2.1 States

For now, we will restrict ourselves to the simplest example of a quantum field: the free real scalar field $\phi (x,t)$ . The label “free” means that excitations of this field – its associated particle – do not interact with one another. States of this field can be understood to describe multiple, noninteracting particles. We will introduce some simple machinery to construct the states of this free scalar field: creation and annihilation operators. Our goal is to see how the formalism of creation and annihilation operators can be used to construct the entire Hilbert space of states associated with this free field $\phi (x,t)$ , to see that these creation and annihilation operators provide a useful representation of the quantum field itself, and to use them to clarify the relationship between the field operators $\phi (x,t)$ and other elements of the algebra $A$ of operators.

Suppose the field $\phi (x,t)$ is in its ground state, that is, its lowest-energy state: a state with no particles. This state is called the vacuum and we will label it $|0⟩$ . To construct the additional states in $H$ , we can introduce creation and annihilation operators $a^{†}$ and $a$ that add or remove a particle from a state and apply them to the vacuum state. A real scalar field has no “internal” degrees of freedom (i.e., no degrees of freedom other than momentum and position), so the creation and annihilation operators can be labeled $a_p^{†}$ and $a_p$ : they create and annihilate particles with 4-momentum $p^{\mu }=\left(E_p,\text{p}\right)$ . In general, creation and annihilation operators add or remove particles with properties, or “quantum numbers,” determined by the associated quantum field; for example, for a fermionic field these operators would be labeled by momentum and spin. (We label particles by their spatial momenta, with the understanding that they satisfy the relativistic mass–energy relation $E_p^2=p^2+m^2$ , where m is the rest mass of the particle. In the rest frame of the particle, where 3-momentum p = 0, this becomes $E=m$ , which is $E=m{c}^2$ in natural units.)

Beginning with the vacuum state $|0⟩$ , one can construct states of the field describing multiple noninteracting particles as follows:

$a_p^{†}|0⟩=|p⟩,a_{p^{\prime }}^{†}{a}_p^{†}|0⟩=|p,p^{\prime }⟩,a_{p^{\prime \prime }}^{†}{a}_{p^{\prime }}^{†}{a}_p^{†}|0⟩=|p,p^{\prime },p^{\prime \prime }⟩,\dots$

The first is a state of the field containing a single particle with momentum p; the second is a state of the field containing two noninteracting particles, one with momentum p and the other with momentum $p^{\prime }$ ; and so on. Similarly, the action of the annihilation operator is defined as follows:

$\dots ,a_{p^{\prime \prime }}|p,p^{\prime },p^{\prime \prime }⟩=|p,p^{\prime }⟩,a_{p^{\prime }}{a}_p|p,p^{\prime }⟩=|0⟩,a_p|0⟩=0.$

A defining property of the vacuum state $|0⟩$ is that it is the unique state in $H$ that returns zero when acted on by any annihilation operator $a_p$ .

As one might suspect, any multiparticle state of the quantum field $\phi (x,t)$ can be constructed by appropriate applications of creation operators to the vacuum state:

$|p_1^{n_1},p_2^{n_2},\dots ,p_r^{n_r}⟩\propto \left(a_p^{†}{)}^{n_1}\right(a_{p_2}^{†}{)}^{n_2}\dots (a_{p_r}^{†}{)}^{n_r}|0⟩,$

where $n_j$ labels the number of particles with momentum $p_j$ . (This representation of the state is called the occupation number representation: $n_j$ labels the number of particles occupying a particular momentum mode $p_j$ .)

Every state of the free real scalar field $\phi (x,t)$ can be constructed via this procedure: one can construct the entire Hilbert space for a free quantum field by acting on its vacuum state with (linear combinations of) creation and annihilation operators. This Hilbert space consisting of states that describe multiple, noninteracting particles is called a Fock space. It plays a central role in the formulation of scattering theory, a central aspect of QFT that we will describe shortly.

The commutation relations that hold between creation and annihilation operators encode information about the physical system with which they are associated. For bosons, they obey the following commutation relations:

$[a_{p\sigma },a_{p^{\prime }{\sigma }^{\prime }}]=0=[a_{p\sigma }^{†},a_{p^{\prime }{\sigma }^{\prime }}^{†}],[a_{p\sigma },a_{p^{\prime }{\sigma }^{\prime }}^{†}]=\delta (p-p^{\prime }){\delta }_{\sigma {\sigma }^{\prime }},$

where the index $\sigma$ stands for any additional quantum numbers that the bosonic field might possess. The creation and annihilation operators for fermions obey anti-commutation relations:

$\{a_{p\sigma },a_{p^{\prime }{\sigma }^{\prime }}\}=0=\{a_{p\sigma }^{†},a_{p^{\prime }{\sigma }^{\prime }}^{†}\},\{a_{p\sigma },a_{p^{\prime }{\sigma }^{\prime }}^{†}\}=\delta (p-p^{\prime }){\delta }_{\sigma {\sigma }^{\prime }}.$

The commutation relations for bosonic creation and annihilation operators reflect the fact that multiple particles associated with a bosonic field can occupy the same state. For the real scalar field $\phi (x,t)$ , this means multiple particles can occupy the same momentum mode p; for bosons with more degrees of freedom, it means that multiple bosons can occupy states with identical quantum numbers. Things are different for fermions: the anti-commutation relations entail that no two particles associated with a fermionic field can occupy the same state, as required by the Pauli exclusion principle.

The Hilbert space $H$ associated with the free scalar field $\phi (x,t)$ is the Fock space we just described. How do we represent operators acting on this Hilbert space?

2.2 Operators

The formalism of creation and annihilation operators proves extremely useful here as well. One can prove that any operator $O$ acting on $H$ can be written as a sum of products of creation and annihilation operators (Reference WeinbergWeinberg, 1995: section 4.2):

$\begin{array}{cc}O& =\sum _{N=0}^{\infty }\sum _{M=0}^{\infty }\int d{p}_1^{\prime }\dots d{p}_N^{\prime }d{p}_1\dots d{p}_M{C}_{NM}(p_1^{\prime }\dots p_N^{\prime }{p}_1\dots p_M)\\ & \times a_{p_1^{\prime }}^{†}\dots a_{p_N^{\prime }}^{†}{a}_{p_M}\dots a_{p_1},\end{array}$

where the $C_{NM}$ are complex-valued coefficients. This means that the action of any operator on states in $H$ can be described by acting with an appropriate sequence of creation and annihilation operators. (Note that all of the creation operators are to the left of the annihilation operators; this is called normal ordering. We will always assume our operators are normal ordered.) It is instructive to see a couple of operators expressed in this form; eventually, we will see how to express the field operators $\phi (x,t)$ themselves this way.

Suppose one wanted an operator that counted the number of particles present in any state of the field. One can construct such an operator using the creation and annihilation operators as follows. First, we were cavalier about normalization when introducing the action of the creation and annihilation operators Section 2.1. If we are more careful, the action of the creation and annihilation operators is

$\begin{array}{c}{a}_{p_j}|p_1^{n_1},\dots ,p_j^{n_j},\dots ,p_r^{n_r}⟩=\sqrt{n_j}|p_1^{n_1},\dots ,p_j^{n_j-1},\dots ,p_r^{n_r}⟩\\ a_{p_j}^{†}|p_1^{n_1},\dots ,p_j^{n_j},\dots ,p_r^{n_r}⟩=\sqrt{n_j+1}|p_1^{n_1},\dots ,p_j^{n_j+1},\dots ,p_r^{n_r}⟩.\end{array}$

The product $a_{p_j}^{†}{a}_{p_j}$ counts the number of particles occupying the momentum mode $p_j$ :

$\begin{array}{cc}{a}_{p_j}^{†}{a}_{p_j}|p_1^{n_1},\dots ,p_j^{n_j},\dots ,p_r^{n_r}⟩& =\sqrt{n_j}{a}_{p_j}^{†}|p_1^{n_1},\dots ,p_j^{n_j-1},\dots ,p_r^{n_r}⟩\\ & =\sqrt{n_j}\sqrt{n_j-1+1}|p_1^{n_1},\dots ,p_j^{n_j},\dots ,p_r^{n_r}⟩\\ & =n_j|p_1^{n_1},\dots ,p_j^{n_j},\dots ,p_r^{n_r}⟩,\end{array}$

We want our operator to count the number of particles occupying any momentum mode, so to construct our operator we integrate over all momentum modes:

$N=\int d^{3}p{a}_p^{†}{a}_p.$

The result is the number operator, an operator that counts the total number of particles in any state of the field.

