Free scalar field theory

3.1 Free scalar field theory

Historically, field theory was in part an attempt to develop local theories rather than those, such as Newtonian gravity, implying action-at-a-distance.¹ The idea is to associate each point in space and time with a value or set of values $ϕ_{a} (x, t)$ , called fields. As long as these fields interact only at the same point in spacetime or, at most, with their immediate neighbors (via their derivatives), the theory is guaranteed to be local. Classic examples include the vector-valued electric and magnetic fields $E (x, t)$ and $B (x, t)$ . The behavior of the fields is encapsulated by the Lagrangian of the system.

In this section, we briefly recap the Lagrangian formulation of classical field theory (Section 3.1.1) and Noether’s theorem connecting symmetries to conserved quantities (Section 3.1.2). We then present the quantized form of the free scalar field, and the interpretation of particles as excitations of these fields, in Section 3.1.3. Finally, we conclude in Section 3.1.4 with a discussion of particle propagators, which are the connection between these abstract fields and the physical observables we can measure in experiments. Further background on classical Lagrangian and Hamiltonian mechanics can be found in Appendix B.1, and on quantization in Appendix B.2.

3.1.1 Classical field theory

The Lagrangian of a classical field $ϕ (x, t)$ is given as a function of the field and its derivatives:

L (t) = \int d^{3} x L (\partial_{μ} ϕ, ϕ),

(3.1.1)

where $L$ is the Lagrangian density. The action is the integral of $L$ over time, or $L$ over spacetime:

S = \int L dt = \int d^{4} x L .

(3.1.2)

The equations of motion (EOMs) of the field is derived from the principle of stationary action, which states that the true path is an extremum of $S$ , yielding the Euler-Lagrange (E-L) equations:

\partial_{μ} (\frac{\partial L}{\partial (\partial_{μ} ϕ)}) - \frac{\partial L}{∂ϕ} = 0 .

(3.1.3)

The Klein-Gordon equation

The Lagrangian for a free, scalar relativistic field $ϕ (x, t)$ is:

L = \frac{1}{2} \partial_{μ} ϕ \partial^{μ} ϕ - \frac{1}{2} m^{2} ϕ^{2},

(3.1.4)

The E-L equation for this Lagrangian is called the Klein-Gordon equation:

\partial_{μ} \partial^{μ} ϕ + m^{2} ϕ \equiv (□ + m^{2}) ϕ = 0,

(3.1.5)

where $□ \equiv \partial_{μ} \partial^{μ} = \partial_{t}^{2} - \nabla^{2}$ is the d’Alembertian operator.

The Klein-Gordon equation is essentially the relativistic generalization of the Schrödinger equation. Just as the Schrödinger equation quantizes the non-relativistic EOM $E = \frac{p^{2}}{2 m}$ , the Klein-Gordon equation converts the relativistic EOM for a free particle

E^{2} = p^{2} c^{2} + m^{2} c^{4}

(3.1.6)

into quantum operator form, with $E \to Ê = i ℏ \partial_{t}$ and $p \to \hat{p} = - i ℏ \nabla$ :

\begin{matrix} \begin{aligned} Ê^{2} & = {\hat{p}}^{2} c^{2} + m^{2} c^{4} \\ - ℏ^{2} \partial_{t}^{2} ϕ & = - ℏ^{2} c^{2} \nabla^{2} ϕ + m^{2} c^{4} ϕ \\ \Rightarrow (\partial_{t}^{2} & - c^{2} \nabla^{2} + \frac{m^{2} c^{4}}{ℏ^{2}}) ψ = 0 . \end{aligned} \end{matrix}

(3.1.7)

Natural units

It is conventional in high energy physics to use natural units:

ℏ = c = 1 .

(3.1.8)

Besides being notationally convenient, this enables all dimensionful physical quantities to be described by the same scale — conventionally, in terms of energy, e.g. in units of electronvolts (eV). For example:

Mass: $E = m c^{2} \to m = E$ ,
Compton wavelength: $λ = \frac{ℏ}{mc} \to λ = \frac{1}{E}$ ,
Momentum: $p = \frac{ℏ}{λ} \to p = E$ .

We define each quantity to have a dimension in terms of energy, i.e. energy, mass, and momentum all have dimension $[E] = [m] = [p] = 1$ , while length has dimension $[λ] = - 1$ . Thus, in natural units Eqs. 3.1.5 and 3.1.7 are identical.

