CERN-TH/2001-052

ZH-TH 06/01

hep-th/0103069

Remarks on Time-Space Noncommutative Field Theories

L. Alvarez-Gaumé,
J.L.F. Barbón ^{1}^{1}1On leave from the
Departmento de Física de Partículas. Universidade de Santiago
de Compostela, Spain. and R. Zwicky

Theory Division, CERN

CH 1211 Geneva 23

Switzerland

Institut für theoretische Physik.
Universitaet Zürich

Winterthurerstrasse 190, 8057 Zürich

Switzerland

Abstract

We propose a physical interpretation of the perturbative breakdown of unitarity in time-like noncommutative field theories in terms of production of tachyonic particles. These particles may be viewed as a remnant of a continuous spectrum of undecoupled closed-string modes. In this way, we give a unified view of the string-theoretical and the field-theoretical no-go arguments against time-like noncommutative theories. We also perform a quantitative study of various locality and causality properties of noncommutative field theories at the quantum level.

CERN-TH/2001-052

March 2001

## Introduction

Quantum field theories in noncommutative spaces [1, 2] with noncommutative time coordinate are notoriously ill-defined. Heuristically, noncommutative particles of momentum can be regarded as rigid, extended dipoles oriented along the four-vector , and interacting through the end-points [3]. It is already clear from this consideration that noncommutative time coordinates imply particles effectively ‘extended in time’. Thus, the breakdown of naive criteria of local causality, in the form of ‘advanced effects’ in tree-level scattering processes, should not come as a surprise (c.f. [4, 5]).

On a more technical level, these theories have no straightforward Hamiltonian quantization (see however [6]) and are defined, in an operational sense, via the Feynman diagram expansion [7, 8]. It is thus necessary to check explicitly the unitarity of the theory. Indeed, in a theory with built-in nonlocality in time, it is hard to imagine an appropriate notion of causality which is at the same time useful and nontrivial. It is more likely that these theories must be interpreted as S-matrix theories, in the same sense as critical string theory. Thus, from this very fundamental point of view, we may regard the existence of a consistent S-matrix as the weakest possible notion of causality.

More specifically, it was shown in [9, 10] that the stringy
regularization of such systems is ill-defined. Following
[11, 12], we can recover certain noncommutative field theories
(NCFTs) as an appropriate low-energy limit
of D-brane dynamics in the background of a constant electromagnetic field.
While this limit is smooth in string perturbation theory for the case of
a background magnetic field, it is problematic for a background electric
field^{1}^{1}1The marginal case of a null electromagnetic
background is also
smooth, c.f. [13]..
This is tied to the well-known instabilities of open-string
dynamics in the presence of electric fields [14],
i.e. beyond a maximal value
of the electric field, the D-brane becomes effectively tachyonic. Since
the time-noncommutativity is directly related to the electric field,
and this must be large (in string units) in the low-energy limit, one
finds that the low-energy limit always lies on the tachyonic regime
of D-brane dynamics. Therefore, consistency of the string background
requires the scale of time noncommutativity to be of the same order
of magnitude as the string scale.

The breakdown of the stringy regularization for time-NCFT is a strong hint at the inconsistency of these models. Still, one can imagine some contrived analytic continuation of the background parameters, so that the open-string perturbation theory does converge in the formal low-energy limit to the series of Feynman diagrams of the time-NCFT. In particular, a formal continuation of the closed string coupling and an exchange of the roles of space and time in the plane of the electric field would do the job (c.f. [10]).

Therefore, it is desirable to find an internal inconsistency of the Feynman expansion in time-NCFTs. Such an inconsistency was found in [15], where a violation of the unitarity cutting rules was reported in a number of examples. The authors of [15] showed that scattering amplitudes have extra singularities that cannot be understood in terms of unitarity thresholds. In itself, this does not furnish a no-go argument for time-NCFT, since Feynman diagrams in ordinary theories are known to present the so-called ‘anomalous thresholds’ whose interpretation in terms of unitarity rules is very problematic (see [16] for a review).

