Symmetry in quantum mechanics

Roughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is. Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator, i.e., a linear map \(u : H \rightarrow H\) satisfying one and hence all of the following equivalent conditions: 
 
 
\(uu^*=u^{*}u=1_H\); 
 
 
u is invertible with \(u^{-1}=u^{*}\); 
 
 
u is a surjective isometry (or, if \(\mathrm{dim}(H)<\infty \), just an isometry); 
 
 
u is invertible and preserves the inner product, i.e., \(\langle u \varphi , u \psi \rangle = \langle \varphi , \psi \rangle (\varphi , \psi \in H).\)


Six basic mathematical structures of quantum mechanics
We first recall the objects just described in a bit more detail. We have: (5.6) The point is that each of these sets has some additional structure that defines what it means to be a symmetry of it, as we now spell out in detail.
Definition 5.1. Let H be a Hilbert space (not necessarily finite-dimensional).

a. A Jordan symmetry is an invertible
Here a • b = 1 2 (ab + ba) (5.14) is the Jordan product on B(H) sa , which turns the (real) vector space B(H) sa into a Jordan algebra, cf. §C.25. b. A weak Jordan symmetry is an invertible weak Jordan map, i.e., a bijection (5.11) of which the restriction J |C sa is a Jordan map for each C ∈ C (B(H)).

A Ludwig symmetry is an affine order isomorphism
L : E (H) → E (H). which satisfies (5.12) for all a, b, as well as with notation as in Proposition 2.6. Conversely, such a Jordan map (5.18) defines a real Jordan map (5.11) by J = J |B(H) sa . Similarly, a weak Jordan symmetry is equivalent to a map (5.18) that satisfies (5.19), preserves squares as in (5.13), and is linear on each subspace C of B(H), with C ∈ C (B(H)). In other words (in the spirit of Bohrification), J C is a homomorphism of C*-algebras on each commutative unital C*-subalgebra C ⊂ B(H). Therefore, either way J and J C are essentially the same thing, and if no confusion may arise we call it J. Note that a weak Jordan map J a priori satisfies (5.12) only for commuting self-adjoint a and b. It follows that weak (and hence ordinary) Jordan symmetries are unital: since for any b, we may pick b = J −1 (1 H ) to find, reading (5.21) from right to left, The special role of unitary operators u now emerges: each such operator defines the relevant symmetry in the obvious way, namely, in order of appearance: W(e) = ueu * ; (5.23) K(ρ) = uρu * ; (5.24) L(a) = uau * ; (5.25) J(a) = uau * ; (5.26) N(e) = ueu * ; (5.27) B(C) = uCu * , (5.28) where a * = a in (5.26). If not, this formula remains valid also for the map J C . Furthermore, in (5.28) the notation uCu * is shorthand for the set {uau * | a ∈ C}, which is easily seen to be a member of C (B(H)). Here, as well as in the other three cases, it is easy to verify that the right-hand side belongs to the required set, that is, ueu * ∈ P 1 (H), uρu * ∈ D(H), uρu * ∈ E (H), (5.29) uau * ∈ B(H) sa , uρu * ∈ P(H), uCu * ∈ C (B(H)), (5.30) respectively, provided, of course, that e ∈ P 1 (H), ρ ∈ D(H), a ∈ E (H) a ∈ B(H) sa , e ∈ P(H), C ∈ C (B(H)).
The first case follows because SU(2) consist of all matrices of the form α β −β α , α, β ∈ C, |α| 2 + |β | 2 = 1. (5.43) The second case is obvious, and the third follows from Proposition 2.9. Assume the third case, so that a = e with e 2 = e * = e and Tr (e) = 1. If a linear map u : C 2 → C 2 is unitary, then simple computations show that e = ueu * is a onedimensional projection, too, given by e = 1 2 ∑ 3 μ=0 x μ σ μ with x 0 = 1, x ∈ R 3 , and x = 1. Writing x = Rx for some map R : S 2 → S 2 , we have u(x · σ )u * = (Rx) · σ , (5.44) where x · σ = ∑ 3 j=1 x j σ j . This also shows that R extends to a linear isometry R : R 3 → R 3 . Using the formula Tr (σ i σ j ) = 2δ i j , the matrix-form of R follows as This implies det(M) = ±1 (as can be seen by diagonalizing M; being a real linear isometry, its eigenvalues can only be ±1, and det(M) is their product). Thus O(3) breaks up into two parts O ± (3) = {R ∈ O(3) | det(R) = ±1}, of which O + ≡ SO(3) consists of rotations. Using an explicit parametrization of SO(3), e.g., through Euler angles, or, using surjectivity of the exponential map (from the Lie algebra of SO(3), which consist of anti-symmetric real matrices), it follows that O ± (3) are precisely the two connected components of O(3), the identity of course lying in O + (3).
To incorporate O − (3), let U a (2) be the set of all anti-unitary 2 × 2 matrices. These do not form a group, as the product of two anti-unitaries is unitary, but the union U(2) ∪U a (2) is a disconnected Lie group with identity component U(2).
Proof. The map u → R in (5.44) sends the anti-unitary operator u = J on C 2 to R = diag(1, −1, 1) ∈ O − (3). Since U a (2) = J ·U(2) and similarly O − (3) = R·SO(3), the last claim follows. The computation of the kernel may now be restricted to U(2), and then follows as in the last step op the proof of the previous proposition.
We now return to Theorem 5.4 and go through its special cases one by one. Part 1 of Theorem 5.4 is Wigner's Theorem, which in the case at hands reads: Theorem 5.7. Each bijection W : P 1 (C 2 ) → P 1 (C 2 ) that satisfies Tr (W(e)W( f )) = Tr (e f ) (5.49) for each e, f ∈ P 1 (C 2 ) takes the form W(e) = ueu * , where u is either unitary or anti-unitary, and is uniquely determined by W up to a phase.
To prove, this we transfer the whole situation to the two-sphere, where it is easy: Proposition 5.8. The pure state space P 1 (C 2 ) corresponds bijectively to the sphere in that each one-dimensional projection e ∈ P 1 (C 2 ) may be expressed uniquely as e(x, y, z) = 1 2 1 + z x− iy x + iy 1 − z , (5.50) where (x, y, z) ∈ R 3 and x 2 + y 2 + z 2 = 1. Under the ensuing bijection

51)
Wigner symmetries W of C 2 turn into orthogonal maps R ∈ O(3), restricted to S 2 .
Proof. With (u 1 , u 2 , u 3 ) the standard basis of R 3 , define a 3 × 3 matrix by R kl = u k , R(u l ) . (5.54) It follows from (5.53) that R −1 (u j ) k = R jk , which implies R −1 (u j ), x = ∑ k R jk x k , or, once again using (5.53), R(x) j = ∑ k R jk x k . Hence the map x → ∑ j,k R jk x k u j , i.e., the usual linear map defined by the matrix (5.54), extends the given bijection R.
Orthogonality of this linear map is, of course, equivalent to (5.53).
Wigner's Theorem then follows by combining Propositions 5.6 and 5.8: given the linear map R just constructed, read (5.44) from right to left, where u exists by surjectivity of the map (5.48), and the precise lack of uniqueness of u as claimed in Theorem 5.4 is just a restatement of the fact that (5.48) has U(1) as its kernel.
Kadison's Theorem is part 2 of Theorem 5.4. Explicitly, for H = C 2 we have: Theorem 5.10. Each affine bijection K : D(C 2 ) → D(C 2 ) is given as K(ρ) = uρu * , where u is unitary or anti-unitary, and is uniquely determined by K up to a phase.
Proof. We once again invoke Proposition 2.9, implying that any density matrix ρ on C 2 takes the form in that a convex sums tρ + (1 − t)ρ of density matrices correspond to convex sums tx + (1 − t)x of the corresponding vectors in R 3 .
Lemma 5.11. Any affine bijection K of the unit ball B 3 in R 3 is given by an orthog- Second, the basis of all further steps is the property This is because 0 is intrinsic to the convex structure of B 3 : it is the unique point with the property that for any x ∈ S 2 there exists a unique x such that 1 2 x + 1 2 x = 0, namely x = −x. Thus 0 must be preserved under affine bijections. For a formal proof (by contradiction), suppose K(0) = 0, and define y = K(0)/ K(0) ∈ S 2 . Then K(0) has an extremal decomposition K(0) = ty + (1 − t)y , with y = −y and t = 1 2 (1 + K(0) ). Applying the affine map K −1 then gives Now y ∈ S 2 and hence K −1 (y) ∈ S 2 by part one of this proof (applied to K −1 ), so that K −1 (y) = 1. But this implies K −1 (y ) > 1, which is impossible because y ∈ S 2 and hence K −1 (y ) = 1. Third, for x ∈ B 3 and t ∈ [0, 1] the preceding point implies that The same then holds for x ∈ B 3 and all t ≥ 0 as long as tx ∈ B 3 : for take t > 1, so that t −1 ∈ (0, 1), and use the previous step with x tx and t t −1 to compute Also, (5.58) and affinity imply that for any x, y ∈ B 3 for which x + y ∈ B 3 , we have With our earlier result (5.57), this also gives For some nonzero x ∈ R 3 , take s ≥ x and t ≥ x . Then by (5.58) we have We may therefore define a map R : R 3 → R 3 by for any choice of s ≥ x . For x ∈ B 3 we may take s = 1, so that R extends K.
To prove that R is linear, for x ∈ R 3 and t ≥ 0 pick some s ≥ t x and compute For t < 0, we first show from (5.60) and (5.62) that Furthermore, for given x, y ∈ B 3 , pick s > 0 such that s ≥ x and s ≥ y , so that s = 2s ≥ x + y by the triangle inequality, and use (5.59) to compute Finally, R is an isometry by (5.62) and step one of the proof. Being also linear and invertible, R must therefore be an orthogonal transformation.
Given step one, an alternative proof derives this lemma from Proposition 5.18 below, which shows that the transition probabilities (5.52) on S 2 are determined by the convex structure of B 3 , so that affine bijections must preserve them. In other words, the boundary map S 2 → S 2 defined by K preserves transition probabilities and hence satisfies the conditions of Lemma 5.9. This reasoning effectively reduces Kadison's Theorem to Wigner's Theorem, a move we will later examine in general.
In any case, Theorem 5.10 now follows from Lemma 5.11 is exactly the same way as Theorem 5.7 followed from the corresponding Lemma 5.9.
We have given this proof in some detail, because step 3 will recur on other occasions where a given affine bijection is to be extended to some linear map. Ludwig's Theorem is part 3 of Theorem 5.4. For H = C 2 , we have: where u is unitary or anti-unitary, and is uniquely fixed by L up to a phase.
Proof. Using the parametrization (5.41), we have a( In particular, we have 0 ≤ x 0 ≤ 2. This easily follows from (2.38), noting that a ∈ E (C 2 ) just means that a * = a and that both eigenvalues of a lie in [0, 1]. Thus E (C 2 ) is isomorphic as a convex set to a convex subset C of R 4 that is fibered over the x 0 -interval [0, 2], where the fiber C x 0 of C over x 0 is the three-ball B 3 x 0 with radius x = x 0 as long as 0 ≤ x 0 ≤ 1, whereas for 1 ≤ x 0 ≤ 2 the fiber is B 3 2−x 0 , so at x 0 = 1 the fiber is C 1 = B 3 ≡ B 3 1 (in one dimension less, this convex body is easily visualizable as a double cone in R 3 , where the fibers are disks). The partial order on C induced from the one on E (C 2 ) is given by which follows from (5.41) and (2.38), noting that for matrices one has a ≤ a iff a − a has positive eigenvalues. A similar argument to the one proving (5.57) then shows that any affine bijection L of C must map the base space [0, 2] to itself (as an affine bijection), and hence either x 0 → x 0 or x 0 → 2 − x 0 . The latter fails to preserve order, so L must fix x 0 . Similarly, L maps each three-ball C x 0 to itself by an affine bijection, which, by the same proof as for Kadison's Theorem above, must be induced by some element R x 0 of O(3). Finally, the order-preserving condition Part 3 of Theorem 5.4 does not carry an official name; it may be attributed to Kadison, too, but the hard part of the proof was given earlier by Jacobson and Rickart. Rather than a contrived (though historically justified) name like "Jacobson-Rickart-Kadison Theorem", we will simply speak of Jordan's Theorem (for H = C 2 ): Theorem 5.13. Each linear bijection J : M 2 (C) sa → M 2 (C) sa that satisfies (5.13) and hence (5.12) takes the form J(a) = uau * , where u is either unitary or antiunitary, and is uniquely determined by J up to a phase.
Proof. First, any Jordan map (and hence a fortiori any Jordan automorphism) trivially maps projections into projections, as it preserves the defining conditions e 2 = e * = e. Second, any Jordan automorphism J maps one-dimensional projections into one-dimensional projections: if e ∈ P 1 (H), then J(e) = 0 and J(e) = 1 2 , both because J is injective in combination with J(0) = 0 and J(1 2 ) = 1 2 , respectively. Hence J(e) ∈ P 1 (H), since this is the only remaining possibility (a more sophisticated argument shows that this is even true for any Hilbert space H). From (5.41) and subsequent text, as in (5.44), by linearity of J we therefore have from some map R : S 2 → S 2 , which is bijective because J is. Linearity of J then allows us to extend R to a linear map R 3 → R 3 , with matrix cf. (5.45). By (5.69), this linear map restricts to the given bijection R : S 2 → S 2 , which also shows that it is isometric. Thus we have a linear isometry on R 3 , which therefore lies in O(3). The proof may then be completed as in Theorem 5.7.
The case H = C 2 was already exceptional in the context of Gleason's Theorem, and it remains so as far as weak Jordan symmetries and Bohr symmetries are concerned.
Proposition 5.14. The poset C (M 2 (C)) is isomorphic to {⊥} ∪ RP 2 , where the real projective plane RP 2 is the quotient S 2 / ∼ under the equivalence relation x ∼ −x, and the only nontrivial ordering is ⊥ ≤ p for any p ∈ RP 2 .
Proof. It is elementary that M 2 (C) has a single one-dimensional unital * -subalgebra, namely C · 1, the multiples of the unit; this gives the singleton ⊥ in C (M 2 (C)). Furthermore, any two-dimensional unital * -subalgebra C of M 2 (C) is generated by a one-dimensional projection e, in that C is the linear span of e and 1 2 . Hence C is also the linear span of (the projection) 1 2 − e and 1 2 . In our parametrization of all one-dimensional projections e on C 2 by S 2 (cf. Proposition 2.9), if e corresponds to x, then 1 − e corresponds to −x. This yields the remainder RP 2 of C (M 2 (C)).
Finally, commutative unital * -subalgebras D of M 2 (C) of dimension > 2 do not exist. For any such algebra D would contain some two-dimensional C just defined, but a simple computation (for example, in a basis were C consists of all diagonal matrices) shows that the only matrices that commute with all elements of C already lie in C (i.e., are diagonal). Hence no commutative extension of C exists.
Bohr symmetries B for C 2 therefore correspond to bijections of RP 2 . Similarly, weak Jordan symmetries J for C 2 corresponds to bijections of S 2 (the difference with Bohr symmetries lies in the fact that J may also map C = span(e, 1 2 ) to itself nontrivially, i.e., by sending e to 1 2 − e, which for B would yield the identity map). In both cases, few of these bijections are (anti-) unitarily implemented.

