2 Path Integral Field Quantization
Until now you will mainly have seen quantum mechanics in terms of non-commuting operators and states. An alternative, equivalent approach, based on path integrals, was proposed by Dirac and formulated by Feynman. Dirac’s observation was that the more familiar approach to QM puts space (an operator) and time (a variable) on different footing, so is hard to reconcile with relativity. Path integrals instead rely on the action \(S\), which is a Lorentz scalar. Hence, it is the more natural approach to quantum mechanics in relativistic settings.
We will first quickly see path integral quantum mechanics, before generalising to quantum fields. This provides an alternative route to building quantum fields, equivalent to the canonical quantization method in Chapter 1.
2.1 Path Integral Quantum Mechanics
2.1.1 Motivation: Young’s Slits
Consider a particle departing from position \(x_{0}\) at time \(t_{0}\), passing through one of two slits located at positions \(A\) and \(B\) at time \(t\), and continuing onto a screen. The amplitude to find the particle at position \(x\) on the screen at time \(T\), which we can denote \(\psi\left(x,T\right)\), is the sum of the amplitudes to take each of the two paths. That is:
\[\begin{aligned} \psi\left(x,T\right) & =\text{Amp}\left(x,T|A,t\right)\text{Amp}\left(A,t|x_{0},t_{0}\right)\nonumber \\ & +\text{Amp}\left(x,T|B,t\right)\text{Amp}\left(B,t|x_{0},t_{0}\right) \end{aligned} \tag{2.1}\]
where \(\text{Amp}\left(x,T|A,t\right)\) denotes the amplitude to find the particle at position \(x\) at time \(T\) given that it was certainly at position \(A\) at time \(t\). Eq 2.1 shows that amplitudes in quantum mechanics play the role that probabilities play in classical systems. We then find quantum probabilities using
the Born rule:
\[\text{Probability}=\left|\text{Amplitude}\right|^{2}.\]
What if there are not two slits to pass through at time \(t\), but an infinite number of slits with nothing between them? In that case, the sum must be replaced by an integral:
\[\psi\left(x,T\right)=\int\text{d}y\text{Amp}\left(x,T|y,t\right)\text{Amp}\left(y,t|x_{0},t_{0}\right).\]
But this should be true at all times between \(t_{0}\) and \(T\). Imagine ‘slicing’ time into \(N\) discrete segments. Then the amplitude to find a particle at position \(x\) at time \(T\), given it started at \(x_{0}\) at time \(t_{0}\), must be:
\[\psi\left(x,T\right)=\int\text{d}y_{N-1}\int\text{d}y_{N-2}\ldots\int\text{d}y_{1}\text{Amp}\left(x,T|y_{N-1},t_{N-1}\right)\text{Amp}\left(y_{N-1},t_{N-1}|y_{N-2},t_{N-2}\right)\ldots\text{Amp}\left(y_{1},t_{1}|x_{0},t_{0}\right). \tag{2.2}\]
Eq 2.2 encodes the idea that a quantum particle can be thought of as taking every possible path between two points. This is the basic idea behind path integral quantum mechanics. The object
\[\int\mathscr{D}\boldsymbol{y}\triangleq\prod^{N}_{n=1}\int\text{d}y_{n}\]
which, in the limit \(N\rightarrow\infty\) contains an infinite number of independent integrals, is called a ‘functional integral’ or ‘path integral’ (not to be confused with the 1D ‘line integrals’ seen in first year).
The posulates of path integral QM (Feynman, 1948)
(i) The Born rule: probability \(=\left|\psi\right|^{2}\). (ii) \(\psi\) is given by a weighted sum over all possible classical trajectories \(\boldsymbol{x}\left(t\right)\). (iii) in this sum, each path is weighted by \(\exp\left(iS\left[\boldsymbol{x}\right]\right)\).
2.1.2 The Propagator approach to QM
In Section @subsec-Time-dependence we met the time evolution operator \(\hat{U}\) and saw its effect on operators in the Heisenberg picture. Similarly it evolves states from one time to another:
\[|\psi\left(t'\right)\rangle=\hat{U}\left(t'-t\right)|\psi\left(t\right)\rangle\]
or, projecting into the position basis:
\[\langle x'|\psi\left(t'\right)\rangle=\langle x'|\hat{U}\left(t'-t\right)|\psi\left(t\right)\rangle.\]
Using a
resolution of the identity (position basis):
\[\hat{\mathbb{I}}=\int\text{d}x|x\rangle\langle x|\]
we can rewrite this as \[\begin{aligned} \langle x'|\psi\left(t'\right)\rangle & =\int\text{d}x\langle x'|\hat{U}\left(t'-t\right)|x\rangle\langle x|\psi\left(t\right)\rangle \end{aligned}\]
or, in wavefunction notation,
\[\psi\left(x',t'\right)=\int\text{d}xK\left(x',t';x,t\right)\psi\left(x,t\right) \tag{2.3}\]
where we have defined the propagator:
\[K\left(x',t';x,t\right)\triangleq\langle x'|\hat{U}\left(t'-t\right)|x\rangle.\]
That is, given a wavefunction at one point in space and time, the propagator ‘propagates’ the solution to any other point in space and time according to Eq 2.3.
2.1.3 The Propagator: free space
In free space we can calculate the propagator exactly.
\[K\left(x,t;x_{0},t_{0}\right)=\langle x|\exp\left(-i\hat{H}\left(t-t_{0}\right)\right)|x_{0}\rangle.\]
In free space the potential \(V=0\) and so
\[\hat{H}=\frac{\hat{p}^{2}}{2m}.\]
Therefore
\[K\left(x,t;x_{0},t_{0}\right)=\langle x|\exp\left(-i\hat{p}^{2}\left(t-t_{0}\right)/\left(2m\right)\right)|x_{0}\rangle.\]
Inserting a resolution of the identity (momentum basis):
\[\hat{\mathbb{I}}=\int^{\infty}_{-\infty}\text{d}p|p\rangle\langle p|\] gives \[K\left(x,t;x_{0},t_{0}\right)=\int^{\infty}_{-\infty}\text{d}p\langle x|\exp\left(-i\hat{p}^{2}\left(t-t_{0}\right)/\left(2m\right)\right)|p\rangle\langle p|x_{0}\rangle.\]
By definition, \(|p\rangle\) is an eigenstate of \(\hat{p}\) with eigenvalue \(p\):
\[\hat{p}|p\rangle=p|p\rangle\]
and so
\[\begin{aligned} K\left(x,t;x_{0},t_{0}\right) & =\int^{\infty}_{-\infty}\text{d}p\langle x|\exp\left(-ip^{2}\left(t-t_{0}\right)/\left(2m\right)\right)|p\rangle\langle p|x_{0}\rangle\\ & =\int^{\infty}_{-\infty}\text{d}p\exp\left(-ip^{2}\left(t-t_{0}\right)/\left(2m\right)\right)\langle x|p\rangle\langle p|x_{0}\rangle \end{aligned}\]
(where the momentum operator has become simply a variable). Recalling that momentum eigenstates are plane waves:
\[\langle x|p\rangle=\frac{1}{\sqrt{2\pi}}\exp\left(ipx\right)\]
We find
\[\begin{aligned} K\left(x,t;x_{0},t_{0}\right) & =\frac{1}{2\pi}\int^{\infty}_{-\infty}\text{d}p\exp\left(i\left(p\left(x-x_{0}\right)-p^{2}\left(t-t_{0}\right)/\left(2m\right)\right)\right). \end{aligned}\]
This is a Gaussian integral. It can be solved to give the propagator in free space:
\[K\left(x,t;x_{0},t_{0}\right)=\sqrt{\frac{m}{2\pi i}}\frac{1}{\sqrt{t-t_{0}}}\exp\left(\frac{im\left(x-x_{0}\right)^{2}}{2\left(t-t_{0}\right)}\right).\]
In free space, a particle can be thought as either taking a straight line between two points, or taking arbitrarily jagged paths between the same points. To see this, note that (for a free propagator only):
\[K\left(x_{3},t_{3};x_{1},t_{1}\right)\equiv\int\text{d}x_{2}K\left(x_{3},t_{3};x_{2},t_{2}\right)K\left(x_{2},t_{2};x_{1},t_{1}\right).\]
Proof:
\[\int\text{d}x_{2}K\left(x_{3},t_{3};x_{2},t_{2}\right)K\left(x_{2},t_{2};x_{1},t_{1}\right)=\int\text{d}x_{2}\langle x_{3}|\hat{U}\left(t_{3}-t_{2}\right)|x_{2}\rangle\langle x_{2}|\hat{U}\left(t_{2}-t_{1}\right)|x_{1}\rangle\]
remove the identity:
\[\begin{aligned} \int\text{d}x_{2}K\left(x_{3},t_{3};x_{2},t_{2}\right)K\left(x_{2},t_{2};x_{1},t_{1}\right) & =\langle x_{3}|\hat{U}\left(t_{3}-t_{2}\right)\hat{U}\left(t_{2}-t_{1}\right)|x_{1}\rangle\\ & =\langle x_{3}|\exp\left(-i\hat{T}\left(t_{3}-t_{2}\right)\right)\exp\left(-i\hat{T}\left(t_{2}-t_{1}\right)\right)|x_{1}\rangle\\ & =\langle x_{3}|\exp\left(-i\hat{T}\left(t_{3}-t_{1}\right)\right)|x_{1}\rangle\\ & =K\left(x_{3},t_{3};x_{1},t_{1}\right). \end{aligned} \tag{2.4}\]
In this simplest case, we can break the propagator between two points into propagators between any intermediate number of points without introducing any approximation. The particle can either be thought of as following a straight line, or a sum over jagged lines.
When a potential \(\hat{V}\) is introduced, we will be forced to sum over jagged lines. The free space calculation is simple because the only operator appearing in Eq 2.4 and Eq 2.4 is the kinetic operator \(\hat{T}\): if a potential is present, \(\left[\hat{T},\hat{V}\right]\ne0\) and the exponentials cannot combine.
2.1.4 The propagator in a general potential: Deriving Feynman (iii)
In general, \[\hat{H}=\hat{T}+\hat{V}=\frac{\hat{p}^{2}}{2m}+V\left(\hat{x}\right)\]
and so
\[K\left(x,t;x_{0},t_{0}\right)=\langle x|\exp\left(-i\left(\hat{T}+\hat{V}\right)\left(t-t_{0}\right)\right)|x_{0}\rangle.\]
The problem with evaluating this is that \(\hat{T}\) and \(\hat{V}\) do not commute, and
\[\exp\left(A+B\right)\ne\exp\left(A\right)\exp\left(B\right)\]
unless \(\left[A,B\right]=0\). Hence we cannot just insert an identity as in the case of a free particle.
