Orientador: Jayme Vaz Jr / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataglin / Made available in DSpace on 2018-08-05T13:26:17Z (GMT). No. of bitstreams: 1 RochaJunior_Roldaoda_D.pdf: 1670364 bytes, checksum: 3d62c507080592c925245e4858fab674 (MD5) Previous issue date: 2005 / Resumo: Investigamos generalizações das álgebras de Clifford (ACs) e suas vastas aplicações na Física. Classificamos o mais novo candidato à descrição da matéria escura como um campo espinorial bandeira, que pertence à classe 5 proposta por Lounesto, de acordo com os valores assumidos pelos seus covariantes bilineares. Decompomos a AC em partes a pares e ímpares relativas a uma dada a graduação automórfica interna, além de descrever suas diversas consequências na decomposição de operadores que agem sobre a álgebra exterior e sobre a AC. Além de escrever a equação de Dirac no contexto dessa decomposição, estendemos os resultados conhecidos sobre uma partícula-teste nas vizinhanças de um buraco negro de Schwarzschild para um buraco negro de Reissner-Nordstrom. Introduzimos as ACs estendidas, construídas sobre duas cópias (quiral e aquiral) de um espaço vetorial de dimensão finita munido de uma métrica de assinatura (p, q). Formulamos a AC sobre uma cópia quiral do contraespaço, mostrando propriedades surpreendentes, tais como: a indefinição do elemento de volume do contraespaço sob o produto regressivo, com a possibilidade de ele ser um escalar ou pseudoescalar, dependendo da dimensão do espaço vetorial; e o fato de que a co-cadeia de de Rham do operador codiferencial ser formada por uma sequência de subespaços homogêneos da álgebra exterior subsequentemente quirais e aquirais. Dessa maneira provamos que a álgebra exterior sobre o espaço e aquela construída sobre o contraespaço são apenas pseudo-duais ao introduzirmos quiralidade. A super álgebra de Poincaré é obtida a partir da introdução de algumas estruturas algébricas sobre o espaço euclidiano R3, a partir da utilização de spinors puros e do Princípio da Trialidade juntamente com sua generalização. Introduzimos os octonions no contexto das ACs e definimos unidades octoniônicas parametrizadas por elementos arbitrários, mas fixos, de uma AC sobre R0,7 e também produtos octoniônicos entre multivetores, além de generalizarmos as identidades de Moufang para esse formalismo. O Modelo Padrão das partículas elementares é rediscutido nesse contexto, além de obtermos uma Teoria de Calibre não-associativa em Cl0,7 , onde o campo espinorial é dado pela soma direta de um quark e um lépton. Finalmente introduzimos as isotopias, associativas e não-associativas, das ACs e em particular a simetria de sabor SU(6) dos quarks se apresenta como uma simetria exata dentro do contexto do levantamento isotópico da AC CL12. Bárions e mésons também são descritos nesse contexto / Abstract: We investigate Clifford algebras (ACs) generalizations and their wide applications in Physics. The candidate for the description of the dark matter is classified as a agpole spinor field, that is in the class 5 spinors proposed by Lounesto according to his spinor field classification by the values assumed by their bilinear covariants. The AC is split in a-even and a-odd components, related to a given inner automorphic a-grading, besides describing various consequences of this decomposition in the splitting of operators acting on the exterior and Clifford algebras. Besides writing the Dirac equation in the spacetime splitting context, we extend the well known results concerning a spinning test particle in a Schwarzschild black hole neighboorhood to a Reissner-Nordstrom black hole. We alsointroduce the extended ACs associated with two copies (chiral and achiral) of a finite-dimensional vector space endowed with a metric of signature (p, q). ACs are formulated on a chiral copy of the counterspace, where we show astounding and astonishing properties such as: the de Rham co-chain associated with the codifferential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove thatthe exterior algebra on the space and the exterior algebra constructed on the counterspace are pseudoduals, if we introduce chirality. The Poincaré superalgebra is obtained from the introduction of some algebraic structures on the Euclidean space R3 , via the pure spinor formalism and the triality principle and its generalization. Octonions are introduced in thecontext of ACs and we define AC-parametrized octonionic units, besides generalizing Moufang identities in this context. The Standard Model of elementary particles is revisited in the octonionic context and we also obtain a gauge theory using the new octonionic products introduced, where a spinor field describes the direct sum of a quark and a lepton. Finally we introduce associative and non-associative isotopies of ACs. In particular we present the avor quark symmetry SU(6) as an exact symmetry in the Cl12 isotopic lifting context. Barions and mesons are also described via isotopic lifting of ACs / Doutorado / Fisica-Matematica / Doutor em Ciências

