Matricial Representation of Rational Power of Operators and Paragrassmann Extension of Quantum Mechanics

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1 Matricial Reresentation of Rational Power of Oerators and Paragrassmann Extension of Quantum Mechanics N. Fleury, M. Rausch de Traubenberg Physique Théorique Centre de Recherches Nucléaires et Université Louis Pasteur B.P. 0 F Strasbourg Cedex France IUFM d Alsace, 00 Route de Colmar, Strasbourg and R.M. Yamaleev Joint Institute for Nuclear Research Dubna-Russia Int. J. Mod. Phys. A9 (1995) 169 1

2 Abstract: Using aragrassmann, or generalized Grassmann algebras, we define rational ower of annihilation and creation oerators, in order to extend suersymmetric quantum mechanics. This extension can be relaced, under some assumtions, in aragrassmann quantum mechanics. We then aly this method to build an equation which generalizes the (+1)D Pauli equation to a article of arbitrary sin s. This is done by means of a hamiltonian defined as a sum of s monomials of degree s+1. I Introduction The basic rinciles of suersymmetric quantum mechanics (SQM) were formulated in [1] to obtain an exlicit examle in which suersymmetry (in less than four dimensions) could be broken. It has been noticed that SQM can also be used to describe systems with hidden symmetries []. However, as it is well known, suersymmetry is the more general non trivial extension of the Poincaré algebra that one can built [3]. Thus, to have no conflict with this no-go theorem and relativistic quantum field theory, it seems to be more easy to extend SQM than symmetry of sace-time. For instance, SQM has been extended to arasuersymmetric quantum mechanics (PSQM) [4], and this work have been develoed in several aers [5,6,7]. All these ossible extensions of SQM are related with a generator of order and involve araquantization of arafermions [8] of order 1, arasuerconformal (Z -conformal) algebras [9]. Other attemts based on aragrassmann calculus have been also undertaken in [10,11,1]. In the resent aer, we roose another way to extend SQM and PSQM by considering fractional ower of momenta, in a matricial reresentation. The models involving fractional creation and annihilation oerators have been studied in [13]. But unlike the general case, we just consider rational owers of these oerators, which can be reresented by aragrassmann [8] or generalized Grassmann [14] algebras. The reresentation of fractionnal ower we use is different of the one defined by seudo-differential oerators, and is given in terms of aroriate matrices. The content of this aer is the following: in sect. we formulate the oscillator model in terms of 1 - ladder oerators. Sect. 3 is devoted to a descrition of these oerators by means of aragrassmann variables, this gives an extension of SQM. Finally, in sect. 4 we use these oerators to construct a aragrassmann extension of the Pauli equation for an arbitrary value of the sin. 1

3 II The 1 -ste ladder oerator In [15] two of us have established a general rocedure for the linearization of any olynomial. Accordingly, the oerator a 1 ( =, 3,...) can be reresented by a matrix, the elements of which are ( ) a 1 = a l i δ i,j+1 i,j = 1,... (.1) ij { 0 if i < whereδ ij isthekroneckersymbolandl i = 1 if i = andtheindicesaredefinedmodulo. Notice that the maing which reresents a 1/ by a matrix is not unique.moreover, for hermitic conjugation consistency, one has to set (a + ) 1/ = (a 1/ ) + So, to define the th root of a, an enlargement of the number of degrees of freedom is needed. Strictly seaking, we take the th root of a.i, where I is the identity matrix, just as one takes the square root of the Klein-Gordon oerator to get the Dirac equation. The vector states associated to a 1 have obviously internal comonents, and endow a structure which is richer than just coies of the vector states associated to a. It has to be noticed that in this formalism, a 1 has nothing to do with the roots calculated in the frame of the theory of seudo-differential oerators. In the sequel, all our oerators have to be understood as matrices, and for scalar oerators the identity matrix will be omitted. Let a be the annihilation oerator of the usual bosonic oscillator. Now the matrix a 1/ is understood to act on the vector state N> = times (.). where the comonent is in the usual Hilbert sace H of the ordinary oscillator. The action of a k 1 = (a ) k (k = 0, 1,..., ) is then 1 a k N> = n... n. n 1 >.. n 1 > k k (.3)

