–1–
A Note on Regular States and Supplementary Conditions.
Hendrik Grundling and C.A. Hurst.
Abstract. We show that linear Hermitian supplementary conditions can never
be imposed in a representation associated with a regular state on the C*–algebra
of the CCR’s. Nevertheless there is a well–defined method for imposing the constraints in an abstract C*–framework, which yields as its final physical algebra a
CCR C*–algebra, on which one can again require its physical states to be regular.
These states derive from states on the original C*–algebra which are “regular up
to nonphysical quantities.”
–2–
1. Introduction.
In heuristic physics, the following problem occurs in the imposition of a supplementary condition. One is given a densely defined operator χ : D(χ) 7→ R(χ) on
a Hilbert space H , typically a Fock space, and then require the space of physical
vectors to be
Hp :=
©
¯
ª−−
|ψi ∈ D(χ) ¯ χ|ψi = 0
.
The operator χ is called a supplementary condition. However, it frequently
is the case that {0} is not in the point spectrum of χ , in which case there
are no normalized eigenvectors associated with {0} , i.e.
Hp = ∅ . The best
known example of this is the quantum theory of the electromagnetic field, where
the nonnormalizability lead to many difficulties [1], which were finally removed,
again at the heuristic level, by the introduction of the indefinite metric [2]. In an
earlier paper by one of the authors, a first attempt was made to avoid the problem
by using a formulation in terms of von Neumann algebras [2], but that approach
lacked rigour, in its failure to establish the weak continuity of the homomorphism
which was central to the method. The same problem reappeared in the more
general setting of providing a quantum theory of Dirac’s treatment of constraints
[3].
Our claim in this note is that the problem above can be overcome if one sets
it in the representation–free framework of abstract C*–algebras. First, to avoid
complications with unbounded operators, assume that either χ or χ∗ χ are
self–adjoint operators on H , in which case we can construct the unitary group
Uλ := exp iλχ , λ ∈ IR , respectively Uλ := exp iλχ∗ χ , λ ∈ IR via Stone’s
theorem. Then
Hp :=
With the notation:
©
¯
ª
|ψi ∈ H ¯ Uλ |ψi = |ψi ∀ λ ∈ IR .
σH (A) is the spectrum of the operator A : H 7→ H , and
P σH (A) is its point spectrum, the problem will occur if 1 6∈ P σH (Uλ ) , for then
Hp = ∅ . In C*–language, we set the problem up as follows [4]: Let F
be a
simple unital C*–algebra, called the field algebra, within which is specified a set of
unitaries U ⊂ Fu , called supplementary conditions, and which correspond to the
–3–
Uλ ’s above. If a physical representation π : F 7→ B(H) is specified, then it is
faithful because F is simple. Denote the C*–spectrum of A ∈ F by σF (A) ,
¡
¢
then the spectra are related as follows: σH (π(A)) ⊆ σF (A) , where σH π(A)
is the union of its point, continuous and residual spectra. Then the problem
¡
¢
above will occur as 1 6∈ P σH π(U ) for some U ∈ U . However, it is possible
(an example is given below) that there are other representations π 0 for which
1 ∈ P σH0 (π 0 (U )) ∀ U ∈ U , and so it is clear that this problem is associated
with the specific representation we have chosen as physical. Now methods have
already been developed for eliminating degeneracies in the abstract C*–framework
cf. [4–7], and so we argue below that this should be done first before imposing
physical criteria on the representations of the resulting physical C*–algebra.
In principle it is also possible to treat the problem by the use of rigged Hilbert
spaces. That is, it could be set up on a Gel’fand triplet [13], Φ ⊂ H ⊂ Φ0 , where
Φ is a nuclear countably Hilbert space which is dense in the Hilbert space H ,
and Φ0 is the dual of Φ . The supplementary condition χ is then defined
as an operator bounded on Φ , and the physical subspace consists of generalized
eigenvectors
Hp :=
©
¯
ª
F ∈ Φ0 ¯ F (χϕ) = 0 ∀ ϕ ∈ Φ ⊂ Φ0 .
We will not further consider this method in the present note.
–4–
2. The C*–algebra of the CCR and its regular states.
Henceforth we restrict our attention to linear boson theories.
The field algebra
F is chosen to be Manuceau’s C*–algebra of the CCR [8] ∆(Q) , over a suitable
test function space Q with symplectic form B(·, ·) . To fix notation, we define
∆(Q) and indicate its heuristic correspondence rules.