The solution is to use time slicing. Break the path into tiny slices \(\delta t=t/N\) with \(N\) large. Then,
\[\exp\left(-i\hat{H}t\right)=\left(\exp\left(-i\hat{H}\delta t\right)\right)^{N}\]
and
\[\exp\left(-i\left(\hat{T}+\hat{V}\right)\delta t\right)\approx\exp\left(-i\hat{T}\delta t\right)\exp\left(-i\hat{V}\delta t\right)\]
which becomes exact in the limit \(\delta t\rightarrow0\) (equivalently \(N\rightarrow\infty\)). Using time slicing we have:
\[K\left(x_{N},t;x_{0},t=0\right)=\langle x_{N}|\exp\left(-i\hat{H}\delta t\right)\times\ldots|x_{0}\rangle\]
where ‘\(\ldots\)’ signifies there are \(N\) identical copies within the bracket. Now we can insert \(N-1\) resolutions of the identity into the position basis:
\[\begin{aligned} K\left(x_{N},t;x_{0},t=0\right) & =\int\text{d}x_{N-1}\ldots\int\text{d}x_{1}\langle x_{N}|\exp\left(-i\hat{H}\delta t\right)|x_{N-1}\rangle\ldots\langle x_{1}|\exp\left(-i\hat{H}\delta t\right)|x_{0}\rangle. \end{aligned}\]
We must evaluate \(N\) integrals of the form
\[K_{n}\triangleq K\left(x_{n+1},\delta t;x_{n},0\right)=\langle x_{n+1}|\exp\left(-i\hat{H}\delta t\right)|x_{n}\rangle=\langle x_{n+1}|\exp\left(-i\left(\hat{T}+\hat{V}\right)\delta t\right)|x_{n}\rangle.\]
We use the fact that the exponent is small to approximate
\[K_{n}\approx\langle x_{n+1}|\exp\left(-i\hat{T}\delta t\right)\exp\left(-i\hat{V}\delta t\right)|x_{n}\rangle\]
and we can now treat this slice like a free particle. That is, we insert a resolution of the identity into the momentum basis:
\[\begin{aligned} K_{n} & \approx\int\text{d}p\langle x_{n+1}|\exp\left(-i\hat{T}\delta t\right)|p\rangle\langle p|\exp\left(-i\hat{V}\delta t\right)|x_{n}\rangle\\ & =\frac{1}{2\pi}\int\text{d}p\exp\left(-ip^{2}\delta t/\left(2m\right)\right)\langle x_{n+1}|p\rangle\exp\left(-iV\left(x_{n}\right)\delta t\right)\langle p|x_{n}\rangle\\ & =\exp\left(-iV\left(x_{n}\right)\delta t\right)K^{\text{free}}_{n}\\ & =\exp\left(-iV\left(x_{n}\right)\delta t\right)\sqrt{\frac{m}{2\pi i\delta t}}\exp\left(im\left(x_{n+1}-x_{n}\right)^{2}/\left(2\delta t\right)\right). \end{aligned}\]
Putting it all together gives
\[\begin{aligned} K\left(x_{N},t;x_{0},0\right) & =\underset{N\rightarrow\infty}{\textrm{lim}}\left(\frac{m}{2\pi i\delta t}\right)^{N/2}\int\text{d}x_{N-1}\ldots\int\text{d}x_{1}\exp\left(i\delta t\sum^{N-1}_{n=1}\left(\frac{m}{2}\left(\frac{x_{n+1}-x_{n}}{\delta t}\right)^{2}-V\left(x_{n}\right)\right)\right). \end{aligned}\]
The exponent in this limit simply becomes an integral:
\[\begin{aligned} \underset{N\rightarrow\infty}{\textrm{lim}}\delta t\sum^{N-1}_{n=1}\left(\frac{m}{2}\left(\frac{x_{n+1}-x_{n}}{\delta t}\right)^{2}-V\left(x_{n}\right)\right) & =\int^{t}_{0}\text{d}t'\left(\frac{m}{2}\dot{x}^{2}-V\left(x\right)\right)\\ & =\int^{t}_{0}\text{d}t'L\\ & =S\left[x\right] \end{aligned}\]
where \(S\left[x\right]\) is the action of the classical trajectory \(x\). Therefore we have derived
\[K\left(x,t;x_{0},t_{0}\right)=\int\mathscr{D}x\exp\left(iS\left[x\right]\right) \tag{2.5}\]
where we have defined the
the functional integral (aka path integral)
\[\int\mathscr{D}x\triangleq\underset{N\rightarrow\infty}{\textrm{lim}}\prod^{N}_{n=1}\int\text{d}x_{n}.\]
Note that the functional integral is a product of an infinite number of normal integrals(!). Eq. 2.5 is Feynman’s postulate (iii): all classical paths, no matter how crazy, appear in the quantum sum over trajectories (‘histories’) with equal magnitude. All that changes is the complex phase assigned to each path.
2.1.5 Wick Rotation to Imaginary Time
Now that we have the basics of path integral QM, it is possible to see an otherwise unexpected connection to classical statistical mechanics (which will immediately become important). Note a resemblance between the time dependent Schroedinger equation of a free particle:
\[i\frac{\partial\psi}{\partial t}=-\frac{1}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}\]
and the diffusion equation describing the classical evolution of heat density \(\rho\left(x,\tau\right)\) in position \(x\) and time \(\tau\):
\[\frac{\partial\rho}{\partial\tau}=D\frac{\partial^{2}\rho}{\partial x^{2}}\]
where \(D\) is the diffusion coefficient. Formally, we can get from one to the other using a
Wick rotation to imaginary time:
\[t\rightarrow-i\tau.\]
If you ever read A Brief History of Time, and wondered what Hawking was on about when he kept referring to imaginary time, now you know! The connection goes further. In QM we now know that the amplitude for a particle to start at \(x_{i}\) and end at \(x_{f}\) is
\[\langle x_{f}|\exp\left(-i\hat{H}t\right)|x_{i}\rangle=\int\mathscr{D}x\exp\left(iS\left[x\right]\right)\]
where
\[S\left[x\right]=\int^{t_{f}}_{t_{i}}\text{d}t\left\{ \frac{1}{2}m\dot{x}^{2}-V\left(x\right)\right\} .\]
In statistical mechanics, we have a similar relation: the probability for a system to evolve from initial state \(x_{i}\) to final state \(x_{f}\), at inverse temperature \(\beta=1/T\), is
\[\langle x_{f}|\exp\left(-\beta\hat{H}\right)|x_{i}\rangle=\int\mathscr{D}x\exp\left(-S_{E}\left[x\right]\right)\]
where
\[S_{E}\left[x\right]=\int^{\beta}_{0}\text{d}\tau\left\{ \frac{m}{2}\left(\frac{\text{d}x}{\text{d}\tau}\right)^{2}+V\left(x\right)\right\} .\]
Here, \(S_{E}\) is called the ‘Euclidean action’. This name derives from the fact that applying a Wick rotation to the Lorentz metric returns a Euclidean metric:
\[\begin{aligned} \text{Lorentz metric:}\quad x^{2} & -t^{2}\\ t=-i\tau & \downarrow\\ \text{Euclidean metric:}\quad x^{2} & +\tau^{2}. \end{aligned}\]
A major use of Wick rotation is in making functional integrals converge. Whether a given path integral is well defined depends on the action. In quantum calculations, trajectories are weighted by complex phases \(\exp\left(iS\right)\): convergence of the functional integrals requires the cancellation of rapidly varying phases for paths away from the paths of extremal action. In classical statistical mechanics, trajectories away from the extremal paths still contribute, but only with exponentially small probabilities \(\exp\left(-S_{E}\right)\). These integrals are much more likely to converge. Hence, it is frequently a useful trick to Wick rotate to obtain convergence, then to analytically continue the solution back to real time.
2.1.6 Commutation Relations in Path Integral QM
Note that path integral QM uses the time slicing relation
\[\lim_{\delta t\rightarrow0}\exp\left(-i\left(\hat{T}+\hat{V}\right)\delta t\right)=\lim_{\delta t\rightarrow0}\exp\left(-i\hat{T}\delta t\right)\exp\left(-i\hat{V}\delta t\right)\]
suggesting
\[\left[\hat{T},\hat{V}\right]=0 \tag{2.6}\]
and therefore
\[\left[\hat{p},\hat{x}\right]=0. \tag{2.7}\]
Clearly, this cannot be the case if we are to reproduce quantum mechanics. In Lagrangian mechanics we deal with positions and velocities, rather than positions and momenta. Hence our question is whether \(x\) and \(\dot{x}\) commute. We have:
\[\begin{aligned} \left[x,\dot{x}\right] & =x\left(t+\delta t\right)\frac{x\left(t+\delta t\right)-x\left(t\right)}{\delta t}-x\left(t\right)\frac{x\left(t+\delta t\right)-x\left(t\right)}{\delta t}\\ & =\left(\frac{x\left(t+\delta t\right)-x\left(t\right)}{\delta t}\right)^{2}\delta t\\ & =\dot{x}^{2}\delta t. \end{aligned} \tag{2.8}\]
In the first line we used the fact that
Operator ordering becomes time ordering in path integral QM.
At first glance this looks like it ought to be zero in the limit \(\delta t\rightarrow0\). But that might not be the case if \(\dot{x}^{2}\) is infinite. Since all paths \(x\left(t\right)\) are included in the functional integral, many (in fact, almost all) will be nowhere differentiable: they will have divergent \(\dot{x}\) at all instants in time. So we have an infinity multiplying a zero.
To see what Eq 2.8 evaluates to, it is easiest to use Wick rotation. Consider the classical statistical mechanics problem of Brownian motion. Here, a pollen grain (say) receives random kicks from water molecules. Its change in velocity at each instant is random. But this does not mean its position at each instant is random, as its position at one instant must be close to where it was the instant before. Specifically, we know that for Brownian motion we have
\[\delta x^{2}\propto\delta t \tag{2.9}\]
(as the pollen grain undergoes a random walk). This is formalised mathematically as Itô’s lemma. Carrying this intuition back, we have that
\[\left[x,\dot{x}\right]=\dot{x}^{2}\delta t=\left(\frac{\delta x}{\delta t}\right)^{2}\delta t=\frac{\delta x^{2}}{\delta t}\]
and so, invoking Eq 2.9, we have
\[\left[x,\dot{x}\right]=1.\]
That is, \(x\) and \(\dot{x}\) do not commute, after all! Wick rotating back, and transforming from velocity to conjugate momentum, you obtain the usual canonical commutation relation
\[\left[x,p\right]=i.\]
The key point is this:
Non-commutation of operators arises in path integral QM from the nowhere-differentiability of typical paths being summed over in the functional integral.
This will be important when we go to quantum fields, because it will allow us to sum over classical, commuting fields, but to arrive at non-commuting field operators.
