+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Physics +--{: .hide} [[!include physicscontents]] =-- #### Fields and quanta +-- {: .hide} [[!include fields and quanta - table]] =-- =-- =-- \tableofcontents ## Idea The [[field (physics)|field]] of [[quantum field theory]] started out as a description of the [[fundamental particles]] that are observed in [[experiment]], such as [[electrons]] and [[photons]]. However, even so, abstractly the formalization of the concept of a _particle_ within [[QFT|QFTs]] is somewhat subtle. If the quantum field theory is on [[Minkowski space]] and comes with a [[Hilbert space|Hilbert]] [[space of states]] on which the [[Poincare group]] of translations, rotations, and boosts in Minkowski space acts, the massive _particle_ excitations of the theory can be found in the discrete spectrum of the time translation operator as the [[irreducible representation|irreducible]] [[unitary representations of the Poincare group]]. For QFTs on [[curved spacetime|curved]] [[spacetimes]], the situation is more subtle. Often, QFTs are considered as [[quantizations]] of given [[Lagrangians]]. In these cases, one often identifies their particle content with that explicitly encoded by the Lagrangian. Notably, when that arises from [[second quantization]] of some 1-dimensional [[sigma-model]], the particles of the theory are those described by these sigma-models. ## What is a particle? {#WhatIsAParticle} The fundamental concept of modern [[physics]] is that of [[quantum field theory]] (QFT); the concept of a particle is derived from that, and need not make sense in every case. ("That's why it's called 'field theory'.") In the perspective of the [[Schrödinger picture]], a $(d+1)$-dimensional [[QFT]] is given by a [[functor]] $Z$ on a [[category of cobordisms]] (possibly with [[geometry|geometric]] structure, such as a [[pseudo-Riemannian metric]] structure) between $d$-dimensional [[manifolds]] ("[[FQFT]]"). It is crucial to notice that one such QFT always has **two different interpretations**: 1. a [first quantized worldvolume perspective](#FirstQuantizedPerspective); 1. a [second quantized spacetime perspective](#SecondQuantizedSpacetimePerspective). ### First quantized worldvolume perspective {#FirstQuantizedPerspective} If we think of a $d$-dimensional manifold as the shape of a some quantum object -- (commonly called a $d$-[[brane]]) -- then a [[cobordism]] between two such is thought of as a piece of [[worldvolume]], a way for parts of such an object to interact with other parts. From this perspective, the functor $Z$ assigns to a manifold the [[space of quantum states]] that the [[brane]] of this shape may have, and to a [[cobordism]] the linear map which is the time evolution along the cobordism. If $d = 0$, then the [[brane]] is a "0-brane" and this is a "particle" (or [[D0-brane]]), the worldvolume is the "[[worldline]]", and the QFT encodes the [[worldline theory]] of the particle, its [[quantum mechanics]]. If instead $d = 1$, then the brane is a 1-brane, for instance a [[string]] of [[D1-brane]]. If $d = 2$, then the brane is a 2-brane also called a [[membrane]], and so on. Given this, one may try to see if this data describes a brane propagating _in_ some [[spacetime]] (the "[[target space]]" of the brane). It is the topic of [[spectral geometry]] (in the sense of [[Alain Connes]]) to try to reconstruct from this data the would-be target spacetime that the brane is propagating in. For instance, for $d = 0$, the data of a QFT in this sense is a [[spectral triple]], and [[noncommutative geometry]] provides a general way to make sense of the target space of the particle. If $d =1$, the QFT data here is that of a [[2-spectral triple]], and so on. ### Second quantized perspective {#SecondQuantizedSpacetimePerspective} On the other hand, we may think of the $d+1$-dimensional [[cobordisms]] themselves already as [[spacetimes]]. In this case, the QFT describes [[field (physics)|fields]] on spacetime. In favorable circumstances, this can arise from the previous case by a process of [[second quantization]], meaning that these fields may be thought of as [[condensates]] of branes/particles in the previous sense. Conversely, one says that these particles are the _quanta_ of the fields that we start with. But generally, given a QFT in this perspective, to extract from it the particle content that it comes from under [[second quantization]] is subtle. One of the common definitions of particle quanta only applies to non [[general covariance|generally-covariant]] [[free field theories]] (e.g. [Haag 92, section VI](#Haag92)). This means that already for quantum field theory on a fixed [[curved