- Email: [email protected]
Quasi-bosons approximate bosons
Vol. 40 (1997)
REPORTS
ON M/ITHEMATICAL
QUASI-BOSONS ANNE Department
KIRSTINE
of Mathematics,
APPROXIMATE
NIELSEN
No. I
PHYSIC.5
BOSONS
and ERIK BJERRUM
NIELSEN*
University of Aarhus, Ny Munkegade, 8000 Aarhus C, Denmark (e-mail: [email protected]) (Received
December
30,
1996)
In the papers [2-51, operators on the Fermi-Fock space that represent a Bose algebra are constructed. Since the operators fulfilling the CCR are unbounded, the Fermi-Fock space is infinite dimensional, and thus the Fermi one-particle space is infinite dimensional. In this paper we consider the construction in [Z] as a limit case of finite dimensional representations of the quasi-bosom. We show that the quasi-Bose operators converge pointwise to the Bose operators. The quasi-Bose operators are naturally divided into two sets, the quasi-annihilation and the quasi-creation operators. The quasi-annihilation operators annihilate the vacuum and commute painvise. We show that the quasi-creation operators are the adjoints of the quasi-annihilation operators, and hence they commute pairwise as well.
1. Preliminaries 1.1. The Fermi-Fock
space and the central extension
dU
To a given complex Hilbert space R, (, ) we attach an algebra POX generated by % (one-particle space) and a unity which will be denoted by o and called the vacuum. We call ~oZ a Fermi algebra [7] if the scalar product from 3-1 is extended over rO’Ft in such a way that for every 5 E 3t the operator a+(x) of multiplication by 2 admits the adjoint a(x) defined on the whole ra’I-& and if the adjoint fulfils the anti-Leibniz rule, i.e., [a(z),
kf, 9) = (f, a(x)g).
a+(y)]-1 = (2,YV
(1)
and a(z)@ = 0, where [A, B]_, := AB + BA. Finally, we assume that o is a unit vector. We shall write l7i for the completion of rati, (,). We denote by 31” the closed span of rb-fold products of vectors from ‘I-k Let us introduce
*Supported
the double-folded
by Carlsberg Foundation
Hilbert
and SNF. P31
space W = (z)
of vectors
(‘i),
64
A. K. NIELSEN and E B. NIELSEN
5, y E X,
with the usual real-linear
conjugation
Q =
operations,
the complex unit i’ =
, the
and the scalar product
(((I),(~)))-ir,ui+(a,y). Let us define the operator P(Z) = P
= a+(4
J: 0 Y
z E w,
+ a(Y),
on TON. In [2] it is shown that given g E @r&V,
we have
jP(x)g(I b/m3+ m2+ llxllgl.
(2)
Consider a bilinear form given by (u, 4 = OQv, 4). The set of linear operators 02,@(X) = {S / (Sx,w) = (x, -SW) for all z, w E W and I~SJIA< CO}, where IlSllA := maWblI, constitutes Let
the complexified A
L
K
-A*
E
IIS22ll}
restricted
+ mdL%2)l~s,
orthogonal
IIS~~IIHS},
Lie algebra [2].
Given an orthonormal
02,3’FI).