2.2 Relativistic single-particle quantum mechanics fails
2.2.1 The Klein Gordon Equation
The Schroedinger equation you have seen in previous years is a single-particle non-relativistic equation. To make a relativistic quantum mechanics, we might try to simply canonically quantize the relativistic dispersion relation
\[E^{2}-\boldsymbol{p}^{2}-m^{2}=0\]
using
\[\begin{aligned} E & \rightarrow\hat{H}\triangleq i\partial_{t}\\ \boldsymbol{p} & \rightarrow\hat{p}\triangleq-i\nabla. \end{aligned}\]
The result is the
Klein Gordon Equation:
\[\left(\partial^{2}_{t}-\nabla^{2}+m^{2}\right)\varphi\left(x\right)=0\]
which can be written compactly using
\[\partial^{2}\triangleq\partial^{\mu}\partial_{\mu}=\partial^{2}_{t}-\nabla^{2}\]
to give:
\[\left(\partial^{2}+m^{2}\right)\varphi=0.\]
Interestingly, Schroedinger devised the Klein Gordon equation before he devised the Schroedinger equation. He rejected it as an equation for the electron for various legitimate reasons. It is, however, a perfectly good quantum relativistic single-particle equation for spin-0 particles. However, there are necessarily fundamental issues with any single-particle relativistic quantum theories, as we will now see.
2.2.2 The Failure of single-particle relativistic quantum theories
A question many students ask around this point is: can’t we just upgrade the time evolution operator to a relativistic version, and create a single-particle relativistic quantum mechanics? Unfortunately this turns out not to work. Let’s see why.
Consider the probability amplitude[^4] \(\mathcal{A}\) for a particle to propagate from position \(\boldsymbol{x}\) at time \(t\) to position \(\boldsymbol{x}'\) at time \(t'\):
\[\mathcal{A}\left(\boldsymbol{x}',t';\boldsymbol{x},t\right)=\langle\boldsymbol{x}'|\exp\left(-i\hat{H}\left(t'-t\right)\right)|\boldsymbol{x}\rangle.\]
We will use the relativistic Klein Gordon equation for the Hamiltonian. Inserting a complete set of momentum states as before,
\[\hat{\mathbb{I}}=\int\text{d}^{3}\boldsymbol{p}|p\rangle\langle\boldsymbol{p}|\]
but this time using
\[\hat{H}|\boldsymbol{p}\rangle=\sqrt{\boldsymbol{p}^{2}+m^{2}}|\boldsymbol{p}\rangle\]
gives
\[\begin{aligned} \mathcal{A}\left(\boldsymbol{x}',t';\boldsymbol{x},t\right) & =\int\text{d}^{3}\boldsymbol{p}\langle\boldsymbol{x}'|\exp\left(-i\hat{H}\left(t'-t\right)\right)|p\rangle\langle\boldsymbol{p}|\boldsymbol{x}\rangle\\ & =\int\text{d}^{3}\boldsymbol{p}\langle\boldsymbol{x}'|\exp\left(-i\sqrt{\boldsymbol{p}^{2}+m^{2}}\left(t'-t\right)\right)|p\rangle\exp\left(-i\boldsymbol{p}\cdot\boldsymbol{x}\right)\\ & =\int\text{d}^{3}\boldsymbol{p}\langle\boldsymbol{x}'|p\rangle\exp\left(-i\sqrt{\boldsymbol{p}^{2}+m^{2}}\left(t'-t\right)\right)\exp\left(-i\boldsymbol{p}\cdot\boldsymbol{x}\right)\\ & =\int\text{d}^{3}\boldsymbol{p}\exp\left(-i\sqrt{\boldsymbol{p}^{2}+m^{2}}\left(t'-t\right)+i\boldsymbol{p}\cdot\left(\boldsymbol{x}'-\boldsymbol{x}\right)\right). \end{aligned}\]
A convenient co-ordinate choice is spherical polars with \(\boldsymbol{p}\cdot\left(\boldsymbol{x}'-\boldsymbol{x}\right)=p\left|x'-x\right|\cos\left(\theta\right)\), where \(p=\left|\boldsymbol{p}\right|\). This gives
\[\begin{aligned} \mathcal{A}\left(\boldsymbol{x}',t';\boldsymbol{x},t\right) & =2\pi\int^{\infty}_{0}\text{d}pp^{2}\int^{1}_{-1}\text{d}\left(\cos\theta\right)\exp\left(-i\sqrt{p^{2}+m^{2}}\left(t'-t\right)+ip\left|\boldsymbol{x}'-\boldsymbol{x}\right|\cos\theta\right)\\ & =\frac{4\pi}{\left|\boldsymbol{x}'-\boldsymbol{x}\right|}\int^{\infty}_{0}\text{d}pp\exp\left(-i\sqrt{p^{2}+m^{2}}\left(t'-t\right)\right)\sin\left(p\left|\boldsymbol{x}'-\boldsymbol{x}\right|\right). \end{aligned}\]
To proceed from here you can either do a contour integral[^5], or do it the old-fashioned way: look it up in Gradshtein & Ryzhik. There you will find (7th edition equation 3.914.6):
\[\int^{\infty}_{0}p\exp\left(-\beta\sqrt{\gamma^{2}+p^{2}}\right)\sin\left(bp\right)\text{d}p=\frac{b\beta\gamma^{2}}{\beta^{2}+b^{2}}K_{2}\left(\gamma\sqrt{\beta^{2}+b^{2}}\right)\]
where \(K_{2}\) is a modified Bessel function of the second kind. This gives
\[\mathcal{A}\left(\boldsymbol{x}',t';\boldsymbol{x},t\right)=\frac{4\pi i\left(t'-t\right)m^{2}}{\left|\boldsymbol{x}'-\boldsymbol{x}\right|^{2}-\left(t'-t\right)^{2}}K_{2}\left(m\sqrt{\left|\boldsymbol{x}'-\boldsymbol{x}\right|^{2}-\left(t'-t\right)^{2}}\right)\]
or
\[\mathcal{A}\left(\boldsymbol{x}',t';\boldsymbol{x},t\right)=\frac{4\pi i\left(t'-t\right)m^{2}}{\Delta s^{2}}K_{2}\left(m\Delta s\right)\]
where
\[\Delta s=\sqrt{\left|\boldsymbol{x}'-\boldsymbol{x}\right|^{2}-\left(t'-t\right)^{2}}\]
is the Lorentz-invariant proper distance between the events. In the case where \(\Delta s\) is very large, i.e. well outside the light cone, we can look up the asymptotic expression for the Bessel function:
\[\lim_{x\rightarrow\infty}K_{2}\left(x\right)\sim\sqrt{\frac{\pi}{2x}}\exp\left(-x\right)\]
which gives
\[\lim_{\Delta s\rightarrow\infty}\mathcal{A}\left(\boldsymbol{x}',t';\boldsymbol{x},t\right)=i\frac{\left(t'-t\right)}{\Delta s}\left(\frac{2\pi m}{\Delta s}\right)^{3/2}\exp\left(-m\Delta s\right).\]
OK, so now say that the particle sets off from some volume \(V=\text{d}^{3}\boldsymbol{x}\) centred on \(\left(\boldsymbol{x},t\right)\), which is, for example, a box somewhere near Andromeda three seconds ago in your reference frame (therefore very much spacelike separated from you). The probability for you to find the particle in front of you now, within a volume \(V'=\text{d}^{3}\boldsymbol{x}'\), is
\[\text{Prob}=VV'\frac{\left(t'-t\right)^{2}}{\Delta s^{2}}\left(\frac{2\pi m}{\Delta s}\right)^{3}\exp\left(-2m\Delta s\right).\]
The probability might be exponentially small... but it is non-zero!
This is the problem with single-particle relativistic quantum mechanics: there is always a finite probability to detect particles released from spacelike separations — that is, a finite probability to signal backwards in time. Since we’ve never seen anything signal backwards in time, we must reject any theory which says it is routine.
While this might seem like a problem which couldn’t have come about before 1905, it was really known much earlier. For example, Michael Faraday took exception to the fact that waggling a magnet here can seemingly affect a magnet over there. Newton took exception to the fact that a planet here can seemingly affect an apple over there. The solution, in both cases, was to posit the existence of an invisible field which connects the observable objects and which is defined to act locally. Waggling the magnet here only affects the electromagnetic field here; a signal propagates through the field to the magnet over there. Faced with action at a distance (particle propagation over spacelike separation), our solution is the same. We introduce quantum fields, which are defined to act locally.
2.3 Path Integral Field Quantization
The switch from single particles to fields is now carried out as follows:
\[\boldsymbol{x}\left(t\right)\rightarrow\varphi\left(\boldsymbol{x},t\right)=\varphi\left(x^{\mu}\right). \tag{2.10}\]
That is, rather than \(\boldsymbol{x}\) specifying the location of a single particle at time \(t\), we instead have a field \(\varphi\) which can host any number of particles at any points in spacetime \(x^{\mu}\). It is interesting to note that one can think of the position in the single-particle QM as a field in its own right (simply relabel \(\boldsymbol{x}\) to \(\boldsymbol{\varphi}\)). In this sense, QM is simply 0+1D QFT! It is both interesting and important to note that the fields in Eq 2.10 are classical, commuting fields! Nevertheless, the functional integral over these fields yields a quantum field theory. This work precisely as it did in single-particle path integral quantum mechanics in Section @subsec-Commutation-Relations-in. The non-commutation of observables appears via the fractal nature of the fields being integrated over: a typical field in the integral is nowhere-differentiable.
We formulate a particular QFT by writing down its action, which is a functional of its (classical, commuting) fields. The action, as before, is the time integral of the Lagrangian:
\[\begin{aligned} S\left[\varphi\right] & \triangleq\int\text{d}tL\left[\varphi,\partial_{\mu}\varphi\right] \end{aligned}\]
or the spacetime integral of the Lagrange density:
\[S\left[\varphi\right]=\int\text{d}^{4}x\mathscr{L}\left(\varphi,\partial_{\mu}\varphi\right).\]
Here, \(\varphi\) is a scalar field. This means it maps a number (well, 4-vector) \(x^{\mu}\) to a scalar \(\varphi\left(x^{\mu}\right)\). Standard notation you may see in textbooks denotes the space of \(x^{\mu}\) as the ‘base manifold’ (recalling that Minkowski space is a manifold since it is locally \(\mathbb{R}^{N}\)), and the space of \(\varphi\left(x^{\mu}\right)\) to be the ‘target manifold’.
The particle excitations of scalar fields are spin-0 bosons. In the standard model the only example is the Higgs boson. But pions are composite spin-0 objects, and many particles in condensed matter physics (e.g. phonons) are spin-0. We can also consider complex scalar fields with little extra work. The standard notation is to consider \(\varphi\) and \(\varphi^{*}\) to be independent fields. Then the action is a functional of both these fields:
\[\text{complex scalar field theory: }S\left[\varphi,\varphi^{*}\right].\]
We will see that the particle excitations of \(\varphi^{*}\) can be interpreted as the antiparticles to those of \(\varphi\).
