Tensor operators and Wigner–Eckart theorem for U q→0 ( sl(2) )V. Marotta and A. Sciarrino Citation: Journal of Mathematical Physics 41, 5735 (2000); doi: 10.1063/1.533435 View online: http://dx.doi.org/10.1063/1.533435 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/41/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the vertex operators of the elliptic quantum algebra U q , p ( s l 2 ) k J. Math. Phys. 49, 043513 (2008); 10.1063/1.2905151 Refined algebraic quantization in the oscillator representation of SL(2, R) J. Math. Phys. 41, 132 (2000); 10.1063/1.533126 Adapted Wigner–Eckart theorem Am. J. Phys. 50, 337 (1982); 10.1119/1.12857 Wigner–Eckart theorem for tensor operators of graded Lie algebras J. Math. Phys. 18, 453 (1977); 10.1063/1.523288 WignerEckart Theorem and Simple Lie Groups J. Math. Phys. 4, 720 (1963); 10.1063/1.1704010

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Tensor operators and Wigner–Eckart theoremfor Uq\0„sl „2……

V. Marottaa) and A. Sciarrinob)

Universitadi Napoli ‘‘Federico II,’’ Dipartimento di Scienze Fisicheand I.N.F.N.—Sezione di Napoli, I-80125 Napoli, Italy

~Received 12 November 1999; accepted for publication 18 April 2000!

Crystal tensor operators, which transform underUq→0(sl~2!), in analogous way asthe vectors of the crystal basis, are introduced. The Wigner–Eckart theorem for thecrystal tensor is defined: the selection rules depend on the initial state and on thecomponent of the tensor operator; the transition amplitudes to the states of the samefinal irreducible representation are all equal. ©2000 American Institute of Phys-ics. @S0022-2488~00!03208-4#

I. INTRODUCTION

Deformation of enveloping Lie algebraUq(G) introduced by Drinfeld1 and Jimbo2 is by nowa subject of standard text books. For the arguments discussed in this paper, see Ref. 3 where anaccurate list of references can be found. In the limitq→0 it has been shown by Kashiwara4 thatUq(G) admits a canonical peculiar basis, calledcrystal basis. Since that paper, crystal bases havebeen the object of very intensive mathematical studies and have also been extended to the case ofdeformation of affine Kac–Moody algebras. However a point is still, to our knowledge, missing:it is possible to introduce the concept ofq-tensor andq-Wigner–Eckart theorem in the limitq→0? Besides the mathematical interest, the question may be interesting in application to physicalor to physics inspired models. It is clear that in this limit we are no longer dealing with thedeformation of a universal enveloping Lie algebra, but it is interesting to study what are the relicsof the symmetry structure described originally by the algebraG and then by the deformation of itsenveloping algebraUq(G). It is, indeed, well known that Wigner–Eckart theorem is one of themilestones in the application of algebraic methods in physics. Let us remark that one of themotivations to study the limitq→0 by Date, Jimbo, and Miwa,5 which firstly discovered thepeculiar behavior ofn-dimensionalUq→0(sl(n,C))-modules, whose axiomatic settlement has beengiven in Ref. 4, was the study of solvable lattice models where the parameterq plays the role ofthe temperature. Moreover, in Ref. 6 the quantum enveloping algebraUq(sl~2!% sl~2!) in the limitq→0 has been proposed as symmetry algebra for the genetic code assigning the~4! nucleotides~elementary constituents of the genetic code! to the fundamental representation and the~64!codons~triplets of nucleotides! to the threefold tensor product of the fundamental representation,using crystal basis.

In the following we will consider only the crystal basis forUq(sl~2!). For completeness in Sec.II we give a short reminder of the main properties of the crystal basis andq-tensor operators,limiting ourself to the case ofUq(sl~2!). In Sec. III theq-tensor operators in the limitq→0, calledcrystal tensor operators, are introduced on the basis of a conjecture. To support the conjecture weconsider a few examples of explicitly known tensor operator and we show that in such cases theconjecture is verified. A discussion of the results and a high-light of possible applications arepresented in Sec. V.

a!Electronic mail: vmarotta@na.infn.itb!Address for correspondences: Mostra d’Oltremare, Pad. 20-I-80125 Napoli, Italy. Electronic mail: sciarrino@na.infn.it

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 8 AUGUST 2000

57350022-2488/2000/41(8)/5735/10/$17.00 © 2000 American Institute of Physics

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II. A REMINDER OF THE CRYSTAL BASIS AND q-TENSOR OPERATORS

To set the notation, let us recall the definition ofUq(sl~2!)

