3.3 Hilbert space and particle interpretation

To obtain a representation of the Heisenberg picture field operators (3.20) on a Hilbert space \(\HH \), we can proceed in analogy to the quantum harmonic oscillator and build a Hilbert space using the annihilation and creation operators \(a(\mathbf {k})\) and \(a^{\dagger }(\mathbf {k})\). Such type of Hilbert space is often called Fock space in the literature.

We start by introducing a ground state \(\ket {0}\in \HH \), called the vacuum state in the context of QFT, which is characterized by the property that it is annihilated by all annihilation operators, i.e.

for all \(\mathbf {k}\in \bbR ^{d-1}\), and the usual normalization condition \(\braket {0}{0}=1\) for states. Note that the vacuum state is an eigenstate of the normal ordered relativistic momentum (3.34) and relativistic angular momentum (3.35) operators with zero eigenvalues,

\begin{flalign} \noor {P^\mu }\ket {0} =0\quad ,\qquad \noor {L^{\nu \rho }}\ket {0} = 0\quad . \end{flalign} In words, the vacuum has zero relativistic momentum and angular momentum, which is a feature resulting from our normal ordering prescription.

Further states are then obtained by acting with the creation operators \(a^\dagger (\mathbf {k})\) on the vacuum state \(\ket {0}\). Acting with \(n\) creation operators results in the state

which we label by a family of relativistic Fourier momenta \(k_a = (\omega _{\mathbf {k}_a},\mathbf {k}_a) \in \bbR ^{d}\), for \(a=1,\dots ,n\), that satisfy the on-shell condition \(k_a^2 = -m^2\) from (3.19). The factors \(\sqrt {2\,\omega _{\mathbf {k}_a}}\) are a convenient choice of normalization. (As we explain below, they are useful to make manifest the Poincaré symmetry of our QFT.) The physical interpretation of (3.41) is that of an \(n\)-particle state. From the fact that any two creation operators commute, i.e. \(a^\dagger (\mathbf {q})\, a^{\dagger }(\mathbf {k}) = a^\dagger (\mathbf {k})\,a^{\dagger }(\mathbf {q})\), we find that \(\ket {k_1,\dots ,k_n}\) is symmetric under the exchange of any pair of \(k\)’s, hence the particles are bosons.

Can one say more about these particles? For instance, what are their masses and spins? To answer these questions, it suffices to analyze a single particle state \(\ket {k} = \sqrt {2\,\omega _{\mathbf {k}}}\,a^{\dagger }(\mathbf {k})\ket {0}\). One easily checks that this is an eigenstate of the normal ordered relativistic momentum operator (3.34) with eigenvalue

\begin{flalign} \noor {P^\mu } \ket {k} = k^\mu \,\ket {k} \end{flalign} returning the relativistic Fourier momentum \(k\). Hence, one interprets \(\ket {k}\) as a state that describes a single quantum particle with fixed relativistic momentum \(k=(\omega _{\mathbf {k}},\mathbf {k})\). (Note that the energy of the particle is positive.) The on-shell condition \(k^2 = -m^2\) then tells us that the parameter \(m\geq 0\) from the Klein-Gordon action defines the mass of the particle. To say something about the spin of the particle, we act with the relativistic angular momentum operator (3.35) on \(\ket {k}\) and observe that the spatial components \(\noor {L^{ij}}\ket {k}\) vanish when we take (informally) the limit \(\mathbf {k}\to \mathbf {0}\). Note that this limit models the scenario in which the particle is not moving, hence it detects the intrinsic angular momentum, a.k.a. the spin. Since the latter vanishes, the particles associated with the free Klein-Gordon quantum field have spin \(0\).