spacetime]] there is in general no longer any concept of particle-quanta of the fields. This situation would only become worse were one to think of the given QFT as incorporating [[quantum gravity]] as well. The concept of a [[field (physics)|field]] is fundamental, while that of particle quanta is not. ### Mixed perspectives Since the formalism of [[FQFT]] does not "know" whether we want to think of a given QFT as a [first-quantized worldvolume theory](#FirstQuantizedPerspective) or as a [second quantized spacetime theory](#SecondQuantizedSpacetimePerspective), in general both perspectives may be sensible at the same time. This is indeed so, but of course this mixing only becomes relevant once one really dares to consider higher-dimensional [[branes]] in the first place, hence in [[string theory]]. Indeed, [[perturbative string theory]] is all set up this way: one starts with a 2-dimensional QFT which one thinks of as the first-quantized [[worldsheet]] theory of a [[string]]. But this means that one may start to ask which "particles" propagate "on the worldvolume". Notably, the "embedding fields" of the string [[sigma-model]] which describes how its worldsheet sits in [[spacetime]] look, from the perspective of the worldsheet theory, like [[scalar fields]]. Their [[superpartners]] look like [[fermion]] fields. If one considers the string worldsheet before gravitational [[gauge fixing]], then there is also a [[graviton]] on the worldsheet, and so on. Hence one may want to/need to consider an intricate pattern of "branes within branes". For instance, the [[worldsheet]] [[gravity]] of the [[string]] may itself arise from quantizing other strings for which that worldsheet is the target spacetime ([[world sheets for world sheets|Green 87]]). ## Examples ### Particles and non-particles in 3d TQFT {#ParticlesAndNonParticlesIn3dTQFT} We consider a [[QFT]] which is a [[3d TQFT]] of [[Chern-Simons theory]] type and discuss some aspects of the notion of a _particle_ in that context. Assume for the sake of argument that we agree to think of the 3d TQFT as a realization of [[3d quantum gravity]], as indicated there. Then this means that, as an [[FQFT]], the system assigns to every closed [[surface]] a [[space of quantum states]] to be thought of as the space of states of the [[observable universe]] in that 2+1-dimensional world. A state encodes the field of [[gravity]]. It would be somewhat subtle to extract from just the [[FQFT|functorial]] [[3d TQFT]] the intrinsic notion of particle-quanta. In general, given a [[TQFT]] in the form of an [[FQFT]], there is essentially no established way to go about determining what the particle-quanta would be that an observer "in this universe" would see. In fact, an argument due to ([Witten 92](A-model#Witten92)) says that if [[Chern-Simons theory]] [[3d TQFT]] is the [[second quantization]] of anything, then it is not of particles but of [[topological strings]] ([[A-model]]). See also ([Costello 06](A-model#Costello06)) and see at _[TCFT -- Worldsheet and effective background theories](TCFT#ActionFunctionals)_ for more on this. Beware, again, that this concerns the quanta for the fields of the QFT regarded as a QFT on a 2+1-dimensional [[spacetime]]. However we may change perspective and instead think of the [[3d TQFT]] here as a first-quantized [[worldvolume]] [[theory (physics)|theory]]. As such it would be a [[membrane]] [[theory (physics)|theory]], often called the _[[topological membrane]]_, naturally. Now if we allow [[boundaries]] of [[worldvolume]], hence consider an [[extended TQFT]] with its [[boundary field theory]], then the boundary theory is a 1+1-dimensional worldsheet theory, hence describes a [[string]]. This way of how a first quantized [[string]] can arise as the [[boundary field theory]] of a first-quantized [[topological membrane]] is an instance of the "[[holographic principle]]" known as _[[AdS3-CFT2 and CS-WZW correspondence]]_. Moreover, going further up in [[codimension]], the [[3d TQFT]] may have [[defect field theory|defects]] of codimension 2, hence have inside it a 0+1-dimensional [[defect field theory]]. This hence may be thought of as a first-quantized particle. (Notice that it is a _first quantized_ particle, not a quantum of a field of the 3d theory regarded as a spacetime theory, for these particles-as-quanta do not have worldlines given by cobordisms,only their first quantized avatars do, but they are not the first quantized 1d defect theory considered now.) Indeed, these first quantized codim-2-defects/0-branes/1d-particles in [[Chern-Simons theory]] are famous as having "[[Wilson line]] [[worldline theory]]". See at _[orbit method -- Nonabelian