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-12-22T13:06:55Z No. of bitstreams: 1 filipedeoliveirasalles.pdf: 461360 bytes, checksum: 0a9c082265ec8357b1badcccd8374bd2 (MD5) / Approved for entry into archive by Diamantino Mayra (mayra.diamantino@ufjf.edu.br) on 2017-01-31T10:20:11Z (GMT) No. of bitstreams: 1 filipedeoliveirasalles.pdf: 461360 bytes, checksum: 0a9c082265ec8357b1badcccd8374bd2 (MD5) / Made available in DSpace on 2017-01-31T10:20:11Z (GMT). No. of bitstreams: 1 filipedeoliveirasalles.pdf: 461360 bytes, checksum: 0a9c082265ec8357b1badcccd8374bd2 (MD5) Previous issue date: 2011-02-15 / FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais / A necessidade de considerar efeitos quânticos e a presença de singularidades nas soluções da Relatividade Geral indicam o desejo indispensável de estudar diferentes modelos de gravitação modificada. Além disso existe uma esperança de utilizar estes modelos para melhor entendimento da aceleração do Universo. É bem conhecido que estes tipos de teorias podem ser equivalentes aos modelos métrico-escalares, com diferentes mecanismos de equivalência. O objetivo principal do nosso trabalho é sugerir uma modificação e generalização desses mecanismos conhecidos que podem ser aplicados a uma grande variedade de teorias gravitacionais. No primeiro capítulo fazemos uma revisão histórica sobre as teorias de gravitação e uma breve introdução à teoria f(R). No segundo capítulo utilizamos funções f(R) para a métrica de Friedmann-Lemaître-Robertson-Walker, calculamos as equações e tentamos encontrar soluções do tipo exponencial. Em seguida, propomos uma generalização dessa teoria, para funções f(Xi), onde Xi pode ser R, R, Gauss-Bonnet, o quadrado do tensor deWeyleetc.Porúltimoanalisamosasperspectivasfuturasdessesmodelosdegravitação modificada. No terceiro capítulo desta dissertação sugerimos uma representação escalar-tensorial equivalente para teorias f(R). Em seguida, calculamos as equações do campo gravitacional pelo método direto, utilizando a ação escalar-tensorial e variando-a em relação à métrica esfericamente simétrica. Logo em seguida encontramos as mesmas equações de campo, porém utilizando o método de transformação conforme. Por fim veremos as perspectivas de continuidade desse trabalho, ainda em fase de conclusão. / The need to consider quantum effects and the presence of singularities in the solutions of general relativity indicate the desire necessary to study different models of modified gravity. In addition there is a hope of using these models to better understand the accelerationoftheUniverse.Itiswellknownthatthesetypesoftheoriescanbeequivalent to the metric-scalar models with different mechanisms of equivalence. The main goal of ourworkistosuggestamodificationandgeneralizationoftheseknownmechanismsthat can be applied to a wide variety of gravitational theories. In the first part we use functions f(R) for the Friedmann-Lemaître-Robertson-Walker equations and calculate the many solutions of exponential type. Then, we propose a generalization of this theory for functions of type f(Xi), where Xi can be R, R, GaussBonnet, the square of the Weyl tensor and so on. Finally we analyze the future prospects of these models of modified gravity. In the second part of this thesis is a brief introduction to the f(R) theory and suggest a scalar-tensor representation equivalent to this theory. Then, we calculate the gravitational field equations by the direct method using the scalar-tensor action and varying with respect to the spherically symmetric metric. Soon after we find the same field equations, but using the method of conformal transformation. Finally we see the prospects for continuing this work, still in the process and conclusion.

