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Exclusion principle and indistinguishability of identical particles in quantum mechanics
Journal of Molecular Structure, 212 (1992) 187-196 Elsevier Science Publishers B.V., Amsterdam
187
Exclusion principle and indistinguishability identical particles in quantum mechanics*
of
Ilya G. Kaplan’ Karpov Institute of Physical Chemistry, ul. Obukha 10, K-64 Moscow, 103064 (Russian Federation) (Received 17 December 1991)
Abstract Two points of view on the exclusion principle are discussed. One is that there are no laws in nature that forbid the existence of particles described by a wavefunction with an intermediate permutation symmetry, and the existence of only symmetric and antisymmetric ones is due exclusively to the properties of known elementary particles. The other is that the existence of only two types of symmetry can be derived from the principle of indistinguishability of identical particles. It is proved that the indistinguishability principle is not sensitive to the symmetry of the wavefunction and cannot be used as a criterion for the correct symmetry. It is further shown that the realization of only two types of permutation symmetry in nature is the only way by which there is no contradiction with the concept of particle identity.
INTRODUCTION
I am very pleased to publish this paper in the special issue of the Journal of Molecular Structure devoted to the scientific activity of Professor M.A. Elyashevitch. An outstanding spectroscopist, M.A. Elyashevitch has over the last few years been investigating very intensively and fruitfully the history of quantum mechanics (see refs 1 and 2). I think that Mikhail Alexandrovich will read with interest the paper devoted to one of the basic principles of quantum mechanics, the Pauli principle. Although it was formulated more than 75 years ago, up to the present time it has had no rigorous theoretical foundations. The exclusion principle, first discovered by Pauli [3] for electrons and later spread to all particles, was based on the analysis of experimental data. Correspondence to: Professor I.G. Kaplan, Institute de Fisica UNAM, Ciudad Universitaria APDO, Postal 20-364, C.P. 01000 Mexico D.F., Mexico. *Dedicated to Professor M.A. Elyashevitch. ’ Permanent address: Karpov Institute of Physical Chemistry, ul. Obukha 10, K-64 Moscow, 103064. Russian Federation.
0022-2860/92/$05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
188
I. Kaplan/J. Mol. Struct., 272 (1992) 187-196
Pauli himself was never satisfied by this. In his Nobel Prize lecture [4] Pauli said, Already in my initial paper, I especially emphasized the fact that I could not find a logical substantiation for the exclusion principle nor derive it from more general assumptions. I always had a feeling, which remains until this day, that this is the fault of some flaw in the theory. All the experimental data known to date agree with the exclusion principle. Several types of experiments on the search for possible small violations of the Pauli principle have been discussed in the literature [!&lo]. A test of the Pauli principle was made earlier by searching for y quanta with energies around ZOMeV, which should be produced with transitions of nucleons in the lzC nucleus from the 2p shell to the occupied 1 s shell. This has given a lower limit for the formation time of a “non Pauli” nucleus z 2 2 x 1020years [ll]. The search for X-ray radiation produced by the transition of an electron in a germanium detector to the occupied (1 s)~ shell, either in the case of the hypothetical spontaneous decay of an electron in this shell or in the case of violation of the Pauli principle, gave an even greater lower limit z > 1.5 x 1O25years [12]. The substantiation of the exclusion principle has been made by Pauli in his famous theorem about the relation between spin and statistics [13]. In this theorem, Pauli has shown that the field operators for particles with integer spin cannot obey fermionic commutation relations since this would lead to violation of the causality principle. However, the field operators of particles with half-integer spin cannot obey the bosonic commutation relations, since this would result in negative values of the total energy of the system. From this Pauli concluded that particles with integer spin have to obey Bose-Einstein statistics, while those with half-integer spin have to obey Fermi-Dirac statistics. In this proof of the theorem, as well as in the others given later (see, for example, Luders and Zumino [14]), it is implicitly postulated that only two types of commutation relations are possible for field operators: bosonic and fermionic. However, in 1953 Green [15], and later on Volkov [16, 171, have shown that the field operators that satisfy the requirements of causality relativistic invariance, and positivity of energy, can obey more general commutation relations than the bosonic and fermionic ones: these are the so-called parabosonic and parafermionic commutation relations. For each of them there is a corresponding parastatistic of rank p characterized by p-fold occupation of single-particle states. At p = 1 the parastatistic becomes that of Fermi-Dirac statistics, while at p = 00 we get Bose Einstein statistics, see ref. 18. Greenberg and Messiah [19] have formulated the selection rules for reactions of elementary paraparticles and have analyzed, from that point of
