On the algebra of symmetries of Laplace and Dirac operators

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by  Hendrik De Bie

On the algebra of symmetries of Laplace and Dirac operators
Hendrik De Bie1, Roy Oste2, Joris Van der Jeugt3
1Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent
University, Krijgslaan 281-S8, 9000 Gent, Belgium
2,3Department of Applied Mathematics, Computer Science and Statistics, Faculty of Sciences,
Ghent University, Krijgslaan 281-S9, 9000 Gent, Belgium.
7 E-mail: [email protected];[email protected]; [email protected]
1
0 Abstract
2
We consider a generalization of the classical Laplace operator, which includes the Laplace-

r Dunkl operator defined in terms of the differential-difference operators associated with finite
p
A reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of
symmetries commuting with it, which are generalized angular momentum operators, and we

4 present the algebraic relations for the symmetry algebra. In this context, the generalizedDirac

operator is then defined as a square root of our Laplace-like operator. We explicitly determine
]
h a family of graded operators which commute or anti-commute with our Dirac-like operator
p depending on their degree. The algebra generated by these symmetry operators is shown to
-
be a generalizationof the standard angular momentum algebra and the recently defined higher
h
t rank Bannai-Ito algebra.
a
m
Keywords: Laplace operator; Dirac operator; Dunkl operator; Symmetry algebra; Bannai-Ito
[ algebra

2
v
1 Introduction
0
6
7
The study of solutions of the Laplace equation or of the Dirac equation, in any context or setting,
5
0 is a major topic of investigation. For that purpose,a crucial role is played by the symmetries of the
. Laplaceoperator∆oroftheDiracoperatorD,i.e.operatorscommutingwith∆or(anti)commuting
1
0 with D. The symmetries involved and the algebras they generate lead to topics such as separation
7 of variables and special functions. Without claiming completeness we refer the reader to [2,3,6,7,
1
10,18].
:
v
For this paper, the context we have in mind is that of Dunkl derivatives [9,23], i.e. where the
i
X ordinaryderivative ∂ isreplacedbyaDunklderivative intheexpressionoftheLaplaceorDirac
∂xi Di
r operator. One often refers to these operators as the Laplace-Dunkl and the Dirac-Dunkl operator.
a
The chief purpose of this paper is to determine the symmetries of the Laplace-Dunkl operator and
of the Dirac-Dunkl operator, and moreover study the algebra generated by these symmetries.
Intheprocessofthisinvestigation, itoccurredtousthatitisadvantageoustotreatthisproblem
in a more general context, which we shall describe here in the introduction. For this purpose, let
us first turn to a standard topic in quantum mechanics: the description of the N-dimensional
(isotropic) harmonic oscillator. TheHamiltonian Hˆ of this oscillator (with the common convention
m = ω = ~= 1 for mass, frequency and the reduced Planck constant) is given by
N N
1 1
Hˆ = pˆ2+ xˆ2. (1.1)
2 j 2 j
j=1 j=1
X X
In canonical quantum mechanics, the coordinate operators xˆ and momentum operators pˆ are
j j
required to be (essentially self-adjoint) operators satisfying the canonical commutation relations
[xˆ ,xˆ ]= 0, [pˆ,pˆ ]= 0, [xˆ ,pˆ ] = iδ . (1.2)
i j i j i j ij
1
So in the “coordinate representation”, where xˆ is represented by multiplication with the variable
j
x , the operator pˆ is represented by pˆ = i ∂ .
