Jakob Schwichtenberg

Demystifying the QCD Vacuum: Part 4 – Physical Implications of $\theta$

In the last parts, we have seen how a new parameter $\theta$ can emerge when we take a closer look at the structure of the QCD vacuum. Here we continue the standard story that was discussed in the first part.

The question, we would like to answer is: How can we incorporate the emergence of $\theta$ into our formalism?

The true QCD ground state: $|\theta\rangle$

Recall that, at least in the temporal gauge, we found a periodic structure of the QCD vacuum. The minima correspond to field configurations with different integer winding numbers. The basic idea is that the correct vacuum state is not one of the configurations with some definite winding number, but instead a superposition of all of them. This superposition emerges because the different degenerate ground states are connected by instanton processes. This simply means that the field can change from a configuration with some winding number $n$ into a configuration with a different winding number, through a tunnel process. This tunneling process through the potential barrier that separates the different ground states is what we call an instanton.

In addition, recall that we found a classification for all gauge transformations that satisfy $U (x) \to 1$ for $|x| \to \infty$. This classification made use of the label “winding number”. The thing is now that a ground state with definite winding number $n$ is not gauge invariant. A gauge transformation with winding number $n’$ changes the ground state $|n\rangle $ into $|n’\rangle $. Thus, if we want a gauge invariant ground state, we need to construct a superposition. This is another possibility to see why our true ground state is not one with a definite winding number, but a superposition.

We write this superposition as

$$ |\theta\rangle = \sum_{n=-\infty}^\infty e^{in\theta} |n\rangle $$

This superposition $ |\theta\rangle$ is known as the theta vacuum. This state only changes by a phase if we act on it with a gauge transformation. For example, when we act with a gauge transformation $g_1$ with winding number $1$ on it, we get

\begin{align}
g_1 |\theta\rangle &= g_1 \sum_{n=-\infty}^\infty e^{in\theta} |n \rangle \notag \\
&= g_1 \sum_{n=-\infty}^\infty e^{in\theta} |n \rangle \notag \\
&= \sum_{n=-\infty}^\infty e^{in\theta} g_1 |n \rangle \notag \\
&= \sum_{n=-\infty}^\infty e^{in\theta} |n +1 \rangle \notag \\
&= \sum_{n’=-\infty}^\infty e^{i(n’-1)\theta} |n’\rangle \notag \\
&= e^{-i\theta} \sum_{n’=-\infty}^\infty e^{i(n’)\theta} |n’\rangle \notag \\
&= e^{-i\theta} |\theta\rangle \notag \\
\end{align}

where we defined $n+1 = n’$ to bring the sum back to its original from. Thus, we can say that $\theta$ is an eigenstate of the operator $g_1$ with eigenvalue $e^{-i\theta}$. Moreover, since the Hamiltonian is invariant under gauge transformations, i.e. $[g_1,H]=0$ holds, this means that $|\theta\rangle$ can be simultaneously an energy eigenstate.

By invoking the analogy with an electron in a periodic potential, one can show that the energy of the ground state depends on $\theta$. In the analogous model, the different values of the Bloch momentum label different energy bands. The energy density of each such energy band is approximately $E_\theta / L = C- e^{-S_0} 2 B \cos(\theta) $, where $L$ denotes the “length of space”. An important observation here is that the band that corresponds to $\theta =0$ has the lowest energy density.

Another important observation is that the value of $\theta$ is fixed and cannot be changed. This can be seen as follows:

Consider a gauge invariant operator $B$. Gauge invariance means that $[g_1 ,B]=0$ holds. Therefore, we can compute

\begin{align}
0 &= \langle \theta | [g_1 ,B] |\theta’\rangle \notag \\
&= \langle \theta | g_1 B |\theta’\rangle – \langle \theta | B g_1 |\theta’\rangle \notag \\
&= e^{-i\theta} \langle \theta | B |\theta’\rangle – e^{-i\theta’} \langle \theta | B |\theta’\rangle \notag \\
&= (e^{-i\theta} – e^{-i\theta’})\langle \theta | B |\theta’\rangle .
\end{align}

This means that $ \langle \theta | B |\theta’\rangle =0$ unless $\theta = \theta ‘$. In other words, the value of $\theta $ cannot be changed by a gauge invariant operator! One such gauge invariant operator is, of course, the Hamiltonian $H$ and therefore our calculation shows that the value $\theta$ does not change as time moves on. Therefore $|\theta\rangle$ is really an energy eigenstate.

The true true QCD ground state: $|\phi\rangle$

We noted above, that $|\theta\rangle$ changes under gauge transformations by a phase: $g_1 |\theta\rangle= e^{-i\theta} |\theta\rangle $. Now want to get rid of the phase change $ e^{-i\theta}$ and use a ground state instead that is completely invariant under gauge transformations.

So, we first need to remember that our state $ | \theta\rangle$ describes a state of the gauge fields $A$. To emphasize this, we write now denote the ground state as $ | \theta [A]\rangle$.

We now recall the definition of the winding number of a field configuration $A$: $W[A]$, which was discussed in the first part.