Orientador: Marcelo Moraes Guzzo / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-24T10:34:02Z (GMT). No. of bitstreams: 1 Jurkovich_HeitordoAmaral_M.pdf: 1327191 bytes, checksum: 56f02396f6172fcf091b0c18db27bd7a (MD5) Previous issue date: 2014 / Resumo: Nessa dissertação foi estudada a quebra espontânea de simetria desde suas bases mais fundamentais. A quebra espontânea de simetria e o único mecanismo conhecido capaz de gerar massas a uma teoria, preservando as simetrias e sendo renormalizável. Tal processo ocorre quando se define um setor escalar para uma teoria com uma certa simetria e se quebra os geradores dessa simetria, criando bosons de Goldstone no caso de simetrias globais e campos de gauge massivos, no caso de simetrias locais. Nessa dissertação começamos o estudo de quebra espontânea de simetria com um estudo de Teoria de Grupos, que e fundamental para o entendimento da mesma seguido por um estudo em simetrias nas lagrangianas, ou seja, como gerar uma lagrangiana invariante por um dado grupo, local ou global; depois estudamos o Teorema de Goldstone e por m aplicamos a quebra espontânea de simetria para diversos modelos, entre eles o campo escalar, campo escalar complexo com simetria global, campo escalar complexo com simetria local, modelo padrão. Depois fazemos um estudo em uma das extensões do modelo padrão, o modelo SU(5) e aplicamos a quebra espontânea de simetria em tal modelo. No m mostramos como a abordagem de minimização de um potencial clássico e valida, pois contribuições quântica são análogas a uma energia de ponto zero ao potencial efetivo / Abstract: A detailed study of the spontaneous symmetry breaking since its fundamental basis is done. The spontaneous symmetry breaking is the only know mechanism capable of generating masses to a theory still preserving its symmetry and being renormalizable. Such process occurs when one defines a scalar sector to a certain theory with a symmetry and then break the generators of this symmetry, creating Goldstone bosons in the case of global symmetries and massive gauge fields in the case of local symmetries. We start this dissertation studying Group Theory, which is fundamental to the understanding of spontaneous symmetry breaking. Then we study local and global symmetries of the lagrangian, Goldstone Theorem, then spontaneous symmetry breaking to several models, including the scalar theory, complex scalar theory with global and local symmetries and the standard model. Then we make a study in one of the standard model extensions, the SU(5) model and we applied spontaneous symmetry breaking to this model. In the end we show that the approach of minimizing que classical potential is valid because quantum contributions are analogous to a zero point energy to the effective potential / Mestrado / Física / Mestre em Física

Cette thèse concerne l'étude des propriétés de champs scalaires classiques et quantiques en présence d'un environnement inhomogène et/ou dépendant du temps. Nous nous concentrerons sur des modèles pouvant être décrits, fondamentalement ou de manière effective, par un espace-temps courbe contenant un horizon des événements. Nous verrons en particulier comment une correspondance mathématique, provenant d'une symétrie de Lorentz effective à basse énergie, permet de relier les comportements des ondes dans un cadre non relativiste à la physique des trous noirs, quelles en sont les limites et dans quelle mesure les résultats ainsi obtenus sont og analogues fg à leurs pendants gravitationnels. Après un premier chapitre d'introduction rappelant quelques bases de relativité générale puis une dérivation de la radiation de Hawking et de la correspondance avec des systèmes non relativistes, je présenterai le détail de quatre travaux effectués durant ma thèse. Les autres articles écrits dans ce cadre sont résumés dans le dernier chapitre, précédant une conclusion générale. Mes collaborateurs et moi nous sommes concentrés sur trois aspects du comportement des champs près de l'analogue d'un horizon des événements dans des modèles avec une symétrie de Lorentz effective à basse énergie. Le premier