j j j − ∂xj
Because the canonical commutation relations are sometimes considered as “unphysical” or “im-
posed without a physical motivation”, more fundamental ways of quantization have been the topic
of various research fields (such as geometrical quantization). One of the pioneers of a more fun-
damental quantization procedure was Wigner, who introduced a method that later became known
as “Wigner quantization” [20,21,24,26,27]. Briefly said, in Wigner quantization one preserves all
axioms of quantum mechanics, except that the canonical commutation relations are replaced by
a more fundamental principle: the compatibility of the (classical) Hamilton equations with the
Heisenberg equations of motion. Concretely, these compatibility conditions read
[Hˆ,xˆ ] = ipˆ , [Hˆ,pˆ ] = ixˆ (j = 1,...,N). (1.3)
j j j j
−
Thus for the quantum oscillator, one keeps the Hamiltonian (1.1), but replaces the relations (1.2)
by (1.3). When the canonical commutation relations (1.2) hold, the compatibility relations (1.3)
are automatically valid (this is a version of the Ehrenfest theorem), but not vice versa. Hence
Wigner quantization is a generalization of canonical quantization, and canonical quantization is
justonepossiblesolution of Wigner quantization. Note that in Wigner quantization the coordinate
operators xˆ (and the momentum operators) in general do not commute, so this is of particular
j
significance in the field of non-commutative quantum mechanics.
In a mathematical context, as in this paper, one usually replaces the physical momentum
operator components pˆ by operators p = ipˆ , and one also denotes the coordinate operators xˆ
j j j j
by x . Then the operator H takes the form
j
N N
1 1
H = p2+ x2. (1.4)
−2 j 2 j
j=1 j=1
X X
So in the canonical case, where x stands for multiplication by the variable x , p is just the
j j j
derivative ∂ , and the first term of H is (up to a factor 1/2) equal to the Laplace operator
∂xj −
N p2 = N ∂2 = ∆. In the more general case, the compatibility conditions (1.3) read
j=1 j j=1 ∂x2
j
P P
[H,x ] = p , [H,p ]= x (j = 1,...,N). (1.5)
j j j j
− −
We are now in a position to describe our problem in a general framework:
Given N commuting operators x ,...,x and N commuting operators p ,...,p , con-
1 N 1 N
N N
1 1
sider the operator H = p2 + x2, and suppose that the compatibility con-
−2 j 2 j
j=1 j=1
X X
ditions (1.5) hold. Classify the symmetries of the generalized Laplace operator, i.e.
classify the operators that commute with N p2.
j=1 j
In other words, we are given N operators x1,..P.,xN and N operators p1,...,pN that satisfy
[x ,x ] = 0, [p ,p ] = 0, (1.6)
i j i j
N N
1 1
p2,x = p , x2,p = x . (1.7)
2 i j j 2 i j − j
(cid:20) i=1 (cid:21) (cid:20) i=1 (cid:21)
X X
Under these conditions, the first problem is: determine the operators that commute with the
generalized Laplace operator
N
∆ = p2. (1.8)
i
i=1
X
2
Our two major examples of systems satisfying (1.6) and (1.7) are the “canonical case” and the
“Dunkl case”.
For the first example, x is just the multiplication by the variable x , and p is the derivative
i i i
with respect to x : p = ∂ . Clearly, these operators satisfy (1.6) and (1.7), and the operator ∆
i i ∂xi
in (1.8) coincides with the classical Laplace operator.
For the second example, x is again the multiplication by the variable x , but p is the Dunkl
i i i
derivative p = , which is a certain differential-difference operator with an underlying reflection
i i
D
group determined by a root system (a precise definition follows later in this paper). Conditions
(1.6) still hold: the commutativity of the operators x is trivial, but the commutativity of the
i
operators p is far from trivial [9,11]. Following [9], also the conditions (1.7) are valid in the Dunkl
i
case. The operator ∆ in (1.8) now takes the form N 2 and is known as the Dunkl Laplacian
i=1Di
or the Laplace-Dunkl operator. By the way, it is no surprise that the operators x and do not
i j
P D
satisfy the canonical commutation relations. It is, however, very surprising that they satisfy the
more general Wigner quantization relations (for a Hamiltonian of oscillator type).
So the solution of the first problem in the general context will in particular lead to the deter-
mination of symmetries of the Laplace-Dunkl operator.
