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Particle representations of canonical commutation relations
Vol. 3 (1972)
REPORTS
PARTICLE
MATHEMATICAL
NAPI~RKOWSKI
of Mathematical
Methods
(Received
COMMUTATION
and WIESLAW Pusz
of Physics, University
November
No. 3
PHYSICS
REPRESENTATIONS OF CANONICAL RELATIONS KAZIMIERZ
Department
ON
of Warsaw, Warsaw
15, 1971)
The aim of the present paper is to find a necessary and sufficient condition for the existence of the operator of the total number of particles in a representation of canonical commutation relations. The result is formulated by means of generating functionals. of representations. It is shown that an irreducible representation possesses a (generalized) number operator if and only if the representation obtained by averaging its generating functional over the group of phase transformations is a factor representation of type I.
Let H be a complex Hilbert space. We say that a mapping W from H to the set of unitary operators on a Hilbert space K is a representation of canonical commutation relations over H (in Weyl’s form) if W(x+y)=e
;
B
(Y.x)
W(x) W(Y)
for
x,y~H,
where B(.;) is the imaginary part of the scalar product in H. It is supposed that W is a continuous mapping in the weak operator topology when x runs across finite dimensional subspaces of H. Let M be a finite dimensional subspace of H and let &M be a von Neumann algebra generated by the set {W(x) :x E M}. A uniformly closed C*-algebra Cd (with unit) generated by the algebras &,, where M runs over all finite dimensional subspaces of H, is called a Weyl algebra ouer H. Segal [5] showed that a Weyl algebra is unique as the abstract C*-algebra (it is independent of the choice of a concrete representation). It is well known from the Gelfand-Najmark-Segal construction ([2], [6]) that every state E (positive and normed linear functional) on ~4 defines a cyclic representation, i.e. there exist (71,K, D) where z is a representation of & by bounded operators in a (complex) Hilbert space K and v E K is a cyclic vector for z normed to the unit and E(A)=(n(A)ulu)
for every AE.&.
However, only regular states give representations of Weyl’s relations if it is weakly continuous on the unit ball of every von Neumann is a finite dimensional subspace of H).
(1) (E is called regular algebra d,, where M
K. NAPIdRKOWSKI
222 Every functional
regular state on & defines for the state E: HEZ+~(Z)
AND W. PUS2
a complex
function
,U on H called
the generating
:=E(W(Z))=(~(W(Z))+)EC~.
Sometimes it is important to know when a complex Segal [4] proved the following statement:
function
on H defines a state on d.
LEMMA 1. A complex function IL on H is a generating functional for a regular E on the Weyl algebra JZZover H tf and only if p satisfies the following conditions:
(i) ,U restricted
to any finite
dimensional
subspace
state
of H is a continuous function;
(ii) p(O)= 1; (iii) if I is a finite
set of indices,
where zj E H and aj are complex Moreover,
then
numbers.
the state E is dejined uniquely.
It is important to know whether a given representation of Weyl relations possesses the operator of the total number of particles (number operator shortly) or not? Following Chaiken [l], we shall adopt a generalized definition of the number operator N. It has the following properties: (i) N is selJadjoint , (ii) exp(itN)
W(z)exp(
- itN)=
W(e”z) for real t and ZE H.
(2)
A representation possessing such a number operator is calied a particle representation. It can be shown that if such an operator N exists it can be chosen to have its spectrum (0, 1, 2, . . .} or otherwise Sp N={O, _t 1, +2, . . .}. A number operator with integer spectrum is said to be normalized. Thus we may suppose that a particle representation has normalized number operator [l]. Chaiken proved the following sufficient condition for a cyclic representation to be a particle one: LEMMA 2.
If ,u is a generating functional for a regular state on the Weyl algebra over H and p is phase-invariant (i.e. p(e”z)=p(z) for real t and z E H), then the cyclic representation defined by ,a (GNS-construction) is a particle one. If ,u is a given generating functional for a regular state on ~2, then it determines a one-parameter family of generating functionals for regular states (this follows directly from Lemma 1): ZEH and tisreal. P,(Z) = P (e”z) , (3) Let us define
a complex
function
on H In
s
;(z)=;~&>dt 0
.
(4)
PARTICLE
From
Lemma
1 it follows
easily that i; is also a generating
on d. Let d, denote the cyclic representation of d v of a regular state on & (GNS-construction);
functional
of a regular
determined by the generating then we have
If d, is an irreducible particle representation
THEOREM 1. entation
223
REPRESENTATlONS
state
functional
then .JzZ;;is a factor
repres-
of type I and &;I = 0~4,.