concerne les effets non linéaires, cruciaux pour comprendre l'évolution de la radiation de Hawking ainsi que pour les réalisations expérimentales mais auparavant peu étudiés. Nous montrerons comment ceux-ci déterminent les possibles comportements aux temps longs pour des systèmes stables ou instables. Le second aspect a trait aux effets linéaires et quantiques, en particulier la radiation de Hawking elle-même, son devenir lorsque l'horizon est continûment effacé, ainsi que les diverses instabilités à même de survenir dans différents modèles. Enfin, nous avons participé à l'élaboration, à l'analyse et à l'étude d’expériences dites de og gravité analogue fg dans des condensats de Bose-Einstein et des systèmes hydrodynamiques ou acoustiques, dont je rapporte les principaux résultats. / The present thesis deals with some properties of classical and quantum scalar fields in an inhomogeneous and/or time-dependent background, focusing on models where the latter can be described as a curved space-time with an event horizon. While naturally formulated in a gravitational context, such models extend to many physical systems with an effective Lorentz invariance at low energy. We shall see how this effective symmetry allows one to relate the behavior of perturbations in these systems to black-hole physics, what are its limitations, and in which sense results thus obtained are “analogous” to their general relativistic counterparts. The first chapter serves as a general introduction. A few notions from Einstein's theory of gravity are introduced and a derivation of Hawking radiation is sketched. The correspondence with low-energy systems is then explained through three important examples. The next four chapters each details one of the works completed during this thesis, updated and slightly reorganized to account for new developments which occurred after their publication. The other articles I contributed to are summarized in the last chapter, before the general conclusion. My collaborators and I focused on three aspects of the behavior of fields close to the (analogue) event horizon in models with an effective low-energy Lorentz symmetry. The first one concerns nonlinear effects, which had been given little attention in view of their crucial importance for understanding the evolution in time of Hawking radiation as well as for experimental realizations. We showed in particular how they determine the late-time behavior in stable and unstable configurations. The second aspect concerns linear and quantum effects. We studied the Hawking radiation itself in several models and what replaces it when continuously erasing the horizon. We also characterized and classified the different types of linear instabilities which can occur. Finally, we contributed to the design and analysis of “analogue gravity” experiments in Bose-Einstein condensates, hydrodynamic flows, and acoustic setups, of which I report the main results.

In dieser Arbeit wird die analytische Fortsetzung von Quantenfeldtheorien auf dem nichtkommutativen Euklidischen Moyal-Raum mit kommutativer Zeit zu entsprechenden Moyal-Minkowski Raumzeit (Wick Rotation) erarbeitet. Dabei sind diese Moyal-Räume durch eine konstante Nichtkommutativiät gegeben. Einerseits wird die Wick Rotation im Kontext der algebraischen Quantenfeldtheorie, ausgehend von einer Arbeit von Schlingemann, hergeleitet. Von einem Netz Euklidischer Observablen wird die Lorentz’sche Theorie durch alle Bilder der fortgesetzten Poincare Gruppenwirkung auf der Zeit-Null Schicht erhalten. Dabei wird gezeigt, dass die Vorgänge der nichtkommutativen Deformation und der Wick Rotation kommutieren. Andererseits ist so eine analytische Fortsetzung ebenfalls für Quantenfeldtheorien, die durch einen Satz von Schwingerfunktionen definiert ist, möglich. Durch die Gültigkeit einer Kombination aus Wachstumsbedinungen, die aus der Wick Rotation von Osterwalder und Schrader bekannt sind, kann der Übergang zu einer deformierten Wightman-Theorie gezeigt werden. Abschließend beinhaltet diese Arbeit ergänzende Resultate zu den physikalischen Eigenschaften der Kovarianz und der Lokalität.