Since we are dealing with these operators in an algebraic context, it is worthwhile to move to
a closely related operator, the Dirac operator. For this purpose, consider a set of N generators of
a Clifford algebra, i.e. N elements e satisfying
i
e ,e = e e +e e = ǫ2δ
i j i j j i ij
{ }
where ǫ is +1 or 1. The generators e are supposed to commute with the general operators x
i j
−
and p . Under the general conditions (1.6) and (1.7), the second problem is now: determine the
j
operators that commute (or anti-commute) with the generalized Dirac operator
N
D = e p . (1.9)
i i
i=1
X
Obviously, this is a refinement of the first problem, since D2 = ǫ∆.
For our two major examples, in the canonical case the operator (1.9) is just the classical Dirac
operator; in the Dunkl case, the operator (1.9) is known as the Dirac-Dunkl operator.
In the present paper we solve both problems in the general framework (1.6)–(1.7), and even go
beyond it by determining the algebraic relations satisfied by the symmetries. In section 2 we con-
siderthegeneralizedLaplaceoperator∆anddetermineallsymmetries,i.e.alloperatorscommuting
with ∆ (Theorem 2.3). Next, in Theorem 2.5 the algebraic relations satisfied by these symmetries
are established. For the generalized Dirac operator D, the determination of the symmetries is com-
putationally far more involved. In section 3, Theorem 3.7 classifies essentially all operators that
commute or anti-commute with D. In the following subsections, we derive the quadratic relations
satisfied by the symmetries of the Dirac operator. The computations of these relations are very
intricate, and involve subtle techniques. Fortunately, there is a case to compare with. For the
Dunkl case, in which the underlyingreflection group is the simplest possible example (namely ZN),
2
the symmetries and their algebraic relations have been determined in [6,7] and give rise to the
so-called (higher rank) Bannai-Ito algebra. Our results can be considered as an extension of these
relations to an arbitrary underlying reflection group, in fact in an even more general context.
2 Symmetries of Laplace operators
We start by formally describing the operator algebra that will contain the desired symmetries of a
generalized Laplace operator (1.8), as brought up in the introduction.
3
Definition 2.1. We define the algebra to be the unital (over the field R or C) associative algebra
A
generated by the 2N elements x ,...,x and p ,...,p subject to the following relations:
1 N 1 N
[x ,x ] = 0, [p ,p ] = 0,
i j i j
N N
1 1
p2,x = p , x2,p = x .
2 i j j 2 i j − j
(cid:20) i=1 (cid:21) (cid:20) i=1 (cid:21)
X X
Note that an immediate consequence of the relations of is
A
[x ,p ]= [x , [H,x ]] = [[x ,H],x ] = [p ,x ]= [x ,p ],
i j i j i j i j j i
− − −
where H is given by (1.4). This reciprocity
[x ,p ]= [x ,p ] (2.1)
i j j i
will be useful for many ensuing calculations, starting with the following theorem.
Theorem 2.2. The algebra contains a copy of the Lie algebra sl(2) generated by the elements
A
x 2 1 N ∆ 1 N 1 N
| | = x2, = p2, E = p ,x , (2.2)
2 2 i − 2 −2 i 2 { i i}
i=1 i=1 i=1
X X X
satisfying the relations
x 2 ∆ x 2 ∆
E,| | = x 2, E, = ∆, | | , = E.
2 | | − 2 − 2 − 2
h i h i h i
Proof. By direct computation we have
N N N
1 1 1 1
[∆, x 2] = [∆,x2]= [∆,x ],x = p ,x .
4 | | 4 i 4 { i i} 2 { i i}
i=1 i=1 i=1
X X X
Using the commutativity of p ,...,p and relation (2.1), we have
1 N
N N N N
1 1
[E,∆] = [ p ,x ,p2] = [ p ,x ,p ],p
2 { i i} j 2 { i i} j j
i=1 j=1 i=1 j=1
XX XX(cid:8) (cid:9)
N N
1
= p (x p p x )+(x p p x )p ,p
i i j j i i j j i i j
2 − −
i=1 j=1
XX(cid:8)(cid:0) (cid:1) (cid:9)
N N
1
= p (x p p x )+(x p p x )p ,p
i j i i j j i i j i j
2 − −
i=1 j=1
XX(cid:8)(cid:0) (cid:1) (cid:9)
N N N
1
= p2,x ,p = p ,p = 2∆.
