Deforming Maps for Lie Group Covariant Creation and Annihilation Operators

26 Pages·0.26 MB·English

by  Gaetano Fiore

Deforming Maps for Lie Group Covariant
Creation & Annihilation Operators
Gaetano Fiore*
8
9
9 Sektion Physik der Ludwig-Maximilians-Universit¨at Mu¨nchen
1
Theoretische Physik — Lehrstuhl Professor Wess
n
a
Theresienstraße 37, 80333 Mu¨nchen
J

8 Federal Republic of Germany

5
v
5
0
0
0
1
6
9
Abstract
/
g
al Any deformation of a Weyl or Clifford algebra can be realized through
A
-
q a ‘deforming map’, i.e. a formal change of generators in . This is true
A
:
v in particular if is covariant under a Lie algebra g and its deformation is
i A
X induced by some triangular deformation U g of the Hopf algebra Ug. We
h
r
a propose a systematic method to construct all the corresponding deforming
maps, together with thecorrespondingrealizations of theaction of U g. The
h
method is then generalized and explicitly applied to the case that U g is the
h
quantum group sl(2). A preliminary study of the status of deforming
h
U
maps at the representation level shows in particular that ‘deformed’ Fock
representations induced by a compact U g can be interpreted as standard
h
‘undeformed’ Fock representations describing particles with ordinary Bose or
Fermi statistics.
PACS: 02.20.-a, 03.65.Fd, 05.30.-d, 11.30.-j
*EU-fellow,TMRgrantERBFMBICT960921. e-mail: Gaetano.Fiore @ physik.uni-muenchen.de
I Introduction
In recent years the idea of noncocommutative Hopf algebras [1] (in particular quan-
tum groups [2]) as candidates for generalized symmetry transformations in quan-
tum physics has raised an increasing interest. One way to implement this idea in
quantum field theory or condensed matter physics would be to deform the canon-
ical commutation relations (CCR) of some system of mode creators/annihilators,
covariant under the action of a Lie algebra g, in such a way that they become
covariant under the action of a noncocommutative deformation U g (with defor-
h
mation parameter h) of the cocommutative Hopf algebra Ug, as it has been done
e.g. in Ref. [3, 4] for the sl(N) covariant Weyl algebra in N dimensions.
h
U
As a toy model for these deformations one can consider the deformed Weyl
algebra in 1 dimension [5] with generators fulfilling the ‘quantum’ commutation
h
A
relation (QCR)
A˜A˜+ = 1+q2A˜+A˜ (I.1)
with q = eh. When q = 1 the above reduces to the classical Weyl algebra
A
aa+ = 1+a+a. (I.2)
If we define n := a+a, (x) = zx−1 and [6]
z z−1
(n) (n)
A := a q2 A+ := q2a+, (I.3)
s n s n
we find out that A,A+ fulfil the QCR (I.1); hence we can define an algebra homo-
morphism f : [[h]], or “deforming map” (in the terminology of Ref. [7, 8]),
h
A → A
starting from
f(A˜) = A f(A˜) = A+. (I.4)
The RHS(I.3) have to be understood as formal power series in the deformation
parameter h.
We are interested in deformed multidimensional Weyl or Clifford algebras A
h
where the QCR:
1. keep a quadratic structure as in eq. (I.1), so that one can represent the
generators as creation or annihilation operators;
2. are covariant under the action ˜⊲ of U g.
h h
1
More precisely, the generators A˜+,A˜i should transform linearly under the action
i
of ˜⊲ ,
h
x˜⊲ A˜+ = ρ˜j(x)A+ (I.5)
h i i j
x˜⊲ A˜i = ρ˜∨i(x)Aj (I.6)
h j
with ρ˜∨ being the the contragradient representation of the representation ρ˜of U g
h
(ρ˜∨ = ρ˜T S , where S is the antipode of U g and T is the operation of matrix
h h h
◦
transposition).