3.1.2 Symmetries and Noether’s theorem

Noether’s theorem states an important consequence of continuous symmetries of a system: they are associated with a physical conserved currents. For example, translational and rotational invariance of the potential energy imply conservation of momentum and angular momentum, respectively.

More precisely, if a continuous transformation on the field

ϕ (x) \to ϕ^{'} (x) = ϕ (x) + 𝜖 Δ ϕ (x)

(3.1.9)

is a symmetry, leaving the EOMs invariant, it can be shown to imply the existence of a conserved current $j^{μ}$ :

\partial_{μ} j^{μ} = 0, j^{μ} = \frac{\partial L}{\partial (\partial_{μ} ϕ)} Δ ϕ - J^{μ},

(3.1.10)

and conserved charge $Q$

Q = \int_{allspace} d^{3} x j^{0} .

(3.1.11)

Example: translation symmetry

Consider a translation-invariant theory, such as for the free scalar field (Eq. 3.1.4). A spacetime translation $x^{μ} \to x^{μ} - a^{μ}$ leads to the transformation $ϕ (x) \to ϕ (x + a) ≃ ϕ (x) + a^{μ} \partial_{μ} ϕ (x)$ , yielding the conserved current:

{(j^{μ})}_{ν} = \frac{\partial L}{\partial (\partial_{μ} ϕ)} \partial_{ν} ϕ - δ_{ν}^{μ} L,

(3.1.12)

which we call the energy-momentum tensor $T_{ν}^{μ}$ . The associated conserved quantities (or “charges”) are the total energy and momentum of the field configuration:

E = \int d^{3} x T^{00}, P^{i} = \int d^{3} x T^{0 i} .

(3.1.13)

For our free scalar field, this turns out to be:

\begin{matrix} \begin{aligned} E & = \int d^{3} x \frac{1}{2} {\dot{ϕ}}^{2} + \frac{1}{2} {(\nabla ϕ)}^{2} + \frac{1}{2} m^{2} ϕ^{2}, \\ P^{i} & = \int d^{3} x \dot{ϕ} \partial^{i} ϕ . \end{aligned} \end{matrix}

(3.1.14)

The interpretation of these as energy and momenta is described further in Appendix B.2.3.

Example: a U(1) internal symmetry

Symmetries such as translational and rotational invariance are spacetime, or external, symmetries. An internal symmetry is a transformation that acts only on the fields, at each point of spacetime. A simple example is the complex scalar field $ψ (x)$ , for which we can write down the free Lagrangian:

L = \partial_{μ} ψ^{*} \partial^{μ} ψ - m^{2} ψ^{*} ψ .

(3.1.15)

This Lagrangian possesses an internal $U (1)$ symmetry: it is invariant under $ψ (x) \to e^{iα} ψ (x)$ for any constant $α$ . Noether’s theorem tells us this too has the important physical consequences of a conserved current and charge:

j^{μ} = i (ψ^{*} \partial^{μ} ψ - ψ \partial^{μ} ψ^{*}), Q = \int d^{3} x i (ψ^{*} \partial^{0} ψ - ψ \partial^{0} ψ^{*}) .

(3.1.16)

Once quantized, we will see this exactly corresponds to the conservation of electric charge!

In fact, we say that a field that transforms as so under a global $U (1)$ rotation

\begin{matrix} \begin{aligned} ψ (x) & \to e^{iqα} ψ (x), \\ ψ^{*} (x) & \to e^{- iqα} ψ^{*} (x), \end{aligned} \end{matrix}

(3.1.17)

is charged under the $U (1)$ symmetry, with charge $q$ (and its complex conjugate with charge $- q$ ).

