3.2 Poincaré symmetry
Our canonical quantization procedure in Section 3.1 heavily relies on choosing a splitting of spacetime \(x = (t,\mathbf {x})\) into time and space. These two concepts are then used very differently in the construction, hence it is a priori unclear if the free quantum Klein-Gordon field inherits the Poincaré symmetries of its classical analog from Example 2.4. Note that there are already some indications that the free quantum Klein-Gordon field is compatible with the laws of special relativity, for instance it satisfies the Poincaré invariant Klein-Gordon equation (3.21) and relativistic causality, but we still have to make more precise in which sense it admits Poincaré symmetries.
Recall that symmetries in quantum theory are realized by unitary operators \(U\) that act on other operators \(A\) via the adjoint action \(A\mapsto U A U^{\dagger }\). Given a continuous family of symmetries \(U_\beta = e^{\ii \,\beta _a X^a}\) that is obtained by exponentiating Hermitian operators \(X^1,\dots , X^N\) (called the generators) with parameters \(\beta = (\beta _1,\dots ,\beta _N) \in \bbR ^N\), we can Taylor expand the adjoint action
\(\seteqnumber{0}{3.}{27}\)\begin{flalign} U_\beta \,A\,U_\beta ^\dagger = e^{\ii \,\beta _a X^a}\,A \,e^{-\ii \,\beta _a X^a} = A + \ii \,\beta _a \,\big [X^a,A\big ] + \mathcal {O}(\beta ^2) \end{flalign} and find that the infinitesimal transformation of \(A\) is given by the commutator
\(\seteqnumber{0}{3.}{28}\)\begin{flalign} \delta _\beta A = \ii \,\beta _a \big [X^a,A\big ]\quad . \end{flalign} When discussing symmetries in QFT, we shall focus mainly on such infinitesimal transformations. Many noninfinitesimal symmetries, but crucially not all, then follow by exponentiating infinitesimal transformations. For Poincaré transformations, one can show that every proper and orthochronous transformation (recall Section 1.4) can be obtained by exponentiating generators, but more general Poincaré transformations such as time-reversal \((t,\mathbf {x})\mapsto (-t,\mathbf {x})\) do not arise in this way.
You might now ask: From where do we get symmetry generators? In classical field theory, Noether’s Theorem 2.6 provides an answer: A continuous family of symmetries gives rise to a conserved Noether current \(J^\mu \) and hence to a conserved charge \(Q = \int _{\bbR ^{d-1}}J^0(x)\,\dd \mathbf {x}\). Passing to the Hamiltonian formalism for classical field theory, one can show that \(Q\) defines the corresponding (classical) symmetry generator that acts on the field and its canonical momentum via the Poisson bracket \(\{Q,-\}\). This suggests that we should try to obtain symmetry generators in QFT by quantizing their associated classical Noether charges.
Let us illustrate how this works for the case of Poincaré transformations. The associated (classical) Noether charges for the free Klein-Gordon field have been computed in Examples 2.9 and 2.10. Let us start with the relativistic momentum \(P^\mu \) from (2.37), which is the conserved charge associated with spacetime translations. When written in terms of the field \(\Phi \) and its conjugate momentum \(\Pi \), the components read as
\(\seteqnumber{0}{3.}{29}\)\begin{flalign} P^0 = \int _{\bbR ^{d-1}} \frac {1}{2}\Big (\Pi ^2 + (\nabla \Phi )^2 + m^2\,\Phi ^2\Big )\,\dd \mathbf {x}\quad ,\qquad P^i = - \int _{\bbR ^{d-1}} \Pi \,\partial ^i\Phi \,\dd \mathbf {x}\quad . \end{flalign} Note that \(P^0 = H\) agrees with the Hamiltonian, whose normal ordered quantization was worked out in Section 3.1 and is given by
\(\seteqnumber{0}{3.}{30}\)\begin{flalign} \noor {P^0} = \noor {H} = \int _{\bbR ^{d-1}} \omega _{\mathbf {k}}~a^\dagger (\mathbf {k})\,a(\mathbf {k})~\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\quad . \end{flalign} Using the same Fourier transformation techniques as in Section 3.1, we can also determine a quantization of the spatial components \(P^i\). Let me illustrate briefly the relevant steps. Inserting the Fourier expansions (3.3) and (3.8) into the formula for \(P^i\) yields
\(\seteqnumber{0}{3.}{31}\)\begin{flalign} \nn P^i &= \int _{\bbR ^{d-1}} \widetilde {\Pi }(\mathbf {k})\,\ii \, k^i\, \widetilde {\Phi }(-\mathbf {k})\,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\\ \nn &=\int _{\bbR ^{d-1}} \frac {k^i}{2}~\Big (a(\mathbf {k}) - a^\dagger (-\mathbf {k})\Big ) ~\Big (a(-\mathbf {k}) + a^\dagger (\mathbf {k})\Big ) \,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\\ \nn &=\int _{\bbR ^{d-1}} \frac {k^i}{2}~ \bigg (a(\mathbf {k})\,a(-\mathbf {k}) - a^\dagger (-\mathbf {k}) \,a(-\mathbf {k}) + a(\mathbf {k})\,a^\dagger (\mathbf {k}) - a^\dagger (-\mathbf {k}) \,a^\dagger (\mathbf {k})\bigg ) \,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\\ &=\int _{\bbR ^{d-1}} \frac {k^i}{2}~ \Big ( a^\dagger (\mathbf {k}) \,a(\mathbf {k}) + a(\mathbf {k})\,a^\dagger (\mathbf {k})\Big )\,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\quad , \end{flalign} where in the last step we have used that \(k^i \,a(\mathbf {k})\,a(-\mathbf {k})\) and \(k^i\,a^\dagger (-\mathbf {k}) \,a^\dagger (\mathbf {k})\) are odd functions in \(\mathbf {k}\), hence the corresponding integrals vanish. Applying normal ordering (see Definition 3.1), we obtain
