J #x$+(x) eikx. For simplicity, we assume that k takes discrete values and that all ki’s are different from each other. Corresponding to the state vector (2.1) we define the wavefunction in the xrepresentation by dxrl,
.xr2,.
=
. t x,,; &,I, ko2, . , k,,)
QW(x,1Mx,2)
’ Throughout the present paper definitions as those in (A) and (B).
... 4(x,&,, we adopt,
unless
3 kc,,, . . , km), otherwise
stated,
the same
(2.2) notations
and
22
OHNUKI
AND
KAMEFUCHI
where r and c are arbitrary elements of the symmetric group S, of order n! In (2.2) the coordinates xi, x2, . . . , x, are the quantities that correspond to dynamical variables in the first-quantized formalism. In our following discussions it is essential to make a clear distinction between the quantum numbers ki that specify state vectors and the variables xi that are introduced as coordinates of individual particles. Obviously, any physical quantities other than momentum and position may be chosen as quantum numbers and variables, respectively (when we retain the same notations k and x for them, most of the following relations remain unaltered).’ For the wavefunction (2.2) we can formally introduce the following four kinds of permutation operators, each corresponding to an element p E S, : ~xW(~(x,t,
xrz, . . . 3 x,,; k,t, ka2, . . . > km)
= dxprt,xpr2, ~,(p)&rt,
. . ..~~rn.ku’c,z,...
,kA
x,2, . . . , x,,; k,, , km2, . . . , k,,)
= cpk, 7 ~72, . . . > x,, ; &,.,I, kpa2, . . . 7 kpan), il,(ddx,,
,
fi,Wdx,,
, x,2, (Ph,
(2.4)
. . . , x,,; k,, , kc,,, . . . , km)
x,2,
(2.5)
=(~(~,~-1,,xr~-12,...,xr~-~n;k,t,k,z,...,k,,),
=
(2.3)
. . . , x,, x72,
; k,,
. . . 3 %I;
, k,, kc+t,
, . . . , km) +a,.
. , k+n).
(2.6)
These are all independent operations for general operands cp,and especially there hold the relations
[I,,
m41 = [fl,(P),m41 = 0.
(2.7)
We refer to ZZ,(p)(II,(p)) as particle permutation_ opera_tors of the variables xi (of the quantum numbers k,), whereas we refer to J7,(p)(A!,(p)) as place permutation operators of the variables xi (of the quantum numbers ki). As was discussed in (A), the place permutation operators fix(p) and n,(p) are not well-defined operations for the wavefunction (2.2) when the field operator 4(x) satisfies certain commutation relations such as those for parafields. In the following we therefore restrict ourselves to the particle permutation operators of both kinds.3 In the second-quantized formalism, there do not originally exist those dynamical variables that label individual particles. As far as we are concerned with identical 2 More generally, we may even choose k such that states specified by different values of k and not necessarily orthogonal. 3 In (A) and (B) both IT,(p) and II,(p) were simply called particle permutation operators. This seems to be the cause of the confusion mentioned in Section 1. As will be seen below, the two kinds of operators play quite different roles, especially in the first-quantized formalism.
IDENTICAL PARTICLES
23
particles there is no physical meaning in the statement such that xi is the variable that describes a certain particle in the state ki. We take this to be one of the implications of indistinguishability. Accordingly, it is expected that the particle permutation operators of variables II,(p) are not physical observables (in fact, this was proved in (B)). On the other hand, we are assuming in the second-quantized formalism that state vectors can be specified by a set of quantum numbers ki, so that the particle permutation operators of quantum numbers II,(p) are expected to be physical observables. This problem is examined more closely in Section 4. Now, the wavefunctions (p(xrl, x,~, . . . , x,,; kcl, ka2, . . . , k,,) can be decomposed into irreducible representations of the symmetric groups S, of II,(p) and of II,(p). The detailed procedure of this decomposition is described in Section 3.
3. CORRESPONDENCE BETWEENTHE FIRST- AND SECOND-QUANTIZED FORMALISMS
For definiteness let us assume that 4(x) is a parafermi (parabose) field of order (equal-time) paracommutation relations :
p, which satisfies the following
[4(x),[4+(x’),4W)lT1= 26’3’(x- x’ww’), [(b(X),[cj+(X'),q5+(X")]r] = 26(3)(x - x')c#J+(x")f 26(3)(x - xp)'(x'),
[4(x),[4(X’)> 4W)l TI = 0,
(3.1)
where the upper (lower) signs correspond to parafermi (parabose) cases. For the vacuum state IO) there holds the relation &x)c)+(x’)lo) = pP’(x
- x’)IO).