3.1.3 Quantization

Details of quantization can be found in Appendix B.2; briefly, the quantized solution to the Klein-Gordon equation for a real field is akin to a superposition of plane waves:

\hat{ϕ} (x, t) = \int \frac{d^{3} p}{{(2 π)}^{3}} \frac{1}{\sqrt{2 ω_{p}}} (â_{p} e^{ip \cdot x} + â_{p}^{†} e^{- ip \cdot x})

(3.1.18)

where $p \cdot x = p_{μ} x^{μ}$ is the 4D spacetime inner product and $p_{μ} = (ω_{p} = \sqrt{{| p |}^{2} + m^{2}}, p)$ . The quantum operators $â_{p}$ and $â_{p}^{†}$ are annihilation and creation operators, respectively, for a particle with momentum $p$ and mass $m$ , just as for states in a quantum harmonic oscillator.²

The solutions for a complex scalar field are (see Appendix B.2.4):

\begin{matrix} \begin{aligned} \hat{ψ} (x, t) & = \int \frac{d^{3} p}{{(2 π)}^{3}} \frac{1}{\sqrt{2 ω_{p}}} ({\hat{b}}_{p} e^{ip \cdot x} + ĉ_{p}^{†} e^{- ip \cdot x}), \\ {\hat{ψ}}^{†} (x, t) & = \int \frac{d^{3} p}{{(2 π)}^{3}} \frac{1}{\sqrt{2 ω_{p}}} ({\hat{b}}_{p}^{†} e^{- ip \cdot x} + ĉ_{p} e^{ip \cdot x}), \end{aligned} \end{matrix}

(3.1.19)

where we now have two sets of creation and annihilation operators, ${\hat{b}}_{p}$ and ${\hat{b}}_{p}^{†}$ , and $ĉ_{p}$ and $ĉ_{p}^{†}$ , again for particles with momentum $p$ and mass $m$ but opposite charges, respectively, under the $U (1)$ symmetry described above. These are interpreted as particles and antiparticles.

Particle eigenstates

The Hilbert space of the free scalar field theory comprises the vacuum — 0-particle — state $| 0 ⟩$ and the states — i.e., particles — created by the creation operators acting on it:

\begin{matrix} \begin{aligned} â_{p} | 0 ⟩ = 0 \forall p & \Rightarrow H | 0 ⟩ = 0, \\ | p ⟩ \propto â_{p}^{†} | 0 ⟩ & \Rightarrow H | p ⟩ = ω_{p} | p ⟩ . \\ | p_{1}, p_{2} ⟩ \propto â_{p_{1}}^{†} â_{p_{2}}^{†} | 0 ⟩ & \Rightarrow H | p_{1}, p_{2} ⟩ = (ω_{p_{1}} + ω_{p_{2}}) | p_{1}, p_{2} ⟩ \\ ⋮ \\ | p_{1}, \dots, p_{n} ⟩ \propto â_{p_{1}}^{†} \dots â_{p_{n}}^{†} | 0 ⟩ & \Rightarrow H | p_{1}, \dots, p_{n} ⟩ = (\sum_{i = 1}^{n} ω_{p_{i}}) | p_{1}, \dots, p_{n} ⟩, \end{aligned} \end{matrix}

(3.1.20)

where $H$ is the Hamiltonian operator of the theory (see Appendix B.2.2), and $| p ⟩$ is a state with momentum $p$ and energy $ω_{p} = \sqrt{{| p |}^{2} + m^{2}}$ : i.e., a single particle of mass $m$ . This is essentially the sum of the Hilbert spaces of an infinite number of QHOs, across all momenta, and is called the Fock space. The momentum eigenstates are normalized such that their inner products are Lorentz scalars:

| p ⟩ = \sqrt{2 E_{p}} â_{p}^{†} | 0 ⟩ \Rightarrow ⟨ q | p ⟩ = 2 E_{p} δ^{3} (q - p) .

(3.1.21)

For a scalar field, the creation operators commute amongst themselves, which means the states $| p_{1}, \dots, p_{n} ⟩$ are symmetric under exchange of particles, and thus describe bosons.³

Interpretation of the field operators

The action of the fields themselves on the vacuum is:

\hat{ϕ} (x) | 0 ⟩ = \int \frac{d^{3} p}{{(2 π)}^{3}} \frac{1}{2 E_{p}} e^{ip \cdot x} | p ⟩ .

(3.1.22)

This is very similar to the Fourier transform of the position eigenstate $| x ⟩$ in nonrelativistic QM, except with an integral measure that is now Lorentz-invariant due to our normalizations above. Thus, we can roughly interpret $ϕ (x)$ as an operator which creates a particle at position $x$ . However, we will see next that $ϕ (x) | 0 ⟩ \equiv | x ⟩$ , unlike in QM, is not exactly localized in position (although it’s pretty close).