The QFT governing scalar fields is called Klein Gordon (KG) theory. For real fields its Lagrange density is defined to be \[\mathscr{L}_{\text{KG}}\left(\varphi,\partial^{\mu}\varphi\right)\triangleq\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi-\frac{1}{2}m^{2}\varphi^{2}.\]
Einstein summation notation is assumed, so there is an implicit sum over \(\mu\). For complex fields it is
\[\mathscr{L}^{\mathbb{C}}_{\text{KG}}\left(\varphi,\varphi^{*},\partial^{\mu}\varphi,\partial^{\mu}\varphi^{*}\right)\triangleq\partial^{\mu}\varphi^{*}\partial_{\mu}\varphi-m^{2}\varphi^{*}\varphi.\]
2.4 Euler Lagrange Equations
In classical mechanics (using fields or otherwise) one constructs the action, considers all possible trajectories through phase space, and finds the trajectory that extremises the action subject to the initial and final boundary conditions. In path integral quantum mechanics, one does the same, except non-extremal trajectories now contribute. The system evolves by taking all possible trajectories through phase space with each trajectory weighted by \(\exp\left(iS\right)\). Still, the extremal trajectories are typically expected to give the biggest overall contribution, since the phase winding is slowest when \(S\) is near an extremum. When we shift to QFT, trajectories (single-particle paths) become field configurations which can contain any number of particles. Otherwise the method is unchanged. The sequences of field configurations which extremise the action again play a special role. In this case, they obey the single-particle quantum wave equation. Let’s see how this works for the Klein Gordon field.
2.4.1 Example: the Klein Gordon field
The action defining the (real) Klein Gordon quantum field theory is
\[S_{\text{KG}}\left[\varphi\right]\triangleq\int\text{d}^{4}x\left(\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi-\frac{1}{2}m^{2}\varphi^{2}\right). \tag{2.11}\]
Here, \(\varphi\left(x^{\mu}\right)\) is a real scalar field. Recall that, in the path integral formalism, fields are commuting: non-commutation appears from the fractal nature of typical field configurations in the functional integral. To find the associated Euler Lagrange equations we must extremise the action:
\[\begin{aligned} S_{\text{KG}}\left[\varphi\right] & =\int\text{d}^{4}x\left(\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi-\frac{1}{2}m^{2}\varphi^{2}\right)\\ & \downarrow\\ S_{\text{KG}}\left[\varphi+\lambda\epsilon\right] & =\int\text{d}^{4}x\left(\frac{1}{2}\left(\partial^{\mu}\varphi+\lambda\partial^{\mu}\epsilon\right)\left(\partial_{\mu}\varphi+\lambda\partial_{\mu}\epsilon\right)-\frac{1}{2}m^{2}\left(\varphi+\lambda\epsilon\right)^{2}\right)\\ & \downarrow\\ \frac{S_{\text{KG}}\left[\varphi+\lambda\epsilon\right]}{\partial\lambda} & =\int\text{d}^{4}x\left(\frac{1}{2}\partial^{\mu}\epsilon\left(\partial_{\mu}\varphi+\lambda\partial_{\mu}\epsilon\right)+\frac{1}{2}\left(\partial^{\mu}\varphi+\lambda\partial^{\mu}\epsilon\right)\partial_{\mu}\epsilon-m^{2}\left(\varphi+\lambda\epsilon\right)\epsilon\right)\\ & \downarrow\\ \left.\frac{S_{\text{KG}}\left[\varphi+\lambda\epsilon\right]}{\partial\lambda}\right|_{\lambda=0} & =\int\text{d}^{4}x\left(\frac{1}{2}\partial^{\mu}\epsilon\partial_{\mu}\varphi+\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\epsilon-m^{2}\varphi\epsilon\right). \end{aligned}\]
We set this equal to zero. Integrating the first two terms by parts, using
\[\partial^{\mu}\partial_{\mu}\varphi=\partial_{\mu}\partial^{\mu}\varphi=\partial^{2}\varphi\]
gives
\[0=\int\text{d}^{4}x\left(-\partial^{2}\varphi-m^{2}\varphi\right)\epsilon \tag{2.12}\]
where we used the fact that \(\epsilon\left(x^{\mu}\right)\) is defined to vanish at the limits of the spacetime integral. The only way Eq 2.12 can be true for any field \(\epsilon\) is if the term in parentheses is zero. This term gives the Euler Lagrange equation corresponding to the Klein Gordon action:
\[\left(\partial^{2}+m^{2}\right)\varphi=0\]
which is the Klein Gordon equation governing a single spin-0 relativistic particle.
2.5 Propagators & Green’s Functions
2.5.1 Green’s Functions for linear differential equations
Given a solution to a homogeneous linear equation – such as the Klein Gordon equation – there is a general method to solve the corresponding inhomogeneous equation. This is the method of Green’s functions. Say we have
\[\hat{L}_{x}\varphi\left(x\right)=j\left(x\right) \tag{2.13}\]
where \(\hat{L}_{x}\) is some differential operator acting on \(x\) (for example, \(\hat{L}_{x}=\partial^{2}+m^{2}\)) and \(j\left(x\right)\) is a source term (a fixed, specified current) which renders the equation inhomogeneous. Working maximally generally like this, it might seem implausible that we could hope for a general solution. But looked at another way, in the finite dimensional case Eq 2.13 is just a matrix equation, and we know how to invert matrices to find solutions. Green’s method generalises this intuition, providing a general solution to a linear differential equation. The principle is simple: if you can find the response a delta function source (an ‘impulse’), you can integrate to get the solution for arbitrary sources.
We therefore begin by seeking to solve the special case
\[\hat{L}_{x}G\left(x,y\right)=\delta^{D+1}\left(x-y\right).\]
This immediately gives the general solution:
\[\begin{aligned} \hat{L}_{x}G\left(x,y\right) & =\delta^{D+1}\left(x-y\right)\\ \int\text{d}^{D+1}yj\left(y\right)\rightarrow & \downarrow\\ \int\text{d}^{D+1}yj\left(y\right)\hat{L}_{x}G\left(x,y\right) & =\int\text{d}^{D+1}yj\left(y\right)\delta^{D+1}\left(x-y\right)\\ \hat{L}_{x}\left[\int\text{d}^{D+1}yj\left(y\right)G\left(x,y\right)\right] & =j\left(x\right). \end{aligned}\]
Since we already know that
\[\hat{L}_{x}\varphi\left(x\right)=j\left(x\right)\]
it follows that
\[\varphi\left(x\right)=\int\text{d}^{D+1}yG\left(x,y\right)j\left(y\right).\]
The Green’s function ‘propagates’ the disturbance caused by the current \(j\left(x\right)\) to give the field \(\varphi\left(x\right)\), and hence the Green’s function is a propagator[^6]. This can be used to obtain a perturbation expansion:
\[\begin{aligned} \varphi\left(x\right) & =\varphi_{0}\left(x\right)+\int\text{d}^{D+1}yG\left(x-y\right)j\left(y\right)\\ & =\varphi_{0}\left(x\right)+\int\text{d}^{D+1}yG\left(x-y\right)\hat{L}_{y}\varphi\left(y\right)\\ & =\varphi_{0}\left(x\right)+\int\text{d}^{D+1}yG\left(x-y\right)\hat{L}_{y}\left[\varphi_{0}\left(y\right)+\int\text{d}^{D+1}zG\left(y-z\right)j\left(z\right)\right]+\ldots \end{aligned}\]
where \(\varphi_{0}\left(x\right)\) is the solution to the homogeneous equation. This is particularly useful later when we look at perturbation theory, where \(j\) becomes a dynamical functional of \(\varphi\). In cases of physical interest, symmetries generally constrain and simplify the problem. In particular, we often care about problems with translational symmetry, in which case
\[\text{translational symmetry}\implies G\left(x,y\right)=G\left(x-y\right).\]
2.5.2 Green’s function for the Klein Gordon equation: momentum space
In practice this works as follows. Adding a delta function source to the Klein Gordon equation (in \(D=3\)) gives
\[\left(\partial^{2}+m^{2}\right)G\left(x,y\right)=-i\delta^{4}\left(x-y\right)\]
where we have added a \(-i\) in the definition purely for convenience (it neatens things later on). The Klein Gordon equation has spacetime translational symmetry, so this simplifies to
\[\left(\partial^{2}+m^{2}\right)G\left(x-y\right)=-i\delta^{4}\left(x-y\right).\]
Further, translational symmetry suggests that we will do well to employ the Fourier transform:
\[G\left(x-y\right)=\int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)\tilde{G}\left(p\right) \tag{2.14}\]
to give
\[\begin{aligned} \left(\partial^{2}+m^{2}\right)\int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)\tilde{G}\left(p\right) & =-i\delta^{4}\left(x\right)\\ \int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\left(-p^{2}+m^{2}\right)\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)\tilde{G}\left(p\right) & =-i\delta^{4}\left(x\right)\\ \int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\left(-p^{2}+m^{2}\right)\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)\tilde{G}\left(p\right) & =-i\int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right) \end{aligned}\]
where we defined
\[p^{2}\triangleq p_{\mu}p^{\mu}\]
and in the last line we used
\[\int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)=\delta^{4}\left(x-y\right).\]
Equating the integrands gives
the momentum-space Klein Gordon Green’s function:
\[\tilde{G}\left(p\right)=\frac{i}{p^{2}-m^{2}}=\frac{i}{E^{2}-\boldsymbol{p}^{2}-m^{2}}.\]
2.5.3 Klein Gordon Equation Green’s function: real space
Using the Fourier transform in Eq 2.14 gives the real-space solution
\[G\left(x-y\right)=i\int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\frac{\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)}{p^{2}-m^{2}}. \tag{2.15}\]
It is simple to check the this indeed obeys the Klein Gordon equation:
\[\begin{aligned} \left(\partial^{2}+m^{2}\right)G\left(x-y\right) & =i\left(\partial^{2}+m^{2}\right)\int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\frac{\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)}{p^{2}-m^{2}}\\ & =i\int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\left(-p^{2}+m^{2}\right)\frac{\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)}{p^{2}-m^{2}}\\ & =-i\int\frac{\text{d}^{4}p}{\left(2\pi\right)^{4}}\exp\left(-ip_{\mu}\left(x^{\mu}-y^{\mu}\right)\right)\\ & =-i\delta^{4}\left(x-y\right). \end{aligned}\]
To find an explicit formula for \(G\left(x\right)\) (we set \(y=0\) for notational convenience), the Klein Gordon Green’s function, we must use contour integration. Expanding the terms in Eq 2.15 gives
\[\begin{aligned} G\left(x\right) & =i\left(2\pi\right)^{-4}\int\text{d}E\text{d}^{3}\boldsymbol{p}\frac{\exp\left(-iEt+i\boldsymbol{p}\cdot\boldsymbol{x}\right)}{E^{2}-\boldsymbol{p}^{2}-m^{2}}\\ & =i\left(2\pi\right)^{-4}\int\text{d}E\text{d}^{3}\boldsymbol{p}\frac{\exp\left(-iEt+i\boldsymbol{p\cdot x}\right)}{\left(E+E_{\boldsymbol{p}}\right)\left(E-E_{\boldsymbol{p}}\right)} \end{aligned}\]
where
\[E_{\boldsymbol{p}}\triangleq\sqrt{\boldsymbol{p}^{2}+m^{2}}.\]
Inspecting the energy integral, we find that it has simple poles at \(E=\pm E_{\boldsymbol{p}}\). Since the integral is along the entire real axis, both poles are intercepted and cause divergences. To proceed, let us temporarily promote \(E\) to a complex variable
\[E\rightarrow z=z_{1}+iz_{2}\]
with \(z_{1,2}\in\mathbb{R}\). Then
\[G\left(x\right)=i\left(2\pi\right)^{-4}\int_{C}\text{d}z\text{d}^{3}\boldsymbol{p}\frac{\exp\left(z_{2}t\right)\exp\left(-iz_{1}t+i\boldsymbol{p\cdot x}\right)}{\left(z+E_{\boldsymbol{p}}\right)\left(z-E_{\boldsymbol{p}}\right)}.\] {#eqG-complex}
If we take the contour \(C\) to be the real axis, this is exactly identical to the previous integral. However, for positive \(t\) we can connect \(z=\infty+i0\) to \(z=-\infty+i0\) by adding an infinite semicircular arc in the lower half plane: since \(z_{2}t\) is negative and infinite along this arc, this gives zero contribution to the integral. In this equivalent integral we now have a closed contour to which we can apply Cauchy’s theorem. Similarly, if \(t<0\), we can add a contour which closes in the upper half plane.
