Communicated by E. P. Wigner, May 20, Introduction.-In a previous note we have given some results on representations The object of the present

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1 REPRESENTATIONS OF THE COMMUTATION RELATIONS BY L. GARDING AND A. WIGHTMAN UNIVERSITY OF LUND AND INSTITUTE FOR ADVANCED STUDY; PRINCETON UNIVERSITY Communicated by E. P. Wigner, May 20, 1954 Introduction.-In a previous note we have given some results on representations The object of the present For a definition of of the anticommutation relations of quantum physics.1 note is to give similar results for the commutation relations. direct integrals of Hilbert spaces which will be used below we refer the reader to our previous note the paper by J. von Neumann.2 Roughly speaking, a representation of the commutation rules is two sets of unbounded self-adjoint operators, {IP " { qk4i on a separable Hilbert space H satisfying Pkqj - qjpk = i 6bk7 qkqj - qjqk = 0, PkpJ - pjpk = 0. (1) In order to avoid difficulties with the domains of these operators, we introduce with H. Weyl3 the operators U(a) = exp i 2akpk V(a) = exp i lakqk, where the numbers ak are real all but a finite number of them vanish if m = oa. Then { U(a)} { V(a)} are two sets of unitary operators on H which are continuous functions of the components ak of a satisfy U(O) = V(0) = 1, U(a)U(a') = U(a + a'), V(a)V(a') = V(a + a'), U(a)V(3) = exp i ac3v(q3)u(a) (a'f3 = Eak3k). (2) Two sets of unitary operators { U(a) } j V(a) I with these properties will be called a representation of the commutation rules. The infinitesimal generators Pk = (1/i) (bu(a)/1ak)a...o qk = (1/i)(6V(a)/1bai)azo are then densely defined self-adjoint, (1) is true on a dense subset of H. The problem of finding all representations was solved by von Neumann4 when m < o. He found that every representation is a direct sum of irreducible representations all equivalent to each other. For m = o, Friedrichs5 has found some essentially inequivalent representations. We shall consider this case more closely, give all representations, state some results on irreducibility unitary equivalence which show that here, as in the-case of the anticommutation relations, there exists a maze of irreducible inequivalent representations. 1. The Representations. On the common domain of definition of Pk qk put 1 ak = (- iqk) ak * = (pk + iqk)- 622

2 VOL. 40, 1954 PHYSICS: GARDING AND WIGHTMAN 623 By virtue of von Neumann's results, ak* is then the actual adjoint of ak, a,** - ak. Moreover, akak* is defined on a dense subset of H has the form akak * = Ei'= O (1 + 1)Pk1, where the PkI are commuting projections which satisfy Pk'Pk' = 8jlPk P~'O E 1"=0 Pk' pk1 1. Let I be the set of all infinite sequences n = (n1, n2,...) with integral components nk > 0. It is a semigroup under component-wise addition. Let A denote the set of "rational" sequences, i.e., those with only a finite number of components different from zero. It is generated by the sequences ak = (0,...,0, 1, 0,... ), with 1 in the kth place. Let I*> be the set of all n in I for which nk = j. By a Borel set in I we mean a set constructed from all the sets Ikj by the usual countable processes. In the following, when speaking of a measure on I, we mean the completion of a totally additive, bounded, not negative, not identically vanishing set function defined on the Borel sets. For a given representation we can always write H as a direct integral over I, f, HndM(n), which diagonalizes the projections Pki in such a fashion that if f corresponds to the function f(n), then P i corresponds to the function 5j, f(n). The measure ju is quasi-invariant in the sense that M&(n) the translated measure u(n + 8) are absolutely continuous with respect to each other for all a in A. The dimension v(n) of the Hilbert space H,, is invariant under A, v(n + 6) = '(n) for almost all n all 6 in A. The operators ak ak* are given by (akf)(n) = akf(n) = -/n+ 1 Ck(f)f(f + 8k) Id(n +k) d/a(n) ak*f(n) = V/CkCk*(n -8k)f(n - 8) f(n -k) (3) d/a(n)(3 Here Ck(n) are measurable unitary operators from Hn to Hn+6, = Hn, satisfying ck(n)cl(n + 8k) = cl(n)ck(n + 8g). (4) The domain of definition Dk of ak ak* consists of allf for which fnkl f(n)12d(n) < c. Conversely, if H = fj' Hndc(n) is a direct integral with a. quasi-invariant measure u an invariant dimension function v, if { Ck (n) } ' are measurable unitary operators satisfying (4), if ak ak* are defined on Dk by (3), then the operators pk qk defined on Dk by Pk = (ak + ak) 17k = (ak - ak*)

