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Home , Hilbert space, Positive element, Representation theorem, Unitary element

Chapter 2

C*-, states and representations

2.1 C*-algebras

Let be an associative over C. is a normed algebra if there is a x x A A A3 7! k k2 R+ such that xy x y . A complete normed algebra is a . A mapping k kk kk k x x of into itself is an involution if 7! ⇤ A

(x⇤)⇤ = x;

(x + y)⇤ = x⇤ + y⇤;

(xy)⇤ = y⇤x⇤;

(x)⇤ = x⇤.

An algebra with an involution is a *-algebra.

Definition 1. A Banach *-algebra is called a C*-algebra if A 2 x⇤x = x ,x. k k k k 2A Proposition 1. Let be a C*-algebra. A 1. x = x ; k ⇤k k k 2. If does not have an identity, let be the algebra obtained from by adjoining an A A A identity 1. Then is a C*-algebra with norm defined by A k·k e y + xy e 1+x =supk k,C. k k y=0 y 2 6 k k Proof. Exercise.

In the following, will always denote a C*-algebras with an identity if not specified other- A wise.

Definition 2. The spectrum Sp(x) of x is the set 2A Sp(x):= C : x 1 is not invertible in . { 2 A}

3 1 n 1 If > x ,thentheseries n N (x/) is norm convergent and sums to (1 x) . | | k k 2 Hence, Sp(x) B x (0). Assume now that x is a self-adjoint element and that a +ib ⇢ k k P 2A 2 Sp(x), a, b R.Thena+i(b+t) Sp(x+it1). Since x+it1 2 = x+it1 x it1 = x2+t21 2 2 k k k kk k k k x 2 + t2, and by the remark above, a +i(b + t) 2 x 2 + t2, and further 2bt x a2 b2 k k | | k k nk k for all t R, so that b = 0. For any polynomial P over C, P (µ) = A i=1(µ zi), and 2 n P (x) 1=A (x zi) for any x . Hence, Sp(P (x)) i↵ zj Sp(x) for a i=1 2A 2A 2 Q 2 1 j n.SinceP (zj)=, we have that Sp(P (x)) i↵ P (Sp(x)). We have proved   Q 2 2 Proposition 2. Let x . 2A

1. Sp(x) B x (0); ⇢ k k 2. if x = x , then Sp(x) [ x , x ]; ⇤ ⇢ k k k k 3. if xx = x x, i.e. x is normal, then x =sup : Sp(x) ; ⇤ ⇤ k k {| | 2 } 4. for any polynomial P , Sp(P (x)) = P (Sp(x));

The proof of 3. is left as an exercise. Note that the condition holds in particular for x = x⇤ An element x is positive if it is self-adjoint and Sp(x) R+. 2A ⇢ Proposition 3. Let x , x =0. The following are equivalent: 2A 6 1. x is positive;

2. there is a self-adjoint z such that x = z2; 2A 3. there is y such that x = y y; 2A ⇤ Proof. (3) (2) by choosing y = z. (3) (1) since z2 is self-adjoint and since, by Proposition 2, ) ) Sp(z2) [0, z 2]. To show (1) (3), we note that1 for any µ>0, ⇢ k k ) 1 1 1 2 µ = 1 p d (2.1) ⇡ + µ  Z0 ✓ ◆

Since x is positive, (x+1) is invertible for all >0 so that z := ⇡ 1 p 1 (x + 1) 1 d 0 is well defined as a norm convergent , and x = z2. Using again (2.1) with µ = 1, we R have that

1/2 1/2 x 1 p 1 1 x 1 z = k k (ˆx + 1) (ˆx 1)d, xˆ = x x k k ⇡ +1 k k Z0 Butx ˆ positive implies Sp(ˆx) [0, 1], hence Sp(1 xˆ) [0, 1] and 1 xˆ 1. Moreover, since ⇢ ⇢ k k Sp((ˆx + 1) 1)=(Sp(ˆx + 1)) 1 [(1 + ) 1, 1], we have (ˆx + 1) 1 1 for >0. ⇢ k k Hence, 1 z x 1/2 1 so that Sp(z) [0, 2 x 1/2] and finally z is positive. k k k k ⇢ k k It remains to prove (2) (1). Since x = y y is self-adjoint, x2 is positive and we denote by ) ⇤ x its positive defined by the integral above. Then x := ( x x)/2 is positive and | | ± | |± x x+ = x+x = 0. Decomposing yx = s +it, with self-adjoint s, t,wehave(yx )⇤(yx )+ 2 2 3 (yx )(yx )⇤ = 2(s +t ) 0. But (yx )⇤(yx )= x ( x +x+)x = x is positive, so that (yx )(yx )⇤ is positive. On the other hand, Sp((yx )(yx )⇤) 0 =Sp((yx )⇤(yx )) 0 [{ } [{ }⇢ R ,hence(yx )⇤(yx ) = 0 so that x = 0, and finally x = x+ is positive. 1 1/2 3/2 Convergence follows from the asymptotics O(s )ass 0andO(s )ass ,whilethechangeof ! !1 variables = µ⇠ yields immediately that the integral is pµ,uptoaconstant.