− 2 i j j − { j j} −
j=1(cid:26)(cid:20)i=1 (cid:21) (cid:27) j=1
X X X
In the same manner, using now the commutativity of x ,...,x , we find [E, x 2] = 2x 2.
1 N
| | | |
In the spirit of Howe duality [16,17], our objective is to determine the subalgebra of which
A
commutes with the Lie algebra sl(2) realized by ∆ and x 2 as appearing in Theorem 2.2. As
| |
mentioned in the introduction, the element ∆ corresponds to a generalized version of the Laplace
operator, which reduces to the classical Laplace operator for a specific choice of the elements
p ,...,p . Inthe(Euclidean)coordinaterepresentation, x 2 ofcourserepresentsthenormsquared.
1 N
| |
4
2.1 Symmetries
As p ,...,p are commuting operators, by definition they also commute with ∆. However, in
1 N
general they are not symmetries of x 2. An immediate first example of an operator which does
| |
commutewithboth∆and x 2 isgivenbytheCasimiroperator(intheuniversalenvelopingalgebra)
| |
of their sl(2) realization
Ω = E2 2E x 2∆ (sl(2)) . (2.3)
− −| | ∈ U ⊂ A
Note that this operator is of the same order in both x ,...,x and p ,...,p as it has to commute
1 N 1 N
with both ∆ and x 2. More precisely it is of fourth order in the generators of , being quadratic in
| | A
x ,...,x andquadraticinp ,...,p . Wenowsetouttoconsiderthemostelementarysymmetries,
1 N 1 N
those which are of second order in the generators of .
A
Theorem 2.3. In the algebra , the elements which are quadratic in the generators x ,...,x
1 N
A
and p ,...,p , and which commute with ∆ and x 2 are spanned by
1 N
| |
L = x p x p , C = [p ,x ]= p x x p (i,j 1,...,N ). (2.4)
ij i j j i ij i j i j j i
− − ∈ { }
Notethatwheni = j wehaveL = 0,whileC = [p ,x ]doesnotnecessarilyvanish. Moreover,
ii ii i i
as L = L , every symmetry L is up to a sign equal to one of the N(N 1)/2 symmetries
ij ji ij
− −
L 1 i < j N . By relation (2.1), C = C and thus we have N(N +1)/2 symmetries
ij ij ji
{ | ≤ ≤ }
C 1 i j N . In total, this gives N2 generically distinct symmetries.
ij
{ | ≤ ≤ ≤ }
Proof. It is trivial that there are no nonzero quadratic elements in x ,...,x that commute with
1 N
∆, and no nonzero quadratic elements in p ,...,p that commute with x 2. Now, as ∆ commutes
1 N
| |
with p ,...,p and using condition (1.7), we have for i,j 1,...,N
1 N
∈ { }
[∆,x p x p ]= x [∆,p ]+[∆,x ]p x [∆,p ] [∆,x ]p = 2p p 2p p = 0.
i j j i i j i j j i j i i j j i
− − − −
In the same manner we have [∆,p x x p ] = 0. The relations for x 2 follow similarly.
i j j i
− | |
2.2 Symmetry algebra
For the following results, wemake explicit useof thesymmetry C = [p ,x ] beingsymmetricin its
ij i j
two indices, by relation (2.1). This is the case for p corresponding to classical partial derivatives,
i
but also for their generalization in the form of Dunkl operators. We will return in detail to these
examples in section 2.3. Another consequence of relation (2.1) pertains to the form of the other
symmetries of Theorem 2.3. By means of x p p x = x p p x we readily observe that
i j j i j i i j
− −
L = x p x p = p x p x . (2.5)
ij i j j i j i i j
− −
Given these symmetry properties, the symmetries of Theorem 2.3 generate an algebraic structure
within whose relations we present after the following lemma.
