PHYSICAL REVIEW D 102, 094014 (2020)

Covariantizing phase space

† Andrew J. Larkoski 1,* and Tom Melia2, 1Physics Department, Reed College, Portland, Oregon 97202, USA 2Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

(Received 21 August 2020; accepted 21 October 2020; published 16 November 2020)

We covariantize calculations over the manifold of phase space, establishing Stokes’ theorem for differential cross sections and providing new definitions of familiar observable properties like infrared and collinear safety. Through the introduction of explicit coordinates and a metric we show phase space is isomorphic to the product space of a simplex and a hypersphere, and we identify geometric phenomena that occur when its dimensions are large. These results have implications for fixed-order subtraction schemes, machine learning in particle physics and high-multiplicity heavy ion collisions.

DOI: 10.1103/PhysRevD.102.094014

Relativistic N-body phase space is the manifold on from unresolved collinear or very low-energy which essentially all calculations in a perturbative quantum particles exactly cancel [1–4]. Making a mathematically field theory take place. S-matrix elements are functions that rigorous statement of precisely how IRC safety constrains live on the phase space manifold and can exhibit diver- observables is known to have problems [5]. Much of the gences on degenerate subspaces. Experiments extrapolate challenge is related to the technical fact that real and virtual smooth probability distributions from discrete, finite data of divergences only need to strictly cancel in the exact soft particles’ momenta. The first step of any Monte Carlo and/or collinear , but a lack of a smoothness can render generator for fixed-order calculation or parton shower perturbative predictions pathological slightly away from simulation involves the sampling of points on phase space. these limits. Nevertheless, some progress has been made by In each of these cases, phase space itself is often treated as either restricting to a smaller class of observables or the background on which the calculations take place, with exploiting smoothness properties of the space of collections little focus on its intrinsic geometry. of particles equipped with a metric [6]. We make a In this paper, we present a covariant description of phase conjecture for a definition of IRC safety based on the space and elucidate some of its novel geometric properties. validity of Stokes’ theorem. Our aim is to demonstrate that a deeper understanding of Establishing the phase space manifold is also important phase space can enable the identification of restrictions on for applications of machine learning in particle physics differentiable functions that can live on it, bring new [7–11]. The space in which the data input to the machine interpretations of fundamental quantities like differential lives can be used to optimize its architecture, as exploited in cross sections, and broaden the questions that can be asked convolutional or recurrent neural networks, and a neural of particle physics data. network equivariant under the Lorentz group has recently We establish the application of Stokes’ theorem to been constructed [12]. Recent work has shown that phase differential cross sections, viewing observables as provid- space in four dimensions is a Stiefel manifold modulo the ing foliations of the phase space manifold. This sheds new little group [13,14], see also [15]. Optimization and machine light on the criteria of infrared and collinear (IRC) safety learning on Stiefel manifolds is well explored in other fields, and additivity of an observable. These properties play a particularly for pattern recognition, e.g., [16–22]. special role in massless gauge theories in four dimensions We introduce explicit global coordinates that enable the —only through the calculation of IRC safe observables do construction of a metric and other quantities on phase space, providing essential input and new ways of organ- *[email protected] izing data to machine learning applications. Another † [email protected] promising application of explicit metrics on phase space is to provide natural distance measures that can act as Published by the American Physical Society under the terms of regularization observables; such observables are employed the Creative Commons Attribution 4.0 International license. in techniques [23–36] to isolate soft and collinear diver- Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, gences in modern efforts to push calculations in perturba- and DOI. Funded by SCOAP3. tion theory to high orders in QCD.