4 Definition 3. A *-morphism between two *-algebras and is a ⇡ : A B A!B such that ⇡(A A )=⇡(A )⇡(A ) and ⇡(A )=⇡(A) ,forallA, A ,A .Itiscalleda 1 2 1 2 ⇤ ⇤ 1 2 2A *- if it is bijective. A *-isomorphism is an automorphism. A!A Proposition 4. Let , be two C*-algebras and ⇡ : a *-morphism. Then ⇡(x) A B A!B k kB  x , and the range ⇡(A):A is a *-subalgebra of . k kA { 2A} B Proof. If x is self-adjoint, so is ⇡(x) and ⇡(x) =sup : Sp(⇡(x)) .Ifx 1isinvertible, k k {| | 2 } then 1 = ⇡((x 1) 1(x 1)) = ⇡((x 1) 1)⇡(x 1) so that ⇡(x) 1isinvertible,whence Sp(⇡(x)) Sp(x), we have that ⇡(x) sup : Sp(x) = x . The general case follows ⇢ k k {| | 2 } k k from ⇡(x) 2 = ⇡(x x) x x = x 2. k k k ⇤ kk ⇤ k k k Let be a locally compact Hausdor↵, and let C0() be the algebra, under pointwise multiplication, of all complex valued continuous functions that vanish at infinity.

Theorem 5. If is a commutative C*-algebra, then there is a locally compact Hausdor↵space A such that is *-isomorphic to C (). A 0 Proof. See Robert’s lectures.

In classical mechanics, the space is usually referred to as the . If is a commutative C*-algebra with an identity, then is isomorphic to C(K), the A A algebra of continuous functions on a compact Hausdor↵space K. Let be a *-subalgebra of ( ). The commutant is the subset of ( ) of operators U L H U 0 L H that commute with every element of , and so forth with := ( ) . In particular, , U U 00 U 0 0 U⇢U00 and further = . U 0 U 000 Definition 4. A or W*-algebra on is a *-subalgebra of ( ) such H U L H that 00 = .Itscenter is ( ):= 0,and is a factor if ( )=C 1. U U Z U U\U U Z U · Theorem 6. Let be a *-subalgebra of ( ) such that = . Then is a von Neumann U L H UH H U algebra i↵ is weakly closed. U Note that = is automatically satisfied if 1 . Furthermore, for any *-subalgebra UH H 2U U of ( ), let be its weak for which 00 = by the theorem. Since ,wehave L H U U U U⇢U 0 . Furthermore, if x , and y ,with y *y,thenx commutes with y for all U ⇢U0 2U0 2 U U3 n n n and hence with y, so that 0. It follows that 0 = ,whence 00 = and so: U 0 ⇢ U U U 0 U U 00 Corollary 7. is weakly dense in , namely = . U U 00 U U 00 2.2 Representations and states

Definition 5. Let be a C*-algebra and a . A representation of in is a A H A H *-morphism ⇡ : ( ). Moreover, A!LH 1. Two representations ⇡,⇡ in , are equivalent if there is a unitary map U : 0 H H0 H!H0 such that U⇡(x)=⇡0(x)U; 2. A representation ⇡ is topologically irreducible if the only closed subspaces that are invari- ant under ⇡( ) are 0 and ; A { } H 3. A representation ⇡ is faithful if it is an isomorphism, namely Ker⇡ = 0 . { }

5 Note that in general ⇡(x) x , with equality if and only if ⇡ is faithful. One can further k kk k show that for any C*-algebra, there exists a faithful representation. n n If ( ,⇡) is a representation of and n N,thenn⇡(A)( i):= ⇡(A) i defines a H A 2 i=1 i=1 representation n⇡ on n . i=1H Proposition 8. Let ⇡ be a representation of in .T.f.a.e A H 1. ⇡ is topologically irreducible;

