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Quantum massless field in 1+1 dimensions Jan Derezin´skia, Krzysztof A. Meissnerb a Dept of Math. Methods in Physics, Warsaw University, Hoz˙a 74, 00-682 Warsaw, Poland, b Institute of Theoretical Physics, Warsaw University Hoz˙a 69, 00-681 Warsaw, Poland WepresentaconstructionofthealgebraofoperatorsandtheHilbertspaceforaquantummassless field in 1+1 dimensions. I. INTRODUCTION In the first construction we obtain a separable Hilbert 5 space and well defined fields, however we do not have a 0 It is usually stated that quantum massless bosonic vacuum vector. In the second construction, the Hilbert 0 fieldsin1+1dimensions(withnoncompactspacedimen- spaceisnon-separable,onlyexponentiatedfieldsarewell 2 sion) do not exist. With massive fields the correlation defined, but there exists a vacuum vector. Thus, neither n function of them satisfies all Wightman axioms. Nevertheless, we a believe that both our constructions are good candidates J dp 11 hΩ|φ(f1)φ(f2)Ωi=Z 2π2Epfˆ1∗(Ep,p)fˆ2(Ep,p) (1) f1o+r1adpihmyesnicsaiollnys.correct massless quantum field theory in Our constructions have supersymmetric extensions, (where Ep = p2+m2) is well defined but in the limit which we describe at the end of our article. 2 m → 0 diverges because of the infrared problem. The Inthe literatureknownto us the onlyplace where one v p 7 limitexistsonlyafteraddinganadditionalnonlocalcon- can find a treatment of massless fields in 1+1 dimension 5 straint on the smearing functions: similar to ours is [1,2] by Acerbi, Morchio and Strocchi. 0 Their construction is equivalent to our second (nonsepa- 8 fˆ(0,0)= dtdxf(t,x)=0. (2) rable) construction. We have never seen our first (sepa- 0 Z rable) construction of massless fields in the literature. 4 0 Underthisconstraintitisnotdifficulttoconstructmass- Acerbi, Morchio and Strocchi start from the C∗- / lessfieldsin1+1dimension(seeeg.[15],wheretheframe- algebraassociatedto the CCRoverthe symplectic space h work of the Haag-Kastleraxioms is used). of solutions of the wave equation parametrized by the p Massless fields are extensively used for example in initial conditions. Then they apply the GNS construc- - h string theory (albeit most often after Wick rotation to tion to the Poincaré invariant quasi-free state obtaining at the space with Euclidean signature). They also appear a non-regular representation of CCR. m asthe scalinglimitofmassivefields[6]. Usually,inthese In our presentation we prefer to use the derivatives of applications, the constraint (2) appears to be present at right and left movers to parametrize fields, rather than : v leastimplicitly. e.g. instringamplitudesoneimposesthe the initial conditions. We also avoid,as long as possible, Xi condition that sum of all momenta is equal to 0. Never- to invoke abstract constructions from the theory of C∗- theless, it seems desirable to have a formalism for mass- algebras, which may be less transparent to some of the r a less 1+1-dimensionalfields free of this constraint. readers. We explain the relationship between our for- In the literature there are many papers that propose malismandthatof[1,2]. Thesymmetrystructureofthis to use an indefinite metric Hilbert space for this pur- theoryissurprisinglyrich. Someoftheobjectsarecovari- pose[10,12,13,9,11,4]. Clearly,anindefinitemetricisnot ant only under Poincarégroup but there are others that physical and in order to determine physical observables are covariant under larger groups: A+(1,R)×A+(1,R), one needs to perform a reduction similar to that of the SL(2,R)×SL(2,R), Diff+(R)×Diff+(R), Diff+(S1)× Gupta-Bleuler formalism used in QED. The outcome of Diff+(S1). this Gupta-Bleuler-like procedure is essentially equivalent to imposing the constraint (2) [11]. Therefore, we II. FIELDS do not find the indefinite metric approach appropriate. In this paper we present two explicit constructions of (positive definite) Hilbert spaces with representations of The action of the 1+1 dimensional free real scalar the massless Poincaréalgebrain 1+1dimensions and lo- massless field theory reads calfields(oratleasttheirexponentials). Weallowalltest 1 functionsf thatbelongtotheSchwartzclassonthe 1+1 S = dtdx (∂ φ)2−(∂ φ)2 . (3) t x dimensionalMinkowskispace,withouttheconstraint(2). 