\(\seteqnumber{0}{3.}{32}\)\begin{flalign} \noor {P^i} = \int _{\bbR ^{d-1}} k^i~ a^\dagger (\mathbf {k}) \,a(\mathbf {k})\,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\quad , \end{flalign} hence the normal ordered quantization of the relativistic momentum is simply given by
with the relativistic Fourier momentum \(k = (\omega _{\mathbf {k}},\mathbf {k})\) introduced in (3.18). With a similar but more involved calculation, one obtains the normal ordered quantization of the relativistic angular momentum \(L^{\rho \nu }\) from (2.44). Its components read explicitly as
\begin{equation} \noor {L^{i0}} = - \noor {L^{0i}} = \int _{\bbR ^{d-1}} a^\dagger (\mathbf {k})\, \bigg (\ii \, \omega _{\mathbf {k}} \,\frac {\partial }{\partial k_i} +\frac {\ii }{2} \, \frac {\partial \omega _\mathbf {k}}{\partial k_i}\bigg ) \, a(\mathbf {k})~\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\quad , \end{equation}
and
\begin{equation} \noor {L^{ij}} = \int _{\bbR ^{d-1}} a^\dagger (\mathbf {k}) \,\bigg (\ii \, k^j\,\frac {\partial }{\partial k_i} - \ii \, k^i\,\frac {\partial }{\partial k_j}\bigg )\,a(\mathbf {k})~\frac {\dd \mathbf {k}}{(2\pi )^{d-1}} \quad . \end{equation}
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Remark 3.3. One can check that these operators satisfy the Poincaré Lie algebra relations
\(\seteqnumber{0}{3.}{35}\)\begin{flalign} \nn \big [\noor {P^\mu },\noor {P^\nu }\big ] &= 0\quad ,\qquad \big [\noor {L^{\mu \nu }}, \noor {P^\rho }\big ] = \ii \,\big (\eta ^{\mu \rho }\,\noor {P^\nu } - \eta ^{\nu \rho }\,\noor {P^\mu }\big )\quad ,\\[5pt] \big [\noor {L^{\mu \nu }},\noor {L^{\rho \sigma }}\big ] &= \ii \,\big (\eta ^{\nu \rho }\,\noor {L^{\mu \sigma }} - \eta ^{\mu \rho }\,\noor {L^{\nu \sigma }} - \eta ^{\nu \sigma }\,\noor {L^{\mu \rho }} +\eta ^{\mu \sigma }\,\noor {L^{\nu \rho }}\big )\quad , \end{flalign} that characterize infinitesimal Poincaré transformations. We shall introduce Lie algebras more formally later in this module, so for the moment this is just a side-remark. More details about the Lorentz and Poincaré Lie algebras can be found e.g. in the textbook by Maggiore (Chapter 2).
With these preparations, we can now investigate how the Heisenberg picture field operator (3.20) transforms under (infinitesimal) Poincaré transformations and thereby answer affirmatively the question whether the quantum Klein-Gordon field inherits the Poincaré symmetries from its classical counterpart. For the transformations induced by the generators \(\noor {P^\mu }\), we find
\(\seteqnumber{0}{3.}{36}\)\begin{flalign} \nn \delta _b\Phi (x) &= \ii \,b_\mu \,\big [\noor {P^\mu },\Phi (x)\big ]\\ \nn &=\ii \,b_\mu \int _{\bbR ^{d-1}}\int _{\bbR ^{d-1}} \frac {k^\mu }{\sqrt {2\,\omega _{\mathbf {q}}}}\, \Big [a^\dagger (\mathbf {k}) \,a(\mathbf {k}), a(\mathbf {q})\,e^{\ii \,q\,x} + a^\dagger (\mathbf {q})\,e^{-\ii \,q\,x}\Big ]\,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\,\frac {\dd \mathbf {q}}{(2\pi )^{d-1}}\\ \nn &= \ii \,b_\mu \int _{\bbR ^{d-1}}\frac {k^\mu }{\sqrt {2\,\omega _{\mathbf {k}}}} \,\Big (-a(\mathbf {k})\,e^{\ii \,k\,x} + a^\dagger (\mathbf {k})\,e^{-\ii \,k\,x}\Big )\, \frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\\ &= -b_\mu \,\partial ^\mu \Phi (x)\quad , \end{flalign} where in the third equality we used the commutation relations (3.10) for the annihilation and creation operators. We recognize this as the infinitesimal spacetime translations from Example 2.9. Hence, the operators \(\noor {P^\mu }\) are the generators for translations. With a more involved calculation, one finds for the transformation induced by the generators \(\noor {L^{\nu \rho }}\) that
\(\seteqnumber{0}{3.}{37}\)\begin{flalign} \delta _\omega \Phi (x) = \frac {\ii }{2}\,\omega _{\rho \nu } \,\big [\noor {L^{\nu \rho }},\Phi (x)\big ] = -\omega ^{\mu }_{~~\nu } \, x^\nu \,\partial _\mu \Phi (x)\quad , \end{flalign} which we recognize as the infinitesimal Lorentz transformations from Example 2.10. Hence, the operators \(\noor {L^{\nu \rho }}\) are the generators for Lorentz transformations.
Summing up, we have found operators \(\noor {P^{\mu }}\) and \(\noor {L^{\nu \rho }}\) that play the role of generators for infinitesimal Poincaré transformations. These generators implement Poincaré symmetries at the level of the quantum field \(\Phi (x)\) in a way that is characteristic for scalar fields.