# What is a quantum field?

- The first, completely reasonable, attitude is: "We will introduce the formalism and show you how to perform computations: that's what's important. What you do with the formalism in the privacy of your own room is your business."
- The second, also completely reasonable, attitude is: "We have thought long and hard about this, and the following is the cleanest and most beautiful axiomatisation of the subject we could find; we wish you the best of luck unpicking this."

I will start by reviewing quantisation in ordinary quantum mechanics, before moving on to the simplest QFT (the Klein-Gordon field) and explaining the wavefunctional picture. Hopefully, once you have seen this intuition, the abstract frameworks built by books like Glimm-Jaffe or Streater-Wightman will make more sense.

## Review of quantisation

When we *quantise*, we start with a classical system and try to
construct a quantum mechanical system which reduces to this classical
system in the \(\hbar\to 0\) limit. Of course, it's not clear that there
is an answer, let alone a unique answer, or that there will be some
prescriptive way to find the answer for an arbitrary classical system,
even if it exists.

Nonetheless, let's review the basic set-up:

- A classical mechanical system comprises a phase space (symplectic manifold \((M,\omega)\)) of positions and momenta; observable quantities are functions on this phase space. There is a distinguished function \(H\) called the Hamiltonian which governs time evolution of the system. Explicitly, the Hamiltonian generates a Hamiltonian vector field \(X_H\) on the phase space via Hamilton's equations \(\omega(X_H,-)=-dH\) and integrating this vector field gives a flow \(\phi_H^t\colon M\to M\) which is the time evolution of the system.
- A quantum mechanical system comprises a Hilbert space together with
a collection of operators whose spectrum (eigenstates/eigenvalues)
you want to compute. There is a distinguished operator \(H\) called
the Hamiltonian which governs time evolution of the system. In the
Schrödinger picture, this time evolution works as follows. The
Hamiltonian generates a one-parameter family of unitary
transformations \(U_t\) satisfying the Schrödinger equation
\[\frac{dU_t}{dt}=\frac{i}{\hbar}HU_t.\] (Unitarity of the
transformation is equivalent to \(H\) being Hermitian). For example,
if \(H\) is time-independent then \(U_t=e^{iHt/\hbar}\). If the
system is in the state \(\psi\) at time \(0\) then at time \(t\) it
will be in the state \(U_t\psi\).

## Canonical quantisation of the free particle

The most basic classical system is a particle living in a 1-dimensional space with coordinate \(q\). The phase space keeps track of the position \(q\) but also the momentum \(p\), so it is \(\mathbf{R}^2\) with coordinates \(p,q\) and symplectic form \(\omega=dp\wedge dq\). We want to turn this into a quantum system by finding a suitable Hilbert space and turning functions of \(q,p\) into operators on this Hilbert space.

We take our Hilbert space to be the space of square-integrable
*wavefunctions*/ of \(q\), written \(L^2(\mathbf{R})\). The two
most famous wavefunctions, whose physical interpretation is clear,
actually don't live in this space (but can be closely approximated by
wavefunctions in \(L^2\)):

- Fix a point \(q_0\in\mathbf{R}\). The delta function \(\delta(q-q_0)\) should represent something like a particle localised at the point \(q_0\). Delta functions are not really functions at all, let alone square-integrable, but one can approximate a delta function arbitrarily closely by a strongly peaked Gaussian.
- Fix a frequency \(\omega\). The function \(e^{i2\pi\omega q}\) is a
wave with pure frequency \(\omega\). This should represent a
particle (like a photon) whose momentum is \(\omega\). To see why,
recall:
- De Broglie's formula \(E=h\omega\) (\(h=2\pi\) in our units) relating the energy and frequency of matter waves,
- the fact that a light wave has energy equal to its momentum (up to a factor of \(c=1\)) since its energy-momentum vector lives on the null-cone in spacetime.

The wavefunction \(\delta(x-q_0)\) is the unique \(q_0\)-eigenfunction of the operator \((\hat{q}\psi)(q)=q\psi(q)\). The wavefunction \(e^{ipx}\) is the unique \(p\)-eigenfunction of the operator \(\hat{p}\psi=-i\partial_q\psi\). The standard guess at a quantisation of this phase space is therefore to replace \(q\) and \(p\) by the operators \(\hat{q}\) and \(\hat{p}\) defined above, acting on the Hilbert space of \(L^2\)-functions of \(q\).

If \(P\) is a polynomial in \(p\) and \(q\) then we can try replacing \(P\) by the corresponding polynomial in the operators \(\hat{p}\) and \(\hat{q}\). Alas, the operators \(\hat{p}\) and \(\hat{q}\) no longer commute, indeed, we have \[ [\hat{q},\hat{p}]=i,\] so there are many choices of how to order the operators in the polynomial \(P\). For an example of how one quantises a particular quadratic Hamiltonian \(H=\frac{1}{2}p^2+\omega^2q^2\), see my pre-QFT post on the simple harmonic oscillator.

## Quantum field theory

Let us now turn to the problem of finding a quantum system to replace
the classical *Klein-Gordon field*. The Klein-Gordon field is a
complex-valued function \(\phi(x)\). The classical Hamiltonian
associated to this field is \[\frac{1}{2}\int
d^3x\left(\pi^2+|\nabla\phi|^2+m^2\phi^2\right).\] The quantity
\(\pi\) is the analogue of "momentum" for a field: in the case of the
Klein-Gordon field, it is just the time-derivative of \(\phi\) (just
as momentum is related to the time derivative of \(q\)).

