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\begin{document}

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\section*{particle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{fields_and_quanta}{}\paragraph*{{Fields and quanta}}\label{fields_and_quanta}

[[!include fields and quanta - table]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{WhatIsAParticle}{What is a particle?}\dotfill \pageref*{WhatIsAParticle} \linebreak
\noindent\hyperlink{FirstQuantizedPerspective}{First quantized worldvolume perspective}\dotfill \pageref*{FirstQuantizedPerspective} \linebreak
\noindent\hyperlink{SecondQuantizedSpacetimePerspective}{Second quantized perspective}\dotfill \pageref*{SecondQuantizedSpacetimePerspective} \linebreak
\noindent\hyperlink{mixed_perspectives}{Mixed perspectives}\dotfill \pageref*{mixed_perspectives} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ParticlesAndNonParticlesIn3dTQFT}{Particles and non-particles in 3d TQFT}\dotfill \pageref*{ParticlesAndNonParticlesIn3dTQFT} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 \emph{particle} within [[QFT]]s is somewhat subtle.

If the quantum field theory is on [[Minkowski space]] and comes with a [[Hilbert space|Hilbert]] [[space of states]] on which thus the [[Poincare group]] of translations, rotations and boosts in Minkowski space acts, the massive \emph{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, however, 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.

\hypertarget{WhatIsAParticle}{}\subsection*{{What is a particle?}}\label{WhatIsAParticle}

The fundamental concept of modern [[physics]] is that of [[quantum field theory]] (QFT); the concept of 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 [[pseudo-Riemannian metric]] structure) between $d$-dimensional [[manifolds]] (``[[FQFT]]''). It is crucial to notice that one such QFT always has \textbf{two different interpretations}:

\begin{enumerate}%
\item a \hyperlink{FirstQuantizedPerspective}{first quantized worldvolume perspective};


\item a \hyperlink{SecondQuantizedSpacetimePerspective}{second quantized spacetime perspective}.



\end{enumerate}
\hypertarget{FirstQuantizedPerspective}{}\subsubsection*{{First quantized worldvolume perspective}}\label{FirstQuantizedPerspective}

If we think of a $d$-dimensional manifold as the shape of a some quantum object -- (as such 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.

Here 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 \emph{in} some [[spacetime]] (the ``[[target space]]'' of the brane). It is the topic of [[spectral geometry]] (in the sense of [[Alain Connes]]`s) 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 here 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.

\hypertarget{SecondQuantizedSpacetimePerspective}{}\subsubsection*{{Second quantized perspective}}\label{SecondQuantizedSpacetimePerspective}

On the other hand, we may think of the $d+1$-dimensional [[cobordisms]] here 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 \emph{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. \hyperlink{Haag92}{Haag 92, section VI}). 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 also [[quantum gravity]]. The concept of [[field (physics)|field]] here is fundamental, that of particle quanta is not.

\hypertarget{mixed_perspectives}{}\subsubsection*{{Mixed perspectives}}\label{mixed_perspectives}

Since the formalism of [[FQFT]] does not ``know'' whether we want to think of a given QFT as a \hyperlink{FirstQuantizedPerspective}{first-quantized worldvolume theory} or as a \hyperlink{SecondQuantizedSpacetimePerspective}{second quantized spacetime theory} 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 in general one may want to/need to consider an intricate pattern of ``branes withing branes''. For instance the [[worldsheet]] [[gravity]] of the [[string]] may itself arises from quantizing other strings for which that worldsheet is target spacetime ([[world sheets for world sheets|Green 87]]).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{ParticlesAndNonParticlesIn3dTQFT}{}\subsubsection*{{Particles and non-particles in 3d TQFT}}\label{ParticlesAndNonParticlesIn3dTQFT}

We consider a [[QFT]] which is a [[3d TQFT]] of [[Chern-Simons theory]] type and discuss some aspects of the notion of \emph{particle} in that context.

Assume for the sake of argument that we agree to think of the 3d TQFT here 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 in here encodes the field of [[gravity]]. It would be somewhat subtle to extract from just the [[FQFT|functorial]] [[3d TQFT]] here the intrinsic notion of particle-quanta. In general, given an [[TQFT]] in the form of an [[FQFT]], there is essentially no established way to going about determining what the particle-quanta would be that an observer ``in this universe'' would see.

In fact, an argument due to (\href{A-model#Witten92}{Witten 92}) 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 (\href{A-model#Costello06}{Costello 06}) and see at \emph{\href{TCFT#ActionFunctionals}{TCFT -- Worldsheet and effective background theories}} 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 \emph{[[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 \emph{[[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 \emph{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 \emph{\href{orbit%20method#GaugeAndGravityWilsonLoops}{orbit method -- Nonabelian charged particle trajectories}} 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]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[sigma-model]]


\item [[worldline]], [[worldline theory]]


\item [[non-relativistic particle]]


\item [[relativistic particle]], [[spinning particle]], [[superparticle]]


\item [[virtual particle]], [[antiparticle]]


\item [[fundamental particle]], [[standard model of particle physics]]


\item [[matter]], [[force]]


\item [[brane]], [[string]], [[membrane]]


\item [[mechanics]]


\item [[vacuum]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

Chapter VI of

\begin{itemize}%
\item [[Rudolf Haag]], \emph{[[Local Quantum Physics]]} (1992)

\end{itemize}
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 \emph{in} (non-[[free field theory|free]]) field theories includes

\begin{itemize}%
\item [[Jonathan Bain]], \emph{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 (\href{http://faculty.poly.edu/~jbain/papers/lsz.pdf}{pdf})

\end{itemize}
[[!redirects particles]]

[[!redirects quantum]] [[!redirects quanta]]



\end{document}