I. Kaplan/J. Mol. Strut.,
272 (1992) 187-196
189
view, all the elementary particles known at the time. Their conclusion was that none of the latter are paraparticles. However, analysis of the commutation relations for operators of quasi-particles in a periodic lattice (molecular Frenkel excitons and magnons) has shown (Kaplan [ZO]) that the latter obey modified parastatistics of rank JV, where JV is the number of lattice sites within the delocalization region of the excitation. Later on, these results were generalized by Nguen and Hoang [21] to the case of Wanier-Mott excitons. For bosons and fermions there is a one-to-one correspondence between the symmetry of the wavefunction and the commutation relations for the field operators, but in the case of parastatistics this is not so. It is no longer possible to establish a unique relation between the type of commutation relations for parafield operators and the permutation symmetry of the wavefunction in the configuration space [19]. This means that due to the problem of the relation between the particle spin and permutation symmetry the Pauli theorem remains unsolved. What Pauli has actually proved is that the states of a system of identical particles with integer spin cannot be described by an antisymmetric wavefunction, while the states for identical particles with half-integer spin cannot be described by a symmetric function. No constraints are imposed upon the realization of states with intermediate types of permutation symmetry. The Schrodinger equation can be satisfied by solutions with a permutation symmetry of arbitrary type. Below we discuss the situation that arises in this case and make a critical analysis of existing points of view on the subject. INDISTINGUISHABILITY
OF IDENTICAL PARTICLES AND THE SYMMETRY
POSTULATE
Since the Schrodinger equation is invariant under any permutation of identical particles, its solutions can belong to any representation of the permutation group, including a degenerate one. However, according to the exclusion principle, a system of identical particles can be only in those states that are not degenerate with respect to permutations: in a symmetric or an antisymmetric state. All other types of symmetry are forbidden. One may now ask whether this limitation on the solutions of the Schrodinger equation follows from the fundamental principles of quantum mechanics or whether it is an independent principle. There are two points of view on this problem. Some physicists, including one of the founders of quantum mechanics, Dirac [22] (see also Shiff [23] and Messiah [24]) believe that there are no laws in nature that forbid the existence of particles described by wavefunctions with more complicated permutation properties than those of bosons and fermions, and that the
existing limitations are only due to the specific properties of the known elementary particles. Messiah [24] has even introduced the term symmetry postulate to emphasize the primary nature of the constraint on the allowed types of the wavefunction pe~utation symmetry. By using the Schur lemma, Messiah and Greenberg [25] have shown that the existence of pe~utation degeneracy should not introduce additional un~e~ainty into the characteristics of a state. This also follows directly from the WignerEckart theorem generalized by Koster [26]. According to formula (4.60) in ref. 27 the matrix element of an operator L, that is symmetric in all the particles, is ~%VNY’~
= a,,
a4lIml>
(1)
where index I‘ labels the basic functions of the representation I?[~’of the permutation group. The double vertical line in the right hand side of this formula means that this expression is independent of the basic function number. Thus, the average of operator 2 is the same for all the functions ‘PFJbelonging to the degenerate state. Another point of view is that the symmetry postulate is not an independent principle, but can be derived from the fund~ental principles of quantum mechanics, in particular, from the principle of indistinguishability of identical particles. The typical argumentation presented in several textbooks and monographs [28-301 is the following. From the requirement that the states of a system obtained by permutation of identical particles must all be physically equivalent, one concludes that the change in the wavefunction resulting from the transposition of any two identical particles should only cause multiplication by an insignificant phase factor: Y(x,, x,) = e”“Y(x,, x2)
c4
where CIis a real constant and x is the set of spatial and spin variables. One more application of the permutation operator PI2 gives Y(x,, x2) = ei2”V-‘(x,,xz)
(3)
or @u = 1,
(4)
eia= & 1
Since all these particles are assumed to be identical, the wavefunction should change in exactly the same way under transposition to any pair of particles, i.e. it should be either totally symmetric or totally antisymmetric. The evident incorrectness of this proof is the following. Equation (2) is valid only for one-dimensional representations. The result of the action of a permutation on a wavefunction belonging to a multidimensional representation is a linear combination of the basic functions of this representation.
