43.5 The scalar field propagator

43.5 The scalar field propagator

As for any quantum field theory, a fundamental quantity to calculate is the propagator, which for a free quantum field theory actually can be used to calculate all amplitudes between multiparticle states. For the free relativistic scalar field theory, we have

Definition (Propagator, Klein-Gordon theory)

The propagator for the relativistic scalar field theory is the amplitude

$$U(t_2,{\mathbf{x}}_2,t_1,{\mathbf{x}}_1)=⟨0|\hat{\Phi }(t_2,{\mathbf{x}}_2)\hat{\Phi }(t_1,{\mathbf{x}}_1)|0⟩$$

By translation invariance, the propagator will only depend on $t_2-t_1$ and ${\mathbf{x}}_2-{\mathbf{x}}_1$ , so we can just evaluate the case $(t_1,{\mathbf{x}}_1)=(0,0),(t_2,{\mathbf{x}}_2)=(t,\mathbf{x})$ , using the formula 43.23 for the time-dependent quantum field to get

$$\begin{array}{rlr}& & U(t,\mathbf{x},0,0)=\frac{1}{(2\pi {)}^3}{\int }_{{\mathbf{R}}^3\times {\mathbf{R}}^3}⟨0|(a(\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+a^{†}(\mathbf{p})e^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\\ & & (a({\mathbf{p}}^{\prime })+a^{†}({\mathbf{p}}^{\prime }))|0⟩\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}\frac{d^3{\mathbf{p}}^{\prime }}{\sqrt{2{\omega }_{{\mathbf{p}}^{\prime }}}}\\ & & =\frac{1}{(2\pi {)}^3}{\int }_{{\mathbf{R}}^3\times {\mathbf{R}}^3}{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}\frac{d^3{\mathbf{p}}^{\prime }}{\sqrt{2{\omega }_{{\mathbf{p}}^{\prime }}}}\\ & & =\frac{1}{(2\pi {)}^3}{\int }_{{\mathbf{R}}^3}{e}^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}\frac{d^3\mathbf{p}}{2{\omega }_{\mathbf{p}}}\\ & & =\frac{1}{(2\pi {)}^3}{\int }_{{\mathbf{R}}^4}\theta (p_0)\delta (p^2+m^2)e^{-i{p}_{0}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}{d}^3\mathbf{p}d{p}_0\end{array}$$

The last line shows that this distribution is the Minkowski space Fourier transform of the delta-function distribution on the positive energy hyperboloid

$$p^2=-p_0^2+|\mathbf{p}{|}^2=-m^2$$

As in the non-relativistic case (see section 12.5), this is a distribution that can be defined as a boundary value of an analytic function of complex time. The integral can be evaluated in terms of Bessel functions, and its properties are discussed in all standard quantum field theory textbooks. These include:

• For $\mathbf{x}=0$ , the amplitude is oscillatory in time, a superposition of terms with positive frequency.

• For $t=0$ , the amplitude falls of exponentially as $e^{-m|\mathbf{x}|}$

The resolution of the potential causality problem caused by the non-zero amplitude at space-like separations between $\left(t_1,{\mathbf{x}}_1\right)$ and $\left(t_2,{\mathbf{x}}_2\right):(e.g.$ , for $t_1=$ $t_2)$ is that the condition really needed on observable operators $\mathcal{O}$ localized at points in space time is that

$$[\mathcal{O}(t_2,{\mathbf{x}}_2),\mathcal{O}(t_1,{\mathbf{x}}_1)]=0$$

for $\left(t_1,{\mathbf{x}}_1\right)$ and $\left(t_2,{\mathbf{x}}_2\right)$ space-like separated (this condition is known as “microcausali ${\mathrm{ty}}^{†})$ . This will ensure that measurement of the observable $\mathcal{O}$ at a point will not afect its measurement at a space-like separated point, avoiding potential conflicts with causality. For the field operator $\Phi (t,\mathbf{x})$ one can calculate the commutator by a similar calculation to the one above for $U(t,\mathbf{x},0,0)$ , with result

$$\begin{array}{rl}[\hat{\Phi }(t,\mathbf{x}),\hat{\Phi }(0,0)]& =\frac{1}{(2\pi {)}^3}{\int }_{{\mathbf{R}}^4}\delta (p^2+m^2)(\theta (p_0)e^{ip\cdot x}-\theta (-p_0)e^{-ip\cdot x})d^3\mathbf{p}d{p}_0\\ & =U(t,\mathbf{x},0,0)-U(-t,-\mathbf{x},0,0)\end{array}$$

This will be zero for space-like $(t,\mathbf{x})$ , something one can see by noting that the result is Lorentz invariant, is equal to 0 at $t=0$ by the canonical commutation relation

$$[\hat{\Phi }({\mathbf{x}}_1),\hat{\Phi }({\mathbf{x}}_2)]=0$$

and any two space-like vectors are related by a Lorentz transformation. Note that the vanishing of this commutator for space-like separations is achieved by cancellation of propagation amplitudes for a particle and antiparticle, showing that the relativistic choice of complex structure for quantization is needed to ensure causality.

One can also study the propagator using Green’s function methods as in the single-particle case of section 12.7, now with

$$D=-\frac{{\partial }^2}{\partial t^2}+{\nabla }^2-m^2$$ $$\hat{D}(p_0,\mathbf{p})=p_0^2-(|\mathbf{p}{|}^2+m^2)$$

and

$$\hat{G}(p_0,\mathbf{p})=\frac{1}{p_0^2-(|\mathbf{p}{|}^2+m^2)}=\frac{1}{(p_0-{\omega }_{\mathbf{p}})(p_0+{\omega }_{\mathbf{p}})}$$

In the non-relativistic case the Green’s function $\hat{G}$ only had one pole, at $p_0=$ $|\mathbf{p}{|}^2/2m$ . Two possible choices of how to extend integration over $p_0$ into the complex plane avoiding the pole gave either a retarded Green’s function and propagation from past to future, or an advanced Green’s function and propagation from future to past. In the Klein-Gordon case there are now two poles, at $p_0=\pm {\omega }_{\mathbf{p}}$ . Micro-causality requires that the pole at positive energy be treated as a retarded Green’s function, with propagation of positive energy particles from past to future, while the one at negative energy must be treated as an advanced Green’s function with propagation of negative energy particles from future to past.

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