(3.2)
As discussed in (A) the dual state vectors (O[~(X,~)&X,~) 1. . 4(x,,) and state vectors Ik,, , k,, , . . . , k,,) can be decomposed into irreducible representations of the symmetric groups S, of II,(p) and of II,(p), respectively. According to the theorem of (A) there appears, in such decompositions, once and only once each irreducible representation whose Young diagram has no more than p columns (rows). For the dual state vectors (O(C$(X,,)$J(X,,) . . . 4(x,,,) we denote by ((x),; 1,sI the s-th basis vector of the I-th such irreducible representation (dimension: d,) of the group S, of n,(p), where s = 1,2, . . . , d,. Similarly, for the state vectors lkal, 42, . . . , k,,) we denote by I(k),; t, S) the corresponding basis vector for the group S, of II,(p), where S = 1,2, . . . , di. The wavefunction (2.2) can thus be written as a linear combination of ((x),; 1, sl(k),; i, S). When both the dual and ordinary state vectors are suitably normalized, the inner products ((x).; 1, sl(k),; 1, S) are given by (3.20) of (B), and become nonvanishing only for 1 = t. As the wavefunctions with definite transformation properties under both II,(p) and ZZ,(p)
24
OHNUKI
AND
KAMEFUCHI
we have thus the following:
where at/(p) is the (s, @-element of the representation matrix, for an element p E S,,, of the l-th irreducible representation, and (pL(x) is the one-particle wavefunction corresponding to the eigenvalue k : (3.4)
(P&X)= +WW~t(WO.
From the way of construction
we see that the wavefunction (3.3) transforms as
~x(~)cp'~W,,s; (k),,51= ? d?W')cp"'[M,
t; (k),, $1,
(3.5)
t=1
and
k(~W~‘[(4,~s; (k),731= : &Wcp”‘[(x),7s; W, >ll.
(3.6)
f=l
We are now in a position to introduce the correspondence between the firstand second-quantized formalisms. In this connection we first note that the right side of (3.3) contains the product of n one-particle wavefunctions, so that through the intermediary of this relation we can construct the ket vectors, in the firstquantized formalism, that correspond to the wavefunctions #‘[(x)~, s: (k),; S] in the second-quantized.4 To this end let us assign a subscript i to all ket vectors 4 In fact, number
the case p 2 n we can
of possible
inequivalent simple
for
#n’s
which
x1(p), such that
we obtain
respect
to [Il,W{
i,
vRkrW)].
since
the
1 df = n!, where 4 is the number of all the possible ,=1 of S, By using a corollary of the second orthogonality theorem concerning i d&r) ,=I
fi e&4 i=l from
(3.3) with
in such a case equals
representations
characters
solve
= n! for p = e (unit
= ,gl jl
(2)
[ ii=,k~xdl n,(p)
fi
cp .( .)
This
“2~‘i’[b)..
element)
s; (4,
and 0 otherwise
[4], we find
sl,
does not mean, however,
i=L
that we may use f1 cpk,(xi)
instead of #r’s as wavefunction of an n-particle system, because, as emphasized of @r’s with different I’s does not in general represent pure states.