I. Kaplan/J. Mol. Struct., 272 (1992) 187-196
191
The common belief that wavefunctions describing the same physical state can differ by no more than a phase factor is evidently not true. For instance, according to eqn. (l), the values of the physical quantities characterizing a system of identical particles are the same for all functions belonging to the same irreducible representation, and consequently all these functions describe the same physical state. By requiring that under permutations the wavefunction should change by no more than a phase factor, one actually postulates that the representation of the permutation group is one-dimensional. In the above proof, the indistinguishability principle is directly related to the behaviour of the wavefunction. However, since the wavefunction is not an observable, the indistinguishability principle is related to it only indirectly via the expressions for measureable quantities. A rigorous proof should be based on a rigorous formulation of the indistinguishability principle for identical particles. One possible formulation is the following: all observable quantities are invariant under permutation of identical particles, and, vice versa, the permutations of identical particles cannot be observed. Since in quantum mechanics the physcial quantities are expressed as bilinear forms of wavefunctions, the indistinguishability principle requires the invariance of these bilinear forms: P(Y!]I;]Y) = (Y]L]Y)
(5)
Often, one limits oneself to the requirement that the probability with which a given con~guration of a system of identical particles is realized must be invariant under permutations [31, 321 WI%,
- * * 9 %-A2 = IWX,, * * . , %412
‘23
Evidently, this is a particular case of Eqn. (5). For a function to satisfy eqn. (6), it is sufficient that under permutations it would change as PY(x,, . . . , xN) = e4(x1~~~~~xN)Y(x1, , . . , xN)
(7)
i.e. unlike the case of the requirement of eqn. (2), in the general case the phase is a function of coordinates and the permutation. Equations (3) and (4) in this case evidently do not hold. Most other proofs of the symmetry postulate contain unjustified eonstraints. A critical survey of such proofs can be found in refs, 25,31 and 33. The proofs of the symmetry postulate without imposing additional constraints have been given by Girardeau [32] and Kaplan [33]. These authors have used the formulation of the indistinguishability principle, respectively, in the form of eqn. (6) [32] and eqn. (5) [33]. As was noted later by the author [34], these two formulations of the indistinguishability principle are correct only when the system is in a non-degenerate state. For instance, the
192
I. Kaplan/J. Mol. Struct., 272 (1992) 187-196
expression for the probability density in eqn. (6) used by Girardeau [32] should be modified if the state is degenerate, since in this case the system can be described by any one of the basic vectors of the degenerate state with equal probability. As a result, we can no longer select a pure state, the one that is described by the wavefunction, and should regard the degenerate state as a mixed one, where each basic vector enters with the same probability. The diagonal element of the density matrix for a degenerate state has the form [35]
@x1, . . .
)
xhr: x1,
. . .
)
XN) = ;, ir
$ Y!“‘(x,. . .
)
xN)* - Yy(x, . . . XN) )
(8)
1
The possibility of expressing the density matrix through only one of the functions ‘I’, implies that the degeneracy with respect to permutations can be eliminated. However, the latter cannot be achieved without violating the identity of the particles. It is not difficult to. check that the probability density, eqn. (8), is a group invariant: PDPl=
D
for all
69
From it follows the probability obeys the guishability principle, in the of multidimensional tations degenerate respect to One comes similar conclusions the case the proof the symmetry suggested by author [33]. of the density matrix a pure the indistinguishability in terms the mean of single-particle should deal the density of a state. As result, as eqn. (a), obtains an invariant under which means the indistinguishability holds for irreducible representation the permutation Thus, the principle is sensitive to symmetry wavefunction and be used a criterion selecting the symmetry. Does mean that viewpoint of authors of 22-25 is and that mechanics allows existence of described by with an permutation symmetry? arguments presented the next [34] show this is so. The of only types of symmetry in (symmetric and is by means occasional, there are reasons why is so.
of
1. Kaplan/J. Mol. Struct., 272 (1992) 187-196
193
CONTRADICTION WITH THE CONCEPT OF PARTICLE IDENTITY IN THE CASE OF PERMUTATION DEGENERACY
The states of a system of identical particles in cases where the number of particles is not conserved can be presented as vectors in the Fock space F [36]. As is known, the latter space is a direct sum of spaces I@‘) corresponding to a fixed number of particles N:
Each of the spaces FCN’ can be presented as a direct product of singleparticle spaces, f: F’N’ = f@f@ L
. # . @f I Y N
The basic vectors of FtN) are the product of single-particle belonging to the space fi I@?? = /VI>I%) * f * IYV)
(11)
vectors ]vk) (12)