2. ⇡( )0 := B ( ):[B,⇡(x)] = 0, for all x = C 1; A { 2LH 2A} · 3. Any ⇠ , ⇠ =0is cyclic: ⇡(x)⇠ = ,or⇡ =0. 2H 6 H Proof. (1) (3) : If ⇡( )⇠ is not dense, then ⇡( )⇠ = 0 . It follows that C⇠ is an invariant ) A A { } subspace, and hence = C⇠ and ⇡ =0 H (3) (1) : Let = 0 be a closed . For any ⇠ , ⇡( )⇠ and since ⇠ ) K6 { } 2K A ⇢K is cyclic, ⇡( )⇠ is dense in A H (2) (1) : Let = 0 be a closed invariant subspace, and let P be the orthogonal projection ) K6 { } K on .ThenP ⇡( )0, since for ⇠ ,⌘ ?, ⇠,⇡(x)⌘ = ⇡(x⇤)⇠,⌘ = 0 so that K K 2 A 2K 2K h i h i ⇡(x)⌘ ? for any x . Hence P = 0 or P = 1, i.e. = 0 or = . 2K 2A K K K { } K H (1) (2) : Let c ⇡( ) be self-adjoint. Then all spectral projectors of c belong to ⇡( ) ,so ) 2 A 0 A 0 that they are all either 0 or 1 by (1), and c is a scalar. If c is not self-adjoint, apply the above to c c . ± ⇤ Atriple( ,⇡,⇠)where⇠ is a cyclic vector is called a cyclic representation. H Recall that ⇤ := ! : C : ! is linear and bounded . For any ⇠ , the map A { A! } 2 2H x ⇠,⇡(x)⇠ is an element of ⇤ since ⇠,⇡(x)⇠ ⇠ x and it is positive: A3 7! h i A |h i| k k k kA If x is positive, then x = y y and ⇠,⇡(x)⇠ = ⇡(y)⇠ 2 0. WeH shall denote it ! .If ⇤ h i k k ⇡,⇠ 0 T 1 is a self-adjoint in and T ⇡( ) ,H then the form x ! is positive,   H 2 A 0 7! ⇡,T⇠ and ! (y y)= ⇡(y)T⇠ 2 = T⇡(y)⇠ 2 ⇡(y)⇠ 2 = ! (y y), so that ! ! . ⇡,T⇠ ⇤ k k k k k k ⇡,⇠ ⇤ ⇡,T⇠  ⇡,⇠ Lemma 9. Let ! be a positive linear on . Then A 2 !(x⇤y)=!(y x), !(x⇤y) !(x⇤x)!(y⇤y). ⇤ | |  Proof. This follows from the positivity of the quadratic form !((x+y) (x+y)) 0. 7! ⇤ In fact, any positive ⌫ bounded above by !⇡,⇠ is of the form above. Indeed,

2 2 2 ⌫(x⇤y) ⌫(x⇤x)⌫(y⇤y) ! (x⇤x)! (y⇤y) ⇡(x)⇠ ⇡(y)⇠ | |   ⇡,⇠ ⇡,⇠ k k k k so that ⇡(x)⇠ ⇡(y)⇠ ⌫(x y) is a densely defined, bounded, symmetric linear form on ⇥ 7! ⇤ . By Riesz , there exists a unique T such that H⇥H ⌫(x y)= ⇡(x)⇠,T⇡(y)⇠ , and 0 T 1. Moreover, ⇤ h i  

⇡(x)⇠,T⇡(z)⇡(y)⇠ = ⌫(x⇤zy)=⌫((z⇤x)⇤y)= ⇡(x)⇠,⇡(z)T⇡(y)⇠ , h i h i so that T (⇡( )) . 2 A 0 Definition 6. A state ! on a C*-algebra is a of such that A A⇤ !(x) ! =sup =1. k k x x 2A k k Astate! is called

6 pure if the only positive linear functionals majorised by ! are !, 0 1, •   faithful if !(x x)=0implies x =0. • ⇤ If ! is normalised and has an identity, then !(1) = 1. Reciprocally, !(x) 2 !(1)!(x x). A | |  ⇤ Since x x 1 x x 0, we further have !(x) 2 x x !(1)2,i.e. ! !(1), which proves: k ⇤ k ⇤ | | k ⇤ k k k Proposition 10. If has an identity, and ! is a positive linear form on , then ! =1if A A k k and only if !(1) = 1. By Corollary 7, ⇡( ) is a von Neumann algebra for any state !. ! is called a factor state A 00 if ⇡( )00 is a factor, i.e. if ⇡( )0 ⇡( )00 = C 1. A A \ A · We shall denote ( ) the set of states over and ( ) the set of pure states. E A A P A Proposition 11. ( ) is a , and it is weakly-* compact i↵ has an identity. In E A A that case, ! ( ) i↵it is an extremal point of ( ). 2P A E A Proof. We only prove the second part, the first part being is a version of the Banach-Alaoglu theorem. Let ! ( ). Assume that ! = ! +(1 )! .Then! ! ,hence! = µ !, 2P A 1 2 1 1 1 0 µ 1 and similarly for ! . Hence ! =(µ + µ )! and ! is extremal. Reciprocally, assume  2  2 1 2 that ! is not pure, in which case there is a linear functional⌫ ˜ = !˜ such that ! ⌫˜ .In 1 6 1 particular, :=⌫ ˜ (1) !(1) = 1. Since ⌫ := 1⌫˜ is a state, ⌫ := (! ⌫ )/(1 )defines 1  1 1 2 1 a state, and ! = ⌫ +(1 )⌫ . Hence ! is not extremal. 1 2 In particular, if !i i I is an arbitrary infinite family of states, then there exists at least one { } 2 weak-* accumulation point. Note that ! *!in the weak-* if ! (x) !(x) for all n n ! x . In fact, it is defined as the weakest topology in which this holds, namely in which the 2A map x : ! !(x) are continuous. 7! Theorem 12. Let be a C*-algebra and ! ( ). Then there exists a cyclic representation A 2E A ( ,⇡,⌦) such that H !(x)= ⌦,⇡(x)⌦ h i for all x . Such a representation is unique up to unitary isomorphism. 2A Proof. We consider only the case where has an identity. Let := a : !(a a)=0. A N { 2A ⇤ } Since, by Lemma 9, 0 !(a x xa) !(a a) x 2 = 0, we have a ,x implies xa is  ⇤ ⇤  ⇤ k k 2N 2A 2N a left ideal. On h := , we denote the equivalence class of x , and the A\N x 2A ( , ) !(x y) is positive and well-defined, since !((x + a) ,y + b)=!(x y)+!(a y)+ x y 7! ⇤ ⇤ ⇤ ⇤ !(x b)+!(a b)=!(x y) for any x, y ; a, b . Let be the Hilbert space completion ⇤ ⇤ ⇤ 2A 2N H of h. For any h,let⇡(y) := . The map ⇡ : (h) is linear and bounded x 2 x yx A!L since ⇡(y) 2 = , = !(x y yx) y 2 x 2 and thus has a bounded closure. It is a k xk h yx yxi ⇤ ⇤ k k k k *-homomorphism since