We shall now investigate the inner product on the Hilbert space. Since the states (3.41) are defined by creation operators, the inner product can be determined from the commutation relations (3.10) and the definition of the vacuum state (3.39). In full generality, one finds that

where \(S_n\) denotes the permutation group on \(n\) letters. In particular, any two states with a different number of particles are orthogonal to each other. In case you find this general formula hard to read, it helps to look at some simple examples

\begin{flalign} \nn \braket {q}{k} &= (2\pi )^{d-1}\,2\,\omega _{\mathbf {k}}\,\delta (\mathbf {k}-\mathbf {q})\quad ,\\ \braket {q_1,q_2}{k_1,k_2} &= (2\pi )^{2(d-1)}\,2\,\omega _{\mathbf {k}_1}\,2\,\omega _{\mathbf {k}_2} \,\Big (\delta (\mathbf {k}_1-\mathbf {q}_1)\,\delta (\mathbf {k}_2-\mathbf {q}_2) + \delta (\mathbf {k}_1-\mathbf {q}_2)\,\delta (\mathbf {k}_2 -\mathbf {q}_1)\Big )\quad . \end{flalign}

Working with the inner products in (3.43) requires some care because of the appearance of Dirac delta functions, which can cause divergences such as e.g. in \(\braket {k}{k} =(2\pi )^{d-1}\,2\,\omega _{\mathbf {0}}\,\delta (\mathbf {0})\). The way how one can make these inner products mathematically precise is by considering so-called wave packet states instead of the states \(\ket {k}\) that, as we have seen, are too singular. Let me illustrate the key idea for \(1\)-particle states. (The \(n\)-particle wave packet states are defined similarly.) Consider the following state

\begin{flalign} \ket {\psi } := \int _{\bbR ^{d-1}} \,\widetilde {\psi }(\mathbf {k})\,\ket {k}~\frac {\dd \mathbf {k}}{2\,\omega _{\mathbf {k}}\,(2\pi )^{d-1}} \end{flalign} that is given by integrating \(\ket {k}\) over a wave packet \(\widetilde {\psi }(\mathbf {k})\) in the Fourier space. (Think, for instance, of a Gaussian that is peaked around your favorite momentum \(\mathbf {k}\).) Physically, the state \(\ket {\psi }\) describes a quantum particle that does not have a sharp relativistic momentum, but rather is in a superposition of different momenta. Note that the integration measure has been chosen such that it is invariant under proper and orthochronous Poincaré transformations, which follows from the same argument used in (3.26). The inner product between two \(1\)-particle wave packet states is then given by

\begin{flalign} \nn \braket {\chi }{\psi } &= \int _{\bbR ^{d-1}} \int _{\bbR ^{d-1}} \widetilde {\chi }(\mathbf {q})^\ast \, \widetilde {\psi }(\mathbf {k})\, \braket {\mathbf {q}}{\mathbf {k}}\,\frac {1}{2\,\omega _{\mathbf {q}}\,2\,\omega _{\mathbf {k}}}~\frac {\dd \mathbf {q}}{(2\pi )^{d-1}}~\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\\ &= \int _{\bbR ^{d-1}} \widetilde {\chi }(\mathbf {k})^\ast \, \widetilde {\psi }(\mathbf {k})\, \frac {1}{2\,\omega _{\mathbf {k}}}~\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\quad ,\label {KGFockinnerproductWP1} \end{flalign} where we recognize again the Poincaré invariant measure. Note that this expression is well-defined, provided that the two wave packets are square-integrable with respect to the measure. We can now finally justify our choice of normalization in (3.41): The factors of \(\sqrt {2\,\omega _{\mathbf {k}_a}}\) are needed in order to make the wave packet inner product (3.46) manifestly Poincaré invariant. To conclude, let us note that the analog of the inner product (3.43) for multiparticle wave packet states reads as

\begin{flalign} \label {KGFockinnerproductWP} \braket {\chi _1,\dots ,\chi _m}{\psi _1,\dots ,\psi _n} = \begin{cases} \sum \limits _{\sigma \in S_n} \prod \limits _{a=1}^n \braket {\chi _{\sigma (a)}}{\psi _{a}}&,~\text {for } n=m\quad ,\\ 0 &,~\text {for } n\neq m\quad . \end {cases} \end{flalign} Again, this expression is well-defined, provided that all wave packets are square-integrable with respect to the measure.