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Commuting functions of the position and momentum observables on locally compact abelian groups
JOURNAL
OF MATHEMATlCAL
ANALYSIS
AND
APPLICATIONS
137, 185-192 (1989)
Commuting Functions of the Position and Momentum Observables Locally Compact Abelian Groups
on
KARI YLINEN Department
of Mathematics, SF-20500 Turku, Submitted
University Finland
uf Turku,
by James S. Howland
Received April 2, 1987
A theorem is proved which yields a characterization of the commuting bounded measurable functions, in particular the commuting spectral projections, of general position and momentum observables. 8 1989 Academic Press, Inc.
1. INTRODUCTION
AND NOTATION
Let G, unless specified, be an arbitrary locally compact abelian group, and r its dual group. The Haar measures II, (= dx) and Ar (= &) are assumed to be so adjusted that the Plancherel theorem holds [7, p. 261. We let 9: L*(G) -+ L*(f) denote the Fourier-Plancherel transformation. The characteristic function of a set X is denoted by xx. If Kc G, K’ is its annihilator in r. In general, the notation of [7] is followed; for measure and integration theory we refer to [3]. 1.1. DEFINITION. Let S= L*(G), and define EY(X) cp= xx’p for all XEB(G), cp~ A?. We call this spectral measure EY: 93(G) -+ L?(X) the (generalized) position observable (related to G). We also define EP( Y)cp = 9-‘(x+y) for all Y~9#(r), VE#, and call this EP: LB(r) -+ L?(X) the (generalized) momentum o@ervable (related to G). In the case of G = R’ (i.e., of a “free particle in one dimension”) the commutation problem of EY and EP has received considerable attention (see, e.g., [ 11). In particular, in [l] it was observed that though Eq and EP are totally noncommutative in the sense that for no nonzero cpE L’(W) is Eq(X) EP( Y)cp = EP( Y) Eq(X)(p for all X, YEB(R), certain nontrivial projections Eq(X) and EP( Y) do commute. More generally, the following characterization is (a slight reformulation of a statement) in [2]. 185 0022-247X/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
186
KARI
YLINEN
1.2. THEOREM [2]. Let f, g E L”(R). Then fF-‘[gF(cp)] = F-‘[g~(f~)]forallrp~L2(R)ifandonly tf one of thefollowing conditions holds: (i) f or g equals a constant a.e.; (ii) both f and g coincide a.e. with periodic functions with minimal periods CIand 8, respectively, such that 2443 is an integer.
The proof in [2] uses the special structure of R in an essential way, but we show that by different techniques one can get a generalization valid for any locally compact abelian group (Theorem 3.3). The generality is motivated by concrete applications: we note in Section 2 that also the study of the position and momentum of a “particle in a box” (Example 2.3) can be subsumed in the scheme of Definition 1.1.
2. Two EXAMPLES The following well-known simple result is recorded for convenience to unify the ensuing discussion. 2.1. LEMMA.
Let D c R be a Bore1 set and u: 93(a) + [0, a] a measure. Denote %0= L*(sZ, g(Q), u, C), and let A be the operator whose domain is 9A = (q E X0 1the function x H xv(x) is in X0} and for which (Aq)(x) = -v(x)9 cPE9AY x ESz. Then A is self-adjoin& and its spectral measure EA satisfies EAWcp=xx,a’p for all XEa(R),
(1)
cpGXO.
The usual quantum mechanical description of a free par2.2. EXAMPLE. ticle in one dimension involves the Hilbert space L*(R) (with respect to the Lebesgue measure m), the position observable Q taken as the self-adjoint operator with domain 5@o= { cpE L*(R) ) the function XH xv(x) is in L*(R)}, and satisfying Qcp(x) = xcp(x) for cpE go. The linear momentum observable is the self-adjoint operator P with domain gP = { cpE L*(R) 1cp is absolutely continuous on every compact interval, and cp’ E L*(R)}, and Pq(x)= -i(d/dx) q(x) (a.e.). (We use units as in [4, p. 901 so that the conversion factor A does not appear.) In this example we take G = lR with the Lebesgue measure as the Haar measure, so that r is R, and its Haar measure is the Lebesgue measure divided by 271. If we regard F as an operator from L*(R) = L*(R, m) onto itself (rather than onto L*(R, (2n))‘m)), then (27c)i/*~ is unitary and we have P=B-‘QF
OBSERVABLESON
187
LCA GROUPS
(see, e.g., [lo, p.441; 11, p. 3153; we now have ~=((27~)‘/~ notation of [ 10, 111). It is also well known (and follows from combined with the formula P= SP’QP) that Eq (resp. EP) in Definition 1.1 is the spectral measure of the self-adjoint (resp. P).