We must then either deform the contour to miss the poles, or equivalently shift the poles to miss the contour. I will use the latter convention. There are four ways to move the poles (either can move up or down). Any choice gives a legitimate Green’s function, but the forms differ. We will see the relationship between possible choices shortly.
For now, we will make the so-called ‘Feynman prescription’. It is shown in Fig 2.2. We shift the pole at \(-E_{\boldsymbol{p}}\) up infinitesimally to \(-E_{\boldsymbol{p}}+i0^{+}\) (where \(0^{+}\) means an infinitesimal positive number) and the pole at \(E_{\boldsymbol{p}}\) down to \(E_{\boldsymbol{p}}-i0^{+}\). With this choice, only the \(-E_{\boldsymbol{p}}+i0^{+}\) pole is enclosed when \(t<0\), and only the \(E_{\boldsymbol{p}}-i0^{+}\) pole is enclosed when \(t>0\).
Overall, we have
\[G\left(x\right)=i\left(2\pi\right)^{-4}\oint_{C}\text{d}z\text{d}^{3}\boldsymbol{p}\frac{\exp\left(-iz_{1}t+i\boldsymbol{p\cdot x}\right)}{\left(z+E_{\boldsymbol{p}}-i0^{+}\right)\left(z-E_{\boldsymbol{p}}+i0^{+}\right)}\exp\left(z_{2}t\right)\] {#eqG-complex-2}
where
\[C=\begin{cases} \begin{array}{c} \infty\text{ clockwise semicircle in LHP},\\ \infty\text{ anticlockwise semicircle in UHP}, \end{array} & \begin{array}{c} t>0\\ t<0 \end{array}\end{cases}\]
!@eqG-complex, along with the Feynman prescription for handling the poles (top row), and the prescriptions leading to the retarded and advanced Green’s functions (bottom row).](contours){#fig-Feynman-prescription}
(the contour directions are fixed by the original integral, which runs from \(-\infty\) to \(\infty\)). Note that this choice of pole-shifts means that \(E_{\boldsymbol{p}}\rightarrow E_{\boldsymbol{p}}-i0^{+}\) throughout; hence, Eq @eqG-complex-2 can be compactly written as the Fourier transform of
\[\tilde{G}\left(p\right)=\frac{i}{p^{2}-m^{2}+i0^{+}}\]
which is a convenient way to remember it. Applying Cauchy’s theorem (remembering that the pole encircled clockwise gives a negative contribution) gives
\[\begin{aligned} G\left(t>0,\boldsymbol{x}\right) & =\left(2\pi\right)^{-3}\int\frac{\text{d}^{3}\boldsymbol{p}}{2E_{\boldsymbol{p}}}\exp\left(-iE_{\boldsymbol{p}}t+i\boldsymbol{p\cdot x}\right)\\ G\left(t<0,\boldsymbol{x}\right) & =\left(2\pi\right)^{-3}\int\frac{\text{d}^{3}\boldsymbol{p}}{2E_{\boldsymbol{p}}}\exp\left(iE_{\boldsymbol{p}}t+i\boldsymbol{p\cdot x}\right). \end{aligned}\] {#eqG-complex-2-2}
Returning to the original argument \(x-y\), this can be written as
\[G\left(x-y\right)=\left(2\pi\right)^{-3}\int\frac{\text{d}^{3}\boldsymbol{p}}{2E_{\boldsymbol{p}}}\exp\left(i\boldsymbol{p\cdot}\left(\boldsymbol{x}-\boldsymbol{y}\right)\right)\left[\Theta\left(x^{0}-y^{0}\right)\exp\left(-iE_{\boldsymbol{p}}\left(x^{0}-y^{0}\right)\right)+\Theta\left(y^{0}-x^{0}\right)\exp\left(iE_{\boldsymbol{p}}\left(x^{0}-y^{0}\right)\right)\right] \tag{2.16}\]
where we have defined
the Heaviside step function
\[\Theta\left(x\right)=\begin{cases} \begin{array}{c} 1,\\ 0, \end{array} & \begin{array}{c} x>0\\ x<0 \end{array}\end{cases}\]
and where \(x^{0}\) is the time component of \(x^{\mu}\). A neater way to rewrite this uses
time ordering \(\mathcal{T}\): earlier events are written to the right of later events.
As with normal ordering, time ordering is not an operator; rather, it is a label indicating a convention that is being followed. Using this symbol, we can write the Green’s function in an elegant form:
\[G\left(x-y\right)=\left(2\pi\right)^{-3}\int\frac{\text{d}^{3}\boldsymbol{p}}{2E_{\boldsymbol{p}}}\mathcal{T}\exp\left(-ip\cdot\left(x-y\right)\right). \tag{2.17}\]
2.5.4 The Feynman Propagator
In Section we saw a formulation of single-particle non-relativistic QM in which the particle propagates from \(\left(\boldsymbol{x},t\right)\) to \(\left(\boldsymbol{x}',t'\right)\) via the propagator \(K\left(\boldsymbol{x}',t';\boldsymbol{x},t\right)\). It is straightforward to construct a relativistic single-particle propagator \(\Delta\left(x-y\right)\) using the Klein Gordon equation. In that case we require
\[\psi\left(x\right)=\int\text{d}^{4}x\Delta\left(x-y\right)\psi\left(y\right).\]
Back in the general theory of Green’s functions in Section @subsec-Green-s-Functions we saw that the general solution to a linear equation
\[\hat{L}_{x}\psi\left(x\right)=f\left(x\right)\]
is given by
\[\psi\left(x\right)=\int\text{d}^{4}yG\left(x,y\right)f\left(y\right).\]
Hence, the definition of the propagator tells us that it must be a Green’s function for the Klein Gordon equation, with a ‘source’ which is the field (wavefunction, in the case of single particles) itself!
Returning to quantum fields, in which particles are created and destroyed, the propagator \(\Delta\left(x-y\right)\) gives the amplitude to annihilate a particle at \(y\) and to create a particle at \(x\). We are no longer restricted to the particle travelling from one event to the other. Instead, we integrate over all possible field configurations compatible with those two events.
In the operator language of Section Chapter 1, we might formalise the stated behaviour of the propagator as follows:
\[\Delta\left(x-y\right)=\langle\Omega|\mathcal{T}\hat{\varphi}_{\boldsymbol{y}}\hat{\varphi}_{\boldsymbol{x}}|\Omega\rangle. \tag{2.18}\]
Writing out each field operator as an integral over creation and annihilation operators, we see that this creates a particle at \(x\) (since \(\hat{a}|\Omega\rangle=0\)) and annihilates one at \(y\) (since \(\langle\Omega|\hat{a}^{\dagger}=0\)). The time ordering \(\mathcal{T}\) ensures that the particle is not created before it is destroyed: we require \(x^{0}>y^{0}\). We have that
\[\mathcal{T}\hat{\varphi}_{\boldsymbol{y}}\hat{\varphi}_{\boldsymbol{x}}=\Theta\left(y^{0}-x^{0}\right)\hat{\varphi}_{\boldsymbol{y}}\hat{\varphi}_{\boldsymbol{x}}+\Theta\left(x^{0}-y^{0}\right)\hat{\varphi}_{\boldsymbol{x}}\hat{\varphi}_{\boldsymbol{y}}.\]
Let’s confirm that the canonical quantization approach of Section Chapter 1 gives the same propagator as the path integral approach. Using the results from Section @subsec-Time-dependence, the time dependence of \(\hat{\varphi}_{\boldsymbol{x}}\) is already contained within it implicitly: we can rewrite it as \(\hat{\varphi}_{x}\) by switching \(\boldsymbol{k}\cdot\boldsymbol{x}\rightarrow-k\cdot x\) in the exponents of its Fourier expansion. We have:
\[\begin{aligned} \langle\Omega|\mathcal{T}\hat{\varphi}_{\boldsymbol{y}}\hat{\varphi}_{\boldsymbol{x}}|\Omega\rangle & =\langle\Omega|\mathcal{T}\hat{\varphi}_{y}\hat{\varphi}_{x}|\Omega\rangle\\ & =\int\frac{\text{d}^{3}\boldsymbol{p}}{\left(2\pi\right)^{3}}\frac{1}{\sqrt{2\omega_{\boldsymbol{p}}}}\int\frac{\text{d}^{3}\boldsymbol{q}}{\left(2\pi\right)^{3}}\frac{1}{\sqrt{2\omega_{\boldsymbol{q}}}}\langle\Omega|\mathcal{T}\left(\hat{a}^{\phantom{\dagger}}_{\boldsymbol{p}}\exp\left(-ip\cdot y\right)+\hat{a}^{\dagger}_{\boldsymbol{p}}\exp\left(ip\cdot y\right)\right)\left(\hat{a}^{\phantom{\dagger}}_{\boldsymbol{q}}\exp\left(-iq\cdot x\right)+\hat{a}^{\dagger}_{\boldsymbol{q}}\exp\left(iq\cdot x\right)\right)|\Omega\rangle\\ & =\int\frac{\text{d}^{3}\boldsymbol{p}}{\left(2\pi\right)^{3}}\frac{1}{\sqrt{2\omega_{\boldsymbol{p}}}}\int\frac{\text{d}^{3}\boldsymbol{q}}{\left(2\pi\right)^{3}}\frac{1}{\sqrt{2\omega_{\boldsymbol{q}}}}\langle\Omega|\mathcal{T}\hat{a}^{\phantom{\dagger}}_{\boldsymbol{p}}\exp\left(-ip\cdot y\right)\hat{a}^{\dagger}_{\boldsymbol{q}}\exp\left(iq\cdot x\right)|\Omega\rangle\\ & =\int\frac{\text{d}^{3}\boldsymbol{p}}{\left(2\pi\right)^{3}}\frac{1}{\sqrt{2\omega_{\boldsymbol{p}}}}\int\frac{\text{d}^{3}\boldsymbol{q}}{\left(2\pi\right)^{3}}\frac{1}{\sqrt{2\omega_{\boldsymbol{q}}}}\langle\boldsymbol{p}|\boldsymbol{q}\rangle\mathcal{T}\exp\left(-ip\cdot y\right)\exp\left(iq\cdot x\right)\\ & =\int\frac{\text{d}^{3}\boldsymbol{p}}{\left(2\pi\right)^{3}}\frac{1}{\sqrt{2\omega_{\boldsymbol{p}}}}\int\frac{\text{d}^{3}\boldsymbol{q}}{\left(2\pi\right)^{3}}\frac{1}{\sqrt{2\omega_{\boldsymbol{q}}}}\left(2\pi\right)^{3}\delta^{3}\left(\boldsymbol{p}-\boldsymbol{q}\right)\mathcal{T}\exp\left(-ip\cdot y\right)\exp\left(iq\cdot x\right)\\ & =\int\frac{\text{d}^{3}\boldsymbol{p}}{\left(2\pi\right)^{3}}\frac{1}{2\omega_{\boldsymbol{p}}}\mathcal{T}\exp\left(-i\left(p^{0}y^{0}-\boldsymbol{p}\cdot\boldsymbol{y}\right)\right)\exp\left(i\left(q^{0}x^{0}-\boldsymbol{p}\cdot\boldsymbol{x}\right)\right)\\ & =\int\frac{\text{d}^{3}\boldsymbol{p}}{\left(2\pi\right)^{3}}\frac{1}{2\omega_{\boldsymbol{p}}}\exp\left(i\boldsymbol{p}\cdot\left(\boldsymbol{y}-\boldsymbol{x}\right)\right)\mathcal{T}\exp\left(-i\left(p^{0}y^{0}-q^{0}x^{0}\right)\right). \end{aligned} \tag{2.19}\]
We can rewrite the time ordering as before to give
\[\begin{aligned} \Delta\left(x-y\right) & =\int\frac{\text{d}^{3}\boldsymbol{p}}{\left(2\pi\right)^{3}}\frac{1}{2\omega_{\boldsymbol{p}}}\exp\left(i\boldsymbol{p}\cdot\left(\boldsymbol{y}-\boldsymbol{x}\right)\right)\left[\Theta\left(y^{0}-x^{0}\right)\exp\left(-i\left(p^{0}y^{0}-q^{0}x^{0}\right)\right)+\Theta\left(x^{0}-y^{0}\right)\exp\left(i\left(p^{0}y^{0}-q^{0}x^{0}\right)\right)\right] \end{aligned}\]
which is equal to the expression obtained using path integrals in Eq 2.16.
