Spacelike deformations: Higher-spin fields from scalar fields

In contrast to Hamiltonian perturbation theory that changes the time evolution,"spacelike deformations"proceed by changing the translations (momentum operators). The free Maxwell theory is the first member of an infinite family of spacelike deformations of the complex massless Klein-Gordon quantum field into fields of higher spin.


Introduction
The basic idea of Hamiltonian perturbation theory is to start from a time zero algebra ("canonical commutation relations") equipped with a free time evolution, and perturb the free Hamiltonian such that the observables at later time Φ(t) := e iHt Φ 0 e −iHt (where H is the perturbed Hamiltonian) deviate from the free ones. We present here a "complementary" deformation scheme for free quantum field theories: fixing the algebra along the time axis, we deform the space translations, so as to obtain a different local quantum field theory in Minkowski space. Similar ideas were previously pursued in two spacetime dimensional models [6].
Despite the apparent similarity, there are many differences, though. Hamiltonian Perturbation Theory (PT) is well-known to be obstructed by Haag's theorem, which implies that the perturbation is possible on the same Hilbert space only locally. Globally, the perturbed vacuum state is not a state in the "free Hilbert space", so that one is forced to change the representation of the time zero algebra. The need of renormalization of the mass also shows that one is even forced to change the time zero algebra itself. More precisely, interacting quantum fields in general do not even exist as distributions at a fixed time (see, e.g., [12,13]). A recent approach [2], designed to avoid these obstructions, uses instead of a CCR time-zero algebra, an abstract "off-shell" C*-algebra of kinematical fields on spacetime that supports a large class of dynamics (one-parameter groups of time-evolution automorphisms). The invariant states under each dynamics, however, annihilate different ideals of the algebra ("field equations"), such that the corresponding GNS Hilbert spaces cannot be identified for any time-zero subalgebra.
In contrast, Wightman quantum fields can always be restricted to the time axis [1]. Our spacelike deformations are globally well-defined on the original Hilbert space. They consist in a redefinition of the generators of the spacelike translations (momentum operators). The perturbed fields away from the time axis are then defined as Φ(t, x) = e ix k P k Φ 0 (t, 0)e −ix k P k , where P k are the deformed generators.
In Hamiltonian PT, the "field content" is fixed by the choice of the free theory. The "particle content" is determined by the spectrum of the (renormalized) perturbed Hamiltonian, and may well change, e.g., when the interacting theory has bound states, or confinement occurs. Yet, the relation to the free particle content is usually not entirely lost.
In contrast, our spacelike deformations are (non-continuous) algebraic deformations that drastically change the field content without changing the Hamiltonian: e.g., one obtains the free Maxwell field by a deformation of a massless free scalar field.
In fact, we know spacelike deformations only for massless free fields, producing other massless free fields of any higher helicity (Sect. 2.3), or massive scalar fields (Sect. 3). The reason is that (a) we work on the one-particle space, from which the deformation passes to the Fock space by "second quantization"; and (b) we exploit conformal invariance in an essential way. The helicity deformations are in fact nonlinear deformations of the representation of the conformal Lie algebra, while the dilation operator enters the mass deformations.
We are therefore far from "interactions via deformation"; but our models illustrate the potential of a new approach, and more sophisticated new ideas may emerge from the present simple prototypes.