UP1 in the Lemma 2.1 the sense of operator Q
Let m be the Lebesgue measure on the interval [0, a], denote X0= L2( [0, a], m). Define Q: So + So by Qcp(x) = xv(x), and P: GSp+ I& by Pq(x) = -i(d/dx) q(x) (a.e.) where ~9~= { cpE X0 ( cp is absolutely continuous, cp’ E X0, and ~(0) = q(a)}. Again, both Q and P are known to be self-adjoint (see, e.g., Lemma 2.1 and [S, p. 3313). Let G be the quotient group R/ah, and choose for G the Haar measure with total mass a. If p: [0, a] -+ R/aZ is the quotient map restricted to [0, a], the mapping f3 for which e(f) =fop is an isometric isomorphism from &? onto X0, where 2 = L*(G) as in Definition 1.1. For each n E Z we define yn: G + C by y,(x + am) = exp(i2za- ‘nx), x E R. Then the dual group r of G is {y, 1n E Z}; we identify 2na-‘n with yn, so that r= 2za-‘Z. Now let EY: 9(G) + Y(X) and EP: S?(f) + Y(X) be as in Definition 1.1. 2.3. EXAMPLE. a E (0, 02L and
2.4. PROPOSITION. satisfies
(a)
The spectral measure EQ of Q in Example 2.3
EQ(X) = BEYp(Xn [0, a]) 8-l
(b)
for XE 99(R).
The spectral measureEp of P in Example 2.5 satisfies E’(X) = BEP(Xn2za-‘Z)
8-l
for XE%?(R).
Proof. (a) We apply Lemma 2.1 to sZ= [0, a] and p=m 1SY([O, a]). In its notation A = Q, so that E”(X)q =x~~~~,+,(P for all XE~(R), cpE So. But BEYp(Xn [0, a]) tV’(p is the same element (equivalence class) in 6 as xXn Co,al~. (b) Now we take 52 = 2na-‘Z (= r), and for p the Haar measure of r (the counting measure divided by a). Let then A, gA, and EA be as in Lemma 2.1. For all m, n E Z we have F”y,(2na-‘m)
= 6 exp(i2zaC’(n-m)x)dx=ad,,,
=n2za-‘7,. On the other so that ASy,, = n2n~~,,,~,,~, and .9-‘A9y, hand, -i(d/dx)(f?y,)(x) = -i(d/dx) exp(i2na-‘nx) = n2na-‘(@,)(x). It follows that A and B= 98-‘PM-’ agree on all finitely supported functions on r=2~a-*H. (Note that t7y,Egp.) Both A and 9;8-‘PtU-’
188
KARI
YLINEN
are self-adjoint (see 2.1 and 2.3)). If g E gA, define g, = x1 ~2ncr~ln,.,,,Zncr~ln~g. Then g, + g and Bg, = Ag, + Ag. Since B is closed, we have g E gB, and Bg = Ag. Thus B is an extension of A, but this implies that A = B, since both A and B are self-adjoint [8, p. 3371. (Compare also the proof in [ll, pp. 31553161.) Thus from Lemma 2.1 it follows that the spectral measure of P = 89 ‘A.98 ~ ’ has as its value for XE B( W) the projection ‘PH W-‘(X
Xn2rrl-lL5FO~‘cp)
= BEP(Xn
27cC’Z)
P”p.
1
2.5. Remark. Proposition 2.4(b) yields, in particular, a quick proof of the fact that the spectrum of P in Example 2.3, i.e., the support supp(EP) of its spectral measure, is 27rna~ ‘Z and (being discrete) consists of eigenvalues.
3. THE MAIN
RESULT
Recall that G is an arbitrary locally compact abelian group. If fe L”(G) and ge L”(T), we define Q(f), Y(g): L*(G) --t L*(G) by @(f)cp =fq and lu(g)rp=9”[g9(q)], (Thus @(xx)=E4(X) for XE~?(G) and Y(v(x,,)= EP( Y) for YE B(f).) For y E G, we say that a function f: G + @ is essentially y-invariant, if f(x + y) =f(x) for locally almost every x E G. For Kc G, f is said to be essentially K-invariant, if f is essentially y-invariant for every YEK. For ~‘EL”(G) we denote Cf= {hall 1 @(f) y(h)= y(‘(h) Q(f)}, and Kf.= {x E G 1f is essentially x-invariant} (and define similarlyK,crforgEL”(T)).Iff:G-,@andg:r~@,wewritef,(y)= fly-x), x,y~G, and g,(6)=g(6-y), Y, Ser. 3.1. LEMMA. (a) (b) (cl
(d) (x, y)=l; (e)
(f)
Let feL”(G),
gEL”(T),
XEG, and YET. Then
WY) = Q(y) Y(g) Q(Y); @(fx) = Ylu((x>.)I @i(f) Vu((x, .I); Kf= {=G I (x, .,+); for x E G, y Er, Q(y) Y((x, .)) = Y((x, .)) D(y) if and only if C, is a weak* closed translation invariant subspaceof L”(r); K, is a closed subgroup of G.
