Types of quantum field thories
However, even so, abstractly the formalization of the concept of particle within QFTs is somewhat subtle.
If the quantum field theory is on Minkowski space and comes with a Hilbert space of states on which thus 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 unitary representations of the Poincare group. For QFTs on 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.
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 -dimensional QFT is given by a functor on a category of cobordisms (possibly with geometric structure, such as pseudo-Riemannian metric structure) between -dimensional manifolds (“FQFT”). It is crucial to notice that one such QFT always has two different interpretations:
If we think of a -dimensional manifold as the shape of a some quantum object – (as such commonly called a -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 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.
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’s) to try to reconstruct from this data the would-be target spacetime that the brane is propagating in. For instance for 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 the QFT data here is that of a 2-spectral triple, and so on.
On the other hand, we may think of the -dimensional cobordisms here themselves already as spacetimes. In this case the QFT describes 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 generally-covariant free field theories (e.g. 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 here is fundamental, that of particle quanta is not.
Since the formalism of FQFT does not “know” whether we want to think of a given QFT as a first-quantized worldvolume theory or as a second quantized spacetime theory in general both perspectives may be sensible at the same time.
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 (Green 87).
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 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 (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 (Costello 06) and see at 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. As such it would be a membrane 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 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 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 0-branes is a famous step in Chern-Simons theory.
Chapter VI of
Further discussion of subtleties of the definition of particles in (non-free) field theories includes