2.6 Retarded and Advanced Green’s Functions
In Section @subsec-Klein-Gordon-Green-realspace we obtained the real-space Green’s function for the Klein Gordon equation using the Feynman prescription: taking one pole below the real axis and one above. The result is termed the Feynman propagator. Feynman’s choice of pole placements precisely produces the time ordering of the field operators in Eq 2.18.
However, there are 4 choices in total for how to offset the poles:
\[G_{\pm_{1}\pm_{2}}\left(x\right)=i\left(2\pi\right)^{-4}\oint_{C}\text{d}z\text{d}^{3}\boldsymbol{p}\frac{\exp\left(-iz_{1}t+i\boldsymbol{p\cdot x}\right)}{\left(z+E_{\boldsymbol{p}}\pm_{1}i0^{+}\right)\left(z-E_{\boldsymbol{p}}\pm_{2}i0^{+}\right)}\exp\left(z_{2}t\right)\] {#eqG-complex-2-1}
where each \(\pm_{1}\) and \(\pm_{2}\) are independent (see Fig 2.2).
Each choice gives a valid Green’s function, solving the defining equation
\[\left(\partial^{2}+m^{2}\right)G\left(x\right)=-i\delta^{4}\left(x\right).\]
But the physical behaviour of each solution is very different. It is convenient to define the Wightman function
\[D\left(x-y\right)=\langle\hat{\varphi}_{x}\hat{\varphi}_{y}\rangle=\int\frac{\text{d}^{3}\boldsymbol{p}}{\left(2\pi\right)^{3}}\exp\left(-ip\cdot\left(x-y\right)\right)\]
i.e. the propagator we have already seen, but without requiring time ordering. This way, we have already seen
The Feynman propagator:
\[G_{+-}\left(x-y\right)=\Delta_{F}\left(x-y\right)=\langle\mathcal{T}\hat{\varphi}_{x}\hat{\varphi}_{y}\rangle=\Theta\left(x^{0}-y^{0}\right)D\left(x-y\right)+\Theta\left(y^{0}-x^{0}\right)D\left(y-x\right).\]
The reverse choice gives
\[G_{-+}\left(x-y\right)=\Theta\left(y^{0}-x^{0}\right)D\left(x-y\right)+\Theta\left(x^{0}-y^{0}\right)D\left(y-x\right).\]
I haven’t seen this one used. Instead choosing \(\left(-,-\right)\) puts both poles below the real axis. Hence, we can close the contour in the upper half plane, where \(t>0\). This is the retarded Green’s function: it propagates a disturbance from the present, \(t=0\), to the future (having support in the future light cone). For this reason it is also called the causal Green’s function. Mathematically,
\[G_{--}\left(x-y\right)=G_{\text{R}}\left(x-y\right)=\Theta\left(x^{0}-y^{0}\right)\left(D\left(x-y\right)-D\left(y-x\right)\right).\]
Choosing \(\left(+,+\right)\) puts both poles above the real axis. The contour now closes in the lower half plane, \(t<0\), having support in the past light cone. This is the advanced Green’s function: it propagates a disturbance from the past to the present. It is useful if you specify as a boundary condition the current situation, and ask what happened in the past to cause this. Mathematically,
\[G_{++}\left(x-y\right)=G_{\text{A}}\left(x-y\right)=\Theta\left(y^{0}-x^{0}\right)\left(D\left(y-x\right)-D\left(x-y\right)\right).\]
2.7 Microcausality
2.7.1 Amplitudes versus signalling
We motivated the need for QFT by observing that relativistic single-particle propagators predict non-zero amplitudes to signal over spacelike separations. We did this by constructing a propagator using the single-particle Klein Gordon equation in Section @subsec-The-Failure-of-single-particle-QM. We subsequently formulated the multi-particle Klein Gordon QFT, then we found its single-particle propagator in Eq 2.16. Has this actually helped? To answer this, we can again do the integral exactly. From Eq 2.16, assuming \(x^{0}>0\), we have
\[\begin{aligned} G\left(x\right) & =\left(2\pi\right)^{-3}\int\frac{\text{d}^{3}\boldsymbol{p}}{2E_{\boldsymbol{p}}}\exp\left(i\boldsymbol{p\cdot}\boldsymbol{x}\right)\exp\left(-iE_{\boldsymbol{p}}x^{0}\right)\\ & =\left(2\pi\right)^{-2}\int\frac{\text{d}\left|\boldsymbol{p}\right|\text{d}\left(\cos\left(\theta\right)\right)\boldsymbol{p}^{2}}{2E_{\boldsymbol{p}}}\exp\left(i\left|\boldsymbol{p}\right|\left|\boldsymbol{x}\right|\cos\left(\theta\right)\right)\exp\left(-iE_{\boldsymbol{p}}x^{0}\right)\\ & =\frac{i}{\left(2\pi\right)^{2}\left|\boldsymbol{x}\right|}\int\frac{\text{d}\left|\boldsymbol{p}\right|\left|\boldsymbol{p}\right|}{E_{\boldsymbol{p}}}\sin\left(\left|\boldsymbol{p}\right|\left|\boldsymbol{x}\right|\right)\exp\left(-iE_{\boldsymbol{p}}x^{0}\right) \end{aligned}\]
on which we can use integration by parts (noting that \(\boldsymbol{\left|p\right|}\exp\left(-iE_{\boldsymbol{p}}x^{0}\right)/E_{\boldsymbol{p}}\) happens to be an exact differential!) to give
\[G\left(x\right)=\frac{i}{\left(2\pi\right)^{2}x^{0}}\int\text{d}\left|\boldsymbol{p}\right|\cos\left(\left|\boldsymbol{p}\right|\left|\boldsymbol{x}\right|\right)\exp\left(-iE_{\boldsymbol{p}}x^{0}\right).\]
We can again look this up in Gradsteyn and Ryzhik (7th Edition, Eq 3.914.1) where we find
\[\int^{\infty}_{0}\text{d}p\exp\left(-\beta\sqrt{\gamma^{2}+p^{2}}\right)\cos\left(bp\right)=\frac{\beta\gamma}{\sqrt{\beta^{2}+b^{2}}}K_{1}\left(\gamma\sqrt{\beta^{2}+b^{2}}\right)\]
and so
\[\begin{aligned} G\left(x-y\right) & =\frac{1}{\left(2\pi\right)^{2}}\frac{m}{\sqrt{\left|\boldsymbol{x}-\boldsymbol{y}\right|^{2}-\left(x_{0}-y_{0}\right)^{2}}}K_{1}\left(m\sqrt{\left|\boldsymbol{x}-\boldsymbol{y}\right|^{2}-\left(x_{0}-y_{0}\right)^{2}}\right)\\ & =\frac{1}{\left(2\pi\right)^{2}}\frac{m}{\Delta s}K_{1}\left(m\Delta s\right) \end{aligned}\]
where
\[\Delta s=\sqrt{\left|\boldsymbol{x}-\boldsymbol{y}\right|^{2}-\left(x_{0}-y_{0}\right)^{2}}\]
is the proper length separating the events. As before, let’s take the case \(y=0\), with \(x\) very spacelike separated. Here we can use the asymptotic expression for \(K_{1}\):
\[\lim_{x\rightarrow\infty}K_{1}\left(x\right)\sim\sqrt{\frac{\pi}{2x}}\exp\left(-x\right)\]
to find
\[\lim_{\Delta s\rightarrow\infty}G\left(x\right)\sim\frac{1}{\left(2\pi\right)^{2}}\frac{\sqrt{\pi m}}{\sqrt{2}\Delta s^{3/2}}\exp\left(-m\Delta s\right)\]
which, as before, is exponentially small but non-zero outside the light cone! So what has gone wrong?
The problem lies with our interpretation. The single-particle propagator
\[G\left(x-y\right)=\langle\Omega|\mathcal{T}\hat{\varphi}_{\boldsymbol{y}}\hat{\varphi}_{\boldsymbol{x}}|\Omega\rangle\]
gives the amplitude to find the particle at \(x\) given it was at \(y\), or equivalently to create a particle at \(x\) and annihilate one at \(y\). In single-particle QM this would mean the particle travelled from \(y\) to \(x\). But in QFT this is not true. A particle does not need to travel from \(y\) to \(x\) to be found at \(y\) at one instant, then \(x\) at the next. Rather, all particles are excitations of an underlying quantum field which existis everywhere and which can create particles anywhere.
The propagator in QFT is simply telling you that there is some correlation between the field amplitudes at \(x\) and \(y\). That is perfectly natural. When I read the headline of my newspaper in the morning, my observation is perfectly correlated with that of my friend who reads the same newspaper at the same instant on the other side of the country, even though those events are spacelike separated. Similarly, the electron field exists across all of spacetime, and so there is a non-zero amplitude for you to find an electron in a box in front of you right now, and for your friend somewhere in the vicinity of Andromeda to find an electron in a box in front of them 3 seconds later. That doesn’t allow you to signal between each other. You might even choose to interpret their electron to be your electron if you so wish, although since all electrons are identical that would be rather arbitrary.