For simplicity, let us consider the case where all the vectors in eqn. (12) are different. There will be no qualitative changes if some of the vectors IQ) coincide. From vector ]<(N’) one can produce N! new vectors by applying to it N! permutations of the state indices. These new vectors will also belong to FCN’ and will form in it a certain invariant subspace, which will be reducible. The basic vectors of the latter, P]c$~‘), make up a regular representation of the permutation group xN. As is known, a regular representation is decomposed into irreducible representations, each of which appears a number of times equal to its dimension. The space .sCN) falls into the direct sum
where $“I is an irreducible subspace of dimension fi drawn over the basic vectors ][;z&? (inNI is a Young diagram with N boxes). The latter are constructed of non-symmetrized basic vectors I<(“‘) by using the Young operator u$] [27] @,]lFt)
=
Wt:“’16(N)
(14)
where K’;p’ (I’) are matrix elements of representation FLAN1 and index t distinguishes between the bases in accordance with the decomposition of .z$@)into f, invariant subspaces. Thus, a space with a fixed number of particles can always be divided into
194
I. Kaplan/J. Mol. Struct., 272 (1992) 187-196
irreducible subspaees E?‘, each of which is characterized by a certain permutation symmetry given by a Young diagram with N boxes. The symmetry postulate requires that the basic vectors of a system of N identical particles belong to one of the subspaces characterized by irreducible onedimensional representations, either [N] or [lN]. All other subspaces are “empty”. Let us examine the situation that arises when no symmetry constraints are imposed. Let the system of N identical particles be described by basic vectors belonging to an arbitrary irreducible subspace eFN1. As is known, one of the consequences of the different symmetry of state vectors for bosons and fermions is the dependence of the system energy on the particle statistics. For the same law of dynamic interaction, the socalled exchange terms enter the expression for the energy of fermions and bosons with opposite signs. Let us examine from this point of view the expression for the energy of a system of particles belonging to an irreducible subspace EP’ with an arbitrary Young diagram [A,]. The energy of the system in a degenerate state is E = Tr(HD)
(15)
where D is the density operator defined, similarly to eqn. (8), as 1 fi 0, = z r&
II4r~>
(33)
and H is the hamiltonian. We will assume that the latter includes only one and two-particle interaction terms: H =
c h, + c gij i
(17)
The calculation of the trace over the functions with symmetry &,] yields
The matrix element in eqn. (18) has been calculated in ref. 37 in the general case of non-orthogonal one-particle vectors. In the case where all the vectors in eqn. (12) are different and orthogonal one gets
P’ = r: + 1 r
(19)
a<6
Thus, the energy of the system in a state with symmetry corresponding to the Young diagram [A] depends not only on [A], but also on the type of symmetry of the basic functions {on index t). Since the transitions between f1 irreducible subspaces, characterized by index t, are allowed, there are fj. split levels with symmetry [A]. Let us now consider states with different symmetry [&,,I as different
I. Kaplan/J. Mol. Struct., 272 Q99.Z’)187-196
195
states of the same system of N identical particles. The energy of such a system depends upon the form of the Young diagram [A,], i.e. [&,,Iplays the role of a dynamic quantum number (e.g. as the angular momentum, J, and its projection, M). Since the classification of states with respect to [A,] is connected exclusively with the particle identity, there must also be other additional particle characteristics which are different for states with different [&,,I (like the integer and half-integer values of the spin for bosons and fermions) and which, in the end, lead to di~erent energies. Considering also that the transitions between states with different i/2,] are strictly forbidden, we come to the conclusion that each type of symmetry [A] corresponds to a certain kind of particle just as the symmetry [N] corresponds to bosons and the symmetry [lN]corresponds to fermions. The number of particle types depends on the number of particles in the system. However, this evidently cannot be so for elementary particles (the situation with quasiparticles is not so clear). Let us consider the case where the number of elementary particles is a variable and trace down the genealogy of irreducible subspaces EP] in the chain Ff’) --+$‘c3)-_, P4)_
The above diagram shows that particles of the same kind in the (N + 1)th generation can originate from particles of different kinds in the Nth generation and, vice versa, particles of different kinds in the (N + l)th generation can originate from particles of the same kind in the Nth generation. It is impossible to make this picture agree with the concept of identity, for it implies that up to a certain N = NOa given kind of particle may obey, say, Fermi statistics, while starting from N = NO+ 1 they obey some other statistics. Evidently, the addition of a particle identical to those in a given system of particles cannot change the statistics of the system. All these contradictions are resolved if we are restricted to only onedimensional irreducible representations. In this case we have two nonintersecting chains of irreducible representations: [iV] + ]N + l] and [P]
