Algebraic Quantum Field Theory

• Observables describe properties of measuring devices (possible measured values, commensurability properties). • States describe properties of prepared ensembles (probability distributions of measured values, correlations between observables) Mathematical description based on Hilbert space formalism, Hilbert space H. • Observables: self-adjoint operators A on H. • States: density matrices ρ on H (i.e. ρ ≥ 0, Tr ρ = 1). • Expectation values A, ρ → TrρA. Remark 1.1 pure states ('optimal information')= rays e iφ φ ∈ H, φ = 1 = orthogonal projections ρ 2 = ρ. ( Question: Why equivalent? Express in a basis, there can be just one eigenvalue with multiplicity one). • Usual framework : fixed by specifying H. E.g. for spin H = C 2 , for particle L 2 (R 3 ). Question: What is the Hilbert space for a particle with spin? L 2 (R 3 ; C 2 ). • Question: Does every s.a. operator A correspond to some measurement? Does every density matrix ρ correspond to some ensamble which can be prepared? In general no. Superselection rules. For example, you cannot superpose two states with different charges. • New point of view: Observables are primary objects (we specify the family of measuring devices). The rest of the theory follows. Heisenberg algebra Quantum Mechanics. Observables: Q j , j = 1, . . . , n and P k , k = 1, . . . , n. (n = N d, N -number of particles, d-dimension of space). We demand that observables form (generate) an algebra. Definition 1.2 The "free (polynomial) * -algebra P" is a complex vector space whose basis vectors are monomials ("words") in Q j , P k (denoted Q j 1 . . . P k 1 . . . Q jn . . . P kn ). 1. Sums: Elements of P have the form c j 1 ...kn Q j 1 . . . P kn . 2 2. Products: The product operation is defined on monomials by (Q j 1 . . . P k 1 . . . Q jn . . . P kn ) · (Q j 1 . . . P k 1 . . . Q j n . . . P k n ) = Q j 1 . . . P k 1 . . . Q jn . . . P kn Q j 1 . . . P k 1 . . . Q j n . . . P k n 3. Adjoints: Q * j = Q j , P * k = P k , c j 1 ...kn Q j 1 . . . P kn * = c j 1 ...kn P kn . . . Q j 1 . 4. Unit: 1. The operations (+, ·, * ) are subject to standard rules (associativity, distributivity, antilinearity etc.) but not commutativity. • Quantum Mechanics requires the following relations: • Consider a two-sided ideal J generated by all linear combinations of for all A, B ∈ P. Definition 1.3 Quotient P\J is again a * -algebra, since J is a two-sided ideal and J * = J . We will call it "Heisenberg algebra". This is the free algebra 'modulo relations' (3). Weyl algebra The elements of polynomial algebra are intrinsically unbounded (values of position and momentum can be arbitrarily large). This causes technical problems. A way out is to consider their bounded functions. For z = u + iv ∈ C n we would like to set W (z) ≈ exp(i k (u k P k + v k Q k )). We cannot do it directly, because exp is undefined for 'symbols' P k , Q k . But we can consider abstract symbols W (z) satisfying the expected relations keeping in mind the formal Baker-Campbell-Hausdorff (BCH) relation. The BCH formula gives We On the other hand: Hence 3 Definition 1.4 The (pre-)Weyl algebra W is the free polynomial * -algebra generated by the symbols W (z), z ∈ C n modulo the relations where z|z = kz k z k is the canonical scalar product in C n . The Weyl algebra has the following properties: 1. We have W (0) = 1 (by the uniqueness of unity). By the above Weyl operators are unitary. 3. We have Thus elements of W are linear combinations of Weyl operators W (z). Representations of the Weyl algebra Definition 1.5 A * -representation π : W → B(H) is a homomorphism i.e. a map which preserves the algebraic structure. That is for W, W 1 , W 2 ∈ W: If in addition π(1) = I, we say that the representation is unital. (In these lectures we consider unital representations unless specified otherwise). (Note that for u = 0 π 1 (W (z) is a multiplication operator and for v = 0 it is a shift). This is Schrödinger representation in configuration space. For the last step note (e iuP f )(x) = (e This is Schrödinger representation in momentum space. Relation between (π 1 , H 1 ), (π 2 , H 2 ) is provided by the Fourier transform ( F is isometric, i.e. Ff, Ff = f, f , (Plancherel theorem) and invertible (Fourier theorem). Hence it is unitary. We have the two representations are said to be (unitarily) equivalent (denoted (π a , H a ) (π b , H b )). As we will see, equivalent representations describe the same set of states. Is any representation of W unitarily equivalent to the Schrödinger representation π 1 ? Certainly not, because we can form direct sums e.g. π = π 1 ⊕ π 1 is not unitarily equivalent to π 1 . We have to restrict attention to representations which cannot be decomposed into "smaller" ones. Definition 1.10 Irreducibility of representations: We say that a closed subspace K ⊂ H is invariant (under the action of π(W)) if π(W)K ⊂ K. We say that a representation of (π, H) of W is irreducible, if the only closed invariant subspaces are H and {0}. Remark 1.11 The Schroedinger representation π 1 is irreducible (Homework). Lemma 1.12 Irreducibility of (π, H) is equivalent to any of the two conditions below: (i.e. if every non-zero vector is cyclic). implies that A ∈ CI ("Schur lemma") (i.e. the commutant of π(W) is trivial). Remark 1.13 Recall that the commutant of π(W) is defined as hence Ψ cannot be cyclic because χ ∆ (B) projects on a subspace which is strictly smaller than H. Question: Are any two irreducible representations of the Weyl algebra unitarily equivalent? Answer: In general, no. After excluding pathologies yes. Example 1.14 Let H 3 be a non-separable Hilbert space with a basis e p , p ∈ R n . Elements of H 3 : (i.e. all c p = 0 apart from some countable set). f |f = p c p c p . We define This representation is irreducible but not unitarily equivalent to (π 1 , H 1 ) (π 2 , H 2 ) because H 1,2 and H 3 have different dimension. Criterion: Representation (π, H) of W is of "physical interest" if for any f ∈ H the expectation values 6 depend continuously on z. Physical meaning of the Criterion: ) is an n-parameter unitary representation of translations on H. Hence, by the Criterion and Stone's theorem where P π,i is a family of commuting s.a operators on (a domain in) H. They can be interpreted as momentum operators in this representation. Analogously, we obtain the position operators Q π,i . By taking derivatives of the Weyl relations w.r.t, u l , v k one obtains [Q π,j , P π,k ] = iδ j,k 1 on a certain domain (on which the derivatives exist). Remark 1.16 This theorem does not generalize to systems with infinitely many degrees of freedom (n = ∞). In particular, it does not hold in QFT. This is one reason why charges, internal ('gauge') symmetries, and groups play much more prominent role in QFT than in QM. As we will see in Section 5, they will be needed to keep track of all these inequivalent representations. States Definition 1.17 A state ω of a physical system is described by 1. specifying a representation (π, H) of W, 2. specifying a density matrix ρ on H. Then ω(W ) = Trρπ(W ). Lemma 1.18 A state is a map ω : W → C which satisfies 2. normalization ω(1) = 1. Proof. The only non-trivial fact is positivity: Then, if the sum is finite, we can write 7 by completing Ψ i to orthonormal bases. In the general case we can use cyclicity of the trace The result is finite (because ρπ(W * W ) is trace-class) and manifestly positive. Proof. GNS construction (we will come to that). 2 ) then the corresponding sets of states coincide. Proof. Let ρ 1 be a density matrix in representation π 1 and W ∈ W. Then Hence it does not matter if we measure W in representation π 1 on ρ 1 or in π 2 on ρ 2 = U −1 ρ 1 U . 1.1.5 Weyl C * -algebra Definition 1.22 We define a seminorm on W: where the supremum extends over all cyclic representations. The completion of W/ ker · is the Weyl C * -algebra which we denoteW. A few remarks about this definition: 1. The supremum is finite because for any representation π we have and thus π(W ) for any W ∈ W is finite. 8 2. We cannot take supremum over all representations because this is not a set. In fact, take the direct sum of all the representations which do not have themselves as a direct summand and call this representation Π. Then we get the Russel's paradox: where π 1 ∈ π 2 means here that π 1 is contained in π 2 as a direct summand. 