Given a canonical pair
qi (x) ,
pi (x)
on a Hilbert space
H , with
some internal tensor or Lie structure indicated by the index i , and ETCR:
£
¤
qi (x), pj (x0 ) x0 =x0 = igij δ 3 (x − x0 ) , smear over a suitable test function space,
0
£
¤
say ⊕i S (i) (IR) to obtain the form (·, ·) and the CCR: qx0 (F ), px0 (G) =
i(F, G) . Let Q be the complexification of ⊕i S (i) (IR) (or equivalently, its
direct sum with itself) with the usual norm. Then a symplectic form B(F, G) =
B(F1 + iF2 , G1 + iG2 ) := (F1 , G2 ) − (F2 , G1 ) can be defined on it. Using
¡
¢
¡
¢
£
¤
W (F ) := exp ipx0 (F1 ) exp iqx0 (F2 ) exp −i(F1 , F2 )/2 ,
this defines a heuristic Weyl system:
£
¤
W (F ) W (F 0 ) = W (F + F 0 ) exp −iB(F, F 0 )/2 ,
which expresses the canonical structure. For commutation relations of the form
£
¤
Aµ (x), Aν (x0 ) = igµν ∆(x − x0 ) we obtain a similar Weyl system. Abstractly,
the procedure is as follows [8]:
Definition. Given a linear topological space Q with a symplectic form B
on it, let ∆(Q) be the normed *–algebra such that
(i) The elements of ∆(Q) are complex valued functions on Q
with support consisting of a finite subset of Q .
(ii) Let ∆(Q) have the obvious linear structure, and the multiplication law:
(f1 f2 )(z) :=
X
£
¤
f1 (z1 ) f2 (z − z1 ) exp −iB(z1 , z)/2 .
z1 ∈Q
The involution is defined by f ∗ (z) := f (−z) .
¯
P ¯¯
(iii) Define a norm in ∆(Q) by kf k1 :=
f (z)¯ . Denote
z∈Q
the completion of ∆(Q) in this norm by ∆1 (Q) .
–5–
The set of functions δz such that δz (z 0 ) = 1 if z = z 0 , and zero otherwise,
forms a linear basis for ∆(Q) . Then the C*–algebra of the CCR, ∆(Q) , is defined as the enveloping C*–algebra of ∆1 (Q) , i.e. the closure of the latter in the
°
°
following C*–norm: kf k := sup °π(f )° , where P denotes the set of all nonπ∈P
degenerate representations of ∆1 (Q) . We also use the notation ∆(Q, B) for
∆(Q) when we want to indicate B explicitly. The enveloping C*–norm above
appears to be different from the norm normally employed to define ∆(Q) [8, 12],
°
°
kf k0 := sup °π(f )° , where Pe denotes the set of all regular representations:
e
π∈P
Pe :=
©
¯
ª
π ∈ P ¯ λ 7→ π(δλF ) is weak operator continuous for fixed F ∈ Q .
However, from Slawny’s uniqueness theorem [10, 9], we see that the two resulting
C*–algebras are the same, and hence so are the norms. In other words, even if
∆(Q) is initially defined from the set Pe of all regular representations, it can
still have representations which are not regular. The motivation for this (failed)
attempt to incorporate regular representations in the abstract definition of ∆(Q)
lies in the result [12] that there is a bijection between Pe , and the set of Weyl
systems on {Q, B} , and this bijection is realized by the relation Wπ (F ) =
π(δF ) . Moreover, when λ 7→ π(δλF ) is weak operator continuous, then by
Stone’s theorem we can recover the smeared canonical fields qx0 (F2 ) , px0 (F1 )
as unbounded operators on Hπ [9]. This is the reason why the set of physical
representations of ∆(Q) is specified to be Pe as an additional requirement, since
it was not possible to incorporate Pe into the abstract structure of F = ∆(Q) .
The regular representations π ∈ Pe arise as GNS–representations of the regular
states [9, 11]:
℘R :=
©
¯
ª
ω ∈ ℘(F) ¯ λ 7→ ω(δλF ) is continuous for fixed F ∈ Q
where ℘(F) denotes the set of states on F . These in turn arise by linearity
and continuity from generating functionals ρ(F ) := ω(δF ) which satisfy:
Definition.
ρ : Q 7→ C is a function such that:
(i) ρ(0) = 1 ,
(ii) ρ(λF + G) is continuous in λ ∈ IR ∀ F, G ∈ Q ,
–6–
P
(iii)
i, j∈K
ρ(Fi − Fj ) λi λj exp −i
2 B(Fi , Fj ) ≥ 0
∀ Fi ∈ Q ,
λi ∈ C , Card(K) < ∞ .
