The Algebra of Block Spin Renormalization Group Transformations
Tadeusz Balaban, Joel Feldman, Horst Knörrer, Eugene Trubowitz
aa r X i v : . [ m a t h - ph ] S e p The Algebra of Block Spin Renormalization GroupTransformations
Tadeusz Balaban , Joel Feldman ∗ , Horst Kn¨orrer , and EugeneTrubowitz Department of MathematicsRutgers, The State University of New [email protected] ∼ feldman/ ∼ knoerrer/ September 6, 2016
Abstract
Block spin renormalization group is the main tool used in our program to seesymmetry breaking in a weakly interacting many Boson system on a threedimensional lattice at low temperature. In this paper, we discuss some of itspurely algebraic aspects in an abstract setting. For example, we derive some“well known” identities like the composition rule and the relation betweencritical fields and background fields. ∗ Research supported in part by the Natural Sciences and Engineering Research Council ofCanada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich. lattice X − of the form Z Q x ∈X − dφ ∗ ( x ) dφ ( x )2 πi e A ( α , ··· ,α s ; φ ∗ ,φ ) (1)with an action A ( α , · · · , α s ; φ ∗ , φ ) that is a function of external complex valued fields α , · · · , α s , and the two complex fields φ ∗ , φ on X − . This scenario occurs in [4, 5],where we use block spin renormalization group maps to exhibit the formation of apotential well, signalling the onset of symmetry breaking in a many particle systemof weakly interacting Bosons in three space dimensions. (For an overview, see [3].)For simplicity, we suppress the external fields in this paper.Under the renormalization group approach to controlling integrals like (1) onesuccessively “integrates out” lower and lower energy degrees of freedom. In theblock spin formalism this is implemented by considering a decreasing sequence ofsublattices of X − . The formalism produces, for each such sublattice, a representationof the integral (1) that is a functional integral whose integration variables are indexedby that sublattice. To pass from the representation associated with one sublattice X ⊂ X − , with integration variables ψ ( x ), x ∈ X , to the representation associated tothe next coarser sublattice X + ⊂ X , with integration variables θ ( y ), y ∈ X + , one • paves X by rectangles centered at the points of X + (this is illustrated in the figurebelow — the dots, both small and large, are the points of X and the large dotsare the points of X + ) and then, • for each y ∈ X + integrates out all values of ψ whose “average value” over therectangle centered at y is equal to θ ( y ). The precise “average value” used isdetermined by an averaging profile. One uses this profile to define an averagingoperator Q from the space H of fields on X to the space H + of fields on X + . Onethen implements the “integrating out” by first, inserting, into the integrand, 1expressed as a constant times the Gaussian integral Z Q y ∈X + dθ ∗ ( y ) dθ ( y )2 πi e − b h θ ∗ − Q ψ ∗ , θ − Q ψ ) i (2)with some constant b >
0, and then interchanging the order of the θ and ψ integrals.For example, in [3, 4, 5] the model is initially formulated as a functional integralwith integration variables indexed by a lattice (cid:0) Z /L tp Z (cid:1) × (cid:0) Z /L sp Z (cid:1) . After n Usually, the finite lattice is a “volume cutoff” infinite lattice and one wants to get bounds thatare uniform in the size of the volume cutoff. In the actions, we treat φ and its complex conjugate φ ∗ as independent variables. The volume cutoff is determined by L tp and L sp . X and X + renormalization group steps this lattice is scaled down to X n = (cid:0) L n Z (cid:14) L tp L n Z (cid:1) × (cid:0) L n Z (cid:14) L sp L n Z (cid:1) . The decreasing family of sublattices is X ( n − j ) j = (cid:0) L j Z (cid:14) L tp L n Z (cid:1) × (cid:0) L j Z (cid:14) L sp L n Z (cid:1) , j = n , n − · · · . The abstract lattices X − , X , X + in the aboveframework correspond to X n , X ( n )0 and X ( n +1) − , respectively.Return to the abstract setting. The integral is often controlled using stationaryphase/steepest descent. The contributions to the integral that come from integrationvariables close to their critical values are called “small field” contributions. At theend of every step, the small field contribution to the original integral (1) is, up to amultiplicative normalization constant , of the form Z Q x ∈X dψ ∗ ( x ) dψ ( x )2 πi e −h ψ ∗ − Q − φ ∗ , Q ( ψ − Q − φ ) i− A ( φ ∗ ,φ )+ E ( ψ ∗ ,ψ ) (cid:12)(cid:12)(cid:12)(cid:12) φ ∗ = φ ∗ bg( ψ ∗ ,ψ ) φ = φ bg( ψ ∗ ,ψ ) (3)where • Q − is an averaging operator that maps the space H − of fields on X − to the space H of fields on X . It is the composition of the averaging operations for all previoussteps. • the exponent h ψ ∗ − Q − φ ∗ , Q ( ψ − Q − φ ) i is a residue of the exponents in theGaussian integrals (2) inserted in the previous steps. The operator Q is boundedand boundedly invertible on L ( X ). See Remark 1 for the core of the recursion responsible for this form. See Remark 1 for the recursion relation that builds Q . the “background fields”( ψ ∗ , ψ ) φ ∗ bg ( ψ ∗ , ψ ) ( ψ ∗ , ψ ) φ bg ( ψ ∗ , ψ )map sufficiently small fields ψ ∗ , ψ on X to fields on X − . They are the concatina-tion of “steepest descent” critical field maps for all previous steps. • A ( φ ∗ , φ ) , the “dominant part” of the action, is an explicit function of φ ∗ , φ ∈ H − • E ( ψ ∗ , ψ ) is the contribution to the action that consists of “perturbative correc-tions”. It is an analytic function of ψ ∗ , ψ ∈ H .The next block spin renormalization group step then consists of • rewriting (3), by inserting 1 expressed as a constant times (2), as Z Q y ∈X + dθ ∗ ( y ) dθ ( y )2 πi Z Q x ∈X dψ ∗ ( x ) dψ ( x )2 πi e − b h θ ∗ − Q ψ ∗ , θ − Q ψ i e −h ψ ∗ − Q − φ ∗ , Q ( ψ − Q − φ ) i− A ( φ ∗ ,φ )+ E ( ψ ∗ ,ψ ) (cid:12)(cid:12)(cid:12)(cid:12) φ ∗ = φ ∗ bg( ψ ∗ ,ψ ) φ = φ bg( ψ ∗ ,ψ ) (4)up to a multiplicative normalization constant, • and performing a stationary phase argument, for the ψ integral, around appropri-ate critical fields ψ ∗ cr ( θ ∗ , θ ), ψ cr ( θ ∗ , θ ) that map sufficiently small fields θ ∗ , θ on X + to fields on X .In this paper, we discuss some purely algebraic aspects of the block spin renormal-ization group in an abstract setting. We derive some “well known” identities like, inProposition 4.c, the composition rule, and, in Proposition 4.a, the relation betweencritical fields and background fields, and, in Lemma 12, a formula for the dominantpart of the action in the fluctuation integral. They are used in Proposition 3.4.b,Proposition 3.4.a, and Lemma 4.1.a of [4], respectively.We use the following abstract environment: • Let H − , H , H + be finite dimensional, real vector spaces with positive definitesymmetric bilinear forms h · , · i − , h · , · i , h · , · i + . These bilinear forms extend tonondegenerate bilinear forms on their complexifications H − , H , H + . Think of H − , H and H + as being the vector spaces of real valued functions on the finite lattices X − , X and X + , respectively, and think of the complexifications H − , H , H + asbeing L ( X − ), L ( X ) and L ( X + ) respectively. • Let dµ H ( φ ∗ , φ ) be the volume form on H determined by its bilinear form. If H = L ( X ), then dµ H ( φ ∗ , φ ) = Q x ∈X dφ ( x ) ∗ ∧ dφ ( x )2 πı . See Proposition 4.c for the recursion relation that builds φ ( ∗ )bg . ψ ∗ cr ( θ ∗ , θ ) and ψ cr ( θ ∗ , θ ) need not be complex conjugates of each other Let Q − : H − → H Q : H → H + be linear maps. They induce C linear maps between H − , H , H + which are denotedby the same letter. We set ˇ Q − = Q ◦ Q − • Fix b > Q , on H . • Let A be a polynomial on H − × H − .Set, for φ ∗ , φ ∈ H − , ψ ∗ , ψ ∈ H and θ ∗ , θ ∈ H + A ( ψ ∗ , ψ ; φ ∗ , φ ) = h ψ ∗ − Q − φ ∗ , Q ( ψ − Q − φ ) i + A ( φ ∗ , φ ) A eff ( θ ∗ , θ ; ψ ∗ , ψ ; φ ∗ , φ ) = b h θ ∗ − Qψ ∗ , θ − Qψ i + + A ( ψ ∗ , ψ ; φ ∗ , φ )ˇ A ( θ ∗ , θ ; φ ∗ , φ ) = (cid:10) θ ∗ − ˇ Q − φ ∗ , ˇ Q (cid:0) θ − ˇ Q − φ (cid:1)(cid:11) + + A ( φ ∗ , φ )where ˇ Q = (cid:0) b H + + Q Q − Q ∗ (cid:1) − (5) Remark 1.
In this setting, the action of the functional integral (3) that appears atthe beginning of the renormalization group step is − h ψ ∗ − Q − φ ∗ , Q ( ψ − Q − φ ) i − A ( φ ∗ , φ ) + E ( ψ ∗ , ψ ) = −A ( ψ ∗ , ψ ; φ ∗ , φ ) + E ( ψ ∗ , ψ )and the action of the functional integral (4) that appears in the middle of the renor-malization group step is − b h θ ∗ − Q ψ ∗ , θ − Q ψ i + − h ψ ∗ − Q − φ ∗ , Q ( ψ − Q − φ ) i − A ( φ ∗ , φ ) + E ( ψ ∗ , ψ )= −A eff ( θ ∗ , θ ; ψ ∗ , ψ ; φ ∗ , φ ) + E ( ψ ∗ , ψ )We show in Proposition 4.b, below, that when one substitutes the critical ψ into A eff one gets ˇ A . Upon scaling (and renormalizing) ˇ A becomes the A for the beginningof the next renormalization group step. Equation (5) is the recursion relation thatbuilds the operator Q in A ( ψ ∗ , ψ ; φ ∗ , φ ). Remark 2. ˇ Q = b (cid:2) H + − bQ (cid:0) bQ ∗ Q + Q (cid:1) − Q ∗ (cid:3) Proof.