In this work we essentially stick to the case that U g is triangular; we treat
h
the general quasitriangular case in Ref. [9]. In the former case one can show easily
that, for arbitrary ρ˜, U g-covariant QCR are given by
h
A˜iA˜j = Rij A˜uA˜v (I.7)
± vu
A˜+A˜+ = RvuA˜+A˜+ (I.8)
i j ± ij u v
A˜iA˜+ = δi1 RuiA˜+A˜v; (I.9)
j j A ± jv u
here the sign refers to the Weyl/Clifford case respectively, and R is the corre-
±
sponding ‘R-matrix’ of U g1.
h
A is a left-module algebra of U g: the ‘quantum’ action ˜⊲ is extended to
h h h
products of the generators as a left-module algebra map ˜⊲ : U
h h h h
× A → A
(i.e. consistently with the QCR) using the coproduct ∆ (x) = xµ xµ of
h µ (¯1) ⊗ (¯2)
Uhg, P
x˜⊲ (a b) = (xµ ˜⊲ a) (xµ ˜⊲ b), (I.10)
h · (¯1) h · (¯2) h
µ
X
because the (I.8) are covariant under (i.e. compatible with) ˜⊲ .
h
The existence of deforming maps for arbitrary (i.e. not necessarily of the kind
described above) deformations of Weyl (or Clifford) algebras is a consequence [10]
of a theorem [11] asserting the triviality of the cohomology groups of the latter
(see Ref. [12, 9] for an effective and concise presentations of these results. See also
Ref. [13], where the problem of stability of quantum mechanics under deformations
was addressed for the first time.). However, no general method for their explicit
construction is available. Actually, using cohomological arguments, one can also
easily show that deforming maps are unique up to a inner automorphism,
f f := αf( )α−1 α = 1 +O(h); (I.11)
α A
→ ·
1One just has to note that R=ρ˜ ρ˜ , where is the universal triangular structure of H ,
h
⊗ R R
and that τ ∆ (x)= ∆ (x) −1 (τ denotes the flip operator).
h h
◦ R R
2
therefore it is enough to construct one to find all of them.
In this work we present a general method which allows, given a triangular Hopf
algebra U g andany U g-covariant deformed Weyl or Cliffordalgebra, to explicitly
h h
construct the corresponding deforming maps f and the corresponding realizations
⊲ of ˜⊲ [ ⊲ is defined by ⊲ := (id f) ˜⊲ (id f−1)]. In a first attempt
h h h h h
⊗ ◦ ◦ ⊗
to generalize our construction procedure to quasitriangular U g, we also generalize
h
the constrution to the case that U g is the quantum group sl(2) and ρ˜ is its
h h
U
fundamental representation. Finally we investigate on the status of deforming maps
at the representation-theoretic level.
Theconstructionmethodisbased(Sect. II.1)onuseoftheDrinfel’d-Reshetikhin
twist [14, 15], intertwining between the coproducts of Ug and U g, and on the
h
F
fact that within [[h]] one can realize both the action ⊲ of U g (Section II.1) and
h
A
the action ⊲ in an ‘adjoint-like’ way. We show first (Section III) that can be
h
F
used in a universal way to construct, within , U g-tensors out of Ug-tensors,
h h
A
and in particular out of a+,ai objects A+,Ai that transform under ⊲ as in formula
i i h
(I.6). Then (Section IV) we verify that the objects A+,Aj really satisfy the QCR
i
(I.8). In Section V we generalize our construction (by means of the Drinfel’d twist
[16]) to the case of deformed Weyl & Clifford algebras with generators belonging to
the fundamental representation of the quantum group sl(2); the deforming map
h
U
is again completely explicit thanks to the semiuniversal expression [8] for . We
F
compare our deforming map with the one previously found in Ref. [17]. At the
representation-theoretic level it would be natural to interpret deforming maps as
“operator maps”, in other words as intertwiners between the representations of
A
and . However we have to expect that, in the role of intertwiners, deforming
h
A
maps may become singular at h = 0, because the representation theories of ,
A
are in general rather different. In Section VI we show that there is always
h
A
a -representation of which is intertwined by f with the Fock representation
h
∗ A
of ; this allows to interpret A˜i,A˜+ as ‘composite’ operators on a classical Fock
i
A
space describing ordinary Bosons and Fermions. We also explicitly show that f−1
α
is ill-defined as an intertwiner from the remaining (if any) unitarily inequivalent
-representations of .
h
∗ A
Onthebasisoftheaboveresult weconcludethatalsoatthequantization-of-field
level noncommutative Hopf algebra symmetries are not necessarily incompatible
with Bose or Fermi statistics (contrary to what is often claimed). We arrived at
3
the same conclusion at the first-quantization level in Ref. [18, 19], where the initial
motiation for the present work has originated. The connection between the two
approaches through second quantization will be described elsewhere.
II Preliminaries and notation
II.1 Twisting groups into quantum groups
Let H = (Ug,m,∆,ε,S) be the cocommutative Hopf algebra associated to the
universal enveloping (UE) algebra Ug of a Lie algebra g. The symbol m denotes
themultiplication(inthesequel itwillbedroppedintheobvious waym(a b) ab,
⊗ ≡
unlessexplicitlyrequired),whereas∆,ε,S thecomultiplication,counitandantipode
respectively.
