Scalar Field Theory in Curved Momentum Space
aa r X i v : . [ h e p - t h ] D ec Scalar Field Theory in Curved Momentum Space
Laurent Freidel and Trevor Rempel
1, 2 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada Department of Physics, University of Waterloo, Waterloo, Ontario, Canada (Dated: December 16, 2013)We derive an action for scalar quantum field theory with cubic interaction in thecontext of relative locality. Beginning with the generating functional for standard ϕ –theory and the corresponding Feynman rules we modify them to account forthe non–trivial geometry of momentum space. These modified rules are then usedto reconstruct the generating functional and extract the action for the theory. Amethod for performing a covariant Fourier transform is then developed and appliedto the action. We find that the transformed fields depend implicitly on a fixed pointin momentum space with fields based at different points being related by a non-localtransformation. The interaction term in the action is also non–local, but the kineticterm can be made local by choosing the base point to be the origin of momentumspace. I. INTRODUCTION
Relative locality (see [1],[2],[3]) considers a paradigm in which one systematically weakensthe notion of absolute locality by allowing momentum space to posses a non–trivial geome-try. In this framework momentum space is taken to be fundamental and spacetime emergentfrom the geometric structure of momentum space. This geometry is codified in terms of ametric and a connection which measure modifications in the energy–momentum relationsand non–linearities in the conservation law, respectively. The most startling prediction ofthe theory is that localization of events becomes an observer dependent phenomenon, withthe degree of the non–locality scaling with an observers distance from the event.As originally formulated relative locality describes a “classical non–gravitational” regimein which ~ and G N are neglected but their ratio m p = p ~ /G N is held fixed. In neglecting ~ the theory offers no formulation of quantum field theory and therefore no insight intoparticle phenomenology. The goal of this paper is to take a first step towards addressingthis issue by “turning ~ back on.” More specifically, we derive an action for scalar fieldtheory with cubic interaction term in the framework of relative locality.We begin with the generating functional for standard ϕ –theory, Fourier transform thisinto momentum space and extract the corresponding Feynman rules. We then deform theserules to account for the non–trivial geometry on momentum space. With modified momen-tum space Feynman rules in hand we write down the corresponding generating functionaland read off the action for our theory. The action will be written in terms of momenta andshould be Fourier transformed into spacetime. However, since momentum space is curvedany transformation we perform should preserve the covariance of the action. As such, itwill be necessary to develop a method for performing a covariant Fourier transform. Wedevelop this method in detail and then apply it to our action. In doing so we find that thetransformed fields depend, implicitly, on a fixed point in momentum space with fields basedat different points being related by a non–local transformation. This implies that there area continuum of quantum field theories, one for each point in momentum space. The trans-formed action is also non-local, although the kinetic term can be made local by choosingthe base point to be the origin of momentum space. In this case the relative locality actionis of the form S RL = 12 Z dν ( x ) (cid:2) ( ˆ ϕ (cid:3) ˆ ϕ ) ( x ) − m ˆ ϕ ˆ ϕ ( x ) (cid:3) + g Z dν ( x ) ( ˆ ϕ ⋆ ( ˆ ϕ ⋆ ˆ ϕ )) ( x ) , (1)where ⋆ denotes a non commutative and non associative product that encodes the non trivialgeometry of momentum space via its deformed addition. II. GEOMETRY OF MOMENTUM SPACE
In what follows we take momentum space to be a non–linear manifold P and phasespace the cotangent bundle T ∗ P . Spacetime then emerges as cotangent planes to points inmomentum space T ∗ p P . We will now embark on a self–contained review of momentum spacegeometry; the presentation will be as general as possible, although in later sections we willbe forced to give up some of this generality for the sake of coherence and ease of calculation. A. Combination of Momenta
Any process in which particles interact with each other is governed by imposing conser-vation of energy and momentum. Our choice of conservation law is an expression of thelocality properties of the theory. Here we want to investigate conservation laws which arenon linear and understand the effects such a modification has on the formulation of fieldtheory. Thus, we postulate a rule, ⊕ , for combining particles momenta. Before we definethis rule let us pause and consider what properties it should posses. First, interaction witha zero momentum object will produce no change in momenta and so 0 should be an identityfor ⊕ . Secondly, we need a method for turning an incoming particle into an outgoing oneand so our rule needs an inverse. As mentioned we will not assume this rule is linear andso there is no reason to demand either commutativity or associativity. In keeping the ruleas general as possible we allow the physics to tell us what properties are mathematicallyacceptable. Formally, we define our rule as a C ∞ map: ⊕ : P × P → P ( p, q ) p ⊕ q, (2)having identity 0 0 ⊕ p = p ⊕ p ∀ p ∈ P , (3)and inverse ⊖ ( ⊖ p ) ⊕ p = p ⊕ ( ⊖ p ) = 0 ∀ p ∈ P . (4)Note that we assume a unique inverse; if p, q ∈ P are such that q ⊕ p = p ⊕ q = 0 then q = ⊖ p .Equipped with this combination rule we can enforce the conservation of energy andmomentum at each interaction. We will write this as K µ ( p I ) = 0 , (5)where I = 1 , , . . . runs over the number of particles participating in the interaction. Forexample, a process with two incoming particles p, q and one outgoing particle k may have K µ = ( p ⊕ ( q ⊖ k )) µ , (6)where we have made use of the obvious notation q ⊖ k = q ⊕ ( ⊖ k ) and have adopted theconvention that all momenta are taken to be incoming. Observe that (6) is just one of twelvepossible choices for K all of which are distinct if ⊕ is neither commutative nor associative.Differences arising from alternate choices of the conservation law are explored in detail in[4].Suppose we are given a generic conservation law p ⊕ ( q ⊕ k ) = 0. For this to be meaningfulit must be possible to solve for any one of the momenta uniquely in terms of the other two.To address this issue we introduce left ( L p ) and right ( R p ) translation operators L p ( q ) ≡ p ⊕ q and R p ( q ) ≡ q ⊕ p, (7)which allow the conservation law to be re–written as R q ⊕ k ( p ) = L p ( R k ( q )) = L p ( L q ( k )) = 0 . (8)The existence of a unique solution for each momenta then reduces to the requirement thatthe left and right translation operators be invertible. It is therefore assumed that L − p and R − p exist for all p ∈ P and so the solutions of our conservation law are given by p = ⊖ ( q ⊕ k ) q = R − k ( ⊖ p ) k = L − q ( ⊖ p ) , (9)where we have used that L − p (0) = R − p (0) = ⊖ p , by the uniqueness of the inverse. Notethat we are not assuming the composition law ⊕ is left or right invertible; doing so wouldbe equivalent to setting L − p = L ⊖ p and R − p = R ⊖ p respectively. B. Curvature and Torsion
The algebra induced on momentum space by our composition rule determines a connec-tion on P via Γ µνρ (0) = ∂∂p µ ∂∂q ν ( p ⊕ q ) ρ (cid:12)(cid:12)(cid:12) p,q =0 . (10)The torsion is the anti-symmetric part of Γ µνρ and measures the extent to which the combi-nation rule fails to commute T µνρ (0) = Γ [ µν ] ρ (0) = ∂∂p µ ∂∂q ν ( p ⊕ q − q ⊕ p ) ρ (cid:12)(cid:12)(cid:12) p,q =0 . (11) In special relativity K µ ( p I ) = P I p Iµ Similarly, the curvature of P is a measure of the lack of associativity of the combination rule R βγδµ (0) = − ∂∂p [ β ∂∂q γ ] ∂∂k δ ( p ⊕ ( q ⊕ k ) − ( p ⊕ q ) ⊕ k ) µ (cid:12)(cid:12)(cid:12) p = q = k =0 . (12)Unlike general relativity the connection Γ µνρ is not necessarily metric compatible and so g µν may fail to be covariantly constant. To measure the extent to which the covariant derivativeof g µν deviates from zero we introduce the non-metricity tensor N µνρ = ∇ µ g νρ = ∂ µ g νρ − Γ νµα g αρ − Γ ρµα g να . (13)Let { µ νρ } denote the standard Levi-Civita connection compatible with the metric g µν .We can then decompose the full connection Γ µνρ in–terms of the Levi–Civita connection, thetorsion and the non-metricity tensor, vizΓ µνρ = { µ νρ } + 12 T µνρ − g ρα ( N µνα + N νµα − N αµν + T αµν + T ανµ ) , (14)where T µνρ = T µνα g αρ . Similarly, we can expand the non–metricity tensor in–terms of thetorsion and the symmetric tensor N µνρ = Γ ( µν ) ρ − { µ νρ } , the result is N µνρ = 12 ( T µνρ + T µρν ) − N νµα g αρ − N ρµα g αν . (15) C. Transport Operators