A particularly important example is the Hamiltonian of the free scalar field $\phi (x,t)$ . Although more frequently expressed using field operators $\phi (x,t)$ , in a form we will see shortly, it is often convenient to write it using creation and annihilation operators:

$H=\int d^{3}p{E}_p{a}_p^{†}{a}_p.$

This Hamiltonian “counts” the total energy in a state of the free scalar field $\phi (x,t)$ . It does this by counting the total number of particles in each momentum mode, with each term weighted by the energy $E_p$ associated with a particle in that mode. It is easy to verify that the Fock space states

$|p_1^{n_1},p_2^{n_2},\dots ,p_r^{n_r}⟩$

are eigenstates of this Hamiltonian, with eigenvalues of the form

$E=E_{p_1}{n}_1+\dots +E_{p_r}{n}_r.$

For a QFT describing free fields, the number operator and the Hamiltonian commute: total particle number is conserved during time evolution. This is only true for free fields; in QFT, interactions can create and annihilate particles. For interacting fields, total energy is still conserved but particle number is not.

We can now turn to quantum fields. Recall that the free real scalar field, being a quantum field, is an assignment of an operator to every point on a spacetime manifold $M$ . The operators $\phi (x,t)$ assigned to each spacetime point are operators and so, by the theorem cited above, can be expressed using creation and annihilation operators:

$\phi (x,t)=\int \widetilde{dp}{a}_p{e}^{ipx}+a_p^{†}{e}^{-ipx},$

where $\widetilde{dp}\equiv \frac{d^{3}p}{(2\pi {)}^{3}2E_p}$ is a Lorentz invariant integration measure. (The integral over 3-momentum does not look Lorentz invariant, but it is Reference SrednickiSrednicki (2007, chapter 3.)

Naively, the physical significance of the field operator $\phi (x,t)$ is that it can create and annihilate particles of any momentum at the spacetime point $(x,t)$ . We will see shortly that this is too naive. Nevertheless, it is true, in a sense, that free field operators act on states to create and annihilate particles and even the too-naive reading can be useful in certain respects. Consider acting on the vacuum with the field operator $\phi (x,t)$ and projecting out any particular momentum mode k as follows:

$\begin{array}{cc}⟨k|\phi (x,t)|0⟩& \propto ⟨0|a_k\int \widetilde{dp}{a}_p{e}^{ipx}+a_p^{†}{e}^{-ipx}|0⟩\\ & =\int \widetilde{dp}\left(e^{ipx}⟨0|a_k{a}_p|0⟩+e^{-ipx}⟨0|a_k{a}_p^{†}|0⟩\right)=e^{-ikx}.\end{array}$

This suggests that acting on the vacuum with the field operator $\phi (x,t)$ creates a particle localized at the spatial point x. One can already see one sense in which this is too naive: the particle created “at x” is in an eigenstate of momentum k. What could it mean to create a particle in an eigenstate of momentum that is also localized at a spatial point? In Section 3, we will return to this and other subtleties concerning our (in)ability in QFT to localize particles in any bounded region of spacetime.

Field operators also obey commutation relations. Bosonic fields like $\phi (x,t)$ obey an analogue of the canonical commutation relations that hold between position and momentum in nonrelativistic quantum mechanics:

$\left[\phi \right(x,t),\phi (y,t\left)\right]=0=\left[\pi \right(x,t),\pi (y,t\left)\right],\left[\phi \right(x,t),\pi (y,t\left)\right]=i{\delta }^3(x-y),$

where $\pi (x,t)=\frac{\partial }{\partial t}\phi (x,t)$ is the canonical momentum associated with the field. For fermionic fields, the field and its associated canonical momentum satisfy analogous anti-commutation relations. In both cases, the commutation relations for the field operators hold if and only if the corresponding commutation relations for the creation and annihilation operators are satisfied. (Note that the canonical momentum is not the same thing as the observable corresponding to momentum.)

We wrote the Hamiltonian for the free real scalar field $\phi (x,t)$ using creation and annihilation operators:

$H=\int d^{3}p{E}_p{a}_p^{†}{a}_p.$

Now that we have constructed the field operators, we can write it in a more common form:

$H=\frac{1}{2}\int d^{3}x\pi (x{)}^2+(\nabla \phi \left(x\right){)}^2+m^2\phi (x{)}^2,$

where $m$ is the mass of the particle associated with the field $\phi (x,t)$ . It is a useful exercise to show that the two formulations are equivalent (Reference SrednickiSrednicki, 2007, chapter 3).

We also require that not only the fields themselves, but also any two operators whatsoever commute when they are localized in spacelike separated regions. That is, suppose that two spacetime points $x^{\mu }$ and $y^{\mu }$ cannot be connected by a light signal. We require that any two operators $O\left(x^{\mu }\right)$ and $O\left(y^{\mu }\right)$ commute; this ensures that actions performed at $x^{\mu }$ cannot influence the statistical distribution of outcomes of measurements of $O\left(y^{\mu }\right)$ . More generally, if no light signal can connect any two points in bounded regions of spacetime $O_1$ and $O_2$ , any two operators localized in those regions will commute. This is called the microcausality condition.

The field operators are, in a sense, simply operators like any other. Nevertheless, they play a privileged role in QFT. Quantum fields are not necessarily observables themselves; the real scalar field $\phi (x,t)$ happens to be Hermitian, but others – complex scalar fields, spinor fields, and so on – are not. However, observables can be constructed out of sums of products of field operators. Furthermore, the field itself is the fundamental dynamical object in the theory, in the sense that the equations of motion (i.e., the Euler-Lagrange equations) of any QFT model describe the dynamical evolution of the fields in that model. Our goal in the remainder of this section is to work up to one of the central uses of QFT in particle physics: the calculation of probability distributions for the outcomes of scattering processes.

2.3 Dynamics

Just as in nonrelativistic quantum mechanics, dynamical evolution in QFT is generated by a Hamiltonian $H$ and is represented by a unitary operator

$U\left(\tau \right)=e^{-iH\tau },$

where $\tau$ is the duration of time that the system is subjected to the conditions described by $H$ . (In practice, one encounters Lagrangians much more frequently than Hamiltonians in QFT. We’ll see the virtue of formulating QFT using Lagrangians when we encounter path integrals.) However, while use of the Schrödinger picture of time evolution is common in nonrelativistic quantum mechanics, it is rare in QFT. It is much more common to use either the Heisenberg picture or the interaction picture.

In the Heisenberg picture, states of the system are stationary and the operators evolve in time. In the interaction picture, the total Hamiltonian $H$ is split into a “free” term $H_0$ and an “interaction” term $H_{int}$ and the time-dependence of the system is distributed across the states and the operators: operators evolve in accord with $H_0$ , while the evolution of states is governed by $H_{int}$ . In our discussion of scattering theory, we will make use of the interaction picture. This is worth noting because Haag’s theorem demonstrates that the interaction picture is mathematically ill-defined in QFT, a fact that has generated interesting discussion about how to justify use of the interaction picture; see Reference Earman and FraserEarman and Fraser (2006), Reference DuncanDuncan (2012, chapter 10.5), and Reference MillerMiller (2018).

In particle physics, very frequently the dynamical setting of interest is scattering theory. (For a clear and careful presentation of assumptions employed by several different formulations of scattering theory, see Reference DuncanDuncan (2012, chapter 9). For a historical perspective on how scattering theory became central to QFT, see Reference BlumBlum (2017).) In scattering theory, one aims to model the following situation: a set of widely separated, noninteracting particles in some initial state $|\alpha ⟩$ approach each other, interact and undergo extremely complicated motion for some finite period of time, and then once again become widely separated and noninteracting in some final state $|\beta ⟩$ . The initial state $|\alpha ⟩$ and the final state $|\beta ⟩$ are both states in the Fock space $H$ composed of states of the free field; this means they are eigenstates of the free Hamiltonian $H_0$ . The justification for this is the assumption that the particles in these initial and final states are sufficiently widely separated as to be noninteracting.

