Commutation Relations, Locality, and Causality

1. Causality: From Particle Amplitudes to Field Observables

First, let's understand why the problem of causality in QFT is framed differently than in single-particle quantum mechanics.

As you may recall from your QM studies, the propagator for a free non-relativistic particle is non-zero for any spatial separation at any non-zero time. This seems to imply that a particle can travel faster than light.

In QFT, we must be more careful. The fundamental objects are fields, not particles. The relevant question is not whether a particle could travel between two points, but whether a measurement at one spacetime point $x$ can affect a measurement at another point $y$.

From quantum mechanics, we know that if two observables, represented by operators $\hat{A}$ and $\hat{B}$, commute (i.e., $[\hat{A},\hat{B}]=0$), they can be measured simultaneously and independently. A measurement of $A$ does not disturb the outcome of a measurement of $B$.

In a relativistic theory, this principle is elevated to a condition on spacetime events. If two spacetime points $x$ and $y$ are spacelike separated, meaning $(x-y{)}^2<0$, a signal cannot travel between them without exceeding the speed of light. Therefore, any two physical observables ${\hat{O}}_1(x)$ and ${\hat{O}}_2(y)$ at these points must commute:

This is known as the principle of microcausality.

Since all local observables (like the energy-momentum tensor ${\hat{T}}^{\mu \nu }$ or charge density ${\hat{j}}^{\mu }$) are constructed from products of the fundamental field operators $\hat{\varphi }(x)$ and their derivatives, this condition will be satisfied if the fields themselves commute at spacelike separations. Our task is therefore to calculate the commutator $[\hat{\varphi }(x),\hat{\varphi }(y)]$ and check if it vanishes when $(x-y{)}^2<0$.

2. Calculating the Field Commutator

Let's compute the commutator $[\hat{\varphi }(x),\hat{\varphi }(y)]$ directly. We start with the mode expansion of the real scalar field that we derived in previous lessons:

Substituting this into the commutator gives a double integral over momenta $\vec{p}$ and $\vec{q}$. Most of the terms vanish due to the commutation relations $[{\hat{a}}_{\vec{p}},{\hat{a}}_{\vec{q}}]=[{\hat{a}}_{\vec{p}}^{†},{\hat{a}}_{\vec{q}}^{†}]=0$. The only surviving terms involve $[{\hat{a}}_{\vec{p}},{\hat{a}}_{\vec{q}}^{†}]$ and $[{\hat{a}}_{\vec{p}}^{†},{\hat{a}}_{\vec{q}}]$.

The following resource provides a concise written derivation of this commutator.

After using the canonical commutation relation $[{\hat{a}}_{\vec{p}},{\hat{a}}_{\vec{q}}^{†}]=(2\pi {)}^3{\delta }^{(3)}(\vec{p}-\vec{q})$, the double integral collapses into a single one, yielding:

where $p^0={\omega }_{\vec{p}}$ in the exponents. A crucial observation is that the result is no longer an operator, but a complex number (a "c-number").

Let's define the function we encountered in the last lesson:

With this definition, the commutator can be written compactly as:

We now need to prove that this expression is zero for a spacelike interval $z=x-y$, i.e., for $z^2<0$.

3. Proof of Causality

The proof hinges on a key property: the function $D(z)$ is Lorentz invariant. Let's briefly see why. The integration measure $\frac{d^{3}p}{2{\omega }_{\vec{p}}}$ is a known Lorentz-invariant measure for integrating over the mass shell. Under a Lorentz transformation $p\to p^{\prime }=\Lambda p$ and $z\to z^{\prime }=\Lambda z$, the dot product in the exponent remains invariant: $p\cdot z=p^{\prime }\cdot z^{\prime }$. Since both the measure and the integrand are invariant, the whole integral $D(z)$ is a Lorentz scalar.

Now, we can prove that $D(x-y)=D(y-x)$ for spacelike separations.