Equivalence between the six symmetry theorems
If dim(H) > 1, the first three claims of Theorem 5.4 are equivalent; if dim(H) > 2, all claims are. We will show this in some detail, if only because the proofs of the various equivalences relate the six symmetry concepts stated in Definition 5.1 in an instructive way. We will do this in the sequence Wigner ↔ Kadison ↔ Jordan, and subsequently Jordan ↔ Ludwig, Jordan ↔ von Neumann, and Jordan ↔ Bohr. Consequently, in principle only one part of Theorem 5.4 requires a proof. Although redundant, we will, in fact, prove both Wigner's Theorem and Jordan's (indeed, no independent proof of the other parts of Theorem 5.4 seems to be known!). The most transparent way to state the various equivalences is to note that in each case the set of symmetries of some given kind (i.e., Wigner, . . . ) forms a group. In all cases, the nontrivial part of the proof is the establishment of a "natural" bijection, from which the group homomorphism property is trivial (and hence will not be proved).
where ρ = ∑ i λ i e υ i is some (not necessarily unique) expansion of ρ ∈ D(H) in terms of a basis of eigenvector υ i with eigenvalues λ i , where λ i ≥ 0 and ∑ i λ i = 1. In particular, (5.70) and (5.71) are well defined.
Proof. It is conceptually important to distinguish between B(H) sa as a Banach space in the usual operator norm · , and B 1 (H) sa , the Banach space of trace-class operators in its intrinsic norm · 1 . Of course, if dim(H) < ∞, then B(H) sa = B 1 (H) sa as vector spaces, but even in that case the two norms do not coincide (although they are equivalent). The proof below has the additional advantage of immediately generalizing to the infinite-dimensional case. We start with (5.70).
1. Since P 1 (H) = ∂ e D(H), by the same argument as in the proof of Lemma 5.11, any affine bijection of the convex set D(H) must preserve its boundary, so that K maps P 1 (H) into itself, necessarily bijectively. The goal of the next two steps is to prove that (5.70) satisfies (5.8), i.e., preserves transition probabilities. 2. An affine bijection K : D(H) → D(H) extends to an isometric isomorphism K 1 : B 1 (H) sa → B 1 (H) sa with respect to the trace-norm · 1 , as follows: a. Put K 1 (0) = 0 and for b ≥ 0, b ∈ B 1 (H), i.e. b ∈ B 1 (H) + , and b = 0, define (5.72) By construction, K 1 is isometric and preserves positivity. For b ∈ B 1 (H) + we have Tr (b) = b 1 , hence b/ b 1 ∈ D(H), on which K is defined. Linearity of K 1 with positive coefficients (as a consequence of the affine property of K) is verified as in the proof of Lemma 5.11; this time, use A.24 (this remains valid in general Hilbert spaces). We then define To show that this makes K 1 linear on all of and since each term is positive, by the previous step. Hence , so that (5.74) is actually independent of the choice of the decomposition of b as long as the operators are positive. Hence for a, b ∈ B 1 (H) sa we may compute since a + + b + and a − + b − are both positive.
The key point in verifying isometry of K 1 is the property |b| = b + + b − , which follows from (A.76) or Theorem B.94. Using this property, we have 3. For any two unit vectors ψ, ϕ in H we have the formula which can easily be proved by a calculation with 2 × 2 matrices (since everything takes place is the two-dimensional subspace spanned by ψ and ϕ, expect when ϕ = zψ, z ∈ T, in which case (5.75) reads 0 = 0 and hence is true also). Since K 1 is linear as well as isometric with respect to the trace-norm, we have and hence, by (5.75), Tr (K 1 (e ψ )K 1 (e ϕ )) = Tr (e ψ e ϕ ). Eq. (5.70) then gives (5.8).
We move on to (5.71). The main concern is that this expression be well defined, since in case some eigenvalue λ > 0 of ρ is degenerate (necessarily with finite multiplicity, even in infinite dimension, since ρ is compact), the basis of the eigenspace H λ that takes part in the sum ∑ i λ i e υ i is far from unique. This is settled as follows: In other words, e ψ ≤ e L iff ∑ j Tr (e υ j e ψ ) = 1. Furthermore, by (5.8) the images W(e υ j ) remain orthogonal; hence ∑ j W(e υ j ) is a projection, and e ≤ ∑ j W(e υ j ) iff ∑ j Tr (W(e υ j )e) = 1. By (5.8), this condition is satisfied for e = W(e υ i ), so that W(e υ i ) ≤ ∑ j W(e υ j ) for each j. Since also the projections W(e υ i ) are orthogonal, this gives ∑ i W(e υ i ) ≤ ∑ j W(e υ j ). Interchanging the roles of the two bases gives the converse, yielding (5.76).
Finally, to prove bijectivity of the correspondence K ↔ W, we need the property since this implies that K is determined by its action on P 1 (H) ⊂ D(H). In finite dimension this follows from convexity of K, and we are done. In infinite dimension, we in addition need continuity of K, as well as convergence of the sum ∑ i λ i e υ i not only in the operator norm (as follows from the spectral theorem for self-adjoint compact operators), but also in the trace norm: for finite n, m, since e υ i 1 = 1. Because ∑ i λ i = 1, the above expression vanishes as n, m → ∞, whence ρ n = ∑ n i=1 λ i e υ i is a Cauchy sequence in B 1 (H), which by completeness of the latter converges (to an element of D(H), as one easily verifies).
The proof of continuity is completed by noting that K is continuous with respect to the trace norm, for it is isometric and hence bounded (see step 2 above).
It is enlightening to give a rather more conceptual proof that K |P 1 (H) satisfies (5.8), which is based on a result to be used more often in the future. In what follows, for any convex set C, the notation A b (K) stands for the real vector space of bounded affine functions f : C → R, that is, bounded functions satisfying It is easily checked that A b (K) with the supremum-norm is a real Banach space. Note that under the identification D(H) ∼ = S n (B(H)) (where in finite dimension the normal state space S n (B(H)) simply coincides with the state space S(B(H))), where ρ ↔ ω as in (2.33), i.e., ω(a) = Tr (ρa), the above isomorphism simply reads This function is clearly bounded on the unit ball of H, as in To check that Q in fact defines a quadratic form on H, we verify the properties (A.8) -(A.9). The first is trivial. The second follows from the easily verified identity where v, w = 0, v = w, and the coefficients s,t are given by (5.90) The affine property (5.78) then immediately yields (A.9). According to Proposition B.79, we obtain a unique operator a ∈ B(H) sa such that Q(ψ) = ψ, aψ , i.e., ψ, aψ = f (e ψ ), ψ ∈ H, ψ = 1. (5.91) Since also ψ, aψ = Tr (e ψ a), we have established (5.81) for each ρ = e ψ , where ψ ∈ H, ψ = 1. To extend this result to general density operators ρ = ∑ i λ i e υ i , we use (A.100) as well as convergence of the above sum in the trace norm · 1 , cf. the proof of Lemma 5.16; the details are analogous to the proof of Theorem B.146.
Proposition 5.18. For any unit vectors ψ, ϕ ∈ H we have The virtue of this formula is that the expression on the left-hand side, which defines the transition probabilities on ∂ e D(H) = P 1 (H), is intrinsically given by the convex structure of D(H). Consequently, any affine bijection of this convex set (which already preserves the boundary) must preserve these probabilities.
We now turn to the equivalence between Jordan's Theorem and Kadison's Theorem.
which is evidently unital, positive, and isometric. Consequently, by Proposition 5.17, K * corresponds to some isomorphism α : B(H) sa → B(H) sa , which necessarily shares the properties of being unital, positive, and isometric; this follows abstractly from the proposition, but may also be verified directly from (5.95). Conversely, such a map α yields a map K directly by (5.95); to see this, we identify D(H) with the normal state space of B(H) through ρ ↔ ω, as usual, cf.
This is possible only if α(a) ≤ a , and hence α is continuous with norm bounded by α ≤ 1. In fact, since a is unital we have α = 1.
Therefore, any unital positive linear map α preserves orthogonality of projec- since e i e j = δ i j e j and by the above comment also α(e i )α(e j ) = δ i j α(e j ). By continuity of α, this property extends to arbitrary a ∈ B(H) sa . Finally, since preserving squares as in (5.100) implies preserving the Jordan product •.
We now turn to the equivalence between Ludwig symmetries and Jordan ones.
Proposition 5.21. There is an isomorphism of groups between: • The group of affine order isomorphism L : Proof. Since L is an order isomorphism, it satisfies L(0) = 0 (as well as L(1 H ) = 1 H ), since 0 is the bottom element of E (H) as a poset (and 1 H is its the top element). As in the proof of Lemma 5.11, one shows that this property plus convexity implies L(ta) = tL(a) and L(a + b) = L(a) + L(b) whenever defined. Defining J by where a > 0 means a ≥ 0 and a = 0, and a < 0 means −a ≥ 0 and a = 0, once again the reasoning near the end of the proof of Lemma 5.11 shows that J is linear; it is a untital order-preserving bijection by construction. Hence J is a Jordan automorphism by Lemma 5.20.2 Of course, instead of (5.104) one could equivalently have defined J on general a ∈ B(H) sa by J(a) = J(a + ) − J(a − ), using the (by now hopefully familiar) decomposition a = a + − a − with a ± ≥ 0 and a + a − = 0. Conversely, once again using Lemma 5.20.2, a Jordan automorphisms (5.11) preserves order as well as the unit, so that the inequality 0 where it preserves order. Convexity is obvious, since L = J |E (H) comes from a linear map.
Corollary 5.22. Let dim(H) > 2. Then an isomorphism N of P(H) as an orthocomplemented lattice has a unique extension to a linear map α : B(H) sa → B(H) sa , which is (automatically) invertible, unital, and positive.
Proof. According to Lemma D.2, N preserves all suprema in P(H). Since we have ∑ i e i = e i for any family of mutually orthogonal projections and since N by definition preserves the orthocomplementation e ⊥ = 1 − e and hence preserves orthogonality of projections, we may compute (5.105) Consequently, for any normal state ω on B(H), the map e → ω •N(e) is a probability measure on P(H), which by Gleason's Theorem has a unique linear extension to B(H) and hence a fortiori to B(H) sa . We use this in order to define α, as follows. First, let a ∈ B(H) sa and suppose a = ∑ j λ j f j for some finite family ( f j ) of projections (not necessarily orthogonal), and some λ j ∈ R. Then ∑ j λ j N( f j ) is independent of the particular decomposition of a that has been chosen, so we may put To see this, put a = ∑ j λ j f j and hence α (a) = ∑ j λ j N( f j ), and suppose α (a) = α(a). By (B.477) there exists a normal state ω such that ω(α (a)) = ω(α(a)); indeed, each element of B 1 (H) is a linear combination of at most four density operators, so that each normal linear functional on B(H) is a linear combination of at most four normal states.
which is a contradiction. Hence α (a) = α(a) and accordingly, (5.106) is well defined. Because it is independent of the decomposition of a into projections, α is Similarly, for any t ∈ R we have We may now extend α to all of B(H) sa by continuity. Indeed, according to the spectral theorem in the form (B.326), the set of all operators of the form a = ∑ j λ j f j with all f j mutually orthogonal (so that a is given by its spectral resolution) is normdense in B(H) sa . Applying (5.106), and noting that a = sup j |λ j |, we may estimate since the N( f j ) are mutually orthogonal and hence sum to some projection, which has norm 1 (unless a = 0). For general a ∈ B(H) sa , we may therefore define N by N(a) = lim n N(a n ), where each a n is of the above (spectral) form and a n − a → 0.
To prove that α is positive, we show that α(a) ≥ 0 whenever a ≥ 0. As in the preceding step, initially suppose that a = ∑ j λ j f j has a finite spectral resolution. Then a ≥ 0 iff λ j ≥ 0 for each j, and hence α(a) ≥ 0 by (5.106), since by orthogonality of the N( f j ) this equation states the spectral resolution of α(a). Now if a n ≥ 0 and a n → a (in norm), then ψ, a n ψ → ψ, aψ , which must remain positive, so that a ≥ 0. Hence positivity of α on all of B(H) sa follows by continuity.
Finally, α inherits invertibility from N, and it is unital by (5.105), taking e i = |υ i υ i | for some basis (υ i ) of H (or using the fact that it preserves = 1 H ).
Subsequently, we use Lemma 5.20 to further extend α by complex linearity to a Jordan isomorphism of B(H); see Definition 5.1.
Finally, the equivalence between weak Jordan symmetries and Bohr symmetries follows from Hamhalter's Theorem 9.4, whereas Theorem 9.7 strengthens this to an equivalence between Jordan symmetries and Bohr symmetries. The proof of these theorems does not seem to simplify in the special case at hand, i.e. A = B(H).