3 624 PHYSICS: GARDING AND WIGHTMAN PROC. N. A. S. have self-adjoint closures generate sets { U(a) { V(a) } of unitary operators which constitute a representation of the commutation relations. Because a representation is determined by the measure 1A, the dimension function v, a set of operators satisfying (4), we shall sometimes denote it by { U(a), V(a)} = {/,V,{Ck(n)} } Postponing for the moment the question of the existence of quasi-invariant measures, let us analyze (4). Put ho = (n; nk = 0) let Tkn = (O...., 0nk,k nk+1,...) be the projection of n upon I'k1 = n Ik1, o. Then it follows from (4) that we can express ck(n) in terms of Yk(n) = ck(tktn). In fact, a straightforward calculation shows that where c*(n) = 7i (n)... 7rk-l(n)ck(Tkn)7rk l (n + bk) * * * 7r*(n + Sk) (5) 7rk(n) = 1 (nk = 0), rk(n) = Ck*(Tkn - Ak)... Ck*(Tk(n - nfk)) (nk > 0) (6) Conversely, if ck(tkn) = 'Yk(n) are arbitrary unitary matrices which are invariant under 61,... I,aik then Ck(n) as defined by (5) (6) will satisfy (4). There are similar formulas expressing ck(n) in terms of y'(r) (n) = c(r,... rk, 1,nk,... ) for an arbitrary but fixed r in I. 2. Quasi-invariant Measures.-When m <ca, the analogue of I is the set Im of all sequences (n1,..., nfl) with integral components nk > 0. It is clear that a measure ju on Im is always discrete, that it is quasi-invariant if only if /A(n) > 0 for all n, that all quasi-invariant measures are equivalent, i.e., absolutely continuous with respect to each other. When m = co, this simple situation no longer holds, there are many inequivalent quasi-invariant measures. For a given r in I let Ar denote its "coset" with respect to A, i.e., the set of all n in I for which nk= rk for all but a finite number of kvs. A discrete measure on I is quasi-invariant if only if j(r) > 0 implies that u(n) > 0 for all n in Ar. In a natural way it splits up into a countable number of discrete measures, each concentrated on some Ar. Any two such measures are equivalent. As a set, I is a direct product of copies of the natural numbers. Hence, putting Ikj = (n; nk = j), we may define a product measure on I, putting i(hkj) = PkJ > 0, where IA(I) = 2,=o Pj = 1. Such a measure is quasi-invariant if only if all PkJ > 0. It is discrete if only if, for some m, HPk,, > 0, then it is concentrated on Ar. The question of equivalence of quasi-invariant product measures is more complicated than in the case of the anticommutation relations. A rather special result is the following: Two measures, 1A IA', for which PJ = pj, Pk' = p/' are independent of k, are inequivalent if the set { pj-1p/} does not have 1 as a limit point. At any rate, this result shows the existence of a large class of inequivalent quasi-invariant measures. All quasi-invariant product measures /2 are ergodic in the sense that if f is a bounded, invariant u-measurable function, it is a constant. 3. Some simple Examples-The simplest solution of (4) is obtained by putting Ck(n) = 1 for all k. Putting v = 1 letting 1A be concentrated on the rationals, we obtain essentially the canonical representation mostly used in quantum physics,