It is worth pausing to be clear about the meaning of “widely separated.” Mathematically, one requires only that the particles be strictly noninteracting when their spatial separation becomes infinite. Philosophers are sometimes alarmed by such infinite idealizations. “The tunnels of particle accelerators are only a few meters wide!” they object. “How could particles possibly become sufficiently separated to justify the assumption that they are non-interacting?” Although there are a number of interesting philosophical issues surrounding the use of idealizations in physics, this particular idealization has a straightforward unpacking: one requires only that the spatial separation between the particles be large compared to the range of the forces by which they interact. Often this can be a surprisingly short distance: in the 1930s, Hideki Yukawa showed that a force mediated by a massive particle decays exponentially with the distance between the bodies. This means that bodies have to be very close together to experience meaningful interactions due to such a force; a meter is so much larger than the range of such a force that it might as well be infinite. (Justifying this assumption for particles that interact via forces mediated by massless particles, like electromagnetism, requires a more subtle mathematical story, since those forces decay only polynomially with distance. See Reference StrocchiStrocchi (2013, chapter 6.3) or Reference DuncanDuncan (2012, chapter 19.1) for discussion.)

A very important fact about scattering theory is that one does not keep track of the detailed dynamical evolution of the particles. One is interested only in the initial and final states long before and long after the scattering event; the scattering itself is treated as a black box. Aside from specifying the interaction in the Hamiltonian, scattering theory says essentially nothing about the extremely complicated behavior that the particles exhibit while interacting. Instead, one is interested in the following question: Given a quantum field in an initial state $|\alpha ⟩$ and a Hamiltonian $H$ , what is the probability distribution over possible final states $|\beta ⟩$ in which one might find the field after the scattering event?

This information is encoded in an object called the S-matrix. The S-matrix consists of matrix elements of the scattering operator $S$ taken between states in the Fock space $H$ , that is, initial and final states $|\alpha ⟩$ and $|\beta ⟩$ . The scattering operator $S$ acts on the initial state $|\alpha ⟩$ to turn it into a “post-scattering” state $S|\alpha ⟩$ , and the matrix element

$⟨\beta |S|\alpha ⟩\equiv S_{\beta \alpha }$

is the probability amplitude for finding the system in the final state $|\beta ⟩$ . (The terms “scattering amplitude” and “S-matrix element” are used interchangeably.) The probability of finding the system in the final state $|\beta ⟩$ , given that it began in initial state $|\alpha ⟩$ and its dynamical evolution was governed by $S$ , is calculated using the Born rule:

${\left(⟨\beta |S|\alpha ⟩\right)}^2.$

The S-matrix is an object of central importance in particle physics, and in QFT more generally. (At least, in asymptotically flat spacetimes like Minkowski. In cosmologically relevant spacetimes like those that are asymptotically de Sitter, for instance, one cannot define an S-matrix at all (Reference WittenWitten, 2001; Reference BoussoBousso, 2005).) Much of the remainder of this Element is focused on philosophical challenges posed by the strategies that physicists have developed for calculating S-matrix elements.

The reader may be wondering about the nature of the scattering operator $S$ . Its action on the initial state $|\alpha ⟩$ evidently represents dynamical evolution, so how does it relate to the Hamiltonian $H$ ? The short answer is that the scattering operator is constructed by taking early-time and late-time limits of operators that are themselves constructed out of the Hamiltonian $H$ , so that the scattering operator takes a state in the asymptotic past $\tau \to -\infty$ and dynamically evolves it to a state in the asymptotic future $\tau \to +\infty$ .

A slightly more detailed answer begins by returning to our assumption that the initial and final states of the scattering process describe particles so widely separated that they do not interact (for a genuinely detailed answer, see Reference DuncanDuncan (2012, chapter 4.3)). This amounts to the assumption that there is a state of the field $|\Psi ⟩$ in the Hilbert space associated with the interacting theory such that at very early times $\tau \ll 0$ its dynamical evolution – determined by the interacting Hamiltonian – coincides with the dynamical evolution of some state $|\alpha ⟩$ in the Fock space associated with the free theory, governed by the free Hamiltonian $H_0$ :

$\underset{\tau \ll 0}{\lim }{e}^{-iH\tau }|{\psi }_{\alpha }〉=e^{-i{H}_0\tau }|\alpha 〉.$

The analogue must be true for very late times

$\underset{\tau \gg 0}{lim}{e}^{-iH\tau }|{\Psi }_{\beta }⟩=e^{-i{H}_0\tau }|\beta ⟩.$

From here, we can define the unitary operator

$U\left(\tau \right)=e^{iH\tau }{e}^{-i{H}_0\tau }.$

This unitary operator is just a time-evolution operator in the interaction picture that evolves a state from an early time $\tau$ to a later $T=0$ :

$U(T,\tau )=e^{i{H}_{0}T}{e}^{-iH(T-\tau )}{e}^{-i{H}_0\tau }\stackrel{T=0}{\to }U(0,\tau )=e^{iH\tau }{e}^{-i{H}_0\tau }.$

The scattering operator $S$ is obtained by taking early-time and late-time limits of this time-evolution operator in the interaction picture. By taking those limits, one can define the two operators (typically called Møller operators)

$U^{\pm }\left(\tau \right)=\underset{\tau \to \pm \infty }{lim}U\left(\tau \right).$

Scattering theory calculates transition amplitudes between states that evolve like eigenstates of the free Hamiltonian $H_0$ at early times and states that evolve like eigenstates of $H_0$ at late times. One can then use the fact that the dynamical evolution of states in the full, interacting theory will “coincide” with the evolution of states in the free, noninteracting theory at early and late times as follows:

$\begin{array}{c}\left({\Psi }_{\beta }|{\Psi }_{\alpha }\right)=\left(\beta |U^{+}\left(\tau {)}^{†}{U}^{-}\right(\tau )|\alpha \right)=\left(\beta |U(+\infty ,-\infty )|\alpha \right).\end{array}$

This gives the result that the scattering operator $S$ is a unitary operator that dynamically evolves a state of noninteracting particles $|\alpha ⟩$ from the asymptotic past into a state of noninteracting particles $U(+\infty ,-\infty )|\alpha ⟩$ in the asymptotic future.

The fact that the initial and final states come from the Fock space associated with a free theory means that we already know them. They are states of the form

$|(p,\sigma {)}_1^{n_1};(p,\sigma {)}_2^{n_2};\dots ;(p,\sigma {)}_r^{n_r}⟩,$

where $\sigma$ represents any additional quantum numbers the system may have. That means that a generic S-matrix element has the following form:

$\left((p^{\prime },{\sigma }^{\prime }{)}_1^{n_1^{\prime }};(p^{\prime },{\sigma }^{\prime }{)}_2^{n_2^{\prime }};\dots ;(p^{\prime },{\sigma }^{\prime }{)}_q^{n_q^{\prime }}|S|(p,\sigma {)}_1^{n_2};(p,\sigma {)}_2^{n_2};\dots ;(p,\sigma {)}_r^{n_r}\right).$

The replacement of $n_r$ with $n_q^{\prime }$ in the final state reflects the fact that while total energy is conserved during a scattering event in QFT, particle number is not.

The take-home points from this section are the following. A quantum field is an assignment of operators to spacetime points. The possible states of a quantum field form a Hilbert space $H$ . Just like in nonrelativistic quantum mechanics, each physical system is associated with an algebra of operators that act on this Hilbert space. Each of these operators can be expressed as a sum of products of creation and annihilation operators. The field operators are no different: they act on states in $H$ to create or annihilate particles. One of the most important operators in QFT, the scattering operator $S$ , allows one to calculate the probability amplitude that a state $|\alpha ⟩$ of noninteracting particles prepared in the asymptotic past will evolve into a state $|\beta ⟩$ of noninteracting particles in the asymptotic future. These scattering amplitudes are directly related to the quantities one measures in particle physics experiments, like scattering cross sections and branching ratios.

We now must think more carefully about the notion of “particle” that we have been so far employing rather naively. In the first part of Section 3, we do this by way of addressing another very important topic in more detail: the constraints that Poincaré invariance place on the structure of QFT. We then conclude Section 3 by discussing several well-known oddities of “particles” in QFT.

3.1 Representations of the Poincaré Group

Recall that the primary empirical constraint imposed by Poincaré invariance is that any two observers related by any combination of spatial rotations, Lorentz boosts, and spacetime translations must agree about probability distributions for the outcomes of scattering experiments. This translates into the mathematical requirement that the S-matrix be invariant under (orthochronous) Lorentz transformations and spacetime translations.