ProoJ Straightforward. (For instance, (a) is used in (e), (b) in (c) and (d), and since XH (x, .) is continuous from G to a(L”(T), L’(T)), K, is closed by (e) and (c).) 1 3.2. LEMMA. Let Kc G be a closed subgroup and f: G + Q=a bounded &measurable function. The following conditions are equivalent:
OBSERVABLE.5
ON
LCA
189
GROUPS
(i) f is essentially K-invariant; (ii) there is a net (h)icX of linear combinations
of characters
in K’
such that liy jG f,(x) 0)
h(x dx
dx = 1 f(x) G
(2)
for all h E L’(G). Proof (i) = (ii): Assume (i). Then there is a Ha r measurable function f’: G/K -+ C such that f(x) = f’(n(x)) 1.a.e. where rr: G + G/K is the quotient map (see [6, p. 821). Clearly, f’ may be assumed to be bounded. Since the dual group of G/K separates L’(G/K), there is a net (f i)ie,p of linear combinations of continuous characters of G/K converging to f’ with respect to o(L”(G/K), L’(G/K)) (see, e.g., [9, p. 1251). The Haar measure of G/K is taken to be normalized in such a way that the Weil formula h(x+y)dy
di=j
h(x)dx
(3)
G
holds for all h E L’(G) ‘means the following: function y H h(x + y) a-x(i) = SK h(x +y) belongs to L’(G/K)), also for x H f ‘(n(x))
(we denote i = X(X)). As explained in [6, p. 701, this There is a negligible set N in G/K such that the is in L’(G) for every x E G with n(x) 4 N, the function dy is (well) defined almost everywhere on G/K, it
and (3) holds. Clearly, all this can be taken to hold to h, and so we get
h(x) in addition
= =s
f ‘(4x))
jK 4x + y) &}
dk
G,K f ‘(i) z(a) dk
and a similar K(i) di=
formula
jGIK f’(i)
(ii)*(i):
for f, in place of f’(n( K(i) dz?, we get (2).
Let (fi)rc,,
.)). Since lim,J,,,
be as in (ii). For any ye K and hE L’(G)
f I(i)
190
KARI YLINEN
= lim G.fifi(x) w ; s
-Y) dx
= liy jGfix +Y)0) dx = liy JGfjf,(x)h(x) dx, and so f(x+~)=f(x)
1.a.e. 1
3.3. THEOREM. Suppose f ELM conditions are equivalent: (i) f~-‘[gS(cp)]=~--‘[go]
and gEL”(T).
for all MEL’,
The following i.e., Q(f)
and Y(g) commute;
(ii) (iii)
K, 1 Kt;
there is a closed subgroup KC G such that f K-invariant and g is essentially K’-invariant.