In order to write the locality equations at each vertex we need to introduce transportoperators that arise from the infinitesimal transformation of the addition law. We definethe left transport operator as (cid:0) U qp ⊕ q (cid:1) µν = (d q L p ) µν = ∂ ( p ⊕ q ) ν ∂q µ , (16)and the right transport operator as (cid:0) V qq ⊕ p (cid:1) µν = (d q R p ) µν = ∂ ( q ⊕ p ) ν ∂q µ . (17)Here the notation d p f ≡ ( ∂ p µ f ( p )) dx µ denotes the differential at p of the function f . Themost general form of the transport operators, U qk and V qk , from point q to k , can be obtainedfrom the ones defined above by setting p = R − q ( k ) and p = L − q ( k ) respectively. It will alsobe useful to give a name to the derivative of the inverse:( I p ) µν = ( d p ⊖ ) µν = ∂ ( ⊖ p ) ν ∂p µ . (18)It turns out that these operators are not independent and can be related by V p = − U ⊖ p I p . (19)The proof of this formula is straight forward and requires only the existence of the inverse ⊖ p : 0 = ∂∂p ( p ⊕ ( ⊖ p ))= ∂∂k ( k ⊕ ( ⊖ p )) (cid:12)(cid:12)(cid:12) k = p + ∂∂k ( p ⊕ k ) (cid:12)(cid:12)(cid:12) k = ⊖ p ∂ ⊖ p∂p = V p + U ⊖ p I p . By considering equations of the form L p ( L − p ( q )) = q and R p ( R − p ( q )) = q we can also deriveformulas for the derivatives of L − p and R − p : ∂L − p ( q ) ∂q = (cid:16) U L − p ( q ) q (cid:17) − , ∂L − p ( q ) ∂p = − (cid:16) U L − p ( q ) q (cid:17) − V pq , (20)and ∂R − p ( q ) ∂q = (cid:16) V R − p ( q ) q (cid:17) − , ∂R − p ( q ) ∂p = − (cid:16) V R − p ( q ) q (cid:17) − U pq . (21)Without demanding certain properties of the composition rule we can not say anythingfurther. For the sake of completeness we now now present a collection of results that areapplicable if the following conditions on ⊕ are fulfilled: • Composition rule is left invertible, i.e. L − p = L ⊖ p : (cid:0) U qp ⊕ q (cid:1) − = U p ⊕ qq and V ⊖ p ⊖ p ⊕ q I p = − U q ⊖ p ⊕ q V pq • Composition rule is right invertible, i.e. R − p = R ⊖ p : (cid:0) V qq ⊕ p (cid:1) − = V q ⊕ pq and U ⊖ pq ⊖ p I p = − V qq ⊖ p U pq D. Metric and Distance Function
It is assumed that the metric on momentum space, g µν ( p ), is known. It is then a standardresult that the distance between two points p , p ∈ P along a path γ ( τ ) is given by: D γ ( p , p ) = Z ba r g µν ( γ ( τ )) dγ µ dτ dγ ν dτ dτ, where γ ( a ) = p and γ ( b ) = p . Of all the paths connecting p and p geodesics will beof principle importance, but here we run into trouble. In relative locality, where the non-metricity tensor does not necessarily vanish, there is more than one viable definition of ageodesic, so it is not immediately clear what one means by a “geodesic.” This ambiguity isdiscussed in Appendix A, where we argue that the most appropriate definition of a geodesicis a path which extremizes D γ ( p , p ). We will adopt this convention for the remainder ofthe paper and note that if γ is a geodesic we write D γ ( p , p ) = D ( p , p ).The standard definition of a particles mass is by means of the dispersion relation p = − m . To account for the geometry of momentum space we deform this relation and assumethat the mass of a particle with momentum p is related to the geodesic distance from p tothe origin, i.e. D ( p ) = − m , (22)where we have used the simplified notation D ( p,
0) = D ( p ). III. ϕ SCALAR FIELD
Having completed our review of the geometric structure of momentum space we willnow examine how this new paradigm alters our understanding of quantum field theory. Inparticular we will consider a quantum scalar field theory with cubic interaction term.
A. Modified Feynman Rules
The starting point for our analysis will be the generating functional, Z ( J ), for standard ϕ –theory: Z ( J ) = Z D ϕ exp (cid:18) i Z d x (cid:20) − ∂ µ ϕ∂ µ ϕ − m ϕ + 13! gϕ + J ϕ (cid:21)(cid:19) . (23)This is the position space representation of Z ( J ) which is ill-suited for our purposes. Relativelocality treats momentum space as fundamental and so we should Fourier transform Z ( J )so that all integrals are over momenta. Denote by F the Fourier transform of the argumentof the exponential, then F = i Z d p (2 π ) (cid:18) − (cid:0) p + m (cid:1) ϕ ( p ) ϕ ( − p ) + J ( p ) ϕ ( − p ) (cid:19) + i (2 π ) g Z d p (2 π ) Z d q (2 π ) Z d k (2 π ) δ ( p + k + q ) ϕ ( p ) ϕ ( q ) ϕ ( k )Following the standard procedure we extract the interaction terms from Z ( J ) and re-writethem as functional derivatives with respect to J acting on the remainder of Z ( J ). We canthen separate out the J dependent terms from the functional by completing the square, inthe end we find Z ( J ) = exp (cid:18) − (2 π ) g Z d p Z d q Z d kδ ( p + q + k ) δδJ ( p ) δδJ ( q ) δδJ ( k ) (cid:19) × exp (cid:18) i Z d p (2 π ) J ( p ) (cid:0) p + m (cid:1) − J ( − p ) (cid:19) × Z D ϕ exp (cid:18) − i Z d p (2 π ) (cid:0) p + m (cid:1) ϕ ( p ) ϕ ( − p ) (cid:19) . (24) Normally we would denote the Fourier transformed fields as ˆ ϕ ( p ), ˆ J ( p ) but since we will be regarding themomentum space representation as fundamental we will drop the hat. Having successfully removed all J dependence from the functional integral we can evaluateit to obtain some C–number. However, if we insist on the normalization Z (0) = 1 we canignore this number and simply impose the normalization by hand. Hence, Z ( J ) ∝ exp (cid:18) − (2 π ) g Z d p Z d q Z d kδ ( p + q + k ) δδJ ( p ) δδJ ( q ) δδJ ( k ) (cid:19) × exp (cid:18) i Z d p (2 π ) J ( p ) (cid:0) p + m (cid:1) − J ( − p ) (cid:19) . (25)This generating functional can now be expanded as a sum of all possible Feynman diagramshaving E external points, P propagators and V vertices where E = 3 V − P . Each diagramis then assigned a value by means of the following Feynman rules:1. To each propagator, p = i (2 π ) ( p + m ) ;2. To each external point, p = J ( p );3. To each vertex, q kp = − g (2 π ) δ ( p + q + k )4. Integrate over all momenta;5. Divide by the symmetry factor.We now consider how these rules are modified in the framework of relative locality intro-duce in the previous section. Let us begin with rule 4), integrate over all momenta. This isequivalent to introducing a measure on momentum space, call it dµ ( p ). For the time beingwe will make no assumptions about the measure other than demanding it reduce to thestandard Lebesgue measure in the limit when momentum space becomes a linear manifold .Given dµ ( p ) we define δ ( p, q ) to be a delta function compatible with this measure, that is: Z dµ ( p ) δ ( p, q ) f ( p ) = f ( q ) (26)for any function f : P → P . Note that this delta function is assumed to be symmetric uponinterchange of its arguments, i.e. δ ( p, q ) = δ ( q, p ).In deriving the original Feynman rules we tacitly assumed that the change of variables p → − p has unit Jacobian. In relative locality the equivalent change of variables is p →⊖ p which has Jacobian | det( d p ⊖ ) | = | det( I p ) | . A priori this quantity could differ fromunity which amounts to breaking the symmetry associated with flipping the direction ofa propagator. Therefore, diagrams which are related by such a transformation should beregarded as inequivalent, see Figure 1. An obvious choice would be dµ ( p ) = p g ( p ) d p . Our assumption of a unique inverse is critical here; it is equivalent to demanding that ⊖ be invertiblewhich in turn is necessary to even define this change of variables. p q p q FIG. 1:
Feynman diagrams related by switching the direction of a propagator are inequivalent.
Diagrams do, however, still posses a symmetry under relabelling of propagators, for examplethe diagrams shown in Figure 2 are equivalent. p qk pk q
FIG. 2:
Feynman diagrams related by relabelling of propagators are equivalent.