2 Z We try to make sure that as many Wightman axioms as (cid:0) (cid:1) This leads to the equations of motion possible are satisfied. 1 (−∂2+∂2)φ=0. (4) with all other commutators vanishing. t x To proceed we choose two real functions σ (x) and R The solution of (4) is the sum of right and left movers, σ (x) satisfying L i.e. functions of (t−x) and (t+x) respectively: σˆ (0)=σˆ (0)=1 (11) φ(t,x)=φ (t−x)+φ (t+x). (5) R L R L We will often used ”smeared” fields in the sense andotherwisearbitrary. Tosimplify furtherformulaewe define the combinations φ(f) = dtdxφ(t,x)f(t,x), a (k):=a (k)−iσˆ (k)χ Z σR R R whereweassumethatf arerealSchwartzfunctions. Be- a† (k):=a†(k)+iσˆ∗(k)χ σR R R cause of (5), they can be written in the form a (k):=a (k)−iσˆ (k)χ σL L L φ(f) = φ(g ,g ), a† (k):=a†(k)+iσˆ∗(k)χ (12) R L σL L L where and therefore gˆR(k)=fˆ(k,k), gˆL(k)=fˆ(k,−k), a (k),a† (k′) =2πkδ(k−k′), σR σR (k ≥ 0) and the Fourier transforms of the test function h i f and g are defined as aσL(k),a†σL(k′) =2πkδ(k−k′), h [a (k),pi] =σˆ (k), fˆ(E,p):= dtdx f(t,x) eiEt−ipx, (6) σR R a† (k),p =−σˆ∗(k), Z σR R h[a (k),pi] =σˆ (k), ∞ σL L gˆ(k):= dtg(t)e−ikt. (7) a† (k),p =−σˆ∗(k). (13) Z−∞ σL L h i The function g corresponds to right movers and g to R L Now we are in a position to introduce the field oper- left movers. Note that ator φ(g ,g ), depending on a pair of functions g , g R L R L satisfying (8). gˆ (0)=gˆ (0)=:gˆ(0). (8) R L gˆ(0) is real, because function f is real. φ(g ,g )= dk (gˆ (k)−gˆ(0)σˆ (k))a† (k) We introduce the notation R L 2πk R R σR Z+(gˆ∗(k(cid:16))−gˆ(0)σˆ∗(k))a (k) (g |g ):= (9) R R σR 1 2 ∞ +(gˆL(k)−gˆ(0)σˆL(k))a†σL(k) 1 dk 2π ǫlցim0 k gˆ1∗(k)gˆ2(k)+ln(ǫ/µ)gˆ1∗(0)gˆ2(0), +(gˆL∗(k)−gˆ(0)σˆL∗(k))aσL(k) Z ǫ +gˆ(0)p. (cid:17) (14)   where µ is a positive constant having the dimension of mass. For functions that satisfy gˆ(0) = 0, (g1|g2) is a The field φ(gR,gL) is hermitian and satisfies the com- (positive) scalar product – otherwise it is not positive mutation relation definite andthereforecannotbe useddirectlyinthecon- structionofaHilbertspace. Suchascalarproductcorre- [φ(gR1,gL1),φ(gR2,gL2)] sponds to quantization of the theory in a constant com- =(g |g )−(g |g ) R1 R2 R2 R1 pensating background. +(g |g )−(g |g ) L1 L2 L2 L1 In view of the infrared divergence we factorize the =i2Im(g |g )+i2Im(g |g ). (15) Hilbert space into two parts – one that is infrared safe R1 R2 L1 L2 and the second that in some sense regularizes the diver- Thecommutatorin(15)doesnotdependonthefunctions gent part. σ , σ . We introduce the creation a†(k), a†(k) and annihila- R L R L tion a (k), a (k) operatorsas well as pairs of operators R L (χ,p†), (χ†,p). They satisfy the commutation relations III. POINCARE´ COVARIANCE a (k),a†(k′) =2πkδ(k−k′), R R Let A (1,R) denote the group of orientation preserv- + ha (k),a†(k′)i =2πkδ(k−k′), ing affine transformations of the real line, that is the L L group of maps t 7→ at + b with a > 0. The proper h [χ,pi] =i (10) 2 Poincaré group in 1+1 dimension can be naturally em- IV. CHANGING THE COMPENSATING bedded in the direct product of two copies of A (1,R), FUNCTIONS + one for the right movers and one for the left movers. The infinitesimal generators of the right A+(1,R) It is importantto discuss the dependence of the whole group will be denoted HR (the right Hamiltonian) and constructiononthe choiceofcompensating functions σR DR (therightgeneratorofdilations)andtheysatisfythe and σL. commutation relations Let σ˜ and σ˜ be another pair of real functions sat- R L isfying (11). Set ξ (x) := σ˜ (x) − σ (x), ξ (x) := R R R L [D ,H ]= iH . R R R σ˜ (x)−σ (x). Note that ξˆ (0)=ξˆ (0)=0. Define L L R L The representation of these operators in terms of the dk creation and annihilation operators is given by U(ξ ,ξ )=exp R L 2πk (cid:18)Z dk HR = 2π a†σR(k)aσR(k), iχξˆR∗(k)aR(k)+iχξˆR(k)a†R(k) Z DR = 2i d2πk a†σR(k)∂kaσR(k)− ∂ka†σR(k) aσR(k) , +(cid:16) 12χ2(ξˆR∗(k)σˆR(k)−ξˆR(k)σˆR∗(k)) Z (cid:16) (cid:0) (cid:1) (1(cid:17)6) +R→L . (18) (cid:17)(cid:17) Their action on fields is given by Using the formula [H ,φ(g ,g )] =−iφ(∂ g ,0), R R L t R 1 eABe−A =B+[A,B]+ [A,[A,B]]+... (19) [DR,φ(gR,gL)] =iφ(∂ttgR,0), 2 and in the exponentiated form by we have eisHRφ(g ,g )e−isHR =φ(g (·−s),g , Ua (k)U−1 =a (k)−iχξˆ (k), R L R L R R R eisDRφ(gR,gL)e−isDR =φ(e−sgR(e−s·),gL). Ua†(k)U−1 =a†(k)+iχξˆ∗(k), R R R For (a,b) ∈ A+(1,R) we set ra,bg(t) := a−1g(a−1(t− UaL(k)U−1 =aL(k)−iχξˆL(k), b)) and Ua†(k)U−1 =a†(k)+iχξˆ∗(k), L L L R (a,b)=eilnaDReibHR. UχU−1 =χ (20) R R is a unitary representation of A (1,R), which acts and R + naturally on the fields: dk UpU−1 =p+ −ξˆ∗(k)a (k)−ξˆ (k)a†(k) R (a,b)φ(g ,g )R (a,b)† 2πk R R R R R R L R Z (cid:16) =φ(ra,bgR,gL). (17) −iχξˆR(k)σˆR∗(k)+iχξˆR∗(k)σˆR(k) +R→L). (21) Note, however, that r does not preserve the indefi- a,b nite scalar product (9) unless we impose the constraint Using these relations we get for example gˆ(0)=0: Uφ (g ,g )U−1 =φ (g ,g ), (r g |r g )=(g |g )−lnagˆ∗(0)gˆ (0). σ R L σ˜ R L a,b 1 a,b 2 1 2 1 2 Ua† U−1 =a† , σR σ˜R Similarly we introduce the left Hamiltonian HL and UH U−1 =H , (22) σR σ˜R the left generator of dilations D satisfying analogous L commutation relations and the representation of the left where we made explicit the dependence of φ and H on R A+(1,R). σ andσ˜. Thusthe twoconstructions—withσ andwith The Poincaré group generators are the Hamiltonian σ˜ — are unitarily equivalent. H = H +H , the momentum P = H −H and the R L R L boostoperatorΛ=D −D (theonlyLorentzgenerator R L in 1+1dimensions). The elements ofthe Poincarégroup V. HILBERT SPACE are of the form (a,b ),(a−1,b )∈A (1,R)×A (1,R). TheHilbertspaceofthesystemistheproductofthree R L + + spaces: H=H ⊗H ⊗H . H isthebosonicFockspace R L 0 R Thescalarproduct(g |g )+(g |g )isinvariantwrt spanned by the creation operators a†(k) acting on the R1 R2 L1 L2 R the proper Poincarégroup. vacuumvector|ΩRi. AnalogouslyHListhebosonicFock 3 space spannedby the creationoperatorsa†(k) acting on But for χ 6= 0, |β ,β i is not well defined as a vector L 1 R L the vacuum vector |Ω i. With the third sector H we in the Hilbert space. To see this we can note that the L 0 have essentially two options. If we take the usual choice normalizing constant C equals zero, because then H = L2(R,dχ) then we can define the vacuum state 0 dk (vacuum expectation value) but there does not exist a χ2 |σˆ (k)|2+|σˆ (k)|2 =∞. vacuum vector. On the other hand, we can take H = 1 2πk R L 0 Z l2(R), i.e. the space with the scalar product (cid:0) (cid:1) (Thefactthatoperatorsoftheform(25)havenoground (f|g)= f∗(χ)g(χ), (23) state is well known in the literature, see eg. [7]). In the nonseparable case H = l2(R), the expectation χ∈R 0 X value which is a nonseparable space. It may sound as a non- hΩ|·|Ωi=:ω(·) standard choice, it has however the advantage of pos- sessing a vacuum vector. The orthonormal basis in the is a Poincaré-invariant state (positive linear functional) latter space consists of the Kronecker delta functions δ for each χ ∈ R. In the nonseparable case, the operatoχr on the algebra of observables. If we take the separable case H = L2(R,dχ), the state ω can also be given a p, and therefore also φ(g ,g ), cannot be defined. But 0 there exist operators eispR, forLs ∈ R, and also eiφ(gR,gL). meaning, even though the vector Ω does not exist (since then δ is not well defined). The commutation relations for these exponential oper- 0 Note thatinthe nonseparablecasethe stateω canact ators follow from the commutation relations for p and onanarbitraryboundedoperatoronH. Intheseparable φ(g ,g ) described above. R L case we have to restrict ω to a smaller algebra of opera- In such a space the vacuum vector is given by tors,say,the algebra(ortheC∗-algebra)spannedbythe |Ωi=|Ω ⊗Ω ⊗δ i. (24) operators of the form eiφ(gR,gL). R L 0 Theexpectationvaluesoftheexponentialsofthe1+1- This vector is invariant under the action of the Poincaré dimensional massless field make sense and can be com- groupandtheactionofthegaugegroupU. Wenowprove puted, both in the separable and nonseparable case: that it is the unique vector with the lowestenergy. Note first that H is diagonal in χ ∈ R. Now for an arbitrary ω(exp(iφ(gR,gL))) Φ∈HR⊗HL and χ1 ∈R, =exp −1 dk |gˆ (k)|2+|gˆ (k)|2 . (27) R L 2 2πk hΦ⊗δχ1|H|Φ⊗δχ1i (cid:18) Z (cid:0) (cid:1)(cid:19) Notethattheintegralintheexponentof(27)istheusual = Φ| a†(k)+iχ σˆ∗(k) a (k)−iχ σˆ (k) Φ R 1 R R 1 R integral of a positive function, and not its regularization Z (cid:16)D (cid:0) (cid:1)(cid:0) (cid:1) E asin(9). Therefore,ifgˆ(0)6=0,thenthisintegralequals +R→L (25) +∞ and (27) equals zero. (cid:17) The “two-point functions” of