Quantum field

Related: quantum particle.
See my paper in Researchgate.

Introduction

Quantum Field Theory (QFT) is not a broader or more general framework than Quantum Mechanics (QM); rather, it is a particular case of a quantum mechanical theory. As is well known, QM can be formulated for systems with continuous variables, such as position $q$ , or discrete variables, such as spin.
In the case of a continuous variable like position, the state of the system is described by a wavefunction $| Ψ ⟩$ , which is an element of an infinite-dimensional Hilbert space $H$ . This Hilbert space is typically spanned by a kind of basis ${| q ⟩}$ , where $q$ represents a specific position in space and $| q ⟩$ is a state of definite position. The state $| Ψ ⟩$ can then be expressed as a linear combination of these basis states:

| Ψ ⟩ = \int Ψ (q) | q ⟩ d x,

where the set of values ${Ψ (q)}$ can be interpreted as a complex-valued function assigning an amplitude to each position $q$ , hence the function part of the word wavefunction.

For discrete systems, such as the spin of a spin- $\frac{1}{2}$ particle, the Hilbert space $H$ is finite-dimensional. Here, the basis states might be $| ↑ ⟩$ and $| ↓ ⟩$ , representing the two possible spin orientations along a chosen axis. The wavefunction in this case takes the form:

| Ψ ⟩ = α | ↑ ⟩ + β | ↓ ⟩,

where $α$ and $β$ are complex numbers encoding the probability amplitudes for the spin states. Here the wavefunction is a function with a discrete domain: the set ${| ↑ ⟩, | ↓ ⟩}$ .

In both the continuous and discrete cases, the wavefunction $| Ψ ⟩$ serves as a mathematical object that assigns complex numbers to elements of a basis in $H$ , capturing the probabilistic nature of quantum systems.

At this point, we can say that a Quantum Field Theory is, in essence, a quantum mechanical theory in which wavefunctions are defined over classical field configurations—that is, the basis of the Hilbert space consists of such configurations. This perspective is sometimes referred to as the field approach to QFT (see Sean Carroll in this video, and the paper The fundamentality of fields; also the book Understanding quantum mechanics section 11.2.).

Initial setup

In order to be as simple as possible (but no simpler), we will consider a flat spacetime with only one spatial dimension and one temporal dimension, that is, $M = S \times T$ , where $S = T = R$ . We will call field configuration to any element of the set $F$ of all smooth functions

ϕ : R \to R,

which are often referred to as matter fields. The set $F$ will play the role analogous to the set of all possible positions of a particle in QM, which is $R$ . That is, their elements are a kind of pure states of the system.

Remark:
More generally, in physical theories, field configurations are often described as sections of fiber bundles rather than simple real-valued functions. This broader perspective naturally leads to gauge theory, where additional geometric structures—such as connections on principal bundles—play a fundamental role. However, in our simplified setting, we restrict attention to real-valued fields to build intuition.

Remark:
It is also worth emphasizing the distinction between the terms field and field configuration. A field, in the physicist’s sense, is a more abstract entity that implicitly includes the choice of an appropriate bundle and the type of sections it admits, so determining a specific set $F$ . In contrast, a field configuration is a particular assignment of values to the field at each point in space, that is, a choice $ϕ \in F$ . This distinction is somewhat analogous to the difference between the concept of position of a one-dimensional particle, determining a specific set of possible values, $R$ , and a particular value $q \in R$ .

Then, we consider a quantum system whose Hilbert space $H$ consists of the linear combinations of field configurations, i.e., given $θ, φ, ξ, \dots \in F$ , we have that the elements of $H$ are of the form

| Ψ ⟩ = α_{θ} | θ ⟩ + α_{φ} | φ ⟩ + α_{ξ} | ξ ⟩ + \dots,

where $α_{θ}, α_{φ}, α_{ξ}, \dots$ are complex numbers, and $| θ ⟩, | φ ⟩, | ξ ⟩, \dots$ are pure states of the system, i.e., states with definite field configurations. This is analogous to the way we represent a state in QM as a linear combination of position eigenstates. The state $| Ψ ⟩$ is a superposition of field configurations, each with a corresponding probability amplitude $α_{θ}, α_{φ}, α_{ξ}, \dots$ .