Proof of Jordan's Theorem
In view of the equivalence between the six parts of Theorem 5.4, we only need to prove one of them. In the literature, one only finds proofs of Jordan's Theorem and of Wigner's Theorem, and we present each of these (surprisingly but instructively, these proofs look completely different). We start with Jordan's Theorem: where u is unitary (and is determined by J C up to a phase), or by where u is anti-unitary (and is determined by J C up to a phase, too).
The difficult part of the proof is Theorem C.175, which implies:

Recall that an automorphism of B(H) is a linear bijection
; an anti-automorphism, on the other hand, satisfies the first property whilst the latter is replaced by Clearly, both automorphisms and anti-automorphisms are Jordan automorphisms.
Granting this result, we may deal with the two cases separately.

(H) → B(H) is an automorphism and e ∈ B(H) is a onedimensional projection, then so is α(e).
Proof. It should be obvious that automorphisms α preserve projections e (whose defining properties are e 2 = e * = e). Furthermore, α preserves order, i.e., if a ≥ 0 (in that, as always, ψ, aψ ≥ 0 for each ψ ∈ H, or, equivalently, a = b * b), then α(a) ≥ 0 (this is clear from the second way of expressing positivity). Consequently, ; see Proposition C.170. With respect to the ordering ≤ the onedimensional projections e are atomic, in the sense that 0 ≤ e (but e = 0) and if 0 ≤ f ≤ e, then either f = 0 or f = e. Now automorphisms of the projection lattice B(H) restrict to isomorphisms of P(H), which preserve atoms (as these are intrinsically defined by the partial order).
We are now ready for the (constructive!) proof of Proposition 5.25.
Proof. For some fixed unit vector χ ∈ H, take the corresponding one-dimensional projection e χ and define a new unit vector ϕ (up to a phase) by Now any ψ ∈ H may be written as ψ = aϕ, for some a ∈ B(H). Attempt to define an operator u by uψ = α(a)χ, i.e., This looks dangerously ill-defined, since many different operators a may give rise to the same ψ. Fortunately, we may compute so that if aϕ = bϕ, then α(a)χ = α(b)χ and hence u is well defined. By this computation u is also isometric and since it is clearly surjective, it is unitary. The property α(a) = uau * is equivalent to ua = α(a)u, which in turn is equivalent to uabϕ = α(a)ubϕ for any b ∈ B(H), which by definition of u is the same as Then α • β is an automorphism, to which Proposition 5.25 applies, so that for some unitaryũ. Hence The precise lack of uniqueness of u is inherited from the unitary case.

Proof of Wigner's Theorem
We recall Wigner's Theorem, i.e. Theorem 5.4.1: Theorem 5.29. Each bijection W : P 1 (H) → P 1 (H) that satisfies where the operator u is either unitary or antiunitary, and is uniquely determined by W up to a phase.
The problem is to lift a given map W : P 1 (H) → P 1 (H) that satisfies (5.115) to either a unitary or an anti-unitary map u : Since e zψ = e ψ for any z ∈ T, and likewise for e ψ , this means that uψ = zψ for some z ∈ T; the problem is to choose the z's coherently all over the unit sphere of H. There are many proofs in the literature, of which the following one-partly based on an earlier proof by Bargmann (1964)-has the advantage of making at least the construction of u explicit (at the cost of opaque proofs of some crucial lemma's). We assume dim(H) > 2, since H = C 2 has already been covered. Fix unit vectors ψ ∈ H and ψ ∈ W(e ψ )H; clearly, ψ is unique up to multiplication by z ∈ T, whose choice turns out to completely determine u (i.e., the ambiguity in ψ is the only one in the entire construction). For a modest start, we put uψ = ψ . (5.117) , then there is a unique k-dimensional linear subspace V ⊂ H with the following property: Proof. Pick a basis (υ 1 ,..., υ k ) of V and find unit vectors υ i ∈ H such that υ i ∈ W(e υ i )H, i = 1,..., k. Then, using (5.115) we compute so that the vectors (υ 1 ,..., υ k ) form an orthonormal set and hence form a basis of their linear span V . Now, as mentioned below (B.214), Since W preserves transition probabilities, a computation similar to one just given yields (5.118) so that both sides do or do not equal unity, and hence ψ ∈ V iff ψ ∈ V .
Wigner's Theorem for H = C 2 (i.e. Theorem 5.7) implies: Lemma 5.31. If V and V are related as in Lemma 5.30, and Proof. A choice of basis for both V and V gives unitary isomorphisms u : ( 5.121) This maps satisfies the hypotheses of Wigner's Theorem in d = 2, and so it is (anti-) Then the operator u V = (u ) −1 vu does the job; its lack of uniqueness stems entirely from v.
Proof. Let (υ 1 , υ 2 , υ 3 ) be some basis of of H (like the usual basis of H = C 3 ). We first show that if W is the identity if restricted to both span(υ 1 , υ 2 ) and span(υ 1 , υ 3 ), then W is the identity on H altogether. To this end, take ψ = ∑ i c i υ i , initially with c 1 ∈ R\{0}. Take a unit vector ψ ∈ W(e ψ ), with ψ = ∑ i c i υ i . By the first assumption on W we have | υ, ψ | = | υ, ψ | for any unit vector υ ∈ span(υ 1 , υ 2 ). Taking gives the equations respectively. By a choice of phase we may and will assume c 1 = c 1 , in which case the only solution is c 2 = c 2 (geometrically, the solution c 2 lies in the intersection of three different circles in the complex plane, which is either empty or consists of a single point). Similarly, the second assumption on W gives c 3 = c 3 , whence ψ = ψ. The case c 1 = 0 may be settled by a straightforward limit argument, since inner products (and hence their absolute values) are continuous on H × H. Given a Wigner symmetry W : P 1 (H) → P 1 (H), we now construct u as follows.
We now finish the proof of Wigner's Theorem. We assume that the outcome of Lemma 5.32 is that each u V is unitary; the anti-unitary case requires obvious modifications of the argument below. The first step is, of course, to define u(λ ψ) = λ uψ, λ ∈ C (so this would have been λ uψ in the anti-unitary case). Let ϕ ∈ H be linearly independent of ψ and consider the two-dimensional space V spanned by ψ and ϕ. Define u(ϕ) = u V ϕ. With (5.117), this defines u on all of H. To prove that u is linear, take ϕ 1 and ϕ 2 linearly independent of each other and of ψ, so that the linear span V 3 of ψ, ϕ 1 , and ϕ 2 is three-dimensional. Let V i be the two-dimensional linear span of ψ and ϕ i , i = 1, 2. Then uϕ i = u V i ϕ i , where the phase of u V i is fixed by (5.117). Let w : V 3 → V 3 be the unitary that implements W according to Lemma 5.33.2, with phase determined by (5.117). Since u V 1 and u V 2 and w are unique up to a phase and this phase has been fixed for each in the same way, we must have u V 1 = w |V 1 and u V 2 = w |V 2 . Finally, we have V 12 spanned by ψ and ϕ 1 + ϕ 2 , and by the same token, u V 12 = w |V 12 . Now w is unitary and hence linear, so since this is how u was defined. Since each u V is unitary, so is u, and similarly it is easy to verify that u implements W, because each u V does so.

