Electric-magnetic duality implies (global) conformal invariance
aa r X i v : . [ h e p - t h ] M a y Electric-magnetic duality implies (global) conformalinvariance
Sung-Pil Moon , Sang-Jin Lee , Ji-Hye Lee and Jae-Hyuk Oh Department of Physics, Hanyang University, Seoul 133-791, Korea
Abstract
We have examined quantum theories of electric magnetic duality invariant vectorfields enjoying classical conformal invariance in 4-dimensional flat spacetime. We extendDirac’s argument about “the conditions for a quantum field theory to be relativistic” to“those for a quantum theory to be conformal”. We realize that electric magnetic dualityinvariant vector theories together with classical conformal invariance defined in 4- d flatspacetime are still conformally invariant theories when they are quantized in a way thatelectric magnetic duality is manifest. [email protected] [email protected] [email protected] e-mail:[email protected] Introduction
Electric magnetic duality is originally observed from Maxwell equations, which describe oneof the fundamental forces in nature. Under switching ~E → ~B and ~B → − ~E , where ~E iselectric field and ~B is magnetic field (without considering any electric and magnetic sources),the Maxwell equations are invariant[1]. The duality is extended to string theory and variouskinds of field theories of free massless fields with various spins, sometimes to those in curvedspacetime e.g. Maxwell system in de Sitter spacetime and to approximate non-Abelian dualities[2, 3, 4, 5, 6].One of the interesting directions of developing electric magnetic duality is a research ifelectric magnetic duality ensures that certain classical symmetry of a system is retained whenthe system is quantized(e.g. see [8]). In [8], the authors argue that electric magnetic dualitycan ensure if a classical vector field theory enjoying Lorentz symmetry is still Lorentz invarianteven when it is quantized.The pioneering argument started from a paper by Dirac[7] in 1962. In his paper, he dis-cussed this issue as follows. It is not manifest if a quantum field theory keeps its classicalsymmetry(symmetry of the classical Lagrangian and equations of motion) because of (e.g.)the ordering issue of the field variables(due to the second quantization rule on them). Sincea state in quantum field theory can change to another representation by unitary transformand its dynamics is described by unitary time evolution, acting symmetry generators(spatialtranslation, rotation and boost, temporal translation) on that state, then if the second quan-tization is consistent with the algebra of the symmetry generators, then this ensures that thesymmetry retains in its quantum field theory.More precisely, he introduces a canonical pair of quantum fields as ξ and η satisfying[ ξ, η ′ ] = δ, (1.1)where prime denotes that the field variable depends on prime coordinate i.e. η ′ = η ( x ′ ), δ = δ d ( x − x ′ ), d -dimensional δ -function and so it is an equal-time commutator . ξ maybecome a field variable in the theory and η is its canonical conjugate. From them, he constructsa momentum density K s and introduces an energy density U , which provide the representationof the symmetry generators, where the index s is space index . It turns out that such symmetrygenerators constructed from K s and U satisfy Poincare algebra if the energy density satisfiesthe following commutation relation:[ U, U ′ ] = K t,t δ + 2 K t δ ,t , (1.2)where A ,s ≡ ∂A∂x s .By using this observation, the authors in [8] discovered the following: Suppose a vectorfield theory in 4- d flat spacetime which enjoys electric magnetic duality and Lorentz symmetryis quantized in a way that electric magnetic duality is manifest, more precisely it is requested For further discussion, even if we develop every mathematical equation in terms of d , in fact we restrictourselves to d = 3 case only. We will use s, t, r, u to be spatial indices running from 1 to 3. B as , B bt ] = ǫ ab ǫ stu δ ,u , (1.3)where a, b = 1 , SO (2) indices related to electric magnetic duality rotation, ǫ is