Nevertheless, we will use the notation

| Ψ ⟩ = Ψ [θ] | θ ⟩ + Ψ [φ] | φ ⟩ + Ψ [ξ] | ξ ⟩ + \dots,

to highlight that the coefficients constitute not a function but a functional, since they assign a complex number to each field configuration (a function itself):

\begin{array}{rccc} Ψ : & F & \to & C \\ ϕ & \mapsto & Ψ [ϕ] := α_{ϕ} . \end{array}

Moreover, the linear combination is not over a countable set of basis elements, but over all possible field configurations, so the sum is actually an integral:

| Ψ ⟩ = \int Ψ [ϕ] | ϕ ⟩ D ϕ,

where $D ϕ$ denotes a measure over all field configurations. This is a subtle point that we will not delve into here, but we refer the interested reader to [simon2005functional] for a more detailed discussion.

The squared modulus of the wavefunctional, $| Ψ [ϕ] |^{2}$ , gives the probability of measuring the field with precisely the configuration $ϕ$ over the whole space. Of course, this measurement does not happen in real life; it is an idealization.

Finally, the inner product between two wavefunctionals $Ψ_{1} [ϕ]$ and $Ψ_{2} [ϕ]$ would be something like

⟨ Ψ_{1} | Ψ_{2} ⟩ = \int Ψ_{1}^{*} [ϕ] Ψ_{2} [ϕ] D ϕ .

Fields promoted to operators

In standard QM for the position of a one-dimensional particle, the variable $q$ is promoted to an operator $\hat{q}$ , which acts on the Hilbert space $H$ by multiplication. In the case of QFT, we will promote the variable field configuration $ϕ \in F$ to an operator-valued function of space.

To better understand this transition, consider first a QM system of three one-dimensional particles, with position labelled as $q_{1}, q_{2}, q_{3}$ respectively. Instead of representing the particles as points on a line, we can think of them as beads on a horizontal line:
Pasted image 20250201110704.png
Do not think of the continuous variables $q_{1}, q_{2}, q_{3}$ as positions on physical space. Instead, assume that we have only three positions, labelled as 1, 2 and 3; and that $q_{1}, q_{2}, q_{3}$ reflects values of a continuous magnitude happening in a kind of internal space.

The Hilbert space of this quantum system is the tensor product of the Hilbert spaces of each particle, that is,

H = L^{2} (R) \otimes L^{2} (R) \otimes L^{2} (R) = L^{2} (R^{3}),

and we have a kind of basis, usually denoted by

{| q_{1}, q_{2}, q_{3} ⟩ : q_{1}, q_{2}, q_{3} \in R} .

However, instead of this notation, we will use

{| v ⟩ : v = (v_{1}, v_{2}, v_{3}) \in R^{3}},

since it is more suggestive for the generalization to fields.

An element $| Ψ ⟩ \in H$ is therefore generically written as

| Ψ ⟩ = \int ψ (v) | v ⟩ d v_{1} d v_{2} d v_{3},

and often $| Ψ ⟩$ is identified with the corresponding function $ψ \in L^{2} (R^{3})$ .

Recall that this system counts with three built-in operators: ${\hat{q}}_{1}$ , ${\hat{q}}_{2}$ , and ${\hat{q}}_{3}$ . For instance, the operator

{\hat{q}}_{1} : H \to H

acts on a basis element $| v ⟩$ as follows:

{\hat{q}}_{1} | v ⟩ = v_{1} | v ⟩,

where $v_{1}$ is the first component of the vector $v$ . This action is extended by linearity to any general state vector $| Ψ ⟩ \equiv ψ (v)$ in $H$ . Thus, the action of ${\hat{q}}_{1}$ on a state $| Ψ ⟩$ can be expressed as:

{\hat{q}}_{1} ψ (v) = v_{1} ψ (v) .

Observe that the input and the output are both functions which depend on the variables $v_{1}, v_{2}, v_{3}$ , the components of the vector $v$ . Also observe that the eigenspace of ${\hat{q}}_{1}$ corresponding to a specific eigenvalue, let's say 4, consists of all states $| Ψ ⟩$ that satisfy:

{\hat{q}}_{1} | Ψ ⟩ = 4 | Ψ ⟩ .