Apply Lemma 13 with V = H , W = H + , q = Q , q ∗ = Q ∗ , f = Q and g = b W . 5 efinition 3. (a) Let N be a domain in H which is invariant under complex conjugation. “Back-ground fields on N ” are maps φ ∗ bg , φ bg : N × N → H − such that, for each( ψ ∗ , ψ ) ∈ N × N , the point (cid:0) φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ) (cid:1) is a critical point of themap ( φ ∗ , φ )
7→ A ( ψ ∗ , ψ ; φ ∗ , φ )That is, it solves Q ∗− Q Q − φ ∗ + ∇ φ A ( φ ∗ , φ ) = Q ∗− Q ψ ∗ Q ∗− Q Q − φ + ∇ φ ∗ A ( φ, φ ) = Q ∗− Q ψ (6)“Formal background fields” are formal power series φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ), in( ψ ∗ , ψ ) with vanishing constant terms, that solve (6).(b) Let N + and N be domains in H + and H , respectively, which are invariantunder complex conjugation. Let φ ∗ bg , φ bg be background fields on N . “Criticalfields on N + with respect to φ ∗ bg , φ bg ” are maps ψ ∗ cr , ψ cr : N + × N + → N suchthat, for each ( θ ∗ , θ ) ∈ N + × N + , the point (cid:0) ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) (cid:1) is a criticalpoint for the map( ψ ∗ , ψ )
7→ A eff ( θ ∗ , θ ; ψ ∗ , ψ ; φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ))That is, it solves ( bQ ∗ Q + Q ) ψ ∗ = bQ ∗ θ ∗ + Q Q − φ ∗ bg ( ψ ∗ , ψ )( bQ ∗ Q + Q ) ψ = bQ ∗ θ + Q Q − φ bg ( ψ ∗ , ψ ) (7)If φ ∗ bg , φ bg are formal background fields, then “formal critical fields with re-spect to φ ∗ bg , φ bg ” are formal power series ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ), in ( θ ∗ , θ ) withvanishing constant terms, that solve (7).(c) Let N + be a domain in H + which is invariant under complex conjugation. “Nextscale background fields on N + ” are maps ˇ φ ∗ bg , ˇ φ bg : N + × N + → H − such that,for each ( θ ∗ , θ ) ∈ N + × N + , the point (cid:0) ˇ φ ∗ bg ( θ ∗ , θ ) , ˇ φ bg ( θ ∗ , θ ) (cid:1) is a critical pointof the map ( φ ∗ , φ ) ˇ A ( θ ∗ , θ ; φ ∗ , φ )That is, it solves ˇ Q ∗− ˇ Q ˇ Q − ˇ φ ∗ + ∇ ˇ φ A ( ˇ φ ∗ , ˇ φ ) = ˇ Q ∗− ˇ Q θ ∗ ˇ Q ∗− ˇ Q ˇ Q − ˇ φ + ∇ ˇ φ ∗ A ( ˇ φ ∗ , ˇ φ ) = ˇ Q ∗− ˇ Q θ (8)6ormal power series ˇ φ ∗ bg ( θ ∗ , θ ), ˇ φ bg ( θ ∗ , θ ), in ( θ ∗ , θ ) with vanishing constantterms, that solve (8) are called “formal next scale background fields”. Proposition 4.
Let N + and N be domains in H + and H , respectively, which areinvariant under complex conjugation. Let φ ∗ bg , φ bg be background fields on N and ψ ∗ cr , ψ cr be critical fields on N + with respect to φ ∗ bg , φ bg . Define the composition ˇ φ ∗ cp ( θ ∗ , θ ) = φ ∗ bg (cid:0) ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) (cid:1) ˇ φ cp ( θ ∗ , θ ) = φ bg (cid:0) ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) (cid:1) (9) Then, for all ( θ ∗ , θ ) ∈ N + × N + ,(a) (cid:0) ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) (cid:1) fulfils the equations ψ ∗ cr ( θ ∗ , θ ) = ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ ∗ + Q Q − ˇ φ ∗ cp ( θ ∗ , θ ) (cid:1) ψ cr ( θ ∗ , θ ) = ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ + Q Q − ˇ φ cp ( θ ∗ , θ ) (cid:1) (b) The effective action A eff (cid:0) θ ∗ , θ ; ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ); ˇ φ ∗ cp ( θ ∗ , θ ) , ˇ φ cp ( θ ∗ , θ ) (cid:1) = ˇ A ( θ ∗ , θ ; ˇ φ ∗ cp ( θ ∗ , θ ) , ˇ φ cp ( θ ∗ , θ )) (c) ˇ φ ∗ cp ( θ ∗ , θ ) , ˇ φ cp ( θ ∗ , θ ) are next scale background fields on N + .(d) For any continuous function E ( ψ ∗ , ψ ) on N × N Z N ×N dµ H ( ψ ∗ , ψ ) e −A ( ψ ∗ ,ψ ; φ ∗ bg ( ψ ∗ ,ψ ) ,φ bg ( ψ ∗ ,ψ ))+ E ( ψ ∗ ,ψ ) = b dim H + (cid:26) Z N + ×N + dµ H + ( θ ∗ , θ ) e − ˇ A ( θ ∗ ,θ ; ˇ φ ∗ cp ( θ ∗ ,θ ) , ˇ φ cp ( θ ∗ ,θ )) e E ( ψ ∗ cr ( θ ∗ ,θ ) ,ψ cr ( θ ∗ ,θ )) F ( θ ∗ , θ )+ Z ( H + ×H + ) \ ( N + ×N + ) dµ H + ( θ ∗ , θ ) Z N ×N dµ H ( ψ ∗ , ψ ) e −A eff ( θ ∗ ,θ ; ψ ∗ ,ψ ; φ ∗ bg ( ψ ∗ ,ψ ) ,φ bg ( ψ ∗ ,ψ ))+ E ( ψ ∗ ,ψ ) (cid:27) where the fluctuation integral F ( θ ∗ , θ ) = Z D ( θ ∗ ,θ ) dµ H ( δψ ∗ , δψ ) e − δ A ( θ ∗ ,θ ; δψ ∗ ,δψ ) e δ E ( θ ∗ ,θ ; δψ ∗ ,δψ ) ere the functions δ A and δ E are given by δ A ( θ ∗ , θ ; δψ ∗ , δψ ) = A eff (cid:0) θ ∗ , θ ; ψ ∗ , ψ ; φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ) (cid:1)(cid:12)(cid:12)(cid:12) ψ ∗ = ψ ∗ cr + δψ ∗ , ψ = ψ cr + δψψ ∗ = ψ ∗ cr , ψ = ψ cr δ E ( θ ∗ , θ ; δψ ∗ , δψ ) = E (cid:0) ψ ∗ , ψ (cid:1)(cid:12)(cid:12)(cid:12) ψ ∗ = ψ ∗ cr + δψ ∗ , ψ = ψ cr + δψψ ∗ = ψ ∗ cr , ψ = ψ cr with ψ ∗ cr = ψ ∗ cr ( θ ∗ , θ ) , ψ cr = ψ cr ( θ ∗ , θ ) , and the domain D ( θ ∗ , θ ) = (cid:8) ( δψ ∗ , δψ ) ∈ H × H (cid:12)(cid:12) ψ ∗ cr ( θ ∗ , θ ) + δψ ∗ = (cid:0) ψ cr ( θ ∗ , θ ) + δψ (cid:1) ∗ ∈ N (cid:9) The formal power series versions of parts (a), (b) and (c) of Proposition 4 are
Proposition 4’.