is essentially
ProoJ: (i) =S (ii): Assume (i). The set C,= (h E L”(T) 1Q(f) Y(h) = Y(h) Q(f)} is a weak* closed translation invariant subspace of L”(T), and a character (x, .), x E G, of r belongs to CY if and only if x E K/ (see Lemma 3.1), i.e., K, is the spectrum of C, in the sense of [7, p. 1843. (We are regarding G as the dual group of r.) Now K, is a closed subgroup of G
(Lemma 3.1(f)), and spectral synthesis is known to hold for closed subgroups. More precisely, combining Theorem 4 in [S, p. 2591 with Theorem 7.8.2 in [7, p. 1843 we see that C, is the weak* closed subspace generated by the continuous characters in C,, and so g (E Cr) can be approximated in o(L”(ZJ, L’(T)) by linear combinations of characters (x, .) where x E K, But Kf= Kk’ [7, p. 361, and applying Lemma 3.2 to g and K/’ in place of f and K we see that g is essentially Kt-invariant, i.e., K, 2 Kf’. (ii) =s.(iii): If (ii) holds, then (iii) holds with K= K, (iii) * (i): Assume (iii). By Lemma 3.2 there is a net ( fj)ies (resp. (g!‘i’-, ) of linear combinations of characters of G in K’ (resp. of r in K -K) such that fi+f in a(L”(T), L’(T)) (resp. gj-+g in o(L”(T), L’(T))). From Lemma 3.1(d) it follows that @(fi) !P(g,) = ul(g,) @(fi) for all i E 4 and j E 9. Using the Plancherel theorem one can easily show that @(fi) + Q(f) and !Y( g,) + Y’(g) in the weak operator topology, with respect to which multiplication is separately continuous, and SO c?(f) Y(g) = Y’(g) a(f), i.e., (i) holds. 1
OBSERVABLES ON LCA GROUPS
191
3.4. EXAMPLE. (a) Suppose G= R in Theorem 3.3. Then K, is {0}, R, or aZ for some CI> 0, so that K,I = R, {0}, or 27~6 ‘Z, respectively. For fe L”(R), Kf= R is equivalent to requiring that for some constant c, f(x) = c a.e. (To see this, note that one may assume f to be Bore1 measurable and apply Fubini’s theorem to the characteristic function of the Bore1 set {(x, y) E R2 1f(x +y) #f(x)}. Now 1.a.e. means the same as a.e., since R is o-compact.) It is even more elementary to see that in the case ZC,= cr7 f agrees a.e. with the function obtained by extending f 1[IO, LY]to be an cc-periodic function on R. From these remarks it easily follows that Theorem 3.3 implies Theorem 1.2. (b) Let CI> 0 and take G = [w/an as in Proposition 2.4 and the discussion preceding it. The closed subgroups of G are {0}, G, and the images of sets of the form (0, n-la, 2n-‘a, .... a}, no N, n>2, under the quotient map from R onto R/aZ. Their annihilators in the dual group r= 2xa ~‘Z of G are r, {0}, and 2nna-‘Z, respectively. Using Proposition 2.4, we get in this case from Theorem 3.3 a result which we formulate as follows: 3.5. COROLLARY. If f, g: R + @ are bounded Bore1 functions, their spectral integrals JRf dE Q and JR g dE ‘, where E Q and EP are the spectral measuresof the position Q and momentum P of a “particle in the box [0, a]” (Example 2.3), commute if and only if one of the following conditions holds: (i)
the restriction
f 1 [0, a] equals a constant a.e. or g 12za-‘Z
is
constant;
(ii) there is an integer n > 2 such that 2na-‘n is a period g 12na- ‘Z, and whenever j= 1, .... n-l, the equality f(x)=f(x+jan-‘) holdsfor almost every x E [0, an ~ ‘1.
of
In particular, we have obtained a characterization of the commuting spectral projections EQ(X), E ‘( Y). Of course only Xn supp( EQ) = Xn [0, a] and Y n supp(EP) = Yn 2na-‘Z are relevant, and if both sets are nontrivial, the condition involves periodicity (for Xn [0, a] modulo the Lebesgue measure) with suitably related periods. ACKNOWLEDGMENTS This research was supported by the Academy of Finland. The author is also grateful for the hospitality at the Department of Mathematics and Computer Science, University of California, Riverside, and at the Institute for Theoretical Physics of the University of Cologne. REFERENCES 1. P. BUSCHAND P. J. LAHTI, To what extent do position and momentum commute? Phys. Lea. A 115 (1986), 259-264.
192
KARI YLINEN
2. P. BUSCH, T. P. SCHONBEK, AND F. E. SCHROECK, JR., Quantum observables: Compatibility versus commutativity and maximal information, J. Math. Phys. 28 (1987) 2866-2872. 3.
4. 5. 6. 7. 8. 9. 10. 11,
E. HEWITT AND K. A. Ross, “Abstract Harmonic Analysis,” Vol. I, Springer-Verlag, New York/Berlin, 1963. G. W. MACKEY, “Induced Representations of Groups and Quantum Mechanics,” Benjamin, New York, 1968. H. REITER, Contributions to harmonic analysis, Acta Math. 96 (1956), 253-263. H. REITER, “Classical Harmonic Analysis and Locally Compact Groups,” Oxford Univ. Press, Oxford, 1968. W. RUDIN, “Fourier Analysis on Groups,” Interscience, New York, 1962. W. RUDIN, “Functional Analysis,” McGraw-Hill, New York, 1973. H. H. SCHAEFER,“Topological Vector Spaces,” Springer-Verlag, New York/Heidelberg/ Berlin, 1971. M. H. STONE,Linear Transformations in Hilbert Space, “Amer. Math. Sot. Colloq. Publ.,” Vol. 15, Amer. Math. Sot., Providence, RI, 1932. K. YOSIDA, “Functional Analysis,” Springer-Verlag, New York/Berlin, 1968.