Some abstract representation theory
Since all symmetries we have considered (named after Wigner, Kadison, Jordan, Ludwig, von Neumann, and Bohr) are implemented by either unitary or anti-unitary operators, which are determined (by the given symmetry) only up to a phase z ∈ T, the quantum-mechanical symmetry group G H of a Hilbert space H is given by where U(H) is the group of unitary operators on H, and U a (H) is the set of antiunitary operators on H; the latter is not a group (since the product of two antiunitaries is unitary) but their union is. Furthermore, T is identified with the normal H) (and also of U(H)) consisting of multiples of the unit operators by a phase; thus the quotient G H is a group. The fact that G H rather than U(H) is the symmetry group of quantum mechanics has profound consequences (one of which is our very existence), which we will study from §5.10 onwards. However, this material relies on the theory of "ordinary" (i.e., non-projective) unitary representations, which we therefore review first.
Namely, let G be a group. In mathematics, the natural kind of action of G on a Hilbert space H is a unitary representation, i.e., a homomorphism As to the possible continuity properties of unitary representations in case that G is a topological group (i.e., a group G that is also a topological space, such that group multiplication G × G → G and inverse G → G are continuous), one should equip U(H) with the strong operator topology (as opposed to the norm topology).
Proposition 5.35. If u : x → u(x) is a unitary representation of some locally compact group G on a Hilbert space H, then the following conditions are equivalent: . This is clearly implied by the first kind of continuity, giving 1 ⇒ 2, so let us prove the nontrivial converse. Suppose x λ → x and ψ μ → ψ; since G is locally compact, x has a compact neighborhood K and we may assume that each x λ ∈ K. If u is strongly continuous, then for any ϕ ∈ H the set {u(y)ϕ, y ∈ K} is compact in H and hence bounded. The Banach-Steinhaus Theorem B.78 gives boundedness of the corresponding operator norms, that is, { u(y) , y ∈ K} < C K for some C K > 0. We now estimate The first term vanishes as ψ μ → ψ since it is bounded by C K ψ μ − ψ , whereas the second vanishes as x λ → x by the (assumed) strong continuity of u.
Since the first kind of continuity is the usual one for group actions, this justifies the choice of strong continuity as the natural one for unitary representations (to which a pragmatic point may be added: norm continuity is quite rare for unitary representations on infinite-dimensional Hilbert spaces). Things further simplify under mild restrictions on G and H, which are satisfied in all examples of physical interest.
Proposition 5.36. If H is separable and G is second countable locally compact (sclc), then each of the two continuity conditions in Proposition 5.35 is in turn equivalent to weak measurability of u, in that for each ϕ, ψ ∈ H the function Proof. This spectacular result is due to von Neumann, who more generally proved that a measurable homomorphism between sclc groups is continuous. This implies the claim: first, if H is separable, then the group U(H) is sclc in its weak operator topology, so that if the map G → U(H), x → u(x) is weakly measurable, then it is continuous in the weak topology on U(H). Second, for any Hilbert space, weak (operator) continuity of a unitary representation implies strong continuity (so that, given the trivial converse, weak and strong continuity of unitary group representations are equivalent). We only prove this last claim: for x, y ∈ G, we compute Weak continuity obviously implies that the function x → ψ, u(x)ψ is continuous at the identity e ∈ G, so if y = x λ → x, then (u(x λ ) − u(x))ψ → 0.
In view of this, it is hardly a restriction for a unitary representation of a locally compact group on a Hilbert space to be continuous in the sense of Proposition 5.35, so we always assume this in what follows. Furthermore, any group we consider is locally compact, so this will be a standing assumption, too. An important consequence of this assumption is the existence of a translation-invariant measure on G.
Theorem 5.37. Each locally compact group G has a canonical nonzero (outer regular Borel) measure μ, called Haar measure, which is left-invariant in that for each f ∈ C c (G) and y ∈ G, where the left translation L y of f by y is defined by This measure is unique up to scalar multiplication. Moreover, if G is compact, then: 1. μ is finite and hence can be normalized to a probability measure, i.e., where the right translation R y of f by y ∈ G is defined by Existence is due to Haar and uniqueness was first proved by von Neumann. One often writes dx ≡ dμ(x) for Haar measure. Here are some examples: • For G = R n , Haar measure equals Lebesgue measure μ L (up to a constant); eqs.
i.e., if au(x) = u(x)a for each x ∈ G implies a = λ · 1 H for some λ ∈ C.
This follows from Theorem C.90, of which the above lemma is a special case: take A = u(G) ≡ (u(G) ) . The second is part of the Peter-Weyl Theorem.
Theorem 5.40. Irreducible representations of compact groups are finite-dimensional.
Proof. We first reduce the situation to the unitary case: if ·, ·, is the given inner product on H, we define a new inner product ·, ·, by averaging with respect to Haar measure dx ≡ dμ(x), i.e., Using (5.141), it is easy to verify that this new inner product makes u unitary. So let u : G → u(H) be an irreducible unitary representation. For each unit vector ϕ ∈ H and x ∈ G, we define the following projection and its G-average: The Weyl operator (5.150) is initially defined as a quadratic form by The integral exists because the integrand is continuous and bounded, defining a bounded quadratic form by the estimate | ψ 1 ,W ϕ ψ 2 | ≤ ψ 1 ψ 2 , where we assumed (5.140) and used e u(x)ϕ = 1, as (5.149) is a nonzero projection. Thus the operator W ϕ may be reconstructed from its matrix elements (5.151), cf. Proposition B.79. It is easy to verify that [W ϕ , u(y)] = 0 for each y ∈ G, so that Schur's Lemma yields W ϕ = λ ϕ · 1 H for some λ ϕ ∈ C. Hence ψ,W ϕ ψ = λ ϕ ψ 2 , in other words, If we now interchange ϕ and ψ and use (5.143) we find λ ϕ ψ 2 = λ ψ ϕ 2 , so that, taking ψ to be a unit vector, too, since ψ and ϕ are arbitrary we obtain λ ϕ = λ ψ ≡ λ , where in fact λ > 0, as follows by taking ψ = ϕ in (5.152). Finally, take n orthornormal vectors (υ 1 ,..., υ n ) in H, so that also (u(x)υ 1 ,..., u(x)υ n ) are orthonormal (since u(x) is unitary), upon which Bessel's inequality (B.212) gives Integrating both sides over G, taking ψ = 1, and using (5.140) gives On the other hand, summing (5.152) over i simply yields nλ , whence nλ ≤ 1, for any n ≤ dim(H). Since λ > 0 this forces dim(H) < ∞.

Representations of Lie groups and Lie algebras
We now assume that G is a Lie group; as in §3.3, for our purposes we may restrict ourselves to linear Lie groups, i.e. closed subgroups of GL n (K) for K = R or C. Let u : G → U(H) be a unitary representation of a Lie group G on some Hilbert space H (assumed strongly continuous). If H is finite-dimensional, the following operation is unproblematic: for A ∈ g (i.e. the Lie algebra of G) we define an operator (5.164) The property of irreducibility of such a representation ρ : g → B(H) is defined in the same way as for groups, namely that the only linear subspaces of H ∼ = C n that are stable under ρ(g) are {0} and H. Equivalently, by Schur's Lemma, ρ(g) is irreducible iff the only operators that commute with all π(A) are multiples of the unit operator. If ρ = u for some unitary representation u(G), it is easy to see that u is irreducible iff u is irreducible. In view of this, it is a reasonable strategy to try and construct irreducible unitary representations u(G) by starting, as it were, from u (g). More precisely, if ρ is some (irreducible) skew-adjoint representation of g, we may ask if there is a (necessarily irreducible) unitary representation u(G) such that ρ = u . Writing exp(ρ) for u, one would therefore hope that u e A ≡ e ρ e A = e ρ(A) , (5.165) as in (5.162). Note that if G is connected, then ρ duly defines u(x) for each x ∈ G through (5.165), since by Lie theory every element x of a connected Lie group is a finite product x = exp(A 1 ) · · ·exp(A n ) of exponentials of elements (A 1 ,..., A n ) of g.
In general, this hope is in vain, since although each operator exp(A) is unitary, the representation property u(x)u(y) = u(xy) may fail for global reasons. For example, if G = SO(3), then g ∼ = R 3 , with basis (J 1 , J 2 , J 3 ), as in (3.66). Define an a priori linear map ρ : g → M 2 (C) by linear extension of where (σ 1 , σ 2 , σ 3 ) are the Pauli matrices (5.42), so that physicists would write cf. (5.159). This is easily checked to give a skew-adjoint representation of g, but it does not exponentiate to a unitary representation of SO (3): as already mentioned after Proposition 5.46, if u is a unit vector in R 3 , then a rotation R θ (u) around the u-axis by an angle θ ∈ [0, 2π] is represented by Consequently, u(R π (u)) = iu · σ , so that u(R π (u)) 2 = −1 2 , although within SO(3) one has R π (u) 2 = e, the unit of SO (3), so that u(R π (u)) 2 = u(R π (u) 2 ). However, ρ does exponentiate to a representation of SU (2), which happens to be the universal covering group of SO(3). This is typical of the general situation, which we state without proofs. We first need a refinement of Lie's Third Theorem: Theorem 5.41. Let G be a connected Lie group G with Lie algebra g. There exists a simply connected Lie groupG, unique up to isomorphism, such that: • The Lie algebra ofG is g.
• G ∼ =G/D, where D is a discrete normal subgroup of the center ofG.
Theorem 5.42. Let G 1 and G 2 be Lie groups, with Lie algebras g 1 and g 2 , respectively, and suppose that G 1 is simply connected. Then every Lie algebra homomorphism ϕ : g 1 → g 2 comes from a unique Lie group homomorphism Φ : G 1 → G 2 through ϕ = Φ , where (realizing G 1 and G 2 as matrices) For example, G = SO (3), the last condition is satisfied for the irreducible representations with integer spins j ∈ N (as well as for j = 0), see §5.8.
A similar construction is possible when H is infinite-dimensional, except for the fact that the derivative in (5.156) may not exist. For example, G = R has its canonical regular representation on H = L 2 (R), defined by u(a)ψ(x) = ψ(x−a), in which case (5.159) gives some multiple of the momentum operator −ihd/dx. This operator is unbounded and hence is not defined on all of H, see also §5.11 and §5.12. As in Stone's Theorem 5.73, this problem is solved by finding a suitable domain in H on which the underlying limit, taken strongly, does exist. This is the Gårding domain where for each f ∈ C ∞ c (G) (or even f ∈ L 1 (G)) the operator u ( f ) is defined by Like the derivative u , this integral is most easily defined weakly, i.e., the (bounded) operator u ( f ) is initially defined as a bounded quadratic form from which the operator u ( f ) may be reconstructed as in Proposition B.79. Note that the function x → ϕ, u(x)ψ is in C b (G), so that the integral (5.173) exists. It can be shown that D G is dense in H, as well as invariant under u (g), in the sense that if ψ ∈ D G , then u (A)ψ ∈ D G for any A ∈ g. Furthermore, for each ϕ ∈ D G the function x → u(x)ϕ from G to H is smooth (if G is unimodular this property even characterizes D G ). The commutation relations (5.157) then hold on D G , but the equalities (5.164) do not: one has to choose between (5.157) and (5.164), since the latter holds for the closure of each π(A) (i.e., each iρ(A) is essentially selfadjoint on D G ), whose domain however depends on A: there is no common domain on which each iρ(A) is self-adjoint and the commutation relations (5.157) hold.

