603:
obtain a rigorous result, one must require that the operators satisfy the exponentiated form of the canonical commutation relations, known as the Weyl relations. The exponentiated operators are bounded and unitary. Although, as noted below, these relations are formally equivalent to the standard canonical commutation relations, this equivalence is not rigorous, because (again) of the unbounded nature of the operators. (There is also a discrete analog of the Weyl relations, which can hold in a finite-dimensional space, namely
2893:
6096:
2641:
4359:, one can then obtain "position" and "momentum" operators satisfying the canonical commutation relations. It is not hard to show that the exponentials of these operators satisfy the Weyl relations and that the exponentiated operators act irreducibly. The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from
3834:
602:
The idea of the Stone–von
Neumann theorem is that any two irreducible representations of the canonical commutation relations are unitarily equivalent. Since, however, the operators involved are necessarily unbounded (as noted above), there are tricky domain issues that allow for counter-examples. To
2888:{\displaystyle {\begin{aligned}P&={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}},&Q&={\begin{bmatrix}0&0&0\\0&0&1\\0&0&0\end{bmatrix}},&z&={\begin{bmatrix}0&0&1\\0&0&0\\0&0&0\end{bmatrix}},\end{aligned}}}
957:
It is important to note that the preceding derivation is purely formal. Since the operators involved are unbounded, technical issues prevent application of the Baker–Campbell–Hausdorff formula without additional domain assumptions. Indeed, there exist operators satisfying the canonical commutation
3142:
is multiplicative is a straightforward calculation. The hard part of the theorem is showing the uniqueness; this claim, nevertheless, follows easily from the Stone–von
Neumann theorem as stated above. We will sketch below a proof of the corresponding Stone–von Neumann theorem for certain
323:
4837:
5039:
3582:
2599:
1659:
Then the Stone–von
Neumann theorem is that, given the standard quantum mechanical scale (effectively, the value of ħ), every strongly continuous unitary representation is unitarily equivalent to the standard representation with position and momentum.
949:
formally differentiating at 0 shows that the two infinitesimal generators satisfy the above canonical commutation relation. This braiding formulation of the canonical commutation relations (CCR) for one-parameter unitary groups is called the
3359:
1797:
155:
4228:
4700:
2073:
4845:
1158:
3948:
2149:
3089:
2271:
939:
1371:
4669:
2485:
4050:
809:
2646:
2379:
160:
3505:
3244:
2326:
1968:
4305:
572:; the left-hand side is zero, the right-hand side is non-zero. Further analysis shows that any two self-adjoint operators satisfying the above commutation relation cannot be both
1590:: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is the
1697:
1640:-structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of
526:
4116:
3829:{\displaystyle \int _{\mathbf {R} ^{n}}e^{-ix\cdot p}e^{i(b\cdot x+hc)}\psi (x+ha)\ dx=e^{i(ha\cdot p+h(c-b\cdot a))}\int _{\mathbf {R} ^{n}}e^{-iy\cdot (p-b)}\psi (y)\ dy.}
5985:
5167:
624:
365:
116:
1598:
value (in physics terms, the Planck constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit).
2275:
It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, all
4146:
5043:
By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von
Neumann theorem for Heisenberg groups
1987:
409:
385:
5377:
1048:
3849:
5648:
460:
440:
343:
150:
2087:
5811:
702:
6135:
1854:, this is Stone's theorem characterizing one-parameter unitary groups. The theorem of Stone–von Neumann can also be restated using similar language.
6130:
5938:
5793:
1800:
619:
One would like to classify representations of the canonical commutation relation by two self-adjoint operators acting on separable
Hilbert spaces,
5769:
2952:
2186:
1240:
4545:
964:). Nevertheless, in "good" cases, we expect that operators satisfying the canonical commutation relation will also satisfy the Weyl relations.
318:{\displaystyle {\begin{aligned}(x_{0})&=x_{0}\psi (x_{0})\\(x_{0})&=-i\hbar {\frac {\partial \psi }{\partial x}}(x_{0})\end{aligned}}}
5588:
5511:
3971:
708:
4832:{\displaystyle \chi (\mathrm {M} (a,b,c))={\begin{cases}|K|^{n}\,\omega (hc)&{\text{if }}a=b=0\\0&{\text{otherwise}}\end{cases}}}
6125:
5661:
5548:
5034:{\displaystyle {\frac {1}{\left|H_{n}(\mathbf {K} )\right|}}\sum _{g\in H_{n}(K)}|\chi (g)|^{2}={\frac {1}{|K|^{2n+1}}}|K|^{2n}|K|=1.}
3433:
5750:
5641:
2382:
1376:
608:
4230:
acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely,
2603:
In fact, using the
Heisenberg group, one can reformulate the Stone von Neumann theorem in the language of representation theory.
841:
6020:
3119:
is a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to the
5665:
5620:
2282:
1453:
1586:
Concretely, by a central representation one means a representation such that the center of the
Heisenberg group maps into the
627:, there is a one-to-one correspondence between self-adjoint operators and (strongly continuous) one-parameter unitary groups.
463:
44:
1908:
5816:
4233:
1673:
5872:
2331:
6120:
6099:
5821:
5806:
5634:
5836:
1229:, which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent to
6081:
5841:
469:
4055:
6035:
5959:
2925:
2276:
1386:
In terms of representation theory, the Stone–von
Neumann theorem classifies certain unitary representations of the
968:
6076:
5141:
2594:{\displaystyle \mathrm {M} (a,b,c)={\begin{bmatrix}1&a&c\\0&1_{n}&b\\0&0&1\end{bmatrix}}.}
6140:
5892:
5136:
5123:
4334:
with respect to the inner product coming from the
Gaussian measure. By taking appropriate linear combinations of
4122:
40:
5826:
5172:
5152:
4396:
4134:
1579:
If the center does not map to zero, one has a more interesting theory, particularly if one restricts oneself to
5928:
5729:
1827:
5801:
4370:
to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operators
6025:
4508:
412:
1594:. Thus the representation of the center of the Heisenberg group is determined by a scale value, called the
1572:
maps to zero, then one simply has a representation of the corresponding abelian group or algebra, which is
6056:
6000:
5964:
5119:
3095:
1234:
604:
557:
4143:
that are square-integrable with respect to a
Gaussian measure. Fock observed in 1920s that the operators
1221:
degrees of freedom. Historically, this result was significant, because it was a key step in proving that
5447:
5372:
5330:
5283:
3519:
3226:
6039:
5386:
5146:
3221:
1606:
1587:
1230:
5615:
Summers, Stephen J. (2001). "On the Stone–von Neumann Uniqueness Theorem and Its Ramifications." In
4747:
1397:
Informally stated, with certain technical assumptions, every representation of the Heisenberg group
6005:
5943:
5657:
5126:
for finite groups to the context of unitary representations of locally compact topological groups.
4532:
666:
32:
3354:{\displaystyle \alpha _{h}:\mathrm {M} (a,b,c)\to \mathrm {M} \left(-h^{-1}b,ha,c-a\cdot b\right)}
348:
99:
6030:
5897:
5467:
5412:
5355:
5308:
3843:
547:
55:
6010:
5584:
5544:
5507:
5430:
5404:
5375:(1930), "Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory",
5347:
5300:
3553:
2628:
1691:
1569:
1222:
993:
which satisfy the Weyl relation on separable Hilbert spaces. The answer is the content of the
130:
123:
75:
48:
6015:
5933:
5902:
5882:
5867:
5862:
5857:
5694:
5495:
5459:
5420:
5394:
5339:
5325:
5292:
5278:
4504:
3839:
3527:
2447:
1387:
1226:
573:
394:
370:
63:
5598:
599:, so that, effectively, it is replaced by 1. We assume this normalization in what follows.
5877:
5831:
5779:
5774:
5745:
5626:
5594:
5580:
5114:
The Stone–von Neumann theorem admits numerous generalizations. Much of the early work of
2082:
1792:{\displaystyle f\mapsto {\hat {f}}(\gamma )=\int _{G}{\overline {\gamma (t)}}f(t)d\mu (t)}
1683:
1591:
1478:
1445:– and was the motivation for the introduction of the Heisenberg group in quantum physics.
577:
83:
5704:
17:
5390:
4503:
one can give a simple proof of the Stone–von Neumann theorem using simple properties of
6066:
5918:
5719:
5425:
3130:
1573:
1471:
the corresponding Heisenberg algebra is a central extension of the abelian Lie algebra
584:
445:
425:
328:
135:
59:
6114:
6071:
5995:
5724:
5709:
5699:
5312:
5157:
5115:
4455:
3212:
2631:
in Heisenberg's original CCRs. The Heisenberg group Lie algebra generators, e.g. for
1442:
87:
6061:
5714:
5684:
5483:
5177:
5162:
3534:
is unitary, this scalar multiple is uniquely determined and hence such an operator
3362:
3216:
1624:, so rather than simply thinking of the group algebra as an algebra over the field
1417:
1377:
Generalizations of Pauli matrices § Construction: The clock and shift matrices
531:
5579:, Grundlehren der Mathematischen Wissenschaften, vol. 220, Berlin, New York:
1634:. As the center of a matrix algebra or operator algebra is the scalar matrices, a
122:. In the Schrödinger representation quantum description of such a particle, the
4223:{\displaystyle a_{j}={\frac {\partial }{\partial z_{j}}},\qquad a_{j}^{*}=z_{j},}
5990:
5980:
5887:
5689:
4479:
3229:
is invariant under the action of the discrete subgroup of the Heisenberg group.