Formally, this requires that S-matrix elements satisfy

$S_{\beta \alpha }=\left(\beta |S|\alpha \right)=\left(\Lambda \beta |S|\Lambda \alpha \right)=S_{\Lambda \beta ,\Lambda \alpha },$

where $|\Lambda \beta ⟩$ and $|\Lambda \alpha ⟩$ are initial and final states obtained by acting on the states $|\beta ⟩$ and $|\alpha ⟩$ with a Poincaré transformation $\Lambda$ . Poincaré transformations must be represented as unitary operators acting linearly on the Fock space $H$ – a fact that will be of central importance momentarily – so we can write this requirement as

$\left(\beta |S|\alpha \right)=\left(\beta |U^{†}\left(\Lambda \right)SU\left(\Lambda \right)|\alpha \right),$

from which it follows immediately that $S=U^{†}\left(\Lambda \right)SU\left(\Lambda \right)$ : the scattering operator $S$ is invariant under Poincaré transformations. This is equivalent to requiring that the action of the scattering operator $S$ on any state $|\Psi ⟩$ of a quantum field commutes with the action of an arbitrary Poincaré transformation $U\left(\Lambda \right)$ .

We said that Poincaré transformations must be represented by unitary operators acting on $H$ . This follows from two facts: (i) we want the probability distributions our QFT predicts for the outcomes of scattering experiments to be invariant under Poincaré transformations, and (ii) a theorem of Wigner stating that the action of any symmetry group in a quantum theory must be represented by a set of unitary or anti-unitary operators acting linearly on $H$ (Reference WeinbergWeinberg, 1995: chapter 2, appendix A). (The anti-unitary operators change the sign of the time coordinate and so these representations are ruled out by the restriction to orthochronous Lorentz transformations.) The fact that Poincaré transformations act unitarily on $H$ entails that they preserve the inner product between states in $H$ :

$\left({\Psi }_{\beta }{U}^{†}\left(\Lambda \right)|U\left(\Lambda \right){\Psi }_{\alpha }\right)=\left({\Psi }_{\beta }|U^{†}\left(\Lambda \right)U\left(\Lambda \right)|{\Psi }_{\alpha }\right)=\left({\Psi }_{\beta }|{\Psi }_{\alpha }\right).$

This means that the isometries of Minkowski spacetime are also isometries of the Hilbert space $H$ . We also consider only irreducible representations, for reasons that will be explained shortly.

The structure of these irreducible unitary representations of the Poincaré group will occupy us for the first portion of this section. (There are many good textbook presentations of this material; in roughly increasing sophistication, see Reference RyderRyder (1996, chapter 2.7), Reference ColemanColeman (2019, chapter 18), or Reference WeinbergWeinberg (1995, chapter 2).) Our interest in these representations stems from the fact that they are used to classify the allowed particles in a Poincaré invariant QFT.

Wigner discovered that one can label all possible irreducible unitary representations of the Poincaré group by the eigenvalues of two operators. The first is the operator $P^{\mu }{P}_{\mu }=P^2$ , where $P^{\mu }$ is the 4-momentum operator. When acting on a single-particle state $|p,\sigma ⟩$ , the eigenvalues of this operator are $M^2$ , where $M$ is the mass of the particle. The second is $W^{\mu }{W}_{\mu }=W^2$ , where $W^{\mu }$ is the Pauli-Lubanski operator. The eigenvalues of this operator are proportional to $J^2$ , where $J$ is the total spin of the particle.

As is familiar from quantum mechanics, the total spin $J$ of a particle can only take half-integer values: $J=0,\frac{1}{2},1,\frac{3}{2},$ and so on. Thus every irreducible unitary representation of the Poincare group can be labeled by a mass $M$ and a total spin $J$ . In our discussion, we will always assume that $M>0$ , that is, we are considering massive particles. (Extending Wigner’s analysis to massless particles involves some important mathematical differences, but no major conceptual surprises.) Note that this already rules out, for example, particles with $J=\frac{1}{3}$ because there is no irreducible representation of the Poincaré group under which the states of such a particle could transform.

It is natural to associate irreducible unitary representations that are labeled by these two eigenvalues with particles for the following reason. The two operators $P^2$ and $W^2$ are Casimir operators of the Poincaré group: they commute with every element of the group. This means that the eigenvalues $M^2$ and $J^2$ of those operators are invariant under Poincaré transformations. This already suggests that one might be able to use irreducible representations to classify particles, since the mass and total spin of a particle are themselves invariant under Poincaré transformations. There is no combination of spatial rotations, Lorentz boosts, or spacetime translations that can turn an electron into a spin-1 particle or make its mass anything but 0.511 MeV.

However, it is also the case that specifying the state of a particle requires more information than just its mass and total spin. At minimum, fully specifying the state of the particle will require specifying its momentum. If the particle has total spin $J>0$ , then the state description will also specify the state of its spin along a chosen spatial axis, traditionally the z-axis. These properties are not Poincaré invariant: they change under general Poincaré transformations. Furthermore, particles usually possess more properties than mass and spin – electric charge, for example. This fact will be important for how we should understand the slogan that particles “are” irreducible representations of the Poincaré group.

We have restricted to irreducible representations. The requirement that our unitary representations of the Poincaré group be irreducible is important for understanding Wigner’s classification of particles, so it is worth pausing to get clear on irreducibility.

A (matrix) representation of a Lie group $G$ consists of a set of $n\times n$ matrices that satisfy the algebraic relations that define the Lie algebra associated with the Lie group. In quantum theories, we typically want to represent the action of a group on states of a physical system, that is, on rays in a Hilbert space $H$ . This makes matrix representations especially useful, since the action of the group elements on the physical system is reduced to matrix multiplication. Recall that the action of a linear operator on $H$ is determined by its action on the states in any basis of $H$ . Let $|{\Psi }_i⟩$ be an element of a basis for $H$ . Then the action of the group on $H$ has the following form:

$U\left(g\right)|{\Psi }_i⟩=\sum _{j}U(g{)}_{ij}|{\Psi }_j⟩,$

where $U\left(g\right)$ is a unitary matrix that represents the the group element $g$ . The action of $U\left(g\right)$ “mixes” the basis states among themselves – each basis state is transformed into a linear combination of other basis states.

This allows us to define irreducibility. A representation of a group $G$ acts irreducibly on a Hilbert space $H$ if there is no nontrivial subspace of states that is left invariant by the action of the group. Intuitively, this means that if one was allowed to move around in $H$ using only Poincaré transformations, they would never find themselves trapped in a subspace of $H$ .

A simple example illustrates the idea. Consider a Hilbert space $H$ with the following two subspaces: one subspace $H_1$ in which all states take the value $Q=1$ for some variable $Q$ , and a second subspace $H_2$ in which all states take the value $Q=2$ . If the action of an arbitrary $U\left(g\right)$ only turns states in $H_1$ into linear combinations of states in $H_1$ and states in $H_2$ into linear combinations of states in $H_2$ , then $U\left(g\right)$ acts reducibly on the full Hilbert space $H=H_1\oplus H_2$ . If one starts in $H_1$ , then one is trapped in $H_1$ , and if one starts in $H_2$ , then one is trapped in $H_2$ . Loosely speaking, a representation of a group $G$ acts irreducibly on a Hilbert space $H$ if, starting in an arbitrary state $|\Psi ⟩$ in $H$ , one can reach any other state in $H$ by acting on $|\Psi ⟩$ with elements of $G$ .

Irreducibility matters for Wigner’s classification of particles for the following reason. Suppose that mass and total spin were an exhaustive list of the invariant properties of particles. Then we could label states of a particle by momentum and the value of the spin along the z-axis $J_z$ , which can take values between $-J$ and $J$ . For a particle with $J=1/2$ , for example, a possible state is

$|\Psi ⟩=|p,J_z=1/2⟩.$

A particle state with momentum $p$ can be transformed into a state of any other momentum $p^{\prime }$ by Lorentz boosts, and a state with spin $J_z=1/2$ can be transformed into a state of any other spin, along any other axis, by a spatial rotation. All Lorentz transformations can be written as a product of Lorentz boosts and rotations, so for a single particle we should be able to transform any particular state into any other state by Lorentz transformations. If there was a subspace of states in the Hilbert space $H$ associated with our particle into which we could not get our particle by Lorentz transformations, those states must be labeled by properties that (i) are invariant under Poincaré transformations and (ii) differ from properties possessed by our particle. The only possibilities are that the states in that subspace of $H$ are labeled by different values of $M$ or $J$ than our particle.