,⇡(z⇤) = , = !(y⇤z⇤x)= , = ⇡(z) , h y xi h y z⇤xi h zy xi h y xi and ⇡(xy) = = ⇡(x)⇡(y) and defines a representation of in . Moreover, ,⇡(x) = z xyz z A H h 1 1i , = !(x), so that ⌦= . Cyclicity follows from ⇡(x)⌦: x = : x ,which h 1 xi 1 { 2A} { x 2A} is the of equivalence classes by construction. Finally, let ( ,⇡, ⌦ ) be another such H0 0 0 representation. Then the map U : defined by ⇡ (x)⌦ = U⇡(x)⌦is a densely defined H!H0 0 0 , since

⇡(y)⌦,⇡(x)⌦ = !(y⇤x)= ⇡0(y)⌦0,⇡0(x)⌦0 = U⇡(y)⌦,U⇡(x)⌦ , h iH h iH0 h iH0 and hence extends to a .

7 Corollary 13. Let be a C*-algebra and ↵ a*-automorphism.If! ( ) is ↵-invariant, A 2EA !(↵(x)) = !(x) for all x , then there is a unique unitary operator U on the GNS Hilbert 2A space such that, for all x , H 2A U⇡(x)=⇡(↵(x))U, and U⌦=⌦.

One says that ↵ is unitarily implementable in the GNS representation.

Proof. The corollary follows from the uniqueness part of Theorem 12 applied to ( ,⇡ ↵, ⌦), H since ⌦,⇡(x)⌦ = !(x)=!(↵(x)) = ⌦,⇡ ↵(x)⌦ . h i h i Proposition 14. Let be a C*-algebra, ! ( ) and ( ,⇡,⌦) the associated representation. A 2E A H Then ⇡ is irreducible and ⇡ =0i↵ ! is a pure state. 6 Proof. Let ⌫ be majorised by ! = ! .Thereisa0 T 1 such that ⌫(x y)= ⇡(x)⇠,T⇡(y)⇠ ⇡,⌦   ⇤ h i with T (⇡( )) .If⇡ is irreducible, then T = p 1 so that ⌫ = !,0 1 and ! is pure. 2 A 0 ·   Reciprocally, if ⌫ is not a multiple of !,thenT is not a multiple of the identity, so that ( ,⇡) H is not irreducible.

Definition 7. Let ( ,⇡) be a representation of .Astate! is ⇡-normal if there exists a H A density ⇢ in such that !(A)=Tr(⇢ ⇡(A)). Two representations ( ,⇡ ), ( ,⇡ ) ! H ! H1 1 H2 2 are quasi-equivalent if every ⇡1-normal state is ⇡2-normal and conversely.

Further, two states !1,!2 are said to be quasi-equivalent if their GNS representations are quasi- equivalent. These correspond to thermodynamically equivalent states.

2.3 Examples: Quantum spin systems, the CCR and CAR al- gebras

2.3.1 Quantum spin systems Let be a . Denote ⇤ the finite sets of and () the set of finite subsets. For b F each x ,let x be a finite dimensional Hilbert space, and assume that supx dim( x) < . 2H H 2 H 1 The Hilbert space of ⇤ b is given by ⇤ := x ⇤ x. The associated algebra of local H ⌦ 2 H is ⇤ := ( ⇤) x ( x). A L H '⌦ 2 L H Inclusion defines a partial order on (), which induces the following imbedding: F

⇤ ⇤0 = ⇢ )A⇤ ⇢A⇤0 where x ⇤ is identified with x 1⇤ ⇤ ⇤ . Note that ⇤ ⇤0 = implies xy = x y = yx 2A ⌦ 0\ 2A 0 \ ; ⌦ for all x ,y . Finally, the algebra of quasi-local observables is given by 2A⇤ 2A⇤0

k·k := k·k A A⇤ ⌘ Aloc ⇤ () 2[F and it is a C*-algebra. Note that has an identity. In other words, is obtained as a A A of finite-dimensional matrix algebras, which is referred to as a uniformly hyperfine algebra (UHF). From the physical point of view, a finite dimensional Hilbert space is the of a physical system with a finite number of degrees of freedom, namely a few-levels atom or 2Sx+1 a spin. In the latter case, x = C is the state space of a spin-S,withS 1/2N, and it H 2