Helicity deformations 2.1 Background
The examples to be demonstrated rely on a recent observation in [7]: For the massless free fields of any integer helicity h > 0, the one-particle spaces H (h) are strict subspaces of the one-particle space H = H (0) of the complex massless free scalar field. More precisely, H (h) as representations U h ⊕ U −h of the Poincaré group extend to representations of the conformal group, whose restriction to the subgroup Möb × SO(3) is given by where D (ℓ) are the spin-ℓ representations of SO(3), and U (d) are the irreducible positiveenergy representations of Möb with lowest eigenvalue d of L 0 . The doubling is due to the "electric" and "magnetic" degrees of freedom. The same decomposition with h = 0 holds for the complex scalar field, where the doubling corresponds to the subspaces of charge ±1.
Möb × SO(3) is the subgroup of the conformal group that fixes the time axis x = 0. The vectors transforming in the displayed subrepresentations are spacelike derivatives of fields on the time axis, that transform like quasiprimary fields under Möb, applied to the vacuum vector Ω. For the scalar field, these fields are simply [3] P ℓ ( ∇)ϕ ( * ) (x)| x=t,0 with harmonic polynomials P ℓ of degree ℓ, transforming like spin-ℓ multiplets of quasiprimary fields of scaling dimension d = ℓ + 1. For h > 0, when the electric and magnetic fields are combined into a complex field tensor F j 1 ...j h , the equations of motion impose linear relations among the fields ∇ i 1 . . . ∇ ir F ( * ) j 1 ...j h (x)| x=t,0 . One finds [7] exactly one quasiprimary spin-ℓ multiplet (both for F and F * ) of scaling dimension ℓ + 1 for each ℓ ≥ h.
In [7], these facts were exploited to estimate the trace of e −βL 0 , whose finiteness then implies the split property for all finite-helicity massles free quantum field theories. Here, we take them as the starting point of spacelike deformation, as already speculated in [7].
To illustrate the idea, consider the case h = 1 (Maxwell). The Maxwell equations for F read The component fields F k (t, 0) on the time axis transform in the same way under Möbius transformations of the time axis and rotations, like the fields ∇ k ϕ(x, 0) of the complex massless Klein-Gordon theory. Similarly, the spin-2 fields (∇ i F j + ∇ j F i )(t, 0) transform in the same way as the fields (∇ i ∇ j − 1 3 δ ij ∆)ϕ(t, 0). Because the representations of the Möbius and rotation groups on the one-particle spaces are the same -except for the absence of the subrepresentation with ℓ = 0 in the Maxwell theory -we can algebraically identify these pairs of fields along the time axis. We get where the Maxwell equations dictate the anti-symmetric part in (2.2) as well as the absence of an ℓ = 0 contribution; the normalizations are fixed via the two-point functions, giving |α| 2 = 12.
The problem is apparent: the left-hand side of (2.2) is the derivative of the left-hand side of (2.1), which is not true for the right-hand sides. The spatial derivatives being implemented by the momentum operators P k , we conclude that while the Möbius and rotation generators of both theories (including the Hamiltonian P 0 ) can be identified, their spatial momentum operators must differ.
We are going to determine the momentum operators P k of the Maxwell theory as polynomials of the conformal and charge generators of the Klein-Gordon theory (and along with them the boosts and the generators of spatial special conformal transformations). Then, starting from the identification (2.1) as a definition of the Maxwell field on the time axis, and acting with U ( x) = e ix k P k on ϕ(t, 0), one obtains the Maxwell field everywhere in Minkowski space. The same works for any helicity h > 0.
As a second instance, we present the spacelike deformation of the massless scalar field into the massive scalar field in Sect. 3.
The mere existence of such deformations should not be too surprising, given that "all Hilbert spaces are the same". The noticeable facts are that the deformations fix parts of the symmetry, and that they can be given on the remaining generators by explicit formulae.

Preliminaries about the conformal Lie algebra
We denote by P µ , M µν , D, K µ the generators of translations, Lorentz transformations, dilations, and special conformal transformations in the conformal Lie algebra so(2, 4), respectively. Their commutators are explicitly In particular, we have the Lie subalgebras möb: and so (3): The conformal transformations of the massless Klein-Gordon field are given by the infinitesimal action of so(2, 4): The generators of m transform like vectors under so (3):
We finally list the commutation relations of the conformal generators and the charge operator with the anti-unitary PCT operator J:

Main result
Let H = H + ⊕ H − the one-particle space of the complex massless Klein-Gordon field, where the superscript ± stands for the eigenvalue ±1 of the charge operator Q. As representations of Möb × SO (3), both H ± decompose as represented on the one-particle space of the complex massless Klein-Gordon field.
The main result defines deformed generators P k (the translations of the deformed QFT) in terms of the generators of the scalar QFT on the subspace H (h) of the one-particle space of the scalar QFT.
where S k := ε kmn (P m M 0n + M 0n P m ) commute with E ℓ , and the coefficients a ℓ , b ℓ are real.
(i) The deformed generators P k , M 0k , K k satisfy the correct commutation relations (2.7) with the PCT operator.
(ii) Together with the undeformed generators P 0 , D, K 0 of möb and M kl of so (3), they satisfy the conformal Lie algebra (2.