Idea of the proof:
Let H, be the representation space for -01,. The representationdP N (we suppose N is normalized). The cyclic vector u for &, need of N and in general
has a number operator not be an eigenvector
where I is a subset of the spectrum in (2) and (5) we have:
of N. Using
Lebesgue’s
theorem,
the second property
2n
i%z)=&
(w,,(e”z)cIu)dt=“~~(~~(z)c~unIcnun). s 0
Now
take
a direct
sum of representations A?= @&x2,; nsl
then
the vector
V= @ c,,u, belongs n‘ZI
to E?= @ H, and nel 1.
((v((q&
It can be shown that v is a cyclic vector for the representation a state
on ~2 the generating
functional
of which
2.
i+,=(@(z)v” 1v)=,TI (W,(z)c,u,1c,u,), We see that E(z)=;(z), tation JZ of & [4]. COROLLARY 1. particle
therefore
the representation
If'd; is not a factor
The vector v determines
is:
ZEH.
-02; is isomorphic
representation,
to the represen-
then -01, (irreducible)
is not a
representation.
COROLLARY 2. tation .d;
is a particle
For any generating functional p of regular state on ~4 the cyclic represen representation.
The proof follows from (4) (i; is phase-invariant) and Lemma 2. The condition of Theorem 1 proves to be also sufficient for an irreducible tation to be a particle.
represen-
K. NAPI6RKOWSKI
224
AND
W. PUSZ
THEOREM2. Let d,, be an irreducible representation. tion of type I, then &,, is a particle representation.
Proof: irreducible
If 59, is a factor
representa-
Since Js;; is a factor representation of type I, it is a direct sum of equivalent representations pi, i.e. &;= 8 ~4~. We shall first prove that di are particle ieJ
representations and next that pi are equivalent to 58,. Let us denote the spaces of the representations .z$; and di by H; and Hi, respectively. From the uniqueness of the Weyl algebra it follows that the mapping [5] 2~23 W(z)-+ w(e”z) E .CZZ induces an automorphism O(t) of &. Since fi is a generating functional of a regular state g and i; is phase-invariant,2 is invariant under the group of automorphisms O(t). From the GNS-construction it follows that O(t) are implementable by a group of unitary operators U(t), i.e. ~~3A~O(t)(A)=U(t)AU(-t)E~~. (6) From (6) it follows of the factor
that the group
O(t) can be extended S=(d;)“=
to a group
of automorphisms
@B(Hi) ieJ
of type I. Since automorphisms
of a factor
B3B+O(t)(B)= Taking
into account
of type I are inner, and
V(t) BV(-t)Ea
the decomposition
of 59, we get a group
we have
v(t)EsT of automorphisms
B (Hi) 3 Bi-r vi(t) Bi Vi( - t) =B,(Bi) E B (Hi).
of B(Hi): (7)
where vi(t) denotes the restriction of V(t) to Hi. The group U(t) is continuous at zero [I] and induces the same automorphism of B as V(t). This way we get a projective representation of the topological group R': R’++Aut
B(Hi).
(8)
From [8] it follows that (8) is induced by a unitary representation t+ c(t) and E(t) =e irNi. It is easy to check that Ni is a number operator for di (not necessarily normalized). For the proof that di are equivalent to _d, we shall need the following b3MMA
f ITKfd,W
3. -a
Let SX?be a C*-algebra, n its representation in a Hilbert space K and K = decomposition of K into a direct integral corresponding to a decomposition
of IL into irreducible representations n,. Let K, be a subspace of K invariant under z such that the representation n restricted to K, is irreducible. Then there is a subset T,, c T with a positive measure such that for t E To zr is equivalent to n restricted to K,,.
PARTICLE Proof of the lemma:
By PO we denote
Let g==((J8)“.