8 carries the (2Sx + 1)-dimensional irreducible representation of the rotation SU(2). A UHF is therefore the algebra of observables of atoms in an optical lattice or of magnetic moments of nuclei in a crystal. A state ! on has the property that it is generated by a a family of density matrices defined A ! by: if x ⇤,then!(x)=Tr ⇤ (⇢⇤x). Such a state is called locally normal. We have: 2A H Proposition 15. If ! is a state of a quantum spin system , then the density matrices ⇢! obey A ⇤ ! ! ! 1. ⇢⇤ ⇤, ⇢⇤ 0 and Tr ⇤ (⇢⇤)=1 2A H ! ! 2. the consistency condition ⇤ ⇤0 and x ⇤, then Tr ⇤ (⇢⇤x)=Tr (⇢⇤ x) ⇢ 2A H H⇤0 0 ⇢ Conversely, given a family ⇢⇤ ⇤ () satisfying (1, 2), there is a unique state ! on . { } 2F A Proof. Since is a finite dimensional matrix algebra, the restriction of ! to is given by a A⇤ A⇤ satisfying (1). (2) follows from the identification ⇤ ⇤ 1⇤ ⇤. Conversely, A 'A ⌦ 0\ a family of ⇢ defines is a bounded linear functional on the dense subalgebra . Hence it ⇤ Aloc extends uniquely to a linear functional on with the same bound. A Theorem 16. Let !1,!2 be two pure states of a quantum spin system. Then !1 and !2 are equivalent if and only if for all ✏>0, there is ⇤ b such that

! (x) ! (x) ✏ x , | 1 2 | k k for all x with ⇤ ⇤ = . 2A⇤0 \ 0 ; In other words, two pure states of a quantum spin system are equivalent if and only if they are ‘equal at infinity’, namely thermodynamically equal. More generally, the theorem holds – with quasi-equivalence – for any two factor states. Note that if a state is pure, then it is irreducible, i.e. ⇡(A)0 = C 1, so that ⇡(A)00 = ( ) and ⇡(A)0 ⇡(A)00 = C 1, hence ! is a factor state. · L H i\ · In practice, one is given a family of vectors ⇤, i = I⇤ an index set, typically the set of thermal/ground states of a finite volume Hamiltonian H on . All states !i := i , i ⇤ H⇤ ⇤ h ⇤ · ⇤i on can be extended to a state on (by Hahn-Banach), that we still denote !i .Theset A⇤ A ⇤ := !i :⇤ ,i I is a subset of ( ), which is weakly-* compact, hence there are weak- S { ⇤ b 2 ⇤} E A * accumulation points, denoted !i , i I . These are usually taken as the thermodynamic 2 thermal/ground states of the quantum spin system. Finally, let = Zd. There is a natural notion of translations on which defines a group of d A automorphisms Z z ⌧z:If⇤ and x ⇤, ⌧z(x) is the same on ⇤+ z.This 3 7! b 2A defines an automorphism on the dense subalgebra , which can be extended by continuity Aloc to ⌧z on all of . If a state is translation invariant, ! ⌧z = ! for all z Z,then⌧z is unitarily A 2 implementable in the GNS representation, namely there is Zd z U(z), where U(z) are 3 7! d unitary operators on with U(z)⌦=⌦, such that ⇡(⌧z(x)) = U(z)⇤⇡(x)U(z) for all z Z H 2 and x . Furthermore, ⌧ = ⌧ ⌧ implies U(z + z )=U(z )U(z ). 2A z1+z2 z1 z1 1 2 1 2 The following proposition is usually referred to as the asymptotic abelianness of A Proposition 17. Let and z ⌧ be as above. Then for each x, y , A 7! z 2A

lim [⌧z(x),y]=0. z | |!1 Proof. Exercise.

9 2.3.2 Fermions: the CAR algebra The algebra of canonical anticommutation relations (CAR) is the algebra of creation and anni- hilation operators of fermions

Definition 8. Let be a prehilbert space. The CAR algebra ( ) is the C*-algebra generated D A+ D by 1 and elements a(f), f satisfying 2D f a(f) is antilinear 7! a(f),a(g) =0, a(f)⇤,a(g)⇤ =0 { } { } a(f)⇤,a(g) = g, f 1 { } h i for all f,g . 2D It follows from the CAR relations that (a(f) a(f))2 = a(f) a(f),a(f) a(f)= f 2a(f) a(f), ⇤ ⇤{ ⇤} k k ⇤ and the C*-property then implies a(f) = f so that f a(f) is a continuous map. k k k k 7! Proposition 18. Let be a prehilbert space with closure = . Then D D H 1. ( )= ( ) A+ D A+ H 2. ( ) is unique: If , both satisfy the above definition, then there exists a unique A+ D A1 A2 *-isomorphism : such that a (f)=(a (f)) for all f A1 !A2 2 1 2D 3. If L is a bounded linear operator in and A a bounded antilinear operator in satisfying2 H H