2443:
1656:, which is the formal way of saying that it sends the center to a chosen scale.
1421:
28:
5605:
5378:
Proceedings of the National Academy of Sciences of the United States of America
2068:{\displaystyle {\widehat {(s\cdot f)}}(\gamma )=\gamma (s){\hat {f}}(\gamma ).}
1490:
the discrete Heisenberg group is a central extension of the free abelian group
5923:
5763:
5759:
5755:
3144:
1602:
1438:(up to scale) non-trivial central strongly continuous unitary representation.
1217:
There is also a straightforward extension of the Stone–von Neumann theorem to
79:
5408:
5351:
5304:
1972:
Under the isomorphism given above, this action becomes the natural action of
1153:{\displaystyle W^{*}U(t)W=e^{itx}\quad {\text{and}}\quad W^{*}V(s)W=e^{isp},}
638:
be two self-adjoint operators satisfying the canonical commutation relation,
2467:
94:
5434:
3943:{\displaystyle (\alpha _{h})^{2}\mathrm {M} (a,b,c)=\mathrm {M} (-a,-b,c).}
3098:; and any irreducible representation which is not trivial on the center of
5399:
3838:
This theorem has the immediate implication that the Fourier transform is
2938:
119:
52:
5499:
5471:
5359:
5296:
3402:
are unitarily equivalent. This means that there is a unitary operator
999:
all such pairs of one-parameter unitary groups are unitarily equivalent
5416:
1210:
must range along the entire real line. The analog argument holds for
5463:
5343:
2144:{\displaystyle C^{*}\left({\hat {G}}\right)\rtimes _{\hat {\rho }}G}
1168:
are the explicit position and momentum operators from earlier. When
1410:
is equivalent to the position operators and momentum operators on
4446:
a prime. This field has the property that there is an embedding
3211:
One representation of the Heisenberg group which is important in
1628:, one may think of it as an algebra over the commutative algebra
5630:
587:). For notational convenience, the nonvanishing square root of
39:
refers to any one of a number of different formulations of the
560:
over both sides of the latter equation and using the relation
5281:(1931), "Die Eindeutigkeit der Schrödingerschen Operatoren",
345:
of infinitely differentiable functions of compact support on
5450:(1932), "On one-parameter unitary groups in Hilbert Space",
3084:{\displaystyle \left\psi (x)=e^{i(b\cdot x+hc)}\psi (x+ha).}
2442:
are identical to the commutation relations that specify the
2337:
2266:{\displaystyle U(s)V(\gamma )U^{*}(s)=\gamma (s)V(\gamma ).}
934:{\displaystyle U(t)V(s)=e^{-ist}V(s)U(t)\qquad \forall s,t,}
4825:
5543:, Graduate Texts in Mathematics, vol. 267, Springer,
1366:{\displaystyle (x)=e^{itx}\psi (x),\qquad (x)=\psi (x+s).}
5118:
was directed at obtaining a formulation of the theory of
5106:
on which the center acts nontrivially arise in this way.
4664:{\displaystyle \left(x)=\omega (b\cdot x+hc)\psi (x+ha).}
2416:
are unitarily equivalent. Specializing to the case where
5617:
John von Neumann and the foundations of quantum physics
4045:{\displaystyle W_{1}U_{h}W_{1}^{*}=U_{h}\alpha ^{2}(g)}
701:.) A formal computation (using a special case of the
5606:"A Selective History of the Stone–von Neumann Theorem"
4507:
of representations. These properties follow from the
2820:
2742:
2664:
2523:
804:{\displaystyle e^{itQ}e^{isP}=e^{-ist}e^{isP}e^{itQ}.}
580:
shows the relation cannot be satisfied by elements of
4848:
4703:
4548:
4236:
4149:
4058:
3974:
3852:
3585:
3436:
3247:
2955:
2644:
2488:
2334:
2285:
2189:
2090:
1990:
1911:
1700:
1416:. Alternatively, that they are all equivalent to the
1243:
1051:
844:
711:
472:
448:
428:
397:
373:
351:
331:
158:
138:
102:
4511:
for characters of representations of finite groups.
118:, there are two important observables: position and
6049:
5973:
5952:
5911:
5850:
5792:
5738:
5673:
3509:Moreover, by irreducibility of the representations
2374:{\displaystyle {\mathcal {K}}\left(L^{2}(G)\right)}
2077:So a covariant representation corresponding to the
813:Conversely, given two one-parameter unitary groups
5986:Spectral theory of ordinary differential equations
5328:(1932), "Ueber Einen Satz Von Herrn M. H. Stone",
5079:Pairwise inequivalence of all the representations
5033:
4831:
4663:
4425:. In this section let us specialize to the field
4299:
4222:
4110:
4044:
3942:
3828:
3499:
3353:
3083:
2887:
2593:
2373:
2320:
2265:
2143:
2067:
1962:
1791:
1365:
1152:
933:
803:
611:in the finite Heisenberg group, discussed below.)
520:
454:
434:
403:
379:
359:
337:
317:
144:
110:
70:Representation issues of the commutation relations
3107:is unitarily equivalent to exactly one of these.
967:The problem thus becomes classifying two jointly
5168:Stone's theorem on one-parameter unitary groups
3500:{\displaystyle WU_{h}(g)W^{*}=U_{h}\alpha (g).}
5486:(1927), "Quantenmechanik und Gruppentheorie",
2434:The above canonical commutation relations for
1023:acting jointly irreducibly on a Hilbert space
5642:
5092:Actually, all irreducible representations of
8:
5385:(2), National Academy of Sciences: 172–175,
3579:in the definition of the Fourier transform,
665:, the corresponding unitary groups given by
5564:The Theory of Unitary Group Representations
5534:
5532:
5530:
5528:
5526:
5524:
5522:
5520:
4674:
4403:Representations of finite Heisenberg groups
3953:
3542:
3150:In particular, irreducible representations
2914:
1509:is a central extension of the free abelian
681:defined above, these are multiplication by
556:vanishes. This is apparent from taking the
5677:
5649:
5635:
5627:
5619:, pp. 135-152. Springer, Dordrecht, 2001,
5504:The Theory of Groups and Quantum Mechanics
1605:of the Heisenberg group over its field of
1545:In all cases, if one has a representation
591:may be absorbed into the normalization of
5577:Elements of the theory of representations
5424:
5398:
5291:, Springer Berlin / Heidelberg: 570–578,
5020:
5012:
5003:
4998:
4989:
4971:
4966:
4957:
4951:
4942:
4937:
4919:
4902:
4891:
4872:
4863:
4849:
4847:
4817:
4788:
4770:
4764:
4759:
4750:
4742:
4710:
4702:
4564:
4558:
4547:
4282:
4264:
4259:
4246:
4235:
4211:
4198:
4193:
4176:
4163:
4154:
4148:
4137:is the space of holomorphic functions on
4066:
4057:
4027:
4017:
4004:
3999:
3989:
3979:
3973:
3905:
3876:
3870:
3860:
3851:
3772:
3760:
3755:
3753:
3698:
3631:
3609:
3597:
3592:
3590:
3584:
3476:
3463:
3444:
3435:
3307:
3290:
3261:
3252:
3246:
3027:
2974:
2965:
2954:
2815:
2737:
2659:
2645:
2643:
2552:
2518:
2489:
2487:
2351:
2336:
2335:
2333:
2321:{\displaystyle C_{0}(G)\rtimes _{\rho }G}
2309:
2290:
2284:
2218:
2188:
2126:
2125:
2106:
2105:
2095:
2089:
2042:
2041:
1992:
1991:
1989:
1910:
1741:
1735:
1708:
1707:
1699:
1391:
1281:
1242:
1135:
1107:
1097:
1084:
1056:
1050:
876:
843:
786:
770:
751:
732:
716:
710:
471:
447:
427:
396:
372:
353:
352:
350:
330:
302:
275:
250:
218:
202:
182:
159:
157:
137:
104:
103:
101:
5939:Group algebra of a locally compact group
5566:, The University of Chicago Press, 1976.
3571:This means that, ignoring the factor of
3176:are unitarily equivalent if and only if
415:, which carries units of action (energy
5270:
5190:
3374:which is the identity on the center of
3167:which are non-trivial on the center of
534:observed that this commutation law was
512:
398:
374:
272:
5149:(for bosons and fermions respectively)
3383:. In particular, the representations
2426:yields the Stone–von Neumann theorem.