3. Using the GNS theorem one can show that Here the supremum extends over the set of states. Indeed: On the other hand 4. In the case of the Weyl algebra ker · = 0 so the seminorm (32) is actually a norm. [5] Apart from standard properties of the norm, it satisfies This is adventageous from the point of view of functional calculus: For W ∈ W we have f (W ) ∈ W for polynomials f , but for more complicated functions there is no guarantee. For W ∈W we have f (W ) ∈W for any continuous function f . Nevertheless, in the next few subsections we will still work with the pre-Weyl algebra W. Symmetries Postulate: Symmetry transformations are described by automorphisms (invertible homomorphisms) of W. Definition 1.23 We say that a map α : W → W is an automorphism if it is a bijection and satisfies Automorphisms of W form a group which we denote Aut W. Example 1.24 If U ∈ W is a unitary, then α U (W ) = U W U −1 is called an inner automorphism. Inner automorphisms form a group In W. For example, for U = W (u 0 ) we have This is translation of coordinates, as one can see in the Schroedinger representation π 1 : Similarly, for is a translation in momentum space. is an automorphism which is not inner. (Set n = 3 and let R be a rotation around the z axis by angle θ. Then, in the Schrödinger representation Clearly, U is not an element of W). Automorphisms which are not inner are called outer automorphisms. They form a set OutW which is not a group. As we have seen above, even if an automorphism is not inner, it can be implemented by a unitary in some given representation. Definition 1.26 Let (π, H) be a representation of W. Then α ∈ AutW is said to be unitarily implementable on H if there exists some unitary U ∈ B(H) s.t. 10 Example 1.27 A large class of automorphisms is obtained as follows where c(z) ∈ C\{0} and S : C n → C n a continuous bijection. Weyl relations impose restrictions on c, S: The latter property means that S is a real-linear symplectic transformation. For continuous c and S such automorphisms are unitarily implementable in all irreducible representations satisfying the Criterion (consequence of the v.N. uniqueness theorem). See Homeworks. Remark 1.28 ω(z 1 , z 2 ) := Im z|z is an example of a symplectic form. In general, we say that a bilinear form ω is symplectic if it is: 2. Non-degenerate: If ω(z 1 , z 2 ) = 0 for all z 2 , then z 1 = 0. Dynamics Definition 1.29 A dynamics on W is a one-parameter group of automorphisms on W i.e. R t → α t s.t. α 0 = id, α t+s = α t • α s . Proposition 1.30 Suppose that the dynamics is unitarily implemented in an irreducible representation π i.e. there exists a family of unitaries s.t. Suppose in addition that t → U (t) continuous (in the sense of matrix elements) and differentiable (i.e. for some 0 = Ψ ∈ H, ∂ t U (t)Ψ exists in norm). Then there exists a continuous group of unitaries t → V (t) (i.e. Remark 1.31 By the Stone's theorem we have V (t) = e itH for some self-adjoint operator H on (a domain in) H (the Hamiltonian). Whereas α t is intrinsic, the Hamiltonian is not. Its properties (spectrum etc.) depend in general on representation. Proof. We have α s • α t = α s+t . Hence 11 By irreducibility of π U (s + t) = η(s, t)U (s)U (t), where |η(s, t)| = 1. By multiplying U by a constant phase e iφ 0 we can assume that U (0) = I, hence Now consider a new family of unitaries V (s) = ξ(s)U (s), |ξ(s)| = 1. We have The task is to obtain η (s, t) = 1 for all s, t for a suitable choice of ξ (depending on η). The key observation is that associativity of addition in R imposes a constraint on η: In fact, we can write s + t)U (r)U (s + t) = η(r, s + t)η(s, t)U (r)U (s)U (t),(57) U (r + s + t) = η(r + s, t)U (r + s)U (t) = η(r + s, t)η(r, s)U (r)U (s)U (t).(58) Hence we get the "cocycle relation" (cohomology theory) η(r, s + t)η(s, t) = η(r + s, t)η(r, s). Using this relation one can show that given η one can find such ξ that η = 1. "cocycle is a coboundary" (Howework). Important intermediate step is to show, using the cocycle relation that η(s, t) = η(t, s). To express ξ as a function of η we will have to differentiate η. By assumption, there is Ψ ∈ H, Ψ =1 s.t. ∂ t U (t)Ψ exists. By (53), we have Hence ∂ t η(s, t) exists and by (60) also ∂ s η(s, t). Example 1.32 Isotropic harmonic oscillator: In the framework of the polynomial algebra P we have (heuristically) 12 In the Weyl setting α t (W (z)) = W (e itω 0 z). This defines a group of automorphisms from Example 1.27 with S t z = e itω 0 z, c(z) = 1. (S t is complex-linear). This dynamics is unitarily implemented in the Schrödinger representation: Example 1.33 Free motion in the framework of