The Planar Thirring Model with K\"ahler-Dirac Fermions

K\"ahler's geometric approach in which relativistic fermion fields are treated as differential forms is applied in three spacetime dimensions. It is shown that the resulting continuum theory is invariant under global U($N)\otimes$U($N)$ field transformations, and has a parity-invariant mass term, both symmetries shared in common with staggered lattice fermions. The formalism is used to construct a version of the Thirring model with contact interactions between conserved Noether currents. Under reasonable assumptions about field rescaling after quantum corrections, a more general interaction term is derived, sharing the same symmetries but now including terms which entangle spin and taste degrees of freedom, which exactly coincides with the leading terms in the staggered lattice Thirring model in the long-wavelength limit. Finally truncated versions of the theory are explored; it is found that excluding scalar and pseudoscalar components leads to a theory of six-component fermion fields describing particles with spin 1, with fermion and antifermion corresponding to states with definite circular polarisation. In the UV limit only transverse states with just four non-vanishing components propagate. Implications for the description of dynamics at a strongly interacting renormalisation-group fixed point are discussed.


Introduction
This paper concerns relativistic fermions interacting strongly in three spacetime dimensions, in the context of a field theory known as the Thirring model with Lagrangian density Here the fields ψ i ,ψ i are reducible spinors, so the Dirac matrices γ µ are 4 × 4.
In Euclidean metric they obey γ µ = γ † µ and {γ µ , γ ν } = 2δ µν . The index i = 1, . . . , N runs over N distinct fermion species. The contact interaction between conserved fermion currentsψγ µ ψ results in a repulsive force between fermions, but attraction between fermion and antifermion. A question of interest is then whether in the massless limit m → 0 a bilinear condensate ψ ψ = 0 forms as a result of strong interactions, leading to the dynamical generation of fermion mass.
It is natural to analyse bilinear condensation in terms of symmetry breaking. In three dimensions there are two elements of the reducible Dirac algebra γ 4 and γ 5 which anticommute with the kinetic term of (1). Accordingly for m → 0 (1) is invariant under the following field rotations: ψ → e iα 1 ψ;ψ →ψe −iα 1 : ψ → e α 45 γ 4 γ 5 ψ;ψ →ψe −α 45 γ 4 γ 5 ; (2) ψ → e iα 4 γ 4 ψ;ψ →ψe iα 4 γ 4 : ψ → e iα 5 γ 5 ψ;ψ →ψe iα 5 γ 5 . ( Together these rotations generate a U(2N) global symmetry which can be broken either explicitly by m = 0 or spontaneously by ψ ψ = 0 to U(N)⊗U(N), when the rotations (3) no longer leave the ground state invariant. Goldstone's theorem implies spontaneous symmetry breaking results in 2N 2 massless bosons in the theory's spectrum. It is suspected that symmetry breaking occurs for sufficiently large interaction strength g 2 and sufficiently small N; it is even possible that the resulting quantum critical point observed at g 2 c (N) might be a UV-stable fixed point of the renormalisation group, implying that a continuum limit at this point is possible. The fixed-point theory is expected to display universal features of the strongly-interacting dynamics characterised by the pattern of symmetry breaking. However, there are no small parameters to enable a systematic investigation of this phenomenon by analytic means. Determination of the critical exponents, and even the critical flavor number N c below which symmetry breaking can occur, are essentially non-perturbative problems.