All of this implies that we must propose a different interpretation of the symmetry factor,rule 5). A bit of thought suggests the following modification: Divide by 2 P , where P is thenumber of propagators appearing in the diagram, then divide by a factor associated with anyresidual symmetries of the diagram. The diagrams in Figures 1, 2 have no residual symmetrywhereas those in Figure 3 have residual symmetry factors of 3! and 2! respectively, given byrelabelling the propagators. pqk pqk FIG. 3:
Relabelling the propagators gives a residual symmetry factor of for the left diagram and for the right. We turn next to Rule 1), the factor associated with the propagator. The propagatormust have a single simple pole at the particles mass which, given the definition of mass inRelative Locality, suggest that we make the following replacement: p + m → D ( p ) + m . (27)where D ( p ), we recall, is the distance of p from the origin, as measured by the momentumspace geometry g ( p ).Rule 2) requires no modification and so we come to rule 3), the factor assigned to a vertex.What properties should the modified factor posses? First, it should reduce to the original inthe case where momentum space is a linear manifold. Second, it should respect the statisticsof our particles. It is well known that in standard QFT scalar particles obey Bose statistics. In what follows we will drop all factors of (2 π ) . In our case since we modify the addition rule and relax the notion of locality, we could alsorelax the bose statistics and investigate non trivial field statistics. In this work we will takethe simplest hypothesis and assume that we have Bose statistics in the present frameworkas well. Therefore, our factor must be symmetric upon interchange of momentum labels.Given that the combination rule is neither associative nor commutative there are severalchoices we could make, we will consider three of them in detail. Assuming all particles areincoming to the vertex the first of these is:∆ = 16 (cid:2) δ ( p ⊕ ( q ⊕ k )) + δ ( p ⊕ ( k ⊕ q )) + δ ( q ⊕ ( p ⊕ k )) + δ ( q ⊕ ( k ⊕ p ))+ δ ( k ⊕ ( p ⊕ q )) + δ ( k ⊕ ( q ⊕ p )) (cid:3) , (28)where we have used the simplified notation δ ( p,
0) = δ ( p ). In this option we always assumethat the second and third terms in the sum are grouped together. The second choiceincludes all possible groupings and we write it as:∆ = 112 X K ( p,q,k ) δ ( K ( p, q, k )) , (29)where K ( p, q, k ) represents a possible ordering of momenta. The final option is similar to∆ but we move the grouped factors to the other side of the delta function, this gives∆ = 16 (cid:2) δ ( p, ⊖ ( q ⊕ k )) + δ ( p, ⊖ ( k ⊕ q )) + δ ( q, ⊖ ( p ⊕ k )) + δ ( q, ⊖ ( k ⊕ p ))+ δ ( k, ⊖ ( p ⊕ q )) + δ ( k, ⊖ ( q ⊕ p )) (cid:3) . (30)The difference between ∆ and ∆ is related to the discrepancy between δ ( p ⊕ q,
0) and δ ( q ⊕ p,
0) whereas the difference between ∆ and ∆ is related to the discrepancy between δ ( p ⊕ q,
0) and δ ( p, ⊖ q ). To gain some understanding of these discrepancies let us integratethese delta functions against an arbitrary function f ( p ), we start with δ ( p ⊕ q ): Z dµ ( p ) δ ( p ⊕ q, f ( p ) = (cid:12)(cid:12) det (cid:0) V ⊖ q (cid:1)(cid:12)(cid:12) − f ( ⊖ q )) . The calculation for δ ( q ⊕ p ) is identical and yields: Z dµ ( p ) δ ( q ⊕ p, f ( p ) = (cid:12)(cid:12) det (cid:0) U ⊖ q (cid:1)(cid:12)(cid:12) − f ( ⊖ q ) . Obviously these results would be interchanged if we had instead integrated over q . It remainsto consider the value obtained from δ ( p, ⊖ q ): Z dµ ( p ) δ ( p, ⊖ q ) f ( p ) = f ( ⊖ q ) . Note that if we interchanged the roles of p and q in the previous integral we would obtain: Z dµ ( p ) δ ( q, ⊖ p ) f ( p ) = | det ( I q ) | f ( ⊖ q ) . Another, nearly equivalent, choice would be to group the first two terms together. i is governed by the extent to which the determi-nant of the left or right transport operator differs from unity.It still remains to choose which ∆ i to use as a vertex factor. To motivate this choice letus imagine conserving momentum at a “two point vertex”, see figure 4. p q FIG. 4:
Conserving momentum at a two point vertex.
Our prescription for conserving momentum should give p = q , i.e. R dµ ( q )∆ i ( p, q ) = 1. Both∆ and ∆ yield a factor of12 Z dµ ( q ) ( δ ( p ⊖ q ) + δ ( ⊖ q ⊕ p )) = 12 | det ( I p ) | − (cid:16)(cid:12)(cid:12) det (cid:0) U ⊖ p (cid:1)(cid:12)(cid:12) − + (cid:12)(cid:12) det (cid:0) V ⊖ p (cid:1)(cid:12)(cid:12) − (cid:17) , whereas ∆ gives Z dµ ( q ) δ ( p, q ) = 1 . This strongly suggests that we adopt ∆ as our vertex factor and we will do so for theremainder of the paper. To keep notation simple we drop the 3 and denote our vertex factorby − g ∆( p, q, k ).In summary, the modified generating functional is expanded as a sum of all Feynamndiagrams with E external points, P propagators and V vertices, where E = 3 v − P . Foreach such diagram we include all possible orientations of propagator momenta that areinequivalent under relabelling. A numerical value is then assigned to these diagrams bymeans of the following Feynman rules:1. To each propagator, p = iD ( p ) + m ;2. To each external point, p = J ( p );3. To each vertex, q kp = − g ∆( p, q, k )4. Integrate over all momenta using the measure dµ ( p );5. Divide by 2 P times the residual symmetry factor. B. Modified Generating Functional and Action
Having derived a set of Feynman rules we can now write down a generating functionalfor our theory. It is a straight forward exercise to see that the generating functional for1 ϕ –theory in relative locality is given by: Z RL ( J ) ∝ exp (cid:18) − g Z dµ ( p ) Z dµ ( q ) Z dµ ( k )∆( p, q, k ) δδJ ( p ) δδJ ( q ) δδJ ( k ) (cid:19) × exp (cid:18) i Z dµ ( p ) J ( p ) (cid:0) D ( p ) + m (cid:1) − J ( ⊖ p ) (cid:19) , (31)where the proportionality constant is fixed by demanding Z RL (0) = 1. The functionalderivatives are defined to yield the delta function introduced in the previous section, viz δδJ ( p ) J ( q ) = δ ( p, q ) . (32)To extract an action from this generating functional we need to evaluate the functionalderivatives. This can be done by re–introducing scalar fields ϕ ( p ) as follows: Z RL ( J ) ∝ exp (cid:18) − g Z dµ ( p ) Z dµ ( q ) Z dµ ( k )∆( p, q, k ) δδJ ( p ) δδJ ( q ) δδJ ( k ) (cid:19) × exp (cid:18) i Z dµ ( p ) J ( p ) (cid:0) D ( p ) + m (cid:1) − J ( ⊖ p ) (cid:19) × Z D ϕ exp (cid:18) − i Z dµ ( p ) (cid:0) D ( p ) + m (cid:1) ϕ ( p ) ϕ ( ⊖ p ) (cid:19) , where we have used that Z RL is only defined up to a numerical factor. We can now bringthe factor containing J into the functional integral and then perform the change of variables ϕ ( p ) → ϕ ( p ) − J ( p )( D ( p ) + m ) − . After some cancellation we find that the argument ofthe