Following Wigner’s analysis, we know this means those states must transform under a different irreducible representation of the Poincaré group than the states of our particle. Since we consider states labeled by distinct masses, or distinct total spins, to be states of distinct types of particles, we can conclude that if two sets of states transform under distinct irreducible representations of the Poincaré group then they must be states of distinct particles.

If mass and total spin exhausted the properties of elementary particles, there would be a one-to-one correspondence between particle types and irreducible representations of the Poincaré group: two particles would be distinct if and only if they transformed under different irreducible representations. In the real world, of course, particles are labeled by many more properties than just mass and spin, and this complicates the correspondence between particle species and irreducible representations of the Poincaré group. Consider two particles we consider distinct: the electron and the positron. These two particles have the same mass and total spin, so their states transform under the same representation of the Poincaré group. The property that distinguishes the two is the electric charge $Q$ : states of the electron have charge $-1$ and states of the positron have charge $+1$ . Electric charge is a quantity of obvious physical importance, but to which the Poincaré group is blind: an irreducible representation of the Poincaré group acts the same way on states with $\pm q$ as long as those states are labeled by the same mass and total spin. This means that, in the real world, different species of elementary particle are not in one-to-one correspondence with distinct irreducible representations of the Poincaré group. (This is not to say that considering irreducible representations of all of the symmetries of the theory is insufficient to distinguish distinct particle species. For example, electrons and positrons are associated with distinct irreducible representations of the group U(1), whose irreducible representations are labeled by the value of $q$ . Including U(1) will, roughly speaking, “split” the irreducible representation of the Poincaré group in a way that distinguishes electron states and positron states.)

This returns us, finally, to the question of how one ought to understand the slogan that an elementary particle is an irreducible representation of the Poincaré group. I will make two points. The first is that we should resist the idea that the slogan entails any kind of structural realism according to which an elementary particle can be metaphysically identified with an irreducible representation of the Poincaré group. (One might prefer structural realism on independent grounds, but the point is that the connection between particles and irreducible representations does not force it on us.) One does encounter the idea that it has this implication in the philosophical literature; see for example, Reference RobertsRoberts (2011, section 2) (who does not endorse it) for discussion. One ought to resist this interpretation for at least two reasons. First, for the simple reason that there is an acceptable, less metaphysically extravagant understanding: that the states of elementary particles transform under irreducible representations of the Poincaré group, while the particles themselves are material bodies. Second, an irreducible representation of the Poincaré group does not fix enough properties on its own to distinguish any two elementary particles. This seems an insurmountable obstacle to metaphysically identifying particular elementary particle types, like electrons or muons, with irreducible representations of the Poincaré group.

The second point builds on this inability to map distinct particle species one-to-one onto distinct irreducible representations of the Poincaré group. It is true that the space of states $H$ associated with any particular particle transforms under some irreducible representation of the Poincaré group. In that sense, the slogan does provide a definition of the general notion of a particle: a Hilbert space $H$ in QFT represents a single type of particle if and only if a representation of the Poincaré group acts irreducibly on $H$ . However, as just mentioned, the association is not strong enough to distinguish any two distinct particle species based solely on the representation theory of the Poincaré group. It is true that if two sets of states transform under different representations, then they are associated with distinct particle species: the two sets of states must be labled by different mass or total spin. However, the only if direction fails: one cannot infer solely from the fact that two particle species are distinct that they transform in distinct representations. It may be that they are only distinguished by quantum numbers that the Poincaré group does not see, like electric charge. The classification of particles afforded by irreducible representations of the Poincaré group is simply not fine-grained enough to support any conceptual or metaphysical identity between particular particle types and particular irreducible representations.

Quantum fields have been absent from our discussion of the Poincaré group thus far. Many conclusions of Wigner’s analysis of particles applies equally well to fields; in particular, the representation theory of the Poincaré group imposes the same constraints on the allowed quantum fields in Minkowski spacetime that it imposed on particle states. (This is unsurprising, given that “particle state” just describes a state of the associated quantum field in which particle(s) are present.) However, it is worth emphasizing a compelling way of understanding the origin of the transformation rules for the field operators: they are inherited from the transformation rules for the states. The logical structure of this is as follows (see Reference WeinbergWeinberg (1995, chapter 5)).We began by requiring that probability distributions for the outcomes of scattering experiments, encoded in an S-matrix, be invariant under Poincaré transformations. This required that Poincaré transformations act unitarily on $H$ . Consider the single particle state

$|p,\sigma ⟩=a_{p,\sigma }^{†}|0⟩,$

where $\sigma$ represents all quantum numbers of the particle. Acting on this state with a Poincaré transformation $U\left(\Lambda \right)$ gives

$U\left(\Lambda \right)|p,\sigma ⟩=U\left(\Lambda \right)a_{p,\sigma }^{†}\left(U^{†}\left(\Lambda \right)U\left(\Lambda \right)\right)|0⟩=U\left(\Lambda \right)a_{p,\sigma }^{†}{U}^{†}\left(\Lambda \right)|0⟩,$

where $U\left(\Lambda \right)|0⟩=|0⟩$ because the vacuum state $|0⟩$ does not change under Poincaré transformations. Taking the Hermitian conjugate of this expression gives the transformation rule for the annihilation operator $a_{p,\sigma }$ .

The action of the Poincaré transformation on creation and annihilation operators is determined by the mass and total spin of the particle they create or annihilate. Operators for scalar particles, with spin 0, transform differently than operators for vector particles, with spin 1, and both transform differently than spinors, representing particles with spin-1/2. The important point for our purposes is that because quantum fields are constructed out of creation and annihilation operators, the transformation rules for the fields are inherited from the transformation rules for those operators. The simplest example of this is the scalar field: the creation and annihilation operators are the only elements of the field operator on which Poincaré transformations act nontrivially (see Reference ColemanColeman (2019, chapter 3) for pedagogical discussion of scalar fields and Poincaré transformations). The definition of the scalar field that accompanied us through Section 2 makes this clear, as they are the only operator-valued objects that appear in the definition of the field:

$\phi (x,t)=\int \widetilde{dp}{a}_p{e}^{ipx}+a_p^{†}{e}^{-ipx}.$

The transformation rules for the quantum field is thus determined by the transformation rules for the creation and annihilation operators, which are in turn determined by the transformation rules for the states.

The point of this brief comment about the origin of the transformation rules for fields has been to highlight a fairly direct inferential path that begins with the empirical requirement that probability distributions for the outcomes of scattering experiments be left invariant by Poincaré transformations, passes through the transformation rules for single-particle states, and concludes with the transformation rules for the quantum fields. There are other ways to justify those transformation rules, but this understanding offers a striking demonstration of the powerful constraints placed on the mathematical structure of QFT simply by the empirical need to ensure that the predictions of a quantum theory respect the structure of Minkowski spacetime.

3.2 Localizability

We set out to sharpen our understanding of the concept of a particle in QFT in two ways. The first was to consider its relationship to irreducible representations of the Poincaré group; we have now done that. The second was to consider a serious challenge to our naive use of the term “particle” to describe the content of Fock-space states like

$|(p,\sigma {)}_1^{n_2};(p,\sigma {)}_2^{n_2};\dots ;(p,\sigma {)}_r^{n_r}⟩.$

It turns out that these “particles” fail to exhibit many of the properties that we would expect particles to exhibit. We will focus on the fact that these “particles” cannot be strictly localized in any bounded region of spacetime, but this is just one of several obstacles to interpreting states of a quantum field as describing particles (both Reference Fraser, Knox and WilsonFraser (2021) and Reference BakerBaker (2016, section 3) include more extensive surveys of these obstacles). We will briefly summarize these other obstacles before turning to the issue of localizability.