A natural approach employs numerical simulations of lattice field theory (a recent review can be found in [1]). Most recent work uses a lattice fermion formulation which seeks to respect the U(2N) symmetry such as the SLAC derivative [2,3], or domain wall fermions [4,5]. However, there is also a substantial body of earlier simulations [6] employing the more primitive staggered formulation, in which fermion fields are represented by singlecomponent Grassmann objects χ,χ located on the sites of a cubic lattice. As well as U(N) flavor rotations, staggered fermions also enjoy a second U(N) global symmetry protecting them from acquiring mass, of the form where ε x = (−1) x 1 +x 2 +x 3 is an alternating sign in effect partitioning the sites x into distinct odd and even sublattices. This time, therefore, bilinear condensation drives a symmetry breaking U(N)⊗U(N) →U(N), resulting in just N 2 Goldstones. For a strongly-interacting system, therefore, we expect distinct fixed-point behaviour, and indeed simulations of the staggered model [7] support a critical N c ≈ 3.3 significantly larger than that found for the U(2N)-symmetric variants [3,4,5]. Moreover, simulation studies of the minimal staggered model with N = 1 1 [9] find critical indices indistinguishable from those of the Gross-Neveu model having the same global symmetries [10], even though in the latter case symmetry breaking can be described analytically using a 1/N expansion. Despite these apparent shortcomings the staggered Thirring model does exhibit interesting behaviour; in particular, the critical exponents characterising the fixed-point are particularly sensitive to the value of N < N c . Could there exist a continuum-based description of the corresponding fixed-point theories? One question which needs addressing is the significance of Nin a weak-coupling long-wavelength limit it is natural to interpret staggered femions in terms of N f = 2N autonomous flavors [8], or in modern parlance, each staggered flavor describes two continuum "tastes". However, even in early staggered Thirring studies [6] the factorisation of interaction currents into mutually-distinct taste sectors was not manifest, and there is no reason a priori to require this in a strongly-coupled setting. In what follows we will refer to the difficulty in separating taste and spin components as "spin/taste entanglement". A related question is how to engineer the U(N)⊗U(N) symmetry in the continuum where we have no lattice partition to help recover (4).
This paper will answer such questions using a framework introduced into lattice field theory by Becher and Joos in 1982 [11], who found that a version of the Dirac equation rooted in concepts of differential geometry originally noted by Kähler [12] in 1962 is in fact the formal continuum limit of staggered lattice fermions. As set out in the next few sections, in the Kähler-Dirac approach fermions are not spinor fields but rather are complexes of p-forms, where p = 0, 1, . . . , d, with d the dimension of spacetime. This is a natural way to prepare for transcribing continuum fields to a lattice [13]. Each pform has d C p components. In the four dimensional case analysed in [11] a fermion field has thus 1 + 4 + 6 + 4 + 1 = 16 components, which are recast as 4 independent tastes of 4-component spinor fields. For the case d = 3 to be developed in what follows, the corresponding field has 8 components recast as 2 tastes of reducible 4-component spinor. The algebraic details very closely mirror the assignment of spin/taste degrees of freedom to staggered lattice fermions in three spacetime dimensions originally set out by Burden and Burkitt [8].
The remainder is organised as follows. Sec. 2 is a brief but hopefully self-contained introduction to the differential geometry machinery required. Readers who are already expert will find our notations and conventions set out; those less familiar might also benefit from the helpful Appendix of [11], or a textbook such as [14]. Sec. 3 derives the equivalence between the free Kähler-Dirac equation, which with suitable notation assumes the same form in any dimension, and a continuum Dirac equation in three Euclidean dimensions describing two tastes of reducible spinor. The same framework is used in Sec. 4, following the introduction of a generalised scalar product between p-forms, to identify the fermion current that will be used in building the Thirring interaction term. Sec. 5 at last introduces the Thirring model action in the Kähler-Dirac language, and identifies both the U(N)⊗U(N) global symmetry and also an important parity symmetry shared in common with staggered lattice fermions. The Euclidean path integral is introduced permitting an explicit derivation of the Noether current associated with the symmetry corresponding to (4).