exponential in the path integral is given by − i Z dµ ( p ) h ϕ ( p ) ϕ ( ⊖ p ) (cid:0) D ( p ) + m (cid:1) − J ( p ) ϕ ( ⊖ p ) − ϕ ( p ) J ( ⊖ p ) D ( p ) + m D ( ⊖ p ) + m + J ( p ) J ( ⊖ p ) (cid:16)(cid:0) D ( ⊖ p ) + m (cid:1) − − (cid:0) D ( p ) + m (cid:1) − (cid:17) i . The non–linear terms in J will cancel if we demand D ( p ) = D ( ⊖ p ). This requirementis physically reasonable since D ( p ) yields the squared mass of a particle with momentum p .On the other hand, ⊖ p simply represents a reversal in the direction of a particles momentum;it turns an incoming particle into an outgoing one and vice versa. This operation shouldnot alter the mass of the particle and so D ( ⊖ p ) = − m = D ( p ). The term quadratic in J now drops out of the integrand and it becomes a simple matter to evaluate the functionalderivatives appearing in (31). In doing so we will make the assumption | det( I p ) | = 1 asassuming otherwise would make the result untenable. After we evaluate the functionalderivatives we can read off the action as the argument of the exponential, we find S RL = − Z dµ ( p ) (cid:0) D ( p ) + m (cid:1) ϕ ( p ) ϕ ( ⊖ p )+ g Z dµ ( p ) Z dµ ( q ) Z dµ ( k )∆( p, q, k ) ϕ ( ⊖ p ) ϕ ( ⊖ q ) ϕ ( ⊖ k ) . (33)2The fields ϕ ( p ) commute and so the six terms in ∆( p, q, k ) collapse to δ ( p, ⊖ ( q ⊕ k ), whichwe can then eliminate by integrating over p to obtain S RL = − Z dµ ( p ) (cid:0) D ( p ) + m (cid:1) ϕ ( p ) ϕ ( ⊖ p )+ g Z dµ ( q ) Z dµ ( k ) ϕ ( q ⊕ k ) ϕ ( ⊖ q ) ϕ ( ⊖ k ) . (34)Finally we require that S RL be real valued and so we impose the reality condition ϕ ( ⊖ p ) = ϕ ∗ ( p ); note though that for this prescription to work we also require ⊖ ( p ⊕ q ) = ( ⊖ p ) ⊕ ( ⊖ q ) , or ⊖ ( p ⊕ q ) = ( ⊖ q ) ⊕ ( ⊖ q ) . (35)The first condition demands that ⊖ is a morphism while the second that it is an anti–morphism. These are the two conditions that respect the reality condition. Thus, the finalform of our action is given by S RL = − Z dµ ( p ) (cid:0) D ( p ) + m (cid:1) ϕ ( p ) ϕ ∗ ( p )+ g Z dµ ( q ) Z dµ ( k ) ϕ ( q ⊕ k ) ϕ ∗ ( q ) ϕ ∗ ( k ) . (36)One key property of the action is its covariance under momentum space diffeomorphisms. Ifone assumes that the integration measure is diffeomorphism invariant, i.e. dµ ( f ( p )) = dµ ( p )for a diffeo f : P → P , that fixes the identity f (0) = 0. Then the Relative locality actionsatisfies S RL ( g, ⊕ , ϕ ) = S RL ( g f , ⊕ f , ϕ f ) (37)where ϕ f ( p ) ≡ ϕ ( f ( p )) , p ⊕ f q ≡ f − ( f ( p ) ⊕ f ( q )) , (38)while g f is the pull backed metric. IV. COVARIANT FOURIER TRANSFORM
To explore the spacetime properties, in particular locality, of S RL we need to computeits Fourier transform. Unfortunately we immediately run into a major impediment, thestandard Fourier kernel exp( ip · x ) is not covariant and therefore its use would break the(momentum space) diffeomorphism covariance of our action. Instead we need to developa generalization of the Fourier kernel which is invariant under such diffeomorphisms. Webegin by introducing Synge’s world–function. A. Synge’s World–Function
The world–function was introduced by Synge (see [5]) in the context of General Relativitybut the results apply equally well to a curved momentum space. Consider two points p, p ′ ∈P and let γ ( τ ) be a geodesic connecting p to p ′ then the world–function, σ ( p, p ′ ) is definedvia σ ( p, p ′ ) = 12 Z dτ g µν ( γ ( τ )) dγ µ ( τ ) dτ dγ ν ( τ ) dτ . (39)3This integral is precisely the one used in deriving the geodesic equation, see Appendix A,and so σ ( p, p ′ ) = 12 D ( p, p ′ ) , (40)which implies that the world–function is half the square of the geodesic distance between p and p ′ .Most properties of the world–function derived in [5] follow from the fact that the integrandappearing in (39) is constant along a geodesic. As this condition holds for our definition ofa geodesic, see Appendix A, we can important these properties directly, the most importantof which is the defining differential equation satisfied by σ :2 σ ( p, p ′ ) = σ µ ( p, p ′ ) σ µ ( p, p ′ ) = σ µ ′ ( p, p ′ ) σ µ ′ ( p, p ′ ) , (41)where we have employed the standard notation ∇ p µ σ ( p, p ′ ) = σ µ ( p, p ′ ) and ∇ p ′ µ σ ( p, p ′ ) = σ µ ′ ( p, p ′ ) . One can also examine the behaviour of the world–function (and its derivatives) as p → p ′ or vice versa. This is known as the “coincidence limit” and is indicated by square brackets,[ . . . ]; e.g. [ σ ] = 0. Besides this rather obvious one, the most common coincidence limits aregiven by [ σ µ ] = [ σ µ ′ ] = 0[ σ µν ] = [ σ µ ′ ν ′ ] = − [ σ µν ′ ] = g µν . The coincidence limit will not be of great important in this paper, but we refer the read to[5] for a complete discussion.The covariant derivatives of σ ( p, p ′ ), being the derivatives of a bi–scalar, behave as con-travariant vectors. In particular, σ µ ( p, p ′ ) transforms as a scalar at p ′ and a contravariantvector at p and vice versa for σ µ ′ ( p, p ′ ). Therefore, if x µ ′ ∈ T ∗ p ′ P then x µ ′ σ µ ′ ( p, p ′ ) transformsas a scalar at both p and p ′ and so a natural definition of the covariant Fourier kernel isexp( ix µ ′ σ µ ′ ( p, p ′ )). This isn’t quite right though. In the limit where the geometry of mo-mentum space is trivial we have exp( ix µ ′ σ µ ′ ( p, p ′ )) → exp( ix µ ( p − p ′ ) µ ) and the dependenceon p ′ persists; an undesirable outcome. The solution is to introduce a translated version ofthe world–function and of its derivative at p ′ : σ R ( p, p ′ ) ≡ σ ( R p ′ ( p ) , p ′ ) , σ Rµ ′ ( p, p ′ ) ≡ (cid:0) ∇ p ′ µ σ ( p, p ′ ) (cid:1) (cid:12)(cid:12)(cid:12) p = R p ′ ( p ) . (42)We could have also defined a left translated version of the world–function, σ L ( p, p ′ ) ≡ σ ( L p ′ ( p ) , p ′ ), but we chose σ R for the sake of definiteness. A graphical comparison of σ µ ′ ( p, p ′ )and σ Rµ ′ ( p, p ′ ) is given in Figure 5. The kernel exp( ix µ ′ σ Rµ ′ ( p, p ′ )) is then covariant and reducesto exp( ix µ p µ ) in the limit of flat momentum space. It will form the basis for defining thecovariant Fourier transform .4 p ′ p R p ′ ( p ) σ µ ′ ( p, p ′ ) σ Rµ ′ ( p, p ′ )FIG. 5: Comparing σ µ ′ ( p, p ′ ) and σ Rµ ′ ( p, p ′ ) . The thick black lines connecting p ′ to p and R p ′ ( p ) represent the unique geodesic interpolating between the two points. Before we continue there are some technical issues regarding the domain of the world–function which need to be discussed. Fix the point p ′ ∈ P . The definition of σ ( p, p ′ ) assumesthe existence of a unique geodesic connecting p to p ′ ; a condition which is not, in general,satisfied for two arbitrary points in P . To ensure the world–function remains single valuedwe need to restrict its domain to a “normal convex neighbourhood” (see [6]) of p ′ , denoted C p ′ . More specifically, C p ′ is a subset of P containing p ′ such that, given another point p ∈ C p ′ there exists a unique geodesic, completely contained in C p ′ , connecting p ′ and p . Our primary interest, however, is in the translated world–function σ R ( p, p ′ ) which will havea domain of definition given by D p ′ = R − p ′ ( C p ′ ). It is important to note that even if thisdomain depends on p ′ it is always a domain centered around the identity, i.e. 0 ∈ D p ′ . SeeFigure 6. p ′ R − p ′ C p ′ D p ′ FIG. 6:
The domain, C p ′ , of σ ( p, p ′ ) is mapped via R − p ′ to the domain, D p ′ , of σ R ( p, p ′ ) . B. Van–Vleck Morette Determinant
In this section we introduce the Van–Vleck Morette determinant ([7],[8],[9]), a quantitywhich will play an important role in our definition of the covariant Fourier transform. The The existence of such a neighbourhood for any p ′ ∈ P is guaranteed by Whiteheads theorem [6]. p µ → Q ′ µ = σ Rµ ′ ( p, p ′ ), where Q ′ ∈ T ∗ p ′ P and g − Q ′ ∈ T p ′ P is the initialvelocity vector of the geodesic going from p ′ to p . It has Jacobian given by d Q ′ = (cid:12)(cid:12)(cid:12) det (cid:16) σ Rµν ′ ( p, p ′ ) (cid:17)(cid:12)(cid:12)(cid:12) d p, where we have employed the notation σ Rµν ′ ( p, p ′ ) = ∇ µ σ Rν ′ ( p, p ′ ) . The Van–Vleck Morette determinant is the bi–scalar obtained from this Jacobian throughmultiplication by the metric determinant, in particular V ( p, p ′ ) ≡ (cid:12)(cid:12) det (cid:0) σ Rµν ′ ( p, p ′ ) (cid:1)(cid:12)(cid:12) √ g p ′ √ g R p ′ ( p ) . (43)It appears naturally in the symplectic measure when we go from the symplectic coordinates( Q ′ , p ′ ) to the end point coordinates ( p, p ′ ) asd Q ′ ∧ d p ′ = V ( p, p ′ )( √ g p ′ d p ′ ) ∧ ( √ g R p ′ ( p ) d p ) . (44)Note that the change of coordinates Q ′ → p = R − p ′ (cid:0) exp p ′ ( g − Q ′ ) (cid:1) from T ∗ p ′ P to P , isthe translated exponential map. And the inverse Van–Vleck Morette determinant is theJacobian for this transformation:( √ g R p ′ ( p ) d p ) = V − ( p, p ′ ) d Q ′ √ g p ′ ! , (45)which highlights an important property of the Van–Vleck Morette determinant. If p ∈ P issuch that V − ( p, p ′ ) = 0 then a change in Q ′ produces no change in p which is equivalentto making a change in the geodesic emanating from p ′ but no change in the point at whichthe geodesic terminates; i.e. p is a caustic. The reverse situation, where V ( p, p ′ ) = 0, isimpossible since one cannot change the terminating point of a geodesic without altering thegeodesics tangent vector at the sourcing point. Therefore, while the Van–Vleck Morettedeterminant is non–zero for all p ∈ P it does diverge at caustics. As a final note we observethat V ( p, p ′ ) satisfies V (0 , p ′ ) = 1 . (46) C. Covariant Fourier Transform
Heuristically, we expect the covariant Fourier transform to take functions on P and mapthem to functions on T ∗ p ′ P . It is natural then to introduce the notation M p ′ ≡ T ∗ p ′ P , (47)which express that the cotangent plane at p ′ acts as a “spacetime” at p ′ for the Fouriertransform. To formalize this initial expectation we fix a point p ′ ∈ P and choose a normal6convex neighbourhood C p ′ giving D p ′ ≡ R − p ′ ( C p ′ ) as the domain of σ R ( p, p ′ ). The measureon momentum space, denoted dµ ( p ) above, and on the dual spacetime are defined by dµ p ′ ( p ) = √ g R p ′ ( p ) d p,dν p ′ ( x ) = g − / p ′ d x, respectively. Let L µ p ′ ( D p ′ ) denote the space of all functions on P which are square integrablewith respect to dµ p ′ and vanish outside of D p ′ . The covariant Fourier transform (see [10, 11]for earlier implementation of this object in a different context) is then the map, F p ′ , givenby F p ′ : L µ p ′ ( D p ′ ) → L ν p ′ ( M p ′ ) f ( p ) ˆ f p ′ ( x ) , where ˆ f p ′ ( x ) ≡ Z D p ′ dµ p ′ ( p ) V / ( p, p ′ ) exp (cid:16) − ix µ ′ σ Rµ ′ ( p, p ′ ) (cid:17) f ( p ) . (48)Unless D p ′ = P , the covariant Fourier transform is not surjective and therefore is notinvertible on all of L ν p ′ ( M p ′ ). This difficulty can be circumvented by restricting to theimage of F p ′ , i.e. ˆ f p ′ ( x ) ∈ F p ′ ( L ˆ µ p ′ ( D p ′ )), which allows us to define the inverse Fouriertransform as F − p ′ ( ˆ f p ′ )( p ) ≡ Z M p ′ dν p ′ ( x ) V / ( p, p ′ ) exp (cid:16) ix µ ′ σ Rµ ′ ( p, p ′ ) (cid:17) ˆ f p ′ ( x ) , (49)for p ∈ D p ′ and zero otherwise. The Fourier transform of a function ˆ f p ′ ( x ) = F p ′ ( f ( p ))( x ),one will notice, depends on the choice of base point p ′ . One does not, therefore, obtain asingle Fourier transform but rather a continuum as the base point p ′ varies throughout P .As an initial application of this formalism we will consider the Fourier representation of δ ( p, q ), the delta function on P . Assuming p, q ∈ D p ′ we posit δ ( p, q ) ≡ Z dν p ′ ( x ) V / ( p, p ′ ) V / ( q, p ′ ) exp h ix µ ′ (cid:0) σ Rµ ′ ( p, p ′ ) − σ Rµ ′ ( q, p ′ ) (cid:1)i . (50)This formula is explicitly verified in Appendix B but we note here that the proof dependscrucially on the fact, left implicit in the above formula, that the integral is taken over allof M p ′ . One can also define a Fourier representation of the delta function on M p ′ , denoted δ p ′ ( x, y ), by putting δ p ′ ( x, y ) = Z D p ′ dµ ( p ) V ( p, p ′ ) exp h iσ Rµ ′ ( p, p ′ ) (cid:16) x µ ′ − y µ ′ (cid:17)i . (51)It is important to note this representation is not the usual delta function unless D p ′ = P .It is a projector under convolution, that is δ p ′ ( x, y ) = Z M p ′ dν p ′ ( z ) δ p ′ ( x, z ) δ p ′ ( z, y ) . (52)It therefore acts as an identity on the image of the Fourier transform i.e. on F p ′ ( L ˆ µ p ′ ( D p ′ )).These properties are shown in Appendix B. Note that a mathematical study of a generalizedFourier transformation has already been done in [12] in the context of non–commutativeSU(2) field theory.7 D. Plane waves
