ap.io/mpip

Unification and String Theory

Recap: GR’s Own UV Catastrophe

Last time, we learned that Einstein’s beautiful geometrical theory of gravity, General Relativity (GR), is a sick theory at high energy. Physicists call this a UV sickness because UV is high-energy. The quantum sickness of GR is not resolvable without changing your theory of gravity or changing quantum mechanics (QM).

Formulating Quantum Gravity

The titanic clash we saw last week between GR and QM is not something you can Band-Aid over. It is a deep theoretical emergency. What this means practically is that while GR is a great long-distance theory of gravity, it must be modified at short distances (high energies) to fix up the probability problem.

We now seek a theory of Gravity 3.0, which should have softer high-energy behaviour than GR does and which must knit together GR and QM. The most important principles are:-

It must reduce to known physics in well-understood regimes. This is called the Correspondence Principle.
It must have no internal mathematical inconsistencies, known as anomalies.
It must describe quantum gravity in a calculationally useful way.
The aim is to be able to calculate physically interesting quantities for black holes and the Big Bang, not just to know that those two places in the universe are super-gnarly.

String as Lego

Relaxing our insistence that particles be the Legos of the universe is actually enough (!) to solve our titanic clash between GR and QM. String theory takes this seriously and hypothesizes that the most elementary Legos in the universe are strings, i.e. one-dimensional strands of energy. This is not like cat string: it is relativistic, and much much smaller. You can think of cat string as made of molecules, which are made of atoms, which are made of quarks and gluons and electrons, which in turn are made of fundamental string.

String theory is versatile because it proposes that the same basic ingredient (fundamental string) can wiggle in different oscillation patterns and therefore can describe subatomic particles of differing masses and spins (and charges). In particular, string theorists know how to represent all the particles of the Standard Model in terms of fundamental strings. String theory is economic because it requires only one basic type of ingredient. Strings can be open, i.e. have endpoints (like a skipping rope), or they can be closed, i.e. have no ends (like a rubberband).

Like guitar strings and organ pipes, you can have the string be in the fundamental mode, which is the lowest possible frequency oscillation. (In the case of the organ pipe, it is the air inside it that is doing the waving; for the string it is the body of the string itself that waves.) The string could alternatively be in an overtone, which is a higher excitation that costs more energy. There is an infinite number of possible overtones, each characterized by a specific energy, but only one fundamental. Typically you will not be able to excite the overtones if you have a tiny energy budget. This is why strings look like particles at low-energy. It is like you took your glasses off and saw a blurry string which looked for all intents and purposes like a particle.

String Tension

How does the string mass scale arise? From tension energy. Let us see how this can be.

Nonrelativistic strings, like guitar or violin strings, have independent tension $T$ and mass per unit length $\mu$. The speed of vibrations on the string is set by the ratio of these two quantities, according to $$ c=\sqrt{\frac{T}{\mu}} \,. $$ We also know that frequency and wavelength are inversely proportional for waves. So at fixed wavelength (e.g. fixed distance between the ends of the guitar string), if we crank up the tension we will get a higher frequency, i.e. a higher-pitched note. You will know this from experience if you have ever tried to tune a guitar, violin, harp, or piano.

Relativistic strings -- the ones used for making fundamental Legos -- have tension proportional to mass density. Their vibrations move at the speed of light.

String Dynamics

Strings cannot oscillate longitudinally. They only wiggle in directions transverse to their bodies. As the string moves through spacetime, it sweeps out a surface called the worldsheet. The shape of a string worldsheet looks like a wiggly strip for an open string and a wiggly tube for a closed string.

[a hand-drawn sketch of the wiggly worldsheet swept out in spacetime by an open string (left) and a closed string (right)]

Quantum fluctuations of string position like to spread out over all space: they are like messy teenagers. The worldsheet, on the other hand, has restricted spatial extent. This competition results in a Casimir energy deficit, which translates into a mass-squared deficit for the string. There is quite a lot of math involved in the details, so we just quote what the result is after the dust settles.