in (B), superposition
IDENTICAL
PARTICLES
25
in the Hilbert space of one-particle states described by the variable xi in the firstquantized formalism, and assume the following correspondence between the second- and first-quantized formalisms : V/ci(xoi)c* IkJoi 3
(3.7)
where (ki)Oi denotes the ket vector, in the first-quantized formalism, which is defined as the operand of the variable xgi and which corresponds to an eigenvalue ki of the quantum number under consideration.’ Thus, for the ket vector I(k),, $4’ with the correspondence
cp”‘lM, s; W, ,sl ++IW,,,9!‘,
(3.8)
we have, in view of (3.3) and (3.7), the following expression :
IW,Ai” = ; Ii2 & af’,‘(p)[P(p)(lk,),Ik,), +.. lkJ,H~ i .I
(3.9)
where the operators P(p), which may be called particle permutation operators of the subscripts, act on the subscripts attached to ket vectors in the following way:
and in particular for cr = e,
P’(d~lkAlk,), . . . lk,)Jl = [PAP-‘Hlk,W,h . . . lkJ,~J
(3.10’)
Here, we have also introduced, in the first-quantized formalism, particle and place permutation operators of the quantum numbers ki, P,(p), and p,Jp), which act on ki in the same way as in (2.4) and (2.6), respectively.6 It is to be noted that although in (3.10’) [(3.10)] the operators P(p) and P,(p- ‘) [P(p) and ii,(p)] are related to each other for the particular operand therein, they are independent operations for general operands such as those appearing in (3.11) and (3.12) below.’ As can be easily seen, the ket vector (3.9) has the following transformation properties under P(p) and P,(p): P(p)l(k),, s,p’ = 2 a%-
‘)I(&, s):“,
(3.11)
t=1
5 State vectors in the first- and second-quantized formalisms are denoted by 1 ) and 1 ), respectively. 6 Our definitions of particle and place permutation operators, P&J) and pk(p), are exactly the same as those of Landshoff and Stapp [2]. The operators P(p) were called label permutation operators by Stolt and Taylor [3]. ’ This is due to the fact that the df quantities j(k),, S):) (s, S = 1,2, , d,) are all linearly independent, since a slight generalization of (3.15) gives “Ii!(( S’l(k),, i):” = 6,1.bJs.~jj..
26
OHNUKI
AND
KAMEFUCHI
(3.12) The correspondence (3.8) indicates that between the permutation operators in the second- and first-quantized formalisms there exists the following correspondence :
fl,(P) 4-bP,(P).
(3.13)
We see from (3.12) above and (3.4) of(B) that the ket vector defined by (3.9) and I(k),; 1,S) are transformed in the same way under P,(p) and II,(p), respectively. However, the correspondence
I(k),; 199 4-bl(k),34!)
(3.14)
is not unique, because the subscript s can take any of the values s = 1,2, . . . , d,. This arbitrariness in the choice of s implies that in the second-quantized formalism particle permutation operators of variables are not physical observables, or in other words, that any state is represented by a generalized ray in the sense of Messiah and Greenberg [5], i.e., J(k),, S):” with s being fixed and s being arbitrary. Since the ket vectors I(k),, S):’ with different subscripts s are transformed among themselves under the operations P(p) as in (3.1 l), we can eliminate the generalized ray by choosing a particular value s = sl, say, thereby obtaining an ordinary ray. This is essentially what we have done in (B). We come back to this problem in the next section. The inner products of the ket vectors I(k),, S)!’ can be obtained from (3.9) and the orthogonality relation of&p) (cf. (3.8) of(B)) as follows : ‘?((k’)n3S’l(k)n7s)y’ = C a$)s(P)Jj 6(ki 3&i), PES”
(3.15)
which is independent of s. Bearing in mind the relation (4.2) of (B), in which xi, xi and the Dirac delta function are replaced by kj, ki and the Kronecker delta, respectively, we further find : “j((k’),, S/l(k),, s)f’ = ((k’),; l,S’l(k),; I, s).
(3.16)
Superposition of two or more ket vectors in the first-quantized formalism can also be determined by referring to the second-quantized. Let there be the correspondence
l(k),; 1,s) - IN,, s>:‘! [(k’),; I’, S’) +-+I(k’),, S’):!“.
(3.17)
IDENTICAL
Then, in the case of 1 = 1’ we have, for an arbitrary cl(k), ; 1,S) + c’l(k’),, 1,a>:-
27
PARTICLES
superposition,
cl(k),, sp + C’l(k’)“, sf’,
(3.18)
where it should be noticed that two ket vectors on the right side have the same subscript s. In the case of 1 # I’, however, it is not possible to consider a similar superposition cl(k), , s)f’ + c’l(k’), , S’)$“, corresponding to cl(k), ; I, s) + c’l(k’), ; I’, s’), because the latter does not represent a pure state owing to the superselection rule concerning symmetry types which holds true for an unperturbed system. This can also be understood from the fact that for I # 1’ the above ket vectors I(k),, s)B) and I(E),, S’)$” are to be represented by different components of a multicomponent wavefunction Qi, which it is therefore meaningless to superpose. The situation is quite different, however, for interacting systems. As shown in Sections 2 and 4 of (B), we may introduce, in the second-quantized formalism, those observables that have non-vanishing matrix elements between states with different l’s satisfying a certain condition. In such cases it is of physical meaning to consider superposition of states with different 1’s. We take up this problem again in the following section.