fullyanti-symmetric tensor, ~ B = ~E and ~ B = ~B . One can define the momentum density and theenergy density from the fields B as as K r = − B as B bt ǫ ab ǫ str and U = f ( h, v ) , (1.4)where h = 12 B as B bt δ ab δ st , v = K r K r (1.5)and f ( h, v ) satisfies the following condition( f ,h ) + 4 f ,h f ,v + 4( f ,v ) = k, (1.6)for some constant k . The momentum density generates Lie derivative along a spatial vectorfield v i as L v Φ( B ) = [Φ , R d d xv s K s ] for some field Φ. It turns out that such an energy densitysatisfies the commutation relation that Dirac suggested in his paper. Therefore, one can findout that the vector field theory is manifestly Lorentz invariant when it is quantized.In this paper, we have extended such discussion to conformal symmetry. Our motivation isthat U (1) vector field theory in 4 − d flat spacetime, whose Lagrangian density is comprised ofits kinetic term only, is conformally invariant, since its stress energy tensor vanishes. Thus, onemay ask if quantum version of such kind of classical field theory is still conformally invariantwhen its second quantization rule manifestly enjoys electric magnetic duality transform .In fact, we have shown that the theory is still conformal by examining conformal algebrawith the similar manner that Dirac studied. In section 2, we develop the conditions that themomentum and the energy densities satisfy for this. It turns out that the energy density stillsatisfies (1.2) and therefore the momentum density and the energy density that Dirac suggestedalso satisfy conformal algebra under one condition that conformal dimension of the energydensity is d + 1. The simplest example for such case is U = h .In section 3, we conclude that since a specific class of the energy density (1.4) whoseconformal dimension is d + 1 obtained in [8] satisfy the same commutation relation (1.2), thenconformal symmetry is retained in such quantum theory of U (1) vector field which is manifestlyinvariant under electric magnetic duality rotation.The final issue to discuss is central charge. There possibly is conformal anomaly which showsup in OPE’s of energy momentum tensors, then transformation rule of the field variables andthe momentum and energy densities will be affected by the anomaly. However, as long as werestrict ourselves in global conformal symmetry, central charge cannot affect the transformationrules. For example, in 2- d CFT, the central charge contribution to the transformation of stressenergy tensor is given by derivative of the transformation parameters.2
Conditions for a 4- d quantum field theory to be con-formal In this section, we extend Dirac’s argument about conditions for a quantum field theory toretain Poincare symmetry to conformal symmetry.
Conformal algebra
Conformal algebra in d + 1-dimensional space time is given by[ D, P µ ] = − P µ , [ D, κ µ ] = κ µ , [ κ µ , P ν ] = − g µν D + L µν ) , (2.7)[ κ ρ , L µν ] = ( g ρµ κ ν − g ρν κ µ ) , [ P ρ , L µν ] = g ρµ P ν − g ρν P µ [ L µν , L ρσ ] = g νρ L µσ + g µσ L νρ − g µρ L νσ − g νσ L µρ , and the others vanish , where D is dilatation, κ µ is special conformal, P µ is translation and L µν is rotation and boostgenerators . g µν is d + 1-dimensional flat spacetime metric, whose signature is chosen as g µν = diag(+ , − , − , ..., − ).The symmetry generators are sorted to two different classes. The first class is a set of thegenerators having the quantum fields transform in spatial directions and the second class isthose forcing them transform in temporal direction. The former provides unitary transform ofthe fields in a given spacelike hypersurface and the later does dynamics of the fields. Momentum density