charged particle trajectories](orbit%20method#GaugeAndGravityWilsonLoops)_ for details on their incarnation as [[prequantum field theory]]. After [[quantization]] these first quantized 0-branes/1d-particles are famously represented in the [[Reshetikhin-Turaev construction]] as ribbon lines labeled by [[objects]] in a [[modular tensor category]]. In conclusion, given a [[3d TQFT]] regarded as [[quantum gravity]] of 2+1-dimensional [[spacetimes]], it is at best subtle to extract from it particles in the sense of "quanta of the fields of the spacetime field theory", while extracting from it first quantized codim-2 [[defect field theory|defect]] 0-[[branes]] is a famous step in [[Chern-Simons theory]]. ## Related concepts * [[sigma-model]] * [[worldline]], [[worldline theory]] * [[non-relativistic particle]] * [[relativistic particle]], [[spinning particle]], [[superparticle]] * [[virtual particle]], [[antiparticle]] * [[fundamental particle]], [[standard model of particle physics]] * [[quasiparticle]], [[anyon]] * [[matter]], [[force]] * [[brane]], [[string]], [[membrane]] * [[mechanics]] * [[vacuum]] ## References {#References} ### General Chapter VI of * {#Haag92} [[Rudolf Haag]], _[[Local Quantum Physics]]_ (1992) discusses how to extract notions of particles from a [[local net of observables]] satisfying the [[Haag-Kastler axioms]]. Further discussion of subtleties of the definition of particles _in_ (non-[[free field theory|free]]) [[quantum field theories]]: * {#Bain00} [[Jonathan Bain]], _Against particle/field duality: Asymptotic particle states and interpolating fields in interacting QFT (or: Who's afraid of Haag's theorem)_, Erkenntnis 53: 375–406, 2000 ([pdf](http://faculty.poly.edu/~jbain/papers/lsz.pdf)) * Benjamin H. Feintzeig, Jonah Librande, Rory Soiffer, *Localizable Particles in the Classical Limit of Quantum Field Theory* ([arXiv:2104.06442](https://arxiv.org/abs/2104.06442)) ### Classical charged particles {#ReferencesClassicalChargedParticles} Tentative discussion of charged particles via [[classical mechanics]] (cf. *[[classical charged particle]]*), goes back to: * [[Max Abraham]]: *Prinzipien der Dynamik des Elektrons*, Annalen der Physik **315** 1 (1902) 105-179 [[doi:10.1002/andp.19023150105](https://doi.org/10.1002/andp.19023150105)] * [[Hendrik Lorentz]]: *Electromagnetic phenomena in a system moving with any velocity smaller than that of light*, Proc. Acad. Science Amsterdam **4** (1904) 669–678 * [[P. A. M. Dirac]]: *Classical theory of radiating electrons*, Proc. A **167** 929 (1938) 148–169 [[doi:10.1098/rspa.1938.0124](https://doi.org/10.1098/rspa.1938.0124), [jstor:97128](https://www.jstor.org/stable/97128)] and motivated early discussion of what is now called a *[[relativistic membrane]] model*: * {#Dirac62} [[Paul Dirac]], *An Extensible Model of the Electron*, Proc. Roy. Soc. A **268** (1962) 57-67 [[jstor:2414316](https://www.jstor.org/stable/2414316)] > (also proposing the [[Dirac-Born-Infeld action]]) * [[Paul Dirac]], _The motion of an Extended Particle in the Gravitational Field_, in: [[Leopold Infeld|L. Infeld]] (ed.), *Relativistic Theories of Gravitation*, Proceedings of a Conference held in Warsaw and Jablonna, July 1962, P. W. N. Publishers, 1964, Warsaw, 163-171; discussion 171-175 [[spire:1623740](http://inspirehep.net/record/1623740), article:[[Dirac-MotionOfAnExtendedParticle.pdf:file]], full proceedings:[pdf](https://cds.cern.ch/record/2282975/files/warsaw-1962.pdf)] * [[Paul Dirac]], _Particles of Finite Size in the Gravitational Field_, Proc. Roy. Soc. A **270** (1962) 354-356 [[doi:10.1098/rspa.1962.0228](https://doi.org/10.1098/rspa.1962.0228)] Claim that technical issues with the old Abraham-Lorentz-Dirac approach can be resolved: * [[Fritz Rohrlich]]: *Classical Charged Particles* (1964), World Scientific (1995, 2007), CRC Press (202) [[doi:10.1142/6220](https://doi.org/10.1142/6220), [doi:10.1201/9780429034947](https://doi.org/10.1201/9780429034947)] * Arthur D. Yaghjian: *Relativistic Dynamics of a Charged Sphere -- Updating the Lorentz-Abraham Model*, Lecture Notes in Physics **686**, Springer (2006) [[doi:10.1007/978-3-031-06067-0](https://doi.org/10.1007/978-3-031-06067-0), [pdf](https://www.prl.res.in/~library/gpdf/e-books/Springer_e-books/Relativistic%20Dynamics%202006.pdf)] * [[Fritz Rohrlich]]: *The dynamics of a charged particle*, Phys. Rev. E **77** (2008) 046609 [[arXiv:0804.4614](https://arxiv.org/abs/0804.4614), [doi:10.1103/PhysRevE.77.046609](https://doi.org/10.1103/PhysRevE.77.046609)] [[!redirects particles]] [[!redirects quantum]] [[!redirects quanta]] [[!redirects classical charged particle]] [[!redirects classical charged particles]]