The first of these obstacles comes from a remarkable phenomenon called the Unruh effect. (See Reference Mukhanov and WinitzkiMukhanov and Winitzki (2007, chapter 8) for a pedagogical derivation of a special case, Reference WaldWald (1994, chapter 5) for a more general treatment, and Reference Crispino, Higuchi and MatsasCrispino et al. (2008) for extensive discussion. For philosophical discussion, see Reference Clifton and HalvorsonClifton and Halvorson (2001), Reference Arageorgis, Earman and RuetscheArageorgis et al. (2003), Reference EarmanEarman (2011), and Reference RuetscheRuetsche (2011, chapter 9).) Consider Alice and Bob, both of whom are observing the same free quantum field. Alice is at rest and sees the field in its vacuum state $|0⟩$ , that is, a state containing no particles. The Unruh effect, roughly speaking, is the discovery that if Bob is accelerating uniformly, he will not see the field in a no-particle state. Instead, he will see the field in a thermal state: a state describing a thermal bath of particles, with a temperature proportional to Bob’s rate of acceleration. People have drawn a number of conclusions about the nature of particles from the Unruh effect, but an almost universal reaction has been that it demonstrates that QFT cannot be fundamentally about particles. Wald gives a concise statement:

The second of these obstacles comes from technical issues that arise in defining quantum fields on more exotic spacetimes than Minkowski spacetime (see Reference WaldWald (1994, chapter 4) or Reference Mukhanov and WinitzkiMukhanov and Winitzki (2007, chapter 6); see Reference RuetscheRuetsche (2011, chapters 10–11) for philosophical discussion). In Section 2, the states of the free quantum field that we described as containing particles were Fock states:

$|(p,\sigma {)}_1^{n_2};(p,\sigma {)}_2^{n_2};\dots ;(p,\sigma {)}_r^{n_r}⟩.$

We did not discuss it, but the ability to represent states of a quantum field using a Fock space requires the underlying spacetime to possess certain symmetries. The symmetry structure of some curved spacetimes means they do not admit a unique definition of this Fock space representation. This gives rise to multiple, inequivalent representations of the space of states of a free quantum field in those spacetimes, with each representation equipped with its own particle structure. In other curved spacetimes, the symmetry structure means they do not admit any Fock space representation at all. Many have taken the same conceptual lesson from this as from the Unruh effect, which Wald again states clearly:

The third obstacle appears if one attempts to extend the particle interpretation of Fock states of a free quantum field to states of interacting quantum fields (Reference FraserFraser, 2008). There are a number of reasons this extension fails. For one, the vacuum state of the interacting quantum fields is not the same as the vacuum state $|0⟩$ of the free field that was understood to contain zero particles. It is hard to understand how that could be the case if QFTs are fundamentally about particles. Perhaps worse, the single particle state $|p,\sigma ⟩$ is not an eigenstate of the Hamiltonian $H_{int}$ of a QFT that includes interactions. According to one standard approach to property attribution in quantum theories, the fact that the state $|p,\sigma ⟩$ is not an eigenstate of $H_{int}$ means it has no definite value of energy at all. As Fraser argues, this third obstacle poses a serious problem for anyone arguing that QFTs are fundamentally describing particles, or that states of quantum fields always admit a particle description.

These three obstacles pose serious problems for anyone who wants to interpret QFTs as being fundamentally about particles. Indeed, they make it all but impossible to argue even that the states of a quantum field always admit a particle interpretation, even if one abandons the idea that particles are the ontologically fundamental objects. One might wonder whether it is really necessary to take up the issue of localizability as well. Doesn’t this coffin already have enough nails?

The previous obstacles all involve, in one way or another, departures from the maximally simple context on which we based our naive talk of particles in Section 2: Minkowski spacetime, inertial observers, and a free scalar field $\phi (x,t)$ whose states admit a unique Fock space representation. The nonlocalizability of particles presents an obstacle to interpreting states of a quantum field as describing particles even in this context, which seems most friendly to a particle interpretation.

First, the most obvious question: Can one define “particles” that are more localized than momentum eigenstates $|p⟩$ , that is, that are localized at all? The answer is yes. It is both physically and mathematically more responsible to represent initial and final states in scattering theory using wavepackets peaked around some momentum $p$ and some spatial point $x$ with respective spreads $\Delta p$ and $\Delta x$ ; a Gaussian wavepacket provides a familiar example. One can do scattering theory with wavepackets perfectly well. However, at the typical resolution with which scattering experiments are conducted, the probabilities for the possible outcomes are not sensitive to the detailed spatial structure of the wavepacket and the spread of momenta $\Delta p$ in the wavepacket is typically much smaller than the momentum of the scattering experiment. This is what licenses the use of momentum eigenstates to approximate the wavepackets. (See Reference Peskin and SchroederPeskin and Schroeder (1995, chapters 4.5, 7.2) and Reference Itzykson and ZuberItzykson and Zuber (1980, chapter 5.1) for discussion; for a more rigorous treatment, see Reference DuncanDuncan (2012, chapters 6, 9.2–9.3).) We will continue to use approximations going forward, but emphasize that it is not an essential feature of scattering theory.

Of course, even a Gaussian wavepacket in nonrelativistic quantum mechanics is not strictly localizable in any bounded region of spacetime for more than an instant: it has “tails” that extend out to spatial infinity. However, one might consider it essential to a satisfactory notion of “particle” that it be strictly localizable within some bounded region of spacetime. There are a number of formal results demonstrating that no such notion is allowed in relativistic quantum theories.

The earliest prominent result in the philosophical literature was Reference Malament and CliftonMalament (1996). Malament asked whether it was possible for a relativistic quantum theory to describe even a single strictly localizable particle. Under physically reasonable assumptions, all of which are satisfied for the initial and final states considered in scattering theory, he demonstrated that no such theory is possible. In more detail, Malament demonstrated the following. Consider two disjoint spatial regions $O$ and $O^{\prime }$ and a single instant t, and assume the following:

• Localizability: The probability of finding the particle in both regions $O$ and $O^{\prime }$ at t is 0. Formally, if $P_O$ and $P_{O^{\prime }}$ are projection operators that check whether the particle is localized in $O$ or $O^{\prime }$ and return 1 for yes and 0 for no, we require $P_O{P}_{O^{\prime }}|\Psi ⟩=0$ for all states $|\Psi ⟩$ of the single particle.

• Translation Covariance: There is a unitary representation of the translation subgroup of the Poincaré group defined on the Hilbert space $H$ .

• Energy Bounded Below: The Hamiltonian operator $H$ has a lowest eigenvalue, that is, the particle has a ground state.

• Microcausality: The measurement of an operator in $O$ cannot affect the statistics of measurements performed in $O^{\prime }$ : for any two operators $A$ and $A^{\prime }$ restricted to the regions $O$ and $O^{\prime }$ , we require $\left(A,A^{\prime }\right)=0$ . In particular, for two projection operators $P$ and $P^{\prime }$ we have $\left(P,P^{\prime }\right)=0$ .

On the basis of those four assumptions, Malament proves that the probability of detecting the single particle localized in any spatial region $O$ is 0: given an arbitrary state $|\Psi ⟩$ in $H$ and any spatial region $O$ , the projection operator $P$ associated to $O$ gives $P|\Psi ⟩=0$ . Malament’s result was subsequently strengthened in several respects by Reference Halvorson and CliftonHalvorson and Clifton (2002).

These results are widely described as no-go theorems for particles. However, what they actually demonstrate is that in a relativistic quantum theory, one cannot define a position operator. An intuitive way to see this (glossing over some important mathematical subtleties) is that any Hermitian operator – like a position operator – can always be expressed as a weighted sum of appropriate orthogonal projection operators, and these results show that for any bounded region $O^{\prime }$ , one cannot define an associated projection operator $P^{\prime }$ whose value reflects the location of a particle.

A natural question is whether there is a different strategy for defining localized particles in QFT. Perhaps the most natural strategy would be to define number operators $N$ that count the number of particles in a spatial region $O$ at any particular instant. After all, a global number operator $N$ was the tool we used in Section 2 to count the total number of particles in a state of a quantum field. Could a local number operator $N$ be defined that would count the number of particles in the state of a quantum field in a spatial region $O$ ?

The answer is no, but the strategy fails because of deep aspects of the structure of QFT. That one cannot even define a local number operator $N$ that counts all and only the particles in the spacetime region $O$ is a well-known consequence of one of the most remarkable theorems of QFT, the Reeh–Schlieder theorem. (See Reference WittenWitten (2018, section II) for a clear presentation of the theorem and Reference RedheadRedhead (1995) and Reference HalvorsonHalvorson (2001) for the difficulty it poses for the notion of a localized particle in QFT.) Substantive discussion of the Reeh–Schlieder theorem is beyond the scope of this Element, but its implications for local number operators are easy to state.

