aa r X i v : . [ m a t h - ph ] M a r Dynamical invariance for random matrices
J´er´emie Unterberger
We consider a general Langevin dynamics for the one-dimensional N-particle Coulomb gas withconfining potential V at temperature β . These dynamics describe for β = 2 the time evolutionof the eigenvalues of N × N random Hermitian matrices. The equilibrium partition function –equal to the normalization constant of the Laughlin wave function in fractional quantum Halleffect – is known to satisfy an infinite number of constraints called Virasoro or loop constraints.We introduce here a dynamical generating function on the space of random trajectories whichsatisfies a large class of constraints of geometric origin. We focus in this article on a subclassinduced by the invariance under the Schr¨odinger-Virasoro algebra. Keywords: random matrices, Coulomb gas, quantum Hall effect, Virasoro constraints, loopconstraints, Schr¨odinger-Virasoro algebra, dynamical invariance.
Mathematics Subject Classification (2010):
Contents N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 The N -particle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Appendix 42 β = 2 . . . . . . . . . . . . . . . 435.2 Solution of equation of motion when β = 2 . . . . . . . . . . . . . . . . . . . 435.3 A technical lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Time derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Let us start with a short preliminary discussion of the model ( § § § We consider the following Langevin dynamics [3, 4, 14] for N particles confined to a linewith positions { λ i } , i = 1 , . . . , Ndλ i = dB i − ∂W∂λ i dt = dB i + X j = i βλ i − λ j − V ′ ( λ i ) dt (0.1)where:(i) the noises ( B , . . . , B N ) = ( B ( t ) , . . . , B N ( t )) are N independent Brownian motions;(ii) W ( λ i ) = − β P i,j = i log | λ i − λ j | + P i V ( λ i ) is the sum of the electrostatic energy of asystem of N identically charged particles and of a one-body confining potential V .In our convention, d h B i , B i i t = 2 dt . Then the probability distribution function P ( λ i ; t )for the positions of the particles satisfies the Fokker-Planck equation, ∂ t P = ∆ P + X i ∂∂λ i (cid:18) ∂W∂λ i P (cid:19) = X i ∂ P ∂λ i − β X i,j = i ∂∂λ i (cid:18) P λ i − λ j (cid:19) + X i ∂∂λ i (cid:0) V ′ ( λ i ) P (cid:1) (0.2)(with the usual normalization of Brownian motion one would get ∆ in the above expres-sion).For V growing sufficiently fast at ∞ , the unique stationary measure is the Gibbs measure P eq ( { λ i } ) = 1 Z N ( V ) e − W ( λ i ) N ! dλ = 1 Z N ( V ) 1 N ! N Y i =1 e − V ( λ i ) Y i,j>i ( λ j − λ i ) β dλ (0.3)which may also be interpreted as a normalization constant for the celebrated fractionalquantum Hall effect Laughlin wave-function [9].For β = 2, the normalization constant is the partition function of the Hermitian ensemblewith potential V , Z N ( V ) = R d M e − Tr V ( M ) , for a suitable normalization of the measure d M on the space of N × N Hermitian matrices. In fact, the measure Z N ( V ) e − Tr V ( M ) d M projectsdown by conjugation invariance to a measure on the spectrum { λ i } of M which is none other2han P eq ( { λ i } ). Following Dyson [3] who originally introduced this model, we may considerour dynamics to be the projection to the spectrum of a conjugation invariant random walkon the space of Hermitian matrices, d M = d B − V ′ ( M ) dt , where by assumption the linearlyindependent entries (cid:16) { B ii ( t ) } i ; { Re B ij ( t ) } i 1) (0.11)and the associated free boson ˆ a ( z ) and energy-momentum tensor ˆ L ( z ),ˆ a ( z ) = X n ∈ Z ˆ a n z − n − , ˆ L ( z ) = 12 : (ˆ a ( z )) := X n ∈ Z ˆ L n z − n − . (0.12)Noting that ˆ L n = + ∞ X k =0 kτ k ∂∂τ k + n + β n X k =0 ∂∂τ k ∂∂τ n − k , n ≥ − L eqn Z [ τ ] = 0 (0.14)4ith L eqn = ˆ L n + β − / " + ∞ X k =0 b k ˆ a n + k +1 + ( β − n + 1)ˆ a n . (0.15)This two-line derivation has its interest, but the spirit of these constraints is really ofgeometric origin, see e.g. [1] or [10]: they reflect the way that the potential V is transformedunder generators of conformal transformations L n = − λ n +1 ∂ λ . The aim of the article is to prove the existence dynamical constraints in the same spirit asthe equilibrium Virasoro constraints discussed in the previous subsection.As pointed out just above, the conventional way to prove Virasoro constraints, see e.g. [1]or [10], is to consider the transformation of the equilibrium measure under a conformaltransformation of the eigenvalues, λ λ + ελ n +1 .In the dynamical case we miss a straightforward analogue of (i) conformal transformations;(ii) the equilibrium measure. Let us discuss these two points.(i) Our first claim is the following. The analogue of the group of conformal transforma-tions in the dynamical case is the group of noise-preserving transformations , brieflyintroduced in section 1 and discussed in full details in section 2.1, see in particu-lar Definition 2.2 for the Lie algebra of this group. The corresponding infinitesimaltransformations are ”causality-preserving” transformations of the set of trajectories { λ ( t ) , t ≥ } , with a condition called noise-invariance condition , see (1.7) or (2.10),ensuring that these preserve the strength of the noise for trajectories satisfying aLangevin equation. This group contains in particular as a subgroup the Schr¨odinger-Virasoro group , an infinite-dimensional group of coordinate transformations studiedin details in the book [13], see also [5]. Briefly said, these are coupled space- andtime-transformations which are affine in space, thus defining an infinite-dimensionalextension of the two Virasoro generators L − , . In 1D the Lie algebra is generated by X f := − f ( t ) ∂ t − 12 ˙ f ( t ) λ∂ λ , Y g := − g ( t ) ∂ λ . (0.16)While ( Y g ) is simply the time-current generated by L − , the ( X f ) are local space-time transformations generalizing the infinitesimal scaling transformation − t∂ t − λ∂ λ with dynamical exponent z = 2, which generates the parabolic scaling transformation( t, λ ) ( a t, aλ ). This scaling originates from the transformation properties of whitenoise. Noise-preserving infinitesimal transformations not belonging to the Schr¨odinger-Virasoro algebra may be seen as the sum of a very general transformation of the λ -coordinate, λ λ + εδλ , with δλ ( t ) = f ( λ ( t ))Φ( t, λ ), where Φ( t, · ) is some time-integrated functional of the past of the trajectory, and of a time-transform dependingon λ , which suggests to introduce the notion of a proper time (see section 3). Thoughwe are here in space dimension 1, the extension to d space-dimensions is more or lessstraightforward; using the conformal invariance of Brownian motion, it is enough torequire that the function f in factor in the λ -coordinate transform should define aconformal transformation. 5ii) Turning to the second point, it is not clear to us if there is a straightforward analogueof the equilibrium measure. Naively, the partition function should be replaced by themeasure on trajectories, which is automatically normalized, and thus cannot be usedas a generating functional. However, perturbing the measure in the way of Adler-VanMoerbeke (see previous subsection), one is led very naturally to an un-normalized per-turbed measure on the trajectories, Q lin [ τ ] = Q lin [ τ ]( { λ i } ) (see Definition 4.1) whoseintegral Z lin [ τ ] := R d Q lin [ τ ]( { λ i } ) may serve as generating functional . The upperindex ”lin” stands for ”linear”, since Q [ τ ] is obtained from the original measure onthe trajectories by linearizing in the τ -parameters and then throwing away quadraticterms produced by the two-body potentieal.Our main result is then Theorem 4.1, stating the invariance of the generating func-tional Z lin [ τ ] under Schr¨odinger-Virasoro transformations . The action of these trans-formations is similar in aspect to the action of Virasoro transformations on the par-tition in the equilibrium measure, see (0.15), with the considerable difference thoughthat we restricted ourselves to indices n = − , but on the other hand we have aninfinite number of constraints because of the arbitrary time-dependence. It exhibitsa sum of linear and of quadratic expressions in terms of a static free boson ˆ φ ( z, t ) –the free boson of usual conformal field theory, with an extra, trivial time-dependence– and of a dynamical free boson ˆ ψ ( z, t ) defined via a kernel K depending on V (seeDefinition 4.2, Definition 4.3 and Definition 4.4). The kernel K = K ( z − , w ), oneof the main ingredients in the computations, is the Green function of the operator D := ∂ t + ( β − d dz − ddz b ( z ) acting on formal series a + a z + a z + . . . Just as equilibrium Virasoro constraints may be used to compute the n -point functionsof the first few so-called linear statistics, π k := P Ni =1 λ ki , formula (4.24), which wereproduce here, D (cid:18)Z dt f ( t ) π k ( t ) (cid:19) · · · (cid:18)Z dt f p ( t ) π k p ( t ) (cid:19) E = p Y q =1 (cid:18) − Z dt f q ( t )( K ∗ [ ∂/∂τ ) k q ( t ) (cid:19) Z lin [ τ ] (cid:12)(cid:12)(cid:12) τ =0 (0.17)shows that n -point functions may be obtained from Z lin [ τ ] by the differentiation”trick” π k ≡ ( K ∗ [ ∂/∂τ ) k ( t ) or (in terms of generating series) π ( z ) ≡ − ( K ∗ [ ∂/∂τ )( z, t ).Note that if one had not linearized the generating functional, we would have to solveinstead a complex Burgers equation, Dπ ( z ) + ( π ) ′ ( z ) = − [ ∂/∂τ ( z ) . In a future article, we plan to extend Schr¨odinger-Virasoro constraints to a much moregeneral class of constraints, one per generator of the Lie algebra of noise-preserving trans-formations. Formulas in Theorem (4.1) being readily generalized to arbitrary n = − , 0, itseems very likely that some of the conformal field theoretic structure uncovered for n = − , § n , and also ) show that the action of more general transformationson the two-body potential produces cubic terms, plus an infinite-number of new terms dueto the particle-dependent time-shifts with unresolved singularity, typically R t ds λ i ( s ) − λ j ( s )( λ i ( t ) − λ j ( t )) i = j ) (see § N , withforeseeable applications to the study of the limit N → ∞ in the microscopic regime. Section 1 is an appetizer for the reader willing to understand the objective of the paper and tohave a flavour of the computations. The noise invariance condition is introduced right fromthe beginning in (1.7), but we postpone the general discussion of this condition and consideronly elementary transformations such as (1.9), which do not close under Lie brackets. Thetransformation of the force term under these transformations is given in (1.10) for N = 1and (1.16,1.23) for general N . The very complicated term (1.23) fortunately