Argument from Lorentz Invariance:

1. For any spacelike vector $z=x-y$, there exists a continuous Lorentz transformation $\Lambda$ such that $\Lambda z=-z$. (For instance, you can boost to a frame where $z^0=0$, perform a 180° rotation on the spatial part, and then boost back).
2. Since $D(z)$ is a Lorentz scalar, it must be unchanged by this transformation: $D(z)=D(\Lambda z)$.
3. Combining these facts: $D(z)=D(\Lambda z)=D(-z)$.
4. Applying this to our interval: $D(x-y)=D(-(x-y))=D(y-x)$.

Therefore, for any spacelike separation $(x-y{)}^2<0$:

This completes the proof. Our theory is causal. The equal-time commutation relations we postulated, which explicitly picked a preferred time slice, have miraculously combined with relativistic dynamics to produce a fully causal, Lorentz-invariant theory.

The following video provides a complete, step-by-step walkthrough of this entire argument.

4. Physical Interpretation: Cancellation and Antiparticles

So, how does QFT achieve this cancellation when single-particle QM fails? The commutator $[\hat{\varphi }(x),\hat{\varphi }(y)]=\hat{\varphi }(x)\hat{\varphi }(y)-\hat{\varphi }(y)\hat{\varphi }(x)$ can be interpreted as the difference between two processes:

1. $\hat{\varphi }(x)\hat{\varphi }(y)$: A particle is created at $y$ and annihilated at $x$. The amplitude for this is related to $D(x-y)$.
2. $\hat{\varphi }(y)\hat{\varphi }(x)$: A particle is created at $x$ and annihilated at $y$. The amplitude for this is related to $D(y-x)$.

For a spacelike separation, both processes are unphysical as they would require faster-than-light travel. However, in QFT, both contribute. Our proof shows that for spacelike separations, the amplitudes for these two FTL processes are exactly equal and, due to the minus sign in the commutator, they perfectly cancel. The net result is that no observable effect can be transmitted.

This cancellation is a profound consequence of the structure of QFT. In a theory with charged particles (like a complex scalar field), the cancellation occurs between a particle traveling $y\to x$ and an antiparticle traveling $x\to y$. The existence of antiparticles is intrinsically woven into the causal structure of the theory.

Conclusion

In this lesson, we have established one of the most important consistency checks of our theory. By starting with the canonical equal-time commutation relations and the dynamics of the Klein-Gordon field, we have demonstrated that the theory respects relativistic causality.

Key Takeaways:

• Microcausality: The requirement for a relativistic quantum theory to be causal is that operators for any physical observables at spacelike separated points must commute.
• Field Commutator: We calculated the commutator of the free scalar field at two arbitrary spacetime points $x$ and $y$, finding $[\hat{\varphi }(x),\hat{\varphi }(y)]=D(x-y)-D(y-x)$, which is a c-number.
• Proof of Causality: Using the Lorentz invariance of the function $D(z)$, we proved that for any spacelike interval $z=x-y$, $D(z)=D(-z)$. This means the commutator vanishes, $[\hat{\varphi }(x),\hat{\varphi }(y)]=0$, for spacelike separations.
• Physical Picture: Causality is preserved due to a perfect cancellation between the amplitude for a particle to propagate from $y$ to $x$ and the amplitude for it to propagate from $x$ to $y$. This mechanism is deeply connected to the existence of antiparticles.

We have now completed our development of the free scalar field theory. We have a relativistic, causal, quantum theory of non-interacting particles. The foundation is solid.

Preview of the Next Lesson:

With the free theory firmly in hand, we are finally ready to explore the more realistic and exciting world of interacting fields. In the next lesson, which begins a new module, we will introduce an interaction term to our Lagrangian ($\lambda {\varphi }^4$ theory) and develop the essential tools—the interaction picture and the Dyson series—to begin calculating physical processes like particle scattering.