In Sec. 6 we begin to take the geometrical form of the theory more seriously by exploring the idea that in a suitably-regularised interacting theory the renormalisation of the field components should depend on p: the Thirring interaction term is modified in order to accommodate this possibility, and it is shown that the resulting terms when recast in a spinor basis exhibit spin/taste entanglement, and are in exact correspondence with the interaction derived from the staggered Thirring model [6] using the formalism of [8]. This demonstrates that the proposed p-dependent field rescaling is perfectly consistent with a properly-regularised lattice model, and also that spin/taste entanglement is not a lattice artifact but rather in fact a feature of an interacting continuum field theory. Finally in Sec. 7 the idea is taken a step further with the exploration of truncated actions resulting from retaining just field components with two consecutive values of p. The most interesting case corresponds to keeping just p = 1, 2, resulting in a theory of six-component spin-1 fermions, whose physical states are transverse, and for which fermion and antifermion are states of opposite polarisation. Sec. 8 summarises the paper's findings and speculates on the applicability of the exotic scenario of Sec. 7 to the physics of a putative renormalisation-group fixed point at strong coupling.

Mathematical Preliminaries
The theory to be developed uses the language of differential forms in threedimensional Euclidean spacetime. We will follow the presentation and notation of [11] closely. In this approach all physical quantities are viewed as p-forms defined in some vector space p Λ, with p = 0, 1, . . . , 3. A suitable basis for p Λ is given by dx H = dx µ 1 ∧ . . . ∧ dx µp with the exterior product satisfying where the sign factor ρ H,K = (−1) s , with s the number of pairs µ, ν ∈ H × K with µ > ν, and ρ ∅,H = ρ H,∅ = 1. With this in place any function Φ can be expanded as follows: The convention is that repeated indices are summed over, and no special significance is attached to whether an index is super-or subscript. In dealing with quantities defined on the whole space Λ = 3 p=0 p Λ, it is convenient to define the main automorphism and the main antiautomorphism where for p ∈ {0, . . . , 3} the combinatoric factor p C 2 takes values {0, 0, 1, 3}.
• Hodge Star where CH is the complement of H. In odd-dimensional Euclidean spacetimes ⋆⋆ = 1.
• Co-derivative and it immediately follows from d 2 = 0, ⋆⋆ = 1 that δ 2 = 0. The co-derivative's sign depends in general on p, d and the signature of the metric [14], which in (11) is captured by the use of the automorphisms (7,8). A convenient representation for its action is where the contraction operator enabling differentiation with respect to a differential is defined by

The Kähler-Dirac Equation
The starting point is the observation that (d − δ) 2 = −(dδ + δd) = ∂ µ ∂ µ = ∆, the Laplacian operator. Hence d − δ is in effect the square-root of the Laplacian, and therefore linear in momentum, while still local. It is thus a candidate for incorporating in a relativistic wave equation, as first written by Kähler [12]: The Kähler-Dirac equation (KDE) takes the same form in any spacetime dimension. The scalar parameter m is the fermion mass. Note that since d and δ implement ∆p = ±1, the equation only makes sense if Φ ∈ Λ, ie. Φ admits an expansion of the form (6), with components ϕ(x, H) having mass dimension 1 in three spacetime dimensions. It is helpful to define the Clifford product between differential forms: with particular instances It immediately follows from (9) and (12) that the KDE can be rewritten Now, the identity is strongly reminiscent of the defining relation {γ µ , γ ν } = 2δ µν for Dirac matrices in Euclidean metric, and suggests the operation dx µ ∨ furnishes a representation of the Dirac algebra in the 8-dimensional space spanned by dx H . The appropriate representation of the algebra in 3 spacetime dimensions was identified in [8] in a study of the staggered lattice fermion operator. It is the direct sum σ H ⊕ τ H of two inequivalent irreducible 2-dimensional representations generated by the Pauli matrices σ µ (µ = 1, 2, 3), and by τ µ = −σ µ . The Pauli matrices have the property σ * µ = σ T µ , where * denotes complex conjugation and T the matrix transpose. Analysis proceeds by identifying a new basis The key result is now where Roman indices a, b, c = 1, 2. The derivation of (20) makes repeated In order to express the KDE in the basis (19) we need the orthogonality relations implying Using (6) we then define where we have introduced fields u, d whose lower index a = 1, 2 will turn out to be associated with spinor degrees of freedom in the non-interacting case, and whose upper index b = 1, 2 will be associated with taste. The field transformations between bases are then: Combining the result (20) with the KDE equation (17) we deduce ie. the free Dirac equation for a two-taste four-component spinor field ψ = u ⊕ d, with Euclidean Dirac matrices defined We will refer to this familar form as the free KDE in the ψ-basis.