Let φ ∗ bg , φ bg be formal background fields and ψ ∗ cr , ψ cr be formalcritical fields with respect to φ ∗ bg , φ bg . Set ˇ φ ( ∗ )cp ( θ ∗ , θ ) = φ ( ∗ )bg (cid:0) ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) (cid:1) (9’) (a) (cid:0) ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) (cid:1) fulfils the equations ψ ( ∗ )cr ( θ ∗ , θ ) = ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ ( ∗ ) + Q Q − ˇ φ ( ∗ )cp ( θ ∗ , θ ) (cid:1) (b) The effective action A eff (cid:0) θ ∗ , θ ; ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ); ˇ φ ∗ cp ( θ ∗ , θ ) , ˇ φ cp ( θ ∗ , θ ) (cid:1) = ˇ A ( θ ∗ , θ ; ˇ φ ∗ cp ( θ ∗ , θ ) , ˇ φ cp ( θ ∗ , θ )) (c) ˇ φ ∗ cp ( θ ∗ , θ ) , ˇ φ cp ( θ ∗ , θ ) are formal next scale background fields. The proof of these Propositions will be given after Lemma 7.
Remark 5. (a) Part (c) of the Proposition is often called the “composition rule”.(b) In applications, the domain N + is chosen so that the second integral on theright hand side of the formula in part (d) is small. In that integral either θ or θ ∗ is bounded away from the origin (“large fields”). We routinely use the “optional ∗ ” notation α ( ∗ ) to denote “ α ∗ or α ”. The equation “ α ( ∗ ) = β ( ∗ ) ”means “ α ∗ = β ∗ and α = β ”. φ ∗ bg , φ bg be formal background fields and ψ ∗ cr , ψ cr beformal critical fields with respect to φ ∗ bg , φ bg . Assume, in addition, that theequations (8), for the next scale background fields, have a unique formal powerseries solution, that we denote ˇ φ ∗ bg , ˇ φ bg . Then by part (c) of Proposition 4’,ˇ φ ( ∗ )bg ( θ ∗ , θ ) = ˇ φ ( ∗ )cp ( θ ∗ , θ ) and, by part (a) of Proposition 4’, ψ ( ∗ )cr ( θ ∗ , θ ) = ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ ( ∗ ) + Q Q − ˇ φ ( ∗ )bg ( θ ∗ , θ ) (cid:1) If, in addition, ˇ φ ( ∗ )bg ( θ ∗ , θ ) are analytic functions on some domain, then so are ψ ( ∗ )cr ( θ ∗ , θ ). So to construct analytical critical fields, it suffices to have • uniqueness of formal power series solutions to the next scale background fieldequations • existence of analytic solutions to the next scale background field equations • formal background fields • formal critical fields with respect to the formal background fieldsLemma 6, below, provides existence and uniqueness for formal power seriessolutions of the critical field equations. Lemma 6.
Let φ ∗ bg , φ bg be formal background fields of the form φ ( ∗ )bg ( ψ ∗ , ψ ) = L ( ∗ ) ψ ( ∗ ) + φ ( ≥ ∗ )bg ( ψ ∗ , ψ ) with φ ( ≥ ∗ )bg ( ψ ∗ , ψ ) being of degree at least two in ( ψ ∗ , ψ ) and with the L ( ∗ ) ’s beinglinear operators. If the linear operators bQ ∗ Q + Q − Q Q − L ( ∗ ) are invertible, thenthere exist unique formal critical fields with respect to φ ∗ bg , φ bg .Proof. Rewrite the equations (7) in the form( bQ ∗ Q + Q − Q Q − L ∗ ) ψ ∗ = bQ ∗ θ ∗ + Q Q − φ ( ≥ ∗ bg ( ψ ∗ , ψ )( bQ ∗ Q + Q − Q Q − L ) ψ = bQ ∗ θ + Q Q − φ ( ≥ ( ψ ∗ , ψ )As ψ ∗ and ψ are to have vanishing constant terms, this provides a “lower triangular”recursion relation for the coefficients of ( ψ ∗ , ψ ). As H and H + are finite dimensional,this recursion relation trivially generates a unique solution.The proof of Proposition 4 is based on By this we mean that each nonzero monomial in φ ( ≥ ∗ )bg has degree at least two. emma 7. For φ ∗ , φ ∈ H − and θ ∗ , θ ∈ H + set ˜ ψ ( ∗ ) ( θ ( ∗ ) , φ ( ∗ ) ) = ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ ( ∗ ) + Q Q − φ ( ∗ ) (cid:1) Then ˇ A (cid:0) θ ∗ , θ ; φ ∗ , φ (cid:1) = A eff (cid:0) θ ∗ , θ ; ˜ ψ ∗ ( θ ∗ , φ ∗ ) , ˜ ψ ( θ, φ ); φ ∗ , φ (cid:1) and ( ∇ φ ( ∗ ) ˇ A )( θ ∗ , θ ; φ ∗ , φ )= ( ∇ φ ( ∗ ) A ) (cid:0) ˜ ψ ∗ ( θ ∗ , φ ∗ ) , ˜ ψ ( θ, φ ); φ ∗ , φ (cid:1) + Q ∗− Q ( bQ ∗ Q + Q ) − (cid:2) ( ∇ ψ ( ∗ ) A eff ) (cid:0) θ ∗ , θ ; ˜ ψ ∗ ( θ ∗ , φ ∗ ) , ˜ ψ ( θ, φ ); φ ∗ , φ (cid:1)(cid:3) (10) Proof.