OF MATHEMATlCAL
ANALYSIS
AND
APPLICATIONS
137, 185-192 (1989)
Commuting Functions of the Position and Momentum Observables Locally Compact Abelian Groups
on
KARI YLINEN Department
of Mathematics, SF-20500 Turku, Submitted
University Finland
uf Turku,
by James S. Howland
Received April 2, 1987
A theorem is proved which yields a characterization of the commuting bounded measurable functions, in particular the commuting spectral projections, of general position and momentum observables. 8 1989 Academic Press, Inc.
1. INTRODUCTION
AND NOTATION
Let G, unless specified, be an arbitrary locally compact abelian group, and r its dual group. The Haar measures II, (= dx) and Ar (= &) are assumed to be so adjusted that the Plancherel theorem holds [7, p. 261. We let 9: L*(G) -+ L*(f) denote the Fourier-Plancherel transformation. The characteristic function of a set X is denoted by xx. If Kc G, K’ is its annihilator in r. In general, the notation of [7] is followed; for measure and integration theory we refer to [3]. 1.1. DEFINITION. Let S= L*(G), and define EY(X) cp= xx’p for all XEB(G), cp~ A?. We call this spectral measure EY: 93(G) -+ L?(X) the (generalized) position observable (related to G). We also define EP( Y)cp = 9-‘(x+y) for all Y~9#(r), VE#, and call this EP: LB(r) -+ L?(X) the (generalized) momentum o@ervable (related to G). In the case of G = R’ (i.e., of a “free particle in one dimension”) the commutation problem of EY and EP has received considerable attention (see, e.g., [ 11). In particular, in [l] it was observed that though Eq and EP are totally noncommutative in the sense that for no nonzero cpE L’(W) is Eq(X) EP( Y)cp = EP( Y) Eq(X)(p for all X, YEB(R), certain nontrivial projections Eq(X) and EP( Y) do commute. More generally, the following characterization is (a slight reformulation of a statement) in [2]. 185 0022-247X/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
186
KARI
YLINEN
1.2. THEOREM [2]. Let f, g E L”(R). Then fF-‘[gF(cp)] = F-‘[g~(f~)]forallrp~L2(R)ifandonly tf one of thefollowing conditions holds: (i) f or g equals a constant a.e.; (ii) both f and g coincide a.e. with periodic functions with minimal periods CIand 8, respectively, such that 2443 is an integer.
The proof in [2] uses the special structure of R in an essential way, but we show that by different techniques one can get a generalization valid for any locally compact abelian group (Theorem 3.3). The generality is motivated by concrete applications: we note in Section 2 that also the study of the position and momentum of a “particle in a box” (Example 2.3) can be subsumed in the scheme of Definition 1.1.
2. Two EXAMPLES The following well-known simple result is recorded for convenience to unify the ensuing discussion. 2.1. LEMMA.
Let D c R be a Bore1 set and u: 93(a) + [0, a] a measure. Denote %0= L*(sZ, g(Q), u, C), and let A be the operator whose domain is 9A = (q E X0 1the function x H xv(x) is in X0} and for which (Aq)(x) = -v(x)9 cPE9AY x ESz. Then A is self-adjoin& and its spectral measure EA satisfies EAWcp=xx,a’p for all XEa(R),
(1)
cpGXO.
The usual quantum mechanical description of a free par2.2. EXAMPLE. ticle in one dimension involves the Hilbert space L*(R) (with respect to the Lebesgue measure m), the position observable Q taken as the self-adjoint operator with domain 5@o= { cpE L*(R) ) the function XH xv(x) is in L*(R)}, and satisfying Qcp(x) = xcp(x) for cpE go. The linear momentum observable is the self-adjoint operator P with domain gP = { cpE L*(R) 1cp is absolutely continuous on every compact interval, and cp’ E L*(R)}, and Pq(x)= -i(d/dx) q(x) (a.e.). (We use units as in [4, p. 901 so that the conversion factor A does not appear.) In this example we take G = lR with the Lebesgue measure as the Haar measure, so that r is R, and its Haar measure is the Lebesgue measure divided by 271. If we regard F as an operator from L*(R) = L*(R, m) onto itself (rather than onto L*(R, (2n))‘m)), then (27c)i/*~ is unitary and we have P=B-‘QF
OBSERVABLESON
187
LCA GROUPS
(see, e.g., [lo, p.441; 11, p. 3153; we now have ~=((27~)‘/~ notation of [ 10, 111). It is also well known (and follows from combined with the formula P= SP’QP) that Eq (resp. EP) in Definition 1.1 is the spectral measure of the self-adjoint (resp. P).