4. OBSERVABILITY
OF PERMUTATION
OPERATORS
We have seen in the preceding section that within the framework of the firstquantized formalism we can introduce mathematically two kinds of particle permutation operators P,(p) and P(p), which correspond, in the second-quantized formalism, to particle permutation operators of quantum numbers and of variables, respectively. The problem we now wish to examine is whether or not these permutation operators are physical observables. Since our first-quantized formalism has been so formulated in (B) as to be completely equivalent to the contents of the second-quantized formalism, the above problem is reduced to that of whether particle permutation operators of both kinds, U,(p) and U,(p), are physical observables in field theory. Let us begin with the case of an unperturbed system of n particles. State vectors in this case are of the form (2.1), Jk,, , k,, , . . . , kp,). For a given set of the quantum numbers k,, k,, . . . , k,, these state vectors are in general independent, and should correspond to different physical situation. Hence, there must exist, in field theory, some operators with which to distinguish them. We have seen in (A) and (B) that the particle permutation operators of quantum numbers n,(p), play the required role, and so they are to be regarded as physical observables. Another theoretical basis supporting this statement lies in the fact that the operators II,(p) can be expressed in terms of the field operators (cf, Appendix A of(B)), and be shown to meet the requirement of physical observables which we have discussed
28
OHNUKI
AND
KAMEFUCHI
in Section 4 of(B). The character operators C, which determine symmetry types of state vectors or the label I, are clearly physical observables, as they are given as functions of n,(p). In this connection we make a remark that field-theoretical expressions for U,(p) are not unique owing to the circumstance that the state vectors Ik,,,k,z, . . . , k,,) are not enough to span the entire state-vector space for n particles. One of such expressions, U(ki) kj), given for IZ,(i, j) in Appendix A of (B), reads U(k,, kj) = NijNji
- Nii,
(4.1)
where Nij z *[u+(kJ, u(kj)]3 f p&k,, kj). We note that U(ki, kj) is Hermitian. Another expression U’(k,, kj) for h’,(i,j) is given by U’(ki, kj) = exp
-iF(Nii
+
Njj)
+ Nji) 1, ] exp [ Zz(Nij ‘71
W’)
which is unitary and satisfies the relations U’(ki, kj)at(ki)U’(ki, k,)+ = a’(kj), U’(ki, kj)ut(kj)U’(ki, k,)+ = u’(k,).
(4.3)
The corresponding expression for general operators D,(p) can be made up of U(ki, kj) or U’(k,, kj) according to the recipe described in Appendix A of(B). It is easy to see that both U(k,, kj) and U’(k,, kj) play the role of particle permutation (transposition of ki and kj) when acting on Ik,, , kp2, . . . , kpn), but cease doing so when acting on arbitrary state vectors such as given below: They even lead to different results.8 We note further that both U(k,, kj) and U’(ki, kj) commute with N,,, a result in favor of our assertion, to be made below, that these operators play the required role only for an unperturbed system with a given set of the quantum numbers k,, k,, . . , k,. We now turn to the particle permutation operators of variables II,(p). First of all it must be recalled that the variables xi are not the quantities of primary importance in the second-quantized formalism, but have been introduced merely as mathematical auxiliaries in order to set up its correspondence with the firstquantized formalism. This makes us suspect that U,(p) might not be physical observables. That this in fact is the case can be seen from the arguments of Sections 3 and 4 of(B). We have shown there that any state vectors and matrix elements of physical observables can be expressed in terms only of such components of the wavefunction as #)[(x)~, s ; (k),, S] with 1 = 1,2, . . . , h, and s being fixed for each 1,i.e., s = s,, say. As mentioned in Section 3 this enables us to eliminate generalized U’(k,,
a For k,)lk;.
example, suppose k;, k;,. k;, , k:) = Ik;. k;, , k;).
,k:, # ki, k,.