With the abbreviation ˜ ψ ( ∗ ) = ˜ ψ ( ∗ ) ( θ ( ∗ ) , φ ( ∗ ) ) θ − Q ˜ ψ = θ − Q ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ + Q Q − φ (cid:1) = (cid:2) − bQ ( bQ ∗ Q + Q ) − Q ∗ (cid:3) θ − ˇ Q − φ + QQ − φ − Q ( bQ ∗ Q + Q ) − Q Q − φ = (cid:2) − bQ ( bQ ∗ Q + Q ) − Q ∗ (cid:3) θ − ˇ Q − φ + Q ( bQ ∗ Q + Q ) − (cid:2) ( bQ ∗ Q + Q ) − Q (cid:3) Q − φ = (cid:2) − bQ ( bQ ∗ Q + Q ) − Q ∗ (cid:3)(cid:0) θ − ˇ Q − φ (cid:1) ˜ ψ − Q − φ = ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ + Q Q − φ (cid:1) − Q − φ = ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ + Q Q − φ − bQ ∗ QQ − φ − Q Q − φ (cid:1) = b ( bQ ∗ Q + Q ) − Q ∗ (cid:0) θ − ˇ Q − φ (cid:1) Thereforeˇ A (cid:0) θ ∗ , θ ; φ ∗ , φ (cid:1) − A eff (cid:0) θ ∗ , θ ; ˜ ψ ∗ , ˜ ψ ; φ ∗ , φ (cid:1) = (cid:10) θ ∗ − ˇ Q − φ ∗ , ˇ Q (cid:0) θ − ˇ Q − φ (cid:1)(cid:11) + − b (cid:10) θ ∗ − Q ˜ ψ ∗ , θ − Q ˜ ψ (cid:11) + − (cid:10) ˜ ψ ∗ − Q − φ ∗ , Q ( ˜ ψ − Q − φ ) (cid:11) = b (cid:10) θ ∗ − ˇ Q − φ ∗ , O (cid:0) θ − ˇ Q − φ (cid:1)(cid:11) + where, by Remark 2, O = (cid:2) − bQ (cid:0) bQ ∗ Q + Q (cid:1) − Q ∗ (cid:3) − (cid:2) − bQ ( bQ ∗ Q + Q ) − Q ∗ (cid:3) − bQ ( bQ ∗ Q + Q ) − Q ( bQ ∗ Q + Q ) − Q ∗ = b (cid:2) − bQ (cid:0) bQ ∗ Q + Q (cid:1) − Q ∗ (cid:3) Q ( bQ ∗ Q + Q ) − Q ∗ − bQ ( bQ ∗ Q + Q ) − Q ( bQ ∗ Q + Q ) − Q ∗ = bQ (cid:2) − (cid:0) bQ ∗ Q + Q (cid:1) − bQ ∗ Q − ( bQ ∗ Q + Q ) − Q (cid:3) ( bQ ∗ Q + Q ) − Q ∗ = 0 10his proves the first statement. The second follows by the chain rule and the obser-vation that ∇ φ ( ∗ ) A eff = ∇ φ ( ∗ ) A . Proof of Propositions 4 and 4’.
The proof of Proposition 4’ is virtually identical tothat of Proposition 4.a,b,c, so we just give the proof of Proposition 4. Part (a)follows immediately from (7) and (9). Now evaluate the conclusions of Lemma 7at φ ( ∗ ) = ˇ φ ( ∗ )cp ( θ ∗ , θ ) (cid:1) . The formula for ˇ A in Lemma 7 directly gives part (b).The right hand side of (10) vanishes upon this evaluation by parts (a) and (b)of Definition 3. This shows that (cid:0) ˇ φ ∗ cp ( θ ∗ , θ ) , ˇ φ cp ( θ ∗ , θ ) (cid:1) is critical for the map( φ ∗ , φ ) ˇ A (cid:0) θ ∗ , θ ; φ ∗ , φ (cid:1) , which proves part (c). Now b − dim H + Z N ×N dµ H ( ψ ∗ , ψ ) e −A ( ψ ∗ ,ψ ; φ ∗ bg ( ψ ∗ ,ψ ) ,φ bg ( ψ ∗ ,ψ ))+ E ( ψ ∗ ,ψ ) = Z dµ H + ( θ ∗ , θ ) Z N ×N dµ H ( ψ ∗ , ψ ) e − b h θ ∗ − Qψ ∗ , θ − Qψ i + −A ( ψ ∗ ,ψ ; φ ∗ bg ( ψ ∗ ,ψ ) ,φ bg ( ψ ∗ ,ψ )) + E ( ψ ∗ ,ψ ) = Z dµ H + ( θ ∗ , θ ) Z N ×N dµ H ( ψ ∗ , ψ ) e −A eff ( θ ∗ ,θ ; ψ ∗ ,ψ ; φ ∗ bg ( ψ ∗ ,ψ ) ,φ bg ( ψ ∗ ,ψ ))+ E ( ψ ∗ ,ψ ) = Z N + ×N + dµ H + ( θ ∗ , θ ) Z N ×N dµ H ( ψ ∗ , ψ ) e −A eff ( θ ∗ ,θ ; ψ ∗ ,ψ ; φ ∗ bg ( ψ ∗ ,ψ ) ,φ bg ( ψ ∗ ,ψ ))+ E ( ψ ∗ ,ψ ) + Z H + ×H + \N + ×N + dµ H + ( θ ∗ , θ ) Z N ×N dµ H ( ψ ∗ , ψ ) e −A eff ( θ ∗ ,θ ; ψ ∗ ,ψ ; φ ∗ bg ( ψ ∗ ,ψ ) ,φ bg ( ψ ∗ ,ψ )) + E ( ψ ∗ ,ψ ) Making the change of variables ψ ∗ = ψ ∗ cr ( θ ∗ , θ ) + δψ ∗ , ψ = ψ cr ( θ ∗ , θ ) + δψ in theinner integral of the upper line and applying part (b) gives part (d).From now on we assume that the function A ( φ ∗ , φ ) in the definitions of A and ˇ A is of the form A ( φ ∗ , φ ) = h φ ∗ , Dφ i − + P ( φ ∗ , φ ) (11)where • P is a polynomial whose nonzero monomials are each of degree at least two and • D a linear operator on H − such that both the operators ( D + Q ∗− Q Q − ) and( D + ˇ Q ∗− ˇ Q ˇ Q − ) are invertible. We define the “Green’s functions” S = ( D + Q ∗− Q Q − ) − ˇ S = ( D + ˇ Q ∗− ˇ Q ˇ Q − ) − (12)We think of D as a differential operator, possibly shifted by a chemical potential. Remark 8.