UP1 in the Lemma 2.1 the sense of operator Q
Let m be the Lebesgue measure on the interval [0, a], denote X0= L2( [0, a], m). Define Q: So + So by Qcp(x) = xv(x), and P: GSp+ I& by Pq(x) = -i(d/dx) q(x) (a.e.) where ~9~= { cpE X0 ( cp is absolutely continuous, cp’ E X0, and ~(0) = q(a)}. Again, both Q and P are known to be self-adjoint (see, e.g., Lemma 2.1 and [S, p. 3313). Let G be the quotient group R/ah, and choose for G the Haar measure with total mass a. If p: [0, a] -+ R/aZ is the quotient map restricted to [0, a], the mapping f3 for which e(f) =fop is an isometric isomorphism from &? onto X0, where 2 = L*(G) as in Definition 1.1. For each n E Z we define yn: G + C by y,(x + am) = exp(i2za- ‘nx), x E R. Then the dual group r of G is {y, 1n E Z}; we identify 2na-‘n with yn, so that r= 2za-‘Z. Now let EY: 9(G) + Y(X) and EP: S?(f) + Y(X) be as in Definition 1.1. 2.3. EXAMPLE. a E (0, 02L and
2.4. PROPOSITION. satisfies
(a)
The spectral measure EQ of Q in Example 2.3
EQ(X) = BEYp(Xn [0, a]) 8-l
(b)
for XE 99(R).
The spectral measureEp of P in Example 2.5 satisfies E’(X) = BEP(Xn2za-‘Z)
8-l
for XE%?(R).
Proof. (a) We apply Lemma 2.1 to sZ= [0, a] and p=m 1SY([O, a]). In its notation A = Q, so that E”(X)q =x~~~~,+,(P for all XE~(R), cpE So. But BEYp(Xn [0, a]) tV’(p is the same element (equivalence class) in 6 as xXn Co,al~. (b) Now we take 52 = 2na-‘Z (= r), and for p the Haar measure of r (the counting measure divided by a). Let then A, gA, and EA be as in Lemma 2.1. For all m, n E Z we have F”y,(2na-‘m)
= 6 exp(i2zaC’(n-m)x)dx=ad,,,
=n2za-‘7,. On the other so that ASy,, = n2n~~,,,~,,~, and .9-‘A9y, hand, -i(d/dx)(f?y,)(x) = -i(d/dx) exp(i2na-‘nx) = n2na-‘(@,)(x). It follows that A and B= 98-‘PM-’ agree on all finitely supported functions on r=2~a-*H. (Note that t7y,Egp.) Both A and 9;8-‘PtU-’
188
KARI
YLINEN
are self-adjoint (see 2.1 and 2.3)). If g E gA, define g, = x1 ~2ncr~ln,.,,,Zncr~ln~g. Then g, + g and Bg, = Ag, + Ag. Since B is closed, we have g E gB, and Bg = Ag. Thus B is an extension of A, but this implies that A = B, since both A and B are self-adjoint [8, p. 3371. (Compare also the proof in [ll, pp. 31553161.) Thus from Lemma 2.1 it follows that the spectral measure of P = 89 ‘A.98 ~ ’ has as its value for XE B( W) the projection ‘PH W-‘(X
Xn2rrl-lL5FO~‘cp)
= BEP(Xn
27cC’Z)
P”p.
1
2.5. Remark. Proposition 2.4(b) yields, in particular, a quick proof of the fact that the spectrum of P in Example 2.3, i.e., the support supp(EP) of its spectral measure, is 27rna~ ‘Z and (being discrete) consists of eigenvalues.
3. THE MAIN
RESULT
Recall that G is an arbitrary locally compact abelian group. If fe L”(G) and ge L”(T), we define Q(f), Y(g): L*(G) --t L*(G) by @(f)cp =fq and lu(g)rp=9”[g9(q)], (Thus @(xx)=E4(X) for XE~?(G) and Y(v(x,,)= EP( Y) for YE B(f).) For y E G, we say that a function f: G + @ is essentially y-invariant, if f(x + y) =f(x) for locally almost every x E G. For Kc G, f is said to be essentially K-invariant, if f is essentially y-invariant for every YEK. For ~‘EL”(G) we denote Cf= {hall 1 @(f) y(h)= y(‘(h) Q(f)}, and Kf.= {x E G 1f is essentially x-invariant} (and define similarlyK,crforgEL”(T)).Iff:G-,@andg:r~@,wewritef,(y)= fly-x), x,y~G, and g,(6)=g(6-y), Y, Ser. 3.1. LEMMA. (a) (b) (cl
(d) (x, y)=l; (e)
(f)
Let feL”(G),
gEL”(T),
XEG, and YET. Then
WY) = Q(y) Y(g) Q(Y); @(fx) = Ylu((x>.)I @i(f) Vu((x, .I); Kf= {=G I (x, .,+); for x E G, y Er, Q(y) Y((x, .)) = Y((x, .)) D(y) if and only if C, is a weak* closed translation invariant subspaceof L”(r); K, is a closed subgroup of G.