2470-0010=2020=102(9)=094014(8) 094014-1 Published by the American Physical Society ANDREW J. LARKOSKI and TOM MELIA PHYS. REV. D 102, 094014 (2020)

Z pffiffiffi We also establish geometric phenomena that occur on the D−1 ˜ ˜ i FðOÞ¼ d x g NiV ; ð3Þ phase space manifold when particle multiplicity is large Σ and derive new geometric test statistics in this limit. Specifically, the “curse of dimensionality” forces the phase where we define the vector space to concentrate at the boundaries of phase — ij∇ space we show in our explicit coordinate system that this i g jhðxÞ V ¼ MðxÞ kl : ð4Þ implies light cone momenta of particles are squeezed to the g ∇khðxÞ∇lhðxÞ boundaries of a simplex. Such high-dimensional geometry has a natural application in heavy ion physics, where This vector Vi describes a flow in phase space along the multiplicities are large. of observable O, Let Π denote the phase space manifold of dimension D. Anticipating a covariant description, we introduce local dxi M x Vi: coordinates xi and metric g on Π and write the phase space ð Þ O ¼ ð5Þ pffiffiffi d measure in terms of the metric as dDx g. We consider an of a function MðxÞ over phase space, restricted to a We will show shortly that such an interpretation is useful in hypersurface Σ, defined by hðxÞ¼O, via a δ-function, classifying properties of observables. ’ Z Equation (3) is now in a form where Stokes theorem, in pffiffiffi covariant form, can be readily applied. Consider a volume FðOÞ¼ dDx gMðxÞδðO − hðxÞÞ: ð1Þ Ω Π ∂Ω Π in with closed boundary defined by two hyper- surfaces Σ1 and Σ2, corresponding to two values of the If MðxÞ is a squared S-matrix element, then FðOÞ is observable h ¼ O1 and h ¼ O2 [see Fig. 1(a)], then we interpreted as a differential cross section for O, as defined have, by Stokes, by the function hðxÞ on phase space. This notation makes Z Z the possible multiple real and virtual contributions to the i i FðO1Þ − FðO2Þ¼ dσiV − dσiV matrix element implicit. This integral form also describes ZΣ1 Σ2 observables defined by smooth weights on phase space, ffiffiffi D p i like the energy-energy correlation function [37]. At this ¼ d x gDiV ; ð6Þ Ω point we also notep thatffiffiffiffiffiffiffiffiffi as MðxÞ ≥ 0, it is possible to 2 interpret the quantity gM as a volume form and a metric where the definition of the surface element dσi can be be associated with the dynamical theory; we return to this inferred from Eq. (3), and the covariant is interesting possibility below. given by The δ-function has the effect of turning Eq. (1) into an i Σ i ∇ i Γj i integral of a vector V over the hypersurface . To see this, DiV ¼ iV þ ijV ; ð7Þ we change to coordinates ðh; x˜aÞ, defining the induced metric on Σ, with the contracted Christoffel symbols pffiffiffi ∂xi ∂xj Γj ∇ log g: 8 g˜ ¼ g : ð2Þ ij ¼ i ð Þ ab ij ∂x˜a ∂x˜b In general, a closed boundary on phase space could involve ˜ hh hh Using the identity detðgijÞ¼detðgabÞ=g , where g ¼ (subsets of) the boundary of phase space—see Fig. 1(b) for ij g ∇ih∇jh, introducing the normal covector to the hyper- an illustration. kl 1=2 surface, Ni ¼ð∇ihÞ=ðg ∇kh∇lhÞ , and performing the As an illustrative example, we consider the foliation of now trivial integral over the δ-function in these coordinates, phase space by the C-parameter [38–40] in eþe− → qqg¯ 2 it follows that Eq. (1) becomes events. Defining the variables xi ¼ 2pi · Q=Q , where pi is

FIG. 1. (a) Stokes’ theorem applied to a subvolume Ω of Π bounded by two hypersurfaces Σ1 and Σ2 that are foliations defined by observable values O1 and O2. (b) The case for which the boundary of subvolume Ω includes part of the boundary ∂Π of the full space. (c) C-parameter foliation of three-body phase space in the xi coordinates introduced in the text.