Thus, it is crucial to interpret physically the new singularities in order to evaluate the viability of time-NCFTs. One interesting possibility would be that NCFTs mimic open-string theory in the sense that unitarity of the S-matrix restricted to open-string initial and final states requires the introduction of closed strings as states in the asymptotic Hilbert space, since they contribute singularities in intermediate channels. In an analogous fashion, it is possible that the S-matrix of time-NCFT becomes unitary once we add appropriate new states to the asymptotic Hilbert space.

In fact, such a possibility is hinted at by the simplest example of a ‘noncommutative singularity’, i.e. the case of the normal-ordering correction to the propagator of massless theory in four dimensions. In the commutative theory (or at the level of planar diagrams in the noncommutative theory) the normal-ordering diagram has no analytic structure, since it contributes a quadratically divergent constant renormalizing the mass. On the other hand, the nonplanar diagram, viewed as a one-to-one scattering amplitude, is given by

(1) |

where and . Thus, we find a contribution to the imaginary part:

(2) |

Let us now suppose the initial energy positive and a single noncommutative plane . Then we can write the analog of the optical theorem for this quantity by introducing and obtain:

(3) |

Thus, we see that the cutting rules can be formally recovered if we introduce new states in the asymptotic Hilbert space with dispersion relation . They mix with the off-shell quanta with an effective coupling . All this is strongly reminiscent of the situation one finds in nonplanar open-string scattering amplitudes. In that case, nonplanar amplitudes show new poles, without a clear interpretation in terms of open-string intermediate states. It turns out that they just represent the amplitude for an open string to mix with closed strings. Therefore, in the particular case of the diagram considered here, the particles are analogous to closed strings. This analogy is sharpened by considering the ‘dipole’ picture of noncommutative particles [3], since this is equivalent to rigid open strings. Then, the new states arise from ‘cutting the dipole’ in the intermediate loop.

Notice that the effective coupling for producing the particles blows-up in the limit , reflecting a non-analiticity in the commutative limit –a typical UV/IR mixing. Another peculiar property of the particles is their lack of propagation in commutative spatial directions. The particles could be regarded as excitations of ‘-fields’, a generalization of those proposed in [17, 18] in order to reconcile the Wilsonian interpretation of the renormalization group and the IR divergences found in spatially noncommutative theories. The main difference is that these fields are associated to propagating particles rather than being Lagrange multipliers. Thus, they cannot be simply considered as formal devices but must be included in any attemp to consistently construct the time-NCFT.

In the following section, we study the general structure of the ‘noncommutative singularities’ at one-loop and attemp to give an interpretation along these lines. Using results from [19], we will show that the unitarity-violating singularities in time-NCFT can be manipulated into a form that is strongly reminiscent of undecoupled closed-string modes. Thus, the cutting rules can be formally satisfied by adding an appropriate set extra asymptotic degrees of freedom. On the other hand, unitarity is not restored in a strict sense, because the extra states are necessarily tachyonic. Thus, time-NCFT appears to be perturbatively inconsistent, even if we try to add new degrees of freedom into the problem. Our discussion closes the conceptual gap between the string-theorerical arguments of [9, 10] and the field-theoretical arguments of [15].

In the last section of the paper we come back to the issue of unitarity versus causality. We study in detail some criteria for microscopic causality at the quantum level. These criteria are relevant in situations where some Lorentz subgroup involving boosts remains unbroken. In particular, we will expose the impact of the quantum UV/IR mixing on the locality properties of the theory.

## Noncommutative Singularities at One-loop Order

For a general one-loop diagram with vertices in -dimensional theory we have

(4) |

where is the symmetry factor of the diagram, and is the momentum running through the ’th propagator. There is an overall momentum conservation delta function that we omit in the following. The Moyal phase can be factorized into the overall phase of the diagram , depending only on the external momenta , and the nontrivial phase depending on the loop momentum , given by

where is the total momentum flowing through the ‘nonplanar channel’.

We can evaluate the loop-momentum integral by introducing Feynman parameters in the usual fashion:

(5) | |||||

where

and we have defined an -dependent effective mass

(6) |

The resulting momentum integral can be evaluated exactly in terms of appropriate Bessel functions (c.f. for example [21]):

(7) | |||||

with the convention that all branch cuts are drawn along the negative real axis.