Using $W[A]$, we can define a ground state that is completely unchanged by gauge transformations:

\begin{align}
|\phi\rangle &= e^{-iW[A]\theta} |\theta \rangle
\end{align}

A crucial property of $W[A]$ is that if we act on it with a gauge transformation of winding number $n$ it gets shifted by $n$:

$$ g_n W[A] g_n^{-1} = W[A] +n $$

This sounds, of course, reasonable, but can also be checked explicitly. (See, for example, Eq. 3.36-3.39 in “Topological investigations of quantized gauge theories” by R. Jackiw. The winding number $W[A]$ is a functional of the gauge field $A$. When we act on it with a gauge transformation, we therefore transform its argument, the gauge field: $g_n W[A] g_n^{-1} = W[A’] $, where $A’$ denotes the gauge transformed gauge field. We then put the gauge transformed gauge field into the formula for the winding number and then notice that this formula is not invariant. Instead we get an additional term. This additional term is exactly the winding number of the gauge transformation $g_n$.)

Now, to see that this new ground state is invariant under all gauge transformations, we act on it with a gauge transformation

\begin{align}
g_n |\phi\rangle &= g_n e^{-i\hat W\theta} |\theta \rangle \notag \\
&= g_n e^{-i\hat W\theta} g_n^{-1} g_n|\theta \rangle \notag \\
&= e^{-i (\hat W +n) \theta} g_n|\theta \rangle \notag \\
&= e^{-i (\hat W +n) \theta} e^{i n \theta} n|\theta \rangle \notag \\
&= e^{-i\hat W\theta} |\theta \rangle \\
&= |\phi\rangle \end{align}

Thus, have now found a truly gauge invariant ground state of the QCD gauge fields. With this construction at hand, we are finally ready to investigate the influence of the phase $\theta$ on the actual physics.

Physics in the $\theta$ vacuum

Now let’s consider the type of object that we usually consider in QFT, i.e. vacuum to vacuum transitions, but in the presence of the $\theta$ vacuum. As noted above, $|\theta \rangle$ is an eigenstate of $H$. In the path integral formalism and recalling that we work in Euclidean spacetime we have write

\begin{align}
e^{iE_\theta \tau} &= \langle \theta | e^{-H \tau} |\theta\rangle \notag \\
&= \mathcal{N} \int (dA_1) (dA_2)\int_{A_1}^{A_2}(DA) \langle \theta[A_2] | e^{-S_E[A]} |\theta \theta[A_1 ]\rangle,
\end{align}

where $S_E$ denotes the Euclidean action and $dA_1$ the integration over time-dependent functions $A_1(x)$. Using a “dilute gas” approximation for the occurrence of instantons, one can get an approximation for $E_\theta$ which is similar to the one quoted for the Bloch energy bands quoted above.

We now want to rewrite this expression in terms of the gauge invariant ground state $|\phi\rangle$. Using $|\phi\rangle = e^{-iW[A]\theta} |\theta \rangle $, we get $ e^{iW[A]\theta}|\phi\rangle = |\theta \rangle $ and therefore

\begin{align}
e^{iE_\theta \tau} &= \langle \theta | e^{-H \tau} |\theta\rangle \notag \\
&= \mathcal{N} \int (dA_1) (dA_2)\int_{A_1}^{A_2}(DA) \langle \phi[A_2] | e^{-iW[A_2]\theta} e^{-S_E[A]} e^{iW[A_2]\theta} |\phi[A_1]\rangle \notag \\
&= \mathcal{N} \int (dA_1) (dA_2)\int_{A_1}^{A_2}(DA) \langle \phi[A_2] | e^{-S_E[A]+ i(W[A_1]-W[A_2])\theta} |\phi[A_1]\rangle  \notag \\
&= \mathcal{N} \int (dA_1) (dA_2)\int_{A_1}^{A_2}(DA) \langle \phi[A_2] | e^{-S_E[A]+ iW[A]\theta} |\phi[A_1]\rangle .
\end{align}

(For an alternative derivation see Eq. 10.70-10.72 and Eq. 10.98 in “Instantons and Solitons” by Rajaraman)

This means that the ingredient that we get if we want to take into account the complex structure of the QCD vacuum is simply a new term $ iW[A]\theta $ that gets added to the action.

In Minkowski space this simply means that we need to add a term

$$\Delta \mathcal L = \frac{\theta}{16 \pi^2} Tr[G_{\mu\nu} \tilde{G}^{\mu\nu}] $$

to the original Lagrangian.

To get a physical understanding of this term, we rewrite it in terms of the more familiar “color-E-field” and “color-B-field”, which are simply analogous to the electrical-E-field and the magnetic-B-field. In terms of these, we have

$$ Tr[G_{\mu\nu} \tilde{G}^{\mu\nu}] = Tr[4E_i B_i] ,$$

where the relationship between the field strength tensor $G_{\mu\nu}$ and the fields $E_i$ and $B_i$ is completely analogous to the definition in the electromagnetic theory.

Now there is something important, we can note immediately:

Under parity transformations, we have $E_i \to – E_i$ and $B_i \to B_i$.
Under a time-reversal transformation, we have $E_i \to E_i$ and $B_i \to – B_i$.