094014-2 COVARIANTIZING PHASE SPACE PHYS. REV. D 102, 094014 (2020) the momentum of particle i ¼ 1, 2, 3 and Q is the total and collinear safe if the quantity FðOÞ in Eq. (3) is momentum vector of the collision, we choose three-body calculable on all hypersurfaces defined by hðxÞ. It follows phase space coordinates (x1, x2). The C-parameter sets that the lhs of Eq. (6) is calculable. The textbook statement of conditions for which Stokes’ 1 − 1 − 1 − Ω 6 ð x1Þð x2Þð x3Þ theorem holds is that the manifold is smooth, and that the hðxÞ¼ ; ð9Þ − 1 ∂Ω ðx1x2x3Þ D form that is integrated over the boundary is smooth and has compact support on Ω. We make the with x3 ¼ 2 − x1 − x2, and we plot its contours in Fig. 1(c). conjecture that the same conditions for Stokes’ theorem to In these coordinates the manifold is flat, gij ¼ δij, the hold in Eq. (6) are those that dictate IRC safety, namely that i i gradient is ∇ ¼ð∂=∂x1; ∂=∂x2Þ, and the vector field V in the D − 1 form that is integrated over the hypersurface Σ in Eq. (5) is straightforwardly calculated, using the squared Eq. (3) is smooth and has compact support on all of the matrix element phase space manifold Π, in the case when MðxÞ is constructed from fixed-order matrix elements. 2 2 x1 þ x2 In the C-parameter example above, this definition of IRC MðxÞ¼ : ð10Þ ð1 − x1Þð1 − x2Þ safety holds because Eq. (12) holds for all values of C,in particular as ∂Ω approaches the boundary (and indeed With these results, one can then verify that Stokes’ theorem when it becomes the boundary, where one should also holds for the C-parameter, where the difference between include the contribution of a virtual matrix element). the differential cross section at two different values of the We leave a detailed study as to whether particularly C-parameter is described by the divergence of the vector Vi pathological IRC safe (unsafe) observables exist that evade over that domain. In fact, using the known value of the the above conjecture [and any potential smoothness tests leading-order differential cross section for the C-parameter for IRC safety that could be performed for a given hðxÞ]to at its maximum value [41], where future work, and instead focus here on a geometric   definition of an important subclass of IRC observables. 3 256 pffiffiffi We say that an IRC safe observable O ¼ hðx⃗Þ is additive π 3 F C ¼ 4 ¼ 243 ; ð11Þ if the trajectory from N-toN þ m-body phase space with fixed total momentum by the emission of m arbitrarily soft the value of the differential cross section at a general value particles flows along a gradient perpendicular to the C is N-body phase space submanifold. Flow along the gradient   Z exclusively in the emitted particle phase space means that 3 ffiffiffi the arbitrarily soft emissions can be thought of as changing − D p i FðCÞ¼F C ¼ d x gDiV : ð12Þ the value of O on a fixed background of N particles. This 4 Ω definition of an additive observable is consistent with a Here, Ω is the region of phase space where the C-parameter form established long ago [42] and generalizes a definition takes values between C and 3=4. This formulation of the from Ref. [5]. For example, the definition of Ref. [5] can be cross section with respect to end point values can be stated in the following way. Let τðfpgÞ be an IRC safe generalized to other observables where the end point value observable that depends on a set of particle momenta fpg. τ ˜ at a given order in perturbation theory can be easily Then, is additive if there is a subset of momentum fpg on τ ˜ 0 calculated or is known to vanish, for example. which ðfpgÞ ¼ and m additional emissions remain in The application of Stokes’ theorem to cross sections the soft and collinear region, which can be accomplished by differential in an observable on phase space also enables scaling their momentum by a parameter v. Then, in the limit → 0 enumeration of properties of that observable that are not that v , the observable takes the form δ obvious in its original and familiar -function form. In Xm particular, for (6) to hold for a given MðxÞ requires the limτðfp˜ g;κ1ðvζ1Þ;κ2ðvζ2Þ;…;κmðvζmÞÞ ¼ v ζi; ð13Þ v→0 function hðxÞ on phase space to be highly restricted. i¼1 Most acutely, if MðxÞ is constructed from fixed-order N-body matrix elements, it generically has divergences where ζi is the functional form that τ takes on particle i and throughout N-body phase space as different numbers of κiðζiÞ is a momentum function that translates the value of τ external particles go unresolved. For MðxÞ itself to be to the realization of momentum of particle i. This definition smooth on a subvolume of phase space requires embedding demonstrates that the hard particle momenta fp˜ g are lower-dimensional phase space into higher-dimensional completely unaffected by the m soft and collinear particles, phase space and then real and virtual divergences can be and so indeed corresponds to flow along a gradient canceled point by point within the larger phase space. The perpendicular to the phase space manifold for momenta functional form of the observable O ¼ hðxÞ must respect fp˜ g. This definition of additivity has the further require- this embedding. In such a case, an observable is infrared ment that soft particles individually contribute to the