3.1.4 Propagators and Green functions

We now discuss briefly the concept of propagators in QFT, primarily because of their importance in relating quantum fields to physical observables of the theory, like scattering amplitudes (Section 3.2.2), but also because of some interesting insights they offer regarding our quantized particle states. The propagator is the amplitude for the associated particle at a spacetime point $y$ to be found at $x$ :

D (x - y) \equiv ⟨ 0 | ϕ (x) ϕ (y) | 0 ⟩ = \int \frac{d^{3} p}{{(2 π)}^{3}} \frac{1}{2 E_{p}} e^{ip \cdot (x - y)} .

(3.1.23)

This is also called the two-point correlation function between $x$ and $y$ .

Interestingly, it can be shown that, for a particle with mass $m > 0$ , for space-like separated points, e.g. $x_{0} = y_{0}, | x - y | \equiv r$ ,

D (r) \sim e^{- mr},

(3.1.24)

i.e., it is not 0!⁴ However, it exponentially decays at rate of $\frac{1}{m}$ , or the Compton wavelength. This tells us that there is a fundamental physical limit in relativistic QM to which a particle can be localized in space (or, at least, to which we can measure its position, related to the uncertainty principle).

Green functions

The propagator is closely related to the Green function $Δ (x)$ of the Klein-Gordon equation, which is the solution (or response) to a delta function source:

(□ + m^{2}) Δ (x) = δ^{4} (x) .

(3.1.25)

The Green function $Δ (x - y)$ effectively describes the effect on the field at $x$ due to a localized source at $y$ ; hence, the connection to the two-point correlation function, or propagator, above.

The form of $Δ (x)$ can be found by Fourier transforming this equation to be:

Δ (x) = \int \frac{d^{4} p}{{(2 π)}^{4}} \frac{i}{p^{2} - m^{2}} e^{- ip \cdot x},

(3.1.26)

This has a pole on the real line at $p^{2} = m^{2} \Leftrightarrow E = \pm \sqrt{p^{2} + m^{2}}$ , which means there is an ambiguity in defining the contour integral.

The choice of contour leads to four different Green functions, each with a different physical interpretation. The one choice we make in QFT is the Feynman prescription, often defined as

Δ_{F} (x) = \int \frac{d^{4} p}{{(2 π)}^{4}} \frac{i}{p^{2} - m^{2} + i𝜀} e^{- ip \cdot x},

(3.1.27)

where the $i𝜀$ term resolves the ambiguity by shifting the poles infinitesimally above and below the real line, $Δ_{F} (x)$ is called the Feynman propagator. It is related to our normal propagator above by

Δ_{F} (x - y) = {\begin{matrix} ⟨ 0 | ϕ (x) ϕ (y) | 0 ⟩ = D (x - y) & if x^{0} > y^{0} \\ ⟨ 0 | ϕ (y) ϕ (x) | 0 ⟩ = D (y - x) & if x^{0} < y^{0} \end{matrix} \equiv ⟨ 0 | Tϕ (x) ϕ (y) | 0 ⟩,

(3.1.28)

where we call $T$ the time-ordering operator.

¹See Weinberg’s notes on a history of QFT [95] for a nice summary of its historical development.

²Indeed, one can view particles simply has excited states of a continuous set of QHOs for different masses and momenta. This is made more precise in Appendix B.2.

³Technically, the individual momentum eigenstates are not “physical”, as they are not normalizable. Instead, particles exist in the form of a wavepacket with some spread in momenta $φ (p)$ , which we typically assume to be smaller than our detector resolution can thus ignore (see Appendix B.2.3.)

⁴Mathematically, this stems from the $\frac{1}{2 E_{p}}$ factor in the integral required for Lorentz invariance. Note, however, that this does not violate causality, since the commutator $[ϕ (x), ϕ (y)] = D (x - y) - D (y - x) = 0$ for spacelike separated points, meaning physically they cannot affect each other. For a complex field, $[ψ (x), ψ^{*} (y)] = 0$ has the interesting interpretation of a particle’s amplitude for $x \to y$ being canceled by its antiparticle’s amplitude for $y \to x$ . Or, inversely, this tells us that causality necessitates the existence of antiparticles (Peskin and Schroeder [81] Chapter 2.4).