@J1 ,J2#5@2J3#q , ~1!

@J3 ,J6#56J6 , ~2!

where

@x#q5qx2q2x

q2q21 . ~3!

In the following we shall omit the lower labelq.For later use let us remind that

@n#q! 5@1#q@2#q¯@n#q . ~4!

The algebraUq(sl~2!) is endowed with an Hopf structure. In particular, we recall that the co-product is defined by

D~J3!5J3^ 111^ Js ,~5!

D~J6!5J6 ^ qJ31q2J3^ J6 .

The Casimir operator can be written as

C5J1J21@J3#@J321#5J2J11@J3#@J311#. ~6!

For q generic, i.e., not a root of unity, the irreducible representations~IR! are laleled by aninteger or half-integer numberj and the action of the generators on the vector basisu jm& (2 j<m< j ), of the IR is

J3u jm&5mu jm&, ~7!

J6u jm&5A@ j 7m#@ j 6m11#u j ,m61&5F6~ j ,m!u j ,m61&. ~8!

From Eqs.~7! and ~8! it follows that

Cu jm&5@ j #@ j 11#u jm&. ~9!

Let us study the behavior of aq-number@x# for q→0. In the following the symbol; in theequations has to be read equal in the limitq→0 modulo, the addition of a function regular inq50. From the definition Eq.~3!, we have

@x#q→0;q2x11, xÞ0. ~10!

So it follows that

F6~ j ,m!q→0;q2 j 11/2, ~11!

@ j #@ j 11#q→0;q22 j 11, ~12!

@x#! q→0;q21/2x~x21!. ~13!

From Eqs.~8! and ~11! it follows that the action of the generatorJ6 is not defined in the limitq→0. Let us define the elementG0 belonging to the center ofUq(sl~2!)

5736 J. Math. Phys., Vol. 41, No. 8, August 2000 V. Marotta and A. Sciarrino

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G05C21/2, ~14!

G0u jm&5~@ j #@ j 11# !21/2u jm&q→0;qj 21/2u jm&. ~15!

Let us define

J65G0J6 . ~16!

These operators are well behaved forq→0. Their action in the limitq→0 will define the crystalbasis:

J1u jm&5u j ,m11& for 2 j <m, j , ~17!

J2um&5u j ,m21& for 2 j ,m< j , ~18!

J1u j j &5 J2u j ,2 j &50. ~19!

The tensor product of two representations in the crystal basis is given by Ref. 4.Theorem: If B1 andB2 are the crystal bases of theM1 andM2 Uq→0(sl~2!)-modules, foru

PB1 andvPB2 , we have

J2~u^ v !5H J2u^ v'n>1 such that J2n uÞ0 and J1

n v50,

u^ J2v otherwise.~20!

J1~u^ v !5H u^ J1v'n>1 such that J1n vÞ0 and J2

n u50,

J1u^ v otherwise.~21!

So the tensor product of the two crystal basis is a crystal basis and the states of the basis of thetensor space are pure states. In other words, in the limitq→0 all theq-Clebsch–Gordan (q-CG)coefficients vanish except one which is equal to61. Let us recall the definition ofq-tensor forUq(sl~2!).7–9,3 An irreducibleq-tensor of rankj is a family of 2j 11 operatorsTm

j (2 j <m< j )which transform under the action of the generators ofUq(sl~2!) as

qJ3~Tmj ![qJ3Tm

j q2J35qmTmj ~22!

or

@J3 ,Tmj #5mTm

j , ~23!

J6~Tmj ![J6Tm

j qJ32q2J361Tmj J65F6~ j ,m!Tm61

j . ~24!

In deriving the above equations, use has been made of the nontrivial co-product Eq.~5!. Theq-Wigner–Eckart (q-WE) theorem now reads10

^JMuTmj u j 1m1&5~21!2 j ^JiTj i j 1&

A@2J11#^ j 1m1 jmuJM&, ~25!

where^JiTj i j 1& is the reduced matrix element of theq-tensorTj and^ j 1m1 jmuJM& is theq-CGcoefficients. In the following we will use the explicit expression of theq-CG of Ref. 10. It isuseful to rewrite theq-WE theorem Eq.~25! in the following form:

5737J. Math. Phys., Vol. 41, No. 8, August 2000 Tensor operators and Wigner–Echart theorem

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Tmj u j 1m1&5~21!2 j (

J5 l j 2 j 1

j 1 j 1 ^JiTj i j 1&

A@2J11#^ j 1m1 jmuJM&uJM&. ~26!