094014-3 ANDREW J. LARKOSKI and TOM MELIA PHYS. REV. D 102, 094014 (2020) observable, while our definition just requires the value of geometrical aspects of phase space that were elucidated in the observable in the soft limit to exclusively be a function Refs. [13–15]. In conventional coordinates and normaliza- of the soft momenta. Modern jet grooming algorithms, for tion, the volume form for four-dimensional, on-shell, example, can enforce correlations between the relative massless, N-body phase space in the center-of-mass angle and/or energy of soft particles, while still retaining frame is all of the nice calculability properties of additivity [43].     Because of their nice properties, additive observables are YN XN Π 2π 4−3N 4 δþ 2 δð4Þ − among the most widely studied and include thrust [44,45], d N ¼ð Þ d pi ðpi Þ Q pi : ð16Þ the C-parameter, (recoil-free) angularities [46–49], N-(sub) i¼1 i¼1 jettiness [50–54], energy correlation functions [5,55,56], Q Q; 0; 0; 0 and energy flow polynomials [57], among others. Here ¼ð Þ represents both the total momentum As a simple example of our definition of additivity, we four-vector and the total energy in the center-of-mass δþ 2 δ 2 Θ 0 consider the angularities τðαÞ measured with respect to the frame, and ðpi Þ¼ ðpi Þ ðpi Þ. Our first step is to final state momentum or thrust axis [46–48], which for rescale all momenta by the center-of-mass energy: → δ Θ three-body phase space in the above coordinates, assuming pi Qpi. The on-shell - and -functions can be trivially enforced by expressing a momentum p as the outer product the ordering x1;x2 ≤ x3, can be expressed as λa λ˜a_   of spinors and , where 1 − α=2 ðαÞ x2   τ ¼ x1 1 − − − aa_ 2 − − _ p0 p3 p1 þ ip2 x1ð x1 x2Þ ðp · σÞaa ¼   − − 1 − α=2 p1 ip2 p0 þ p3 1 − x1   þ x2 ; ð14Þ 1 ˜1_ 1 ˜2_ aa_ x2ð2 − x1 − x2Þ _ λ λ λ λ ¼ λaλ˜a ¼ : ð17Þ λ2λ˜1_ λ2λ˜2_ ðαÞ for parameter α > 0. In the soft limit of x1 → 0, τ → 0, demonstrating the infrared and collinear safety of the Reality of momentum p requires that λa and λ˜a_ are complex angularities. For this class of observables, additivity means conjugates: λ˜ ¼ λ. τðαÞ → 0 that the expansion of for x1 is proportional to its In these spinor coordinates, the on-shell integration with respect to x1, in the same limit. That is, measure for momentum p becomes