In this section we introduce the notion of plane waves which turn out to be an efficientmethod for representing the covaiant Fourier transform. Formally, we define a plane wave,based at the point p ′ ∈ P , to be the function of p ∈ D p ′ and x ∈ M p ′ given by e p ′ ( p, x ) = V / ( R p ′ ( p ) , p ′ ) exp (cid:16) − ix µ ′ σ Rµ ′ ( p, p ′ ) (cid:17) . (53)Recalling the defining differential equation for the world–function, equation (41), a simplecalculation shows that e p ′ ( p, x ) is an eigenfunction of the Laplacian on M p ′ , g µ ′ ν ′ ( p ′ ) ∂∂x µ ′ ∂∂x ν ′ e p ′ ( p, x ) = − g µ ′ ν ′ ( p ′ ) σ Rµ ′ ( p, p ′ ) σ ν ′ ( p, p ′ ) e p ′ ( p, x )= − σ R ( p, p ′ ) e p ′ ( p, x )= − D ( R p ′ ( p ) , p ′ ) e p ′ ( p, x ) . In particular, putting p ′ = 0 we find D ( p ) e ( p, x ) = − (cid:3) x e ( p, x ); (54)a result which will be important in the sequel since it is D ( p ) which appears in the action, S RL . Returning to the definition of e p ′ ( p, x ) we see that the covariant Fourier transform, itsinverse and the delta functions introduced in the previous section can be re–written asˆ f p ′ ( x ) = Z D p ′ dµ p ′ ( p ) e p ′ ( p, x ) f ( p ) , (55) f ( p ) = Z M p ′ dν p ′ ( x ) e ∗ p ′ ( p, x ) ˆ f p ′ ( x ) , (56) δ ( p, q ) = Z M p ′ dν p ′ ( x ) e p ′ ( p, x ) e ∗ p ′ ( q, x ) , (57) δ p ′ ( x, y ) = Z D p ′ dµ p ′ ( p ) e ∗ p ′ ( p, x ) e p ′ ( p, y ) . (58)The advantage of this notation becomes apparent when we attempt to prove the Plancherelformula, which states that Z M p ′ dν p ′ ( x ) ˆ f p ′ ( x ) ˆ f ∗ p ′ ( x ) = Z D p ′ dµ p ′ ( p ) f ( p ) f ∗ ( p ) , (59)provided δ p ′ ◦ ˆ f p ′ = ˆ f p ′ , which ensures that ˆ f p ′ is in the image of the Fourier transform. Theproof proceeds as follows, let ˆ f p ′ ( x ) ∈ F p ′ ( L ˆ µ p ′ ( D p ′ )) then Z dν p ′ ( x ) ˆ f p ′ ( x ) ˆ f ∗ p ′ ( x ) = Z dν p ′ ( x ) dµ p ′ ( p ) dµ p ′ ( q ) e p ′ ( p, x ) e ∗ p ′ ( q, x ) f ( p ) f ∗ ( q )= Z dµ p ′ ( p ) dµ p ′ ( q ) δ ( p, q ) f ( p ) f ∗ ( q )= Z dµ p ′ ( p ) f ( p ) f ∗ ( p ) , F p ′ .Observe that the Fourier transform of a function lives in a particular cotangent spacedesignated by p ′ . To understand the relationship between different choices of p ′ we define atransport operator T p ′ ,q ′ ( x, y ) which satisfiesˆ f p ′ ( x ) ≡ Z M q ′ dν q ′ ( y ) T p ′ ,q ′ ( x, y ) ˆ f q ′ ( y ) . (60)In other words, T p ′ ,q ′ maps the Fourier transform in one cotangent space to the Fouriertransform in another. We can derive an explicit expression for the transport operator bytaking the transform of a particular function twice, i.e.ˆ f p ′ ( x ) = Z D p ′ dµ p ′ ( p ) e p ′ ( p, x ) f ( p )= Z D p ′ ∩ D q ′ dµ p ′ ( p ) Z M q ′ dν q ′ ( y ) e p ′ ( p, x ) e ∗ q ′ ( p, y ) ˆ f q ′ ( x ) . In the second line we took the Fourier transform at q ′ which requires f ( p ) to vanish outside D q ′ and so we obtain the stated domain of integration D p ′ ∩ D q ′ . Comparison with thedefinition of T p ′ ,q ′ in (60) yields T p ′ ,q ′ ( x, y ) = Z D p ′ ∩ D q ′ dµ p ′ ( p ) e p ′ ( p, x ) e ∗ q ′ ( p, y ) . (61)In the limit where p ′ = q ′ this transport operator is simply the delta function δ p ′ ( y, x ), inall other cases T p ′ ,q ′ is a non–local operator. E. Star Product
As a final piece of machinery we define a star product on F p ′ ( L µ p ′ ( D p ′ )) as follows( ˆ f p ′ ⋆ p ′ ˆ g p ′ )( x ) ≡ Z M p ′ ×M p ′ dν p ′ ( y ) dν p ′ ( z ) ω p ′ ( x, y, z ) ˆ f p ′ ( y )ˆ g p ′ ( z ) , where the kernel ω p ′ ( x, y, z ) is given by ω p ′ ( x, y, z ) ≡ Z D p ′ × D p ′ dµ p ′ ( p ) dµ p ′ ( q ) e p ′ ( p ⊕ q, x ) e ∗ p ′ ( p, y ) e ∗ p ′ ( q, z ) . (62)Note that the star product is defined only on functions living in the same cotangent spaces M p ′ = T ∗ p ′ P . Let’s take a moment to explore some of the properties this product possesses.First, the product of two plane waves yields the rather pleasing result (see [13, 14] for similarproperties in quantum gravity) e p ′ ( p, x ) ⋆ p ′ e p ′ ( q, x ) = e p ′ ( p ⊕ q, x ) . f p ′ ⋆ p ′ ˆ g p ′ )( x ), we find (cid:16) ˆ f p ′ ⋆ p ′ ˆ g p ′ (cid:17) ( x ) = Z dµ p ′ ( p ) dµ p ′ ( q ) e p ′ ( p ⊕ q, x ) f ( p ) g ( q ) , (63)where f ( p ) and g ( p ) have Fourier transforms ˆ f p ′ and ˆ g p ′ respectively. Furthermore, since ⊕ is not commutative we can see that ⋆ p ′ will also fail to commute. Finally, taking theconvolution product of three functions (cid:16) ˆ f p ′ ⋆ p ′ (cid:16) ˆ g p ′ ⋆ p ′ ˆ h p ′ (cid:17)(cid:17) ( x ) = Z dµ p ′ ( p ) dµ p ′ ( q ) dµ p ′ ( k ) e p ′ ( p ⊕ ( q ⊕ k ) , x ) f ( p ) g ( q ) h ( k ) , (64)which demonstrates that the failure of ⊕ to associate propagates a similar failure into ⋆ p ′ .Let us now investigate the relationship between the star product and the standard point–wise product. Noting that e p ′ (0 , x ) = 1 we can integrate (63) over x to find Z dν p ′ ( x ) (cid:16) ˆ f p ′ ⋆ p ′ ˆ g p ′ (cid:17) ( x ) = Z dµ p ′ ( p ) (cid:12)(cid:12) det (cid:0) V ⊖ p (cid:1)(cid:12)(cid:12) − f ( ⊖ p ) g ( p ) (65)On the other hand, if we compute the integral over the point–wise product f p ′ ( x ) g ∗ p ′ ( x )the Plancherel theorem will give the same result, less the factor of det( V ). By setting | det( V p ) | = 1 for all p ∈ P it follows that (the integral of) the star product and point–wiseproduct match. In this sense, we can say the star product of two functions is a local object.Performing a similar computation for the star product of three functions we find Z dν p ′ ( x ) (cid:16) ˆ f p ′ ⋆ p ′ (cid:16) ˆ g p ′ ⋆ p ′ ˆ h p ′ (cid:17)(cid:17) ( x ) = Z dµ p ′ ( p ) dµ p ′ ( q ) f ( p ⊕ q ) g ( ⊖ p ) h ( ⊖ q ) , (66)where we have also made the change of variables p, q → ⊖ p, ⊖ q . A bit of thought shouldconvince the reader that (66) bears little relation to the integral over the point–wise productof three functions, implying that the star product of three functions is a non–local object.This concludes the technical developments and we are now prepared to apply our formalismto the action S RL . F. Action in Spacetime
For ease of notation we will not explicitly display the domain of integration in any integralsoccurring in this section. Recall that S RL , the momentum space action for our scalar fieldtheory, is given by S RL = − Z dµ p ′ ( p ) (cid:0) D ( p ) + m (cid:1) ϕ ( p ) ϕ ∗ ( p )+ g Z dµ p ′ ( q ) Z dµ p ′ ( k ) ϕ ( q ⊕ k ) ϕ ∗ ( q ) ϕ ∗ ( k ) . (67)Comparing the terms appearing above with equations (65) and (66), and recalling that ϕ ( ⊖ p ) = ϕ ∗ ( p ), we can make the following replacements m Z dµ p ′ ( p ) ϕ ( p ) ϕ ∗ ( p ) = m Z dν p ′ ( x ) ( ˆ ϕ p ′ ⋆ p ′ ˆ ϕ p ′ ) ( x ) , (68) By virtue of (19) it follows that | det( U p ) | = 1 for all p ∈ P as well. Z dµ p ′ ( q ) dµ p ′ ( k ) ϕ ( q ⊕ k ) ϕ ∗ ( q ) ϕ ∗ ( k ) = Z dν p ′ ( x ) ( ˆ ϕ p ′ ⋆ p ′ ( ˆ ϕ p ′ ⋆ p ′ ˆ ϕ p ′ )) ( x ) . (69)As discussed in the previous section the integral appearing in equation (68) is local whereasthe one appearing in equation (69) is not.The D ( p ) term is more complex and we can not make the simple replacements usedabove. We proceed by taking the covariant Fourier transform of ϕ ( p ) and ϕ ∗ ( p ) Z dµ p ′ ( p ) D ( p ) ϕ ( p ) ϕ ∗ ( p ) = Z dµ p ′ ( p ) dν p ′ ( x ) dν p ′ ( y ) D ( p ) e ∗ p ′ ( p, x ) e p ′ ( p, y ) ˆ ϕ p ′ ( x ) ˆ ϕ ∗ p ′ ( y ) . (70)To proceed we would like to use equation (54) and exchange D ( p ) for derivatives of a planewave, but doing so requires a plane wave based at p ′ = 0. As such we shift e p ′ ( p, y ) to e ( p, z ) by introducing the translation operator T p ′ , ( y, z ), viz D ( p ) e p ′ ( p, y ) = Z dν ( z ) D ( p ) T p ′ , ( y, z ) e ( p, z ) = − Z dν ( z ) T p ′ , ( y, z ) (cid:3) z e ( p, z )Integrating