Let Ug[[h]] Ug[[h]] (we will write = (1) (2), in a Sweedler’s
F ∈ ⊗ F F ⊗ F
(1) (2)
notation with upper indices; in the RHS a sum of many terms is
i i i
F ⊗ F
implicitly understood) be a ‘twist’, i.e. an elemenPt satisfying the relations
(ε id) = 1 = (id ε) (II.1)
⊗ F ⊗ F
= 1 1+O(h) (II.2)
F ⊗
(h C is the ‘deformation parameter’, and 1 the unit in Ug; from the second
∈
condition it follows that is invertible as a power series). It is well known [14] that
F
if also satisfies the relation
F
( 1)[(∆ id)( )] = (1 )[(id ∆)( ), (II.3)
F ⊗ ⊗ F ⊗F ⊗ F
and (U g,m ) is an algebra isomorphic to Ug[[h]] with isomorphism, say, ϕ :
h h h
U g Ug[[h]] [in particular, if U g = Ug[[h]]] and ϕ = id (mod h), or even
h h h
→
ϕ = id], then one can construct a triangular non-cocommutative Hopf algebra
h
H = (U g,m ,∆ ,ε ,S , ) having an isomorphic algebra structure [m = ϕ−1
h h h h h h R h h ◦
m (ϕ ϕ )], an isomorphic counit ε := ε ϕ−1, comultiplication and antipode
◦ h ⊗ h h ◦ h
defined by
∆ (a) = (ϕ−1 ϕ−1) ∆[ϕ (a)] −1 , S (a) = ϕ−1 γ−1S[ϕ (a)]γ ,
h h ⊗ h {F h F } h h { h }
(II.4)
where
γ := S −1(1) −1(2), γ−1 = (1) S (2), (II.5)
F ·F F · F
4
and (triangular) universal R-matrix
:= [ϕ−1 ϕ−1]( −1), := (2) (1). (II.6)
R h ⊗ h F21F F21 F ⊗F
Condition (II.3) ensures that ∆ is coassociative as ∆. The inverse of S is given
h h
by S−1(a) = ϕ−1 γ′S[ϕ (a)]γ′−1 , where
h h { h }
γ′ := (2) S (1) γ′−1 = S −1(2) −1(1); (II.7)
F · F F ·F
γ−1γ′ Centre(Ug), and Sγ = γ′−1.
∈
Conversely, given a h-deformation H = (U g,m ,∆ ,ε ,S , ) of H in the
h h h h h h
R
form of a triangular Hopf algebra, one can find [14] and an isomorphism ϕ : U
h h
→
Ug[[h]] an invertible satisfying conditions (II.1), (II.2), (II.3) such that H can
h
F
be obtained from H through formulae (II.4),(II.5),(II.7).
Examples of ’s satisfying conditions (II.3), (II.1), (II.2) are provided e.g. by
F
the socalled ‘Reshetikhin twists’ [15]
:= ehωijhi⊗hj, (II.8)
F
where h is a basis of the Cartan subalgebra of gand ω = ω C. A less
i ij ji
{ } − ∈
obvious example is for instance the ‘Jordanian’ deformation of Ref. [20].
A similar result to the above holds for genuine quantum groups. A well-known
theorem by Drinfel’d, Proposition 3.16 in Ref. [16] proves, for any quasitriangular
deformation H = (U g ,m ,∆ ,ε ,S , ) [2, 21] of Ug, with ga simple finite-
h h h h h h
R
dimensional Lie algebra, the existence of an algebra isomorphism ϕ : U g
h h
→
Ug[[h]] andan invertible satisfying condition (II.1) such that H can be obtained
h
F
from H through formulae (II.4),(II.5),(II.7), as well, after identifying h = lnq. This
does not satisfy condition (II.18), however the (nontrivial) coassociator
F
φ := [(∆ id)( −1)]( −1 1)(1 )[(id ∆)( ) (II.9)
⊗ F F ⊗ ⊗F ⊗ F
still commutes with ∆(2)(Ug),
[φ,∆(2)(Ug)] = 0, (II.10)
thus explaining why ∆ is coassociative in this case, too. The corresponding uni-
h
versal (quasitriangular) R-matrix is related to by
R F
R = [ϕ−h1 ⊗ϕ−h1](F21q2tF−1), (II.11)
5
wheret := ∆( ) 1 1isthecanonicalinvariantelementinUg Ug ( isthe
C − ⊗C−C⊗ ⊗ C
quadratic Casimir). The twist is defined (and unique) up to the transformation
F
T, (II.12)
F → F
where T is a g-invariant [i.e. commuting with ∆(Ug)] element of Ug[[h]]⊗2 such
that
T = 1 1+O(h), (ε id)T = 1 = (id ε)T. (II.13)
⊗ ⊗ ⊗
A function
T = T (1 , 1,∆( )) (II.14)
i i i
⊗C C ⊗ C
of the Casimirs Ug of Ug and of their coproducts clearly is g-invariant.