In this setting, the background field equations (6) become φ ( ∗ ) = S ( ∗ ) Q ∗− Q ψ ( ∗ ) − S ( ∗ ) P ′ ( ∗ ) ( φ ∗ , φ ) (6’)11here P ′∗ ( φ ∗ , φ ) = ∇ φ P ( φ ∗ , φ ) and P ′ ( φ ∗ , φ ) = ∇ φ ∗ P ( φ ∗ , φ ). Similarly, the next scalebackground field equations (8) becomeˇ φ ( ∗ ) = ˇ S ( ∗ ) ˇ Q ∗− ˇ Q θ ( ∗ ) − ˇ S ( ∗ ) P ′ ( ∗ ) ( ˇ φ ∗ , ˇ φ ) (8’)We now continue with our study of the critical field, following the plan of Remark5.c. To describe the leading part of the critical field, we set∆ = Q − Q Q − SQ ∗− Q : H −→ H (13)From now on we assume that ∆ + bQ ∗ Q is invertible and define the “covariance” C = (∆ + bQ ∗ Q ) − : H −→ H (14)
Proposition 9.
Assume that in the setting (11) , each nonzero monomial of P isof degree at least three. Then there exist unique formal background fields φ ( ∗ )bg andunique formal next scale background fields ˇ φ ( ∗ )bg . They are of the form φ ( ∗ )bg ( ψ ∗ , ψ ) = S ( ∗ ) Q ∗− Q ψ ( ∗ ) + φ ( ≥ ∗ )bg ( ψ ∗ , ψ )ˇ φ ( ∗ )bg ( θ ∗ , θ ) = ˇ S ( ∗ ) ˇ Q ∗− ˇ Q θ ( ∗ ) + ˇ φ ( ≥ ∗ )bg ( θ ∗ , θ ) with φ ( ≥ ∗ )bg ( ψ ∗ , ψ ) and ˇ φ ( ≥ ∗ )bg ( θ ∗ , θ ) being of degree at least two. Furthermore, thereare unique formal critical fields with respect to φ ( ∗ )bg . They are of the form ψ ( ∗ )cr ( θ ∗ , θ ) = ( bQ ∗ Q + Q ) − (cid:0) bQ ∗ θ ( ∗ ) + Q Q − ˇ φ ( ∗ )bg ( θ ∗ , θ ) (cid:1) = bC ( ∗ ) Q ∗ θ ( ∗ ) + ψ ( ≥ ∗ )cr ( θ ∗ , θ ) (cid:1) with ψ ( ≥ ∗ )cr being of degree at least two.Proof. The existence, uniqueness and forms of the formal background and next scalebackground fields are proven as Lemma 6 was proven. The existence and uniquenessof the formal critical field now follows from Lemma 6. The first representation ofthe critical fields follows from parts (a) and (c) of Proposition 4’. For the secondrepresentation, rewrite the equations (7) as( bQ ∗ Q + Q ) ψ ( ∗ ) = bQ ∗ θ ( ∗ ) + Q Q − S ( ∗ ) Q ∗− Q ψ ( ∗ ) + Q Q − φ ( ≥ ∗ )bg ( ψ ∗ , ψ )or ψ ( ∗ ) = bC ( ∗ ) Q ∗ θ ( ∗ ) + C ( ∗ ) Q Q − φ ( ≥ ∗ )bg ( ψ ∗ , ψ ) We shall show, in Lemma 12, below, that C is the covariance for the fluctuation integral. ψ cr , given in Proposition 9, combinedwith the representation of ˇ φ bg , suggest a formula for bCQ ∗ . In Remark 10, below,we give an algebraic proof of this formula, together with a number of representationsfor the Green’s functions, S and ˇ S , and covariance C . Then, in Lemma 12 below,we analyze the fluctuation integral of Proposition 4.d in more detail. Remark 10.
Assume that D is invertible.(a) ∆ = (cid:0) H + Q Q − D − Q ∗− (cid:1) − Q = Q (cid:0) H + Q − D − Q ∗− Q (cid:1) − (b) Let R : H − → H and R ∗ : H → H − be linear maps such that R D − R ∗ = Q − D − Q ∗− and such that D + R ∗ Q R is invertible. Then[ D + R ∗ Q R ] − = D − − D − R ∗ ∆ R D − In particular S = D − − D − Q ∗− ∆ Q − D − (c) ˇ S = (cid:2) S − − Q ∗− Q ( Q + bQ ∗ Q ) − Q Q − (cid:3) − = S + SQ ∗− Q C Q Q − S (d) C = (cid:0) bQ ∗ Q + Q (cid:1) − + ( bQ ∗ Q + Q ) − Q Q − ˇ SQ ∗− Q ( bQ ∗ Q + Q ) − (e) bC ( ∗ ) Q ∗ = (cid:0) bQ ∗ Q + Q (cid:1) − h bQ ∗ + Q Q − ˇ S ( ∗ ) ˇ Q ∗− ˇ Q i Proof. (a) By Lemma 13, with V = H − , W = H , q = Q − , q ∗ = Q ∗− , f = D and g = Q (cid:8)
1l + Q Q − D − Q ∗− (cid:9) − Q = (cid:8) − Q Q − ( D + Q ∗− Q Q − ) − Q ∗− (cid:9) Q = ∆ Q (cid:8)