1390:. This is discussed in more detail in
5338:(3), Annals of Mathematics: 567–573,
1963:{\displaystyle (s\cdot f)(t)=f(t+s).}
1799:extends to a C*-isomorphism from the
1505:the discrete Heisenberg group modulo
1452:The continuous Heisenberg group is a
1424:) on a symplectic space of dimension
1180:-representation, it is evident that
958:relation but not the Weyl relations (
7:
4421:is defined for any commutative ring
4300:{\displaystyle \left=\delta _{j,k}.}
3129:and multiplication by a function of
1233:'s wave mechanical formulation (see
93:For a single particle moving on the
1664:Reformulation via Fourier transform
1176:in this equation, so, then, in the
1001:. In other words, for any two such
4711:
4565:
4169:
4165:
3906:
3877:
3291:
3262:
2975:
2490:
916:
286:
278:
82:are represented mathematically by
25:
5612:. American Mathematical Society.
5541:Quantum Theory for Mathematicians
3233:Relation to the Fourier transform
1382:Representation theory formulation
835:satisfying the braiding relation
6136:Theorems in mathematical physics
6095:
6094:
6021:Topological quantum field theory
4873:
3756:
3593:
2466:a positive integer. This is the
2328:are unitarily equivalent to the
703:Baker–Campbell–Hausdorff formula
6131:Theorems in functional analysis
4489:. For finite Heisenberg group
4188:
1308:
1102:
1096:
915:
653:two real parameters. Introduce
521:{\displaystyle =xp-px=i\hbar .}
45:canonical commutation relations
5021:
5013:
4999:
4990:
4967:
4958:
4938:
4933:
4927:
4920:
4914:
4908:
4877:
4869:
4783:
4774:
4760:
4751:
4736:
4733:
4715:
4707:
4655:
4640:
4634:
4613:
4604:
4598:
4587:
4569:
4309:In 1961, Bargmann showed that
4111:{\displaystyle (x)=\psi (-x).}
4102:
4093:
4084:
4078:
4075:
4059:
4039:
4033:
3934:
3910:
3899:
3881:
3867:
3853:
3811:
3805:
3797:
3785:
3744:
3741:
3723:
3702:
3679:
3664:
3656:
3635:
3491:
3485:
3456:
3450:
3287:
3284:
3266:
3094:All these representations are
3075:
3060:
3052:
3031:
3017:
3011:
3000:
2997:
2979:
2971:
2920:For each non-zero real number
2512:
2494:
2363:
2357:
2302:
2296:
2257:
2251:
2245:
2239:
2230:
2224:
2211:
2205:
2199:
2193:
2131:
2111:
2059:
2053:
2047:
2038:
2032:
2023:
2017:
2007:
1995:
1954:
1942:
1933:
1927:
1924:
1912:
1786:
1780:
1771:
1765:
1753:
1747:
1725:
1719:
1713:
1704:
1441:This was later generalized by
1357:
1345:
1336:
1330:
1327:
1321:
1315:
1309:
1302:
1296:
1271:
1265:
1262:
1256:
1250:
1244:
1122:
1116:
1071:
1065:
1027:, there is a unitary operator
912:
906:
900:
894:
866:
860:
854:
848:
669:. (For the explicit operators
485:
473:
464:canonical commutation relation
391:real number—in quantum theory
308:
295:
256:
243:
240:
231:
224:
211:
188:
175:
172:
163:
1:
5817:Uniform boundedness principle
5604:Rosenberg, Jonathan (2004)
5506:, Dover Publications, 1950,
5334:, Second Series (in German),
4129:Example: Segal–Bargmann space
1674:locally compact abelian group
971:one-parameter unitary groups
530:Already in his classic book,
2936:acting on the Hilbert space
2482:square matrices of the form
2151:is a unitary representation
1757:
1692:Fourier–Plancherel transform
1392:the Heisenberg group section
960:
687:and pullback by translation
615:Uniqueness of representation
360:{\displaystyle \mathbb {R} }
111:{\displaystyle \mathbb {R} }
4325:is actually the adjoint of
4052:is the reflection operator
2277:irreducible representations
1184:is unitarily equivalent to
6157:
5960:Invariant subspace problem
4531:on the finite-dimensional
4522:define the representation
4395:. This unitary map is the
2926:irreducible representation
2895:and the central generator
2623:. However, this center is
1434:More formally, there is a
1374:
152:are respectively given by
6126:Mathematical quantization
6090:
5680:
5608:Contemporary Mathematics
5137:Oscillator representation
4684:, the character function
4123:Fourier inversion formula
1456:of the abelian Lie group
995:Stone–von Neumann theorem
621:up to unitary equivalence
37:Stone–von Neumann theorem
18:Stone-von Neumann theorem
5929:Spectrum of a C*-algebra
5575:Kirillov, A. A. (1976),
5122:developed originally by
4397:Segal–Bargmann transform
3158:of the Heisenberg group
2606:Note that the center of
609:clock and shift matrices
6026:Noncommutative geometry
5562:Mackey, G. W. (1976).
5245:‖ ‖
5215:‖ ‖
5120:induced representations
4509:orthogonality relations
3225:, so named because the
2396:. Therefore, all pairs
576:(in fact, a theorem of
413:reduced Planck constant
6082:Tomita–Takesaki theory
6057:Approximation property
6001:Calculus of variations
5494:(1927) pp. 1–46,
5488:Zeitschrift für Physik
5035:
4833:
4665:
4458:into the circle group
4301:
4224:
4112:
4046:
3944:
3830:
3501:
3355:
3096:unitarily inequivalent
3085:
2889:
2595:
2375:
2322:
2267:
2145:
2069:
1964:
1793:
1568:is an algebra and the
1367:
1206:, and the spectrum of
1154:
935:
805:
522:
456:
436:
405:
404:{\displaystyle \hbar }
381:
380:{\displaystyle \hbar }
361:
339:
319:
146:
112:
6077:Banach–Mazur distance
6040:Generalized functions
5452:Annals of Mathematics
5400:10.1073/pnas.16.2.172
5331:Annals of Mathematics
5284:Mathematische Annalen
5142:Wigner–Weyl transform
5036:
4834:
4680:For a fixed non-zero
4666:
4407:The Heisenberg group
4302:
4225:
4113:
4047:
3945:
3831:
3502:
3356:
3227:Jacobi theta function
3086:
2910:is not the identity.
2890:
2615:consists of matrices
2596:
2376:
2323:
2268:
2146:
2070:
1965:
1877:by right translation
1794:
1588:center of the algebra
1368:
1155:
936:
806:
538:for linear operators
536:impossible to satisfy
523:
457:
437:
406:
382:
362:
340:
320:
147:
113:
5822:Kakutani fixed-point
5807:Riesz representation
5173:Hille–Yosida theorem
5153:Segal–Bargmann space
5147:CCR and CAR algebras
4846:
4701:
4546:
4234:
4147:
4135:Segal–Bargmann space
4056:
3972:
3850:
3842:, also known as the
3583:
3434:
3245:
3222:theta representation
2953:
2642:
2486:
2332:
2283:
2187:
2088:
1988:
1909:
1698:
1241:
1049:
952:Weyl form of the CCR
842:
709:
470:
446:
426:
395:
371:
349:
329:
156:
136:
100:
58:. It is named after
6121:Functional analysis
6006:Functional calculus
5965:Mahler's conjecture
5944:Von Neumann algebra
5658:Functional analysis
5539:Hall, B.C. (2013),
5391:1930PNAS...16..172S
4678: —
4533:inner product space
4505:character functions
4269:
4203:
4121:From this fact the
4009:
3957: —
3546: —
3522:, such an operator
3417:such that, for any
3147:Heisenberg groups.
2918: —
1601:More formally, the
1235:Schrödinger picture
941: (
667:functional calculus
33:theoretical physics
6031:Riemann hypothesis
5730:Topological vector
5500:10.1007/BF02055756
5297:10.1007/BF01457956
5068:Irreducibility of
5031:
4918:
4829:
4824:
4676:
4661:
4297:
4255:
4220:
4189:
4108:
4042:
3995:
3955:
3940:
3844:Plancherel theorem
3826:
3544:
3518:, it follows that
3497:
3351:
3215:and the theory of
3081:
2916:
2885:
2883:
2872:
2794:
2716:
2591:
2582:
2371:
2318:
2263:
2141:
2065:
1960:
1789:
1363:
1150:
931:
801:
548:finite-dimensional
518:
452:
432:
401:
377:
357:
335:
315:
313:
142:
108:
6108:
6107:
6011:Integral operator
5788:
5787:
5590:978-0-387-07476-4
5512:978-1-163-18343-4
4987:
4887:
4885:
4820:
4791:
4514:For any non-zero
4183:
3816:
3684:
3554:Fourier transform
3237:For any non-zero
3199:in the center of
2629:identity operator
2383:compact operators
2134:
2114:
2050:
2014:
1848:is the real line
1760:
1716:
1583:representations.