i
C ∈
In general, as a consequence of the existence of an isomorphism ϕ , representa-
h
tions ρ˜,ρ of deformed and undeformed algebrae are in one-to-one correspondence
(except for special values of h making it singular) through
ρ = ρ˜ ϕ . (II.15)
h
◦
A special case of interest is when Ug is a -Hopf algebra and is unitary,
∗ F
∗⊗∗ = −1; (II.16)
F F
note that in this case
γ′ = γ∗. (II.17)
One can show [22] that can always be made unitary if g is compact.
F
We will often use a ‘tensor notation’ for our formulae: eq. (II.3) will read
= , (II.18)
12 12,3 23 1,23
F F F F
and definition (II.9) φ φ = −1 −1 , for instance; the comma sepa-
≡ 123 F12,3F12F23F1,23
rates the tensor factors not stemming from the coproduct. For practical purposes
it will be often convenient in the sequel to use the Sweedler’s notation with lower
indices ∆(x) x x for the cocommutative coproduct (in the RHS a sum
(1) (2)
≡ ⊗
xi xi of many terms is implicitly understood); similarly, we will use the
i (1) ⊗ (2)
SPweedler’s notation ∆(n−1)(x) x(1) ... x(n) for the (n 1)-fold coproduct. For
≡ ⊗ ⊗ −
the non-cocommutative coproducts ∆ , instead, we will use a Sweedler’s notation
h
with barred indices: ∆ (x) x x .
h (¯1) (¯2)
≡ ⊗
6
II.2 Classical Ug-covariant creators and annihilators
Let be the unital algebra generated by 1 and elements a+ and aj
A i i∈I j∈I
A { } { }
satisfying the (anti)commutation relations
[ai, aj] = 0
±
[a+, a+] = 0 (II.19)
i j ±
[ai, a+] = δi1
j ± j A
(the sign denotes commutators and anticommutators respectively), belonging
±
respectively to some representation ρ and to its contragradient ρ∨ = ρT S of H (T
◦
is the transpose):
x⊲a+ = ρ(x)la+
i i l x Ug, ρ(x)i C. (II.20)
x⊲ai = ρ(Sx)ial ∈ j ∈
l
Equivalently, one says that a+,ai are “covariant” under ⊲, or that they span two
i
(left) modules of Ug:
(xy)⊲a = x⊲(y ⊲a), x,y Ug, (II.21)
∈
with either a = ai or a+.
i
is a (left) module algebra of (H,⊲), if the action ⊲ is extended on the whole
A
by means of the (cocommutative) coproduct:
A
x⊲(ab) = (x ⊲a)(x ⊲b). (II.22)
(1) (2)
Then property (II.21) holds for all a .
∈ A
Setting
σ(X) := ρ(X)ia+aj (II.23)
j i
for all X g, one finds that σ : g is a Lie algebra homomorphism, so that
∈ → A
σ can be extended to all of Ug as an algebra homomorphism σ : Ug ; on
→ A
the unit element we set σ(1 g) := 1 . σ can be seen as the generalization of the
U A
Jordan-Schwinger realization of g = su(2) [23] [formula (V.8)].
Then it is easy to check the following
Proposition 1 The (left) action ⊲ : Ug can be realized in an ‘adjoint-
×A → A
like’ way:
x⊲a = σ(x )aσ(Sx ), x Ug, a . (II.24)
(1) (2)
∈ ∈ A
7
In the specially intersting case of a compact section g (with -structure “ ”)
∗ ∗
one can introduce in a -structure, the ‘hermitean conjugation’ (which we will
A ∗
denote by ⋆), such that
(ai)⋆ = a+. (II.25)
i
Correspondingly, ρ is a -representation (ρ ⋆ = ρT) and σ becomes a -
∗ ◦ ∗ ◦ ∗
homomorphism, i.e. σ = ⋆ σ.