What we do require is that we cannot signal at spacelike separation. This is encoded in QFT as the postulate of
microcausality: operators \(\hat{\mathcal{O}}_{x}\) and \(\mathcal{\hat{O}}_{y}\) corresponding to spacelike-separated (bosonic) observables \(\left(x-y\right)^{2}<0\) must commute:
\[\left[\hat{\mathcal{O}}_{x},\hat{\mathcal{O}}_{y}\right]=0.\]
Hence, \(G=\langle\Omega|\mathcal{T}\hat{\varphi}_{\boldsymbol{y}}\hat{\varphi}_{\boldsymbol{x}}|\Omega\rangle\) is simply not the correct object to consider when asking about signalling in QFT (although it is in single-particle QM). In QFT we instead need to consider
\[\langle\Omega|\left[\hat{\varphi}_{\boldsymbol{y}},\hat{\varphi}_{\boldsymbol{x}}\right]|\Omega\rangle.\]
You can expand the field operators in terms of creation and annihilation operators (in the Heisenberg picture) to show explicitly that this vanishes for spacelike separations. However, there ought to be some more fundamental argument as to why relativistic QFTs disallow spacelike signalling. Peskin & Schroeder give a neat argument for real fields, as follows. Define
\[D\left(x-y\right)\triangleq\langle\Omega|\hat{\varphi}_{\boldsymbol{x}}\hat{\varphi}_{\boldsymbol{y}}|\Omega\rangle\]
so that
\[\langle\Omega|\left[\hat{\varphi}_{\boldsymbol{x}},\hat{\varphi}_{\boldsymbol{y}}\right]|\Omega\rangle=D\left(x-y\right)-D\left(y-x\right).\]
In 3+1D, all spacelike points can be smoothly transformed into one another with an appropriate Lorentz transformation (see Fig 2.4). Hence, whenever \(\left(x-y\right)^{2}<0\), we can always find some Lorentz transformation \(\Lambda^{\mu}_{\phantom{\mu}\nu}\) such that \(\Lambda^{\mu}_{\phantom{\mu}\nu}\left(x^{\nu}-y^{\nu}\right)=y^{\mu}-x^{\mu}\) (note that this is not possible for timelike separated events, as the timelike region contains two disconnected regions – past and future). This means that in some frame of reference \(D\left(x-y\right)=D\left(y-x\right)\), and so \(\langle\Omega|\left[\hat{\varphi}_{\boldsymbol{y}},\hat{\varphi}_{\boldsymbol{x}}\right]|\Omega\rangle=0\). But this is a Lorentz invariant quantity, and so must be independent of reference frame. Hence, \(\langle\Omega|\left[\hat{\varphi}_{\boldsymbol{y}},\hat{\varphi}_{\boldsymbol{x}}\right]|\Omega\rangle=0\) whenever \(\left(x-y\right)^{2}<0\). This is microcausality, and it is the statement of no signalling in QFT.
2.8 2-point functions
We saw in the previous section that the interpretation of the Green’s function / propagator is a bit different in QFT compared to single-particle QM, since we no longer have to associate seeing a particle at \(x\) then at \(y\) with the particle propagating between the points. An alternative name for the object
\[G\left(x-y\right)=\langle\Omega|\mathcal{T}\hat{\varphi}_{\boldsymbol{x}}\hat{\varphi}_{\boldsymbol{y}}|\Omega\rangle\]
is the ‘2-point function’. This emphasises that it really just gives the amplitude for particles to be found at two spacetime points. The nomenclature will prove helpful when we look at interacting quantum fields in the next section, where we will see general \(N\)-point functions (correlation amplitudes for multiple particles). Returning to the path integral formalism, in which fields are commuting scalars but where quantum non-commutation arises from the jaggedness of the functional field integral, the 2-point function is given by
\[\langle\varphi_{y}\varphi_{x}\rangle\triangleq\frac{\int\mathscr{D}\varphi\varphi_{y}\varphi_{x}\exp\left(iS\left[\varphi\right]\right)}{\int\mathscr{D}\varphi\exp\left(iS\left[\varphi\right]\right)}.\]
There is a neat trick for calculating 2-point functions which we will turn to now.
2.8.1 The Partition Function
A key quantity of interest when it comes to practical QFT calculations is
the partition function:
\[\mathscr{Z}\triangleq\int\mathscr{D}\varphi\exp\left(iS\left[\varphi\right]\right).\]
Recall that in statistical mechanics, while the partition function itself is a slightly abstract object, it is incredibly useful since all functions of state can be obtained from it (typically by taking derivatives). The same is true in QFT: all physical observables can be obtained from the partition function. This perhaps makes sense since the QFT is entirely specified by its action.
To proceed, let’s look at a specific example.
2.8.2 The Klein Gordon Partition Function
The partition function for the real Klein Gordon field is
\[\mathscr{Z}=\int\mathscr{D}\varphi\exp\left(\frac{i}{2}\int\text{d}^{4}x\left(\partial^{\mu}\varphi_{x}\partial_{\mu}\varphi_{x}-m^{2}\varphi^{2}_{x}\right)\right).\]
Integrating by parts we can rewrite this as
\[\mathscr{Z}=\int\mathscr{D}\varphi\exp\left(-\frac{i}{2}\int\text{d}^{4}x\varphi_{x}\left(\partial^{2}+m^{2}\right)\varphi_{x}\right).\]
We can Fourier transform to give
\[\mathscr{Z}=\int\mathscr{D}\varphi\exp\left(-\frac{i}{2}\int\frac{\text{d}^{D+1}p}{\left(2\pi\right)^{D+1}}\tilde{\varphi}_{p}\left(-p^{2}+m^{2}\right)\tilde{\varphi}_{-p}\right)\]
But recall that the Green’s function of the Klein Gordon equation in Fourier space is
\[\tilde{G}_{p}=\frac{i}{p^{2}-m^{2}}\]
and so
\[\mathscr{Z}=\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\frac{\text{d}^{D+1}p}{\left(2\pi\right)^{D+1}}\tilde{\varphi}_{p}\tilde{G}^{-1}_{p}\tilde{\varphi}_{-p}\right).\]
We can return to real space, at least formally, with
\[\mathscr{Z}=\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\text{d}^{4}x\int\text{d}^{4}y\varphi_{x}G{}^{-1}_{x-y}\varphi_{y}\right)\]
where \(G^{-1}_{x}\) is the inverse[^7] of the differential operator \(i\left(\partial^{2}+m^{2}\right)\) (we will neglect the hat on \(G\) for notational ease, and since it will always either be in either th position or momentum basis). This turns out to be a general rule for QFTs: the action for the free (non-interacting) field is simply two fields sandwiching the inverse of the Green’s function.
Much of the utility of the partition function comes from the fact it is a Gaussian functional integral, and these are basically the only types of functional integrals we can do analytically. In fact it is this reason that we can do QFT at all! Let’s remind ourselves about Gaussian integrals before putting this idea to use.
2.8.3 Gaussian integrals
1D Gaussian integral
Recall the form of a 1D Gaussian integral:
\[I\left(a\right)=\int^{\infty}_{-\infty}\text{d}x\exp\left(-\frac{1}{2}ax^{2}\right),\quad a\in\mathbb{C},\,\,\mathfrak{Re}\left(a\right)\ge0.\]
This can be evaluated using an elegant trick, which is worth remembering:
\[\begin{aligned} I\left(a\right)^{2} & =\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\text{d}x\text{d}y\exp\left(-\frac{1}{2}a\left(x^{2}+y^{2}\right)\right)\\ & =\int^{2\pi}_{0}\text{d}\theta\int^{\infty}_{0}\text{d}rr\exp\left(-\frac{1}{2}ar^{2}\right)\\ & =\frac{2\pi}{a} \end{aligned}\]
and so
\[I\left(a\right)=\sqrt{\frac{2\pi}{a}}.\]
1D Gaussian integral with a source
Now consider
\[I\left(a,b\right)=\int^{\infty}_{-\infty}\text{d}x\exp\left(-\frac{1}{2}ax^{2}+bx\right).\]
This can be done by completing the square:
\[\begin{aligned} I\left(a,b\right) & =\int^{\infty}_{-\infty}\text{d}x\exp\left(-\frac{1}{2}a\left(x^{2}-2\frac{b}{a}x\right)\right)\\ & =\int^{\infty}_{-\infty}\text{d}x\exp\left(-\frac{1}{2}a\left(\left(x-\frac{b}{a}\right)^{2}-\left(\frac{b}{a}\right)^{2}\right)\right)\\ & =\exp\left(\frac{b^{2}}{2a}\right)\int^{\infty}_{-\infty}\text{d}x\exp\left(-\frac{1}{2}a\left(x-\frac{b}{a}\right)^{2}\right)\\ & =\exp\left(\frac{b^{2}}{2a}\right)\int^{\infty}_{-\infty}\text{d}x\exp\left(-\frac{1}{2}ax^{2}\right)\\ & =\sqrt{\frac{2\pi}{a}}\exp\left(\frac{b^{2}}{2a}\right) \end{aligned}\]
where we changed integration variables to \(x'=x-b/a\) and used the fact that the integral has an infinite range.
This calculation turns out to be very handy. If we denote the expectation value of \(x^{n}\) to be
\[\left\langle x^{n}\right\rangle =\int^{\infty}_{-\infty}\text{d}xx^{n}\exp\left(-\frac{1}{2}ax^{2}+bx\right)\]
then we have
\[\begin{aligned} \left\langle x^{n}\right\rangle & =\left(\frac{\partial}{\partial b}\right)^{n}\int^{\infty}_{-\infty}\text{d}x\exp\left(-\frac{1}{2}ax^{2}+bx\right)\\ & =\left(\frac{\partial}{\partial b}\right)^{n}\sqrt{\frac{2\pi}{a}}\exp\left(\frac{b^{2}}{2a}\right) \end{aligned}\]
which is trivial to evaluate. We will shortly adopt this trick in QFT.