Interaction Current
In order to develop an interacting theory we will need a definition of a current in the Kähler-Dirac formalism. This requires the definition of a generalised scalar product (, ) p : Λ × Λ → 3−p Λ [12,11]. The two cases we will need have p = 0: and p = 1: where ε is the volume 3-form dx 1 ∧ dx 2 ∧ dx 3 . In components these are expressed and The following Green's formula identity is useful [12]: Next defineΦ = AΦ * as the solution of the adjoint KDE: A current 1-form is then given by Current conservation follows using (32,14,33): In the ψ-basis the current 1-form thus reads

Action and Symmetries
Now we have enough equipment to define the action and hence the Euclidean path intgeral. The action for free fields is For the Thirring model this is supplemented by a contact interaction of the form − g 2 4 j µ j µ , where the normalisation of the coupling strength, which has mass dimension -1, is somewhat conventional. In the language of forms this reads Using (⋆Φ, ⋆Ξ) 0 = (Φ, Ξ) 0 we arrive at the Thirring model action As a consequence of its construction from (Φ, Φ) bilinears the action (40) has two manifest global symmetries. First: This symmetry correponds to the conservation of fermion charge, and the corresponding Noether current is given by (34). Second, in the limit m → 0: which follows because d, δ both yield ∆p = ±1, and by inspection of the component expansion of (Φ, Φ) 1 (31). This is analogous to the chiral symmetry protecting fermions from additive mass renormalisation in d = 4. The corresponding Noether current is In order to translate to the ψ-basis, observe that the action of A in effect exchanges σ H and τ H in (25). It then follows straightforwardly that where we introduce two new hermitian γ-matrices obeying {γ 4 , γ µ } = {γ 5 , γ µ } = {γ 4 , γ 5 } = 0: From here it is straightforward to extend the model by introducing Finally, consider discrete parity inversion. In odd spacetime dimensions this is conveniently represented by inversion of all spacetime axes: (47) Note that the Noether currents (34,45), along with all bilinears of the form (Φ, Ξ) 1 , are parity-odd.
The Euclidean path integral is defined by where Φ,Φ are now Grassmann-valued andΦ is considered independent of Φ. We illustrate its use via a derivation of the Ward Identity for the divergence of the current j A ; for simplicity we consider only the free action (38). Consider the impact of the field transformation (43) where ω(x) is infinitesimal but now spacetime-dependent.