by parts moves the derivatives onto T p ′ , which allows us to translate the planewave back to p ′ by introducing another translation operator D ( p ) e p ′ ( p, y ) = − Z dν ( z ) dν p ′ ( a ) e p ′ ( p, a ) T ,p ′ ( z, a ) (cid:3) z T p ′ , ( y, z ) . (71)We can now substitute this back into (70) and integrate over p to obtain the delta function δ p ′ ( a, x ), an integration over a then gives Z dµ p ′ ( p ) D ( p ) ϕ ( p ) ϕ ∗ ( p ) = − Z dν p ′ ( x ) dν p ′ ( y ) dν ( z ) T ,p ′ ( z, x ) (cid:3) z T p ′ , ( y, z ) ˆ ϕ p ′ ( x ) ˆ ϕ ∗ p ′ ( y )= − Z dν p ′ ( y ) dν ( z ) ( (cid:3) z T p ′ , ( y, z )) ˆ ϕ ( z ) ˆ ϕ ∗ p ′ ( y )= − Z dν p ′ ( y ) dν ( z ) T p ′ , ( y, z ) (cid:3) z ˆ ϕ ( z ) ˆ ϕ ∗ p ′ ( y ) . In the special case p ′ = 0 the translation operator becomes a delta function and integratingover z we obtain the expected (and local) result − R dν ( y ) ˆ ϕ ∗ ( y ) (cid:3) y ˆ ϕ ( y ). On the otherhand, if p ′ = 0 the transport operator will be de–localized and the overall result non–local.For ease of notation we will denote ( (cid:3) y ˆ ϕ ) p ′ ( y ) = R dν ( z ) T p ′ , ( y, z ) (cid:3) z ˆ ϕ ( z ) and so the D ( p )term can be written as Z dµ p ′ ( p ) D ( p ) ϕ ( p ) ϕ ∗ ( p ) = − Z dν p ′ ( x ) ( ˆ ϕ p ′ ⋆ p ′ ( (cid:3) ˆ ϕ ) p ′ ) ( x ) , (72)recalling that the integral over the point–wise product of two functions is identical to theintegral over the star product of two functions.1Putting the results of this section together we find that the action for our scalar fieldtheory, in the spacetime M p ′ , is given by S p ′ RL = 12 Z dν p ′ ( x ) (cid:2) ( ˆ ϕ p ′ ⋆ p ′ ( (cid:3) ˆ ϕ ) p ′ ) ( x ) − m ( ˆ ϕ p ′ ⋆ p ′ ˆ ϕ p ′ ) ( x ) (cid:3) (73)+ g Z dν p ′ ( x ) ( ˆ ϕ p ′ ⋆ p ′ ( ˆ ϕ p ′ ⋆ p ′ ˆ ϕ p ′ )) ( x ) . (74)Observe that the interaction term is non–local for any choice of p ′ and the m term is localfor any choice of p ′ . The kinetic term on the other hand is local for p ′ = 0 but non–local forany other choice of the base point. This shows that if we denote ˆ ϕ ≡ ˆ ϕ , dν ( x ) ≡ dν ( x )and ⋆ ≡ ⋆ , the relative locality action becomes, simply S RL = 12 Z dν ( x ) (cid:2) ( ˆ ϕ (cid:3) ˆ ϕ ) ( x ) − m ˆ ϕ ˆ ϕ ( x ) (cid:3) + g Z dν ( x ) ( ˆ ϕ ⋆ ( ˆ ϕ ⋆ ˆ ϕ )) ( x ) . (75) V. CONCLUSION
Starting from the generating functional for standard ϕ –theory we wrote down the cor-responding momentum space Feynman rules which were then deformed to incorporate thenon–linear structure of momentum space. We then derived the modified generating func-tional from which we were able to extract the action for our theory. A method for im-plementing a covariant Fourier transform was then developed along with a notion of planewaves and a star product. We found that the Fourier transform of a function on momentumspace depended, implicitly, on a fixed point p ′ in momentum space. Different choices offixed point yielded different Fourier transforms with two such transforms being related by anon–local translation operator.Having developed this formalism in detail we used it to Fourier transform our action intospacetime. The resulting action depended, of course, on the choice of fixed point p ′ . The m term in the action was found to be local for all choices of p ′ and the interaction termnon–local for all choices of p ′ . The kinetic term, however, was found to be local for p ′ = 0and non–local for all other choices of p ′ .This paper represents the first step towards developing quantum field theory in curvedmomentum space. To make phenomenological predictions though we need to incorporatefermions and gauge bosons into this framework, a task which will be the focus of futureresearch. Acknowledgement
We would like to thank G. Amelino-Camelia and the quantum gravity group at PI forfeedback on a talk given on this subject. This research was supported in part by Perime-ter Institute for Theoretical Physics. Research at Perimeter Institute is supported by theGovernment of Canada through Industry Canada and by the Province of Ontario throughthe Ministry of Research and Innovation. This research was also partly supported by grantsfrom NSERC.2
Appendix A: Geodesics
A geodesic can be defined as a path, p ( τ ), which parallel transports its own tangentvector. This requires ˙ p α ∇ α ˙ p µ = 0 and so the geodesic equation is given by: d p µ dτ + Γ αβµ dp α dτ dp β dτ = 0 . (A1)Alternatively, we can define a geodesic as a path which extremizes the distance between twopoints on the manifold. In general relativity, where the connection is metric compatible,these definitions are equivalent. This is not the case in relative locality where the connectionis derived, not from a metric, but from the addition rule ⊕ . In choosing between thesedefinitions we note that the distance function D γ ( p , p ) is tied to the notion of mass andfeatures prominently in the structure of relative locality. As such, it is natural to have adefinition of geodesic which extremizes D γ , and so we make this choice. We will now presenta detailed derivation of the geodesic equation and explore some of its properties.Following the argument given in [5], suppose we have two points P, Q ∈ P and an infinityof curves, p µ ( u, v ) interpolating between P and Q . The parameter v indicates which curve isbeing considered while u parametrizes the selected curve. We assume that u varies between u and u so that P, Q have coordinates p µ ( u , v ) and p µ ( u , v ) respectively. A geodesic isthen a curve which gives a stationary value to the following integral for variations whichleave the endpoints fixed : I ( v ) = 12 Z u u g µν dp µ du dp ν du du. (A2)Introduce the tangent vectors U µ = ∂p µ /∂u and V µ = ∂p µ /∂u , where V µ vanishes at u = u , u . We then define the covariant derivative along the path p µ by DA µ du = dA µ du + Γ αβµ A α U β and DA µ dv = dA µ dv + Γ αβµ A α V β , (A3)where these definitions are extended to arbitrary tensors in the standard way. A briefcalculation shows that DU µ /dv = DV µ /du , which we will make use of shortly. Demandingthat I ( v ) be stationary under variations which leave the end–points fixed is equivalent tothe condition: dI ( v ) /dv = 0 for V µ arbitrary, except at the end–points. Thus we proceed bydifferentiating I ( v ), making use of the fact that d/dv and D/dv are interchangeable whenapplied to a scalar: dI ( v ) dv = 12 ( u − u ) Z u u (cid:18) ∇ ρ g µν V ρ U µ U ν + 2 g µν U ν DU µ dv (cid:19) du (A4)= 12 ( u − u ) Z u u (cid:18) [ N ρµν − N µρν ] V ρ U µ U ν − g µν V µ DU ν du (cid:19) du. (A5)Setting this to zero and expanding DU ν /du using (A3) we find the geodesic equation: dU α du = 12 g ρα [ N ρµν − N µρν ] U µ U ν − Γ µνα U µ U ν . (A6) Such a curve will also give a stationary value to D γ so we are justified in considering the simpler function I ( v ). N ρµν − N µρν ] U µ U ν = 2 [ T ρµν + N µνα g αρ ] U µ U ν . Substituting this back into (A6), noting that Γ µνρ U µ U ν = Γ ( µν ) ρ U µ U ν and using N µνα = Γ ( µν ) α −{ µ να } we find dU α du = (cid:16) g ρα T ρµν − { µ να } (cid:17) U µ U ν , (A7)which is the final form of the geodesic equation.A particularly useful feature of geodesics in the case of a metric compatible connectionis that the quantity L = g µν U µ U ν is constant along a geodesic. It turns out that this holdsfor our definition as well: ddu ( g µν U µ U ν ) = ∂ ρ g µν U ρ U µ U ν + 2 g µν U ν dU µ du = (cid:0) ∂ ρ g βν + 2 T βρν − g µβ { ρ νµ } (cid:1) U β U ν U ρ = (cid:0) ∂ ρ g βν − g µβ { ρ νµ } (cid:1) U β U ν U ρ = 0 . This is extremely fortunate because it allows us to relate the distance function D p ( τ ) ( P, Q )directly to the integral I ( v ), in particular I = 12 D p ( τ ) ( P, Q ) . (A8) Appendix B: Fourier Transform and its Inverse