L'objectif de ce travail est une construction perturbative rigoureuse de la théorie de la Yang-Mills SU(2) dans l'espace euclidien à quatre dimensions. La technique d'intégration fonctionnelle donne une basemathématique pour établir les équations de flot différentielles du groupe de renormalisation pour l'action efficace. Si l'introduction de régulateurs dans l'espace de moments permet de donner une définition mathématique des fonctions de Schwinger, la difficulté importante de l'approche est le fait que cesrégulateurs brisent l'invariance de jauge. Ainsi, le travail principal est alors de prouver à tous les ordres en perturbation l'existence de ces fonctions de correlation et la validité des identités de Slavnov-Taylor pour la théorie renormalisée. / The goal of this work is a rigorous perturbative construction of the SU(2) Yang-Mills theory in four dimensional Euclidean space. The functional integration technique gives a mathematical basis for establishing the differential Flow Equations of the renormalization group for the effective action. While the introduction of momentum space regulators permits to give a mathematical definition of the Schwinger functions, the important difficulty of the approach is the fact that these regulators break gauge invariance. Thus the main part of the work is to prove at all loop orders the existence of the vertex functions and the restoration of the Slavnov-Taylor identities in the renormalised theory.

An investigation into open problems related to black hole evaporation in the semi-classical framework, concerning the existence of quantum field theories on spacetimes modelling evaporating black holes as well as the existence of black hole solutions to the semi-classical Einstein equations. / Eine Untersuchung offener Probleme zur Verdampfung schwarzer Löcher im semi-klassischen Modell, bezüglich der Existenz von Quantenfeldtheorien auf Raumzeiten, die verdampfende schwarze Löcher beschreiben, sowie der Existenz von Lösungen der semi-klassischen Einstein Gleichungen, welche schwarze Löcher darstellen.

Negative energy densities are an abundant and necessary feature of quantum field theory (QFT) and may lead to surprising measurable effects. Some of these stand in contrast to classical physics, so that the accumulation of negative energy, also in quantum field theory, must be subject to some constraints. One class of such constraints is commonly referred to as quantum energy inequalities (QEI). These are lower bounds on the averaged stress-energy tensor which have been established quite generically in quantum field theory, however, mostly excluding models with self-interaction. A rich but mathematically tractable class of interacting models are those subject to integrability. In this thesis, we give an overview of the construction of integrable models via the inverse scattering approach, extending previous results on the char- acterization of local observables to models with more than one particle species and inner degrees of freedom. We apply these results to the stress-energy tensor, leading to a characterization of the stress-energy tensor at one-particle level. In models with simple interaction, where the S-matrix is independent of the particles’ momenta, this suffices to con- struct the full stress-energy tensor and provide a state-independent QEI. In models with generic interaction, we obtain QEIs at the one-particle level and find that they substantially constrain the choice of reasonable stress-energy tensors, in some cases fixing it uniquely.:Acknowledgements 4 Contents 5 1 Introduction 7 2 Constructive aspects of integrable quantum field theories 13 2.1 General notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Particle spectrum and one-particle space . . . . . . . . . . . . . . . . 15 2.3 The scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Full state space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Asymptotic completeness; closing the circle . . . . . . . . . . . . . . . 29 2.6 Connection to algebraic quantum field theory . . . . . . . . . . . . . 33 3 Locality and the form factor series 37 3.1 Locality and the form factor series . . . . . . . . . . . . . . . . . . . 38 3.2 Local commutativity theorem for one- and two-particle form factors . 44 3.3 Transformation properties of the form factors . . . . . . . . . . . . . 58 3.3.1 Form factors of invariant operators and derivatives . . . . . . 62 4 Structure of form factors and the minimal solution 64 4.1 Classification of two-particle form factors . . . . . . . . . . . . . . . . 64 4.2 Existence of the minimal solutions and asymptotic growth . . . . . . 68 4.3 Computing a characteristic function . . . . . . . . . . . . . . . . . . . 74 5 The stress-energy tensor 77 5.1 The stress-energy tensor from first principles . . . . . . . . . . . . . . 77 5.2 The stress-energy tensor at one-particle level . . . . . . . . . . . . . . 83 5.3 Characterization at one-particle level . . . . . . . . . . . . . . . . . . 88 6 State-independent QEI for constant scattering functions 94 6.1 Candidate for the stress-energy tensor . . . . . . . . . . . . . . . . . 