dU : 02,&i)
basis (en}nEz in ?I, we let
-+ B(Wi)
be defined by dU
A
L
K
-A’
=
=
dFA - ia+ c
a+(Aen)a(e,)
nGz -t $ C a( hJa(e,). nEZ
+ ia - i c
a+(e,)a’(Le,>
nE%
(3)
QUASI-BOSONS APPROXIMATE BOSONS
The operators
dUS have the following property. ]duS,
For all 2 E W,
~(~11= 14s~)
holds on roH. 1.2. The Bose algebra
To a given complex Hilbert space %, (, ) we attach an algebra FiiX generated by 1-I (one-particle space) and a unity denoted by 8 and called the vacuum. We call rg:Ft a Bose algebra [7] if the scalar product from 3-1 is extended over rati in such a way that for every z E ‘Ft the operator a+(x) of multiplication by 3: admits the adjoint a(x) defined on the whole Ta’Ft, and if the adjoint fulfils the Leibniz rule, i.e., [a(z),
(zf, 9) = (fl a(G),
a+tv>l~= h Y)I
and a(z)0
=
0,
where [A, B], := AB - BA. Finally, we assume that Q) is a unit vector. 2. Quasi-bosons Let us take a sequence of finite dimensional Hilbert subspaces ‘& c ‘FI. Observe that r7-1, are identical with subspaces of F’FI. The operators dUS, defined on r”‘NrrL we extend naturally to the larger space ra7-t by taking in (3) the sum only over the basis {e,} in ?tFt,. The main result of this paper is that the operators dUS,, can be chosen so as to approximate the Bose creation and annihilation operators pointwise on the space ra3-t. It is well known that the limit operators can represent a Bose algebra (cf. [2-51). Let R be an infinite dimensional Hilbert space, {en}nEz an orthonormal basis, and let ‘Fld denote the subspace spanned by {en}nE[-d,dl. We consider operators S,,.d and PC1on F& defined by Sn,d =
c
(ej, ‘)%+n
for n = 0,1,...,2d,
j=-d
d
S
n,d
=
c
(ej,.)ej+n
for
n =
-2d ,...,
-1,
j=-d-n
Let us introduce the conjugation - on tid by letting e, be real elements. double-folded space W, we define the operators Sri,,, and Td setting
On the
66
A. K. NIELSEN and E. B. NIELSEN
and
(I -
pd
Td =
(I-Pd)Finally, let S,, P and &,T {&,d},
{pd}
represent LEMMA
and
{%,d),
pd)-
’
Pd
denote the d = cc limit of the sequences of operators respectively. It is shown in [2] that dU&(Ts,Tel)
{Td}>
Bose creations for n > 0 and Bose annihilations for n < 0. 1. Using the notation jkom above, we have Td,!?~,d(Td)-’ = T$,dT-l.
Proof: It is sufficient to show that Pd&,d
=
PSn,d,
I&,d
=
I&d,
‘%,dPd sn,dI
= =
sn,dp,
&,dI
and that - commutes with I, P, I, P, and Sn,d. We show only the first formula. Since the scalar product is linear in the second variable, we get d
03
PSn,d
=
c
c
(ej!
‘)(%,
ej+n)%
=
LEMMA
0
2. Let Bddz-+B
Then the sequence (dUBd)
E 02,c6'-0,
tr(B - Bd)f2 + d’cc
0.
converges pointwise to dUB on T07-l.
Proof: Let Ad = Bd - B. We show that
IdU(&)fl Consider an arbitrary element Then
+ 0
for d --) DC)
f f &‘R written in the form f = C,“=, n,“=, p(zj,)P).
QUASI-BOSONS
APPROXIMATE
6’7
BOSONS
and it suffices to show that IdU(A&(q)
(5)
for n ---$xj.
. . p(q)@/ 4 0
From (4) we get
[dUS,P(Zl)
...
P(ZJ)] =
kP(Zl)P(SZJ).
. P(Q)
(0)
J=l
for all S E 02.c(X), and hence IdiZl(A&(z~)
p(z~)oI = ( &+I) j=l
I,et Ad =
’
. p(&q)
Pi
+ P(ZI)
. . P(Q )dWA
Then
and IdU(A
.p(z~)@I I 2
IP(+)
. . P(AG,)
. .P(zJ)~~ + $&).
P(Q)~~L.~1
J=l
Using (2) we find Ip(q)
. P(&zi)
. . p(q)01
5 KI . (~1LIP I Kz . /a
. P(&G) . IA&
.
. . Pi!
1~~1,
and since lA+zil -+ 0, we get
IP(ZI) . . ?‘(&zj)
. . P(a)01
From (2) and the identity lhLd/’ = -2 tr(L$ /P(Q). . ~r4Z.&,12
Since tr(Ls) + 0, we get
Ma)
4
0.
we obtain
L
h-3 .