Now use (29) together with (Φ, Ξ) p = (−1) pC2 (Ξ, Φ) p and the definition (44) to write where in the second step we have integrated the first term by parts. Since the path-integral measure DΦDΦ = x,H dϕ(x, H)dφ(x, H) is formally invariant under the field transformation, the change of variables has no impact on the path integral, and we conclude Since (51) holds for any ω(x), we conclude the expectation value of the 3form in square brackets is identically zero, which is the Ward Identity. In the ψ-basis it has the familiar form

Impact of Quantum Corrections
Our treatment up to this point has been either classical or formal. In any application to a genuine interacting quantum field theory, it is inevitable that the theory will need to be regularised somehow in order to control the calculation of quantum corrections. As a concrete example, we have already discussed the close parallels between the KDE continuum formalism and staggered lattice fermions, and will assume without further discussion that the proof of [11] that the KDE is the formal continuum limit of staggered fermions continues to apply in 3 dimensions. Regularisation is essentially some kind of truncation of the degrees of freedom present in the classical field theory, and inevitably violates some of the symmetries of the classical theory. In many cases this leads to the requirement of renormalisation of both the fields and the coupling parameters in the theory, which depends on some physical scale. As a concrete example, consider the Thirring action in the ψ-basis (41), where the rotations (42,43) take the form ψ → e iθ ψ;ψ →ψe −iθ : ψ → e iθγ 5 ψ;ψ →ψe iθγ 5 Eqn. (41) also looks to be invariant under a U(2) rotation among the tastes indexed by b, c. Beyond that, in the limit m → 0 there is an additional symmetry ψ → e iθγ 4 ψ;ψ →ψe iθγ 4 as well as ψ → e θγ 4 γ 5 ψ;ψ →ψe −θγ 4 γ 5 , valid for any m. Rotations (54,55) combined with (53) and the taste rotations would generate a U(4N) global symmetry broken to U(2N)⊗U(2N) by a fermion mass. Our viewpoint is that this symmetry is not fundamental and can only be recovered in certain limits, such as long wavelength or weak coupling. We will proceed on the assumption that the geometric description employed in the KDE is more natural, so that after quantum corrections the field expansion of eqn. (6) is modified: Here a renormalised field Φ r is defined in terms of bare components ϕ(x, H) via wavefunction renormalisation constants Z p which depend on the interaction strength, the renormalisation scale and, crucially in this context, on the form degree p. This correction is covariant, in the sense that Z p is insensitive to rotations acting on the spacetime indices specific to ϕ(x, H), and the key symmetries (42) and (43) continue to be respected by Φ r even with Z p = 1. The form of (56) motivates a more general exploration of possible interaction currrents. In d = 3 the space of bilinear currents consistent with the four renormalisation constants Z p is spanned by ⋆(Φ, Φ) 1 , ⋆(Φ, AΦ) 1 , ⋆(BΦ, Φ) 1 and ⋆(BΦ, AΦ) 1 . Transcription to the ψ-basis for the first two of these is given in (34,45), and eg.
Now observe the following identities for the components of σ H : Recalling τ H = (−1) p σ H , we deduce a particularly convenient combination: Here the second component of the tensor product is a 2 × 2 matrix acting on taste indices. Similarly, In either case what emerges is an interaction current which although parity-odd and respecting the U(1)⊗U A (1) symmetries (42,43) no longer treats fermion tastes as independent degrees of freedom but rather entangles taste and spacetime rotations, contrary to what is expected for particle flavor degrees of freedom. Remarkably, the currents ⋆(Φ, Φ 1 ), ⋆(Φ, AΦ) 1 , ⋆([1 + 2B]Φ, Φ) 1 and ⋆([1 + 2B]Φ, AΦ) 1 all feature in equal weight contact interactions in the Thirring model formulated with staggered fermions on a 3d cubic lattice as derived in a basis with explicit spinor and taste indices using the formalism of [8], and given in Eqn. (2.12) of [6]. In view of the equivalence [11] between Kähler-Dirac fermions and the formal continuum limit of staggered lattice fermions, this result is not surprising.
It is now clear the interactions survive the long-wavelength a → 0 limit, where the lattice spacing a furnishes an explicit UV cutoff. Other terms entangling spinor and taste degrees of freedom which formally vanish as O(a) are also present in the lattice formulation [6]. The current analysis demonstrates that spin/taste entanglement is not a lattice artifact, but is rooted in a continuum action of the form (40) with U(N)⊗U A (N) symmetry. However, it is significant that such terms also emerge from a well-defined regularisation capable of exploring strongly-interacting dynamics.

Reduced Kähler-Dirac Fermions
Kähler-Dirac fermions offer a new language with which to discuss relativistic fermion dynamics. To quote Becher and Joos, "This differential geometric description of fermions might be a basis for the construction of different finds of field theoretic model," [11]. Once the differential geometric scaffolding has been removed, what kind of stories will we be able to tell? With motivation coming from a desire to understand novel structures at strongly-interacting fixed points, in this section we will hazard some speculations.