3) on H (h) if and only if
and all coefficients b ℓ have the same sign.
(iii) The generators P k as specified by (ii) satisfy the mass-shell condition on H (h) : Proof: (i) is immediate by (2.7). Because S k transform as a vector under so (3), they can change ℓ by at most one. Inspection of the operators P k and M 0k shows that they both change ℓ by exactly one (see below). Hence S k commute with E ℓ . To prove (ii), we start with a Lemma. The generators of the scalar QFT act as follows on the one-particle space. In a schematic notation, where "×" indicates the deletion of an index j a or j b . In order to pass to the (improper) spin-ℓ vectors |j 1 . . . j ℓ ℓ t , one has to subtract contractions of derivatives, which due to the wave equation ∆ϕ = ∂ 2 t ϕ are given by where ". . . " stands for higher contractions. Thus, if X k = P k , M 0k , K k are written in the form with differential operators A, B s , C w.r.t. the time t as in (2.14e-g), then The higher contractions in (2.15) do not contribute because the vector operators X k can change ℓ by at most one. We refrain from displaying the explicit formulae. They are exactly the same for the oppositely charged vectors |j 1 . . . j s * t := ∇ j 1 . . . ∇ js ϕ * (t, 0)Ω and |j 1 . . . j ℓ * ℓ t in H − . From these expressions, it can be seen that the deformed generators make only transitions to spin ℓ + 1 (the A-term) and to ℓ − 1 (the B 0 and C 0 terms). One may then evaluate the commutator (2.13) on arbitrary vectors |j 1 . . . j ℓ ℓ t and equate the result with the desired right-hand side. This gives several conditions on the coefficients a ℓ and b ℓ in (2.9), that turn out to be equivalent to the system  The definition is unambiguous because if

Field algebras
is conformally covariant.
Because for any pair of spacelike separated doublecones, there is a conformal transformation g mapping the doublecones into the forward and backward lightcones, respectively, locality follows from the Huygens property of the scalar field (A(V + ) commutes with A(V − )) by covariance (see, e.g., [8]).

Field equations
At the level of fields, we identify (with the appropriate normalization factor) F ( * ) j 1 ...j h (t, 0) with the derivative fields p j 1 ...j h ( ∇)ϕ ( * ) (t, 0) (harmonic polynomials of spin ℓ = h) restricted to F (h) , and hence F ( * ) x) by the adjoint action of e ix k P k such that F In order to make the identification of the deformed field with the free helicity-h field, we have to establish the equation of motion [7] where F = E + iB, and E j 1 ...j h and B j 1 ...j h are symmetric traceless "electric" and "magnetic" tensors. On the time axis, we have by construction (with the appropriate normalization) This immediately implies Because 2(h + 1)b h = 1, the higher Maxwell equations hold on the time axis and on the vacuum vector: The complex conjugate higher Maxwell equations for F * = E − iB are guaranteed by the presence of the operator Q in (2.9), that switches the sign of i in the right-hand side of (2.18) for the vectors of charge −1.
At this point, it becomes apparent how the charge of the scalar field is re-interpreted as the sign of the helicity of the higher Maxwell field.
By applying the spacelike translations U ( x) to (2.20), we conclude that the higher Maxwell equations on the vacuum vector hold everywhere in Minkowski space. Because F ( * ) are local fields on the time axis, by conformal covariance they are local on Minkowski spacetime. Then the Reeh-Schlieder theorem ensures that the higher Maxwell equations hold as operator equations.