The importance of regular states is much discussed in [11, 9], the chief one of
which is mentioned above, i.e. we can recapture the canonical fields from the representations associated with these. Hence the admission of nonregular states into
the theory would be tantamount to the abandonment of the existence of canonical
field operators. Nevertheless, as we will demonstrate below, linear hermitian supplementary conditions can never be imposed in the regular representations, and
this forces us to do physics in the abstract C*–picture, which is still possible.
3. Supplementary Conditions.
Suppose we need to impose linear Hermitian supplementary conditions on the
boson field above. This is done by specifying a linear space C ⊂ Q which will
correspond to the supplementary conditions cf. [4], and to select the set of physical
states by
℘D :=
©
¯
ª
ω ∈ ℘(F) ¯ ω(δC ) = 1 ∀ C ∈ C ,
the Dirac states. This corresponds to the supplementary condition
hψ| exp iχ |ψi = 1 of the heuristic theory. The theory stemming from this selection condition of ℘D has been extensively developed in [4–6], and we present
some of the results here. The nontriviality condition is:
℘D 6= ∅
iff B(C, C) = 0
iff 11 6∈ C ∗ (δC − 11) ,
where C ∗ (·) denotes the C*–algebra generated by its argument in the larger
¯
©
ª
C*–algebra F , and δC − 11 := δC − 11 ¯ C ∈ C . Henceforth we assume
¯
©
ª
nontriviality. Let p := F ∈ Q ¯ B(F, C) = 0 ∀ C ∈ C , with degenerate part
p0 :=
©
¯
ª
F ∈ p ¯ B(F, P ) = 0 ∀ P ∈ p ⊇ C .
then the final physical algebra is
±
e ,
Rc := C ∗ (δp ) C ∗ (δp ) C ∗ (δp0 − 11) ∼
= ∆(p/p0 , B)
–7–
e is the nondegenerate symplectic form on p/p0 , obtained from B .
where B
Clearly Rc is simple. Previously C ∗ (δp ) ≡ Oc has been called the observable
algebra, and C ∗ (δp ) C ∗ (δp0 − 11) ≡ Dc the constraint algebra.
Theorem 3.1.
Proof: Let
℘R ∩ ℘D = ∅ , i.e. no Dirac state is regular.
F ∈ Q\p , i.e.
∃C ∈ C
such that
B(F, C) 6= 0 . Then
by [4] we have ω(δF δλC ) = ω(δF ) = ω(δλC δF ) ∀ ω ∈ ℘D , λ ∈ IR ,
i.e.
ω(δF +λC ) exp 2i B(F, λC) = ω(δF ) = ω(δF +λC ) exp − 2i B(F, λC) .
Hence either λ B(F, C) = −λ B(F, C) mod 4π ∀ λ ∈ IR , in which case
ω(δF +λC ) = ω(δF ) , or ω(δF ) = ω(δF +λC ) = 0 . The first alternative gives B(F, C) = 0 , which contradicts our assumption. Hence
ω(δF ) = 0 ∀ F ∈ Q\p .
Since
λF ∈ Q\p ∀ λ ∈ IR\{0} , we see
ω(δλF ) = 0 ∀ λ ∈ IR\{0} . But ω(δ0 ) = 1 , hence λ 7→ ω(δλF ) is
discontinuous at 0 for F ∈ Q\p , ω ∈ ℘D .
Note that in the proof we showed that λ 7→ ω(δλF ) is discontinuous on nonphysical objects, Q\p . On physical objects p , we only have that ω(δF +λC ) =
ω(δF ) , i.e. regularity on F ∈ p is not excluded.
Theorem 3.2.
(i)
(ii)
Proof:
ω ∈ ℘D
π ∈ Pe
⇒
⇒
¡
¢
1 ∈ P σHω πω (δC ) ∀ C ∈ C ,
¡
¢
1 6∈ P σHπ π(δC ) for some C ∈ C .
(i) is clear from [4] 2.19, πω (D)Ωω = 0 ∀ ω ∈ ℘D , using δC − 11 ⊂
D , and hence πω (δC )Ωω = Ωω ∀ C ∈ C , i.e. Ωω is an eigenvector of
πω (δC ) with eigenvalue 1 , and hence 1 is in the point spectrum of
πω (δC ) .
(ii) Let π ∈ Pe be the GNS–representation of ω ∈ ℘R . All the vector states associated with ω are also in ℘R , because from the weak
¡
¢
operator continuity of λ 7→ π(δλF ) , we have that λ 7→ ξ, π(δλF )η
¡
¢
is continuous for all ξ, η ∈ Hπ . If 1 ∈ P σHπ π(δC ) ∀ C ∈ C , then
there are eigenvectors ξC ∈ Hπ such that π(δC )ξC = ξC ∀ C ∈ C .