1l + Q − D − Q ∗− Q (cid:9) − = Q (cid:8) − Q − ( D + Q ∗− Q Q − ) − Q ∗− Q (cid:9) = ∆(b) By part (a) (cid:2) D + R ∗ Q R (cid:3) (cid:2) D − − D − R ∗ ∆ R D − (cid:3) = 1l + R ∗ (cid:2) Q − (1l + Q R D − R ∗ ) ∆ (cid:3) R D − = 1l + R ∗ (cid:2) Q − (1l + Q Q − D − Q ∗− ) ∆ (cid:3) R D − = 1l(c) By Remark 2 Q ∗ ˇ Q Q = bQ ∗ Q (cid:2) − ( bQ ∗ Q + Q ) − bQ ∗ Q (cid:3) = bQ ∗ Q (cid:2) ( bQ ∗ Q + Q ) − ( bQ ∗ Q + Q ) − ( bQ ∗ Q + Q ) − bQ ∗ Q (cid:3) = ( Q + bQ ∗ Q − Q )( bQ ∗ Q + Q ) − Q = Q − Q ( Q + bQ ∗ Q ) − Q S − − ˇ S − = Q ∗− Q Q − − Q ∗− Q ∗ ˇ Q QQ − = Q ∗− Q ( Q + bQ ∗ Q ) − Q Q − which gives the first representation of ˇ S . For the proof of the second representation,first observe that, by (13) and (14), C − ( Q + bQ ∗ Q ) − = ( Q + bQ ∗ Q − Q Q − SQ ∗− Q )( Q + bQ ∗ Q ) − = 1l − Q Q − SQ ∗− Q ( Q + bQ ∗ Q ) − so that C = ( Q + bQ ∗ Q ) − (cid:8) − Q Q − SQ ∗− Q ( Q + bQ ∗ Q ) − (cid:9) − (15)Hence, by the first representation of ˇ S , (cid:2) S + SQ ∗− Q C Q Q − S (cid:3) ˇ S − − (cid:2)
1l + SQ ∗− Q C Q Q − (cid:3)(cid:2) − SQ ∗− Q ( Q + bQ ∗ Q ) − Q Q − (cid:3) − SQ ∗− Q h C (cid:8) − Q Q − SQ ∗− Q ( Q + bQ ∗ Q ) − (cid:9) − ( Q + bQ ∗ Q ) − i Q Q − = 0which implies the second representation of ˇ S .(d) By Lemma 13 with q = Q Q − , q ∗ = Q ∗− Q , f = S − and g = − ( Q + bQ ∗ Q ) − (cid:8) − Q Q − SQ ∗− Q ( Q + bQ ∗ Q ) − (cid:9) − = 1l + Q Q − (cid:2) S − − Q ∗− Q ( Q + bQ ∗ Q ) − Q Q − (cid:3) − Q ∗− Q ( Q + bQ ∗ Q ) − = 1l + Q Q − ˇ SQ ∗− Q ( Q + bQ ∗ Q ) − (16)The second equality follows by the first representation of ˇ S in part (c). Substituting(16) into (15) gives the desired representation of C .(e) By Remark 2 ˇ Q ∗− ˇ Q = bQ ∗− Q ∗ (cid:2) − bQ ( bQ ∗ Q + Q ) − Q ∗ (cid:3) = bQ ∗− (cid:2) − bQ ∗ Q ( bQ ∗ Q + Q ) − (cid:3) Q ∗ = bQ ∗− Q ( bQ ∗ Q + Q ) − Q ∗ bC ( ∗ ) Q ∗ = (cid:0) bQ ∗ Q + Q (cid:1) − (cid:2) bQ ∗ + b Q Q − ˇ S ( ∗ ) Q ∗− Q ( bQ ∗ Q + Q ) − Q ∗ (cid:3) = (cid:0) bQ ∗ Q + Q (cid:1) − h bQ ∗ + Q Q − ˇ S ( ∗ ) ˇ Q ∗− ˇ Q i Define, in the setting of Proposition 4, δφ ( ∗ )bg (cid:0) ψ ∗ , ψ, δψ ∗ , δψ (cid:1) by φ ( ∗ )bg (cid:0) ψ ∗ + δψ ∗ , ψ + δψ (cid:1) = φ ( ∗ )bg (cid:0) ψ ∗ , ψ (cid:1) + δφ ( ∗ )bg (cid:0) ψ ∗ , ψ, δψ ∗ , δψ (cid:1) (17.a)and set δ ˇ φ ( ∗ )bg (cid:0) θ ∗ , θ, δψ ∗ , δψ (cid:1) = δφ ( ∗ )bg (cid:0) ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) , δψ ∗ , δψ (cid:1) (17.b)With the ˇ φ ( ∗ )bg ( θ ∗ , θ ) of Proposition 4 and (9), φ ( ∗ )bg (cid:0) ψ ∗ cr ( θ ∗ , θ )+ δψ ∗ , ψ cr ( θ ∗ , θ )+ δψ (cid:1) = ˇ φ ( ∗ )bg ( θ ∗ , θ ) + δ ˇ φ ( ∗ )bg (cid:0) θ ∗ , θ ; δψ ∗ , δψ (cid:1) (18)Also define δ ˇ φ (+)( ∗ ) (cid:0) θ ∗ , θ ; δψ ∗ , δψ (cid:1) by δ ˇ φ ( ∗ )bg (cid:0) θ ∗ , θ ; δψ ∗ , δψ (cid:1) = S ( ∗ ) Q ∗− Q δψ ( ∗ ) + δ ˇ φ (+)( ∗ ) (cid:0) θ ∗ , θ ; δψ ∗ , δψ (cid:1) (19) Remark 11.
By Remark 8, the fields δ ˇ φ ( ∗ )bg (cid:0) θ ∗ , θ, δψ ∗ , δψ (cid:1) introduced in (17) obey δ ˇ φ ( ∗ )bg = S ( ∗ ) Q ∗− Q δψ ( ∗ ) − S ( ∗ ) P ′ ( ∗ ) ( φ ∗ , φ ) (cid:12)(cid:12)(cid:12) φ ( ∗ ) = ˇ φ ( ∗ )bg ( θ ∗ ,θ )+ δ ˇ φ ( ∗ )bg φ ( ∗ ) = ˇ φ ( ∗ )bg ( θ ∗ ,θ ) In particular, if P = 0, then δ ˇ φ ( ∗ )bg = S ( ∗ ) Q ∗− Q δψ ( ∗ ) . This is the motivation for thedefinition of δ ˇ φ (+)( ∗ ) in (19). Lemma 12.