α ∂τð Þ 2 1 2 2 ðαÞ λ λ lim τ ∝ x1 : 15 4 2 d d ð Þ d pδ p Θ p0 ; x1→0 ∂x1 ð Þ ð Þ¼ ð18Þ x1→0 Uð1Þ

This can only hold if the second term in the expression of where the division by Uð1Þ represents implicit restriction to Eq. (14), which quantifies the recoil of the harder particle 2 one element of the little group action on the spinors. The away from the thrust axis, is subdominant in the x1 → 0 limit. momentum conserving δ-functions can be expressed most Only for α > 1 are these angularities additive and the simply through construction of two N-dimensional com- emission of soft particle 1 corresponds to flow perpendicular plex vectors to the two-body phase space manifold. Thrust corresponds to α ¼ 2 and is therefore indeed classified as an additive 1 1 1 2 2 2 u⃗¼ðλ1λ2 λ Þ; v⃗¼ðλ1λ2 λ Þ; ð19Þ observable. N N O The value of an additive observable is proportional to where λa is the ath (a ¼ 1, 2) component of the spinor for i i the distance along the flow defined by V of the particles to the ith particle. In terms of u⃗and v⃗, the phase space volume a lower-body phase space manifold. This property is one of element is compactly the reasons that makes additive observables especially well suited for the application of regularization of infrared dNudNv divergences. N-jettiness subtraction [32,33] is an example dΠ ¼ð2πÞ4−3NQ2N−4 N Uð1ÞN of an additive observable for regularization. Additionally, it is known that some additive observables can be interpreted × δð1 − ju⃗j2Þδð1 − jv⃗j2Þδð2Þðu⃗†v⃗Þ: ð20Þ as a metric distance from a lower-body phase space manifold [6]. Our covariant definition of additivity dem- The phase space measure describes two orthonormal onstrates that all additive observables enjoy this property. N-dimensional complex vectors u⃗and v⃗. The orthonormal Having presented a number of covariant statements constraints are invariant under the action of UðNÞ and about observables on phase space, we now turn to con- further UðN − 2Þ acts on a subspace without affecting u⃗or structing an explicit coordinate system and metric for the v⃗. Thus the phase space manifold, which we denote by ΠN, phase space manifold. We begin with a brief review of can be expressed as the quotient space