III. DEFINITION OF CRYSTAL TENSOR OPERATORS

Our strategy is to define the (q→0)-tensor and then (q→0)-WE is thefollowing.

~1! Let us write Eq.~24! in the form

J6Tmj qJ35q2J361Tm

j J61F6~ j,m!Tm61j . ~27!

~2! We multiply both sides of Eq.~27! from left and right by an elementG belonging to the centerof the algebra and define

Tmj 5GTm

j G. ~28!

Let us remark thatTmj is still a q-tensor operator of the same rank asTm

j . Indeed it transformsunder the action ofJ6,3 according to Eqs.~22!, ~24! or ~27! which have been derived byapplication of the co-product Eq.~5!.

~3! We make theconjecturethat an element, in general not unique,G exists such thatTmj has a

smooth and defined behavior in the limitq→0. We will discuss below some explicit examplesin which Tm

j is not defined in the limitq→0 and its reduced matrix element diverges, whileon the contrary it is possible to defineTm

j with a well-defined limit.~4! We apply theJ6 ,J3 generators to Eq.~26! written for Tm

1/2 and, subsequently, we study thelimit q→0 of both sides of the equation, assuming that^JiTj i j 1& has a well-defined behaviorin the limit.

~5! From the study of~4! we deduce the action of the generatorsJ6 ,J3 in the limit q→0 on Tm1/2.

~6! From the tensor product we can infer the action for the generic tensor.

To perform the third and fourth steps, we need to compute theq→0 limit of ^ j 1m112muJ,M &. The

results are reported in Table I, where we have used the expressions of^ j 1m112muJM& given in

Appendix B of Ref. 10 and Eq.~10!. For a study of the behavior forq→0 of the genericq-CG thefollowing formulas are useful:

@x#!

@x2a#!.q→0q21/2a~2x2a21!, ~29!

@x1a#!

@x2a#!.q→0q2a~2x21!, ~30!

@x#!

@x2a#!.q→0q21/2a~2x2a21!, ~31!

@x1a#

@x2a#.q→0q22a, ~32!

TABLE I. Behavior of theq-CG^ j 1m112muJM& for q→0.

J m51/2 m521/2

j 111/2 qj 12m1 1j 121/2 21 qj 12m1 q

5738 J. Math. Phys., Vol. 41, No. 8, August 2000 V. Marotta and A. Sciarrino

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@x#!

@x2a#!.q→0q21/2a~2x2a21!, ~33!

@x1a#!

@x2a#!.q→0q2a~2x21!, ~34!

@x#!

@x2a#!.q→0q21/2a~2x2a21!, ~35!

@x1a#

@x2a#.q→0q22a. ~36!

Using the results of Table I, denoting bytm1/2 theq-tensor operatorTm

1/2 in the limit q→0, and

S ^JiTj i j 1&

A@2J11#D

q→0

;^Jit j i j 1&, ~37!

we get

t1/21/2u j 1m1&5~21!d j 1 ,m1

^ j 111/2it1/2i j 1&u j 111/2,m111/2&

1^ j 121/2it1/2i j 1&u j 121/2,m111/2&, ~38!

t21/21/2 u j 1m1&5~21!^ j 111/2it1/2i j 1&u j 111/2,m121/2&. ~39!

Inspection of Eqs.~38! and~39! shows that the rhs of the equations has the structure of the tensorproduct of1

2^ j in the crystal basis, see the above quoted Kashiwara’s theorem. Note that theorderof the factors in the tensor productis important. Here and in the following the order of the factorsis read from left to right~western reading prescription!.

So we can write the action of the generatorsJ6 ,J3 on tm1/2, as

J3~tm1/2![mtm

1/2, J6~tm1/2![tm61

1/2 . ~40!