Then,
U(ki,
k,)lk;,
k;,
, k:)
= 0,
and
IDENTICAL
29
PARTICLES
rays. Since, on the other hand, Jl,(p))s are, by definition, such operators as to transform the labels s among themselves, we arrive at the conclusion mentioned above. So far we have been concerned with an unperturbed system of n particles, and paid attention only to the states in which those particles have a fixed set of quantum numbers k, , k,, . . . , k,. The situation for an interacting system of particles is very different from the above case. For simplicity let us assume that the particle number is conserved. Then, a general state vector If) for n particles can be written in field theory as If> = C Cf~‘(k,,k,,...,k,)l(k),;I,S),
(4.4)
(k)mLJ
or
= C C 4f:f ‘(k, 7 k,, . . . , k,W,; (kh 1
(4.4’)
k s,>,
where c means the respective summations over all possible values of k, , k, , . . . , k,. Wn
The wavefunction for this state vector in the x-representation following components :
consists of the
l/2
4yw,, sl = 2 . (Cd,; 1,a->, i I
I=
1,2 )...) h.
Bearing in mind the correspondence (3.8) we see that the ket vector in the firstquantized formalism that corresponds to each component # in (4.5) is given by 1’2 c Cf$“(k,, Wn s
k,, . . . , k,) c al’s(p)lk,-,,),lk,-,,), P-=%
... lk,-,,),,(4.6)
or =
-5; 1’2 ~4f~‘(k,, k,, . . . , k,,) 1 al’,i(p)lk,-1,),Ik,-1,), . . . I+n)n. (4.6’) 0 Wn P& Now, since we are summing over all possible values of k, , k,, . . . , k, on the right sides of (4.4), (4.4’), (4.6), and (4.6’), it is clear that the particle permutation operators n,(p) or P,(p), defined for certain fixed values of kls, do no longer play the role of permutation when applied to If) or If)!*‘. Hence, we cannot regard them as physical observables with the required role in the present case. Quite contrary, the particle permutation operators of variables n,(p) or P(p) still retain the role of permutation in the mathematical sense, when applied to #[(x),, s] or If):‘. However, for the same reason as explained above these operators cannot in general be regarded as physical observables. At this point we hasten to add that even in the present case the character operators C remain physical observables. In fact, we have shown in (B) that it is
30
OHNUKI
AND
KAMEFUCHI
possible to construct, in field theory, the operators C as functions of the field operator 4 and that such operators play the required role when applied to general state vectors such as (4.4). Now, according to Lemma A, of (A), the characters of the group S, of U,(p) are shown to take the same values as those of the group S, of H,(p) as far as nonvanishing wavefunctions are concerned. (The same result applies to-the characters of the groups S, of P,(p) and of P(p): This can be seen from (3.9) and (4.6).) Consequently, we can say that the character operators C of the groups &, of ~,(~)(f’&)) and of ~,(PM’(PN are physical observables in both formalisms. Lastly, we wish to make a comment on the conventional form of the indistinguishability condition in the first-quantized formalism, that any physical observable % must commute with P(p): P(p - ‘)%P(p) = 8.
(4.7)
As repeatedly emphasized above, the variables xi are not essential quantities in field theory, and we do not believe that it is possible to derive (4.7) from any other basic postulates in quantum field theory (contrary to some authors [5] who claim that (4.7) can somehow be proved). In Section 4 of(B) we have discussed in detail general properties of physical observables. There we have required that physical observables be given as functionals of the field operator $.and satisfy the locality condition (4.5) of (B). Further, we have shown that the observables in the firstquantized formalism, that correspond to those satisfying the above requirements in field theory, can be classified into two categories, observables of the first and of the second kinds : The former satisfy the condition (4.7) whereas the latter do not necessarily satisfy the same. It should be noted, incidentally, that this classification of physical observables is only specific to the first-quantized formalism and not of any essential significance in the second-quantized. In fact, the example given in Section 2 of(B) shows that one and the same operator in field theory can give rise to an observable of the first kind for n = 2, and to that of the second kind for n = 3. Now, Theorem 2 of(B) tells us that ifan observable F in field theory depends on 4 only through the combination [4’, 41, then the corresponding observables % in the first-quantized formalism are always (for all n) of the first kind. We conclude, therefore, that unless there can be found some reasonable physical conditions to be further imposed on quantum field theory, which enable us, for example, to restrict physical observables to functionals only of [4+, 41, (4.7) cannot be regarded as a condition of universal validity. Whether or not the condition (4.7) holds true, the first-quantized formalism we have constructed in (B) is completely equivalent to the corresponding second-quantized. Summarizing the results obtained above, we may say as follows: For the case of an unperturbed system of n identical particles, the particle permutation operators of quantum numbers U,(p) or P,(p) are physical observables, whereas
IDENTICAL PARTICLES
31
the particle permutation operators of variables II,(p) or P(p) are not. For the case of an interacting system, both kinds of permutation operators are not, in general, physical observables. In either of the above cases the character operators C remain physical observables.