Id . . I~J/~I~~L~I~
=
K4
1~~1~
12~1~tr(Lz).
P(a)hLdI + 0.
(7)
68
A. K. NIELSEN and E. B. NIELSEN
The following theorem THEOREM
is the main result of the paper.
3. For each n the sequence on I’o’FI.
of operators dU(Td%+(Td)-‘)
converges
pointwise to dU(T&T-‘)
which are the Bose creation The theorem establishes that dU(T&T-l), nihilation operators, are pointwise approximated by dU(T&+(T~)-l). Proof: From Lemma
and an-
1 we get
and hence Ad = T&d
- -S,)T-l.
Let z E H. Since ((T(( = 1, it follows that - k)T-‘21
IT&+
5 IlTll .I<%,, - %JY( = l(%,d - bYI>
where y = T-lz.
Let y =
<%d- &)y12
x 0W
((s;d_iz,d)-( “,--us_)) (If
=
/
, then
=
I(&,d
-
S&12+ I(sn,d - ‘%L)*W~~.
Taking d > 2n and m E [-$, %] , we get (S, - &,d)em = 0. We have
1(&d
-
%)#
=
1
c
k(s,d
-
Sn)‘%&12
5
xmemtn
c
I2
-ST-$,$]
me[-4341 =
1
~x,~2+Oford-+co,
c
-@[-$>%I which proves that l&w1 + 0 where Ad = ?$&T,j-l Moreover, Ld
- T%,T-1
=
(T(k,d
=
P(i?n,d
-
for d --+ CO;
(9)
and L,j = (A&.
%)T-l),z i&)(1
-
P)-
-
(I
-
P)-&,d
-
g,)?.
QUASI-BOSONS
APPROXIMATE
BOSONS
For d > /nl one has
Jez,d,(eJ, m=l =
F
c
e
(ek,‘)(eJ,ek)(e,,,eJ+.)e,,,
k=-m
m=lj@[-d,d] =
5 (ek,.)ek)ei+rL)c,,,
k=-m
0.
and hence Ld
For n > 0, the operators
=
0.
l-i
A,+ = dU (AT,!?n,dT,l)
and A, = dU (&T’S_,,,dT.‘)
deserve the name quasi-Bose creations and quasi-Bose annihilations, ‘respectively. is easy to show that these operators fulfil the identities A,+ = A;,
[A,+,A;]
= [A,,A,]
I
= 0,
it
A,@ = 0.
All the three properties are well known in the limit case. Let us consider a bosonic state 4. Using the operators AZ and A, we can construct a sequence C#I~such that: (a) {&I} converges to 4, (b) & belongs to a fermionic Fock space, (c) ad is finite dimensional. Hence the fermionic nature of the states $d will diminish with growing d. The sequence the coherent
~,$d= etA:@ 1
-1
I
etA:8 of fermionic
squeezed
states will approximate
bosonic state +4= e-+ite112ete1.
REFERENCES The M&rod of Second Quanrizarion, Academic Press, New York, London 1966. [2] 0. R. Jensen and E. B. Nielsen: Representation of quadratic Hamiltonians by quantization of intinite dimensional Lie algebras and a local, non-unitary, projective representation of the corresponding Lie groups. PhD thesis. [3] V. G. Kac: Infinite Dimensional Lie Algebras, Cambridge University Press, Cambridge 19YO. [4] L.-E. Lundberg: Rev. Math. Phys. 6, No.1 (1994) 1. [5] A. Pressley and G. Segal: Loop Groups, Oxford Science Publications, Oxford 1986. [6] D. Shale and W. F. Stinespring: J. Math. Mech. 14 (1965), 315. [7] W. Slowikowski, Infinite dimensional Lie algebras and Lie groups of operators for paired quantum particles, to appear (e-mail: [email protected]).