225
REPRESENTATIONS
the projection
onto KO and by
Pits central
support in &?. It is quite easy to check that the restriction of z to PK is a factor representation of type I. Let g1 be the maximal abelian subalgebra of & corresponding In the to the decomposition of 71. We have P E ~8~) because P is a central projection. direct integral P is represented by the characteristic function of a set T,, with a positive measure. The representations 71, for t E To give a decomposition of the restriction of 7~ to PK into irreducible representations. It is well known that any decomposition of a factor representation of type I contains representations from the same class of equivalence odly. Because P,,KcPK, it follows from the above that 7cn,are equivalent to the restriction of 7t to POK for (almost) all t E TO. This ends the proof of Lemma 3. To complete the proof of Theorem 2 we notice that &;; is equivalent to a subrepresentation of the direct integral of representations dPt. To see this we take a measurable vector
field o, such that
(W(z)o, / v,)=p,(z) v=
j
and set
v,dt,
co.Znl
VE j
HWtdt.
co, 2x1
5 JzZ,,~dt acting in the subspace gen[0,2771 where -oZi are equivalent irreducible representations, erated from 0,. Since d; = @di, it follows from Lemma 3 that there exists a set TO with positive Lebesgue measure such that for t E To the representation &,,, is equivalent to pi. For each pair t, t’e To we have &,,t equivalent to dPt,. This means that O(t-t’) is unitarily implementable in the representation dPt (and in each .dPt,, for t” E To.) Since To has a positive Lebesgue measure, the set {t-t’ ( t, t’ E TO) contains a neighbourhood of zero [3]. From the group properties of O(t) it follows now that all the representations ~4~~ are equivalent, in particular, they are equivalent to .z&‘~~=&~. The representations “4, are, therefore, equivalent to ZZ?~. The proof is complete. As a corollary from Theorems 1 and 2 we get Then _G?; is equivalent
THEOREM 3. if&;
is a jhctor
to the subrepresentation
An irreducible representation
representation
of
SZCZ’~ is a particle
representation
if and only
of type I.
REFERENCES [l] Chaiken, J. M., Commun. Math. Phys. 8 (1968). [2] Gelfand, I. M., M. A. Najmark, Mat. Sbornik 12 (1943). [3] Halmos, P. R., Measure theory, Toronto-New York-London, 1950. [4] Najmark, M. A., Normed rings, Moscow, 1968 (in Russian). [5] Segal, 1. E., Mat. Fys. Medd. Danske Vid. Selsk. 31.12 (1959). 161 -, Bull. Amer. Math. Sot. 53 (1947). [7] -, Canad. J. Math. 13 (1962). [8] Simms, D. J., Projective representations, sympletic manifolds and extensions seille 69.
of Lie algebras,
Mar-
REPORTS
PARTICLE
MATHEMATICAL
NAPI~RKOWSKI
of Mathematical
Methods
(Received
COMMUTATION
and WIESLAW Pusz
of Physics, University
November
No. 3
PHYSICS
REPRESENTATIONS OF CANONICAL RELATIONS KAZIMIERZ
Department
ON
of Warsaw, Warsaw
15, 1971)
The aim of the present paper is to find a necessary and sufficient condition for the existence of the operator of the total number of particles in a representation of canonical commutation relations. The result is formulated by means of generating functionals. of representations. It is shown that an irreducible representation possesses a (generalized) number operator if and only if the representation obtained by averaging its generating functional over the group of phase transformations is a factor representation of type I.
Let H be a complex Hilbert space. We say that a mapping W from H to the set of unitary operators on a Hilbert space K is a representation of canonical commutation relations over H (in Weyl’s form) if W(x+y)=e
;
B
(Y.x)
W(x) W(Y)
for
x,y~H,
where B(.;) is the imaginary part of the scalar product in H. It is supposed that W is a continuous mapping in the weak operator topology when x runs across finite dimensional subspaces of H. Let M be a finite dimensional subspace of H and let &M be a von Neumann algebra generated by the set {W(x) :x E M}. A uniformly closed C*-algebra Cd (with unit) generated by the algebras &,, where M runs over all finite dimensional subspaces of H, is called a Weyl algebra ouer H. Segal [5] showed that a Weyl algebra is unique as the abstract C*-algebra (it is independent of the choice of a concrete representation). It is well known from the Gelfand-Najmark-Segal construction ([2], [6]) that every state E (positive and normed linear functional) on ~4 defines a cyclic representation, i.e. there exist (71,K, D) where z is a representation of & by bounded operators in a (complex) Hilbert space K and v E K is a cyclic vector for z normed to the unit and E(A)=(n(A)ulu)
for every AE.&.
However, only regular states give representations of Weyl’s relations if it is weakly continuous on the unit ball of every von Neumann is a finite dimensional subspace of H).