94 6.2 A generic estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.3 Derivation of the QEI . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4 Discussion of the QEI . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5 Supplementary computations . . . . . . . . . . . . . . . . . . . . . . 108 7 QEIs at one-particle level for generic scattering functions 110 7.1 Derivation of the QEI at one-particle level . . . . . . . . . . . . . . . 111 7.2 Extending the scope of the QEI result . . . . . . . . . . . . . . . . . 117 7.3 A general recipe to obtain QEIs at one-particle level . . . . . . . . . 119 8 Examples 123 8.1 Models with one scalar particle type without bound states . . . . . . 123 8.2 Generalized Bullough-Dodd model . . . . . . . . . . . . . . . . . . . 125 8.3 Federbush model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.4 O(n)-nonlinear sigma model . . . . . . . . . . . . . . . . . . . . . . . 130 9 Conclusion, discussion, and outlook 134 A Constructive aspects of integrable quantum field theory 137 A.1 Representation theory of the Poincaré group in 1+1d . . . . . . . . . 137 A.2 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.3 S-function and ZF operators in a basis . . . . . . . . . . . . . . . . . 143 A.4 Improper rapidity eigenstates . . . . . . . . . . . . . . . . . . . . . . 145 A.5 Bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 B Literature survey: Form factor conventions 158 C Stress-energy tensor 159 C.1 Stress-energy tensors for the free scalar field . . . . . . . . . . . . . . 159 C.2 A weaker notion for the density property . . . . . . . . . . . . . . . . 163 C.3 Stress-energy tensor at one-particle level generating the boosts . . . . 164 Bibliography 166

This thesis explores systems that exhibit strong coupling between an optical cavity field and a many-particle system. To treat the drive and dissipative nature of the cavity on the same footing as the dynamics of the many-particle system, we use a non-equilibrium field theoretic approach. The first system considered is an ultracold bosonic gas trapped inside a cavity. The dispersive coupling between the cavity field and the atoms' motion leads to the formation of a polariton. We show how a modulation of the pump laser on the energy scale of the transverse cavity mode splitting can be used to create effective interactions between different cavity modes. This effective interaction results in the polariton acquiring a multimode nature, exemplified by avoided crossings in the cavity spectrum. As the laser power is increased, the polariton softens and at a critical power becomes unstable. This instability signals the transition into a superradiant state. If the multimode polariton contains a cavity mode with an effective negative detuning, then the transition does not happen through a mode softening but at a finite frequency. To investigate this, classical non-linear equations are constructed from the action and from these we derive the critical couplings and frequencies. It is shown how the superradiant transition happening at a finite frequency is a consequence of a competition between the negatively and the positively detuned cavity modes making up the polariton. The finite-frequency transition is found to be equivalent to a Hopf bifurcation and leads to the emergence of limit cycles. Our analysis shows that the system can exhibit both bistabilities and evolution constricted to a two-torus. We end the investigation by showing how interactions among the atoms combined with the emerging limit cycle open new phonon scattering channels. The second system considered in the thesis is inspired by the recent experiments on gated Transition-metal dichalcogenides (TMD) monolayers inside cavities. An exciton within the TMD can couple strongly to the cavity and, due to the electronic gating, also interact strongly with the conduction electrons. To treat the strong interactions of the excitons with both cavity and electrons, we solve the coupled equations for the correlation functions non-perturbatively within a ladder approximation. The strong interactions give rise to new quasiparticles known as polaron-polaritons. By driving the system through the cavity, we show how the competition between electron-induced momentum relaxation and cavity loss leads to the accumulation of polaritons at a small but finite momentum, which is accompanied by significant decrease of the polariton linewidth Due to the hybrid nature of the polaron-polariton, we show that this behavior can by qualitatively modified by changing the cavity detuning.

We calculate tree-level scattering amplitudes in φ^3 theory and Yang-Mills theory by means of the perturbiner expansion. This involves solving the Euler-Lagrange equations of motion perturbatively via a multi-particle ansatz, and using Berends-Giele recursion relations to extract the solution from simple on-shell data. The results are Berends-Giele currents which are then used to calculate the scattering amplitudes. The theoretical calculations are implemented into a Mathematica script which effectively handles recursive calculations and allows us to calculate amplitudes for an arbitrary number of particles.