massless fields in 1+1 Foranyχ ,theexpression(25)isnonnegative. Ifχ =0, 1 1 dimension, even smeared out ones, are not well defined. it has a unique ground state |Ω ⊗Ω i. R L Formally, they are introduced as If χ 6=0, then (25) has no ground state. In fact, it is 1 well known that a ground state of a quadratic Hamilto- ω(φ(g ,g )φ(g ,g )). (28) R,1 L,1 R,2 L,2 nian is a coherent state, that is given by a vector of the form If we use the nonseparable H = l2(R), then field op- 0 erators φ(g ,g ) is not well defined if gˆ(0) 6= 0, and R L |βR,βLi= thus (28) is not defined. If we use the separable Hilbert dk spaceH =L2(R,dχ), then φ(g ,g )φ(g ,g )are Cexp β (k)a†(k)+β (k)a†(k) |Ω ⊗Ω i 0 R,1 L,1 R,2 L,2 2πk R R L L R L unbounded operators and there is no reason why the (cid:18)Z (cid:16) (cid:17)(cid:19) state ω could act on them. Thus (28) a priori does not and C is the normalizing constant make sense. In the usual free quantum field theory, if mass is positive or dimension more than 2, the expec- 1 dk C =exp − |β (k)|2+|β (k)|2 . (26) tation values the exponentials depend on the smeared R L 2 2πk fields analytically, and by taking their second derivative (cid:18) Z (cid:19) (cid:0) (cid:1) at (g ,g ) = (0,0), one can introduce the 2-point func- If we set Φ=|β ,β i in (25), then we obtain R L R L tion. This is not the case for massless field in 1+1 di- hΦ⊗δ |H|Φ⊗δ i=|β (k)+iχ σ (k)|2+R→L. mension. χ1 χ1 R 1 R The above discussion shows that the problem of the that takes the minimum for non-positivedefinitenessofthetwo-pointfunction,soex- tensively discussed in the literature [4,12,13,9,11] does β (k)=−iχ σˆ (k), β (k)=−iχ σˆ (k). not exist in our formalism. R 1 R L 1 L 4 It should be noted that massless fields in 1+1 dimen- Y :=exp iχ dtσ (t)ψ (t) R R R sion do not satisfy the Wightman axioms [14]. In the (cid:18) Z (cid:19) separable case there is no vacuum vector in the Hilbert Note that Y is not a well defined operator, since space;inthenonseparablecasethereisavacuumvector, R σˆ (0)6=0. but there are no fields φ(f), only the “Weyl operators” R eiφ(f). Expressed in position representation the fields are given by VI. FIELDS IN POSITION REPRESENTATION φ(g ,g )= dt(g (t)−gˆ(0)σ (t))ψ (t) R L R R σR Z +R→L+gˆ(0)p. (31) So far in our discussion we found it convenient to use the momentum representation. The position representa- Since (g (t)−gˆ(0)σ (t))dt=0 the whole expressionis R R tion is, however, better suited for many purposes. well defined. Let W(t) denote the Fourier transform of the appro- R The commutator of two fields equals priately regularized distribution θ(k), that is 2πk [φ(f ),φ(f )]=i dt dt dx dx f (t ,x )f (t ,x ) 1 dk 1 2 1 2 1 2 1 1 1 2 2 2 W(t)= lim eikt+ln(ǫ/µ) (29) Z 2π ǫ→0(cid:18)Zk>ǫ k (cid:19) ×(sgn(t1−t2+x1−x2)+sgn(t1−t2−x1+x2)) 1 iπ = −γ −ln|µt|+ sgn(t) =W∗(−t) Notethatthecommutatoroffieldsiscausal–itvanishes E 2π 2 (cid:18) (cid:19) if the supports of f1 and f2 are spatially separated. where γ is the Euler’s constant. We can rewrite (9) as E ∞ VII. THE SL(2,R)×SL(2,R) COVARIANCE (g |g )= dtdsg∗(t)W(t−s)g (s) (30) 1 2 1 2 Z−∞ Massless fields in 1+1 dimension satisfying the con- To describe massless field in the position representa- straint (2) actually possess much bigger symmetry than tion we introduce the operators ψ (t) defined as just the A (1,R)×A (1,R) symmetry, they are covari- R + + ant wrt the action of SL(2,R)×SL(2,R) (for right and dk ψ (t)= a†(k)e−ikt+a (k)eikt , left movers). R 2πk R R WewillrestrictourselvestotheactionofSL(2,R)for, Z (cid:0) (cid:1) say,rightmovers. Firstweconsideritonthe leveloftest andsimilarlyforR→L. Notethatitisallowedtosmear functions. ψ (t) and ψ (t) only with test functions satisfying R L We assume that test functions satisfy gˆ(0)=0 and g(t)dt =0. g(t)=O(1/t2), |t|→∞. (32) Z Let Note that W(t−s) and isgn(t−s) are the correlator 2 and the commutator functions for ψR(t): C = a b ∈SL(2,R) (33) c d hΩ |ψ (t)ψ (s)Ω i=W(t−s), (cid:20) (cid:21) R R R R (i.e. ad−bc=1). We define the action of C on g by i [ψ (t),ψ (s)]= sgn(t−s), R R 2 dt−b (r g)(t)=(−ct+a)−2g . (34) C −ct+a and similarly for R→L. (cid:18) (cid:19) We introduce also Note that (34) preserves (32) and the scalar product dk dk ψσR(t)= 2πka†σR(k)e−ikt+ 2πkaσR(k)eikt (rCg1|rCg2)= (g1|g2), Z Z χ and is a representation, that is r r =r . =ψ (t)+ dsσ (s)sgn(t−s), C1 C2 C1C2 R 2 R We secondquantize r by introducing the unitary op- C Z erator R (C) on H fixed uniquely by the conditions R R as well as R→L. It is perhaps useful to note that formally we can write R (C)Ω =Ω , R R R ψ (t)=Y ψ (t)Y†, σR R R R at+b R (C)ψ (t)R (C)† =ψ . (35) where R R R R ct+d (cid:18) (cid:19) 5 Note that From the general representation of any classical solution R (C) dtg (t)ψ (t) R (C)† R R R R φ(t,x)=φ (t−x)+φ (t+x) (39) (cid:18)Z (cid:19) R L = dt(r g )(t)ψ (t). (36) C R R we get (in notation where f(±∞) stands for Z lim f(t)) t→±∞ C 7→ R (C) is a representation in H . Thus the oper- R R ators R (C) act naturally on fields satisfying (2) (and φ(t,∞)+φ(t,−∞)= R hence also gˆR(0)=0). φR(−∞)+φL(∞)+φR(∞)+φL(−∞) The fields without the constraint(2)are notcovariant =φ(∞,x)+φ(−∞,x) (40) with respect to SL(2,R)×SL(2,R), since this symmetry fails even at the classical level. What remains is the ItwillbeconvenienttodenotebythespaceofSchwartz A+(1,R)×A+(1,R) symmetry described in (17). Note functions on R by S and by ∂−1S the space of functions that A (1,R) can be viewed as a subgroup of SL(2,R): 0 + whose derivatives belong to S and satisfy the condition A (1,R)∋(a,b)7→C f(∞)=−f(−∞). + Weareinterestedonlyinthosesolutionsthatrestricted = a1/2 ba−1/2 ∈SL(2,R). (37) to lines of constant time and lines of constant position 0 a−1/2 belongto∂−1S (wewilldenotethemasF ). Neglecting (cid:20) (cid:21) 0 11 a possible globalconstantshift we therefore assume that Clearly, on the restricted Hilbert space, under the iden- they satisfy tification (37), R (a,b) coincides with R (C). R R φ(t,∞)+φ(t,−∞)=φ(∞,x)+φ(−∞,x)=0 (41) VIII. NORMAL ORDERING F is characterizedby two numbers 11 Inthetheorywithoutthecompensatingsectorthenor- lim φ(t,x) =− lim φ(t,x)=:c0, t→∞ t→−∞ mal ordering can be introduced in a standard way. In lim φ(t,x)=− lim φ(t,x)=:c . particular we have 1 x→∞ x→−∞ :eiφ(gR,gL): = e21(gR|gR)+21(gL|gL)eiφ(gR,gL) (38) It is natural to distinguish the following subclasses of solutions to (4): If the compensating sector is present then the theory does not act in the Fock space any longer and we do not • F – solutions that restricted to lines of constant 00 haveaninvariantparticlenumberoperator. Itishowever time and to lines of constant position belong to S possible (and useful) to introduce the notion of the nor- i.e. c =c =0. 0 1 malordering. ForWeyloperatorsitisbydefinitiongiven • F – solutions that restricted to lines of constant by (38). For an arbitrary operator, we first decompose 10 time belongtoS andrestrictedtolines ofconstant it in terms of Weyl operators, and then we apply (38). position belong to ∂−1S i.e. c =0. Notethatourdefinitionhasaninvariantmeaningwrtthe 0 1 change of the compensating function: in the notation of • F – solutions that restricted to lines of constant 01 (22) we have position belong to S and restricted to lines of constant time belong to ∂−1S i.e. c =0. U :eıφσ(gR,gL) :U−1 =:eıφσ˜(gR,gL) :. 0 0 There are several useful ways to parametrize elements Normal ordering is Poincaré invariant but suffers of F . anomaliesunderremainingA (1,R)×A (1,R)transfor- 11 + + mations(becausetheprefactorontherhsof(38)isinvari- 1. Initial conditions at t=0: antonlyunderthe Poincarégroup). Ifthe constraint(2) is satisfied, then normal ordering is SL(2,R)×SL(2,R) f0(x) = φ(0,x), (42) covariant. f1(x) := ∂tφ(0,x). Here, f ∈∂−1S, f ∈S. Note that 0 0 1 IX. CLASSICAL FIELDS 1 c = f (x)dx, c =f (∞). 0 1 1 0 In order to better understand massless quantum fields 2 Z in1+1dimensionitisusefultostudytheunderlyingclas- sicalsystem,thatisthe waveequationin1+1dimension (4). 6 2. Derivatives of right/left movers: {φ(g ,g ),φ(g ,g )} R1 L1 R2 L2 1 1 = f (x)f (x)dx− f (x)f (x)dx, (47) g (t):=− f′(−t)+ f (−t), 01 12 02 11 R 2 0 2 1 Z Z 1 1 g (t):= f′(t)+ f (t). = gR1(t)sgn(s−t)gR1(s)dtds L 2 0 2 1 Z + g (t)sgn(s−t)g (s)dtds Note that g ,g ∈S and they satisfy L1 L1 R L Z =Im(g |g )+Im(g |g ) (48) R1 R2 L1 L2 gR(t)dt=c0−c1, gL(t)dt=c0+c1. (43) 1 = ∂ φ (t)φ (t)dt Z Z t R1 R2 2 Z 1 3. Right/left movers: + ∂ φ (t)φ (t)dt. (49) t L1 L2 2 Z 1 φR(t)= 2 gR(t−u)sgn(u)du Above, (f0i,f1i) and (φRi,φLi) correspond to (gRi,gLi). Z The formula in (47) is the usual Poisson bracket for 1 φ (t)= g (t−u)sgn(u)du. (44) the space of solutions of relativistic 2nd order