5. FIRST-QUANTIZED APPROACH TO PARTICLESNOT OBEYING PARASTATISTICS
In our previous papers [6,1], we have briefly discussed, within the framework of the second-quantized formalism, a theoretical possibility of identical particles which do not obey the usual parastatistics. One of the most characteristic features of this kind of particles is that when n-particle state vectors (dual vectors) are decomposed into irreducible representations of the group S,, of II,(p)(II,(p)), the multiplicities g of certain irreducible representations become greater than unity, g > 1. (This is to be contrasted with g = 1 for all possible irreducible representations in the case of paraparticles; cf., the theorem of (A).) In the present section we wish to investigate whether and how a system of such particles can be described in terms of the first-quantized formalism. In order to study general properties of an unperturbed system of n such particles let us accept, for the time being, the conventional assumption in the firstquantized formalism that all the ket vectors necessary to describe n particles with momenta k,, k,, . . . , km can be obtained by applying to lk,),l/& .I- Ik,), the particle permutation operators P,(p) or P(p), as defined in Section 3. It is to be noted that owing to (3.10’) the ket vectors generated by applying P,(p) are not independent of those generated by applying P(p). Thus, in order to obtain the set of all n! independent ket vectors we have only to consider ket vectors generated by either of the above operations. Let us first pay attention to P(p) and construct the basis vectors for irreducible representations of the group S, of P(p) as linear combinations of (k,),,lk,),, . I. Ik,),, . According to Appendix B of(B), they are given in the form (apart from normalization) :
[A’% W4lk,h
. . . lk,),)l,
(5.1)
where
Now, from the relation (B.lO) of(B) we see that
P(p)A")(s,S)= 1 ay;
(5.3)
32
OHNUKI
AND
KAMEFUCHI
This implies that for a given 4 the ket vectors (5.1) transform, under P(p), as basis vectors for the I-th irreducible representation, and therefore that with (5.1) we have altogether d, equivalent representations corresponding to the Z-th Young diagram, since S = 1,2, . . . , d,. In order to see the meaning of the label S, let us now apply P,(p) to (5.1). By use of the relation [P,(p), P(a)] = 0 and (3.10’) we obtain MPW’h
~HlklMk,h
. . . IU”H
= [A”‘(%w-
‘Hl~1hl~2h. . . IWI.
(5.4)
On the other hand, (B.lO) of(B) gives us A”‘(S, s)P(p- ‘) = 1 u$‘,(p)A’“(s, S’). 3’
Combining
(5.5)
(5.4) and (5.5) we have
which implies that S is the label to specify the basis vectors for the I-th irreducible representation of the group S, of P,(p). We see, therefore, that when fixing s in (5.1), we are thereby selecting those basis vectors under the operations P,(p), which altogether form just one of the d, irreducible representations, for the group S, of P(p), corresponding to the I-th Young diagram. A similar situation occurs when the roles of P,(p) and P(p) are interchanged in the above argument. Such a result appears, at first sight, to imply that it is not possible to construct a theory of identical particles under consideration within the framework of the first-quantized formalism since as already mentioned above, the multiplicities g of equivalent representations of the group S, of P,(p) or of P(p) should in general be greater than unity. In other words we may say that the Hilbert space consisting of those ket vectors that we can obtain by applying P,(p) or P(p) to (k,),(k,), . -. Ik,), is not large enough to accommodate our particles. In order to construct a firstquantized formalism we have, therefore, to enlarge the Hilbert space. We study this problem by referring to the corresponding situation in field theory. Let at(k) be the creation operator for a particle with momentum k, which, together with u(k), does not necessarily satisfy the paracommutation relations such as (3.1). By taking suitable linear combinations of those state vectors that are generated by applying n,(p) to at(k . . . at(kn) we can construct the basis vectors J(k),; I, S,g) for the I-th irreducible representation of the group S, ofn,(p),whereg(= 1,2,. . . , g1 5 d,) is the label to specify one of the g1 equivalent representations corresponding to the I-th Young diagram. For these basis vectors we have the following relations:
(0%; I’,s’,g’lM,; 4s,s> = 4,4~~~,,~,
(5.7)
IDENTICAL
33
PARTICLES
and ok.