Irreducible representations of SU(2)
One of the most important groups in quantum physics is SU(2), both as an internal symmetry group-e.g. of the Heisenberg model of ferromagnetism, of the weak nuclear interaction, and possibly also of (loop) quantum gravity-and as a spatial symmetry group in disguise (all projective unitary representations of SO(3) come from unitary representations of SU(2), preserving irreducibility, cf. Corollary 5.61). In this section we review the well-known classification and construction of its unitary irreducible representations. Since SU(2) is compact, by Theorem 5.40 all its unitary irreducible representations are finite-dimensional. Since G = SU(2) is also simply connected, by Corollary 5.43 its irreducible finite-dimensional (unitary) representations u bijectively correspond to the irreducible finite-dimensional skew-adjoint representations ρ = u of its Lie algebra g. Hence our job is to find the latter.
We already encountered the basis (3.66) of the Lie algebra so(3) ∼ = R 3 of SO (3); the corresponding basis of the Lie algebra su (2) (5.174) and the σ k are the Pauli matrices given in (5.42); linear extension of the map J k → S k defines an isomorphism between so(3) and su (2). These matrices satisfy where ε i jk is the totally anti-symmetric symbol with ε 123 = 1 etc., so that (5.175) comes down to [S 1 , S 2 ] = S 3 , [S 3 , S 1 ] = S 2 , and [S 2 , S 3 ] = S 1 . By linearity, finding ρ is the same as finding n × n matrices i.e., [L 1 , L 2 ] = iL 3 , etc., and L * k = L k . (5.178) It turns out to be convenient to introduce the ladder operators By Schur's lemma, in any irreducible representation we therefore must have where c ∈ R (in fact, c ≥ 0). We will also use the additional algebraic relations The simple idea is now to diagonalize L 3 , which is possible as L * 3 = L 3 . Hence where σ (L 3 ) is the spectrum of L 3 (which in this finite-dimensional case consists of its eigenvalues), and H λ is the eigenspace of L 3 for eigenvalue λ (i.e., if υ ∈ H λ , then L 3 υ = λ υ). The structure of (5.187) in irreducible representations is as follows. Proof. For any λ ∈ σ (L 3 ) and nonzero υ λ ∈ H λ , we have: • either λ + 1 ∈ σ (L 3 ) and L + υ λ ∈ H λ +1 (as a nonzero vector); • or L + υ λ = 0.
Since H is finite-dimensional by assumption, there must be some k ∈ N 0 = N ∪ {0} such that L k+1 + υ λ 0 = 0, whereas all vectors L l + υ λ 0 for l = 0,..., k are nonzero (and lie in H λ 0 +l ). With c defined as in (5.184), it then follows from (5.185) -(5.186) that c − λ 0 (λ 0 − 1) = 0; (5.189) These relations imply λ 0 = −k/2, so that by the above bullet points we also have To prove equality, as in (5.188), consider the vector space (5.192) this is just the subspace of H with basis (υ λ 0 , L + υ λ 0 ,..., L k−1 + υ λ 0 , L k + υ λ 0 ). By the previous arguments following from (5.180), we see that the operators L + and L − never leave H , and the same is trivially true for L 3 . Therefore, if ρ is irreducible, then we must have H = H (and conversely). All claims of the lemma are now trivially verified on H .
It should be clear from this proof that the actions of L + , L − , and L 3 (and hence of all elements of su (2)) on H = H) are fixed, so that ρ is determined by its dimension dim(H) = 2 j + 1, (5.193) from which it follows that j can only take the values 0, 1/2, 1, 3/2,.... It remains to fix an inner product on H in which ρ is skew-adjoint, i.e., in which L * 3 = L 3 and L * + = L − (which implies that L * 1 = L 1 and L * 2 = L 2 , which jointly imply ρ(X * ) = −ρ(X) for any X ∈ g). This may be done in principle by starting with any inner product, integrating ρ to a unitary representation of SU(2), and using the construction explained at the beginning of the proof of Theorem 5.40. In practice, it is easier to just calculate: take H = C n with n = 2 j + 1, standard inner product, and standard orthonormal basis (u l ), labeled as l = 0, 1,..., 2 j). Then put L 3 u l = (l − j)u l ; (5.194) L + u l = (l + 1)(n − l − 1)u l+1 ; (5.195) (5.196) Note that (5.195) is even formally correct for l = 2 j, since in that case n−2 j −1 = 0, and similarly, (5.196) formally holds even for l = 0. The commutation relations (5.180) -(5.181) as well as the above conditions for skew-adjointness may be explicitly verified, from which it follows that for any prescribed dimension (5.193) we have found a skew-adjoint realization of ρ. Clearly, u l = υ l− j . In view of Theorem 5.40 and Corollary 5.43 we have therefore proved: Physicists typically label these irreducible representations by j (called the spin of the given representation) rather than by n, or even by c = j( j + 1), cf. (5.184).
Corollary 5.43 shows that one may pass from ρ(su (2)) to a unitary representation u(SU(2)), of which one may give a direct realization. For j ∈ N 0 /2, define H j as the complex vector space of all homogeneous polynomials p in two variables z = (z 1 , z 2 ) of degree 2 j. A basis of H j is given by (z 2 j 1 , z 2 j−1 1 z 2 ,..., z 1 z 2 j−1 2 , z 2 j 2 ), which has 2 j + 1 elements. So dim(H j ) = 2 j + 1. Then consider the map so D j (e) = 1, and Hence D j is a representation of SU (2). We now compute L 3 = − 1 2 iS 3 on this space. From (5.156) with u D j , we have Similarly, we obtain Hence f 2 j (z) = z 2 j 1 gives L 3 f 2 j = j f 2 j , and f 0 (z) = z 2 j 2 gives L 3 f 0 = − j f 0 . In general, f l (z) = z l 1 z 2 j−l 2 spans the eigenspace H λ of L 3 with eigenvalue λ = − j + l. Since l = 0, 1,..., 2 j, this confirms (5.188), as well as the fact that the corresponding eigenspaces are all one-dimensional. The rest is easily checked, too, except for the unitarity of the representation, for which we refer to the proof of Theorem 5.40.
Finally, we return to SO(3). Either explicit exponentiation (5.165), as done for j = 1/2 in (5.168), or the above construction of D j , allows one to verify the crucial condition stated in Corollary 5.43, namely that D j (δ ) = 1 H j for δ ∈ D = Z 2 , which comes down to D j (−1 2 ) = 1 H j . This is easily seen to be the case iff j ∈ N 0 .