The most significant property of this representation is the occurrence of a generic branch-point singularity at . The noncommutativity matrix can be invariantly characterized as space-like or ‘magnetic’, light-like or ‘null’ and time-like or ‘electric’ [13]. In the first two cases, we have for real momenta, and we can only access the singularity at , i.e. the so-called UV/IR singularities of [17].

On the other hand, in the ‘electric’ case we can access the full branch cuts along with real momenta in the physical region, i.e. these singularities resemble particle-production cuts that are characteristic of time-NCFT. However, the examples studied in [15] show that the singularities at do not satisfy the standard cutting rules, i.e. they do not have the standard interpretation in terms of production of -field quanta.

Unitarity thresholds in Feynman diagrams are associated to particles in a number of internal lines going on-shell. If a Schwinger parameter is introduced for each propagator:

normal thresholds of one-loop diagrams correspond to exactly two of the Schwinger integrals being dominated by the region . Alternatively, we can replace the set of parameters by Feynman parameters , plus a global Schwinger parameter , defined by , with and . Then and any cutting of the diagram produces a singularity associated to the limit . Therefore, the parametric representation of (7) with respect to the variables is useful in disentangling the new ‘noncommutative singularities’ from the usual ones associated with unitarity cuts.

Taking advantage of the analysis in ref. [19] we compactify one spatial direction on a circle of length , which we assume to be commutative. We expect this to produce an effective mass in the nonplanar channel, reminiscent of massess of closed-string winding modes. This will also allow us to make more precise the interpretation of as an invariant mass-squared.

The effect of the compactification at the level of the previous diagram is simply to discretize the momenta in that direction , leading to a measure

(8) |

A convenient way of writing this measure uses the identity to rewrite the diagram as

(9) |

where is a shifted momentum given by

A more technical motivation for introducing the compactification is apparent in (9). Namely if we concentrate on the sectors, the finite size of the circle acts as an ultraviolet cutoff for the diagram. This is very convenient, since we would like to disentangle the occurrence of singularities related to particle production, from those at , inherent to the UV/IR effects of the theory. By selecting the sectors we can effectively do so.

We can now introduce Feynman and Schwinger parameters as above:

(10) |

Next, we evaluate the gaussian integral over and perform a modular transformation of the Schwinger parameter to obtain:

(11) |

where we have used , with playing the expected role of an effective ‘winding mass’, with dimension length-squared. This expression may also be obtained directly from (7) using the integral representation of the Bessel function.

The singularity structure of this expression is non-standard. As stressed before, normal one-loop thresholds must be associated with on-shell intermediate quanta of the field. However, these contributions come from the region of integration of large proper times or, equivalently, short dual proper times . On the other hand, the integral (11) shows singularity structure in the opposite limit: , which is the ultraviolet domain in terms of the original -field quanta.

The analytic structure of the amplitudes can be inferred from the representation (11). For ordinary theories, one defines the amplitude by analytic continuation from the euclidean region for all momentum invariants (c.f. [22]). In this domain and the proper-time integral admits a Wick rotation that makes it convergent at . The same Wick rotation renders the integral convergent in the ultraviolet, , provided a convenient cutoff is in place (in our case, ). The only novelty in the noncommutative case is the requirement of excluding the real branch cut from the ordinary euclidean domain, in order to keep the integral convergent in the limit.

This argument shows that any singularity at is of ‘ordinary type’, since it is in fact due to the limit. Thus, ‘noncommutative singularities’ appear as real branch cuts for . Notice that the Wick rotation of the proper-time parameter is equivalent the regularization of the the large- oscillatory phase by adding a small positive imaginary part to the winding mass, i.e. . Then the behaviour in the vicinity of the branch points is given by a series of terms of the form:

(12) |

where the integer comes from the Taylor expansion of the phase containing , and the logarithm is present whenever the number is a positive integer or zero. In this formula, all branch cuts are conventionally drawn along the negative real axis.

What is the physical interpretation of these branch cuts? Guided by the example of the normal-ordering diagram in the introduction, together with the dipole picture, we would expect to find a series of pole singularities corresponding to the exchange of an infinite tower of ‘winding’ particles of ‘mass’ . These states would naturally descend from closed-string winding modes in the low-energy limit.