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1 Π ≅ UðNÞ The manifold that remains after explicitly accounting for N N − 1 Δ Uð1Þ UðN − 2Þ the little group is the N simplex N−1 with unit base, which represents the conservation of þ-component light 1 UðNÞ UðN − 1Þ cone momentum. We can then express the integration ¼ 1 N − 1 − 2 Uð Þ UðN Þ UðN Þ measure for u⃗as 1 ¼ S2N−1 × S2N−3: ð21Þ Uð1ÞN N d u 2 1 δð1 − ju⃗j Þ¼ dΔ −1; ð26Þ Uð1ÞN 2N N The quotient space UðNÞ=UðN − 2Þ is the Stiefel manifold of complex two-frames in CN: where dΔN−1 represents the flat measure on the simplex. The other factor in the phase space measure that depends N UðNÞ V2ðC Þ¼ : ð22Þ on the vector v⃗can be manifestly expressed as the measure UðN − 2Þ for the sphere S2N−3. We just outline the procedure here. δ ⃗ ⃗ The quotient space UðNÞ=UðN − 1Þ¼S2N−1, the 2N − 1 First, the -function that enforces orthogonality of u and v sphere, so the phase space manifold has topology of a can be used to eliminate the component vN so that product of spheres modulo the action of the little group. N−1 Our development of these ideas from hereon is twofold. N 2 ð2Þ † d v 2 2 d vδð1−jv⃗j Þδ ðu⃗v⃗Þ¼ δð1−jv⃗j −jvNj Þ; ð27Þ First, we establish the topology of phase space when the ρN little group redundancy is eliminated. This is important for applications of machine learning, so as to remove the need where the factor of 1=ρN is the resulting Jacobian and for learning of redundant directions on the manifold. we have left v component implicit in the remaining Second, we provide explicit global coordinates and con- N δ-function. In general, vN now has dependence on all struct a metric on phase space. N − 1 other components of v⃗as well as all coordinates of The little group action can be explicitly accounted for by the simplex, so the measure is not manifestly that of the “gauge fixing” (see also [15] where this was done in sphere. We can transform it into the desired form by conjunction with fixing the Lorentz frame, whereupon changing variables from v⃗ to v⃗0, under which the real phase space has topology of a Grassmann manifold). We and imaginary parts of v⃗mix only among themselves via a focus on the action of the little group on vector u⃗for which real, symmetric matrix. Conservation of the ρ coordinates its ith entry is of the simplex ensures that this transformation has the ρ 1 ϕ Jacobian J ¼ N, rendering the measure exactly that of the λ i i ui ¼ i ¼ rie : ð23Þ sphere. That is, the measure for the v⃗coordinates can be expressed as We can then express the integration measure for u⃗mod the little group as dNvδð1 − jv⃗j2Þδð2Þðu⃗†v⃗Þ Q   XN −1 2 2 −3 dNu N r dr dϕ ¼ dN v0δð1 − jv⃗0j Þ ≡ dS N ; ð28Þ δð1 − ju⃗j2Þ¼ i¼1 i i i δ 1 − r2 U 1 N U 1 N i ð Þ ð Þ i¼1 Z     the measure of the 2N − 3 sphere. Explicit coordinates for YN dr2 XN i ϕ δ ϕ δ 1 − 2 v⃗0 that ensure normalization jv⃗0j2 ¼ 1 are ¼ 2 d i ð iÞ ri i¼1 i¼1   YN XN 0 −iξ1 1 v e η1; ρ δ 1 − ρ 1 ¼ cos ¼ N ½d i i : ð24Þ 2 0 −iξ2 i¼1 i¼1 v2 ¼ e sin η1 cos η2; . In the first equation, we express the integration measure in . ⃗ polar coordinates for the components of u. In the second 0 −iξ −2 v e N sin η1 sin η −3 cos η −2; line, we use the little group invariance to explicitly fix the N−2 ¼ N N − ξ ϕ 0 0 i N−1 η η η phases i ¼ , and then on the third line we make the vN−1 ¼ e sin 1 sin N−3 sin N−2: ð29Þ 2 ρ ρ ri ¼ i. The variable i is just a light cone component of momentum, from the mapping in These generalize coordinates for the Hopf fibration to the Eq. (17), embedding of S2N−3 in CN−1. The parameters have ranges ξ ∈ ½0; 2π, and η ∈ ½0; π=2, and the volume form in these ρ 2 λ1 2 − ≡ þ i i i ¼ ri ¼ð i Þ ¼ p0;i p3;i pi : ð25Þ coordinates is