+ [lN+y.
196
I. Kaplan/J. Mol. Strut.,
272 (1992) 187-196
Thus, the existence of states of a system of identical elementary particles described by multidimensional irreducible representations would contradict the concept of particle identity; the symmetry postulate should no longer be regarded as a postulate for it can be substantiated within quantum mechanics. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
M.A. Elyashevitch, Contribution of Einstein in the development of quantum conceptions, Usp. Fiz. Nauk, 128 (1979) 503. M.A. Elyasheviteh, The development by Niels Bohr of the atomic quantum theory. Usp. Fiz. Nauk, 147 (1985) 253. W. Pauli, Z. Phys., 31 (1925) 765. W. Pauli, Nobel lecture, in M. Fierx and V.P. Weisskopf (Eds.) Theoretical Physics in the Twentieth Century, Cambridge, MA, 1960. A. Yu. Ignatiev and V.A. Kuzmin, Yad. Fiz., 46 (1987) 786. A. Yu. Ignatiev and V.A. Kuzmin, Pis’ma Zh. Exp. Theor. Fiz., 47 (1988) 6. V.N. Garvin, A. Yu. Ignatiev and V.A. Kuzmin, Phys. Lett. B, 206 (1988) 343. O.W. Greenberg and R.N. Mohapatra, Phys. Rev. Lett., 59 (1987) 2507. L.B. Okun, Pigma Zh. Exp. Theor. Fiz., 46 (1987) 420. L.B. Okun, Usp. Fiz. Nauk, (1989) 293. B.A. Logan and A. Ljubicic, Phys. Rev. C, 20 (1979) 1957. F.T. Avignone III et al., Phys. Rev. D, 34 (1986) 97. W. Pauli, Phys. Rev., 58 (1940) 716. G. Luders and B. Zumino, Phys. Rev., 110 (1958) 1450. H.S. Green, Phys. Rev., 90 (1953) 270. D.V. Volkov, Zh. Eksp. Theor. Fiz., 36 (1959) 1560. D.V. Volkov, Zh. Eksp. Theor. Fiz., 38 (1980) 518. A. Isihara, Statistical Physics, Academic Press, New York, 1971. O.W. Greenberg and A.M. Messiah, Phys. Rev. B, 138 (1965) 1155. I.G. Kaplan, Teor. Mat. Fiz., 27 (1976) 254. B.A. Nguen and N.C. Hoang, J. Phys: Condens. Matter, 2 (1990) 4127. P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1958. L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York 1955. A.M. Messiah, Quantum Mechanics, Vol. 2, North-Holland, Amsterdam, 1962. A.M. Messiah and O.W. Greenberg, Phys. Rev. B, 136 (1964) 248. G.F. Koster, Phys. Rev., 109 (1968) 227. I.G. Kaplan, Symmetry of Many-electron Systems, Academic Press, New York, 1975. L.D. Landau and E.M. Lifschitz, Quantum Mechanics, Addison-Wesley, Massachusetts, 1965. E.M. Corson, Perturbation Methods in Quantum Mechanics of Electron Systems, Glasgow, 1951. D.I. Blokhintzev, Principles of Quantum Mechnics, Allyn and Bacon, Boston, MA, 1949. M.D. Girardeau, Phys. Rev. B, 139 (1965) 500. M.D. Girardeau, J. Math. Phys., 10 (1969) 1302. I.G. Kaplan, Usp. Fiz. Nauk, 117 (1975) 691. I.G. Kaplan, Group Theoretical Methods in Physics. Vol. 1, Nauka, Moscow, 1980, p. 175. J.V. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932. S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row Peterson, New York, 1961. I.G. Kaplan and O.B. Rodimova, Int. J. Quantum. Chem., 7 (1973) 1203.
187
Exclusion principle and indistinguishability identical particles in quantum mechanics*
of
Ilya G. Kaplan’ Karpov Institute of Physical Chemistry, ul. Obukha 10, K-64 Moscow, 103064 (Russian Federation) (Received 17 December 1991)
Abstract Two points of view on the exclusion principle are discussed. One is that there are no laws in nature that forbid the existence of particles described by a wavefunction with an intermediate permutation symmetry, and the existence of only symmetric and antisymmetric ones is due exclusively to the properties of known elementary particles. The other is that the existence of only two types of symmetry can be derived from the principle of indistinguishability of identical particles. It is proved that the indistinguishability principle is not sensitive to the symmetry of the wavefunction and cannot be used as a criterion for the correct symmetry. It is further shown that the realization of only two types of permutation symmetry in nature is the only way by which there is no contradiction with the concept of particle identity.
INTRODUCTION
I am very pleased to publish this paper in the special issue of the Journal of Molecular Structure devoted to the scientific activity of Professor M.A. Elyashevitch. An outstanding spectroscopist, M.A. Elyashevitch has over the last few years been investigating very intensively and fruitfully the history of quantum mechanics (see refs 1 and 2). I think that Mikhail Alexandrovich will read with interest the paper devoted to one of the basic principles of quantum mechanics, the Pauli principle. Although it was formulated more than 75 years ago, up to the present time it has had no rigorous theoretical foundations. The exclusion principle, first discovered by Pauli [3] for electrons and later spread to all particles, was based on the analysis of experimental data. Correspondence to: Professor I.G. Kaplan, Institute de Fisica UNAM, Ciudad Universitaria APDO, Postal 20-364, C.P. 01000 Mexico D.F., Mexico. *Dedicated to Professor M.A. Elyashevitch. ’ Permanent address: Karpov Institute of Physical Chemistry, ul. Obukha 10, K-64 Moscow, 103064. Russian Federation.