One of the properties we wanted for a total number operator is that, when applied to the vacuum state $|0⟩$ of a quantum field, we have $N|0⟩=0$ . After all, the vacuum state is the state of the quantum field with no particles present. We would like the same property to hold for a local number operator $N$ . We immediately encounter the problem: it is a consequence of the Reeh–Schlieder theorem that any operator $A^{\prime }$ restricted to a bounded region of spacetime $O^{\prime }$ satisfies $A^{\prime }|0⟩=0$ if and only if $A^{\prime }=0$ , that is, the operator is the zero operator.

In fact, Reference Halvorson and CliftonHalvorson and Clifton (2002, section 6) show that the culprit is a microcausality assumption: the requirement that $\left(N,N^{\prime }\right)=0$ for disjoint regions of spacetime $O$ and $O^{\prime }$ . If two operators in disjoint spacetime regions do not commute, then measurements in region $O$ can superluminally affect the long-run statistics of measurement outcomes in $O^{\prime }$ ; this would allow for superluminal signaling. Indeed, if one attempts to ignore the implication of the Reeh–Schlieder theorem and define local number operators anyway, one quickly finds that two number operators restricted to disjoint spacetime regions do not commute, precisely as Halvorson and Clifton claimed they could not (Reference DuncanDuncan, 2012, chapter 6.5).

I said above that a widespread reaction to the results discussed in this section is that particles are not part of the fundamental ontology of QFT. This raises a question that is, to my mind, more pressing: What is the status of particles in QFT? After all, experimental physicists design particle accelerators, collect data using particle detectors, and so on. What are those accelerators accelerating? What are the detectors detecting? What notion of “particle” is employed by engineers and experimentalists in designing their detection apparatus, and how can one locate that notion in QFT? We have seen that states of a quantum field cannot always be interpreted as describing particles and even in the restricted contexts in which they can, those “particles” do not behave like one might expect. However, considerations like those just mentioned have led to consensus that there must be some role for a notion of nonfundamental particles. (See Reference Fraser, Knox and WilsonFraser (2021, section 21.3) and Reference BakerBaker (2016, section 3) for interpretive options.) What that role is, exactly, strikes me as an interesting and challenging question for those interested in ontological implications of QFT.

One natural response to the obstacles to a fundamental ontology of particles is exemplified by the quotations from Wald: conclude that the ontologically fundamental objects in QFT are quantum fields. Particles are emergent, or approximate, entities in QFT: they are descriptions of certain behaviors of quantum fields that can be usefully deployed in a restricted set of circumstances. This too faces significant obstacles. Baker has shown that the most natural interpretation of QFT along these lines faces many of the same problems as particle interpretations (Reference BakerBaker, 2009) (though Reference SebensSebens (2022) identifies a potential loophole). Furthermore, the choice of a particular operator-valued field $\phi$ is highly nonunique: a large class of fields $\phi ,{\phi }^{\prime },{\phi }^{\prime \prime },\dots$ , each apparently exhibiting quite different structure, can be shown to generate empirically equivalent models. A number of mathematically rigorous results capture aspects of this underdetermination; perhaps the most famous example, proven by Borchers, demonstrates that any two fields $\phi$ and ${\phi }^{\prime }$ that satisfy mild conditions will generate identical S-matrices (Reference Streater and WightmanStreater and Wightman, 1964, chapter 4.6). This suggests that the structure of a quantum field itself is in principle underdetermined by the empirical evidence we can acquire about it. This is a variety of underdetermination that has been notoriously challenging for any would-be scientific realist. The ontological standing of quantum fields is, in my opinion, perhaps the most pressing open question about the metaphysics of QFT.

4.1 Correlation Functions and Path Integrals

The clearest way to illustrate the significance of renormalization methods is to sketch a calculation of an S-matrix element using our perturbative strategy. In fact, there are several ways to carry out such a calculation; in this section, we will introduce a particularly powerful approach that uses path integral methods. Path integral methods have come to be preferred in QFT, not least because of their ability to handle certain technical subtleties of gauge theories like quantum electrodynamics or quantum chromodynamics. We will not need to avail ourselves of the technical power of the framework, but path integral methods also make the physical meaning of renormalization methods particularly clear. (This is especially true for the renormalization group (RG) methods that we will discuss in Section 4.3.) It will therefore be useful to develop an understanding of path integrals.

We will stick with the scalar field $\phi (x,t)$ . Our first course of action is to add interactions to the theory’s dynamics. We treat this interaction as a small perturbation of the dynamics of the noninteracting theory. We add the term $\lambda \phi (x,t{)}^4$ describing the interaction, producing the following Hamiltonian:

$H=\frac{1}{2}\int d^{3}x\pi (x{)}^2+(\nabla \phi \left(x\right){)}^2+m^2\phi (x{)}^2+\frac{\lambda }{4!}\phi (x{)}^4,$

where $\lambda$ is a real number that satisfies $\lambda \ll 1$ . This is essential to our perturbative strategy for studying QFTs with interactions: if  $\lambda \underset{˜}{>}1$ , our assumption that adding interactions does not dramatically alter the structure of the noninteracting theory is no longer justified.

To emphasize the distinction between the dynamics of the noninteracting theory and the perturbing interaction, one typically splits the Hamiltonian into two terms:

$\begin{array}{cc}{H}_0& =\frac{1}{2}{\int }^{\text{}}{d}^{3}x\pi {\left(x\right)}^2+{(\nabla \phi (x\left)\right)}^2+m^2\phi {\left(x\right)}^2\\ H_{int}& =\frac{\lambda }{4!}\phi {\left(x\right)}^4,\end{array}$

with the total Hamiltonian for the interacting system being given by $H=H_0+H_{int}$ .

We have described the dynamics of our scalar field using a Hamiltonian operator, but in practice it is much more common to use a Lagrangian operator. (The main reason for this is that verifying Poincaré invariance is much simpler in a Lagrangian formalism than a Hamiltonian formalism, since one has to choose a foliation of spacetime to even write down the Hamiltonian.) The Lagrangian density for our interacting scalar field is

$L=\frac{1}{2}{\partial }_{\mu }\phi (x,t){\partial }^{\mu }\phi (x,t)-\frac{m^2}{2}\phi (x,t{)}^2-\frac{\lambda }{4!}\phi (x,t{)}^4$

and the Lagrangian itself is obtained by integrating the Lagrangian density over the entirety of space:

$L=\int d^{3}xL.$

The Lagrangian formalism will be central to our discussion of QFT. Also, for notational simplicity we will start writing $\phi (x,t)\equiv \phi$ on occasion.

Path integrals provide a powerful method for the calculation of S-matrix elements, but they do so in a slightly roundabout way. What they enable one to calculate directly is a time-ordered correlation function, a very important mathematical object in QFT. Roughly speaking, the correlation function encodes the probability amplitude that if a quantum field is excited out of its vacuum state by the creation of particles localized around spatial points $x_1,x_2,\dots ,x_n$ at time $t$ , the field will be found at a later time $t^{\prime }$ in a state containing particles localized around spatial points $y_1,y_2,\dots ,y_m$ . The Lehmann–Symanzik–Zimmerman (LSZ) reduction formula tells us how to systematically relate these correlation functions to scattering amplitudes. Details of the LSZ reduction formula lie outside the scope of this Element (see Reference ColemanColeman (2019, chapter 14) for a lucid exposition), but the important conceptual point is that it reduces calculating S-matrix elements to calculating time-ordered correlation functions. This justifies our focus on the latter.

In fact, we can restrict our attention even further: a very useful result called Wick’s theorem demonstrates that any time-ordered correlation function involving $n$ particles – a so-called $n$ -point function – can always be written as a sum of products of time-ordered correlation functions involving only two particles – so-called two-point functions or propagators. We will therefore focus on the structure of two-point functions for the moment.