The general expressions just derived imply that this picture is too naive. Namely, the leading singularity is a pole at only for very special diagrams with . In general, it is a softer branch-point singularity, which would suggest a multi-particle threshold, rather than the single-particle exchange implied by the dipole picture.

One possible interpretation of the cut uses a trick developed in [18], based on the simple observation that a branch cut can be viewed as a higher-dimensional pole, i.e. the negative half-integer powers of in (11) may be traded by a gaussian integral over ‘transverse’ momenta . In this way, we can approximate the amplitude in the vicinity of the singularities as:

(13) |

for an appropriate ‘coupling’ function that should be related to the couplings of particles to the in and out states in the nonplanar channel. This function has a complicated momentum dependence, which makes this interpretation rather cumbersome. In addition, the number of extra ‘transverse’ dimensions is completely ad hoc and depending on the particular diagram we consider, unlike the true number of transverse dimensions to a D-brane.

It is perhaps more appropriate to interpret the structure of the singularity in terms of a continuous spectrum of particles, so that the amplitude is approximated by

(14) |

for an appropriate ‘spectral density’ that should be roughly proportional to the product , and will have a complicated structure due to the breakdown of Lorentz invariance. Its explicit form can be worked out in particular examples from the general expression (7).

However, even if the spectral density had the right properties to be consistent with unitarity, the required on-shell condition of the particles, , would be inconsistent with a positive-energy asymptotic Fock space. Solving it in the simplified situation of two orthogonal ‘electric and magnetic’ noncommutative planes:

(15) |

We see that, in general, the particles have tachyonic excitations for nonvanishing ‘electric’ noncommutativity . The massless dispersion relation of the single particle in the example of the Introduction generalizes to a continuum of tachyons. In addition, as soon as , the energy squared at fixed ‘mass’ is unbounded from below.

In our view, this is the ultimate reason for the breakdown of unitarity in time-NCFT. Namely, the new singularities really represent production of tachyonic states, even if the cutting rules could eventually be satisfied in a formal sense.

### Connection with String Theory

The upshot of the discussion in the previous section is that, although the interpretation of the extra singularities, present for , in terms of new ‘closed-string’ particles is very suggestive and even precise in some simple examples, the general structure is rather involved. In addition, it is found that unitarity is not preserved even if the cutting rules are formally restored, because the added part of the asymptotic Hilbert space contains tachyonic states.

Given the intuitive ‘dipole’ picture and the occurrence of ‘winding-like masses’ for the effective particles in compact space, it would be desirable to establish a link between these ad hoc degrees of freedom and true closed strings in a model that would arise from a low-energy limit of string theory. The main challenge for such a discussion would be obtaining the required continuous spectrum of the particles for each value of the ‘winding number’ , as well as the explanation of the sick dispersion relation (15).

Recall that NCFT amplitudes, when written in terms of Feynman parameters and Schwinger parameter , descend directly from the string counterparts in the Seiberg–Witten (SW) limit. For example, nonplanar annulus amplitudes are integrals over the annulus modular parameter , defined so that the length of the annulus is , and Koba–Nielsen parameters . It was shown in a series of papers [23] that the string amplitude descends in the SW limit to the parametric representation (10) of the set of low-energy diagrams associated to the given string diagram. In this process, Koba–Nielsen moduli map to Feynman parameters according to and the annulus modular parameter maps to the Schwinger parameter via .

Under a modular transformation, we can write the string amplitude in the closed-string channel as an overlap of the closed string propagator between D-brane boundary states.

(16) |

where runs over all closed-string oscillator states, plus the momentum variables transverse to the D-brane, and the winding quantum numbers contributing to :

In (16), the dependence of the boundary states on Koba–Nielsen moduli has been obviated. The factor of comes from the two boundaries of the cylinder and the determinant factor comes from the canonical normalization of closed-string states propagating in a ten-dimensional bulk with metric . We choose this metric to be unity in the space transverse to the D-brane. On the other hand, both and the world-volume components of the closed-string metric are determined by formulas in [12]:

(17) |

where is the effective string coupling, which in turn determines the NCFT coupling in the SW limit.