[L] F. A. Berezin:
REPORTS
ON M/ITHEMATICAL
QUASI-BOSONS ANNE Department
KIRSTINE
of Mathematics,
APPROXIMATE
NIELSEN
No. I
PHYSIC.5
BOSONS
and ERIK BJERRUM
NIELSEN*
University of Aarhus, Ny Munkegade, 8000 Aarhus C, Denmark (e-mail: [email protected]) (Received
December
30,
1996)
In the papers [2-51, operators on the Fermi-Fock space that represent a Bose algebra are constructed. Since the operators fulfilling the CCR are unbounded, the Fermi-Fock space is infinite dimensional, and thus the Fermi one-particle space is infinite dimensional. In this paper we consider the construction in [Z] as a limit case of finite dimensional representations of the quasi-bosom. We show that the quasi-Bose operators converge pointwise to the Bose operators. The quasi-Bose operators are naturally divided into two sets, the quasi-annihilation and the quasi-creation operators. The quasi-annihilation operators annihilate the vacuum and commute painvise. We show that the quasi-creation operators are the adjoints of the quasi-annihilation operators, and hence they commute pairwise as well.
1. Preliminaries 1.1. The Fermi-Fock
space and the central extension
dU
To a given complex Hilbert space R, (, ) we attach an algebra POX generated by % (one-particle space) and a unity which will be denoted by o and called the vacuum. We call ~oZ a Fermi algebra [7] if the scalar product from 3-1 is extended over rO’Ft in such a way that for every 5 E 3t the operator a+(x) of multiplication by 2 admits the adjoint a(x) defined on the whole ra’I-& and if the adjoint fulfils the anti-Leibniz rule, i.e., [a(z),
kf, 9) = (f, a(x)g).
a+(y)]-1 = (2,YV
(1)
and a(z)@ = 0, where [A, B]_, := AB + BA. Finally, we assume that o is a unit vector. We shall write l7i for the completion of rati, (,). We denote by 31” the closed span of rb-fold products of vectors from ‘I-k Let us introduce
*Supported
the double-folded
by Carlsberg Foundation
Hilbert
and SNF. P31
space W = (z)
of vectors
(‘i),
64
A. K. NIELSEN and E B. NIELSEN
5, y E X,
with the usual real-linear
conjugation
Q =
operations,
the complex unit i’ =
, the
and the scalar product
(((I),(~)))-ir,ui+(a,y). Let us define the operator P(Z) = P
= a+(4
J: 0 Y
z E w,
+ a(Y),
on TON. In [2] it is shown that given g E @r&V,
we have
jP(x)g(I b/m3+ m2+ llxllgl.
(2)
Consider a bilinear form given by (u, 4 = OQv, 4). The set of linear operators 02,@(X) = {S / (Sx,w) = (x, -SW) for all z, w E W and I~SJIA< CO}, where IlSllA := maWblI, constitutes Let
the complexified A
L
K
-A*
E
IIS22ll}
restricted
+ mdL%2)l~s,
orthogonal
IIS~~IIHS},
Lie algebra [2].
Given an orthonormal
02,3’FI).
