Quantization

3.2 Quantization

In this section, we briefly sketch canonical quantization, a process of turning a classical field theory into a QFT. It is based on the Hamiltonian formalism, in close analogy to the quantization of classical mechanics $\to$ QM. The result makes manifest the connection between quantum fields and their associated particles.

An alternative quantization approach not discussed here is based on the path integral formulation (see Section 3.1.1.0). As with most alternative mathematical prescriptions of the same physics, it provides useful insight into the theory and can simplify certain calculations. Further detail can be found, for example, in Peskin and Schroeder [10] Chapter 9.

3.2.1 Canonical quantization

The process of quantizing a classical system in QM can be summarized as (1) promoting the canonical coordinates to quantum operators, and (2) imposing the canonical Poisson bracket relations as quantum commutator relations:

\begin{matrix} \begin{aligned} x \to \hat{x}, & p \to \hat{p}, \\ {x, p} = 1 & \to [\hat{x}, \hat{p}] = i ℏ . \end{aligned} \end{matrix}

(3.41)

Canonical quantization of a field theory is done analogously, with fields becoming operator-valued and obeying their own canonical commutation relations based on Eq. 3.34. For our free scalar field theory (Eq. 3.10), this means promoting the integration constants in the classical solution (Eq. 3.17 and 3.39) to operators:

\begin{matrix} \begin{aligned} 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}), \\ π (x, t) & = - i \int \frac{d^{3} p}{{(2 π)}^{3}} \sqrt{\frac{ω_{p}}{2}} (â_{p} e^{ip \cdot x} - â_{p}^{†} e^{- ip \cdot x}), \end{aligned} \end{matrix}

(3.42)

where again $p \cdot x = p_{μ} x^{μ}$ is the 4D spacetime inner product and $p_{μ} = (ω_{p} = \sqrt{{| p |}^{2} + m^{2}}, p)$ . Recall that the integration constants $a (p)$ and $a^{*} (p)$ arose from a SHO equation for each momentum $p$ (Eq. 3.16); thus, quantized, we expect them to correspond to the raising ( $â^{†}$ ) and lowering ( $â$ ) operators of a quantum harmonic oscillator (QHO), again one for each momentum mode $p$ . We can check this by deriving their commutation relations. Indeed, imposing the canonical commutation relationships:

[ϕ (x, t), ϕ (y, t)] = [π (x, t), π (y, t)] = 0, [ϕ (x, t), π (y, t)] = i δ^{3} (x - y),

(3.43)

reproduces (continuous versions of) the raising and lowering operator commutation relationships for a QHO:

[â_{p}, â_{q}] = [â_{p}^{†}, â_{q}^{†}] = 0, [â_{p}, â_{q}^{†}] = {(2 π)}^{3} δ^{3} (p - q) .

(3.44)

Next, we look at the commutators with the Hamiltonian and the resulting Hilbert space.

3.2.2 The Hamiltonian and the vacuum catastrophe

The quantized Hamiltonian, from Eq. 3.40, can be found to be:

\begin{matrix} \begin{aligned} H & = \int d^{3} x (\frac{1}{2} π^{2} + \frac{1}{2} {(\nabla ϕ)}^{2} + \frac{1}{2} m^{2} ϕ^{2}) \\ = \int \frac{d^{3} p}{{(2 π)}^{3}} ω_{p} [â_{p}^{†} â_{p} + \frac{1}{2} {(2 π)}^{3} δ^{3} (0)] . \end{aligned} \end{matrix}

(3.45)

This looks a lot like the Hamiltonian for a QHO, $H = ω (a^{†} a + \frac{1}{2})$ , for each momenta, but with an unwieldy delta function. This latter term is called the zero-point energy and represents the energy of the vacuum state. It is infinite, and, indeed, is one of the many infinities that have to be dealt with in QFT. In this case, since it is a constant energy term, it does not affect the dynamics of the system and can simply be ignored / subtracted for our purposes.³ However, the vacuum energy density does affect Einstein’s equations of general relativity, and the disagreement between the large zero-point energy we expect from QFT and the small observed value is known as the cosmological constant problem (or, more dramatically, the vacuum catastrophe) [26, 27].

An infinity of harmonic oscillators

Subtracting away the zero-point energy gives us

H = \int \frac{d^{3} p}{{(2 π)}^{3}} ω_{p} â_{p}^{†} â_{p},

(3.46)

whose commutators with the raising and lowering operators are:

[H, â_{p}^{†}] = ω_{p} â_{p}^{†}, [H, â_{p}] = - ω_{p} â_{p},

(3.47)

just as for a QHO. This tells us that given an eigenstate of $H$ , $| E ⟩$ , with eigenvalue $E$ , $â_{p}^{†} | E ⟩$ and $â_{p} | E ⟩$ are also eigenstates with eigenvalues $E + ω_{p}$ and $E - ω_{p}$ , respectively:

\begin{matrix} \begin{aligned} H â_{p}^{†} | E ⟩ & = (â_{p}^{†} H + ω_{p} â_{p}^{†}) | E ⟩ = (E + ω_{p}) â_{p}^{†} | E ⟩, \\ H â_{p} | E ⟩ & = (â_{p} H - ω_{p} â_{p}) | E ⟩ = (E - ω_{p}) â_{p} | E ⟩ . \end{aligned} \end{matrix}

(3.48)

To derive the Hilbert space of this theory, we first define the vacuum $| 0 ⟩$ as the state for which

â_{p} | 0 ⟩ = 0 \forall p \Rightarrow H | 0 ⟩ = 0 .

(3.49)

The remaining eigenstates are then obtained by (repeatedly) acting with $â_{p}^{†}$ on the vacuum:

\begin{matrix} \begin{aligned} | 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}

(3.50)

This is essentially the sum of the Hilbert spaces of an infinite number of QHOs, across all momenta. It is called the Fock space.

3.2.3 Particles

To understand these states further, we can also quantize the total momentum of the field density we found from Noether’s theorem (Eq. 3.24):⁴

P = \int d^{3} x π \nabla ϕ = \int \frac{d^{3} p}{{(2 π)}^{3}} p â_{p}^{†} â_{p} .

(3.51)

Acting with $P$ on $| p ⟩$ gives us:

\begin{matrix} \begin{aligned} P | p ⟩ & = \int \frac{d^{3} k}{{(2 π)}^{3}} k â_{k}^{†} â_{k} â_{p}^{†} | 0 ⟩ \\ = \int \frac{d^{3} k}{{(2 π)}^{3}} k â_{k}^{†} (â_{p}^{†} â_{k} - {(2 π)}^{3} δ^{3} (k - p)) | 0 ⟩ \\ = p | p ⟩ . \end{aligned} \end{matrix}

(3.52)

Thus the states $| p ⟩$ are eigenstates of $P$ as well, with eigenvalues $p$ . Putting this together, we have a Hilbert space spanned by the states $| p ⟩$ , which each have momentum $p$ and energy $ω_{p} = | \sqrt{p^{2} + m^{2}} |$ , i.e. the relativistic energy-momentum relation for a free particle. Thus, we see $| p ⟩$ exactly corresponds to the momentum eigenstate for a single particle of mass $m$ and momentum $p$ ! One can similarly quantize the total angular momentum of the field $J$ and show that $J | p = 0 ⟩ = 0$ , i.e. the particle has spin 0.

This is one of the miracles of QFT: what from quantizing a free, relativistic field looked bizarrely like an infinite series of QHOs, actually gives the intuitive physical result of discrete particle states. The Fock space hence is the space spanned by different numbers of discrete particles per each continuous momentum mode $p$ . The number of particles $n$ in a particular state of the Fock space is given by the number operator $N$ , essentially the Hamiltonian density divided by $ω_{p}$ :

N = \int \frac{d^{3} p}{{(2 π)}^{3}} â_{p}^{†} â_{p} \Rightarrow N | p_{1}, \dots, p_{n} ⟩ = n | p_{1}, \dots, p_{n} ⟩ .

(3.53)

Note that the number operator $N$ commutes with the Hamiltonian $H$ , $[N, H] = 0$ , which means particle number is conserved; however, this will not be the case for interacting theories in the next section.

Finally, observe that for our scalar field, the creation operators commute amongst themselves. This means the states $| p_{1}, \dots, p_{n} ⟩$ are symmetric under exchange of particles, and thus describe bosons.

3.2.4 The complex scalar field and antiparticles

The complex scalar field Lagrangian from Eq. 3.26 has the EOMs:

\begin{matrix} \begin{aligned} (\partial_{μ} \partial^{μ} + m^{2}) ψ & = 0, \\ (\partial_{μ} \partial^{μ} + m^{2}) ψ^{*} & = 0, \end{aligned} \end{matrix}

(3.54)

with solutions:

\begin{matrix} \begin{aligned} ψ (x) & = \int \frac{d^{3} p}{{(2 π)}^{3}} \frac{1}{\sqrt{2 ω_{p}}} (b (p) e^{ip \cdot x} + c^{*} (p) e^{- ip \cdot x}) . \end{aligned} \end{matrix}

(3.55)

Note that because the field is complex, the coefficients $b$ and $c^{*}$ need not be complex conjugates of each other as for a real field. This field can be quantized analogously to above:

\begin{matrix} \begin{aligned} 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}), \\ 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}

(3.56)

where we now have two sets of creation and annihilation operators, ${{\hat{b}}^{†}, \hat{b}}$ and ${ĉ^{†}, ĉ}$ . One can check each pair individually satisfies the canonical commutation relations from Eq. 3.44, and mutually commutes with each other. Thus, they are interpreted as corresponding to two different particles, with the same mass $m$ and spin 0, but, as we saw, with opposite charges under the $U (1)$ internal symmetry. Such pairs are considered particles and antiparticles.

Finally, let us revisit and quantize the conserved charge associated with the $U (1)$ symmetry (Eq. 3.27):

Q = \int d^{3} x i (ψ^{*} \partial^{0} ψ - ψ \partial^{0} ψ^{*}) \to \int \frac{d^{3} p}{{(2 π)}^{3}} ({\hat{b}}_{p}^{†} {\hat{b}}_{p} - ĉ_{p}^{†} ĉ_{p}) = N_{b} - N_{c} .

(3.57)

This is saying the difference in the number of particles and antiparticles is conserved, which for a single charged particle-antiparticle pair, is equivalent to charge conservation. This will be more significant for interacting theories, in which $N_{b}$ and $N_{c}$ are not individually conserved but as long as the interactions retain the $U (1)$ symmetry, $Q$ is.

Negative energy states?

Note that the full, time-dependent formula for the field $ψ (x, t)$ contains both the $e^{- ip \cdot x} \propto e^{- iEt}$ and $e^{ip \cdot x} \propto e^{iEt}$ terms. As single-particle plane-wave solutions to the nonrelativistic Schrödinger equation, these would correspond to positive and negative energy states, the latter of which does not make physical sense.⁵ Our solution is to refer to these states instead as positive- and negative-frequency modes, which, as we saw, are always associated to operators that create and destroy positive-energy (anti)particles, respectively.

3.2.5 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 3.3.2), but also because of some interesting insights they offer regarding our quantized particle states.

Normalization of states and wavepackets

First, note that we have not chosen a normalized the momentum eigenstates in Eq. 3.48. We cannot do this simply as $⟨ q | p ⟩ = δ^{3} (q - p)$ , as in nonrelativistic QM, because the delta function alone is not Lorentz-invariant. Instead, we choose

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

(3.58)

which is a Lorentz scalar.

Like in QM, however, these momentum eigenstates are not normalized to 1: $⟨ p | p ⟩ = 2 E_{p} δ^{3} (0)$ , so they are not exactly physical one-particle states. Physical particles must exist in the form of a wavepacket:

| φ ⟩ = \int d^{3} p φ (p) | p ⟩,

(3.59)

with some spread in momenta $φ (p)$ . However, as long as this variation is smaller than the resolution of our detector (as we will assume), for all practical purposes and calculations we can continue to treat particles as momentum eigenstates. This assumption is further motivated in Peskin and Schroeder [10] Chapter 4.5.

Interpretation of $\hat{ϕ} (x)$

Let us consider the field $ϕ (x)$ itself for a moment. We have understood the meaning of the creation, annihilation, momentum, and other operators, but what does the field $ϕ (x)$ do? If we look at its action on the vacuum, plugging in its quantized form (Eq. 3.42) and our new normalization of the momentum eigenstates:

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

(3.60)

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).

Propagators

We can now discuss the propagator of a field, which 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)} .

(3.61)

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},

(3.62)

i.e., it is not 0!⁶ However, it exponentially decays at rate of $\frac{1}{m}$ , or the Compton wavelength. Thus, this is telling us 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).

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 fun 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 [10] Chapter 2.4).

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) .

(3.63)

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},

(3.64)

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. What’s cool is that the choice of contour leads to four different Green functions, each with a different physical interpretation.

The choice we care most about in QFT is the Feynman prescription, which is 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},

(3.65)

where the $i𝜀$ term resolves the ambiguity by shifting the poles infinitesimally above and below the real line, and $Δ_{F} (x)$ 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 ⟩,

(3.66)

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

³Equivalently, we can consider the normal-ordered Hamiltonian (see, e.g., Tong QFT [3] Chapter 2.3). An alternative way of resolving the infinity is to introduce an ultra-violet cut-off scale $Λ$ in the integral over momenta.

⁴Technically, we show here the normal-ordered momentum.

⁵This is related to the problem Dirac faced in developing his relativistic quantum theory of the electron, except we are dealing with bosons instead of fermions, so we cannot rely on the fermionic Dirac sea “solution”.

⁶Mathematically, this stems from the $\frac{1}{2 E_{p}}$ factor in the integral required for Lorentz invariance.