0022-2860/92/$05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
188
I. Kaplan/J. Mol. Struct., 272 (1992) 187-196
Pauli himself was never satisfied by this. In his Nobel Prize lecture [4] Pauli said, Already in my initial paper, I especially emphasized the fact that I could not find a logical substantiation for the exclusion principle nor derive it from more general assumptions. I always had a feeling, which remains until this day, that this is the fault of some flaw in the theory. All the experimental data known to date agree with the exclusion principle. Several types of experiments on the search for possible small violations of the Pauli principle have been discussed in the literature [!&lo]. A test of the Pauli principle was made earlier by searching for y quanta with energies around ZOMeV, which should be produced with transitions of nucleons in the lzC nucleus from the 2p shell to the occupied 1 s shell. This has given a lower limit for the formation time of a “non Pauli” nucleus z 2 2 x 1020years [ll]. The search for X-ray radiation produced by the transition of an electron in a germanium detector to the occupied (1 s)~ shell, either in the case of the hypothetical spontaneous decay of an electron in this shell or in the case of violation of the Pauli principle, gave an even greater lower limit z > 1.5 x 1O25years [12]. The substantiation of the exclusion principle has been made by Pauli in his famous theorem about the relation between spin and statistics [13]. In this theorem, Pauli has shown that the field operators for particles with integer spin cannot obey fermionic commutation relations since this would lead to violation of the causality principle. However, the field operators of particles with half-integer spin cannot obey the bosonic commutation relations, since this would result in negative values of the total energy of the system. From this Pauli concluded that particles with integer spin have to obey Bose-Einstein statistics, while those with half-integer spin have to obey Fermi-Dirac statistics. In this proof of the theorem, as well as in the others given later (see, for example, Luders and Zumino [14]), it is implicitly postulated that only two types of commutation relations are possible for field operators: bosonic and fermionic. However, in 1953 Green [15], and later on Volkov [16, 171, have shown that the field operators that satisfy the requirements of causality relativistic invariance, and positivity of energy, can obey more general commutation relations than the bosonic and fermionic ones: these are the so-called parabosonic and parafermionic commutation relations. For each of them there is a corresponding parastatistic of rank p characterized by p-fold occupation of single-particle states. At p = 1 the parastatistic becomes that of Fermi-Dirac statistics, while at p = 00 we get Bose Einstein statistics, see ref. 18. Greenberg and Messiah [19] have formulated the selection rules for reactions of elementary paraparticles and have analyzed, from that point of
I. Kaplan/J. Mol. Strut.,
272 (1992) 187-196
189
view, all the elementary particles known at the time. Their conclusion was that none of the latter are paraparticles. However, analysis of the commutation relations for operators of quasi-particles in a periodic lattice (molecular Frenkel excitons and magnons) has shown (Kaplan [ZO]) that the latter obey modified parastatistics of rank JV, where JV is the number of lattice sites within the delocalization region of the excitation. Later on, these results were generalized by Nguen and Hoang [21] to the case of Wanier-Mott excitons. For bosons and fermions there is a one-to-one correspondence between the symmetry of the wavefunction and the commutation relations for the field operators, but in the case of parastatistics this is not so. It is no longer possible to establish a unique relation between the type of commutation relations for parafield operators and the permutation symmetry of the wavefunction in the configuration space [19]. This means that due to the problem of the relation between the particle spin and permutation symmetry the Pauli theorem remains unsolved. What Pauli has actually proved is that the states of a system of identical particles with integer spin cannot be described by an antisymmetric wavefunction, while the states for identical particles with half-integer spin cannot be described by a symmetric function. No constraints are imposed upon the realization of states with intermediate types of permutation symmetry. The Schrodinger equation can be satisfied by solutions with a permutation symmetry of arbitrary type. Below we discuss the situation that arises in this case and make a critical analysis of existing points of view on the subject. INDISTINGUISHABILITY
OF IDENTICAL PARTICLES AND THE SYMMETRY
POSTULATE
Since the Schrodinger equation is invariant under any permutation of identical particles, its solutions can belong to any representation of the permutation group, including a degenerate one. However, according to the exclusion principle, a system of identical particles can be only in those states that are not degenerate with respect to permutations: in a symmetric or an antisymmetric state. All other types of symmetry are forbidden. One may now ask whether this limitation on the solutions of the Schrodinger equation follows from the fundamental principles of quantum mechanics or whether it is an independent principle. There are two points of view on this problem. Some physicists, including one of the founders of quantum mechanics, Dirac [22] (see also Shiff [23] and Messiah [24]) believe that there are no laws in nature that forbid the existence of particles described by wavefunctions with more complicated permutation properties than those of bosons and fermions, and that the
existing limitations are only due to the specific properties of the known elementary particles. Messiah [24] has even introduced the term symmetry postulate to emphasize the primary nature of the constraint on the allowed types of the wavefunction pe~utation symmetry. By using the Schur lemma, Messiah and Greenberg [25] have shown that the existence of pe~utation degeneracy should not introduce additional un~e~ainty into the characteristics of a state. This also follows directly from the WignerEckart theorem generalized by Koster [26]. According to formula (4.60) in ref. 27 the matrix element of an operator L, that is symmetric in all the particles, is ~%VNY’~
= a,,
a4lIml>
(1)
where index I‘ labels the basic functions of the representation I?[~’of the permutation group. The double vertical line in the right hand side of this formula means that this expression is independent of the basic function number. Thus, the average of operator 2 is the same for all the functions ‘PFJbelonging to the degenerate state. Another point of view is that the symmetry postulate is not an independent principle, but can be derived from the fund~ental principles of quantum mechanics, in particular, from the principle of indistinguishability of identical particles. The typical argumentation presented in several textbooks and monographs [28-301 is the following. From the requirement that the states of a system obtained by permutation of identical particles must all be physically equivalent, one concludes that the change in the wavefunction resulting from the transposition of any two identical particles should only cause multiplication by an insignificant phase factor: Y(x,, x,) = e”“Y(x,, x2)
c4
where CIis a real constant and x is the set of spatial and spin variables. One more application of the permutation operator PI2 gives Y(x,, x2) = ei2”V-‘(x,,xz)
(3)
or @u = 1,
(4)
eia= & 1
Since all these particles are assumed to be identical, the wavefunction should change in exactly the same way under transposition to any pair of particles, i.e. it should be either totally symmetric or totally antisymmetric. The evident incorrectness of this proof is the following. Equation (2) is valid only for one-dimensional representations. The result of the action of a permutation on a wavefunction belonging to a multidimensional representation is a linear combination of the basic functions of this representation.