4 It can be ointed out that the Hilbert sace obtained here, because of the secial structure of a 1/, has a richer structure than the one obtained with coies of the bosonic states, but less than H. Indeed, the relation (.3) leads to the natural definition of the states N k > and N + l > A basis of the Fock sace is then N k. > = n 1 >. n 1 > N + l > = n+1 >.. n+1 >.. k, (.4.a) k l l. (.4.b) 0>, 1 >, >,... 1>, 1+ 1 >,... The eigenstate relation (.3) suggests to introduce the convenient notation k = diag( 0,..., 0,1,..., 1), }{{}}{{} ( k) times k times leading to and ( a k N> = 1+( n 1) k ) N k >, ( ( n+1 a +k N >= 1 k ) + k ) N +1 k >. So, the oerator a 1 aears as the basic tool, and allows to construct (+1) monomials Nk having a diagonal form in our Hilbert sace H Nk = a k a + a k ; k = 0, 1,...,. (.5), 3

5 Their matricial exression is Nk = diag(a + a,..., a + a,aa +,..., aa + ) (.6). }{{}}{{} k times k times The usual terms aa + and a + a are resectiveley obtained for the values 0 and of k. We notice that Nk a + k is hermitian, and therefore, there is no need to introduce terms like aa +k/. Its action on the eigenstate N> is ( Nk N> = n+ k ) N > (.7) Moreover, one can check the commutation relations [ ] a k,nk = k ak (.8.a) [ ] a +k,nk = a +k k (.9.a) When k =, one retrieves the usual case. More general relations like a k l N ± > or [ ] a k, N l can be calculated, but the general case is tedious because one has to comare the relative values of k, l, k +l,,... Now, we define in a natural way the hamiltonian as the sum of the monomials (.5), the k = 0 and k = terms exceted H 1 = k=1 Nk ( 1) (.10) The usual terms a + a and aa + are not considered because we do not want to describe here the usual bosonic oscillator. One has the relation H 1 N k ( > = (n k )+ 1 1 ) N k 1 S 1 > (.11) where the integer or half integer sin matrix S l is defined by S l = diag(l, l 1,..., (l 1), l). l = 1, 1, 3... The matrix elements of the hamiltonian H 1/ are ( H 1 ) ij = δ ij 1 [a+ a( i)+aa + (i 1)] (.1) 4

6 Remembering that the usual bosonic oscillator has the hamiltonian H B = 1 (a+ a+aa + ) one obtains from (.1) H 1 = H B 1 1 S 1 (.13) This is an interesting result: the introduction of the 1 -ste ladder oerator leads to an additionnal sin term to the ordinary bosonic hamiltonian. This clearly indicates a connection between the -root of the lowering and raising oerators and 1 sin oerator. This relation will be exhibited in the next section with our extension of the Witten s model of SQM. III Paragrassmann extension of suersymmetric quantum mechanics For =, the hamiltonian (.10) is reduced to H1 = a1 a + a 1 = HB S1 This is exlicitly the hamiltonian of the suersymmetric oscillator. The fermionic oerators, θ + for creation and θ for annihilation, are in fact hidden in a 1. Indeed, we have where, from (.1), θ + = a 1/ = aθ + +θ (3.1) ( ) ( ) and θ = One gets the basic relations { (θ + ) = θ = 0 {θ +, θ} = 1 and in addition, the connection with the 1 -sin oerator is obtained through [θ, θ + ] = S1 (3.) (3.3) In SQM, if x is a bosonic coordinate and ( x = i ) x its conjugate ( momentum, ) Witten has introduced the two suercharges Q + = and Q π 0 π+ = with π 0 0 ± = ±iw and W the suerotential. They obey the graded algebra { Q, Q +} = H [ Q ±, H ] = 0 ( Q ± ) = 0 (3.4) 5

7 The hamiltonian has the form H = 1 {Q, Q + } = H 0 W S1 (3.5) where W is the derivative of the suerotential with resect to x and H 0 = 1 ( +W ). Now let us introduce the (π ) 1 oerator just as in the revious section. As in (.10), write H 1 = k=1 (π ) k π+ (π ) k k (3.6.a) With the same remark as in the revious section, we notice that one has only to consider terms like (π ) k/ π + (π ) k in (.6.a). This can be develoed to get H 1 = H 0 W 1 S 1 (3.6.b) which is (3.5) extended to the value 1 of the sin. To make a close connection with SQM, we reresent a 1/ in a aragrassmann algebra, which is related to arafermions of order 1. The generator θ fulfils the condition θ = 0 (3.7) Note that all the oerators satisfying (3.7) are equivalent through a Jordan transformation. So θ is equivalent to the arafermionic annihilation oerator, and this shows the corresondance between aragrassmann and generalized Grassmann algebras. A matricial reresentation of the oerator θ can be found in [15]. In relation with the revious section, it is easily seen that one can write a 1 = a(θ + ) 1 +θ (3.8) The identification of (3.8) with (.1) gives θ = (3.9) Moreover, one gets more generally a k = a(θ + ) k +θ k (3.10) 6