dU : 02,&i)
basis (en}nEz in ?I, we let
-+ B(Wi)
be defined by dU
A
L
K
-A’
=
=
dFA - ia+ c
a+(Aen)a(e,)
nGz -t $ C a( hJa(e,). nEZ
+ ia - i c
a+(e,)a’(Le,>
nE%
(3)
QUASI-BOSONS APPROXIMATE BOSONS
The operators
dUS have the following property. ]duS,
For all 2 E W,
~(~11= 14s~)
holds on roH. 1.2. The Bose algebra
To a given complex Hilbert space %, (, ) we attach an algebra FiiX generated by 1-I (one-particle space) and a unity denoted by 8 and called the vacuum. We call rg:Ft a Bose algebra [7] if the scalar product from 3-1 is extended over rati in such a way that for every z E ‘Ft the operator a+(x) of multiplication by 3: admits the adjoint a(x) defined on the whole Ta’Ft, and if the adjoint fulfils the Leibniz rule, i.e., [a(z),
(zf, 9) = (fl a(G),
a+tv>l~= h Y)I
and a(z)0
=
0,
where [A, B], := AB - BA. Finally, we assume that Q) is a unit vector. 2. Quasi-bosons Let us take a sequence of finite dimensional Hilbert subspaces ‘& c ‘FI. Observe that r7-1, are identical with subspaces of F’FI. The operators dUS, defined on r”‘NrrL we extend naturally to the larger space ra7-t by taking in (3) the sum only over the basis {e,} in ?tFt,. The main result of this paper is that the operators dUS,, can be chosen so as to approximate the Bose creation and annihilation operators pointwise on the space ra3-t. It is well known that the limit operators can represent a Bose algebra (cf. [2-51). Let R be an infinite dimensional Hilbert space, {en}nEz an orthonormal basis, and let ‘Fld denote the subspace spanned by {en}nE[-d,dl. We consider operators S,,.d and PC1on F& defined by Sn,d =
c
(ej, ‘)%+n
for n = 0,1,...,2d,
j=-d
d
S
n,d
=
c
(ej,.)ej+n
for
n =
-2d ,...,
-1,
j=-d-n
Let us introduce the conjugation - on tid by letting e, be real elements. double-folded space W, we define the operators Sri,,, and Td setting
On the
66
A. K. NIELSEN and E. B. NIELSEN
and
(I -
pd
Td =
(I-Pd)Finally, let S,, P and &,T {&,d},
{pd}
represent LEMMA
and
{%,d),
pd)-
’
Pd
denote the d = cc limit of the sequences of operators respectively. It is shown in [2] that dU&(Ts,Tel)
{Td}>
Bose creations for n > 0 and Bose annihilations for n < 0. 1. Using the notation jkom above, we have Td,!?~,d(Td)-’ = T$,dT-l.
Proof: It is sufficient to show that Pd&,d
=
PSn,d,
I&,d
=
I&d,
‘%,dPd sn,dI
= =
sn,dp,
&,dI
and that - commutes with I, P, I, P, and Sn,d. We show only the first formula. Since the scalar product is linear in the second variable, we get d
03
PSn,d
=
c
c
(ej!
‘)(%,
ej+n)%
=
LEMMA
0
2. Let Bddz-+B
Then the sequence (dUBd)
E 02,c6'-0,
tr(B - Bd)f2 + d’cc
0.
converges pointwise to dUB on T07-l.
Proof: Let Ad = Bd - B. We show that
IdU(&)fl Consider an arbitrary element Then
+ 0
for d --) DC)
f f &‘R written in the form f = C,“=, n,“=, p(zj,)P).
QUASI-BOSONS
APPROXIMATE
6’7
BOSONS
and it suffices to show that IdU(A&(q)
(5)
for n ---$xj.
. . p(q)@/ 4 0
From (4) we get
[dUS,P(Zl)
...
P(ZJ)] =
kP(Zl)P(SZJ).
. P(Q)
(0)
J=l
for all S E 02.c(X), and hence IdiZl(A&(z~)
p(z~)oI = ( &+I) j=l
I,et Ad =
’
. p(&q)
Pi
+ P(ZI)
. . P(Q )dWA
Then
and IdU(A
.p(z~)@I I 2
IP(+)
. . P(AG,)
. .P(zJ)~~ + $&).
P(Q)~~L.~1
J=l
Using (2) we find Ip(q)
. P(&zi)
. . p(q)01
5 KI . (~1LIP I Kz . /a
. P(&G) . IA&
.
. . Pi!
1~~1,
and since lA+zil -+ 0, we get
IP(ZI) . . ?‘(&zj)
. . P(a)01
From (2) and the identity lhLd/’ = -2 tr(L$ /P(Q). . ~r4Z.&,12
Since tr(Ls) + 0, we get
Ma)
4
0.
we obtain
L
h-3 .
