What equation do we use to measure the energy level of a string, to determine it's "particle correlation"

Question

If string theory happened to be correct, and a point-particle is replaced with a string, there is a direct correlation between the vibrating frequency of the string and the particle it produces. I was wondering if anyone knows what equation is used to determine what particle the string will transform to based upon its frequency? I'm sure there has to be any equation floating around somewhere.

Answer 1

For any particle, we can define a continuous quantum number - its momentum $k^\mu$ and a discrete internal quantum number - for instance, its spin or charge under some symmetry group (note spin is also the charge under Lorentz transformations). The continuous quantum number defines the mass of the particle via $k^2 = -m^2$. In any actual theory, $m^2$ is known to be "quantized" in that more often than not, there isn't a continuous spectrum of masses.

To describe to you the correlation between strings and particles, I will need to describe to you how these quantum numbers arise from the perspective of the string. Let me do so in the simple case of bosonic string theory.

The former is easy. A string also has a center mass momentum, $k^\mu$. In addition to this a string may also have vibrational excitations such as the ones you have described. The description of these vibrational excitations can be encoded in an integer $N$ which is roughly the "level" of these excitations. At each level, there are multiple possible vibrational modes. $N$ is also related to the "frequency" of the string.

Let us be more concrete. As you expected, the level (frequency) does determine the mass of the state to which the string corresponds to by $$ m^2 = \frac{4}{\alpha'} (N-1) $$ $\alpha'$ is a string coupling (some constant). So for instance, when there is no vibrational excitation, $N=0$ and we find a state of mass $$ m^2 = - \frac{4}{\alpha'} $$ PS - Don't be alarmed by negative mass squared. Bosonic string theory is known to be inconsistent due to the presence of such a state (called tachyon). The correct theory - superstring theory does not have such tachyons. However, it has some extra complications that only makes the argument more complicated. For simplicity, I'm describing this in bosonic string theory, but the argument can easily be generalized to superstrings.

The next question is at each level $N$ what is the spin of the particle? There is also a theory for that and while there is no "equation", there is a systematic algorithm to determine the spin of particles at each $N$. The algorithm determines $N=0$ state is a scalar (spin 0), level $N=1$ state contains a graviton (spin 2) a dilaton (spin 0) and a 2-form gauge field (closed strings). In open strings, $N=1$ contains a gauge field (spin 1). Similarly, level $N=2$ state contains spin 3 particles, etc. The highest spin particle present at level $N$ is $N+1$. Note that all spins described here are integers and fermions do not appear. This also is an artefact of bosonic string theory. Superstring theory also rectifies this and half-integer spins appear in the spectrum.

Now, so far all I have described is spin. Particles should also have other charges. How do they appear? They appear in several ways. For instance, electromagnetic charges appear when we compactify string theory on a circle (or on more complicated things). Non-abelian charges appear when some Chan-Paton factors are introduced on open strings.

Unfortunately that's all I can say at a superficial level. To truly understands how these things appear in string theory, you have to understand some basics (but not all). There is no "equation" that tells you exactly what particle each frequency corresponds to. However, there is a systematic algorithm to determine the same.

Answer 2

What equation do we use to measure the energy level of a string, to determine it's “particle correlation”

We have already measured the particles. We have studied their properties and "measured" their quantum numbers as expressed in this table

Measured within ( using the tools of) the standard theory of quantum mechanics and special relativity. There is a Lagrangian describing the standard model of particle physics which encapsulates measurements/data for the last hundred years. It has been validated well and there are very few discrepancies which point to the need to go beyond the standard model. The last validation coming from the measurements of the LHC experiments, including the discovery of theHiggs meson.

A larger theory that would embed the standard model is required for the unification of the three forces within the standard model, electromagnetic, weak and strong and for eliminating imbalances in the standard model for the high energies needed to describe cosmological observations, and more so if gravity is included in the unification.

String models at present time are the only models that can consistently embed the standard model and offer quantization of gravity within a Lorenz invariant frame for flat spaces. That is the reason why a large number of very competent theorists are exploring string theories for particle physics.-

In the successful low energy standard model , particles are point particles. In string models a generic particle is a string, i.e. it has one dimension more. This engenders a lot of extra dimensions to the four (space time) that are used in the parametrization and the calculations of the standard model and offer a plethora of parameters so that thousands of string theories can be envisaged. None has been proposed as the "standard one" at the moment although many phenomenological models exist.

What does it mean "a particle is not a point but a vibration on a string". In standard second quantization field theory a particle is an excitation on the undelying field, at a point in space, that is what all those creation and annihilation operators are about, exciting a specific particle. String theory changes the point to a tiny vibrating string.

In our current standard model the group structure observed/measured that the particles in the table follow, SU(3)xSU(2)xU(1), is imposed on the Lagrangian by hand. The group symmetries are not an inherent expression of the model because field theory on point fields does not have emergent group structures that can be identified with the measured ones.

String theories, due to the extra dimension, point to string, and the hidden dimensions , allow for group structures/symmetries of the vibrations. Thus, in the finally proposed standard string model, the observed group structures will be emergent from the theory and not put in by hand. The measured quantum numbers of the particles will be assigned a one to one correspondence with the group structure of a vibrating quantized string, in the sought for theory. It is the possibility of doing this that attracted theorists to the subject .

So the question puts the cart in front of the horse. The equation is chosen because the particles are measured. (if they manage to settle on a particular string theory model).

That said, phenomenologists are theorists who make specific models and give predictions for new measurements that will point the way to the standard string theory model. The search for supersymmetry is one of these, also of various string excitations as described in this talk as an example, or in this recent publication.