A
Lemma 2.4. In the algebra , the symmetries (2.4) satisfy the following relations for all i,j,k
A ∈
1,...,N
{ }
[C ,p ] = [C ,p ], and [C ,x ] = [C ,x ].
ij k kj i ij k kj i
Moreover, we also have
L p +L p +L p =0 = p L +p L +p L ,
ij k ki j jk i k ij j ki i jk
and
x L +x L +x L =0 = L x +L x +L x .
k ij j ki i jk ij k ki j jk i
5
Proof. For the first relation, writing out the commutators in [p ,x ],p [p ,x ],p we find
k j i i j k
−
(cid:2) (cid:3) (cid:2) (cid:3)
p x p x p p p p x +p x p p x p +x p p +p p x p x p .
k j i j k i i k j i j k i j k j i k k i j k j i
− − − −
We see that all terms cancel due to the mutual commutativity of the operators p ,...,p . The
1 N
other relation of the first line follows in the same way.
For the other two relations, the identities follow immediately by choosing the appropriate ex-
pressionforL of (2.5)andmakinguseofthecommutativity ofeitherx ,...,x orp ,...,p .
ij 1 N 1 N
Theorem 2.5. In the algebra , the symmetries (2.4) satisfy the following relations for all i,j
A ∈
1,...,N ,
{ }
[L ,L ]= L C +L C +L C +L C (2.6)
ij kl il jk jk il ki lj lj ki
= C L +C L +C L +C L ,
jk il il jk lj ki ki lj
and
L ,L + L ,L + L ,L = 0, (2.7)
ij kl ki jl jk il
{ } { } { }
and
[L ,C ]+[L ,C ]+[L ,C ]= 0. (2.8)
ij kl ki jl jk il
Proof. We will prove the first line of the first relation, i.e. (2.6), the second line follows in a similar
manner. We have
[x p x p ,x p x p ] = [x p ,x p ] [x p ,x p ] [x p ,x p ]+[x p ,x p ]
i j j i k l l k i j k l i j l k j i k l j i l k
− − − −
= x [p ,x ]p +x [x ,p ]p x [p ,x ]p x [x ,p ]p
i j k l k i l j i j l k l i k j
− −
x [p ,x ]p x [x ,p ]p +x [p ,x ]p +x [x ,p ]p
j i k l k j l i j i l k l j k i
− −
= x C p x C p x C p +x C p
i jk l k li j i jl k l ki j
− −
x C p +x C p +x C p x C p .
j ik l k lj i j il k l kj i
− −
Swapping all operators p with C , we find
l jk
[L ,L ]= x p C +x [C ,p ] x p C x [C ,p ] x p C x [C ,p ]
ij kl i l jk i jk l k j li k li j i k jl i jl k
− − − −
+x p C +x [C ,p ] x p C x [C ,p ]+x p C
l j ki l ki j j l ik j ik l k i lj
− −
+x [C ,p ]+x p C +x [C ,p ] x p C x [C ,p ]
k lj i j k il j il k l i kj l kj i
− −
= x p C x p C x p C +x p C x p C +x p C +x p C x p C
i l jk k j li i k jl l j ki j l ik k i lj j k il l i kj
− − − −
+x [C ,p ] [C ,p ] +x [C ,p ]+[C ,p ]
i jk l jl k k il j lj i
− −
x [C ,p ]+[C ,p ] x [C ,p ] [C ,p ]
l(cid:0) ki j kj (cid:1)i (cid:0) j ik l il k(cid:1)
− − − −
= L C +L C +L C +L C ,
il j(cid:0)k jk il ki lj (cid:1) lj k(cid:0)i (cid:1)
where we used C = C and Lemma 2.4.
jk kj
The identities (2.7) and (2.8) follow by making explicit use of both expressions of (2.5) for L .
ij
For the left-hand side of (2.7) we have
L L +L L +L L +L L +L L +L L
ij kl kl ij ki jl jl ki jk il il jk
= (x p x p )(p x p x )+(x p x p )(p x p x )+(x p x p )(p x p x )
i j j i l k k l k l l k j i i j k i i k l j j l
− − − − − −
+(x p x p )(p x p x )+(x p x p )(p x p x )+(x p x p )(p x p x ),
j l l j i k k i j k k j l i i l i l l i k j j k
− − − − − −
where again one observes that all terms vanish due to the commutativity of p ,...,p .