ProoJ Straightforward. (For instance, (a) is used in (e), (b) in (c) and (d), and since XH (x, .) is continuous from G to a(L”(T), L’(T)), K, is closed by (e) and (c).) 1 3.2. LEMMA. Let Kc G be a closed subgroup and f: G + Q=a bounded &measurable function. The following conditions are equivalent:
OBSERVABLE.5
ON
LCA
189
GROUPS
(i) f is essentially K-invariant; (ii) there is a net (h)icX of linear combinations
of characters
in K’
such that liy jG f,(x) 0)
h(x dx
dx = 1 f(x) G
(2)
for all h E L’(G). Proof (i) = (ii): Assume (i). Then there is a Ha r measurable function f’: G/K -+ C such that f(x) = f’(n(x)) 1.a.e. where rr: G + G/K is the quotient map (see [6, p. 821). Clearly, f’ may be assumed to be bounded. Since the dual group of G/K separates L’(G/K), there is a net (f i)ie,p of linear combinations of continuous characters of G/K converging to f’ with respect to o(L”(G/K), L’(G/K)) (see, e.g., [9, p. 1251). The Haar measure of G/K is taken to be normalized in such a way that the Weil formula h(x+y)dy
di=j
h(x)dx
(3)
G
holds for all h E L’(G) ‘means the following: function y H h(x + y) a-x(i) = SK h(x +y) belongs to L’(G/K)), also for x H f ‘(n(x))
(we denote i = X(X)). As explained in [6, p. 701, this There is a negligible set N in G/K such that the is in L’(G) for every x E G with n(x) 4 N, the function dy is (well) defined almost everywhere on G/K, it
and (3) holds. Clearly, all this can be taken to hold to h, and so we get
h(x) in addition
= =s
f ‘(4x))
jK 4x + y) &}
dk
G,K f ‘(i) z(a) dk
and a similar K(i) di=
formula
jGIK f’(i)
(ii)*(i):
for f, in place of f’(n( K(i) dz?, we get (2).
Let (fi)rc,,
.)). Since lim,J,,,
be as in (ii). For any ye K and hE L’(G)
f I(i)
190
KARI YLINEN
= lim G.fifi(x) w ; s
-Y) dx
= liy jGfix +Y)0) dx = liy JGfjf,(x)h(x) dx, and so f(x+~)=f(x)
1.a.e. 1
3.3. THEOREM. Suppose f ELM conditions are equivalent: (i) f~-‘[gS(cp)]=~--‘[go]
and gEL”(T).
for all MEL’,
The following i.e., Q(f)
and Y(g) commute;
(ii) (iii)
K, 1 Kt;
there is a closed subgroup KC G such that f K-invariant and g is essentially K’-invariant.