1454:central extension
1100:
705:) readily yields
455:{\displaystyle p}
435:{\displaystyle x}
338:{\displaystyle V}
293:
145:{\displaystyle p}
131:momentum operator
124:position operator
76:quantum mechanics
16:(Redirected from
6148:
6141:John von Neumann
6098:
6097:
6016:Jones polynomial
5934:Operator algebra
5678:
5651:
5644:
5637:
5628:
5601:
5567:
5560:
5554:
5553:
5536:
5515:
5481:
5475:
5474:
5444:
5438:
5437:
5428:
5402:
5369:
5363:
5362:
5322:
5316:
5315:
5275:
5258:
5256:
5250:
5244:
5232:
5230:
5220:
5214:
5206:
5195:
5105:
5087:
5076:
5064:, particularly:
5063:
5040:
5038:
5037:
5032:
5024:
5016:
5011:
5010:
5002:
4993:
4988:
4986:
4985:
4984:
4970:
4961:
4952:
4947:
4946:
4941:
4923:
4917:
4907:
4906:
4886:
4884:
4880:
4876:
4868:
4867:
4850:
4842:It follows that
4838:
4836:
4835:
4830:
4828:
4827:
4821:
4818:
4792:
4789:
4769:
4768:
4763:
4754:
4714:
4696:
4687:
4683:
4679:
4670:
4668:
4667:
4662:
4597:
4593:
4568:
4563:
4562:
4541:
4530:
4521:
4517:
4502:
4488:
4477:
4463:
4453:
4449:
4445:
4441:
4424:
4420:
4394:
4393:
4392:
4378:
4369:
4358:
4357:
4356:
4342:
4333:
4324:
4323:
4322:
4306:
4304:
4303:
4298:
4293:
4292:
4274:
4270:
4268:
4263:
4251:
4250:
4229:
4227:
4226:
4221:
4216:
4215:
4202:
4197:
4184:
4182:
4181:
4180:
4164:
4159:
4158:
4142:
4125:easily follows.
4117:
4115:
4114:
4109:
4071:
4070:
4051:
4049:
4048:
4043:
4032:
4031:
4022:
4021:
4008:
4003:
3994:
3993:
3984:
3983:
3967:
3958:
3949:
3947:
3946:
3941:
3909:
3880:
3875:
3874:
3865:
3864:
3835:
3833:
3832:
3827:
3814:
3801:
3800:
3767:
3766:
3765:
3764:
3759:
3748:
3747:
3682:
3660:
3659:
3626:
3625:
3604:
3603:
3602:
3601:
3596:
3578:
3566:
3551:
3547:
3537:
3533:
3525:
3517:
3506:
3504:
3503:
3498:
3481:
3480:
3468:
3467:
3449:
3448:
3429:
3420:
3416:
3405:
3401:
3391:
3382:
3373:
3360:
3358:
3357:
3352:
3350:
3346:
3315:
3314:
3294:
3265:
3257:
3256:
3240:
3207:
3198:
3194:
3175:
3166:
3157:
3153:
3141:
3128:
3118:
3106:
3090:
3088:
3087:
3082:
3056:
3055:
3007:
3003:
2978:
2970:
2969:
2948:
2935:
2923:
2919:
2909:
2904:(0, 0, 1) = exp(
2894:
2892:
2891:
2886:
2884:
2877:
2876:
2799:
2798:
2721:
2720:
2637:
2622:
2614:
2600:
2598:
2597:
2592:
2587:
2586:
2557:
2556:
2493:
2481:
2465:
2461:
2448:Heisenberg group
2441:
2437:
2430:Heisenberg group
2425:
2415:
2395:
2380:
2378:
2377:
2372:
2370:
2366:
2356:
2355:
2341:
2340:
2327:
2325:
2324:
2319:
2314:
2313:
2295:
2294:
2272:
2270:
2269:
2264:
2223:
2222:
2182:
2176:
2165:
2161:
2150:
2148:
2147:
2142:
2137:
2136:
2135:
2127:
2120:
2116:
2115:
2107:
2100:
2099:
2080:
2074:
2072:
2071:
2066:
2052:
2051:
2043:
2016:
2015:
2010:
1993:
1983:
1975:
1969:
1967:
1966:
1961:
1904:
1892:
1888:
1884:
1880:
1876:
1864:
1860:
1853:
1847:
1843:
1837:
1825:
1813:
1809:
1801:group C*-algebra
1798:
1796:
1795:
1790:
1761:
1756:
1742:
1740:
1739:
1718:
1717:
1709:
1689:
1681:
1671:
1655:
1645:
1639:
1633:
1627:
1623:
1617:
1567:
1561:
1540:
1527:
1512:
1508:
1501:
1495:
1486:
1476:
1467:
1461:
1430:
1415:
1409:
1388:Heisenberg group
1372:
1370:
1369:
1364:
1292:
1291:
1227:matrix mechanics
1220:
1213:
1209:
1205:
1183:
1179:
1175:
1171:
1167:
1163:
1159:
1157:
1156:
1151:
1146:
1145:
1112:
1111:
1101:
1098:
1095:
1094:
1061:
1060:
1044:
1026:
1022:
1011:
992:
981:
944:
940:
938:
937:
932:
890:
889:
834:
823:
810:
808:
807:
802:
797:
796:
781:
780:
765:
764:
743:
742:
727:
726:
700:
686:
680:
674:
664:
658:
652:
648:
644:
637:
633:
598:
594:
590:
571:
555:
545:
541:
527:
525:
524:
519:
461:
459:
458:
453:
441:
439:
438:
433:
410:
408:
407:
402:
386:
384:
383:
378:
366:
364:
363:
358:
356:
344:
342:
341:
336:
324:
322:
321:
316:
314:
307:
306:
294:
292:
284:
276:
255:
254:
223:
222:
207:
206:
187:
186:
151:
149:
148:
143:
128:
117:
115:
114:
109:
107:
84:linear operators
64:John von Neumann
21:
6156:
6155:
6151:
6150:
6149:
6147:
6146:
6145:
6111:
6110:
6109:
6104:
6086:
6050:Advanced topics
6045:
5969:
5948:
5907:
5873:Hilbert–Schmidt
5846:
5837:Gelfand–Naimark
5784:
5734:
5669:
5655:
5591:
5581:Springer-Verlag
5574:
5571:
5570:
5561:
5557:
5551:
5538:
5537:
5518:
5482:
5478:
5464:10.2307/1968538
5446:
5445:
5441:
5371:
5370:
5366:
5344:10.2307/1968535
5326:von Neumann, J.
5324:
5323:
5319:
5279:von Neumann, J.
5277:
5276:
5272:
5267:
5262:
5261:
5246:
5240:
5234:
5226:
5216:
5210:
5208:
5197:
5196:
5192:
5187:
5182:
5132:
5112:
5110:Generalizations
5098:
5093:
5085:
5080:
5074:
5069:
5049:
5044:
4997:
4965:
4956:
4936:
4898:
4859:
4858:
4854:
4844:
4843:
4840:
4823:
4822:
4815:
4809:
4808:
4786:
4758:
4743:
4699:
4698:
4694:
4689:
4685:
4681:
4677:
4554:
4553:
4549:
4544:
4543:
4535:
4528:
4523:
4519:
4515:
4495:
4490:
4482:
4478:is finite with
4470:
4465:
4459:
4451:
4447:
4443:
4426:
4422:
4413:
4408:
4405:
4391:
4386:
4385:
4384:
4380:
4376:
4371:
4360:
4355:
4350:
4349:
4348:
4344:
4340:
4335:
4331:
4326:
4321:
4316:
4315:
4314:
4310:
4278:
4242:
4241:
4237:
4232:
4231:
4207:
4172:
4168:
4150:
4145:
4144:
4138:
4131:
4119:
4062:
4054:
4053:
4023:
4013:
3985:
3975:
3970:
3969:
3966:
3960:
3956:
3866:
3856:
3848:
3847:
3846:. Moreover,
3768:
3754:
3749:
3694:
3627:
3605:
3591:
3586:
3581:
3580:
3572:
3569:
3557:
3549:
3545:
3535:
3531:
3526:is unique (cf.