Id . . I~J/~I~~L~I~
=
K4
1~~1~
12~1~tr(Lz).
P(a)hLdI + 0.
(7)
68
A. K. NIELSEN and E. B. NIELSEN
The following theorem THEOREM
is the main result of the paper.
3. For each n the sequence on I’o’FI.
of operators dU(Td%+(Td)-‘)
converges
pointwise to dU(T&T-‘)
which are the Bose creation The theorem establishes that dU(T&T-l), nihilation operators, are pointwise approximated by dU(T&+(T~)-l). Proof: From Lemma
and an-
1 we get
and hence Ad = T&d
- -S,)T-l.
Let z E H. Since ((T(( = 1, it follows that - k)T-‘21
IT&+
5 IlTll .I<%,, - %JY( = l(%,d - bYI>
where y = T-lz.
Let y =
<%d- &)y12
x 0W
((s;d_iz,d)-( “,--us_)) (If
=
/
, then
=
I(&,d
-
S&12+ I(sn,d - ‘%L)*W~~.
Taking d > 2n and m E [-$, %] , we get (S, - &,d)em = 0. We have
1(&d
-
%)#
=
1
c
k(s,d
-
Sn)‘%&12
5
xmemtn
c
I2
-ST-$,$]
me[-4341 =
1
~x,~2+Oford-+co,
c
-@[-$>%I which proves that l&w1 + 0 where Ad = ?$&T,j-l Moreover, Ld
- T%,T-1
=
(T(k,d
=
P(i?n,d
-
for d --+ CO;
(9)
and L,j = (A&.
%)T-l),z i&)(1
-
P)-
-
(I
-
P)-&,d
-
g,)?.
QUASI-BOSONS
APPROXIMATE
BOSONS
For d > /nl one has
Jez,d,(eJ, m=l =
F
c
e
(ek,‘)(eJ,ek)(e,,,eJ+.)e,,,
k=-m
m=lj@[-d,d] =
5 (ek,.)ek)ei+rL)c,,,
k=-m
0.
and hence Ld
For n > 0, the operators
=
0.
l-i
A,+ = dU (AT,!?n,dT,l)
and A, = dU (&T’S_,,,dT.‘)
deserve the name quasi-Bose creations and quasi-Bose annihilations, ‘respectively. is easy to show that these operators fulfil the identities A,+ = A;,
[A,+,A;]
= [A,,A,]
I
= 0,
it
A,@ = 0.
All the three properties are well known in the limit case. Let us consider a bosonic state 4. Using the operators AZ and A, we can construct a sequence C#I~such that: (a) {&I} converges to 4, (b) & belongs to a fermionic Fock space, (c) ad is finite dimensional. Hence the fermionic nature of the states $d will diminish with growing d. The sequence the coherent
~,$d= etA:@ 1
-1
I
etA:8 of fermionic
squeezed
states will approximate
bosonic state +4= e-+ite112ete1.
REFERENCES The M&rod of Second Quanrizarion, Academic Press, New York, London 1966. [2] 0. R. Jensen and E. B. Nielsen: Representation of quadratic Hamiltonians by quantization of intinite dimensional Lie algebras and a local, non-unitary, projective representation of the corresponding Lie groups. PhD thesis. [3] V. G. Kac: Infinite Dimensional Lie Algebras, Cambridge University Press, Cambridge 19YO. [4] L.-E. Lundberg: Rev. Math. Phys. 6, No.1 (1994) 1. [5] A. Pressley and G. Segal: Loop Groups, Oxford Science Publications, Oxford 1986. [6] D. Shale and W. F. Stinespring: J. Math. Mech. 14 (1965), 315. [7] W. Slowikowski, Infinite dimensional Lie algebras and Lie groups of operators for paired quantum particles, to appear (e-mail: [email protected]).
[L] F. A. Berezin: