## Microcausality

Consider a scalar field in ${3+1}$ dimensions with the standard decomposition into creation and annihilation operators,

$\displaystyle \phi(x)=\int\frac{\mathrm{d}^3k}{\sqrt{2\omega(2\pi)^3}}\left( a_ke^{-ik\cdot x}+a_k^\dagger e^{ik\cdot x}\right)~. \ \ \ \ \ (1)$

Then the commutation relations between the creation/annihilation operators,

$\displaystyle \left[a_k,a_{k'}^\dagger\right]=2\omega(2\pi)^3\delta^3\left(\mathbf{k}-\mathbf{k}'\right)~, \;\;\; \left[a_k,a_{k'}\right]= \left[a_k^\dagger,a_{k'}^\dagger\right]= 0~, \ \ \ \ \ (2)$

where ${\omega^2=\mathbf{k}^2+m^2}$, imply exact commutativity of spacelike separated fields:

\displaystyle \begin{aligned} \left[\phi(x),\phi(y)\right]&=\int\frac{\mathrm{d}^3k\mathrm{d}^3k'}{(2\pi)^64\omega\omega'} \left( a_ka_{k'}^\dagger e^{-ikx+ik'y}+a_k^\dagger a_{k'} e^{ikx-ik'y} -a_{k'}a_{k}^\dagger e^{-ik'y+ikx}-a_{k'}^\dagger a_{k} e^{ik'y-ikx}\right)\\ &=\int\frac{\mathrm{d}^3k\mathrm{d}^3k'}{(2\pi)^64\omega\omega'}\left(\left[a_k,a_{k'}^\dagger\right]e^{-ikx+ik'y}-\left[a_{k'},a_k^\dagger\right]e^{ikx-ik'y}\right)\\ &=\int\frac{\mathrm{d}^3k}{(2\pi)^32\omega}\left( e^{-ik(x-y)}-e^{ik(x-y)}\right)=0~, \end{aligned} \ \ \ \ \ (3)

where the last step follows from the fact that since ${(x-y)^2<0}$, there exists a continuous Lorentz transformation that interchanges the order of events; in particular, this allows us to take ${(x-y)\rightarrow-(x-y)}$ in one of the two terms, whereupon the commutator vanishes. A similar argument can be made for commutators of the form ${\left[\phi^\dagger(x),\phi(y)\right]}$.

The vanishing of the commutator between spacelike separated observables is what is technically meant by microcausality in QFT. The prefix “micro” is to distinguish this quantum concept from macrocausality, which refers to our classical notion that no effects propagate faster than light. Since the latter is almost always taken for granted in such discussions, one often takes the unqualified “causality” to refer to the former, but we shall refrain from following this convention here for the sake of exactness.

Note that the correlation function ${\left<\phi^\dagger(x)\phi(y)\right>}$ does not vanish, even for ${(x-y)^2<0}$. This does not imply any superluminal violations of causality. Rather, this is simply the statement that the fields share some small correlation; i.e., that their past lightcones overlap. (Ignoring such issues as cosmological expansion, in flat space this will always be true if one goes back far enough). In other words, it is the commutator, and not the correlator, that provides the correct diagnostic of microcausality in field theories.

Ironically, the exact microcausality of fields (inherently nonlocal entities defined on the whole space) does not extend to particles (the local excitations thereof). Consider the number operator density ${\mathcal{N}(x)}$, defined via

$\displaystyle N_V\equiv\int_V\mathrm{d}^3\,ka_k^\dagger a_k=\int\mathrm{d}^3x\,\mathcal{N}(\mathbf{x},t)~, \ \ \ \ \ (4)$

where ${N_V}$ is the number operator that counts the number of particles in a state within a given spatial region ${V}$. It is then straightforward to show (see, e.g., Anthony Duncan’s The Conceptual Framework of Quantum Field Theory, 2012, sec. 6.5) that

$\displaystyle \left[\mathcal{N}(\mathbf{x},t),\mathcal{N}(\mathbf{y},t)\right]\neq0~, \ \ \ \ \ (5)$

and therefore measurements of the number of particles in two non-overlapping volumes, ${N_{V_1}}$, ${N_{V_2}}$, (where ${\mathbf{x}\in V_1}$ and ${\mathbf{y}\in V_2}$) will exhibit an interference that falls off exponentially with the minimum spacelike separation between ${V_1}$ and ${V_2}$. In other words, a one-particle state can be localized with respect to the number density operator only with an energy density that falls off exponentially at a rate determined by the Compton wavelength, i.e., ${\sim e^{-2m|x|}}$. This is why it is meaningless to speak of “particles” below their Compton wavelength. Incidentally, lest one worry, the above is a uniquely quantum phenomenon: one recovers ${\left[N_{V_1},N_{V_2}\right]=0}$ in the non-relativistic limit, ${c\rightarrow\infty}$. (In fact, the discussion is even more subtle: technically speaking, number operators localized within finite subregions do not exist).

As should now be apparent, the concepts of causality and locality are intimately linked, but technically distinct. In particular, the impossibility of exactly localizing particles reflects the fact that one cannot localize physical attributes (e.g., energy, momentum, charge) at a dimensionless spacetime point. Indeed, the point-like nature of elementary particles is merely a statement about their interaction (via a Hamiltonian which multiplies them at the same spacetime point), and is therefore epistemic. At best, one can localize particles as wavepackets. This is sufficient to ensure clustering: the factorization of the S-matrix for groups of separated particles such that, in the limit that the separation distance goes to ${\infty}$, the total scattering amplitude approaches the product of the independent scattering amplitudes of the particles in each region. Ultimately, this is what we observe in the laboratory, and as such one could consider this an operational definition of locality. But it is microcausality that ensures the analyticity of the S-matrix (via the singularity structure, which inhibits future interactions from influencing the past) on which the covariant formulation of clustering rests.

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