L⇤L + A⇤A = LL⇤ + AA⇤ =1,

LA⇤ + AL⇤ = L⇤A + A⇤L =0,

there is a unique *-automorphism of ( ) such that (a(f)) = a(Lf)+a(Af) . L,A A+ H L,A ⇤ Proof. Since is a subset of , we have that ( ) ( ). Moreover, if f ,thereisa D H A+ D ⇢A+ H 2H f such that f f. By linearity and continuity, a(f) a(f ) = a(f f ) = n 2D n ! k n k k n k f fn 0, showing that a(f) +( ), and +( ) ( ), proving (1). k k! 2A D An H ⇢AD Assume now that dim < , and that fi i=1 is an orthonormal basis. Then the map n H 1 { } : ( ) ⌦ defined by I A+ H !M2 k k (a(fk)a(fk)⇤)=e11 (Vk 1a(fk)) = e12 I I k k (Vk 1a(fk)⇤)=e21 (a(fk)⇤a(fk)) = e22 I I k n where eij is the canonical matrix in 2⌦ which is non-trivial on the k-th factor, and k M Vk = i=1(1 2a(fi)⇤a(fi)), is an algebra isomorphism. In particular, the CAR imply that k k k k l e e = jae and [e ,e ]=0ifk = l as it should. Furthermore, it is invertible with inverse ij ab Q ib ij ab 6 k 1 1 i i k a(fk)= (e11 e22)e12 . I ! Yi=1 This proves (2) for the finite dimensional case. If is infinite dimensional, there is a basis H f↵ ↵ A of , not necessarily countable, and the above construction can be made with any { } 2 H finite subset of A. We conclude in this case by (1) since the of finite linear combinations of f is dense in . ↵ H 2 By definition, f,Ag = g, A⇤f for an antilinear operator h i h i

10 Finally,

a(Lf)+a(Af)⇤,a(Lg)⇤ + a(Ag) = Lf, Lg + Ag, Af = f,g , { } h i h i h i and similar computations for other anticommutators show that 1 and a(Lf)+a(Af)⇤ for all f also generate ( ), conluding the proof by (2). 2H A+ H Note that the proof of (2) shows that the CAR algebra is a UHF algebra. The transformation L,A is called a Bogoliubov transformation. Its unitary implementability in a given representation is a separate question, which can be completely answered in the case of so-called quasi-free representations. A particularly simple case is given by A = 0 and a unitary L, corresponding to the non-interacting evolution of single particles under L.

2.3.3 Bosons: the CCR algebra The algebra of canonical commutation relations (CCR) is the algebra of creation and annihila- tion operators of bosons. Being unbounded operators, they do not form a C*-algebra, but their exponentials do so and it is usually referred to, in this form, as the Weyl algebra.

Definition 9. Let be a prehilbert space. The Weyl algebra ( ) is the C*-algebra generated D A D by W (f), f satisfying 2D

W ( f)=W (f)⇤ i W (f)W (g)=exp Im f,g W (f + g) 2 h i ✓ ◆ for all f,g . 2D Note the commutation relation W (f)W (g)=exp( iIm f,g )W (g)W (f). h i Proposition 19. Let be a prehilbert space with closure = . Then D D H 1. ( )= ( ) if and only if = A D A H D H 2. ( ) is unique: If 1, 2 both satisfy the above definition, then there exists a unique A D A A *-isomorphism : such that W (f)=(W (f)) for all f A1 !A2 2 1 2D 3. W (0) = 1, W (f) is a and W (f) 1 =2for all f , f =0 k k 2D 6 4. If S is a real linear, invertible operator in such that Im Sf,Sg =Imf,g , then there D h i h i is a unique *-automorphism S of ( ) such that S(W (f)) = W (Sf). A D In fact, only needs to be a real linear vector space equipped with a symplectic form, and S D is a symplectic map. This is the natural structure of phase space and its Hamiltonian dynamics in classical mechanics, and the map f W (f) is called the Weyl quantisation3. (3) shows in 7! particular that it is a discontinuous map. We only prove (3) and (4). The di↵erence between the CAR and CCR algebra with respect to closure of the underlying space is due to the lack of continuity of f W (f). 7! Proof. The definition implies that W (f)W (0) = W (f)=W (0)W (f) so that W (0) = 1. More- over, W (f)W ( f)=W ( f)W (f)=W (0) = 1 so that W (f) is unitary. In turn, this implies

W (g)W (f)W (g)⇤ =exp(iImf,g )W (f). h i 3 In fact, it is also an algebra isomorphism between = C1(X) equipped with a and ( ) D A D

11 Hence, the spectrum of W (f) is invariant under arbitrary rotations for any f = 0, so that 6 Sp(W (f)) = S1. Hence, sup : Sp(W (f) 1) = 2, which concludes the proof of (3) {| | 2 } since W (f) 1 is a . Finally, (4) follows again from (2) and the invariance of the Weyl relations.