(1) (E is called regular algebra d,, where M
K. NAPIdRKOWSKI
222 Every functional
regular state on & defines for the state E: HEZ+~(Z)
AND W. PUS2
a complex
function
,U on H called
the generating
:=E(W(Z))=(~(W(Z))+)EC~.
Sometimes it is important to know when a complex Segal [4] proved the following statement:
function
on H defines a state on d.
LEMMA 1. A complex function IL on H is a generating functional for a regular E on the Weyl algebra JZZover H tf and only if p satisfies the following conditions:
(i) ,U restricted
to any finite
dimensional
subspace
state
of H is a continuous function;
(ii) p(O)= 1; (iii) if I is a finite
set of indices,
where zj E H and aj are complex Moreover,
then
numbers.
the state E is dejined uniquely.
It is important to know whether a given representation of Weyl relations possesses the operator of the total number of particles (number operator shortly) or not? Following Chaiken [l], we shall adopt a generalized definition of the number operator N. It has the following properties: (i) N is selJadjoint , (ii) exp(itN)
W(z)exp(
- itN)=
W(e”z) for real t and ZE H.
(2)
A representation possessing such a number operator is calied a particle representation. It can be shown that if such an operator N exists it can be chosen to have its spectrum (0, 1, 2, . . .} or otherwise Sp N={O, _t 1, +2, . . .}. A number operator with integer spectrum is said to be normalized. Thus we may suppose that a particle representation has normalized number operator [l]. Chaiken proved the following sufficient condition for a cyclic representation to be a particle one: LEMMA 2.
If ,u is a generating functional for a regular state on the Weyl algebra over H and p is phase-invariant (i.e. p(e”z)=p(z) for real t and z E H), then the cyclic representation defined by ,a (GNS-construction) is a particle one. If ,u is a given generating functional for a regular state on ~2, then it determines a one-parameter family of generating functionals for regular states (this follows directly from Lemma 1): ZEH and tisreal. P,(Z) = P (e”z) , (3) Let us define
a complex
function
on H In
s
;(z)=;~&>dt 0
.
(4)
PARTICLE
From
Lemma
1 it follows
easily that i; is also a generating
on d. Let d, denote the cyclic representation of d v of a regular state on & (GNS-construction);
functional
of a regular
determined by the generating then we have
If d, is an irreducible particle representation
THEOREM 1. entation
223
REPRESENTATlONS
state
functional
then .JzZ;;is a factor
repres-
of type I and &;I = 0~4,.
Idea of the proof:
Let H, be the representation space for -01,. The representationdP N (we suppose N is normalized). The cyclic vector u for &, need of N and in general
has a number operator not be an eigenvector
where I is a subset of the spectrum in (2) and (5) we have:
of N. Using
Lebesgue’s
theorem,
the second property
2n
i%z)=&
(w,,(e”z)cIu)dt=“~~(~~(z)c~unIcnun). s 0
Now
take
a direct
sum of representations A?= @&x2,; nsl
then
the vector
V= @ c,,u, belongs n‘ZI
to E?= @ H, and nel 1.
((v((q&
It can be shown that v is a cyclic vector for the representation a state
on ~2 the generating
functional
of which
2.
i+,=(@(z)v” 1v)=,TI (W,(z)c,u,1c,u,), We see that E(z)=;(z), tation JZ of & [4]. COROLLARY 1. particle
therefore
the representation
If'd; is not a factor
The vector v determines
is:
ZEH.
-02; is isomorphic
representation,
to the represen-
then -01, (irreducible)
is not a
representation.
COROLLARY 2. tation .d;
is a particle
For any generating functional p of regular state on ~4 the cyclic represen representation.
The proof follows from (4) (i; is phase-invariant) and Lemma 2. The condition of Theorem 1 proves to be also sufficient for an irreducible tation to be a particle.
represen-
K. NAPI6RKOWSKI
224
AND
W. PUSZ
THEOREM2. Let d,, be an irreducible representation. tion of type I, then &,, is a particle representation.