equations L L 2 (both wave and Klein-Gordon equations). (48) we have Z already seen in (15). Note that φR,φL ∈∂0−1S and they satisfy The Poisson bracket in F11 is invariant wrt to the conformal group fixing the infinities preserving sepa- 1 φ (∞)=−φ (−∞)= (c −c ), rately the orientation of right and left movers, that is R R 0 1 2 Diff (R)×Diff (R). In the case of F we can extend + + 00 1 this action to the full orientation preserving conformal φ (∞) =−φ (−∞)= (c +c ). (45) L L 0 1 2 group, that is Diff (S1)×Diff (S1), where we identify + + R togetherwith the pointatinfinity withthe unit circle. We can go back from (g ,g ) to (f ,f ) by R L 0 1 1 X. ALGEBRAIC APPROACH f (x)= g (s−x)sgn(−s)ds 0 R 2 Z 1 Among mathematical physicists, it is popular to use + g (s+x)sgn(−s)ds, 2 R the formalism of C∗-algebras to describe quantum sys- Z f (x)=g (−x)+g (x). tems. A description of massless fields in 1+1 dimension 1 R L within this formalism is sketched in this section. We can go back from (φ ,φ ) to (g ,g ) by To quantize the space F , we consider formal expres- R L R L 11 sions g = φ′ , g = φ′ . R R L L eiφ(gR,gL) (50) The unique solution of (4) with the initial conditions (42) equals equipped with the relations eiφ(gR1,gL1)eiφ(gR2,gL2) =eiIm(gR1|gR2)+iIm(gL1|gL2) φ(t,x)=φ (t−x)+φ (t+x). R L ×eiφ(gR1+gR2,gL1+gL2); It will be sometimes denoted by φ(gR,gL). (eiφ(gR,gL))† =eiφ(−gR,−gL). In the literature, one can find all three parametrizations of solutions of the wave equation. In particular, Linear combinations of (50) form a ∗-algebra, which we note that 3. is especially useful in the case of F00, since willdenoteWeyl(F11). (Ifwewant,wecantakeits com- then φR,φL ∈S. pletion in the natural norm and obtain a C∗-algebra). Note that in our paper we use 2. as the standard Note that the group Diff (R) × Diff (R) act on + + parametrization of solutions of the wave equation. We Weyl(F )by∗-automorphisms. Inotherwords,wehave 11 are interested primarily in the space F10. Note that F10 two actions are the solutions to the wave equation with f ,f ∈ S. Equivalently, for F10, the functions gR, gL sati0sfy1 Diff+(R)∋F 7→αR(F)∈Aut(Weyl(F11)), Diff (R)∋F 7→α (F)∈Aut(Weyl(F )), + L 11 g (t)dt= g (t)dt. (46) R L commuting with one another given by Z Z We equip the space F11 with the Poisson bracket, αR(F) eiφ(gR,gL) =eiφ(rFgR,gL), (51) which we write for all three parametrizations: αL(F)(cid:0)eiφ(gR,gL)(cid:1)=eiφ(gR,rFgL). (cid:0) (cid:1) 7 Above, Aut(Weyl(F )) denotes the group of ∗- for left movers. Suppose that the complex numbers 11 automorphisms of the algebra Weyl(F)) and r g(t) := β ,...,β , and β ,...,β denote the correspond- F R1 Rn L1 Lm 1 g(F−1(t)). ing insertion amplitudes and satisfy F′(t) Similarly Diff (S1)×Diff (S1) acts on Weyl(F ) by + + 00 β = β . ∗-automorphisms. Ri Lj The state ω given by (27) is invariant wrt A+(1,R)× Then the corresponXding verteXx operator is formally de- A+(1,R) on Weyl(F11) and wrt SL(2,R)×SL(2,R) on fined as Weyl(F ). 00 V(t ,β ;...;t ,β ;t ,β ;...;t ,β ) In our paper we restricted ourselves to Weyl(F10). R1 R1 Rn Rn L1 L1 Lm Lm The constructions presented in this paper give repre- := :exp(iφ(g ,g )):, (54) R L sentations of Weyl(F ) in a Hilbert space H and two 10 where commutingwithoneanotherstronglycontinuousunitary representations g =β δ +···+β δ , (55) R R1 tR1 Rn tRn A+(1,R)∋(a,b)7→RR(a,b)∈U(H), gL =βL1δtL1 +···+βLmδtLm. (56) A (1,R)∋(a,b)7→R (a,b)∈U(H). Strictly speaking, the rhs of (54) does not make sense + L asanoperatorintheHilbertspace. Infact,inorderthat implementing the automorphisms (51): eiφ(gR,gL) be a well defined operator, we need that αR(a,b)(A)=RR(a,b)ARR(a,b)†, (52) 2dπkk|gˆR(k)−gˆ(0)σˆR(k)|2 αL(a,b)(A)=RL(a,b)ARL(a,b)†. (53) +R 2dπkk |gˆL(k)−gˆ(0)σˆL(k)|2<∞. (57) In the caseof the algebraWeyl(F00) the same is true for This is notRsatisfied if gR or gL are as in (55) and (56). SL(2,R). Nevertheless,proceedingformally,wecandeducevari- InsectionVwedescribedtworepresentationsthatsat- ousidentities. Forinstance,wehavethePoincarécovari- isfy the above mentioned conditions. The first, call it ance: πI, represents Weyl(F10) in a separable Hilbert space. R (a,b )R (a−1,b ) R R L L Its drawback is the absence of a vacuum vector – a ×V(t ,β ;...;t ,β ;t ,β ;...) Poincaré invariant vector. The second call it π , rep- R1 