Definition 10. A representation ( ,⇡) of ( ) is regular if t ⇡(W (tf)) is a strongly H A D 7! continuous map on for all f . H 2D In a regular representation, R t ⇡(W (tf)) is a strongly continuous group of uni- 3 7! taries by the Weyl relations, so that Stone’s theorem yields the existence of a densely defined, self-adjoint generator (f) such that ⇡(W (tf)) = exp(it (f)) for all f . In fact, for ⇡ ⇡ 2D any finite dimensional subspace there is a common dense space of analytic vectors of K⇢D n n (f), (if),f , namely for which 1 t /n! < for t small enough. The { ⇡ ⇡ 2K} n=0 k ⇡ k 1 creation and annihilation operators can be defined P 1/2 1/2 a⇤ (f):=2 ( (f) i (if)) ,a(f):=2 ( (f)+i (if)) ⇡ ⇡ ⇡ ⇡ ⇡ ⇡ on D(a (f)) = D(a (f)) = D( (f)) D( (if)), which is dense. Note that a (f) a (f) . ⇡⇤ ⇡ ⇡ \ ⇡ ⇡⇤ ⇢ ⇡ ⇤ In fact, equality holds. By construction f (f) is real linear, so that f a (f) is antilinear and f a (f) 7! ⇡ 7! ⇡ 7! ⇡⇤ is linear. Now, taking the second of the Weyl relations applied on any vector ⇠ 2 D( (f)) D( (g)) at t = t = 0, one obtains ( (f) (g) (g) (f)) ⇠ =iImf,g ⇠,so ⇡ \ ⇡ 0 ⇡ ⇡ ⇡ ⇡ h i that (a (f)a⇤ (g) a⇤ (g)a (f)) ⇠ = f,g ⇠ ⇡ ⇡ ⇡ ⇡ h i the usual form of the canonical commutation relations (CCR). Finally, we prove that the creation/annihilation operators are closed. Indeed, (f)⇠ 2 + (if)⇠ 2 = a (f)⇠ 2 + k ⇡ k k ⇡ k k ⇡ k a (f)⇠ 2, while the commutation relations yield a (f)⇠ 2 a(f)⇠ 2 = f 2 ⇠ 2. Together, k ⇡⇤ k k ⇤ k k k k k k k (f)⇠ 2 + (if)⇠ 2 =2a (f)⇠ 2 + f 2 ⇠ 2. Hence, for any sequence D(a (f)) k ⇡ k k ⇡ k k ⇡ k k k k k n 2 ⇡ such that and a (f) converges, we have that (f) , (if) converge. Since n ! ⇡ n ⇡ n ⇡ n are self-adjoint and hence closed, we have that D(a (f)), and (f) (f) and ⇡ 2 ⇡ ⇡ n ! ⇡ (if) (if) . By the norm equality again, a (f) a (f) and a (f) is closed. ⇡ n ! ⇡ ⇡ n ! ⇡ ⇡ 2.3.4 Fock spaces and the Fock representation The set n carries an action ⇧of the permutation group S D⌦ n ⇧⇡ : 1 n 1 1 ⌦···⌦ 7! ⇡ (1) ⌦···⌦ ⇡ (n) (n) (n) n (n) sgn⇡ (n) for any ⇡ Sn and we denote := ⌦ :⇧⇡ =( 1) , namely the 2 D± { 2D n ±(0) } symmetric, respectively antisymmetric subspace of ⌦ . Let also := C. The bosonic, D D(n±) respectively fermionic over is denoted ( ):= n1=0 . That is, a vector ± D (n) F D (n) D±(n) (n) 2 ( ) can be represented as a sequence ( )n N such that ,with n < F± D 2 2D± N k k . The vector ⌦:= (1, 0,...) is called the . We further denote fin( ):= 2 ( ): 1 F± D P{ 2F± D N N with (n) =0, n N , which is dense. Note that the probability to find more than 9 2 8 } N particles in any vector vanishes as N , !1 (n) 2 P N ( ) := 0, (N ), k k ! !1 n N X which we interpret as follows: In Fock space, there is an arbitrarily large but finite number of particles. In particular, there is no vector representing a gas at non-zero density in the thermodynamic limit. We define N : fin( ) fin( )byN =n whenever (n). F± D !F± D 2D±