1 N
6
Working out the commutators, the left-hand side of (2.8) becomes
L [p ,x ] [p ,x ]L +L [p ,x ] [p ,x ]L +L [p ,x ] [p ,x ]L
ij l k l k ij ki l j l j ki jk l i l i jk
− − −
= L p x p x L +L p x p x L +L p x p x L
ij l k l k ij ki l j l j ki jk l i l i jk
− − −
L x p +x p L L x p +x p L L x p +x p L .
ij k l k l ij ki j l j l ki jk i l i l jk
− − −
Hence, plugging in suitable choices for the symmetries L , this becomes
ij
(x p x p )p x p x (x p x p )+(x p x p )p x p x (x p x p )
i j j i l k l k i j j i k i i k l j l j k i i k
− − − − − −
+(x p x p )p x p x (x p x p ) (p x p x )x p +x p (p x p x )
j k k j l i l i j k k j j i i j k l k l j i i j
− − − − − −
(p x p x )x p +x p (p x p x ) (p x p x )x p +x p (p x p x ).
i k k i j l j l i k k i k j j k i l i l k j j k
− − − − − −
One observes that all terms vanish due to the commutativity of x ,...,x and p ,...,p respec-
1 N 1 N
tively.
2.3 Examples
Example 2.1. As a first example, we consider N mutually commuting variables x ,...,x ,
1 N
doubling as operators acting on functions by left multiplication with the respective variable and
p being just the derivative ∂/∂x for j 1,...,N . In this case obviously p ,...,p mutually
j j 1 N
∈ { }
commute and the operators of interest are
N ∂2 N 1 1
∆ = , x 2 = x2, H = ∆+ x 2,
∂x2 | | i −2 2| |
i=1 i i=1
X X
which satisfy
1 ∂ 1
[∆,x ] = = p , x 2,p = x .
i i i i
2 ∂x 2 | | −
i
By Theorem 2.3, we have the following symmetries(cid:2): (cid:3)
∂ ∂ 1 if i = j
L = x x , C = δ =
ij i j ij ij
∂xj − ∂xi (0 if i = j.
6
WhileC isascalarforeveryi,j,theL symmetriesarethestandardangularmomentumoperators
ij ij
whose symmetry algebra is the Lie algebra so(N):
[L ,L ]= L δ +L δ +L δ +L δ .
ij kl il jk jk il ki lj lj ik
This is in accordance with Theorem 2.5 as in this case evidently C = C .
ij ji
Note that ∆ and x 2 are also invariant under O(N), the group of orthogonal transformations
| |
on RN, but these transformations are not contained in the algebra .
A
Example 2.2. A more intriguing example is given by a generalization of partial derivatives to
differential-difference operators associated to a Coxeter or Weyl group W. Let R be a (reduced)
root system and k a multiplicity function which is invariant under the natural action of the Weyl
group W consisting of all reflections associated to R,
σ (x)= x 2 x,α α/ α 2, α R,x RN.
α
− h i k k ∈ ∈
For ξ RN, the Dunkl operator [9,23] is defined as
∈
∂ f(x) f(σ (x))
α
f(x):= f(x)+ k(α) − α,ξ ,
ξ
D ∂ξ α,x h i
αX∈R+ h i
7
where the summation is taken over all roots in R , a fixed positive subsystem of R. For a fixed
+
root system and function k, the Dunkl operators associated to any two vectors commute, see [9].
Hence, they form potential candidates for the operators p ,...,p satisfying condition (1.6). The
1 N
operator of interest is the Laplace-Dunkl operator ∆ , which can be written as
k
N
∆ = ( )2
k Dξi
i=1
X
for any orthonormal basis ξ ,...,ξ of RN. For the orthonormal basis associated to the coordi-
1 N
{ }
nates x ,...,x we use the notation
1 N
∂ f(x) f(σ (x))
α
f(x):= f(x)+ k(α) − α i 1,...,N . (2.9)
i i
D ∂x α,x ∈ { }
i
αX∈R+ h i
For our purpose, let x again stand for multiplication by the variable x and take now p =
j j j j
D
for j 1,...,N . Besides condition (1.6), condition (1.7) is also satisfied (see for instance [9,23]).