Let's start by expressing the free action (38) in the ϕ-basis (6), with Lagrangian density Each term in (61) is separately invariant under U(1)⊗U A (1) and parity (47). Now consider a reduced action containing just a subset of the p-form fields ϕ(x, H). The motivation comes from Eqn. (56), where we envisage a partition of {0, 1, 2, 3} into sets P, Q with Z p∈P ≫ Z p∈Q ≈ 0 arising, say, as a consequence of large anomalous scaling dimensions at a renormalisation group fixed point. Clearly only cases retaining consecutive values of p will result in propagating states. We consider two examples.

P = {0, 1}
If we truncate the field content to just p = 0, 1 there are four components φ ≡ (ϕ ∅ , ϕ µ ) T to keep track of. The Lagrangian density is Mφ.
which therefore vanishes identically for massless fermions. The propagator M −1 has components where the transverse projector such that P µν ∂ µ = 0, P µλ P λν = P µν , and trP = 2. In momentum space, all components manifest a particle pole at k 2 = −m 2 , but asymptotically scale differently: ∅∅ ∼ k −2 ; µν ∼ k −2 + k 0 ; ∅µ ∼ k −1 . We conclude that L 01 describes particles of mass m, and that the resulting effective theory is well-behaved in the IR regime k 2 m 2 . The singular part in the m → 0 limit has vanishing longitudinal component.

P = {1, 2}
In this case there are 6 field components ϕ µ , ϕ µν with Lagrangian density After a field redefinition L 12 can be rewritten Unlike Dirac matrices the λ µ don't obey a Clifford algebra, so the 6 × 6 matrix M in (69) is less straightforward to invert than a conventional Dirac operator. We start by checking its determinant, introducing the notation ∂ µ λ µ ≡ ∂ · λ: Now use (∂ · λ) 2 = −∆P and trP = 2 to write Again, the determinant vanishes if m = 0. The propagator exists for m = 0 and is given by The fact that M in the massless limit is invertible when acting on a transverse subspace is reminiscent of gauge theories, where the same issue occurs due to the redundancy of the field description as a consequence of an underlying invariance of the action under local gauge transformations of the form A µ → A µ + ∂ µ Λ. We can trace this to the invariance of (68), after integration by parts, under Here the ϑ(x) are Grassmann-valued fields, and the subscripts emphasise that independent shifts are applied to each fermi field. For this reason the mass term is not in general invariant under (75), consistent with the fact that M is invertible once m = 0. Further note that the textbook solution to defining a gauge-field propagator, namely to fix a gauge by adding a covariant term of the form ζ −1 (∂ µ A µ ) 2 to the action, would in this case yield terms of the form , eg. ∼ (ǫ µνλ ∂ µφνλ )(∂ ρ ϕ ρ ), consistent with U(1) A but violating parity. Rather, it makes more sense to regard the term m(χξ +ξχ) as the "gauge-fixing term".
We conclude that L 12 describes a fermion field transforming in the spin-1 representation of the rotation group, with some features reminiscent of a gauge field, namely that in the UV limit the only remaining degrees of freedom are transverse, ie. helicity eigenstates, so that six components are reduced to four. The Noether currents corresponding to symmetries (42,43) are given by Assigning 3 as the timelike direction, we identify a fermion charge operator −iλ 3 ⊗ σ 3 with ± restframe eigenstates F,F = (1, ∓i, 0, 1, ±i, 0) T , ie. fermions (antifermions) correspond to left(right)-and right(left)-handed circularly polarised χ(ξ)-states, which remain transverse under SO(3) rotations. Asymptotically the propagator scales as k −1 , as expected for a relativistic fermion. The propagator pole at k 2 = −m 2 again corresponds to a physical particle. Finally, we can use the fermion current of (76) to write the Lagrangian for the Thirring model based on L 12 , using the Υ-basis: In the same basis the invariances (42,43) read while parity is

Discussion
This paper has developed the description of relativistic fermions in the language of differential geometry, originally set out in [12], to three spacetime dimensions. The principal result is the specification of a continuum field theory sharing the same parity and global U(N)⊗U(N) invariances as the "staggered Thirring model" originally studied numerically using lattice field theory simulations in [6]. In our view this puts the staggered Thirring model on a firm footing as an interacting quantum field theory distinct from the U(2N)invariant version based on the continuum action (1), which is the focus of much recent numerical work. This result is entirely consistent with Becher and Joos' demonstration that Kähler-Dirac fermions are the correct continuum limit for staggered lattice fermions [11]. Beyond the weak-coupling and long-wavelength limits, we've seen that spin/taste entanglement is not merely a lattice artifact, but a genuine feature of an interacting continuum field theory: tastes are not the same as flavors.