Clearly, if umu.1/2 thentm1/2 has to be considered vanishing. Equation~40! shows that the spino-

rial operatorstm1/2 transform under the action ofJ3,6 as the crystal basis vectorsu1/2,m&. It has

been proven by Rittenberg–Scheunert9 that for quasitriangular Hopf algebra@Uq~sl~2!! is ‘‘al-most’’ quasitriangular which does not affect the following considerations# the tensor product oftensor operators is a tensor operator. So by applying the Rittenberg–Scheunert’s theorem and inthe limit q→0 the Kashiwara’s theorem we can extend Eq.~40! to any valuej. So we definecrystal tensorof rank j a set of operator which transform underJ6 ,J3 according to Eq.~40!. Asan explicit check and a further example, we compute the (q→0)-WE theorem forT1. We need tocompute theq→0 limit of ^ j 1m11muJM&. The results are reported in Table II, where we haveused the expressions of^ j 1m11muJM& given in Appendix B of Ref. 10 and Eq.~10! and Eqs.~29!–~36!.

TABLE II. Behavior of theq-CG^ j 1m11muJM& for q→0.

J m51 m50 m521

j 111 q2( j 12m1) qj 12m1 1j 1 2q21qj 12m1 q2q2( j 12m1)211d j 1m1

qqj 12m1

j 121 1 2qj 12m1 qq2( j 12m1)

5739J. Math. Phys., Vol. 41, No. 8, August 2000 Tensor operators and Wigner–Echart theorem

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Using the results of Table II, we obtain in the limitq→0:

t11u j 1m1&5^J5 j 111it1i j 1&uJ,m111& if m15 j 1

52^J5 j 1it1i j 1&uJ,m111& if m15 j 121,s

5^J5 j 121it1i j 1&uJ,m111& if m1, j 121, ~41!

t01u j 1m1&5^J5 j 111it1i j 1&u j ,m1& if m15 j 1

52^J5 j 15 j 1it1i j 1&uJ,m1& if m1, j 1 , ~42!

t211 u j 1m1&5^ j 111it1i j 1&uJ,m121&. ~43!

Let us now prove the following statement.Proposition 1: If the q-tensors Tˆ r 1 and Tr 2 have a well-defined behavior for q→0, i.e., the

crystal tensorst r 1 and t r 2 are defined, than the q-tensors TR, obtained by the tensor product ofTr 1 and Tr 2 has a well-defined limit for q→0.

Proof: Let us define

TKR5 (

k1 ,k2

^r 1k1r 2k2uRK&Tk1

r 1Tk2

j 2. ~44!

Take the matrix element of the rhs and lhs of Eq.~44! between the initial stateu j 1m1& and the finalstateuJM&. Insert the identity

15(j ,m

u jm&^ jmu ~45!

in the rhs and apply theq-WE theorem Eq.~25! for Tr 1 and Tr 2. We get

^JMuTKRu j 1m1&5 (

k1 ,k2 , j ,m^r 1k1r 2k2uRK&^ jmr1k1uJM&^ j 1m1r 2k2u jm&

^JiTr 1i j &

A2J11

^ j iTJ2i j 1&

A2 j 11.

~46!

If we apply theq-WE to the lhs of the above equation and make the limitq→0, as by assumptionthe rhs of Eq.~46! has a limit, it follows that

S ^JiTr i j 1&

A@2J11#D

q→0

;^Jit j i j 1&. ~47!

Use of Eq.~45! requires at least a comment. The completeness of the basisujm& for su~2! is aparticular case of the completeness of the IRs of a compact group. ForqÞ1 we cannot appeal tothis general property as we are no longer dealing with a Lie group. However, the completeness oftheq-coherent states11 for theq-bosons12,13gives us an argument for the completeness of the statesu jm&, as a realization of the deformed enveloping algebraUq~su~2!!, for q generic, and of itsrepresentations can be written in terms ofq-bosons. See below for comments about the use ofq-bosons in theq→0 limit. However, should the above argument not convince the reader, a lesselegant, but straightforward proof can be carried on using the explicit expression of a genericq-CG coefficients, see Eq.~5.18! of Ref. 10.

Let us remark that the knowledge of the elementsG r 1and G r 2

, which allow to define,respectively, the crystal tensorst r 1 and t r 2 from q-tensorsTr 1 and Tr 2, does not determine the

5740 J. Math. Phys., Vol. 41, No. 8, August 2000 V. Marotta and A. Sciarrino

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elementGR , which allows to define the crystal tensortR from theq-tensorTR, obtained from thetensor product ofTr 1 andTr 2, as the elements of the center of the algebra do not commute withthe genericq-tensorsTj , for j Þ0.