R1 Rn Rn L1 L1 II resents Weyl(F10) in a non-separable Hilbert space. It ×RL†(a−1,bL)RR†(a,bR) has an invariant vector |Ωi. =V(at +b ,β ;...;a−1t +b ,β ; We can perform the GNS construction with ω. As R1 L R1 Rn R Rn a−1t +b ,β ;...). a result we obtain the representation π together with L1 L L1 II the cyclic invariant vector Ω. The description of this If in addition β = 0, then a similar identity is true Ri construction for massless fields in 1+1 dimension can be for SL(2,R)×SL(2,R). found in [1], Sect. III D, and [2] Sect. 4. Note, however, Clearly, we hPave that we have not seen the representation π in the liter- I ature,eventhoughone canargue thatit is in some ways ω(V(tR1,βR1;...;tRn,βRn;tL1,βL1;...;tLm,βLm)) superior to π . II Let us make a remark concerning the role played by 1, βRi =0; = . (58) the functions (σ ,σ ). We note that F is a subspace R L 00 ( 0, Pβ 6=0. of F of codimension 1. Fixing (σ ,σ ) satisfying (11) Ri 10 R L allowsustoidentifyF10 withF00⊕R. Thusany(gR,gL) The folPlowing identities are often used in string the- satisfying(46)isdecomposedintothedirectsumof(gR− ory for the calculation of on–shell amplitudes. Suppose gˆ(0)σR,gL−gˆ(0)σL) and gˆ(0)(σR,σL). that tR1,...,tRn are distinct, and the same is true for Of course, similar constructions can be performed for t ,...,t . Then, using (29), we obtain L1 Ln thealgebraWeyl(F )orWeyl(F ). Intheliterature,al- 11 01 V(t ,β ;t ,β )···V(t ,β ;t ,β ) gebrasofobservablesbasedonF appearinthecontext R1 R1 L1 L1 Rn Rn Ln Ln 01 of “Doplicher-Haag-Robertscharged sectors” in [15,5,6]. W(tRi−tRj))βRiβRj+W(tLi−tLj)βLiβLj =e i<j ! P XI. VERTEX OPERATORS ×V(t ,β ;...;t ,β ;t ,β ;...;t ,β ) R1 R1 Rn Rn L1 L1 Ln Ln Finally,letusmakesomecommentsabouttheso-called tRi−tRj −βRiβRj/2π tLi−tLj −βLiβLj/2π = vertex operators, often used in string theory [8]. Let δy iµeγE iµeγE denote the delta function at y ∈R. Yi<j(cid:16) (cid:17) (cid:16) (cid:17) moLveetrstRa1n,d...t,tR,n..∈.,Rt cor∈resRpocnodrrteospinosnedrtitoonsinfsoerrtriiognhst ×V(tR1,βR1;...;tRn,βRn;tL1,βL1;...;tLn,βLn). L1 Lm 8 dk XII. FERMIONS Hf = kb†(k)b (k) R 2π R R Z Massless fermions in 1+1 dimension do not pose such Df = i dk b†(k)k∂ b (k)− k∂ b†(k) b (k) . problems as bosons. The fields are spinors, they will be R 2 2π R k R k R R λ (t,x) Z (cid:16) (cid:0) (cid:1) (cid:17) written as λRL(t,x) . They satisfy the Dirac equation WehavetheusualcommutationrelationsforHRf andDRf (cid:20) (cid:21) and their action on the fields is anomaly-free: ∂ −∂ 0 λ (t,x) t x R =0. Hf ,λ (g ) =−iλ(∂ g ), 0 ∂t+∂x λL(t,x) R R R t R (cid:20) (cid:21)(cid:20) (cid:21) (cid:2)DRf ,λR(gR)(cid:3) =iλ (t∂t+1/2)gR . We will also use the fields smeared with real functions f, where the condition (2) is not needed any more: We have(cid:2)also the cov(cid:3)arianc(cid:0)e with respect(cid:1)to the conformal group SL(2,R)×SL(2,R). We need to assume λR(f) = λR(t,x) f(t,x)dtdx. that test functions satisfy λ (f) λ (t,x) L L (cid:20) (cid:21) Z (cid:20) (cid:21) g(t)=O(1/t), |t|→∞. (62) Because of the Dirac equation, they can be written as We define the action of λ (f)=λ (g ), λ (f)=λ (g ), R R R L L L a b C = ∈SL(2,R) (63) where gR and gL where introduced when we discussed c d (cid:20) (cid:21) bosons. For k >0, we introduce fermionic operators (for right on g by and left sectors)b (k) and b (k) satisfying the anticom- R L mutation relations (rf g)(t)=(−ct+a)−1g dt−b . (64) C −ct+a {b (k),b†(k′)}=2πδ(k−k′), (cid:18) (cid:19) R R Note that (64) has a different power than (34). It is a {b (k),b†(k′)}=2πδ(k−k′), (59) L L unitary representation for the scalar product (·|·) . f Wesecondquantizerf onthe fermionicFockspaceby with all other anticommutators vanishing. Now C introducing the unitary operator Rf (C) fixed uniquely R dk by the conditions λ (g )= g∗(k)b (k)+g (k)b†(k) R R 2π R R R R Z dk (cid:16) (cid:17) RRf (C)ΩR =ΩR, λ (g )= g∗(k)b (k)+g (k)b†(k) (60) L L 2π L L L L Z (cid:16) (cid:17) at+b The anticommutation relations for the smeared fields Rf (C)λ (t)Rf (C)† =(ct+d)−1λ . (65) R R R R ct+d read (cid:18) (cid:19) Note that C 7→Rf (C) is a unitary representationand it {λ†(g ),λ (g )}= g∗ (t)g (t)dt R R R1 R R2 R1 R2 acts naturally on fields: Z dk = 2πgˆR∗1(k)gˆR2(k) (61) RRf (C)λR(gR)RRf (C)† =λR(rCf gR), (66) Z and similarly for the left sector. Note the difference