vanishes forSchr¨odinger-Virasoro transformations, for which all time shifts are equal.We present the noise invariance condition in whole generality in the strictly algebraically-minded section 2 and define the Lie algebra of noise-preserving transformations F NP inDefinition 2.2. We also compute the Lie brackets of elementary transformations in thenatural basis of iterated integrals .The very short Section 3 lies a general geometric foundation to these sets of transformations.It is also the occasion to introduce the Schr¨odinger-Virasoro transformations.The main section is Section 4. We present the key ingredients in § § The purpose in this section is to introduce and motivate the fundamental noise-invariancecondition in a simplified setting, and to show some preliminary computations in the case N = 1 and in the general case, paving the way to the more involved computations of section4. N = 1 For pedagogical reasons we start from the case N = 1, a one-dimensional general Langevinequation, dλ t = dB t − V ′ ( λ t ) dt. (1.1)We look for infinitesimal transformations of the set of trajectories, { λ ( t ) , t ≥ } 7→ { λ ( t ) + ε (¯ δλ )( t ) , t ≥ } (1.2)7hat preserve the general structure of the equation. We actually restrict to causality-preserving, first-order transformation, namely, we assume that ε (¯ δλ )( t ) = ε (cid:16) φ ( t, λ ) − ψ ( t, λ ) ˙ λ ( t ) (cid:17) (1.3)where φ ( t, λ ) , ψ ( t, λ ) depend only on the values of ( λ s ) s ≤ t . Since the trajectory t λ ( t ) isnot differentiable, this should be understood (to order one in ε ) as the composition of twotransformations, λ λ + εδλ, t t + εδt (1.4)where ( δλ )( t ) = φ ( t, λ ) , δt = ψ ( t, λ ) . (1.5)Put in another way, we look for the dynamical law satisfied by the transformed trajectory˜ λ ( t + εδt ) := ( λ + εδλ )( t ), or (to order 1 in ε ) ˜ λ ( t ) = ( λ + εδλ )( t − εδt ). Since dB t − εδt =(1 − ε ( ˙ δt )) / d ˜ B t where ˜ B has the same law as B , we get to order 1 in ε , taking into accountthe Itˆo correction written as ”Ito” in the following formula, d ˜ λ = ε (cid:18) ∂ ( δλ ) ∂t + Ito (cid:19) dt + (1 + ε ∂ ( δλ ) ∂λ ) n − (1 − ε ( ˙ δt )) V ′ ( λ ) dt + (1 − ε δt )) d ˜ B o . (1.6)Under the fundamental noise invariance condition ∂ ( δλ ) ∂λ = 12 ( ˙ δt ) . (1.7)(1.1) is turned into a similar Langevin equation with transformed force − ( V ′ + εδV ′ ) definedto order one in ε by( V ′ + εδV ′ )( λ + εδλ ) = − ε ∂ ( δλ ) ∂t + (cid:20) ε (cid:18) ∂ ( δλ ) ∂λ − ( ˙ δt ) (cid:19)(cid:21) V ′ ( λ ) − ε Ito= V ′ ( λ + εδλ ) − ε (cid:26) ∂ ( δλ ) ∂t + ∂ ( δλ ) ∂λ V ′ ( λ + εδλ ) + V ′′ ( λ + εδλ ) δλ + Ito (cid:27) = V ′ ( λ + εδλ ) − ε (cid:26) ∂ ( δλ ) ∂t + ∂∂λ (cid:0) δλ V ′ ( λ + εδλ ) (cid:1) + Ito (cid:27) (1.8)Looking for specific examples, we now specialize to the transformations where φ ( t, λ ) de-pends only on the value of φ at time t , namely (for n ≥ − δλ ( t ) = − λ n +1 ( t ) ˙ a ( t ) , δt = 2 Z t ds ∂ ( δλ ) ∂λ ( s ) = − n + 1) Z t ds ˙ a ( s ) λ n ( s ) , (1.9)For n = − , coordinatetransformations ; they generate the Schr¨odinger-Virasoro algebra introduced in section 3.For n ≥ ∂ ( δλ ( t )) ∂λ ( t ) = − ( n + 1) nλ n − ( t ) ˙ a ( t ) in this specific case, we getour first important formula, 8 N = 1 force change) δV ′ = λ n +1 ¨ a + X k ≥ b k ( n + 1 + k ) λ n + k + ( n + 1) nλ n − ˙ a . (1.10)All these terms extend trivially to the case of N particles when β = 0, i.e. in absence oftwo-body potential, yielding N terms, δV ′ i , where V ′ i := − ∂W∂λ i = X j = i βλ i − λ j − V ′ ( λ i ) (1.11)is the force felt by the i -th particle . The first term in (1.10) reflects the time-dependence ofthe transformation. The last term is the Itˆo’s correction. The second term expresses simplythe action of the Virasoro vector field − ˙ a ( λ n +1 ∂∂λ + ( n + 1) λ n ) on the confining force − V ′ . N -particle model General transformations leaving invariant the form of the equation for N ≥ N particles located at { λ i } , as is immediately seen from thenoise invariance condition (1.7); see section 3 for general geometric considerations. Thismakes in general the transformation of the two-body force more complicated, though (as weshall see later on) the change in the action remains surprisingly simple.Generally speaking the change of the force felt by the i -th particle (see (1.11)) or simply force change , δV ′ i , is the sum of two terms. The first one (thereafter called simultaneous ), δ simul V ′ i , is the more or less straightforward of the N = 1 force change written in theprevious subsection, taking also into account the action of the coordinate change on thetwo-body force. The second (called delayed ), δ delay V ′ i , takes into account the difference oftime-shifts between two trajectories ( λ i ( t )) t ≥ and ( λ j ( t )) t ≥ , i = j . A. Simultaneous force change Compared with the N = 1 case, we must now write down the effect on the dynamics of λ i of the coordinate change. In addition to the term (1.9), one has an extra term due to thetransformation of the two-body force, which must also take into account the transformationof the other eigenvalues { λ j } j = i , − ˙ a N X i ′ =1 λ n +1 i ′ ∂∂λ i ′ + ( n + 1) λ ni ! X j = i βλ i − λ j = − β X j = i λ n +1 i − λ n +1 j ( λ i − λ j ) − ( n + 1) X j = i λ ni λ i − λ j ˙ a ( t ) ≡ − " n X k =0 A n,k ˙ a ( t ) , (1.12)where A n,k := β X j = i λ ki λ n − kj − λ ni λ i − λ j = βλ ki n − − k X p =0 λ pi X j = i λ n − − k − pj , (1.13)9rom which n X k =0 A n,k = β n − X q =0 ( q + 1) λ qi X j = i λ n − − qj = β n − X q =0 ( q + 1) λ qi π n − − q − 12 ( n + 1) nλ n − i . (1.14)Changing sign, we may interpret (1.14) as an additive contribution to δV ′ i . Combiningwith the Itˆo term, see third term in (1.9), we get n X k =0 A n,k + ( n + 1) nλ n − i = β n − X q =0 ( q + 1) λ qi π n − − q + (1 − β n + 1) nλ n − i . (1.15)The other terms in the action transform as in section 1 (compare with (1.9)), yielding atotal variation (simultaneous force change for general N) δ simul V ′ i = λ n +1 i ¨ a + ( + ∞ X l =0 b l ( n + l + 1) λ n + li + β n − X q =0 ( q + 1) λ qi π n − − q − ( β − n + 1) nλ n − i ˙ a. (1.16)We return to these computations in section 4 after a more detailed discussion of thenoise-preserving condition. B. Delayed force change Consider only the part of the variation ¯ δλ due to the time-shifts, δt i := 2( n + 1) Z t ds ˙ a ( s ) λ ni ( s ) . (1.17)Letting ˜ λ i ( t ) := λ i ( t − εδt i ), the system of coupled equations for the particles becomes (tofirst order in ε ) d ˜ λ i ( t ) = (cid:16) − ε ( n + 1) ˙ a ( t )˜ λ ni ( t ) (cid:17) d ˜ B i ( t ) − (cid:16) − ε ( n + 1) ˙ a ( t )˜ λ ni ( t ) (cid:17) V ′ (˜ λ i ( t )) dt + (cid:16) − ε ( n + 1) ˙ a ( t )˜ λ ni ( t ) (cid:17) X j = i β ˜ λ i ( t ) − λ j ( t − εδt i ) dt (1.18)where √ ( ˜ B i ) i are standard Brownian motions. Adding to this variation the one due to δλ compensates the change of noise strength due to the noise invariance condition. The lastterm in the r.-h.s. of (1.18) brings to light a new effect due to the different time-shift. Since λ j ( t − εδt i ) = ˜ λ j ( t + ε ( δt j − δt i )), we have to order 1 in εβ ˜ λ i ( t ) − λ j ( t − εδt i ) = β ˜ λ i ( t ) − ˜ λ j ( t ) + ε β (˜ λ i ( t ) − ˜ λ j ( t )) · d ˜ λ j ( t ) dt ( δt j − δt i ) (1.19)10hus (combining with the effect of the δλ -variation studied in A.), λ ′ i = λ i + (¯ δλ ) i , i =1 , . . . , N follow the modified system of equations to first order in εdλ ′ i dt = d ˜ B i ( t ) − ( V ′ i ( t, λ ′ ) + εδ simul V ′ i ( t, λ ′ )) dt + ε X j = i ( δt j − δt i ) β ( λ ′ i ( t ) − λ ′ j ( t )) dλ ′ j dt . (1.20)Replacing dλ ′ j dt by the 0-th order term d ˜ B j ( t ) − ∂W∂λ ′ j ( t ) dt in the right-hand side of (1.20), weget dλ ′ i dt = dB ′ i ( t ) − ( V ′ i ( t, λ ′ ) + εδ simul V ′ i ( t, λ ′ )) dt − ε β X j = i ∂W∂λ ′ j ( t ) δt j − δt i ( λ ′ i ( t ) − λ ′ j ( t )) dt, (1.21)where dB ′ i ( t ) := d ˜ B i ( t ) + ε X j = i ( δt j − δt i ) β ( λ ′ i ( t ) − λ ′ j ( t )) d ˜ B j ( t ) , i = 1 , . . . , n (1.22)have same law as the original Brownians since the ε -term defines an infinitesimal rotationand white noise is invariant by rotation. Thus we have found δ delay V ′ i ( t ) = − β X j = i ∂W∂λ ′ j ( t ) δt j − δt i ( λ ′ i ( t ) − λ ′ j ( t )) . (1.23) C. Change of measure Let us finally discuss the change of measure on the trajectories – we shall return to thisin section 4 with a modified, τ -dependent measure.Comparing the measure Q ( V ′ ), resp. Q ( V ′ + δV ′ ) ≡ Q ( V ′ ) + δ Q on the space oftrajectories of (0.1) with confining forces {− V ′ ( λ i ) } i , resp. {− ( V ′ ( λ i ) + δV ′ i ) } i , we see from(0.4) or rather from the rigorous Girsanov formula (0.5) that δ Q ( { λ i } ) = Q ( { λ i } ) X i Z δV ′ i dB i ( t )= Q ( { λ i } ) X i (cid:16) Z δV ′ i dλ i ( t ) + Z δV ′ i ∂W∂λ i ( t ) dt (cid:17) . (1.24)The main technical task in section 4 is to compute the terms appearing in (1.24) in thecase of Schr¨odinger-Virasoro transformations, for which δV ′ i = δ simul V ′ i simply. We shall now construct the Lie algebra generated by the transformations (1.9). Definition 2.1 Let F be the space of functionals Φ = Φ( t, λ ) generated (as as vector space)by functionals of the form ˙ a ( t ) Z t ds ˙ a ( s ) λ k ( s ) Z s ds ˙ a ( s ) λ k ( s ) · · · Z s p − ds p ˙ a p ( s p ) λ k p ( s p ) ( p ≥ , k , . . . , k p ≥ where ˙ a, ˙ a , . . . , ˙ a p are smooth functions of time. ( k ,...,k p ) ( ˙ a , . . . , ˙ a p ; λ )( t ) := Z t ds ˙ a ( s ) λ k ( s ) Z s ds ˙ a ( s ) λ k ( s ) · · · Z s p − ds p ˙ a p ( s p ) λ k p ( s p )(2.2)are called iterated integrals . As a prominent example, a completely factorized functional Q pi =1 (cid:16)R t ds i ˙ a i ( s i ) λ k i ( s i ) (cid:17) is a sum of p ! iterated integrals since (denoting by Σ p the groupof permutations of a set of p elements) Z t ds Z t ds · · · Z t ds p ( · · · ) = X σ ∈ Σ p Z t ds σ (1) Z s σ (1) ds σ (2) · · · Z s σ ( p − ds σ ( p ) ( · · · ) . (2.3)The class F is stable by multiplication because of the shuffle relation, (cid:20)Z t ds Z t ds · · · Z t ds p ( · · · ) (cid:21) (cid:20)Z t ds ¯1 Z t ds ¯2 · · · Z t ds ¯ q ( · · · ) (cid:21) == X σ (cid:20)Z t ds σ (1) Z t ds σ (2) · · · Z t ds σ ( p + q ) ( · · · ) (cid:21) (2.4)where σ ranges over shuffles of the lists (1 , . . . , p ), (¯1 , . . . , ¯ q ), i.e. over all re-orderings of thecompound list (1 , . . . , p, ¯1 , . . . , ¯ q ) preserving the orderings of the two sub-lists. In particular,the prefactor ˙ a ( t ) in (2.1) may be interpreted as a multiplication by R t ds ¨ a ( s ) and absorbedinto an iterated integral of order p + 1. Finally, ”polarizing” a p -th iterated integral byreplacing λ with p independent copies λ , . . . , λ p , namely,Φ ( k ,...,k p ) ( ˙ a , . . . , ˙ a p ; λ , . . . , λ p )( t ) := Z t ds ˙ a ( s ) λ k ( s ) Z s ds ˙ a ( s ) λ k ( s ) · · · Z s p − ds p ˙ a p ( s p ) λ k p p ( s p )(2.5)and permuting the order of integration by use of Fubini’s theorem, we obtain after somecomputations (see [15] or [16])Φ ( k ,...,k p ) ( ˙ a , . . . , ˙ a p ; λ , . . . , λ p )( t ) == X σ ∈ Σ p ε ( σ ) Z t ds ˙ a σ (1) ( s ) λ k σ (1) σ (1) ( s ) Z s ds ˙ a σ (2) ( s ) λ k σ (2) σ (2) ( s ) · · · Z s p − ds p ˙ a σ ( p ) ( s p ) λ k σ ( p ) σ ( p ) ( s p )(2.6)for some universal coefficients ε ( σ ) ∈ Z . Alternatively, if Φ( λ ) ≡ Φ ( k ,...,k p ) ( ˙ a , . . . , ˙ a p ; λ ),then we get Φ( λ , . . . , λ p )( t ) = X σ ∈ Σ p ε ( σ )Φ σ ( λ σ (1) , . . . , λ σ ( p ) )( t ) (2.7)by defining Φ σ ( λ , . . . , λ p )( t ) to be the polarization of Φ σ ( λ )( t ) := Φ ( k σ (1) ,...,k σ ( p ) ) ( ˙ a σ (1) , . . . , ˙ a σ ( p ) ; λ )( t ) . This polarization trick will allow us later on to evaluate ddε (cid:12)(cid:12) ε =0 Φ( λ + ε ¯ δλ )( t ) as the sum X σ ∈ Σ p ε ( σ ) ddε (cid:12)(cid:12) ε =0 Φ σ ( λ + ε ¯ δλ, λ, . . . , λ )( t ) . (2.8)12ote that, by completing the tensor product, we may also choose to replace ˙ a ( s ) ˙ a ( s ) · · · ˙ a p ( s p )with a general time coefficient g ( s , . . . , s p ) in Definition 2.1.Now comes our main definition. Definition 2.2 (noise-preserving transformations) Let F NP be the Lie algebra gener-ated (as a vector space) by infinitesimal trajectory transformations of the type (¯ δλ )( t ) = λ ( t ) n +1 Φ( t, λ ) − ˙ λ ( t )Ψ( t, λ ) (2.9) with Φ ∈ F and Ψ( t, λ ) = 2( n + 1) Z t ds λ ( s ) n Φ( s, λ ) . (2.10)Replacing as in the previous section the infinitesimal transformation λ λ + ε ¯ δλ by thecomposition of λ λ + εδλ = λ + ελ n +1 Φ( · , λ ) with the time-transformation t t + ε Ψ( t, λ ),we see that (2.10) generalizes (1.7) in an obvious way to transformations depending on thepast of the trajectory. For the sequel we note that:12 ∂ t Ψ( t, λ ) = ( n + 1) λ ( t ) n Φ( t, λ ) = ∂∂λ ( t ) Φ( t, λ ) , (2.11)where the partial derivative ∂∂λ ( t ) (to be distinguished from the functional derivative δδλ ( t ) )acts on the function λ ( t ) n +1 but vanishes on the integrated functional Φ( t, λ ).We want to compute the Lie bracket of two noise-preserving transformations (2.9) andcheck that it is still a noise-preserving transformation. We start by specializing to the casewhen (¯ δ i λ )( t ) = λ ( t ) n i +1 ˙ a i ( t ) − n i + 1) ˙ λ ( t ) Z t ds ˙ a i ( s ) λ n i ( s ) , i = 1 , . (2.12)Then (cid:0) [¯ δ , ¯ δ ] λ (cid:1) ( t ) = ∂ ∂ε ∂ε (cid:12)(cid:12) ε = ε =0 (cid:2) ( λ + ε ¯ δ λ ) + ε ¯ δ ( λ + ¯ δ λ ) − ( λ + ε ¯ δ λ ) − ε ¯ δ ( λ + ¯ δ λ ) (cid:3) = ∂ ∂ε ∂ε (cid:12)(cid:12) ε = ε =0 ( ε "(cid:18) λ ( t ) + ε λ ( t ) n +1 ˙ a ( t ) − ε ( n + 1) ˙ λ ( t ) Z t ds ˙ a ( s ) λ n ( s ) (cid:19) n +1 ˙ a ( t ) − n + 1) ∂ t (cid:18) λ ( t ) + ε λ ( t ) n +1 ˙ a ( t ) − ε ( n + 1) ˙ λ ( t ) Z t ds ˙ a ( s ) λ n ( s ) (cid:19) ·· Z t ds ˙ a ( s ) (cid:18) λ ( s ) + ε λ ( s ) n +1 ˙ a ( s ) − ε ( n + 1) ˙ λ ( s ) Z s ds ′ ˙ a ( s ′ ) λ n ( s ′ ) (cid:19) n (cid:21) − (1 ↔ (cid:27) (2.13)Easy computations give (cid:0) [¯ δ , ¯ δ ] λ (cid:1) ( t ) = h F I + F II + ( F + F + F + F ) ˙ λ ( t ) i ( n , ˙ a ; n , ˙ a ) − h F I + F II + ( F + F + F + F ) ˙ λ ( t ) i ( n , ˙ a ; n , ˙ a ), with (abbreviating F · ( n , ˙ a ; n , ˙ a ) to F · ): F I := ˙ a ( n + 1) λ ( t ) n + n +1 ˙ a ( t ); (2.14)13 II := − n + 1) λ ( t ) n +1 ¨ a ( t ) Z t ds ˙ a ( s ) λ ( s ) n ; (2.15) F := − a ( t )( n + 1)( n + 1) λ ( t ) n Z t ds ˙ a ( s ) λ ( s ) n ; (2.16) F := 2( n + 1)( n + 1) (cid:18)Z t ds ˙ a ( s ) λ ( s ) n (cid:19) λ ( t ) n ˙ a ( t ); (2.17) F := − n + 1) n Z t ds ˙ a ( s ) λ ( s ) n + n ˙ a ( s ); (2.18)and (integrating by parts) F := 4( n +1) n ( n +1) Z t ds ˙ λ ( s ) ˙ a ( s ) λ ( s ) n − Z s ds ′ ˙ a ( s ′ ) λ ( s ′ ) n ≡ n +1)( n +1)( F + F + F ) , (2.19)with F := λ ( t ) n ˙ a ( t ) Z t ds ′ ˙ a ( s ′ ) λ ( s ′ ) n , F := − Z t ds λ n ( s )¨ a ( s ) Z s ds ′ ˙ a ( s ′ ) λ n ( s ′ )(2.20) F := − Z t ds λ n + n ( s ) ˙ a ( s ) ˙ a ( s ) (2.21)There is also a term F ∗ in ¨ λ , but due to symmetry F ∗ ( n , ˙ a ; n , ˙ a ) − F ∗ ( n , ˙ a ; n , ˙ a ) =0. For the same reason, the two F -terms cancel. Finally, one remarks that F , F , F areproportional and sum up to 0, while ∂ t F = ∂∂λ ( t ) F II and ∂ t F = ∂∂λ ( t ) F I .Concluding, (cid:0) [¯ δ , ¯ δ ] λ (cid:1) ( t ) = (cid:16) λ ( t ) n + n +1 Φ δ ,δ ] ( t, λ ) − ˙ λ ( t )Ψ δ ,δ ] ( t, λ ) (cid:17) + (cid:16) λ ( t ) n +1 Φ δ ,δ ] ( t, λ ) − ˙ λ ( t )Ψ δ ,δ ] ( t, λ ) (cid:17) − (cid:16) λ ( t ) n +1 Φ δ ,δ ] ( t, λ ) − ˙ λ ( t )Ψ δ ,δ ] ( t, λ ) (cid:17) (2.22)is a noise-preserving transformation, withΦ δ ,δ ] ( t, λ ) = ( n − n ) ˙ a ( t ) ˙ a ( t ); (2.23)Φ δ ,δ ] ( t, λ ) = − n +1)¨ a ( t ) Z t ds ˙ a ( s ) λ ( s ) n , Φ δ ,δ ] ( t, λ ) = − n +1)¨ a ( t ) Z t ds ˙ a ( s ) λ ( s ) n (2.24)and Ψ δ ,δ ] , resp. Ψ δ ,δ ] , Ψ δ ,δ ] associated to Φ δ ,δ ] , resp. Φ δ ,δ ] , Φ δ ,δ ] by (2.10).The above formulas for Φ δ ,δ ] , Φ δ ,δ ] show clearly the necessity to extend the set of noise-preserving transformations by allowing iterated integrals.Consider now two general noise-preserving transformations ¯ δ i with¯ δ i λ ( t ) = λ ( t ) n i +1 Φ i ( t, λ ) − n i + 1) ˙ λ ( t ) Z t ds λ ( s ) n i Φ i ( s , λ ) , (2.25)14ith Φ ( t, λ ) = Z t ds ˙ a ( s ) λ ( s ) k Z s ds ˙ a ( s ) λ ( s ) k · · · Z s p − ds p ˙ a p ( s p ) λ ( s p ) k p , Φ ( t, λ ) = Z t ds ˙ b ( s ) λ ( s ) k ′ Z s ds ˙ b ( s ) λ ( s ) k ′ · · · Z s p ′− ds p ′ ˙ b p ′ ( s p ′ ) λ ( s p ′ ) k p ′ . (2.26)Let Φ σ ( λ σ (2) , λ . . . , λ )( t ) =: R t ds ˙ a σ (2) ( s )( λ σ (2) ( s )) k σ (2) B (Φ σ )( s , λ ) and similarly,Φ σ ( λ σ (2) , λ, . . . , λ )( t ) =: R t ds ˙ b σ (2) ( s )( λ σ (2) ( s )) k ′ σ (2) B (Φ σ )( s , λ ). Then we get (cid:0) [¯ δ , ¯ δ ] λ (cid:1) ( t ) = (cid:16) λ ( t ) n + n +1 Φ δ ,δ ] ( t, λ ) − ˙ λ ( t )Ψ δ ,δ ] ( t, λ ) (cid:17) + (cid:16) λ ( t ) n +1 Φ δ ,δ ] ( t, λ ) − ˙ λ ( t )Ψ δ ,δ ] ( t, λ ) (cid:17) − (cid:16) λ ( t ) n +1 Φ δ ,δ ] ( t, λ ) − ˙ λ ( t )Ψ δ ,δ ] ( t, λ ) (cid:17) + h F I ′ + F II ′ + ( F ′ + F ′ ) ˙ λ ( t ) i ( n , Φ ; n , Φ ) − h F I ′ + F II ′ + ( F ′ + F ′ ) ˙ λ ( t ) i ( n , Φ ; n , Φ ) , (2.27)where (substituting Φ i ( s, λ ) to ˙ a i ( s ) with respect to the previous computations)Φ δ ,δ ] ( t, λ ) = ( n − n )Φ ( t, λ )Φ ( t, λ ); (2.28)Φ δ ,δ ] ( t, λ ) = − n + 1) ∂ t Φ ( t, λ ) Z t ds Φ ( s, λ ) λ ( s ) n , (2.29)Φ δ ,δ ] ( t, λ ) = − n + 1) ∂ t Φ ( t, λ ) Z t ds Φ ( s, λ ) λ ( s ) n (2.30)and F I ′ ( n , Φ ; n , Φ ) ≡ F I ′ , F II ′ ( n , Φ ; n , Φ ) ≡ F II ′ , F ′ ( n , Φ ; n , Φ ) ≡ F ′ , F ′ ( n , Φ ; n , Φ ) ≡ F ′ are new terms obtained by letting the derivative ∂∂ε (cid:12)(cid:12) ε =0 act on the λ -dependent termsΦ ( t, λ ) , Φ ( s, λ ) found instead of ˙ a ( t ), resp. ˙ a ( s ) in the straightforward generalization of(2.13). The polarization trick (2.8) applied to Φ yields F · ′ ≡ P σ ∈ Σ p − ε ( σ ) F · ′ ( σ ), with F I ′ ( σ ) = λ ( t ) n +1 Z t ds ˙ b σ (2) ( s ) k ′ σ (2) λ ( s ) k ′ σ (2) + n Φ ( s , λ ) B (Φ σ )( s , λ ) (2.31) F II ′ ( σ ) = − n +1) λ ( t ) n +1 Z t ds ˙ b σ (2) ( s ) B (Φ σ )( s , λ ) k ′ σ (2) λ ( s ) k ′ σ (2) − ˙ λ ( s ) Z s ds λ ( s ) n Φ ( s , λ )(2.32) F ′ ( σ ) = − n + 1) Z t ds λ ( s ) n Z s ds ˙ b σ (2) ( s ) B (Φ σ )( s , λ ) k ′ σ (2) λ ( s ) k ′ σ (2) + n Φ ( s , λ )(2.33) F ′ ( σ ) = 4( n + 1)( n + 1) Z t ds λ ( s ) n Z s ds ˙ b σ (2) ( s ) B (Φ σ )( s , λ ) k ′ σ (2) λ ( s ) k ′ σ (2) − ˙ λ ( s ) Z s ds λ ( s ) n Φ ( s , λ ) (2.34)15ne checks straightforwardly that ∂ t F ′ = ∂∂λ ( t ) F I ′ and ∂ t F ′ = ∂∂λ ( t ) F II ′ . Hence[¯ δ , ¯ δ ] is indeed a noise-preserving transformation.The next task is obviously to express the above Lie brackets in some appropriate basis.The natural basis here is ( L ˙ a, ( a ,...,a p ) n, ( n ,...,n p ) ) where n ≥ − n ∈ Z ), n , . . . , n p ≥ a, a , . . . , a k are chosen in some fixed basis of time functions (forinstance among t l , l = 0 , , . . . ). By definition, L ˙ a, ( a ,...,a p ) n, ( n ,...,n p ) acts on the trajectories ( λ ( t )) t ≥ as the infinitesimal transformation λ λ + εδλ , with( δλ )( t ) := ˙ a ( t ) λ ( t ) n +1 Φ ( n ,...,n p ) ( ˙ a , . . . , ˙ a p ; λ )( t ) − n +1) ˙ λ ( t )Φ ( n,n ,...,n p ) ( ˙ a , ˙ a , . . . , ˙ a p ; λ )( t )(2.35)(see noise-preserving condition in Definition 2.2) where ˙ a is the constant function ≡ L ˙ a, ( a ,...,a p ) n, ( n ,...,n p ) , L ˙ a ′ , ( a ′ ,...,a ′ p ′ ) n ′ , ( n ′ ,...,n ′ p ′ ) ] for general indices p, p ′ and corre-spond only elementary transformations L ˙ a i n i of the type (2.12), corresponding to p, p ′ = 0.Computing the bracket in the above basis yields[ L ˙ a n , L ˙ a n ] = ( n − n ) L ˙ a ˙ a n + n − (cid:8) ( n + 1) L ¨ a ,a n ,n − ( n + 1) L ¨ a ,a n ,n (cid:9) . (2.36)The second and last terms in the above equation become very simple for n , n = − , L ˙ a,bn, = L ˙ abn , which explains why the Schr¨odinger-Virasoro algebra is closed under brackets.For n , n ≥ 1, on the other hand, we get iterated integrals of higher order and generalformulas become very involved, exhibiting sums over shuffles and permutations. Let ussimply remark at this point that the linear span of the L · , ( ··· ) n, ( ··· ) with n = − , , L − , L , L ) is. We consider in this article space-time transformations such as (1.9) whose form is dictatedby the noise invariance condition (1.7). Briefly said, these are obtained by integrating time-dependent infinitesimal conformal transformations and considering an associated transfor-mation of the time parameter. As shown in the previous section, such transformations donot constitute a group for the composition of space-time transformations, except if one re-stricts to indices n = − , 0, obtaining in this way the Schr¨odinger-Virasoro group [13]. Letus introduce the latter smoothly in a pleasant geometric framework.It turns out that these transformations may be described in a coordinate-independentsetting on an arbitrary manifold R + ×M , where ( M , g ) is any Riemannian manifold with itsmetric two-form g . The applications we have in view in the context of random matrices are( M , g ) = ( R , dλ ), resp. ( M , g ) = ( C ∪{∞} , dλ d ¯ λ ), where λ is an eigenvalue of a Hermitian,resp. normal matrix. Let Φ t : M → M ( t ≥ 0) be a C family of conformal diffeomorphismsof M : by definition, the Jacobian matrix J (Φ t ( m )) ≡ D Φ t ( m ) Dm is scalar. Restricting totransformations (Φ t ) t ≥ such that Φ ≡ Id , we get a set C = C ( R + , Conf( M )). Considernow a world-line ( m t , t ) := (Φ t ( m ) , t ) in M × R + . A non-relativistic particle living on thisworld-line has proper time T m ( t ) ≡ T ( m, t ) = Z t ds | J (Φ s ( m )) | α , (3.1)16here the dynamical scaling exponent α equals 2 if we want (1.7) to be satisfied. Thisrelation defines an extended space-time diffeomorphism ˜Φ : M × R + → M × R + , ˜Φ( m, t ) =( T m ( t ) , Φ t ( m )) such that˜Φ( m, 0) = ( m, , ddt T ( m, t ) = (Φ t ) ∗ ( g ij dx i dx j )( m ) g ij dx i dx j ( m ) = | J (Φ t ( m )) | . (3.2)Alternatively, (Φ t ) t ≥ is characterized by the velocity field v ( m, t ) ≡ d Φ t ( m,t ) dt , a time-dependent vector field on M such that v ( · , t ) belongs for every fixed t to the conformalLie algebra conf ( M ) . Keeping to the one-dimensional case, let us now introduce the Schr¨odinger-Virasoro group inthis context. Let M be flat space R d for some d ≥ 1. Global conformal transformations aresimply affine transformations, i.e. compositions of rotations, scale changes x ax ( a ∈ R )and translations, x x + v ( v ∈ R d ).