094014-5 ANDREW J. LARKOSKI and TOM MELIA PHYS. REV. D 102, 094014 (2020) dS2N−3 dimension of the simplex, and the concentration of phase   NY−2 space at the boundary implies that a number of particles will 2kþ1 have close to zero pþ. ¼ cos ηksin ηk dξ1 dξN−1dη1 dηN−2: i k¼1 This can be made more quantitative, and general state- ments about how light cone momenta are distributed ð30Þ around zero at large particle multiplicity can be derived The phase space manifold is the product of the simplex and purely from geometrical features of the high-dimensional the sphere, phase space manifold. We assume a flat matrix element on phase space and that the number of particles N is large, ≫ 1 þ Π ≅ Δ 2N−3 N . The probability of m particles with p less than N N−1 × S : ð31Þ ρ ρ ≪ 1 minQ for min =N is   The dimension of phase space is reconstructed as the N mρm 1 − ρ N−m sum of the simplex and sphere dimensions, ðN − 1Þþ pm ¼ N ð N minÞ : ð34Þ m min ð2N − 3Þ¼3N − 4. Similarly the phase space volume μ 2ρ can be especially easily derived in this framework, As a binomial distribution, its mean is ¼ N min and Δ 1 − 1 ! 2 2 using well-known formulas Volð N−1Þ¼ =ðN Þ and variance is σ ¼ N ρ ð1 − Nρ Þ. This novel “large N” 2 −3 −1 min min VolðS N Þ¼2πN =ðN − 2Þ!. limit is still consistent with σ ≪ μ in which a significant We can further construct the line element (metric) on the number of particles have very small light cone momenta. phase space manifold. As phase space is a product mani- The assumption of a flat matrix element on phase space fold, its line element can be constructed from the individual is motivated by strongly coupled systems like heavy ions line elements of the simplex and sphere, requiring the for which the matrix elements are expected to be smooth, resulting line element to be positive definite and produce nonsingular distributions on phase space. The large event the correct volume form for phase space. The line element multiplicities suggests that the probabilities pm given in of the simplex is just the Euclidean metric in the ρ Eq. (34) could make interesting test statistics. coordinates, Relaxing the assumption of a flat matrix element to one that is slowly varying means that a harmonic expansion on XN−1 the Stiefel manifold and phase space (see [13,14]) would 2 ρ2 dsΔ ¼ d i : ð32Þ quickly converge, and a small number of coefficients of that 1 i¼ expansion would quantify interesting correlations at differ- In the Hopf-like coordinates, the line element of the sphere ent angular and energy scales, as would deviations from the satisfies a recursive relationship, pm given in Eq. (34). This scenario also suggests itself as one well suited to the interesting possibility we mentioned 2 η2 2 η ξ2 2 η 2 under Eq. (1)—of incorporating the slowly varying matrix ds 2N−3 ¼ d 1 þ cos 1d 1 þ sin 1ds 2N−5 ; ð33Þ S S element MðxÞ itself into the metric. The resulting geometry 1 2 2 would encapsulate both phase space and dynamics, and it and the line element on S is flat, ds 1 ¼ dξ . It is trivial to S 1 would be fascinating to study geodesics and the observable extend this for systems in which energy conservation is not flows per Eq. (5) in this space. This would be the ultimate assumed, but note that we can always work in the frame in promotion of phase space—from a background upon which which the net three-momentum is zero. A metric on calculations take place to being geometrically entwined unordered and arbitrary collections of particle momenta with a theory’s dynamics. has been proposed [6,58], but to our knowledge, this is the first that is directly constructed from the phase space manifold. ACKNOWLEDGMENTS We end by presenting some observations about the A. L. thanks the IPMU for support and hospitality where geometry of phase space we derived in Eq. (31) in the case this work was initiated. We thank Ben Nachman for where the dimension 3N − 4 is large. One of the phenomena emphasizing the importance of the manifold of input data associated with the curse of dimensionality is that the in machine learning, Patrick Komiske, Eric Metodiev, and volume of a manifold with boundary becomes increasingly Jesse Thaler for comments on the manuscript and dis- concentrated at its boundary. For the n-ball, for example, cussions regarding the relationship to the metric of Ref. [6], when n ¼ 1000, more than 99.99% of its volume lies within and Peter Cox, Ian Moult, Duff Neill, and Mihoko Nojiri 1% of the surface. When little group redundancy is removed, for useful discussions and comments. T. M. is supported by phase space has a boundary, and in the coordinates we the World Premier International Research Center Initiative introduced, this is the boundary of the simplex of light cone (WPI) MEXT, Japan and by JSPS KAKENHI Grants þ momenta pi , see Eq. (25). On the boundary, one or more of No. JP18K13533, No. JP19H05810, No. JP20H01896, þ the pi are zero. As particle multiplicity grows, so does the and No. JP20H00153.

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