Thus, while the original Lagrangian $\mathcal L$ is invariant under both parity and time-reversal transformations, this new term is not!

This already hints towards dramatic physical implications of these new terms. However, to actually understand these implications, we need to “transport” $\theta$ into the fermion sector. This will be discussed in the next part. The “transport” of $\theta$ into the fermionic sector is possible through a so-called chiral rotation. Then, after performing this rotation, we can see that $\theta$ actually implies a non-zero electric dipole moment of the neutron $D_n$. This yields a direct possibility to measure $\theta$, because

$$ D_n\approx 5.2 \times 10 ^{-16} \theta \text { cm}. $$

This non-zero dipole moment is possible via a CP violating pion-nucleon coupling, which is directly related to $\theta$. In this sense, $\theta$ implies CP violation in QCD interactions!

The perturbation series in QFT is an “invention of the devil” and this is actually a good thing

In quantum field theory, we can’t calculate things exactly but instead must use a perturbative approach. This means, we calculate observables in terms of a perturbative series: $\mathcal{O}= \sum_n c_n \alpha^n$.

This approach works amazingly well. The most famous example is the magnetic dipole moment of the electron $a_e \equiv (g – 2)/2$. It was calculated to order $\alpha^5$ in Phys. Rev. Lett. 109, 111807 and this result agrees perfectly with the measured value:

$$a_e(exp) – a_e(theory) = −1.06 \ (0.82) \times 10^{-12}$$

However, there is one thing that seems to cast a dark shadow over this rosy picture: there are good reasons to believe that if we would sum up all the terms in the perturbative series, we wouldn’t get a perfectly accurate result, but instead infinity: $ \sum_n^\infty c_n \alpha^n = \infty$. This is not some esoteric thought but widely believed among experts. For example,

“The only difficulty is that these expansions will at best be asymptotic expansions only; there is no reason to expect a finite radius of convergence.”

G. ‘t Hooft, “Quantum Field Theory for Elementary Particles. Is quantum field theory a theory?”, 1984

“Quantum field theoretic divergences arise in several ways. First of all, there is the lack of convergence of the perturbation series, which at best is an asymptotic series.”

R. Jackiw, “The Unreasonable Effectiveness of Quantum Field Theory”, 1996

 

“It has been known for a long time that the perturbation expansions in QED and QCD, after renormalization, are not convergent series.”

G. Altarelli, “Introduction to Renormalons”, 1995

And this is really just a small sample of people you could quote.

This is puzzling, because as is “well known”:

“Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.”

Niels Hendrik Abel, 1828

In this sense, the perturbation series in QFT is an “invention of the devil” and we need to wonder $ \sum_n^\infty c_n \alpha^n = \infty \quad \Rightarrow $ ???

But, of course, before we deal with that, we need to talk about why this divergence of the perturbation series is so widely believed.

Does the perturbation series converge?

$$\mathcal{O}(\alpha)= c_0 + \alpha c_1 + \alpha^2 c_2 + \ldots $$

To answer this question, Dyson already in 1952 published a paper titled “Divergence of perturbation theory in quantum electrodynamics” in which he put forward the clever idea to exploit one of the basic theorems of analysis.

The theorem is: If $0 < r < \infty$, the series converges absolutely for every real number $\alpha$ such that $|\alpha|<r$ and diverges outside of this radius. Here, $r$ is called the radius of convergence and is a non-negative real number or $\infty$ such that the series converges if $|\alpha| < r$.

The important subtlety implied by this theorem that Dyson focused on is that if the radius of convergence is finite $\neq 0$, according to the theorem, the series would also converge for small negative $\alpha$.

In other words: If a series converges it always converges on a disk!

Dyson idea to answer the question “Does the perturbation series converge?” is that we should check if the theory makes sense for a negative value of the coupling constant $\alpha$. If we can argue somehow that the theory explodes for any negative $\alpha$ then we know immediately that $r =0$ and therefore that the perturbation series diverges.

Does QED make sense with negative $\alpha$?

Before we discuss Dyson’s argument why the theory explodes for negative $\alpha$ in detail, here is a short summary of the main line of thought:

We consider a “fictitious world” with negative $\alpha$. In such a world, equal charges attract each other, and opposite charges repel each other.

With some further thought, we will discuss in a moment, this means that the energy is no longer bounded from below. Therefore, in a world with negative $\alpha$, there is no stable ground state.

For our perturbation series, this means, that it is non-analytic around $\alpha = 0$. In other words, electrodynamics with negative $\alpha$, cannot be described by well-defined analytic functions. Therefore we can conclude that the radius of convergence is zero $r=0$, which implies that the perturbation series in QFT diverges also for a positive value of $\alpha$.

In other words, the physics as soon as $\alpha$ becomes negative is so dramatically different that we expect a singularity at $\alpha =0$. Consequently, there doesn’t exist a convergent perturbation series.

After this short summary, let’s discuss how this comes about in more detail.

The important change as soon as $\alpha$ becomes negative is that equal charges start to attract each other. In the “normal” world with positive $\alpha$ a pair of, say, electron and positron that are created from the vacuum attract each other and therefore annihilate immediately, In a world with negative $\alpha$ they repel each other and therefore fly away from each other instead of annihilating.