I. Kaplan/J. Mol. Struct., 272 (1992) 187-196
191
The common belief that wavefunctions describing the same physical state can differ by no more than a phase factor is evidently not true. For instance, according to eqn. (l), the values of the physical quantities characterizing a system of identical particles are the same for all functions belonging to the same irreducible representation, and consequently all these functions describe the same physical state. By requiring that under permutations the wavefunction should change by no more than a phase factor, one actually postulates that the representation of the permutation group is one-dimensional. In the above proof, the indistinguishability principle is directly related to the behaviour of the wavefunction. However, since the wavefunction is not an observable, the indistinguishability principle is related to it only indirectly via the expressions for measureable quantities. A rigorous proof should be based on a rigorous formulation of the indistinguishability principle for identical particles. One possible formulation is the following: all observable quantities are invariant under permutation of identical particles, and, vice versa, the permutations of identical particles cannot be observed. Since in quantum mechanics the physcial quantities are expressed as bilinear forms of wavefunctions, the indistinguishability principle requires the invariance of these bilinear forms: P(Y!]I;]Y) = (Y]L]Y)
(5)
Often, one limits oneself to the requirement that the probability with which a given con~guration of a system of identical particles is realized must be invariant under permutations [31, 321 WI%,
- * * 9 %-A2 = IWX,, * * . , %412
‘23
Evidently, this is a particular case of Eqn. (5). For a function to satisfy eqn. (6), it is sufficient that under permutations it would change as PY(x,, . . . , xN) = e4(x1~~~~~xN)Y(x1, , . . , xN)
(7)
i.e. unlike the case of the requirement of eqn. (2), in the general case the phase is a function of coordinates and the permutation. Equations (3) and (4) in this case evidently do not hold. Most other proofs of the symmetry postulate contain unjustified eonstraints. A critical survey of such proofs can be found in refs, 25,31 and 33. The proofs of the symmetry postulate without imposing additional constraints have been given by Girardeau [32] and Kaplan [33]. These authors have used the formulation of the indistinguishability principle, respectively, in the form of eqn. (6) [32] and eqn. (5) [33]. As was noted later by the author [34], these two formulations of the indistinguishability principle are correct only when the system is in a non-degenerate state. For instance, the
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I. Kaplan/J. Mol. Struct., 272 (1992) 187-196
expression for the probability density in eqn. (6) used by Girardeau [32] should be modified if the state is degenerate, since in this case the system can be described by any one of the basic vectors of the degenerate state with equal probability. As a result, we can no longer select a pure state, the one that is described by the wavefunction, and should regard the degenerate state as a mixed one, where each basic vector enters with the same probability. The diagonal element of the density matrix for a degenerate state has the form [35]
@x1, . . .
)
xhr: x1,
. . .
)
XN) = ;, ir
$ Y!“‘(x,. . .
)
xN)* - Yy(x, . . . XN) )
(8)
1
The possibility of expressing the density matrix through only one of the functions ‘I’, implies that the degeneracy with respect to permutations can be eliminated. However, the latter cannot be achieved without violating the identity of the particles. It is not difficult to. check that the probability density, eqn. (8), is a group invariant: PDPl=
D
for all
69
From it follows the probability obeys the guishability principle, in the of multidimensional tations degenerate respect to One comes similar conclusions the case the proof the symmetry suggested by author [33]. of the density matrix a pure the indistinguishability in terms the mean of single-particle should deal the density of a state. As result, as eqn. (a), obtains an invariant under which means the indistinguishability holds for irreducible representation the permutation Thus, the principle is sensitive to symmetry wavefunction and be used a criterion selecting the symmetry. Does mean that viewpoint of authors of 22-25 is and that mechanics allows existence of described by with an permutation symmetry? arguments presented the next [34] show this is so. The of only types of symmetry in (symmetric and is by means occasional, there are reasons why is so.