We first examine the generators having the fields transform in spatialdirections. For this, we decompose these generators into spatial and temporal parts as P µ → P s , P , L µν → L st , L t , (2.9) κ µ → κ s , κ and D → D ( s ) + D ( t ) , where we have defined the spatial parts of the symmetry generators in terms of a momentumdensity, K s as P t = Z K t d d x, L rs = Z ( x r K s − x s K r ) d d x (2.10) D ( s ) = − Z x s K s d d x, κ t = Z ( − x t x r K r + x r x r K t ) d d x To specify field variables, V (1) t , V (2) t and the momentum density in our vector theory, weintroduce variables ξ s and η s as V (1) t = η t , V (2) t = ξ t , and K t = η u ξ u,t − ( η u ξ t ) ,u , (2.11) The generators are given by D = x µ P µ , L µν = x µ P ν − x ν P µ , κ µ = 2 x µ x ν P ν − x ν x ν P µ , (2.8)in terms of translation generator, P µ . η s and ξ s form a canonical pair as[ ξ t , η ′ s ] = δ ts δ, (2.12)where δ ts is Kronecker’s delta whereas δ is d -dimensional delta function.By using canonical commutation relation of ξ s and η s , transformation rules of the fieldvariables are obtained as[ V (1) t , P r ] = V (1) t,r (2.13)[ V (1) t , L rs ] = x r V (1) t,s − x s V (1) t,r + ( − δ rt V (1) s + δ st V (1) r )[ V (1) t , D ( s ) ] = − x s V (1) t,s + (∆ V (1) t ) , [ V (1) t , κ s ] = − x s x r V (1) t,r + x r x r V (1) t,s + (2 δ ts x r V (1) r − x t V (1) s + 2∆ x s V (1) t )and [ V (2) t , P r ] = V (1) t,r (2.14)[ V (2) t , L rs ] = x r V (1) t,s − x s V (1) t,r + ( − δ rt V (1) s + δ st V (1) r )[ V (2) t , D ( s ) ] = − x s V (1) t,s + (∆ V (1) t ) , [ V (2) t , κ s ] = − x s x r V (2) t,r + x r x r V (2) t,s + (2 δ ts x r V (2) r − x t V (2) s + 2∆ x s V (2) t )where ∆ = d − = 1, which are conformal dimensions of the field variables, V (1) t and V (2) t respectively.From these we can obtain the following relations:[ K t , P r ] = K t,r , (2.15)[ K t , L rs ] = x r K t,s − x s K t,r − δ rt K s + δ st K r , [ K t , D ( s ) ] = − x r K t,r + (∆ + ∆ + 1) K t [ K t , κ s ] = − x s x r K t,r + x r x r K t,s + 2 δ st x r K r + 2(∆ + ∆ + 1) x s K t − x t K s , which provide the commutation relations of conformal algebra for the spacetime indices µ and ν to be restricted in µ, ν = 1 , ..., d . Energy density
To complete the conformal algebra(2.7), we need to examine the temporalparts of the generators. To do this, we define a local quantity, “energy density” U and expressthese generators by it as P = Z U d d x, L t = Z x t U d d x, (2.16) D ( t ) = 0 , κ ( t )0 = Z x s x s U d d x. This energy density is scalar under spatial parts of symmetry transforms and we suppose thatit has conformal dimension ∆ E , so it might transform as below:[ U, P t ] = U ,t , [ U, L st ] = x s U ,t − x t U ,s , (2.17)[ U, D ( s ) ] = − x s U ,s +∆ E U , [ U, κ s ] = − x s x r U ,r + x r x r U ,s +2∆ E x s U P , P t ] = 0 , [ P , L st ] = 0 , [ P s , L t ] = − δ st P , [ L t , L rs ] = δ ts L r − δ tr L s (2.18)[ D ( s ) , P ] = − P , [ D ( s ) , L t ] = 0 , [ κ r , L t ] = δ rt κ , [ κ , L st ] = 0[ D ( s ) , κ ] = κ ( t )0 , [ κ , P t ] = − L t , [ κ s , P ] = − L s , [ κ , κ s ] = 0 , under one condition that the conformal dimension of the energy density ∆ E = d + 1 . (2.19)They are the commutation relations between temporal and spatial parts of the generators.Finally, we request the commutation relations between temporal parts of the generators tocomplete our discussion. They are given by[ P , P ] = 0 , [ L t , L s ] = L st , [ P , L t ] = P t , [ κ , L t ] = κ t , (2.20)[ κ , P ] = − D ( s ) , [ κ , κ ] = 0 , These are translated to the following equations by using (2.16):
Z Z [ U, U ′ ] d d xd d x ′ = 0 (2.21) Z Z x t x ′ s [ U, U ′ ] d d xd d x ′ = Z ( x s K t − x t K s ) d d x (2.22) Z Z x t [ U, U ′ ] d d xd d x ′ = Z K t d d x (2.23) Z Z x s x s x ′ t [ U, U ′ ] d d xd d x ′ = Z (2 x t x s K s − x s x s K t ) d d x (2.24) Z Z x s x s [ U, U ′ ] d d xd d x ′ = 2 Z x s K s d d x (2.25) Z Z x u x u x ′ s x ′ s [ U, U ′ ] d d xd d x ′ = 0 (2.26)The remaining task is to find out commutation relation between energy densities satisfyingthe above relations. We start with the most general form of the energy density commutationrelation as Dirac suggested[7]. It is[ U, U ′ ] = aδ + b r δ ,r + c rs δ ,rs + d rst δ ,rst + ..., (2.27)where the coefficients in front of δ -functions are functions of x s only. If we switch U and U ′ ,by its anti commuting nature we have[ U ′ , U ] = aδ − b ′ r δ ,r + c ′ rs δ ,rs − d ′ rst δ ,rst + ..., (2.28)= aδ − ( b r δ ) ,r + ( c rs δ ) ,rs − ( d rst δ ) ,rst + ... = δ ( a − b r,r + c rs,rs − d rst,rst + ... )+ δ ,r ( − b r + 2 c ru,u − d rsu,su + ... )+ δ ,rs ( c rs − d rsu,u + ... ) . a − b r,r + c rs,rs − d rst,rst + ... (2.29)0 = 2 c rs,s − d rst,st + ... (2.30)0 = 2 c rs − d rsu,u + ... (2.31) ... (2.29) gives a solution of a as a = α r,r , where 2 α r = b r − c rs,s + d rst,st − ..., (2.32)and (2.30) means that c ru,u is indeed second derivative, then Z (2 α r − b r ) d d x = 0 , and Z x s (2 α r − b r ) d d x = 0 , (2.33)since 2 α r − b r = − c rs,s + d rst,st − ... → (second derivative and higher)By using these, we derive more useful relations as Z [ U, U ′ ] d d x ′ = α r,r (2.34) Z x ′ s [ U, U ′ ] d d x = x s α r,r − b s (2.35)After all, we plug (2.27) into (2.22-2.25) to fix coefficients of δ -functions(and derivative ofthem) on the right hand side of (2.27). The relation(2.34) directly solves (2.21). (2.23) gives Z K t d d x = Z x t α r,r d d x = Z α t d d x = 12 Z b t d d x, (2.36)where we have used (2.34). From this, we get the most general form of the solutions α r and β r as α t = K t + β tr,r + ζ ,t and b t = 2 K t + ¯ β tr,r + ¯ ζ ,t , (2.37)where β t , ¯ β t , ζ and ¯ ζ are arbitrary functions of x s . (2.22) provides Z ( x s K t − x t K s ) d d x = Z x t ( x s α u,u − b s ) = 12 Z d d x ( x s b t − x t b s ) (2.38)= 12 Z d d x (2 x s K t − x t K s + x s ¯ β tr,r − x t ¯ β sr,r + x s ¯ ζ ,t − x t ¯ ζ ,s ) , This relation restricts ¯ β st to be Z ( ¯ β ts − ¯ β st ) d d x = 0 . (2.39)and similarly Z ( β ts − β st ) d d x = 0 . (2.40)6ext, consider (2.24), which is given by2 Z x s K s d d x = Z x s x s α t,t d d x = Z x t ( K t + β tr,r + ζ ,t ) , (2.41)which provides conditions for β st and ζ as Z ( β tt + dζ ) d d x = 0 , (2.42)Moreover, (2.25) becomes Z (2 x t x s K s − x s x s K t ) d d x = Z x s x s ( x t α r,r − b t ) (2.43)= Z (2 x t x s K s − x s x s K t ) d d x + Z { x t (2 β rr + 2( d + 2) ζ − ζ ) + 2 x s ( β st + β ts − ¯ β ts ) } d d x, Then, from this we get Z { x t (2 β rr + 2( d + 2) ζ − ζ ) + 2 x s ( β st + β ts − ¯ β ts ) } d d x = 0 (2.44)Finally we examine (2.26). (2.37) satisfies this under a condition that Z { x t x u (2 β ut − ¯ β ut ) + x s x s (2 β tt − ¯ β tt + 2(2 + d ) ζ − (2 + d ) ¯ ζ + c uu ) } = 0 . (2.45)Minimal solutions of the coefficients in front of δ -functions on the right hand side of (2.27)are given by 2 α t = b t = 2 K t , and β st = ¯ β st = ζ = ¯ ζ = c rs ... = 0 (2.46)Therefore, the minimal solution of the commutation relation between the energy densitieswhich satisfies conformal algebra becomes[ U, U ′ ] = K t,t δ + 2 K t δ ,t . (2.47) d vector theories The main consequence of the last section is (2.47). Once we quantize our vector field theoryas (1.3) and define the momentum and the energy densities as (1.4), then this satisfies[8][
U, U ′ ] = − εδ st ( K s + K ′ s ) δ ,t , (3.48)where ε = 0 or −
1. Conformal algebra is consistently constructed from the energy densityonly when the conformal dimension of energy density is ∆ E = 4 in 4-dimensional spacetime.The simplest candidate for this is U = h , since B as has conformal dimension 2.7 cknowledgement We would like to thank Alfred D. Shapere for the useful discussion. J.H.O thanks his W .J .This work is supported by the research fund of Hanyang University(HY-2013) only. References [1] S. Deser and C. Teitelboim, Phys. Rev. D 13, 1592(1976), S. Deser, J. Phys. A 15, 1053(1982).[2] M. Henneaux and C. Teitelboim, Phys. Rev. D , 024018 (2005) [gr-qc/0408101].[3] S. Deser and D. Seminara, Phys. Lett. B , 317 (2005) [hep-th/0411169].[4] S. Deser and A. Waldron, Phys. Rev. D , 087702 (2013) [arXiv:1301.2238 [hep-th]].[5] S. Deser and D. Seminara, Phys. Rev. D , 081502 (2005) [hep-th/0503030].[6] D. P. Jatkar and J. -H. Oh, JHEP , 077 (2012) [arXiv:1203.2106 [hep-th]].[7] P. A. M. Dirac, REVIEW OF MODERN PHYSICS, VOLUME 34, NUMBER 4, 592-596,“The Condition for a Quantum Field Theory to be Relativistic”.[8] C. Bunster and M. Henneaux, Phys. Rev. Lett.110