4 VOL. 40, 1954 PHYSICS: GARDING AND WIGHTMAN 625 A different choice of j, as long as it is concentrated on the rationals, does not change the unitary equivalence class of the representation, because equivalent measures give unitary equivalent representations (see the next section). The canonical representation is the only one for which the number of particles operator 2 ak*ak exists. Letting M be concentrated on a Ar, we obtain a translated canonical representation. We can also satisfy (4) by putting ck(n) = Ok, with 1 k = 1, combine this choice with any v ja. 4. Unitary Equivalence.-Let two representations of the commutation relations R = {U (a), V(O)} = {A, V, C{Ck(n)}I } R' = U'(a), V'(a)} = {/', V', {Ck'(n) } 1} be given. They are said to be unitary equivalent if a unitary operator W from H to H' exists so that U'(a) = WU(a)W*, V'(a) = WV(a)W*. (7) If, for example, R R' differ only with respect to the measures 1u 1u',,u is equivalent to MA', then, putting Sf(n) - d ~n), di='(n) f(n), we find that S is unitary that {SU(a)S*, SV(a)S*} = {, v, {Ck(n)} } = R'. so that the two representations are unitary equivalent. Consider any two representations R R' which are unitary equivalent. Then (7) implies that PTk = WPk Wy for this it is necessary sufficient that,u IM' are equivalent that v = v' a.e. If this condition is satisfied, we can use the mapping S to map H upon H',, considering {S*U'(aC)S, S*V'(a)S} = {j1s, v, {Ck (n)}1} instead of R', we may suppose from the beginning, without loss of generality, that I = AI', v = v', so thatpk' = PY. In this case, W is given by Wf(n) = w(n)f(n), (8) where w(n) is a measurable unitary operator from Hn to Hn, satisfying cki(n) = w(n)ck(n)w*(n + tk) (9) for almost all n all k. Conversely, if a set I w(n) } of measurable unitary operators satisfies this equation, W as defined by (8) will satisfy (7), so that the two representations are unitary equivalent. For k < I we can satisfy (9) by putting w(n) = wi(n) = 7r1'(n).....rl.l'(n) 7r,-*(n)...7r*(n) where 7rk(n) is given by (6) analogously for 7rk'(n). If now IA is concentrated on the rationals, the operators w1(n) converge trivially to a unitary operator as

5 626 PHYSICS: GARDING AND WIGHTMAN PROC. N. A. S. 1- a). Hence, choosing in particular ck' = 1, we find that the representation is equivalent to a direct sum of copies of the canonical representation. More generally, it can be shown that any representation with a discrete sp is a direct sum of translated canonical representations. When,u is not discrete, the unitary equivalence class not only depends on p v but may also depend on the terminal behavior of the operators Ck. Let us, for example, consider the case where v = 1 the measure,u is a product measure given by A(Ikj) = 2-J-1. Then, for the two representations given by ck(n) = Ok Ck'(n) = Ok', respectively, to be unitary equivalent, it is necessary that _ Ok' 00 as k -o oo sufficient that Ok - Ok' converge. More generally, with the same,u constant v < a, let two representations be given with constant Ck(Tkn) = Yk Ck'(Tkn) = 'Yk', respectively. If [Yk, Yk'] = inf UzkU* - vzk'v*i (u, v unitary, operator norm), then [Yk, Yk'] -_ O as k Xo is necessary, I - UYk'U* < co for some fixed unitary u is sufficient for unitary equivalence. The same results are true where ia is replaced by a product measure for which p(ikj)/p(ikj+1) is bounded away from 0 o for all k j. 5. Irreducible Representations.-A representation is called irreducible if any bounded operator which commutes with all U(a) V(a) is a constant. Any such operator has the form Tf(n) = t(n)f(n), (10) where t(n) is a measurable bounded operator from Hn to H. satisfies t(n)ck(n) = ck(n)t(n + AX) (11) for almost all n all k. Conversely, if T is defined by (10), where t(n) is measurable bounded satisfies (11), T commutes with all U(a) V(a). It is clear that any invariant, bounded, complex-valued function satisfies (11), hence, for a representation to be irreducible, it is necessary that pa be ergodic. Moreover, because v-l(n) is invariant bounded, it is also necessary that v be a constant. If v = 1, then t(n) has complex values, it follows from (11) that the ergodicity of A is sufficient to guarantee irreducibility. If p is a product measure for which P(Ikj)/P(Ikj+ 1) is bounded, we can always construct irreducible representations with a given v < o by choosing unitary matrices hi... h8 which generate all v X v matrices putting ck+z(tk+ln) = 'Yk+I = hi (1 = 1,..., s) when k runs through a sequence ki, k2...- o for which kj+l - kj > s. This, combined with the results of the preceding section, shows the existence of a large number of irreducible inequivalent representations with a suitable given pu any finite v. It is not known whether irreducible representations with v = oo exist. 1 L. G&rding A. Wightman, these PROCEEDINGS, 40, , J. von Neumann, Ann. Math., 50,401, H. Weyl, Z. Phys., 46, 1, J. von Neumann, Math. Ann., 104, 570, K. 0. Friedrichs, Mathematical Aspects of the Quantum Theory of Fields (New York: Interscience Publishers, Inc., 1953).