◦∗ ◦
III Quantum covariant creators and annihilators
Let H and ϕ be as in section II.1. Clearly, σ := σ ϕ is an algebra homomor-
h h ϕh ◦ h
phism σ : U g [[h]]. Inspired by proposition 3 we are led to define
ϕh h → A
x⊲ a := σ (x )aσ (S x ). (III.1)
h ϕh (¯1) ϕh h (¯2)
Using the Hopf algebra axioms it is straightforward to prove the relations [cfr.
relations (I.10)]
(xy)⊲ a = x⊲ (y ⊲ a)
h h h
x,y Ug[[h]], a,b [[h]].
x⊲ (ab) = (x ⊲ a)(x ⊲ b), ∀ ∈ ∀ ∈ A
h (¯1) h (¯2) h
(III.2)
In other words
Proposition 2 The definition (III.1) realizes ˜⊲ (the left action of H ) on the
h h
algebra [[h]].
A
However, a+,aj are not covariant w.r.t. to ⊲ . One may ask whether there exist
i h
some objects A+,Aj that are covariant under ⊲ and transform as in eq. (I.8).
i h
∈ A
The answer comes from the crucial
Proposition 3 The elements
A+ := σ( (1))a+σ(S (2)γ) [[h]] (III.3)
i F i F ∈ A
Ai := σ(γ′S −1(2))Aiσ( −1(1)) [[h]] (III.4)
F F ∈ A
are “covariant” under ⊲ , more precisely belong respectively to the representation ρ˜
h
and to its quantum contragredient ρ˜∨ = ρ˜T S of U g acting through ⊲ :
h h h
◦
x⊲ A+ = ρ˜(x)lA+ x⊲ Ai = ρ˜(S x)i Am. (III.5)
h i i l h h m
8
Proof. Due to relation (II.4), is an intertwiner between ∆ and ∆ (in this proof
h
F
we drop the symbol ϕ ):
h
x (1) x (2) = (1)x (2)x . (III.6)
(¯1′) (¯2′) (1′) (2′)
F ⊗ F F ⊗F
Applying id S on both sides of the equation and multiplying the result by 1 γ
⊗ ⊗
from the right we find [with the help of relation (II.5)]
x (1) (S (2))γS x = (1)x (Sx )(S (2))γ. (III.7)
(¯1′) h (¯2′) (1′) (2′)
F ⊗ F F ⊗ F
Applying σ σ to both sides and sandwiching a+ between the two tensor factors
i
⊗
we find
σ(x )A+σ(S x ) = σ( (1))σ(x )a+σ(Sx )σ[(S (2))γ], (III.8)
(¯1′) i h (¯2′) F (1′) i (2′) F
which, in view of formula (III.1), proves the first relation.
To prove the second relation, let usnote that relation(II.4) implies ananalogous
relation
∆ (a) = ∆(a), with := [γ′S −1 (2) γ′S −1 (1)]∆(Sγ).
h
F F F F ⊗ F
(III.9)
e e e
This can be shown by applying in the order the following operations to both sides
of eq. (II.4): multiplying by −1 from the left and from the right, applying S S,
F ⊗
multiplying by γ′ γ′ fromthe left andby ∆(Sγ) fromthe right, replacing a S x,
h
⊗ →
using the properties (II.4) and (S S ) ∆ = τ ∆ S . Next, we observe that
h h h h h
⊗ ◦ ◦ ◦
Ai can be rewritten as
Ai = σ[ (1)S(γ−1) ]aiσ[(γ−1) S (2)γ] = σ( (1))alσ(S (2)γ)ρ(γ−1)i; (III.10)
F (1) (2) F F F l
e e e e
whence, reasoning as for the first relation,
σ(x )Aiσ(S x ) (II=I.10) σ( (1))σ(x )alσ(Sx )σ[(S (2))γ]ρ(γ−1)i
(¯1′) h (¯2′) F (1′) (2′) F l
(II=.24) σ( (1))alσ[(S (2))γ]ρ(γ−1Sx)i
Fe F le
(I=I.4) σ( (1))alσ[(S (2))γ]ρ(S x γ−1)i
Fe Fe h · l
(II=I.10) ρ(S x)iAl
eh l e
2
which proves the second relation.
Remark 1 The proposition clearly holds even if one chooses in formulae (III.3),
(III.4) two ’s differing by a tranformation II.12. We shall exploit this freedom
F
when U g is genuinely quasitriangular.
h
9