Mass deformation
A second, and much simpler, instance of spacelike deformation is the construction of the massive Klein-Gordon field as a deformation of the spacelike translations and boosts of the massless Klein-Gordon field. It is "complementary" to the corresponding Hamiltonian deformation treated in [5].
In this instance, we can just write down the deformed generators. Because the deformations turn out to concern the differential operators w.r.t. t, it will be advantageous to pass to the Fourier transform on the time axis: | . . . ω = dt e −iωt | . . . t . Because the massive oneparticle vectors have energy ω ≥ m, the spectral projection E m = θ(P 0 − m) will play the role of the projection E (h) in Sect. 2.3.
The Lie algebra of the Poincaré group again has a symmetric space decomposition h ⊕ m, where h = t ⊕ so(3) (t = time translations), and m is spanned by the spacelike momenta and the boost generators. By definition, the rotations M kl and the Hamiltonian P 0 remain undeformed. Thus, the deformed boosts M 0k also determine the deformed momenta P k = −i[P 0 , M 0k ].
We return to the real scalar field, and denote by ϕ 0 and ϕ m the massless and massive fields. Whenever P , P ′ are homogeneous polynomials of degree s, s ′ , the scalar product in spherical coordinates is where p 0 = p 2 + m 2 , p = p n, and dσ is the invariant measure on the sphere. Passing to the integration variable ω = p 0 , one has p 2+s+s ′ dp If we write |j 1 . . . j s t := ∇ j 1 . . . ∇ js ϕ 0 (t, 0)Ω as before, and m |j 1 . . . j s t := ∇ j 1 . . . The massive Poincaré generators act on m |j 1 . . . j s t in exactly the same way as the massless generators on |j 1 . . . j s t in (2.14a,d-f). In particular, the deformation preserves the Hamiltonian P 0 and the generators M kl of rotations. The deformation of the spacelike momenta gives immediately The mass-shell condition P 2 k = P 2 0 − m 2 is trivially fulfilled by (3.3). For the boosts, one gets Using (2.14a,b,e,f), this can be seen to be equivalent to (where the operator ordering has been adjusted so as to match the coefficient s + 3 2 ). Because the generators on the subspace E m H 0 arise by the unitary conjugation (3.2) of the generators on H m , they are self-adjoint and satisfy the Poincaré commutation relations. Indeed, the hermiticity, as well as the commutator i[P 0 , M 0k ] = − P k , can be verified without much effort. The explicit verification of the commutation relation i[ M 0k , M 0l ] = M kl requires another cumbersome computation, which gives This equals M kl on E m H 0 because M 0k P l +M l0 P k +M kl P 0 = ε klj W j , and the Pauli-Lubanski operator W µ = 1 2 ε µνκλ M νκ P λ vanishes in the massless scalar representation.

Summary
We have given two families of examples of spacelike deformations that allow to construct new quantum field theories by fixing the restriction of a given QFT to the time axis, and deforming only the "transverse" symmetry generators. The remarkable feature is that the scheme admits the change of discrete quantum numbers (the helicity in our first example). In both cases, it is true that we knew the expected deformation from the outset. But only in the mass deformation case did we use this knowledge to compute the deformed generators. In contrast, Prop. 2.2 is a uniqueness result, once the subspace is specified on which the deformation is supposed to be defined.
Both instances of spacelike deformation presented here make essential use of the envelopping algebra of the Lie algebra of the respective spacetime symmetry group (conformal, resp. Poincaré). It is a noticeable feature that in both cases, one extra element (the charge Q in the higher helicity case, the dilation operator D in the massive case) is needed for the deformation.
From the underlying pattern of inclusions of Hilbert spaces, we expect that one can deform any given helicity h ′ ≥ 0 to a helicity h > h ′ , and any given mass m ≥ 0 to a mass m > m ′ . On the other hand, increasing mass and spin simultaneously might not be possible by lack of an inclusion of one-particle representations of the subgroup fixing the time axis.
The case of interacting theories will need methods going beyond representation theory of spacetime symmetry groups.

Outlook
Our constructions may give insights into the modular theory of local algebras for massive theories [11], that is not as well known as for massless theories. Let us explain what we have in mind. The situation is very different in the mass deformation of Sect. 3. Because the spectral projection E m does not commute with the dilations, the latter are not defined on the subspace H m , and the BGL construction is not possible. Indeed, it is well-known that in the massive case, H m (R + ) (to be identified with H m (V + ) in the net on Minkowski spacetime) has trivial symplectic complement ( [9,10]), in contrast to the duality (iH 0 (R + )) ⊥ = H 0 (R − ) in the massless case. On the other hand, we know that the massive local subspace H m (I) of an interval I on the time axis coincides with the local subspace H m (O I ) for the doublecone O I spanned by I; and by the work [5] of Eckmann and Fröhlich, we have a local unitary equivalence between the massive and massless time-zero algebras. Specifically, there is a unitary operator U R such that for intervals I r = (−r, r) ⊂ R symmetric around t = 0 and r < R, one has H m (O r ) = U R H 0 (O r ) where O r is the causal completion of the time-zero ball of radius r. Thus, the modular groups of H m (O r ) are, for r < R, conjugate to the known modular groups of H 0 (O r ) by U R . Increasing R, the unitary U R will change, but the subspaces H(I r ) for r < R and their modular groups remain unchanged. Thus, the modular groups for r < R 1 < R 2 commute with U R 2 U * R 1 , and a more detailed investigation of the unitaries U R would be worthwhile to get a first insight into the hitherto unknown massive modular groups.
This information about the modular groups then passes to arbitary doublecones via the adjoint action of the deformed translations and boosts, as constructed in Sect. 3.