However, from the nontriviality assumption B(C, C) = 0 , we have
£
¤
that π(δC ), π(δC 0 ) = 0 ∀ C, C 0 ∈ C and hence there is a simultaneous eigenvector ξ ∈ Hπ such that π(δC )ξ = ξ ∀ C ∈ C . Thus
¡
¢
ξ, π(δC )ξ = 1 ∀ C ∈ C , i.e. the vector state defined by ξ is a Dirac
–8–
state, but since it is also regular, this contradicts 3.1, and hence our
¡
¢
hypothesis is wrong, i.e. 1 6∈ P σHπ π(δC ) for some C ∈ C .
From this, we see that it is not possible to impose the given supplementary conditions in a regular representation, although it can easily be done in a Dirac state
representation, which happens to be nonregular on nonphysical objects. Since
regular states are identified with the existence of field operators, this suggests
that supplementary conditions can only be imposed in the abstract C*–approach,
not in the concrete field operator approach. Nevertheless, though regular states
must be relinquished on the field algebra of constrained systems, since the final
e is a CCR C*–algebra, we can
constraint–free physical algebra Rc = ∆(p/p0 , B)
once more on this algebra specify physical states as being regular, and so recover
field operators after elimination of degeneracies.
The states on
Rc
are derived from those states on
F
which have
C ∗ (δp0 − 11) in their kernels, and are obtained first by restriction to C ∗ (δp ) , and
±
then taken through the factorization C ∗ (δp ) 7→ C ∗ (δp ) C ∗ (δp ) C ∗ (δp0 − 11) . In
terms of generating functionals ρ , this is a restriction to p , and then we can
define generating functionals ρe([F ]) := ρ(F ) = ρ(F + C) ∀ C ∈ p0 , F ∈ p ,
because ω(δF ) = ω(δF +C ) ∀ F ∈ p, C ∈ p0 . (Note the small extension of the
constraint set from C to p0 ). Clearly the regular states on Rc arise from
nonregular states on F , but deviate from regularity only in nonphysical objects
Q\p . To those who insist on regularity for physically acceptable states on F ,
we ask why a physicality condition should be imposed on nonphysical objects.
In more general settings, we have given various methods for the imposition of
constraints in a C*–framework [4–7], and in the light of the analysis above, that
now seems the natural setting for the elimination of quantum degeneracies.
–9–
4. The Failure of Weak Continuity.
As mentioned in the introduction, an earlier attempt to construct an algebraic
theory of constraints for the electromagnetic field in terms of von Neumann algebras failed, because it was not possible to show that the relevant homomorphism
Oc 7→ Rc , where these algebras were interpreted as von Neumann algebras, was
weak operator continuous. We show in this section under more general conditions, that this homomorphism is necessarily discontinuous in the weak operator
topology as a consequence of the initial assumptions.
Suppose therefore that all the physical representations are in Pe . Since
π ∈ Pe
iff
λ 7→ π(δλF )
is continous in the weak operator topology for fixed
F ∈ Q , this suggests that the weak operator topology is more appropriate from a
physical point of view in the representation π ∈ Pe , than the uniform topology.
So, given an ω ∈ ℘R , the weak closure N := πω (F)00 is more appropriate as
a field algebra than is πω (F) . This is the von Neumann algebra point of view.
Any method for imposing the constraints in this framework, would have to respect
the weak operator topology.
Theorem 4.1. Given the system Dc := C ∗ (δp ) C ∗ (δC − 11) / C ∗ (δp ) ⊂ ∆(Q) =
F
as above, and a regular state ω ∈ ℘R , then the canonical
homomorphism
¡
¢
¡
¢±
ρ : πω C ∗ (δp ) 7→ πω C ∗ (δp ) πω (Dc )
is not continuous in the weak operator topology on B(Hω ) .
Proof: Assume πω (Dc ) acts degenerately on Hω , i.e. ∃ ξ ∈ Hω \0 such that
πω (Dc )ξ = 0 . Then the vector state ωξ (·) := (ξ, ·ξ) is a Dirac state
¡
¢
because ξ, πω (Dc )ξ = 0 . However, all the associated vector states of
a regular state are also regular, and so we have ωξ ∈ ℘R ∩ ℘D . Now
by theorem 3.1, ℘R ∩ ℘D = ∅ , and so ωξ 6 ∃ , i.e. πω (Dc ) acts nondegenerately on Hω .