Now let us discuss in some explict examples ourconjecturethat it is possible to find anelementG in the center of the algebra such that the operatorGTjG is well defined in the limitq→0.

Let us consider the vector operator constructed with the generators10

T61 56

1

A@2#q2J3J6 ,

T015

1

@2#~q21@2J3#1~q2q21!J1J2!

51

@2#~q21@2J3#1~q2q21!~C2@J321/2#2!!.

~48!

The reduced matrix element^ j 1iT1i j 1& ~which is the only nonvanishing! for theq-vector operatorEq. ~48! can be computed from Eq.~25! and we get

^ j 1iT1i j 1&5A@2 j 1#@2 j 111#@2 j 112#

@2#. ~49!

The expression Eq.~48! has no defined meaning in the limitq→0 and

^ j 1iT1i j 1&q→0;q23 j 111. ~50!

If we multiply Eq. ~48! by the elementG5Aq1/2G03. we have, from Eqs.~14!–~22!:

^ j 1iT1i j 1&q→0;1. ~51!

Let us remark that the multiplication ofG by a real number, the addition of any element of thecenter vanishing forq→0 as well as any functional construction ofG0 behaving in the limitq→0 asq23 j 1

11 does not modify our conclusion. Our choice is theminimal one.The Uq~sl~2!! can be realized in terms ofq-bosons12,13 defined by

aiaj12qd i j aj

1ai5d i j q2Ni, ~52!

@Ni ,aj1#5d i j aj

1 , @Ni ,aj #52d i j aj , @Ni ,Nj #50. ~53!

Using q-boson,q-spinorial operators have been constructed7

T1/21/25a1

1qN2/2, T21/21/2 5a2

1q2N1/2. ~54!

As it can be seen from the defining expression Eq.~3! or from the relation betweenq-bosons andstandard bosonic operators,14 the q-bosons have no well-defined behavior in the limitq→0.

However, it is always possible to compute theq-spinor reduced matrix using Eq.~54!, even ifin the limit q→0 the explicit realization of the tensor operator in terms ofq-bosons is meaning-less. Indeed, in the case of theq-vector operator previously discussed, the same results are ob-tained using the definition Eq.~48! in terms of the abstract generators ofUq~sl~2!! or making useof the explicit realization of the algebra generators in terms of theq-bosons. From Eq.~25! and theexpression ofq-CG we get

^ j 111/2iT1/2i j 1&52A@2 j 111#@2 j 112# ~55!

5741J. Math. Phys., Vol. 41, No. 8, August 2000 Tensor operators and Wigner–Echart theorem

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and

^ j 111/2iT1/2i j 1&1&q→0;q22~ j 111/4!. ~56!

ChoosingG5AqG0 , we obtain from Eqs.~14!–~22!

^ j 111/2iT1/2i j 1&q→0;21. ~57!

If we consider theq-spinorial operators, Hermitian conjugate to Eq.~54!,3 given by

T1/2†,1/252a2q2@N121#/2, T21/2

†,1/25a1q~N211!/2. ~58!

From Eq.~25! and the expression ofq-CG we get

^ j 121/2iT†,1/2i j 1&52A@2 j 1#@2 j 111# ~59!

and

^ j 121/2iT†,1/2i j 1&q→0;q22~ j 121/4!. ~60!

Multiplying by the same elementG5AqG0 , we used to buildT1/2 from T1/2, we obtain, using Eqs.~14!–~22!

^ j 121/2iT†,1/2i j 1&q→0;21. ~61!

IV. DISCUSSION

In conclusion we have introduced (q→0)-tensor operators, which we call crystal tensoroperators, which transform underUq→0~sl~2!! according to Eq.~40!. We have made the conjecturethat an elementG of the center of the deformedq-algebra exists such that the limit, forq→0, oftheq-tensor operator forUq~sl~2!!, multiplied to the right and to the left byG, is well defined andthe expressions, in this limit, are the crystal tensor operators. To support our conjecture we haveconsidered a few examples of explicitly knownq-tensor operators and we have shown that, in theconsidered cases, a~not unique! G element can be consistently found and its minimal~up a factor!form has been explicitly obtained.

The transformation law for the generic crystal tensor operators is

J3~tmj ![mtm

j , J6~tmj ![tm61

j . ~62!