In a proper-time representation, we write

where the modular parameter of the cylinder is related to that of the annulus by . On the other hand, the dual proper time for the particles is . Therefore, the SW limit, involving at fixed , takes , which is not the region where the amplitude can be approximated by a few low-lying closed-string states. Rather, in this scaling the whole tower of closed-string oscillator states contributes coherently. Using the equation (17), the proper-time kernel can be written as:

(18) |

which is precisely the expression found in the proper-time integral (11): , when we send at fixed and . The effective gap induced by oscillator states of closed string on the spectrum of the particles is of order

so that, in the SW limit, the closed-string oscillator spectrum becomes effectively continuous on the scale of the particles. It follows that we cannot simply set in (18) because an infinite set of modes, labelled by and the transverse momentum to the D-brane, contribute on the length scale relevant to the particles. This is the interpretation of the integral over the continuous ‘mass parameter’

in equation (14). In a proper treatment one would write a sum rule for the effective coupling of the particles, as in [19], involving a trace over all the closed-string spectrum, and using the detailed string amplitude and the known low-energy reduction to the expression (11). This trace generates the extra powers of and the detailed structure found in (11) that define the properties of the particles, including the effective transverse dimension , which is not directly related to the true number of transverse dimensions of the D-brane.

According to this picture, the fields are effective formal devices that represent the coherent coupling of an infinite number of closed-string states. The fact that they still resemble ordinary fields in some respects is a nontrivial property of NCFT, having a degenerate version of the string-theory’s channel duality.

This construction also explains the tachyonic character of the particles for . From the formula (17) for the closed-string metric, we see that only degenerates at the NCOS boundary , [9, 10], which is the only point where we can argue for the full decoupling of closed strings. For , including in particular the SW limit defining the time-NCFT, the role of space and time in the timelike noncommutative plane is exchanged [10]. This is exactly what is required to match the ‘empirical’ dispersion relations (15) introduced above for the particles.

Therefore, we conclude that the SW limit of strings with timelike noncommutativity does not decouple closed-string states, even in the low-energy limit, since they show up as production of tachyonic states in nonplanar amplitudes. This closes the circle of arguments in favour of the inconsistency of these theories. In this sense, our arguments bridge the gap between the criteria of [9, 10] and the purely field-theoretical one of [15].

## Tests of Locality and Causality at the Quantum Level

In ordinary local quantum field theory, various technical concepts such as analiticity, microscopic causality and unitarity are roughly interchangable, due to some underlying ‘physical’ requirements, such as Lorentz invariance and locality. NCFT violates these two physical conditions in a peculiar way, keeping still a controlable (and interesting) structure. Thus, NCFT is a nice laboratory to disentangle the relationships between the technical criteria cited above.

In particular, the breakdown of analiticity is to be expected generically in the case of noncommutative time, since the Moyal phases in amplitudes are not analytic functions of the energy in the upper half plane. This is related to the observed violations of classical causality criteria in [4, 5], and also renders invalid most derivations of dispersion relations by means of Cauchy’s theorem (except for the two-point function, since the global Moyal phase is trivial in this case). Thus, we observe a close parallel between the violations of analiticity and those of unitarity.

On the other hand, the standard criterion of microcausality (local observables
commute outside the relative light-cone) is more intuitive than the
‘technical’ criterion based on analiticity, but it is clearly tied to
the underlying Lorentz invariance of the theory. In fact, it loses much
of its significance in situations where the light-cone itself has no
dynamical meaning. Thus, we expect the microcausality criterion to
break down ‘trivially’ already for purely spatial noncommutativity.
Still, since free propagation is Lorentz invariant in NCFT, the breakdown
occurs necessarily as a result of the interactions and, in view of the
UV/IR effects, it is an interesting
question to determine the size of the ‘causality violation’ in perturbation
theory^{2}^{2}2For other related discussions in different contexts see
[20]..

A related interesting question is the following. One can turn on spatial and/or time noncommutativity while still preserving a certain Lorentz ‘little group’. For example, in four dimensions, the frame satisfying preserves a subgroup of the four-dimensional Lorentz group. Thus, boosts along the axis are still a symmetry even for , and we can define a ‘two-dimensional’ light-cone by the equation . In this situation, the microcausality criterion with respect to the four-dimensional light-cone has no particular meaning. However, the same criterion referred to the two-dimensional light-cone is still meaningful.