Irreducible representations of compact Lie groups
Because of its importance for the classical-quantum correspondence (cf. §7.1) we first reformulate the main result of the previous section (i.e. the classification the irreducible representations of SU(2)) and on that basis generalize this result to arbitrary compact Lie groups. This gives a classification of great simplicity and beauty.
We already encountered the coadjoint representation (3.100) of a Lie group G on g * , given by ( The orbits under this action are called coadjoint orbits. If G = SO(3), we have g ∼ = R 3 under the map where the matrices J k are given in (3.66). Hence also g * ∼ = R 3 under the map Writing R ∈ SO(3) for a generic element x ∈ G, analogously to (5.44), we can compute the adoint action R : A → RAR −1 , seen as an action on R 3 , through Using the fact that the angular momentum matrices transform as vectors, i.e., we find that the adjoint action of SO(3) on g, seen as R 3 , is its defining action. In general, if g ∼ = R n and also g * ∼ = R n under the usual pairing of R n and R n through the Euclidean inner product, the coadjoint action of G on g * , seen as an action on R n , is given by the inverse transpose of the adjoint action on g ∼ = R n . For SO(3) we have (R −1 ) T = R, so the coadjoint action of SO(3) on R 3 is just its defining action, too, and hence the coadjoint orbits are the 2-spheres S r with radius r ≥ 0. Turning to SU(2), we now make the identification of g * with R 3 slightly differently, namely by replacing the 3 × 3 real matrices J i in (5.205) by the 2 × 2 matrices S i in (5.174), but the computation is similar: using (5.44) -(5.45), we find that the coadjoint action of u ∈ SU(2) on R 3 is given by the defining action ofπ(u) ∈ SO(3), cf. (5.46). It follows that the coadjoint orbits for SU(2) are the same as for SO(3).
Returning to general Lie groups G for the moment, assumed connected for simplicity, we take some coadjoint orbit O ⊂ g * , fix a point θ ∈ O (so that O = G · θ ≡ G θ ), and look at the stabilizer G θ and its Lie algebra g θ . Since the derivative Ad of the adjoint action Ad of G on g-defined as in (5.156)-is given by 209) it follows that the "infinitesimal stabilizer" g θ is given by Consequently, the restriction of θ : g → R to g θ ⊂ g is a Lie algebra homomorphism (where R is obviously endowed with the zero Lie bracket). Consider a character χ : G θ → T, which is the same thing as a one-dimensional unitary representation of G θ . If we regard T as a closed subgroup of GL 1 (C), its Lie algebra t is given by iR ⊂ M 1 (C) = C. It is conventional (at least among physicists) to take −i as the basis element of t, so that t ∼ = R under −it ↔ t, so that the exponential map exp : t → T (which is the usual one), seen as a map from R to T, is given by t → exp(−it). Defining the derivative χ : g θ → C as in (5.156), it follows that actually χ : g θ → iR, so that iχ maps g θ to R and is a Lie algebra homomorphism. In the simplest case where G = T, the coadjoint action on t * is evidently trivial, so that G θ = G = T for any θ ∈ t * ∼ = R. Furthermore, any character on T takes the form χ n (z) = z n , where n ∈ Z, cf. (C.351). As explained above, if t ∼ = R and hence also t * ∼ = R, the identification of λ ∈ t * with λ ∈ R is made by λ (−i) ↔ λ , where −i ∈ t. If χ = χ n , the right-hand side of (5.211) evaluated at A = −i equals n, so that (5.211) holds iff θ = n for some n ∈ Z. Thus the integral coadjoint orbits in t * are the integers Z ⊂ R. Similarly, if G = T d , the characters are elements of Z d , as in (5.212) and the integral coadjoint orbits in g * ∼ = R d are the points of the lattice Z d ⊂ R d . For G = SU(2) we take a coadjoint orbit S 2 r ⊂ R 3 and fix θ r = (0, 0, r). If r = 0, then G θ = G and (5.211) holds for the trivial character χ ≡ 1, so the orbit {(0, 0, 0)} is integral. Let r > 0. Then G θ r ≡ G r consist of the pre-image of SO(2) in SU(2) under the projectionπ in (5.46), where SO(2) ⊂ SO(3) is the group of rotations around the z-axis. This is the abelian group This group is isomorphic to T under diag(z, z) → z and hence its characters are given by χ n (diag(z, z)) = z n , where n ∈ Z. The identification g * ∼ = R 3 is made by identifying θ ∈ g * with (θ 1 , θ 2 , θ 3 ), where θ 1 = θ (S i ). Putting A = S 3 in (5.211), see (5.174), therefore gives r = n/2 for some n ∈ N. We conclude that the coadjoint orbits for SU(2) are given by the two-spheres S 2 r ⊂ R 3 with r ∈ N 0 /2. Similarly, for G = SO(3) the stabilizer of (0, 0, r) is SO(2) ∼ = T itself, and putting A = J 3 in (5.211) one finds that the coadjoint orbits are the spheres S 2 r with r ∈ N 0 .
For any (Lie) group G, let the unitary dualĜ be the set whose elements are equivalence classes of unitary irreducible representations of G, where we say: The examples G = T d as well as for G = SU(2) now suggest the following theorem: Theorem 5.49. If G is a compact connected Lie group, then the unitary dualĜ is parametrized by the set of integral coadjoint orbits in g * .
Furthermore, there is an explicit (geometric) procedure to a construct an irreducible representation u O corresponding to such an orbit, namely by the method of geometric quantization. We will not explain this method, which would require some reasonably advanced differential geometry, but instead we outline the connection between coadjoint orbits and the well-known method of the highest weight.
Let G be a compact connected Lie group and pick a maximal torus T ⊂ G. Let . Note that all maximal tori in compact connected Lie groups are conjugate, so that the specific choice of T is irrelevant. For example, for SU(2) we take (5.213), in which case N(T ) is generated by T and σ 1 ∈ SU(2), so that W ∼ = S 2 , i.e., the permutation group on two variables. In general the Weyl group inherits the adjoint action of N(T ) on T , so that W T acts on T and hence also acts on t and t * ; for SU(2) the action of the nontrivial element of W T , i.e., image [σ 1 ] of σ 1 ∈ N(T ) in N(T )/T ), on T is given by so that its action on T ∼ = T is z → z, which gives rise to actions A → −A of W T on t and hence λ → −λ of W T on t * . This is a special case of the following bijection: (5.216) where the G-action on g * is the coadjoint one; globally, one has G/Ad(G) ∼ = T /W T . Indeed, for SU(2) the left-hand side of (5.216) is the set of spheres S 2 r in R 3 , r ≥ 0, whereas the right-hand side is R/S 2 (where S 2 acts on R by θ → −θ ).
In general, a given coadjoint orbit O ⊂ g * defines a Weyl group orbit O W in t * as follows: O contains a point θ for which T ⊆ G θ , and we take O W to be the orbit through θ |t . Conversely, any G-invariant inner product on g induces a decomposition (5.217) which yields an extension of λ ∈ t * to θ λ ∈ g * that vanishes on t ⊥ . Let Λ ⊂ t * be the set of integral elements in t * (as explained after Definition 5.47). Elements of Λ are called weights. Theorem 5.51 below gives a parametrization (5.218) which, restricting (5.216) to the integral part Λ ⊂ t * , implies Theorem 5.49. Instead of with the quotient Λ /W T , one may prefer to work with Λ itself, as follows: we say that λ ∈ t * is regular if w · λ for w ∈ W T iff w = e; this is the case iff λ = θ |t with G θ = T . For SU(2) all weights λ ∈ Z are regular except λ = 0. The set t * r of regular elements of t * falls apart into connected components C, called Weyl chambers, which are mapped into each other by W T . For SU(2) one has t * = (−∞, 0) ∪ (0, ∞), so that the Weyl chambers are (−∞, 0) and (0, ∞).
One picks an arbitrary Weyl chamber C d (for SU(2) this is (0, ∞)) and forms Elements of Λ d are called dominant weights. For each element of Λ /W T there is a unique dominant weight representing it in Λ , so that instead of (5.218) we may also write what Theorem 5.51 actually gives, viz. (5.220) To explain this in some detail, we need further preparation. Any (unitary) representation u : G → U(H) on some finite-dimensional Hilbert space H restricts to T , and since T is abelian, we may simultaneously diagonalize all operators u(z), z ∈ T . The operators iu (A), where A ∈ t, commute as well, so that we may decompose where Λ H ⊂ Λ contains the weights that occur in u |T , so that for each ψ ∈ H μ , where the character χ μ : T → T corresponding to the weight μ ∈ Λ is defined as in (5.212) with μ = (n 1 ,..., n d ) and z = (z 1 ,..., For example, we have seen that the irreducible representations D j (SU (2)) on In particular, take H = g C with some G-invariant inner product, cf. (5.148), and take u = Ad, given by Ad(x)B = xBx −1 , so that Ad (A)(B) = [A, B], extended from g to g C : we write g C = g + ig and hence put Ad [A,C], where A, B,C ∈ g. We assume that the inner product ·, ·, on g C is obtained from a real inner product on g by complexification. This inner product on g may be restricted to t ⊂ g and hence induces an inner product on t * , also denoted by ·, ·, . For example, if G is semi-simple (like SU(2)), one may take the inner product on g and hence on g C to be the Cartan-Killing form A, B = − 1 2 Tr (Ad (A)Ad (B)), which is nondegenerate because G is semi-simple, and positive definite since G is compact. For SU(2) or SO(3) this gives the usual inner product on R 3 and C 3 .
Definition 5.50. The roots of g are the nonzero weights of the adjoint representation u = Ad on H = g C . That is, writing Δ ⊂ Λ for the set of roots, we have α ∈ Δ iff α : t → R is not identically zero and there is some E α ∈ g C such that for each Z ∈ t, cf. (5.223). Furthermore, subject to the choice of a preferred Weyl chamber C d in t * r , we say α ∈ Δ is positive, denoted by α ∈ Δ + , if α, λ > 0 for each λ ∈ C d .
Since α, λ is real and nonzero for each α ∈ Δ and λ ∈ C d , one has either α ∈ Δ + or −α ∈ Δ + , i.e., α ∈ Δ − = −Δ + . Since t is maximal abelian in g, it can also be shown that each root is nondegenerate. Writing g α = C · E α , this gives a decomposition (5.225) (2), the single generator of t is S 3 , and taking E ± = i(S 1 ± iS 2 ), we see from (5.180) that i[S 3 , E ± ] = ±E ± . Hence the roots are α ± , given by α ± (S 3 ) = ±1, and with (0, ∞) as the Weyl chamber of choice, the root α + is the positive one. We now define a partial ordering ≤ on Λ by putting μ ≤ λ iff λ − μ = ∑ i n i α i for some n i ∈ N 0 and α i ∈ Δ + . This brings us to the theorem of the highest weight: Theorem 5.51. Let G be a connected compact Lie group. There is a parametrization G ∼ = Λ d , such that any unitary irreducible representation u λ : G → H λ in the class λ ∈Ĝ defined by a given dominant weight λ ∈ Λ d has the following properties: 1. H λ contains a unit vector υ λ , unique up to a phase, such that iu λ (Z)υ λ = λ (Z)υ λ (Z ∈ t); (5.226) iu λ (E α )υ λ = 0 (α ∈ Δ + ). where θ λ ∈ g * was defined after (5.217) by λ ∈ Λ d ⊂ t * . Since each operator u λ (x) is unitary, each vector u λ (x)υ λ is a unit vector, so we may form the G-orbit is a G-equivariant diffeomorphism (in fact, a symplectomorphism) from O λ to O λ . This amplifies Theorem 5.49 by making the the bijective correspondence between the set Λ d of dominant weights and the set of integral coadjoint orbits explicit.

Symmetry groups and projective representations
Despite the power and beauty of unitary group representations in mathematics, in the context of e.g. Wigner's Theorem we have seen that in physics one should look at homomorphisms x → W(x), where W(x) is a symmetry of P 1 (H). In view of Theorems 5.4, this is equivalent to considering a single homomorphism h : G → G H , cf.
(5.136). To simplify the discussion, we now drop U a (H) from consideration and just deal with the connected component G H 0 = U(H)/T of the identity. This restriction may be justified by noting that in what follows we will only deal with symmetries given by connected Lie groups, which have the property that each element is a product of squares x = y 2 . In that case, h(x) = h(y) 2 is always a square and hence it cannot lie in the component U a (H)/T (the anti-unitary case does play a role as soon as discrete symmetries are studied, such as time inversion, parity, or charge conjugation). Thus in what follows we will study continuous homomorphisms where U(H)/T has the quotient topology inherited from the strong operator topology on U(H), as explained above. Since it is inconvenient to deal with such a quotient, we try to lift h to some map ( Indeed, since π and h are homomorphisms, we may compute
where b : G → T satisfies b(e) = 1 (and is measurable and smooth near e as appropriate), then c is called a 2-coboundary or an exact multiplier. The set of trivial multipliers forms a (normal) subgroup B 2 (G, T) of Z 2 (G, T), and the quotient is called the second cohomology group of G with coefficients in T.
The reason 2-coboundaries and the ensuing group H 2 (G, T) are interesting for our problem is as follows. Given a map x → u(x) from G to U(H) with (5.238), suppose we change u(x) to u(x) = b(x)u(x). The associated multiplier then changes to in that u(x) u(y) = c (x, y)u xy . In particular, a multiplier of the form (5.242) may be removed by such a transformation, and is accordingly called exact. This is true by construction. By the same token, if H 2 (G, T) is non-trivial, then G will have projective representations that cannot be turned into ordinary ones by a change of phase (for it can be shown that any multiplier c ∈ Z 2 (G, T) is realized by some projective representation). Thus it is important to compute H 2 (G, T) for any given (physically relevant) group G, and see what can be done if it is non-trivial.
To this end we present the main results of practical use. In order to state one of the main results (Whitehead's Lemma), we need to set up a cohomology theory for g (which we only need with trivial coefficients). Let C k (g, R) be the abelian group of all k-linear totally antisymmetric maps ϕ : g k → R, with coboundary maps (5.245) where the hat means that the corresponding entry is omitted. For example, we have These maps satisfy "δ 2 = 0", or, more precisely, and hence we may define the following abelian groups: B k (g, R) = ran(δ (k−1) ); (5.248) Z k (g, R) = ker(δ (k) ); (5.249) .
(5.250) Note that B k (g, R) ⊆ H k (g, R) because of (5.247). In particular, for k = 2 the group Z 2 (g, R) of all 2-cocycles on g consists of all bilinear maps ϕ : g×g → R that satisfy and its subgroup B 2 (g, R) of all 2-coboundaries comprises all ϕ taking the form For example, for g = R any antisymmetric bilinear map ϕ : R 2 → 0 is zero, so that This has nothing to with the fact that the Lie bracket on g vanishes. Indeed, g = R 2 does admit a unique nontrivial 2-cocycle, given by (half) the symplectic form, i.e., ϕ 0 ((p, q), (p , q )) = 1 2 (pq − qp ). (5.255) Since B 2 (R 2 , R) = 0, this cannot be removed, hence (5.255) generates H 2 (R 2 , R): As far as cohomology is concerned, each Lie group and each Lie algebra has its own story, although in some cases a group of stories may be collected into a single narrative. As a case in point, a Lie algebra g is called simple when it has no proper ideals, and semi-simple when it has no commutative ideals. A Lie algebra is semisimple iff it is a direct sum of simple Lie algebras. If a Lie group G is (semi-) simple, then so is its Lie algebra g. A basic result, often called Whitehead's Lemma, is: Lemma 5.54. If g is semi-simple, then H 2 (g, R) = 0.
Theorem 5.55. Let G be a connected and simply connected Lie group. Then H 2 (G, T) ∼ = H 2 (g, R).