This means, the naive empty vacuum state starts to fill up with electrons and positrons when $\alpha$ is negative.

Wait, is this energetically possible? What’s the energy of this new state full of electrons and positrons?

To answer this question, let’s consider a ball with radius $R$ full of electrons and calculate its energy content.

On the one hand, we have the positive rest + kinetic energy, which are proportional to the number of particles inside the ball $\propto N$.

On the other hand, we have the negative potential energy, which is given by the usual formula for a Coulomb potential $\propto N^2 R^{-1}$.

To compare them with each other, we need to know the number of electrons inside the ball. According to the Pauli principle, this number is proportional to the volume of the ball and therefore we conclude $\propto V \propto R^3 \propto N$.

Therefore, the positive part of the energy is proportional to $R^3$, and the negative energy to $\propto N^2 R^{-1} \propto R^5$. The negative part wins.

The total energy of the electrons inside the ball is negative. Most disturbingly the energy is unbounded from below and becomes arbitrarily negative as we make the ball bigger: $R,N \to \infty$.

Take note that this is only the case in our fictitious world with negative $\alpha$. In our normal world with positive $\alpha$, opposite charges attract each other and this repulsion sets a lower bound on the energy. The analogous situation to the ball above in our fictitious world is a ball full of electron-positron pairs.

The crucial thing is now that we can’t lower the energy indefinitely because when we try to group more and more electron-positron pairs together, we necessarily bring electrons close to other electrons and positrons close to other positrons. These equal charged particles repel each other in our normal world and this sets a lower bound on the energy.

So, to conclude: In contrast to our normal world with positive $\alpha$, in our fictitious world with negative $\alpha$, a bound state of many electrons or positrons has a large negative energy. This means that our energy isn’t bounded from below because it can become arbitrarily negative. The most dramatic effect of this is what happens to the ground state in such a world, as already mentioned above. If we would start with a naive vacuum with no particles, it would spontaneously turn into a state with lots of electrons on one side and lots of positrons on the other side.

“Although this state is separated from the usual vacuum by a high potential barrier (of the order of the rest energy of the 2N particles being created), quantum-mechanical tunneling from the vacuum to the pathological state would be allowed, and would lead to an explosive disintegration of the vacuum by spontaneous polarization.”

S. Adler, “Short-Distance Behavior of Quantum Electrodynamics and an Eigenvalue Condition for $\alpha$”, 1972

This process would never stop. When the vacuum state isn’t stable against decay, no state is. Therefore, in a world with a negative coupling constant, every state could decay into pairs of electrons and positrons indefinitely.

So, as claimed earlier, physics truly becomes dramatically different as soon as the coupling constant becomes negative.

“This instability means that electrodynamics with negative $\alpha$, cannot be described by well-defined analytic functions; hence the perturbation series of electrodynamics must have zero radius of convergence.”

Adler, “Short-Distance Behavior of Quantum Electrodynamics and an Eigenvalue Condition for $\alpha$”, 1972

For example, an observable like the magnetic dipole moment of the electron will have completely different value as soon as $\alpha$ becomes negative. Or it is even possible that such a property wouldn’t even make any sense in a world with negative $\alpha$.

This leads us to the conclusion that we have a singularity at $\alpha=0$, which means we can’t write down a convergent perturbation series for observables.

It is certainly fun to think about a world with a negative coupling constant and Dyson’s argument makes a lot of sense. Nevertheless, it is important to keep in mind that this is by no means a proof. It’s just a heuristic argument, but neither general nor rigorous.

Yet, many people are convinced by it and further arguments that point in the same direction.

One such further argument is the observation, already made in 1952 and later refined by Bender and Wu that the number of Feynman diagrams grows rapidly at higher orders of perturbation theory.

At order $n$, we get $n!$ Feynman diagrams. For our sum $\sum_n^\infty c_n \alpha^n$ this means that $c_n \propto n!$. Thus, no matter how small $\alpha$ is, at some order $N$ the factor $N!$ wins.

Now that I have hopefully convinced you that $ \sum_n^\infty c_n \alpha^n = \infty$, we can start asking:

What does $ \sum_n^\infty c_n \alpha^n = \infty$ mean?

The best way to understand what $ \sum_n^\infty c_n \alpha^n = \infty$ really means and how we can nevertheless get good predictions out of the perturbation series is to consider toy models.

As already mentioned in my third post about the QCD vacuum, one of my favorite toy models is the quantum pendulum. It is the perfect toy model to understand the structure of the QCD vacuum and the electroweak vacuum and will be now invaluable again.

The Schrödinger equation for the quantum pendulum is

$$ – \frac{d^2 \psi}{d x^2} + g(1-cos x) \psi = E \psi . $$

We want to calculate things explicitly and therefore consider a closely related, simpler model and will come back to the full pendulum later. For small motions of the pendulum, we can approximate the potential ( $\cos(x) \approx 1-x^2/2+x^4/4! – \ldots$) of the quantum pendulum and end up with the Schrödinger equation for the anharmonic oscillator

$$ – \frac{d^2 \psi}{d x^2} – (x^2+ g x^4 )\psi = E \psi . $$

Now, the first thing we can do with this toy model is to understand Dyson’s argument from another perspective.