3523:
3515:
3510:
3472:
3459:
3440:
3432:
3431:
3427:
3422:
3418:
3407:
3403:
3398:
3393:
3389:
3384:
3380:
3375:
3371:
3366:
3303:
3299:
3295:
3248:
3243:
3242:
3238:
3235:
3205:
3200:
3196:
3177:
3173:
3168:
3164:
3159:
3155:
3151:
3139:
3134:
3124:
3116:
3111:
3104:
3099:
3092:
3023:
2961:
2960:
2956:
2951:
2950:
2937:
2933:
2928:
2921:
2917:
2896:
2882:
2881:
2871:
2870:
2865:
2860:
2854:
2853:
2848:
2843:
2837:
2836:
2831:
2826:
2816:
2808:
2803:
2793:
2792:
2787:
2782:
2776:
2775:
2770:
2765:
2759:
2758:
2753:
2748:
2738:
2730:
2725:
2715:
2714:
2709:
2704:
2698:
2697:
2692:
2687:
2681:
2680:
2675:
2670:
2660:
2652:
2640:
2639:
2632:
2616:
2612:
2607:
2581:
2580:
2575:
2570:
2564:
2563:
2558:
2548:
2546:
2540:
2539:
2534:
2529:
2519:
2484:
2483:
2471:
2463:
2460:
2450:
2446:of the general
2439:
2435:
2432:
2417:
2397:
2386:
2347:
2346:
2342:
2330:
2329:
2305:
2286:
2281:
2280:
2214:
2185:
2184:
2178:
2167:
2163:
2152:
2121:
2101:
2091:
2086:
2085:
2083:crossed product
2078:
1994:
1986:
1985:
1977:
1973:
1907:
1906:
1898:
1894:
1890:
1886:
1882:
1878:
1870:
1866:
1862:
1858:
1849:
1845:
1839:
1831:
1819:
1815:
1811:
1803:
1743:
1731:
1696:
1695:
1687:
1684:Pontryagin dual
1677:
1669:
1666:
1647:
1641:
1635:
1629:
1625:
1619:
1613:
1592:scalar matrices
1563:
1556:
1546:
1529:
1514:
1510:
1506:
1497:
1491:
1482:
1481:) by a copy of
1479:trivial bracket
1472:
1463:
1457:
1425:
1411:
1408:
1398:
1384:
1379:
1277:
1239:
1238:
1218:
1211:
1207:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1131:
1103:
1080:
1052:
1047:
1046:
1028:
1024:
1013:
1002:
983:
972:
947:
942:
872:
840:
839:
825:
814:
782:
766:
747:
728:
712:
707:
706:
688:
682:
676:
670:
660:
654:
650:
646:
639:
635:
631:
625:Stone's theorem
617:
596:
592:
588:
561:
551:
543:
539:
468:
467:
444:
443:
424:
423:
393:
392:
369:
368:
347:
346:
327:
326:
312:
311:
298:
285:
277:
259:
246:
228:
227:
214:
198:
191:
178:
154:
153:
134:
133:
126:
98:
97:
72:
23:
22:
15:
12:
11:
5:
6154:
6152:
6144:
6143:
6138:
6133:
6128:
6123:
6113:
6112:
6106:
6105:
6103:
6102:
6091:
6088:
6087:
6085:
6084:
6079:
6074:
6069:
6067:Choquet theory
6064:
6059:
6053:
6051:
6047:
6046:
6044:
6043:
6033:
6028:
6023:
6018:
6013:
6008:
6003:
5998:
5993:
5988:
5983:
5977:
5975:
5971:
5970:
5968:
5967:
5962:
5956:
5954:
5950:
5949:
5947:
5946:
5941:
5936:
5931:
5926:
5921:
5919:Banach algebra
5915:
5913:
5909:
5908:
5906:
5905:
5900:
5895:
5890:
5885:
5880:
5875:
5870:
5865:
5860:
5854:
5852:
5848:
5847:
5845:
5844:
5842:Banach–Alaoglu
5839:
5834:
5829:
5824:
5819:
5814:
5809:
5804:
5798:
5796:
5790:
5789:
5786:
5785:
5783:
5782:
5777:
5772:
5770:Locally convex
5767:
5753:
5748:
5742:
5740:
5736:
5735:
5733:
5732:
5727:
5722:
5717:
5712:
5707:
5702:
5697:
5692:
5687:
5681:
5675:
5671:
5670:
5656:
5654:
5653:
5646:
5639:
5631:
5625:
5624:
5613:
5602:
5589:
5569:
5568:
5555:
5550:978-1461471158
5549:
5516:
5476:
5458:(3): 643–648,
5439:
5364:
5317:
5269:
5268:
5266:
5263:
5260:
5259:
5189:
5188:
5186:
5183:
5181:
5180:
5175:
5170:
5165:
5160:
5155:
5150:
5144:
5139:
5133:
5131:
5128:
5111:
5108:
5096:
5090:
5089:
5083:
5077:
5072:
5047:
5030:
5027:
5023:
5019:
5015:
5009:
5006:
5001:
4996:
4992:
4983:
4980:
4977:
4974:
4969:
4964:
4960:
4955:
4950:
4945:
4940:
4935:
4932:
4929:
4926:
4922:
4916:
4913:
4910:
4905:
4901:
4897:
4894:
4890:
4883:
4879:
4875:
4871:
4866:
4862:
4857:
4853:
4826:
4816:
4814:
4811:
4810:
4807:
4804:
4801:
4798:
4795:
4787:
4785:
4782:
4779:
4776:
4773:
4767:
4762:
4757:
4753:
4749:
4748:
4746:
4741:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4717:
4713:
4709:
4706:
4692:
4672:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4596:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4567:
4561:
4557:
4552:
4526:
4493:
4468:
4456:additive group
4411:
4404:
4401:
4387:
4374:
4351:
4338:
4329:
4317:
4296:
4291:
4288:
4285:
4281:
4277:
4273:
4267:
4262:
4258:
4254:
4249:
4245:
4240:
4219:
4214:
4210:
4206:
4201:
4196:
4192:
4187:
4179:
4175:
4171:
4167:
4162:
4157:
4153:
4130:
4127:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4069:
4065:
4061:
4041:
4038:
4035:
4030:
4026:
4020:
4016:
4012:
4007:
4002:
3998:
3992:
3988:
3982:
3978:
3964:
3951:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3908:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3879:
3873:
3869:
3863:
3859:
3855:
3825:
3822:
3819:
3813:
3810:
3807:
3804:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3771:
3763:
3758:
3752:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3697:
3693:
3690:
3687:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3630:
3624:
3621:
3618:
3615:
3612:
3608:
3600:
3595:
3589:
3540:
3520:up to a scalar
3513:
3496:
3493:
3490:
3487:
3484:
3479:
3475:
3471:
3466:
3462:
3458:
3455:
3452:
3447:
3443:
3439:
3425:
3396:
3387:
3378:
3369:
3349:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3313:
3310:
3306:
3302:
3298:
3293:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3264:
3260:
3255:
3251:
3241:, the mapping
3234:
3231:
3203:
3171:
3162:
3137:
3131:absolute value
3114:
3102:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3026:
3022:
3019:
3016:
3013:
3010:
3006:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2977:
2973:
2968:
2964:
2959:
2931:
2912:
2880:
2875:
2869:
2866:
2864:
2861:
2859:
2856:
2855:
2852:
2849:
2847:
2844:
2842:
2839:
2838:
2835:
2832:
2830:
2827:
2825:
2822:
2821:
2819:
2814:
2811:
2809:
2807:
2804:
2802:
2797:
2791:
2788:
2786:
2783:
2781:
2778:
2777:
2774:
2771:
2769:
2766:
2764:
2761:
2760:
2757:
2754:
2752:
2749:
2747:
2744:
2743:
2741:
2736:
2733:
2731:
2729:
2726:
2724:
2719:
2713:
2710:
2708:
2705:
2703:
2700:
2699:
2696:
2693:
2691:
2688:
2686:
2683:
2682:
2679:
2676:
2674:
2671:
2669:
2666:
2665:
2663:
2658:
2655:
2653:
2651:
2648:
2647:
2610:
2590:
2585:
2579:
2576:
2574:
2571:
2569:
2566:
2565:
2562:
2559:
2555:
2551:
2547:
2545:
2542:
2541:
2538:
2535:
2533:
2530:
2528:
2525:
2524:
2522:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2492:
2454:
2431:
2428:
2369:
2365:
2362:
2359:
2354:
2350:
2345:
2339:
2317:
2312:
2308:
2304:
2301:
2298:
2293:
2289:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2221:
2217:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2140:
2133:
2130:
2124:
2119:
2113:
2110:
2104:
2098:
2094:
2064:
2061:
2058:
2055:
2049:
2046:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2013:
2009:
2006:
2003:
2000:
1997:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1896:
1868:
1817:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1759:
1755:
1752:
1749:
1746:
1738:
1734:
1730:
1727:
1724:
1721:
1715:
1712:
1706:
1703:
1665:
1662:
1574:Fourier theory
1550:
1543:
1542:
1503:
1488:
1469:
1402:
1383:
1380:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1307:
1304:
1301:
1298:
1295:
1290:
1287:
1284:
1280:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1149:
1144:
1141:
1138:
1134:
1130:
1127:
1124:
1121:
1118:
1115:
1110:
1106:
1093:
1090:
1087:
1083:
1079:
1076:
1073:
1070:
1067:
1064:
1059:
1055:
930:
927:
924:
921:
918:
914:
911:
908:
905:
902:
899:
896:
893:
888:
885:
882:
879:
875:
871:
868:
865:
862:
859:
856:
853:
850:
847:
837:
800:
795:
792:
789:
785:
779:
776:
773:
769:
763:
760:
757:
754:
750:
746:
741:
738:
735:
731:
725:
722:
719:
715:
616:
613:
585:normed algebra
550:spaces unless
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
451:
431:
422:The operators
400:
387:to be a fixed
376:
355:
334:
325:on the domain
310:
305:
301:
297:
291:
288:
283:
280:
274:
271:
268:
265:
262:
260:
258:
253:
249:
245:
242:
239:
236:
233:
230:
229:
226:
221:
217:
213:
210:
205:
201:
197:
194:
192:
190:
185:
181:
177:
174:
171:
168:
165:
162:
161:
141:
106:
88:Hilbert spaces
71:
68:
60:Marshall Stone
24:
14:
13:
10:
9:
6:
4:
3:
2:
6153:
6142:
6139:
6137:
6134:
6132:
6129:
6127:
6124:
6122:
6119:
6118:
6116:
6101:
6093:
6092:
6089:
6083:
6080:
6078:
6075:
6073:
6072:Weak topology
6070:
6068:
6065:
6063:
6060:
6058:
6055:
6054:
6052:
6048:
6041:
6037:
6034:
6032:
6029:
6027:
6024:
6022:
6019:
6017:
6014:
6012:
6009:
6007:
6004:
6002:
5999:
5997:
5996:Index theorem
5994:
5992:
5989:
5987:
5984:
5982:
5979:
5978:
5976:
5972:
5966:
5963:
5961:
5958:
5957:
5955:
5953:Open problems
5951:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5920:
5917:
5916:
5914:
5910:
5904:
5901:
5899:
5896:
5894:
5891:
5889:
5886:
5884:
5881:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5855:
5853:
5849:
5843:
5840:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5799:
5797:
5795:
5791:
5781:
5778:
5776:
5773:
5771:
5768:
5765:
5761:
5757:
5754:
5752:
5749:
5747:
5744:
5743:
5741:
5737:
5731:
5728:
5726:
5723:
5721:
5718:
5716:
5713:
5711:
5708:
5706:
5703:
5701:
5698:
5696:
5693:
5691:
5688:
5686:
5683:
5682:
5679:
5676:
5672:
5667:
5663:
5659:
5652:
5647:
5645:
5640:
5638:
5633:
5632:
5629:
5622:
5618:
5614:
5611:
5607:
5603:
5600:
5596:
5592:
5586:
5582:
5578:
5573:
5572:
5565:
5559:
5556:
5552:
5546:
5542:
5535:
5533:
5531:
5529:
5527:
5525:
5523:
5521:
5517:
5513:
5509:
5505:
5501:
5497:
5493:
5489:
5485:
5480:
5477:
5473:
5469:
5465:
5461:
5457:
5453:
5449:
5443:
5440:
5436:
5432:
5427:
5422:
5418:
5414:
5410:
5406:
5401:
5396:
5392:
5388:
5384:
5380:
5379:
5374:
5368:
5365:
5361:
5357:
5353:
5349:
5345:
5341:
5337:
5333:
5332:
5327:
5321:
5318:
5314:
5310:
5306:
5302:
5298:
5294:
5290:
5286:
5285:
5280:
5274:
5271:
5264:
5254:
5249:
5243:
5238:
5229:
5224:
5219:
5213:
5205:
5201:
5194:
5191:
5184:
5179:
5176:
5174:
5171:
5169:
5166:
5164:
5161:
5159:
5158:Moyal product
5156:
5154:
5151:
5148:
5145:
5143:
5140:
5138:
5135:
5134:
5129:
5127:
5125:
5121:
5117:
5116:George Mackey
5109:
5107:
5103:
5099:
5086:
5078:
5075:
5067:
5066:
5065:
5061:
5058:
5054:
5050:
5041:
5028:
5025:
5017:
5007:
5004:
4994:
4981:
4978:
4975:
4972:
4962:
4953:
4948:
4943:
4930:
4924:
4911:
4903:
4899:
4895:
4892:
4888:
4881:
4864:
4860:
4855:
4851:
4839:
4812:
4805:
4802:
4799:
4796:
4793:
4780:
4777:
4771:
4765:
4755:
4744:
4739:
4730:
4727:
4724:
4721:
4718:
4704:
4697:is given by:
4695:
4671:
4658:
4652:
4649:
4646:
4643:
4637:
4631:
4628:
4625:
4622:
4619:
4616:
4610:
4607:
4601:
4594:
4590:
4584:
4581:
4578:
4575:
4572:
4559:
4555:
4550:
4539:
4534:
4529:
4512:
4510:
4506:
4500:
4496:
4486:
4481:
4475:
4471:
4464:. Note that
4462:
4457:
4440:
4437:
4433:
4429:
4418:
4414:
4402:
4400:
4398:
4390:
4383:
4377:
4367:
4363:
4354:
4347:
4341:
4332:
4320:
4313:
4307:
4294:
4289:
4286:
4283:
4279:
4275:
4271:
4265:
4260:
4256:
4252:
4247:
4243:
4238:
4217:
4212:
4208:
4204:
4199:
4194:
4190:
4185:
4177:
4173:
4160:
4155:
4151:
4141:
4136:
4128:
4126:
4124:
4118:
4105:
4099:
4096:
4090:
4087:
4081:
4072:
4067:
4063:
4036:
4028:
4024:
4018:
4014:
4010:
4005:
4000:
3996:
3990:
3986:
3980:
3976:
3963:
3959:The operator
3950:
3937:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3902:
3896:
3893:
3890:
3887:
3884:
3871:
3861:
3857:
3845:
3841:
3836:
3823:
3820:
3817:
3808:
3802:
3794:
3791:
3788:
3782:
3779:
3776:
3773:
3769:
3761:
3750:
3738:
3735:
3732:
3729:
3726:
3720:
3717:
3714:
3711:
3708:
3705:
3699:
3695:
3691:
3688:
3685:
3676:
3673:
3670:
3667:
3661:
3653:
3650:
3647:
3644:
3641:
3638:
3632:
3628:
3622:
3619:
3616:
3613:
3610:
3606:
3598:
3587:
3576:
3568:
3564:
3560:
3555:
3548:The operator
3539:
3529:
3528:Schur's lemma
3521:
3516:
3507:
3494:
3488:
3482:
3477:
3473:
3469:
3464:
3460:
3453:
3445:
3441:
3437:
3428:
3414:
3410:
3400:
3390:
3381:
3372:
3364:
3347:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3311:
3308:
3304:
3300:
3296:
3281:
3278:
3275:
3272:
3269:
3258:
3253:
3249:
3232:
3230:
3228:
3224:
3223:
3218:
3217:modular forms
3214:
3213:number theory
3209:
3206:
3192:
3188:
3184:
3180:
3174:
3165:
3148:
3146:
3140:
3132:
3127:
3122:
3117:
3108:
3105:
3097:
3091:
3078:
3072:
3069:
3066:
3063:
3057:
3049:
3046:
3043:
3040:
3037:
3034:
3028:
3024:
3020:
3014:
3008:
3004:
2994:
2991:
2988:
2985:
2982:
2966:
2962:
2957:
2946:
2942:
2941:
2934:
2927:
2911:
2907:
2903:
2899:
2878:
2873:
2867:
2862:
2857:
2850:
2845:
2840:
2833:
2828:
2823:
2817:
2812:
2810:
2805:
2800:
2795:
2789:
2784:
2779:
2772:
2767:
2762:
2755:
2750:
2745:
2739:
2734:
2732:
2727:
2722:
2717:
2711:
2706:
2701:
2694:
2689:
2684:
2677:
2672:
2667:
2661:
2656:
2654:
2649:
2635:
2630:
2626:
2620:
2613:
2604:
2601:
2588:
2583:
2577:
2572:
2567:
2560:
2553:
2549:
2543:
2536:
2531:
2526:
2520:
2515:
2509:
2506:
2503:
2500:
2497:
2479:
2475:
2469:
2458:
2453:
2449:
2445:
2429:
2427:
2424:
2420:
2413:
2409:
2405:
2401:
2393:
2389:
2384:
2367:
2360:
2352:
2348:
2343:
2315:
2310:
2306:
2299:
2291:
2287:
2278:
2273:
2260:
2254:
2248:
2242:
2236:
2233:
2227:
2219:
2215:
2208:
2202:
2196:
2190:
2181:
2174:
2170:
2159:
2155:
2138:
2128:
2122:
2117:
2108:
2102:
2096:
2092:
2084:
2075:
2062:
2056:
2044:
2035:
2029:
2026:
2020:
2011:
2004:
2001:
1998:
1981:
1970:
1957:
1951:
1948:
1945:
1939:
1936:
1930:
1921:
1918:
1915:
1902:
1874:
1855:
1852:
1842:
1838:is precisely
1835:
1829:
1823:
1807:
1802:
1783:
1777:
1774:
1768:
1762:
1750:
1744:
1736:
1732:
1728:
1722:
1710:
1701:
1693:
1685:
1680:
1675:
1663:
1661:
1657:
1654:
1650:
1644:
1638:
1632:
1622:
1618:, has center
1616:
1611:
1608:
1604:
1603:group algebra
1599:
1597:
1593:
1589:
1584:
1582:
1577:
1575:
1571:
1566:
1560:
1554:
1549:
1539:
1536:
1532:
1528:by a copy of
1525:
1522:
1518:
1504:
1500:
1496:by a copy of
1494:
1489:
1485:
1480:
1475:
1470:
1466:
1462:by a copy of
1460:
1455:
1451:
1450:
1449:
1446:
1444:
1443:Mackey theory
1439:
1437:
1432:
1429:
1423:
1419:
1414:
1406:
1401:
1395:
1393:
1389:
1381:
1378:
1373:
1360:
1354:
1351:
1348:
1342:
1339:
1333:
1324:
1318:
1312:
1305:
1299:
1293:
1288:
1285:
1282:
1278:
1274:
1268:
1259:
1253:
1247:
1236:
1232:
1228:
1224:
1215:
1204:
1200:
1196:
1192:
1188:
1147:
1142:
1139:
1136:
1132:
1128:
1125:
1119:
1113:
1108:
1104:
1091:
1088:
1085:
1081:
1077:
1074:
1068:
1062:
1057:
1053:
1043:
1039:
1035:
1031:
1020:
1016:
1009:
1005:
1000:
996:
990:
986:
979:
975:
970:
965:
963:
962:
955:
953:
946:
928:
925:
922:
919:
909:
903:
897:
891:
886:
883:
880:
877:
873:
869:
863:
857:
851:
845:
836:
832:
828:
821:
817:
811:
798:
793:
790:
787:
783:
777:
774:
771:
767:
761:
758:
755:
752:
748:
744:
739:
736:
733:
729:
723:
720:
717:
713:
704:
699:
695:
691:
685:
679:
673:
668:
663:
657:
643:
628:
626:
622:
614:
612:
610:
606:
600:
586:
583:
579:
575:
569:
565:
559:
554:
549:
537:
533:
528:
515:
509:
506:
503:
500:
497:
494:
491:
488:
482:
479:
476:
466:Lie algebra,
465:
449:
429:
420:
418:
414:
390:
332:
303:
299:
289:
281:
269:
266:
263:
261:
251:
247:
237:
234:
219:
215:
208:
203:
199:
195:
193:
183:
179:
169:
166:
139:
132:
125:
121:
96:
91:
89:
85:
81:
77:
69:
67:
65:
61:
57:
54:
50:
46:
42:
38:
34:
30:
19:
6062:Balanced set
6036:Distribution
5974:Applications
5827:Krein–Milman
5812:Closed graph
5616:
5609:
5576:
5563:
5558:
5540:
5503:
5502:; Weyl, H.,
5491:
5487:
5479:
5455:
5451:
5448:Stone, M. H.
5442:
5382:
5376:
5373:Stone, M. H.
5367:
5335:
5329:
5320:
5288:
5282:
5273:
5252:
5247:
5241:
5236:
5227:
5222:
5217:
5211:
5203:
5199:
5193:
5178:C0-semigroup
5163:Weyl algebra
5113:
5101:
5094:
5091:
5081:
5070:
5059:
5056:
5052:
5045:
5042:
4841:
4690:
4673:
4537:
4524:
4513:
4498:
4491:
4484:
4473:
4466:
4460:
4438:
4435:
4431:
4427:
4416:
4409:
4406:
4388:
4381:
4372:
4365:
4361:
4352:
4345:
4336:
4327:
4318:
4311:
4308:
4139:
4132:
4120:
3961:
3952:
3837:
3574:
3570:
3562:
3558:
3541:
3511:
3508:
3423:
3412:
3408:
3394:
3385:
3376:
3367:
3363:automorphism
3236:
3220:
3210:
3201:
3190:
3186:
3182:
3178:
3169:
3160:
3149:
3135:
3133:1. To show
3125:
3120:
3112:
3109:
3100:
3093:
2944:
2939:
2929:
2924:there is an
2913:
2905:
2901:
2897:
2633:
2624:
2618:
2608:
2605:
2602:
2477:
2473:
2456:
2451:
2433:
2422:
2418:
2411:
2407:
2403:
2399:
2391:
2387:
2274:
2179:
2172:
2168:
2157:
2153:
2076:
1979:
1971:
1900:
1872:
1861:acts on the
1856:
1850:
1840:
1833:
1821:
1805:
1678:
1667:
1658:
1652:
1648:
1642:
1636:
1630:
1620:
1614:
1609:
1600:
1596:quantization
1595:
1585:
1580:
1578:
1564:
1558:
1552:
1547:
1544:
1537:
1534:
1530:
1523:
1520:
1516:
1498:
1492:
1483:
1473:
1464:
1458:
1447:
1440:
1435:
1433:
1427:
1418:Weyl algebra
1412:
1404:
1399:
1396:
1385:
1216:
1202:
1198:
1194:
1190:
1186:
1041:
1037:
1033:
1029:
1018:
1014:
1007:
1003:
998:
994:
988:
984:
977:
973:
966:
959:
956:
951:
948:
838:
830:
826:
819:
815:
812:
697:
693:
689:
683:
677:
671:
661:
655:
641:
629:
620:
618:
601:
581:
567:
563:
552:
535:
532:Hermann Weyl
529:
462:satisfy the
421:
416:
388:
92:
73:
36:
26:
5991:Heat kernel
5981:Hardy space
5888:Trace class
5802:Hahn–Banach
5764:Topological
5251:‖ ≥
5239:: 2‖
5233:, so that,
5221:‖ ≥
4480:cardinality
3538:is unique.
2444:Lie algebra
1826:, i.e. the
1694:defined by
1448:In detail:
1422:CCR algebra
1231:Schrödinger
969:irreducible
80:observables
78:, physical
29:mathematics
6115:Categories
5924:C*-algebra
5739:Properties
5265:References
5225:ℏ ‖
3968:such that
3530:). Since
3110:Note that
2183:such that
1865:*-algebra
1857:The group
1646:-algebras
1612:, written
1375:See also:
1223:Heisenberg
566:) = Trace(
546:acting on
367:. Assume
41:uniqueness
5898:Unbounded
5893:Transpose
5851:Operators
5780:Separable
5775:Reflexive
5760:Algebraic
5746:Barrelled
5409:0027-8424
5352:0003-486X
5313:120528257
5305:0025-5831
5209:2‖
5124:Frobenius
4925:χ
4896:∈
4889:∑
4819:otherwise
4772:ω
4705:χ
4638:ψ
4620:⋅
4611:ω
4591:ψ
4536:ℓ(
4280:δ
4266:∗
4200:∗
4170:∂
4166:∂
4097:−
4091:ψ
4073:ψ
4025:α
4006:∗
3923:−
3914:−
3858:α
3803:ψ
3792:−
3783:⋅
3774:−
3751:∫
3736:⋅
3730:−
3712:⋅
3662:ψ
3642:⋅
3620:⋅
3611:−
3588:∫
3483:α
3465:∗
3341:⋅
3335:−
3309:−
3301:−
3288:→
3250:α
3058:ψ
3038:⋅
3009:ψ
2468:Lie group
2311:ρ
2307:⋊
2255:γ
2237:γ
2220:∗
2209:γ
2132:^
2129:ρ
2123:⋊
2112:^
2097:∗
2057:γ
2048:^
2030:γ
2021:γ
2012:^
2002:⋅
1919:⋅
1778:μ
1758:¯
1745:γ
1733:∫
1723:γ
1714:^
1705:↦
1394:, below.
1343:ψ
1325:ψ
1294:ψ
1260:ψ
1109:∗
1058:∗
917:∀
878:−
753:−
605:Sylvester
513:ℏ
498:−
399:ℏ
375:ℏ
287:∂
282:ψ
279:∂
273:ℏ
267:−
238:ψ
209:ψ
170:ψ
95:real line
56:operators
6100:Category
5912:Algebras
5794:Theorems
5751:Complete
5720:Schwartz
5666:glossary
5484:Weyl, H.