∈ { }
We note that the sl(2) relations in this context were already obtained by [14].
By Theorem 2.3 we have as symmetries, on the one hand, the Dunkl version of the angular
momentum operators
L = x x .
ij i j j i
D − D
On the other hand, the symmetries
C = [ ,x ]= δ + 2k(α)α α σ
ij i j ij i j α
D
αX∈R+
consist of linear combinations of the reflections in the Weyl group, with coefficients determined by
the multiplicity function k and the roots of the root system. This is of course in agreement with
∆ being W-invariant [23]. The Weyl group is a subgroup of O(N), and in this case the algebra
k
A
does contain these reflections in W.
Note that indeed C = C , in accordance with relation (2.1). Theorem 2.5 now yields the
ij ji
Dunkl version of the angular momentum algebra for an arbitrary Weyl group or root system:
[L ,L ] = L C +L C +L C +L C
ij kl il jk jk il ki lj lj ki
= L δ +L δ +L δ +L δ + 2k(α) L α α +L α α +L α α +L α α σ .
il jk jk il ki lj lj ki il j k jk i l ki l j lj k i α
αX∈R+ (cid:0) (cid:1)
This relation states the interaction of the L symmetries amongst one another. The interaction
ij
between the symmetries C is governed by the group multiplication of the Weyl group, while the
kl
relations for the symmetries L and C follow from the action of a reflection σ for a root α:
ij kl α
σ = σ , hence σ L = L σ ,
αDξ Dσα(ξ) α α ij σα(ξi)σα(ξj) α
where ξ ,...,ξ is the orthonormal basis associated to the coordinates x ,...,x . This allows
1 N 1 N
{ }
us to interchange any two symmetries of the form L and C .
ij kl
Specific cases of this result have been considered before, namely for W = (Z )3 in [13] and
2
W = S in [12].
N
8
3 Symmetries of Dirac operators
We now turn to a closely related operator of the generalized Laplace operator considered in the
preceding section, namely the Dirac operator. For an operator of the form (1.8), one can construct
a “square root” by introducing a set of elements e ,...,e which commute with x and p for all
1 N i i
i 1,...,N and which satisfy the following relations
∈{ }
e ,e = e e +e e = ǫ2δ , (3.1)
i j i j j i ij
{ }
where ǫ = 1, or thus for i = j
± 6
(e )2 = ǫ = 1, e e +e e = 0.
i i j j i
±
We use these elements to define the following two operators
N N
D = e p , x = e x ,
i i i i
i=1 i=1
X X
whose squares equal
N N
D2 = ǫ (p )2 = ǫ∆, x2 = ǫ (x )2 =ǫ x 2,
i i
| |
i=1 i=1
X X
bymeansoftheanti-commutation relations (3.1)ofe ,...,e andcondition(1.6). Fortheclassical
1 N
case where p is the ith partial derivative, the operator D is the standard Dirac operator.
i
The elements e ,...,e in fact generate what is known as a Clifford algebra [22], which we will
1 N
denote as = ℓ(RN). A general element in this algebra is a linear combination of products of
C C
e ,...,e . The standard convention is to denote for instance e e e simply as e . Hereto, we
1 N 1 2 3 123
introduce the concept of a ‘list’ for use as index of Clifford numbers.
Definition 3.1. We define a list to indicate a finite sequence of distinct elements of a given set,
in our case the set 1,...,N . For a list A = a a of 1,...,N with 0 n N, we will use
1 n
{ } ··· { } ≤ ≤
the notation
e = e e e . (3.2)
A a1 a2··· an
Remark 3.2. Note that in a list the order matters as the Clifford generators e ,...,e anti-
1 N
commute. Moreover, duplicate elements would cancel out as they square to ǫ = 1, so we consider
±
only lists containing distinct elements. For a set A = a ,...,a 1,...,N , the notation e
1 n A
{ } ⊂ { }
stands for e e e with a < a < < a .
a1 a2··· an 1 2 ··· n
The collection e A 1,...,N forms a basis of the Clifford algebra , where for the
A
{ | ⊂ { }} C
empty set we put e = 1.