When we quantised a free particle, our Hamiltonian was a function on the space of possible positions and momenta of the particle.

Now we are quantising a field, the Hamiltonian is a function on the space of possible configurations and momenta of the field.

When we quantised a free particle, we took as our Hilbert space the space of wavefunctions on \(\mathbf{R}^3\), where \(\mathbf{R}^3\) is the space of possible configurations (positions) of our particle.

Now we are quantising a field, we will take as our Hilbert space the space of wavefunctions on \(\mathcal{F}\), the space of possible configurations of the field.

A configuration of the field is just a complex-valued function on
\(\mathbf{R}^3\). In other words, \(\mathcal{F}\) is a suitable space
of functions \(\phi(x)\). A wavefunction will be a map
\(\Psi\colon\mathcal{F}\to\mathbf{C}\); something which eats a
function \(\phi\) and outputs a number, for example: \[\Psi(\phi)=\int
d^3x\phi(x).\] To distinguish the functions
\(\phi\colon\mathbf{R}^3\to\mathbf{C}\) from the wavefunctions
\(\Psi\colon\mathcal{F}\to\mathbf{C}\), we will call \(\Psi\) a
*wavefunctional*. The suffix *-al* denotes a function which
eats functions and outputs numbers.

Just as a wavefunction in quantum mechanics describes some kind of superposition of particles at different points, the wavefunctional in QFT describes a superposition of field configurations.

I am not going to address the questions "which wavefunctionals do we allow?" or "what is the inner product making them into a Hilbert space?", nor will I talk about normalising wavefunctionals; therein lie some analytical issues. Instead, here are some examples of wavefunctionals.

- Given a function \(f\in\mathcal{F}\), there is a delta-functional \(\delta(\phi-f)\) concentrated at \(f\) (which vanishes unless \(\phi\equiv f\)). This probably shouldn't be allowed in the rigorous theory, in the same way that the delta function \(\delta(q-q_0)\) is not an \(L^2\)-function; nonetheless, it's a convenient storytelling device. You can imagine this as a QFT state where the field has a definite value \(f(x)\) at each point \(x\) (in the same way that the delta function \(\delta(q-q_0)\) is a quantum mechanical state where the particle has definite position \(q_0\)).
- Given a function \(\lambda(x)\), the wavefunctional
\(\Psi(\phi)=\exp(i\int d^3x\lambda(x)\phi(x))\) is going to be a
QFT state where the field has definite "momentum"
\(\pi=\lambda\). This is analogous to the quantum mechanical "plane
wave" state \(\psi(x)=e^{ipx}\), which has momentum \(p\).

*operator-valued distribution*.

- The terms \(\int\int\int d^3kd^3xd^3e^{ik\cdot(x-y)}yi\pi(y)\omega_k\phi(x)\) and \(-\int\int\int d^3kx^3xd^3ye^{ik\cdot(x-y)}i\pi(x)\omega_k\phi(y)\) cancel. To see this, take the second term, switch \(x\leftrightarrow y\) and change \(k\leftrightarrow -k\). The integrand becomes the integrand in the first term, the volume form \(d^3kd^xd^3y\) is invariant, so the terms cancel because one appears with a minus sign.
- The term \[\int\int\int\frac{d^3k}{(2\pi)^3}d^3xd^3ye^{ik\cdot(x-y)}\pi(x)\pi(y),\] becomes \[\int d^3x\pi(x)^2,\] using the identity \[\delta(x-y)=\int\frac{d^3k}{(2\pi)^3}e^{ik\cdot(x-y)}.\]
- The term
\[\int\int\int\frac{d^3k}{(2\pi)^3}d^3xd^3ye^{ik\cdot(x-y)}\left(|k|^2+m^2\right)\phi(x)\phi(y),\]
becomes \[\int d^3x\left(|\nabla\phi|^2+m^2\right)\phi^2,\] using
the standard trick in Fourier theory of exchanging derivatives for
factors of \(k\).

*lowest-energy (or vacuum) state*\(\Psi_0\) which is annihilated by the operator \(\widehat{a(k)}\), and therefore satisfies \[\hat{H}\Psi_0=\int\frac{d^3k}{(2\pi)^3}a(k)^\dagger a(k)\Psi_0=0.\] One can generate other states by acting on the vacuum with \(\widehat{a(k)^\dagger}\), \(\widehat{ev_x}\) or combinations of these operators. The state \[\widehat{a(k)}^\dagger\Psi_0\] has the interpretation of a single particle with momentum \(k\). The state \(\widehat{ev_x}\Psi_0\) has the interpretation of a single particle at \(x\).

## Conclusion

The aim of all this was to show that, if you understand quantum mechanics, you can also understand quantum field theory (in principle!). The ideas are the same, but the wavefunctionals of QFT are on a different plane of difficulty: they are functions on the infinite-dimensional space of field configurations. To solve an eigenvalue problem in this context means solving an infinite-dimensional differential equation (like \(\widehat{a(k)}\Psi_0=0\) in the final theorem).

For more on this, there is an excellent physics textbook which covers this material (thereby taking neither of the attitudes described in the first paragraph). Chapter 10 covers the material I described above:

- Hatfield "The quantum field theory of point particles and
strings." Frontiers in Physics, Perseus, 1998.

- Glimm and Jaffe, "Quantum physics: a functional integral point of
view", Springer Verlag, 1987.

- David Tong's lecture notes on QFT.
- Peskin and Schroeder, "An introduction to quantum field theory", Avalon, 1995.
- Ryder, "Quantum field theory", Cambridge University Press, Second Edition 1996.