(5.259)
Proof. This is really a conjunction of two isomorphisms: where R is the usual additive group, and Z 2 (G, R), B 2 (G, R), and hence H 2 (G, R) are defined analogously to Z 2 (G, T) etc. The first isomorphism is simply induced by which preserves exactness and induces an isomorphism in cohomology (but note that (5.262) -(5.263) may not itself define an isomorphism). The isomorphism (5.261) is induced at the cochain level, too. Given a cocycle ϕ ∈ Z 2 (G, R), we construct a new Lie algebra g ϕ (called a central extension of g) by taking g ϕ = g ⊕ R as a vector space, equipped though with the unusual bracket (5.264) the condition ϕ ∈ Z 2 (G, R) guarantees that this is a Lie bracket. Furthermore, g ϕ is isomorphic (as a Lie algebra) to a direct sum iff ϕ ∈ B 2 (g, R); indeed, if (5.253) holds, then (X, v) → (X, v + θ (X)) yields the desired isomorphism g ϕ → g ⊕ R.
By Lie's Third Theorem, there is a connected and simply connected Lie group G ϕ (again called a central extension of G), with Lie algebra g ϕ , As a manifold, G ϕ = G × R, but the group laws are given, in terms of a function Γ : (5.266) The group axioms then imply (indeed, they are equivalent to) the condition Γ ∈ Z 2 (G, R). Furthermore, two such extensions G ϕ and G ϕ are isomorphic iff the corresponding cocycles Γ and Γ are related by (5.244), and in particular, Γ ∈ B 2 (G, R) iff G ϕ is isomorphic (as a Lie group) to a direct product G × R, which in turn is the case iff ϕ ∈ B 2 (g, R). Conversely, given Γ ∈ Z 2 (G, R), we define the central extension G ϕ by (5.265) -(5.266), to find that the associated Lie algebra g ϕ takes the above form, defining ϕ ∈ B 2 (g, R) through (5.264). Explicitly, Lie's Third Theorem thus implies that the map ϕ ↔ Γ (which is not necessarily a bijection) descends to an isomorphism H 2 (g, R) → H 2 (G, R) in cohomology.
Given (5.254), Theorem 5.55 immediately gives In particular, if R is the relevant symmetry group, which is the case e.g. with time translation, by Proposition 5.53 we may restrict ourselves to unitary representations.
Once again, this has nothing to do with abelianness or topological triviality of R. Indeed, for G = g = R 2 , the Heisenberg cocycle (5.255) comes from the multiplier c 0 ((p, q), (p , q )) = e i(pq −qp )/2 , (5.269) where R 2 is seen as the group of translations in the phase space R 2 of a particle moving on R. Accordingly, this multiplier is realized by the following projective representation of R 2 on L 2 (R): If R 2 is the configuration space of some particle, and the group R 2 produces translations in the latter (i.e., of position), then the appropriate unitary representation would rather be on L 2 (R 2 ) and would have trivial multiplier, viz.
Similarly, G = R 2 , now seen as generating translations of momentum in the phase space R 4 of the latter example would appropriately be represented on L 2 (R 2 ) as Corollary 5.56. Let G be a connected and simply connected semi-simple Lie group. Then H 2 (G, T) is trivial.
Finally, even if h is continuous, it is a priori unclear ifũ is, since the crosssections s ands appearing in the above construction typically fail to be continuous. Fortunately, since they are assumed measurable, there is no question about measurability ofũ, and if H is separable, continuity follows from Proposition 5.36.

) is super-admissible iff j is integral, in which case it defines a unitary irreducible representation of SO(3).
Indeed, the assumption H 2 (g, R) = 0 in Theorem 5.59 is satisfied for SO(3) because of Whitehead's Lemma 5.54. The case where H 2 (g, R) = 0 occurs e.g. for the Galilei group (cf. §7.6). It can be shown that H 2 (g, R) has finitely many generators, for which one finds pre-images (ϕ 1 ,..., ϕ M ) in Z 2 (g, R), with corresponding elements (Γ 1 ,...,Γ M ) of Z 2 (G, R), cf. the proof of Theorem 5.55. Of these, a subset (Γ 1 ,...,Γ N ), N ≤ M, satisfies the relation Γ i (δ ,x) = Γ i (x, δ ) for any δ ∈ D (cf. Theorem 5.41) andx ∈G. This yields a map Γ :G ×G → R N given by Γ (x,ỹ) = (Γ 1 (x,ỹ),...,Γ N (x,ỹ)), which in turn equips the seť with a group multiplication (x, v) · (ỹ, w) = (xỹ, v + w + Γ (x,ỹ)). We then have the following generalization of Theorem 5.59, in which a unitary representation u ofǦ is called admissible if u(δ , v) ∈ T · 1 H for any δ ∈ D and v ∈ R N . As we only apply this to the Galilei group (where N = 1), basically only for illustrative purposes, we omit the proof. The correct (and natural) notion of equivalence of projective representations is as follows: we say that two such homomorphisms (5.295) where Ad w : This induces the following notion forǦ: two admissible unitary representations u 1 ,ũ 2 ofG on Hilbert spaces H 1 , H 2 are equivalent if there is a unitary w : H 1 → H 2 and a map b :Ǧ → T such that wu 1 (x)w * = b(x)u 2 (x), for anyx ∈Ǧ. It can be shown that such a map b always comes from a character χ : To close this long and difficult section, in relief it should be mentioned that the above theory vastly simplifies if H is finite-dimensional. By Theorem 5.40, this is true, for example, if G is compact and u is irreducible. Suppose u : G → U(H) is merely a projective unitary representation of G, so that instead of (5.157) one has

Position, momentum, and free Hamiltonian
The three basic operators of non-relativistic quantum mechanics are position, denoted q, momentum, p, and the free Hamiltonian h 0 . Assuming for simplicity that the particle moves in one dimension, these are informally given on H = L 2 (R) by where m is the mass of the particle under consideration. We puth = 1 and m = 1/2. The issue is that these operators are unbounded; see §B.13. In general, quantummechanical observables are supposed to be represented by self-adjoint operators, and examples like (5.300) -(5.302) show that these may not be bounded. The Hellinger-Toeplitz Theorem B.68 then shows that it makes no sense to try and extend the above expressions to all of L 2 (R), so we have to live with the fact that some crucial operators a : D(a) → H are merely defined on a dense subspace D(a) ⊂ H.
Each such operator has an adjoint a * : D(a * ) → H, whose domain D(a * ) ⊂ H consists of all ψ ∈ H for which the functional ϕ → ψ, aϕ is bounded on D(a), and hence (since D(a) is dense in H) can be extended to all of H by continuity through the unique "Riesz-Fréchet vector" χ for which ψ, aϕ = χ, ϕ . Writing χ = a * ψ, for each ψ ∈ D(a * ) and ϕ ∈ D(a) we therefore have a * ψ, ϕ = ψ, aϕ . (5.303) Assuming that D(a) is dense in H, we say that a is self-adjoint, written a * = a, if aϕ, ψ = ϕ, aψ , (5.304) for each ψ, ϕ ∈ D(a) and D(a * ) = D(a). A self-adjoint operator a is automatically closed, in that its graph G(a) = {(ψ, aψ) | ψ ∈ D(a)} is a closed subspace of the Hilbert space H ⊕ H (indeed, the adjoint of any densely defined operator is closed, see Proposition B.72). In practice, self-adjoint operators often arise as closures of essentially self-adjoint operators a, which by definition satisfy a * * = a * . Equivalently, such an operator is closable, in that the closure of its graph is the graph of some (uniquely defined) operator, called the closure a − of a, and furthermore this closure is self-adjoint, so that a − = a * . If a is closable, the domain D(a − ) of its closure consists of all ψ ∈ H for which there exists a sequence (ψ n ) in D(a) such that ψ n → ψ and aψ n converges, on which we define a − by a − ψ = lim n aψ n . The simplest case is the position operator.
Theorem 5.63. The operator q is self-adjoint on the domain Each · n,m happens to be a norm, but positive definiteness is nowhere used in the theory below (which therefore works for families of seminorms, which satisfy the axioms of a norm expect perhaps for positive definiteness). Since there are countably many such (semi)norms defining the topology, we may equivalently say that S (R) is a metric space defined by Definition 5.65. A tempered distribution is a continuous linear map ϕ : S (R) → C. The space of all such maps, equipped with the topology of pointwise convergence It can be shown that (because of the metrizability of S (R)) continuity is the same as sequential continuity, i.e., some linear map ϕ : S (R) → C belongs to S (R) iff lim N ϕ( f N ) = ϕ( f ) for each convergent sequence f N → f in S (R). Like S (R), the tempered distributions S (R) form a (locally convex) topological vector space, that is, a vector space with a topology in which addition and scalar multiplication are continuous. The topology of S (R) is given by a family of seminorms, namely ϕ f = |ϕ( f )|, f ∈ S (R), and hence a simple way to prove that ϕ ∈ S (R) is to find some (n, m) for which |ϕ( f ))| ≤ C f n,m for each f ∈ S (R), since in that case f N → f , which means that f N − f n,m → 0 for all n, m ∈ N, certainly implies that ϕ( f N ) → ϕ( f ), so that ϕ is continuous. For example, the evaluation maps δ x defined by δ x ( f ) = f (x) are continuous (take n = m = 0). Similarly, each finite measure on R defines a tempered distribution. Taking the (0, m) seminorm shows that the maps f → f (m) (x) for fixed m ∈ N and x ∈ R are tempered distributions.
A less obvious example (defining a so-called Gelfand triple) is as follows: Proposition 5.66. We have continuous dense inclusions where the second inclusion identifies ϕ ∈ L 2 (R) with the map (5.309) Proof. As vector spaces, the first inclusion is obvious. For f ∈ S (R) we estimate so that, noting that · 0,0 = · ∞ , we have This shows that the first inclusion in (5.308) is continuous. Density may be proved in two steps. First, take some fixed positive function h ∈ C ∞ c (−1, 1) with the property dx h(x) = 1, and define h n (x) = nh(nx), so that informally h n ∈ C ∞ c (R) converges to a δ -function as n → ∞. For each ψ ∈ L 2 (R), we consider the convolution h n * ψ, where for suitable f , g, (5.313) Then h n * ψ ∈ C ∞ (R) ∩ L 2 (R) and, from elementary analysis, h n * ψ − ψ → 0. Second, for ψ ∈ C c (R), the functions h n * ψ lie in C ∞ c (R) and hence in S (R). Since C c (R) is dense in L 2 (R) by Theorem B.30, for ψ ∈ L 2 (R) and ε > 0 we can find ϕ ∈ C c (R) such that ψ − ϕ < ε/2, and (as just shown) find n such that ϕ − ϕ n < ε/2, whence ψ − ϕ n < ε. This proves that S (R) is dense in L 2 (R).
The second inclusion is continuous by Cauchy-Schwarz, which gives to be combined with (5.312). It should be noted that also the second inclusion in (5.308) is indeed an injection, i.e., that ϕ( f ) = 0 for each f ∈ S (R) implies ϕ = 0 in L 2 (R); this is true because S (R) is dense in L 2 (R), plus the standard fact that, in any Hilbert space H, if ϕ, f = 0 for all f in some dense subspace of H, then ϕ = 0. Finally, the fact that L 2 (R) is dense in the seemingly huge space S (R) follows from the even more remarkable fact that S (R) is dense in S (R). On top of the functions h n just defined, also employ a function χ ∈ C ∞ c (R) such that χ(x) = 1 on (−1, 1), and define χ n (x) = χ(x/n), so that informally lim n→∞ χ(x) = 1 (as opposed to the h n , which converge to a δ -function as n → ∞). If for any g ∈ S (R) and any ϕ ∈ S (R) we define gϕ as the distribution that maps f ∈ S (R) to ϕ( f g), and similarly define g * ϕ as the distribution that maps f to ϕ(g * f ), we may define a sequence of distributions ϕ n = h n * (χ n ϕ). From the point of view of (5.308), these correspond to functions ϕ n ∈ S (R) in the sense that ϕ n ( f ) = dx ϕ n (x) f (x), where f ∈ S (R). Using similar analysis as above, it then follows that for any f ∈ S (R) we have ϕ n ( f ) → ϕ( f ), so that ϕ n → ϕ in S (R).
For our purposes, the point of all this is that we can define generalized derivatives of (tempered) distributions, and hence, because of (5.308), of functions in L 2 (R).
Definition 5.67. For ϕ ∈ S (R) and m ∈ N, the m'th generalized derivative ϕ (m) is defined by The idea is that under (5.308) this is an identity if ϕ ∈ S (R) (partial integration).
Like the constructions at the end of the proof of Proposition 5.66, this is a special case of a more general construction: whenever we have a continuous linear map T : S (R) → S (R), we obtain a dual continuous linear map T : Sometimes a slight change in the definition (as in (5.314), or as in the Fourier transform below) is appropriate so that the restriction of T to S (R) coincides with T .
Theorem 5.68. The momentum operator p = −id/dx is self-adjoint on the domain where the derivative ψ is taken in the distributional sense (i.e., letting ψ ∈ S (R)).
Proof. We first show that p is symmetric, or p ⊆ p * . This comes down to for each ψ, ϕ ∈ D(p), where both derivates are "generalized". The most elegant proof (though perhaps not the shortest) uses the Sobolev space H 1 (R), which equals D(p) as a vector space, now equipped, however, with the new inner product ψ, ϕ (1) = ψ, ϕ + ψ , ϕ , (5.318) with both inner products on the right-hand side in L 2 (R); the associated norm is Similar to the Gelfand triple (5.308), we have dense continuous inclusions with analogous proof. All we need for Theorem 5.68 is the first inclusion of the triple (5.320): for ψ ∈ H 1 (R) we now have h n * ψ ∈ C ∞ (R) ∩ H 1 (R) as well as h n * ψ → ψ in H 1 (R), both of which follow from the L 2 -case plus the identity (h n * ψ) = h n * ψ . (5.321) Using the same cutoff function χ as in the L 2 case, we have χ n ψ → ψ and χ n ψ → 0 in L 2 (R), so that (χ n ψ) → ψ in L 2 (R) and hence χ n ψ → ψ also in H 1 (R).
Furthermore, the functions ψ n = h n * (χ n ψ) lie in C ∞ c (R) and hence in S (R); using the above facts we obtain ψ n → ψ in H 1 (R). In sum, for each ψ ∈ H 1 (R) we can find a sequence (ψ n ) in S (R) such that ψ n → ψ and ψ n → ψ in L 2 (R). Hence (5.322) For the converse, let ψ ∈ D(p * ), so that by definition for each ϕ ∈ D(p) we have Since S (R) ⊂ D(p), this is true in particular for each ϕ ∈ S (R), in which case the right-hand side equals −iψ (ϕ), where the derivative is distributional. But this equals p * ψ, ϕ and so the distribution −iψ is given by taking the inner product with p * ψ ∈ L 2 (R). Hence −iψ = p * ψ ∈ L 2 (R), and in particular ψ ∈ L 2 (R), so that ψ ∈ D(p). This proves that D(p * ) ⊆ D(p), and since from the first step we have the oppositie inclusion, we find D(p * ) = D(p) and p * = p.
For the free Hamiltonian h 0 = −Δ with Δ = d 2 /dx 2 , we similarly have: Theorem 5.69. The free Hamiltonian h 0 = −Δ is self-adjoint on the domain where the double derivative ψ is taken in the distributional sense.
Although this may be proved in an analogous way, such proofs are increasingly burdensome if the number of derivatives gets higher. It is easier to use the Fourier transform (which also provided an alternative way of proving Theorem 5.68).
Theorem 5.70. The formulaê In all three cases we have the identities (in a distributional sense if appropriate) Thus we may now reformulate Theorems 5.68 and 5.69 as follows: Theorem 5.71. The momentum operator p is self-adjoint on the domain (5.330).
The free Hamiltonian h 0 = −Δ is self-adjoint on the domain (5.331).
Proof. Denoting multiplication by x n by the symbol k n , we have Much is known about regularity properties of functions in such domains, e.g., These are the most elementary cases of the famous Sobolev Embedding Theorem.
Finally, we give a common domain of essential self-adjointness for q, p, and h 0 .
Proposition 5.72. The operators q, p, and h 0 are essentially self-adjoint on S (R).
Proof. We see from (5.332) that the cases of p and q are similar, so we only explain the case of q. Denoting the operator of multiplication by x on the domain S (R) by q 0 , as in the proof of Proposition B.73 it is easy to see that D(q * 0 ) = D(q). Fouriertransforming, the fact that S (R) is dense in H 1 (R) (cf. the proof of Theorem 5.68) shows that D(q − 0 ) = D(q),so that D(q * 0 ) = D(q − 0 ). The actions of q * 0 and q − 0 obviously being given by multiplication by x in both cases, we have q * 0 = q − 0 . The proof for h 0 is similar; in the second step we now use the fact that S (R) is dense in H 2 (R), defined as D(Δ ), as in (5.324), but now seen as a Hilbert space in the inner product ψ, ϕ (2) = ψ, ϕ + ψ , ϕ , with corresponding norm given by ψ 2 (2) = ψ 2 + ψ 2 . This is proved just as in the case of a single derivative.
We also say that S (R) is a core for the operators in question. For example, the canonical commutation relations [q, p] = ih · 1 H rigorously hold on this domain.