∅
Remark 3.3. In general, the square of each individual element e (i 1,...,N ) can indepen-
i
∈ { }
dently be chosen equal to either +1 or 1. This corresponds to an underlying space with arbitrary
−
signature defined by the specified signs. The original Dirac operator was constructed as a square
root of the wave operator by means of the gamma or Dirac matrices which form a matrix realization
of the Clifford algebra for N = 4 with Minkowski signature.
To simplify notations in the following, we have chosen the square of all e (i 1,...,N ) to
i
∈ { }
be equal to ǫ which can be either +1 or 1. One can generalize all results to arbitrary signature by
−
making the appropriate substitutions.
9
In order to consider symmetries of the generalized Dirac operator (1.9) we will work in the
tensor product with the algebra as defined in Definition 2.1. To avoid overloading on
A ⊗ C A
notations, we will omit the tensor symbol when writing down elements of and use regular
⊗ A⊗C
productnotation. Inthisnotation e ,...,e indeedcommutes withx andp foralli 1,...,N .
1 N i i
∈ { }
Akin to the realization of the Lie algebra sl(2) in the algebra given by Theorem 2.2, we have
A
something comparable in this case.
Theorem 3.4. The algebra contains a copy of the Lie superalgebra osp(12) generated by
A⊗C |
the (odd) elements D and x satisfying the relations
x,x = ǫ2x 2 D,D =ǫ2∆ x,D = ǫ2E
{ } | | { } { }
x 2,x = 0 x 2,D = 2D [E,x] = x
| | | | −
[∆,x]= 2x [∆,D] =0 [E,D] = D
(cid:2) (cid:3) (cid:2) (cid:3)
−
and containing as an even subalgebra the Lie algebra sl(2) in the algebra given by Theorem 2.2
A
with relations
x 2 ∆ x 2 ∆
E,| | = x 2 E, = ∆ | | , = E.
2 | | − 2 − 2 − 2
h i h i h i
Proof. The relations follow by straightforward computations. By means of the anti-commutation
relations (3.1) one finds that
N N N N
x,D = x e p e + p e x e
i i j j j j i i
{ }
i=1 j=1 j=1 i=1
X X X X
N
= ǫ(x p +p x )+ (x p p x x p +p x )e e .
i i i i i j j i j i i j i j
− −
i=1 1≤i<j≤N
X X
Looking back at (2.2), the first summation is precisely ǫ2E, while the second summation vanishes
by relation (2.1). Moreover, by relation (2.1) we have
N N N N
1 1
[E,D] = [ p ,x ,p ]e = p ,[x ,p ] e
i i j j i i j j
2 { } 2 { }
i=1 j=1 i=1 j=1
XX XX
N N N N
1 1
= p ,[x ,p ] e = p2,x e = D,
2 { i j i } j −2 i j j −
i=1 j=1 j=1(cid:20)i=1 (cid:21)
XX X X
and in the same manner [E,x]= x.
3.1 Symmetries
We wish to determine symmetries in the algebra for the Dirac operator D which are linear
A⊗C
in both x ,...,x and p ,...,x . Given the Lie superalgebra framework, it is natural to consider
1 N 1 N
operators which either commute or anti-commute with D. Indeed, the Lie superalgebra osp(12)
|
has both a Scasimir and a Casimir element in its universal enveloping algebra [1]. The Scasimir
operator
1
= ([D,x] ǫ) (osp(12)) , (3.3)
S 2 − ∈U | ⊂ A⊗C
anti-commutes with odd elements and commutes with even elements. In the classical case, the
Scasimir operator is up to a constant term equal to the angular Dirac operator Γ, i.e. D restricted
to the sphere. The Scasimir is a symmetry which is linear in both x ,...,x and p ,...,x and
1 N 1 N
S
10