\(N\)-dimensional Gaussian integral
Now consider an \(N\)-dimenional Gaussian integral
\[I\left(A\right)=\int\text{d}^{N}\boldsymbol{x}\exp\left(-\frac{1}{2}\sum^{N}_{i=1}\sum^{N}_{j=1}x^{T}_{i}A_{ij}x_{j}\right)\]
where \(A\) is a symmetric positive definite matrix (meaning all its eigenvalues are strictly positive). To do this integral, note that we can diagonalise \(A\):
\[A=ODO^{T}\]
where \(O\) is an orthogonal matrix:
\[O^{T}O=\mathbb{I} \tag{2.20}\]
and
\[D=\text{diag}\left(a_{1},a_{2},\ldots a_{N}\right).\]
Therefore, defining
\[\tilde{\boldsymbol{x}}\triangleq O^{T}\boldsymbol{x}\]
gives
\[\begin{aligned} \boldsymbol{x}^{T}A\boldsymbol{x} & =\tilde{\boldsymbol{x}}^{T}D\tilde{\boldsymbol{x}}\\ & =\sum^{N}_{i=1}a_{i}\tilde{x}^{2}_{i}. \end{aligned}\]
In this diagonal basis the integral reduces to
\[I\left(A\right)=\prod^{N}_{i=1}\int\text{d}\tilde{x}_{i}\exp\left(-\frac{1}{2}a_{i}\tilde{x}^{2}_{i}\right) \tag{2.21}\]
where the orthogonality condition, Eq 2.20, ensures that the Jacobian of the transformation (the equivalent of the \(r\) that appeared when switching to plane polar co-ordinates in 1D) is unity. Eq 2.21 is just a product of 1D Gaussian integrals, and so
\[I\left(A\right)=\left(2\pi\right)^{N/2}\left(a_{1}a_{2}\ldots a_{N}\right)^{-1/2}.\]
Recalling that the determinant of a matrix is the product of its eigenvalues, this gives
\[I\left(A\right)=\sqrt{\frac{\left(2\pi\right)^{N}}{\text{det}\left(A\right)}.}\]
Gaussian Functional Integral
The limit \(N\rightarrow\infty\) can be taken in the previous result without issue. Recalling that functions are infinite-dimensional vectors, and differential operators can be thought of as infinity-by-infinity matrices, we have
\[\begin{aligned} I\left(\hat{A}\right) & =\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\varphi_{x}\hat{A}_{xy}\varphi_{y}\right)\\ & =\sqrt{\frac{\left(2\pi\right)^{\infty}}{\text{det}\left(\hat{A}\right)}} \end{aligned}\]
where the determinant of an operator is again the product of its eigenvalues. The infinity does not prove to be a problem in practice, as we end up dividing throught by it later.
Notation: the symbol \(\hat{A}_{xy}\) is intended to show the connection to the \(N\times N\) matrix \(A_{ij}\) above. Operators that act locally (i.e. not at multiple spacetime events) must take the form
\[\hat{A}_{xy}=\delta^{D+1}\left(x-y\right)\hat{A}_{x}\]
so that
\[I\left(\hat{A}\right)=\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\varphi_{x}\hat{A}_{x}\varphi_{x}\right).\]
It is sometimes convenient to use \(\hat{A}_{xy}\), sometimes \(\hat{A}_{x}\). We will use both below.
Gaussian Functional Integral with a source
As in 1D, we can introduce a source to our functional integral.
\[\begin{aligned} I\left[J\right] & =\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\varphi_{x}\hat{A}_{xy}\varphi_{y}+\int\text{d}^{D+1}x\varphi_{x}J_{x}\right). \end{aligned}\]
Taking inspiration from 1D, we can make a change of variables
\[\varphi_{x}\rightarrow\varphi_{x}-\int\text{d}^{D+1}z\hat{A}^{-1}_{xz}J_{z}\]
to give
\[\begin{aligned} I\left[J\right] & =\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\left(\varphi_{x}-\int\text{d}^{D+1}z\hat{A}^{-1}_{xz}J_{z}\right)\hat{A}_{xy}\left(\varphi_{y}-\int\text{d}^{D+1}z\hat{A}^{-1}_{yz}J_{z}\right)\right.\\ & \left.+\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}\hat{A}^{-1}_{xy}J_{y}\right)\nonumber \\ & =\exp\left(\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}\hat{A}^{-1}_{xy}J_{y}\right)I\left[0\right]. \end{aligned}\]
Hence
\[\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\varphi_{x}\hat{A}_{xy}\varphi_{y}+\int\text{d}^{D+1}x\varphi_{x}J_{x}\right)=\exp\left(\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}\hat{A}^{-1}_{xy}J_{y}\right)\sqrt{\frac{\left(2\pi\right)^{\infty}}{\text{det}\left(\hat{A}\right)}} \tag{2.22}\]
which proves invaluable for calculating \(N\)-point functions.
2.9 2-point functions from the generating functional
The term \(J\left(x\right)\) can be thought of as a fixed, specified function which acts as a current (source) of \(\varphi\) particles. Returning to our generic free field theory, the partition function with a source is now
\[\mathscr{Z}\left[J\right]=\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\varphi_{x}\hat{G}^{-1}_{xy}\varphi_{y}+i\int\text{d}^{D+1}x\varphi_{x}J_{x}\right). \tag{2.23}\]
This is called the generating functional. Following the working above gives the exact result
\[\mathscr{Z}\left[J\right]=\left(2\pi\right)^{\infty/2}\sqrt{\text{det}\left(\hat{G}\right)}\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}\hat{G}_{xy}J_{y}\right). \tag{2.24}\]
It is customary to drop the hat on the Green’s function for notational convenience. The importance of this is that we can readily use it to generate \(N\)-point functions. For example, the two-point function (propagator) is given by
\[\langle\mathcal{T}\varphi_{x_{1}}\varphi_{x_{2}}\rangle=\mathscr{Z}^{-1}\left.\left(-i\frac{\delta}{\delta J_{x_{2}}}\right)\left(-i\frac{\delta}{\delta J_{x_{1}}}\right)\mathscr{Z}\left[J\right]\right|_{J=0}.\]
Let’s check this explicitly. I will define
\[\begin{aligned} \varphi_{1} & \triangleq\varphi_{x_{1}}\\ G_{12} & \triangleq G_{x_{1}x_{2}}=G\left(x_{1},x_{2}\right) \end{aligned}\] and so on, to save the notation getting too messy (this notation is used in some textbooks). Then
\[\begin{aligned} \mathscr{Z}^{-1}\left(-i\frac{\delta}{\delta J_{2}}\right)\left(-i\frac{\delta}{\delta J_{1}}\right)\mathscr{Z}\left[J\right] & =\mathscr{Z}^{-1}\int\mathscr{D}\varphi\left(-i\frac{\delta}{\delta J_{2}}\right)\left(-i\frac{\delta}{\delta J_{1}}\right)\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\varphi_{x}G{}^{-1}_{xy}\varphi_{y}+i\int\text{d}^{D+1}x\varphi_{x}J_{x}\right)\\ & =\mathscr{Z}^{-1}\int\mathscr{D}\varphi\varphi_{1}\varphi_{2}\exp\left(-\frac{i}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\varphi_{x}G{}^{-1}_{xy}\varphi_{y}+i\int\text{d}^{D+1}x\varphi_{x}J_{x}\right) \end{aligned}\]
and so
\[\begin{aligned} \left.\mathscr{Z}^{-1}\left(-i\frac{\delta}{\delta J_{2}}\right)\left(-i\frac{\delta}{\delta J_{1}}\right)\mathscr{Z}\left[J\right]\right|_{J=0} & =\frac{\int\mathscr{D}\varphi\varphi_{1}\varphi_{2}\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\varphi_{x}G{}^{-1}_{xy}\varphi_{y}\right)}{\int\mathscr{D}\varphi\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}y\varphi_{x}G{}^{-1}_{xy}\varphi_{y}\right)}\\ & =\frac{\int\mathscr{D}\varphi\varphi_{2}\varphi_{1}\exp\left(iS_{0}\right)}{\int\mathscr{D}\varphi\exp\left(iS_{0}\right)} \end{aligned}\]
as required. We can now calculate this using the exact solution in Eq 2.23:
\[\begin{aligned} \left.\mathscr{Z}^{-1}\left(-i\frac{\delta}{\delta J_{2}}\right)\left(-i\frac{\delta}{\delta J_{1}}\right)\mathscr{Z}\left[J\right]\right|_{J=0} & =\mathscr{Z}^{-1}\left(2\pi\right)^{\infty/2}\sqrt{\text{det}\left(G\right)}\left.\left(-i\frac{\delta}{\delta J_{2}}\right)\left(-i\frac{\delta}{\delta J_{1}}\right)\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}G_{xy}J_{y}\right)\right|_{J=0}\\ & =\left.\left(-i\frac{\delta}{\delta J_{2}}\right)\left(-i\frac{\delta}{\delta J_{1}}\right)\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}G_{xy}J_{y}\right)\right|_{J=0}\\ & =\left.\left(-i\frac{\delta}{\delta J_{2}}\right)\left(\frac{i}{2}\int\text{d}^{D+1}xJ_{x}G_{x1}+\frac{i}{2}\int\text{d}^{D+1}xJ_{x}G_{1x}\right)\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}G_{xy}J_{y}\right)\right|_{J=0}\\ & =\left(\frac{1}{2}\left(G_{12}+G_{21}\right)+\left(\frac{i}{2}\right)^{2}\left(\int\text{d}^{D+1}xJ_{x}\left(G_{x1}+G_{1x}\right)\right)\left(\int\text{d}^{D+1}xJ_{x}\left(G_{x2}+G_{2x}\right)\right)\right)\left.\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}G_{xy}J_{y}\right)\right|_{J=0}\\ & =G_{12} \end{aligned}\]
noting in the last line that \(G_{12}=G_{21}\). This confirms that the 2-point function is the Green’s function (up to a normalisation).
2.10 Wick’s Theorem
Another vital tool in QFT is Wick’s theorem. To understand it, let’s calculate some higher \(N\)-point functions from the generating functional. First, we can see that any \(2N+1\)-point function must vanish, since this involves an infinite integral over a product of an odd and even functional.
Let’s look at the 4-point function, which corresponds (for example) to 2-particle scattering. Then
\[\begin{aligned} \langle\mathcal{T}\varphi_{4}\varphi_{3}\varphi_{2}\varphi_{1}\rangle & =\mathscr{Z}^{-1}\left(-i\right)^{4}\left.\frac{\delta}{\delta J_{1}}\frac{\delta}{\delta J_{2}}\frac{\delta}{\delta J_{3}}\frac{\delta}{\delta J_{4}}\mathscr{Z}\left[J\right]\right|_{J=0}\\ & =\left(-i\right)^{2}\frac{\delta}{\delta J_{1}}\frac{\delta}{\delta J_{2}}\left(G_{34}+i^{2}\left(\int\text{d}^{D+1}xJ_{x}G_{x3}\right)\left(\int\text{d}^{D+1}xJ_{x}G_{x4}\right)\right)\left.\exp\left(-\frac{1}{2}\int\text{d}^{D+1}x\int\text{d}^{D+1}yJ_{x}G_{xy}J_{y}\right)\right|_{J=0}\\ & =G_{12}G_{34}+G_{13}G_{24}+G_{14}G_{23}. \end{aligned}\]
That is, the 4-point function is simply the sum of all possible products of pairs of 2-point functions! In general, we have
Wick’s Theorem: the \(N\)-point function of a Gaussian theory is given by all possible products of pairings of 2-point functions.
Wick’s theorem follows from the fact that the action is quadratic in the fields, and so our partition function is Gaussian. In statistics, you may have heard of this result as the fact that Normal (Gaussian) distributions are entirely characterised by their mean and variance. In the end, this is why QFT describes reality as harmonic oscillators at every point in spacetime: if it were anything other than a harmonic oscillator at each point, we would not be able to apply Wick’s theorem.
A convenient representation of this pairing is shown in Fig 2.6. These are examples of Feynman diagrams, of which we will see much more in the next section.