(i) Let φ : R + → R + be a C -diffeomorphism with φ (0) = 0. Define a ( t ) ≡ 12 ¨ φ ( t )˙ φ ( t ) and v ( x, t ) := a ( t ) x. Then˜Φ( x, t ) = (cid:16) e R t ds a ( s ) x, Z t ds e R s ds ′ a ( s ′ ) (cid:17) = ( q ˙ φ ( t ) x, φ ( t )) . (3.3)(ii) Let v ( x, t ) := ˙ b ( t ) for some function b : R + → R d . Then˜Φ( x, t ) = (cid:18) x + Z t ds b ( s ) , t (cid:19) . (3.4)(iii) Let v ( x, t ) := R ( t ) x where R ( t ) ∈ so ( d ) is an antisymmetric matrix (infinitesimalrotation). Then ˜Φ( t, x ) = (cid:18) t, −→ exp (cid:18)Z t ds R ( s ) (cid:19) x (cid:19) (3.5)where −→ exp is the time-ordered exponential.Composing these transformations one obtains a zero-mass representation of the so-calledSchr¨odinger-Virasoro group. Considering infinitesimal transformations and restricting to theone-dimensional case for simplicity, one gets ddε F (cid:16) e − ε R t ds ¨ f ( s ) x Z t ds e − ε R s ds ′ ¨ f ( s ′ ) (cid:17)(cid:12)(cid:12)(cid:12) ε =0 = ( X f F )( x, t ) (3.6)and ddε F (cid:16) x − εg ( t ) , t (cid:17)(cid:12)(cid:12)(cid:12) ε =0 = ( Y g F )( x, t ) (3.7)where the vector fields X f := − f ( t ) ∂ t − 12 ˙ f ( t ) x∂ x , Y g := − g ( t ) ∂ x (3.8)17ake up a zero mass representation of the Schr¨odinger-Virasoro algebra, namely,[ X f , X g ] = X ˙ fg − f ˙ g (3.9)[ Y f , X g ] = Y ˙ fg − f ˙ g , [ Y f , Y g ] = 0 . (3.10)One recognizes elementary transformations as in (2.12), with X f ≡ L ˙ f and Y g ≡ L g − (see end of § If we search for a dynamical analogue of the equilibrium constraints L eqn Z [ τ ] = 0, we mustchoose a dynamical functional replacing the partition function. Clearly a substitute for theequilibrium measure is the measure Q on trajectories. However (whatever its precise de-pendence on the parameters of the potential), Q is always normalized, viz. Q [1] = 1. Usingthe trivial identity δ Q [1] = 0 does give non-trivial identities, but Q [1] is not a generatingfunctional. Instead we consider a perturbed evolution, dλ i = dB i − ∂W [ τ ] ∂λ i dt, (4.1)where W [ τ ]( { λ i } ) ≡ W ( { λ i } ) + P k ≥ τ k P i λ ki . Copying the change-of-measure leading to(0.5), and throwing away the second-order term in τ completing the square, e − P i ( ∂ λi ( P k ≥ τ k λ ki )) ,we get a new measure Q [ τ ]( { λ i } ) = Q ( { λ i } ) e − P + ∞ k =1 P i R + ∞ τ k ( t ) kλ k − i ( t ) (cid:18) dλ i ( t )+ (cid:18) V ′ ( λ i ( t )) − P j = i βλi ( t ) − λj ( t ) (cid:19) dt (cid:19) (4.2)Using Itˆo’s formula, kλ k − i dλ i ( t ) = d ( λ ki )( t ) − k ( k − λ k − i ( t ) dt, (4.3)and the identity β X i,j = i λ k − i λ i − λ j = β k − X q =0 ( π q π k − − q − π k − ) , (4.4)we obtain a second, more useful expression for Q [ τ ], Q [ τ ]( { λ i } ) = Q ( { λ i } ) e − S [ τ ] , (4.5)where S [ τ ] ≡ X k ≥ Z S k ( t ) τ k ( t ) dt, (4.6) S k ( t ) = ˙ π k ( t )+( β − k ( k − π k − ( t )+ k X l ≥ b l π l + k − ( t ) − β k k − X q =0 π q ( t ) π k − − q ( t ) . (4.7)18he action S [ τ ] is the sum of a linear term (linearized action) S lin [ τ ] ≡ X k ≥ Z S link ( t ) τ k ( t ) dt, S link ( t ) = ˙ π k ( t )+( β − k ( k − π k − ( t )+ k X l ≥ b l π l + k − ( t )(4.8)and of a quadratic term, S quadr [ τ ].Throwing this quadratic term in turn, we finally get a linearized functional. Definition 4.1 (generating functional) Let Q lin [ τ ]( { λ i } ) := Q ( { λ i } ) e − S lin [ τ ] (4.9) and Z lin [ τ ] := Q lin [ τ ][1] = Z d Q lin [ τ ]( { λ i } ) . (4.10)For simplicity we shall from now on write h · i instead of Q ( · ), viz. h · i τ instead of Q lin [ τ ]( · )Differentiating with respect to τ yields δδτ k ( t ) e − S [ τ ] = − e − S [ τ ] ˙ π k ( t ) + ( β − k ( k − π k − ( t ) + k X l ≥ b l π l + k − ( t ) − β k k − X q =0 π q ( t ) π k − − q ( t ) , (4.11)formally,˙ π k ( t ) + ( β − k ( k − π k − ( t ) + k X l ≥ b l π l + k − ( t ) − k β k − X q =0 π q ( t ) π k − − q ( t ) = − δδτ k ( t ) (4.12)in average (i.e. when inserted into an expectation value).We need to invert the linearized version of this equation,˙ π k ( t ) + ( β − k ( k − π k − ( t ) + k X l ≥ b l π l + k − ( t ) = − δδτ k ( t ) , k ≥ π k ( t ) = − X l Z t ds K kl ( t − s ) δδτ l ( s ) , (4.14)or, in a mixed operator/convolutional notation, π k ( t ) = − ( K ∗ δδτ ) k ( t ). Note that K ( t − s ) =( K kl ( t − s ) kl is an upper-triangular matrix, so the sum in (4.14) really ranges over l ≥ k .At this point we introduce the generating series [ ∂/∂τ ( z, t ) ≡ X k ≥ z − k − ∂∂τ k ( t ) , π ( z, t ) = X k ≥ π k ( t ) z − k − , (4.15)19 ( z, t ) = X k ≥ kτ k ( t ) z k − , b ( z ) = X l ≥ b l z l (4.16)Note that the zero mode π ( t ) of the field π ( z, t ) is a constant, π ( t ) = N . In somewhatabstract terms, we use the canonical splitting of the formal series algebra C [[ z, z − ]] into A + ⊕ A − ≡ C [[ z ]] ⊕ z − C [[ z − ]]; each of these two subalgebras is isotropic for the scalarproduct ( u, v ) = I u ( z ) v ( z ) dz := 12i π Z C u ( z ) v ( z ) dz (4.17)given by the residue integral, where C is any counterclockwise simple contour circling aroung0, and A + = A ∗− . Then [ ∂/∂τ ( z ) , π ( z ) ∈ A − , while b ( z ) ∈ A + . We write quite generally u − ( z ) = ( P − u )( z ) := X n ≤− u n z n , u + ( z ) = ( P + u )( z ) := X n ≥ u n z n (4.18)if u ( z ) = P n ∈ Z u n z n ∈ C [[ z, z − ]] (see Appendix B). Also, we write : u ( z ) = v ( z ) mod A + if P − u = P − v . With these notations, letting ˙ π = ∂π∂t , π ′ = ∂π∂z , we see that (4.13) is equivalentto the following Definition 4.2 (equation of motion) ˙ π ( z ) + ( β − π ′′ ( z ) − ( bπ ) ′ ( z ) = − [ ∂/∂τ ( z ) mod A + . (4.19)As mentioned in the Introduction, the original non-linearized equation of motion (4.12)contributes to (4.19) an additive term ( π ) ′ ( z ) which turns it into a Burgers equation, butwe shall not pursue along this road.The solution of (4.19) is π ( z, t ) ≡ − z − Z t ds I dw K t − s ( z − , w ) [ ∂/∂τ ( w, s ) (4.20)or more schematically, π ( z, t ) ≡ − ( K ∗ [ ∂/∂τ )( z, t ) , (4.21)where (comparing with (4.14)) K t − s ( z − , w ) = X k,l ≥ z − k w l K kl ( t − s ) . (4.22)When t → K t ( z − , w ) → − w/z , so that z − H dw K ( z − , w ) f ( w − ) = f ( z − ) for every f = f ( w − ) = a + a w − + a w − + . . . .When β = 2, the equation of motion (4.19) is a transport equation which may be solvedexplicitly in terms of the characteristics; as proved in Appendix B, K t ( z − , w ) = 11 − w ( t ) /z (4.23)where w ( t ) ∈ C [[ w ]] is the solution at time t of the ordinary differential equation ˙ w t = − b ( w t )with initial condition w ≡ w . When β = 2, semi-explicit but complicated formulas for K t ∂ z : π 7→ − π ′′ with the semi-group generated by the transport equation B : π → ( bπ ) ′ ( z ) through the use of Trotter’sformula, exp t (cid:16) − ( β − ∂ z + B (cid:17) = lim n →∞ (cid:16) exp( − tn ( β − ∂ z ) exp( tn B ) (cid:17) n , resulting in aFeynman-Kac type formula which looks awful. Hence we do not write it down, but thereader should be able to reproduce it by looking at the computations in Appendix B.As mentioned in the introduction, we see that n-point functions of the functions { π k ( t ) } may be obtained by differentiating Z , D (cid:18)Z dt f ( t ) π k ( t ) (cid:19) · · · (cid:18)Z dt f p ( t ) π k p ( t ) (cid:19) E = p Y q =1 (cid:18) − Z dt f q ( t )( K ∗ [ ∂/∂τ ) k q ( t ) (cid:19) Z lin [ τ ] (cid:12)(cid:12)(cid:12) τ =0 . (4.24)The kernel K satisfies the semi-group properties, X l ≥ K kl ( t − t ′ ) K lm ( t ′ − t ′′ ) = K lm ( t − t ′′ ) , t > t ′ > t ′′ (4.25)or equivalently I dz ′ z ′ K t − t ′ ( z − , z ′ ) K t ′ − t ′′ (( z ′ ) − , z ′′ ) = K t − t ′′ ( z − , z ′′ ); (4.26)letting t ′ → t or t ′ → t ′′ and differentiating we get the following formulas, ∂∂t K km ( t ) = − X l ≥ kb l − k +1 K lm ( t ) = − X l ≥ K kl ( t ) lb m − l +1 (4.27)or equivalently ∂∂t K t ( z − , z ′′ ) = − I dz ′ z ′ b ( z ′ ) z − z ′ /z ) K t (( z ′ ) − , z ′′ ) (4.28)= − I dz ′ z ′ K t ( z − , z ′ ) b ( z ′′ ) z ′ − z ′′ /z ′ ) . (4.29)In the last two equalities we used the following expression for the generator, P k,l ≥ z − k w l kb l − k +1 = b ( w ) z − w/z ) . Following the probabilists’ convention we shall referto (4.28), resp. (4.29) as the forward , resp. backward Kolmogorov equation .We now define the two bosonic fields. Definition 4.3 (static and dynamic free bosons) (i) (static free boson) Let, for k ≥ , ˆ φ − k ( t ) := β − / kτ k ( t ) , ˆ φ k ( t ) := β / δδτ k ( t ) (4.30) and ˆ φ := 0 , ˆ φ ( z, t ) := X k ∈ Z ˆ φ k ( t ) z − k − . (4.31)21 ii) (dynamic free boson) Let, for k ≥ , ˆ ψ − k ( t ) := β − / kτ k ( t ) , ˆ ψ k ( t ) := β / ( K ∗ δδτ ) k ( t ) (4.32) and ˆ ψ := − β / N , ˆ ψ ( z, t ) ≡ X k ∈ Z ˆ ψ k z − k − ( t ) . (4.33)Since ( K ∗ δδτ ) k ( t ) identifies with − π k ( t ) for k ≥ 1, and π ( t ) = P i ≡ N , the definitionof the zero mode ˆ ψ is coherent. Thenˆ ψ ( z, t ) ≡ ˆ ψ + ( t, z ) + ˆ ψ − ( t, z ) (4.34)where ˆ ψ + ( z, t ) = β − X k ≥ kτ k ( t ) z k − = β − / τ ( z, t ) ∈ A + , (4.35)ˆ ψ − ( z, t ) = − β / N z − − X k ≥ ( K ∗ δδτ ) k ( t ) z − k − = β / n − N z − + ( K ∗ [ ∂/∂τ )( z, t ) o = β / (cid:26) − N z − + z − Z t ds I dζ K t − s ( z − , ζ ) [ ∂/∂τ ( ζ, s ) (cid:27) . (4.36)Similarly, ˆ φ ( z, t ) ≡ ˆ φ + ( t, z ) + ˆ φ − ( z, t ) (4.37)where ˆ φ + ≡ ˆ ψ + , ˆ φ − ( z, t ) = β / [ ∂/∂τ ( z, t ) . (4.38)For further use we write down a formula regarding the time-derivative of ˆ ψ , ∂ t ( ˆ ψ − ( z, t )) = β / (cid:26) [ ∂/∂τ ( z, t ) + 1 z Z t ds I dζ ∂ t ( K t − s ( z − , ζ )) [ ∂/∂τ ( ζ, s ) (cid:27) . (4.39)Alternatively, from (5.25), ∂ t ( ˆ ψ − ( z, t )) = ˆ φ − ( z, t ) − β / Z t ds I dζ b ( ζ ) G + t − s ( z − , ζ ) [ ∂/∂τ ( ζ, s ) . (4.40)Taking commutators, we get: Definition 4.4 (Dynamic free boson algebra) Let G + ( t, z − ; t ′ , w ) = G + t − t ′ ( z − , w ) bethe retarded propagator , G + t − t ′ ( z − , w ) := t>t ′ z ∂∂w K t − t ′ ( z − , w ) , (4.41) G − ( t ′ , z ; t, w − ) := G + ( t, w − ; t ′ , z ) the