To analyze the issue we would like to compute the perturbative corrections to the commutator function

(19) |

In fact, it is technically more convenient to consider the related function given by the difference of retarded and advanced commutators

(20) |

which in turn is related to the imaginary part of the Feynman propagator in position space:

(21) |

We can then define microscopic causality by the requirement that be supported ‘inside’ the light cone, i.e. . This is certainly satisfied at the level of free fields, since the bare propagator is -independent. The obvious advantage of this definition for our purposes is that it extends naturally to the case, where a Hamiltonian construction of the commutator from its definition (19) is absent. In this case, we only have the Feynman rules as an operational definition of the theory, and one can readily compute perturbative corrections to .

The dressed propagator takes the form

(22) |

where is the 1PI self-energy, a function of the two invariants of the problem:

(23) |

In terms of these quantities we have

(24) |

In (22), the definition of is such that gives the ‘physical’ mass after renormalization by the planar diagrams. Thus, includes the finite part of planar diagrams and the contribution from nonplanar diagrams, which breaks Lorentz invariance.

On general grounds, we expect the analytic structure of the self-energy to present the normal thresholds for , and the ‘noncommutative thresholds’ for . Namely, poles at real positive values of if the theory develops bound states, the usual multi-particle cuts for , and the noncommutative cuts for in the case .

In what follows, we will assume that these singularities in the real axis exhaust all the singularities of the self-energy function in the physical sheet as a function of . This assumption is motivated by various considerations. The planar contribution to the self-energy is exactly equal to that in the commutative theory, the overall Moyal phase being trivial due to momentum conservation. In addition, the nonplanar contributions have a better high-energy behaviour than the planar ones, and all examples considered show the nonplanar singularities accumulating in the real line. Finally, the intuition from string theory points in the same direction, since is nothing but the ‘Mandelstam variable’ in the closed-string channel.

In performing the Fourier transform to compute it is convenient to use the corresponding ‘polar coordinates’ with respect to the and groups. Thus, for the ‘magnetic’ plane we change variables from to the invariant and angle, and we can write:

(25) |

for a general function of the magnetic invariant . In this expression, stands for the zeroth-order Bessel function.

On the other hand, the polar decomposition in the ‘electric’ plane parametrizes the momenta in terms of the invariant and the rapidity. In this case the contributions to the integral from the different signs of must be considered separately. The complete expression also depends on the sign of . Thus, for one finds:

(26) |

where denotes the zeroth-order Neumann function and is the zeroth-order McDonald function. We may simplify this expression under the assumption that the otherwise arbitrary function admits a ‘Wick rotation’ in the evaluation of the integral. Under this analytic continuation, the Bessel function transforms

and the complete integral simplifies, since the functions cancel out. The final reduction formula is

(27) |

Entirely analogous manipulations give a similar reduction formula for . In this case, the elimination of the Neumann functions requires the opposite Wick rotation , so that . The result is

(28) |

With these preliminaries, we are ready to write down a general formula for the modified commutator function, using the general expression for the dressed propagator in momentum space.

The explicit assumptions that we need for the analytic structure of the self-energy can be summarized by demanding that

(29) |

has only singularities in the real axis, as a function of the complex variable . These singularities include the usual poles and cuts for , , as well as the ‘noncommutative singularities at . In particular, these conditions can be explicitly checked for all the one-loop examples considered in the literature.

There is a component respecting microcausality given by:

(30) |

In addition, the component that violates the microcausality is given by

(31) |

From these general expressions we see that, as expected, four-dimensional microcausality is violated as a result of the breakdown of Lorentz invariance. There is no particular structure depending on . On the other hand, there is room for violations of the microcausality criterion coming from the term proportional to .

In general, a non-vanishing imaginary part of in the previous expression implies a violation of two-dimensional microcausality. Since in terms of two-dimensional momenta (23), this corresponds, after the analytic continuation that leads to (31), to having a contribution to the imaginary part coming from the cut starting at and extending along the negative real axis. In (31), .