12 n n 1 For f ,letb (f): ⌦ ⌦ be defined by 2D ± D !D

b (f)( 1,... n)=pn f, 1 ( 2,... n), ± h i (n) (n 1) (0) which maps to ,withb (f) = 0, and hence b (f): ( ) ( ). Its D± D± (n 1) ±(n) D± ± F± D !F± D adjoint b⇤ (f):=b (f)⇤ : such that ± ± D± !D± n (n 1) 1 k 1 (n 1) b⇤ (f) = ( 1) ⇧⇡k f ± pn ± ⌦ Xk=1 1 ˜ (n) where ⇡k =(k, 1, 2,...,k 1,k+1,...,n). Indeed, the right hand side isin :(⇡(k) 1 1 D± ⇡k )(1) = 1 and the signature of the permutation is (k 1) + sgn()+((k) 1), (n 1) sgn()+((k) k) (n 1) ˜ so that ⇧⇡ 1 ⇧⇧⇡k (f )=( 1) f , which implies that ⇧ = (k) ⌦ ± ⌦ ( 1)sgn() ˜ . Moreover, for any ⌥(n) (n), ± 2D± n (n) (n 1) (n) (n 1) 1 (n) (n 1) (n) b (f)⌥ , = pn ⌥ ,f = ⇧⇡ ⌥ , ⇧⇡ (f ) = ⌥ , ˜ h ± i h ⌦ i pn h k k ⌦ i h i Xk=1 ˜ (n 1) where we used that ⇧( ) is unitary, proving that =b⇤ (f) . · ± Proposition 20. 1. f b (f) is antilinear, f b⇤ (f) is linear 7! ± 7! ± 2. Nb (f)=b (f)(N 1) ± ± 3. b (f),b⇤ (g) satisfy the canonical commutation, resp. anticommutation relations ± ± (n 1) Proof. We denote [A, B] := AB BA and prove [b (f),b⇤ (g)] = f,g . Indeed, for = ± ⌥ ± ± ± h i 1 n 1, ⌦···⌦ 1 (n 1) 1 (n 1) b (f)⇧⇡ (g )= f, 1 2 k g n 1 = ⇧⇡ (g b (f) ), pn ± k+1 ⌦ h i ⌦···⌦ ⌦ ⌦··· pn 1 k ⌦ ± so that n (n 1) 1 k 1 (n 1) b (f)b (g) = ( 1) b (f)⇧⇡ (g ) ± ± pn ± ± k ⌦ Xk=1 n 1 (n 1) 1 k 1 (n 1) = f,g ( 1) ⇧⇡ (g b (f) ) h i ± pn 1 ± k ⌦ ± Xk=1 (n 1) (n 1) = f,g b (g)b (f) , h i ± ± ± where the second equality follows by extracting the first term in the sum and using the obser- vation above in the remaining terms.

In particular, b (f):f form a representation of the CAR algebra. Furthermore, The { 2D} operators (f):=2 1/2(b (f)+b (f)) are symmetric on fin( ) and extend to self-adjoint + + +⇤ F+ D operators, so that W (f):=exp(i (f)) are well-defined unitary operators on ( ), yielding + + F+ D a representation of the Weyl algebra. They are the fermionic and bosonic Fock representations associated to the

CAR CAR !F (a(f)⇤a(g)) := ⌦,b⇤ (f)b (g)⌦ = 0 and !F (a(f)) := 0 (fermions) CCR f 2/4 (! (W (f)) := ⌦,W+(f)⌦ =ek k (bosons) F h ⌦ i ↵ 13 In other words, Fock spaces are the GNS Hilbert spaces for the Fock states. Quantum mechanics in one for one particle is usually associated with the Schr¨odinger representation, defined on the Hilbert space L2(R). It arises as the regular representation of the Weyl algebra (C) given by A i st ⇡S(W (s +it)) := e 2 U(s)V (t), where (U(s) )(x)=eist (x), (V (t) )(x)= (x + t), with self-adjoint generators X := S(1) and P := S(i) = i@x. 2 1/2 In fact, L (R) carries a Fock space structure, obtained by introducing aS := 2 (X +iP ) 1/2 and aS⇤ := 2 (X iP ), which satisfy the CCR (strongly on a dense set such as Cc1(R). The 2 vacuum vector ⌦S is the L -normalised solution of aS⌦S = 0, namely

1/4 x2/2 (x + @x)⌦S(x)=0, i.e. ⌦S(x)=⇡ e .

n n 2 With := span (aS⇤ ) ⌦S , namely the span of the nth Hermite function, one obtains L (R)= Hn { } 2 1 . In other words, L (R) +(C) and the Schr¨odinger and Fock representations are n=0H 'F equivalent, the unitary map being (a )n⌦ (b )n⌦. S⇤ S 7! +⇤ Hence, the dimension of has the interpretation of ‘the number of degrees of freedom’ of D the system and N-body quantum mechanics in Rd corresponds to the algebra (CNd), which A has a Schr¨odinger representation, namely the (Nd)-fold product representation of that given above. In fact, this is the only one:

Theorem 21. Let be a finite dimensional Hilbert space, dim = n. Then, any irreducible H H representation of ( ) is equivalent to the Schr¨odinger representation. A H In other words, the algebraic machinery is useless in quantum mechanics. Whenever dim = , H 1 typically = L2(Rd) itself, there are truly inequivalent representations: these are in particular H those arising in quantum statistical mechanics.

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