8 and using (a 1/ ) = a, one has the identities ( k) times {}}{ { θ,...,θ, k times {}}{ θ +( 1),...,θ +( 1) } = ( 1)!δ k1 (3.11), where {...} reresents the comletely symmetric roduct. This can be interreted as the extension of (3.) to the > case. These relations are of course different from the araquantization ones[8]. In this direction, the comarison between the hamiltonians (3.5) and (3.6) allows a generalization of (3.3) to an arbitrary value 1 of the sin. The result is 1 S 1 = k=1 [θ k,(θ + ) k ] (3.1) This relation can be used to rewrite (3.6.a). Define the charge oerator Q m deending on the aragrassmann generator θ m times {}}{ Q m = θdiag( 1,... 1, π +, which are used to build the ( 1) hamiltonians H m = k=1 The summation over m gives H 1 Now consider the eigenvalue roblem for H m. As in the Witten s model, the eigenvalues of ( m 1) times {}}{ 1,... 1 ) m = 1,..., 1 (3.13) Q k m (Q + m) 1 Q k 1 m = (Q m +(Q + m) 1 ) (3.14) H 1 = m=1 H m H m,k = Q k m (Q + m) 1 Q k 1 m, (3.15.a) which can be rooved to be hermitian, coincide excet for the lowest eigenstate. Thus, if we have an eigenvector which satisfies H m,k ψ (m) k,n = λ nψ (m) k,n (3.15.b) one can decrease the value of the index k by the action of Q m. This is shown using the relation (Q m ) l H m,k = H m,k l (Q m ) l l < k 7

9 As a result, one gets H m,k l ψ (m) k l,n = λ nψ (m) k l,n (3.16) where ψ (m) k l,n = (Q m) l ψ (m) k,n. Usingthehermitianroerty,H + m,k = (Q+ m) k 1 (Q m ) 1 (Q + m) k andbysimilararguments as before, one can easily check that (Q + m) l l < k 1, is an increasing oerator for the eigenstates. These raising and lowering oerators Q + m, Q m have the status of charges and commute with the Hamiltonian H m. Similarly as in the context of SQM [16], one can interret the constituents H m,k (k = 1,...) as arasuersymmetric artners: one may consider that H m,1 corresonds to bosons whereas H m,i (i =,... ) will describe arafermions. Obviously, the charges Q m do not commute with the full Hamiltonian (3.6.b). However, these 1 charges can be used to build the ( 1) arasuercharges defined in [7] Q 1 = 1 m=1 Q m ( )θ + Q r = m rq m Q r ( 4)θ + +h i,i+1 r =,, 1 (3.17) where h ij is the matrix having the coefficient one at the intersection of the i-th row with the j-th column. These h matrices can be exressed in terms of the basic elements θ and θ + of the algebra because (θ + ) a θ a = h a, a +h a+1, a+1 + +h, and h i,i+1 = h i,i θ + These conserved charges commute with the Hamiltonian and generate a arasuersymmetric extension of (3.4), rovided that W is submitted to a consistency condition which leads to W = αx + β for = 3 [4,17] for instance. Using these oerators together with the arafermionic one ( which is equivalent to θ through a Jordan transformation) one constructs [17] a arasuersymmetric extension of Sch(1) Su() U(1), where Sch(1) is the one dimensionnal Schrödinger algebra. Notice that such a reresentation is not faithful because the n-exterior algebra [18] is infinite dimensional, but a finite dimensional reresentation can be obtained in terms of the generalized Grassmann algebras[14,15]. 8