Proof: irreducible
If 59, is a factor
representa-
Since Js;; is a factor representation of type I, it is a direct sum of equivalent representations pi, i.e. &;= 8 ~4~. We shall first prove that di are particle ieJ
representations and next that pi are equivalent to 58,. Let us denote the spaces of the representations .z$; and di by H; and Hi, respectively. From the uniqueness of the Weyl algebra it follows that the mapping [5] 2~23 W(z)-+ w(e”z) E .CZZ induces an automorphism O(t) of &. Since fi is a generating functional of a regular state g and i; is phase-invariant,2 is invariant under the group of automorphisms O(t). From the GNS-construction it follows that O(t) are implementable by a group of unitary operators U(t), i.e. ~~3A~O(t)(A)=U(t)AU(-t)E~~. (6) From (6) it follows of the factor
that the group
O(t) can be extended S=(d;)“=
to a group
of automorphisms
@B(Hi) ieJ
of type I. Since automorphisms
of a factor
B3B+O(t)(B)= Taking
into account
of type I are inner, and
V(t) BV(-t)Ea
the decomposition
of 59, we get a group
we have
v(t)EsT of automorphisms
B (Hi) 3 Bi-r vi(t) Bi Vi( - t) =B,(Bi) E B (Hi).
of B(Hi): (7)
where vi(t) denotes the restriction of V(t) to Hi. The group U(t) is continuous at zero [I] and induces the same automorphism of B as V(t). This way we get a projective representation of the topological group R': R’++Aut
B(Hi).
(8)
From [8] it follows that (8) is induced by a unitary representation t+ c(t) and E(t) =e irNi. It is easy to check that Ni is a number operator for di (not necessarily normalized). For the proof that di are equivalent to _d, we shall need the following b3MMA
f ITKfd,W
3. -a
Let SX?be a C*-algebra, n its representation in a Hilbert space K and K = decomposition of K into a direct integral corresponding to a decomposition
of IL into irreducible representations n,. Let K, be a subspace of K invariant under z such that the representation n restricted to K, is irreducible. Then there is a subset T,, c T with a positive measure such that for t E To zr is equivalent to n restricted to K,,.
PARTICLE Proof of the lemma:
By PO we denote
Let g==((J8)“.
225
REPRESENTATIONS
the projection
onto KO and by
Pits central
support in &?. It is quite easy to check that the restriction of z to PK is a factor representation of type I. Let g1 be the maximal abelian subalgebra of & corresponding In the to the decomposition of 71. We have P E ~8~) because P is a central projection. direct integral P is represented by the characteristic function of a set T,, with a positive measure. The representations 71, for t E To give a decomposition of the restriction of 7~ to PK into irreducible representations. It is well known that any decomposition of a factor representation of type I contains representations from the same class of equivalence odly. Because P,,KcPK, it follows from the above that 7cn,are equivalent to the restriction of 7t to POK for (almost) all t E TO. This ends the proof of Lemma 3. To complete the proof of Theorem 2 we notice that &;; is equivalent to a subrepresentation of the direct integral of representations dPt. To see this we take a measurable vector
field o, such that
(W(z)o, / v,)=p,(z) v=
j
and set
v,dt,
co.Znl
VE j
HWtdt.
co, 2x1
5 JzZ,,~dt acting in the subspace gen[0,2771 where -oZi are equivalent irreducible representations, erated from 0,. Since d; = @di, it follows from Lemma 3 that there exists a set TO with positive Lebesgue measure such that for t E To the representation &,,, is equivalent to pi. For each pair t, t’e To we have &,,t equivalent to dPt,. This means that O(t-t’) is unitarily implementable in the representation dPt (and in each .dPt,, for t” E To.) Since To has a positive Lebesgue measure, the set {t-t’ ( t, t’ E TO) contains a neighbourhood of zero [3]. From the group properties of O(t) it follows now that all the representations ~4~~ are equivalent, in particular, they are equivalent to .z&‘~~=&~. The representations “4, are, therefore, equivalent to ZZ?~. The proof is complete. As a corollary from Theorems 1 and 2 we get Then _G?; is equivalent
THEOREM 3. if&;
is a jhctor
to the subrepresentation
An irreducible representation
representation
of
SZCZ’~ is a particle
representation
if and only
of type I.
REFERENCES [l] Chaiken, J. M., Commun. Math. Phys. 8 (1968). [2] Gelfand, I. M., M. A. Najmark, Mat. Sbornik 12 (1943). [3] Halmos, P. R., Measure theory, Toronto-New York-London, 1950. [4] Najmark, M. A., Normed rings, Moscow, 1968 (in Russian). [5] Segal, 1. E., Mat. Fys. Medd. Danske Vid. Selsk. 31.12 (1959). 161 -, Bull. Amer. Math. Sot. 53 (1947). [7] -, Canad. J. Math. 13 (1962). [8] Simms, D. J., Projective representations, sympletic manifolds and extensions seille 69.
of Lie algebras,
Mar-