An important consequence of regarding the ϕ-basis as more fundamental than the more familiar ψ-basis is the response to quantum corrections encapsulated in the proposed relation (56) relating renormalised to bare fields, in which multiplicative renormlisation depends solely on p, consistent with U(N)⊗U(N) symmetry. This was demonstrated explicitly in Sec. 6 through the recovery of interaction currents entangling spin and taste originally found in the staggered Thirring model. However, a more spectacular, if speculative consequence was worked out in Sec. 7, where the assumption of a strong hierarchy of the Z p arising due to large anomalous scaling dimensions in the vicinity of a renormalisation-group fixed point motivated the investigation of truncated actions retaining just two p-values. In particular the Lagrangian L rThir (77) was found to be particularly compelling, describing six-component spin-1 fermions, with fermions/antifermions being states of well-defined polarisation, and dynamics dominated by the four components lying in the transverse subspace in the UV limit. Could this exotica form the basis for a description of strongly interacting fixed-point dynamics? The answer must await a controlled non-perturbative investigation.
We conclude with a brief discussion of spin and statistics. The Lagrangian (77) describes spin-1 fermions which in the canonical approach to field quantisation would be represented by field operators with anticommutator {Υ α ( x, t), Υ † β ( x ′ , t)} = δ 2 ( x − x ′ )δ αβ . An immediate concern is the apparent contradiction with the spin-statistics theorem requiring Lorentzinvariant theories of anti-commuting fields to be quantised with half-integer spin representations of the Lorentz group. A symptom of the problem is revealed through the ground state expectation of the anticommutator of fields at arbitrary spacetime separation [15]: Hereλ µ are Minkowski space versions of the λ-matrices, we have assumed that all states are defined in the transverse subspace, and for field quantisation with the "wrong" statistics the P CT theorem dictates the appearance on the RHS of the symmetric solution of the Klein-Gordon equation (or its generalisation): where for free fields ω(k) = √ k 2 + m 2 . Now specialise to the case of a spacelike interval x = xx with |x| = 1. We find (83) The non-vanishing of the RHS of (81) outside the lightcone signals a violation of microcausality. This is a general result independent of the detailed form of the dispersion ω(k). For free fields the asymptotic properties of the modified Bessel functions in (83) can be used to to find lim x→∞ 0|{Υ(0),Ῡ( x))}|0 = i 2π and lim x→0 0|{Υ(0),Ῡ( x))}|0 = i 2πx 2 (ix ·λ ⊗ σ 3 ) + im 2πx (1 ⊗ σ 1 ); that is, the causality violation is localised to within roughly a Compton wavelength of the lightcone, but diverges as x → 0, although less severely than the x −3 behaviour of 3+1d [15]. Since microcausality is a desirable property for a fundamental theory, the correct relation between spin and statistics is a necessary ingredient of a complete quantum field theory. By hypothesis, however, the spin-1 action (77) serves only as an effective description of the dynamics near a UV fixed point, in the deep Euclidean regime k 2 → ∞ very far from the lightcone. The question of whether the spin-statistics linkage compromises the fixed-point description remains open.