The potential of the anharmonic oscillator is $ V= x^2+ g x^4$ and let’s say we want to calculate the energy levels by using the usual quantum mechanical perturbation theory $E(g) = \sum_n c_n g^n $. (More precisely: The energy levels of the harmonic oscillator are well known and we are using the Rayleigh-Schrödinger perturbation theory to calculate corrections to them which come from the anharmonic term $\propto x^4$ in the potential. )

For positive values of $g$ the potential is quite boring and looks almost like for the harmonic oscillator. However, for negative values of $g$ the situation becomes much more interesting.

The energy is no longer bounded from below. The states inside the potential are no longer stable but can decay indefinitely by tunneling through the potential barrier. This is exactly the same situation that we discussed earlier for QED with negative $\alpha$.

Thus, according to Dyson’s argument, we suspect that the perturbation series for the energy levels is not convergent.

This was confirmed by Bender and Wu, who treated the “anharmonic oscillator to order 150 in perturbation theory“. (Phys. Rev. 184, 1231 (1969); Phys. Rev. D 7, 1620 (1973))

We can already see from the first terms in the perturbation theory how the series explodes:

$$ \rightarrow E_0 \propto \frac{1}{2} + \frac{3}{4}g – \frac{21}{8}g^2 + \frac{333}{16}g^3 + \ldots $$

This gives further support to Dyson’s conjecture that a dramatically different physics for negative values of the coupling constant means that the perturbation theory does not converge.

Yet, the story of this toy model does not end here. There is much more we can do with it.

The Anharmonic Oscillator in “QFT”

Let’s have a look how we would treat the anharmonic oscillator in quantum field theory. (This example is adapted from the excellent https://arxiv.org/abs/1201.2714). For this purpose, we consider the following “action” integral

$$\mathcal{Z}(g) = \int_{-\infty}^\infty \mathrm{d}x \mathrm{e}^{-x^2-g x^4} .$$

The cool thing is now that we can compute for this toy model the exact answer, for example, using Mathematica. Then, in a second step we can treat the same integral was we usually do in QFT and then compare the perturbative result with the exact result. Then in the last step, we can understand at what order and why the perturbation series diverges.

The full, exact solution isn’t pretty, but no problem for Mathematica:

$$ \mathcal{Z}(g) = \frac{\mathrm{e}^{\frac{1}{8g}}K_{1/4}(1/8g}{2 \sqrt{g}}, $$

where $K_n$ denotes modified Bessel function of the second kind. This solution yields a finite positive number for each value $g$.

Next, we do what we usually do in QFT. We split the “kinetic” and the “interaction” part and expand the interaction part as a power series

\begin{align}
\mathcal{Z}(g) &= \int_{-\infty}^\infty \mathrm{d}x \mathrm{e}^{-x^2-g x^4} = \int_{-\infty}^\infty \mathrm{d}x \mathrm{e}^{-x^2} \sum_{k=0}^\infty \frac{(-gx^4)^k}{k!} \notag \\
& \text{“}= \text{“} \sum_{k=0}^\infty \int_{-\infty}^\infty \mathrm{d}x \mathrm{e}^{-x^2} \frac{(-gx^4)^k}{k!} \notag
\end{align}

Take note that the exchange of sum and integral is the “root of all evil”, but necessary to interpret the theory in terms of a power series of Feynman diagrams. That’s why the last equal sign is put in quotes. (This exchange is actually a “forbidden” step that changes the behaviour at $\pm \infty$. )

So, with this approach to what extend can we get a good approximation?

Using a bit of algebra, we can solve the polyonimian times Gaussian integral and get

\begin{align}
\mathcal{Z}(g) & \text{“}= \text{“} \sum_{k=0}^\infty \int_{-\infty}^\infty \mathrm{d}x \mathrm{e}^{-x^2} \frac{(-gx^4)^k}{k!} \notag \\
&= \sum_{k=0}^\infty \sqrt{\pi} \frac{(-g)^k (4k)!}{2^{4k}(2k)! k!}
\end{align}

This perturbative answer is a series that diverges! (For more details, see the excellent detailed discussion in https://arxiv.org/abs/1201.2714)

Is this perturbative answer, although divergent, useful anyway?

Let’s have a look.

The thing is that in QFT we can only compute a finite number of Feynman diagrams. This means we can only evaluate the first few terms of the perturbation series. Thus we consider the “truncated” series, instead of the complete series, which simply means we stop at some finite order $N$:

$$ \Rightarrow \text{Truncated Series: } \mathcal{Z}_N(g) = \sum_{k=0}^N \sqrt{\pi} \frac{(-g)^k (4k)!}{2^{4k}(2k)! k!} $$

For definiteness let’s choose some value for the coupling constant, say $g=\frac{1}{50}$. How good is the perturbative answer from the truncated series compared to the full exact answer?

\begin{align}
\text{Exact: } \mathcal{Z}(1/50) &= 1.7478812\ldots \notag \\
\text{Perturbatively: } \mathcal{Z}_5(1/50) &= 1.7478728\ldots \notag \\
\mathcal{Z}_{10}(1/50) &= 1.7478818\ldots \notag
\end{align}

This is astoundingly good! The complete series is divergent, which means if we would sum up all the terms, we would get infinity. Nevertheless, if we only consider the first
few terms, we get an excellent approximation of the correct answer!