There are important differences between the structure of a propagator for a free QFT and an interacting QFT. We will start with free QFT: when we discuss renormalization, we will see that as long as $\lambda \ll 1$ – the essential assumption of our perturbative strategy – the free propagator serves as a “starting point” of sorts for the propagator in the QFT with interactions. For a free scalar field $\phi (x,t)$ , the two-point function is

$\left(0|T\phi (x,t)\phi (y,t^{\prime })|0\right),$

where $|0⟩$ is the ground state of the noninteracting Hamiltonian $H_0$ , that is, $|0⟩$ is the state of the free field $\phi (x,t)$ in which no particles are present. The function T is the time-ordering function: it cares only about the temporal coordinates $t$ and $t^{\prime }$ and is defined to be:

$T\phi (x,t)\phi (y,t^{\prime })=\left(\begin{array}{c}\phi (x,t)\phi (y,t^{\prime })t>t^{\prime }\\ \phi (y,t^{\prime })\phi (x,t)t^{\prime }>t\end{array}\right).$

T acts on a product of field operators to arrange the fields localized at the earlier spacetime points on the right.

Why is this mathematical object called a “propagator”? Recall that the free field $\phi (x,t)$ has the form

$\phi (x,t)=\int \widetilde{dp}{a}_p{e}^{ipx}+a_p^{†}{e}^{-ipx},$

and that its action on the vacuum state $\phi (x,t)|0⟩$ produces a state containing a single particle with some momentum p, and localized around the spatial point x at time $t$ . (This isn’t strictly true: it describes a wavepacket and technically $\phi$ creates a momentum eigenstate. However, we discussed earlier how the two are related in scattering theory.) Similarly, the state $⟨0|\phi (y,t^{\prime })$ describes a single particle with some momentum q and localized around the spatial point y at time $t^{\prime }$ . When we sandwich those two states together, we get a transition amplitude encoding the probability that if, at time $t$ , the quantum field was in a state describing a particle of momentum p localized around x, it will be found at a later time $t^{\prime }$ in a state describing a particle of momentum q localized around y (or vice versa for $t^{\prime }$ earlier than $t$ ). (Conservation of energy requires that $p=q$ for the free propagator, since no interactions will change the particle’s momentum between $(x,t)$ and $(y,t^{\prime })$ .) Speaking loosely, the propagator encodes the probability that a particle initially at spacetime point $(x,t)$ will “propagate” to $(y,t^{\prime })$ .

A simple calculation shows that the propagator for a free scalar field takes the form

$\left(0|T\phi (x,t)\phi (y,t^{\prime })|0\right)=\int \frac{d^{4}p}{(2\pi {)}^4}{e}^{-ip(x-y)}\frac{1}{p^2-m^2+i\epsilon }.$

One is typically interested in more complicated time-ordered correlation functions that encode the probability that if, at $t$ , a quantum field is in a state describing multiple particles localized around spatial points $x_1,x_2,\dots ,x_n$ , it will be found at a later $t^{\prime }$ in a state containing particles localized around spatial points $y_1,y_2,\dots ,y_m$ . (In a free QFT, $m=n$ since there are no interactions to create or annihilate particles.) Wick’s theorem lets us decompose these $n$ -point functions into sums of products of two-point functions, that is, propagators, so it is sufficient to focus on those.

We now turn to the path integral methods that one typically uses to calculate time-ordered correlation functions. Path integrals were initially introduced into quantum mechanics by Feynman (Reference FeynmanFeynman, 1948), following up on an earlier idea proposed in an obscure paper by Dirac (Reference Dirac and BrownDirac, 1933/2005). Our strategy will be to first develop an understanding of the core ideas of the path integral approach in non-relativistic quantum mechanics by way of a great example due to Feynman himself.

Suppose you have the following problem: a source emits a particle that propagates some distance through space and slams into a detection screen. In between the source and the screen is a wall with two slits, labeled $A_1$ and $A_2$ . You want to calculate the probability that a particle emitted from the source will be detected at a particular point $x$ on the detection screen. This is easy enough: the particle must pass through either $A_1$ or $A_2$ to reach the screen, so the total probability amplitude for the particle being detected at $x$ is the sum of two terms: the amplitude for the particle passing through $A_1$ and being detected at $x$ , and the amplitude for the particle passing through $A_2$ and being detected at $x$ . If we represent the probability amplitude for a particle passing through slit $A_j$ and being detected at $x$ as $M(A_j\to x)$ , we can write the total amplitude for detecting the particle at $x$ as

$M\left(x\right)=M(A_1\to x)+M(A_2\to x),$

where the probability of the particle being detected at $x$ is given by $|M\left(x\right){|}^2$ .

Suppose one adds an additional slit $A_3$ to the wall. The probability amplitude for detecting the particle at $x$ then becomes

$M\left(x\right)=M(A_1\to x)+M(A_2\to x)+M(A_3\to x).$

Now suppose that between the first wall and the detector, one adds a second wall with two slits $B_1$ and $B_2$ . Reasoning as we just did, the probability amplitude for detecting the particle at $x$ now becomes

$\begin{array}{c}M\left(x\right)=M(A_1\to B_1\to x)+M(A_2\to B_1\to x)+M(A_3\to B_1\to x)\\ +M(A_1\to B_2\to x)+M(A_2\to B_2\to x)+M(A_3\to B_2\to x).\end{array}$

If one adds more walls and more slits in each wall, this increases the number of “paths” $A_i\to B_j\to C_k\to \dots \to Z_n\to x$ that the particle could take between the source and the point $x$ on the detection screen:

$M\left(x\right)=\sum _i\sum _j\sum _k\dots \sum _{n}M(A_i\to B_j\to C_k\to \dots \to Z_n\to x).$

As one approaches infinitely many walls, each with infinitely many slits, one is eventually left with empty space between the source and the detection screen! At this point, there will be infinitely many “paths” between the source and the point $x$ on the detection screen, and we will have to add up all of them to calculate the probability amplitude $M\left(x\right)$ . The number of paths has become continuously infinite, so this sum becomes an integral over all paths the particle could take between the source and the detection screen: a path integral.

It is worth sketching this more formally. (See Reference Peskin and SchroederPeskin and Schroeder (1995, chapter 9.1).) Suppose one wants to calculate the probability amplitude for detecting a particle at a spatial point $q_D$ on the detection screen, given that it was emitted from the source at the spatial point $q_S$ at an initial time $t=0$ and its time evolution during that interval was described by $U\left(T\right)=e^{-iHT}$ for some Hamiltonian $H$ . This probability amplitude is

$M\left(qs\to qD\right)=〈qD\left|e^{-iHT}\right|qs〉.$

We can divide the time interval $T$ into $N$ segments of length $T/N=dt$ . That lets us split the time evolution operator $U\left(T\right)$ into a product of $N$ time evolution operators as follows:

$U\left(T\right)=e^{-iHT}=\underset{N\text{copies}}{\underbrace{e^{-iHdt}{e}^{-iHdt}\dots e^{-iHdt}}}.$

Between each of these new time evolution operators, we can insert copies of the identity operator $\int d{q}_j|q_j⟩⟨q_j|$ . (We will use the identity operator expressed in the basis provided by the eigenstates $|q_i⟩$ of the position operator, but in general one can use any appropriate basis.) Each copy of the identity operator corresponds to inserting a new wall in the example we already discussed, and the fact that each copy of the identity contains continuously many eigenstates of position corresponds to the fact that each “wall” has infinitely many slits.

After making these insertions, we get

$\begin{array}{c}M(qs\to qD)=\int d{q}_a\int d{q}_b\dots \int dqN\\ 〈q_D\left|e^{-iHdt}\right|q_q〉〈q_a\left|e^{-iHdt}\right|q_b〉〈q_b\left|e^{-iHdt}\right|q_c〉\dots 〈q_N\left|e^{-iHdt}\right|q_S〉.\end{array}$

The final step in this process is to take the limit $N\to \infty$ , which makes the time intervals $dt$ arbitrarily short. In the example above, this corresponds to inserting infinitely many walls between the source and the detection screen (captured by the presence of $N\to \infty$ copies of the identity operator in the above expression), thereby making the time that the particle spends between walls arbitrarily short. The final result for our probability amplitude $M(q_S\to q_D)$ , expressed as a path integral, is

$\begin{array}{c}M(q_S\to q_D)=\underset{N\to \infty }{lim}\int d{q}_a\int d{q}_b\dots \int d{q}_N\\ \left(q_D|e^{-iHdt}|q_q\right)\left(q_a|e^{-iHdt}|q_b\right)\left(q_b|e^{-iHdt}|q_c\right)\dots \left(q_N|e^{-iHdt}|q_S\right)\end{array}$

or, in more elegant form,