P: In the framework of W: We have that S t (z) = Rez + (t/m + i)Im z is a symplectic transformation, but only real linear. This dynamics is unitarily implemented in the Schrödinger representation: By generalizing the above discussion, one can show that dynamics governed by Hamiltonians which are quadratic in P i , Q j correspond to groups of automorphisms of W. But there are many other interesting Hamiltonians, for example: where n = 1, V ∈ C ∞ 0 (R) R (smooth, compactly supported, real). implies that V = 0. Proof. See [3]. Thus AutW does not contain dynamics corresponding to Hamiltonians (69). A recently proposed solution to this problem is to pass from exponentials W (z) = e i(uP +vQ) to resolvents R(λ, z) = (iλ − uP − vQ) −1 and work with an algebra generated by these resolvents [4]. 13 1.1.8 Resolvent algebra Definition 1.35 The pre-resolvent algebra R is the free polynomial * -algebra generated by symbols R(λ, z), λ ∈ R\{0}, z ∈ C n modulo the relations where λ, µ, ν ∈ R\{0} and in Realtions One can check that this prescription defines a representation of R which is irreducible. Definition 1.38 We define a seminorm on R where the supremum is over all cyclic representations of R. (A cyclic representation is a one containing a cyclic vector. In particular, irreducible representations are cyclic). The resolvent C * -algebra R is defined as the completion of R/ ker · . Remark 1.39 The supremum is finite because for any representation π we have . Then all c i 1 ,...in = 0. Definition 1.40 A representation (π, H) ofR is regular if there exist self-adjoint For example, the Schrödinger representation π 1 (of R) is regular. Fact: Any regular irreducible representation π of R is faithful Proof. (Idea). Use the Laplace transformation to construct a regular representation ofR out of a regular representation ofW. Remark 1.42 The Laplace transform can also be useful in checking if ker · is trivial. Up to now, we found no essential difference between the Weyl algebra and the resolvent algebra. An important difference is that the Weyl C * -algebra W is simple, i.e. it has no non-trivial two sided ideals. The resolvent C * -algebra has many ideals. They help to accommodate interesting dynamics. Theorem 1.43 There is a closed two-sided ideal J ⊂ R s.t. in any irreducible regular representation (π, H) one has π(J ) = K(H) where K(H) is the algebra of compact operators on H. Remark 1.44 We recall: • A is a compact operator if it maps bounded operators into pre-compact operators. (On a separable Hilbert space if it is a norm limit of a sequence of finite rank operators). 15 • A is Hilbert-Schmidt (A ∈ K 2 (H)) if A 2 := Tr(A * A) 1/2 < ∞. HilbertSchmidt operators are compact. • A convenient way to show that an operator on L 2 (R n ) is Hilbert-Schmidt is to study its integral kernel K, defined by the relation: • For example, consider A = f (Q)g(P ). Its integral kernel in momentum space is determined as follows: Hence the integral kernel of Proof. (Idea). By the von Neumann uniqueness theorem we can assume that π is the Schrödinger representation π 1 . Then it is easy to show that π( R) contains some compact operators: For example, set u i = (0, . . . , 1 i , . . . , 0) and v i = (0, . . . , 1 i , . . . , 0). Then the operator is Hilbert-Schmidt for all λ i , µ i ∈ R\{0}. (This can be shown by checking that it has a square-integrabe kernel). In particular it is compact. Now it is a general fact in the theory of C * -algebras that if the image of an irreducibe representation contains one non-zero compact operator then it contains all of them (Howework or Corollary 4.1.10 of 16 Theorem 1.45 Let n = 1, H = P 2 + V (Q), where V ∈ C 0 (R) R real, continuous vanishing at infinity and U (t) = e itH . Then Remark 1.46 Since π 1 is faithful, we can define the group of automorphisms of R which is the dynamics governed by the Hamiltonian H. Remark 1.47 For simplicity, we assume that V ∈ S(R) R and dx V (x) = 0. General case follows from the fact that such functions are dense in C 0 (R) R in supremum norm. Proof. Let U 0 (t) = e itH 0 , where H 0 = P 2 . Since this is a quadratic Hamiltonian, we have Now we consider Γ V (t) := U (t)U 0 (t) −1 . It suffices to show that Γ V (t) − 1 are compact for all V ∈ C 0 (R) R since then Γ V (t) ∈ π 1 ( R) by Theorem 1.43 and hence using Γ V (t) −1 = Γ V (t) * ∈ π 