This behavior can be understood nicely by a plot

The first few terms are okay, then the approximation becomes really good, but at some point, the perturbative answer explodes. A series that behaves like this is known as an asymptotic series.

So now, the question we need to answer is:

When and why does the series diverge?

Again, I will give you a short summary of the answer first, and afterward discuss it in more detail.

The reason that the series explodes at some point is that a perturbative treatment in terms of a Taylor series misses completely factors of the form $ \mathrm{e}^{-c/g} \sim 0 + 0 g + 0 g^2 + \ldots $. The Taylor expansion of such a factor yields zero at all order, although the function obviously isn’t zero. This is a severe limitation of the usual perturbative approach.

You may wonder, why we should care about such funny looking factors. The thing is that tunneling effects in a quantum theory are described precisely by such factors! Remember, for example, the famous quantum mechanical answer of a particle that encounters a potential barrier. It will tunnel through the barrier, although classically forbidden. Inside the potential barrier, we don’t get an oscillating wave function, but instead, an exponentially damping, described by factors of the form $ \mathrm{e}^{-c/g}$.

To summarize: Our perturbative approach misses tunneling effects completely and this is why our perturbation series explodes!

We will see in a moment that this means that the divergence starts around order $N=\mathcal{O}(1/g)$. For example, in QED the perturbative approach is expected to work up to order 137.

We can understand this, by going back to our toy model. Have a look again at the quantum pendulum.

Usually, we consider small oscillations around the ground state, which means low energy states. However in a quantum theory, even at low energies, the pendulum can do something completely different. It can rotate once around its suspension. As it classically does not have the energy to do this, we have here a tunneling phenomenon. This kind of effect is what our usual perturbative approach misses completely and this is why the perturbation series explodes.

After this heuristic discussion, let’s have a more mathematical look how this comes about.

There is a third method, how we can treat our integral $\mathcal{Z}(g) = \int_{-\infty}^\infty \mathrm{d}x \ \mathrm{e}^{-x^2-g x^4}$. This third method is known as the method of steepest descend and it shows nicely what goes wrong with when we use the usual perturbative method.

First, we substitute $x^2\equiv \frac{u^2}{g}$ and then have

$$\mathcal{Z}(g) = \int_{-\infty}^\infty \mathrm{d}x \ \mathrm{e}^{-x^2-g x^4} = \frac{1}{\sqrt{g}} \int_{-\infty}^\infty \mathrm{d}u \ \mathrm{e}^{- \frac{u^2 + u^4}{g}}$$

Now, we deal with small values of the coupling $g$ and thus the integrand is large. The crucial idea behind the method of steepest descend is that the main contributions to the integral come from the extrema of the integrand $\phi(u)\equiv u^2+u^4$.

As usual, we can calculate the extrema by solving $\phi'(u_0) =0$. Take note that in QFT these extrema of our action integrand are simply the solutions to the equations of motion! (This is how we calculate the equations of motion: we look for extrema of the action. This means, $\phi'(u_0) =0$ are in QFT simply our equations of motion and the solutions $u_0$ are solutions of the equations of motion.

To approximate the integral, we then expand the integrand around these extrema. In our example, the extrema are $u=0$ an $u=\pm i/\sqrt{2}$. (For more details, see https://arxiv.org/abs/1201.2714)

This method tells us exactly what goes wrong in the usual approach. The standard perturbation theory corresponds to the expansion around $u=0$.

The other extrema yield contributions $\propto \mathrm{e}^{-\frac{1}{4g}}$ $\rightarrow$ and as already discussed earlier, these are missed completely by a Taylor series around $g=0$.

With this explicit result, we can calculate when these “non-perturbative” contributions become important.

This question in mathematical terms is: When is $\mathrm{e}^{-\frac{1}{4g}} \approx g^k $?

First, we use the log on both sides, which yields the question: When is $-\frac{1}{4g} \approx k \ \mathrm{ln}(g) $?

Now, if we have a look at some explicit numbers: $ \mathrm{ln}(1/50)\approx -3.9$, $ \mathrm{ln}(1/100)\approx -4.6$, $ \mathrm{ln}(1/150)\approx -5$, we see that the answer is: for $ k \approx \frac{1}{g}$!

Thus, as claimed earlier, the nonperturbative effects that are missed by the Taylor expansion treatment become important at order $\approx \frac{1}{g}$ and this is exactly where our perturbation series stops to make sense.

(For a nice discussion of this method of steepest descent, see page 2 here )

Before we summarize what we have found out and learned here there is one last thing.

One Last Thing

There is an amusing empirical rule related to such asymptotic series:

“(Carrier’s Rule). Divergent series converge faster than convergent
series because they don’t have to converge.” from The Devil’s Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series by John P. Boyd

The thing is that convergence is a concept relating to the behavior at $n \to \infty$. This is not what we are really interested in. We want a good approximation by calculating just a few terms of the perturbation series, not all of them.