10 IV Paragrassmann extension of the Pauli equation for s > 1 values of the sin The Klein-Gordon equation and the Pauli [19] Dirac [0] equations are the only ones describing articles of sin 0 and 1/ resectively. The situation is quite different for higher values of the sin and the way to construct s > 1 equations is not unique (see for instance [1,]). In addition, the difference between Pauli and Dirac equations and the others is that they have led to a great variety of detailed redictions which have been exerimentally confirmed allready at the level of the first quantization. In this section we develo a new aroach to construct a non relativistic lanar equation for s > 1 sin articles. This result is connected to the formula (3.1). In fact, the fermionic oerators θ and θ + can be used to reresent the hamiltonian of the Pauli equation in the SQM formalism. Write with ± = x ±i y. It can be rewritten equivalently H H ( s = 1 ) = 1 m ( θ θ) (4.1) ( s = 1 ) = 1 m (Q+ Q+h c) (4.) where we defined Q + = θ +, Q = + θ. This equation is the one allowing the generalization of the Pauli equation using the extended oscillator model of the revious section. To this urose, set m {}}{ Q m = θdiag( 1,..., 1, +, 1..., 1) Note that Q 1 m is indeendent of m. Consider also H m = k=1 Qm 1 (Q + m) 1 Q k 1 m (4.3.a) and the hamiltonian H(s) = 1 m m=1 H m (4.3.b) Now the question is: what value of s is associated to this hamiltonian? Using the minimal couling to describe the interaction of a article of charge e with an electromagnetic field A, one makes the substitution i i ea i 9

11 Eq.(4.3.b), together with (3.1), gives H(s) = H 0 e m 1 1 S 1 F 1 (4.4) where H 0 = 1 m ( x+ y+e (A x+a y)) and F 1 = A y x A x is the magnetic comonent y of the Maxwell field. Hence, the question is answered and s = 1 because we have exhibited the couling between the magnetic moment of a sin 1 and the magnetic field through the minimal couling. In other words, one can state that the hamiltonian (4.3.b) describes a article of sin s = 1 where is the order of the aragrassmann algebra. Note, so far, in this descrition, that the value of the gyromagnetic ratio ( at the classical level) is connected with the value of the sin of the article for any sin for g = 1 s. Of course, the hamiltonian (4.3.b) for an arbitrary sin s = 1 can be rewritten in a form analogous to (.10) H(s) = 1 4ms s k=1 ( ) k s+1 + ( ) s+1 k s+1 (4.5) and when s = 1, one gets the Pauli equation with the form (4.1). So, in this sense, (4.5) is the generalization of the Pauli equation for a higher sin in a two dimensional geometric sace. As a remark, note that (4.3.a) is the binomial exansion of and the sum over m gives H m = [ (Q + m) 1 +Q m ], H(s) = 1 4ms s m=1 [ (Q + m ) s +Q m ] s+1 (4.6) This shows that the roblems encountered with a sin s seem to be related with olynomials of degree s + 1. In this context, even if we ut no hysics behind this mathematical evidence, it has something intriguing. The roerties of each term H m (s) = 1 [ (Q + 4ms m ) s ] s+1 +Q m m = 1,...,s under a lanar rotation are obviously different of those of the sum. Does it have any hysical significance for the sin s article as a comound object? 10

12 The Hamiltonian (4.4) is equivalent to (3.6.b) through the suitable change of variable[17] (in the case = 3 ) π 1 = i x, π W with W = F 1 x + β. So, our construction, based on matricial reresentation of th root of oerators, with the constraints on W, is clearly related to arasueralgebras and corresonds to the two dimensional arasuersymmetric model of[17]. All this is basically connected to Clifford algebras of olynomials or n-exterior algebras [15,18]. On the other hand, the fact that a sin s article is connected to a s + 1 aragrassmann algebra is not really surrising because it underlies the Z s+1 structure that aragrassmann algebras ossess. In addition arafermions are nothing else but reresentations of SU() [8] exactly as the sin is. Notice that an attemt of extension of the Dirac rocedure of linearization has been investigated, in connection with sums of monomials[3]. V Remarks and conclusion Starting from the usual oscillator model, we have defined the notion of fractional ste ladder oerator, in a matricial reresentation. The 1 -ste ladder oerator corresonds to a value s = 1 of the sin and the corresonding Fock sace is a tensor roduct of the oscillator basis with an irreducible sin basis of dimension s+1. We have shown, in this formalism, that aragrassmann oerators aear in a hidden way and that we get an extension of SQM. Our method using matricial realization of roots of oerators has been related to arasueralgebras. All this can be understood, in fact, in the frame of Clifford algebras of olynomials. In this formalism, it turns out that a article with sin s has an hamiltonian which is the sum of s terms of degree s+1. The question is oen whether this fact reflects some dee connection between the value of the sin of the described article and the degree of the olynomial which has to be used in the hamiltonian. Of course, our formalism of 1 -ste ladder oerators can be alied in all the quantum models where raising and lowering oerators aear. It is exected that it can aly in atomic and nuclear sectroscoy, in interactions with magnetic field. (+1). The next ste is now to generalize our results to a sace of dimension higher than 11

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