of the fermionic scalar product (61) and the bosonic one XIII. SUPERSYMMETRY (·|·). In terms of space-time smearing functions these anti- In this section we consider both bosons and fermions. commutation relations read Thus our Hilbert space is the tensor product of the {λ†(f ),λ (f )}=2 dtdxδ(t+x)f (t,x)f (t,x), bosonic andfermionic part. We assume that the bosonic R 1 R R 1 2 and fermionic operators commute with one another. Z Clearly, our theory is A (1,R)×A (1,R) covariant. In {λ†(f ),λ (f )}=2 dtdxδ(t−x)f (t,x)f (t,x). + + L 1 R L 1 2 fact, the right Hamiltonian and the generator of dila- Z tions forthe combinedtheoryareequalto H +Hf and R R Fermionicfieldsarecovariantwithrespecttothegroup D +Df . R R A (1,R)×A (1,R). We will restrict ourselves to dis- In the case of the theory with the constraint (2), we + + cussing the covariance for say, right movers. The right have also the SL(2,R)×SL(2,R). covariance. Hamiltonian and the right dilation generator are 9 On top of that, the combined theory is supersymmet- [4] S. De Bièvre and J. Renaud: A conformally covariant ric. The supersymmetry generators Q , Q are defined quantum field in 1+1 dimension, J.Phys. A34 (2001) R L as 10901-10919. [5] Buchholz, D.: Quarks, gluons, colour: facts or fiction?, dk Q = a† (k)b (k)+a (k)b†(k) Nucl. Phys. B469 (1996) 333-356. R 2π σR R σR R [6] D. Buchholz and R. Verch, Scaling algebras and renor- Z (cid:16) (cid:17) dk malization group in algebraic quantum field theory. II. Q = a† (k)b (k)+a (k)b†(k) (67) L 2π σL L σL L Instructive examples, Rev. Math. Phys. 10 (1998) 775- Z (cid:16) (cid:17) 800. Theysatisfythebasicsupersymmetryalgebrarelations [7] J. Derezin´ski: Van Hove Hamiltonians – exactly solvable without the central charge models of the infrared and ultraviolet problem, Ann. H. Poincaré 4 (2003) 713-738. {QR,QR}=2(HR+HRf ), [8] M.B. Green, J.H. Schwarz and E. Witten: Superstring {Q ,Q }=2(H +Hf), theory, Cambridge Univ.Press, Cambridge 1987. L L L L [9] G.W. Greenberg, J.K. Kang and C.H. Woo: Infrared {Q ,Q }=0. (68) R L regularization of the massless scalar free field in two- The action of the supersymmetric charge transforms dimensional space-time via Lorentz expansion, Phys. Lett. 71B (1977) 363-366. bosons into fermions and vice versa: [10] C. Itzykson and J.B. Zuber: Quantum Field Theory, [Q ,φ(g ,g )] =λ (g ), McGraw-Hill, 1980, Chap. 11. R R L R R [11] G. Morchio, D. Pierotti and F. Strocchi: Infrared and [Q ,φ(g ,g )] =λ (g ), L R L L L vacuum structure in two-dimensional local quantum field [QR,λR(gR)] =φ(∂tgR,0), theory models. The massless scalar field, Journ. Math. [Q ,λ (g )] =φ(0,∂ g ). Phys. 31 (1990) 1467-1477. L L L t L [12] N. Nakanishi: Free massless scalar field in two- Q dimensional space-time, Prog. Theor. Phys. 57 (1977) Thepairofoperators R behaveslikeaspinorun- Q 269-278. L der the Poincaré group.(cid:20)Even(cid:21)more is true: we have the [13] N. Nakanishi: Free massless scalar field in two- covariance under the group A (1,R)×A (1,R), which dimensional space-time: revisited, Z.Physik C. Particles + + for the right movers can be expressed in terms of the and Fields 4 (1980) 17-25. [14] R.F.StreaterandA.S.Wightman: PCT,spinandstatis- following commutation relations: tics and all that, W.A.Benjamin, New York-Amsterdam [H ,Q ]=0, 1964. R R [15] R.F. Streater and I.F. Wilde: Fermion states of a boson i [D ,Q ]=− Q . field, Nucl.Phys. B24 (1970) 561-575. R R R 2 E-mail addresses: AcknowledgementJ.D.waspartlysupportedbytheEuro- [email protected], pean Postdoctoral Training Program HPRN-CT-2002-0277, [email protected] thePolish KBNgrantSPUB127and2P03A02725. K.A.M. was partially supported bythePolish KBN grant 2P03B 001 25 and theEuropean Programme HPRN–CT–2000–00152. J.D. would like to thank S. DeBièvre, C. Gérard and C. Jäkel for useful discussions. [1] Acerbi, F., Morchio, G., Strocchi, F.: Infrared singular fields and nonregular representations of canonical com- mutationrelationalgebras,Journ.Math.Phys.34(1993) 899-914 [2] Acerbi, F., Morchio, G., Strocchi, F.: Theta vacua, charge confinement and charged sectors from nonregular representations of CCR algebras, Lett. Math. Phys. 27 (1993) 1-11 [3] Brattelli, O., Robinson D. W.: Operator Algebras and Quantum Statistical Mechanics, Volume 2, Springer- Verlag, Berlin, second edition 1996. 10