This kind of behavior is often observed in divergent series. They often yield good approximation at a low order, which, in contrast, is unusual for convergent power series. This is a numerical, empirical statement that was found in many explicit examples, see: Bender, C. M., and Orszag, S. A.: Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978, p. 594

Thus, instead of Abel’s perspective

“Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.“

we should prefer Heaviside’s attitude

“The series is divergent; therefore we may be able to do something with it“

Summary, Conclusions, and Outlook

The thing to take away is nicely summarized by the following picture adapted from a presentation by Aleksey Cherman:

The perturbation series in QFT diverge $\sum_n^\infty c_n g^n =\infty$, but are expected to yield meaningful results up to order $N=\mathcal{O}(1/g)$.

This observation is a great reminder that perturbative Feynman diagrams don’t tell the whole story: tunneling effect, which is proportional to $\mathrm{e}^{1/g}$ are missed completely.

Dyson published his argument in 1952 so all this is known for a long time.

However, there is still a lot of research going on.

One concept people talk about all the time when it comes to this is Borel summation. This is a cool mathematical trick to improve the convergence of divergent series. For the anharmonic oscillator, this works perfectly. By performing a Borel transformation, we can tame the divergence. However, in realistic quantum field theoretical examples this does not work.

The main reason is singularities of the Borel sum. One source of these singularities are the tunneling effects we already talked about. However, much more severe are singularities coming from so-called “renormalons”. This word is used to describe the singularities coming from the renormalization procedure and thus in some sense from the running of the coupling constants.

An active field of research in this direction is “resurgence theory“. People working on this try to use a more general perturbation ansatz
$$\mathcal{O}= \sum_n c_n^{(0)} \alpha^n + \sum_i \mathrm{e}^{S_i/g} \sum_n c_n^{(i)} \alpha^n $$
called a trans-series expansion. The crucial thing is, of course, that they explicitly include the factors $\mathrm{e}^{S_i/g}$ that are missed by the usual ansatz. Thus, in some sense they try to describe “non-perturbative” effects with a new perturbation series.

At the other end of the spectrum are people working on completely non-perturbative results for observables. The most famous example is the amplituhedron, which was proposed a few years ago. This is a geometric object and the people working on it hope that it paves the way to a “nice general solution which would provide all-loop order results.” (J. Trnka)

PS: Many thanks to Marcel Köpke who spotted several typos in the original version.

Unfortunately, repetition is a convincing argument.

I recently wrote about the question “When do you understand?“. In this post, I outlined a pattern that I observed how I end up with a deep understanding of a given topic. However, there is also a second path that I totally missed in this post.

The path to understanding that I outlined requires massive efforts to get to the bottom of things. I argued that you only understand something when you are able to explain it in simple terms.

The second path that I missed in my post doesn’t really lead to understanding. Yet, the end result is quite similar. Oftentimes, it’s not easy to tell if someone got to his understanding via path 1 or path 2. Even worse, oftentimes you can’t tell if you got to your own understanding via path 1 or path 2.

So what is this second path?

It consists of reading something so often that you start to accept it as a fact. The second path makes use of repetition as a strong argument.

Once you know this, it is shocking to observe how easy oneself gets convinced by mere repetition.

However, this isn’t as bad as it may sound. If dozens of experts that you respect, repeat something, it is a relatively safe bet to believe them. This isn’t a bad strategy. At least, not always. Especially when you are starting, you need orientation. If you want to move forward quickly, you can’t get to the bottom of every argument.
Still, there are instances where this second path is especially harmful. Fundamental physics is definitely one of them. If we want to expand our understanding of nature at the most fundamental level, we need to constantly ask ourselves:

Do we really understand this? Or have we simply accepted it, because it got repeated often enough?

The thing is that physics is not based on axioms. Even if you could manage to condense our current state of knowledge into a set of axioms, it would be a safe bet that at least one of them will be dropped in the next century.

Here’s an example.

Hawking and the Expanding Universe

In 1983 Stephen Hawking gave a lecture about cosmology, in which he explained

“The De Sitter example was useful because it showed how one could solve the Wheeler-DeWitt equation and apply the boundary conditions in a simple case. […] However, if there are two facts about our universe which we are reasonably certain, one is that it is not exponentially expanding and the other is that it contains matter.”

Only 15 years later, physicists were no longer “reasonably certain” that the universe isn’t exponentially expanding. On the contrary, we are now reasonably certain of the exact opposite. By observing the most distant supernovae two experimental groups established the accelerating expansion as an experimental fact. This was a big surprise for everyone and rightfully led to a Nobel prize for its discoverers.

The moral of this example isn’t, of course, that Hawking is stupid. He only summarized what everyone at this time believed to know. This example shows how quickly our most basic assumptions can change. Although most experts were certain that the expansion of the universe isn’t accelerating, they were all wrong.

Theorems in Physics and the Assumptions Behind Them

If you want further examples, just have a look at almost any theorem that is commonly cited in physics.

Usually, the short final message of the theorem is repeated over and over. However, you almost never hear about the assumptions that are absolutely crucial for the proof.
This is especially harmful, because, as the example above demonstrated, our understanding of nature constantly changes.