; 1,s, s> = 1 &P)l(~)” ; 1,f, s>. t
(5.8)
As in the case of paraparticles, we introduce, by means of the transformation function (p,Jx), the state vectors Itx)n; k soS> - 1 lj Cp~(xi)l(k)n; I, % S>, (k).
(5.9)
i= 1
for which we then have
The corresponding, (3.3), by
nonvanishing
wavefunctions are defined, in a way similar to
which are then transformed in the same way as in (3.5) and (3.6) under II,(p) and 17,(p), respectively. It should be noticed here that the right side of (5.11) does not depend on g. In constructing the corresponding first-quantized formalism, we cannot, however, adopt (n!/dl)“2[A(“(s, S){lkl)llk2)2 . .. lk,),}] as the ket vector corresponding to the wavefunction (5.11). This is because as already mentioned above, such ket vectors with s being fixed lead only to the I-th irreducible representation, for the group S, of P(p), of multiplicity equal to unity. In order to introduce a new degree of freedom corresponding to g we must therefore assume that the Hilbert space 4$, for n-particle states is given as a direct sum of those subspaces !$‘7g’ that consist of the ket vectors corresponding to the wavefunction &rSg’: sj, = c @ !p). (5.12) L&T Thus, @n,g)consist of the ket vector l(k),, S),(Lo)which has the correspondence q+‘*“‘[(x) “Ys *7 (k),) S] c* I(k)“7 Sps P and which can be written in the form
(5.13)
34
OHNUKI
AND
KAMEFUCHI
where the subscript g is to indicate that the ket vector concerned belongs to @‘*g). In the case of paraparticles g = 1 for all 1, and (5.12) becomes a single summation c @ $3”*“. From the correspondence (5.13) it is expected that the quantities I and Q (also S for an unperturbed system) are physical observables, but s is not.9 As in the case of paraparticles any state belonging to !$‘*g) can be described by a generalized ray l(k), , ~)i’.g)with S being fixed and s being arbitrary. For a system of interacting particles ket vectors belonging to sj(‘,g) take the form If):‘*“’ = c f(‘*g)(kI, k,, . . . , k&k),,
S)f*g).
(5.15)
(k)n
Thus, most general states can be described in terms of a set of ket vectors {Ifp’; 1 = 1,2, . . . , h;g= L2,...,g,), and physical observables show the corresponding matrix-structure when acting on such sets of ket vectors. Superposition, inner product, etc. of two sets of ket vectors { Ifi)tsg’} and { 1fz)il*g)} can be defined in a way analogous to the case of paraparticles.
REFERENCES AND S. KAMEFUCHI, Ann. Phys. New York 51(1969), 3377358, to be referred to as (A) ; 57 (1970), 543-578, to be referred to as (B). 2. P. V. LANDSHOFF AND H. P. STAPP, Ann. Phys. New York 45 (1967). 72-92. 3. R. H. STOLT AND J. R. TAYLOR, Nucl. Phys. B19 (1970), I-19. 4. See for example, H. BOERNER, “Representations of Groups”, Chap. III, Section 7, p. 78, North Holland, Amsterdam, 1963. 5. A. M. L. MBSIAH AND 0. W. GREENBERG, Phys. Rev. 136 (1964), B248-B267. 6. Y. OHNUKI, Progr. Theorei. Phys. (Kyoto) Suppl., 37,38 (1966), 285-296. 1. Y. OHNUKI
spond
’ We are assuming here that there exists, in field theory, an operator to 9 = 1,2, , gr. We do not know, however, its explicit expression
G whose eigenvalues correin terms of field operators.