Clearly, if umu. j thentmj has to be considered vanishing. The (q→0)-Wigner–Eckart theorem

can be written

tmj u j 1m1&5~21!2 j (

a50

2 j

^ j 11 j 1ait j i j 1&u j 11 j 1a,m11m&

3~dm1 , j 12a1d2m, j 2a2dm1 j 12adm, j 2a!. ~63!

Let us emphasize that in order to discuss more generally our conjecture or to compute thereduced matrix element,^ j 2it j i j 1&, one should know an explicit realization of the generators ofUq→0~sl~2!! which, at our knowledge, is still missing.

Let us stress that while theq-WE theorem~for q generic! has the same form as the usual WEtheorem, roughly speaking one has to replace the numerical expression byq-numerical expression,so its content~selection rules, relation between the transition amplitudes! is of the same form, the(q→0)-WE theorem has a completely different structure. The IR to which the final state belongs,

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depends not only from the rank of the tensor and from initial IR, but in a crucial way from theinitial state and from the component of the tensor in consideration. In Table III we report the finalstates, such that^ j 2m2utm

j u j 1m1&Þ0, for j 51,1/2 and forj 15 12,1,32,2.

In particular the highest weight state of the initial IRj 1 is always transformed under action oft j into a state of the final IRJ5 j 11 j , while the lowest weight state is transformed into a state ofany final IR~exactly one state ifj 1> j with J5 j 11m). Let us remark that the peculiar feature isthat no vector crystal operator can be build up with the generatorsJ6 ,J3 . Indeed, as a generator,it should connect any initial state to a state of the same IR, but there is no crystal vector operatorwith such a property as one can realize from Table III or from the general form of the transfor-mation law Eq.~63!. The transitions between an initial state, belonging to IRj 1 , and any finalstate, belonging to the IR,J, are all equal, the nonvanishingq-CG in the limit q→0 being equalto 61.

An interesting consequence of the above-discussed mathematical structure is the lack ofreversibility, in some sense, of the Wigner–Eckart theorem in the limitq→0. In quantum me-chanics, it is a well-known and a straightforward consequence of the Wigner–Eckart theorem thatif a perturbation, described by a tensor operatorTjm , connects the initial statec i to the final statec f , then the statec f is connected to the statec i by Tj ,2m , with the same transition amplitude.For qÞ1 this statement is true up to a possibleq factor, as a consequence of the symmetryproperty10

^ j 1m1 jmuJM&5~21! j 11 j 2J~21! j 2mqmS @2J11#

@2J111# D1/2

^JM jmu j 1m1&. ~64!

In the crystal basis the reversibility is completely lost. For example, from Table III, we see that,under the action of the crystal vector operatort1

1, the initial stateu1/2, 1/2& is transformed~up toa factor! into the final stateu3/2, 3/2&. The stateu3/2, 3/2&, under the action oftm

1 , is transformed~up to a factor! into the final stateu5/2,3/21m&.

This amazing feature can be relevant in the mathematical model for biophysics. Indeed it wasrealized for a long time, see e.g., the lectures by Schro¨dinger in 1943, ‘‘What is life?,’’ 15 thatmodels able to mimick ‘‘elementary’’ life process, as the mutations of the genetic code, have to

TABLE III. Selection rules fort1, on the left, andt1/2, on the right. In thecentral entries the value of the finalJ(M5m11m), in function of the com-ponent oft and of the initialj 1 , m1 /m entries, forj 151/2,1,3/2,2

m1 /m 1 0 2112 2

12

12

32

32

32 1 1

212

12

12

32 0 1

1 2 2 232

32

0 1 1 212

32

21 0 1 212

32

32

52

52

52 2 2

12

32

32

52 1 2

212

12

32

52 1 2

232

12

32

52 1 2

2 3 3 352

52

1 2 2 332

52

0 1 2 332

52

21 1 2 332

52

22 1 2 332

52

5743J. Math. Phys., Vol. 41, No. 8, August 2000 Tensor operators and Wigner–Echart theorem

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reproduce the irreversibility of such a process. Therefore mathematical models able to describe atleast some simple features of the extremely complicated elementary biophysics process have toincorporate structures suitable to reproduce the irreversible character of the process.

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5744 J. Math. Phys., Vol. 41, No. 8, August 2000 V. Marotta and A. Sciarrino

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