is essentially
ProoJ: (i) =S (ii): Assume (i). The set C,= (h E L”(T) 1Q(f) Y(h) = Y(h) Q(f)} is a weak* closed translation invariant subspace of L”(T), and a character (x, .), x E G, of r belongs to CY if and only if x E K/ (see Lemma 3.1), i.e., K, is the spectrum of C, in the sense of [7, p. 1843. (We are regarding G as the dual group of r.) Now K, is a closed subgroup of G
(Lemma 3.1(f)), and spectral synthesis is known to hold for closed subgroups. More precisely, combining Theorem 4 in [S, p. 2591 with Theorem 7.8.2 in [7, p. 1843 we see that C, is the weak* closed subspace generated by the continuous characters in C,, and so g (E Cr) can be approximated in o(L”(ZJ, L’(T)) by linear combinations of characters (x, .) where x E K, But Kf= Kk’ [7, p. 361, and applying Lemma 3.2 to g and K/’ in place of f and K we see that g is essentially Kt-invariant, i.e., K, 2 Kf’. (ii) =s.(iii): If (ii) holds, then (iii) holds with K= K, (iii) * (i): Assume (iii). By Lemma 3.2 there is a net ( fj)ies (resp. (g!‘i’-, ) of linear combinations of characters of G in K’ (resp. of r in K -K) such that fi+f in a(L”(T), L’(T)) (resp. gj-+g in o(L”(T), L’(T))). From Lemma 3.1(d) it follows that @(fi) !P(g,) = ul(g,) @(fi) for all i E 4 and j E 9. Using the Plancherel theorem one can easily show that @(fi) + Q(f) and !Y( g,) + Y’(g) in the weak operator topology, with respect to which multiplication is separately continuous, and SO c?(f) Y(g) = Y’(g) a(f), i.e., (i) holds. 1
OBSERVABLES ON LCA GROUPS
191
3.4. EXAMPLE. (a) Suppose G= R in Theorem 3.3. Then K, is {0}, R, or aZ for some CI> 0, so that K,I = R, {0}, or 27~6 ‘Z, respectively. For fe L”(R), Kf= R is equivalent to requiring that for some constant c, f(x) = c a.e. (To see this, note that one may assume f to be Bore1 measurable and apply Fubini’s theorem to the characteristic function of the Bore1 set {(x, y) E R2 1f(x +y) #f(x)}. Now 1.a.e. means the same as a.e., since R is o-compact.) It is even more elementary to see that in the case ZC,= cr7 f agrees a.e. with the function obtained by extending f 1[IO, LY]to be an cc-periodic function on R. From these remarks it easily follows that Theorem 3.3 implies Theorem 1.2. (b) Let CI> 0 and take G = [w/an as in Proposition 2.4 and the discussion preceding it. The closed subgroups of G are {0}, G, and the images of sets of the form (0, n-la, 2n-‘a, .... a}, no N, n>2, under the quotient map from R onto R/aZ. Their annihilators in the dual group r= 2xa ~‘Z of G are r, {0}, and 2nna-‘Z, respectively. Using Proposition 2.4, we get in this case from Theorem 3.3 a result which we formulate as follows: 3.5. COROLLARY. If f, g: R + @ are bounded Bore1 functions, their spectral integrals JRf dE Q and JR g dE ‘, where E Q and EP are the spectral measuresof the position Q and momentum P of a “particle in the box [0, a]” (Example 2.3), commute if and only if one of the following conditions holds: (i)
the restriction
f 1 [0, a] equals a constant a.e. or g 12za-‘Z
is
constant;
(ii) there is an integer n > 2 such that 2na-‘n is a period g 12na- ‘Z, and whenever j= 1, .... n-l, the equality f(x)=f(x+jan-‘) holdsfor almost every x E [0, an ~ ‘1.
of
In particular, we have obtained a characterization of the commuting spectral projections EQ(X), E ‘( Y). Of course only Xn supp( EQ) = Xn [0, a] and Y n supp(EP) = Yn 2na-‘Z are relevant, and if both sets are nontrivial, the condition involves periodicity (for Xn [0, a] modulo the Lebesgue measure) with suitably related periods. ACKNOWLEDGMENTS This research was supported by the Academy of Finland. The author is also grateful for the hospitality at the Department of Mathematics and Computer Science, University of California, Riverside, and at the Institute for Theoretical Physics of the University of Cologne. REFERENCES 1. P. BUSCHAND P. J. LAHTI, To what extent do position and momentum commute? Phys. Lea. A 115 (1986), 259-264.
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2. P. BUSCH, T. P. SCHONBEK, AND F. E. SCHROECK, JR., Quantum observables: Compatibility versus commutativity and maximal information, J. Math. Phys. 28 (1987) 2866-2872. 3.
4. 5. 6. 7. 8. 9. 10. 11,
E. HEWITT AND K. A. Ross, “Abstract Harmonic Analysis,” Vol. I, Springer-Verlag, New York/Berlin, 1963. G. W. MACKEY, “Induced Representations of Groups and Quantum Mechanics,” Benjamin, New York, 1968. H. REITER, Contributions to harmonic analysis, Acta Math. 96 (1956), 253-263. H. REITER, “Classical Harmonic Analysis and Locally Compact Groups,” Oxford Univ. Press, Oxford, 1968. W. RUDIN, “Fourier Analysis on Groups,” Interscience, New York, 1962. W. RUDIN, “Functional Analysis,” McGraw-Hill, New York, 1973. H. H. SCHAEFER,“Topological Vector Spaces,” Springer-Verlag, New York/Heidelberg/ Berlin, 1971. M. H. STONE,Linear Transformations in Hilbert Space, “Amer. Math. Sot. Colloq. Publ.,” Vol. 15, Amer. Math. Sot., Providence, RI, 1932. K. YOSIDA, “Functional Analysis,” Springer-Verlag, New York/Berlin, 1968.