Open strings can have vibration patterns called standing waves, like the waves you can make with a skipping rope being held at both ends by children. For open strings, $$ {\frac{m^2_{\rm open}}{m_s^2}} = \left(N_O-1\right) \geq 0 \,. $$ where $m_s$ is the string energy scale set by the tension, and $N_O$ is an integer denoting oscillation number.

In other words, closed strings have travelling waves, with independent left- and right-moving vibrations. For closed strings:- $$ {\frac{m^2_{\rm closed}}{m_s^2}} = \left(N_R+N_L-2\right) \geq 0 \,. $$ where $m_s$ is the string energy scale, $N_L$ is the left-moving oscillator number, and $N_R$ is the right-moving oscillator number. Momentum balance for closed strings insists that $N_R=N_L$.

In all cases, we still have the relativistic mass shell relation $$ E^2 - |{\vec{p}}|^2 = (mc^2)^2 \,. $$

Very generally, the motion of a string is composed of two parts:

motion of the centre of mass with energy $E$ and momentum $\vec{p}$,
oscillations about the centre of mass.

The oscillations are quantized: only an integer number of wiggles can fit around the string, whether it is open or closed.

Groundstates of Open and Closed Strings

The above mass formulae are perhaps a bit daunting for this audience. Do not worry. We do not actually need the details for getting across the meaty point of today’s discussion. What we need the most is to work out what the lowest state of the string would be.

For the open string, the groundstate has zero mass and one oscillator. This single oscillator can point in any direction, which gives what physicists call a vector (something that points in one direction). The resulting object has zero mass and spin one. In other words, open string groundstates describes massless messenger bosons of the Standard Model like the photon and the gluon.

For the closed string, the groundstate has zero mass and two oscillators: one left-moving and one right-moving. These two oscillators can point in any directions. The resulting object has mass zero and spin two. This is the graviton missing from the Standard Model. The fact that closed strings have graviton groundstates is why string theorists can claim with a straight face that string theory is a theory of quantum gravity -- it is not some random gravity model from our imagination but reduces to Einstein's gravity theory at low energy.

For those interested in working out further details: note that the frequency of the $k$th harmonic is $k$ times the frequency of the fundamental. A higher oscillator mode number corresponds to a more massive string state. As you increase the mass, there is a rapidly growing number of ways that you can distribute your oscillator energy budget.

Smoothness of String Interactions

Recall our Ice Skater Analogy for messenger bosons. What we saw when we discussed that analogy was that interactions happen at points in spacetime: i.e., at a specific place in space at a specific point in time.

[Feynman diagrams denoting elementary particle scattering with electromagnetic, weak, and strong nuclear forces operating, as well as gravity]

What about strings? String interactions are naturally and inherently spread out in spacetime. Strings interact by smoothly splitting or joining. This softening makes Gravity 3.0 calculable! Booyah!!

[the string theory diagram that can reduce to any of the above particle diagrams, in various limits -- and it's smooth, no pointy edges or anything.]

Resolving Power

Particles are pointy, or hard in physicists’ terms. Strings are extended, which makes their high-energy behaviour softer.

In particle QM, we learned that higher momentum means shorter quantum (de Broglie) wavelength, and therefore greater sensitivity. So more money means better resolution. Schematically, $$ \Delta x_{\rm min, particle} = {\frac{\hbar}{\Delta p}} $$ For strings, the behaviour has a qualitatively new ingredient:- $$ \Delta x_{\rm min, string} = {\frac{\hbar}{\Delta p}} + {\frac{\hbar c}{m_s^2}} \Delta p $$

String resolution follows particle resolution at low energy (this must happen, by the Correspondence Principle). But it then worsens again at high energy. Why? With a big enough energy budget, the string’s oscillator energy beats the tension energy and the probe string gets floppier and fatter. This weakens its effectiveness as a probe.

[a cartoon plot depicting the above two equations for minimum spatial resolution graphically]

The most important qualitative conclusion from this is that there is a law of diminishing returns in string theory: if you keep cranking up the energy higher and higher beyond the turnover point (the valley in the red curve in the figure), you will not make a better experiment. Maybe the buck really does stop at string theory! There might not be any more onion layers.