Stone's Theorem
We now come to a central result on symmetries in quantum mechanics "explaining" the Hamiltonian. Recall that a continuous unitary representation of R (as an additive group) on a Hilbert space H is a map t → u t , where t ∈ R and each u t ∈ B(H) is unitary, such that the associated map R × H → H, (t, ψ) → u t ψ, is continuous, and u s u t = u s+t , s,t ∈ R; (5.336) Note that according to Proposition 5.36 continuity may be replaced by weak measurability. Probably the simplest nontrivial example is given by H = L 2 (R) and u t ψ(x) = ψ(x − t). (5.340) To prove (5.338), we use a routine ε/3 argument. We first prove (5.338) for ψ ∈ C c (R), where it is elementary in the sup-norm, i.e., lim t→0 u t ψ − ψ ∞ = 0 by continuity and hence (given compact support) uniform continuity of ψ. But then the (ugly) estimate ψ 2 2 ≤ |K| ψ ∞ , where K ⊂ R is any compact set containing the support of ψ, also yields lim t→0 u t ψ − ψ 2 = 0. Hence for ε > 0 we may find δ > 0 such that u t ψ − ψ 2 < ε/3 whenever |t| < δ . For general ψ ∈ H, we find ψ ∈ C c (R) such that ψ − ψ < ε/3, and, using unitarity of u t , estimate u t ψ − ψ ≤ u t ψ − u t ψ + u t ψ − ψ + ψ − ψ ≤ ε/3 + ε/3 + ε/3 = ε.
In the context of quantum mechanics, physicists formally write where a is usually thought of as the Hamiltonian of the system, although in the previous example it is rather the momentum operator. In any case, we avoid the notation h instead of a here, partly in order to rightly suggest far greater generality of the construction and partly to avoid confusion with the notation in §B. Proof. We use the setting of §B.21, so that b is the bounded transform of a.
1. Eqs. (5.336) -(5.337) are immediate from Theorem B.158, which also yields unitarity of each operator u t . To prove (5.338) we first take ϕ ∈ C * c (b)H, which means that ϕ is a finite linear combinations of vectors of the type ϕ = h(a)ψ, where h ∈ C c (σ (a)) and ψ ∈ H. Using (5.342) and (B.573), we have where K is the (compact) support of h in σ (b). Since the exponential function is uniformly convergent on any compact set, this gives lim t→0 u t ϕ − ϕ = 0. Taking finite linear combinations of such vectors ϕ gives the same result for any ϕ ∈ C * c (b)H (with an extra step this could have been done on C * 0 (b)H, too). Thus for ε > 0 we can find δ > 0 so that u t ϕ − ϕ < ε/3 whenever |t| < δ . For general ψ ∈ H, we find ϕ ∈ C * 0 (b)H such that ϕ − ψ < ε/3, and estimate since u t ψ − u t ϕ = ψ − ϕ by unitarity of u t . This is equivalent to (5.338). 2. For any ψ ∈ H and n ∈ N, define ψ n ∈ H by ψ n = n ∞ 0 ds e −ns u s ψ, (5.346) either as a Riemann-type integral (whose approximants converge in norm) or as a functional ϕ → n ∞ 0 ds e −ns u s ψ, ϕ , which is obviously continuous and hence is represented by a unique vector ψ n ∈ H. Then simple computations show that lim s→0 u s − 1 s ψ n = n(ψ n − ψ), so that ψ n ∈ D(a). The proof that ψ n → ψ starts with the elementary estimate ψ n − ψ ≤ n ∞ 0 ds e −ns u s ψ − ψ , in which we split up the ∞ 0 as δ 0 · · · + ∞ δ · · ·, where δ > 0. Using strong continuity of the map t → u t , i.e., (5.338), for any n the first integral vanishes as δ → 0. In the second integral we estimate u s ψ − ψ ≤ 2 ψ and take the limit n → ∞. Thus ψ n → ψ, so that D(a) is dense in H. To prove self-adjointness of a, we need a tiny variation on Theorem B.93: Lemma 5.74. Let a be symmetric. Then a is self-adjoint (i.e. a * = a) iff ran(a + i) = ran(a − i) = H.
(5.347) so that by definition of the (strong) derivative we obtain du t dt ϕ = lim s→0 u t+s ϕ − u t ϕ s = −iau t ϕ, (5.348) initially for any ϕ of the said form h(a)ψ, and hence, taking finite sums, for any ϕ ∈ D(a 0 ). The existence of this limit shows that, on the assumption ψ ∈ D(a 0 ), we have ψ ∈ D(a ), and we also see that a = a on D(a 0 ), or, in other words, that a 0 ⊆ a . Since a is self-adjoint (by part 2 of the theorem) and hence closed, we have a − 0 ⊆ a . Since a 0 is essentially self-adjoint by Theorem B.159, this gives a ⊆ a . Taking adjoints reverses the inclusion, and since both operators are self-adjoint this gives a = a . • Given u t and hence (5.343) -(5.344) defining a, we change notation from u t to u t in (5.341) and need to show that u t = u t . Indeed, let (5.349) and similarly ψ t = u t ψ. If ψ ∈ D(a), then by definition of a we have which also shows that ψ t ∈ D(a). Similarly, idψ t /dt = aψ t , so that ψ t and ψ t satisfy the same differential equation with the same initial condition ψ (0) = (ψ (0) ) = ψ.
For our specificψ t we have ψ 0 = 0 and hence ψ t = ψ t , that is, u t = u t .
Corollary 5.75. With t → u t and a defined and related as in Theorem 5.73, if ψ ∈ D(a), for each t ∈ R the vector ψ t defined by (5.349) lies in D(a) and satisfies aψ t = i dψ t dt , (5.351) whence t → ψ t is the unique solution of (5.351) with initial value ψ (0) = ψ.
This follows from the proof of part 3 of Theorem 5.73. With a = h/h (as above), this is just the famous time-dependent Schrödinger equation hψ t = ih dψ t dt . (5.352) If dim(H) ≥ 3, the conclusion of Wigner's Theorem follows if W merely preserves orthogonality (Uhlhorn, 1963). See also Cassinelli et al (2004). This, in turn, has been generalized in various directions, e.g. to indefinite inner product spaces (Molnár, 2002) as well as to certain Banach spaces, where one says that x is orthogonal to y if for all λ ∈ C one has x + λ y ≥ x (Blanco & Turnšek, 2006). §5.6. Some abstract representation theory Among numerous books on representation theory, our personal favourite is Barut & Raçka (1977), and also Gaal (1973) and Kirillov (1976) are classics at least for the abstract theory. An interesting recent paper on the unitary group on infinitedimensional Hilbert space is Schottenloher (2013).

§5.7. Representations of Lie groups and Lie algebras
This section was inspired by Hall (2013) and Knapp (1988). For Lie's Third Theorem, see, for example, Duistermaat & Kolk (2000), §1.14. To obtain Theorem 5.41, consider the canonical projectionπ :G → G and define D =π −1 ({e}). This is a discrete normal subgroup ofG, and it is an easy fact that a discrete normal subgroup of any connected topological group must lie in its center. Note that a discrete subgroup of the center ofG is automatically normal.
The exponentiation problem for skew-adjoint representations of g is considerably more complicated than in finite dimension. Let H be an infinite-dimensional Hilbert space with dense subspace D and let ρ :