Another source of -causality violations is the possibility of having zeroes in the denominator of the integrand in (31) on the real axis: . Viewed as a two-dimensional dispersion relation this corresponds to the presence of tachyon poles.

The general structure can be understood by recalling that, under our analyticity assumptions, the function in (29) admits a dispersion relation of the form

(32) |

where the ‘spectral function’

(33) |

splits naturally in two pieces, , respectively associated to the normal and ‘noncommutative’ thresholds.

Incidentally,
we notice that has the interpretation of a honest spectral
function in a Källen–Lehmann representation, with positivity ensured
by the unitarity relation at the
‘normal thresholds’.
However, a glance at the explicit examples below shows that
has no definite sign for the ‘noncommutative thresholds’, so that the
spectral interpretation of is problematic^{3}^{3}3
This particular
case of (14) makes it explicit that the spectral
interpretation of
the function appearing in that equation
will necessarily involve indefinite norm in
the asymptotic Hilbert space of the particles..

Taking the appropriate Fourier transforms we can give a two-dimensional ‘spectral representation’ of the commutator function , where:

(34) |

with

and denoting the two-dimensional free commutator functions with mass squared :

(35) |

Thus, we see that violations of two-dimensional microcausality are associated to the contribution of the ‘spectral function’ in the tachyonic branch, i.e. the noncommutative thresholds for emission of particles.

This shows that our -invariant causality criterion is equivalent to unitarity. In particular, NCFTs with space-like noncommutativity are ‘causal’ by the two-dimensional criterion.

### Some Examples

It is instructive to check the previous general statements with some simple examples. We consider the one-loop normal-ordering correction in massless theory, with renormalization conditions so that the ‘physical’ mass vanishes after planar renormalization. Then, the nonplanar graph is given by

(36) |

There are no finite contributions to , and the only source for comes from the pole terms. In the case of purely spatial noncommutativity one finds

(37) |

where .

It is interesting to notice the ‘duality’ in the argument of the second Bessel function; a rather transparent manifestation of the UV/IR effects. One can also check, using identities of Bessel functions, that for , i.e. the modified commutator at is supported on the four-dimensional light-cone. This corresponds either to the free-field case, , or to the case of infinite noncommutativity, .

As expected, the violations of microscopic causality for purely spatial noncommutativity are tied to the breaking of Lorentz invariance. Thus, is non-zero outside the four-dimensional light-cone . On the other hand, the two-dimensional microcausality is not violated, since vanishes for . It is also interesting to notice that for any value of .

An asymptotic expansion of (37) for large at constant ratio may be obtained by a saddle-point approximation:

modulated by an oscillatory phase of frequency . We see that the non-locality is of long range, presumably as a consequece of the UV/IR mixing.

In the case , the special choice is convenient to simplify the analysis, while maintaining all the qualitative features intact. The contributions to are again of pole type. However, now there is also a term proportional to :

(38) |

where , and is still given as above.

If we interpret the pole in two-dimensional terms, the term proportional to comes from a ‘tachyonic’ excitation. One can calculate exactly the integral for in terms of Bessel and Thomson functions. The piece that violates causality yields

(39) |

where . The leading asymptotic behaviour for and large is

modulated by an oscillating phase of frequency , so that the violation of two-dimensional microcausality is also of long range in this case.

Another instructive case is that of two-dimensional massive scalars. Here, Lorentz invariance is never broken, but time is always noncommutative. The general formula for the commutator function is derived along similar lines and one obtains

(40) |

Taking for the self-energy the nonplanar normal-ordering graph one finds

(41) |

In fact, in this case one may evaluate the commutator in a power series in , since the corresponding integrals are convergent. The leading correction outside the light-cone is given by the function

(42) |

This function increases away from the light cone to reach a maximum around . Then it decreases with exponential asymptotics controlled just by the mass of the field. Therefore, we have the expected behaviour, with breakdown of microcausality in spite of the preservation of Lorentz invariance. In this case however the light-cone fuzziness at is of short range, with a width of order .

For completeness, we quote here the result of the same calculation for the light-like case [13], with , so that . Defining and , we find the exact result