of
1. Kaplan/J. Mol. Struct., 272 (1992) 187-196
193
CONTRADICTION WITH THE CONCEPT OF PARTICLE IDENTITY IN THE CASE OF PERMUTATION DEGENERACY
The states of a system of identical particles in cases where the number of particles is not conserved can be presented as vectors in the Fock space F [36]. As is known, the latter space is a direct sum of spaces I@‘) corresponding to a fixed number of particles N:
Each of the spaces FCN’ can be presented as a direct product of singleparticle spaces, f: F’N’ = f@f@ L
. # . @f I Y N
The basic vectors of FtN) are the product of single-particle belonging to the space fi I@?? = /VI>I%) * f * IYV)
(11)
vectors ]vk) (12)
For simplicity, let us consider the case where all the vectors in eqn. (12) are different. There will be no qualitative changes if some of the vectors IQ) coincide. From vector ]<(N’) one can produce N! new vectors by applying to it N! permutations of the state indices. These new vectors will also belong to FCN’ and will form in it a certain invariant subspace, which will be reducible. The basic vectors of the latter, P]c$~‘), make up a regular representation of the permutation group xN. As is known, a regular representation is decomposed into irreducible representations, each of which appears a number of times equal to its dimension. The space .sCN) falls into the direct sum
where $“I is an irreducible subspace of dimension fi drawn over the basic vectors ][;z&? (inNI is a Young diagram with N boxes). The latter are constructed of non-symmetrized basic vectors I<(“‘) by using the Young operator u$] [27] @,]lFt)
=
Wt:“’16(N)
(14)
where K’;p’ (I’) are matrix elements of representation FLAN1 and index t distinguishes between the bases in accordance with the decomposition of .z$@)into f, invariant subspaces. Thus, a space with a fixed number of particles can always be divided into
194
I. Kaplan/J. Mol. Struct., 272 (1992) 187-196
irreducible subspaees E?‘, each of which is characterized by a certain permutation symmetry given by a Young diagram with N boxes. The symmetry postulate requires that the basic vectors of a system of N identical particles belong to one of the subspaces characterized by irreducible onedimensional representations, either [N] or [lN]. All other subspaces are “empty”. Let us examine the situation that arises when no symmetry constraints are imposed. Let the system of N identical particles be described by basic vectors belonging to an arbitrary irreducible subspace eFN1. As is known, one of the consequences of the different symmetry of state vectors for bosons and fermions is the dependence of the system energy on the particle statistics. For the same law of dynamic interaction, the socalled exchange terms enter the expression for the energy of fermions and bosons with opposite signs. Let us examine from this point of view the expression for the energy of a system of particles belonging to an irreducible subspace EP’ with an arbitrary Young diagram [A,]. The energy of the system in a degenerate state is E = Tr(HD)
(15)
where D is the density operator defined, similarly to eqn. (8), as 1 fi 0, = z r&
II4r~>
(33)
and H is the hamiltonian. We will assume that the latter includes only one and two-particle interaction terms: H =
c h, + c gij i
(17)
The calculation of the trace over the functions with symmetry &,] yields
The matrix element in eqn. (18) has been calculated in ref. 37 in the general case of non-orthogonal one-particle vectors. In the case where all the vectors in eqn. (12) are different and orthogonal one gets
P’ = r:
(19)
a<6
Thus, the energy of the system in a state with symmetry corresponding to the Young diagram [A] depends not only on [A], but also on the type of symmetry of the basic functions {on index t). Since the transitions between f1 irreducible subspaces, characterized by index t, are allowed, there are fj. split levels with symmetry [A]. Let us now consider states with different symmetry [&,,I as different
I. Kaplan/J. Mol. Struct., 272 Q99.Z’)187-196
195
states of the same system of N identical particles. The energy of such a system depends upon the form of the Young diagram [A,], i.e. [&,,Iplays the role of a dynamic quantum number (e.g. as the angular momentum, J, and its projection, M). Since the classification of states with respect to [A,] is connected exclusively with the particle identity, there must also be other additional particle characteristics which are different for states with different [&,,I (like the integer and half-integer values of the spin for bosons and fermions) and which, in the end, lead to di~erent energies. Considering also that the transitions between states with different i/2,] are strictly forbidden, we come to the conclusion that each type of symmetry [A] corresponds to a certain kind of particle just as the symmetry [N] corresponds to bosons and the symmetry [lN]corresponds to fermions. The number of particle types depends on the number of particles in the system. However, this evidently cannot be so for elementary particles (the situation with quasiparticles is not so clear). Let us consider the case where the number of elementary particles is a variable and trace down the genealogy of irreducible subspaces EP] in the chain Ff’) --+$‘c3)-_, P4)_
The above diagram shows that particles of the same kind in the (N + 1)th generation can originate from particles of different kinds in the Nth generation and, vice versa, particles of different kinds in the (N + l)th generation can originate from particles of the same kind in the Nth generation. It is impossible to make this picture agree with the concept of identity, for it implies that up to a certain N = NOa given kind of particle may obey, say, Fermi statistics, while starting from N = NO+ 1 they obey some other statistics. Evidently, the addition of a particle identical to those in a given system of particles cannot change the statistics of the system. All these contradictions are resolved if we are restricted to only onedimensional irreducible representations. In this case we have two nonintersecting chains of irreducible representations: [iV] + ]N + l] and [P]
+ [lN+y.
196
I. Kaplan/J. Mol. Strut.,
272 (1992) 187-196
Thus, the existence of states of a system of identical elementary particles described by multidimensional irreducible representations would contradict the concept of particle identity; the symmetry postulate should no longer be regarded as a postulate for it can be substantiated within quantum mechanics. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
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