In terms of wavefunctions, this means:

v_{1} ψ (v) = 4 ψ (v), \forall v_{1}, v_{2}, v_{3} \in R,

or otherwise

ψ (v_{1}, v_{2}, v_{3}) = δ_{4} (v_{1}) ϕ (v_{2}, v_{3}),

where $ϕ (v_{2}, v_{3})$ is an arbitrary square-integrable function of $v_{2}$ and $v_{3}$ , and $δ_{4}$ is the corresponding Dirac delta centered at 4, i.e., $δ_{4} (y) = δ (y - 4)$ . We have then that the eigenspace corresponding to 4 is generated by the basis elements ${| (4, v_{2}, v_{3}) ⟩ : v_{2}, v_{3} \in R}$ .

From here, we can move on to QFT by substituting $q_{i}$ by $ϕ (x)$ , where $ϕ \in F$ denotes field configuration. That is, we have a continuous amount of particles, indexed by $x$ instead of $i$ , and the quantity under study is denoted by $ϕ$ instead of $q$ .
Pasted image 20250201105303.png
This new quantum system has a corresponding Hilbert space $H$ , which we can consider to be generated by the elements of the set

{| φ ⟩ : φ \in F},

in analogy with the previous case. You can think of $φ (x), x \in R,$ as the continuum version of $v_{i}, i = 1, 2, 3$ ; and $| φ ⟩$ corresponds to $| v ⟩$ . So, instead of the discrete case, we now have

| Ψ ⟩ = \int Ψ [φ] | φ ⟩ D φ,

for a general element $| Ψ ⟩ \in H$ .

And now, we don't have three operators ${\hat{q}}_{i}$ , indexed by $i = 1, 2, 3$ ; but a continuous family of operators $\hat{ϕ} (x)$ indexed by $x \in R$ . How do they work? Well, given a fixed $x \in R$ , we have an operator

\hat{ϕ} (x) : H \to H

such that when applied to the basis vectors $| φ ⟩$ it yields

\hat{ϕ} (x) (| φ ⟩) = φ (x) | φ ⟩,

in complete analogy with the discrete case.

Given a general state $| Ψ ⟩ \in H$ , identified with the wavefunctional $ψ$ , these operators act as follows

\hat{ϕ} (x) ψ [φ] = φ (x) ψ [φ],

which is similar to the equation for the discrete case.

It is important to emphasize that the input and the output of the operator $\hat{ϕ} (x)$ , for a particular $x \in R$ , are functionals which depend on the entire field configuration $φ$ . In particular, the output depends explicitly on the value of $φ$ on the specific value $x$ , in the same way that the output in the discrete case is a function which depends on the vector $v$ and with an additional dependence on the component number $1$ of $v$ .

Observe also that, for a fixed $x \in R$ , the eigenspace of a particular eigenvalue of $\hat{ϕ} (x)$ , let's say 4, is generated by the states

{| φ ⟩ : φ \in F such that φ (x) = 4} .

The dynamics

To understand the dynamics of our brand new quantum system we will start, again, with our discrete model of a finite number of particles, as done in \cite{eigenchrisSpinorsForBeginners21}. For example, we may suppose that the three one-dimensional particles of the system from the previous section were connected by springs:
Pasted image 20250309191538.png

In Classical Mechanics, the Hamiltonian for these three coupled harmonic oscillators, assuming unitary mass, is given by

H = \sum_{i = 1}^{3} (\frac{p_{i}^{2}}{2} + \frac{1}{2} k q_{i}^{2}) + \sum_{i = 1}^{2} \frac{1}{2} κ (q_{i + 1} - q_{i})^{2},

\label{hamiltonian3beads}
where $p_{i}$ is the conjugate momentum of $q_{i}$ , $k$ is the spring constant, and $κ$ is the spring constant of the springs connecting the particles \cite{MERDACI2020126134}.

Recall that the quantized version of this Hamiltonian is given by the operator defined as

\hat{H} = \sum_{i = 1}^{3} (\frac{{\hat{p}}_{i}^{2}}{2} + \frac{1}{2} k {\hat{q}}_{i}^{2}) + \sum_{i = 1}^{2} \frac{1}{2} κ ({\hat{q}}_{i + 1} - {\hat{q}}_{i})^{2},

\label{tag2}
where ${\hat{p}}_{i}$ and ${\hat{q}}_{i}$ are the operators associated with the conjugate momentum and the displacement of the $i$ -th particle, respectively.

Finally, the evolution of an initial state

| Ψ (0) ⟩ = \int g (v) | v ⟩ d v_{1} d v_{2} d v_{3}

is given by the time-dependent state

| Ψ (t) ⟩ = \int ψ (v, t) | v ⟩ d v_{1} d v_{2} d v_{3},

with $ψ (v, t)$ satisfying the Schr"odinger equation

\frac{d}{d t} ψ (v, t) = - \frac{i}{ℏ} \hat{H} ψ (v, t),

and initial condition

ψ (v, 0) = g (v) .

To move on to QFT, we add to this initial setup more coupled oscillators, a continuous amount of them, as we did in the previous section.
Pasted image 20250309191632.png
But now, the (still classical) field $ϕ$ inherits the dynamics of the particles, so it is governed by a Hamiltonian corresponding to a continuous version of \eqref{hamiltonian3beads}, i.e.,

H = \int h (x) d x,

\label{tag3}
where $h (x)$ is the Hamiltonian density given by

h (x) = \frac{π (x)^{2}}{2} + \frac{1}{2} k ϕ (x)^{2} + \frac{1}{2} κ {(\frac{\partial ϕ (x)}{\partial x})}^{2} .

Here, $π (x)$ is the conjugate momentum associated with $ϕ (x)$ , for each $x \in R$ . And the term $\frac{1}{2} κ {(\frac{\partial ϕ (x)}{\partial x})}^{2}$ reflects the coupling of each oscillator with its neighbors, in the same way that the term $\frac{1}{2} κ (q_{i + 1} - q_{i})^{2}$ in \eqref{hamiltonian3beads} does.

At this point, we proceed by quantizing \eqref{tag3} to obtain the Hamiltonian operator

\hat{H} = \int \hat{h} (x) d x,

\label{tag4}
where the Hamiltonian density operator is given by

\hat{h} (x) = \frac{\hat{π} (x)^{2}}{2} + \frac{1}{2} k \hat{ϕ} (x)^{2} + \frac{1}{2} κ {(\frac{\partial \hat{ϕ} (x)}{\partial x})}^{2} .

This operator is applied to the Hilbert space $H$ , whose elements are wavefunctionals of the form \eqref{generalstatefield}. To describe the effect of $\hat{H}$ on a state $| Ψ ⟩ \equiv Ψ [φ]$ in $H$ , we only have to understand how the operators $\hat{π} (x)$ and $\frac{\partial \hat{ϕ} (x)}{\partial x}$ act on $| Ψ ⟩$ :

The conjugate momentum operator $\hat{π} (x)$ acts on a wavefunctional $Ψ [φ]$ analogously to the discrete conjugate momentum ${\hat{p}}_{i}$ acting on a state $| v ⟩$ , by differentiating with respect to $φ (x)$ , instead of with respect to $v_{i}$ . This corresponds to the functional derivative \cite{olver86}:

\hat{π} (x) Ψ [φ] = - i ℏ \frac{\partial Ψ [φ]}{\partial φ (x)} = - i ℏ lim_{ϵ \to 0} \frac{Ψ [φ + ϵ δ_{x}] - Ψ [φ]}{ϵ},

where $δ_{x} (y) = δ (y - x)$ is the Dirac delta function.

The field derivative operator $\frac{\partial \hat{ϕ} (x)}{\partial x}$ acts as

\begin{aligned} \frac{\partial \hat{ϕ} (x)}{\partial x} Ψ [φ] & = lim_{ϵ \to 0} \frac{\hat{ϕ} (x + ϵ) Ψ [φ] - \hat{ϕ} (x) Ψ [φ]}{ϵ} \\ = lim_{ϵ \to 0} \frac{φ (x + ϵ) Ψ [φ] - φ (x) Ψ [φ]}{ϵ} \\ = \frac{\partial φ}{\partial x} (x) Ψ [φ] . \end{aligned}

The evolution of a given state $Ψ [φ]$ is determined via the Schrödinger equation:

\frac{\partial}{\partial t} Ψ [φ, t] = - \frac{i}{ℏ} \hat{H} Ψ [φ, t] .

How do particles appear?

See preprint at Researchgate

Generalization

See preprint at Researchgate