In this appendix we explicitly verify some of the technical details discussed in the paper.Let us begin with equation (50) which gives the Fourier representation for δ ( p, q ); the deltafunction on P . Assuming p, q ∈ D p ′ and f ( p ) ∈ L µ p ′ ( D p ′ ) we put˜ f ( q ) ≡ Z D p ′ dµ p ′ ( p ) δ ( p, q ) f ( p )= Z D p ′ dµ p ′ ( p ) Z M p ′ dν p ′ ( x ) V / ( p, p ′ ) V / ( q, p ′ ) × exp h ix µ ′ (cid:0) σ Rµ ′ ( p, p ′ ) − σ Rµ ′ ( q, p ′ ) (cid:1)i f ( p ) . (B1)The integral over x covers the entire cotangent space M p ′ and therefore turns the exponentialinto δ (4) ( σ Rµ ′ ( p, p ′ ) − σ Rµ ′ ( q, p ′ )) which can be decomposed in the standard fashion. To do thiswe note that the uniqueness of the geodesic connecting p ′ to p and p ′ to q implies that σ Rµ ′ ( p, p ′ ) = σ Rµ ′ ( q, p ′ ) if an only if p = q , and so δ (4) (cid:0) σ Rµ ′ ( p, p ′ ) − σ Rµ ′ ( q, p ′ ) (cid:1) = δ (4) ( p − q ) 1 | det( σ Rµν ′ ) | = δ (4) ( p − q ) V ( p, p ′ ) √ g p ′ √ g R p ′ ( p ) , (B2)4where the definition of the Van–Vleck Morette determinant together with | g p ′ det( σ Rµν ′ ) | = | det( σ Rµν ′ ) | was used in the last equality. Substituting into our expression for ˜ f ( q ) andnoting that the presence of δ (4) ( p, q ) allows us to replace all occurrences of q with p we find˜ f ( q ) = Z D p ′ dµ p ′ ( p ) δ (4) ( p, q ) √ g R p ′ ( p ) f ( p ) = f ( q ) , (B3)where we used p, q ∈ D p ′ in the second equality. This demonstrates the validity of (50) asa representation of the delta function.The Fourier representation of δ p ′ ( x, y ), the “delta function” on M p ′ , is given in equation(51). There are two important properties of this representation which we would like toverify:1. δ p ′ ( x, y ) is a projector.2. The image of δ p ′ ( x, y ) is identical to the image of F p ′ .To demonstrate the first item recall (51), δ p ′ ( x, y ) = Z D p ′ dµ p ′ ( p ) V ( p, p ′ ) exp h iσ Rµ ′ ( p, p ′ ) (cid:16) x µ ′ − y µ ′ (cid:17)i . Making the change of variables p µ → Q ′ µ = σ Rµ ′ ( p, p ′ ), the Jacobian of which is given in(45), we find δ p ′ ( x, y ) = Z Σ p ′ d Q ′ √ g p ′ exp h iQ ′ µ ′ (cid:16) x µ ′ − y µ ′ (cid:17)i , (B4)where D p ′ → Σ p ′ under the coordinate change and Q ′ µ ′ ≡ Q ′ µ g µ ′ ν ′ ( p ′ ). Unless Σ p ′ = R theintegral over Q ′ µ ′ does not give the usual delta function δ (4) ( x − y ). However, taking theconvolution product of δ p ′ with itself we find: Z M p ′ d ν p ′ ( y ) δ p ′ ( x, y ) δ p ′ ( y, z ) = Z D p ′ × D p ′ d Q ′ d K ′ | g p ′ | Z M p ′ d ν p ′ ( y ) e iy ν ′ ( Q ′ µ ′ − K µ ′ ) ! × e iK µ ′ x µ ′ e − iQ ′ µ ′ z µ ′ = Z D p ′ × D p ′ d Q ′ d K ′ √ g p ′ δ (4) ( Q ′ − K ′ ) e iK µ ′ x µ ′ e − iQ ′ µ ′ z µ ′ = δ p ′ ( x, z ) , which confirms that δ p ′ ( x, y ) is a projector, i.e. identity onto its image. For the second item,suppose ˆ f p ′ ( x ) ∈ F p ′ ( L µ p ′ ( D p ′ )) so there exists a function f ( p ) ∈ L µ p ′ ( D p ′ ) such thatˆ f p ′ ( x ) = Z D p ′ dµ p ′ ( p ) V / ( p, p ′ ) exp (cid:16) − ix µ ′ σ Rµ ′ ( p, p ′ ) (cid:17) f ( p ) . (B5)5Evaluating the convolution of δ p ′ with ˆ f p ′ we find( δ p ′ ◦ ˆ f p ′ )( x ) = Z M p ′ dν p ′ ( x ) Z D p ′ dµ p ′ ( p ) V ( p, p ′ ) exp h iσ Rµ ′ ( p, p ′ ) (cid:16) x µ ′ − y µ ′ (cid:17)i × Z D p ′ dµ p ′ ( q ) V / ( q, p ′ ) exp (cid:16) − ix µ ′ σ Rµ ′ ( q, p ′ ) (cid:17) f ( q )= Z D p ′ dµ p ′ ( p ) dµ p ′ ( q ) V / ( p, p ′ ) exp (cid:16) − iσ Rµ ′ ( p, p ′ ) y µ ′ (cid:17) × Z M p ′ dν p ′ ( x ) V / ( p, p ′ ) V / ( q, p ′ ) exp h ix µ ′ (cid:0) σ Rµ ′ ( p, p ′ ) − σ Rµ ′ ( q, p ′ ) (cid:1)i f ( q )= Z D p ′ dµ p ′ ( p ) dµ p ′ ( q ) V / ( p, p ′ ) exp (cid:16) − iσ Rµ ′ ( p, p ′ ) y µ ′ (cid:17) δ ( p, q ) f ( q )= Z D p ′ dµ p ′ ( p ) V / ( p, p ′ ) exp (cid:16) − iσ Rµ ′ ( p, p ′ ) y µ ′ (cid:17) f ( p )= ˆ f p ′ ( x ) , where we have used the Fourier representation of δ ( p, q ) in going from the third line to thefourth. This shows that the image of δ p ′ under convolution is identical with the image of F p ′ , verifying the second item above. [1] Giovanni Amelino-Camelia, Laurent Freidel, Jerzy Kowalski-Glikman, and Lee Smolin. Theprinciple of relative locality. Phys.Rev. , D84:084010, 2011.[2] Laurent Freidel and Lee Smolin. Gamma ray burst delay times probe the geometry of mo-mentum space. 2011.[3] Giovanni Amelino-Camelia, Laurent Freidel, Jerzy Kowalski-Glikman, and Lee Smolin. Rela-tive locality: A deepening of the relativity principle.
Gen.Rel.Grav. , 43:2547–2553, 2011.[4] Jose Ricardo Camoes de Oliveira. Relative localization of point particle interactions. 2011.[5] J.L. Synge.
Relativity: The General Theory . North-Holland Publishing Company, 1966.[6] M.M. Postnikov.
The Variational Theory of Geodesics . Dover Publications, Inc., 1967.[7] J.H. Van Vleck. The Correspondence Principle in the Statistical Interpretation of QuantumMechanics.
Proc.Nat.Acad.Sci. , 14:178–188, 1928.[8] C. Morette. On the definition and approximation of Feynman’s path integrals.
Phys.Rev. ,81:848–852, 1951.[9] Eric Poisson, Adam Pound, and Ian Vega. The Motion of point particles in curved spacetime.
Living Rev.Rel. , 14:7, 2011.[10] I.G. Avramidi. The covariant technique for the calculation of one loop effective action.
Nucl.Phys. , B355:712–754, 1991.[11] T.S. Bunch and L. Parker. Feynman Propagator in Curved Space-Time: A Momentum SpaceRepresentation.
Phys.Rev. , D20:2499–2510, 1979.[12] Laurent Freidel and Shahn Majid. Noncommutative harmonic analysis, sampling theory andthe Duflo map in 2+1 quantum gravity.
Class.Quant.Grav. , 25:045006, 2008. [13] Laurent Freidel and Etera R. Livine. Ponzano-Regge model revisited III: Feynman diagramsand effective field theory. Class.Quant.Grav. , 23:2021–2062, 2006.[14] Laurent Freidel and Etera R. Livine. Effective 3-D quantum gravity and non-commutativequantum field theory.