advanced propagator , and G +0 ( z − , w ) := lim t → ,t> G + t ( z − , w ) = 1 z (1 − w/z ) , (4.42)22 − ( z, w − ) := lim t → ,t< G − t ( z, w − ) = 1 w (1 − z/w ) . (4.43) Then [ ˆ φ ( z, t ) , ˆ φ ( w, t ′ )] = δ ( t − t ′ ) (cid:8) G +0 ( z − , w ) − G − ( z, w − ) (cid:9) (4.44)[ ˆ ψ ( z, t ) , ˆ ψ ( w, t ′ )] = G + t − t ′ ( z − , w ) − G − t − t ′ ( z, w − ) (4.45)[ ˆ ψ ( z, t ) , ˆ φ ( w, t ′ )] = G + t − t ′ ( z − , w ) − δ ( t − t ′ ) G − ( z, w − ) . (4.46)Let us give a sketchy proof. First, if k, l ≥ ψ k ( t ) , ˆ ψ − l ( t ′ )] = t>t ′ lK kl ( t − t ′ ) . (4.47)Summing over Fourier components yields for t > t ′ [ ˆ ψ ( z, t ) , ˆ φ ( w, t ′ )] = [ ˆ ψ ( z, t ) , ˆ ψ ( w, t ′ )] = X k,l ≥ lw l − z − k − K kl ( t − t ′ ) = G + t − t ′ ( z − , w ) (4.48)Other commutators either vanish identically or involve a δ -function. Example (Hermite polynomials). Assume b = 1 /σ and b i = 0, i = 1. Then theequation (4.19) reduces to ˙ π + ( β − π ′′ = σ ( π + z dπdz ) − [ ∂/∂τ ( z ).Consider first the case β = 2. The equation is diagonal when written in Fourier modes,˙ π k = − σ kπ k − ∂∂τ k . The solution is π ( z, t ) = Z t ds X k ≥ e − k ( t − s ) /σ ∂∂τ ( s ) z − k − = Z t ds z I dw X k ≥ ( e − ( t − s ) /σ w/z ) k [ ∂/∂τ ( w, s ) . Hence for t ≥ K t ( z − , w ) = 11 − e − t/σ w/z , G + t ( z − , w ) = t> e − t/σ z (1 − e − t/σ w/z ) . (4.49)Note that G +0 ( z − , w ) = z (1 − w/z ) ; one retrieves the equal-time, equilibrium OPE φ ( z, t ) φ ( w, t ′ ) ∼ δ ( t − t ′ ) z − w ) .When β = 2, an explicit expression for K t , loosely related to the Mehler kernel, is givenin Appendix B, see (5.13).We may now state our main result. We denote by C ∞ c ( R ∗ + ) the space of smooth functionswith compact support ⊂ (0 , + ∞ ). (In particular, a function in C ∞ c ( R ∗ + ) vanishes to arbitraryorder at 0). 23 heorem 4.1 (dynamical constraints) Let, for a ∈ C ∞ c ( R ∗ + ) , L a − := β − / Z dt (cid:26) ¨ a ( t ) I z ˆ ψ t ( z ) dz − a ( t ) I (( β − b ′′ + b ′ b )( z ) ˆ ψ t ( z ) dz (cid:27) − Z dt (cid:20) 12 ˙ a ( t ) I : ( ˆ ψ ( z, t )) : dz − a ( t ) (cid:26)I b ′ ( z ) : ( ˆ ψ ( z, t )) : dz + I : ( ˆ φ ( z, t )) : dz (cid:27)(cid:21) (4.50) and L a := − a ( t ) ∂ t + 12 β − / Z dt (cid:26) 12 ... a ( t ) I z ˆ ψ t ( z ) dz − ˙ a ( t ) I (( β − zb ( z )) ′′ + ( zb ( z )) ′ b ( z )) ˆ ψ t ( z ) dz (cid:27) − Z dt (cid:20) 12 ¨ a ( t ) I : ( ˆ ψ ( z, t )) : z dz − 12 ˙ a ( t ) (cid:26)I ( zb ( z )) ′ : ( ˆ ψ ( z, t )) : dz + I z : ( ˆ φ ( z, t )) : dz (cid:27)(cid:21) . (4.51) Then L an Z [ τ ] = 0 , n = − , . (4.52)The proof is elementary but somewhat lengthy. It will take up the rest of the section.As it happens, see (1.9), the Schr¨odinger-Virasoro transformation Y ˙ a , δλ ( t ) = − ˙ a ( t ) , δt = 0generates L ˙ a − , while the transformation X a , δλ ( t ) = − λ ˙ a ( t ) , δt = − a ( t )generates L a .The action of the time derivation a ( t ) ∂ t is made explicit in the Appendix. We show here that the variation δ Q lin [ τ ] := δ n Q lin [ τ ] of the action under the change ofcoordinates (1.9) is the sum of four terms, δ ( i ) Q lin [ τ ] , . . . , δ ( iv ) Q lin [ τ ] which we evaluate oneby one.First, a straightforward extension of Girsanov’s formula yields δ Q lin [ τ ] = Q lin [ τ ] (cid:18)Z δV ′ i ( t ) dλ i ( t ) + Z δV ′ i ( t ) ∂W∂λ i ( t ) dt (cid:19) − Q lin [ τ ] δ + ∞ X k =0 τ k ( t ) S k ( t ) ! = δ ( i ) Q lin [ τ ] + δ ( ii ) Q lin [ τ ] + δ ( iii ) Q lin [ τ ] + δ ( iv ) Q lin [ τ ] (4.53)with δ ( i ) Q lin [ τ ] = Q lin [ τ ] Z δV ′ i ( t ) dλ i ( t ) , δ ( ii ) Q lin [ τ ] = Q lin [ τ ] Z δV ′ i ( t ) ∂W∂λ i ( t ) dt, (4.54)24 ( iii ) Q lin [ τ ] = −Q lin [ τ ] Z δV ′ i ( t ) X j = i βλ i ( t ) − λ j ( t ) dt, δ ( iv ) Q lin [ τ ] = −Q lin [ τ ] δ + ∞ X k =0 τ k ( t ) S k ( t ) ! . (4.55)We consider separately each of these four terms.(i) The quantity δV ′ i is given by the total variation formula (1.16) as a sum of four terms.Though the third and the fourth one vanish for n = − , 0, we evaluate them to somepoint and shall use those computations in another article. By Itˆo’s formula, λ n +1 i ( t ) dλ i ( t ) = 1 n + 2 d ( λ n +2 i )( t ) − ( n + 1) λ ni ( t ) dt (4.56)hence (by integration by parts) X i Z ¨ a ( t ) λ n +1 i ( t ) dλ i ( t ) = − n + 2 Z ... a ( t ) π n +2 ( t ) dt − ( n + 1) Z ¨ a ( t ) π n ( t ) dt = − n + 2 ... a ( t ) I π ( z ) z n +2 dz + ¨ a ( t ) I π ′ ( z ) z n +1 dz. (4.57)The second and fourth terms are similar, X i Z ˙ a X l ≥ b l ( n + l + 1) λ n + li − ( β − n + 1) nλ n − i dλ i = − Z ¨ a X l ≥ b l π n + l +1 − ( β − n + 1) π n dt + − Z ˙ a X l ≥ b l ( n + l + 1)( n + l ) π n + l − − ( β − n + 1) n ( n − π n − dt = − ¨ a (cid:18)I b ( z ) π ( z ) z n +1 dz + ( β − I z n +1 π ′ ( z ) dz (cid:19) − ˙ a (cid:18)I b ( z ) π ′′ ( z ) z n +1 dz + ( β − I z n +1 π ′′′ ( z ) dz (cid:19) (4.58)For the third term, we remark similarly that d n − X q =0 π q π n − q = 2 n − X q =0 X i dλ i ( q + 1) λ qi π n − q − + n − X q =0 X i ∂ ∂λ i ( π q π n − q ) dt. (4.59)We compute n − X q =0 X i ∂ ∂λ i ( π q π n − q ) = 2 n − X q =1 q ( q − π q − π n − q + q ( n − q ) π q − π n − q − = 2( n + 1) n − X q =1 ( q − π q − π n − q (4.60)25ence β n − X q =0 X i Z ˙ a ( t )( q + 1) λ qi π n − − q dλ i ( t ) = − β Z ¨ a ( t ) n − X q =0 π q ( t ) π n − q ( t ) dt −− β ( n + 1) Z ˙ a ( t ) n − X q =1 ( q − π q − ( t ) π n − q ( t ) dt = − β a I π ( z ) z n +1 dz − β ˙ a I ( ππ ′ ) ′ ( z ) z n +1 dz. (4.61)(ii) We now evaluate R dt δV ′ i V ′ ( λ i ) . It is a linear combination of terms of the type (with φ ( t ) = ˙ a ( t ) or ¨ a ( t )) b l R dt φ P i λ m + li = b l R dt φπ m + l and b l R dt φ P i λ k + li π q − k = b l R dt φπ q + l . Summing up all four terms, we get a β -independent contribution,¨ a X l ≥ b l π l + n +1 + ˙ a X l,l ′ ≥ b l b l ′ ( l ′ + n + 1) π l + l ′ + n + β ˙ a X l ≥ b l n − X q =0 ( q + 1) π n + l − − ˙ a X l ≥ b l · ( β − n + 1) nπ n + l − = ¨ a X l ≥ b l π l + n +1 + ˙ a X l,l ′ ≥ b l b l ′ ( l ′ + n + 1) π l + l ′ + n + β n + 1) n X l ≥ b l π n + l − = ¨ a I b ( z ) π ( z ) z n +1 dz + ˙ a (cid:18) − I b ( z )( bπ ) ′ ( z ) z n +1 dz + β I ( bπ ) ′′ ( z ) z n +1 dz (cid:19) . (4.62)Comparing (4.62) with (4.58), we see that the first terms sum up to zero.(iii) We now evaluate − P i,i ′ = i R dt δV ′ i βλ i − λ i ′ . First − β X i,i ′ = i Z dt φ λ mi − λ mi ′ λ i − λ i ′ = − β Z dt φ ( t ) m − X q =0 ( π q ( t ) π m − − q ( t ) − π m − ( t ))= β Z dt φ ( t ) mπ m − ( t ) − m − X q =0 π q ( t ) π m − − q ( t ) ;(4.63)summing up the contributions of the terms 1,2,4 in (1.16), we get26 a ( n + 1) π n − n X q =0 π q π n − q + β a X l ≥ b l ( l + n + 1) ( l + n ) π l + n − − l + n − X q =0 π q π l + n − − q −− β a · ( β − n + 1) n ( n − π n − − n − X q =0 π q π n − − q = − β a I ( π ′ ( z ) + π ( z )) z n +1 dz + β a (cid:18)I b ( z )( π ′′ ( z ) + ( π ( z )) ′ ) z n +1 dz + β β − I ( π ′′′ ( z ) + ( π ( z )) ′′ ) z n +1 dz (cid:19) . (4.64)The third term contributes − β Z dt ˙ a n − X q =0 ( q + 1) X i,j,i ′ = i ( λ qi − λ qi ′ ) λ n − − qj λ i − λ i ′ = − β Z dt ˙ a n − X q =1 ( q + 1) q − X p =0 X i,j,i ′ = i λ pi λ q − − pi ′ λ n − − qj = − β Z dt ˙ a n X q + r + s = n − ( q + 1) π q π r π s − n − X k =0 ( k + 2)( k + 1) π k π n − − k o = − β Z dt ˙ a (cid:26) − I π ′ ( z ) π ( z ) z n +1 dz − I π ′′ ( z ) π ( z ) z n +1 dz (cid:27) , (4.65)including a term of order 3 (which does not appear for n ≤ − δ n (cid:16)R + ∞ dt P + ∞ k =0 τ k ( t ) S k ( t ) (cid:17) = − δS lin un-der the change of coordinates (1.9) for n = − , . First we have δ n ( π k ( t )) = − ˙ a ( t ) kπ k + n ( t ) , δ n ( ˙ π k ) = ddt ( δ n ( π k )) = − ˙ ak ˙ π k + n − ¨ akπ k + n . (4.66)Recall we have defined τ ( z ) ≡ P k ≥ kτ k z k − . Thus, for n = − − δ − S lin = ˙ a X k ≥ kτ k X l ≥ b l ( l + k − π l + k − + X k ≥ kτ k ˙ π k − + ( β − X k ≥ k ( k − k − τ k π k − +¨ a X k ≥ kτ k π k − = ˙ a (cid:26) − I τ ( z ) b ( z ) π ′ ( z ) dz + I τ ( z ) ˙ π ( z ) + ( β − I τ ( z ) π ′′ ( z ) dz dz (cid:27) + ¨ a I τ ( z ) π ( z ) dz. (4.67)27hen n = 0 we must add to a term similar to (4.67) a contribution δ time ( S ) due tothe time change; letting ˜ t = t − εδt = t + 2 εa be the new time coordinate, we get − εδ time ( Z dt τ k ( t ) S link ( t )) = − εδ time (cid:18)Z dt τ k ( t ) (cid:18) ˙ π k ( t ) + ( β − k ( k − π k − ( t ) (cid:19) + k X l ≥ b l Z dt τ k ( t ) π l + k − ( t ) = Z dt τ k ( t ) S link ( t ) − Z d ˜ t ( τ k (˜ t ) − εa ˙ τ k (˜ t )) dπ k d ˜ t (˜ t ) − Z d ˜ t (1 − ε ˙ a (˜ t ))( τ k (˜ t ) − εa ˙ τ k (˜ t )) k X l ≤ b l π l + k − (˜ t ) + ( β − k ( k − π k − (˜ t ) = 2 ε (cid:18)Z dt a ( t ) ˙ τ k ( t ) ˙ π k ( t ) + Z dt ( ˙ a ( t ) τ k ( t ) + a ( t ) ˙ τ k ( t )) ·· k X l ≥ b l π l + k − ( t ) + ( β − k ( k − π k − ( t ) . (4.68)Hence − δ S lin = ˙ a X k ≥ kτ k X l ≥ b l ( l + k − π l + k − + X k ≥ kτ k ˙ π k + ( β − X k ≥ k ( k − k − τ k π k − +¨ a X k ≥ kτ k π k + 2 ˙ a X k ≥ kτ k X l ≥ b l π l + k − + ( β − X k ≥ k ( k − τ k π k − +2 a X k ≥ ˙ τ k ( ˙ π k + ( β − k ( k − π k − ) + X k ≥ k ˙ τ k X l ≥ b l π l + k − = ¨ a I τ ( z ) π ( z ) z dz + ˙ a (cid:26) − I τ ( z ) b ( z )( zπ ( z )) ′ dz + I τ ( z )( ˙ π ( z ) + ( β − π ′′ ( z )) z dz +2 I τ ( z ) b ( z ) π ( z ) dz (cid:27) − a∂ t (4.69)where a∂ t = a ( t ) ∂ t is the time derivation acting on the coefficients of the func-tional Z [ τ ] (see Appendix B). The term H τ ( z ) π ′′ ( z ) z dz in (4.69) is equal to thesum P k ≥ k ( k − k − τ k π k − + 2 P k ≥ k ( k − τ k π k − . We may finally collect all contributions to obtain generators of transformations, denoted by L ˙ a − and L a . Recall π ( z, t ) ≡ − β − / ˆ ψ − ( z, t ).28i) ( n = − 1) Collecting all terms in δ ( i ) , δ ( ii ) , δ ( iii ) and δ ( iv ) , we get a term L − ,lin linearin ( τ, ∂/∂τ ), plus a term L − ,quadr which is quadratic, L ˙ a − ≡ Z dt (cid:8) L ˙ a − ,lin ( t ) + L ˙ a − ,quadr ( t ) (cid:9) , (4.70) L ˙ a − ,lin ( t ) = − ... a π + ( β − 1) ˙ a I b ( z ) π ′′ ( z ) dz − ˙ a I b ( z )( bπ ) ′ ( z ) dz = β − / (cid:26) ... a I z ˆ ψ ( z ) dz − ˙ a I (( β − b ′′ ( z ) + b ( z ) b ′ ( z )) ˆ ψ ( z ) dz (cid:27) (4.71) L ˙ a − ,quadr ( t ) = ˙ a (cid:26) β I b ( z )( π ) ′ ( z ) dz − I τ ( z ) b ( z ) π ′ ( z ) dz + I τ ( z ) ˙ π ( z ) dz +( β − I τ ( z ) π ′′ ( z ) dz (cid:27) + ¨ a I τ ( z ) π ( z ) dz. (4.72)Using (4.19) and taking into account the fact that the subalgebras A + , A − areisotropic, we get L ˙ a − ,quadr ( t ) = ˙ a (cid:26) − β I b ′ ( z )( π )( z ) dz − I τ ( z ) b ( z ) π ′ ( z ) dz + ( β − I τ ( z ) π ′′ ( z ) dz + I τ ( z )(( bπ ) ′ ( z ) − [ ∂/∂τ ( z ) − ( β − π ′′ ( z )) dz (cid:27) + ¨ a I τ ( z ) π ( z ) dz = ˙ a (cid:26) − I b ′ ( z )( β π )( z ) − τ ( z ) π ( z )) dz − I τ ( z ) [ ∂/∂τ ( z ) dz (cid:27) + ¨ a I τ ( z ) π ( z ) dz = − ˙ a (cid:26) I b ′ ( z ) ( β / π ( z ) − β − / τ ( z )) dz + 12 I : ( β / [ ∂/∂τ ( z ) + β − / τ ( z )) : dz (cid:27) − 12 ¨ a I ( β / π ( z ) − β − / τ ( z )) dz = − ˙ a (cid:18) I b ′ ( z ) : ( ˆ ψ ( z )) : dz + 12 I : ( ˆ φ ( z )) : dz (cid:19) − 12 ¨ a I : ( ˆ ψ ( z )) : dz (4.73)(ii) ( n = 0) One finds L a ≡ − a∂ t + Z dt (cid:8) L a ,lin ( t ) + L a ,quadr ( t ) (cid:9) , (4.74) L a ,lin ( t ) = ( β − aπ − 12 ... a π + ˙ a (cid:26) ( β − I b ( z ) π ′′ ( z ) z dz − I b ( z )( bπ ) ′ z dz (cid:27) = ( β − N ¨ a + β − / (cid:26) 12 ... a I z ˆ ψ ( z ) dz − ˙ a I (( β − zb ( z )) ′′ + ( zb ( z )) ′ b ( z )) ˆ ψ t ( z ) dz (cid:27) (4.75)29note that the first term, a total derivative, disappears after integration in (4.51)); L a ,quadr ( t ) = ¨ a I z ( τ π − β π )( z ) dz + ˙ a (cid:26) β I zb ( z )( π ) ′ ( z ) dz + ( β − I τ ( z ) π ′′ ( z ) z dz + I zτ ( z )(( bπ ) ′ ( z ) − [ ∂/∂τ ( z ) − ( β − π ′′ ( z )) dz − I zτ ( z ) b ( z ) π ′ ( z ) dz + I τ ( z ) b ( z ) π ( z ) dz (cid:27) = ¨ a I z ( τ π − β π )( z ) dz + ˙ a (cid:26)I ( zb ( z )) ′ ( τ π − β π )( z ) dz − I zτ ( z ) [ ∂/∂τ ( z ) dz (cid:27) = − 12 ¨ a I : ( ˆ ψ ( z )) : z dz − 12 ˙ a (cid:26)I ( zb ( z )) ′ : ( ˆ ψ ( z )) : dz + I z : ( ˆ φ ( z )) : dz (cid:27) . (4.76) We prove in this paragraph that ( L a − ,quadr , L a ,quadr ) a ∈ C ∞ c ( R ∗ + ) provide a zero mass represen-tation of the Schr¨odinger-Virasoro algebra, see (3.10): Theorem 4.2 h Z L f ,quadr ( t ) dt, Z L g ,quadr ( t ′ ) dt ′ i = Z L ˙ fg − f ˙ g ,quadr ( t ) dt, (4.77) h Z L f − ,quadr ( t ) dt, Z L g ,quadr ( t ′ ) dt ′ i = Z L ˙ fg − f ˙ g − ,quadr ( t ) dt, h Z L f − ,quadr ( t ) dt, Z L g − ,quadr ( t ′ ) dt ′ i = 0 . (4.78)In order to keep computations to a reasonable length, we consider commutators of thefunctionals A quadrv ( f ) := − v ′ ( z ) f ( t ) ∂ t − Z dt ¨ f ( t ) I : ( ˆ ψ t ( z )) : v ( z ) dz − Z dt ˙ f ( t ) (cid:26)I ( v ( z ) b ( z )) ′ : ( ˆ ψ t ( z )) : dz + I v ( z ) : ( ˆ φ t ( z )) : dz (cid:27) (4.79)with v ( z ) = 1 or z . Note that A quadrv ( f ) = L ˙ f − ,quadr for v ( z ) = 1, and A quadrv ( f ) = L f ,quadr for v ( z ) = z . The non-differential part of A quadrv ( f ), ¯ A quadrv ( f ) := A quadrv ( f ) + 2 v ′ ( z ) f ( t ) ∂ t ,is by definition A quadrv ( f ) shorn of its differential part − v ′ ( z ) f ( t ) ∂ t . The first computationsare valid for an arbitrary function v ∈ C [[ z ]], but at some point we must restrict to v ( z ) = 1or z , which are the only cases needed. We want to prove:[ A quadrz ( f ) , A quadrz ( g )] = 4 Z L ˙ fg − f ˙ g ,quadr ( t ) dt, (4.80)[ A quadr ( f ) , A quadrz ( g )] = 2 Z L ¨ fg − ˙ f ˙ g − ,quadr ( t ) dt, [ A quadr ( f ) , A quadr ( g )] = 0 . (4.81)30sing the commutation relations of the boson algebra, we find for u, v ∈ C [[ z ]] (cid:20) I u ( z ) : ( ˆ ψ t ( z )) : dz, I v ( w ) : ( ˆ φ t ′ ( w )) : dw (cid:21) = I I dzdw u ( z ) v ( w ) : ˆ ψ t ( z ) ˆ φ t ′ ( w ) : G + t − t ′ ( z − , w ) − δ ( t − t ′ ) I I dzdw u ( z ) v ( w ) : ˆ ψ t ( z ) ˆ φ t ( w ) : G − ( z, w − ); (4.82) (cid:20) I u ( z ) : ( ˆ ψ t ( z )) : dz, I v ( w ) : ( ˆ ψ t ′ ( w )) : dw (cid:21) = I I dzdw u ( z ) v ( w ) : ˆ ψ t ( z ) ˆ ψ t ′ ( w ) : ( G + t − t ′ ( z − , w ) − G − t − t ′ ( z, w − ))(4.83) (cid:20) I u ( z ) : ( ˆ φ t ( z )) : dz, I v ( w ) : ( ˆ φ t ′ ( w )) : dw (cid:21) = I dz I dw u ( z ) v ( w ) : ˆ φ t ( z ) ˆ φ t ′ ( w ) : ( G +0 ( z − , w ) − G − ( z, w − ))(4.84)hence [ A quadru ( f ) , A quadrv ( g )] = X i =1 C i ( f, g ) , (4.85)where:(i) (contribution of the commutator [ ψ, ψ ]) C ( f, g ) = I dzdw Z dt Z t dt ′ ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) v ( w )¨ g ( t ′ ) : ˆ ψ t ( z ) ˆ ψ t ′ ( w ) : G + t − t ′ ( z − , w ) − I dzdw Z dt ′ Z t ′ dt u ( z ) ¨ f ( t )( ˙ g ( t ′ )( vb ) ′ ( w ) + v ( w )¨ g ( t ′ )) : ˆ ψ t ( z ) ˆ ψ t ′ ( w ) : G − t − t ′ ( z, w − ) ≡ C , ( f, g ) + C , ( f, g ) + C , ( f, g ) (4.86)where (by integrating by parts with respect to t ′ or t , and using the fundamentalrelations (5.1,5.2,5.3))(1) C , ( f, g ) == − I dz Z dt ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˙ g ( t ) ˆ ψ t ( z )( P − ( v ˆ ψ t )) ′ ( z )+ I dw Z dt ′ ˙ f ( t ′ )( ˙ g ( t ′ )( vb ) ′ ( w ) + v ( w )¨ g ( t ′ )) ˆ ψ t ′ ( w )( P − ( u ˆ ψ t ′ )) ′ ( w )(4.87)31f u = v then the two terms in ˙ f ( t ) ˙ g ( t ) cancel each other, and there remains only C , ( f, g ) = − I dz Z dt ( ¨ f ( t ) ˙ g ( t ) − ˙ f ( t )¨ g ( t ))( P + ( u ˆ ψ t ))( z )( P − ( u ˆ ψ t )) ′ ( z ) . (4.88)Otherwise we may assume that u ( z ) = 1, v ( w ) = w , from which ( P − ( v ˆ ψ t )) ′ ( z ) =( v P − ( ˆ ψ t )) ′ ( z ) = z ( P − ˆ ψ t ) ′ ( z ) + ( P t ˆ ψ t )( z ) (observe that the first equality is wrong if v ( w ) = w n +1 , n ≥ 1) and C , ( f, g ) = − I dz z Z dt ( ¨ f ( t ) ˙ g ( t ) − ˙ f ( t )¨ g ( t ))( P + ( ˆ ψ t ))( z )( P − ( ˆ ψ t )) ′ ( z ) (4.89) − Z dt ˙ f ( t ) ˙ g ( t ) (cid:26)I dz b ′ ( z ) ˆ ψ t ( z )( P − ˆ ψ t )( z ) − I dz b ( z ) ˆ ψ t ( z )( P − ˆ ψ t ) ′ ( z ) (cid:27) − Z dt ¨ f ( t ) ˙ g ( t ) I dz : ( ˆ ψ t ( z )) : (4.90)In the last line we have used: I dz : ( ˆ ψ t ( z )) : = 2 I dz ( P + ˆ ψ t )( z )( P − ˆ ψ t )( z ) = 2 I dz ˆ ψ t ( z )( P − ˆ ψ t )( z ) . (2) C , ( f, g ) = − β / I dzdw Z dt Z t dt ′ ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) v ( w ) ˙ g ( t ′ ) ˆ ψ t ( z ) G + t − t ′ ( z − , w ) [ ∂/∂τ ( w, t ′ ) − sym. = − I dzdw Z dt Z t dt ′ ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) v ( w ) ˙ g ( t ′ ) ˆ ψ t ( z ) G + t − t ′ ( z − , w ) ˆ φ t ′ ( w ) − sym., (4.91)a contribution due to the first term in the right-hand side of (4.39); ”-sym”. indicates,here as in the following computations, a similar term with the kernel G − t − t ′ in factor;(3) C , ( f, g ) = − β / Z dt Z t dt ′ Z t ′ ds I dz ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˙ g ( t ′ ) ˆ ψ t ( z )1 z ∂ t ′ (cid:18)I dww v ( w ) ∂ w ( K t − t ′ ( z − , w )) K t ′ − s ( w − , ζ ) (cid:19) [ ∂/∂τ ( ζ, s ) − sym. = − β / Z dt Z t dt ′ Z t ′ ds I dz ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˙ g ( t ′ ) ˆ ψ t ( z )1 z ∂ t ′ I dww ( v ( w ) b ′ ( w ) − v ′ ( w ) b ( w )) ∂ w ( K t − t ′ ( z − , w )) K t ′ − s ( w − , ζ ) [ ∂/∂τ ( ζ, s ) − sym. = − Z dt Z t dt ′ I dzdw ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˙ g ( t ′ )( v ( w ) b ′ ( w ) − v ′ ( w ) b ( w )) : ˆ ψ t ( z ) ˆ ψ t ′ ( w ) : G + t − t ′ ( z − , w ) − sym. (4.92)by (5.15); 32ii) (contribution of the commutator [ ψ, ψ ], continued) C ( f, g ) = Z dt Z t dt ′ I dzdw ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˙ g ( t ′ )( vb ) ′ ( w ) : ˆ ψ t ( z ) ˆ ψ t ′ ( w ) : G + t − t ′ ( z − , w ) − Z dt ′ Z t ′ dt I dzdw ˙ f ( t )( ub ) ′ ( z )( ˙ g ( t ′ )( vb ) ′ ( w ) + v ( w )¨ g ( t ′ )) : ˆ ψ t ( z ) ˆ ψ t ′ ( w ) : G − t − t ′ ( z, w − )(4.93)(iii) (non- δ contribution of the commutator [ ψ, φ ] for t = t ′ ) C ( f, g ) = Z dt Z t dt ′ I dzdw ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) v ( w ) ˙ g ( t ′ ) : ˆ ψ t ( z ) ˆ φ t ′ ( w ) : G + t − t ′ ( z − , w ) − Z dt ′ Z t ′ dt I dzdw ˙ f ( t ) u ( z )( ˙ g ( t ′ )( vb ) ′ ( w ) + v ( w )¨ g ( t )) : ˆ φ t ( z ) ψ t ′ ( w ) : G − t − t ′ ( z, w − )(4.94)(iv) ( δ -contribution of the commutator [ ψ, φ ]) C ( f, g ) = − Z dt I dzdw ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) v ( w ) ˙ g ( t ) ˆ φ t ( w ) ˆ ψ t ( z ) G − ( z, w − )+ Z dt I dzdw u ( z ) ˙ f ( t )( ˙ g ( t )( vb ) ′ ( w ) + v ( w )¨ g ( t )) ˆ φ t ( z ) ˆ ψ t ( w ) G +0 ( z − , w )= − Z dt I dz ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˙ g ( t )( P + ( v ˆ φ t )) ′ ( z ) ˆ ψ t ( z )+ Z dt I dw ˙ f ( t )( ˙ g ( t )( vb ) ′ ( w ) + v ( w )¨ g ( t ))( P + ( u ˆ φ t )) ′ ( w ) ˆ ψ t ( w ) (4.95)This term is parallel to the term C , ( f, g ), to the analysis of which we refer. In thefollowing expressions, we use the fact that P + ˆ φ t = P + ˆ ψ t . When u = v we find C ( f, g ) = − I dz Z dt ( ¨ f ( t ) ˙ g ( t ) − ˙ f ( t )¨ g ( t ))( P + ( u ˆ ψ t )) ′ ( z )( P − ( u ˆ ψ t ))( z ) . (4.96)Otherwise we may assume that u ( z ) = 1, v ( w ) = w , from which C ( f, g ) = − I dz z Z dt ( ¨ f ( t ) ˙ g ( t ) − ˙ f ( t )¨ g ( t ))( P + ( ˆ ψ t )) ′ ( z )( P − ( ˆ ψ t ))( z ) (4.97) − Z dt ˙ f ( t ) ˙ g ( t ) (cid:26)I dz b ′ ( z )( P + ˆ ψ t )( z ) ˆ ψ t ( z ) − I dz b ( z )( P + ˆ ψ t ) ′ ( z ) ˆ ψ t ( z ) (cid:27) − Z dt ¨ f ( t ) ˙ g ( t ) I dz : ( ˆ ψ t ( z )) : (4.98)(v) ( δ -contribution due to the commutator [ φ, φ ])33his term clearly vanishes when u = v . Hence we may assume that u ( z ) = 1, v ( w ) = w ,in which case C ( f, g ) = Z dt ˙ f ( t ) ˙ g ( t ) I dz I dw w : ˆ φ t ( z ) ˆ φ t ( w ) : ( G +0 ( z − , w ) − G − ( z, w − ))= Z dt ˙ f ( t ) ˙ g ( t ) I dw w : ˆ φ t ( w ) n ( P + ˆ φ t ) ′ ( w ) + ( P − ˆ φ t ) ′ ( w ) o := − Z dt ( ˙ f ˙ g )( t ) I dw : ( ˆ φ t ( w )) : (4.99)(vi) asssume v ( w ) = w : by the results of Appendix B, in particular, (5.20, 5.22, 5.23,5.26) C ( f, g ) = 2 h ¯ A quadru ( f ) , − g ( t ) ∂ t i = − Z dt I dz ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )): ˆ ψ t ( z ) · n ( g ( t ) ∂ t · ˆ ψ − ( z, t )) + ( g ( t ) ∂ t · ˆ ψ + ( z, t )) o : − Z dt I dz u ( z ) ˙ f ( t ) : ˆ φ t ( z ) n ( g ( t ) ∂ t · ˆ φ − ( z, t )) + ( g ( t ) ∂ t · ˆ φ + ( z, t )) o := − (cid:18)Z dt I dz ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t ))( ˆ ψ + ( z, t ) + ˆ ψ − ( z, t )) g ( t )( ∂ t ˆ ψ − )( z, t )+ Z dt I dz ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˆ ψ t ( z ) Z t dt ′ ˙ g ( t ′ ) I dw b ( w ) G + t − t ′ ( z − , w ) ˆ ψ t ′ ( w ) (cid:19) − (cid:18) − Z dt I dz g ( t )( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˆ ψ + ( z, t )( ∂ t ˆ ψ − )( z, t ) − Z dt I dz (cid:26) ddt ( g ˙ f )( t )( ub ) ′ ( z ) + u ( z ) ddt ( g ¨ f )( t ) (cid:27) ˆ ψ + ( z, t ) ˆ ψ − ( z, t ) (cid:19) − (cid:18) − Z dt I dz g ( t ) u ( z ) ˙ f ( t )( ∂ t ˆ φ )( z, t ) ˆ φ − ( z, t ) − Z dt I dz g ( t ) u ( z ) ¨ f ( t ) ˆ φ t ( z ) ˆ φ − ( z, t ) (cid:19) − (cid:18) − Z dt I dz g ( t ) u ( z ) ˙ f ( t ) ˆ φ + ( z, t )( ∂ t ˆ φ )( z, t ) − Z dt I dz u ( z ) ddt ( g ˙ f )( t ) ˆ φ + ( z, t ) ˆ φ t ( z ) (cid:19) =: C nonloc ( f, g ) + C loc ( f, g ) (4.100)where C nonloc ( f, g ) := − Z dt I dz ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˆ ψ t ( z ) Z t dt ′ ˙ g ( t ′ ) I dw b ( w ) G + t − t ′ ( z − , w ) ˆ ψ t ′ ( w ) (4.101)34s a non-local term, and C loc ( f, g ) = Z dt I dz (cid:26) ddt ( g ˙ f )( t )( ub ) ′ ( z ) + u ( z ) ddt ( g ¨ f )( t ) (cid:27) : ( ˆ ψ t ( z )) : − Z dt I dz u ( z ) ddt ( g ˙ f )( t ) : ( ˆ φ t ( z )) :+2 Z dt I dz g ( t ) ¨ f ( t ) u ( z ) : ( ˆ φ t ( z )) :+2 Z dt I dz u ( z ) ˙ g ( t ) ˙ f ( t ) ˆ φ + ( z, t ) ˆ φ − ( z, t )= Z dt I dz (cid:26) ddt ( g ˙ f )( t )( ub ) ′ ( z ) + u ( z ) ddt ( g ¨ f )( t ) (cid:27) : ( ˆ ψ t ( z )) :+ Z dt I dz u ( z )( g ¨ f )( t ) : ( ˆ φ t ( z )) : (4.102)(vii) if u ( z ) = z , C ( f, g ) := 2[ − f ( t ) ∂ t , ¯ A quadrv ( g )] is ”sym.” of (vi);(viii) finally, assuming u ( z ) = z and v ( w ) = w , C ( f, g ) = 4[ − f ( t ) ∂ t , − g ( t ) ∂ t ] = 4( f ( t ) ˙ g ( t ) − ˙ f ( t ) g ( t )) ∂ t . (4.103)Let us now sum up the different contributions. We leave out the differential term (viii)which is as expected. Note that C , + C ≡ C , ( f, g ) + C ( f, g ) = 2 Z dt Z t dt ′ I dzdw ( ˙ f ( t )( ub ) ′ ( z ) + u ( z ) ¨ f ( t )) ˙ g ( t ′ )( v ′ b )( w ): ˆ ψ t ( z ) ˆ ψ t ′ ( w ) : G + t − t ′ ( z − , w ) − sym. (4.104)is exactly compensated by C nonloc ( f, g ) + C nonloc ( f, g ).Now all remaining terms ( C , , C , C , C loc , C loc ) are local functionals of the fields ˆ ψ, ˆ φ ,i.e. are expressed as some integral R dt H dzF ( z, t, ˆ ψ ( z, t ) , ˆ φ ( z, t )).Assume first u = v . Then C = 0, C , + C = 0 (see (4.88),(4.96)). Thus [ A quadr ( f ) , A quadr ( g )] =0, [ A quadrz ( f ) , A quadrz ( g )] = C loc ( f, g ) + C loc ( f, g )= Z dt I dz (cid:26) ddt ( g ˙ f − f ˙ g )( t )( zb ( z )) ′ + z ddt ( g ¨ f − f ¨ g )( t ) (cid:27) : ( ˆ ψ t ( z )) :+ Z dt I dz z ( g ¨ f − f ¨ g )( t ) : ( ˆ φ t ( z )) := 4 Z dt L ˙ fg − f ˙ g ,quadr ( t ) . (4.105)35ssume now u ( z ) = 1 , v ( w ) = w . Then (4.89,4.97) sum up to − Z dt ( ¨ f ( t ) ˙ g ( t ) − ˙ f ( t )¨ g ( t )) I dz z : n ˆ ψ t ( z )(( P − ˆ ψ t ) ′ ( z ) + ( P + ˆ ψ t ) ′ ( z )) o := 12 Z dt ( ¨ f ( t ) ˙ g ( t ) − ˙ f ( t )¨ g ( t )) I dz : ( ˆ ψ t ( z )) : (4.106)Adding this to (4.90,4.98) yields − Z dt ddt ( ˙ f ˙ g )( t ) : ( ˆ ψ t ( z )) : − Z dt ˙ f ( t ) ˙ g ( t ) (cid:26)I dz b ′ ( z ) : ( ˆ ψ t ( z )) : − I dz b ( z ) : ˆ ψ t ( z )( ˆ ψ t ) ′ ( z ) : (cid:27) = − Z dt ddt ( ˙ f ˙ g )( t ) : ( ˆ ψ t ( z )) : − Z dt ˙ f ( t ) ˙ g ( t ) I dz b ′ ( z ) : ( ˆ ψ t ( z )) : (4.107)To the latter expression we must still add C ( f, g ) = − Z dt I dz ( ˙ f ˙ g )( t ) : ( ˆ φ t ( z )) : (4.108)and C loc ( f, g ) = Z dt I dz (cid:26) ddt ( g ˙ f )( t ) b ′ ( z ) + ddt ( g ¨ f )( t ) (cid:27) : ( ˆ ψ t ( z )) :+ Z dt I dz ( g ¨ f )( t ) : ( ˆ φ t ( z )) : (4.109)Adding all terms yields as expected2 L ¨ fg − ˙ f ˙ g − ,quadr = − Z dt I dz (cid:26) ( ¨ f g − 12 ˙ f ˙ g )( t ) b ′ ( z ) + ddt ( ¨ f g − 12 ˙ f ˙ g ) (cid:27) : ( ˆ ψ t ( z )) : − Z dt I dz ( ¨ f g − 12 ˙ f ˙ g )( t ) : ( ˆ φ t ( z )) : (4.110) Copying what we did in the last subsection, we set A linu ( f ) := β − / Z dt (cid:26) ... f ( t ) I ( Z u )( z ) ˆ ψ t ( z ) dz − ˙ f ( t ) I (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˆ ψ t ( z ) dz (cid:27) (4.111)for u ( z ) = 1 , z , with ( R u )( z ) = z , resp. z when u ( z ) = 1, resp. z . In coherence with thequadratic parts, A linu ( f ) = L ˙ f − ,lin for u ( z ) = 1, and A linu ( f ) = L f ,lin for u ( z ) = z . Sinceobviously [ A linu ( f ) , A linv ( g )] = 0, we must prove:[ A linz ( f ) , A quadrz ( g )] − ( f ↔ g ) = 4 Z L ˙ fg − f ˙ g ,lin ( t ) dt, (4.112)36 A lin ( f ) , A quadrz ( g )] − [ A linz ( f ) , A quadr ( g )] = 2 Z L ¨ fg − ˙ f ˙ g − ,lin ( t ) dt, [ A lin ( f ) , A quadr ( g )] − ( f ↔ g ) = 0 . (4.113)For u ( z ) = 1 , z , v ( w ) = 1 , w , we find in general:[ A linu ( f ) , A quadrv ( g )] = X i =1 D i ( f, g ) , (4.114)with:(i) D ( f, g ) = (cid:20) A linu ( f ) , − Z dt ′ I dw v ( w )¨ g ( t ′ ) : ( ˆ ψ ( w, t ′ )) : (cid:21) = − β − / Z dt Z t dt ′ I dz dw (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ¨ g ( t ′ ) v ( w ) ˆ ψ t ′ ( w ) G + t − t ′ ( z − , w )= − Z dt Z t dt ′ Z t ′ ds I dz dw dζ (cid:18) z ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ¨ g ( t ′ ) v ( w ) G + t − t ′ ( z − , w ) 1 w K t ′ − s ( w − , ζ ) [ ∂/∂τ ( ζ, s )= D , ( f, g ) + D , ( f, g ) + D , ( f, g ) , (4.115)where (by integrating by parts)(1) D , ( f, g ) = − β − / Z dt I dz dw (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ˙ g ( t ) v ( w ) G +0 ( z − , w ) ˆ ψ t ( w )= β − / Z dt I dz (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) u + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ˙ g ( t ) (cid:16) P − ( v ˆ ψ t ) (cid:17) ′ ( z ) (4.116)(2) D , ( f, g ) = Z dt Z t ds I dz dw dζ (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ˙ g ( s ) v ( w ) G + t − s ( z − , w ) 1 w K ( w − , ζ ) [ ∂/∂τ ( ζ, s )= Z dt Z t ds I dz I dζ (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ˙ g ( s ) v ( ζ ) G + t − s ( z − , ζ ) [ ∂/∂τ ( ζ, s ); (4.117)373) D , ( f, g ) = Z dt Z t ds Z ts dt ′ I dz dw dζ (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) z ˙ g ( t ′ ) ∂ t ′ (cid:18)I dww v ( w ) ∂ w ( K t − t ′ ( z − , w )) K t ′ − s ( w − , ζ ) (cid:19) [ ∂/∂τ ( ζ, s )= Z dt Z t dt ′ I dz dζ (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ˙ g ( t ′ ) I dww ( v ( w ) b ′ ( w ) − v ′ ( w ) b ( w )) G + t − t ′ ( z − , w ) ˆ ψ t ′ ( w ) (4.118)using (5.15);(ii) D ( f, g ) = (cid:20) A linu ( f ) , − Z dt ′ I dw ˙ g ( t ′ )( vb ) ′ ( w ) : ( ˆ ψ ( w, t ′ )) : (cid:21) = − Z dt Z t dt ′ I dz dw dζ (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ( vb ) ′ ( w )˙ g ( t ′ ) G + t − t ′ ( z − , w ) 1 w ˆ ψ t ′ ( w ); (4.119)(iii) D ( f, g ) = (cid:20) A linu ( f ) , − Z dt ′ I dw ˙ g ( t ′ ) v ( w ) : ( ˆ φ ( w, t ′ )) : (cid:21) = − Z dt Z t dt ′ I dzdw (cid:18) ( Z u )( z )... f ( t ) − (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˙ f ( t ) (cid:19) ˙ g ( t ′ ) v ( w ) G + t − t ′ ( z − , w ) [ ∂/∂τ ( w, t ′ ); (4.120)(iv) D ( f, g ) = [ A linu ( f ) , − g ( t ) ∂ t ] =: D loc , ( f, g ) + D , ( f, g ) + D nonloc ( f, g ) , (4.121)where: D loc , ( f, g ) = 2 β − / Z dt ... f ( t ) I ( Z u )( z ) g ( t )( ∂ t ˆ ψ − )( z, t ) dz = − β − / Z dt ddt ( g ... f )( t ) I ( Z u )( z ) ˆ ψ − ( z, t ) dz ; (4.122)38 loc , ( f, g ) = − β − / Z dt ˙ f ( t ) I (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) g ( t )( ∂ t ˆ ψ − )( z, t ) dz = 2 β − / Z dt ddt ( g ˙ f )( t ) I (( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) ˆ ψ − ( z, t ) dz ; (4.123) D nonloc ( f, g ) = 2 Z dt Z t dt ′ I dz Z dw (cid:26) ... f ( t )( Z u )( z ) − ˙ f ( t )(( β − ub ) ′′ ( z ) + ( ub ) ′ ( z ) b ( z )) (cid:27) ˙ g ( t ′ ) b ( w ) G + t − t ′ ( z − , w ) ˆ ψ t ′ ( w ) . (4.124)Let us now add the different contributions. First, D , + D ≡ D , + D + D nonloc ≡ D , ( f, g ) , D loc ( f, g ) and their symmetric counterparts.Assume first u ( z ) = v ( z ) = 1 or z . Then D , ( f, g ) − ( f ↔ g ) = β − / Z dt (... f ( t ) ˙ g ( t ) − ˙ f ( t )... g ( t )) I dz ( Z u )( z )( P − ( u ˆ ψ t )) ′ ( z )= − β − / Z dt (... f ( t ) ˙ g ( t ) − ˙ f ( t )... g ( t )) I dz u ( z ) ˆ ψ t ( z ) . (4.125)In particular, if u ( z ) = v ( z ) = 1, this is equal to − N Z dt (... f ( t ) ˙ g ( t ) − ˙ f ( t )... g ( t )) = − N Z dt ddt ( ¨ f ( t ) ˙ g ( t ) − ˙ f ( t )¨ g ( t )) = 0 . (4.126)Since there is not D -term in that case, we have proved: [ A lin ( f ) , A quadr ( g )] − ( f ↔ g ) = 0.The reader may easily check that one also gets the correct formula for [ A linz ( f ) , A quadrz ( g )] − ( f ↔ g ).There remains the case u ( z ) = 1, v ( w ) = w . Then D , ( f, g ) − sym. = β − / Z dt (cid:26) (... f ˙ g )( t ) I z ( P − ( z ˆ ψ t )) ′ ( z ) dz − ( ˙ f ... g )( t ) I z P − ˆ ψ t ) ′ ( z ) dz (cid:27) − β − / Z dt ( ˙ f ˙ g )( t ) (cid:26) ( β − b ′′ ( z ) + b ′ ( z ) b ( z ))( P − ( z ˆ ψ t )) ′ ( z ) −− (( β − zb ( z )) ′′ + ( zb ( z )) ′ b ( z ))( P − ˆ ψ t ) ′ ( z ) (cid:27) = − β − / Z dt (... f ˙ g − ˙ f ... g )( t ) I z ˆ ψ t ( z ) dz − β − / Z dt ( ˙ f ˙ g )( t )(( β − b ′′ ( z ) + b ′ ( z ) b ( z )) ˆ ψ t ( z ) dz, (4.127)from which the reader may easily check the remaining bracket, [ A lin ( f ) , A quadrz ( g )] − [ A linz ( f ) , A quadr ( g )]. 39 .6 A detailed example: the Hermite case We compute once again commutators for the sake of the reader in a simple case (Hermitepolynomials, β = 2) using Fourier modes. Assume as in Example 1 that b = 1 /σ and b i = 0, i = 1 and let β = 2. Then L a − = Z dt (cid:26) ( ˙ aσ − ... a )( t ) π ( t ) − 12 ( ˙ aσ + ¨ a )( t ) I : ( ˆ ψ ( z, t )) : dz − 12 ˙ a ( t ) I : ( ˆ φ ( z, t )) : dz (cid:27) (4.128)and L a = Z dt (cid:26) ( ˙ aσ − 12 ... a )( t ) π ( t ) + 4 Nσ ˙ a − a ( t ) ∂ t − 12 (2 ˙ aσ + ¨ a )( t ) I : ( ˆ ψ ( z, t )) : z dz − 12 ˙ a ( t ) I z : ( ˆ φ ( z, t )) : dz (cid:27) (4.129)with H ˆ φ ( z, t ) = P k ≥ kτ k ( t ) δδτ k − ( t ) and12 I ˆ ψ ( z, t ) dz = − N τ ( t ) + X k ≥ kτ k ( t ) Z t ds e − ( k − t − s ) /σ δδτ k − ( s ) (4.130)We first compute Lie brackets and prove that ( L a − , L a ) a ∈ C ∞ provide a zero mass repre-sentation of the Schr¨odinger-Virasoro algebra. Let L − ,lin ( a ) , L − ,quadr ( a ), resp. L ,lin ( a ) , L ,quadr ( a )be the linear and quadratic parts of L a − , resp. L a as in the previous paragraph. Using therelations in the dynamic boson algebra, we find (cid:20)I ˆ ψ ( z, t ) dz, I ˆ φ ( w, t ′ ) dw (cid:21) = t>t ′ X k ≥ k ( k − τ k ( t ) e − ( k − t − t ′ ) /σ δδτ k − ( t ′ )+ δ ( t − t ′ ) N τ ( t ) − X k ≥ k ( k − τ k ( t ) Z t ds e − ( k − t − s ) /σ δδτ k − ( s ) ;(4.131)for t > t ′ , (cid:20)I ˆ ψ ( z, t ) dz, I ˆ ψ ( w, t ′ ) dw (cid:21) = − N τ ( t ) e − ( t − t ′ ) /σ + X k ≥ k ( k − τ k ( t ) e − ( k − t − t ′ ) /σ Z t ′ ds e − ( k − t ′ − s ) /σ δδτ k − ( s ) . (4.132)From this we get[ L − ,quadr ( f ) , L − ,quadr ( g )] ≡ X i =1 ( C i ( f, g ) − C i ( g, f )) , (4.133)with (following the same scheme as in the previous subsection):40i) (contribution of the commutator [ ψ, ψ ]) C ( f, g ) = Z dt ( ˙ f ( t ) σ + ¨ f ( t )) X k k ( k − τ k ( t ) e − ( k − t/σ Z t dt ′ e ( k − t ′ /σ ¨ g ( t ′ ) Z t ′ ds e − ( k − t ′ − s ) /σ δδτ k − ( s ) ! ≡ C , ( f, g ) + C , ( f, g ) + C , ( f, g ) (4.134)where (by integration by parts) C , ( f, g ) = Z dt ( ˙ f ( t ) σ + ¨ f ( t )) X k k ( k − τ k ( t ) ˙ g ( t ) Z t ds e − ( k − t − s ) /σ δδτ k − ( s ) ;(4.135) C , ( f, g ) = − Z dt ( ˙ f ( t ) σ + ¨ f ( t )) X k k ( k − τ k ( t ) Z t dt ′ e − ( k − t − t ′ ) /σ ˙ g ( t ′ ) δδτ k − ( t ′ ) ;(4.136) C , ( f, g ) = − Z dt ( ˙ f ( t ) σ + ¨ f ( t )) X k k ( k − τ k ( t ) Z t dt ′ e − ( k − t − t ′ ) /σ ˙ g ( t ′ ) σ Z t ′ ds e − ( k − t ′ − s ) /σ δδτ k − ( s ) ! (4.137)(ii) (contribution of the commutator [ ψ, ψ ], continued) C ( f, g ) = Z dt ( ˙ f ( t ) σ + ¨ f ( t )) X k k ( k − τ k ( t ) e − ( k − t/σ Z t dt ′ e ( k − t ′ /σ ˙ g ( t ′ ) σ Z t ′ ds e − ( k − t ′ − s ) /σ δδτ k − ( s ) ! (4.138)(iii) (contribution of the commutator [ ψ, φ ] for t = t ′ ) C ( f, g ) = Z dt ( ˙ f ( t ) σ + ¨ f ( t )) X k k ( k − τ k ( t ) Z t dt ′ e − ( k − t − t ′ ) /σ ˙ g ( t ′ ) δδτ k − ( t ′ )(4.139)(iv) ( δ -contribution) C ( f, g ) = − Z dt ( ˙ f ( t ) σ + ¨ f ( t )) ˙ g ( t ) X k k ( k − τ k ( t ) Z t ds e − ( k − t − s ) /σ δδτ k − ( s )(4.140)41v) (zero-momentum