Physics is never as definitive as mathematics. Even theorems aren’t bulletproof in physics because the assumptions can turn out to be wrong with new experimental findings. What we currently think to be true about physics will be completely obsolete in 100 years. That’s what history teaches us.

An example, closely related to the accelerating universe example from above, is the Coleman-Mandula theorem. There is probably no theorem that is cited more often. Most talks related to supersymmetry mention it at some point. It is no exaggeration when I say that I have heard at least 100 talks that mentioned the final message of the proof: “space-time and internal symmetries cannot be combined in any but a trivial way”.

Yet, so far I’ve found no one who was able to discuss the assumptions of the theorem. The theorem got repeated so often in the last decades that it is almost universally accepted to be true. And yes, the proof is, of course, correct.
However, what if one of the assumptions that go into the proof isn’t valid?

Let’s have a look.

An important condition, already mentioned in the abstract of the original paper is Poincare symmetry. This original paper was published in 1967 and then it was reasonably certain we are living in a universe with Poincare symmetry.
However, as already mentioned above, we know since 1998 that this isn’t correct. The expansion of the universe is accelerating. This means the cosmological constant is nonzero. The correct symmetry group that preserves the constant speed of light and the value of a nonzero cosmological constant is the De Sitter group and not the Poincare group. In the limit of a vanishing cosmological constant, the De Sitter group contracts to the Poincare group. The cosmological constant is indeed tiny and therefore we aren’t too wrong if we use the Poincare group instead of the De Sitter group.
Yet, for a mathematical proof like the one proposed by Coleman and Mandula, whether we use De Sitter symmetry or Poincare symmetry makes all the difference in the world.

The Poincare group is a quite ugly group and consist of Lorentz transformations and translations: $ \mathbb{R}(3,1) \rtimes SL(2,\mathbb{C}) .$ The Coleman-Mandula proof makes crucial use of the inhomogeneous translation part of this group $\mathbb{R}(3,1)$. In contrast, the De Sitter group is a simple group. There is no inhomogeneous part. As far as I know, there is no Coleman-Mandula theorem if we replace the assumption: “Poincare symmetry” with “De Sitter symmetry”.

This is an example where repetition is the strongest argument. The final message of the Coleman-Mandula theorem is universally accepted as a fact. Yet, almost no one had a look at the original paper and its assumptions. The strongest argument for the Coleman-Mandula theorem seems to be that is was repeated so often in the last decades.

Maybe you think: what’s the big deal?

Well, if the Coleman-Mandula no-go theorem is no longer valid, because we live in a universe with De Sitter symmetry, a whole new world would open up in theoretical physics. We could start thinking about how spacetime symmetries and internal symmetries fit together.

The QCD Vacuum

Here is another example of something people take for given only because it was repeated often enough: The structure of the CP vacuum. I’ve written about this at great length here.

I talked to several Ph.D. students who work on problems related to the strong CP problem and the vacuum in quantum field theory. Few knew the assumptions that are necessary to arrive at the standard interpretation of the QCD vacuum. No one knew where the assumptions actually come from and if they are really justified. The thing is that when you dig deep enough you’ll notice that the restriction to gauge transformations that satisfy $U \to 1$ at infinity is not based on something bulletproof, but simply an assumption. This is a crucial difference and if you want to think about the QCD vacuum and the strong CP problem you should know this. However, most people take this restriction for granted, because it has been repeated often enough.

Progress in Theoretical Physics without Experimental Guidance

The longer I study physics the more I become convinced that people should be more careful about what they think is definitely correct. Actually, there are very few things we know for certain and it never hurts to ask: what if this assumption everyone uses is actually wrong?

For a long time, physics was strongly guided by experimental findings. From what I’ve read these must have been amazing exciting times. There was tremendous progress after each experimental finding. However, in the last decades, there were no experimental results that have helped to understand nature better at a fundamental level. (I’ve written about the status of particle physics here).

So currently a lot of people are asking: How can there be progress without experimental results that excite us?

I think a good idea would be to take a step back and talk openly, clearly and precisely about what we know and understand and what we don’t.

Already in 1996, Nobel Prize winner Sheldon Lee Glashow noted:

“[E]verybody would agree that we have right now the standard theory, and most physicists feel that we are stuck with it for the time being. We’re really at a plateau, and in a sense, it really is a time for people like you, philosophers, to contemplate not where we’re going, because we don’t really know and you hear all kinds of strange views, but where we are. And maybe the time has come for you to tell us where we are. ‘Cause it hasn’t changed in the last 15 years, you can sit back and, you know, think about where we are.”

The first step in this direction would be that more people were aware that while repetition is a strong argument, it is not a good one when we try to make progress. The examples above hopefully made clear that only because many people state that something is correct, does not mean that it is actually correct. The message of a theorem can be invalid, although the proof is correct, simply because the assumptions are no longer up to date.

This is what science is all about. We should always question what we take for given. As for many things, Feynman said it best:

“Science alone of all the subjects contains within itself the lesson of the danger of belief in the infallibility of the greatest teachers in the preceding generation. . . Learn from science that you must doubt the experts. As a matter of fact, I can also define science another way: Science is the belief in the ignorance of experts.“