5435:16587545
5231:‖
5207:, hence
5130:See also
4790:if
3195:for any
2617:M(0, 0,
2476:+ 2) × (
1828:spectrum
1562:, where
1045:so that
1032: :
578:Wielandt
389:non-zero
120:momentum
53:momentum
49:position
47:between
5903:Unitary
5883:Nuclear
5868:Compact
5863:Bounded
5858:Adjoint
5832:Min–max
5725:Sobolev
5710:Nuclear
5700:Hilbert
5695:Fréchet
5660: (
5599:0407202
5472:1968538
5426:1075964
5387:Bibcode
5360:1968535
4675:Theorem
3954:Theorem
3840:unitary
3552:is the
3543:Theorem
3219:is the
2915:Theorem
1844:. When
1682:be the
1607:scalars
1581:central
1513:-group
1193:
1189:
574:bounded
419:time).
411:is the
43:of the
31:and in
5878:Normal
5715:Orlicz
5705:Hölder
5685:Banach
5674:Spaces
5662:topics
5621:online
5597:
5587:
5547:
5510:
5470:
5433:
5423:
5415:
5407:
5358:
5350:
5311:
5303:
4454:as an
3815:
3683:
3361:is an
3145:finite
2900:= log
2638:, are
2381:, the
1881:: for
1690:. The
1570:center
1477:(with
1436:unique
1160:where
645:, and
562:Trace(
35:, the
5690:Besov
5468:JSTOR
5417:85485
5413:JSTOR
5356:JSTOR
5309:S2CID
5185:Notes
2908:) − 1
1672:be a
1502:, and
623:. By
558:trace
417:times
6038:(or
5756:Dual
5585:ISBN
5545:ISBN
5508:ISBN
5431:PMID
5405:ISSN
5348:ISSN
5301:ISSN
4442:for
4379:and
4343:and
4133:The
3392:and
3185:) =
3121:left
2627:the
2611:2n+1
2480:+ 2)
2462:for
2166:and
1889:and
1814:and
1676:and
1668:Let
1420:(or
1164:and
1040:) →
1012:and
982:and
824:and
675:and
659:and
649:and
634:and
630:Let
595:and
129:and
62:and
51:and
5610:365
5496:doi
5460:doi
5421:PMC
5395:doi
5340:doi
5293:doi
5289:104
4688:of
4542:by
4518:in
4450:of
3556:on
3421:in
3406:on
3365:of
3123:by
2949:by
2636:= 1
2625:not
2470:of
2414:)}
2406:),
2385:on
2279:of
2177:of
2162:of
1978:C*(
1976:on
1893:in
1885:in
1832:C*(
1830:of
1810:of
1804:C*(
1686:of
1555:+ 1
1407:+ 1
1237:),
1225:'s
1172:is
1099:and
607:'s
582:any
86:on
74:In
27:In
6117::
5664:–
5595:MR
5593:,
5583:,
5519:^
5492:46
5490:,
5466:,
5456:33
5454:,
5429:,
5419:,
5411:,
5403:,
5393:,
5383:16
5381:,
5354:,
5346:,
5336:33
5307:,
5299:,
5287:,
5204:nx
5202:ℏ
5198:=
5029:1.
4430:=
4399:.
3573:(2
3567:.
3430:,
3208:.
3187:π′
3156:π′
3154:,
3126:ha
2459:+1
2438:,
2421:=
2394:))
2081:*-
1984::
1905:,
1651:→
1576:.
1557:→
1431:.
1214:.
1201:+
1197:=
997::
961:E1
954:.
945:)
943:E1
696:+
692:→
640:=
568:BA
564:AB
542:,
442:,
90:.
66:.
6042:)
5766:)
5762:/
5758:(
5668:)
5650:e
5643:t
5636:v
5623:.
5514:.
5498::
5462::
5397::
5389::
5342::
5295::
5257:.
5255:ℏ
5253:n
5248:x
5242:p
5237:n
5235:∀
5228:x
5223:n
5218:x
5212:p
5200:i
5104:)
5102:K
5100:(
5097:n
5095:H
5088:.
5084:h
5082:U
5073:h
5071:U
5062:)
5060:Z
5057:p
5055:/
5053:Z
5051:(
5048:n
5046:H
5026:=
5022:|
5018:K
5014:|
5008:n
5005:2
5000:|
4995:K
4991:|
4982:1
4979:+
4976:n
4973:2
4968:|
4963:K
4959:|
4954:1
4949:=
4944:2
4939:|
4934:)
4931:g
4928:(
4921:|
4915:)
4912:K
4909:(
4904:n
4900:H
4893:g
4882:|
4878:)
4874:K
4870:(
4865:n
4861:H
4856:|
4852:1
4813:0
4806:0
4803:=
4800:b
4797:=
4794:a
4784:)
4781:c
4778:h
4775:(
4766:n
4761:|
4756:K
4752:|
4745:{
4740:=
4737:)
4734:)
4731:c
4728:,
4725:b
4722:,
4719:a
4716:(
4712:M
4708:(
4693:h
4691:U
4686:χ
4682:h
4659:.
4656:)
4653:a
4650:h
4647:+
4644:x
4641:(
4635:)
4632:c
4629:h
4626:+
4623:x
4617:b
4614:(
4608:=
4605:)
4602:x
4599:(
4595:]
4588:)
4585:c
4582:,
4579:b
4576:,
4573:a
4570:(
4566:M
4560:h
4556:U
4551:[
4540:)
4538:K
4527:h
4525:U
4520:K
4516:h
4501:)
4499:K
4497:(
4494:n
4492:H
4487:|
4485:K
4483:|
4476:)
4474:K
4472:(
4469:n
4467:H
4461:T
4452:K
4448:ω
4444:p
4439:Z
4436:p
4434:/
4432:Z
4428:K
4423:K
4419:)
4417:K
4415:(
4412:n
4410:H
4389:j
4382:a
4375:j
4373:a
4368:)
4366:R
4364:(
4362:L
4353:j
4346:a
4339:j
4337:a
4330:j
4328:a
4319:j
4312:a
4295:.
4290:k
4287:,
4284:j
4276:=
4272:]
4261:k
4257:a
4253:,
4248:j
4244:a
4239:[
4218:,
4213:j
4209:z
4205:=
4195:j
4191:a
4186:,
4178:j
4174:z
4161:=
4156:j
4152:a
4140:C
4106:.
4103:)
4100:x
4094:(
4088:=
4085:)
4082:x
4079:(
4076:]
4068:1
4064:W
4060:[
4040:)
4037:g
4034:(
4029:2
4019:h
4015:U
4011:=
4001:1
3997:W
3991:h
3987:U
3981:1
3977:W
3965:1
3962:W
3938:.
3935:)
3932:c
3929:,
3926:b
3920:,
3917:a
3911:(
3907:M
3903:=
3900:)
3897:c
3894:,
3891:b
3888:,
3885:a
3882:(
3878:M
3872:2
3868:)
3862:h
3854:(
3824:.
3821:y
3818:d
3812:)
3809:y
3806:(
3798:)
3795:b
3789:p
3786:(
3780:y
3777:i
3770:e
3762:n
3757:R
3745:)
3742:)
3739:a
3733:b
3727:c
3724:(
3721:h
3718:+
3715:p
3709:a
3706:h
3703:(
3700:i
3696:e
3692:=
3689:x
3686:d
3680:)
3677:a
3674:h
3671:+
3668:x
3665:(
3657:)
3654:c
3651:h
3648:+
3645:x
3639:b
3636:(
3633:i
3629:e
3623:p
3617:x
3614:i
3607:e
3599:n
3594:R
3577:)
3575:π
3565:)
3563:R
3561:(
3559:L
3550:W
3536:W
3532:W
3524:W
3514:h
3512:U
3495:.
3492:)
3489:g
3486:(
3478:h
3474:U
3470:=
3461:W
3457:)
3454:g
3451:(
3446:h
3442:U
3438:W
3426:n
3424:H
3419:g
3415:)
3413:R
3411:(
3409:L
3404:W
3399:α
3397:h
3395:U
3388:h
3386:U
3379:n
3377:H
3370:n
3368:H
3348:)
3344:b
3338:a
3332:c
3329:,
3326:a
3323:h
3320:,
3317:b
3312:1
3305:h
3297:(
3292:M
3285:)
3282:c
3279:,
3276:b
3273:,
3270:a
3267:(
3263:M
3259::
3254:h
3239:h
3204:n
3202:H
3197:z
3193:)
3191:z
3189:(
3183:z
3181:(
3179:π
3172:n
3170:H
3163:n
3161:H
3152:π
3138:h
3136:U
3115:h
3113:U
3103:n
3101:H
3079:.
3076:)
3073:a
3070:h
3067:+
3064:x
3061:(
3053:)
3050:c
3047:h
3044:+
3041:x
3035:b
3032:(
3029:i
3025:e
3021:=
3018:)
3015:x
3012:(
3005:]
3001:)
2998:)
2995:c
2992:,
2989:b
2986:,
2983:a
2980:(
2976:M
2972:(
2967:h
2963:U
2958:[
2947:)
2945:R
2943:(
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