1 ( R). We use the Dyson perturbation series of Γ V (t): where V t := U 0 (t)V (Q)U 0 (t) −1 and the integrals are defined in the strong-operator topology, that is exist on any fixed vector. (Cf. Proposition 1.50 below). The key observation is that t 0 ds V s are Hilbert-Schmidt. To this end compute the integral kernel K s of V s : This is clearly not Hilbert-Schmidt. Now let us compute the integral kernelK s of t 0 ds V s : 17 This is Hilbert-Schmidt. In fact: dp 1 dp 2 |( Since (FV )(0) = 0 we have (FV )(q 1 ) ≤ c|q 1 | near zero so the integral exists. Consequently, the strong-operator continuous functions have values in the Hilbert-Schmidt class and their Hilbert-Schmidt (HS) norms are bounded by (since AB 2 ≤ A 2 B ). The integral of any strong-operator continuous HSvalued function with uniformly bounded (on compact sets) HS norm is again HS. (See Lemma 1.49 below). So each term in the Dyson expansion (apart from n = 0) is in π 1 ( R) and the expansion converges uniformly in norm. So Γ V (t) − 1 is a compact operator. Remark 1.48 The resolvent algebra admits dynamics corresponding to H = P 2 + V (Q). But there are other interesting Hamiltonians which are not covered e.g. So there remain open questions... In the above proof we used two facts, which we will now verify: be continuous in the strong operator topology and suppose that for some compact set K ⊂ R n we have where is again Hilbert-Schmidt. 18 Proof. We have Since the summands/integrals are positive, I can exchange the order of integration/summation. By Cauchy-Schwarz inequality: Where in the last step we use the assumption (96). Lemma 1.50 (Special case of Theorem 3.1.33 of [1]) Let R t → U 0 (t) be a strongly continuous group of unitaries on H with generator H 0 (i.e. U 0 (t) = e itH 0 , above we had H 0 = P 2 ) and let V be a bounded s.a. operator on H. Then H 0 + V generates a strongly continuous group of unitaries U s.t. For any Ψ ∈ H. (To get the expression for Γ V (t) it suffices to set Ψ = U 0 (t) −1 Ψ ). Proof. Strategy: we will treat (100) as a definition of a t ≥ 0 dependent family of operators t → U (t). We will use this definition to show that it can be naturally extended to a group of unitaries parametrized by t ∈ R. Then, by differentiation, we will check that its generator is H 0 + V . Hence, by Stone's theorem we will have U (t) = e it(H 0 +V ) . Let U (n) (t) be the n-th term of the series of U . We have, by a change of variables, 19 Iteratively, one can show that all U (n) (t) are well defined and strongly continuous. It is easy to check that this is a series of bounded operators which converges in norm: In fact By taking the sum of both sides of the recursion relation (101), we get Now we want to show the (semi-)group property: Now t 1 +t 2 t 1 part of the last integral cancels the integral (change of variables). We are left with U (t 1 )U (t 2 ) − U (t 1 + t 2 ) = t 1 0 ds U 0 (s)iV U (t 1 − s)U (t 2 ) − U (t 1 + t 2 − s)). is real-analytic. By (105) we get Clearly, F t 1 (0) = 0. Using this, and differentiating the above equation w.r.t. λ at 0, we get ∂ λ F t 1 (0) = 0. By iterating we get that all the Taylor series coefficients of F t 1 at zero are zero and thus F t 1 (λ) = 0 by analyticity. We conclude that the semigroup property holds i.e. U (t 1 + t 2 ) = U (t 1 )U (t 2 ). 20 Now we want to show that U (t) are unitaries. A candidate for an inverse of U (t) is U (t) defined by replacing H 0 with H 0 := −H 0 and V by V = −V . (JUMP DOWN). We also set U 0 (t) = e i(−H 0 )t . Let t 2 ≥ t 1 . Then U (t 1 )U (t 2 ) = U 0 (t 1 )U (t 2 ) + t 1 0 ds U 0 (s)iV U (t 1 − s)U (t 2 ) = U 0 (t 1 − t 2 ) + t 2 0 ds U 0 (−t 1 + s)iV U (t 2 − s) = U (t 2 − t 1 ) + t 2 0 ds U 0 (−t 1 + s)iV U (t 2 − s) In the last integral the part − −t 1 0 combines with the second line and − −t 1 +t 2 −t 1 cancels the first line. Thus we get U (t 1 )U (t 2 ) − U (t 2 − t 1 ) = (JUMP TO HERE). By an analogous argument as above we obtain U (t 1 )U (t 2 ) = U (t 2 − t 1 ), In particular, U (t)U (t) = 1 and we can consistently set U (−t) := U (t) for t ≥ 0. Moreover, it is easily seeen from (100), by a change of variables, that U (t) = U (t) * . Thus we have a group of unitaries. By Stone's theorem it has a generator which can be obtained by differentiation: Clearly we ha

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