The following result is stated in Pedersen [14] 2.2.5: Let A ⊂ B(Hω )
be a C*–algebra with strong operator closure (= weak operator closure)
M . Then M = A00 if A acts nondegenerately on Hω .
–10–
Apply this result to the present situation; let
¡
¢−−w
M = πω (Dc )
= πω (Dc )00 3 11 .
A = πω (Dc ) , then
Assume ρ is weak operator continuous. Then it can be extended
¢00
¡
to πω C ∗ (δp ) , and Ker ρ is weak operator closed. That is,
¡
¢−−w
11 ∈ πω (Dc )
= πω (Dc )00 ⊆ Ker ρ . But this means that Ker ρ
¡
¢00
cannot be a proper two-sided ideal of πω C ∗ ((δp ) , and hence
³ ¡
¢00 ´
∗
ρ πω C (δp )
= 0 . However, since F is simple, πω is faith³ ¡
¢´
¡
¢±
ful, and hence ρ πω C ∗ (δp ) = πω C ∗ (δp ) πω (Dc ) ∼
= Rc 6= 0 , and
since this contradicts the previous statement, the assumption is wrong.
Thus ρ is not weak operator continuous.
Notice that the proof generalizes to any homomorphism
where
¡
¢
ρe : πω C ∗ (δp ) 7→ A
A
is any nontrivial image C*–algebra, for which Ker ρe ⊃ πω (Dc ) .
¡
¢00
Hence there are no normal morphisms ρe : πω C ∗ (δp )
7→ M such that
πω (Dc ) ⊂ Ker ρe , and so the von Neumann algebra point of view is inappropriate for the imposition of constraints. This is in agreement with our previous
results that the canonical field operator point of view must be abandoned in the
presence of constraints. It is possible to ask if ρ is weak operator continuous
when ω ∈ ℘D , but since λ 7→ πω (δλF ) is not weak operator continuous in this
case, there is no reason here why the weak operator topology should be considered
physically relevant.
–11–
4. Conclusion.
In the presence of linear Hermitian constraints on linear boson theories, any attempt to impose them as eigenvalue conditions on physical states will fail because
such states do not exist. Similarly the use of von Neumann algebras will also not
be successful if weak continuity is demanded.
However if instead an abstract C*–algebra framework is used in order to
construct the final physical algebra from a CCR–algebra, then all these difficulties
disappear. For the final physical algebra suitable regular states exist, and all
evidence of unacceptable discontinuities will be confined to nonphysical elements.
Finally, we make two remarks about nonlinear situations. First, a method for
imposing quadratic constraints on a linear boson theory has been developed in [6],
and in that situation the same problem will exist although we will not consider
it here. Suffice it to say that the abstract C*–framework provided a well–defined
method for eliminating these constraints, and that the final physical C*–algebra
was a CCR–algebra. Second, in a fully nonlinear boson theory, if it is canonical
and it has linear hermitian constraints, then the situation described in this note
will be embedded in the nonlinear theory, and our findings are still valid.
–12–
Bibliography.
1. Jauch, J.M., Rohrlich, F.: The theory of photons and electrons, Reading,
Addison–Wesley 1955.
2. Hurst, C.A., Il Nuovo Cim. 21, 274 (1961).
3. Dirac, P.A.M.: Lectures on quantum mechanics, New York, Belfer Graduate
School of Science, Yeshiva University 1964.
4. Grundling, H.B.G.S., Hurst, C.A., Commun. Math. Phys. 98, 369 (1985).
5. Grundling, H., Hurst, C.A., J. Math. Phys. 28, 559 (1987).
6. Grundling, H.: Systems with outer constraints. Gupta–Bleuler electromagnetism as an algebraic field theory. Commun. Math. Phys. to appear.
7. Grundling, H., Hurst, C.A.: The quantum theory of second class constraints.
(to appear)
8. Manuceau, J., Ann. Inst. H. Poincaré 8, 139 (1968).
9. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical
mechanics vol. 2. Berlin, Heidelberg, New York, Springer 1979.
10. Slawny, J., Commun. Math. Phys. 24, 151 (1971).
11. Segal, I.E., Cargése lectures in theoretical physics 1967, ed. F. Lurcat, London, Gordon and Breach Science Publishers 1967.
12. Emch G.G.: Algebraic methods in statistical mechanics and quantum field
theory, New York, Wiley 1972.
13. Gel’fand, I.M., Vilenkin, N.Ya.: Generalised functions Vol. 4, New York,
Academic Press 1964.
14. Pedersen, G.K.: C*–algebras and their automorphism groups, London, Academic Press 1979.