The function δ A appearing in the exponent of the fluctuation integral F ( θ ∗ , θ ) of Proposition 4.d is δ A ( θ ∗ , θ ; δψ ∗ , δψ ) = (cid:10) δψ ∗ , C − δψ (cid:11) − Z dt (cid:10) δψ ∗ , Q Q − δ ˇ φ (+) (cid:0) θ ∗ , θ ; t δψ ∗ , t δψ (cid:1)(cid:11) − Z dt (cid:10) Q Q − δ ˇ φ (+) ∗ (cid:0) θ ∗ , θ ; t δψ ∗ , t δψ (cid:1) , δψ (cid:11) roof. Set B ( ψ ∗ , ψ ) = A (cid:0) ψ ∗ , ψ ; φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ) (cid:1) . As (cid:0) ∇ φ ∗ A (cid:1)(cid:0) ψ ∗ , ψ ; φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ) (cid:1) = (cid:0) ∇ φ A (cid:1)(cid:0) ψ ∗ , ψ ; φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ) (cid:1) = 0we have (cid:0) ∇ ψ ∗ B (cid:1)(cid:0) ψ ∗ , ψ (cid:1) = (cid:0) ∇ ψ ∗ A (cid:1)(cid:0) ψ ∗ , ψ ; φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ) (cid:1) = Q (cid:0) ψ − Q − φ bg ( ψ ∗ , ψ ) (cid:1)(cid:0) ∇ ψ B (cid:1)(cid:0) ψ ∗ , ψ (cid:1) = (cid:0) ∇ ψ A (cid:1)(cid:0) ψ ∗ , ψ ; φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ) (cid:1) = Q (cid:0) ψ ∗ − Q − φ ∗ bg ( ψ ∗ , ψ ) (cid:1) Therefore B ( ψ ∗ + δψ ∗ , ψ + δψ ) − B ( ψ ∗ , ψ )= Z dt h(cid:10) δψ ∗ , ( ∇ ψ ∗ B )( ψ ∗ + tδψ ∗ , ψ + tδψ ) (cid:11) + (cid:10) ( ∇ ψ B )( ψ ∗ + tδψ ∗ , ψ + tδψ ) , δψ (cid:11)i = Z dt (cid:10) δψ ∗ , Q ( ψ + tδψ ) − Q Q − φ bg ( ψ ∗ + tδψ ∗ , ψ + tδψ ) (cid:11) + Z dt (cid:10) Q ( ψ ∗ + tδψ ∗ ) − Q Q − φ ∗ bg ( ψ ∗ + tδψ ∗ , ψ + tδψ ) , δψ (cid:11) = (cid:10) δψ ∗ , Q δψ (cid:11) + (cid:10) δψ ∗ , Q ψ (cid:11) + (cid:10) ψ ∗ , Q δψ (cid:11) − I where I = Z dt (cid:10) δψ ∗ , Q Q − φ bg ( ψ ∗ cr + tδψ ∗ , ψ cr + tδψ ) (cid:11) + Z dt (cid:10) Q Q − φ ∗ bg ( ψ ∗ cr + tδψ ∗ , ψ cr + tδψ ) , δψ (cid:11) Since A eff (cid:0) θ ∗ , θ ; ψ ∗ , ψ ; φ ∗ bg ( ψ ∗ , ψ ) , φ bg ( ψ ∗ , ψ ) (cid:1) = b h θ ∗ − Qψ ∗ , θ − Qψ i + + B ( ψ ∗ , ψ )we get, using Proposition 4, δ A = b h Q δψ ∗ , Q δψ i + − b h Q δψ ∗ , θ − Qψ cr i + − b h θ ∗ − Qψ ∗ cr , Q δψ i + + (cid:10) δψ ∗ , Q δψ (cid:11) + (cid:10) δψ ∗ , Q ψ cr (cid:11) + (cid:10) ψ ∗ cr , Q δψ (cid:11) − I = h δψ ∗ , ( bQ ∗ Q + Q ) δψ i + h δψ ∗ , ( bQ ∗ Q + Q ) ψ cr − bQ ∗ θ i + h ( bQ ∗ Q + Q ) ψ ∗ cr − bQ ∗ θ ∗ , δψ i − I = h δψ ∗ , ( bQ ∗ Q + Q ) δψ i + (cid:10) δψ ∗ , Q Q − ˇ φ bg (cid:11) + (cid:10) Q Q − ˇ φ ∗ bg , δψ (cid:11) − I h δψ ∗ , ( bQ ∗ Q + Q ) δψ i − Z dt (cid:10) δψ ∗ , Q Q − (cid:2) φ bg ( ψ ∗ cr + tδψ ∗ , ψ cr + tδψ ) − ˇ φ bg (cid:3)(cid:11) − Z dt (cid:10) Q Q − (cid:2) φ ∗ bg ( ψ ∗ cr + tδψ ∗ , ψ cr + tδψ ) − ˇ φ ∗ bg (cid:3) , δψ (cid:11) = (cid:10) δψ ∗ , ( bQ ∗ Q + Q − Q Q − SQ ∗− Q ) δψ (cid:11) − Z dt (cid:10) δψ ∗ , Q Q − δ ˇ φ (+) (cid:0) θ ∗ , θ ; tδψ ∗ , tδψ (cid:1)(cid:11) − Z dt (cid:10) Q Q − δ ˇ φ (+) ∗ (cid:0) θ ∗ , θ ; tδψ ∗ , tδψ (cid:1) , δψ (cid:11) By the definition of C in (14), this is the desired representation.In the course of the arguments above the following simple algebraic observationwas used several times. Lemma 13.
Let V and W be vector spaces and let q : V → W , q ∗ : W → V , f : V → V and g : W → W be linear maps. Assume that f and f + q ∗ g q areinvertible. Then W + gq f − q ∗ and W + q f − q ∗ g are also invertible and (cid:0) W + gq f − q ∗ (cid:1) − = 1l W − gq ( f + q ∗ gq ) − q ∗ (cid:0) W + q f − q ∗ g (cid:1) − = 1l W − q ( f + q ∗ gq ) − q ∗ g Proof.
Replacing q by gq for the first line and q ∗ by q ∗ g for the second, we mayassume that g = 1l W . Write 1l W = 1l. Then (cid:0) − q ( f + q ∗ q ) − q ∗ (cid:1)(cid:0)
1l + qf − q ∗ (cid:1) = 1l + q (cid:2) − ( f + q ∗ q ) − f − ( f + q ∗ q ) − q ∗ q (cid:3) f − q ∗ = 1land similarly (cid:0)
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