contribution) C ( f, g ) = − N Z dt ( ˙ f ( t ) σ + ¨ f ( t )) τ ( t ) Z t dt ′ e − ( t − t ′ ) /σ ( ˙ g ( t ′ ) σ + ¨ g ( t ′ ))= − N Z dt ( ˙ f ( t ) σ + ¨ f ( t )) ˙ g ( t ) τ ( t ) (4.141)by integration by parts;(vi) (zero-momentum contribution, continued) C ( f, g ) = 2 N Z dt ( ˙ f ( t ) σ + ¨ f ( t )) ˙ g ( t ) τ ( t ) . (4.142)Then one sees that C , + C = 0, C , + C = 0, C , + C = 0, C + C = 0. Consequently,[ L − ,quadr ( f ) , L − ,quadr ( g )] = 0.The contribution of L − ,lin to the bracket [ L f − , L g − ] is easily computed, [ L − ,lin ( f ) , L − ,lin ( g )] =0 clearly while by integration by parts[ L − ,lin ( f ) , L − ,quadr ( g )] − ( f ↔ g )= " − Z dt ( ˙ fσ − ... f )( t ) Z t ds e − ( t − s ) /σ δδτ ( s ) , N Z dt ′ ( ˙ g ( t ′ ) σ + ¨ g ( t ′ )) τ ( t ′ ) − ( f ↔ g )= − N "Z dt ( ˙ fσ − ... f )( t ) Z t ds e − ( t − s ) /σ ( ˙ g ( s ) σ + ¨ g ( s )) − ( f ↔ g ) = N Z dt (... f ˙ g − ˙ f ... g )( t ) = N Z dt ddt ( ¨ f ˙ g − ˙ f ¨ g ) = 0 . (4.143)Thus finally: [ A ( f ) , A ( g )] = 0. We prove here a certain number of explicit expressions given in section 4. At some pointwe use in the proofs the following fundamental relations , P − f ( z ) = 1 z I dw − w/z f ( w ) , P + f ( z ) = I dww (1 − z/w ) f ( w ) (5.1)if f ( z ) = P n ∈ Z a n z n ∈ C [[ z − , z ]], where P − f , resp. P + f , is the projection onto A − parallelto A + , resp. onto A + parallel to A − , namely, P − f ( z ) = P n ≤− a n z n , P + f ( z ) = P n ≥ a n z n .Differentiating with respect to z we also get( P − f ) ′ ( z ) = − I dwz (1 − w/z ) f ( w ) = − I G +0 ( z − , w ) f ( w ) , (5.2)( P + f ) ′ ( z ) = I dww (1 − z/w ) f ( w ) = I G − ( z, w − ) f ( w ) . (5.3)Note that, by construction,( P ± ˆ ψ t )( z ) = ˆ ψ ± ( z, t ) , ( P ± ˆ φ t )( z ) = ˆ φ ± ( z, t ) . (5.4)42 .1 Explicit solution of equation of motion when β = 2 We prove here formula (4.23). Let ˜ w ≡ w ( t ) ∈ C [[ w ]], resp. z ( t ) ∈ C [[ z ]] be the solution attime t ∈ R of the ODE ˙ w t = − b ( w ( t )), resp. ˙ z t = − b ( z ( t )) with initial condition w (0) = w ,resp. z (0) = z . Let ˜ K t ( z − , w ) := − w ( t ) /z . Then ∂∂t (cid:18) z I dw ˜ K t ( z − , w ) π ( w ) (cid:19) = 1 z I dw ˙ w ( t )(1 − w ( t ) /z ) π ( w )= − I dw b ( w ( t )) z (1 − w ( t ) /z ) π ( w )= − I d ˜ wz (1 − ˜ w/z ) b ( ˜ w ) π ( ˜ w ( − t )) ∂ ˜ w/∂w = P − (cid:18)(cid:18) b ( z ) π ( z ( − t )) ∂z/∂z ( − t ) (cid:19) ′ ( z ) (cid:19) (5.5)We used (5.1) in the last step. Similarly, P − (cid:18) ∂∂z (cid:18) b ( z ) z I dw − w ( t ) /z π ( w ) (cid:19)(cid:19) = P − (cid:18) b ′ ( z ) I dw π ( w ) z (1 − w ( t ) /z ) − b ( z ) I dw π ( w ) z (1 − w ( t ) /z ) (cid:19) = P − (cid:18) b ′ ( z ) P − (cid:18) π ( z ( − t )) ∂z/∂z ( − t ) (cid:19) + b ( z ) P − (cid:18)(cid:18) π ( z ( − t )) ∂z/∂z ( − t ) (cid:19) ′ ( z ) (cid:19)(cid:19) = P − (cid:18)(cid:18) b ( z ) π ( z ( − t )) ∂z/∂z ( − t ) (cid:19) ′ ( z ) (cid:19) (5.6) β = 2 Let us now consider the equation of motion for β = 2. To start with, let π ( z, t ) :=exp( t∂ z ) π ( z, ≡ P k ≥ π k ( t ) z − k − be the image of π ( z, ≡ P k ≥ π k z − k − by the semi-group generated by ∂ z . One may check by inspection that π ( z, t ) = 1 z ∞ X l =0 π l z − l l ! ∞ X m =0 ( l + 2 m )!(2 m )! (cid:18) tz (cid:19) m ; (5.7)in Fourier modes one gets π k ( t ) = ⌊ k/ ⌋ X m =0 (cid:18) k m (cid:19) π k − m t m . (5.8)Next we compute K t ( z − , w ) in the Hermite case (see example in Section 1). By def-inition π ( z, t ) = exp( t D ) π ( z, 0) = z H dw K t ( z − , w ) π ( w, D π )( z ) := σ ( π ( z ) + zπ ′ ( z )) − ( β − π ′′ ( z ). In order to exponentiate the semi-group D , we consider the formalseries ρ ( ζ, t ) := X k ≥ π k ( t ) k ! ζ k (5.9)related to π ( ζ, t ) by a Mellin transform. Through this non-local transform ∂ z becomesthe multiplication by − ζ , and the multiplication by z becomes the derivative ∂ ζ , hence43 ≡ − ( β − ζ − σ ζ∂ ζ is a first-order operator. Looking for a solution of the form ρ ( ζ, t ) ≡ e f ( t ) ζ ρ ( g ( t ) ζ, g = − σ g, ˙ f = − β − σ f − ( β − 1) (5.10)which can be solved straightforwardly, yielding ρ ( ζ, t ) = e f ( t ) ζ ρ ( e − t/σ ζ, 0) (5.11)with f ( t ) = − σ (1 − e − ( β − t/σ ) . Inverting now the Mellin transform, we remark that ρ ( ζ, t ) is given in (5.11) as the image by exp( f ( t ) ζ ) ≡ exp( f ( t ) ∂ z ) of a transformed initialcondition ˜ ρ ( e − t/σ ζ ) = P k ≥ e − kt/σ π k k ! ζ k , associated to ˜ π ( z, 0) = P k ≥ e − kt/σ π k z − k − .Hence we get π ( z, t ) = 1 z X k ≥ e − kt/σ π k z − k k ! X m ≥ (cid:18) k + 2 m m (cid:19) (cid:18) f ( t ) z (cid:19) m (5.12)from which we finally obtain an explicit formula for K t in the Hermite case,ˆ K t ( z − , w ) = X k ≥ ( e − t/σ w/z ) k X m ≥ (cid:18) k + 2 m m (cid:19) (cid:18) f ( t ) z (cid:19) m (5.13)extending (4.49). In Fourier modes this is π k ( t ) = ⌊ k/ ⌋ X m =0 e − ( k − m ) t/σ (cid:18) k m (cid:19) ( f ( t )) m π k − m . (5.14) Let u ∈ A + . We prove here the following result, ∂ t ′ (cid:18)I dww u ( w ) ∂ w ( K t − t ′ ( z − , w )) K t ′ − s ( w − , ζ ) (cid:19) == I dww ( u ( w ) b ′ ( w ) − u ′ ( w ) b ( w )) ∂ w ( K t − t ′ ( z − , w )) K t ′ − s ( w − , ζ ) (5.15)Namely, ∂ t ′ (cid:0) ∂ w ( K t − t ′ ( z − , w )) K t ′ − s ( w − , ζ ) (cid:1) == ∂ w ( ∂ t ′ ( K t − t ′ ( z − , w )) K t ′ − s ( w − , ζ ) + ∂ w ( K t − t ′ ( z − , w ))) ∂ t ′ ( K t ′ − s ( w − , ζ ))(5.16)is the sum of two terms. We use the second Kolmogorov formula (4.29) for the the first one,and the first Kolmogorov formula (4.28) for the second one; using the fundamental relations445.1, 5.2, 5.3) yields I dww u ( w ) ∂ w ( ∂ t ′ ( K t − t ′ ( z − , w ))) K t ′ − s ( w − , ζ ) = I dww u ( w ) ∂ w (cid:18) b ( w ) I dαα (1 − w/α ) K t − t ′ ( z − , α ) (cid:19) K t ′ − s ( w − , ζ )= I dww u ( w ) ∂ w (cid:0) b ( w ) ∂ w ( K t − t ′ ( z − , w )) (cid:1) K t ′ − s ( w − , ζ ) (5.17)and I dww u ( w ) ∂ w ( K t − t ′ ( z − , w )) ∂ t ′ ( K t ′ − s ( w − , ζ )) = − I dww u ( w ) ∂ w ( K t − t ′ ( z − , w )) I dαw (1 − α/w ) b ( α ) α K t ′ − s ( α − , ζ )= I dw u ( w ) ∂ w ( K t − t ′ ( z − , w ))) P − (cid:18)(cid:18) b ( w ) w K t ′ − s ( w − , ζ ) (cid:19)(cid:19) ′ ( w )= I dw u ( w ) ∂ w ( K t − t ′ ( z − , w ))) ∂ w (cid:18) b ( w ) w K t ′ − s ( w − , ζ ) (cid:19) = − I dw b ( w ) w K t ′ − s ( w − , ζ ) ∂ w (cid:0) u ( w ) ∂ w ( K t − t ′ ( z − , w ))) (cid:1) (5.18)Hence the result. One finds in the formula (4.51) for L f the time-derivation f ( t ) ∂ t . By definition, it acts onlocal functionals of { ( τ k ( t )) t ≥ } k ∈ Z as an infinitesimal change of coordinates, f ( t ) ∂ t · Z F ( s, ( τ k ( s )) k ) ds := X k ≥ Z f ( s ) ∂∂y k F ( s, ( y l ) l ) (cid:12)(cid:12) y = τ ( s ) ˙ τ k ( s ) ds. (5.19)In particular, for a linear functional, one finds f ( t ) ∂ t · Z g ( s ) τ k ( s ) ds = Z g ( s ) f ( s ) ˙ τ k ( s ) ds = − Z dds ( f ( s ) g ( s )) τ k ( s ) ds. (5.20)This action of f ( t ) ∂ t extends in a natural way (by duality) to an action on local func-tionals of ( τ k ( · )) k ≥ and ( δδτ k ( · ) ) k ≥ . Restricting to local functionals, we impose0 ≡ f ( t ) ∂ t · (cid:18)D Z γ ( s ) ∂/∂τ k ( s ) ds, Z g ( s ) τ k ( s ) ds E(cid:19) = D f ( t ) ∂ t · Z γ ( s ) ∂/∂τ k ( s ) ds, Z g ( s ) τ k ( s ) ds E + D Z γ ( s ) ∂/∂τ k ( s ) ds, f ( t ) ∂ t · Z g ( s ) τ k ( s ) ds E (5.21)so f ( t ) ∂ t · Z γ ( s ) ∂/∂τ k ( s ) ds = − Z f ( s ) ˙ γ ( s ) ∂/∂τ k ( s ) ds. (5.22)45n particular, f ( t ) ∂ t · ˆ ψ − ( z, t ) = − β / · z I dζ Z ds f ( s ) ∂∂s ( [0 ,t ] ( s ) K t − s ( z − , ζ )) [ ∂/∂τ ( ζ, s )= β / (cid:26) f ( t ) [ ∂/∂τ ( z, t ) + 1 z Z t ds f ( s ) I dζ ∂ t ( K t − s ( z − , ζ )) [ ∂/∂τ ( ζ, s ) (cid:27) , (5.23)compare with the straightforward time-derivative formula (4.39). Obviously the two formu-las coincide when f ≡ K , we may express (5.23) somewhat differ-ently. First f ( t ) ∂ t · ˆ ψ − ( z, t ) = f ( t )( ∂ t ˆ ψ − )( z, t )+ β / z Z t ds ( f ( s ) − f ( t )) I dζ ∂ t ( K t − s ( z − , ζ )) [ ∂/∂τ ( ζ, s ) . (5.24)Then, by (4.39),( ∂ t ˆ ψ − )( z, t ) = ˆ φ − ( z, t ) + β / z Z t ds I dζ ∂ t ( K t − s ( z − , ζ )) [ ∂/∂τ ( ζ, s )= ˆ φ − ( z, t ) − β / z Z t ds I dζ I dαα (1 − ζ/α ) K t − s ( z − , α ) b ( ζ ) [ ∂/∂τ ( ζ, s ) using (4 . φ − ( z, t ) + β / z Z t ds I dα K t − s ( z − , α ) (cid:16) P − (cid:16) b ( ζ ) [ ∂/∂τ ( ζ, s ) (cid:17)(cid:17) ′ ( ζ = α )= ˆ φ − ( z, t ) − β / Z t ds I dζ G + t − s ( z − , ζ ) b ( ζ ) [ ∂/∂τ ( ζ, s ) (5.25)and1 z β / Z t ds ( f ( s ) − f ( t )) I dζ ∂ t ( K t − s ( z − , ζ )) [ ∂/∂τ ( ζ, s )= − z β / Z t ds ˙ f ( s ) Z s dt ′ I dζ ∂ t ( K t − t ′ ( z − , ζ )) [ ∂/∂τ ( ζ, t ′ )= − z β / Z t ds ˙ f ( s ) Z s dt ′ I dζ I dξξ ∂ t ( K t − s ( z − , ξ )) K s − t ′ ( ξ − , ζ ) [ ∂/∂τ ( ζ, t ′ ) using (4 . − z Z t ds ˙ f ( s ) I dξ ∂ t ( K t − s ( z − , ξ ))( P − ˆ ψ s )( ξ )= − z Z t ds ˙ f ( s ) I dξ ∂ t ( K t − s ( z − , ξ )) ˆ ψ s ( ξ )= 1 z Z t ds ˙ f ( s ) I dξ ˆ ψ s ( ξ ) I dζζ (1 − ξ/ζ ) b ( ξ ) K t − s ( z − , ζ ) using (4 . − z Z t ds ˙ f ( s ) I dζ K t − s ( z − , ζ )( P − ( b ˆ ψ s )) ′ ( ζ )= Z t ds ˙ f ( s ) I dζ b ( ζ ) G + t − s ( z − , ζ ) ˆ ψ s ( ζ ) . (5.26)46 eferences [1] M. Adler, P. van Moerbeke. Hermitian, symmetric and symplectic random ensembles:PDEs for the distribution of the spectrum , Annals of Math. 153 (2001), 149.[2] P. Di Francesco, P. Ginsparg, J. Zinn-Justin. 2D gravity and random matrices , Phys.Rep. 254, 1-133 (1995).[3] F. J. Dyson. A Brownian motion model for the eigenvalues of a random matrix , J.Math. Phys. 3, 1191 (1962).[4] P. J. Forrester, Log-gases and random matrices , Princeton University Press (2010).[5] M. Henkel. J. Stat. Phys 75, 1023 (1994) and Nucl. Phys. B641, 405 (2002).[6] Johansson. On fluctuations of eigenvalues of random Hermitian matrices, Duke MathJ 91, 151-204 (1998).[7] V. G. Kac, A. K. Raina. Bombay Lectures on Highest weight representations of infinitedimensional algebras , World Scientific (1987).[8] I. K. Kostov. Conformal field theory techniques in random matrix models, arXiv:hep-th/997060.[9] R. B. Laughlin, Phys. Rev. Lett. , 1395-8 (1983).[10] A. Mironov, A. Morozov, Phys. Lett. B 252 (19990) 47.[11] P.C. Martin, E. D. Siggia, H. A. Rose, Phys. Rev . A 8, 423 (1973).[12] D. Revuz, M. Yor. Continuous martingales and Brownian motion , Springer-Verlag(1991).[13] C. Roger, J. Unterberger. The Schr¨odinger-Virasoro algebra , Theoretical and Mathe-matical Physics, Springer (2012).[14] T. Tao’s blog on Brownian motion, https://terrytao.wordpress.com/2010/01/18/254a-notes-3b-brownian-motion-and-dyson-brownian-motion/[15] J. Unterberger. H¨older-continuous paths by Fourier normal ordering , Comm. Math.Phys. (1), 16636 (2010).[16] J. Unterberger. Mode d’emploi de la th´eorie constructive des champs bosoniques, avecune application aux chemins rugueux , Confluentes Mathematici4